Finite Group Databasebeta

← #29669376

Subgroups — #29669376

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
1D4×#11758201C211yesmaximal, characteristic, complemented
2D8×D244C211yesmaximal, characteristic, complemented
3#21457439C211yesmaximal, characteristic, complemented
4#21445487C211yesmaximal, characteristic, complemented
5#21445286C211yesmaximal, characteristic, complemented
6#21445253C211yesmaximal, characteristic, complemented
7#21445179C211yesmaximal, characteristic, complemented
8#21445172C211yesmaximal, characteristic, complemented
9#21444997C211yesmaximal, characteristic, complemented
10#21444922C211yesmaximal, characteristic, complemented
11#21444884C211yesmaximal, characteristic, complemented
12#21444872C211yesmaximal, characteristic, complemented
13#21444649C211yesmaximal, characteristic, complemented
14#21444629C211yesmaximal, characteristic, complemented
15C61×#3358C211yesmaximal, characteristic, complemented
16C2×#14519802—21no—
17C22×#11758201—21no—
18C2×D8×D61—21no—
19D4×D244C2211yescharacteristic
20D4×D244C2211yescharacteristic
21#16840139C2211yescharacteristic
22#16840138C2211yescharacteristic, complemented
23#16840138C2211yescharacteristic, complemented
24#16840136C2211yescharacteristic, complemented
25#16840110C2211yescharacteristic, complemented
26#16840013—21no—
27#16839999—21no—
28#16839993—21no—
29#16839964C2211yescharacteristic
30#16839962C2211yescharacteristic, complemented
31#16839962C2211yescharacteristic, complemented
32#16839957C2211yescharacteristic
33#16839957C2211yescharacteristic
34#16839954C2211yescharacteristic
35#16839912—21no—
36#16839907—21no—
37#16839901—21no—
38#16839897—21no—
39#16839897—21no—
40#16839895—21no—
41C4×#11758201C2211yescharacteristic
42#16839840C2211yescharacteristic, complemented
43C8×D244C2211yescharacteristic
44D4×#92418C2211yescharacteristic, complemented
45#16839515C2211yescharacteristic, complemented
46#16839514C2211yescharacteristic, complemented
47#16839508C2211yescharacteristic, complemented
48#16839506C2211yescharacteristic, complemented
49#16839505C2211yescharacteristic, complemented
50#16839505C2211yescharacteristic, complemented
51Q8⋊#92418C2211yescharacteristic, complemented
52D4⋊#92418C2211yescharacteristic, complemented
53D4⋊#92418C2211yescharacteristic, complemented
54#16839489C2211yescharacteristic
55#16839484C2211yescharacteristic, complemented
56C61×#550C2211yescharacteristic, complemented
57D4×Q8×C61C2211yescharacteristic, complemented
58C61×#486—21no—
59C61×#486C2211yescharacteristic, complemented
60C61×#474—21no—
61C61×#474C2211yescharacteristic, complemented
62C61×#459—21no—
63C61×#459C2211yescharacteristic, complemented
64C8×Q8×C61C2211yescharacteristic, complemented
65C4×D8×C61—21no—
66C4×D8×C61C2211yescharacteristic
67C22×D244—21no—
68C22×D244—21no—
69C2×D4×D61—21no—
70C2×D4×D61—21no—
71#14519802—44no—
72C2×#11758201—21no—
73C2×#11758201—21no—
74C2×#11758201—21no—
75C2×#11758201—21no—
76C2×#11758201—21no—
77C2×#11758201—44no—
78C2×#11758201D411yescharacteristic
79C2×D488—21no—
80C2×D488—21no—
81C2×#11758199—21no—
82C2×#11758199—21no—
83C2×#11758199—21no—
84C2×#11758199—21no—
85C2×#11758199—21no—
86C2×#11758199—41no—
87D8×D61—44no—
88C2×#11758194—21no—
89C22×#92418—21no—
90C2×C8×D61—21no—
91#14519774C2311yescharacteristic
92#14519774C2311yescharacteristic
93#14519752—21no—
94#14519752—21no—
95#14519751—21no—
96#14519751—21no—
97#14519747—41no—
98#14519747—41no—
99C4×D244C2311yescharacteristic
100#14519708D411yescharacteristic
101C4⋊#92418C2311yescharacteristic
102C4⋊#92418C2311yescharacteristic
103C4⋊#92418C2311yescharacteristic
104#14519701—21no—
105#14519701D411yescharacteristic
106#14519698—21no—
107#14519697—21no—
108#14519697—21no—
109#14519697—21no—
110#14519697—21no—
111#14519697—41no—
112#14519697D411yescharacteristic
113C22⋊#92418—21no—
114C22⋊#92418—21no—
115#14519689—21no—
116C4×#92418C2311yescharacteristic
117C4×#92418C2311yescharacteristic
118C2×#41×C61—21no—
119C22×Q8×C61—21no—
120C2×D8×C61—21no—
121C2×D8×C61D411yescharacteristic
122C4⋊Q8×C61—21no—
123C4⋊Q8×C61C2311yescharacteristic
124#127×C61—21no—
125#127×C61C2311yescharacteristic
126C22⋊Q8×C61—21no—
127C22⋊Q8×C61—41no—
128#121×C61—21no—
129#121×C61—41no—
130C4×Q8×C61C2311yescharacteristic
131C4×D4×C61—21no—
132C4×D4×C61—21no—
133C4×D4×C61C2311yescharacteristic
134C4×D4×C61C2311yescharacteristic
135#107×C61—21no—
136#107×C61D411yescharacteristic
137C4⋊C8×C61—21no—
138C4⋊C8×C61C2311yescharacteristic
139#102×C61—21no—
140#102×C61—21no—
141#102×C61—21no—
142#102×C61—21no—
143#102×C61D411yescharacteristic
144#102×C61D411yescharacteristic
145C4×C8×C61—21no—
146C4×C8×C61C2311yescharacteristic
147C23×D61—41no—
148C23×D61—41no—
149#41×C61—42no—
150#41×C61—81no—
151C2×Q8×C61—21no—
152C2×Q8×C61—41no—
153C2×Q8×C61—42no—
154C2×Q8×C61—81no—
155C2×Q8×C61C2×D411yescharacteristic
156C2×D4×C61—21no—
157C2×D4×C61—21no—
158C2×D4×C61—21no—
159C2×D4×C61—41no—
160C2×D4×C61—41no—
161C2×D4×C61C2×D411yescharacteristic
162C2×D4×C61C2×D411yescharacteristic
163C22×C4×C61—21no—
164C22×C4×C61—21no—
165C22×C4×C61—41no—
166C22×C4×C61—41no—
167C2×C22⋊D61—41no—
168C2×C22⋊D61—41no—
169D4×D61—42no—
170D4×D61—42no—
171D4×D61—81no—
172D4×D61—81no—
173C2×D244—21no—
174C2×D244—21no—
175C2×D244—21no—
176C2×D244—21no—
177C2×D244—21no—
178C2×D244—21no—
179C2×D244—42no—
180C2×D244—42no—
181C2×D244—41no—
182C2×D244—41no—
183C2×D244—81no—
184C2×D244—81no—
185C2×D244C2×D411yescharacteristic
186C2×C4×D61—21no—
187D8×C61—22no—
188D8×C61—42no—
189C2×C8×C61—21no—
190C2×C8×C61C2×D411yescharacteristic
191C2×C8×C61C2×D411yescharacteristic
192C4⋊C4×C61—21no—
193C4⋊C4×C61—21no—
194C4⋊C4×C61—21no—
195C4⋊C4×C61—41no—
196C4⋊C4×C61C2×D411yescharacteristic
197C4⋊C4×C61C2×D411yescharacteristic
198C4⋊C4×C61C2×D411yescharacteristic
199C22⋊C4×C61—21no—
200C22⋊C4×C61—21no—
201C22⋊C4×C61—41no—
202C22⋊C4×C61—41no—
203C42×C61—21no—
204C42×C61C2411yescharacteristic
205#11758201—42no—
206#11758201—42no—
207#11758201—42no—
208#11758201—42no—
209#11758201—81no—
210#11758201D814yescomplemented
211#11758199—42no—
212#11758199—42no—
213#11758199—42no—
214#11758199—42no—
215#11758199—81no—
216#11758199—82no—
217#11758198—21no—
218#11758198—41no—
219#11758198—41no—
220C4⋊Dic61C2×D411yescharacteristic
221#11758194—42no—
222C2×#92418—21no—
223C2×#92418—21no—
224C2×#92418—21no—
225C2×#92418—21no—
226C2×#92418—42no—
227C2×#92418C2×D411yescharacteristic
228C2×#92418C2×D411yescharacteristic
229D488—42no—
230D488—81no—
231C8×D61—42no—
232C23×C61—21no—
233C23×C61—21no—
234C23×C61—41no—
235C22×D61—21no—
236C22×D61—41no—
237C22×D61—41no—
238C22×D61—82no—
239C22×D61—82no—
240C22×D61—82no—
241C22×D61—82no—
242C22×D61—161no—
243C22×D61—161no—
244Q8×C61—21no—
245Q8×C61—41no—
246Q8×C61—41no—
247Q8×C61—82no—
248Q8×C61C2×D812yescomplemented
249D4×C61—21no—
250D4×C61—21no—
251D4×C61—21no—
252D4×C61—41no—
253D4×C61—41no—
254D4×C61—41no—
255D4×C61—41no—
256D4×C61—82no—
257D4×C61—81no—
258D4×C61—161no—
259D4×C61AGL(1,ℤ/8)12yescomplemented
260C2×C4×C61—21no—
261C2×C4×C61—21no—
262C2×C4×C61—21no—
263C2×C4×C61—21no—
264C2×C4×C61—41no—
265C2×C4×C61—41no—
266C2×C4×C61—41no—
267C2×C4×C61—41no—
268C2×C4×C61—41no—
269C2×C4×C61—82no—
270C2×C4×C61—82no—
271C2×C4×C61C22×D411yescharacteristic
272C2×C4×C61C22×D411yescharacteristic
273C2×C4×C61H(5,𝔽2)11yescharacteristic
274C22⋊D61—82no—
275C22⋊D61—82no—
276C2×Dic61—21no—
277D244—21no—
278D244—41no—
279D244—41no—
280D244—41no—
281D244—41no—
282D244—41no—
283D244—41no—
284D244—41no—
285D244—41no—
286D244—81no—
287D244—81no—
288D244—82no—
289D244—82no—
290D244C2×D812yes—
291C4×D61—41no—
292C4×D61—41no—
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—41no—
305C22×C61—41no—
306C22×C61—41no—
307C22×C61—41no—
308C22×C61—41no—
309C22×C61—81no—
310C22×C61—81no—
311C22×C61—81no—
312C22×C61D4211yescharacteristic
313C2×D61—22no—
314C2×D61—41no—
315C2×D61—41no—
316C2×D61—81no—
317C2×D61—81no—
318C2×D61—81no—
319C2×D61—81no—
320C2×D61—81no—
321C2×D61—81no—
322C2×D61—164no—
323C2×D61—164no—
324C4×C61—22no—
325C4×C61—21no—
326C4×C61—41no—
327C4×C61—41no—
328C4×C61—41no—
329C4×C61—81no—
330C4×C61C22×D811yescharacteristic
331C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
332C4×C61#57611yescharacteristic
333C4×C61#57711yescharacteristic
334Dic61—41no—
335#3358D6111yescharacteristic, complemented
336D4×SD16—611no—
337D4×D8—611no—
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—81no—
352C2×C61—161no—
353C2×C61#326511yescharacteristic
354C2×C61D4×D811yescharacteristic
355C2×C61#328811yescharacteristic
356D61—42no—
357D61—161no—
358D61—161no—
359C2×AGL(1,ℤ/8)—1221no—
360C22×SD16—1221no—
361C22×D8—1221no—
362#550C2×D6111yescharacteristic, complemented
363D4×Q8C2×D6111yescharacteristic, complemented
364D42—611no—
365D42—611no—
366#493—611no—
367#493—611no—
368#493—611no—
369#492—611no—
370#486—21no—
371#486—611no—
372#486C2×D6111yescharacteristic, complemented
373#474—21no—
374#474C2×D6111yescharacteristic, complemented
375#469—1221no—
376#466—1221no—
377#465—1221no—
378#461—611no—
379#460—611no—
380#460—611no—
381#459—21no—
382#459—611no—
383#459—611no—
384#459—611no—
385#459C2×D6111yescharacteristic, complemented
386#450—1221no—
387#449—1221no—
388#448—1221no—
389#447—1221no—
390#447—1221no—
391#447—1221no—
392C8×Q8C2×D6111yescharacteristic, complemented
393C4×SD16—611no—
394C4×D8—21no—
395C4×D8C2×D6111yescharacteristic
396#436—611no—
397C8×D4—611no—
398C8×D4—611no—
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—21no—
412C22×Q8—21no—
413C22×D4—1221no—
414C22×D4—1221no—
415C22×D4—1221no—
416C22×D4—1221no—
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—1221no—
427C2×D8—1221no—
428C2×D8—1221no—
429C2×D8—1221no—
430C2×D8—1221no—
431C2×D8—1221no—
432C2×D8—1221no—
433C2×D8—2441no—
434C2×D8—2444no—
435C2×D8C22⋊D6111yescharacteristic
436C2×M16—1221no—
437C22×C8—1221no—
438C22×C8—1221no—
439C4⋊Q8—21no—
440C4⋊Q8C22×D6111yescharacteristic
441#127—21no—
442#127—611no—
443#127—611no—
444#127C22×D6111yescharacteristic
445C22⋊Q8—21no—
446C22⋊Q8—41no—
447#121—21no—
448#121—41no—
449#121—1221no—
450#121—1221no—
451#121—1221no—
452#121—1221no—
453#120—2441no—
454#120—2441no—
455C4×Q8C22×D6111yescharacteristic
456C4×D4—21no—
457C4×D4—21no—
458C4×D4—611no—
459C4×D4C22×D6111yescharacteristic
460C4×D4C22×D6111yescharacteristic
461#107—21no—
462#107C22⋊D6111yescharacteristic
463#106—611no—
464C4⋊C8—21no—
465C4⋊C8—611no—
466C4⋊C8—611no—
467C4⋊C8—611no—
468C4⋊C8C22×D6111yescharacteristic
469#103—611no—
470#103—1221no—
471#102—21no—
472#102—21no—
473#102—21no—
474#102—21no—
475#102—611no—
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—1221no—
485C22⋊C8—1221no—
486C22⋊C8—1221no—
487C4×C8—21no—
488C4×C8—611no—
489C4×C8—611no—
490C4×C8C22×D6111yescharacteristic
491C24—2441no—
492C24—2441no—
493#41—42no—
494#41—81no—
495C2×Q8—21no—
496C2×Q8—41no—
497C2×Q8—42no—
498C2×Q8—81no—
499C2×Q8D4×D6111yescharacteristic
500C2×D4—21no—
501C2×D4—21no—
502C2×D4—21no—
503C2×D4—41no—
504C2×D4—41no—
505C2×D4—611no—
506C2×D4—1221no—
507C2×D4—1221no—
508C2×D4—1221no—
509C2×D4—1221no—
510C2×D4—1221no—
511C2×D4—1221no—
512C2×D4—4881no—
513C2×D4—4881no—
514C2×D4—4881no—
515C2×D4—4881no—
516C2×D4—2441no—
517C2×D4—2441no—
518C2×D4—2442no—
519C2×D4—2442no—
520C2×D4—2441no—
521C2×D4—2441no—
522C2×D4—2442no—
523C2×D4—2442no—
524C2×D4C2×C22⋊D6111yescharacteristic
525C2×D4C2×C22⋊D6111yescharacteristic
526C22×C4—21no—
527C22×C4—21no—
528C22×C4—41no—
529C22×C4—41no—
530C22×C4—1221no—
531SD16—614no—
532SD16—4881no—
533SD16—2442no—
534SD16—2442no—
535SD16—2442no—
536SD16—2442no—
537D8—22no—
538D8—42no—
539D8—4881no—
540D8—4881no—
541D8—4882no—
542D8—2442no—
543D8—2442no—
544D8—2442no—
545D8—2442no—
546D8—2442no—
547M16—2442no—
548C2×C8—21no—
549C2×C8—611no—
550C2×C8—611no—
551C2×C8—1221no—
552C2×C8—1221no—
553C2×C8—1221no—
554C2×C8—1221no—
555C2×C8—2442no—
556C2×C8—2442no—
557C2×C8C2×C22⋊D6111yescharacteristic
558C2×C8C2×C22⋊D6111yescharacteristic
559C4⋊C4—21no—
560C4⋊C4—21no—
561C4⋊C4—21no—
562C4⋊C4—41no—
563C4⋊C4—611no—
564C4⋊C4D4×D6111yescharacteristic
565C4⋊C4C2×C22⋊D6111yescharacteristic
566C4⋊C4C2×C22⋊D6111yescharacteristic
567C22⋊C4—21no—
568C22⋊C4—21no—
569C22⋊C4—41no—
570C22⋊C4—41no—
571C22⋊C4—1221no—
572C22⋊C4—2441no—
573C22⋊C4—2441no—
574C42—21no—
575C42C23×D6111yescharacteristic
576C23—21no—
577C23—21no—
578C23—41no—
579C23—1221no—
580C23—9761no—
581C23—9761no—
582C23—4882no—
583C23—4882no—
584C23—4882no—
585C23—4882no—
586C23—2441no—
587C23—2441no—
588Q8—21no—
589Q8—41no—
590Q8—41no—
591Q8—82no—
592Q8D8×D6112yescomplemented
593D4—21no—
594D4—21no—
595D4—21no—
596D4—41no—
597D4—41no—
598D4—41no—
599D4—41no—
600D4—82no—
601D4—81no—
602D4—161no—
603D4—612no—
604D4—1221no—
605D4—4881no—
606D4—4881no—
607D4—4882no—
608D4—4882no—
609D4—4882no—
610D4—4882no—
611D4—2441no—
612D4—2441no—
613D4—2441no—
614D4—2441no—
615D4—2441no—
616D4—2441no—
617D4—2441no—
618D4—2441no—
619D4#1451980212yescomplemented
620C2×C4—21no—
621C2×C4—21no—
622C2×C4—21no—
623C2×C4—21no—
624C2×C4—41no—
625C2×C4—41no—
626C2×C4—41no—
627C2×C4—41no—
628C2×C4—41no—
629C2×C4—82no—
630C2×C4—82no—
631C2×C4—1221no—
632C2×C4—2441no—
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—41no—
648C22—41no—
649C22—41no—
650C22—41no—
651C22—41no—
652C22—81no—
653C22—81no—
654C22—81no—
655C22—1222no—
656C22—9764no—
657C22—9764no—
658C22—4881no—
659C22—4881no—
660C22—4881no—
661C22—4881no—
662C22—4881no—
663C22—4881no—
664C22—2441no—
665C22—2441no—
666C22D4×C22⋊D6111yescharacteristic
667C4—22no—
668C4—21no—
669C4—41no—
670C4—41no—
671C4—41no—
672C4—81no—
673C4—2441no—
674C4C2×D8×D6111yescharacteristic
675C4C2×#1451980211yescharacteristic
676C4#1684056311yescharacteristic
677C4#1684057011yescharacteristic
678C2—42no—
679C2—81no—
680C2—161no—
681C2—9761no—
682C2—9761no—
683C2—2442no—
684C2#2145731511yescharacteristic
685C2D8×C22⋊D6111yescharacteristic
686C2#2145758111yescharacteristic
subgroupG/H
centerC22D4×C22⋊D61
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#3358C2
socleC22×C61D42
hypercenter#3358D61
hypocenterC61#19756
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ #3358

Complements

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

NH ≅ G/Ntype
1. D4×#11758201C2semidirect
2. D8×D244C2semidirect
3. #21457439C2semidirect
4. #21445487C2semidirect
5. #21445286C2semidirect
6. #21445253C2semidirect
7. #21445179C2semidirect
8. #21445172C2semidirect
9. #21444997C2semidirect
10. #21444922C2semidirect
11. #21444884C2semidirect
12. #21444872C2semidirect
13. #21444649C2semidirect
14. #21444629C2semidirect
15. C61×#3358C2semidirect
22. #16840138C22semidirect
23. #16840138C22semidirect
24. #16840136C22semidirect
25. #16840110C22semidirect
30. #16839962C22semidirect
31. #16839962C22semidirect
42. #16839840C22semidirect
44. D4×#92418C22semidirect
45. #16839515C22semidirect
46. #16839514C22semidirect
47. #16839508C22semidirect
48. #16839506C22semidirect
49. #16839505C22semidirect
50. #16839505C22semidirect
51. Q8⋊#92418C22semidirect
52. D4⋊#92418C22semidirect
53. D4⋊#92418C22semidirect
55. #16839484C22semidirect
56. C61×#550C22semidirect
57. D4×Q8×C61C22semidirect
59. C61×#486C22semidirect
61. C61×#474C22semidirect
63. C61×#459C22semidirect
64. C8×Q8×C61C22semidirect
210. #11758201D8semidirect
248. Q8×C61C2×D8semidirect
259. D4×C61AGL(1,ℤ/8)semidirect
301. #92418C2×D8semidirect
335. #3358D61semidirect
362. #550C2×D61semidirect
363. D4×Q8C2×D61semidirect
372. #486C2×D61semidirect
374. #474C2×D61semidirect
385. #459C2×D61semidirect
392. C8×Q8C2×D61semidirect
410. C61#19756semidirect
592. Q8D8×D61semidirect
619. D4#14519802semidirect

Maximal subgroups

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

subgroupindex
D4×#117582012
D8×D2442
#214574392
#214454872
#214452862
#214452532
#214451792
#214451722
#214449972
#214449222
#214448842
#214448722
#214446492
#214446292
C61×#33582
#1975661