Finite Group Databasebeta

← #29721533

Subgroups — #29721533

Order 15616.

658 conjugacy classes of subgroups.

The subgroup lattice has 618 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21455642C211yesmaximal, characteristic
2C2×#16840069C211yesmaximal, characteristic, complemented
3C4×#14519808C211yesmaximal, characteristic
4#21454487C211yesmaximal, characteristic
5#21447268C211yesmaximal, characteristic
6#21447263C211yesmaximal, characteristic
7#21447118C211yesmaximal, characteristic
8#21447118C211yesmaximal, characteristic
9#21446811C211yesmaximal, characteristic, complemented
10#21446603C211yesmaximal, characteristic, complemented
11#21446598C211yesmaximal, characteristic, complemented
12#21446592C211yesmaximal, characteristic, complemented
13#21446476C211yesmaximal, characteristic, complemented
14#21446476C211yesmaximal, characteristic, complemented
15C61×#1801C211yesmaximal, characteristic, complemented
16C2×#14519808C2211yescharacteristic
17C2×#14519769C2211yescharacteristic
18C4×#11758222C2211yescharacteristic
19#16840070—21no—
20#16840070—21no—
21#16840069—21no—
22#16840069—21no—
23#16840069—21no—
24#16840069—21no—
25#16840063—21no—
26#16840063—21no—
27#16839867—21no—
28#16839860—21no—
29C4×#11758202—21no—
30C4×#11758201—21no—
31#16839822—21no—
32#16839822—21no—
33#16839818—21no—
34#16839814—21no—
35C2×#14519712C2211yescharacteristic
36C2×#14519712C2211yescharacteristic
37#16839804C2211yescharacteristic
38#16839804C2211yescharacteristic
39#16839802C2211yescharacteristic
40C2×#14519710C2211yescharacteristic
41C2×#14519708C2211yescharacteristic
42C2×C4⋊#92418C2211yescharacteristic
43#16839771C2211yescharacteristic
44#16839764C2211yescharacteristic
45#16839764C2211yescharacteristic
46#16839761C2211yescharacteristic
47C2×#14519701C2211yescharacteristic
48C2×#14519701C2211yescharacteristic
49#16839730C2211yescharacteristic
50#16839730C2211yescharacteristic
51#16839720C2211yescharacteristic
52#16839720C2211yescharacteristic
53C4×#11758194C2211yescharacteristic
54#16839530C2211yescharacteristic
55#16839530C2211yescharacteristic
56#16839530C2211yescharacteristic
57#16839530C2211yescharacteristic
58#16839526C2211yescharacteristic
59#16839526C2211yescharacteristic
60C2×C4×Q8×C61C2211yescharacteristic
61C61×#423C2211yescharacteristic
62C61×#423C2211yescharacteristic
63C2×#101×C61C2211yescharacteristic
64C2×#101×C61C2211yescharacteristic
65C61×#343C2211yescharacteristic
66C61×#343C2211yescharacteristic
67C2×#11758222C2311yescharacteristic
68#14519808—22no—
69#14519808—22no—
70C2×#11758202—21no—
71C2×#11758201—21no—
72C2×#11758194C2311yescharacteristic
73C2×#11758194C2311yescharacteristic
74C22×#92418C2311yescharacteristic
75C22×#92418C2311yescharacteristic
76#14519769—21no—
77#14519769—21no—
78#14519769—21no—
79#14519769—21no—
80C4×Dic122D411yescharacteristic
81C4×Dic122D411yescharacteristic
82C4×D244D411yescharacteristic
83C4×D244D411yescharacteristic
84C4×C22⋊D61—21no—
85C4×C22⋊D61—21no—
86#14519736C2311yescharacteristic
87#14519734—21no—
88C2×C4⋊Dic61C2311yescharacteristic
89C2×C4⋊Dic61C2311yescharacteristic
90C2×#11758196—21no—
91C2×#11758196—21no—
92C42×D61—21no—
93#14519712—22no—
94#14519712—22no—
95#14519711—21no—
96#14519711—21no—
97#14519710—21no—
98#14519710—21no—
99#14519710—21no—
100#14519709—21no—
101#14519709—21no—
102#14519708—21no—
103#14519708—21no—
104#14519708—21no—
105C4⋊#92418D411yescharacteristic
106C4⋊#92418D411yescharacteristic
107C4⋊#92418D411yescharacteristic
108C4⋊#92418D411yescharacteristic
109#14519702—21no—
110#14519702—21no—
111#14519701—21no—
112#14519701—21no—
113#14519701—21no—
114#14519701—21no—
115#14519701—21no—
116#14519701—21no—
117#14519700—21no—
118#14519699—21no—
119#14519699—21no—
120#14519697—21no—
121#14519689—21no—
122#14519689—21no—
123#14519689—21no—
124#14519689—21no—
125#14519688—21no—
126#14519687—21no—
127#14519687—21no—
128C4×#92418—21no—
129C8×Dic61—21no—
130C8×Dic61—21no—
131C22×Q8×C61C2311yescharacteristic
132C2×M16×C61C2311yescharacteristic
133C2×M16×C61C2311yescharacteristic
134C2×M16×C61C2311yescharacteristic
135C2×M16×C61C2311yescharacteristic
136C4×Q8×C61—21no—
137C4×Q8×C61—21no—
138C4×Q8×C61—21no—
139C4×Q8×C61—21no—
140C2×C4⋊C4×C61—21no—
141C2×C4⋊C4×C61C2311yescharacteristic
142C2×C42×C61—21no—
143C2×C42×C61C2311yescharacteristic
144C4⋊C8×C61—21no—
145C4⋊C8×C61—21no—
146C4⋊C8×C61—21no—
147C4⋊C8×C61—21no—
148#101×C61—22no—
149#101×C61—22no—
150C2×Q8×C61—21no—
151C2×Q8×C61—21no—
152C2×Q8×C61—21no—
153C2×Q8×C61—21no—
154C2×Q8×C61—21no—
155C2×Q8×C61#4112yes—
156C22×C4×C61—21no—
157C22×C4×C61—21no—
158C22×C4×C61#4111yescharacteristic
159C22×C4×C61#4111yescharacteristic
160C22×C4×C61C2411yescharacteristic
161C2×C22⋊D61—21no—
162C22×Dic61—21no—
163C22×Dic61—21no—
164#11758222—21no—
165#11758222—21no—
166#11758222—21no—
167#11758222—21no—
168C2×D244C2×D411yescharacteristic
169C2×C4×D61—21no—
170C2×C4×D61—21no—
171C2×C4×D61—21no—
172C2×Dic122C2×D411yescharacteristic
173M16×C61—21no—
174M16×C61—21no—
175M16×C61—21no—
176M16×C61—21no—
177M16×C61—21no—
178M16×C61—21no—
179M16×C61—21no—
180M16×C61—21no—
181C2×C8×C61—21no—
182C2×C8×C61—21no—
183C2×C8×C61—21no—
184C2×C8×C61—21no—
185C4⋊C4×C61—21no—
186C4⋊C4×C61—21no—
187C4⋊C4×C61—41no—
188C4⋊C4×C61—41no—
189C42×C61—41no—
190C42×C61—41no—
191C42×C61C2×D411yescharacteristic
192C42×C61C2×D411yescharacteristic
193C42×C61C2×D411yescharacteristic
194C42×C61C2×D411yescharacteristic
195C22⋊Dic61—21no—
196#11758202—24no—
197#11758201—24no—
198#11758198—21no—
199#11758198—21no—
200C4⋊Dic61—21no—
201C4⋊Dic61—21no—
202C4⋊Dic61—21no—
203C4⋊Dic61—21no—
204C4⋊Dic61C2×D411yescharacteristic
205C4⋊Dic61C2×D411yescharacteristic
206#11758196—21no—
207#11758196—21no—
208#11758196—41no—
209#11758196—41no—
210#11758196—41no—
211#11758196—41no—
212C4×Dic61—21no—
213C4×Dic61—21no—
214C4×Dic61—21no—
215#11758194—21no—
216#11758194—21no—
217#11758194—21no—
218#11758194#4112yes—
219C2×#92418—21no—
220C2×#92418—21no—
221C2×#92418—21no—
222C2×#92418—21no—
223C2×#92418—21no—
224C2×#92418—21no—
225C2×#92418C2×D411yescharacteristic
226C2×#92418C2×D411yescharacteristic
227C2×#92418C2×D411yescharacteristic
228C2×#92418C2×D411yescharacteristic
229C23×C61C2×#4111yescharacteristic
230C22×D61—21no—
231Q8×C61—22no—
232Q8×C61—22no—
233Q8×C61—41no—
234Q8×C61—41no—
235C2×C4×C61—22no—
236C2×C4×C61—41no—
237C2×C4×C61—41no—
238C2×C4×C61—41no—
239C2×C4×C61—41no—
240C2×C4×C61—41no—
241C2×C4×C61—81no—
242C2×C4×C61#12111yescharacteristic
243C2×C4×C61#12111yescharacteristic
244C2×C4×C61#12111yescharacteristic
245C2×C4×C61#12111yescharacteristic
246C2×C4×C61C22×D411yescharacteristic
247C2×C4×C61C22×D411yescharacteristic
248C2×C4×C61C2×#4111yescharacteristic
249C2×C4×C61C2×#4111yescharacteristic
250C2×C4×C612−1+411yescharacteristic
251C2×C4×C612−1+411yescharacteristic
252C22⋊D61—41no—
253C22⋊D61—41no—
254C2×Dic61—21no—
255C2×Dic61—21no—
256C2×Dic61—21no—
257C2×Dic61—41no—
258C2×Dic61—41no—
259C2×Dic61—81no—
260C2×Dic61—81no—
261D244—21no—
262D244—21no—
263C4×D61—42no—
264C4×D61—41no—
265C4×D61—41no—
266C4×D61—81no—
267Dic122—21no—
268Dic122—21no—
269C8×C61—41no—
270C8×C61—41no—
271C8×C61—41no—
272C8×C61—41no—
273#92418—22no—
274#92418—21no—
275#92418—21no—
276#92418—21no—
277#92418—21no—
278#92418—41no—
279#24725—611nomaximal, Sylow
280C22×C61—21no—
281C22×C61—21no—
282C22×C61C2×#12111yescharacteristic
283C22×C61#55211yescharacteristic
284C22×C61#55611yescharacteristic
285C2×D61—41no—
286C2×D61—41no—
287C2×D61—41no—
288C4×C61—21no—
289C4×C61—21no—
290C4×C61—21no—
291C4×C61—21no—
292C4×C61—42no—
293C4×C61—81no—
294C4×C61#52511yescharacteristic
295C4×C61#53611yescharacteristic
296C4×C61#54711yescharacteristic
297C4×C61#54811yescharacteristic
298Dic61—41no—
299Dic61—41no—
300Dic61—41no—
301Dic61—81no—
302Dic61—81no—
303#3080—611no—
304C2×#479—611no—
305C4×#137—611no—
306#2925—611no—
307#2079—611no—
308#2078—611no—
309#2052—611no—
310#2052—611no—
311#1998—611no—
312#1952—611no—
313#1951—611no—
314#1950—611no—
315#1930—611no—
316#1930—611no—
317#1801D6111yescharacteristic, complemented
318C2×C61—21no—
319C2×C61—21no—
320C2×C61#241511yescharacteristic
321C2×C61#306711yescharacteristic
322C2×C61#306911yescharacteristic
323D61—81no—
324C2×#137—611no—
325C2×#125—611no—
326C4×#41—611no—
327C2×C4×Q8C2×D6111yescharacteristic
328#479—1221no—
329#479—1221no—
330#479—1221no—
331#479—1221no—
332#479—1221no—
333#479—1221no—
334#478—1221no—
335#478—1221no—
336#441—1221no—
337#440—1221no—
338C4×Q16—1221no—
339C4×SD16—1221no—
340#433—1221no—
341#433—1221no—
342#432—1221no—
343#431—1221no—
344C2×#108—611no—
345C2×#108—611no—
346#428—611no—
347#428—611no—
348#428—611no—
349C2×#107—611no—
350C2×#106—611no—
351#423C2×D6111yescharacteristic
352#423C2×D6111yescharacteristic
353C2×C4⋊C8—611no—
354#419—611no—
355#417—611no—
356#417—611no—
357#417—611no—
358C2×#103—611no—
359C2×#103—611no—
360C2×#101C2×D6111yescharacteristic
361C2×#101C2×D6111yescharacteristic
362#408—611no—
363#408—611no—
364#405—611no—
365#405—611no—
366C4×M16—611no—
367#343—611no—
368#343—611no—
369#343—611no—
370#343—611no—
371#343C2×D6111yescharacteristic
372#343C2×D6111yescharacteristic
373#341—611no—
374#341—611no—
375C61#2472511yescharacteristic, complemented, Sylow
376C2×#41—611no—
377C22×Q8C22×D6111yescharacteristic
378#137—1222no—
379#137—1222no—
380C2×Q16—1221no—
381C2×SD16—1221no—
382C2×M16—611no—
383C2×M16—611no—
384C2×M16C22×D6111yescharacteristic
385C2×M16C22×D6111yescharacteristic
386C2×M16C22×D6111yescharacteristic
387C2×M16C22×D6111yescharacteristic
388C22×C8—611no—
389C22×C8—611no—
390#125—1221no—
391#125—1221no—
392#125—1221no—
393#125—1221no—
394C4×Q8—21no—
395C4×Q8—21no—
396C4×Q8—21no—
397C4×Q8—21no—
398C4×Q8—611no—
399C4×Q8—611no—
400C4×D4—611no—
401C4×D4—611no—
402C4×D4—1221no—
403C4×D4—1221no—
404#117—611no—
405#117—1221no—
406C2×C4⋊C4—21no—
407C2×C4⋊C4—611no—
408C2×C4⋊C4—611no—
409C2×C4⋊C4—1221no—
410C2×C4⋊C4—1221no—
411C2×C4⋊C4C22×D6111yescharacteristic
412C2×C42—21no—
413C2×C42—1221no—
414C2×C42C22×D6111yescharacteristic
415#108—1222no—
416#108—1222no—
417#107—1221no—
418#107—1221no—
419#107—1221no—
420#107—1221no—
421#107—1221no—
422#106—1221no—
423#106—1221no—
424#106—1221no—
425#106—1221no—
426#106—1221no—
427C4⋊C8—21no—
428C4⋊C8—21no—
429C4⋊C8—21no—
430C4⋊C8—21no—
431C4⋊C8—611no—
432C4⋊C8—611no—
433C4⋊C8—611no—
434C4⋊C8—611no—
435#103—1221no—
436#103—1221no—
437#103—1221no—
438#103—1221no—
439#103—1221no—
440#103—1221no—
441#103—1221no—
442#103—1221no—
443#103—1221no—
444#102—1221no—
445#102—1221no—
446#102—1221no—
447#101—22no—
448#101—22no—
449C22⋊C8—1221no—
450C22⋊C8—1221no—
451C22⋊C8—1221no—
452C22⋊C8—1221no—
453#97—1221no—
454#97—1221no—
455#97—1221no—
456C4×C8—1221no—
457C4×C8—1221no—
458C4×C8—1221no—
459#41—1221no—
460#41—1221no—
461#41—1221no—
462#41—1221no—
463C2×Q8—21no—
464C2×Q8—21no—
465C2×Q8—21no—
466C2×Q8—21no—
467C2×Q8—21no—
468C2×Q8—611no—
469C2×Q8#1175822412yes—
470C2×D4—611no—
471C2×D4—1221no—
472C22×C4—21no—
473C22×C4—21no—
474C22×C4—1221no—
475C22×C4—1221no—
476C22×C4—1221no—
477C22×C4—1221no—
478C22×C4—1221no—
479C22×C4#1175822211yescharacteristic
480C22×C4#1175822211yescharacteristic
481C22×C4C23×D6111yescharacteristic
482Q16—1224no—
483SD16—1224no—
484M16—21no—
485M16—21no—
486M16—21no—
487M16—21no—
488M16—21no—
489M16—21no—
490M16—21no—
491M16—21no—
492M16—612no—
493M16—1221no—
494M16—1221no—
495M16—1221no—
496C2×C8—21no—
497C2×C8—21no—
498C2×C8—21no—
499C2×C8—21no—
500C2×C8—611no—
501C2×C8—611no—
502C2×C8—611no—
503C2×C8—611no—
504C2×C8—1221no—
505C2×C8—1221no—
506C2×C8—1221no—
507C2×C8—1221no—
508C2×C8—1221no—
509C2×C8—1221no—
510C4⋊C4—21no—
511C4⋊C4—21no—
512C4⋊C4—41no—
513C4⋊C4—41no—
514C4⋊C4—611no—
515C4⋊C4—611no—
516C4⋊C4—1221no—
517C4⋊C4—1221no—
518C4⋊C4—1221no—
519C4⋊C4—1221no—
520C4⋊C4—1221no—
521C4⋊C4—1221no—
522C4⋊C4—2441no—
523C4⋊C4—2441no—
524C4⋊C4—2441no—
525C4⋊C4—2441no—
526C22⋊C4—1221no—
527C22⋊C4—1221no—
528C22⋊C4—1221no—
529C42—41no—
530C42—41no—
531C42—1221no—
532C42—1221no—
533C42—1221no—
534C42D4×D6111yescharacteristic
535C42D4×D6111yescharacteristic
536C42D4×D6111yescharacteristic
537C42D4×D6111yescharacteristic
538C23—1221no—
539C23C2×#1175822211yescharacteristic
540Q8—22no—
541Q8—22no—
542Q8—41no—
543Q8—41no—
544Q8—1221no—
545Q8—1221no—
546D4—1221no—
547D4—1221no—
548D4—2441no—
549D4—2441no—
550C2×C4—22no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—81no—
557C2×C4—1221no—
558C2×C4—1221no—
559C2×C4—1221no—
560C2×C4—4881no—
561C2×C4—4881no—
562C2×C4—4881no—
563C2×C4—2442no—
564C2×C4—2441no—
565C2×C4—2441no—
566C2×C4—2441no—
567C2×C4—2441no—
568C2×C4#1451975011yescharacteristic
569C2×C4#1451975011yescharacteristic
570C2×C4#1451975011yescharacteristic
571C2×C4#1451975011yescharacteristic
572C2×C4C2×D4×D6111yescharacteristic
573C2×C4C2×D4×D6111yescharacteristic
574C2×C4#41×D6111yescharacteristic
575C2×C4C2×#1175822411yescharacteristic
576C2×C4#1451982411yescharacteristic
577C2×C4#1451982511yescharacteristic
578C8—41no—
579C8—41no—
580C8—41no—
581C8—41no—
582C8—1222no—
583C8—1221no—
584C8—1221no—
585C8—1221no—
586C8—1221no—
587C8—2441no—
588C22—21no—
589C22—21no—
590C22—2441no—
591C22—2441no—
592C22—2441no—
593C22C2×#1451975011yescharacteristic
594C22#1684044711yescharacteristic
595C22#1684048011yescharacteristic
596C4—21no—
597C4—21no—
598C4—21no—
599C4—21no—
600C4—42no—
601C4—81no—
602C4—4881no—
603C4—4881no—
604C4—2441no—
605C4—2441no—
606C4—2441no—
607C4#1684027211yescharacteristic
608C4#1684033511yescharacteristic
609C4#1684041411yescharacteristic
610C4#1684041811yescharacteristic
611C2—21no—
612C2—21no—
613C2—4881no—
614C2#2144925011yescharacteristic
615C2#2145553811yescharacteristic
616C2#2145555511yescharacteristic
subgroupG/H
centerC22C2×#14519750
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#1801C2
socleC22×C61C2×#121
prime radicalC2×#11758222C23
hypercenter#1801D61
hypocenterC61#24725
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1801

Complements

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

NH ≅ G/Ntype
2. C2×#16840069C2semidirect
9. #21446811C2semidirect
10. #21446603C2semidirect
11. #21446598C2semidirect
12. #21446592C2semidirect
13. #21446476C2semidirect
14. #21446476C2semidirect
15. C61×#1801C2semidirect
317. #1801D61semidirect
375. C61#24725semidirect

Maximal subgroups

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

subgroupindex
#214556422
C2×#168400692
C4×#145198082
#214544872
#214472682
#214472632
#214471182
#214468112
#214466032
#214465982
#214465922
#214464762
C61×#18012
#2472561