Finite Group Databasebeta

← #30613545

Subgroups — #30613545

Order 15616.

624 conjugacy classes of subgroups.

The subgroup lattice has 578 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21462751C211yesmaximal, characteristic
2#21462722C211yesmaximal, characteristic, complemented
3#21461462C211yesmaximal, characteristic, complemented
4#21461455C211yesmaximal, characteristic, complemented
5#21461448C211yesmaximal, characteristic, complemented
6#21461444C211yesmaximal, characteristic, complemented
7#21461432C211yesmaximal, characteristic, complemented
8#21461428C211yesmaximal, characteristic, complemented
9#21461412C211yesmaximal, characteristic, complemented
10Q16×#92420C211yesmaximal, characteristic, complemented
11#21461383C211yesmaximal, characteristic, complemented
12#21461150C211yesmaximal, characteristic
13#21461147C211yesmaximal, characteristic, complemented
14#21461038C211yesmaximal, characteristic, complemented
15#137×Dic61C211yesmaximal, characteristic
16#16840813—21no—
17#16840813—21no—
18#16840813C2211yescharacteristic
19#16840813C2211yescharacteristic
20Q8×#92420C2211yescharacteristic
21Q8×#92420C2211yescharacteristic
22Q8×#92420C2211yescharacteristic
23Q8×#92420C2211yescharacteristic
24#16840797—21no—
25#16840797—21no—
26#16840797C2211yescharacteristic
27#16840797C2211yescharacteristic
28#16840783—21no—
29#16840783C2211yescharacteristic
30#16840779—21no—
31#16840779—21no—
32#16840779C2211yescharacteristic
33#16840779C2211yescharacteristic
34#16840765—21no—
35#16840765C2211yescharacteristic
36#16840626C2211yescharacteristic
37#16840625C2211yescharacteristic
38#16840624C2211yescharacteristic
39#16840623C2211yescharacteristic
40Q8⋊#92420C2211yescharacteristic
41Q8⋊#92420C2211yescharacteristic
42Q8⋊#92420C2211yescharacteristic
43Q8⋊#92420C2211yescharacteristic
44Q8⋊#92420C2211yescharacteristic
45Q8⋊#92420C2211yescharacteristic
46D4⋊#92420C2211yescharacteristic
47D4⋊#92420C2211yescharacteristic
48#92420⋊C8C2211yescharacteristic
49#92420⋊C8C2211yescharacteristic
50#16840603C2211yescharacteristic
51C8×#92420C2211yescharacteristic
52#41×Dic61C2211yescharacteristic
53C2×Q8×Dic61C2211yescharacteristic
54#16839869C411yescharacteristic
55#16839869C411yescharacteristic
56#16839863C411yescharacteristic
57#16839863C411yescharacteristic
58Q16×Dic61C2211yescharacteristic
59Q16×Dic61C2211yescharacteristic
60SD16×Dic61C2211yescharacteristic
61SD16×Dic61C2211yescharacteristic
62#16839804C411yescharacteristic
63#16839771C411yescharacteristic
64#16839763C411yescharacteristic
65M16×Dic61C2211yescharacteristic
66C2×#137×C61C411yescharacteristic
67C2×#11758217—21no—
68C2×#11758217—21no—
69C2×#11758217—21no—
70#14519844—21no—
71#14519844—21no—
72#14519844—21no—
73#14519844—21no—
74#14519844D411yescharacteristic
75#14519844D411yescharacteristic
76C4⋊#92420—21no—
77C4⋊#92420—21no—
78C4⋊#92420—21no—
79C4⋊#92420—21no—
80C4⋊#92420C2311yescharacteristic
81C4⋊#92420C2311yescharacteristic
82C4⋊#92420C2311yescharacteristic
83C4⋊#92420C2311yescharacteristic
84C22⋊#92420—21no—
85C22⋊#92420—21no—
86C22⋊#92420—21no—
87C22⋊#92420—21no—
88C22⋊#92420D411yescharacteristic
89C22⋊#92420D411yescharacteristic
90#14519832—21no—
91#14519832—21no—
92#14519832—21no—
93C4×#92420—21no—
94C4×#92420—21no—
95C4×#92420—21no—
96C4×#92420—21no—
97C4×#92420C2311yescharacteristic
98C4×#92420C2311yescharacteristic
99C4×#92420C2311yescharacteristic
100C4×#92420C2311yescharacteristic
101C2×#11758194C2×C411yescharacteristic
102Q8×Dic61—21no—
103Q8×Dic61—21no—
104Q8×Dic61C2×C411yescharacteristic
105Q8×Dic61C2×C411yescharacteristic
106Q8×Dic61C2×C411yescharacteristic
107Q8×Dic61C2311yescharacteristic
108Q8×Dic61C2311yescharacteristic
109Q8×Dic61C2311yescharacteristic
110D4×Dic61—21no—
111D4×Dic61—21no—
112D4×Dic61C2×C411yescharacteristic
113D4×Dic61C2311yescharacteristic
114#14519736—21no—
115#14519736C2×C411yescharacteristic
116C2×C4⋊Dic61—21no—
117C2×C4⋊Dic61C2×C411yescharacteristic
118C2×C4×Dic61—21no—
119C2×C4×Dic61—21no—
120C2×C4×Dic61C2311yescharacteristic
121#14519711C2×C411yescharacteristic
122#14519711C2×C411yescharacteristic
123#14519709C2×C411yescharacteristic
124#14519709C2×C411yescharacteristic
125#14519701C2×C411yescharacteristic
126#14519701C2×C411yescharacteristic
127#14519701C2×C411yescharacteristic
128#14519701C2×C411yescharacteristic
129#14519701C2×C411yescharacteristic
130#14519701C2×C411yescharacteristic
131#14519698C2×C411yescharacteristic
132#14519698C2×C411yescharacteristic
133#14519687C2×C411yescharacteristic
134#14519687C2×C411yescharacteristic
135C8×Dic61C2311yescharacteristic
136C8×Dic61C2311yescharacteristic
137C2×#41×C61C2×C411yescharacteristic
138C22×Q8×C61C2×C411yescharacteristic
139#137×C61—24no—
140C2×Q16×C61C2×C411yescharacteristic
141C2×Q16×C61C2×C411yescharacteristic
142C2×SD16×C61C2×C411yescharacteristic
143C2×SD16×C61C2×C411yescharacteristic
144C2×M16×C61C2×C411yescharacteristic
145#41×C61—22no—
146#41×C61—41no—
147C2×Q8×C61—21no—
148C2×Q8×C61—22no—
149C2×Q8×C61—41no—
150C2×Q8×C61C22×C411yescharacteristic
151C2×Q8×C61C22×C411yescharacteristic
152C2×Q8×C61C22×C411yescharacteristic
153C2×D4×C61—21no—
154C2×D4×C61C22×C411yescharacteristic
155C22×C4×C61—21no—
156C22×C4×C61—21no—
157C22×C4×C61C22×C411yescharacteristic
158C22×Dic61—21no—
159C22×Dic61—21no—
160C22×Dic61—21no—
161C22×Dic61C2×D411yescharacteristic
162C22×Dic61#4111yescharacteristic
163#11758217—81no—
164#11758217—81no—
165#11758217—81no—
166C2×#92420—21no—
167C2×#92420—21no—
168C2×#92420—21no—
169C2×#92420—21no—
170C2×#92420—21no—
171C2×#92420—21no—
172C2×#92420C2×D411yescharacteristic
173C2×#92420C2×D411yescharacteristic
174C2×#92420C2×D411yescharacteristic
175C2×#92420C2×D411yescharacteristic
176Q16×C61—22no—
177Q16×C61M1614yescomplemented
178SD16×C61—22no—
179SD16×C61—22no—
180M16×C61—22no—
181C2×C8×C61C22×C411yescharacteristic
182C2×C8×C61C22×C411yescharacteristic
183C22⋊Dic61—21no—
184C22⋊Dic61—21no—
185C22⋊Dic61—21no—
186C4⋊Dic61—21no—
187C4⋊Dic61—21no—
188C4⋊Dic61—21no—
189C4⋊Dic61—21no—
190C4⋊Dic61—21no—
191C4⋊Dic61—21no—
192C4⋊Dic61—21no—
193C4⋊Dic61C22×C411yescharacteristic
194C4⋊Dic61C22×C411yescharacteristic
195C4⋊Dic61C22×C411yescharacteristic
196C4⋊Dic61C22×C411yescharacteristic
197C4×Dic61—21no—
198C4×Dic61—21no—
199C4×Dic61—21no—
200C4×Dic61—21no—
201C4×Dic61—21no—
202C4×Dic61—21no—
203C4×Dic61—21no—
204C4×Dic61C22×C411yescharacteristic
205C4×Dic61C2×D411yescharacteristic
206C4×Dic61#4111yescharacteristic
207C4×Dic61C2411yescharacteristic
208#11758194—41no—
209C2×#92418C22×C411yescharacteristic
210C2×#92418C22×C411yescharacteristic
211C23×C61—21no—
212C23×C61C4×D411yescharacteristic
213Q8×C61—21no—
214Q8×C61—21no—
215Q8×C61—21no—
216Q8×C61—21no—
217Q8×C61—42no—
218Q8×C61C2×M1612yes—
219Q8×C61C2×M1612yes—
220D4×C61—21no—
221D4×C61—21no—
222D4×C61—42no—
223C2×C4×C61—21no—
224C2×C4×C61—21no—
225C2×C4×C61—21no—
226C2×C4×C61—21no—
227C2×C4×C61—21no—
228C2×C4×C61—42no—
229C2×C4×C61—42no—
230C2×C4×C61C4×D411yescharacteristic
231C2×C4×C61C23×C411yescharacteristic
232C2×Dic61—21no—
233C2×Dic61—21no—
234C2×Dic61—21no—
235C2×Dic61—21no—
236C2×Dic61—21no—
237C2×Dic61—21no—
238C2×Dic61—41no—
239C2×Dic61—42no—
240C2×Dic61—42no—
241C2×Dic61—41no—
242C2×Dic61—41no—
243C2×Dic61C4×D411yescharacteristic
244C2×Dic61C4×D411yescharacteristic
245C2×Dic61C22×D411yescharacteristic
246C2×Dic61C2×#4111yescharacteristic
247#92420—21no—
248#92420—21no—
249#92420—21no—
250#92420—21no—
251#92420—41no—
252#92420—41no—
253#92420—41no—
254#92420—41no—
255#92420—41no—
256#92420—41no—
257C8×C61—21no—
258C8×C61C2×M1612yes—
259#92418—21no—
260#92418—21no—
261#18053—611nomaximal, Sylow
262C22×C61—21no—
263C22×C61—21no—
264C22×C61—21no—
265C22×C61—41no—
266C22×C61—41no—
267C22×C61—41no—
268C22×C61C2×C4×D411yescharacteristic
269C4×C61—21no—
270C4×C61—22no—
271C4×C61—22no—
272C4×C61—41no—
273C4×C61C22×M1611yescharacteristic
274C4×C61#56811yescharacteristic
275Dic61—21no—
276Dic61—21no—
277Dic61—21no—
278Dic61—41no—
279Dic61—41no—
280Dic61—41no—
281Dic61—41no—
282Dic61#57711yescharacteristic
283Dic61#57811yescharacteristic
284#2991—611no—
285#2984—611no—
286C4×#137—611no—
287#1592—611no—
288#1590—611no—
289#1588—611no—
290#1587—611no—
291#1584—611no—
292#1583—611no—
293#1579—611no—
294C8×Q16—611no—
295#1569—611no—
296#1488—611no—
297#1487—611no—
298#1453—611no—
299C2×C61—41no—
300C2×C61—81no—
301C2×C61D4×M1611yescharacteristic
302C2×C61#294711yescharacteristic
303C2×C61#295511yescharacteristic
304C2×#137#692311yescharacteristic
305C4×#41—611no—
306C2×C4×Q8—611no—
307#446—611no—
308#446—611no—
309#446—1221no—
310#446—1221no—
311C8×Q8—611no—
312C8×Q8—611no—
313C8×Q8—611no—
314C8×Q8—611no—
315#441—611no—
316#441—611no—
317#440—611no—
318#440—611no—
319C4×Q16—611no—
320C4×Q16—611no—
321C4×SD16—611no—
322C4×SD16—611no—
323#436—611no—
324#436—611no—
325#436—1221no—
326#436—1221no—
327#432—611no—
328#432—1221no—
329#431—611no—
330#431—611no—
331#431—1221no—
332#431—1221no—
333#428—611no—
334#423—611no—
335#423—1221no—
336#419—611no—
337#417—611no—
338C4×M16—611no—
339#335—611no—
340#335—611no—
341#334—611no—
342#334—611no—
343Q8⋊C8—611no—
344Q8⋊C8—611no—
345Q8⋊C8—611no—
346Q8⋊C8—611no—
347Q8⋊C8—611no—
348Q8⋊C8—611no—
349D4⋊C8—611no—
350D4⋊C8—611no—
351#322—611no—
352#322—611no—
353#322—611no—
354C82—611no—
355C61#1805311yescharacteristic, complemented, Sylow
356C2×#41C2×#692311yescharacteristic
357C22×Q8C2×#692311yescharacteristic
358#137—24no—
359C2×Q16C2×#692311yescharacteristic
360C2×Q16C2×#692311yescharacteristic
361C2×SD16C2×#692311yescharacteristic
362C2×SD16C2×#692311yescharacteristic
363C2×M16—611no—
364C2×M16—1221no—
365C2×M16—1221no—
366C2×M16—1221no—
367C2×M16C2×#692311yescharacteristic
368C4×Q8—611no—
369C4×Q8—611no—
370C4×Q8—611no—
371C4×Q8—611no—
372C4×Q8—611no—
373C4×Q8—611no—
374C4×Q8—1221no—
375C4×Q8—1221no—
376C4×D4—611no—
377C4×D4—611no—
378C4×D4—1221no—
379C4×D4—1221no—
380#117—611no—
381#117—1221no—
382C2×C4⋊C4—611no—
383C2×C4⋊C4—1221no—
384C2×C42—611no—
385C2×C42—1221no—
386C2×C42—1221no—
387#107—611no—
388#107—611no—
389#106—611no—
390#106—611no—
391C4⋊C8—611no—
392C4⋊C8—611no—
393C4⋊C8—611no—
394C4⋊C8—611no—
395C4⋊C8—611no—
396C4⋊C8—611no—
397C4⋊C8—1221no—
398C4⋊C8—1221no—
399C4⋊C8—1221no—
400C4⋊C8—1221no—
401C4⋊C8—1221no—
402C4⋊C8—1221no—
403C4⋊C8—1221no—
404C4⋊C8—1221no—
405#103—611no—
406#103—611no—
407#103—611no—
408#103—611no—
409#103—611no—
410#103—611no—
411#102—611no—
412#102—611no—
413C22⋊C8—611no—
414C22⋊C8—611no—
415C22⋊C8—1221no—
416C22⋊C8—1221no—
417C22⋊C8—1221no—
418C22⋊C8—1221no—
419#97—611no—
420#97—611no—
421#97—1221no—
422#97—1221no—
423#97—1221no—
424C4×C8—611no—
425C4×C8—611no—
426C4×C8—611no—
427C4×C8—611no—
428C4×C8—611no—
429C4×C8—611no—
430C4×C8—1221no—
431C4×C8—1221no—
432C4×C8—1221no—
433C4×C8—1221no—
434#41—22no—
435#41—41no—
436C2×Q8—21no—
437C2×Q8—22no—
438C2×Q8—41no—
439C2×Q8C22×#692311yescharacteristic
440C2×Q8C22×#692311yescharacteristic
441C2×Q8C22×#692311yescharacteristic
442C2×D4—21no—
443C2×D4C22×#692311yescharacteristic
444C22×C4—21no—
445C22×C4—21no—
446C22×C4—611no—
447C22×C4—611no—
448C22×C4—1221no—
449C22×C4—1221no—
450C22×C4—1221no—
451C22×C4C22×#692311yescharacteristic
452Q16—22no—
453Q16#1175821714yescomplemented
454SD16—22no—
455SD16—22no—
456M16—22no—
457M16—4881no—
458M16—4881no—
459M16—4881no—
460M16—2441no—
461C2×C8—611no—
462C2×C8—611no—
463C2×C8—611no—
464C2×C8—611no—
465C2×C8—611no—
466C2×C8—611no—
467C2×C8—1221no—
468C2×C8—1221no—
469C2×C8—1221no—
470C2×C8—1221no—
471C2×C8—1221no—
472C2×C8—1221no—
473C2×C8C22×#692311yescharacteristic
474C2×C8C22×#692311yescharacteristic
475C4⋊C4—611no—
476C4⋊C4—611no—
477C4⋊C4—611no—
478C4⋊C4—611no—
479C4⋊C4—1221no—
480C4⋊C4—1221no—
481C4⋊C4—1221no—
482C4⋊C4—1221no—
483C4⋊C4—1221no—
484C4⋊C4—1221no—
485C4⋊C4—1221no—
486C22⋊C4—1221no—
487C22⋊C4—1221no—
488C22⋊C4—1221no—
489C42—611no—
490C42—611no—
491C42—611no—
492C42—611no—
493C42—1221no—
494C42—1221no—
495C42—1221no—
496C42—1221no—
497C42—1221no—
498C42—1221no—
499C42—1221no—
500C23—21no—
501C23D4×#692311yescharacteristic
502Q8—21no—
503Q8—21no—
504Q8—21no—
505Q8—21no—
506Q8—42no—
507Q8C2×#1175821712yes—
508Q8C2×#1175821712yes—
509D4—21no—
510D4—21no—
511D4—42no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—611no—
520C2×C4—611no—
521C2×C4—611no—
522C2×C4—611no—
523C2×C4—1221no—
524C2×C4—1221no—
525C2×C4—1221no—
526C2×C4—1221no—
527C2×C4—1221no—
528C2×C4—1221no—
529C2×C4—2441no—
530C2×C4—2441no—
531C2×C4—2441no—
532C2×C4—2442no—
533C2×C4—2442no—
534C2×C4D4×#692311yescharacteristic
535C2×C4C23×#692311yescharacteristic
536C8—21no—
537C8—1221no—
538C8—1221no—
539C8—1221no—
540C8—1221no—
541C8—1221no—
542C8—1221no—
543C8—2441no—
544C8—2441no—
545C8—2441no—
546C8—2441no—
547C8—2441no—
548C8—2441no—
549C8C2×#1175821712yes—
550C22—21no—
551C22—21no—
552C22—21no—
553C22—41no—
554C22—41no—
555C22—41no—
556C22C2×D4×#692311yescharacteristic
557C4—21no—
558C4—22no—
559C4—22no—
560C4—41no—
561C4—611no—
562C4—611no—
563C4—1221no—
564C4—1221no—
565C4—1221no—
566C4—2441no—
567C4—2441no—
568C4—2441no—
569C4—2441no—
570C4C22×#1175821711yescharacteristic
571C4#1684084011yescharacteristic
572C2—41no—
573C2—81no—
574C2D4×#1175821711yescharacteristic
575C2#2146257711yescharacteristic
576C2#2146259211yescharacteristic
subgroupG/H
centerC22C2×D4×#6923
commutatorC2×C4×C61C23×C4
FrattiniC2×C4C23×#6923
FittingC2×#137×C61C4
socleC22×C61C2×C4×D4
prime radicalC2×#41×C61C2×C4
cosocleC4×Dic61C24
hypercenterC2×#137#6923
hypocenterC61#18053
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#137

Complements

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

NH ≅ G/Ntype
2. #21462722C2semidirect
3. #21461462C2semidirect
4. #21461455C2semidirect
5. #21461448C2semidirect
6. #21461444C2semidirect
7. #21461432C2semidirect
8. #21461428C2semidirect
9. #21461412C2semidirect
10. Q16×#92420C2semidirect
11. #21461383C2semidirect
13. #21461147C2semidirect
14. #21461038C2semidirect
177. Q16×C61M16semidirect
355. C61#18053semidirect
453. Q16#11758217semidirect

Maximal subgroups

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

subgroupindex
#214627512
#214627222
#214614622
#214614552
#214614482
#214614442
#214614322
#214614282
#214614122
Q16×#924202
#214613832
#214611502
#214611472
#214610382
#137×Dic612
#1805361