Finite Group Databasebeta

← #29669371

Subgroups — #29669371

Order 15616.

808 conjugacy classes of subgroups.

The subgroup lattice has 688 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21458452C211yesmaximal, characteristic, complemented
2D4×#11758200C211yesmaximal, characteristic, complemented
3#21457435C211yesmaximal, characteristic, complemented
4#21445484C211yesmaximal, characteristic, complemented
5#21445282C211yesmaximal, characteristic, complemented
6#21445256C211yesmaximal, characteristic, complemented
7#21445176C211yesmaximal, characteristic, complemented
8#21445172C211yesmaximal, characteristic, complemented
9#21444998C211yesmaximal, characteristic, complemented
10#21444921C211yesmaximal, characteristic, complemented
11#21444883C211yesmaximal, characteristic, complemented
12#21444873C211yesmaximal, characteristic, complemented
13#21444649C211yesmaximal, characteristic, complemented
14#21444631C211yesmaximal, characteristic, complemented
15D4×D8×C61C211yesmaximal, characteristic, complemented
16C2×#14519803—21no—
17C22×#11758200—21no—
18#16840433C2211yescharacteristic, complemented
19D4×Dic122C2211yescharacteristic, complemented
20#16840138C2211yescharacteristic, complemented
21#16840138C2211yescharacteristic, complemented
22#16840134C2211yescharacteristic, complemented
23#16840109C2211yescharacteristic, complemented
24#16840108C2211yescharacteristic, complemented
25#16840107—21no—
26#16840042C2211yescharacteristic, complemented
27#16840041—21no—
28#16840013—21no—
29#16839989—21no—
30#16839967C2211yescharacteristic
31#16839959C2211yescharacteristic, complemented
32#16839959C2211yescharacteristic, complemented
33#16839957C2211yescharacteristic
34#16839957C2211yescharacteristic, complemented
35#16839955—21no—
36#16839913—21no—
37#16839905—21no—
38#16839900—21no—
39#16839896—21no—
40C8×Dic122C2211yescharacteristic, complemented
41C4×#11758200C2211yescharacteristic
42#16839842—21no—
43#16839840C2211yescharacteristic, complemented
44D4×#92418C2211yescharacteristic, complemented
45#16839515C2211yescharacteristic, complemented
46#16839514C2211yescharacteristic, complemented
47#16839507C2211yescharacteristic, complemented
48#16839506C2211yescharacteristic, complemented
49#16839506C2211yescharacteristic, complemented
50#16839505C2211yescharacteristic, complemented
51#16839492C2211yescharacteristic, complemented
52D4⋊#92418C2211yescharacteristic
53D4⋊#92418C2211yescharacteristic, complemented
54D4⋊#92418C2211yescharacteristic, complemented
55#16839484C2211yescharacteristic, complemented
56C22×D8×C61—21no—
57D42×C61C2211yescharacteristic
58D42×C61C2211yescharacteristic
59C61×#493C2211yescharacteristic
60C61×#466—21no—
61C61×#459C2211yescharacteristic
62C61×#459C2211yescharacteristic
63C61×#447—21no—
64C61×#447—21no—
65C4×D8×C61C2211yescharacteristic
66C8×D4×C61C2211yescharacteristic
67C2×#11758222—21no—
68C22×Dic122—21no—
69#14519803—44no—
70C2×#11758200—21no—
71C2×#11758200—21no—
72C2×#11758200—21no—
73C2×#11758200—21no—
74C2×#11758200—21no—
75C2×#11758200—44no—
76C2×#11758200D411yescharacteristic
77C2×#11758199—21no—
78C2×#11758199—21no—
79C2×#11758199—21no—
80C2×#11758199—21no—
81C2×#11758199—41no—
82C2×#11758194—21no—
83C22×#92418—21no—
84#14519777C2311yescharacteristic
85#14519776—21no—
86#14519774C2311yescharacteristic
87#14519773—21no—
88#14519756—21no—
89#14519754—41no—
90#14519751—21no—
91#14519750—41no—
92C4×Dic122C2311yescharacteristic
93D4×Dic61—21no—
94#14519739—21no—
95#14519710—21no—
96#14519708D411yescharacteristic
97C4⋊#92418C2311yescharacteristic
98C4⋊#92418C2311yescharacteristic
99C4⋊#92418C2311yescharacteristic
100#14519706—21no—
101#14519700—21no—
102#14519700D411yescharacteristic
103#14519699—21no—
104#14519699—21no—
105#14519698—21no—
106#14519698—21no—
107#14519698—21no—
108#14519698—41no—
109#14519698D411yescharacteristic
110#14519697—21no—
111#14519697—21no—
112C22⋊#92418—21no—
113C22⋊#92418—21no—
114C4×#92418C2311yescharacteristic
115C4×#92418C2311yescharacteristic
116C8×Dic61—21no—
117C22×D4×C61—21no—
118C22×D4×C61—21no—
119C22×D4×C61—21no—
120C22×D4×C61—21no—
121C2×D8×C61—21no—
122C2×D8×C61—21no—
123C2×D8×C61—21no—
124C2×D8×C61—21no—
125C2×D8×C61—44no—
126C2×D8×C61D411yescharacteristic
127C22×C8×C61—21no—
128#127×C61C2311yescharacteristic
129#127×C61C2311yescharacteristic
130#121×C61—21no—
131#121×C61—21no—
132#121×C61—21no—
133#121×C61—21no—
134#120×C61—41no—
135#120×C61—41no—
136C4×D4×C61C2311yescharacteristic
137C4×D4×C61C2311yescharacteristic
138C4×D4×C61C2311yescharacteristic
139#107×C61D411yescharacteristic
140C4⋊C8×C61C2311yescharacteristic
141#102×C61—21no—
142#102×C61—21no—
143#102×C61D411yescharacteristic
144#102×C61D411yescharacteristic
145C22⋊C8×C61—21no—
146C4×C8×C61C2311yescharacteristic
147C24×C61—41no—
148C24×C61—41no—
149C2×D4×C61—21no—
150C2×D4×C61—21no—
151C2×D4×C61—21no—
152C2×D4×C61—21no—
153C2×D4×C61—21no—
154C2×D4×C61—21no—
155C2×D4×C61—41no—
156C2×D4×C61—41no—
157C2×D4×C61—41no—
158C2×D4×C61—41no—
159C2×D4×C61—42no—
160C2×D4×C61—42no—
161C2×D4×C61—42no—
162C2×D4×C61—42no—
163C2×D4×C61—81no—
164C2×D4×C61—81no—
165C2×D4×C61—81no—
166C2×D4×C61—81no—
167C2×D4×C61C2×D411yescharacteristic
168C2×D4×C61C2×D411yescharacteristic
169C2×D4×C61C2×D411yescharacteristic
170C22×C4×C61—21no—
171C22×C4×C61—21no—
172C22×C4×C61—21no—
173C2×C22⋊D61—41no—
174C2×C22⋊D61—41no—
175C22×Dic61—41no—
176#11758222—42no—
177#11758222—81no—
178C2×D244—21no—
179C2×D244—21no—
180C2×D244—21no—
181C2×C4×D61—41no—
182C2×Dic122—21no—
183C2×Dic122—42no—
184C2×Dic122—41no—
185C2×Dic122—81no—
186C2×Dic122C2×D411yescharacteristic
187D8×C61—22no—
188D8×C61—42no—
189D8×C61—42no—
190D8×C61—42no—
191D8×C61—81no—
192C2×C8×C61—21no—
193C2×C8×C61—42no—
194C2×C8×C61C2×D411yescharacteristic
195C2×C8×C61C2×D411yescharacteristic
196C4⋊C4×C61C2×D411yescharacteristic
197C4⋊C4×C61C2×D411yescharacteristic
198C4⋊C4×C61C2×D411yescharacteristic
199C22⋊C4×C61—21no—
200C22⋊C4×C61—21no—
201C22⋊C4×C61—21no—
202C22⋊C4×C61—41no—
203C22⋊C4×C61—41no—
204C42×C61C2411yescharacteristic
205C22⋊Dic61—41no—
206#11758200—42no—
207#11758200—42no—
208#11758200—42no—
209#11758200—42no—
210#11758200—81no—
211#11758200D814yescomplemented
212#11758199—42no—
213#11758199—42no—
214#11758199—42no—
215#11758199—81no—
216#11758199—82no—
217#11758198—41no—
218C4⋊Dic61—21no—
219C4⋊Dic61—21no—
220C4⋊Dic61C2×D411yescharacteristic
221#11758196—21no—
222#11758196—41no—
223C4×Dic61—21no—
224#11758194—42no—
225C2×#92418—21no—
226C2×#92418—21no—
227C2×#92418—21no—
228C2×#92418—21no—
229C2×#92418—42no—
230C2×#92418C2×D411yescharacteristic
231C2×#92418C2×D411yescharacteristic
232C23×C61—21no—
233C23×C61—21no—
234C23×C61—21no—
235C23×C61—41no—
236C23×C61—41no—
237C23×C61—82no—
238C23×C61—82no—
239C23×C61—82no—
240C23×C61—82no—
241C23×C61—161no—
242C23×C61—161no—
243C22×D61—41no—
244D4×C61—21no—
245D4×C61—21no—
246D4×C61—21no—
247D4×C61—21no—
248D4×C61—41no—
249D4×C61—41no—
250D4×C61—41no—
251D4×C61—41no—
252D4×C61—41no—
253D4×C61—41no—
254D4×C61—41no—
255D4×C61—41no—
256D4×C61—81no—
257D4×C61—81no—
258D4×C61—82no—
259D4×C61—82no—
260D4×C61—82no—
261D4×C61—82no—
262D4×C61C2×D812yes—
263D4×C61AGL(1,ℤ/8)12yescomplemented
264C2×C4×C61—21no—
265C2×C4×C61—21no—
266C2×C4×C61—21no—
267C2×C4×C61—41no—
268C2×C4×C61—41no—
269C2×C4×C61—41no—
270C2×C4×C61—41no—
271C2×C4×C61—41no—
272C2×C4×C61—41no—
273C2×C4×C61C22×D411yescharacteristic
274C2×C4×C61C22×D411yescharacteristic
275C2×C4×C61H(5,𝔽2)11yescharacteristic
276C22⋊D61—82no—
277C22⋊D61—161no—
278C2×Dic61—21no—
279C2×Dic61—21no—
280C2×Dic61—41no—
281C2×Dic61—82no—
282D244—41no—
283D244—41no—
284D244—41no—
285D244—41no—
286D244—81no—
287C4×D61—82no—
288Dic122—21no—
289Dic122—41no—
290Dic122—41no—
291Dic122—82no—
292Dic122C2×D812yescomplemented
293C8×C61—21no—
294C8×C61—21no—
295C8×C61—41no—
296#92418—21no—
297#92418—41no—
298#92418—41no—
299#92418—41no—
300#92418—41no—
301#92418C2×D812yescomplemented
302#19756—611nomaximal, Sylow
303C22×C61—22no—
304C22×C61—22no—
305C22×C61—41no—
306C22×C61—41no—
307C22×C61—41no—
308C22×C61—41no—
309C22×C61—41no—
310C22×C61—41no—
311C22×C61—41no—
312C22×C61—81no—
313C22×C61—81no—
314C22×C61—81no—
315C22×C61—81no—
316C22×C61—81no—
317C22×C61—81no—
318C22×C61—164no—
319C22×C61—164no—
320C22×C61D4211yescharacteristic
321C2×D61—81no—
322C2×D61—81no—
323C2×D61—81no—
324C4×C61—21no—
325C4×C61—41no—
326C4×C61—41no—
327C4×C61—41no—
328C4×C61C22×D811yescharacteristic
329C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
330C4×C61#57611yescharacteristic
331C4×C61#57711yescharacteristic
332Dic61—22no—
333Dic61—41no—
334Dic61—81no—
335#3358—611no—
336D4×SD16—611no—
337D4×D8D6111yescharacteristic, complemented
338#3274—611no—
339#1721—611no—
340#1685—611no—
341#1681—611no—
342#1668—611no—
343#1667—611no—
344#1639—611no—
345#1627—611no—
346#1621—611no—
347#1619—611no—
348#1581—611no—
349SD16⋊C8—611no—
350C2×C61—42no—
351C2×C61—42no—
352C2×C61—81no—
353C2×C61—161no—
354C2×C61—161no—
355C2×C61#326511yescharacteristic
356C2×C61D4×D811yescharacteristic
357C2×C61#328811yescharacteristic
358D61—161no—
359C2×AGL(1,ℤ/8)—1221no—
360C22×SD16—1221no—
361C22×D8—21no—
362#550—611no—
363D4×Q8—611no—
364D42C2×D6111yescharacteristic
365D42C2×D6111yescharacteristic
366#493—611no—
367#493—611no—
368#493C2×D6111yescharacteristic
369#492—611no—
370#486—611no—
371#486—611no—
372#486—1221no—
373#474—611no—
374#474—1221no—
375#469—1221no—
376#466—21no—
377#465—1221no—
378#461—611no—
379#460—611no—
380#460—611no—
381#459—611no—
382#459—611no—
383#459—1221no—
384#459C2×D6111yescharacteristic
385#459C2×D6111yescharacteristic
386#450—1221no—
387#449—1221no—
388#448—1221no—
389#447—21no—
390#447—21no—
391#447—1221no—
392C8×Q8—611no—
393C4×SD16—611no—
394C4×D8—1221no—
395C4×D8C2×D6111yescharacteristic
396#436—611no—
397C8×D4—611no—
398C8×D4C2×D6111yescharacteristic
399#335—611no—
400#335—611no—
401#332—611no—
402#331—611no—
403#331—611no—
404#331—611no—
405Q8⋊C8—611no—
406D4⋊C8—611no—
407D4⋊C8—611no—
408D4⋊C8—611no—
409#322—611no—
410C61#1975611yescharacteristic, complemented, Sylow
411C2×#41—1221no—
412C22×Q8—1221no—
413C22×D4—21no—
414C22×D4—21no—
415C22×D4—21no—
416C22×D4—21no—
417AGL(1,ℤ/8)—2444no—
418C2×SD16—611no—
419C2×SD16—1221no—
420C2×SD16—1221no—
421C2×SD16—1221no—
422C2×SD16—1221no—
423C2×SD16—1221no—
424C2×SD16—2444no—
425C2×D8—21no—
426C2×D8—21no—
427C2×D8—21no—
428C2×D8—21no—
429C2×D8—44no—
430C2×D8—1221no—
431C2×D8—1221no—
432C2×D8—1221no—
433C2×D8—1221no—
434C2×D8—2441no—
435C2×D8C22⋊D6111yescharacteristic
436C2×M16—1221no—
437C22×C8—21no—
438C22×C8—1221no—
439C4⋊Q8—611no—
440C4⋊Q8—1221no—
441#127—611no—
442#127—1221no—
443#127C22×D6111yescharacteristic
444#127C22×D6111yescharacteristic
445C22⋊Q8—1221no—
446C22⋊Q8—2441no—
447#121—21no—
448#121—21no—
449#121—21no—
450#121—21no—
451#121—1221no—
452#121—2441no—
453#120—41no—
454#120—41no—
455C4×Q8—611no—
456C4×D4—1221no—
457C4×D4—1221no—
458C4×D4C22×D6111yescharacteristic
459C4×D4C22×D6111yescharacteristic
460C4×D4C22×D6111yescharacteristic
461#107—1221no—
462#107C22⋊D6111yescharacteristic
463#106—611no—
464C4⋊C8—611no—
465C4⋊C8—611no—
466C4⋊C8—611no—
467C4⋊C8—1221no—
468C4⋊C8C22×D6111yescharacteristic
469#103—611no—
470#103—1221no—
471#102—21no—
472#102—21no—
473#102—611no—
474#102—1221no—
475#102—1221no—
476#102—1221no—
477#102—1221no—
478#102—1221no—
479#102—1221no—
480#102—1221no—
481#102—2441no—
482#102C22⋊D6111yescharacteristic
483#102C22⋊D6111yescharacteristic
484C22⋊C8—21no—
485C22⋊C8—1221no—
486C22⋊C8—1221no—
487C4×C8—611no—
488C4×C8—611no—
489C4×C8—1221no—
490C4×C8C22×D6111yescharacteristic
491C24—41no—
492C24—41no—
493#41—4881no—
494#41—2442no—
495C2×Q8—611no—
496C2×Q8—1221no—
497C2×Q8—4881no—
498C2×Q8—2441no—
499C2×Q8—2442no—
500C2×D4—21no—
501C2×D4—21no—
502C2×D4—21no—
503C2×D4—21no—
504C2×D4—21no—
505C2×D4—21no—
506C2×D4—41no—
507C2×D4—41no—
508C2×D4—41no—
509C2×D4—41no—
510C2×D4—42no—
511C2×D4—42no—
512C2×D4—42no—
513C2×D4—42no—
514C2×D4—81no—
515C2×D4—81no—
516C2×D4—81no—
517C2×D4—81no—
518C2×D4—1221no—
519C2×D4—1221no—
520C2×D4—1221no—
521C2×D4—2441no—
522C2×D4—2441no—
523C2×D4D4×D6111yescharacteristic
524C2×D4C2×C22⋊D6111yescharacteristic
525C2×D4C2×C22⋊D6111yescharacteristic
526C22×C4—21no—
527C22×C4—21no—
528C22×C4—21no—
529C22×C4—2441no—
530C22×C4—2441no—
531SD16—614no—
532SD16—4881no—
533SD16—2442no—
534SD16—2442no—
535SD16—2442no—
536SD16—2442no—
537D8—22no—
538D8—42no—
539D8—42no—
540D8—42no—
541D8—81no—
542D8—4881no—
543D8—4882no—
544D8—2442no—
545D8—2442no—
546D8—2442no—
547M16—2442no—
548C2×C8—21no—
549C2×C8—42no—
550C2×C8—611no—
551C2×C8—611no—
552C2×C8—1221no—
553C2×C8—1221no—
554C2×C8—1221no—
555C2×C8—1221no—
556C2×C8—2442no—
557C2×C8C2×C22⋊D6111yescharacteristic
558C2×C8C2×C22⋊D6111yescharacteristic
559C4⋊C4—611no—
560C4⋊C4—1221no—
561C4⋊C4—1221no—
562C4⋊C4—1221no—
563C4⋊C4—2441no—
564C4⋊C4D4×D6111yescharacteristic
565C4⋊C4C2×C22⋊D6111yescharacteristic
566C4⋊C4C2×C22⋊D6111yescharacteristic
567C22⋊C4—21no—
568C22⋊C4—21no—
569C22⋊C4—21no—
570C22⋊C4—41no—
571C22⋊C4—41no—
572C22⋊C4—2441no—
573C22⋊C4—2441no—
574C42—1221no—
575C42C23×D6111yescharacteristic
576C23—21no—
577C23—21no—
578C23—21no—
579C23—41no—
580C23—41no—
581C23—82no—
582C23—82no—
583C23—82no—
584C23—82no—
585C23—161no—
586C23—161no—
587C23—2441no—
588Q8—612no—
589Q8—1221no—
590Q8—4882no—
591Q8—2441no—
592Q8—2441no—
593D4—21no—
594D4—21no—
595D4—21no—
596D4—21no—
597D4—41no—
598D4—41no—
599D4—41no—
600D4—41no—
601D4—41no—
602D4—41no—
603D4—41no—
604D4—41no—
605D4—81no—
606D4—81no—
607D4—82no—
608D4—82no—
609D4—82no—
610D4—82no—
611D4—9761no—
612D4—4881no—
613D4—4882no—
614D4—2441no—
615D4—2441no—
616D4—2441no—
617D4—2441no—
618D4D8×D6112yes—
619D4#1451980312yescomplemented
620C2×C4—21no—
621C2×C4—21no—
622C2×C4—21no—
623C2×C4—41no—
624C2×C4—41no—
625C2×C4—41no—
626C2×C4—41no—
627C2×C4—41no—
628C2×C4—41no—
629C2×C4—1221no—
630C2×C4—1221no—
631C2×C4—4882no—
632C2×C4—4882no—
633C2×C4—2441no—
634C2×C4C2×D4×D6111yescharacteristic
635C2×C4C22×C22⋊D6111yescharacteristic
636C2×C4#1451982211yescharacteristic
637C8—21no—
638C8—21no—
639C8—41no—
640C8—612no—
641C8—1221no—
642C8—2441no—
643C8—2441no—
644C8—2441no—
645C8—2441no—
646C22—22no—
647C22—22no—
648C22—41no—
649C22—41no—
650C22—41no—
651C22—41no—
652C22—41no—
653C22—41no—
654C22—41no—
655C22—81no—
656C22—81no—
657C22—81no—
658C22—81no—
659C22—81no—
660C22—81no—
661C22—164no—
662C22—164no—
663C22—4881no—
664C22—4881no—
665C22—4881no—
666C22D4×C22⋊D6111yescharacteristic
667C4—21no—
668C4—41no—
669C4—41no—
670C4—41no—
671C4—1222no—
672C4—4881no—
673C4—2441no—
674C4C2×D8×D6111yescharacteristic
675C4C2×#1451980311yescharacteristic
676C4#1684056311yescharacteristic
677C4#1684057311yescharacteristic
678C2—42no—
679C2—42no—
680C2—81no—
681C2—161no—
682C2—161no—
683C2—9761no—
684C2#2145731011yescharacteristic
685C2D8×C22⋊D6111yescharacteristic
686C2#2145758311yescharacteristic
subgroupG/H
centerC22D4×C22⋊D61
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingD4×D8×C61C2
socleC22×C61D42
hypercenterD4×D8D61
hypocenterC61#19756
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ D4×D8

Complements

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

NH ≅ G/Ntype
1. #21458452C2semidirect
2. D4×#11758200C2semidirect
3. #21457435C2semidirect
4. #21445484C2semidirect
5. #21445282C2semidirect
6. #21445256C2semidirect
7. #21445176C2semidirect
8. #21445172C2semidirect
9. #21444998C2semidirect
10. #21444921C2semidirect
11. #21444883C2semidirect
12. #21444873C2semidirect
13. #21444649C2semidirect
14. #21444631C2semidirect
15. D4×D8×C61C2semidirect
18. #16840433C22semidirect
19. D4×Dic122C22semidirect
20. #16840138C22semidirect
21. #16840138C22semidirect
22. #16840134C22semidirect
23. #16840109C22semidirect
24. #16840108C22semidirect
26. #16840042C22semidirect
31. #16839959C22semidirect
32. #16839959C22semidirect
34. #16839957C22semidirect
40. C8×Dic122C22semidirect
43. #16839840C22semidirect
44. D4×#92418C22semidirect
45. #16839515C22semidirect
46. #16839514C22semidirect
47. #16839507C22semidirect
48. #16839506C22semidirect
49. #16839506C22semidirect
50. #16839505C22semidirect
51. #16839492C22semidirect
53. D4⋊#92418C22semidirect
54. D4⋊#92418C22semidirect
55. #16839484C22semidirect
211. #11758200D8semidirect
263. D4×C61AGL(1,ℤ/8)semidirect
292. Dic122C2×D8semidirect
301. #92418C2×D8semidirect
337. D4×D8D61semidirect
410. C61#19756semidirect
619. D4#14519803semidirect

Maximal subgroups

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

subgroupindex
#214584522
D4×#117582002
#214574352
#214454842
#214452822
#214452562
#214451762
#214451722
#214449982
#214449212
#214448832
#214448732
#214446492
#214446312
D4×D8×C612
#1975661