Finite Group Databasebeta

← #29671294

Subgroups — #29671294

Order 15616.

830 conjugacy classes of subgroups.

The subgroup lattice has 710 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21458791C211yesmaximal, characteristic, complemented
2D4×#11758199C211yesmaximal, characteristic, complemented
3#21457440C211yesmaximal, characteristic, complemented
4#21445433C211yesmaximal, characteristic, complemented
5#21445298C211yesmaximal, characteristic, complemented
6#21445295C211yesmaximal, characteristic, complemented
7#21445177C211yesmaximal, characteristic, complemented
8#21445172C211yesmaximal, characteristic, complemented
9#21444884C211yesmaximal, characteristic, complemented
10#21444882C211yesmaximal, characteristic, complemented
11#21444873C211yesmaximal, characteristic, complemented
12#21444872C211yesmaximal, characteristic, complemented
13#21444702C211yesmaximal, characteristic, complemented
14SD16×#92418C211yesmaximal, characteristic, complemented
15C61×#3274C211yesmaximal, characteristic, complemented
16C2×#14519804—21no—
17C22×#11758199—21no—
18#16840434C2211yescharacteristic, complemented
19D4×D244C2211yescharacteristic
20#16840139C2211yescharacteristic, complemented
21#16840138—21no—
22#16840138C2211yescharacteristic
23#16840138C2211yescharacteristic, complemented
24#16840138C2211yescharacteristic, complemented
25#16840138C2211yescharacteristic, complemented
26#16840016—21no—
27#16839996—21no—
28#16839962C2211yescharacteristic, complemented
29#16839957—21no—
30#16839957C2211yescharacteristic
31#16839957C2211yescharacteristic
32#16839957C2211yescharacteristic, complemented
33#16839954—21no—
34#16839954C2211yescharacteristic
35#16839954C2211yescharacteristic, complemented
36#16839910—21no—
37#16839897—21no—
38#16839896—21no—
39#16839895—21no—
40Q8×#92418C2211yescharacteristic, complemented
41C4×#11758199—21no—
42C4×#11758199C2211yescharacteristic
43#16839841C2211yescharacteristic, complemented
44D4×#92418C2211yescharacteristic
45#16839514C2211yescharacteristic, complemented
46#16839513C2211yescharacteristic, complemented
47#16839506C2211yescharacteristic, complemented
48#16839506C2211yescharacteristic, complemented
49#16839505C2211yescharacteristic, complemented
50#16839505C2211yescharacteristic, complemented
51Q8⋊#92418C2211yescharacteristic, complemented
52D4⋊#92418C2211yescharacteristic
53D4⋊#92418C2211yescharacteristic, complemented
54#16839489C2211yescharacteristic, complemented
55C8×#92418C2211yescharacteristic, complemented
56C2×AGL(1,ℤ/8)×C61—21no—
57C61×#550C2211yescharacteristic, complemented
58D42×C61C2211yescharacteristic
59C61×#493C2211yescharacteristic, complemented
60C61×#469—21no—
61C61×#460C2211yescharacteristic, complemented
62C61×#459C2211yescharacteristic
63C61×#449—21no—
64C61×#447—21no—
65C4×SD16×C61C2211yescharacteristic
66C61×#436C2211yescharacteristic, complemented
67C2×#11758226—21no—
68C22×D244—21no—
69C2×D4×D61—21no—
70#14519804—44no—
71C2×#11758201—21no—
72C2×D488—21no—
73C2×D488—21no—
74C2×#11758199—21no—
75C2×#11758199—21no—
76C2×#11758199—21no—
77C2×#11758199—21no—
78C2×#11758199—21no—
79C2×#11758199—21no—
80C2×#11758199—21no—
81C2×#11758199—21no—
82C2×#11758199—21no—
83C2×#11758199—41no—
84C2×#11758199—41no—
85C2×#11758199—44no—
86C2×#11758199D411yescharacteristic
87C2×#11758189—21no—
88C22×#92418—21no—
89#14519774—21no—
90#14519774C2311yescharacteristic
91#14519774C2311yescharacteristic
92#14519752—21no—
93#14519752—21no—
94#14519752—41no—
95#14519751—21no—
96#14519747—41no—
97C4×D244—21no—
98C4×D244C2311yescharacteristic
99#14519710D411yescharacteristic
100C4⋊#92418—21no—
101C4⋊#92418C2311yescharacteristic
102C4⋊#92418C2311yescharacteristic
103C4⋊#92418C2311yescharacteristic
104#14519698—21no—
105#14519698—21no—
106#14519698D411yescharacteristic
107#14519697—21no—
108#14519697—21no—
109#14519697—21no—
110#14519697—21no—
111#14519697D411yescharacteristic
112C22⋊#92418—21no—
113#14519689—21no—
114C4×#92418—21no—
115C4×#92418C2311yescharacteristic
116C4×#92418C2311yescharacteristic
117C2×#41×C61—21no—
118C22×D4×C61—21no—
119C22×D4×C61—21no—
120AGL(1,ℤ/8)×C61—44no—
121C2×SD16×C61—21no—
122C2×SD16×C61D411yescharacteristic
123C2×D8×C61—21no—
124C2×D8×C61—21no—
125C2×D8×C61—21no—
126C2×M16×C61—21no—
127#127×C61—21no—
128#127×C61C2311yescharacteristic
129#127×C61C2311yescharacteristic
130#121×C61—21no—
131#121×C61—21no—
132#121×C61—21no—
133#121×C61—41no—
134#120×C61—41no—
135C4×Q8×C61C2311yescharacteristic
136C4×D4×C61—21no—
137C4×D4×C61C2311yescharacteristic
138C4×D4×C61C2311yescharacteristic
139#106×C61D411yescharacteristic
140C4⋊C8×C61C2311yescharacteristic
141#103×C61D411yescharacteristic
142#102×C61—21no—
143#102×C61—21no—
144#102×C61D411yescharacteristic
145C22⋊C8×C61—21no—
146C4×C8×C61C2311yescharacteristic
147C24×C61—41no—
148C23×D61—41no—
149#41×C61—42no—
150#41×C61—81no—
151C2×Q8×C61C2×D411yescharacteristic
152C2×D4×C61—21no—
153C2×D4×C61—21no—
154C2×D4×C61—21no—
155C2×D4×C61—21no—
156C2×D4×C61—21no—
157C2×D4×C61—21no—
158C2×D4×C61—41no—
159C2×D4×C61—41no—
160C2×D4×C61—41no—
161C2×D4×C61—41no—
162C2×D4×C61—42no—
163C2×D4×C61—42no—
164C2×D4×C61—81no—
165C2×D4×C61—81no—
166C2×D4×C61C2×D411yescharacteristic
167C2×D4×C61C2×D411yescharacteristic
168C22×C4×C61—21no—
169C22×C4×C61—21no—
170C22×C4×C61—41no—
171C2×C22⋊D61—41no—
172#11758226—42no—
173#11758226—81no—
174D4×D61—42no—
175D4×D61—81no—
176C2×D244—21no—
177C2×D244—21no—
178C2×D244—21no—
179C2×D244—21no—
180C2×D244—21no—
181C2×D244—21no—
182C2×D244—42no—
183C2×D244—41no—
184C2×D244—41no—
185C2×D244—41no—
186C2×D244—81no—
187C2×D244C2×D411yescharacteristic
188C2×C4×D61—21no—
189C2×C4×D61—41no—
190SD16×C61—42no—
191SD16×C61D814yescomplemented
192D8×C61—42no—
193D8×C61—42no—
194D8×C61—81no—
195M16×C61—42no—
196C2×C8×C61—21no—
197C2×C8×C61C2×D411yescharacteristic
198C2×C8×C61C2×D411yescharacteristic
199C4⋊C4×C61—21no—
200C4⋊C4×C61C2×D411yescharacteristic
201C4⋊C4×C61C2×D411yescharacteristic
202C4⋊C4×C61C2×D411yescharacteristic
203C22⋊C4×C61—21no—
204C22⋊C4×C61—21no—
205C22⋊C4×C61—41no—
206C22⋊C4×C61—41no—
207C42×C61—21no—
208C42×C61C2411yescharacteristic
209#11758201—42no—
210#11758199—22no—
211#11758199—42no—
212#11758199—42no—
213#11758199—42no—
214#11758199—42no—
215#11758199—42no—
216#11758199—81no—
217#11758199—81no—
218#11758199—82no—
219#11758199—82no—
220#11758199—81no—
221#11758199—81no—
222#11758198—21no—
223#11758198—41no—
224#11758198—41no—
225C4⋊Dic61C2×D411yescharacteristic
226C2×#92418—21no—
227C2×#92418—21no—
228C2×#92418—21no—
229C2×#92418—21no—
230C2×#92418—42no—
231C2×#92418C2×D411yescharacteristic
232C2×#92418C2×D411yescharacteristic
233D488—42no—
234D488—81no—
235#11758189—42no—
236C23×C61—21no—
237C23×C61—21no—
238C23×C61—41no—
239C23×C61—41no—
240C23×C61—82no—
241C23×C61—82no—
242C23×C61—161no—
243C22×D61—21no—
244C22×D61—41no—
245C22×D61—41no—
246C22×D61—82no—
247C22×D61—82no—
248C22×D61—161no—
249Q8×C61—21no—
250Q8×C61C2×D812yescomplemented
251D4×C61—21no—
252D4×C61—21no—
253D4×C61—21no—
254D4×C61—41no—
255D4×C61—41no—
256D4×C61—41no—
257D4×C61—41no—
258D4×C61—41no—
259D4×C61—41no—
260D4×C61—41no—
261D4×C61—41no—
262D4×C61—82no—
263D4×C61—81no—
264D4×C61—81no—
265D4×C61—82no—
266D4×C61—82no—
267D4×C61—161no—
268D4×C61C2×D812yes—
269C2×C4×C61—21no—
270C2×C4×C61—21no—
271C2×C4×C61—21no—
272C2×C4×C61—21no—
273C2×C4×C61—41no—
274C2×C4×C61—41no—
275C2×C4×C61—41no—
276C2×C4×C61—41no—
277C2×C4×C61—82no—
278C2×C4×C61C22×D411yescharacteristic
279C2×C4×C61C22×D411yescharacteristic
280C2×C4×C61H(5,𝔽2)11yescharacteristic
281C22⋊D61—82no—
282C2×Dic61—21no—
283D244—21no—
284D244—21no—
285D244—41no—
286D244—41no—
287D244—41no—
288D244—41no—
289D244—41no—
290D244—41no—
291D244—41no—
292D244—41no—
293D244—82no—
294D244—81no—
295D244—81no—
296D244—82no—
297D244—161no—
298C4×D61—41no—
299C4×D61—41no—
300C4×D61—82no—
301C8×C61—21no—
302C8×C61—41no—
303C8×C61C2×D812yescomplemented
304#92418—22no—
305#92418—21no—
306#92418—41no—
307#92418—41no—
308#92418—41no—
309#92418AGL(1,ℤ/8)12yescomplemented
310#19885—611nomaximal, Sylow
311C22×C61—22no—
312C22×C61—41no—
313C22×C61—41no—
314C22×C61—41no—
315C22×C61—41no—
316C22×C61—41no—
317C22×C61—81no—
318C22×C61—81no—
319C22×C61—81no—
320C22×C61—81no—
321C22×C61—81no—
322C22×C61—81no—
323C22×C61—164no—
324C22×C61D4211yescharacteristic
325C2×D61—41no—
326C2×D61—41no—
327C2×D61—41no—
328C2×D61—81no—
329C2×D61—81no—
330C2×D61—81no—
331C2×D61—81no—
332C2×D61—81no—
333C2×D61—81no—
334C2×D61—164no—
335C4×C61—22no—
336C4×C61—21no—
337C4×C61—41no—
338C4×C61—41no—
339C4×C61—41no—
340C4×C61C22×D811yescharacteristic
341C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
342C4×C61#57611yescharacteristic
343C4×C61#57611yescharacteristic
344Dic61—41no—
345#3391—611no—
346D4×D8—611no—
347#3274—611no—
348#3274D6111yescharacteristic, complemented
349#1712—611no—
350#1687—611no—
351#1687—611no—
352#1668—611no—
353#1667—611no—
354#1621—611no—
355#1621—611no—
356#1619—611no—
357#1619—611no—
358#1589—611no—
359C8×SD16—611no—
360C2×C61—42no—
361C2×C61—81no—
362C2×C61—161no—
363C2×C61—161no—
364C2×C61#327411yescharacteristic
365C2×C61D4×D811yescharacteristic
366C2×C61#329211yescharacteristic
367D61—81no—
368D61—161no—
369D61—161no—
370C2×AGL(1,ℤ/8)—21no—
371C2×AGL(1,ℤ/8)—1221no—
372C22×D8—1221no—
373#550—611no—
374#550C2×D6111yescharacteristic, complemented
375D42—611no—
376D42C2×D6111yescharacteristic
377#493—611no—
378#493—611no—
379#493—611no—
380#493—611no—
381#493—611no—
382#493—1221no—
383#493C2×D6111yescharacteristic, complemented
384#469—21no—
385#469—1221no—
386#466—1221no—
387#460—611no—
388#460C2×D6111yescharacteristic, complemented
389#459—611no—
390#459—611no—
391#459—611no—
392#459—611no—
393#459—611no—
394#459—1221no—
395#459—1221no—
396#459C2×D6111yescharacteristic
397#449—21no—
398#449—1221no—
399#447—21no—
400#447—1221no—
401#447—1221no—
402#447—1221no—
403C8×Q8—611no—
404C4×SD16C2×D6111yescharacteristic
405C4×D8—611no—
406C4×D8—1221no—
407#436—611no—
408#436C2×D6111yescharacteristic, complemented
409C8×D4—611no—
410#335—611no—
411#334—611no—
412#331—611no—
413#331—611no—
414#331—611no—
415#331—611no—
416Q8⋊C8—611no—
417D4⋊C8—611no—
418D4⋊C8—611no—
419D4⋊C8—611no—
420C82—611no—
421C61#1988511yescharacteristic, complemented, Sylow
422C2×#41—21no—
423C2×#41—1221no—
424C22×D4—21no—
425C22×D4—21no—
426C22×D4—1221no—
427C22×D4—1221no—
428AGL(1,ℤ/8)—44no—
429AGL(1,ℤ/8)—2444no—
430C2×SD16—21no—
431C2×SD16—1221no—
432C2×SD16C22⋊D6111yescharacteristic
433C2×D8—21no—
434C2×D8—21no—
435C2×D8—21no—
436C2×D8—611no—
437C2×D8—1221no—
438C2×D8—1221no—
439C2×D8—1221no—
440C2×D8—1221no—
441C2×D8—1221no—
442C2×D8—1221no—
443C2×D8—1221no—
444C2×D8—1221no—
445C2×D8—1221no—
446C2×D8—1221no—
447C2×D8—1221no—
448C2×D8—2441no—
449C2×D8—2441no—
450C2×D8—2444no—
451C2×M16—21no—
452C2×M16—1221no—
453C22×C8—1221no—
454#127—21no—
455#127—611no—
456#127—611no—
457#127—1221no—
458#127C22×D6111yescharacteristic
459#127C22×D6111yescharacteristic
460#121—21no—
461#121—21no—
462#121—21no—
463#121—41no—
464#121—1221no—
465#121—1221no—
466#121—1221no—
467#121—2441no—
468#120—41no—
469#120—2441no—
470C4×Q8C22×D6111yescharacteristic
471C4×D4—21no—
472C4×D4—611no—
473C4×D4—1221no—
474C4×D4C22×D6111yescharacteristic
475C4×D4C22×D6111yescharacteristic
476#107—611no—
477#106C22⋊D6111yescharacteristic
478C4⋊C8—611no—
479C4⋊C8—611no—
480C4⋊C8—611no—
481C4⋊C8—1221no—
482C4⋊C8C22×D6111yescharacteristic
483#103C22⋊D6111yescharacteristic
484#102—21no—
485#102—21no—
486#102—611no—
487#102—611no—
488#102—1221no—
489#102—1221no—
490#102—1221no—
491#102—1221no—
492#102—1221no—
493#102—1221no—
494#102C22⋊D6111yescharacteristic
495C22⋊C8—21no—
496C22⋊C8—1221no—
497C22⋊C8—1221no—
498C4×C8—611no—
499C4×C8—611no—
500C4×C8—1221no—
501C4×C8C22×D6111yescharacteristic
502C24—41no—
503C24—2441no—
504#41—42no—
505#41—81no—
506#41—4881no—
507#41—2442no—
508C2×Q8C2×C22⋊D6111yescharacteristic
509C2×D4—21no—
510C2×D4—21no—
511C2×D4—21no—
512C2×D4—21no—
513C2×D4—21no—
514C2×D4—21no—
515C2×D4—41no—
516C2×D4—41no—
517C2×D4—41no—
518C2×D4—41no—
519C2×D4—42no—
520C2×D4—42no—
521C2×D4—81no—
522C2×D4—81no—
523C2×D4—611no—
524C2×D4—1221no—
525C2×D4—1221no—
526C2×D4—1221no—
527C2×D4—1221no—
528C2×D4—1221no—
529C2×D4—1221no—
530C2×D4—4881no—
531C2×D4—4881no—
532C2×D4—2441no—
533C2×D4—2441no—
534C2×D4—2441no—
535C2×D4—2442no—
536C2×D4—2441no—
537C2×D4—2442no—
538C2×D4D4×D6111yescharacteristic
539C2×D4C2×C22⋊D6111yescharacteristic
540C22×C4—21no—
541C22×C4—21no—
542C22×C4—41no—
543C22×C4—1221no—
544C22×C4—2441no—
545SD16—42no—
546SD16—2442no—
547SD16#1175819914yescomplemented
548D8—42no—
549D8—42no—
550D8—81no—
551D8—1222no—
552D8—4881no—
553D8—4881no—
554D8—4881no—
555D8—4882no—
556D8—4882no—
557D8—4881no—
558D8—4881no—
559D8—2442no—
560D8—2442no—
561D8—2442no—
562D8—2442no—
563D8—2442no—
564D8—2442no—
565M16—42no—
566M16—2442no—
567C2×C8—21no—
568C2×C8—611no—
569C2×C8—611no—
570C2×C8—1221no—
571C2×C8—1221no—
572C2×C8—1221no—
573C2×C8—1221no—
574C2×C8—2442no—
575C2×C8C2×C22⋊D6111yescharacteristic
576C2×C8C2×C22⋊D6111yescharacteristic
577C4⋊C4—21no—
578C4⋊C4—611no—
579C4⋊C4D4×D6111yescharacteristic
580C4⋊C4C2×C22⋊D6111yescharacteristic
581C4⋊C4C2×C22⋊D6111yescharacteristic
582C22⋊C4—21no—
583C22⋊C4—21no—
584C22⋊C4—41no—
585C22⋊C4—41no—
586C22⋊C4—1221no—
587C22⋊C4—2441no—
588C22⋊C4—2441no—
589C42—21no—
590C42C23×D6111yescharacteristic
591C23—21no—
592C23—21no—
593C23—41no—
594C23—41no—
595C23—82no—
596C23—82no—
597C23—161no—
598C23—1221no—
599C23—9761no—
600C23—4882no—
601C23—4882no—
602C23—2441no—
603C23—2441no—
604Q8—21no—
605Q8C2×#1175819912yescomplemented
606D4—21no—
607D4—21no—
608D4—21no—
609D4—41no—
610D4—41no—
611D4—41no—
612D4—41no—
613D4—41no—
614D4—41no—
615D4—41no—
616D4—41no—
617D4—82no—
618D4—81no—
619D4—81no—
620D4—82no—
621D4—82no—
622D4—161no—
623D4—1221no—
624D4—1221no—
625D4—9761no—
626D4—4881no—
627D4—4881no—
628D4—4882no—
629D4—4882no—
630D4—4882no—
631D4—2441no—
632D4—2441no—
633D4—2441no—
634D4—2441no—
635D4—2441no—
636D4—2441no—
637D4—2441no—
638D4—2441no—
639D4C2×#1175819912yes—
640C2×C4—21no—
641C2×C4—21no—
642C2×C4—21no—
643C2×C4—21no—
644C2×C4—41no—
645C2×C4—41no—
646C2×C4—41no—
647C2×C4—41no—
648C2×C4—82no—
649C2×C4—1221no—
650C2×C4—4882no—
651C2×C4—2441no—
652C2×C4—2441no—
653C2×C4C2×D4×D6111yescharacteristic
654C2×C4C22×C22⋊D6111yescharacteristic
655C2×C4#1451982211yescharacteristic
656C8—21no—
657C8—41no—
658C8—612no—
659C8—1222no—
660C8—1221no—
661C8—2441no—
662C8—2441no—
663C8—2441no—
664C8C2×#1175819912yescomplemented
665C22—22no—
666C22—41no—
667C22—41no—
668C22—41no—
669C22—41no—
670C22—41no—
671C22—81no—
672C22—81no—
673C22—81no—
674C22—81no—
675C22—81no—
676C22—81no—
677C22—164no—
678C22—9764no—
679C22—4881no—
680C22—4881no—
681C22—4881no—
682C22—4881no—
683C22—4881no—
684C22—4881no—
685C22—2441no—
686C22—2441no—
687C22—2441no—
688C22D4×C22⋊D6111yescharacteristic
689C4—22no—
690C4—21no—
691C4—41no—
692C4—41no—
693C4—41no—
694C4—2441no—
695C4C22×#1175819911yescharacteristic
696C4AGL(1,ℤ/8)×D6111yescharacteristic
697C4#1684056511yescharacteristic
698C4#1684056611yescharacteristic
699C2—42no—
700C2—81no—
701C2—161no—
702C2—161no—
703C2—9761no—
704C2—9761no—
705C2—4881no—
706C2#2145743311yescharacteristic
707C2D4×#1175819911yescharacteristic
708C2#2145761611yescharacteristic
subgroupG/H
centerC22D4×C22⋊D61
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#3274C2
socleC22×C61D42
hypercenter#3274D61
hypocenterC61#19885
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ #3274

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. #21458791C2semidirect
2. D4×#11758199C2semidirect
3. #21457440C2semidirect
4. #21445433C2semidirect
5. #21445298C2semidirect
6. #21445295C2semidirect
7. #21445177C2semidirect
8. #21445172C2semidirect
9. #21444884C2semidirect
10. #21444882C2semidirect
11. #21444873C2semidirect
12. #21444872C2semidirect
13. #21444702C2semidirect
14. SD16×#92418C2semidirect
15. C61×#3274C2semidirect
18. #16840434C22semidirect
20. #16840139C22semidirect
23. #16840138C22semidirect
24. #16840138C22semidirect
25. #16840138C22semidirect
28. #16839962C22semidirect
32. #16839957C22semidirect
35. #16839954C22semidirect
40. Q8×#92418C22semidirect
43. #16839841C22semidirect
45. #16839514C22semidirect
46. #16839513C22semidirect
47. #16839506C22semidirect
48. #16839506C22semidirect
49. #16839505C22semidirect
50. #16839505C22semidirect
51. Q8⋊#92418C22semidirect
53. D4⋊#92418C22semidirect
54. #16839489C22semidirect
55. C8×#92418C22semidirect
57. C61×#550C22semidirect
59. C61×#493C22semidirect
61. C61×#460C22semidirect
66. C61×#436C22semidirect
191. SD16×C61D8semidirect
250. Q8×C61C2×D8semidirect
303. C8×C61C2×D8semidirect
309. #92418AGL(1,ℤ/8)semidirect
348. #3274D61semidirect
374. #550C2×D61semidirect
383. #493C2×D61semidirect
388. #460C2×D61semidirect
408. #436C2×D61semidirect
421. C61#19885semidirect
547. SD16#11758199semidirect
605. Q8C2×#11758199semidirect
664. C8C2×#11758199semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#214587912
D4×#117581992
#214574402
#214454332
#214452982
#214452952
#214451772
#214451722
#214448842
#214448822
#214448732
#214448722
#214447022
SD16×#924182
C61×#32742
#1988561