Finite Group Databasebeta

← #29732165

Subgroups — #29732165

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#21456368C211yesmaximal, characteristic, complemented
2#21455616C211yesmaximal, characteristic
3#21454762C211yesmaximal, characteristic, complemented
4#21447330C211yesmaximal, characteristic, complemented
5#21447325C211yesmaximal, characteristic, complemented
6#21447177C211yesmaximal, characteristic
7#21447177C211yesmaximal, characteristic
8#21446813C211yesmaximal, characteristic, complemented
9#21446602C211yesmaximal, characteristic, complemented
10#21446597C211yesmaximal, characteristic, complemented
11#21446595C211yesmaximal, characteristic, complemented
12#21446478C211yesmaximal, characteristic
13#21446478C211yesmaximal, characteristic
14C4×#14519694C211yesmaximal, characteristic
15C2×C61×#478C211yesmaximal, characteristic
16#16840319C2211yescharacteristic
17#16840172—21no—
18#16840172—21no—
19#16840161—21no—
20#16840161—21no—
21#16840062—21no—
22#16840062—21no—
23#16840052—21no—
24#16840042—21no—
25#16839891—21no—
26#16839891—21no—
27#16839889—21no—
28#16839886—21no—
29C2×#14519713C2211yescharacteristic
30C2×#14519713C2211yescharacteristic
31#16839802C2211yescharacteristic
32#16839802C2211yescharacteristic
33#16839802C2211yescharacteristic
34#16839802C2211yescharacteristic
35#16839802C2211yescharacteristic
36#16839802C2211yescharacteristic
37#16839788C2211yescharacteristic
38#16839788C2211yescharacteristic
39#16839786C2211yescharacteristic
40#16839767C2211yescharacteristic
41#16839767C2211yescharacteristic
42C2×#14519694C2211yescharacteristic
43C2×#14519694C2211yescharacteristic
44#16839717C2211yescharacteristic
45#16839717C2211yescharacteristic
46C4×#11758194C2211yescharacteristic
47C4×#11758194C2211yescharacteristic
48#16839531C2211yescharacteristic
49#16839531C2211yescharacteristic
50#16839531C2211yescharacteristic
51#16839531C2211yescharacteristic
52#16839531C2211yescharacteristic
53#16839531C2211yescharacteristic
54#16839527C2211yescharacteristic
55#16839527C2211yescharacteristic
56C2×#125×C61C2211yescharacteristic
57C2×C4×D4×C61C2211yescharacteristic
58C61×#478—21no—
59C61×#478—21no—
60C61×#478—21no—
61C61×#478—21no—
62C2×#107×C61C2211yescharacteristic
63C2×#106×C61C2211yescharacteristic
64C2×C4⋊C8×C61C2211yescharacteristic
65C2×#102×C61C2211yescharacteristic
66C2×#102×C61C2211yescharacteristic
67C2×#11758194C2311yescharacteristic
68C2×#11758194C2311yescharacteristic
69C2×#11758194C2311yescharacteristic
70C2×#11758194C2311yescharacteristic
71C2×#11758194C2311yescharacteristic
72C2×#11758194C2311yescharacteristic
73#14519777D411yescharacteristic
74#14519777D411yescharacteristic
75#14519769D411yescharacteristic
76#14519769D411yescharacteristic
77#14519756—21no—
78#14519756—21no—
79C4×Dic122—21no—
80C4×Dic122—21no—
81#14519736C2311yescharacteristic
82#14519736C2311yescharacteristic
83#14519713—22no—
84#14519713—22no—
85#14519710—21no—
86#14519710—21no—
87#14519710—21no—
88#14519710—21no—
89#14519710—21no—
90#14519710—21no—
91#14519708—21no—
92#14519708—21no—
93#14519708—21no—
94#14519708—21no—
95#14519708—21no—
96#14519708—21no—
97C4⋊#92418—21no—
98C4⋊#92418—21no—
99#14519706—21no—
100#14519706—21no—
101#14519706—21no—
102#14519706—21no—
103#14519698—21no—
104#14519698—21no—
105#14519698—21no—
106#14519698—21no—
107#14519694—22no—
108#14519694Q814yescomplemented
109#14519688—21no—
110#14519688—21no—
111#14519687—21no—
112#14519687—21no—
113C4×#92418—21no—
114C4×#92418—21no—
115C8×Dic61—21no—
116C8×Dic61—21no—
117C22×D4×C61C2311yescharacteristic
118C23×C4×C61—21no—
119C22×C8×C61C2311yescharacteristic
120C22×C8×C61C2311yescharacteristic
121#125×C61—21no—
122#125×C61—21no—
123#125×C61—21no—
124#125×C61—21no—
125C4×D4×C61—21no—
126C4×D4×C61—21no—
127C4×D4×C61—21no—
128C4×D4×C61—21no—
129C2×C4⋊C4×C61—21no—
130C2×C4⋊C4×C61—21no—
131C2×C4⋊C4×C61C2311yescharacteristic
132C2×C4⋊C4×C61C2311yescharacteristic
133C2×C4⋊C4×C61C2311yescharacteristic
134C2×C22⋊C4×C61—21no—
135C2×C42×C61C2311yescharacteristic
136#107×C61—21no—
137#107×C61—21no—
138#106×C61—21no—
139#106×C61—21no—
140C4⋊C8×C61D411yescharacteristic
141C4⋊C8×C61D411yescharacteristic
142C4⋊C8×C61D411yescharacteristic
143C4⋊C8×C61D411yescharacteristic
144#102×C61—21no—
145#102×C61—21no—
146#102×C61—21no—
147#102×C61—21no—
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×C61—21no—
157C22×C4×C61—21no—
158C22×C4×C61—41no—
159C22×C4×C61—41no—
160C22×C4×C61—82no—
161C22×C4×C61#4111yescharacteristic
162C22×C4×C61#4111yescharacteristic
163C22×C4×C61C2411yescharacteristic
164C2×Dic122—21no—
165C2×Dic122—21no—
166C2×C8×C61—21no—
167C2×C8×C61—21no—
168C2×C8×C61—21no—
169C2×C8×C61—21no—
170C2×C8×C61C2×D411yescharacteristic
171C2×C8×C61C2×D411yescharacteristic
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—21no—
179C4⋊C4×C61—21no—
180C4⋊C4×C61—41no—
181C4⋊C4×C61—41no—
182C4⋊C4×C61—41no—
183C4⋊C4×C61—41no—
184C22⋊C4×C61—41no—
185C22⋊C4×C61—41no—
186C42×C61C2×D411yescharacteristic
187C42×C61C2×D411yescharacteristic
188C42×C61C2×D411yescharacteristic
189C42×C61C2×D411yescharacteristic
190C22⋊Dic61—21no—
191C22⋊Dic61—21no—
192C4⋊Dic61—21no—
193C4⋊Dic61—21no—
194C4⋊Dic61C2×D411yescharacteristic
195C4⋊Dic61C2×D411yescharacteristic
196C4⋊Dic61C2×D411yescharacteristic
197C4⋊Dic61C2×D411yescharacteristic
198#11758196—21no—
199#11758196—21no—
200#11758196—21no—
201#11758196—21no—
202C4×Dic61—21no—
203C4×Dic61—21no—
204#11758194—21no—
205#11758194—21no—
206#11758194—21no—
207#11758194—21no—
208#11758194—21no—
209#11758194—21no—
210#11758194—21no—
211#11758194—21no—
212#11758194—21no—
213#11758194C2×Q812yes—
214#11758194C2×Q812yes—
215#11758194#4112yes—
216C2×#92418—21no—
217C2×#92418—21no—
218C2×#92418—21no—
219C2×#92418—21no—
220C2×#92418—21no—
221C2×#92418—21no—
222C23×C61—22no—
223C23×C61—41no—
224C23×C61—41no—
225C23×C61—81no—
226C23×C61—81no—
227C23×C61C2×#4111yescharacteristic
228D4×C61—22no—
229D4×C61—22no—
230D4×C61—41no—
231D4×C61—41no—
232C2×C4×C61—41no—
233C2×C4×C61—41no—
234C2×C4×C61—41no—
235C2×C4×C61—41no—
236C2×C4×C61—41no—
237C2×C4×C61—82no—
238C2×C4×C61—82no—
239C2×C4×C61—81no—
240C2×C4×C61—81no—
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×C61C22×Q811yescharacteristic
249C2×C4×C61C2×#4111yescharacteristic
250C2×C4×C612−1+411yescharacteristic
251C2×C4×C612−1+411yescharacteristic
252C2×Dic61—21no—
253C2×Dic61—21no—
254C2×Dic61—21no—
255C2×Dic61—21no—
256Dic122—42no—
257Dic122—81no—
258C8×C61—21no—
259C8×C61—21no—
260C8×C61—21no—
261C8×C61—21no—
262#92418—22no—
263#92418—22no—
264#92418—22no—
265#92418—41no—
266#92418—41no—
267#92418—41no—
268#25493—611nomaximal, Sylow
269C22×C61—21no—
270C22×C61—21no—
271C22×C61—42no—
272C22×C61—82no—
273C22×C61—82no—
274C22×C61—81no—
275C22×C61—81no—
276C22×C61C2×#12111yescharacteristic
277C22×C61#53311yescharacteristic
278C22×C61#54111yescharacteristic
279C4×C61—21no—
280C4×C61—21no—
281C4×C61—21no—
282C4×C61—21no—
283C4×C61—81no—
284C4×C61—81no—
285C4×C61—81no—
286C4×C61#54711yescharacteristic
287C4×C61#54811yescharacteristic
288C4×C61D4×Q811yescharacteristic
289C4×C61D4×Q811yescharacteristic
290Dic61—41no—
291Dic61—41no—
292Dic61—41no—
293Dic61—41no—
294#3166—611no—
295#3077—611no—
296C2×#478D6111yescharacteristic
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—82no—
312C2×C61#242911yescharacteristic
313C2×C61#306711yescharacteristic
314C2×C61#306811yescharacteristic
315#533—611no—
316C2×#125C2×D6111yescharacteristic
317C2×C4×D4C2×D6111yescharacteristic
318#501—1221no—
319#501—1221no—
320#499—1221no—
321#499—1221no—
322#478—21no—
323#478—21no—
324#478—21no—
325#478—21no—
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#428—611no—
339#428—611no—
340#428—611no—
341#428—611no—
342C2×#107C2×D6111yescharacteristic
343C2×#106C2×D6111yescharacteristic
344#424—611no—
345#424—611no—
346#423—611no—
347C2×C4⋊C8C2×D6111yescharacteristic
348#418—611no—
349#418—611no—
350C2×#102C2×D6111yescharacteristic
351C2×#102C2×D6111yescharacteristic
352C2×#100—611no—
353C2×#100—611no—
354#405—611no—
355#405—611no—
356C4×M16—611no—
357C4×M16—611no—
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×D4C22×D6111yescharacteristic
368C23×C4—21no—
369C2×M16—611no—
370C2×M16—611no—
371C2×M16—611no—
372C2×M16—611no—
373C2×M16—611no—
374C2×M16—611no—
375C22×C8C22×D6111yescharacteristic
376C22×C8C22×D6111yescharacteristic
377C4⋊Q8—611no—
378C4⋊Q8—611no—
379#125—21no—
380#125—21no—
381#125—21no—
382#125—21no—
383#125—611no—
384#125—611no—
385C22⋊Q8—1221no—
386C22⋊Q8—1221no—
387C4×Q8—1221no—
388C4×Q8—1221no—
389C4×D4—21no—
390C4×D4—21no—
391C4×D4—21no—
392C4×D4—21no—
393#117—611no—
394#117—611no—
395C2×C4⋊C4—21no—
396C2×C4⋊C4—21no—
397C2×C4⋊C4C22×D6111yescharacteristic
398C2×C4⋊C4C22×D6111yescharacteristic
399C2×C4⋊C4C22×D6111yescharacteristic
400C2×C22⋊C4—21no—
401C2×C42C22×D6111yescharacteristic
402#108—1222no—
403#108—1222no—
404#107—21no—
405#107—21no—
406#107—1221no—
407#107—1221no—
408#107—1221no—
409#107—1221no—
410#107—1221no—
411#107—1221no—
412#106—21no—
413#106—21no—
414#106—1221no—
415#106—1221no—
416#106—1221no—
417#106—1221no—
418#106—1221no—
419#106—1221no—
420C4⋊C8—1221no—
421C4⋊C8—1221no—
422C4⋊C8—1221no—
423C4⋊C8—1221no—
424C4⋊C8—1221no—
425C4⋊C8—1221no—
426C4⋊C8C22⋊D6111yescharacteristic
427C4⋊C8C22⋊D6111yescharacteristic
428C4⋊C8C22⋊D6111yescharacteristic
429C4⋊C8C22⋊D6111yescharacteristic
430#102—21no—
431#102—21no—
432#102—21no—
433#102—21no—
434#102—1221no—
435#102—1221no—
436#102—1221no—
437#102—1221no—
438#100—614no—
439#100—1222no—
440#97—1221no—
441#97—1221no—
442#97—1221no—
443#97—1221no—
444C4×C8—1221no—
445C4×C8—1221no—
446C4×C8—1221no—
447C4×C8—1221no—
448C24—21no—
449C2×Q8—1221no—
450C2×Q8—1221no—
451C2×D4—21no—
452C2×D4—21no—
453C2×D4—21no—
454C2×D4—21no—
455C2×D4—21no—
456C2×D4Q8×D6112yes—
457C22×C4—21no—
458C22×C4—21no—
459C22×C4—21no—
460C22×C4—41no—
461C22×C4—41no—
462C22×C4—82no—
463C22×C4#1175822411yescharacteristic
464C22×C4#1175822411yescharacteristic
465C22×C4C23×D6111yescharacteristic
466M16—612no—
467M16—612no—
468M16—612no—
469M16—1221no—
470M16—1221no—
471M16—1221no—
472M16—1221no—
473M16—1221no—
474M16—1221no—
475M16—1221no—
476M16—1221no—
477M16—1221no—
478C2×C8—21no—
479C2×C8—21no—
480C2×C8—21no—
481C2×C8—21no—
482C2×C8—1221no—
483C2×C8—1221no—
484C2×C8—1221no—
485C2×C8—1221no—
486C2×C8—1221no—
487C2×C8—1221no—
488C2×C8C2×C22⋊D6111yescharacteristic
489C2×C8C2×C22⋊D6111yescharacteristic
490C2×C8C2×C22⋊D6111yescharacteristic
491C2×C8C2×C22⋊D6111yescharacteristic
492C4⋊C4—21no—
493C4⋊C4—21no—
494C4⋊C4—21no—
495C4⋊C4—21no—
496C4⋊C4—21no—
497C4⋊C4—21no—
498C4⋊C4—41no—
499C4⋊C4—41no—
500C4⋊C4—41no—
501C4⋊C4—41no—
502C4⋊C4—611no—
503C4⋊C4—611no—
504C4⋊C4—611no—
505C4⋊C4—611no—
506C4⋊C4—1221no—
507C4⋊C4—1221no—
508C4⋊C4—1221no—
509C4⋊C4—1221no—
510C4⋊C4—1221no—
511C4⋊C4—1221no—
512C22⋊C4—41no—
513C22⋊C4—41no—
514C22⋊C4—1221no—
515C22⋊C4—1221no—
516C42—1221no—
517C42—1221no—
518C42D4×D6111yescharacteristic
519C42D4×D6111yescharacteristic
520C42C2×C22⋊D6111yescharacteristic
521C42C2×C22⋊D6111yescharacteristic
522C23—22no—
523C23—41no—
524C23—41no—
525C23—81no—
526C23—81no—
527C23C2×#1175822411yescharacteristic
528Q8—4881no—
529Q8—2442no—
530D4—22no—
531D4—22no—
532D4—41no—
533D4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—82no—
540C2×C4—82no—
541C2×C4—81no—
542C2×C4—81no—
543C2×C4—81no—
544C2×C4—1221no—
545C2×C4—1221no—
546C2×C4—1221no—
547C2×C4—1221no—
548C2×C4#1451974811yescharacteristic
549C2×C4#1451974811yescharacteristic
550C2×C4#1451974811yescharacteristic
551C2×C4#1451974811yescharacteristic
552C2×C4C2×D4×D6111yescharacteristic
553C2×C4C22×C22⋊D6111yescharacteristic
554C2×C4C2×Q8×D6111yescharacteristic
555C2×C4#41×D6111yescharacteristic
556C2×C4#1451982411yescharacteristic
557C2×C4#1451982511yescharacteristic
558C8—21no—
559C8—21no—
560C8—21no—
561C8—21no—
562C8—1222no—
563C8—1222no—
564C8—1222no—
565C8—2441no—
566C8—2441no—
567C8—2441no—
568C22—21no—
569C22—21no—
570C22—42no—
571C22—82no—
572C22—82no—
573C22—81no—
574C22—81no—
575C22C2×#1451974811yescharacteristic
576C22#1684031411yescharacteristic
577C22#1684037211yescharacteristic
578C4—21no—
579C4—21no—
580C4—21no—
581C4—21no—
582C4—81no—
583C4—81no—
584C4—81no—
585C4—2441no—
586C4—2441no—
587C4—2441no—
588C4—2441no—
589C4#1684041111yescharacteristic
590C4#1684042211yescharacteristic
591C4#1684042511yescharacteristic
592C4Q8×C22⋊D6111yescharacteristic
593C2—21no—
594C2—21no—
595C2—82no—
596C2#2144941411yescharacteristic
597C2#2145553511yescharacteristic
598C2#2145554311yescharacteristic
subgroupG/H
centerC22C2×#14519748
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC2×C61×#478C2
socleC22×C61C2×#121
prime radicalC22×D4×C61C23
hypercenterC2×#478D61
hypocenterC61#25493
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#478

Complements

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

NH ≅ G/Ntype
1. #21456368C2semidirect
3. #21454762C2semidirect
4. #21447330C2semidirect
5. #21447325C2semidirect
8. #21446813C2semidirect
9. #21446602C2semidirect
10. #21446597C2semidirect
11. #21446595C2semidirect
108. #14519694Q8semidirect
366. C61#25493semidirect

Maximal subgroups

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

subgroupindex
#214563682
#214556162
#214547622
#214473302
#214473252
#214471772
#214468132
#214466022
#214465972
#214465952
#214464782
C4×#145196942
C2×C61×#4782
#2549361