Finite Group Databasebeta

← #29841115

Subgroups — #29841115

Order 15616.

2198 conjugacy classes of subgroups.

The subgroup lattice has 552 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840238C211yesmaximal, characteristic
2#21448786C213yesmaximal, complemented
3#21448774C213yesmaximal, complemented
4Q8×C4⋊Dic61C211yesmaximal, characteristic, complemented
5#21448711C213yesmaximal
6#21448703C213yesmaximal
7Q8×#11758198C211yesmaximal, characteristic
8Q8×#11758198C211yesmaximal, characteristic
9#21448466C213yesmaximal, complemented
10#21448429C213yesmaximal
11#21448312C213yesmaximal
12#21448290C213yesmaximal, complemented
13#21448184C211yesmaximal, characteristic, complemented
14C4×Q8×Dic61C211yesmaximal, characteristic, complemented
15C61×#2351C211yesmaximal, characteristic, complemented
16C22×Q8×D61C2211yescharacteristic
17#16840238C4116yescomplemented
18C2×Q8×Dic61C2211yescharacteristic
19C2×Q8×Dic61C2211yescharacteristic
20C2×Q8×Dic61C2211yescharacteristic
21C2×C4×Dic122C2213yes—
22C2×#14519742C2213yes—
23C2×#14519735C2213yes—
24C2×#14519734C2213yes—
25#16839668C2213yes—
26#16839668C2213yes—
27#16839667C2213yes—
28#16839667C2213yes—
29#16839661C2213yes—
30#16839661C2213yes—
31#16839654C2216yes—
32#16839654C2211yescharacteristic
33#16839654C2213yes—
34#16839652C2213yes—
35#16839652C2213yes—
36#16839649C2216yes—
37#16839649C2213yes—
38#16839649C2216yes—
39#16839649C2213yes—
40#16839643C2213yes—
41#16839643C2213yes—
42#16839632C2213yes—
43#16839629C2216yes—
44#16839627C2216yes—
45#16839619C2213yes—
46#16839619C2213yes—
47C4⋊C4×Dic61C2216yes—
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#16839608C2213yes—
58#16839607C2216yes—
59C42×Dic61C2213yes—
60C2×C4×Q8×C61C2211yescharacteristic
61C2×C4×Q8×C61C2211yescharacteristic
62C2×C4×Q8×C61C2211yescharacteristic
63C61×#389C2213yes—
64C61×#389C2213yes—
65C61×#382C2213yes—
66C4×C4⋊C4×C61C2213yes—
67C22×Dic122C2313yes—
68C2×Q8×D61—24no—
69C2×Q8×D61—24no—
70C2×Q8×D61C2×C418yes—
71C22×C4×D61C2313yes—
72Q8×Dic61—24no—
73Q8×Dic61—24no—
74Q8×Dic61C2×C418yes—
75C4×Dic122C2×C4124yes—
76#14519742C2×C4124yes—
77#14519735C2×C4124yes—
78#14519734C2×C4124yes—
79C2×C4⋊Dic61C2313yes—
80C2×C4⋊Dic61C2313yes—
81C2×C4⋊Dic61C2311yescharacteristic
82C2×C4⋊Dic61C2313yes—
83C2×C4⋊Dic61C2316yes—
84C2×C4⋊Dic61C2313yes—
85C2×#11758196C2313yes—
86C2×#11758196C2313yes—
87C2×#11758198C2311yescharacteristic
88C2×#11758198C2311yescharacteristic
89C2×#11758198C2313yes—
90C2×#11758198C2313yes—
91C2×#11758198C2311yescharacteristic
92C2×#11758198C2313yes—
93C2×C4×Dic61C2313yes—
94C2×C4×Dic61C2313yes—
95C2×C4×Dic61C2311yescharacteristic
96C2×C4×Dic61C2313yes—
97C2×C4×Dic61C2313yes—
98C2×C4×Dic61C2313yes—
99C2×C4×Dic61C2316yes—
100C2×C4×Dic61C2313yes—
101C2×C4×Dic61C2316yes—
102#14519684C2313yes—
103#14519684C2313yes—
104#14519684C2316yes—
105#14519684C2311yescharacteristic
106#14519684C2313yes—
107#14519684C2313yes—
108#14519684C2313yes—
109#14519684C2313yes—
110#14519684C2313yes—
111#14519684C2313yes—
112#14519684C2316yes—
113#14519684C2313yes—
114#14519684C2316yes—
115C22×Q8×C61C2311yescharacteristic
116C4×Q8×C61—24no—
117C4×Q8×C61—24no—
118C4×Q8×C61C2×C418yes—
119C2×C4⋊C4×C61C2313yes—
120C2×C4⋊C4×C61C2313yes—
121C2×C4⋊C4×C61C2313yes—
122C2×C4⋊C4×C61C2313yes—
123C2×C4⋊C4×C61C2313yes—
124C2×C42×C61C2313yes—
125C2×C42×C61C2313yes—
126C2×C42×C61C2313yes—
127#95×C61C2311yescharacteristic
128#95×C61C2313yes—
129#95×C61C2316yes—
130C23×D61C2411yescharacteristic
131C2×Q8×C61C22×C414yes—
132C2×Q8×C61#4114yes—
133C2×Q8×C61#4114yes—
134C22×C4×C61C2413yes—
135C22×C4×C61C2413yes—
136C22×C4×C61C2411yescharacteristic
137C22×C4×C61C2411yescharacteristic
138C22×C4×C61C2413yes—
139C22×C4×C61C2411yescharacteristic
140C22×C4×C61C2413yes—
141C22×Dic61C2411yescharacteristic
142C22×Dic61C2413yes—
143C22×Dic61C2413yes—
144C22×Dic61C2413yes—
145C22×Dic61C2411yescharacteristic
146C22×Dic61C2413yes—
147C22×Dic61C2411yescharacteristic
148Q8×D61—232no—
149C2×C4×D61—26no—
150C2×C4×D61—26no—
151C2×C4×D61C22×C4112yes—
152C2×Dic122—26no—
153C2×Dic122—26no—
154C2×Dic122C22×C4112yes—
155C4⋊C4×C61—26no—
156C4⋊C4×C61—26no—
157C4⋊C4×C61—26no—
158C4⋊C4×C61—43no—
159C4⋊C4×C61C22×C4112yes—
160C42×C61—26no—
161C42×C61—26no—
162C42×C61C22×C4112yes—
163#11758198—22no—
164#11758198—22no—
165#11758198—43no—
166#11758198—43no—
167#11758198C22×C414yes—
168#11758198C22×C4112yes—
169C4⋊Dic61—26no—
170C4⋊Dic61—26no—
171C4⋊Dic61—22no—
172C4⋊Dic61—43no—
173C4⋊Dic61—46no—
174C4⋊Dic61C22×C4112yes—
175#11758196C22×C4112yes—
176#11758196C22×C4112yes—
177C4×Dic61—26no—
178C4×Dic61—26no—
179C4×Dic61—22no—
180C4×Dic61—26no—
181C4×Dic61—46no—
182C4×Dic61—43no—
183C4×Dic61—43no—
184C4×Dic61C22×C4112yes—
185C4×Dic61C22×C4124yes—
186C23×C61C2511yescharacteristic
187C22×D61—21no—
188C22×D61—21no—
189C22×D61—21no—
190C22×D61—21no—
191C22×D61—21no—
192C22×D61—21no—
193C22×D61C23×C412yes—
194Q8×C61#117116yes—
195C2×C4×C61—21no—
196C2×C4×C61—21no—
197C2×C4×C61—23no—
198C2×C4×C61—21no—
199C2×C4×C61—21no—
200C2×C4×C61—23no—
201C2×C4×C61—21no—
202C2×C4×C61—23no—
203C2×C4×C61—23no—
204C2×C4×C61—23no—
205C2×C4×C61—21no—
206C2×C4×C61—23no—
207C2×C4×C61—23no—
208C2×C4×C61—23no—
209C2×C4×C61—21no—
210C2×C4×C61C23×C416yes—
211C2×C4×C61C23×C412yes—
212C2×C4×C61C23×C416yes—
213C2×C4×C61C2×#4116yes—
214C2×C4×C61C2×#4116yes—
215C2×C4×C612−1+412yes—
216C2×Dic61—21no—
217C2×Dic61—23no—
218C2×Dic61—23no—
219C2×Dic61—23no—
220C2×Dic61—21no—
221C2×Dic61—21no—
222C2×Dic61—21no—
223C2×Dic61—23no—
224C2×Dic61—21no—
225C2×Dic61—23no—
226C2×Dic61—23no—
227C2×Dic61—21no—
228C2×Dic61—21no—
229C2×Dic61—23no—
230C2×Dic61—23no—
231C2×Dic61—23no—
232C2×Dic61—23no—
233C2×Dic61C23×C412yes—
234C2×Dic61C23×C416yes—
235C2×Dic61C23×C416yes—
236C2×Dic612−1+412yes—
237C4×D61—224no—
238Dic122—224no—
239#34262—611nomaximal, Sylow
240C22×C61C24×C411yescharacteristic
241C22×C61C22×#4111yescharacteristic
242C22×C61C22×#4111yescharacteristic
243C22×C61C2×2−1+411yescharacteristic
244C22×C61C2×2−1+411yescharacteristic
245C22×C61#58511yescharacteristic
246C22×C61#58511yescharacteristic
247C2×D61—22no—
248C2×D61—22no—
249C2×D61—22no—
250C2×D61—41no—
251C2×D61—41no—
252C2×D61—41no—
253C2×D61—41no—
254C4×C61—22no—
255C4×C61—26no—
256C4×C61—22no—
257C4×C61—43no—
258C4×C61—41no—
259C4×C61—43no—
260C4×C61C2×#117112yes—
261Dic61—26no—
262Dic61—26no—
263Dic61—22no—
264Dic61—22no—
265Dic61—41no—
266Dic61—43no—
267Dic61—43no—
268C2×#519—611no—
269#2356—613no—
270#2355—613no—
271#2351D6111yescharacteristic, complemented
272Q8×C4⋊C4—611no—
273#2347—613no—
274#2346—613no—
275Q8×C22⋊C4—611no—
276Q8×C22⋊C4—611no—
277#2320—613no—
278#2314—613no—
279#2298—613no—
280#2295—613no—
281#2276—611no—
282C42×Q8—611no—
283C2×C61C22×#11711yescharacteristic
284C2×C61C4×2−1+411yescharacteristic
285C2×C61C4×2−1+411yescharacteristic
286C2×C61#351611yescharacteristic
287C2×C61#351811yescharacteristic
288C2×C61#352311yescharacteristic
289C2×C61#352411yescharacteristic
290D61—42no—
291C23×Q8—611no—
292#519—6116no—
293C2×C4×Q8—611no—
294C2×C4×Q8—611no—
295C2×C4×Q8—611no—
296C2×C4×Q8—613no—
297C2×C4×Q8—613no—
298C2×C4×Q8C2×D6111yescharacteristic
299C2×C4×Q8C2×D6111yescharacteristic
300C2×C4×Q8C2×D6111yescharacteristic
301C2×#117—613no—
302C2×#117—613no—
303#391—613no—
304#391—613no—
305#391—613no—
306#391—613no—
307#389—613no—
308#389—613no—
309#389C2×D6113yes—
310#389C2×D6113yes—
311#388—613no—
312#388—613no—
313#388—613no—
314#388—616no—
315#388—611no—
316#387—613no—
317#387—613no—
318#387—616no—
319#387—616no—
320#387—613no—
321#387—613no—
322#384—613no—
323#383—616no—
324#382—616no—
325#382C2×D6113yes—
326#380—613no—
327#380—613no—
328C4×C4⋊C4—613no—
329C4×C4⋊C4—613no—
330C4×C4⋊C4—613no—
331C4×C4⋊C4—616no—
332C4×C4⋊C4—613no—
333C4×C4⋊C4—613no—
334C4×C4⋊C4—613no—
335C4×C4⋊C4—613no—
336C4×C4⋊C4C2×D6113yes—
337C4×C22⋊C4—613no—
338C4×C22⋊C4—613no—
339#376—613no—
340#376—616no—
341C43—613no—
342C61#3426211yescharacteristic, complemented, Sylow
343C22×Q8—618no—
344C22×Q8—613no—
345C22×Q8—1224no—
346C22×Q8—1224no—
347C22×Q8C22×D6111yescharacteristic
348C23×C4—613no—
349C4×Q8—24no—
350C4×Q8—24no—
351C4×Q8—618no—
352C4×Q8—6124no—
353C4×Q8—6124no—
354C4×Q8—1224no—
355C4×Q8—1224no—
356C4×Q8C4×D6118yes—
357#117—6124no—
358#117—6124no—
359C2×C4⋊C4—613no—
360C2×C4⋊C4—613no—
361C2×C4⋊C4—613no—
362C2×C4⋊C4—613no—
363C2×C4⋊C4—613no—
364C2×C4⋊C4—611no—
365C2×C4⋊C4—616no—
366C2×C4⋊C4—613no—
367C2×C4⋊C4C22×D6113yes—
368C2×C4⋊C4C22×D6113yes—
369C2×C4⋊C4C22×D6113yes—
370C2×C4⋊C4C22×D6113yes—
371C2×C4⋊C4C22×D6113yes—
372C2×C22⋊C4—613no—
373C2×C22⋊C4—611no—
374C2×C22⋊C4—611no—
375C2×C22⋊C4—611no—
376C2×C22⋊C4—613no—
377C2×C22⋊C4—613no—
378C2×C42—613no—
379C2×C42—616no—
380C2×C42—613no—
381C2×C42—613no—
382C2×C42—611no—
383C2×C42—613no—
384C2×C42—613no—
385C2×C42—613no—
386C2×C42—616no—
387C2×C42C22×D6113yes—
388C2×C42C22×D6113yes—
389C2×C42C22×D6113yes—
390#95—611no—
391#95—613no—
392#95—613no—
393#95—613no—
394#95—613no—
395#95—613no—
396#95—616no—
397#95—613no—
398#95—613no—
399#95—613no—
400#95—616no—
401#95—616no—
402#95—613no—
403#95C22×D6116yes—
404#95C22×D6111yescharacteristic
405#95C22×D6113yes—
406C24—611no—
407C2×Q8—6112no—
408C2×Q8—12232no—
409C2×Q8—1226no—
410C2×Q8—1226no—
411C2×Q8C2×C4×D6114yes—
412C2×Q8#1175822414yes—
413C2×Q8#1175822614yes—
414C22×C4—613no—
415C22×C4—611no—
416C22×C4—613no—
417C22×C4—611no—
418C22×C4—6112no—
419C22×C4—611no—
420C22×C4—613no—
421C22×C4—613no—
422C22×C4—1226no—
423C22×C4—1226no—
424C22×C4C23×D6111yescharacteristic
425C22×C4C23×D6113yes—
426C22×C4C23×D6113yes—
427C22×C4C23×D6113yes—
428C22×C4C23×D6111yescharacteristic
429C22×C4C23×D6111yescharacteristic
430C22×C4C23×D6113yes—
431C4⋊C4—26no—
432C4⋊C4—26no—
433C4⋊C4—26no—
434C4⋊C4—43no—
435C4⋊C4—6112no—
436C4⋊C4—6112no—
437C4⋊C4—6112no—
438C4⋊C4—1222no—
439C4⋊C4—1226no—
440C4⋊C4—1226no—
441C4⋊C4—2446no—
442C4⋊C4—2443no—
443C4⋊C4C2×C4×D61112yes—
444C22⋊C4—6112no—
445C22⋊C4—614no—
446C22⋊C4—1222no—
447C22⋊C4—1222no—
448C22⋊C4—2443no—
449C22⋊C4—2443no—
450C42—26no—
451C42—26no—
452C42—6112no—
453C42—6124no—
454C42—1226no—
455C42—1226no—
456C42—1226no—
457C42—1222no—
458C42—2443no—
459C42—2443no—
460C42—2446no—
461C42C2×C4×D61112yes—
462C23—612no—
463C23—1221no—
464C23—1221no—
465C23—1221no—
466C23—1221no—
467C23—1221no—
468C23—1221no—
469C23C24×D6111yescharacteristic
470Q8—12224no—
471Q8#14519735116yes—
472C2×C4—21no—
473C2×C4—21no—
474C2×C4—23no—
475C2×C4—23no—
476C2×C4—21no—
477C2×C4—21no—
478C2×C4—21no—
479C2×C4—23no—
480C2×C4—23no—
481C2×C4—23no—
482C2×C4—21no—
483C2×C4—23no—
484C2×C4—23no—
485C2×C4—21no—
486C2×C4—23no—
487C2×C4—616no—
488C2×C4—612no—
489C2×C4—616no—
490C2×C4—612no—
491C2×C4—1223no—
492C2×C4—1221no—
493C2×C4—1223no—
494C2×C4—1223no—
495C2×C4—1221no—
496C2×C4—1221no—
497C2×C4—1221no—
498C2×C4—1223no—
499C2×C4—12224no—
500C2×C4—1221no—
501C2×C4—1223no—
502C2×C4—1223no—
503C2×C4—1221no—
504C2×C4—1221no—
505C2×C4—1223no—
506C2×C4—1223no—
507C2×C4—1223no—
508C2×C4—1223no—
509C2×C4C22×C4×D6116yes—
510C2×C4C22×C4×D6112yes—
511C2×C4C22×C4×D6116yes—
512C2×C4C2×#1175822416yes—
513C2×C4C2×#1175822616yes—
514C2×C4#1451982412yes—
515C22—1222no—
516C22—1222no—
517C22—1222no—
518C22—2441no—
519C22—2441no—
520C22—2441no—
521C22—2441no—
522C22C23×C4×D6111yescharacteristic
523C22C22×#1175822411yescharacteristic
524C22C22×#1175822611yescharacteristic
525C222−1+4×D6111yescharacteristic
526C22C2×#1451982411yescharacteristic
527C22#1684059811yescharacteristic
528C22#1684059911yescharacteristic
529C4—22no—
530C4—26no—
531C4—22no—
532C4—43no—
533C4—41no—
534C4—43no—
535C4—1226no—
536C4—1226no—
537C4—1222no—
538C4—1222no—
539C4—2441no—
540C4—2443no—
541C4—2443no—
542C4C2×#14519735112yes—
543C2—2442no—
544C2C22×#1451973511yescharacteristic
545C2#2145902811yescharacteristic
546C2C4×#1451982411yescharacteristic
547C2#2145980111yescharacteristic
548C2#2145982711yescharacteristic
549C2#2145991611yescharacteristic
550C2#2145993111yescharacteristic
subgroupG/H
center, FrattiniC23C24×D61
commutatorC22×C61C24×C4
FittingC61×#2351C2
socle, cosocleC23×C61C25
prime radicalC23×D61C24
hypercenter#2351D61
hypocenterC61#34262
LCSG ▷ C22×C61 ▷ C61
UCSC1 ◁ C23 ◁ #2351

Complements

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

NH ≅ G/Ntype
2. #21448786C2semidirect
3. #21448774C2semidirect
4. Q8×C4⋊Dic61C2semidirect
9. #21448466C2semidirect
12. #21448290C2semidirect
13. #21448184C2semidirect
14. C4×Q8×Dic61C2semidirect
15. C61×#2351C2semidirect
17. #16840238C4semidirect
271. #2351D61semidirect
342. C61#34262semidirect

Maximal subgroups

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

subgroupindex
C2×#168402382
#214487862
#214487742
Q8×C4⋊Dic612
#214487112
#214487032
Q8×#117581982
#214484662
#214484292
#214483122
#214482902
#214481842
C4×Q8×Dic612
C61×#23512
#3426261