Finite Group Databasebeta

← #30613149

Subgroups — #30613149

Order 15616.

602 conjugacy classes of subgroups.

The subgroup lattice has 558 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1Q8×#11758217C211yesmaximal, characteristic
2#21462614C211yesmaximal, characteristic
3#21461459C211yesmaximal, characteristic, complemented
4#21461457C211yesmaximal, characteristic, complemented
5#21461433C211yesmaximal, characteristic, complemented
6#21461432C211yesmaximal, characteristic, complemented
7#21461432C211yesmaximal, characteristic, complemented
8#21461430C211yesmaximal, characteristic, complemented
9#21461406C211yesmaximal, characteristic, complemented
10Q16×#92420C211yesmaximal, characteristic, complemented
11#21461388C211yesmaximal, characteristic
12#21461167C211yesmaximal, characteristic
13C2×Q8⋊#92420C211yesmaximal, characteristic
14#21461020C211yesmaximal, characteristic
15#21455853C211yesmaximal, characteristic
16#16840814—21no—
17#16840813—21no—
18#16840813—21no—
19#16840813C2211yescharacteristic
20#16840813C2211yescharacteristic
21#16840813C2211yescharacteristic
22#16840812—21no—
23#16840812C2211yescharacteristic
24#16840811C2211yescharacteristic
25Q8×#92420C2211yescharacteristic
26Q8×#92420C2211yescharacteristic
27Q8×#92420C2211yescharacteristic
28#16840768—21no—
29#16840765—21no—
30#16840765C2211yescharacteristic
31C2×C4⋊#92420C2211yescharacteristic
32#16840726—21no—
33C4×#11758217—21no—
34C4×#11758217C2211yescharacteristic
35C2×C4×#92420C2211yescharacteristic
36#16840626C2211yescharacteristic
37#16840626C2211yescharacteristic
38#16840625C2211yescharacteristic
39#16840625C2211yescharacteristic
40#16840612C2211yescharacteristic
41#16840612C2211yescharacteristic
42Q8⋊#92420C2211yescharacteristic
43Q8⋊#92420C2211yescharacteristic
44Q8⋊#92420C2211yescharacteristic
45Q8⋊#92420C2211yescharacteristic
46Q8⋊#92420C2211yescharacteristic
47Q8⋊#92420C2211yescharacteristic
48#92420⋊C8C2211yescharacteristic
49#92420⋊C8C2211yescharacteristic
50C8×#92420C2211yescharacteristic
51C8×#92420C2211yescharacteristic
52#16840314C2211yescharacteristic
53C2×Q8×Dic61C2211yescharacteristic
54#16840098C411yescharacteristic
55#16840095C411yescharacteristic
56#16840055C411yescharacteristic
57#16840055C411yescharacteristic
58#16839970C411yescharacteristic
59#16839970C411yescharacteristic
60#16839918C411yescharacteristic
61Q16×Dic61C2211yescharacteristic
62Q16×Dic61C2211yescharacteristic
63#16839854C2211yescharacteristic
64#16839854C2211yescharacteristic
65#16839812C2211yescharacteristic
66C61×#451C411yescharacteristic
67C2×#11758217—21no—
68C2×#11758217D411yescharacteristic
69C2×#11758217D411yescharacteristic
70C22×#92420—21no—
71C22×#92420—21no—
72#14519844—21no—
73#14519844—21no—
74#14519844—21no—
75#14519844—21no—
76#14519844—21no—
77C4⋊#92420—21no—
78C4⋊#92420—21no—
79C4⋊#92420—21no—
80C4⋊#92420—21no—
81C4⋊#92420—21no—
82C4⋊#92420C2311yescharacteristic
83C4⋊#92420C2311yescharacteristic
84C4⋊#92420C2311yescharacteristic
85C4⋊#92420C2311yescharacteristic
86#14519832—21no—
87#14519832—21no—
88#14519832—21no—
89#14519832—21no—
90#14519832D411yescharacteristic
91#14519832D411yescharacteristic
92C4×#92420—21no—
93C4×#92420—21no—
94C4×#92420—21no—
95C4×#92420—21no—
96C4×#92420—21no—
97C4×#92420C2311yescharacteristic
98C4×#92420C2311yescharacteristic
99C4×#92420C2311yescharacteristic
100C4×#92420C2311yescharacteristic
101C2×#11758202C2×C411yescharacteristic
102C2×#11758202C2×C411yescharacteristic
103#14519776C2×C411yescharacteristic
104#14519775C2×C411yescharacteristic
105#14519767—21no—
106#14519756C2×C411yescharacteristic
107#14519754—21no—
108Q8×Dic61—21no—
109Q8×Dic61—21no—
110Q8×Dic61C2×C411yescharacteristic
111Q8×Dic61C2×C411yescharacteristic
112Q8×Dic61C2311yescharacteristic
113Q8×Dic61C2311yescharacteristic
114Q8×Dic61C2311yescharacteristic
115#14519742—21no—
116#14519742C2311yescharacteristic
117#14519733—21no—
118C2×C4⋊Dic61—21no—
119C2×C4⋊Dic61C2×C411yescharacteristic
120C2×C4×Dic61—21no—
121C2×C4×Dic61C2311yescharacteristic
122#14519711C2×C411yescharacteristic
123#14519711C2×C411yescharacteristic
124#14519710C2×C411yescharacteristic
125#14519710C2×C411yescharacteristic
126#14519706C2×C411yescharacteristic
127#14519706C2×C411yescharacteristic
128#14519702C2×C411yescharacteristic
129#14519702C2×C411yescharacteristic
130#14519701C2×C411yescharacteristic
131#14519701C2×C411yescharacteristic
132#14519701C2×C411yescharacteristic
133#14519701C2×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
152C22×Dic61—21no—
153C22×Dic61C2×D411yescharacteristic
154C22×Dic61#4111yescharacteristic
155C2×Dic122—21no—
156C2×Dic122C22×C411yescharacteristic
157#11758217—22no—
158#11758217—42no—
159#11758217Q1614yescomplemented
160C2×#92420—21no—
161C2×#92420—21no—
162C2×#92420—21no—
163C2×#92420—21no—
164C2×#92420—21no—
165C2×#92420—21no—
166C2×#92420—42no—
167C2×#92420—42no—
168C2×#92420C2×D411yescharacteristic
169C2×#92420C2×D411yescharacteristic
170C2×#92420C2×D411yescharacteristic
171C2×#92420C2×D411yescharacteristic
172Q16×C61—22no—
173Q16×C61—41no—
174C2×C8×C61C22×C411yescharacteristic
175C2×C8×C61C22×C411yescharacteristic
176C4⋊C4×C61—21no—
177C4⋊C4×C61C22×C411yescharacteristic
178C22⋊C4×C61—21no—
179C22⋊Dic61—21no—
180#11758202—22no—
181#11758202—41no—
182C4⋊Dic61—21no—
183C4⋊Dic61—21no—
184C4⋊Dic61—21no—
185C4⋊Dic61—21no—
186C4⋊Dic61—21no—
187C4⋊Dic61—21no—
188C4⋊Dic61C22×C411yescharacteristic
189C4⋊Dic61C22×C411yescharacteristic
190C4⋊Dic61C22×C411yescharacteristic
191#11758196—21no—
192#11758196—21no—
193#11758196—21no—
194#11758196—21no—
195#11758196—21no—
196C4×Dic61—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
209Q8×C61—21no—
210Q8×C61—21no—
211Q8×C61—21no—
212Q8×C61—21no—
213Q8×C61—21no—
214Q8×C61—42no—
215Q8×C61#13112yes—
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—42no—
223C2×C4×C61C4×D411yescharacteristic
224C2×C4×C61C23×C411yescharacteristic
225C2×Dic61—21no—
226C2×Dic61—21no—
227C2×Dic61—21no—
228C2×Dic61—21no—
229C2×Dic61—41no—
230C2×Dic61—81no—
231C2×Dic61C4×D411yescharacteristic
232C2×Dic61C4×D411yescharacteristic
233C2×Dic61C2×Q1612yes—
234C2×Dic61#13512yes—
235C2×Dic61C22×D411yescharacteristic
236C2×Dic61C2×#4111yescharacteristic
237Dic122—21no—
238Dic122—21no—
239Dic122—81no—
240#92420—22no—
241#92420—21no—
242#92420—21no—
243#92420—41no—
244#92420—41no—
245#92420—41no—
246#92420—41no—
247#92420—41no—
248#92420C2×Q1612yes—
249#92420C2×Q1612yes—
250C8×C61—21no—
251C8×C61—21no—
252#92418—21no—
253#92418—21no—
254#17998—611nomaximal, Sylow
255C22×C61—21no—
256C22×C61—21no—
257C22×C61C4×Q1612yes—
258C22×C61C2×C4×D411yescharacteristic
259C4×C61—21no—
260C4×C61—22no—
261C4×C61—41no—
262C4×C61—41no—
263C4×C61—41no—
264C4×C61C2×#13111yescharacteristic
265C4×C61#56811yescharacteristic
266Dic61—21no—
267Dic61—21no—
268Dic61—41no—
269Dic61—41no—
270Dic61—41no—
271Dic61—41no—
272Dic61C4×Q1612yes—
273Dic61C22×Q1611yescharacteristic
274Dic61C2×#13511yescharacteristic
275#3104—611no—
276Q8×M16—611no—
277#2962—611no—
278#1591—611no—
279#1591—611no—
280#1584—611no—
281#1584—611no—
282#1584—611no—
283#1584—611no—
284C8×Q16—611no—
285C8×Q16—611no—
286#1571—611no—
287#1493—611no—
288C2×Q8⋊C8—611no—
289C8×M16—611no—
290C2×C61—22no—
291C2×C61#293311yescharacteristic
292C2×C61C2×C4×Q1611yescharacteristic
293C2×C61#295611yescharacteristic
294#533—611no—
295C2×C4×Q8—611no—
296#484—611no—
297#484—611no—
298#477—611no—
299#477—611no—
300#462—611no—
301#462—611no—
302#451—611no—
303#451#692311yescharacteristic
304#446—611no—
305#446—611no—
306#446—611no—
307#446—611no—
308#446—1221no—
309#446—1221no—
310#446—1221no—
311#446—1221no—
312C8×Q8—611no—
313C8×Q8—611no—
314C8×Q8—611no—
315C8×Q8—611no—
316C4×Q16—611no—
317C4×Q16—611no—
318C4×Q16—611no—
319C4×Q16—611no—
320#431—611no—
321#424—1221no—
322#423—611no—
323#423—1221no—
324C2×C4⋊C8—611no—
325#405—1221no—
326C4×M16—611no—
327C4×M16—1221no—
328C2×C4×C8—611no—
329#335—611no—
330#335—611no—
331#335—611no—
332#335—611no—
333Q8⋊C8—611no—
334Q8⋊C8—611no—
335Q8⋊C8—611no—
336Q8⋊C8—611no—
337Q8⋊C8—611no—
338Q8⋊C8—611no—
339Q8⋊C8—611no—
340Q8⋊C8—611no—
341#322—611no—
342#322—611no—
343C82—611no—
344C82—611no—
345C61#1799811yescharacteristic, complemented, Sylow
346C22×Q8C2×#692311yescharacteristic
347C2×Q16—611no—
348C2×Q16—611no—
349C2×Q16C2×#692311yescharacteristic
350C2×Q16C2×#692311yescharacteristic
351C2×M16—611no—
352C2×M16—611no—
353C2×M16—1221no—
354C22×C8—1221no—
355C22×C8—1221no—
356C4⋊Q8—611no—
357C4⋊Q8—611no—
358#125—1221no—
359C22⋊Q8—611no—
360C22⋊Q8—1221no—
361C22⋊Q8C2×#692311yescharacteristic
362C4×Q8—611no—
363C4×Q8—611no—
364C4×Q8—611no—
365C4×Q8—611no—
366C4×Q8—611no—
367C4×Q8—611no—
368C4×Q8—1221no—
369C4×Q8—1221no—
370C4×Q8—1221no—
371#117—1221no—
372C2×C4⋊C4—611no—
373C2×C4⋊C4—1221no—
374C2×C42—611no—
375C2×C42—1221no—
376#107—611no—
377#107—611no—
378#107—611no—
379#107—611no—
380C4⋊C8—611no—
381C4⋊C8—611no—
382C4⋊C8—611no—
383C4⋊C8—611no—
384C4⋊C8—611no—
385C4⋊C8—611no—
386C4⋊C8—1221no—
387C4⋊C8—1221no—
388C4⋊C8—1221no—
389C4⋊C8—1221no—
390C4⋊C8—1221no—
391C4⋊C8—1221no—
392C4⋊C8—1221no—
393C4⋊C8—1221no—
394C4⋊C8—1221no—
395C4⋊C8—1221no—
396#103—611no—
397#103—611no—
398#103—611no—
399#103—611no—
400#103—611no—
401#103—611no—
402#103C2×#692311yescharacteristic
403#103C2×#692311yescharacteristic
404C22⋊C8—611no—
405C22⋊C8C2×#692311yescharacteristic
406#97—611no—
407#97—611no—
408#97—1221no—
409#97—1221no—
410#97—1221no—
411#97—1221no—
412C4×C8—611no—
413C4×C8—611no—
414C4×C8—611no—
415C4×C8—611no—
416C4×C8—611no—
417C4×C8—611no—
418C4×C8—1221no—
419C4×C8—1221no—
420C4×C8—1221no—
421C4×C8—1221no—
422C4×C8—1221no—
423C2×Q8—21no—
424C2×Q8—22no—
425C2×Q8—41no—
426C2×Q8—611no—
427C2×Q8—1221no—
428C2×Q8C22×#692311yescharacteristic
429C2×Q8C22×#692311yescharacteristic
430C2×Q8C22×#692311yescharacteristic
431C22×C4—21no—
432C22×C4—611no—
433C22×C4—611no—
434C22×C4—1221no—
435C22×C4C22×#692311yescharacteristic
436Q16—22no—
437Q16—41no—
438Q16—1222no—
439Q16—2441no—
440M16—614no—
441M16—1222no—
442M16—2442no—
443C2×C8—611no—
444C2×C8—611no—
445C2×C8—611no—
446C2×C8—611no—
447C2×C8—611no—
448C2×C8—611no—
449C2×C8—1221no—
450C2×C8—1221no—
451C2×C8—1221no—
452C2×C8—1221no—
453C2×C8—1221no—
454C2×C8—1221no—
455C2×C8—2442no—
456C2×C8—2442no—
457C2×C8C22×#692311yescharacteristic
458C2×C8C22×#692311yescharacteristic
459C4⋊C4—21no—
460C4⋊C4—611no—
461C4⋊C4—611no—
462C4⋊C4—611no—
463C4⋊C4—1221no—
464C4⋊C4—1221no—
465C4⋊C4—1221no—
466C4⋊C4—1221no—
467C4⋊C4—1221no—
468C4⋊C4—1221no—
469C4⋊C4—1221no—
470C4⋊C4—1221no—
471C4⋊C4—1221no—
472C4⋊C4—1221no—
473C4⋊C4—1221no—
474C4⋊C4C22×#692311yescharacteristic
475C22⋊C4—21no—
476C22⋊C4—1221no—
477C42—611no—
478C42—611no—
479C42—611no—
480C42—611no—
481C42—1221no—
482C42—1221no—
483C42—1221no—
484C42—1221no—
485C42—1221no—
486C42—1221no—
487C23D4×#692311yescharacteristic
488Q8—21no—
489Q8—21no—
490Q8—21no—
491Q8—21no—
492Q8—21no—
493Q8—42no—
494Q8—1221no—
495Q8—1221no—
496Q8—4881no—
497Q8#1451986412yes—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—42no—
505C2×C4—612no—
506C2×C4—612no—
507C2×C4—611no—
508C2×C4—611no—
509C2×C4—611no—
510C2×C4—611no—
511C2×C4—1221no—
512C2×C4—1221no—
513C2×C4—1221no—
514C2×C4—1221no—
515C2×C4—4881no—
516C2×C4—2441no—
517C2×C4D4×#692311yescharacteristic
518C2×C4C23×#692311yescharacteristic
519C8—21no—
520C8—21no—
521C8—612no—
522C8—612no—
523C8—1221no—
524C8—1221no—
525C8—1222no—
526C8—1221no—
527C8—1221no—
528C8—2441no—
529C8—2441no—
530C8—2441no—
531C8—2441no—
532C8—2441no—
533C22—21no—
534C22—21no—
535C22Q16×#692312yes—
536C22C2×D4×#692311yescharacteristic
537C4—21no—
538C4—22no—
539C4—41no—
540C4—41no—
541C4—41no—
542C4—611no—
543C4—611no—
544C4—612no—
545C4—1221no—
546C4—1221no—
547C4—2441no—
548C4—2441no—
549C4—2441no—
550C4—2441no—
551C4C2×#1451986411yescharacteristic
552C4#1684084011yescharacteristic
553C2—22no—
554C2#2146255111yescharacteristic
555C2C2×Q16×#692311yescharacteristic
556C2#2146259411yescharacteristic
subgroupG/H
centerC22C2×D4×#6923
commutatorC2×C4×C61C23×C4
FrattiniC2×C4C23×#6923
FittingC61×#451C4
socleC22×C61C2×C4×D4
prime radicalC23×C61C4×D4
cosocleC4×Dic61C24
hypercenter#451#6923
hypocenterC61#17998
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
3. #21461459C2semidirect
4. #21461457C2semidirect
5. #21461433C2semidirect
6. #21461432C2semidirect
7. #21461432C2semidirect
8. #21461430C2semidirect
9. #21461406C2semidirect
10. Q16×#92420C2semidirect
159. #11758217Q16semidirect
345. C61#17998semidirect

Maximal subgroups

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

subgroupindex
Q8×#117582172
#214626142
#214614592
#214614572
#214614332
#214614322
#214614302
#214614062
Q16×#924202
#214613882
#214611672
C2×Q8⋊#924202
#214610202
#214558532
#1799861