Finite Group Databasebeta

← #29665536

Subgroups — #29665536

Order 15616.

616 conjugacy classes of subgroups.

The subgroup lattice has 576 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456951C211yesmaximal, characteristic, complemented
2#21455981C211yesmaximal, characteristic, complemented
3#21454930C211yesmaximal, characteristic, complemented
4#21445261C211yesmaximal, characteristic, complemented
5#21445254C211yesmaximal, characteristic, complemented
6#21445233C211yesmaximal, characteristic, complemented
7#21445226C211yesmaximal, characteristic, complemented
8#21445221C211yesmaximal, characteristic, complemented
9#21445216C211yesmaximal, characteristic, complemented
10#21445203C211yesmaximal, characteristic, complemented
11#21445196C211yesmaximal, characteristic, complemented
12#21444597C211yesmaximal, characteristic, complemented
13#21444514C211yesmaximal, characteristic, complemented
14#21444474C211yesmaximal, characteristic, complemented
15C61×#3106C211yesmaximal, characteristic
16#16840319C2211yescharacteristic
17#16840165C2211yescharacteristic
18#16840158C2211yescharacteristic
19#16840108C2211yescharacteristic
20#16840108C2211yescharacteristic
21#16840103—21no—
22#16840083—21no—
23#16840083—21no—
24#16840062—21no—
25#16840062—21no—
26#16840057C2211yescharacteristic
27#16840052C2211yescharacteristic
28#16839972C2211yescharacteristic
29#16839967C2211yescharacteristic
30#16839924—21no—
31Q8×#92418C2211yescharacteristic
32Q8×#92418C2211yescharacteristic
33#16839867—21no—
34#16839862—21no—
35D4×#92418—21no—
36D4×#92418—21no—
37D4×#92418C2211yescharacteristic
38D4×#92418C2211yescharacteristic
39#16839819C2211yescharacteristic
40#16839815—21no—
41#16839815C2211yescharacteristic
42C2×C4⋊#92418—21no—
43C2×C4⋊#92418C2211yescharacteristic
44#16839514C2211yescharacteristic
45#16839514C2211yescharacteristic
46#16839512C2211yescharacteristic
47#16839512C2211yescharacteristic
48#16839511C2211yescharacteristic
49#16839511C2211yescharacteristic
50#16839510C2211yescharacteristic
51#16839510C2211yescharacteristic
52#16839509C2211yescharacteristic
53#16839509C2211yescharacteristic
54#16839506C2211yescharacteristic
55#16839506C2211yescharacteristic
56C61×#532C2211yescharacteristic
57C4×#41×C61C2211yescharacteristic
58C61×#482—21no—
59C61×#477C2211yescharacteristic
60C61×#476C2211yescharacteristic
61C61×#460C2211yescharacteristic
62C61×#459C2211yescharacteristic
63C61×#449—21no—
64C4×SD16×C61—21no—
65C4×D8×C61—21no—
66C61×#431C2211yescharacteristic
67C2×#11758202—21no—
68C2×#11758200—21no—
69C22×#92418—21no—
70C22×#92418—21no—
71C22×#92418—21no—
72#14519777C2311yescharacteristic
73#14519777C2311yescharacteristic
74#14519769—21no—
75#14519756—21no—
76#14519756—21no—
77C4×Dic122—21no—
78C4×Dic122—21no—
79#14519736—21no—
80#14519710—21no—
81#14519710—21no—
82#14519710—21no—
83#14519708—21no—
84#14519708—21no—
85#14519708—21no—
86C4⋊#92418—21no—
87C4⋊#92418—21no—
88C4⋊#92418—21no—
89C4⋊#92418D411yescharacteristic
90C4⋊#92418D411yescharacteristic
91C4⋊#92418C2311yescharacteristic
92C4⋊#92418C2311yescharacteristic
93C4⋊#92418C2311yescharacteristic
94C4⋊#92418C2311yescharacteristic
95#14519701—21no—
96#14519700—21no—
97#14519700—21no—
98#14519698—21no—
99#14519698—21no—
100#14519698—21no—
101#14519698—21no—
102#14519698—21no—
103C22⋊#92418—21no—
104C22⋊#92418—21no—
105C22⋊#92418—21no—
106C22⋊#92418D411yescharacteristic
107C22⋊#92418D411yescharacteristic
108#14519688—21no—
109C4×#92418—21no—
110C4×#92418—21no—
111C4×#92418C2311yescharacteristic
112C4×#92418C2311yescharacteristic
113C2×#41×C61—21no—
114C2×#41×C61D411yescharacteristic
115C2×SD16×C61—21no—
116C2×D8×C61—21no—
117C4⋊Q8×C61C2311yescharacteristic
118#127×C61C2311yescharacteristic
119#124×C61—21no—
120#121×C61—21no—
121#121×C61—21no—
122C4×Q8×C61D411yescharacteristic
123C4×Q8×C61C2311yescharacteristic
124C4×D4×C61—21no—
125C4×D4×C61—21no—
126C4×D4×C61—21no—
127C4×D4×C61—21no—
128C4×D4×C61D411yescharacteristic
129C4×D4×C61C2311yescharacteristic
130#117×C61—21no—
131#117×C61D411yescharacteristic
132C2×C42×C61—21no—
133C2×C42×C61C2311yescharacteristic
134#107×C61—21no—
135#106×C61—21no—
136C4⋊C8×C61C2311yescharacteristic
137C4⋊C8×C61C2311yescharacteristic
138#103×C61—21no—
139#102×C61—21no—
140#102×C61—21no—
141#102×C61—21no—
142C22⋊C8×C61—21no—
143C4×C8×C61—21no—
144#41×C61—22no—
145#41×C61—22no—
146#41×C61—82no—
147C2×Q8×C61—21no—
148C2×Q8×C61C2×D411yescharacteristic
149C2×D4×C61—21no—
150C2×D4×C61—21no—
151C2×D4×C61—21no—
152C2×D4×C61—21no—
153C2×D4×C61—41no—
154C2×D4×C61C2×D411yescharacteristic
155C22×C4×C61—21no—
156C22×C4×C61—21no—
157C22×C4×C61—21no—
158C22×C4×C61—41no—
159C22×C4×C61C2×D411yescharacteristic
160C22×C4×C61C2×D411yescharacteristic
161C22×C4×C61#4111yescharacteristic
162C2×Dic122—21no—
163C2×Dic122—21no—
164SD16×C61—42no—
165D8×C61—42no—
166C2×C8×C61—21no—
167C2×C8×C61—21no—
168C4⋊C4×C61—21no—
169C4⋊C4×C61—21no—
170C4⋊C4×C61—21no—
171C4⋊C4×C61—21no—
172C4⋊C4×C61C2×D411yescharacteristic
173C4⋊C4×C61C2×D411yescharacteristic
174C22⋊C4×C61—21no—
175C22⋊C4×C61—21no—
176C22⋊C4×C61—21no—
177C22⋊C4×C61—41no—
178C22⋊C4×C61—41no—
179C42×C61—21no—
180C42×C61—21no—
181C42×C61—21no—
182C42×C61C2×D411yescharacteristic
183C42×C61C2×D411yescharacteristic
184C42×C61#4111yescharacteristic
185C42×C61C2411yescharacteristic
186C22⋊Dic61—41no—
187#11758202—42no—
188#11758200—42no—
189C4⋊Dic61—21no—
190C4⋊Dic61—21no—
191C4⋊Dic61—21no—
192C4⋊Dic61—21no—
193#11758196—41no—
194#11758196—41no—
195C4×Dic61—41no—
196C2×#92418—21no—
197C2×#92418—21no—
198C2×#92418—21no—
199C2×#92418—21no—
200C2×#92418—81no—
201C2×#92418—81no—
202C2×#92418—81no—
203C2×#92418C2×D411yescharacteristic
204C2×#92418C2×D411yescharacteristic
205C2×#92418C2×D411yescharacteristic
206C2×#92418C2×D411yescharacteristic
207C23×C61—21no—
208C23×C61—41no—
209C23×C61#12111yescharacteristic
210Q8×C61—21no—
211Q8×C61—81no—
212Q8×C61#13512yes—
213D4×C61—24no—
214D4×C61—21no—
215D4×C61—41no—
216D4×C61—41no—
217D4×C61—41no—
218D4×C61—41no—
219D4×C61—81no—
220D4×C61—82no—
221D4×C61#13512yes—
222C2×C4×C61—24no—
223C2×C4×C61—21no—
224C2×C4×C61—21no—
225C2×C4×C61—21no—
226C2×C4×C61—21no—
227C2×C4×C61—21no—
228C2×C4×C61—21no—
229C2×C4×C61—21no—
230C2×C4×C61—41no—
231C2×C4×C61—41no—
232C2×C4×C61—41no—
233C2×C4×C61—41no—
234C2×C4×C61—81no—
235C2×C4×C61—82no—
236C2×C4×C61#12111yescharacteristic
237C2×C4×C61#12111yescharacteristic
238C2×C4×C61#12111yescharacteristic
239C2×C4×C61C22×D411yescharacteristic
240C2×C4×C61C22×D411yescharacteristic
241C2×C4×C61C2×#4111yescharacteristic
242C2×Dic61—41no—
243C2×Dic61—41no—
244Dic122—41no—
245Dic122—41no—
246Dic122—81no—
247C8×C61—41no—
248C8×C61—41no—
249#92418—21no—
250#92418—21no—
251#92418—21no—
252#92418—21no—
253#92418—41no—
254#92418—41no—
255#92418—41no—
256#92418—41no—
257#19419—611nomaximal, Sylow
258C22×C61—21no—
259C22×C61—21no—
260C22×C61—22no—
261C22×C61—21no—
262C22×C61—41no—
263C22×C61—41no—
264C22×C61—81no—
265C22×C61—81no—
266C22×C61—81no—
267C22×C61C2×#12111yescharacteristic
268C4×C61—21no—
269C4×C61—21no—
270C4×C61—22no—
271C4×C61—21no—
272C4×C61—21no—
273C4×C61—41no—
274C4×C61—41no—
275C4×C61—81no—
276C4×C61C2×#13511yescharacteristic
277C4×C61C2×#13511yescharacteristic
278C4×C61#57711yescharacteristic
279C4×C61#57811yescharacteristic
280Dic61—81no—
281Dic61—81no—
282#3228—611no—
283#3116—611no—
284#3106D6111yescharacteristic
285#2988—611no—
286#1682—611no—
287#1681—611no—
288#1678—611no—
289#1677—611no—
290#1676—611no—
291#1675—611no—
292#1672—611no—
293#1671—611no—
294#1572—611no—
295#1551—611no—
296#1542—611no—
297C2×C61—41no—
298C2×C61—42no—
299C2×C61—161no—
300C2×C61#304611yescharacteristic
301C2×C61#305011yescharacteristic
302C2×C61#306811yescharacteristic
303#533—611no—
304#532C2×D6111yescharacteristic
305C4×#41C2×D6111yescharacteristic
306#500—611no—
307#498—611no—
308#486—611no—
309#486—611no—
310#485—1221no—
311#482—21no—
312#482—1221no—
313#482—1221no—
314#478—1221no—
315#478—1221no—
316#477—611no—
317#477C2×D6111yescharacteristic
318#476—611no—
319#476C2×D6111yescharacteristic
320#462—611no—
321#461—611no—
322#460C2×D6111yescharacteristic
323#459C2×D6111yescharacteristic
324#452—1221no—
325#449—21no—
326C8×Q8—611no—
327C8×Q8—611no—
328#441—1221no—
329#440—1221no—
330C4×SD16—21no—
331C4×D8—21no—
332C8×D4—611no—
333C8×D4—611no—
334C8×D4—1221no—
335C8×D4—1221no—
336#432—611no—
337#431—611no—
338#431—1221no—
339#431C2×D6111yescharacteristic
340C2×C4⋊C8—611no—
341C2×C4⋊C8—1221no—
342#335—611no—
343#335—611no—
344#334—611no—
345#334—611no—
346#333—611no—
347#333—611no—
348#333—611no—
349#333—611no—
350#332—611no—
351#332—611no—
352#331—611no—
353#331—611no—
354C61#1941911yescharacteristic, complemented, Sylow
355C2×#41—21no—
356C2×#41C22⋊D6111yescharacteristic
357C2×Q16—1221no—
358C2×SD16—21no—
359C2×SD16—1221no—
360C2×D8—21no—
361C22×C8—1221no—
362C22×C8—1221no—
363C22×C8—1221no—
364C4⋊Q8—611no—
365C4⋊Q8—611no—
366C4⋊Q8C22×D6111yescharacteristic
367#127C22×D6111yescharacteristic
368#125—1221no—
369#124—21no—
370C22⋊Q8—1221no—
371C22⋊Q8—1221no—
372#121—21no—
373#121—21no—
374C4×Q8—1221no—
375C4×Q8—1221no—
376C4×Q8C22⋊D6111yescharacteristic
377C4×Q8C22×D6111yescharacteristic
378C4×D4—21no—
379C4×D4—21no—
380C4×D4—21no—
381C4×D4—21no—
382C4×D4C22⋊D6111yescharacteristic
383C4×D4C22×D6111yescharacteristic
384#117—21no—
385#117—1221no—
386#117C22⋊D6111yescharacteristic
387C2×C42—21no—
388C2×C42C22×D6111yescharacteristic
389#107—21no—
390#107—1221no—
391#107—1221no—
392#107—1221no—
393#106—21no—
394#106—1221no—
395#106—1221no—
396#106—1221no—
397C4⋊C8—611no—
398C4⋊C8—611no—
399C4⋊C8—611no—
400C4⋊C8—611no—
401C4⋊C8—611no—
402C4⋊C8—611no—
403C4⋊C8—1221no—
404C4⋊C8—1221no—
405C4⋊C8—1221no—
406C4⋊C8C22×D6111yescharacteristic
407C4⋊C8C22×D6111yescharacteristic
408#103—21no—
409#103—1221no—
410#103—1221no—
411#103—1221no—
412#102—21no—
413#102—21no—
414#102—21no—
415#102—1221no—
416#102—1221no—
417#102—1221no—
418#102—1221no—
419#102—1221no—
420C22⋊C8—21no—
421C22⋊C8—611no—
422C22⋊C8—611no—
423C22⋊C8—1221no—
424C22⋊C8—1221no—
425C22⋊C8—1221no—
426#97—1221no—
427C4×C8—21no—
428C4×C8—611no—
429C4×C8—611no—
430C4×C8—1221no—
431C4×C8—1221no—
432#41—22no—
433#41—22no—
434#41—82no—
435C2×Q8—21no—
436C2×Q8—1221no—
437C2×Q8—1221no—
438C2×Q8C2×C22⋊D6111yescharacteristic
439C2×D4—21no—
440C2×D4—21no—
441C2×D4—21no—
442C2×D4—21no—
443C2×D4—41no—
444C2×D4C2×C22⋊D6111yescharacteristic
445C22×C4—21no—
446C22×C4—21no—
447C22×C4—21no—
448C22×C4—41no—
449C22×C4D4×D6111yescharacteristic
450C22×C4#1175822411yescharacteristic
451C22×C4C2×C22⋊D6111yescharacteristic
452Q16—2442no—
453SD16—42no—
454SD16—2442no—
455D8—42no—
456C2×C8—21no—
457C2×C8—21no—
458C2×C8—611no—
459C2×C8—611no—
460C2×C8—611no—
461C2×C8—611no—
462C2×C8—1221no—
463C2×C8—1221no—
464C2×C8—1221no—
465C2×C8—1221no—
466C2×C8—4881no—
467C2×C8—4881no—
468C2×C8—4881no—
469C4⋊C4—21no—
470C4⋊C4—21no—
471C4⋊C4—21no—
472C4⋊C4—21no—
473C4⋊C4—1221no—
474C4⋊C4—1221no—
475C4⋊C4—1221no—
476C4⋊C4—1221no—
477C4⋊C4—2441no—
478C4⋊C4—2441no—
479C4⋊C4C2×C22⋊D6111yescharacteristic
480C4⋊C4C2×C22⋊D6111yescharacteristic
481C22⋊C4—21no—
482C22⋊C4—21no—
483C22⋊C4—21no—
484C22⋊C4—41no—
485C22⋊C4—41no—
486C22⋊C4—2441no—
487C42—21no—
488C42—21no—
489C42—21no—
490C42—2441no—
491C42D4×D6111yescharacteristic
492C42#1175822411yescharacteristic
493C42C2×C22⋊D6111yescharacteristic
494C42C23×D6111yescharacteristic
495C23—21no—
496C23—41no—
497C23#1451974911yescharacteristic
498Q8—21no—
499Q8—81no—
500Q8—4881no—
501Q8—2441no—
502Q8—2441no—
503Q8#1451979712yes—
504D4—24no—
505D4—21no—
506D4—41no—
507D4—41no—
508D4—41no—
509D4—41no—
510D4—81no—
511D4—82no—
512D4#1451979712yes—
513C2×C4—24no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—81no—
526C2×C4—82no—
527C2×C4—2441no—
528C2×C4—2441no—
529C2×C4#1451974911yescharacteristic
530C2×C4#1451974911yescharacteristic
531C2×C4#1451974911yescharacteristic
532C2×C4C2×D4×D6111yescharacteristic
533C2×C4C22×C22⋊D6111yescharacteristic
534C2×C4C2×#1175822411yescharacteristic
535C8—41no—
536C8—41no—
537C8—1221no—
538C8—1221no—
539C8—1221no—
540C8—1221no—
541C8—2441no—
542C8—2441no—
543C8—2441no—
544C8—2441no—
545C22—21no—
546C22—21no—
547C22—22no—
548C22—21no—
549C22—41no—
550C22—41no—
551C22—81no—
552C22—81no—
553C22—81no—
554C22C2×#1451974911yescharacteristic
555C4—21no—
556C4—21no—
557C4—22no—
558C4—21no—
559C4—21no—
560C4—41no—
561C4—41no—
562C4—81no—
563C4—4881no—
564C4—4881no—
565C4C2×#1451979711yescharacteristic
566C4C2×#1451979711yescharacteristic
567C4#1684056911yescharacteristic
568C4#1684057611yescharacteristic
569C2—41no—
570C2—42no—
571C2—161no—
572C2#2145536711yescharacteristic
573C2#2145539711yescharacteristic
574C2#2145554411yescharacteristic
subgroupG/H
centerC22C2×#14519749
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
Fitting, prime radicalC61×#3106C2
socleC22×C61C2×#121
hypercenter#3106D61
hypocenterC61#19419
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3106

Complements

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

NH ≅ G/Ntype
1. #21456951C2semidirect
2. #21455981C2semidirect
3. #21454930C2semidirect
4. #21445261C2semidirect
5. #21445254C2semidirect
6. #21445233C2semidirect
7. #21445226C2semidirect
8. #21445221C2semidirect
9. #21445216C2semidirect
10. #21445203C2semidirect
11. #21445196C2semidirect
12. #21444597C2semidirect
13. #21444514C2semidirect
14. #21444474C2semidirect
354. C61#19419semidirect

Maximal subgroups

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

subgroupindex
#214569512
#214559812
#214549302
#214452612
#214452542
#214452332
#214452262
#214452212
#214452162
#214452032
#214451962
#214445972
#214445142
#214444742
C61×#31062
#1941961