Finite Group Databasebeta

← #29663632

Subgroups — #29663632

Order 15616.

636 conjugacy classes of subgroups.

The subgroup lattice has 444 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21457161C211yesmaximal, characteristic, complemented
2#21456762C211yesmaximal, characteristic
3#21456195C211yesmaximal, characteristic
4#21454762C211yesmaximal, characteristic, complemented
5#21445390C212yesmaximal, complemented
6#21445362C212yesmaximal, complemented
7#21445266C212yesmaximal, complemented
8#21445238C212yesmaximal, complemented
9#21444572C211yesmaximal, characteristic, complemented
10C2×#16839505C211yesmaximal, characteristic
11C61×#1542C211yesmaximal, characteristic, complemented
12#16840319C2211yescharacteristic
13C2×#14519774C2211yescharacteristic
14#16840135—21no—
15#16840128—22no—
16#16840110C2212yes—
17#16840110C2212yes—
18#16840109—21no—
19#16840057—21no—
20#16840054C2212yescomplemented
21#16840001—22no—
22#16839972—21no—
23#16839969C2212yescomplemented
24#16839963C2212yes—
25#16839963C2212yes—
26#16839915—22no—
27#16839891—22no—
28#16839889C2212yescomplemented
29#16839886C2212yescomplemented
30#16839788—21no—
31C2×C4⋊#92418C2211yescharacteristic
32#16839766—21no—
33#16839766—21no—
34#16839717—21no—
35C4×#11758194C2211yescharacteristic
36#16839514C2212yescomplemented
37#16839512C2212yescomplemented
38#16839505C2212yes—
39#16839505C2212yes—
40C61×#533C2211yescharacteristic
41C61×#431C2211yescharacteristic
42C2×C4⋊C8×C61C2211yescharacteristic
43C61×#333C2211yescharacteristic, complemented
44C61×#333C2211yescharacteristic, complemented
45C61×#332C2211yescharacteristic, complemented
46C61×#332C2211yescharacteristic, complemented
47C22×D244—21no—
48C22×D244—21no—
49C22×D244—21no—
50C2×#11758202—22no—
51C2×#11758190—22no—
52C2×#11758190—22no—
53C2×#11758194D411yescharacteristic
54C2×#11758194D411yescharacteristic
55C22×#92418—21no—
56#14519777D411yescharacteristic
57#14519777D411yescharacteristic
58#14519774—21no—
59#14519774—21no—
60#14519774—21no—
61#14519774C2312yes—
62#14519769—21no—
63#14519756—21no—
64#14519756D412yes—
65C4×Dic122—21no—
66C4×Dic122C2312yes—
67#14519736—21no—
68#14519710—22no—
69C4⋊#92418—21no—
70C4⋊#92418C2312yes—
71#14519706—22no—
72#14519706—22no—
73#14519701—22no—
74#14519700—22no—
75#14519699—22no—
76#14519699—22no—
77#14519697—22no—
78#14519697—22no—
79#14519697—22no—
80#14519697—22no—
81#14519688D412yes—
82#14519687—22no—
83C4×#92418C2312yes—
84C8×Dic61—22no—
85C22×C8×C61—21no—
86C4⋊Q8×C61C2311yescharacteristic
87C4⋊Q8×C61C2311yescharacteristic
88#125×C61—21no—
89C22⋊Q8×C61—21no—
90C22⋊Q8×C61—21no—
91C4×Q8×C61—21no—
92C4×Q8×C61—21no—
93#117×C61—21no—
94C2×C42×C61C2311yescharacteristic
95C4⋊C8×C61—21no—
96C4⋊C8×C61C2311yescharacteristic
97C4⋊C8×C61C2311yescharacteristic
98C4⋊C8×C61C2311yescharacteristic
99C4⋊C8×C61C2311yescharacteristic
100C22⋊C8×C61—21no—
101C4×C8×C61—21no—
102C23×D61—41no—
103C2×Q8×C61—21no—
104C2×Q8×C61—21no—
105C22×C4×C61C2×D411yescharacteristic
106C22×C4×C61C2×D411yescharacteristic
107C22×C4×C61#4111yescharacteristic
108C2×D244—22no—
109C2×D244—22no—
110C2×D244—22no—
111C2×D244—42no—
112C2×D244—42no—
113C2×D244—41no—
114C2×D244—41no—
115C2×D244—41no—
116C2×D244—81no—
117C2×D244—81no—
118C2×D244—82no—
119C2×Dic122—21no—
120C2×Dic122C2×D412yes—
121C2×C8×C61—21no—
122C2×C8×C61—21no—
123C2×C8×C61—21no—
124C2×C8×C61—21no—
125C2×C8×C61—42no—
126C4⋊C4×C61—21no—
127C4⋊C4×C61—21no—
128C4⋊C4×C61—21no—
129C4⋊C4×C61—21no—
130C4⋊C4×C61—41no—
131C4⋊C4×C61—41no—
132C22⋊C4×C61—41no—
133C42×C61—41no—
134C42×C61C2×D411yescharacteristic
135C42×C61C2×D411yescharacteristic
136C42×C61#4111yescharacteristic
137C42×C61C2411yescharacteristic
138C22⋊Dic61—22no—
139#11758202—82no—
140C4⋊Dic61—21no—
141C4⋊Dic61—21no—
142C4⋊Dic61—21no—
143C4⋊Dic61C2×D412yes—
144#11758196—22no—
145#11758196—22no—
146C4×Dic61—22no—
147#11758194—22no—
148#11758194—22no—
149C2×#92418—22no—
150C2×#92418—81no—
151C2×#92418C2×D412yes—
152C2×#92418C2×D412yes—
153#11758190—82no—
154#11758190—82no—
155C23×C61#12111yescharacteristic
156C22×D61—42no—
157C22×D61—82no—
158C22×D61—82no—
159C22×D61—82no—
160Q8×C61—41no—
161Q8×C61—41no—
162Q8×C61—81no—
163C2×C4×C61—21no—
164C2×C4×C61—21no—
165C2×C4×C61—41no—
166C2×C4×C61—41no—
167C2×C4×C61—41no—
168C2×C4×C61#12111yescharacteristic
169C2×C4×C61#12111yescharacteristic
170C2×C4×C61#12111yescharacteristic
171C2×C4×C61#13512yes—
172C2×C4×C61#13712yes—
173C2×C4×C61C22×D411yescharacteristic
174C2×C4×C61C22×D411yescharacteristic
175C2×C4×C61C2×#4111yescharacteristic
176C2×Dic61—22no—
177C2×Dic61—22no—
178D244—42no—
179D244—42no—
180D244—42no—
181D244—42no—
182D244—82no—
183D244—82no—
184D244—82no—
185D244—82no—
186Dic122—22no—
187Dic122—22no—
188Dic122—81no—
189C8×C61—41no—
190C8×C61—41no—
191C8×C61—41no—
192C8×C61—41no—
193#92418—22no—
194#92418—22no—
195#92418—42no—
196#19242—611nomaximal, Sylow
197C22×C61—21no—
198C22×C61—21no—
199C22×C61#46412yes—
200C22×C61C2×#12111yescharacteristic
201C2×D61—82no—
202C2×D61—82no—
203C2×D61—82no—
204C2×D61—164no—
205C4×C61—21no—
206C4×C61—21no—
207C4×C61—21no—
208C4×C61—81no—
209C4×C61—81no—
210C4×C61#46412yes—
211C4×C61C2×#13511yescharacteristic
212C4×C61C2×#13711yescharacteristic
213C4×C61#57611yescharacteristic
214C4×C61#57711yescharacteristic
215Dic61—42no—
216Dic61—42no—
217#3249—611no—
218#3208—611no—
219#3138—611no—
220#2962—611no—
221#1702—612no—
222#1698—612no—
223#1682—612no—
224#1678—612no—
225#1565—611no—
226#1542D6111yescharacteristic, complemented
227C2×#331—611no—
228C2×C61—22no—
229C2×C61C2×#46411yescharacteristic
230C2×C61#306211yescharacteristic
231C2×C61#306811yescharacteristic
232D61—162no—
233#533—611no—
234#533C2×D6111yescharacteristic
235C2×#127—611no—
236#492—1221no—
237#490—1222no—
238#486—612no—
239#486—612no—
240#486—1221no—
241#477—612no—
242#477—1221no—
243#467—1222no—
244#462—612no—
245#462—1221no—
246#460—612no—
247#460—612no—
248#450—1222no—
249#446—612no—
250#446—1222no—
251C8×Q8—612no—
252#431C2×D6111yescharacteristic
253#424—1221no—
254C2×C4⋊C8—611no—
255C2×C4⋊C8C2×D6111yescharacteristic
256#418—1221no—
257#418—1221no—
258#405—1221no—
259C4×M16—611no—
260#335—612no—
261#334—612no—
262#333C2×D6111yescharacteristic, complemented
263#333C2×D6111yescharacteristic, complemented
264#332C2×D6111yescharacteristic, complemented
265#332C2×D6111yescharacteristic, complemented
266#331—612no—
267#331—612no—
268C61#1924211yescharacteristic, complemented, Sylow
269C22×D4—1221no—
270C22×D4—1221no—
271C22×D4—1221no—
272C2×Q16—1222no—
273C2×SD16—1222no—
274C2×SD16—1222no—
275C2×M16—611no—
276C2×M16—611no—
277C22×C8—21no—
278C22×C8—1221no—
279C4⋊Q8—611no—
280C4⋊Q8—611no—
281C4⋊Q8C22×D6111yescharacteristic
282C4⋊Q8C22×D6111yescharacteristic
283#127—612no—
284#127—1221no—
285#127—1221no—
286#127—1221no—
287#125—21no—
288#125—1221no—
289C22⋊Q8—21no—
290C22⋊Q8—21no—
291C22⋊Q8—612no—
292C22⋊Q8—1221no—
293C4×Q8—21no—
294C4×Q8—21no—
295C4×Q8—612no—
296C4×Q8—1221no—
297#117—21no—
298#117—1221no—
299C2×C42C22×D6111yescharacteristic
300#107—1222no—
301C4⋊C8—21no—
302C4⋊C8—612no—
303C4⋊C8—1222no—
304C4⋊C8—1222no—
305C4⋊C8—1221no—
306C4⋊C8C22×D6111yescharacteristic
307C4⋊C8C22×D6111yescharacteristic
308C4⋊C8C22×D6111yescharacteristic
309C4⋊C8C22×D6111yescharacteristic
310#103—1222no—
311#103—1222no—
312#102—1222no—
313#102—1222no—
314#102—1222no—
315#102—1222no—
316#102—1222no—
317#102—1222no—
318C22⋊C8—21no—
319#97—612no—
320#97—1222no—
321C4×C8—21no—
322C4×C8—612no—
323C4×C8—1222no—
324C24—2441no—
325C2×Q8—21no—
326C2×Q8—21no—
327C2×Q8—612no—
328C2×Q8—1221no—
329C2×D4—1222no—
330C2×D4—1222no—
331C2×D4—1222no—
332C2×D4—4881no—
333C2×D4—4881no—
334C2×D4—4882no—
335C2×D4—2442no—
336C2×D4—2441no—
337C2×D4—2441no—
338C2×D4—2441no—
339C2×D4—2442no—
340C22×C4#1175822211yescharacteristic
341C22×C4D4×D6111yescharacteristic
342C22×C4D4×D6111yescharacteristic
343Q16—4882no—
344SD16—4882no—
345SD16—4882no—
346M16—1222no—
347M16—1222no—
348C2×C8—21no—
349C2×C8—21no—
350C2×C8—21no—
351C2×C8—21no—
352C2×C8—42no—
353C2×C8—612no—
354C2×C8—612no—
355C2×C8—1222no—
356C2×C8—4881no—
357C4⋊C4—21no—
358C4⋊C4—21no—
359C4⋊C4—21no—
360C4⋊C4—21no—
361C4⋊C4—41no—
362C4⋊C4—41no—
363C4⋊C4—612no—
364C4⋊C4—1222no—
365C4⋊C4—1222no—
366C4⋊C4—1221no—
367C4⋊C4—1221no—
368C4⋊C4—1221no—
369C22⋊C4—41no—
370C22⋊C4—1222no—
371C42—41no—
372C42—1222no—
373C42#1175822211yescharacteristic
374C42D4×D6111yescharacteristic
375C42D4×D6111yescharacteristic
376C42C23×D6111yescharacteristic
377C23—4882no—
378C23—4882no—
379C23—4882no—
380C23—2442no—
381C23#1451975011yescharacteristic
382Q8—41no—
383Q8—41no—
384Q8—81no—
385Q8—1222no—
386Q8—1222no—
387Q8—4881no—
388D4—4882no—
389D4—4882no—
390D4—4882no—
391D4—4882no—
392D4—2442no—
393D4—2442no—
394D4—2442no—
395D4—2442no—
396C2×C4—21no—
397C2×C4—21no—
398C2×C4—41no—
399C2×C4—41no—
400C2×C4—41no—
401C2×C4—1222no—
402C2×C4—1222no—
403C2×C4#1451975011yescharacteristic
404C2×C4#1451975011yescharacteristic
405C2×C4#1451975011yescharacteristic
406C2×C4#1451979812yes—
407C2×C4#1451980912yes—
408C2×C4C2×D4×D6111yescharacteristic
409C2×C4C2×D4×D6111yescharacteristic
410C2×C4C2×#1175822211yescharacteristic
411C8—41no—
412C8—41no—
413C8—41no—
414C8—41no—
415C8—1222no—
416C8—1222no—
417C8—2442no—
418C22—21no—
419C22—21no—
420C22—9764no—
421C22—4882no—
422C22—4882no—
423C22—4882no—
424C22#1683998712yes—
425C22C2×#1451975011yescharacteristic
426C4—21no—
427C4—21no—
428C4—21no—
429C4—81no—
430C4—81no—
431C4—2442no—
432C4—2442no—
433C4#1683998712yes—
434C4C2×#1451979811yescharacteristic
435C4C2×#1451980911yescharacteristic
436C4#1684056411yescharacteristic
437C4#1684056911yescharacteristic
438C2—22no—
439C2—9762no—
440C2C2×#1683998711yescharacteristic
441C2#2145550211yescharacteristic
442C2#2145555011yescharacteristic
subgroupG/H
centerC22C2×#14519750
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#1542C2
socleC22×C61C2×#121
prime radicalC2×#14519774C22
hypercenter#1542D61
hypocenterC61#19242
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1542

Complements

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

NH ≅ G/Ntype
1. #21457161C2semidirect
4. #21454762C2semidirect
5. #21445390C2semidirect
6. #21445362C2semidirect
7. #21445266C2semidirect
8. #21445238C2semidirect
9. #21444572C2semidirect
11. C61×#1542C2semidirect
20. #16840054C22semidirect
23. #16839969C22semidirect
28. #16839889C22semidirect
29. #16839886C22semidirect
36. #16839514C22semidirect
37. #16839512C22semidirect
43. C61×#333C22semidirect
44. C61×#333C22semidirect
45. C61×#332C22semidirect
46. C61×#332C22semidirect
226. #1542D61semidirect
262. #333C2×D61semidirect
263. #333C2×D61semidirect
264. #332C2×D61semidirect
265. #332C2×D61semidirect
268. C61#19242semidirect

Maximal subgroups

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

subgroupindex
#214571612
#214567622
#214561952
#214547622
#214453902
#214453622
#214452662
#214452382
#214445722
C2×#168395052
C61×#15422
#1924261