Finite Group Databasebeta

← #30648207

Subgroups — #30648207

Order 16256.

584 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 584 conjugacy classes of subgroups — too many to draw. Only its 133 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16861236C212yesmaximal, complemented
2#127×D127C211yesmaximal, characteristic, complemented
3C4×#14523249C211yesmaximal, characteristic, complemented
4#22550076C211yesmaximal, characteristic, complemented
5#22550074C212yesmaximal, complemented
6#22550072C211yesmaximal, characteristic, complemented
7#22550070C212yesmaximal, complemented
8#22550068C211yesmaximal, characteristic, complemented
9#22549898C211yesmaximal, characteristic, complemented
10#22549879C212yesmaximal, complemented
11#460×C127C211yesmaximal, characteristic, complemented
12C2×#14523249C2211yescharacteristic
13C2×D4×D127—21no—
14C2×D4×D127—21no—
15C2×D4×D127C2212yes—
16#16861236—28no—
17C2×#14523235C2212yescomplemented
18C2×D1016C2212yescomplemented
19C2×#14523233C2212yescomplemented
20C2×#14523231C2212yes—
21#16861206C2211yescharacteristic
22#16861205—21no—
23#16861205—21no—
24#16861205C2212yes—
25Q8×Dic127C2211yescharacteristic, complemented
26C4×D508—21no—
27C4×D508C2211yescharacteristic, complemented
28#16861171—21no—
29#16861171C2211yescharacteristic, complemented
30#16861167—21no—
31#16861167C2211yescharacteristic
32C42×D127—21no—
33C42×D127C2211yescharacteristic
34C4⋊(C127⋊C8)C2211yescharacteristic, complemented
35Dic127⋊C8C2212yescomplemented
36#16861131C2212yescomplemented
37#16861130C2212yescomplemented
38#16861129C2212yescomplemented
39C2×SD16×C127C2212yescomplemented
40#127×C127C2211yescharacteristic
41C4×Q8×C127C2211yescharacteristic, complemented
42C4⋊C8×C127C2211yescharacteristic, complemented
43#102×C127C2212yescomplemented
44C23×D127—22no—
45#14523249—22no—
46#14523249D414yescomplemented
47C2×D508—21no—
48C2×D508—21no—
49C2×D508—21no—
50C2×D508C2312yes—
51C2×D508C2311yescharacteristic
52C2×C22⋊D127—22no—
53C2×C22⋊D127—22no—
54C2×C22⋊D127—22no—
55D4×D127—24no—
56D4×D127—24no—
57D4×D127—44no—
58D4×D127—44no—
59C2×C4×D127—21no—
60C2×C4×D127—21no—
61C2×C4×D127—21no—
62C2×C4×D127D411yescharacteristic
63C2×C4×D127D411yescharacteristic
64C2×C4×D127C2311yescharacteristic
65#14523235—24no—
66D1016—24no—
67#14523233—24no—
68#14523231—24no—
69C2×C127⋊C8C2312yes—
70C4⋊Dic127—21no—
71C4⋊Dic127C2311yescharacteristic
72(C2×D127)⋊C4—21no—
73(C2×D127)⋊C4—21no—
74(C2×D127)⋊C4—21no—
75C4×Dic127—21no—
76C4×Dic127—21no—
77C4×Dic127D411yescharacteristic
78C4×Dic127D411yescharacteristic
79C4×Dic127C2311yescharacteristic
80C2×Q8×C127C2311yescharacteristic
81C2×D4×C127—21no—
82C2×D4×C127—21no—
83C2×D4×C127C2312yes—
84SD16×C127—24no—
85C2×C8×C127C2312yes—
86C4⋊C4×C127—21no—
87C4⋊C4×C127C2311yescharacteristic
88C42×C127—21no—
89C42×C127C2311yescharacteristic
90C22×D127—21no—
91C22×D127—22no—
92C22×D127—44no—
93C22×D127—44no—
94C22×D127—44no—
95C22×D127C2×D411yescharacteristic
96D508—24no—
97D508—22no—
98D508—21no—
99D508—22no—
100D508—42no—
101D508—42no—
102D508C2×D412yescomplemented
103C22⋊D127—44no—
104C22⋊D127—44no—
105C22⋊D127—44no—
106C2×Dic127—21no—
107C2×Dic127C2×D411yescharacteristic
108C2×Dic127C2×D411yescharacteristic
109C2×Dic127C2×D411yescharacteristic
110C4×D127—22no—
111C4×D127—24no—
112C4×D127—22no—
113C4×D127—42no—
114C4×D127—42no—
115C4×D127C2×D412yes—
116C4×D127#4112yes—
117C127⋊C8—22no—
118C23×C127—22no—
119Q8×C127—21no—
120Q8×C127C2×D412yescomplemented
121D4×C127—22no—
122D4×C127—22no—
123D4×C127—42no—
124D4×C127—42no—
125C2×C4×C127—21no—
126C2×C4×C127—21no—
127C2×C4×C127C2×D411yescharacteristic
128C2×C4×C127C2×D411yescharacteristic
129C2×C4×C127C2411yescharacteristic
130C8×C127—22no—
131C22×C127—42no—
132C22×C127—42no—
133C22×C127—42no—
134C22×C127C22×D411yescharacteristic
135C2×D127—21no—
136C2×D127—21no—
137C2×D127—22no—
138C2×D127—41no—
139C2×D127—41no—
140C2×D127—42no—
141C2×D127—42no—
142C2×D127—42no—
143C2×D127—88no—
144C2×D127#12112yes—
145C4×C127—22no—
146C4×C127—21no—
147C4×C127—41no—
148C4×C127AGL(1,ℤ/8)12yescomplemented
149C4×C127C22×D411yescharacteristic
150C4×C127C2×#4111yescharacteristic
151Dic127—21no—
152Dic127—41no—
153Dic127#12112yes—
154Dic127AGL(1,ℤ/8)12yescomplemented
155C2×C127—82no—
156C2×C127C2×#12111yescharacteristic
157C2×C127C2×AGL(1,ℤ/8)11yescharacteristic
158C2×C127C2×AGL(1,ℤ/8)11yescharacteristic
159D127—22no—
160D127—42no—
161D127—82no—
162#3038—1271nomaximal, Sylow
163C127#303811yescharacteristic, complemented, Sylow
164C2×AGL(1,ℤ/8)—1272no—
165C2×#127—1271no—
166C4×#41—1271no—
167#460—1272no—
168#460—1271no—
169#460D12711yescharacteristic, complemented
170#459—1271no—
171#459—1271no—
172#459—1272no—
173#423—1271no—
174#418—1272no—
175C2×#41—1271no—
176C22×D4—1272no—
177C22×D4—2541no—
178C22×D4—2541no—
179AGL(1,ℤ/8)—2548no—
180C2×SD16—1272no—
181C2×SD16C2×D12712yescomplemented
182C2×D8—1272no—
183C2×D8—1272no—
184C2×M16—1272no—
185#127—1272no—
186#127—1271no—
187#127—2541no—
188#127—2541no—
189#127C2×D12711yescharacteristic
190C4×Q8—1271no—
191C4×Q8C2×D12711yescharacteristic, complemented
192C4×D4—1271no—
193C4×D4—1271no—
194C4×D4—2541no—
195C4×D4—2541no—
196#117—1271no—
197#117—2541no—
198C2×C42—1271no—
199C2×C42—2541no—
200C4⋊C8—1272no—
201C4⋊C8—1271no—
202C4⋊C8C2×D12711yescharacteristic, complemented
203#102—1272no—
204#102—1272no—
205#102—1272no—
206#102C2×D12712yescomplemented
207C24—2542no—
208#41—1274no—
209#41—2542no—
210C2×Q8C22×D12711yescharacteristic
211C2×D4—21no—
212C2×D4—21no—
213C2×D4—1271no—
214C2×D4—1272no—
215C2×D4—5084no—
216C2×D4—5084no—
217C2×D4—2541no—
218C2×D4—2544no—
219C2×D4—2544no—
220C2×D4—2542no—
221C2×D4—2542no—
222C2×D4—2542no—
223C2×D4—2541no—
224C2×D4—2541no—
225C2×D4C22×D12712yes—
226C22×C4—1271no—
227C22×C4—1271no—
228C22×C4—1271no—
229C22×C4—2541no—
230C22×C4—2541no—
231C22×C4—2541no—
232SD16—24no—
233SD16—2544no—
234D8—2544no—
235D8—2544no—
236M16—2544no—
237C2×C8—1272no—
238C2×C8C22×D12712yes—
239C4⋊C4—21no—
240C4⋊C4—1271no—
241C4⋊C4—2541no—
242C4⋊C4C22×D12711yescharacteristic
243C22⋊C4—2541no—
244C22⋊C4—2541no—
245C22⋊C4—2541no—
246C42—21no—
247C42—1271no—
248C42—1271no—
249C42—1271no—
250C42—2541no—
251C42—2541no—
252C42C22×D12711yescharacteristic
253C23—22no—
254C23—1271no—
255C23—5084no—
256C23—5084no—
257C23—5084no—
258C23—2542no—
259C23—2541no—
260Q8—21no—
261Q8D4×D12712yescomplemented
262D4—22no—
263D4—22no—
264D4—42no—
265D4—42no—
266D4—1272no—
267D4—5082no—
268D4—5084no—
269D4—5084no—
270D4—5084no—
271D4—5082no—
272D4—2544no—
273D4—2542no—
274D4—2542no—
275D4—2541no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—1271no—
279C2×C4—1271no—
280C2×C4—1272no—
281C2×C4—1272no—
282C2×C4—1271no—
283C2×C4—5082no—
284C2×C4—5082no—
285C2×C4—2542no—
286C2×C4—2544no—
287C2×C4—2541no—
288C2×C4—2542no—
289C2×C4D4×D12711yescharacteristic
290C2×C4D4×D12711yescharacteristic
291C2×C4C23×D12711yescharacteristic
292C8—22no—
293C8—2542no—
294C22—42no—
295C22—42no—
296C22—42no—
297C22—1272no—
298C22—10168no—
299C22—5082no—
300C22—5082no—
301C22—5082no—
302C22—5081no—
303C22—5081no—
304C22—2542no—
305C22—2541no—
306C22—2541no—
307C22C2×D4×D12711yescharacteristic
308C4—22no—
309C4—21no—
310C4—41no—
311C4—1272no—
312C4—1272no—
313C4—5081no—
314C4—2541no—
315C4#1686123612yescomplemented
316C4C2×D4×D12711yescharacteristic
317C4#41×D12711yescharacteristic
318C2—82no—
319C2—10162no—
320C2—5082no—
321C2—2542no—
322C2#121×D12711yescharacteristic
323C2AGL(1,ℤ/8)×D12711yescharacteristic
324C2C2×#1686123611yescharacteristic
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#460×C127C2
socleC22×C127C22×D4
hypercenter#460D127
hypocenterC127#3038
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C42 ◁ #460

Complements

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

NH ≅ G/Ntype
1. C2×#16861236C2semidirect
2. #127×D127C2semidirect
3. C4×#14523249C2semidirect
4. #22550076C2semidirect
5. #22550074C2semidirect
6. #22550072C2semidirect
7. #22550070C2semidirect
8. #22550068C2semidirect
9. #22549898C2semidirect
10. #22549879C2semidirect
11. #460×C127C2semidirect
17. C2×#14523235C22semidirect
18. C2×D1016C22semidirect
19. C2×#14523233C22semidirect
25. Q8×Dic127C22semidirect
27. C4×D508C22semidirect
29. #16861171C22semidirect
34. C4⋊(C127⋊C8)C22semidirect
35. Dic127⋊C8C22semidirect
36. #16861131C22semidirect
37. #16861130C22semidirect
38. #16861129C22semidirect
39. C2×SD16×C127C22semidirect
41. C4×Q8×C127C22semidirect
42. C4⋊C8×C127C22semidirect
43. #102×C127C22semidirect
46. #14523249D4semidirect
102. D508C2×D4semidirect
120. Q8×C127C2×D4semidirect
148. C4×C127AGL(1,ℤ/8)semidirect
154. Dic127AGL(1,ℤ/8)semidirect
163. C127#3038semidirect
169. #460D127semidirect
181. C2×SD16C2×D127semidirect
191. C4×Q8C2×D127semidirect
202. C4⋊C8C2×D127semidirect
206. #102C2×D127semidirect
261. Q8D4×D127semidirect
315. C4#16861236semidirect

Maximal subgroups

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

subgroupindex
C2×#168612362
#127×D1272
C4×#145232492
#225500762
#225500742
#225500722
#225500702
#225500682
#225498982
#225498792
#460×C1272
#3038127