Finite Group Databasebeta

← #30612668

Subgroups — #30612668

Order 15616.

624 conjugacy classes of subgroups.

The subgroup lattice has 570 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21462633C211yesmaximal, characteristic
2Q8×D61⋊C8C211yesmaximal, characteristic, complemented
3#21461433C211yesmaximal, characteristic, complemented
4#21461432C211yesmaximal, characteristic, complemented
5#21461429C211yesmaximal, characteristic, complemented
6#21461428C211yesmaximal, characteristic, complemented
7#21461413C211yesmaximal, characteristic, complemented
8#21461412C211yesmaximal, characteristic, complemented
9#21461410C211yesmaximal, characteristic, complemented
10SD16⋊#92420C211yesmaximal, characteristic, complemented
11#21461393C211yesmaximal, characteristic, complemented
12#21461162C211yesmaximal, characteristic, complemented
13#21461135C211yesmaximal, characteristic
14#21461031C211yesmaximal, characteristic, complemented
15#21455898C211yesmaximal, characteristic
16#16840813C2211yescharacteristic
17#16840811—21no—
18#16840811—21no—
19#16840811C2211yescharacteristic
20#16840811C2211yescharacteristic
21Q8×#92420C2211yescharacteristic
22Q8×#92420C2211yescharacteristic
23Q8×#92420C2211yescharacteristic
24#16840797C2211yescharacteristic
25#16840794—21no—
26#16840789—21no—
27D4×#92420C2211yescharacteristic
28#16840786—21no—
29#16840784C2211yescharacteristic
30D61⋊(C4⋊C8)—21no—
31D61⋊(C4⋊C8)C2211yescharacteristic
32#16840726—21no—
33C4×#11758213C2211yescharacteristic
34C4×D61⋊C8—21no—
35C4×D61⋊C8C2211yescharacteristic
36#16840626C2211yescharacteristic
37#16840625C2211yescharacteristic
38#16840624C2211yescharacteristic
39#16840623C2211yescharacteristic
40#16840612C2211yescharacteristic
41#16840612C2211yescharacteristic
42Q8⋊#92420C2211yescharacteristic
43Q8⋊#92420C2211yescharacteristic
44Q8⋊#92420C2211yescharacteristic
45Q8⋊#92420C2211yescharacteristic
46D4⋊#92420C2211yescharacteristic
47D4⋊#92420C2211yescharacteristic
48#92420⋊C8C2211yescharacteristic
49#92420⋊C8C2211yescharacteristic
50#16840603C2211yescharacteristic
51#16840603C2211yescharacteristic
52#16840291C2211yescharacteristic
53C4×Q8×D61C2211yescharacteristic
54#16840091C411yescharacteristic
55#16840074C411yescharacteristic
56#16840070C411yescharacteristic
57#16840069C411yescharacteristic
58#16839981C411yescharacteristic
59#16839916C411yescharacteristic
60#16839899C411yescharacteristic
61Q16×Dic61C2211yescharacteristic
62#16839854C2211yescharacteristic
63#16839853C2211yescharacteristic
64SD16×Dic61C2211yescharacteristic
65#16839814C2211yescharacteristic
66C61×#464C411yescharacteristic
67C2×#11758213—21no—
68C2×#11758217—21no—
69C2×D61⋊C8—21no—
70C2×D61⋊C8D411yescharacteristic
71C2×D61⋊C8D411yescharacteristic
72C22×#92420—21no—
73#14519844—21no—
74#14519844—21no—
75#14519844—21no—
76#14519844—21no—
77C4⋊#92420—21no—
78C4⋊#92420—21no—
79C4⋊#92420—21no—
80C4⋊#92420C2311yescharacteristic
81C4⋊#92420C2311yescharacteristic
82C4⋊#92420C2311yescharacteristic
83C4⋊#92420C2311yescharacteristic
84#14519834—21no—
85C22⋊#92420—21no—
86C22⋊#92420—21no—
87#14519832—21no—
88#14519832—21no—
89#14519832—21no—
90C4×#92420—21no—
91C4×#92420—21no—
92C4×#92420—21no—
93C4×#92420—21no—
94C4×#92420—21no—
95C4×#92420D411yescharacteristic
96C4×#92420D411yescharacteristic
97C4×#92420C2311yescharacteristic
98C4×#92420C2311yescharacteristic
99C4×#92420C2311yescharacteristic
100C4×#92420C2311yescharacteristic
101C2×Q8×D61C2×C411yescharacteristic
102C2×#11758202C2×C411yescharacteristic
103C2×#11758200C2×C411yescharacteristic
104#14519770—21no—
105#14519769C2×C411yescharacteristic
106#14519759—21no—
107#14519753C2×C411yescharacteristic
108#14519748C2×C411yescharacteristic
109Q8×Dic61C2311yescharacteristic
110Q8×Dic61C2311yescharacteristic
111C4×Dic122—21no—
112C4×Dic122C2×C411yescharacteristic
113#14519742—21no—
114#14519742C2311yescharacteristic
115D4×Dic61C2311yescharacteristic
116#14519737—21no—
117#14519733—21no—
118C4⋊C4×D61—21no—
119C4⋊C4×D61C2×C411yescharacteristic
120C42×D61—21no—
121C42×D61C2311yescharacteristic
122#14519711C2×C411yescharacteristic
123#14519710C2×C411yescharacteristic
124#14519709C2×C411yescharacteristic
125#14519708C2×C411yescharacteristic
126C4⋊#92418C2×C411yescharacteristic
127#14519702C2×C411yescharacteristic
128#14519702C2×C411yescharacteristic
129#14519701C2×C411yescharacteristic
130#14519701C2×C411yescharacteristic
131#14519701C2×C411yescharacteristic
132#14519698C2×C411yescharacteristic
133#14519689C2×C411yescharacteristic
134#14519689C2×C411yescharacteristic
135C8×Dic61C2311yescharacteristic
136C8×Dic61C2311yescharacteristic
137C2×Q16×C61C2×C411yescharacteristic
138C2×SD16×C61C2×C411yescharacteristic
139#124×C61C2×C411yescharacteristic
140C4×Q8×C61C2×C411yescharacteristic
141C4⋊C8×C61C2×C411yescharacteristic
142#103×C61C2×C411yescharacteristic
143#102×C61C2×C411yescharacteristic
144C2×Q8×C61C22×C411yescharacteristic
145C2×Q8×C61C22×C411yescharacteristic
146C2×D4×C61C22×C411yescharacteristic
147C2×C22⋊D61—21no—
148C22×Dic61—21no—
149Q8×D61—22no—
150Q8×D61—22no—
151C2×C4×D61—21no—
152C2×C4×D61—21no—
153C2×C4×D61C22×C411yescharacteristic
154C2×C4×D61C2×D411yescharacteristic
155C2×C4×D61#4111yescharacteristic
156C2×Dic122—21no—
157C2×Dic122C22×C411yescharacteristic
158#11758217—42no—
159C2×#92420—21no—
160C2×#92420—21no—
161C2×#92420—21no—
162C2×#92420—21no—
163C2×#92420—21no—
164C2×#92420—21no—
165C2×#92420—42no—
166C2×#92420C2×D411yescharacteristic
167C2×#92420C2×D411yescharacteristic
168C2×#92420C2×D411yescharacteristic
169C2×#92420C2×D411yescharacteristic
170#11758213—42no—
171D61⋊C8—22no—
172D61⋊C8—41no—
173D61⋊C8—42no—
174Q16×C61—22no—
175SD16×C61—22no—
176C2×C8×C61C22×C411yescharacteristic
177C2×C8×C61C22×C411yescharacteristic
178C4⋊C4×C61—21no—
179C4⋊C4×C61C22×C411yescharacteristic
180C22⋊C4×C61—21no—
181C22⋊C4×C61—21no—
182C42×C61—21no—
183C42×C61C22×C411yescharacteristic
184C22⋊Dic61—21no—
185C22⋊Dic61—21no—
186#11758202—22no—
187#11758200—22no—
188#11758198—21no—
189C4⋊Dic61—21no—
190C4⋊Dic61—21no—
191C4⋊Dic61C22×C411yescharacteristic
192C4⋊Dic61C22×C411yescharacteristic
193C4⋊Dic61C22×C411yescharacteristic
194#11758196—21no—
195#11758196—21no—
196#11758196—21no—
197#11758196—21no—
198#11758196—21no—
199#11758196—21no—
200C4×Dic61—21no—
201C4×Dic61—21no—
202C4×Dic61—21no—
203C4×Dic61—21no—
204C4×Dic61C2×D411yescharacteristic
205C4×Dic61#4111yescharacteristic
206C4×Dic61C2411yescharacteristic
207C2×#92418C22×C411yescharacteristic
208C2×#92418C22×C411yescharacteristic
209C23×C61—21no—
210C22×D61C4×D411yescharacteristic
211Q8×C61—21no—
212Q8×C61—21no—
213Q8×C61—21no—
214Q8×C61#13112yes—
215D4×C61—21no—
216D4×C61—21no—
217C2×C4×C61—21no—
218C2×C4×C61—21no—
219C2×C4×C61—21no—
220C2×C4×C61C4×D411yescharacteristic
221C2×C4×C61C4×D411yescharacteristic
222C2×C4×C61C23×C411yescharacteristic
223C22⋊D61—42no—
224C2×Dic61—21no—
225C2×Dic61—21no—
226C2×Dic61—21no—
227C2×Dic61—21no—
228C2×Dic61—41no—
229C2×Dic61—41no—
230C2×Dic61C4×D411yescharacteristic
231C2×Dic61C22×D411yescharacteristic
232C2×Dic61C2×#4111yescharacteristic
233C4×D61—24no—
234C4×D61—22no—
235C4×D61—21no—
236C4×D61—21no—
237C4×D61—21no—
238C4×D61—21no—
239C4×D61—42no—
240Dic122—24no—
241Dic122—21no—
242Dic122#13112yes—
243#92420—21no—
244#92420—21no—
245#92420—21no—
246#92420—41no—
247#92420—41no—
248#92420—41no—
249#92420—41no—
250#92420—41no—
251#92420—41no—
252#92420#13712yes—
253C8×C61—21no—
254C8×C61—21no—
255#92418—21no—
256#92418—21no—
257#17915—611nomaximal, Sylow
258C22×C61—41no—
259C22×C61—41no—
260C22×C61—41no—
261C22×C61C2×C4×D411yescharacteristic
262C2×D61—21no—
263C2×D61—21no—
264C2×D61#44412yes—
265C4×C61—22no—
266C4×C61—21no—
267C4×C61—21no—
268C4×C61—41no—
269C4×C61—41no—
270C4×C61C2×#13111yescharacteristic
271C4×C61C2×#13111yescharacteristic
272Dic61—21no—
273Dic61—22no—
274Dic61—21no—
275Dic61—41no—
276Dic61—41no—
277Dic61—41no—
278Dic61C2×#13711yescharacteristic
279Dic61#57511yescharacteristic
280#3108—611no—
281#2966—611no—
282C2×C8×Q8—611no—
283#1584—611no—
284#1584—611no—
285#1583—611no—
286#1583—611no—
287#1579—611no—
288#1579—611no—
289SD16⋊C8—611no—
290SD16⋊C8—611no—
291#1573—611no—
292#1491—611no—
293#1483—611no—
294#1451—611no—
295C2×C61—81no—
296C2×C61#292911yescharacteristic
297C2×C61C4×#13711yescharacteristic
298C2×C61C2×#44411yescharacteristic
299D61—22no—
300#529—611no—
301C2×C4×Q8—611no—
302#483—611no—
303#480—611no—
304#479—611no—
305#479—611no—
306#464—611no—
307#464#692311yescharacteristic
308#451—611no—
309#448—611no—
310#446—611no—
311C8×Q8—611no—
312C8×Q8—611no—
313C8×Q8—611no—
314C8×Q8—611no—
315C8×Q8—611no—
316C8×Q8—1221no—
317C8×Q8—1221no—
318C4×Q16—611no—
319C4×Q16—611no—
320C4×SD16—611no—
321C4×SD16—611no—
322#436—611no—
323#435—1221no—
324C8×D4—611no—
325C8×D4—1221no—
326#433—1221no—
327#432—611no—
328#431—611no—
329C2×C4⋊C8—611no—
330C2×C4⋊C8—1221no—
331#405—1221no—
332C4×M16—611no—
333C2×C4×C8—611no—
334C2×C4×C8—1221no—
335#335—611no—
336#335—611no—
337#334—611no—
338#334—611no—
339Q8⋊C8—611no—
340Q8⋊C8—611no—
341Q8⋊C8—611no—
342Q8⋊C8—611no—
343Q8⋊C8—611no—
344Q8⋊C8—611no—
345D4⋊C8—611no—
346D4⋊C8—611no—
347#322—611no—
348#322—611no—
349#322—611no—
350#322—611no—
351C61#1791511yescharacteristic, complemented, Sylow
352C22×Q8—611no—
353C2×Q16—611no—
354C2×Q16C2×#692311yescharacteristic
355C2×SD16—611no—
356C2×SD16C2×#692311yescharacteristic
357C2×M16—1221no—
358C2×M16—1221no—
359C22×C8—611no—
360C22×C8—611no—
361C22×C8—1221no—
362C22×C8—1221no—
363#126—1221no—
364#125—611no—
365#124C2×#692311yescharacteristic
366#123—1221no—
367C22⋊Q8—611no—
368#121—611no—
369C4×Q8—611no—
370C4×Q8—611no—
371C4×Q8—611no—
372C4×Q8—611no—
373C4×Q8—1221no—
374C4×Q8—1221no—
375C4×Q8C2×#692311yescharacteristic
376C4×D4—611no—
377C4×D4—1221no—
378#117—1221no—
379C2×C4⋊C4—611no—
380C2×C4⋊C4—1221no—
381C2×C42—611no—
382C2×C42—1221no—
383#107—611no—
384#107—611no—
385#106—611no—
386#106—611no—
387C4⋊C8—611no—
388C4⋊C8—611no—
389C4⋊C8—611no—
390C4⋊C8—611no—
391C4⋊C8—611no—
392C4⋊C8—1221no—
393C4⋊C8—1221no—
394C4⋊C8—1221no—
395C4⋊C8—1221no—
396C4⋊C8—1221no—
397C4⋊C8—1221no—
398C4⋊C8—1221no—
399C4⋊C8C2×#692311yescharacteristic
400#103—611no—
401#103—611no—
402#103—611no—
403#103—611no—
404#103—611no—
405#103C2×#692311yescharacteristic
406#102—611no—
407#102C2×#692311yescharacteristic
408C22⋊C8—611no—
409C22⋊C8—611no—
410C22⋊C8—1221no—
411C22⋊C8—1221no—
412C22⋊C8—1221no—
413#97—1221no—
414#97—1221no—
415#97—1221no—
416C4×C8—611no—
417C4×C8—611no—
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—
428C4×C8—1221no—
429C2×Q8—611no—
430C2×Q8—1221no—
431C2×Q8—1222no—
432C2×Q8—1222no—
433C2×Q8C22×#692311yescharacteristic
434C2×Q8C22×#692311yescharacteristic
435C2×D4—1221no—
436C2×D4C22×#692311yescharacteristic
437C22×C4—611no—
438C22×C4—611no—
439C22×C4—611no—
440C22×C4—1221no—
441C22×C4—1221no—
442C22×C4—1221no—
443Q16—22no—
444Q16—1222no—
445SD16—22no—
446SD16—1222no—
447M16—2442no—
448M16—2442no—
449C2×C8—611no—
450C2×C8—611no—
451C2×C8—611no—
452C2×C8—611no—
453C2×C8—611no—
454C2×C8—611no—
455C2×C8—1222no—
456C2×C8—1221no—
457C2×C8—1221no—
458C2×C8—1221no—
459C2×C8—1221no—
460C2×C8—1221no—
461C2×C8—1221no—
462C2×C8—2441no—
463C2×C8—2442no—
464C2×C8—2442no—
465C2×C8C22×#692311yescharacteristic
466C2×C8C22×#692311yescharacteristic
467C4⋊C4—21no—
468C4⋊C4—611no—
469C4⋊C4—611no—
470C4⋊C4—611no—
471C4⋊C4—1221no—
472C4⋊C4—1221no—
473C4⋊C4—1221no—
474C4⋊C4—1221no—
475C4⋊C4—1221no—
476C4⋊C4—1221no—
477C4⋊C4—1221no—
478C4⋊C4—1221no—
479C4⋊C4C22×#692311yescharacteristic
480C22⋊C4—21no—
481C22⋊C4—21no—
482C22⋊C4—1221no—
483C22⋊C4—1221no—
484C22⋊C4—1221no—
485C42—21no—
486C42—611no—
487C42—611no—
488C42—611no—
489C42—1221no—
490C42—1221no—
491C42—1221no—
492C42—1221no—
493C42C22×#692311yescharacteristic
494C23—21no—
495C23—611no—
496Q8—21no—
497Q8—21no—
498Q8—21no—
499Q8—612no—
500Q8—1224no—
501Q8—1221no—
502Q8#1451986412yes—
503D4—21no—
504D4—21no—
505D4—2442no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—611no—
510C2×C4—611no—
511C2×C4—611no—
512C2×C4—1221no—
513C2×C4—1221no—
514C2×C4—1222no—
515C2×C4—1224no—
516C2×C4—1221no—
517C2×C4—1221no—
518C2×C4—1221no—
519C2×C4—1221no—
520C2×C4—1221no—
521C2×C4—1221no—
522C2×C4—2442no—
523C2×C4—2441no—
524C2×C4—2441no—
525C2×C4D4×#692311yescharacteristic
526C2×C4D4×#692311yescharacteristic
527C2×C4C23×#692311yescharacteristic
528C8—21no—
529C8—21no—
530C8—612no—
531C8—1221no—
532C8—1221no—
533C8—1221no—
534C8—1221no—
535C8—1221no—
536C8—2441no—
537C8—2441no—
538C8—2441no—
539C8—2441no—
540C8—2441no—
541C8—2441no—
542C22—41no—
543C22—41no—
544C22—41no—
545C22—612no—
546C22—1221no—
547C22—1221no—
548C22C2×D4×#692311yescharacteristic
549C4—22no—
550C4—21no—
551C4—21no—
552C4—41no—
553C4—41no—
554C4—611no—
555C4—611no—
556C4—1221no—
557C4—1222no—
558C4—1221no—
559C4—2441no—
560C4—2441no—
561C4—2441no—
562C4C2×#1451986411yescharacteristic
563C4D61⋊#13111yescharacteristic
564C2—81no—
565C2—1222no—
566C2#2146253311yescharacteristic
567C2#137×#692311yescharacteristic
568C2D61⋊#44411yescharacteristic
subgroupG/H
centerC22C2×D4×#6923
commutatorC2×C4×C61C23×C4
FrattiniC2×C4C23×#6923
FittingC61×#464C4
socleC22×C61C2×C4×D4
prime radical#14519748C2×C4
cosocleC4×Dic61C24
hypercenter#464#6923
hypocenterC61#17915
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ #464

Complements

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

NH ≅ G/Ntype
2. Q8×D61⋊C8C2semidirect
3. #21461433C2semidirect
4. #21461432C2semidirect
5. #21461429C2semidirect
6. #21461428C2semidirect
7. #21461413C2semidirect
8. #21461412C2semidirect
9. #21461410C2semidirect
10. SD16⋊#92420C2semidirect
11. #21461393C2semidirect
12. #21461162C2semidirect
14. #21461031C2semidirect
351. C61#17915semidirect

Maximal subgroups

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

subgroupindex
#214626332
Q8×D61⋊C82
#214614332
#214614322
#214614292
#214614282
#214614132
#214614122
#214614102
SD16⋊#924202
#214613932
#214611622
#214611352
#214610312
#214558982
#1791561