Finite Group Databasebeta

← #29667599

Subgroups — #29667599

Order 15616.

568 conjugacy classes of subgroups.

The subgroup lattice has 554 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21457156C211yesmaximal, characteristic
2#21456963C211yesmaximal, characteristic, complemented
3#21454979C211yesmaximal, characteristic
4#21445403C211yesmaximal, characteristic, complemented
5#21445396C211yesmaximal, characteristic, complemented
6#21445327C211yesmaximal, characteristic, complemented
7#21445320C211yesmaximal, characteristic, complemented
8#21445279C211yesmaximal, characteristic, complemented
9#21445272C211yesmaximal, characteristic, complemented
10#21445203C211yesmaximal, characteristic, complemented
11#21445196C211yesmaximal, characteristic, complemented
12#21444605C211yesmaximal, characteristic, complemented
13#21444520C211yesmaximal, characteristic, complemented
14C2×#16839511C211yesmaximal, characteristic, complemented
15C61×#3238C211yesmaximal, characteristic
16#16840319C2211yescharacteristic
17#16840158C2211yescharacteristic
18#16840158C2211yescharacteristic
19#16840114C2211yescharacteristic
20#16840114C2211yescharacteristic
21#16840092—21no—
22#16840092—21no—
23#16840066—21no—
24#16840047C2211yescharacteristic
25#16840044—21no—
26#16840044—21no—
27#16840042C2211yescharacteristic
28#16840001—21no—
29#16839989—21no—
30#16839981—21no—
31#16839972C2211yescharacteristic
32#16839967C2211yescharacteristic
33#16839893C2211yescharacteristic
34Q8×#92418C2211yescharacteristic
35#16839840C2211yescharacteristic
36#16839834—21no—
37D4×#92418—21no—
38D4×#92418C2211yescharacteristic
39#16839823—21no—
40#16839815C2211yescharacteristic
41#16839789—21no—
42C2×C4⋊#92418C2211yescharacteristic
43C2×C4⋊#92418C2211yescharacteristic
44#16839514C2211yescharacteristic
45#16839514C2211yescharacteristic
46#16839512C2211yescharacteristic
47#16839512C2211yescharacteristic
48#16839511C2211yescharacteristic
49#16839511C2211yescharacteristic
50#16839511C2211yescharacteristic
51#16839511C2211yescharacteristic
52#16839509C2211yescharacteristic
53#16839509C2211yescharacteristic
54#16839507C2211yescharacteristic
55#16839507C2211yescharacteristic
56C2×C4⋊Q8×C61C2211yescharacteristic
57C61×#529C2211yescharacteristic
58C61×#501—21no—
59C61×#489—21no—
60C61×#483—21no—
61C61×#480—21no—
62C61×#477C2211yescharacteristic
63C61×#477C2211yescharacteristic
64C61×#476C2211yescharacteristic
65C61×#476C2211yescharacteristic
66C61×#432C2211yescharacteristic
67C2×#11758202—21no—
68C2×#11758200—21no—
69C2×#11758194—21no—
70C22×#92418—21no—
71C22×#92418—21no—
72C22×#92418—21no—
73#14519777C2311yescharacteristic
74#14519777C2311yescharacteristic
75#14519769—21no—
76#14519756—21no—
77#14519756—21no—
78C4×Dic122—21no—
79C4×Dic122—21no—
80#14519736—21no—
81#14519710—21no—
82#14519708—21no—
83#14519708—21no—
84#14519708—21no—
85#14519708—21no—
86#14519708—21no—
87C4⋊#92418—21no—
88C4⋊#92418—21no—
89C4⋊#92418—21no—
90C4⋊#92418—21no—
91C4⋊#92418D411yescharacteristic
92C4⋊#92418D411yescharacteristic
93C4⋊#92418C2311yescharacteristic
94C4⋊#92418C2311yescharacteristic
95C4⋊#92418C2311yescharacteristic
96C4⋊#92418C2311yescharacteristic
97#14519701—21no—
98#14519700—21no—
99#14519700—21no—
100#14519700—21no—
101#14519700—21no—
102#14519700—21no—
103#14519700—21no—
104#14519698—21no—
105C22⋊#92418—21no—
106C22⋊#92418—21no—
107C22⋊#92418D411yescharacteristic
108C22⋊#92418D411yescharacteristic
109#14519688—21no—
110C4×#92418—21no—
111C4×#92418C2311yescharacteristic
112C4×#92418C2311yescharacteristic
113C22×Q8×C61—21no—
114C4⋊Q8×C61—21no—
115C4⋊Q8×C61—21no—
116C4⋊Q8×C61—21no—
117C4⋊Q8×C61C2311yescharacteristic
118C4⋊Q8×C61C2311yescharacteristic
119#126×C61—21no—
120#125×C61D411yescharacteristic
121#124×C61D411yescharacteristic
122#123×C61—21no—
123C22⋊Q8×C61D411yescharacteristic
124#121×C61D411yescharacteristic
125C4×Q8×C61C2311yescharacteristic
126C4×D4×C61—21no—
127C4×D4×C61C2311yescharacteristic
128#117×C61—21no—
129C2×C4⋊C4×C61—21no—
130C2×C4⋊C4×C61—21no—
131C2×C42×C61C2311yescharacteristic
132#107×C61—21no—
133#107×C61—21no—
134#106×C61—21no—
135#106×C61—21no—
136C4⋊C8×C61C2311yescharacteristic
137C4⋊C8×C61C2311yescharacteristic
138#103×C61—21no—
139#103×C61—21no—
140#102×C61—21no—
141#102×C61—21no—
142C22⋊C8×C61—21no—
143#97×C61—21no—
144C2×Q8×C61—21no—
145C2×Q8×C61—21no—
146C2×Q8×C61—41no—
147C2×Q8×C61—82no—
148C2×Q8×C61C2×D411yescharacteristic
149C2×D4×C61—21no—
150C2×D4×C61C2×D411yescharacteristic
151C22×C4×C61—21no—
152C22×C4×C61—41no—
153C22×C4×C61C2×D411yescharacteristic
154C22×C4×C61C2×D411yescharacteristic
155C22×C4×C61#4111yescharacteristic
156C2×Dic122—21no—
157C2×Dic122—21no—
158C2×C8×C61—21no—
159C2×C8×C61—21no—
160C4⋊C4×C61—21no—
161C4⋊C4×C61—21no—
162C4⋊C4×C61—21no—
163C4⋊C4×C61—21no—
164C4⋊C4×C61—21no—
165C4⋊C4×C61—21no—
166C4⋊C4×C61—21no—
167C4⋊C4×C61—21no—
168C4⋊C4×C61—41no—
169C4⋊C4×C61—41no—
170C4⋊C4×C61C2×D411yescharacteristic
171C4⋊C4×C61C2×D411yescharacteristic
172C22⋊C4×C61—21no—
173C22⋊C4×C61—21no—
174C22⋊C4×C61—21no—
175C22⋊C4×C61—21no—
176C22⋊C4×C61—21no—
177C42×C61—21no—
178C42×C61C2×D411yescharacteristic
179C42×C61C2×D411yescharacteristic
180C42×C61#4111yescharacteristic
181C42×C61C2411yescharacteristic
182C22⋊Dic61—41no—
183#11758202—81no—
184#11758200—81no—
185C4⋊Dic61—21no—
186C4⋊Dic61—21no—
187C4⋊Dic61—21no—
188C4⋊Dic61—21no—
189#11758196—41no—
190#11758196—41no—
191C4×Dic61—41no—
192#11758194—81no—
193C2×#92418—21no—
194C2×#92418—21no—
195C2×#92418—21no—
196C2×#92418—21no—
197C2×#92418—81no—
198C2×#92418—81no—
199C2×#92418—81no—
200C2×#92418C2×D411yescharacteristic
201C2×#92418C2×D411yescharacteristic
202C2×#92418C2×D411yescharacteristic
203C2×#92418C2×D411yescharacteristic
204C23×C61—21no—
205C23×C61#12111yescharacteristic
206Q8×C61—21no—
207Q8×C61—21no—
208Q8×C61—82no—
209Q8×C61—81no—
210Q8×C61—81no—
211D4×C61—21no—
212D4×C61—21no—
213D4×C61—42no—
214C2×C4×C61—22no—
215C2×C4×C61—21no—
216C2×C4×C61—21no—
217C2×C4×C61—21no—
218C2×C4×C61—21no—
219C2×C4×C61—21no—
220C2×C4×C61—21no—
221C2×C4×C61—21no—
222C2×C4×C61—41no—
223C2×C4×C61—41no—
224C2×C4×C61—41no—
225C2×C4×C61—41no—
226C2×C4×C61—82no—
227C2×C4×C61#12111yescharacteristic
228C2×C4×C61#12111yescharacteristic
229C2×C4×C61#12111yescharacteristic
230C2×C4×C61C22×D411yescharacteristic
231C2×C4×C61C22×D411yescharacteristic
232C2×C4×C61C2×#4111yescharacteristic
233C2×Dic61—41no—
234C2×Dic61—41no—
235Dic122—41no—
236Dic122—41no—
237Dic122—41no—
238Dic122—41no—
239C8×C61—41no—
240C8×C61—41no—
241#92418—21no—
242#92418—21no—
243#92418—21no—
244#92418—21no—
245#92418—41no—
246#92418—41no—
247#92418—41no—
248#92418—41no—
249#19588—611nomaximal, Sylow
250C22×C61—21no—
251C22×C61—21no—
252C22×C61—41no—
253C22×C61—41no—
254C22×C61—41no—
255C22×C61#47112yes—
256C22×C61C2×#12111yescharacteristic
257C4×C61—21no—
258C4×C61—21no—
259C4×C61—21no—
260C4×C61—21no—
261C4×C61—41no—
262C4×C61—41no—
263C4×C61—41no—
264C4×C61—81no—
265C4×C61—81no—
266C4×C61#57711yescharacteristic
267C4×C61#57711yescharacteristic
268C4×C61#57811yescharacteristic
269C4×C61#57811yescharacteristic
270Dic61—81no—
271Dic61—81no—
272#3248—611no—
273#3238D6111yescharacteristic
274#3229—611no—
275#2994—611no—
276#1704—611no—
277#1703—611no—
278#1692—611no—
279#1691—611no—
280#1684—611no—
281#1683—611no—
282#1672—611no—
283#1671—611no—
284#1573—611no—
285#1552—611no—
286C2×#333—611no—
287C2×C61—22no—
288C2×C61—81no—
289C2×C61#304611yescharacteristic
290C2×C61#306311yescharacteristic
291C2×C61C2×#47111yescharacteristic
292#533—611no—
293C2×C4⋊Q8C2×D6111yescharacteristic
294#529C2×D6111yescharacteristic
295#501—21no—
296#498—611no—
297#498—611no—
298#489—21no—
299#487—611no—
300#487—611no—
301#483—21no—
302#483—1221no—
303#483—1221no—
304#480—21no—
305#479—1221no—
306#477C2×D6111yescharacteristic
307#477C2×D6111yescharacteristic
308#476C2×D6111yescharacteristic
309#476C2×D6111yescharacteristic
310#475—611no—
311#475—1221no—
312#475—1221no—
313#474—611no—
314#467—1221no—
315#465—1221no—
316#464—1221no—
317#462—611no—
318#461—611no—
319#446—611no—
320C8×Q8—611no—
321#436—611no—
322#435—1221no—
323C8×D4—611no—
324C8×D4—1221no—
325#433—1221no—
326#432C2×D6111yescharacteristic
327#431—611no—
328#424—1221no—
329C2×C4⋊C8—611no—
330C2×C4⋊C8—611no—
331#335—611no—
332#335—611no—
333#334—611no—
334#334—611no—
335#333—611no—
336#333—611no—
337#333—611no—
338#333—611no—
339#332—611no—
340#332—611no—
341#332—611no—
342#332—611no—
343C61#1958811yescharacteristic, complemented, Sylow
344C22×Q8—21no—
345C2×Q16—1221no—
346C2×SD16—1221no—
347C2×M16—1221no—
348C22×C8—1221no—
349C22×C8—1221no—
350C22×C8—1221no—
351C4⋊Q8—21no—
352C4⋊Q8—21no—
353C4⋊Q8—21no—
354C4⋊Q8—611no—
355C4⋊Q8—611no—
356C4⋊Q8C22×D6111yescharacteristic
357C4⋊Q8C22×D6111yescharacteristic
358#126—21no—
359#125—1221no—
360#125C22⋊D6111yescharacteristic
361#124C22⋊D6111yescharacteristic
362#123—21no—
363C22⋊Q8—1221no—
364C22⋊Q8—1221no—
365C22⋊Q8C22⋊D6111yescharacteristic
366#121C22⋊D6111yescharacteristic
367C4×Q8—1221no—
368C4×Q8—1221no—
369C4×Q8C22×D6111yescharacteristic
370C4×D4—21no—
371C4×D4C22×D6111yescharacteristic
372#117—21no—
373#117—1221no—
374C2×C4⋊C4—21no—
375C2×C4⋊C4—21no—
376C2×C42C22×D6111yescharacteristic
377#107—21no—
378#107—21no—
379#107—1221no—
380#106—21no—
381#106—21no—
382#106—1221no—
383#106—1221no—
384#106—1221no—
385#106—1221no—
386#106—1221no—
387C4⋊C8—611no—
388C4⋊C8—611no—
389C4⋊C8—611no—
390C4⋊C8—611no—
391C4⋊C8—611no—
392C4⋊C8—611no—
393C4⋊C8—1221no—
394C4⋊C8—1221no—
395C4⋊C8—1221no—
396C4⋊C8—1221no—
397C4⋊C8C22×D6111yescharacteristic
398C4⋊C8C22×D6111yescharacteristic
399#103—21no—
400#103—21no—
401#103—1221no—
402#103—1221no—
403#103—1221no—
404#103—1221no—
405#103—1221no—
406#103—1221no—
407#103—1221no—
408#102—21no—
409#102—21no—
410#102—1221no—
411C22⋊C8—21no—
412C22⋊C8—611no—
413C22⋊C8—611no—
414C22⋊C8—1221no—
415C22⋊C8—1221no—
416#97—21no—
417#97—1221no—
418C4×C8—611no—
419C4×C8—611no—
420C4×C8—1221no—
421C2×Q8—21no—
422C2×Q8—21no—
423C2×Q8—41no—
424C2×Q8—82no—
425C2×Q8—1221no—
426C2×Q8—1221no—
427C2×Q8C2×C22⋊D6111yescharacteristic
428C2×D4—21no—
429C2×D4C2×C22⋊D6111yescharacteristic
430C22×C4—21no—
431C22×C4—41no—
432C22×C4D4×D6111yescharacteristic
433C22×C4#1175822411yescharacteristic
434C22×C4C2×C22⋊D6111yescharacteristic
435Q16—4881no—
436SD16—4881no—
437M16—4881no—
438C2×C8—21no—
439C2×C8—21no—
440C2×C8—611no—
441C2×C8—611no—
442C2×C8—611no—
443C2×C8—611no—
444C2×C8—1221no—
445C2×C8—1221no—
446C2×C8—1221no—
447C2×C8—1221no—
448C2×C8—4881no—
449C2×C8—4881no—
450C2×C8—4881no—
451C4⋊C4—21no—
452C4⋊C4—21no—
453C4⋊C4—21no—
454C4⋊C4—21no—
455C4⋊C4—21no—
456C4⋊C4—21no—
457C4⋊C4—21no—
458C4⋊C4—21no—
459C4⋊C4—41no—
460C4⋊C4—41no—
461C4⋊C4—1221no—
462C4⋊C4—1221no—
463C4⋊C4—1221no—
464C4⋊C4—1221no—
465C4⋊C4—2441no—
466C4⋊C4—2441no—
467C4⋊C4C2×C22⋊D6111yescharacteristic
468C4⋊C4C2×C22⋊D6111yescharacteristic
469C22⋊C4—21no—
470C22⋊C4—21no—
471C22⋊C4—21no—
472C22⋊C4—21no—
473C22⋊C4—21no—
474C22⋊C4—2441no—
475C42—21no—
476C42—2441no—
477C42D4×D6111yescharacteristic
478C42#1175822411yescharacteristic
479C42C2×C22⋊D6111yescharacteristic
480C42C23×D6111yescharacteristic
481C23—21no—
482C23#1451974911yescharacteristic
483Q8—21no—
484Q8—21no—
485Q8—81no—
486Q8—81no—
487Q8—82no—
488Q8—2441no—
489Q8—2441no—
490Q8—2441no—
491Q8—2441no—
492D4—21no—
493D4—21no—
494D4—42no—
495C2×C4—22no—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—41no—
504C2×C4—41no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—82no—
508C2×C4—2441no—
509C2×C4—2441no—
510C2×C4#1451974911yescharacteristic
511C2×C4#1451974911yescharacteristic
512C2×C4#1451974911yescharacteristic
513C2×C4C2×D4×D6111yescharacteristic
514C2×C4C22×C22⋊D6111yescharacteristic
515C2×C4C2×#1175822411yescharacteristic
516C8—41no—
517C8—41no—
518C8—1221no—
519C8—1221no—
520C8—1221no—
521C8—1221no—
522C8—2441no—
523C8—2441no—
524C8—2441no—
525C8—2441no—
526C22—21no—
527C22—21no—
528C22—41no—
529C22—41no—
530C22—41no—
531C22#1684002312yes—
532C22C2×#1451974911yescharacteristic
533C4—21no—
534C4—21no—
535C4—21no—
536C4—21no—
537C4—41no—
538C4—41no—
539C4—41no—
540C4—81no—
541C4—81no—
542C4—4881no—
543C4—4881no—
544C4#1684057011yescharacteristic
545C4#1684057111yescharacteristic
546C4#1684057711yescharacteristic
547C4#1684057811yescharacteristic
548C2—22no—
549C2—81no—
550C2#2145537311yescharacteristic
551C2#2145550611yescharacteristic
552C2C2×#1684002311yescharacteristic
subgroupG/H
centerC22C2×#14519749
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#3238C2
socleC22×C61C2×#121
prime radical#121×C61D4
hypercenter#3238D61
hypocenterC61#19588
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3238

Complements

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

NH ≅ G/Ntype
2. #21456963C2semidirect
4. #21445403C2semidirect
5. #21445396C2semidirect
6. #21445327C2semidirect
7. #21445320C2semidirect
8. #21445279C2semidirect
9. #21445272C2semidirect
10. #21445203C2semidirect
11. #21445196C2semidirect
12. #21444605C2semidirect
13. #21444520C2semidirect
14. C2×#16839511C2semidirect
343. C61#19588semidirect

Maximal subgroups

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

subgroupindex
#214571562
#214569632
#214549792
#214454032
#214453962
#214453272
#214453202
#214452792
#214452722
#214452032
#214451962
#214446052
#214445202
C2×#168395112
C61×#32382
#1958861