Finite Group Databasebeta

← #29738919

Subgroups — #29738919

Order 15616.

674 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#21456353C211yesmaximal, characteristic, complemented
2#21455679C211yesmaximal, characteristic
3#21454417C211yesmaximal, characteristic
4#21447330C211yesmaximal, characteristic, complemented
5#21447329C211yesmaximal, characteristic, complemented
6#21447167C211yesmaximal, characteristic, complemented
7#21447166C211yesmaximal, characteristic, complemented
8#21446601C211yesmaximal, characteristic, complemented
9#21446564C211yesmaximal, characteristic, complemented
10#21446052C211yesmaximal, characteristic
11#21446050C211yesmaximal, characteristic
12#21445980C211yesmaximal, characteristic, complemented
13#21445881C211yesmaximal, characteristic
14#21445659C211yesmaximal, characteristic
15C61×#3084C211yesmaximal, characteristic
16C2×#14519784C2211yescharacteristic
17#16840390C2211yescharacteristic
18#16840245C2211yescharacteristic
19#16840172—21no—
20#16840168—21no—
21#16840165—21no—
22#16840161—21no—
23#16840067—21no—
24#16840064—21no—
25#16840050—21no—
26#16840045—21no—
27C2×#14519712C2211yescharacteristic
28#16839805—21no—
29#16839805C2211yescharacteristic
30#16839805C2211yescharacteristic
31#16839805C2211yescharacteristic
32#16839802—21no—
33#16839802C2211yescharacteristic
34#16839802C2211yescharacteristic
35#16839798—21no—
36#16839798C2211yescharacteristic
37C2×#14519710—21no—
38C2×#14519710C2211yescharacteristic
39C2×#14519710C2211yescharacteristic
40C2×#14519709C2211yescharacteristic
41#16839788C2211yescharacteristic
42C2×#14519704C2211yescharacteristic
43#16839763C2211yescharacteristic
44#16839718C2211yescharacteristic
45C2×C8×Dic61C2211yescharacteristic
46#16839627C2211yescharacteristic
47#16839627C2211yescharacteristic
48#16839531C2211yescharacteristic
49#16839531C2211yescharacteristic
50#16839530C2211yescharacteristic
51#16839530C2211yescharacteristic
52#16839527C2211yescharacteristic
53#16839523C2211yescharacteristic
54#16839521C2211yescharacteristic
55#16839521C2211yescharacteristic
56C61×#544C2211yescharacteristic
57C61×#520C2211yescharacteristic
58C61×#479—21no—
59C61×#478—21no—
60C61×#476—21no—
61C61×#475—21no—
62C61×#428C2211yescharacteristic
63C2×#106×C61C2211yescharacteristic
64C61×#424C2211yescharacteristic
65C61×#417C2211yescharacteristic
66C61×#416C2211yescharacteristic
67#14519784—22no—
68#14519784Q814yescomplemented
69C2×#11758194—21no—
70C2×#11758194C2311yescharacteristic
71C2×#11758194C2311yescharacteristic
72C2×#11758194C2311yescharacteristic
73C22×#92418—21no—
74C22×#92418C2311yescharacteristic
75#14519777—21no—
76#14519776—21no—
77#14519769—21no—
78#14519768—21no—
79#14519754—41no—
80Q8×Dic61—21no—
81D4×Dic61—21no—
82D4×Dic61—41no—
83#14519736—21no—
84#14519736C2311yescharacteristic
85C2×C4⋊Dic61—21no—
86C2×C4⋊Dic61C2311yescharacteristic
87C2×C4⋊Dic61C2311yescharacteristic
88C2×C4×Dic61C2311yescharacteristic
89#14519712—22no—
90#14519710—21no—
91#14519710—21no—
92#14519710—21no—
93#14519710—21no—
94#14519710—21no—
95#14519710—21no—
96#14519710—21no—
97#14519710—21no—
98#14519710—21no—
99#14519710D411yescharacteristic
100#14519710D411yescharacteristic
101#14519709—21no—
102#14519709—21no—
103#14519708—21no—
104#14519708—21no—
105#14519708—21no—
106#14519708—21no—
107#14519708—21no—
108#14519708D411yescharacteristic
109#14519708D411yescharacteristic
110#14519706—21no—
111#14519706—21no—
112#14519704Q814yescomplemented
113#14519701—21no—
114#14519698—21no—
115#14519688—21no—
116C4×#92418—21no—
117C8×Dic61—22no—
118#14519684—41no—
119#14519684—41no—
120C2×#41×C61C2311yescharacteristic
121C2×M16×C61C2311yescharacteristic
122C22×C8×C61C2311yescharacteristic
123C4⋊Q8×C61—21no—
124C4⋊Q8×C61—21no—
125#125×C61—21no—
126#125×C61—21no—
127C22⋊Q8×C61—41no—
128C4×Q8×C61—21no—
129C4×D4×C61—21no—
130C4×D4×C61—41no—
131#117×C61—21no—
132#117×C61C2311yescharacteristic
133#117×C61C2311yescharacteristic
134C2×C4⋊C4×C61—21no—
135C2×C4⋊C4×C61C2311yescharacteristic
136C2×C4⋊C4×C61C2311yescharacteristic
137#107×C61—21no—
138#106×C61—21no—
139#106×C61—21no—
140#106×C61—21no—
141C4⋊C8×C61—21no—
142C4⋊C8×C61—21no—
143#103×C61—21no—
144#103×C61D411yescharacteristic
145#103×C61D411yescharacteristic
146#102×C61—21no—
147#102×C61D411yescharacteristic
148#102×C61D411yescharacteristic
149#41×C61—21no—
150#41×C61—21no—
151#41×C61C2×Q812yes—
152#41×C61C2×Q812yes—
153C2×Q8×C61—41no—
154C2×Q8×C61C2×D411yescharacteristic
155C2×D4×C61—21no—
156C2×D4×C61C2×D411yescharacteristic
157C22×C4×C61—21no—
158C22×C4×C61—21no—
159C22×C4×C61—21no—
160C22×C4×C61—41no—
161C22×C4×C61C2411yescharacteristic
162C22×Dic61—21no—
163C22×Dic61—21no—
164C22×Dic61—21no—
165C22×Dic61—41no—
166C2×Dic122—41no—
167M16×C61—21no—
168M16×C61—21no—
169C2×C8×C61—21no—
170C2×C8×C61—21no—
171C2×C8×C61—21no—
172C2×C8×C61C2×D411yescharacteristic
173C2×C8×C61C2×D411yescharacteristic
174C4⋊C4×C61—21no—
175C4⋊C4×C61—21no—
176C4⋊C4×C61—21no—
177C4⋊C4×C61—21no—
178C4⋊C4×C61—41no—
179C4⋊C4×C61—41no—
180C4⋊C4×C61—41no—
181C4⋊C4×C61—41no—
182C4⋊C4×C61C2×D411yescharacteristic
183C4⋊C4×C61C2×D411yescharacteristic
184C4⋊C4×C61C2×D411yescharacteristic
185C4⋊C4×C61C2×D411yescharacteristic
186C22⋊C4×C61—21no—
187C22⋊C4×C61—21no—
188C22⋊C4×C61—41no—
189C42×C61—21no—
190C42×C61—21no—
191C42×C61—41no—
192C22⋊Dic61—21no—
193C22⋊Dic61—41no—
194C4⋊Dic61—21no—
195C4⋊Dic61—21no—
196C4⋊Dic61—21no—
197C4⋊Dic61—21no—
198C4⋊Dic61—41no—
199C4⋊Dic61—41no—
200C4⋊Dic61C2×D411yescharacteristic
201C4⋊Dic61C2×D411yescharacteristic
202#11758196—41no—
203#11758196—41no—
204C4×Dic61—21no—
205C4×Dic61—21no—
206C4×Dic61—41no—
207C4×Dic61C2×Q812yes—
208#11758194—21no—
209#11758194—21no—
210#11758194—24no—
211#11758194—21no—
212#11758194—21no—
213#11758194C2×Q812yes—
214#11758194C2×Q812yes—
215C2×#92418—21no—
216C2×#92418—21no—
217C2×#92418—24no—
218C2×#92418—21no—
219C2×#92418C2×D411yescharacteristic
220C2×#92418C2×D411yescharacteristic
221C2×#92418C2×Q812yes—
222C2×#92418#4112yes—
223C23×C61—21no—
224C23×C61H(5,𝔽2)11yescharacteristic
225Q8×C61—21no—
226Q8×C61—161no—
227Q8×C61C4⋊Q812yes—
228D4×C61—22no—
229D4×C61—22no—
230D4×C61—21no—
231D4×C61C4⋊Q812yes—
232C2×C4×C61—21no—
233C2×C4×C61—22no—
234C2×C4×C61—22no—
235C2×C4×C61—21no—
236C2×C4×C61—21no—
237C2×C4×C61—21no—
238C2×C4×C61—21no—
239C2×C4×C61—41no—
240C2×C4×C61—41no—
241C2×C4×C61—81no—
242C2×C4×C61—81no—
243C2×C4×C61—161no—
244C2×C4×C61C22×D411yescharacteristic
245C2×C4×C61C22×D411yescharacteristic
246C2×C4×C61C22×Q811yescharacteristic
247C2×C4×C61C22×Q811yescharacteristic
248C2×C4×C61C2×#4111yescharacteristic
249C2×C4×C612−1+411yescharacteristic
250C2×Dic61—21no—
251C2×Dic61—21no—
252C2×Dic61—22no—
253C2×Dic61—41no—
254C2×Dic61—41no—
255C2×Dic61—41no—
256C2×Dic61—41no—
257C2×Dic61—81no—
258C2×Dic61—81no—
259C2×Dic61—161no—
260Dic122—161no—
261C8×C61—21no—
262C8×C61—21no—
263C8×C61—41no—
264#92418—22no—
265#92418—22no—
266#92418—41no—
267#92418C22⋊Q812yes—
268#92418C22⋊Q812yes—
269#26082—611nomaximal, Sylow
270C22×C61—21no—
271C22×C61—21no—
272C22×C61—22no—
273C22×C61—41no—
274C22×C61—41no—
275C22×C61#54411yescharacteristic
276C22×C61D4211yescharacteristic
277C22×C61#55411yescharacteristic
278C4×C61—22no—
279C4×C61—41no—
280C4×C61—41no—
281C4×C61—41no—
282C4×C61—41no—
283C4×C61—81no—
284C4×C61—81no—
285C4×C61C2×C22⋊Q811yescharacteristic
286C4×C61C2×C4⋊Q811yescharacteristic
287C4×C61#54711yescharacteristic
288C4×C61D4×Q811yescharacteristic
289Dic61—42no—
290Dic61—41no—
291Dic61—41no—
292Dic61—81no—
293Dic61—81no—
294#3164—611no—
295#3084—611no—
296#3084D6111yescharacteristic
297#2913—611no—
298#2090—611no—
299#2090—611no—
300#2061—611no—
301#2061—611no—
302#1952—611no—
303#1945—611no—
304#1860—611no—
305#1860—611no—
306#1846—611no—
307#1816—611no—
308#1764—611no—
309C2×C61—21no—
310C2×C61—21no—
311C2×C61—42no—
312C2×C61#244011yescharacteristic
313C2×C61#329011yescharacteristic
314C2×C61#329011yescharacteristic
315C2×#131—611no—
316#544—611no—
317#544C2×D6111yescharacteristic
318#520—611no—
319#520C2×D6111yescharacteristic
320#501—1221no—
321#501—1221no—
322#500—1221no—
323#499—1221no—
324#479—21no—
325#479—1221no—
326#478—21no—
327#478—1221no—
328#476—21no—
329#476—1221no—
330#475—21no—
331#475—1221no—
332C2×#108—611no—
333#428—611no—
334#428—611no—
335#428—611no—
336#428—611no—
337#428—611no—
338#428—1221no—
339#428—1221no—
340#428C2×D6111yescharacteristic
341#427—611no—
342#427—1221no—
343C2×#107—611no—
344C2×#107—611no—
345C2×#107—1221no—
346C2×#106—611no—
347C2×#106C2×D6111yescharacteristic
348#424—611no—
349#424C2×D6111yescharacteristic
350C2×#104—611no—
351#417—611no—
352#417C2×D6111yescharacteristic
353#416C2×D6111yescharacteristic
354#405—611no—
355C2×C4×C8—611no—
356#382—611no—
357#382—611no—
358#343—611no—
359#343—611no—
360#343—611no—
361#343—611no—
362#341—611no—
363#339—611no—
364#339—611no—
365#339—611no—
366C61#2608211yescharacteristic, complemented, Sylow
367C2×#41C22×D6111yescharacteristic
368#131—614no—
369#131—1222no—
370C2×M16—611no—
371C2×M16—611no—
372C2×M16—611no—
373C2×M16—1221no—
374C2×M16C22×D6111yescharacteristic
375C22×C8—611no—
376C22×C8—1221no—
377C22×C8C22×D6111yescharacteristic
378C4⋊Q8—21no—
379C4⋊Q8—21no—
380C4⋊Q8—1221no—
381C4⋊Q8—1221no—
382#125—21no—
383#125—21no—
384#125—1221no—
385#125—1221no—
386C22⋊Q8—41no—
387C22⋊Q8—2441no—
388C4×Q8—21no—
389C4×Q8—1221no—
390C4×D4—21no—
391C4×D4—41no—
392C4×D4—1221no—
393C4×D4—2441no—
394#117—21no—
395#117—611no—
396#117—1221no—
397#117C22×D6111yescharacteristic
398#117C22×D6111yescharacteristic
399C2×C4⋊C4—21no—
400C2×C4⋊C4—611no—
401C2×C4⋊C4—611no—
402C2×C4⋊C4—1221no—
403C2×C4⋊C4C22×D6111yescharacteristic
404C2×C4⋊C4C22×D6111yescharacteristic
405C2×C42—611no—
406#108—1222no—
407#107—21no—
408#107—611no—
409#107—611no—
410#107—1221no—
411#107—1221no—
412#107—1221no—
413#107—1221no—
414#107—1221no—
415#107—1221no—
416#107—1221no—
417#107—1221no—
418#107—1221no—
419#106—21no—
420#106—21no—
421#106—21no—
422#106—611no—
423#106—611no—
424#106—1221no—
425#106—1221no—
426#106—1221no—
427#106—1221no—
428#106—1221no—
429#106—1221no—
430#106—1221no—
431C4⋊C8—21no—
432C4⋊C8—21no—
433C4⋊C8—1221no—
434C4⋊C8—1221no—
435#104—614no—
436#103—21no—
437#103—1221no—
438#103C22⋊D6111yescharacteristic
439#103C22⋊D6111yescharacteristic
440#102—21no—
441#102—1221no—
442#102C22⋊D6111yescharacteristic
443#102C22⋊D6111yescharacteristic
444#97—1221no—
445C4×C8—1221no—
446C4×C8—1222no—
447#95—2441no—
448#95—2441no—
449#41—21no—
450#41—21no—
451#41Q8×D6112yes—
452#41Q8×D6112yes—
453C2×Q8—41no—
454C2×Q8—2441no—
455C2×Q8C2×C22⋊D6111yescharacteristic
456C2×D4—21no—
457C2×D4C2×C22⋊D6111yescharacteristic
458C22×C4—21no—
459C22×C4—21no—
460C22×C4—21no—
461C22×C4—41no—
462C22×C4—1221no—
463C22×C4—1221no—
464C22×C4—1221no—
465C22×C4—2441no—
466C22×C4C23×D6111yescharacteristic
467M16—21no—
468M16—21no—
469M16—612no—
470M16—612no—
471M16—1224no—
472M16—1221no—
473M16—1221no—
474M16—1221no—
475M16—1221no—
476C2×C8—21no—
477C2×C8—21no—
478C2×C8—21no—
479C2×C8—611no—
480C2×C8—611no—
481C2×C8—612no—
482C2×C8—612no—
483C2×C8—1224no—
484C2×C8—1221no—
485C2×C8—1221no—
486C2×C8—1221no—
487C2×C8C2×C22⋊D6111yescharacteristic
488C2×C8C2×C22⋊D6111yescharacteristic
489C4⋊C4—21no—
490C4⋊C4—21no—
491C4⋊C4—21no—
492C4⋊C4—21no—
493C4⋊C4—41no—
494C4⋊C4—41no—
495C4⋊C4—41no—
496C4⋊C4—41no—
497C4⋊C4—611no—
498C4⋊C4—611no—
499C4⋊C4—1221no—
500C4⋊C4—1221no—
501C4⋊C4—1221no—
502C4⋊C4—1221no—
503C4⋊C4—2441no—
504C4⋊C4—2441no—
505C4⋊C4—2441no—
506C4⋊C4—2441no—
507C4⋊C4D4×D6111yescharacteristic
508C4⋊C4D4×D6111yescharacteristic
509C4⋊C4C2×C22⋊D6111yescharacteristic
510C4⋊C4C2×C22⋊D6111yescharacteristic
511C22⋊C4—21no—
512C22⋊C4—21no—
513C22⋊C4—41no—
514C22⋊C4—1221no—
515C22⋊C4—2441no—
516C42—21no—
517C42—21no—
518C42—41no—
519C42—612no—
520C42—1221no—
521C42—1221no—
522C42—2441no—
523C23—21no—
524C23#1451982211yescharacteristic
525Q8—21no—
526Q8—161no—
527Q8—9761no—
528Q8#1451977512yes—
529D4—22no—
530D4—22no—
531D4—21no—
532D4#1451977512yes—
533C2×C4—21no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—81no—
543C2×C4—81no—
544C2×C4—161no—
545C2×C4—1222no—
546C2×C4—1221no—
547C2×C4—1221no—
548C2×C4—9761no—
549C2×C4—4881no—
550C2×C4—4881no—
551C2×C4—2441no—
552C2×C4—2441no—
553C2×C4—2441no—
554C2×C4—2441no—
555C2×C4C2×D4×D6111yescharacteristic
556C2×C4C22×C22⋊D6111yescharacteristic
557C2×C4C2×Q8×D6111yescharacteristic
558C2×C4C2×Q8×D6111yescharacteristic
559C2×C4#41×D6111yescharacteristic
560C2×C4#1451982411yescharacteristic
561C8—21no—
562C8—21no—
563C8—41no—
564C8—612no—
565C8—612no—
566C8—1222no—
567C8—1222no—
568C8—2441no—
569C22—21no—
570C22—21no—
571C22—22no—
572C22—41no—
573C22—41no—
574C22#1684039111yescharacteristic
575C22D4×C22⋊D6111yescharacteristic
576C22#1684046611yescharacteristic
577C4—22no—
578C4—41no—
579C4—41no—
580C4—41no—
581C4—41no—
582C4—81no—
583C4—81no—
584C4—4881no—
585C4—4881no—
586C4—2442no—
587C4—2441no—
588C4—2441no—
589C4C22⋊Q8×D6111yescharacteristic
590C4C2×#1451977511yescharacteristic
591C4#1684041111yescharacteristic
592C4#1684042511yescharacteristic
593C2—21no—
594C2—21no—
595C2—42no—
596C2#2144951811yescharacteristic
597C2#2145759911yescharacteristic
598C2#2145759911yescharacteristic
subgroupG/H
centerC22D4×C22⋊D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#3084C2
socleC22×C61D42
prime radicalC2×#41×C61C23
hypercenter#3084D61
hypocenterC61#26082
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ #3084

Complements

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

NH ≅ G/Ntype
1. #21456353C2semidirect
4. #21447330C2semidirect
5. #21447329C2semidirect
6. #21447167C2semidirect
7. #21447166C2semidirect
8. #21446601C2semidirect
9. #21446564C2semidirect
12. #21445980C2semidirect
68. #14519784Q8semidirect
112. #14519704Q8semidirect
366. C61#26082semidirect

Maximal subgroups

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

subgroupindex
#214563532
#214556792
#214544172
#214473302
#214473292
#214471672
#214471662
#214466012
#214465642
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