Finite Group Databasebeta

← #29657604

Subgroups — #29657604

Order 15616.

926 conjugacy classes of subgroups.

The subgroup lattice has 542 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21459620C211yesmaximal, characteristic
2#21457161C211yesmaximal, characteristic, complemented
3#21455612C211yesmaximal, characteristic
4#21445065C212yesmaximal, complemented
5#21445030C212yesmaximal, complemented
6#21444981C212yesmaximal, complemented
7#21444950C212yesmaximal, complemented
8#21444509C211yesmaximal, characteristic, complemented
9#21444275C211yesmaximal, characteristic, complemented
10C2×#16839491C211yesmaximal, characteristic
11C61×#3170C211yesmaximal, characteristic, complemented
12#16840473C2212yes—
13#16840472—21no—
14#16840472—22no—
15#16840468—22no—
16Q8×D244—21no—
17#16840427C2212yes—
18#16840390—21no—
19#16840387—22no—
20#16840319C2211yescharacteristic
21C22⋊Q8×D61—22no—
22#16840236—21no—
23C2×C4×D244C2211yescharacteristic
24#16840059—22no—
25#16840057—21no—
26#16840054C2212yescomplemented
27#16840049C2212yes—
28#16840039C2212yes—
29#16840001—22no—
30#16839972—21no—
31#16839969C2212yescomplemented
32#16839805—21no—
33#16839786C2211yescharacteristic
34C2×C4⋊#92418C2211yescharacteristic
35#16839766—21no—
36#16839511C2212yescomplemented
37#16839508C2212yescomplemented
38#16839494C2212yescomplemented
39#16839494C2212yescomplemented
40#16839491C2212yes—
41#16839491C2212yes—
42C61×#533C2211yescharacteristic
43C61×#533C2211yescharacteristic
44C61×#500C2211yescharacteristic, complemented
45C61×#494C2211yescharacteristic, complemented
46C61×#487C2211yescharacteristic, complemented
47C61×#487C2211yescharacteristic, complemented
48C61×#467—22no—
49C4×Q16×C61—22no—
50C2×C4×C8×C61C2211yescharacteristic
51C2×Q8×D61—22no—
52C22×D244D411yescharacteristic
53C22×C4×D61—21no—
54C2×#11758202—22no—
55C2×#11758194—21no—
56C22×#92418—21no—
57#14519777—22no—
58#14519777D411yescharacteristic
59#14519777D411yescharacteristic
60#14519776—22no—
61#14519769—22no—
62#14519769—21no—
63#14519768—22no—
64#14519757—24no—
65#14519757—22no—
66#14519756—21no—
67#14519756D412yes—
68#14519755—24no—
69#14519755—24no—
70#14519755—24no—
71#14519754—24no—
72#14519753—24no—
73#14519753—22no—
74C4×Dic122—21no—
75C4×Dic122C2312yes—
76C4×D244—22no—
77C4×D244—22no—
78C4×D244—21no—
79C4×D244—21no—
80C4×D244D412yes—
81C4×D244C2312yes—
82#14519739—22no—
83#14519739—22no—
84#14519736—21no—
85C2×C4⋊Dic61D411yescharacteristic
86C4⋊C4×D61—24no—
87C4⋊C4×D61—22no—
88C2×#11758198—21no—
89C22⋊C4×D61—24no—
90#14519710—22no—
91#14519710—22no—
92#14519708—22no—
93C4⋊#92418—21no—
94C4⋊#92418—21no—
95C4⋊#92418C2312yes—
96C4⋊#92418C2312yes—
97#14519701—22no—
98#14519700—22no—
99#14519697—22no—
100#14519697—22no—
101C2×Q16×C61—22no—
102C22×C8×C61—21no—
103C4⋊Q8×C61C2311yescharacteristic
104C4⋊Q8×C61C2311yescharacteristic
105C4⋊Q8×C61C2311yescharacteristic
106C4⋊Q8×C61C2311yescharacteristic
107#125×C61—21no—
108#125×C61—21no—
109C22⋊Q8×C61—21no—
110C22⋊Q8×C61—21no—
111C22⋊Q8×C61—21no—
112C22⋊Q8×C61D412yes—
113C4×Q8×C61—21no—
114C4×Q8×C61—21no—
115C4×Q8×C61—21no—
116C4×Q8×C61D412yes—
117#117×C61—21no—
118#117×C61—21no—
119C2×C42×C61C2311yescharacteristic
120#107×C61—22no—
121#103×C61—22no—
122#103×C61—22no—
123C4×C8×C61—21no—
124C4×C8×C61C2311yescharacteristic
125C4×C8×C61C2311yescharacteristic
126C23×D61—21no—
127C2×Q8×C61—21no—
128C2×Q8×C61—21no—
129C2×Q8×C61—21no—
130C2×Q8×C61C2×D412yes—
131C22×C4×C61C2×D411yescharacteristic
132C22×C4×C61C2×D411yescharacteristic
133C22×C4×C61C2×D411yescharacteristic
134C22×Dic61—21no—
135Q8×D61—82no—
136Q8×D61—82no—
137C2×D244—21no—
138C2×D244—22no—
139C2×D244Q1614yescomplemented
140C2×D244C2×D412yes—
141C2×C4×D61—22no—
142C2×C4×D61—24no—
143C2×C4×D61—24no—
144C2×C4×D61—22no—
145C2×C4×D61—81no—
146C2×C4×D61—81no—
147C2×Dic122—24no—
148C2×Dic122—21no—
149C2×Dic122C2×D412yes—
150Q16×C61—44no—
151C2×C8×C61—21no—
152C2×C8×C61—21no—
153C2×C8×C61—42no—
154C4⋊C4×C61—22no—
155C4⋊C4×C61—22no—
156C4⋊C4×C61—21no—
157C4⋊C4×C61—21no—
158C4⋊C4×C61—21no—
159C4⋊C4×C61—21no—
160C4⋊C4×C61—21no—
161C4⋊C4×C61—21no—
162C4⋊C4×C61—21no—
163C4⋊C4×C61—41no—
164C4⋊C4×C61—41no—
165C4⋊C4×C61C2×D412yes—
166C22⋊C4×C61—22no—
167C22⋊C4×C61—41no—
168C42×C61—22no—
169C42×C61—41no—
170C42×C61C2×D411yescharacteristic
171C42×C61C2×D411yescharacteristic
172C42×C61C2×D411yescharacteristic
173C42×C61C2411yescharacteristic
174C22⋊Dic61—22no—
175#11758202—82no—
176#11758198—24no—
177#11758198—24no—
178#11758198—22no—
179#11758198—22no—
180C4⋊Dic61—24no—
181C4⋊Dic61—21no—
182C4⋊Dic61—21no—
183C4⋊Dic61—21no—
184C4⋊Dic61—21no—
185C4⋊Dic61C2×D412yes—
186C4⋊Dic61C2×D412yes—
187#11758196—24no—
188#11758196—24no—
189#11758196—22no—
190#11758196—22no—
191C4×Dic61—22no—
192#11758194—42no—
193C2×#92418—22no—
194C2×#92418—22no—
195C2×#92418—81no—
196C23×C61#12011yescharacteristic
197C22×D61—22no—
198C22×D61—24no—
199C22×D61—81no—
200C22×D61—81no—
201Q8×C61—42no—
202Q8×C61—42no—
203Q8×C61—41no—
204Q8×C61—41no—
205Q8×C61—41no—
206Q8×C61—41no—
207C2×C4×C61—21no—
208C2×C4×C61—21no—
209C2×C4×C61—22no—
210C2×C4×C61—22no—
211C2×C4×C61—41no—
212C2×C4×C61—41no—
213C2×C4×C61—41no—
214C2×C4×C61#12011yescharacteristic
215C2×C4×C61#12011yescharacteristic
216C2×C4×C61#12011yescharacteristic
217C2×C4×C61C2×Q1612yes—
218C2×C4×C61#13712yes—
219C2×C4×C61C22×D411yescharacteristic
220C2×C4×C61C22×D411yescharacteristic
221C2×C4×C61C22×D411yescharacteristic
222C2×Dic61—22no—
223C2×Dic61—22no—
224C2×Dic61—22no—
225C2×Dic61—81no—
226D244—22no—
227D244—24no—
228D244C2×Q1614yes—
229C4×D61—42no—
230C4×D61—42no—
231C4×D61—82no—
232C4×D61—84no—
233C4×D61—84no—
234Dic122—22no—
235Dic122—22no—
236Dic122—81no—
237Dic122—84no—
238C8×C61—22no—
239C8×C61—41no—
240#92418—42no—
241#92418—42no—
242#18665—611nomaximal, Sylow
243C22×C61—21no—
244C22×C61—21no—
245C22×C61#45112yes—
246C22×C61C2×#12011yescharacteristic
247C2×D61—24no—
248C2×D61—42no—
249C2×D61—42no—
250C2×D61—84no—
251C4×C61—21no—
252C4×C61—21no—
253C4×C61—21no—
254C4×C61—42no—
255C4×C61—42no—
256C4×C61—81no—
257C4×C61—81no—
258C4×C61#45112yes—
259C4×C61C22×Q1611yescharacteristic
260C4×C61C2×#13711yescharacteristic
261C4×C61#57611yescharacteristic
262C4×C61#57711yescharacteristic
263Dic61—42no—
264Dic61—42no—
265Dic61—42no—
266#3501—611no—
267#3249—611no—
268#3170D6111yescharacteristic, complemented
269#3077—611no—
270#1649—612no—
271#1644—612no—
272#1637—612no—
273#1632—612no—
274#1550—611no—
275#1493—611no—
276C2×D4⋊C8—611no—
277C2×C61—22no—
278C2×C61C2×#45111yescharacteristic
279C2×C61#301211yescharacteristic
280C2×C61#301811yescharacteristic
281D61—44no—
282#554—612no—
283#554—1222no—
284#554—1221no—
285#554—1222no—
286D4×Q8—612no—
287D4×Q8—1221no—
288#544—1221no—
289#543—1222no—
290#533—611no—
291#533C2×D6111yescharacteristic
292#533C2×D6111yescharacteristic
293C2×C22⋊Q8—1222no—
294#518—1221no—
295C2×C4×D4—611no—
296#500C2×D6111yescharacteristic, complemented
297#494C2×D6111yescharacteristic, complemented
298#487C2×D6111yescharacteristic, complemented
299#487C2×D6111yescharacteristic, complemented
300#478—1222no—
301#477—612no—
302#477—1221no—
303#476—612no—
304#474—612no—
305#467—22no—
306#467—1222no—
307#462—612no—
308#462—1221no—
309C4×Q16—22no—
310#428—1221no—
311#423—611no—
312C2×C4⋊C8—611no—
313#418—1221no—
314C2×C4×C8C2×D6111yescharacteristic
315#333—612no—
316#332—612no—
317Q8⋊C8—612no—
318Q8⋊C8—612no—
319D4⋊C8—612no—
320D4⋊C8—612no—
321C61#1866511yescharacteristic, complemented, Sylow
322C22×Q8—1222no—
323C22×D4—611no—
324C23×C4—1221no—
325C2×Q16—22no—
326C2×Q16—1222no—
327C2×M16—1221no—
328C22×C8—21no—
329C22×C8—1221no—
330C4⋊Q8—611no—
331C4⋊Q8—611no—
332C4⋊Q8—1222no—
333C4⋊Q8—1222no—
334C4⋊Q8C22×D6111yescharacteristic
335C4⋊Q8C22×D6111yescharacteristic
336C4⋊Q8C22×D6111yescharacteristic
337C4⋊Q8C22×D6111yescharacteristic
338#125—21no—
339#125—21no—
340#125—1222no—
341#125—1222no—
342#125—1221no—
343C22⋊Q8—21no—
344C22⋊Q8—21no—
345C22⋊Q8—21no—
346C22⋊Q8—612no—
347C22⋊Q8—1224no—
348C22⋊Q8—1222no—
349C22⋊Q8—1224no—
350C22⋊Q8—1224no—
351C22⋊Q8—1224no—
352C22⋊Q8—1222no—
353C22⋊Q8—1224no—
354C22⋊Q8—1221no—
355C22⋊Q8—1224no—
356C22⋊Q8C22⋊D6112yes—
357C4×Q8—21no—
358C4×Q8—21no—
359C4×Q8—21no—
360C4×Q8—612no—
361C4×Q8—1221no—
362C4×Q8C22⋊D6112yes—
363C4×D4—612no—
364C4×D4—612no—
365C4×D4—1222no—
366C4×D4—1222no—
367C4×D4—1222no—
368C4×D4—1222no—
369C4×D4—1221no—
370C4×D4—1221no—
371#117—21no—
372#117—21no—
373#117—1221no—
374C2×C4⋊C4—611no—
375C2×C4⋊C4—1222no—
376C2×C4⋊C4—1224no—
377C2×C22⋊C4—1224no—
378C2×C22⋊C4—1221no—
379C2×C42C22×D6111yescharacteristic
380#107—22no—
381#107—1222no—
382#107—1222no—
383#106—1222no—
384C4⋊C8—612no—
385C4⋊C8—612no—
386C4⋊C8—1221no—
387C4⋊C8—1221no—
388#103—22no—
389#103—22no—
390#103—1222no—
391#103—1222no—
392#102—1222no—
393#102—1222no—
394C4×C8—21no—
395C4×C8C22×D6111yescharacteristic
396C4×C8C22×D6111yescharacteristic
397C24—1221no—
398C2×Q8—21no—
399C2×Q8—21no—
400C2×Q8—21no—
401C2×Q8—612no—
402C2×Q8—1224no—
403C2×Q8—1221no—
404C2×Q8—4882no—
405C2×Q8—4882no—
406C2×Q8C2×C22⋊D6112yes—
407C2×D4—612no—
408C2×D4—614no—
409C2×D4—1222no—
410C2×D4—1221no—
411C22×C4—1224no—
412C22×C4—1224no—
413C22×C4—1221no—
414C22×C4—1222no—
415C22×C4—1222no—
416C22×C4—4881no—
417C22×C4—4881no—
418C22×C4D4×D6111yescharacteristic
419C22×C4D4×D6111yescharacteristic
420C22×C4C2×C22⋊D6111yescharacteristic
421Q16—44no—
422Q16—4882no—
423M16—2442no—
424C2×C8—21no—
425C2×C8—21no—
426C2×C8—42no—
427C2×C8—1222no—
428C2×C8—1222no—
429C2×C8—4881no—
430C4⋊C4—21no—
431C4⋊C4—21no—
432C4⋊C4—21no—
433C4⋊C4—22no—
434C4⋊C4—22no—
435C4⋊C4—21no—
436C4⋊C4—21no—
437C4⋊C4—21no—
438C4⋊C4—21no—
439C4⋊C4—41no—
440C4⋊C4—41no—
441C4⋊C4—612no—
442C4⋊C4—612no—
443C4⋊C4—1224no—
444C4⋊C4—1224no—
445C4⋊C4—1224no—
446C4⋊C4—1222no—
447C4⋊C4—1221no—
448C4⋊C4—1222no—
449C4⋊C4—1221no—
450C4⋊C4—1221no—
451C4⋊C4—1221no—
452C4⋊C4C2×C22⋊D6112yes—
453C22⋊C4—22no—
454C22⋊C4—41no—
455C22⋊C4—1224no—
456C22⋊C4—1224no—
457C22⋊C4—1222no—
458C22⋊C4—1222no—
459C22⋊C4—1222no—
460C42—22no—
461C42—41no—
462C42—1222no—
463C42D4×D6111yescharacteristic
464C42D4×D6111yescharacteristic
465C42C2×C22⋊D6111yescharacteristic
466C42C23×D6111yescharacteristic
467C23—1224no—
468C23—1222no—
469C23—4881no—
470C23—4881no—
471C23#1451974511yescharacteristic
472Q8—42no—
473Q8—42no—
474Q8—41no—
475Q8—41no—
476Q8—41no—
477Q8—41no—
478Q8—1222no—
479Q8—1222no—
480Q8—4881no—
481Q8—4884no—
482D4—614no—
483D4—1222no—
484D4—1224no—
485C2×C4—21no—
486C2×C4—21no—
487C2×C4—22no—
488C2×C4—22no—
489C2×C4—41no—
490C2×C4—41no—
491C2×C4—41no—
492C2×C4—1222no—
493C2×C4—1222no—
494C2×C4—1222no—
495C2×C4—4882no—
496C2×C4—4884no—
497C2×C4—4884no—
498C2×C4—4881no—
499C2×C4—2442no—
500C2×C4—2442no—
501C2×C4#1451974511yescharacteristic
502C2×C4#1451974511yescharacteristic
503C2×C4#1451974511yescharacteristic
504C2×C4Q16×D6112yes—
505C2×C4#1451980912yes—
506C2×C4C2×D4×D6111yescharacteristic
507C2×C4C2×D4×D6111yescharacteristic
508C2×C4C22×C22⋊D6111yescharacteristic
509C8—22no—
510C8—41no—
511C8—2442no—
512C8—2442no—
513C22—21no—
514C22—21no—
515C22—1224no—
516C22—4884no—
517C22—2442no—
518C22—2442no—
519C22#1683991612yes—
520C22C2×#1451974511yescharacteristic
521C4—21no—
522C4—21no—
523C4—21no—
524C4—42no—
525C4—42no—
526C4—81no—
527C4—81no—
528C4—2442no—
529C4—2442no—
530C4—2442no—
531C4#1683991612yes—
532C4C2×Q16×D6111yescharacteristic
533C4C2×#1451980911yescharacteristic
534C4#1684056411yescharacteristic
535C4#1684056911yescharacteristic
536C2—22no—
537C2—2444no—
538C2C2×#1683991611yescharacteristic
539C2#2145511911yescharacteristic
540C2#2145516711yescharacteristic
subgroupG/H
centerC22C2×#14519745
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#3170C2
socleC22×C61C2×#120
prime radicalC22×D244D4
hypercenter#3170D61
hypocenterC61#18665
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3170

Complements

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

NH ≅ G/Ntype
2. #21457161C2semidirect
4. #21445065C2semidirect
5. #21445030C2semidirect
6. #21444981C2semidirect
7. #21444950C2semidirect
8. #21444509C2semidirect
9. #21444275C2semidirect
11. C61×#3170C2semidirect
26. #16840054C22semidirect
31. #16839969C22semidirect
36. #16839511C22semidirect
37. #16839508C22semidirect
38. #16839494C22semidirect
39. #16839494C22semidirect
44. C61×#500C22semidirect
45. C61×#494C22semidirect
46. C61×#487C22semidirect
47. C61×#487C22semidirect
139. C2×D244Q16semidirect
268. #3170D61semidirect
296. #500C2×D61semidirect
297. #494C2×D61semidirect
298. #487C2×D61semidirect
299. #487C2×D61semidirect
321. C61#18665semidirect

Maximal subgroups

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

subgroupindex
#214596202
#214571612
#214556122
#214450652
#214450302
#214449812
#214449502
#214445092
#214442752
C2×#168394912
C61×#31702
#1866561