Finite Group Databasebeta

← #29650118

Subgroups — #29650118

Order 15616.

598 conjugacy classes of subgroups.

The subgroup lattice has 564 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21455890C211yesmaximal, characteristic
2#21454913C211yesmaximal, characteristic, complemented
3#21444690C211yesmaximal, characteristic, complemented
4#21444676C211yesmaximal, characteristic, complemented
5#21444669C211yesmaximal, characteristic, complemented
6#21444655C211yesmaximal, characteristic, complemented
7#21444636C211yesmaximal, characteristic, complemented
8#21444631C211yesmaximal, characteristic, complemented
9C8×#11758202C211yesmaximal, characteristic, complemented
10C8×#11758200C211yesmaximal, characteristic, complemented
11#21444602C211yesmaximal, characteristic, complemented
12#21444253C211yesmaximal, characteristic
13#21444221C211yesmaximal, characteristic, complemented
14#21444119C211yesmaximal, characteristic, complemented
15C61×#2988C211yesmaximal, characteristic
16#16840319C2211yescharacteristic
17#16840103C411yescharacteristic
18#16840103C411yescharacteristic
19#16840047C411yescharacteristic
20#16840042C411yescharacteristic
21#16839972C411yescharacteristic
22#16839967C411yescharacteristic
23#16839924C411yescharacteristic
24#16839924C411yescharacteristic
25#16839892C2211yescharacteristic
26#16839892C2211yescharacteristic
27#16839891—21no—
28#16839891—21no—
29C8×Dic122C2211yescharacteristic
30C8×Dic122C2211yescharacteristic
31C4×#11758202C2211yescharacteristic
32C4×#11758202C2211yescharacteristic
33C4×#11758200C2211yescharacteristic
34C4×#11758200C2211yescharacteristic
35#16839820—21no—
36#16839820—21no—
37#16839819C2211yescharacteristic
38#16839815C2211yescharacteristic
39#16839815C2211yescharacteristic
40#16839514C2211yescharacteristic
41#16839514C2211yescharacteristic
42#16839512C2211yescharacteristic
43#16839512C2211yescharacteristic
44Q8⋊#92418C2211yescharacteristic
45Q8⋊#92418C2211yescharacteristic
46#16839492C2211yescharacteristic
47#16839492C2211yescharacteristic
48#16839492C2211yescharacteristic
49#16839492C2211yescharacteristic
50D4⋊#92418C2211yescharacteristic
51D4⋊#92418C2211yescharacteristic
52#16839484C2211yescharacteristic
53#16839484C2211yescharacteristic
54#16839483C2211yescharacteristic
55C8×#92418C2211yescharacteristic
56C4×#41×C61C2211yescharacteristic
57C8×Q8×C61C2211yescharacteristic
58C8×Q8×C61C2211yescharacteristic
59C8×D4×C61—21no—
60C8×D4×C61—21no—
61C8×D4×C61C2211yescharacteristic
62C8×D4×C61C2211yescharacteristic
63C61×#431—21no—
64C61×#431C2211yescharacteristic
65C2×C4⋊C8×C61—21no—
66C2×C4⋊C8×C61C2211yescharacteristic
67C2×#11758202C2×C411yescharacteristic
68C2×#11758202C2×C411yescharacteristic
69C2×#11758200C2×C411yescharacteristic
70C2×#11758200C2×C411yescharacteristic
71#14519777C2×C411yescharacteristic
72#14519777C2×C411yescharacteristic
73#14519769—21no—
74#14519756—21no—
75#14519756C2×C411yescharacteristic
76#14519756C2×C411yescharacteristic
77C4×Dic122—21no—
78C4×Dic122C2311yescharacteristic
79C4×Dic122C2311yescharacteristic
80#14519736—21no—
81#14519710C2×C411yescharacteristic
82#14519710C2×C411yescharacteristic
83#14519708C2×C411yescharacteristic
84#14519708C2×C411yescharacteristic
85C4⋊#92418—21no—
86C4⋊#92418C2×C411yescharacteristic
87C4⋊#92418C2×C411yescharacteristic
88C4⋊#92418C2311yescharacteristic
89C4⋊#92418C2311yescharacteristic
90#14519706—21no—
91#14519706—21no—
92#14519706—21no—
93#14519706—21no—
94#14519701C2×C411yescharacteristic
95#14519701C2×C411yescharacteristic
96#14519700C2×C411yescharacteristic
97#14519700C2×C411yescharacteristic
98#14519700C2×C411yescharacteristic
99#14519700C2×C411yescharacteristic
100#14519698C2×C411yescharacteristic
101#14519698C2×C411yescharacteristic
102C22⋊#92418—21no—
103C22⋊#92418—21no—
104C22⋊#92418C2×C411yescharacteristic
105C22⋊#92418C2×C411yescharacteristic
106#14519688—21no—
107#14519687—21no—
108#14519687—21no—
109C4×#92418C2311yescharacteristic
110C4×#92418C2311yescharacteristic
111C4×#92418C2311yescharacteristic
112C4×#92418C2311yescharacteristic
113C8×Dic61—21no—
114C8×Dic61—21no—
115C2×#41×C61C2×C411yescharacteristic
116C22×C8×C61—21no—
117C22×C8×C61—21no—
118C22×C8×C61—21no—
119C4×Q8×C61C2×C411yescharacteristic
120C4×Q8×C61C2311yescharacteristic
121C4×D4×C61—21no—
122C4×D4×C61—21no—
123C4×D4×C61C2×C411yescharacteristic
124C4×D4×C61C2311yescharacteristic
125#117×C61—21no—
126#117×C61C2×C411yescharacteristic
127C2×C42×C61—21no—
128C2×C42×C61C2311yescharacteristic
129C4⋊C8×C61—21no—
130C4⋊C8×C61—21no—
131C4⋊C8×C61—21no—
132C4⋊C8×C61D411yescharacteristic
133C4⋊C8×C61D411yescharacteristic
134C4⋊C8×C61C2311yescharacteristic
135C4⋊C8×C61C2311yescharacteristic
136C22⋊C8×C61—21no—
137C22⋊C8×C61—21no—
138C22⋊C8×C61D411yescharacteristic
139C22⋊C8×C61D411yescharacteristic
140C4×C8×C61—21no—
141C4×C8×C61—21no—
142C4×C8×C61C2311yescharacteristic
143C4×C8×C61C2311yescharacteristic
144#41×C61—22no—
145#41×C61—22no—
146C2×Q8×C61C22×C411yescharacteristic
147C2×D4×C61—21no—
148C2×D4×C61C22×C411yescharacteristic
149C22×C4×C61—21no—
150C22×C4×C61—21no—
151C22×C4×C61—21no—
152C22×C4×C61C22×C411yescharacteristic
153C22×C4×C61C2×D411yescharacteristic
154C22×C4×C61#4111yescharacteristic
155C2×Dic122—21no—
156C2×Dic122C22×C411yescharacteristic
157C2×Dic122C22×C411yescharacteristic
158C2×C8×C61—21no—
159C2×C8×C61—21no—
160C2×C8×C61—81no—
161C2×C8×C61—81no—
162C2×C8×C61—81no—
163C2×C8×C61C2×D411yescharacteristic
164C2×C8×C61C2×D411yescharacteristic
165C2×C8×C61C2×D411yescharacteristic
166C2×C8×C61C2×D411yescharacteristic
167C4⋊C4×C61—21no—
168C4⋊C4×C61—21no—
169C4⋊C4×C61C22×C411yescharacteristic
170C4⋊C4×C61C22×C411yescharacteristic
171C22⋊C4×C61—21no—
172C22⋊C4×C61—21no—
173C22⋊C4×C61—21no—
174C42×C61—21no—
175C42×C61—21no—
176C42×C61—21no—
177C42×C61C22×C411yescharacteristic
178C42×C61C2×D411yescharacteristic
179C42×C61#4111yescharacteristic
180C42×C61C2411yescharacteristic
181C22⋊Dic61—21no—
182C22⋊Dic61—21no—
183#11758202—22no—
184#11758202—22no—
185#11758200—22no—
186#11758200—22no—
187C4⋊Dic61—21no—
188C4⋊Dic61—21no—
189C4⋊Dic61—21no—
190C4⋊Dic61C22×C411yescharacteristic
191C4⋊Dic61C22×C411yescharacteristic
192#11758196—21no—
193#11758196—21no—
194#11758196—21no—
195#11758196—21no—
196C4×Dic61—21no—
197C4×Dic61—21no—
198C2×#92418—21no—
199C2×#92418—21no—
200C2×#92418—21no—
201C2×#92418—21no—
202C2×#92418C22×C411yescharacteristic
203C2×#92418C22×C411yescharacteristic
204C2×#92418C22×C411yescharacteristic
205C2×#92418C22×C411yescharacteristic
206C23×C61—21no—
207C23×C61C4×D411yescharacteristic
208Q8×C61—21no—
209Q8×C61#13112yes—
210D4×C61—24no—
211D4×C61—21no—
212D4×C61#13112yes—
213C2×C4×C61—24no—
214C2×C4×C61—21no—
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—41no—
222C2×C4×C61—41no—
223C2×C4×C61—41no—
224C2×C4×C61—81no—
225C2×C4×C61C4×D411yescharacteristic
226C2×C4×C61C4×D411yescharacteristic
227C2×C4×C61C4×D411yescharacteristic
228C2×C4×C61C23×C411yescharacteristic
229C2×C4×C61C22×D411yescharacteristic
230C2×C4×C61C2×#4111yescharacteristic
231C2×Dic61—21no—
232C2×Dic61—21no—
233C2×Dic61—21no—
234C2×Dic61—21no—
235Dic122—21no—
236Dic122—21no—
237Dic122—21no—
238Dic122—21no—
239Dic122—81no—
240C8×C61—21no—
241C8×C61—21no—
242C8×C61—21no—
243C8×C61—21no—
244C8×C61—41no—
245C8×C61—41no—
246#92418—21no—
247#92418—21no—
248#92418—21no—
249#92418—21no—
250#92418—41no—
251#92418—41no—
252#92418—41no—
253#92418—41no—
254#17921—611nomaximal, Sylow
255C22×C61—21no—
256C22×C61—22no—
257C22×C61—21no—
258C22×C61—21no—
259C22×C61—41no—
260C22×C61—41no—
261C22×C61C2×C4×D411yescharacteristic
262C4×C61—21no—
263C4×C61—22no—
264C4×C61—21no—
265C4×C61—21no—
266C4×C61—21no—
267C4×C61—41no—
268C4×C61—41no—
269C4×C61C2×#13111yescharacteristic
270C4×C61C2×#13111yescharacteristic
271C4×C61#57711yescharacteristic
272C4×C61#57811yescharacteristic
273Dic61—41no—
274Dic61—41no—
275Dic61—41no—
276Dic61—41no—
277#3107—611no—
278#2988D6111yescharacteristic
279#2985—611no—
280#1587—611no—
281#1585—611no—
282#1584—611no—
283#1582—611no—
284#1579—611no—
285SD16⋊C8—611no—
286C8×Q16—611no—
287C8×SD16—611no—
288#1573—611no—
289#1488—611no—
290#1481—611no—
291#1454—611no—
292C2×C61—41no—
293C2×C61—42no—
294C2×C61#292911yescharacteristic
295C2×C61#294711yescharacteristic
296C2×C61#295711yescharacteristic
297#533—611no—
298C4×#41C2×D6111yescharacteristic
299#485—611no—
300#485—611no—
301#475—611no—
302#474—611no—
303#462—611no—
304#461—611no—
305#452—611no—
306#452—611no—
307#446—611no—
308#446—611no—
309#446—1221no—
310#446—1221no—
311C8×Q8—611no—
312C8×Q8—611no—
313C8×Q8C2×D6111yescharacteristic
314C8×Q8C2×D6111yescharacteristic
315C4×Q16—611no—
316C4×Q16—611no—
317C4×SD16—611no—
318C4×SD16—611no—
319C8×D4—21no—
320C8×D4—21no—
321C8×D4C2×D6111yescharacteristic
322C8×D4C2×D6111yescharacteristic
323#433—1221no—
324#433—1221no—
325#432—611no—
326#431—21no—
327#431—611no—
328#431—611no—
329#431C2×D6111yescharacteristic
330C2×C4⋊C8—21no—
331C2×C4⋊C8C2×D6111yescharacteristic
332#335—611no—
333#335—611no—
334#334—611no—
335#334—611no—
336Q8⋊C8—611no—
337Q8⋊C8—611no—
338Q8⋊C8—611no—
339Q8⋊C8—611no—
340Q8⋊C8—611no—
341Q8⋊C8—611no—
342D4⋊C8—611no—
343D4⋊C8—611no—
344#322—611no—
345#322—611no—
346#322—611no—
347C82—611no—
348C61#1792111yescharacteristic, complemented, Sylow
349C2×#41C4×D6111yescharacteristic
350C2×Q16—611no—
351C2×Q16—611no—
352C2×SD16—611no—
353C2×SD16—611no—
354C22×C8—21no—
355C22×C8—21no—
356C22×C8—21no—
357C4⋊Q8—611no—
358C4⋊Q8—611no—
359#125—1221no—
360C22⋊Q8—611no—
361C22⋊Q8—611no—
362C22⋊Q8—1221no—
363C4×Q8—611no—
364C4×Q8—611no—
365C4×Q8—1221no—
366C4×Q8C4×D6111yescharacteristic
367C4×Q8C22×D6111yescharacteristic
368C4×D4—21no—
369C4×D4—21no—
370C4×D4C4×D6111yescharacteristic
371C4×D4C22×D6111yescharacteristic
372#117—21no—
373#117—1221no—
374#117C4×D6111yescharacteristic
375C2×C42—21no—
376C2×C42C22×D6111yescharacteristic
377#107—611no—
378#107—611no—
379#106—611no—
380#106—611no—
381C4⋊C8—21no—
382C4⋊C8—21no—
383C4⋊C8—21no—
384C4⋊C8—611no—
385C4⋊C8—611no—
386C4⋊C8—611no—
387C4⋊C8—611no—
388C4⋊C8—1221no—
389C4⋊C8—1221no—
390C4⋊C8—1221no—
391C4⋊C8—1221no—
392C4⋊C8—1221no—
393C4⋊C8C22⋊D6111yescharacteristic
394C4⋊C8C22⋊D6111yescharacteristic
395C4⋊C8C22×D6111yescharacteristic
396C4⋊C8C22×D6111yescharacteristic
397#103—611no—
398#103—611no—
399#103—611no—
400#103—611no—
401#103—611no—
402#103—611no—
403#102—611no—
404#102—611no—
405C22⋊C8—21no—
406C22⋊C8—21no—
407C22⋊C8—611no—
408C22⋊C8—611no—
409C22⋊C8—1221no—
410C22⋊C8—1221no—
411C22⋊C8C22⋊D6111yescharacteristic
412C22⋊C8C22⋊D6111yescharacteristic
413#97—1221no—
414#97—1221no—
415#97—1221no—
416C4×C8—21no—
417C4×C8—21no—
418C4×C8—611no—
419C4×C8—611no—
420C4×C8—611no—
421C4×C8—611no—
422C4×C8—1221no—
423C4×C8—1221no—
424C4×C8C22×D6111yescharacteristic
425C4×C8C22×D6111yescharacteristic
426#41—22no—
427#41—22no—
428C2×Q8—611no—
429C2×Q8—611no—
430C2×Q8—1221no—
431C2×Q8C2×C4×D6111yescharacteristic
432C2×D4—21no—
433C2×D4C2×C4×D6111yescharacteristic
434C22×C4—21no—
435C22×C4—21no—
436C22×C4—21no—
437C22×C4C2×C4×D6111yescharacteristic
438C22×C4#1175822211yescharacteristic
439C22×C4C2×C22⋊D6111yescharacteristic
440Q16—1222no—
441Q16—1222no—
442SD16—1222no—
443SD16—1222no—
444C2×C8—21no—
445C2×C8—21no—
446C2×C8—81no—
447C2×C8—81no—
448C2×C8—81no—
449C2×C8—611no—
450C2×C8—611no—
451C2×C8—611no—
452C2×C8—611no—
453C2×C8—1221no—
454C2×C8—1221no—
455C2×C8—1221no—
456C2×C8—1221no—
457C2×C8C2×C22⋊D6111yescharacteristic
458C2×C8C2×C22⋊D6111yescharacteristic
459C2×C8C2×C22⋊D6111yescharacteristic
460C2×C8C2×C22⋊D6111yescharacteristic
461C4⋊C4—21no—
462C4⋊C4—21no—
463C4⋊C4—611no—
464C4⋊C4—611no—
465C4⋊C4—1221no—
466C4⋊C4—1221no—
467C4⋊C4—1221no—
468C4⋊C4—1221no—
469C4⋊C4—1221no—
470C4⋊C4—1221no—
471C4⋊C4—1221no—
472C4⋊C4C2×C4×D6111yescharacteristic
473C4⋊C4C2×C4×D6111yescharacteristic
474C22⋊C4—21no—
475C22⋊C4—21no—
476C22⋊C4—21no—
477C22⋊C4—1221no—
478C22⋊C4—1221no—
479C42—21no—
480C42—21no—
481C42—21no—
482C42—1221no—
483C42—1221no—
484C42C2×C4×D6111yescharacteristic
485C42#1175822211yescharacteristic
486C42C2×C22⋊D6111yescharacteristic
487C42C23×D6111yescharacteristic
488C23—21no—
489C23C4×C22⋊D6111yescharacteristic
490Q8—21no—
491Q8—1221no—
492Q8—1221no—
493Q8—1221no—
494Q8—1221no—
495Q8—4881no—
496Q8#1451978312yes—
497D4—24no—
498D4—21no—
499D4#1451978312yes—
500C2×C4—24no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—81no—
512C2×C4—1221no—
513C2×C4—1221no—
514C2×C4—1221no—
515C2×C4—1221no—
516C2×C4C4×C22⋊D6111yescharacteristic
517C2×C4C4×C22⋊D6111yescharacteristic
518C2×C4C4×C22⋊D6111yescharacteristic
519C2×C4C22×C4×D6111yescharacteristic
520C2×C4C22×C22⋊D6111yescharacteristic
521C2×C4C2×#1175822211yescharacteristic
522C8—21no—
523C8—21no—
524C8—21no—
525C8—21no—
526C8—41no—
527C8—41no—
528C8—1221no—
529C8—1221no—
530C8—1221no—
531C8—1221no—
532C8—2441no—
533C8—2441no—
534C8—2441no—
535C8—2441no—
536C22—21no—
537C22—21no—
538C22—21no—
539C22—22no—
540C22—41no—
541C22—41no—
542C22C2×C4×C22⋊D6111yescharacteristic
543C4—21no—
544C4—21no—
545C4—21no—
546C4—21no—
547C4—22no—
548C4—41no—
549C4—41no—
550C4—2441no—
551C4—2441no—
552C4—2441no—
553C4—2441no—
554C4C2×#1451978311yescharacteristic
555C4C2×#1451978311yescharacteristic
556C4#1684057311yescharacteristic
557C4#1684057711yescharacteristic
558C2—41no—
559C2—42no—
560C2#2145451311yescharacteristic
561C2#2145465811yescharacteristic
562C2#2145473411yescharacteristic
subgroupG/H
centerC2×C4C22×C22⋊D61
commutatorC2×C4×C61C23×C4
FrattiniC42C23×D61
FittingC61×#2988C2
socleC22×C61C2×C4×D4
prime radicalC2×#41×C61C2×C4
cosocleC42×C61C24
hypercenter#2988D61
hypocenterC61#17921
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ #431 ◁ #2988

Complements

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

NH ≅ G/Ntype
2. #21454913C2semidirect
3. #21444690C2semidirect
4. #21444676C2semidirect
5. #21444669C2semidirect
6. #21444655C2semidirect
7. #21444636C2semidirect
8. #21444631C2semidirect
9. C8×#11758202C2semidirect
10. C8×#11758200C2semidirect
11. #21444602C2semidirect
13. #21444221C2semidirect
14. #21444119C2semidirect
348. C61#17921semidirect

Maximal subgroups

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

subgroupindex
#214558902
#214549132
#214446902
#214446762
#214446692
#214446552
#214446362
#214446312
C8×#117582022
C8×#117582002
#214446022
#214442532
#214442212
#214441192
C61×#29882
#1792161