Finite Group Databasebeta

← #30613249

Subgroups — #30613249

Order 15616.

592 conjugacy classes of subgroups.

The subgroup lattice has 568 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21462652C211yesmaximal, characteristic
2Q8×#11758217C211yesmaximal, characteristic, complemented
3#21461466C211yesmaximal, characteristic, complemented
4#21461453C211yesmaximal, characteristic, complemented
5#21461444C211yesmaximal, characteristic, complemented
6#21461437C211yesmaximal, characteristic, complemented
7#21461432C211yesmaximal, characteristic, complemented
8#21461426C211yesmaximal, characteristic, complemented
9#21461409C211yesmaximal, characteristic, complemented
10Q16×#92420C211yesmaximal, characteristic, complemented
11#21461395C211yesmaximal, characteristic, complemented
12#21461176C211yesmaximal, characteristic
13#21461121C211yesmaximal, characteristic, complemented
14#21461036C211yesmaximal, characteristic, complemented
15#21455900C211yesmaximal, characteristic
16#16840813—21no—
17#16840813—21no—
18#16840813C2211yescharacteristic
19#16840813C2211yescharacteristic
20#16840813C2211yescharacteristic
21Q8×#92420C2211yescharacteristic
22Q8×#92420C2211yescharacteristic
23Q8×#92420C2211yescharacteristic
24#16840796C2211yescharacteristic
25#16840795—21no—
26#16840794—21no—
27#16840790C2211yescharacteristic
28#16840787—21no—
29#16840778C2211yescharacteristic
30#16840778C2211yescharacteristic
31#16840765—21no—
32#16840765C2211yescharacteristic
33#16840727—21no—
34C4×#11758217—21no—
35C4×#11758217C2211yescharacteristic
36#16840626C2211yescharacteristic
37#16840625C2211yescharacteristic
38#16840624C2211yescharacteristic
39#16840623C2211yescharacteristic
40Q8⋊#92420C2211yescharacteristic
41Q8⋊#92420C2211yescharacteristic
42Q8⋊#92420C2211yescharacteristic
43Q8⋊#92420C2211yescharacteristic
44Q8⋊#92420C2211yescharacteristic
45Q8⋊#92420C2211yescharacteristic
46#16840610C2211yescharacteristic
47#16840610C2211yescharacteristic
48#92420⋊C8C2211yescharacteristic
49#92420⋊C8C2211yescharacteristic
50#16840603C2211yescharacteristic
51C8×#92420C2211yescharacteristic
52#16840298C2211yescharacteristic
53C2×Q8×Dic61C2211yescharacteristic
54#16840090C411yescharacteristic
55#16840076C411yescharacteristic
56#16840071C411yescharacteristic
57#16840071C411yescharacteristic
58#16839986C411yescharacteristic
59#16839986C411yescharacteristic
60#16839901C411yescharacteristic
61Q16×Dic61C2211yescharacteristic
62Q16×Dic61C2211yescharacteristic
63#16839847C2211yescharacteristic
64#16839847C2211yescharacteristic
65#16839812C2211yescharacteristic
66C61×#451C411yescharacteristic
67C2×#11758213—21no—
68C2×#11758217—21no—
69C2×#11758217D411yescharacteristic
70C2×#11758217D411yescharacteristic
71C2×D61⋊C8—21no—
72#14519844—21no—
73#14519844—21no—
74#14519844—21no—
75#14519844—21no—
76C4⋊#92420—21no—
77C4⋊#92420—21no—
78C4⋊#92420—21no—
79C4⋊#92420—21no—
80C4⋊#92420C2311yescharacteristic
81C4⋊#92420C2311yescharacteristic
82C4⋊#92420C2311yescharacteristic
83C4⋊#92420C2311yescharacteristic
84#14519834—21no—
85#14519834—21no—
86C22⋊#92420—21no—
87C22⋊#92420—21no—
88#14519832—21no—
89#14519832—21no—
90#14519832—21no—
91#14519832D411yescharacteristic
92#14519832D411yescharacteristic
93C4×#92420—21no—
94C4×#92420—21no—
95C4×#92420—21no—
96C4×#92420—21no—
97C4×#92420C2311yescharacteristic
98C4×#92420C2311yescharacteristic
99C4×#92420C2311yescharacteristic
100C4×#92420C2311yescharacteristic
101C2×#11758201C2×C411yescharacteristic
102C2×#11758201C2×C411yescharacteristic
103#14519772—21no—
104#14519768C2×C411yescharacteristic
105#14519766C2×C411yescharacteristic
106#14519761—21no—
107#14519751C2×C411yescharacteristic
108Q8×Dic61—21no—
109Q8×Dic61—21no—
110Q8×Dic61C2×C411yescharacteristic
111Q8×Dic61C2×C411yescharacteristic
112Q8×Dic61C2311yescharacteristic
113Q8×Dic61C2311yescharacteristic
114Q8×Dic61C2311yescharacteristic
115#14519739C2311yescharacteristic
116#14519737—21no—
117#14519735—21no—
118C2×C4⋊Dic61—21no—
119C2×C4⋊Dic61C2×C411yescharacteristic
120C2×C4×Dic61—21no—
121C2×C4×Dic61C2311yescharacteristic
122#14519711C2×C411yescharacteristic
123#14519711C2×C411yescharacteristic
124#14519708C2×C411yescharacteristic
125#14519708C2×C411yescharacteristic
126#14519706C2×C411yescharacteristic
127#14519706C2×C411yescharacteristic
128#14519701C2×C411yescharacteristic
129#14519701C2×C411yescharacteristic
130#14519701C2×C411yescharacteristic
131#14519701C2×C411yescharacteristic
132#14519699C2×C411yescharacteristic
133#14519699C2×C411yescharacteristic
134C22⋊#92418C2×C411yescharacteristic
135C8×Dic61C2311yescharacteristic
136C8×Dic61C2311yescharacteristic
137C22×Q8×C61C2×C411yescharacteristic
138C2×Q16×C61C2×C411yescharacteristic
139C2×Q16×C61C2×C411yescharacteristic
140C22⋊Q8×C61C2×C411yescharacteristic
141#103×C61C2×C411yescharacteristic
142#103×C61C2×C411yescharacteristic
143C22⋊C8×C61C2×C411yescharacteristic
144C2×Q8×C61—21no—
145C2×Q8×C61—22no—
146C2×Q8×C61—41no—
147C2×Q8×C61C22×C411yescharacteristic
148C2×Q8×C61C22×C411yescharacteristic
149C2×Q8×C61C22×C411yescharacteristic
150C22×C4×C61—21no—
151C22×C4×C61C22×C411yescharacteristic
152C2×C22⋊D61—21no—
153C22×Dic61—21no—
154C22×Dic61C2×D411yescharacteristic
155C22×Dic61#4111yescharacteristic
156C2×D244C22×C411yescharacteristic
157C2×C4×D61—21no—
158#11758217—22no—
159#11758217—41no—
160#11758217—42no—
161C2×#92420—21no—
162C2×#92420—21no—
163C2×#92420—21no—
164C2×#92420—21no—
165C2×#92420—21no—
166C2×#92420—21no—
167C2×#92420C2×D411yescharacteristic
168C2×#92420C2×D411yescharacteristic
169C2×#92420C2×D411yescharacteristic
170C2×#92420C2×D411yescharacteristic
171#11758213—42no—
172D61⋊C8—42no—
173Q16×C61—22no—
174Q16×C61—41no—
175C2×C8×C61C22×C411yescharacteristic
176C2×C8×C61C22×C411yescharacteristic
177C4⋊C4×C61—21no—
178C4⋊C4×C61C22×C411yescharacteristic
179C22⋊C4×C61—21no—
180#11758201—22no—
181#11758201—41no—
182#11758198—21no—
183#11758198—21no—
184#11758198—21no—
185#11758198—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
196#11758196—21no—
197C4×Dic61—21no—
198C4×Dic61—21no—
199C4×Dic61—21no—
200C4×Dic61—21no—
201C4×Dic61—21no—
202C4×Dic61C22×C411yescharacteristic
203C4×Dic61C2×D411yescharacteristic
204C4×Dic61#4111yescharacteristic
205C4×Dic61C2411yescharacteristic
206C2×#92418C22×C411yescharacteristic
207C2×#92418C22×C411yescharacteristic
208C23×C61C4×D411yescharacteristic
209C22×D61—21no—
210Q8×C61—21no—
211Q8×C61—21no—
212Q8×C61—21no—
213Q8×C61—21no—
214Q8×C61—21no—
215Q8×C61—42no—
216Q8×C61#13112yes—
217C2×C4×C61—21no—
218C2×C4×C61—21no—
219C2×C4×C61—21no—
220C2×C4×C61—21no—
221C2×C4×C61—21no—
222C2×C4×C61—21no—
223C2×C4×C61—42no—
224C2×C4×C61C4×D411yescharacteristic
225C2×C4×C61C23×C411yescharacteristic
226C22⋊D61—81no—
227C2×Dic61—21no—
228C2×Dic61—21no—
229C2×Dic61—21no—
230C2×Dic61—21no—
231C2×Dic61—21no—
232C2×Dic61—41no—
233C2×Dic61—81no—
234C2×Dic61C4×D411yescharacteristic
235C2×Dic61C4×D411yescharacteristic
236C2×Dic61C22×D411yescharacteristic
237C2×Dic61C2×#4111yescharacteristic
238D244—21no—
239D244—21no—
240C4×D61—41no—
241C4×D61—41no—
242#92420—21no—
243#92420—21no—
244#92420—21no—
245#92420—41no—
246#92420—41no—
247#92420—41no—
248#92420—41no—
249#92420—41no—
250#92420—41no—
251#92420#13712yes—
252C8×C61—21no—
253C8×C61—21no—
254#92418—21no—
255#92418—21no—
256#18012—611nomaximal, Sylow
257C22×C61—21no—
258C22×C61—21no—
259C22×C61—21no—
260C22×C61C2×C4×D411yescharacteristic
261C2×D61—41no—
262C2×D61—41no—
263C2×D61—41no—
264C4×C61—21no—
265C4×C61—22no—
266C4×C61—41no—
267C4×C61—41no—
268C4×C61—41no—
269C4×C61C2×#13111yescharacteristic
270C4×C61#56811yescharacteristic
271Dic61—21no—
272Dic61—21no—
273Dic61—21no—
274Dic61—41no—
275Dic61—41no—
276Dic61—41no—
277Dic61C2×#13711yescharacteristic
278Dic61#57511yescharacteristic
279#3108—611no—
280#2969—611no—
281Q8×M16—611no—
282#1593—611no—
283#1590—611no—
284#1587—611no—
285#1585—611no—
286#1584—611no—
287#1583—611no—
288SD16⋊C8—611no—
289C8×Q16—611no—
290#1574—611no—
291#1495—611no—
292#1479—611no—
293#1453—611no—
294C2×C61—41no—
295C2×C61#293311yescharacteristic
296C2×C61C4×#13711yescharacteristic
297C2×C61#295711yescharacteristic
298D61—81no—
299#529—611no—
300C2×C4×Q8—611no—
301#483—611no—
302#480—611no—
303#479—611no—
304#479—611no—
305#464—611no—
306#464—611no—
307#451#692311yescharacteristic
308#448—611no—
309#446—611no—
310#446—611no—
311#446—611no—
312#446—1221no—
313#446—1221no—
314C8×Q8—611no—
315C8×Q8—611no—
316C8×Q8—611no—
317C4×Q16—611no—
318C4×Q16—611no—
319C4×SD16—611no—
320C4×SD16—611no—
321#436—611no—
322#436—1221no—
323#435—1221no—
324C8×D4—611no—
325#433—1221no—
326#431—611no—
327#431—611no—
328#431—611no—
329#423—611no—
330#423—1221no—
331#405—1221no—
332C4×M16—611no—
333C4×M16—1221no—
334#335—611no—
335#335—611no—
336#334—611no—
337#334—611no—
338Q8⋊C8—611no—
339Q8⋊C8—611no—
340Q8⋊C8—611no—
341Q8⋊C8—611no—
342Q8⋊C8—611no—
343Q8⋊C8—611no—
344D4⋊C8—611no—
345D4⋊C8—611no—
346#322—611no—
347#322—611no—
348#322—611no—
349C82—611no—
350C61#1801211yescharacteristic, complemented, Sylow
351C22×Q8C2×#692311yescharacteristic
352C2×Q16C2×#692311yescharacteristic
353C2×Q16C2×#692311yescharacteristic
354C2×SD16—611no—
355C2×SD16—611no—
356C2×M16—611no—
357C2×M16—611no—
358C2×M16—1221no—
359C2×M16—1221no—
360C22×C8—1221no—
361#126—1221no—
362#125—611no—
363#124—611no—
364#123—1221no—
365C22⋊Q8C2×#692311yescharacteristic
366#121—611no—
367C4×Q8—611no—
368C4×Q8—611no—
369C4×Q8—611no—
370C4×Q8—611no—
371C4×Q8—611no—
372C4×Q8—1221no—
373C4×Q8—1221no—
374C4×D4—611no—
375C4×D4—1221no—
376#117—1221no—
377C2×C4⋊C4—611no—
378C2×C4⋊C4—1221no—
379C2×C42—611no—
380C2×C42—1221no—
381#107—611no—
382#107—611no—
383#106—611no—
384#106—611no—
385C4⋊C8—611no—
386C4⋊C8—611no—
387C4⋊C8—611no—
388C4⋊C8—611no—
389C4⋊C8—611no—
390C4⋊C8—611no—
391C4⋊C8—1221no—
392C4⋊C8—1221no—
393C4⋊C8—1221no—
394C4⋊C8—1221no—
395C4⋊C8—1221no—
396C4⋊C8—1221no—
397C4⋊C8—1221no—
398C4⋊C8—1221no—
399#103—611no—
400#103—611no—
401#103—611no—
402#103—611no—
403#103C2×#692311yescharacteristic
404#103C2×#692311yescharacteristic
405#102—611no—
406#102—611no—
407C22⋊C8—611no—
408C22⋊C8—1221no—
409C22⋊C8—1221no—
410C22⋊C8—1221no—
411C22⋊C8—1221no—
412C22⋊C8C2×#692311yescharacteristic
413#97—611no—
414#97—611no—
415#97—1221no—
416#97—1221no—
417#97—1221no—
418C4×C8—611no—
419C4×C8—611no—
420C4×C8—611no—
421C4×C8—611no—
422C4×C8—611no—
423C4×C8—611no—
424C4×C8—1221no—
425C4×C8—1221no—
426C4×C8—1221no—
427C4×C8—1221no—
428C2×Q8—21no—
429C2×Q8—22no—
430C2×Q8—41no—
431C2×Q8C22×#692311yescharacteristic
432C2×Q8C22×#692311yescharacteristic
433C2×Q8C22×#692311yescharacteristic
434C2×D4—611no—
435C2×D4—1221no—
436C22×C4—21no—
437C22×C4—611no—
438C22×C4—611no—
439C22×C4—1221no—
440C22×C4—1221no—
441C22×C4C22×#692311yescharacteristic
442Q16—22no—
443Q16—41no—
444SD16—1222no—
445SD16—2441no—
446M16—1222no—
447M16—2442no—
448M16—2441no—
449M16—2442no—
450C2×C8—611no—
451C2×C8—611no—
452C2×C8—611no—
453C2×C8—611no—
454C2×C8—611no—
455C2×C8—611no—
456C2×C8—1221no—
457C2×C8—1221no—
458C2×C8—1221no—
459C2×C8—1221no—
460C2×C8—1221no—
461C2×C8—1221no—
462C2×C8—2442no—
463C2×C8C22×#692311yescharacteristic
464C2×C8C22×#692311yescharacteristic
465C4⋊C4—21no—
466C4⋊C4—611no—
467C4⋊C4—611no—
468C4⋊C4—611no—
469C4⋊C4—1221no—
470C4⋊C4—1221no—
471C4⋊C4—1221no—
472C4⋊C4—1221no—
473C4⋊C4—1221no—
474C4⋊C4—1221no—
475C4⋊C4—1221no—
476C4⋊C4—1221no—
477C4⋊C4C22×#692311yescharacteristic
478C22⋊C4—21no—
479C22⋊C4—1221no—
480C22⋊C4—1221no—
481C22⋊C4—1221no—
482C22⋊C4—1221no—
483C42—611no—
484C42—611no—
485C42—611no—
486C42—611no—
487C42—1221no—
488C42—1221no—
489C42—1221no—
490C42—1221no—
491C42—1221no—
492C23—1221no—
493C23D4×#692311yescharacteristic
494Q8—21no—
495Q8—21no—
496Q8—21no—
497Q8—21no—
498Q8—21no—
499Q8—42no—
500Q8#1451986412yes—
501D4—1221no—
502D4—1221no—
503D4—4881no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—42no—
511C2×C4—611no—
512C2×C4—611no—
513C2×C4—611no—
514C2×C4—611no—
515C2×C4—1221no—
516C2×C4—1221no—
517C2×C4—1221no—
518C2×C4—1221no—
519C2×C4—1221no—
520C2×C4—4881no—
521C2×C4—2441no—
522C2×C4—2441no—
523C2×C4—2441no—
524C2×C4D4×#692311yescharacteristic
525C2×C4C23×#692311yescharacteristic
526C8—21no—
527C8—21no—
528C8—612no—
529C8—1221no—
530C8—1221no—
531C8—1221no—
532C8—1221no—
533C8—1221no—
534C8—2441no—
535C8—2441no—
536C8—2441no—
537C8—2441no—
538C8—2441no—
539C8—2441no—
540C22—21no—
541C22—21no—
542C22—21no—
543C22—2441no—
544C22—2441no—
545C22—2441no—
546C22C2×D4×#692311yescharacteristic
547C4—21no—
548C4—22no—
549C4—41no—
550C4—41no—
551C4—41no—
552C4—611no—
553C4—611no—
554C4—1221no—
555C4—1221no—
556C4—1221no—
557C4—2441no—
558C4—2441no—
559C4—2441no—
560C4C2×#1451986411yescharacteristic
561C4#1684084011yescharacteristic
562C2—41no—
563C2—4881no—
564C2#2146255111yescharacteristic
565C2#137×#692311yescharacteristic
566C2#2146259911yescharacteristic
subgroupG/H
centerC22C2×D4×#6923
commutatorC2×C4×C61C23×C4
FrattiniC2×C4C23×#6923
FittingC61×#451C4
socleC22×C61C2×C4×D4
prime radical#14519751C2×C4
cosocleC4×Dic61C24
hypercenter#451#6923
hypocenterC61#18012
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ #451

Complements

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

NH ≅ G/Ntype
2. Q8×#11758217C2semidirect
3. #21461466C2semidirect
4. #21461453C2semidirect
5. #21461444C2semidirect
6. #21461437C2semidirect
7. #21461432C2semidirect
8. #21461426C2semidirect
9. #21461409C2semidirect
10. Q16×#92420C2semidirect
11. #21461395C2semidirect
13. #21461121C2semidirect
14. #21461036C2semidirect
350. C61#18012semidirect

Maximal subgroups

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

subgroupindex
#214626522
Q8×#117582172
#214614662
#214614532
#214614442
#214614372
#214614322
#214614262
#214614092
Q16×#924202
#214613952
#214611762
#214611212
#214610362
#214559002
#1801261