Finite Group Databasebeta

← #29665420

Subgroups — #29665420

Order 15616.

704 conjugacy classes of subgroups.

The subgroup lattice has 500 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456760C211yesmaximal, characteristic, complemented
2#21455607C211yesmaximal, characteristic, complemented
3#21454930C211yesmaximal, characteristic, complemented
4#21445233C212yesmaximal, complemented
5#21445226C212yesmaximal, complemented
6#21445165C212yesmaximal, complemented
7#21445158C212yesmaximal, complemented
8#21444591C211yesmaximal, characteristic, complemented
9#21444510C211yesmaximal, characteristic, complemented
10C2×#16839506C211yesmaximal, characteristic, complemented
11C61×#3038C211yesmaximal, characteristic
12#16840319C2211yescharacteristic
13#16840134C2212yes—
14#16840108C2212yes—
15#16840069—21no—
16#16840062—21no—
17#16840057C2211yescharacteristic, complemented
18#16840052C2211yescharacteristic, complemented
19#16840047C2211yescharacteristic, complemented
20#16840042C2211yescharacteristic, complemented
21#16839959—21no—
22#16839959—21no—
23#16839913—22no—
24Q8×#92418C2212yescomplemented
25D4×#92418—22no—
26D4×#92418C2212yescomplemented
27#16839815—21no—
28#16839815C2211yescharacteristic
29#16839798—21no—
30C2×C4⋊#92418—21no—
31C2×C4⋊#92418C2211yescharacteristic
32C2×C4⋊#92418C2211yescharacteristic
33#16839758—21no—
34#16839512C2212yescomplemented
35#16839512C2212yescomplemented
36#16839510C2212yescomplemented
37#16839507C2212yescomplemented
38#16839506C2212yes—
39#16839506C2212yes—
40C2×AGL(1,ℤ/8)×C61—21no—
41C2×#127×C61C2211yescharacteristic
42C4×#41×C61C2211yescharacteristic
43C61×#460—21no—
44C61×#460C2212yes—
45C61×#459—21no—
46C61×#459C2212yes—
47C61×#423C2211yescharacteristic
48C61×#418—21no—
49C2×#11758200—22no—
50C2×#11758200—22no—
51C22×#92418—21no—
52C22×#92418—21no—
53C22×#92418—22no—
54#14519777C2311yescharacteristic
55#14519777C2311yescharacteristic
56#14519769—21no—
57#14519756—21no—
58#14519756—21no—
59C4×Dic122—21no—
60C4×Dic122—21no—
61#14519736—21no—
62#14519710—21no—
63#14519710—21no—
64#14519708—21no—
65#14519708—21no—
66C4⋊#92418—21no—
67C4⋊#92418—21no—
68C4⋊#92418—22no—
69C4⋊#92418D411yescharacteristic
70C4⋊#92418D411yescharacteristic
71C4⋊#92418C2312yes—
72C4⋊#92418C2311yescharacteristic
73C4⋊#92418C2311yescharacteristic
74#14519701—21no—
75#14519701—21no—
76#14519698—21no—
77#14519698—21no—
78#14519698—22no—
79#14519698—22no—
80C22⋊#92418—22no—
81C22⋊#92418D412yes—
82C4×#92418—22no—
83C4×#92418C2312yes—
84C2×#41×C61D411yescharacteristic
85C22×D4×C61—21no—
86C22×D4×C61—21no—
87C22×D4×C61—21no—
88AGL(1,ℤ/8)×C61—44no—
89C2×SD16×C61—22no—
90C2×D8×C61—22no—
91C2×M16×C61—21no—
92#127×C61—21no—
93#127×C61—21no—
94#127×C61—21no—
95#127×C61C2312yes—
96C4×Q8×C61D411yescharacteristic
97C4×Q8×C61C2311yescharacteristic
98C4×D4×C61—21no—
99C4×D4×C61—21no—
100C4×D4×C61D411yescharacteristic
101C4×D4×C61C2311yescharacteristic
102#117×C61—21no—
103#117×C61D411yescharacteristic
104C2×C42×C61—21no—
105C2×C42×C61C2311yescharacteristic
106C4⋊C8×C61—21no—
107C4⋊C8×C61C2312yes—
108#102×C61—22no—
109#102×C61—22no—
110C24×C61—41no—
111#41×C61—22no—
112#41×C61SD1614yescomplemented
113C2×Q8×C61C2×D411yescharacteristic
114C2×D4×C61—21no—
115C2×D4×C61—22no—
116C2×D4×C61—22no—
117C2×D4×C61—22no—
118C2×D4×C61—41no—
119C2×D4×C61—41no—
120C2×D4×C61—41no—
121C2×D4×C61—42no—
122C2×D4×C61—42no—
123C2×D4×C61—81no—
124C2×D4×C61—81no—
125C2×D4×C61—82no—
126C2×D4×C61C2×D411yescharacteristic
127C22×C4×C61—21no—
128C22×C4×C61—21no—
129C22×C4×C61—21no—
130C22×C4×C61C2×D411yescharacteristic
131C22×C4×C61C2×D411yescharacteristic
132C22×C4×C61#4111yescharacteristic
133C2×Dic122—21no—
134C2×Dic122—21no—
135SD16×C61—44no—
136D8×C61—44no—
137M16×C61—42no—
138C2×C8×C61—22no—
139C4⋊C4×C61—21no—
140C4⋊C4×C61—21no—
141C4⋊C4×C61C2×D411yescharacteristic
142C4⋊C4×C61C2×D411yescharacteristic
143C22⋊C4×C61—21no—
144C22⋊C4×C61—21no—
145C22⋊C4×C61—21no—
146C42×C61—21no—
147C42×C61—21no—
148C42×C61—21no—
149C42×C61C2×D411yescharacteristic
150C42×C61C2×D411yescharacteristic
151C42×C61#4111yescharacteristic
152C42×C61C2411yescharacteristic
153C22⋊Dic61—41no—
154#11758200—82no—
155#11758200—82no—
156C4⋊Dic61—21no—
157C4⋊Dic61—21no—
158C4⋊Dic61—21no—
159C4⋊Dic61—21no—
160#11758196—41no—
161#11758196—41no—
162C4×Dic61—41no—
163C2×#92418—22no—
164C2×#92418—21no—
165C2×#92418—21no—
166C2×#92418—42no—
167C2×#92418—82no—
168C2×#92418—81no—
169C2×#92418C2×D412yes—
170C2×#92418C2×D412yes—
171C23×C61—21no—
172C23×C61—42no—
173C23×C61—82no—
174C23×C61—82no—
175C23×C61—82no—
176C23×C61#12111yescharacteristic
177Q8×C61—21no—
178Q8×C61C2×SD1612yes—
179D4×C61—24no—
180D4×C61—21no—
181D4×C61—42no—
182D4×C61—42no—
183D4×C61—42no—
184D4×C61—42no—
185D4×C61—82no—
186D4×C61—82no—
187D4×C61—82no—
188D4×C61—82no—
189D4×C61C2×SD1612yes—
190C2×C4×C61—21no—
191C2×C4×C61—21no—
192C2×C4×C61—21no—
193C2×C4×C61—24no—
194C2×C4×C61—21no—
195C2×C4×C61—21no—
196C2×C4×C61—41no—
197C2×C4×C61—41no—
198C2×C4×C61—41no—
199C2×C4×C61—81no—
200C2×C4×C61#12111yescharacteristic
201C2×C4×C61#12111yescharacteristic
202C2×C4×C61#12111yescharacteristic
203C2×C4×C61C2×SD1612yes—
204C2×C4×C61#13512yes—
205C2×C4×C61C22×D411yescharacteristic
206C2×C4×C61C22×D411yescharacteristic
207C2×C4×C61C2×#4111yescharacteristic
208C2×Dic61—41no—
209C2×Dic61—41no—
210Dic122—41no—
211Dic122—41no—
212Dic122—81no—
213C8×C61—42no—
214#92418—22no—
215#92418—22no—
216#92418—41no—
217#92418—41no—
218#92418—42no—
219#19409—611nomaximal, Sylow
220C22×C61—21no—
221C22×C61—21no—
222C22×C61—22no—
223C22×C61—41no—
224C22×C61—41no—
225C22×C61—82no—
226C22×C61—82no—
227C22×C61—82no—
228C22×C61—164no—
229C22×C61#46512yes—
230C22×C61C2×#12111yescharacteristic
231C4×C61—21no—
232C4×C61—22no—
233C4×C61—21no—
234C4×C61—21no—
235C4×C61—41no—
236C4×C61—41no—
237C4×C61#46512yes—
238C4×C61C22×SD1611yescharacteristic
239C4×C61C2×#13511yescharacteristic
240C4×C61#57611yescharacteristic
241C4×C61#57711yescharacteristic
242Dic61—81no—
243Dic61—81no—
244#3208—611no—
245#3076—611no—
246#3038D6111yescharacteristic
247#2988—611no—
248#1678—612no—
249#1677—612no—
250#1666—612no—
251#1665—612no—
252#1570—611no—
253#1550—611no—
254C2×#331—611no—
255C2×C61—22no—
256C2×C61—42no—
257C2×C61—162no—
258C2×C61#304511yescharacteristic
259C2×C61C2×#46511yescharacteristic
260C2×C61#306811yescharacteristic
261C2×AGL(1,ℤ/8)—21no—
262#533—611no—
263C2×#127C2×D6111yescharacteristic
264C4×#41C2×D6111yescharacteristic
265#492—612no—
266#486—612no—
267#479—1221no—
268#478—1221no—
269#477—611no—
270#476—611no—
271#475—611no—
272#474—611no—
273#460—21no—
274#460—1221no—
275#460—1221no—
276#460C2×D6112yes—
277#459—21no—
278#459C2×D6112yes—
279#450—1222no—
280C8×Q8—612no—
281C8×D4—612no—
282C8×D4—1222no—
283#431—611no—
284#431—1221no—
285#427—1221no—
286#423C2×D6111yescharacteristic
287C2×C4⋊C8—611no—
288C2×C4⋊C8—611no—
289C2×C4⋊C8—1221no—
290#418—21no—
291#416—1221no—
292#334—612no—
293#334—612no—
294#333—612no—
295#332—612no—
296#331—612no—
297#331—612no—
298C61#1940911yescharacteristic, complemented, Sylow
299C2×#41C22⋊D6111yescharacteristic
300C22×D4—21no—
301C22×D4—21no—
302C22×D4—21no—
303AGL(1,ℤ/8)—44no—
304C2×SD16—22no—
305C2×SD16—1222no—
306C2×SD16—1222no—
307C2×D8—22no—
308C2×M16—21no—
309C22×C8—1222no—
310C22×C8—1221no—
311C22×C8—1221no—
312C4⋊Q8—611no—
313C4⋊Q8—611no—
314#127—21no—
315#127—21no—
316#127—21no—
317#127C22×D6112yes—
318#125—1221no—
319C22⋊Q8—1221no—
320C22⋊Q8—1221no—
321C4×Q8—1221no—
322C4×Q8—1221no—
323C4×Q8C22⋊D6111yescharacteristic
324C4×Q8C22×D6111yescharacteristic
325C4×D4—21no—
326C4×D4—21no—
327C4×D4C22⋊D6111yescharacteristic
328C4×D4C22×D6111yescharacteristic
329#117—21no—
330#117—1221no—
331#117C22⋊D6111yescharacteristic
332C2×C42—21no—
333C2×C42C22×D6111yescharacteristic
334#107—1221no—
335#107—1221no—
336#106—1221no—
337#106—1221no—
338C4⋊C8—21no—
339C4⋊C8—611no—
340C4⋊C8—611no—
341C4⋊C8—612no—
342C4⋊C8—611no—
343C4⋊C8—611no—
344C4⋊C8—1222no—
345C4⋊C8—1221no—
346C4⋊C8—1221no—
347C4⋊C8C22×D6112yes—
348#103—1221no—
349#103—1221no—
350#102—22no—
351#102—22no—
352#102—1222no—
353#102—1222no—
354#102—1221no—
355#102—1221no—
356C22⋊C8—612no—
357C22⋊C8—1222no—
358C4×C8—612no—
359C4×C8—1222no—
360C24—41no—
361#41—22no—
362#41#1175820014yescomplemented
363C2×Q8—1221no—
364C2×Q8—1221no—
365C2×Q8C2×C22⋊D6111yescharacteristic
366C2×D4—21no—
367C2×D4—22no—
368C2×D4—22no—
369C2×D4—22no—
370C2×D4—41no—
371C2×D4—41no—
372C2×D4—41no—
373C2×D4—42no—
374C2×D4—42no—
375C2×D4—81no—
376C2×D4—81no—
377C2×D4—82no—
378C2×D4C2×C22⋊D6111yescharacteristic
379C22×C4—21no—
380C22×C4—21no—
381C22×C4—21no—
382C22×C4D4×D6111yescharacteristic
383C22×C4#1175822411yescharacteristic
384C22×C4C2×C22⋊D6111yescharacteristic
385SD16—44no—
386SD16—4882no—
387SD16—4882no—
388D8—44no—
389M16—42no—
390C2×C8—22no—
391C2×C8—612no—
392C2×C8—612no—
393C2×C8—1222no—
394C2×C8—1221no—
395C2×C8—1221no—
396C2×C8—4882no—
397C2×C8—4881no—
398C2×C8—2442no—
399C4⋊C4—21no—
400C4⋊C4—21no—
401C4⋊C4—1221no—
402C4⋊C4—1221no—
403C4⋊C4—1221no—
404C4⋊C4—1221no—
405C4⋊C4—2441no—
406C4⋊C4—2441no—
407C4⋊C4C2×C22⋊D6111yescharacteristic
408C4⋊C4C2×C22⋊D6111yescharacteristic
409C22⋊C4—21no—
410C22⋊C4—21no—
411C22⋊C4—21no—
412C22⋊C4—2441no—
413C42—21no—
414C42—21no—
415C42—21no—
416C42—2441no—
417C42D4×D6111yescharacteristic
418C42#1175822411yescharacteristic
419C42C2×C22⋊D6111yescharacteristic
420C42C23×D6111yescharacteristic
421C23—21no—
422C23—42no—
423C23—82no—
424C23—82no—
425C23—82no—
426C23#1451974911yescharacteristic
427Q8—21no—
428Q8—4881no—
429Q8—2441no—
430Q8—2441no—
431Q8C2×#1175820012yes—
432D4—24no—
433D4—21no—
434D4—42no—
435D4—42no—
436D4—42no—
437D4—42no—
438D4—82no—
439D4—82no—
440D4—82no—
441D4—82no—
442D4C2×#1175820012yes—
443C2×C4—21no—
444C2×C4—24no—
445C2×C4—21no—
446C2×C4—21no—
447C2×C4—21no—
448C2×C4—21no—
449C2×C4—41no—
450C2×C4—41no—
451C2×C4—41no—
452C2×C4—81no—
453C2×C4—2441no—
454C2×C4—2441no—
455C2×C4#1451974911yescharacteristic
456C2×C4#1451974911yescharacteristic
457C2×C4#1451974911yescharacteristic
458C2×C4C2×#1175820012yes—
459C2×C4#1451979712yes—
460C2×C4C2×D4×D6111yescharacteristic
461C2×C4C22×C22⋊D6111yescharacteristic
462C2×C4C2×#1175822411yescharacteristic
463C8—42no—
464C8—1222no—
465C8—1222no—
466C8—2441no—
467C8—2441no—
468C8—2442no—
469C22—21no—
470C22—21no—
471C22—22no—
472C22—41no—
473C22—41no—
474C22—82no—
475C22—82no—
476C22—82no—
477C22—164no—
478C22#1683998912yes—
479C22C2×#1451974911yescharacteristic
480C4—21no—
481C4—22no—
482C4—21no—
483C4—21no—
484C4—41no—
485C4—41no—
486C4—4881no—
487C4—4881no—
488C4#1683998912yes—
489C4C22×#1175820011yescharacteristic
490C4C2×#1451979711yescharacteristic
491C4#1684056611yescharacteristic
492C4#1684057211yescharacteristic
493C2—22no—
494C2—42no—
495C2—162no—
496C2#2145536211yescharacteristic
497C2C2×#1683998911yescharacteristic
498C2#2145554411yescharacteristic
subgroupG/H
centerC22C2×#14519749
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
Fitting, prime radicalC61×#3038C2
socleC22×C61C2×#121
hypercenter#3038D61
hypocenterC61#19409
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3038

Complements

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

NH ≅ G/Ntype
1. #21456760C2semidirect
2. #21455607C2semidirect
3. #21454930C2semidirect
4. #21445233C2semidirect
5. #21445226C2semidirect
6. #21445165C2semidirect
7. #21445158C2semidirect
8. #21444591C2semidirect
9. #21444510C2semidirect
10. C2×#16839506C2semidirect
17. #16840057C22semidirect
18. #16840052C22semidirect
19. #16840047C22semidirect
20. #16840042C22semidirect
24. Q8×#92418C22semidirect
26. D4×#92418C22semidirect
34. #16839512C22semidirect
35. #16839512C22semidirect
36. #16839510C22semidirect
37. #16839507C22semidirect
112. #41×C61SD16semidirect
298. C61#19409semidirect
362. #41#11758200semidirect

Maximal subgroups

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

subgroupindex
#214567602
#214556072
#214549302
#214452332
#214452262
#214451652
#214451582
#214445912
#214445102
C2×#168395062
C61×#30382
#1940961