Finite Group Databasebeta

← #29732175

Subgroups — #29732175

Order 15616.

648 conjugacy classes of subgroups.

The subgroup lattice has 600 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21455612C211yesmaximal, characteristic
2C2×#16840059C211yesmaximal, characteristic
3#21454768C211yesmaximal, characteristic, complemented
4#21447332C211yesmaximal, characteristic, complemented
5#21447327C211yesmaximal, characteristic, complemented
6#21447176C211yesmaximal, characteristic
7#21447176C211yesmaximal, characteristic
8#21446813C211yesmaximal, characteristic, complemented
9#21446601C211yesmaximal, characteristic, complemented
10#21446596C211yesmaximal, characteristic, complemented
11#21446594C211yesmaximal, characteristic, complemented
12#21446480C211yesmaximal, characteristic
13#21446480C211yesmaximal, characteristic
14C4×#14519693C211yesmaximal, characteristic
15C61×#3166C211yesmaximal, characteristic, complemented
16C2×C4×D244C2211yescharacteristic
17#16840059—21no—
18#16840059—21no—
19#16840059—21no—
20#16840059—21no—
21#16840059—21no—
22#16840059—21no—
23#16840049—21no—
24#16840039—21no—
25#16839893—21no—
26#16839893—21no—
27#16839893—21no—
28Q8×#92418—21no—
29C2×#14519712C2211yescharacteristic
30C2×#14519712C2211yescharacteristic
31#16839805C2211yescharacteristic
32#16839805C2211yescharacteristic
33C2×#14519710C2211yescharacteristic
34C2×#14519708C2211yescharacteristic
35#16839789C2211yescharacteristic
36#16839789C2211yescharacteristic
37#16839786C2211yescharacteristic
38C2×C4⋊#92418C2211yescharacteristic
39#16839766C2211yescharacteristic
40#16839766C2211yescharacteristic
41C2×#14519697C2211yescharacteristic
42C2×#14519697C2211yescharacteristic
43C2×#14519693C2211yescharacteristic
44C2×#14519693C2211yescharacteristic
45#16839718C2211yescharacteristic
46#16839718C2211yescharacteristic
47C4×#11758194C2211yescharacteristic
48#16839531C2211yescharacteristic
49#16839531C2211yescharacteristic
50#16839531C2211yescharacteristic
51#16839531C2211yescharacteristic
52#16839530C2211yescharacteristic
53#16839530C2211yescharacteristic
54#16839526C2211yescharacteristic
55#16839526C2211yescharacteristic
56C61×#533C2211yescharacteristic
57C2×#125×C61C2211yescharacteristic
58C61×#501—21no—
59C61×#501—21no—
60C61×#499—21no—
61C61×#499—21no—
62C61×#428C2211yescharacteristic
63C61×#428C2211yescharacteristic
64C61×#428C2211yescharacteristic
65C61×#428C2211yescharacteristic
66C4×M16×C61C2211yescharacteristic
67C22×D244C2311yescharacteristic
68C22×C4×D61—21no—
69C2×#11758194C2311yescharacteristic
70C2×#11758194C2311yescharacteristic
71C2×#11758194C2311yescharacteristic
72C2×#11758194C2311yescharacteristic
73C22×#92418C2311yescharacteristic
74C22×#92418C2311yescharacteristic
75C4×D244—21no—
76C4×D244—21no—
77C4×D244—21no—
78C4×D244—21no—
79C2×C4⋊Dic61C2311yescharacteristic
80C2×#11758198—21no—
81#14519712—22no—
82#14519712—22no—
83#14519710—21no—
84#14519710—21no—
85#14519710—21no—
86#14519710—21no—
87#14519708—21no—
88#14519708—21no—
89#14519708—21no—
90#14519708—21no—
91C4⋊#92418—21no—
92C4⋊#92418—21no—
93C4⋊#92418—21no—
94C4⋊#92418—21no—
95C4⋊#92418—21no—
96C4⋊#92418—21no—
97C4⋊#92418D411yescharacteristic
98C4⋊#92418D411yescharacteristic
99C4⋊#92418D411yescharacteristic
100C4⋊#92418D411yescharacteristic
101#14519697—21no—
102#14519697—21no—
103#14519697—21no—
104#14519697—21no—
105#14519697—21no—
106#14519697—21no—
107#14519697—21no—
108#14519697—21no—
109#14519693—22no—
110#14519693Q814yescomplemented
111#14519688—21no—
112#14519688—21no—
113#14519688—21no—
114C4×#92418—21no—
115C4×#92418—21no—
116C4×#92418—21no—
117C2×M16×C61C2311yescharacteristic
118C2×M16×C61C2311yescharacteristic
119C4⋊Q8×C61D411yescharacteristic
120C4⋊Q8×C61D411yescharacteristic
121#125×C61—21no—
122#125×C61—21no—
123#125×C61—21no—
124#125×C61—21no—
125#125×C61D411yescharacteristic
126#125×C61D411yescharacteristic
127C22⋊Q8×C61—21no—
128C22⋊Q8×C61—21no—
129C4×Q8×C61—21no—
130C4×Q8×C61—21no—
131#117×C61C2311yescharacteristic
132#117×C61C2311yescharacteristic
133C2×C4⋊C4×C61—21no—
134C2×C4⋊C4×C61—21no—
135C2×C4⋊C4×C61C2311yescharacteristic
136C2×C4⋊C4×C61C2311yescharacteristic
137C2×C42×C61C2311yescharacteristic
138#107×C61—21no—
139#107×C61—21no—
140#107×C61—21no—
141#107×C61—21no—
142#106×C61—21no—
143#106×C61—21no—
144#106×C61—21no—
145#106×C61—21no—
146#97×C61—21no—
147C4×C8×C61—21no—
148C23×D61—21no—
149C2×Q8×C61—21no—
150C2×Q8×C61—21no—
151C22×C4×C61—21no—
152C22×C4×C61—21no—
153C22×C4×C61#4111yescharacteristic
154C22×C4×C61#4111yescharacteristic
155C22×C4×C61C2411yescharacteristic
156C22×Dic61—21no—
157C2×D244—21no—
158C2×D244—21no—
159C2×D244—21no—
160C2×D244—21no—
161C2×D244—21no—
162C2×D244C2×Q812yes—
163C2×C4×D61—41no—
164C2×C4×D61—41no—
165C2×C4×D61—82no—
166M16×C61—21no—
167M16×C61—21no—
168M16×C61C2×Q812yes—
169M16×C61#4112yes—
170C2×C8×C61—21no—
171C2×C8×C61—21no—
172C4⋊C4×C61—21no—
173C4⋊C4×C61—21no—
174C4⋊C4×C61—21no—
175C4⋊C4×C61—21no—
176C4⋊C4×C61—21no—
177C4⋊C4×C61—21no—
178C4⋊C4×C61—21no—
179C4⋊C4×C61—21no—
180C4⋊C4×C61—21no—
181C4⋊C4×C61—21no—
182C4⋊C4×C61—41no—
183C4⋊C4×C61—41no—
184C4⋊C4×C61—41no—
185C4⋊C4×C61—41no—
186C4⋊C4×C61C2×D411yescharacteristic
187C4⋊C4×C61C2×D411yescharacteristic
188C4⋊C4×C61C2×D411yescharacteristic
189C4⋊C4×C61C2×D411yescharacteristic
190C22⋊C4×C61—21no—
191C22⋊C4×C61—21no—
192C42×C61—21no—
193C42×C61—21no—
194C42×C61C2×D411yescharacteristic
195C42×C61C2×D411yescharacteristic
196C42×C61C2×D411yescharacteristic
197C42×C61C2×D411yescharacteristic
198#11758198—41no—
199#11758198—41no—
200C4⋊Dic61—21no—
201C4⋊Dic61—21no—
202#11758194—21no—
203#11758194—21no—
204#11758194—21no—
205#11758194—21no—
206#11758194—21no—
207#11758194—21no—
208#11758194—21no—
209#11758194C2×Q812yes—
210C2×#92418—21no—
211C2×#92418—21no—
212C2×#92418—21no—
213C2×#92418—21no—
214C2×#92418—21no—
215C2×#92418—21no—
216C2×#92418—21no—
217C2×#92418—21no—
218C2×#92418C2×D411yescharacteristic
219C2×#92418C2×D411yescharacteristic
220C2×#92418C2×D411yescharacteristic
221C2×#92418C2×D411yescharacteristic
222C23×C61C2×#4111yescharacteristic
223C22×D61—22no—
224C22×D61—41no—
225C22×D61—41no—
226C22×D61—81no—
227C22×D61—81no—
228Q8×C61—42no—
229Q8×C61—81no—
230C2×C4×C61—21no—
231C2×C4×C61—21no—
232C2×C4×C61—21no—
233C2×C4×C61—21no—
234C2×C4×C61—41no—
235C2×C4×C61—41no—
236C2×C4×C61—41no—
237C2×C4×C61—41no—
238C2×C4×C61—81no—
239C2×C4×C61—81no—
240C2×C4×C61#12111yescharacteristic
241C2×C4×C61#12111yescharacteristic
242C2×C4×C61#12111yescharacteristic
243C2×C4×C61#12111yescharacteristic
244C2×C4×C61C22×D411yescharacteristic
245C2×C4×C61C22×D411yescharacteristic
246C2×C4×C61C22×Q811yescharacteristic
247C2×C4×C61C2×#4111yescharacteristic
248C2×C4×C612−1+411yescharacteristic
249C2×C4×C612−1+411yescharacteristic
250C2×Dic61—41no—
251C2×Dic61—81no—
252D244—22no—
253D244—22no—
254D244—41no—
255D244—41no—
256C4×D61—82no—
257C4×D61—82no—
258C8×C61—22no—
259C8×C61—22no—
260#92418—21no—
261#92418—21no—
262#92418—21no—
263#92418—21no—
264#92418—22no—
265#92418—41no—
266#92418—41no—
267#92418—41no—
268#25493—611nomaximal, Sylow
269C22×C61—21no—
270C22×C61—21no—
271C22×C61C2×#12111yescharacteristic
272C22×C61#53311yescharacteristic
273C22×C61#54111yescharacteristic
274C2×D61—42no—
275C2×D61—82no—
276C2×D61—81no—
277C2×D61—81no—
278C2×D61—82no—
279C4×C61—21no—
280C4×C61—21no—
281C4×C61—21no—
282C4×C61—21no—
283C4×C61—41no—
284C4×C61—41no—
285C4×C61—41no—
286C4×C61—41no—
287C4×C61—81no—
288C4×C61—81no—
289C4×C61#54711yescharacteristic
290C4×C61#54811yescharacteristic
291C4×C61D4×Q811yescharacteristic
292C4×C61D4×Q811yescharacteristic
293Dic61—81no—
294#3166D6111yescharacteristic, complemented
295#3077—611no—
296C2×#478—611no—
297#2962—611no—
298#2090—611no—
299#2089—611no—
300#2063—611no—
301#2063—611no—
302#1998—611no—
303#1952—611no—
304#1951—611no—
305#1950—611no—
306#1931—611no—
307#1931—611no—
308C4×#100—611no—
309C2×C61—21no—
310C2×C61—21no—
311C2×C61#242911yescharacteristic
312C2×C61#306711yescharacteristic
313C2×C61#306811yescharacteristic
314D61—82no—
315#533C2×D6111yescharacteristic
316C2×#125C2×D6111yescharacteristic
317C2×C4×D4—611no—
318#501—21no—
319#501—21no—
320#499—21no—
321#499—21no—
322#478—1221no—
323#478—1221no—
324#478—1221no—
325#478—1221no—
326#478—1221no—
327#478—1221no—
328#476—1221no—
329#474—1221no—
330#446—1221no—
331#446—1221no—
332#446—1221no—
333C8×Q8—1221no—
334C2×#108—611no—
335C2×#108—611no—
336#428—611no—
337#428—611no—
338#428C2×D6111yescharacteristic
339#428C2×D6111yescharacteristic
340#428C2×D6111yescharacteristic
341#428C2×D6111yescharacteristic
342C2×#107—611no—
343C2×#106—611no—
344#424—611no—
345#424—611no—
346#423—611no—
347C2×C4⋊C8—611no—
348#418—611no—
349#418—611no—
350C2×#102—611no—
351C2×#102—611no—
352C2×#100—611no—
353C2×#100—611no—
354#405—611no—
355#405—611no—
356C4×M16—611no—
357C4×M16C2×D6111yescharacteristic
358#343—611no—
359#343—611no—
360#343—611no—
361#343—611no—
362#343—611no—
363#343—611no—
364#341—611no—
365#341—611no—
366C61#2549311yescharacteristic, complemented, Sylow
367C22×D4—611no—
368C23×C4—1221no—
369C2×M16—611no—
370C2×M16—611no—
371C2×M16—611no—
372C2×M16—611no—
373C2×M16C22×D6111yescharacteristic
374C2×M16C22×D6111yescharacteristic
375C22×C8—611no—
376C22×C8—611no—
377C4⋊Q8C22⋊D6111yescharacteristic
378C4⋊Q8C22⋊D6111yescharacteristic
379#125—21no—
380#125—21no—
381#125—21no—
382#125—21no—
383#125C22⋊D6111yescharacteristic
384#125C22⋊D6111yescharacteristic
385C22⋊Q8—21no—
386C22⋊Q8—21no—
387C4×Q8—21no—
388C4×Q8—21no—
389C4×D4—1221no—
390C4×D4—1221no—
391C4×D4—1221no—
392C4×D4—1221no—
393#117C22×D6111yescharacteristic
394#117C22×D6111yescharacteristic
395C2×C4⋊C4—21no—
396C2×C4⋊C4—21no—
397C2×C4⋊C4—611no—
398C2×C4⋊C4C22×D6111yescharacteristic
399C2×C4⋊C4C22×D6111yescharacteristic
400C2×C22⋊C4—1221no—
401C2×C42C22×D6111yescharacteristic
402#108—1222no—
403#108—1222no—
404#107—21no—
405#107—21no—
406#107—21no—
407#107—21no—
408#107—1221no—
409#107—1221no—
410#107—1221no—
411#107—1221no—
412#106—21no—
413#106—21no—
414#106—21no—
415#106—21no—
416#106—1221no—
417#106—1221no—
418#106—1221no—
419#106—1221no—
420C4⋊C8—611no—
421C4⋊C8—611no—
422C4⋊C8—611no—
423C4⋊C8—611no—
424C4⋊C8—1221no—
425C4⋊C8—1221no—
426C4⋊C8—1221no—
427C4⋊C8—1221no—
428C4⋊C8—1221no—
429C4⋊C8—1221no—
430#102—1221no—
431#102—1221no—
432#102—1221no—
433#102—1221no—
434#102—1221no—
435#102—1221no—
436#102—1221no—
437#102—1221no—
438#100—614no—
439#100—1222no—
440#97—21no—
441#97—1221no—
442#97—1221no—
443#97—1221no—
444C4×C8—21no—
445C4×C8—1221no—
446C4×C8—1221no—
447C4×C8—1221no—
448C24—1221no—
449C2×Q8—21no—
450C2×Q8—21no—
451C2×D4—612no—
452C2×D4—1221no—
453C2×D4—1221no—
454C2×D4—1221no—
455C2×D4—1221no—
456C2×D4—1221no—
457C22×C4—21no—
458C22×C4—21no—
459C22×C4—1221no—
460C22×C4—4882no—
461C22×C4—2441no—
462C22×C4—2441no—
463C22×C4#1175822411yescharacteristic
464C22×C4#1175822411yescharacteristic
465C22×C4C23×D6111yescharacteristic
466M16—21no—
467M16—21no—
468M16—612no—
469M16—1221no—
470M16—1221no—
471M16—1221no—
472M16—1221no—
473M16—1221no—
474M16—1221no—
475M16—1221no—
476M16Q8×D6112yes—
477M16#1175822612yes—
478C2×C8—21no—
479C2×C8—21no—
480C2×C8—611no—
481C2×C8—611no—
482C2×C8—611no—
483C2×C8—611no—
484C2×C8—1221no—
485C2×C8—1221no—
486C2×C8—1221no—
487C2×C8—1221no—
488C2×C8—1221no—
489C2×C8—1221no—
490C2×C8—1221no—
491C2×C8—1221no—
492C4⋊C4—21no—
493C4⋊C4—21no—
494C4⋊C4—21no—
495C4⋊C4—21no—
496C4⋊C4—21no—
497C4⋊C4—21no—
498C4⋊C4—21no—
499C4⋊C4—21no—
500C4⋊C4—21no—
501C4⋊C4—21no—
502C4⋊C4—41no—
503C4⋊C4—41no—
504C4⋊C4—41no—
505C4⋊C4—41no—
506C4⋊C4—1221no—
507C4⋊C4—1221no—
508C4⋊C4C2×C22⋊D6111yescharacteristic
509C4⋊C4C2×C22⋊D6111yescharacteristic
510C4⋊C4C2×C22⋊D6111yescharacteristic
511C4⋊C4C2×C22⋊D6111yescharacteristic
512C22⋊C4—21no—
513C22⋊C4—21no—
514C22⋊C4—2441no—
515C22⋊C4—2441no—
516C42—21no—
517C42—21no—
518C42D4×D6111yescharacteristic
519C42D4×D6111yescharacteristic
520C42C2×C22⋊D6111yescharacteristic
521C42C2×C22⋊D6111yescharacteristic
522C23—1222no—
523C23—4881no—
524C23—4881no—
525C23—2441no—
526C23—2441no—
527C23C2×#1175822411yescharacteristic
528Q8—42no—
529Q8—81no—
530D4—1222no—
531D4—1222no—
532D4—2441no—
533D4—2441no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—81no—
543C2×C4—81no—
544C2×C4—4882no—
545C2×C4—4882no—
546C2×C4—4881no—
547C2×C4—2441no—
548C2×C4#1451974911yescharacteristic
549C2×C4#1451974911yescharacteristic
550C2×C4#1451974911yescharacteristic
551C2×C4#1451974911yescharacteristic
552C2×C4C2×D4×D6111yescharacteristic
553C2×C4C22×C22⋊D6111yescharacteristic
554C2×C4C2×Q8×D6111yescharacteristic
555C2×C4C2×#1175822611yescharacteristic
556C2×C4#1451982511yescharacteristic
557C2×C4#1451982511yescharacteristic
558C8—22no—
559C8—22no—
560C8—1221no—
561C8—1221no—
562C8—1221no—
563C8—1221no—
564C8—1222no—
565C8—2441no—
566C8—2441no—
567C8—2441no—
568C22—21no—
569C22—21no—
570C22—4882no—
571C22—4882no—
572C22—4881no—
573C22—4881no—
574C22—2442no—
575C22C2×#1451974911yescharacteristic
576C22#1684031711yescharacteristic
577C22#1684037311yescharacteristic
578C4—21no—
579C4—21no—
580C4—21no—
581C4—21no—
582C4—41no—
583C4—41no—
584C4—41no—
585C4—41no—
586C4—81no—
587C4—81no—
588C4—4881no—
589C4#1684041511yescharacteristic
590C4#1684042311yescharacteristic
591C4Q8×C22⋊D6111yescharacteristic
592C4#1684042711yescharacteristic
593C2—21no—
594C2—21no—
595C2—4882no—
596C2#2144942111yescharacteristic
597C2#2145553611yescharacteristic
598C2#2145554911yescharacteristic
subgroupG/H
centerC22C2×#14519749
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#3166C2
socleC22×C61C2×#121
prime radicalC22×D244C23
hypercenter#3166D61
hypocenterC61#25493
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3166

Complements

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

NH ≅ G/Ntype
3. #21454768C2semidirect
4. #21447332C2semidirect
5. #21447327C2semidirect
8. #21446813C2semidirect
9. #21446601C2semidirect
10. #21446596C2semidirect
11. #21446594C2semidirect
15. C61×#3166C2semidirect
110. #14519693Q8semidirect
294. #3166D61semidirect
366. C61#25493semidirect

Maximal subgroups

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

subgroupindex
#214556122
C2×#168400592
#214547682
#214473322
#214473272
#214471762
#214468132
#214466012
#214465962
#214465942
#214464802
C4×#145196932
C61×#31662
#2549361