Finite Group Databasebeta

← #29732129

Subgroups — #29732129

Order 15616.

664 conjugacy classes of subgroups.

The subgroup lattice has 606 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456363C211yesmaximal, characteristic, complemented
2#21455613C211yesmaximal, characteristic
3C2×#16840050C211yesmaximal, characteristic
4#21454765C211yesmaximal, characteristic, complemented
5#21447333C211yesmaximal, characteristic, complemented
6#21447329C211yesmaximal, characteristic, complemented
7#21447175C211yesmaximal, characteristic
8#21446806C211yesmaximal, characteristic, complemented
9#21446600C211yesmaximal, characteristic, complemented
10#21446596C211yesmaximal, characteristic, complemented
11#21446591C211yesmaximal, characteristic, complemented
12#21446484C211yesmaximal, characteristic
13#21446478C211yesmaximal, characteristic
14#100×Dic61C211yesmaximal, characteristic
15C61×#2063C211yesmaximal, characteristic
16#16840315C2211yescharacteristic
17C2×#14519776C2211yescharacteristic
18C2×D4×Dic61C2211yescharacteristic
19#16840170—21no—
20#16840169—21no—
21#16840166—21no—
22#16840159—21no—
23#16840064—21no—
24#16840060—21no—
25#16840050—21no—
26#16840050—21no—
27#16840050—21no—
28#16840050—21no—
29#16840050—21no—
30#16840040—21no—
31#16839891—21no—
32#16839891—21no—
33#16839889—21no—
34#16839886—21no—
35C2×#14519712C2211yescharacteristic
36#16839805C2211yescharacteristic
37#16839805C2211yescharacteristic
38#16839804C2211yescharacteristic
39#16839804C2211yescharacteristic
40#16839802C2211yescharacteristic
41C2×#14519709C2211yescharacteristic
42C2×#14519708C2211yescharacteristic
43#16839790C2211yescharacteristic
44#16839785C2211yescharacteristic
45C2×#14519706C2211yescharacteristic
46#16839767C2211yescharacteristic
47C2×#14519698C2211yescharacteristic
48C2×#14519694C2211yescharacteristic
49#16839718C2211yescharacteristic
50#16839717C2211yescharacteristic
51M16×Dic61C2211yescharacteristic
52M16×Dic61C2211yescharacteristic
53#16839531C2211yescharacteristic
54#16839530C2211yescharacteristic
55#16839530C2211yescharacteristic
56#16839530C2211yescharacteristic
57#16839530C2211yescharacteristic
58#16839527C2211yescharacteristic
59#16839526C2211yescharacteristic
60C2×#108×C61C2211yescharacteristic
61C61×#428C2211yescharacteristic
62C61×#424C2211yescharacteristic
63C61×#418C2211yescharacteristic
64C2×#102×C61C2211yescharacteristic
65C2×#100×C61C2211yescharacteristic
66C61×#343C2211yescharacteristic
67C22×Dic122—21no—
68C23×Dic61—21no—
69C2×#11758194C2311yescharacteristic
70C2×#11758194C2311yescharacteristic
71C2×#11758194C2311yescharacteristic
72C22×#92418C2311yescharacteristic
73#14519776—21no—
74#14519776—21no—
75#14519776—21no—
76#14519776—21no—
77#14519776D411yescharacteristic
78#14519776D411yescharacteristic
79#14519768D411yescharacteristic
80#14519768D411yescharacteristic
81#14519754—21no—
82#14519754—21no—
83C4×Dic122—21no—
84C4×Dic122—21no—
85D4×Dic61—21no—
86D4×Dic61—21no—
87D4×Dic61—21no—
88D4×Dic61—21no—
89#14519736C2311yescharacteristic
90C2×C4⋊Dic61C2311yescharacteristic
91C2×C4⋊Dic61C2311yescharacteristic
92C2×#11758196—21no—
93C2×C22⋊Dic61—21no—
94C2×C4×Dic61C2311yescharacteristic
95#14519712—22no—
96#14519711—21no—
97#14519711—21no—
98#14519710—21no—
99#14519710—21no—
100#14519710—21no—
101#14519709—21no—
102#14519709—21no—
103#14519709—21no—
104#14519709—21no—
105#14519708—21no—
106#14519708—21no—
107#14519708—21no—
108#14519708—21no—
109#14519708—21no—
110#14519706—21no—
111#14519706—21no—
112#14519706—21no—
113#14519706—21no—
114#14519706D411yescharacteristic
115#14519706D411yescharacteristic
116#14519706D411yescharacteristic
117#14519706D411yescharacteristic
118#14519698—21no—
119#14519698—21no—
120#14519698—21no—
121#14519698—21no—
122#14519694Q814yescomplemented
123#14519688—21no—
124#14519687—21no—
125#14519687—21no—
126#14519687—21no—
127C4×#92418—21no—
128C8×Dic61—21no—
129C8×Dic61—21no—
130C8×Dic61—21no—
131C22×D4×C61C2311yescharacteristic
132C2×M16×C61C2311yescharacteristic
133C2×M16×C61C2311yescharacteristic
134C2×M16×C61C2311yescharacteristic
135C22×C8×C61C2311yescharacteristic
136#117×C61C2311yescharacteristic
137C2×C4⋊C4×C61C2311yescharacteristic
138#108×C61—22no—
139#107×C61—21no—
140#106×C61—21no—
141C4⋊C8×C61—21no—
142C4⋊C8×C61—21no—
143#102×C61—21no—
144#102×C61—21no—
145#102×C61—21no—
146#102×C61—21no—
147#100×C61—22no—
148C24×C61—21no—
149C2×D4×C61—21no—
150C2×D4×C61—21no—
151C2×D4×C61—21no—
152C2×D4×C61—21no—
153C2×D4×C61—21no—
154C2×D4×C61C2×Q812yes—
155C22×C4×C61—21no—
156C22×C4×C61C2411yescharacteristic
157C22×Dic61—21no—
158C22×Dic61—21no—
159C22×Dic61—41no—
160C22×Dic61—41no—
161C22×Dic61—82no—
162C22×Dic61#4111yescharacteristic
163C22×Dic61#4111yescharacteristic
164C2×Dic122—21no—
165C2×Dic122—21no—
166C2×Dic122—41no—
167C2×Dic122—41no—
168C2×Dic122—82no—
169M16×C61—21no—
170M16×C61—21no—
171M16×C61—21no—
172M16×C61—21no—
173M16×C61—21no—
174M16×C61#4112yes—
175C2×C8×C61—21no—
176C2×C8×C61—21no—
177C2×C8×C61—21no—
178C2×C8×C61—21no—
179C2×C8×C61—21no—
180C2×C8×C61C2×D411yescharacteristic
181C2×C8×C61C2×D411yescharacteristic
182C4⋊C4×C61—21no—
183C4⋊C4×C61—21no—
184C4⋊C4×C61C2×D411yescharacteristic
185C4⋊C4×C61C2×D411yescharacteristic
186C22⋊C4×C61—21no—
187C42×C61—21no—
188C22⋊Dic61—21no—
189C22⋊Dic61—41no—
190C22⋊Dic61—41no—
191C4⋊Dic61—21no—
192C4⋊Dic61—21no—
193C4⋊Dic61—21no—
194C4⋊Dic61—21no—
195C4⋊Dic61—21no—
196C4⋊Dic61—21no—
197C4⋊Dic61C2×D411yescharacteristic
198C4⋊Dic61C2×D411yescharacteristic
199#11758196—21no—
200#11758196—21no—
201#11758196—21no—
202#11758196—21no—
203#11758196—41no—
204#11758196—41no—
205C4×Dic61—21no—
206C4×Dic61C2×D411yescharacteristic
207C4×Dic61C2×D411yescharacteristic
208C4×Dic61C2×D411yescharacteristic
209C4×Dic61C2×D411yescharacteristic
210#11758194—21no—
211#11758194—21no—
212#11758194—21no—
213#11758194—21no—
214#11758194C2×Q812yes—
215#11758194C2×Q812yes—
216C2×#92418—21no—
217C2×#92418—21no—
218C2×#92418—21no—
219C2×#92418—21no—
220C2×#92418—21no—
221C2×#92418C2×D411yescharacteristic
222C2×#92418C2×D411yescharacteristic
223C23×C61—22no—
224C23×C61—41no—
225C23×C61—41no—
226C23×C61—81no—
227C23×C61—81no—
228C23×C61C2×#4111yescharacteristic
229D4×C61—22no—
230D4×C61—22no—
231D4×C61—41no—
232D4×C61—41no—
233C2×C4×C61—21no—
234C2×C4×C61—21no—
235C2×C4×C61—41no—
236C2×C4×C61—81no—
237C2×C4×C61C22×D411yescharacteristic
238C2×C4×C61C22×D411yescharacteristic
239C2×C4×C61C22×Q811yescharacteristic
240C2×C4×C61C2×#4111yescharacteristic
241C2×C4×C612−1+411yescharacteristic
242C2×C4×C612−1+411yescharacteristic
243C2×Dic61—22no—
244C2×Dic61—21no—
245C2×Dic61—21no—
246C2×Dic61—41no—
247C2×Dic61—41no—
248C2×Dic61—41no—
249C2×Dic61—82no—
250C2×Dic61—82no—
251C2×Dic61—81no—
252C2×Dic61—81no—
253C2×Dic61#12111yescharacteristic
254C2×Dic61#12111yescharacteristic
255C2×Dic61#12111yescharacteristic
256C2×Dic61#12111yescharacteristic
257Dic122—42no—
258Dic122—81no—
259Dic122—82no—
260Dic122—82no—
261C8×C61—21no—
262C8×C61—21no—
263C8×C61—22no—
264C8×C61—41no—
265C8×C61—41no—
266#92418—21no—
267#92418—21no—
268#92418—22no—
269#92418—22no—
270#92418—41no—
271#25490—611nomaximal, Sylow
272C22×C61—21no—
273C22×C61—21no—
274C22×C61—42no—
275C22×C61—82no—
276C22×C61—82no—
277C22×C61—81no—
278C22×C61—81no—
279C22×C61C2×#12111yescharacteristic
280C22×C61#53311yescharacteristic
281C22×C61#54111yescharacteristic
282C4×C61—41no—
283C4×C61—41no—
284C4×C61—81no—
285C4×C61#54711yescharacteristic
286C4×C61#54811yescharacteristic
287C4×C61D4×Q811yescharacteristic
288C4×C61D4×Q811yescharacteristic
289Dic61—21no—
290Dic61—21no—
291Dic61—21no—
292Dic61—41no—
293Dic61—41no—
294Dic61—81no—
295Dic61—81no—
296Dic61#47112yes—
297#3165—611no—
298#3077—611no—
299C2×#476—611no—
300#2962—611no—
301#2090—611no—
302#2090—611no—
303#2063—611no—
304#2063D6111yescharacteristic
305#1997—611no—
306#1951—611no—
307#1951—611no—
308#1950—611no—
309#1931—611no—
310#1931—611no—
311C4×#100—611no—
312C2×C61—21no—
313C2×C61—21no—
314C2×C61—82no—
315C2×C61#242911yescharacteristic
316C2×C61C2×#47111yescharacteristic
317C2×C61#306811yescharacteristic
318#533—611no—
319C2×C4⋊Q8—611no—
320C2×C4×D4—611no—
321#501—1221no—
322#501—1221no—
323#500—1221no—
324#498—1221no—
325#478—1221no—
326#478—1221no—
327#476—1221no—
328#476—1221no—
329#476—1221no—
330#476—1221no—
331#476—1221no—
332#474—1221no—
333#446—1221no—
334#446—1221no—
335#446—1221no—
336C8×Q8—1221no—
337C2×#108—611no—
338C2×#108C2×D6111yescharacteristic
339#428—611no—
340#428—611no—
341#428—611no—
342#428—611no—
343#428—611no—
344#428C2×D6111yescharacteristic
345C2×#106—611no—
346C2×#106—611no—
347#424—611no—
348#424C2×D6111yescharacteristic
349#423—611no—
350C2×C4⋊C8—611no—
351#418—611no—
352#418C2×D6111yescharacteristic
353C2×#102—611no—
354C2×#102C2×D6111yescharacteristic
355C2×#100—611no—
356C2×#100C2×D6111yescharacteristic
357#405—611no—
358#405—611no—
359C4×M16—611no—
360C4×M16—611no—
361#343—611no—
362#343—611no—
363#343—611no—
364#343—611no—
365#343—611no—
366#343C2×D6111yescharacteristic
367#341—611no—
368#341—611no—
369C61#2549011yescharacteristic, complemented, Sylow
370C22×Q8—1221no—
371C22×D4C22×D6111yescharacteristic
372C23×C4—1221no—
373C2×M16—611no—
374C2×M16—611no—
375C2×M16—611no—
376C2×M16C22×D6111yescharacteristic
377C2×M16C22×D6111yescharacteristic
378C2×M16C22×D6111yescharacteristic
379C22×C8—611no—
380C22×C8C22×D6111yescharacteristic
381C4⋊Q8—611no—
382C4⋊Q8—611no—
383C4⋊Q8—1221no—
384C4⋊Q8—1221no—
385C4⋊Q8—1221no—
386C4⋊Q8—1221no—
387#125—611no—
388#125—611no—
389C22⋊Q8—1221no—
390C22⋊Q8—1221no—
391C4×Q8—1221no—
392C4×Q8—1221no—
393C4×D4—1221no—
394C4×D4—1221no—
395C4×D4—1221no—
396C4×D4—1221no—
397#117—611no—
398#117C22×D6111yescharacteristic
399C2×C4⋊C4—611no—
400C2×C4⋊C4—611no—
401C2×C4⋊C4—1221no—
402C2×C4⋊C4C22×D6111yescharacteristic
403C2×C22⋊C4—1221no—
404C2×C42—611no—
405#108—22no—
406#108—1222no—
407#107—21no—
408#107—1221no—
409#107—1221no—
410#107—1221no—
411#107—1221no—
412#107—1221no—
413#106—21no—
414#106—1221no—
415#106—1221no—
416#106—1221no—
417#106—1221no—
418#106—1221no—
419#106—1221no—
420#106—1221no—
421#106—1221no—
422#106—1221no—
423C4⋊C8—21no—
424C4⋊C8—21no—
425C4⋊C8—611no—
426C4⋊C8—611no—
427C4⋊C8—611no—
428C4⋊C8—611no—
429C4⋊C8—1221no—
430C4⋊C8—1221no—
431C4⋊C8—1221no—
432C4⋊C8—1221no—
433#102—21no—
434#102—21no—
435#102—21no—
436#102—21no—
437#102—1221no—
438#102—1221no—
439#102—1221no—
440#102—1221no—
441#100—22no—
442#100—614no—
443#97—1221no—
444#97—1221no—
445#97—1221no—
446#97—1221no—
447C4×C8—1221no—
448C4×C8—1221no—
449C4×C8—1221no—
450C4×C8—1221no—
451C24—21no—
452C2×Q8—1221no—
453C2×Q8—1221no—
454C2×Q8—4882no—
455C2×Q8—2441no—
456C2×Q8—2441no—
457C2×D4—21no—
458C2×D4—21no—
459C2×D4—21no—
460C2×D4—21no—
461C2×D4—21no—
462C2×D4Q8×D6112yes—
463C22×C4—21no—
464C22×C4—611no—
465C22×C4—611no—
466C22×C4—1221no—
467C22×C4—1221no—
468C22×C4—4882no—
469C22×C4—2441no—
470C22×C4—2441no—
471C22×C4C23×D6111yescharacteristic
472M16—21no—
473M16—21no—
474M16—21no—
475M16—21no—
476M16—21no—
477M16—612no—
478M16—612no—
479M16—1221no—
480M16—1221no—
481M16—1221no—
482M16—1221no—
483M16#1175822412yes—
484C2×C8—21no—
485C2×C8—21no—
486C2×C8—21no—
487C2×C8—21no—
488C2×C8—21no—
489C2×C8—611no—
490C2×C8—611no—
491C2×C8—1221no—
492C2×C8—1221no—
493C2×C8—1221no—
494C2×C8—1221no—
495C2×C8—1221no—
496C2×C8D4×D6111yescharacteristic
497C2×C8D4×D6111yescharacteristic
498C4⋊C4—21no—
499C4⋊C4—21no—
500C4⋊C4—611no—
501C4⋊C4—611no—
502C4⋊C4—1221no—
503C4⋊C4—1221no—
504C4⋊C4—1221no—
505C4⋊C4—1221no—
506C4⋊C4—1221no—
507C4⋊C4—1221no—
508C4⋊C4—1221no—
509C4⋊C4—1221no—
510C4⋊C4—1221no—
511C4⋊C4—1221no—
512C4⋊C4—2441no—
513C4⋊C4—2441no—
514C4⋊C4D4×D6111yescharacteristic
515C4⋊C4D4×D6111yescharacteristic
516C22⋊C4—21no—
517C22⋊C4—1221no—
518C22⋊C4—2441no—
519C22⋊C4—2441no—
520C42—21no—
521C42—611no—
522C42—611no—
523C42—611no—
524C42—611no—
525C42—1221no—
526C23—22no—
527C23—41no—
528C23—41no—
529C23—81no—
530C23—81no—
531C23#41×D6111yescharacteristic
532Q8—4881no—
533Q8—4882no—
534Q8—4882no—
535Q8—2442no—
536D4—22no—
537D4—22no—
538D4—41no—
539D4—41no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—41no—
543C2×C4—81no—
544C2×C4—611no—
545C2×C4—611no—
546C2×C4—611no—
547C2×C4—611no—
548C2×C4—1222no—
549C2×C4—1221no—
550C2×C4—1221no—
551C2×C4—4882no—
552C2×C4—4882no—
553C2×C4—4881no—
554C2×C4—4881no—
555C2×C4—2441no—
556C2×C4—2441no—
557C2×C4—2441no—
558C2×C4C2×D4×D6111yescharacteristic
559C2×C4C2×D4×D6111yescharacteristic
560C2×C4C2×Q8×D6111yescharacteristic
561C2×C4C2×#1175822411yescharacteristic
562C2×C4#1451982411yescharacteristic
563C2×C4#1451982411yescharacteristic
564C8—21no—
565C8—21no—
566C8—22no—
567C8—41no—
568C8—41no—
569C8—1221no—
570C8—1221no—
571C8—1222no—
572C8—1222no—
573C8—2441no—
574C22—21no—
575C22—21no—
576C22—42no—
577C22—82no—
578C22—82no—
579C22—81no—
580C22—81no—
581C22#121×D6111yescharacteristic
582C22#1684031411yescharacteristic
583C22#1684036911yescharacteristic
584C4—41no—
585C4—41no—
586C4—81no—
587C4—612no—
588C4—1221no—
589C4—1221no—
590C4—1221no—
591C4—4881no—
592C4—4881no—
593C4—2441no—
594C4—2441no—
595C4#1684041211yescharacteristic
596C4#1684041811yescharacteristic
597C4#1684042511yescharacteristic
598C4#1684042511yescharacteristic
599C2—21no—
600C2—21no—
601C2—82no—
602C2#2144941111yescharacteristic
603C2#471×D6111yescharacteristic
604C2#2145554511yescharacteristic
subgroupG/H
centerC22#121×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2063C2
socleC22×C61C2×#121
prime radicalC22×D4×C61C23
hypercenter#2063D61
hypocenterC61#25490
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ #2063

Complements

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

NH ≅ G/Ntype
1. #21456363C2semidirect
4. #21454765C2semidirect
5. #21447333C2semidirect
6. #21447329C2semidirect
8. #21446806C2semidirect
9. #21446600C2semidirect
10. #21446596C2semidirect
11. #21446591C2semidirect
122. #14519694Q8semidirect
369. C61#25490semidirect

Maximal subgroups

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

subgroupindex
#214563632
#214556132
C2×#168400502
#214547652
#214473332
#214473292
#214471752
#214468062
#214466002
#214465962
#214465912
#214464842
#214464782
#100×Dic612
C61×#20632
#2549061