Finite Group Databasebeta

← #29665416

Subgroups — #29665416

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#21456762C211yesmaximal, characteristic, complemented
2#21455607C211yesmaximal, characteristic, complemented
3#21455299C211yesmaximal, characteristic
4#21445235C212yesmaximal, complemented
5#21445228C212yesmaximal, complemented
6#21445167C212yesmaximal, complemented
7#21445160C212yesmaximal, complemented
8#21444590C211yesmaximal, characteristic, complemented
9#21444507C211yesmaximal, characteristic, complemented
10C2×#16839505C211yesmaximal, characteristic, complemented
11C61×#2988C211yesmaximal, characteristic, complemented
12C2×#14519802—21no—
13#16840319C2211yescharacteristic
14C2×#14519774C2211yescharacteristic
15#16840135C2212yes—
16#16840109C2212yes—
17#16840069—21no—
18#16840062—21no—
19#16840057C2211yescharacteristic, complemented
20#16840052C2211yescharacteristic, complemented
21#16840047C2211yescharacteristic, complemented
22#16840042C2211yescharacteristic, complemented
23#16839963—21no—
24#16839963—21no—
25#16839962—21no—
26#16839962C2212yes—
27#16839957—21no—
28#16839957C2212yes—
29#16839915—22no—
30#16839798—21no—
31#16839786C2211yescharacteristic
32C2×C4⋊#92418C2211yescharacteristic
33#16839766—21no—
34#16839758—21no—
35#16839513C2212yescomplemented
36#16839513C2212yescomplemented
37#16839510C2212yescomplemented
38#16839509C2212yescomplemented
39#16839505C2212yes—
40#16839505C2212yes—
41C4×#41×C61C2211yescharacteristic
42C8×Q8×C61C2212yescomplemented
43C8×D4×C61—22no—
44C8×D4×C61C2212yescomplemented
45C61×#431—21no—
46C61×#431C2211yescharacteristic
47C2×C4⋊C8×C61—21no—
48C2×C4⋊C8×C61C2211yescharacteristic
49C22×D244—21no—
50C22×D244—21no—
51C22×D244—21no—
52#14519802—44no—
53C2×#11758190—22no—
54C2×#11758190—22no—
55C2×#11758201—22no—
56C2×#11758199—22no—
57C2×#11758194—21no—
58C22×#92418—21no—
59#14519777C2311yescharacteristic
60#14519777C2311yescharacteristic
61#14519774—21no—
62#14519774—21no—
63#14519774—21no—
64#14519774C2312yes—
65#14519769—21no—
66#14519756—21no—
67#14519756—21no—
68C4×Dic122—21no—
69C4×Dic122—21no—
70#14519736—21no—
71#14519710—21no—
72#14519710—21no—
73#14519708—21no—
74#14519708—21no—
75C4⋊#92418—21no—
76C4⋊#92418—21no—
77C4⋊#92418C2312yes—
78C4⋊#92418C2311yescharacteristic
79C4⋊#92418C2311yescharacteristic
80#14519701—21no—
81#14519701—21no—
82#14519699—22no—
83#14519699—22no—
84#14519698—21no—
85#14519698—21no—
86#14519697—22no—
87#14519697—22no—
88C2×#41×C61D411yescharacteristic
89C22×C8×C61—21no—
90C22×C8×C61—22no—
91C4×Q8×C61D411yescharacteristic
92C4×Q8×C61C2311yescharacteristic
93C4×D4×C61—21no—
94C4×D4×C61—21no—
95C4×D4×C61D411yescharacteristic
96C4×D4×C61C2311yescharacteristic
97#117×C61—21no—
98#117×C61D411yescharacteristic
99C2×C42×C61—21no—
100C2×C42×C61C2311yescharacteristic
101C4⋊C8×C61—22no—
102C4⋊C8×C61—21no—
103C4⋊C8×C61D411yescharacteristic
104C4⋊C8×C61D411yescharacteristic
105C4⋊C8×C61C2312yes—
106C22⋊C8×C61—22no—
107C22⋊C8×C61D412yes—
108C4×C8×C61—22no—
109C4×C8×C61C2312yes—
110C23×D61—41no—
111#41×C61—22no—
112#41×C61SD1614yescomplemented
113C2×Q8×C61C2×D411yescharacteristic
114C2×D4×C61—21no—
115C2×D4×C61C2×D411yescharacteristic
116C22×C4×C61—21no—
117C22×C4×C61—21no—
118C22×C4×C61—21no—
119C22×C4×C61C2×D411yescharacteristic
120C22×C4×C61C2×D411yescharacteristic
121C22×C4×C61#4111yescharacteristic
122C2×D244—22no—
123C2×D244—22no—
124C2×D244—22no—
125C2×D244—42no—
126C2×D244—42no—
127C2×D244—41no—
128C2×D244—41no—
129C2×D244—41no—
130C2×D244—81no—
131C2×D244—81no—
132C2×D244—82no—
133C2×Dic122—21no—
134C2×Dic122—21no—
135C2×C8×C61—22no—
136C2×C8×C61—82no—
137C2×C8×C61—81no—
138C2×C8×C61C2×D412yes—
139C2×C8×C61C2×D412yes—
140C4⋊C4×C61—21no—
141C4⋊C4×C61—21no—
142C4⋊C4×C61C2×D411yescharacteristic
143C4⋊C4×C61C2×D411yescharacteristic
144C22⋊C4×C61—21no—
145C22⋊C4×C61—21no—
146C22⋊C4×C61—21no—
147C42×C61—21no—
148C42×C61—21no—
149C42×C61—21no—
150C42×C61C2×D411yescharacteristic
151C42×C61C2×D411yescharacteristic
152C42×C61#4111yescharacteristic
153C42×C61C2411yescharacteristic
154C22⋊Dic61—41no—
155#11758201—44no—
156#11758199—44no—
157C4⋊Dic61—21no—
158C4⋊Dic61—21no—
159C4⋊Dic61—21no—
160C4⋊Dic61—21no—
161#11758196—41no—
162#11758196—41no—
163C4×Dic61—41no—
164#11758194—42no—
165C2×#92418—22no—
166C2×#92418—21no—
167C2×#92418—21no—
168C2×#92418—42no—
169#11758190—82no—
170#11758190—82no—
171C23×C61—21no—
172C23×C61#12111yescharacteristic
173C22×D61—42no—
174C22×D61—82no—
175C22×D61—82no—
176C22×D61—82no—
177Q8×C61—21no—
178Q8×C61C2×SD1612yes—
179D4×C61—24no—
180D4×C61—21no—
181D4×C61C2×SD1612yes—
182C2×C4×C61—21no—
183C2×C4×C61—21no—
184C2×C4×C61—21no—
185C2×C4×C61—24no—
186C2×C4×C61—21no—
187C2×C4×C61—21no—
188C2×C4×C61—41no—
189C2×C4×C61—41no—
190C2×C4×C61—41no—
191C2×C4×C61—81no—
192C2×C4×C61#12111yescharacteristic
193C2×C4×C61#12111yescharacteristic
194C2×C4×C61#12111yescharacteristic
195C2×C4×C61C2×SD1612yes—
196C2×C4×C61#13512yes—
197C2×C4×C61C22×D411yescharacteristic
198C2×C4×C61C22×D411yescharacteristic
199C2×C4×C61C2×#4111yescharacteristic
200C2×Dic61—41no—
201C2×Dic61—41no—
202D244—42no—
203D244—42no—
204D244—42no—
205D244—42no—
206D244—82no—
207D244—82no—
208D244—82no—
209D244—82no—
210Dic122—41no—
211Dic122—41no—
212Dic122—81no—
213C8×C61—22no—
214C8×C61—22no—
215C8×C61—42no—
216#92418—42no—
217#92418—41no—
218#92418—41no—
219#19409—611nomaximal, Sylow
220C22×C61—21no—
221C22×C61—21no—
222C22×C61—22no—
223C22×C61—41no—
224C22×C61—41no—
225C22×C61#46512yes—
226C22×C61C2×#12111yescharacteristic
227C2×D61—82no—
228C2×D61—82no—
229C2×D61—82no—
230C2×D61—164no—
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#3038—611no—
247#2988D6111yescharacteristic, complemented
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#304511yescharacteristic
258C2×C61C2×#46511yescharacteristic
259C2×C61#306811yescharacteristic
260D61—162no—
261C2×AGL(1,ℤ/8)—1221no—
262#533—611no—
263C2×#127—611no—
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—612no—
274#460—1221no—
275#460—1221no—
276#460—1221no—
277#459—612no—
278#459—1221no—
279#450—1222no—
280C8×Q8C2×D6112yescomplemented
281C8×D4—22no—
282C8×D4C2×D6112yescomplemented
283#431—21no—
284#431C2×D6111yescharacteristic
285#427—1221no—
286#423—611no—
287C2×C4⋊C8—21no—
288C2×C4⋊C8—611no—
289C2×C4⋊C8C2×D6111yescharacteristic
290#418—1221no—
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×#41D24411yescharacteristic
300C22×D4—1221no—
301C22×D4—1221no—
302C22×D4—1221no—
303AGL(1,ℤ/8)—2444no—
304C2×SD16—1222no—
305C2×SD16—1222no—
306C2×SD16—1222no—
307C2×D8—1222no—
308C2×M16—1221no—
309C22×C8—21no—
310C22×C8—22no—
311C22×C8—1221no—
312C4⋊Q8—611no—
313C4⋊Q8—611no—
314#127—612no—
315#127—1221no—
316#127—1221no—
317#127—1221no—
318#125—1221no—
319C22⋊Q8—1221no—
320C22⋊Q8—1221no—
321C4×Q8—1221no—
322C4×Q8—1221no—
323C4×Q8D24411yescharacteristic
324C4×Q8C22×D6111yescharacteristic
325C4×D4—21no—
326C4×D4—21no—
327C4×D4D24411yescharacteristic
328C4×D4C22×D6111yescharacteristic
329#117—21no—
330#117—1221no—
331#117D24411yescharacteristic
332C2×C42—21no—
333C2×C42C22×D6111yescharacteristic
334#107—1221no—
335#107—1221no—
336#106—1221no—
337#106—1221no—
338C4⋊C8—22no—
339C4⋊C8—21no—
340C4⋊C8—611no—
341C4⋊C8—611no—
342C4⋊C8—612no—
343C4⋊C8—1221no—
344C4⋊C8—1221no—
345C4⋊C8C22⋊D6111yescharacteristic
346C4⋊C8C22⋊D6111yescharacteristic
347C4⋊C8C22×D6112yes—
348#103—1221no—
349#103—1221no—
350#102—1221no—
351#102—1221no—
352#102—1222no—
353#102—1222no—
354#102—1222no—
355#102—1222no—
356C22⋊C8—22no—
357C22⋊C8C22⋊D6112yes—
358C4×C8—22no—
359C4×C8C22×D6112yes—
360C24—2441no—
361#41—22no—
362#41#1175819014yescomplemented
363C2×Q8—1221no—
364C2×Q8—1221no—
365C2×Q8C2×D24411yescharacteristic
366C2×D4—21no—
367C2×D4—1222no—
368C2×D4—1222no—
369C2×D4—1222no—
370C2×D4—4881no—
371C2×D4—4881no—
372C2×D4—4882no—
373C2×D4—2441no—
374C2×D4—2441no—
375C2×D4—2441no—
376C2×D4—2442no—
377C2×D4—2442no—
378C2×D4C2×D24411yescharacteristic
379C22×C4—21no—
380C22×C4—21no—
381C22×C4—21no—
382C22×C4C2×D24411yescharacteristic
383C22×C4#1175822211yescharacteristic
384C22×C4C2×C22⋊D6111yescharacteristic
385SD16—4882no—
386SD16—4882no—
387SD16—2444no—
388D8—2444no—
389M16—2442no—
390C2×C8—22no—
391C2×C8—82no—
392C2×C8—81no—
393C2×C8—1222no—
394C2×C8—1221no—
395C2×C8—1221no—
396C2×C8—2442no—
397C2×C8C2×C22⋊D6112yes—
398C2×C8C2×C22⋊D6112yes—
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×D24411yescharacteristic
408C4⋊C4C2×D24411yescharacteristic
409C22⋊C4—21no—
410C22⋊C4—21no—
411C22⋊C4—21no—
412C22⋊C4—2441no—
413C42—21no—
414C42—21no—
415C42—21no—
416C42—2441no—
417C42C2×D24411yescharacteristic
418C42#1175822211yescharacteristic
419C42C2×C22⋊D6111yescharacteristic
420C42C23×D6111yescharacteristic
421C23—21no—
422C23—4882no—
423C23—4882no—
424C23—4882no—
425C23—2442no—
426C23#1451975111yescharacteristic
427Q8—21no—
428Q8—4881no—
429Q8—2441no—
430Q8—2441no—
431Q8C2×#1175819012yes—
432D4—24no—
433D4—21no—
434D4—4882no—
435D4—4882no—
436D4—4882no—
437D4—4882no—
438D4—2442no—
439D4—2442no—
440D4—2442no—
441D4—2442no—
442D4C2×#1175819012yes—
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#1451975111yescharacteristic
456C2×C4#1451975111yescharacteristic
457C2×C4#1451975111yescharacteristic
458C2×C4C2×#1175819012yes—
459C2×C4#1451979912yes—
460C2×C4C22×C22⋊D6111yescharacteristic
461C2×C4C22×D24411yescharacteristic
462C2×C4C2×#1175822211yescharacteristic
463C8—22no—
464C8—22no—
465C8—42no—
466C8—2442no—
467C8—2441no—
468C8—2441no—
469C22—21no—
470C22—21no—
471C22—22no—
472C22—41no—
473C22—41no—
474C22—9764no—
475C22—4882no—
476C22—4882no—
477C22—4882no—
478C22#1683999112yes—
479C22C2×#1451975111yescharacteristic
480C4—22no—
481C4—21no—
482C4—21no—
483C4—21no—
484C4—41no—
485C4—41no—
486C4—4881no—
487C4—4881no—
488C4#1683999112yes—
489C4C22×#1175819011yescharacteristic
490C4C2×#1451979911yescharacteristic
491C4#1684056511yescharacteristic
492C4#1684057011yescharacteristic
493C2—22no—
494C2—42no—
495C2—9762no—
496C2#2145535811yescharacteristic
497C2C2×#1683999111yescharacteristic
498C2#2145554711yescharacteristic
subgroupG/H
centerC22C2×#14519751
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#2988C2
socleC22×C61C2×#121
prime radical#21455299C2
hypercenter#2988D61
hypocenterC61#19409
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2988

Complements

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

NH ≅ G/Ntype
1. #21456762C2semidirect
2. #21455607C2semidirect
4. #21445235C2semidirect
5. #21445228C2semidirect
6. #21445167C2semidirect
7. #21445160C2semidirect
8. #21444590C2semidirect
9. #21444507C2semidirect
10. C2×#16839505C2semidirect
11. C61×#2988C2semidirect
19. #16840057C22semidirect
20. #16840052C22semidirect
21. #16840047C22semidirect
22. #16840042C22semidirect
35. #16839513C22semidirect
36. #16839513C22semidirect
37. #16839510C22semidirect
38. #16839509C22semidirect
42. C8×Q8×C61C22semidirect
44. C8×D4×C61C22semidirect
112. #41×C61SD16semidirect
247. #2988D61semidirect
280. C8×Q8C2×D61semidirect
282. C8×D4C2×D61semidirect
298. C61#19409semidirect
362. #41#11758190semidirect

Maximal subgroups

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

subgroupindex
#214567622
#214556072
#214552992
#214452352
#214452282
#214451672
#214451602
#214445902
#214445072
C2×#168395052
C61×#29882
#1940961