Finite Group Databasebeta

← #29841130

Subgroups — #29841130

Order 15616.

2390 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
1C2×#16840244C211yesmaximal, characteristic
2#21448790C213yesmaximal, complemented
3#21448770C213yesmaximal, complemented
4#21448740C211yesmaximal, characteristic, complemented
5#21448695C213yesmaximal, complemented
6#21448687C213yesmaximal, complemented
7#21448657C211yesmaximal, characteristic
8#21448657C211yesmaximal, characteristic
9#21448458C213yesmaximal, complemented
10#21448427C213yesmaximal, complemented
11#21448312C213yesmaximal, complemented
12#21448277C213yesmaximal, complemented
13#21448184C211yesmaximal, characteristic, complemented
14C4×Q8×Dic61C211yesmaximal, characteristic, complemented
15Q8×C4⋊C4×C61C211yesmaximal, characteristic, complemented
16C22×#11758226C2211yescharacteristic
17#16840244C4116yescomplemented
18C2×Q8×Dic61C2211yescharacteristic
19C2×Q8×Dic61C2211yescharacteristic
20C2×Q8×Dic61C2211yescharacteristic
21C2×C4×D244C2213yes—
22C2×#14519739C2213yes—
23C2×#14519735C2216yes—
24C2×#14519734C2213yes—
25C2×C4⋊C4×D61C2213yes—
26#16839664C2213yes—
27#16839664C2213yes—
28#16839663C2213yes—
29#16839663C2213yes—
30#16839661C2213yes—
31#16839661C2213yes—
32#16839654C2213yes—
33#16839654C2213yes—
34#16839652C2216yes—
35#16839652C2213yes—
36#16839652C2216yes—
37#16839652C2213yes—
38#16839649C2213yes—
39#16839649C2213yes—
40#16839643C2211yescharacteristic
41#16839643C2213yes—
42#16839643C2216yes—
43#16839628C2213yes—
44#16839624C2216yes—
45#16839619C2216yes—
46#16839619C2213yes—
47#16839619C2213yes—
48C4⋊C4×Dic61C2213yes—
49C4⋊C4×Dic61C2213yes—
50C4⋊C4×Dic61C2213yes—
51C4⋊C4×Dic61C2213yes—
52C4×C4⋊Dic61C2213yes—
53C4×C4⋊Dic61C2213yes—
54C4×C4⋊Dic61C2213yes—
55C4×#11758198C2213yes—
56C4×#11758198C2213yes—
57#16839609C2216yes—
58#16839608C2213yes—
59C42×Dic61C2213yes—
60C2×C4×Q8×C61C2211yescharacteristic
61C2×C4×Q8×C61C2211yescharacteristic
62C2×C4×Q8×C61C2211yescharacteristic
63C61×#389C2213yes—
64C61×#389C2213yes—
65C61×#384C2213yes—
66C4×C4⋊C4×C61C2213yes—
67C2×#11758226—24no—
68C2×#11758226—24no—
69C2×#11758226C2×C418yes—
70C22×D244C2313yes—
71C22×C4×D61C2313yes—
72C22×C4×D61C2316yes—
73Q8×Dic61—24no—
74Q8×Dic61—24no—
75Q8×Dic61C2×C418yes—
76C4×D244C2×C4124yes—
77#14519739C2×C4124yes—
78#14519735—412no—
79#14519734C2×C4124yes—
80C2×C4⋊Dic61C2313yes—
81C2×C4⋊Dic61C2313yes—
82C2×C4⋊Dic61C2313yes—
83C2×C4⋊Dic61C2313yes—
84C2×C4⋊Dic61C2313yes—
85C2×#11758196C2313yes—
86C2×#11758196C2311yescharacteristic
87C4⋊C4×D61C2×C4124yes—
88C2×#11758198C2313yes—
89C2×#11758198C2313yes—
90C2×#11758198C2316yes—
91C2×#11758198C2316yes—
92C2×#11758198C2313yes—
93C2×#11758198C2313yes—
94C2×#11758198C2313yes—
95C2×#11758198C2313yes—
96C2×C4×Dic61C2313yes—
97C2×C4×Dic61C2313yes—
98C2×C4×Dic61C2313yes—
99C2×C4×Dic61C2311yescharacteristic
100C2×C4×Dic61C2316yes—
101C2×C4×Dic61C2313yes—
102C2×C4×Dic61C2313yes—
103C2×C4×Dic61C2313yes—
104#14519684C2311yescharacteristic
105#14519684C2313yes—
106#14519684C2313yes—
107#14519684C2313yes—
108#14519684C2316yes—
109#14519684C2311yescharacteristic
110#14519684C2311yescharacteristic
111#14519684C2313yes—
112#14519684C2313yes—
113#14519684C2311yescharacteristic
114#14519684C2316yes—
115#14519684C2313yes—
116C22×Q8×C61C2311yescharacteristic
117C4×Q8×C61—24no—
118C4×Q8×C61—24no—
119C4×Q8×C61C2×C418yes—
120C2×C4⋊C4×C61C2311yescharacteristic
121C2×C4⋊C4×C61C2313yes—
122C2×C4⋊C4×C61C2313yes—
123C2×C4⋊C4×C61C2316yes—
124C2×C4⋊C4×C61C2313yes—
125C2×C4⋊C4×C61C2313yes—
126C2×C42×C61C2313yes—
127C2×C42×C61C2313yes—
128C2×C42×C61C2313yes—
129#95×C61C2313yes—
130#95×C61C2313yes—
131C23×D61C2413yes—
132C2×Q8×C61C22×C414yes—
133C2×Q8×C61#4114yes—
134C2×Q8×C61#4114yes—
135C22×C4×C61C2411yescharacteristic
136C22×C4×C61C2413yes—
137C22×C4×C61C2413yes—
138C22×C4×C61C2413yes—
139C22×C4×C61C2411yescharacteristic
140C22×C4×C61C2411yescharacteristic
141C22×C4×C61C2413yes—
142C22×Dic61C2411yescharacteristic
143C22×Dic61C2413yes—
144C22×Dic61C2413yes—
145C22×Dic61C2411yescharacteristic
146C22×Dic61C2413yes—
147C22×Dic61C2411yescharacteristic
148C22×Dic61C2411yescharacteristic
149#11758226—232no—
150C2×D244—26no—
151C2×D244—26no—
152C2×D244C22×C4112yes—
153C2×C4×D61—26no—
154C2×C4×D61—26no—
155C2×C4×D61—46no—
156C2×C4×D61—46no—
157C2×C4×D61C22×C4112yes—
158C2×C4×D61C22×C4124yes—
159C4⋊C4×C61—26no—
160C4⋊C4×C61—26no—
161C4⋊C4×C61—22no—
162C4⋊C4×C61—43no—
163C4⋊C4×C61—46no—
164C4⋊C4×C61C22×C4112yes—
165C42×C61—26no—
166C42×C61—26no—
167C42×C61C22×C4112yes—
168#11758198—26no—
169#11758198—26no—
170#11758198—43no—
171#11758198—46no—
172#11758198—43no—
173#11758198—46no—
174#11758198C22×C4112yes—
175#11758198C22×C4112yes—
176C4⋊Dic61—26no—
177C4⋊Dic61—26no—
178C4⋊Dic61—26no—
179C4⋊Dic61—43no—
180C4⋊Dic61C22×C4112yes—
181#11758196C22×C4112yes—
182#11758196C22×C414yes—
183C4×Dic61—26no—
184C4×Dic61—26no—
185C4×Dic61—22no—
186C4×Dic61—26no—
187C4×Dic61—43no—
188C4×Dic61—43no—
189C4×Dic61—46no—
190C4×Dic61C22×C4112yes—
191C23×C61C2511yescharacteristic
192C22×D61—23no—
193C22×D61—23no—
194C22×D61—23no—
195C22×D61—23no—
196C22×D61—23no—
197C22×D61—23no—
198C22×D61C23×C416yes—
199Q8×C61#117116yescomplemented
200C2×C4×C61—21no—
201C2×C4×C61—23no—
202C2×C4×C61—21no—
203C2×C4×C61—21no—
204C2×C4×C61—23no—
205C2×C4×C61—21no—
206C2×C4×C61—21no—
207C2×C4×C61—23no—
208C2×C4×C61—23no—
209C2×C4×C61—23no—
210C2×C4×C61—23no—
211C2×C4×C61—21no—
212C2×C4×C61—23no—
213C2×C4×C61—21no—
214C2×C4×C61—23no—
215C2×C4×C61C23×C412yes—
216C2×C4×C61C23×C416yes—
217C2×C4×C61C23×C416yes—
218C2×C4×C61C2×#4116yes—
219C2×C4×C61C2×#4116yes—
220C2×C4×C612−1+412yes—
221C2×Dic61—21no—
222C2×Dic61—21no—
223C2×Dic61—21no—
224C2×Dic61—23no—
225C2×Dic61—21no—
226C2×Dic61—21no—
227C2×Dic61—21no—
228C2×Dic61—23no—
229C2×Dic61—21no—
230C2×Dic61—23no—
231C2×Dic61—23no—
232C2×Dic61—23no—
233C2×Dic61—23no—
234C2×Dic61—23no—
235C2×Dic61—23no—
236C2×Dic61—21no—
237C2×Dic61—21no—
238C2×Dic61C23×C412yes—
239C2×Dic61C23×C416yes—
240C2×Dic61C23×C412yes—
241C2×Dic61H(5,𝔽2)12yes—
242D244—224no—
243C4×D61—224no—
244C4×D61—424no—
245#34263—611nomaximal, Sylow
246C22×C61C24×C411yescharacteristic
247C22×C61C22×#4111yescharacteristic
248C22×C61C22×#4111yescharacteristic
249C22×C61C2×H(5,𝔽2)11yescharacteristic
250C22×C61C2×2−1+411yescharacteristic
251C22×C61#58511yescharacteristic
252C22×C61#58511yescharacteristic
253C2×D61—26no—
254C2×D61—26no—
255C2×D61—26no—
256C2×D61—43no—
257C2×D61—43no—
258C2×D61—43no—
259C2×D61—43no—
260C4×C61—26no—
261C4×C61—22no—
262C4×C61—22no—
263C4×C61—41no—
264C4×C61—43no—
265C4×C61—43no—
266C4×C61C2×#117112yes—
267Dic61—22no—
268Dic61—26no—
269Dic61—22no—
270Dic61—22no—
271Dic61—43no—
272Dic61—41no—
273Dic61—43no—
274C2×#520—611no—
275#2357—613no—
276#2354—613no—
277#2351—611no—
278Q8×C4⋊C4D6111yescharacteristic, complemented
279#2345—613no—
280#2344—613no—
281#2341—611no—
282#2341—611no—
283#2319—613no—
284#2314—613no—
285#2298—613no—
286#2293—613no—
287#2276—611no—
288C42×Q8—611no—
289C2×C61C22×#11711yescharacteristic
290C2×C61C4×H(5,𝔽2)11yescharacteristic
291C2×C61C4×2−1+411yescharacteristic
292C2×C61#351611yescharacteristic
293C2×C61#351711yescharacteristic
294C2×C61#352211yescharacteristic
295C2×C61#352411yescharacteristic
296D61—46no—
297C22×#41—611no—
298#520—6116no—
299C2×C4×Q8—611no—
300C2×C4×Q8—611no—
301C2×C4×Q8—611no—
302C2×C4×Q8C2×D6111yescharacteristic
303C2×C4×Q8C2×D6111yescharacteristic
304C2×C4×Q8C2×D6111yescharacteristic
305C2×C4×D4—613no—
306C2×C4×D4—613no—
307C2×#117—616no—
308C2×#117—613no—
309C22×C4⋊C4—613no—
310#390—613no—
311#390—613no—
312#390—613no—
313#390—613no—
314#389—613no—
315#389—613no—
316#389C2×D6113yes—
317#389C2×D6113yes—
318#388—616no—
319#388—616no—
320#388—613no—
321#388—613no—
322#388—613no—
323#388—613no—
324#387—613no—
325#387—616no—
326#387—611no—
327#387—613no—
328#387—613no—
329#384C2×D6113yes—
330#382—613no—
331#381—616no—
332#380—613no—
333#380—613no—
334#380—616no—
335C4×C4⋊C4—613no—
336C4×C4⋊C4—613no—
337C4×C4⋊C4—613no—
338C4×C4⋊C4—613no—
339C4×C4⋊C4—613no—
340C4×C4⋊C4—613no—
341C4×C4⋊C4—613no—
342C4×C4⋊C4C2×D6113yes—
343C4×C22⋊C4—613no—
344C4×C22⋊C4—613no—
345C4×C22⋊C4—616no—
346#376—613no—
347C43—613no—
348C61#3426311yescharacteristic, complemented, Sylow
349C2×#41—618no—
350C2×#41—1224no—
351C2×#41—1224no—
352C22×Q8C22×D6111yescharacteristic
353C22×D4—613no—
354C23×C4—613no—
355C23×C4—616no—
356C4×Q8—24no—
357C4×Q8—24no—
358C4×Q8—618no—
359C4×Q8—1224no—
360C4×Q8—1224no—
361C4×Q8C4×D6118yes—
362C4×D4—6124no—
363C4×D4—6124no—
364#117—6124no—
365#117—24412no—
366C2×C4⋊C4—613no—
367C2×C4⋊C4—613no—
368C2×C4⋊C4—611no—
369C2×C4⋊C4—6124no—
370C2×C4⋊C4—613no—
371C2×C4⋊C4—613no—
372C2×C4⋊C4—613no—
373C2×C4⋊C4—613no—
374C2×C4⋊C4C22×D6113yes—
375C2×C4⋊C4C22×D6113yes—
376C2×C4⋊C4C22×D6111yescharacteristic
377C2×C4⋊C4C22×D6116yes—
378C2×C4⋊C4C22×D6113yes—
379C2×C4⋊C4C22×D6113yes—
380C2×C22⋊C4—613no—
381C2×C22⋊C4—613no—
382C2×C22⋊C4—613no—
383C2×C22⋊C4—613no—
384C2×C22⋊C4—616no—
385C2×C22⋊C4—613no—
386C2×C22⋊C4—616no—
387C2×C22⋊C4—613no—
388C2×C42—613no—
389C2×C42—613no—
390C2×C42—613no—
391C2×C42—613no—
392C2×C42—611no—
393C2×C42—616no—
394C2×C42—613no—
395C2×C42—613no—
396C2×C42C22×D6113yes—
397C2×C42C22×D6113yes—
398C2×C42C22×D6113yes—
399#95—611no—
400#95—613no—
401#95—613no—
402#95—613no—
403#95—611no—
404#95—611no—
405#95—613no—
406#95—613no—
407#95—616no—
408#95—616no—
409#95—611no—
410#95—613no—
411#95C22×D6113yes—
412#95C22×D6113yes—
413C24—613no—
414#41—12232no—
415C2×Q8C2×C4×D6114yes—
416C2×Q8#1175822414yes—
417C2×Q8#1175822614yes—
418C2×D4—6112no—
419C2×D4—1226no—
420C2×D4—1226no—
421C22×C4—611no—
422C22×C4—611no—
423C22×C4—613no—
424C22×C4—611no—
425C22×C4—6112no—
426C22×C4—6124no—
427C22×C4—611no—
428C22×C4—613no—
429C22×C4—613no—
430C22×C4—1226no—
431C22×C4—1226no—
432C22×C4—2446no—
433C22×C4—2446no—
434C22×C4C23×D6111yescharacteristic
435C22×C4C23×D6113yes—
436C22×C4C23×D6113yes—
437C22×C4C23×D6111yescharacteristic
438C22×C4C23×D6113yes—
439C22×C4C23×D6111yescharacteristic
440C22×C4C23×D6113yes—
441C4⋊C4—26no—
442C4⋊C4—26no—
443C4⋊C4—22no—
444C4⋊C4—46no—
445C4⋊C4—43no—
446C4⋊C4—6112no—
447C4⋊C4—6112no—
448C4⋊C4—614no—
449C4⋊C4—1226no—
450C4⋊C4—1226no—
451C4⋊C4—1226no—
452C4⋊C4—2443no—
453C4⋊C4C2×C4×D61112yes—
454C22⋊C4—6112no—
455C22⋊C4—6112no—
456C22⋊C4—1226no—
457C22⋊C4—1226no—
458C22⋊C4—2446no—
459C22⋊C4—2443no—
460C22⋊C4—2446no—
461C22⋊C4—2443no—
462C42—26no—
463C42—26no—
464C42—6112no—
465C42—1226no—
466C42—1226no—
467C42—1226no—
468C42—1222no—
469C42—2443no—
470C42—2443no—
471C42—2446no—
472C42C2×C4×D61112yes—
473C23—616no—
474C23—1223no—
475C23—1223no—
476C23—1223no—
477C23—1223no—
478C23—1223no—
479C23—1223no—
480C23C24×D6111yescharacteristic
481Q8#14519735116yescomplemented
482D4—12224no—
483C2×C4—21no—
484C2×C4—23no—
485C2×C4—21no—
486C2×C4—21no—
487C2×C4—21no—
488C2×C4—23no—
489C2×C4—21no—
490C2×C4—23no—
491C2×C4—23no—
492C2×C4—23no—
493C2×C4—23no—
494C2×C4—21no—
495C2×C4—23no—
496C2×C4—23no—
497C2×C4—21no—
498C2×C4—612no—
499C2×C4—616no—
500C2×C4—612no—
501C2×C4—612no—
502C2×C4—1221no—
503C2×C4—1221no—
504C2×C4—1221no—
505C2×C4—1223no—
506C2×C4—1221no—
507C2×C4—1221no—
508C2×C4—1221no—
509C2×C4—1223no—
510C2×C4—12224no—
511C2×C4—1221no—
512C2×C4—1223no—
513C2×C4—1223no—
514C2×C4—1223no—
515C2×C4—1223no—
516C2×C4—1223no—
517C2×C4—1221no—
518C2×C4—1223no—
519C2×C4—1221no—
520C2×C4—24424no—
521C2×C4C22×C4×D6112yes—
522C2×C4C22×C4×D6116yes—
523C2×C4C22×C4×D6116yes—
524C2×C4C2×#1175822416yes—
525C2×C4C2×#1175822616yes—
526C2×C4#1451982512yes—
527C22—1226no—
528C22—1226no—
529C22—1226no—
530C22—2443no—
531C22—2443no—
532C22—2443no—
533C22—2443no—
534C22C23×C4×D6111yescharacteristic
535C22C22×#1175822411yescharacteristic
536C22C22×#1175822611yescharacteristic
537C22H(5,𝔽2)×D6111yescharacteristic
538C22C2×#1451982511yescharacteristic
539C22#1684059811yescharacteristic
540C22#1684059911yescharacteristic
541C4—26no—
542C4—22no—
543C4—22no—
544C4—41no—
545C4—43no—
546C4—43no—
547C4—1222no—
548C4—1226no—
549C4—1222no—
550C4—1222no—
551C4—2443no—
552C4—2441no—
553C4—2443no—
554C4C2×#14519735112yes—
555C2—2446no—
556C2C22×#1451973511yescharacteristic
557C2#2145902511yescharacteristic
558C2C4×#1451982511yescharacteristic
559C2#2145981311yescharacteristic
560C2#2145981611yescharacteristic
561C2#2145990511yescharacteristic
562C2#2145993111yescharacteristic
subgroupG/H
center, FrattiniC23C24×D61
commutatorC22×C61C24×C4
FittingQ8×C4⋊C4×C61C2
socle, cosocleC23×C61C25
prime radicalC22×#11758226C22
hypercenterQ8×C4⋊C4D61
hypocenterC61#34263
LCSG ▷ C22×C61 ▷ C61
UCSC1 ◁ C23 ◁ Q8×C4⋊C4

Complements

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

NH ≅ G/Ntype
2. #21448790C2semidirect
3. #21448770C2semidirect
4. #21448740C2semidirect
5. #21448695C2semidirect
6. #21448687C2semidirect
9. #21448458C2semidirect
10. #21448427C2semidirect
11. #21448312C2semidirect
12. #21448277C2semidirect
13. #21448184C2semidirect
14. C4×Q8×Dic61C2semidirect
15. Q8×C4⋊C4×C61C2semidirect
17. #16840244C4semidirect
199. Q8×C61#117semidirect
278. Q8×C4⋊C4D61semidirect
348. C61#34263semidirect
481. Q8#14519735semidirect

Maximal subgroups

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

subgroupindex
C2×#168402442
#214487902
#214487702
#214487402
#214486952
#214486872
#214486572
#214484582
#214484272
#214483122
#214482772
#214481842
C4×Q8×Dic612
Q8×C4⋊C4×C612
#3426361