Finite Group Databasebeta

← #30648313

Subgroups — #30648313

Order 16256.

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

normal subgroups only

This group has 520 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×#16861232C211yesmaximal, characteristic, complemented
2C22×C8×D127C211yesmaximal, characteristic, complemented
3#22550389C212yesmaximal, complemented
4#22550114C211yesmaximal, characteristic, complemented
5#22550113C211yesmaximal, characteristic, complemented
6#22550112C211yesmaximal, characteristic, complemented
7#22550108C212yesmaximal, complemented
8#22550107C212yesmaximal, complemented
9#22549913C211yesmaximal, characteristic, complemented
10#22549871C212yesmaximal, complemented
11#467×C127C211yesmaximal, characteristic, complemented
12C2×#14523249C2212yes—
13C22×C4×D127C2211yescharacteristic
14#16861232—24no—
15C2×#14523235C2212yescomplemented
16C2×D1016C2211yescharacteristic, complemented
17C22×C127⋊C8C2211yescharacteristic
18C2×C8×D127—24no—
19C2×C8×D127C2211yescharacteristic
20C2×C8×D127C2211yescharacteristic
21C2×C8×D127C2211yescharacteristic
22C2×C8×D127C2211yescharacteristic
23#16861194—22no—
24#16861185C2212yes—
25#16861184C2212yes—
26#16861183C2212yes—
27#16861179—22no—
28#16861171—22no—
29(C22⋊D127)⋊C4—22no—
30#16861167C2212yes—
31#16861143C2211yescharacteristic, complemented
32#16861140C2212yescomplemented
33#16861133C2212yescomplemented
34#16861131C2212yescomplemented
35#16861129C2212yescomplemented
36C2×Q16×C127C2211yescharacteristic, complemented
37C22×C8×C127C2211yescharacteristic
38C22⋊Q8×C127C2212yes—
39#107×C127C2211yescharacteristic, complemented
40#103×C127C2212yescomplemented
41C23×D127D411yescharacteristic
42#14523249—24no—
43#14523249—24no—
44C2×D508—22no—
45C2×D508—22no—
46C2×D508C2312yes—
47C2×C22⋊D127—22no—
48C22×Dic127D411yescharacteristic
49C2×C4×D127—22no—
50C2×C4×D127—22no—
51C2×C4×D127—22no—
52C2×C4×D127D411yescharacteristic
53C2×C4×D127D411yescharacteristic
54C2×C4×D127C2311yescharacteristic
55C2×C4×D127C2311yescharacteristic
56#14523235—24no—
57D1016—22no—
58C2×C127⋊C8—22no—
59C2×C127⋊C8C2311yescharacteristic
60C2×C127⋊C8C2311yescharacteristic
61C8×D127—22no—
62C8×D127—22no—
63C8×D127—22no—
64C8×D127D414yescomplemented
65C4⋊Dic127C2312yes—
66Dic127⋊C4—22no—
67(C2×D127)⋊C4—22no—
68(C2×D127)⋊C4—22no—
69(C2×D127)⋊C4—22no—
70(C2×D127)⋊C4—22no—
71C4×Dic127—22no—
72C2×Q8×C127C2312yes—
73C22×C4×C127C2311yescharacteristic
74Q16×C127—22no—
75C2×C8×C127—22no—
76C2×C8×C127C2311yescharacteristic
77C2×C8×C127C2311yescharacteristic
78C4⋊C4×C127—22no—
79C4⋊C4×C127C2312yes—
80C22⋊C4×C127—22no—
81C22×D127—22no—
82C22×D127—22no—
83C22×D127—22no—
84C22×D127—22no—
85C22×D127C2×D411yescharacteristic
86C22×D127C2×D411yescharacteristic
87D508—22no—
88D508—22no—
89D508—44no—
90D508—44no—
91C22⋊D127—44no—
92C2×Dic127—22no—
93C2×Dic127—22no—
94C2×Dic127C2×D411yescharacteristic
95C2×Dic127C2×D411yescharacteristic
96C4×D127—22no—
97C4×D127—21no—
98C4×D127—21no—
99C4×D127—22no—
100C4×D127—44no—
101C4×D127C2×D412yes—
102C4×D127#4112yes—
103C127⋊C8—21no—
104C127⋊C8C2×D412yes—
105C23×C127C2×D411yescharacteristic
106Q8×C127—22no—
107Q8×C127—22no—
108C2×C4×C127—21no—
109C2×C4×C127—21no—
110C2×C4×C127—22no—
111C2×C4×C127—22no—
112C2×C4×C127C2×D411yescharacteristic
113C2×C4×C127C2411yescharacteristic
114C8×C127—21no—
115C8×C127C2×D412yes—
116C22×C127—21no—
117C22×C127—21no—
118C22×C127#13512yescomplemented
119C22×C127C22×D411yescharacteristic
120C2×D127—28no—
121C2×D127—21no—
122C2×D127—21no—
123C2×D127—21no—
124C2×D127—21no—
125C2×D127—42no—
126C2×D127—42no—
127C2×D127—42no—
128C2×D127#12112yes—
129C2×D127#13512yescomplemented
130C4×C127—21no—
131C4×C127—42no—
132C4×C127—42no—
133C4×C127C22×D411yescharacteristic
134C4×C127C2×#4111yescharacteristic
135Dic127—21no—
136Dic127—42no—
137Dic127#12112yes—
138C2×C127—22no—
139C2×C127C2×#12111yescharacteristic
140C2×C127C2×#13511yescharacteristic
141C2×C127C2×#13511yescharacteristic
142D127—22no—
143D127—22no—
144D127—82no—
145#3050—1271nomaximal, Sylow
146C127#305011yescharacteristic, complemented, Sylow
147C2×#135—1271no—
148C23×C8—1271no—
149#525—1272no—
150#467—1271no—
151#467D12711yescharacteristic, complemented
152#466—1271no—
153#466—1271no—
154#465—1272no—
155#465—1272no—
156#427—1271no—
157#416—1272no—
158C2×#41—1272no—
159C23×C4—1271no—
160#135—2544no—
161C2×Q16C2×D12711yescharacteristic, complemented
162C2×SD16—1272no—
163C2×D8—1271no—
164C22×C8—1271no—
165C22×C8—1271no—
166C22×C8—1271no—
167C22×C8—1271no—
168C22×C8—1271no—
169C22×C8—2544no—
170C22×C8C2×D12711yescharacteristic
171#123—2542no—
172C22⋊Q8—1272no—
173C22⋊Q8C2×D12712yes—
174#121—1272no—
175#121—1272no—
176#120—2542no—
177C4×D4—2542no—
178C4×D4—2542no—
179#117—1272no—
180#107—1271no—
181#107C2×D12711yescharacteristic, complemented
182#106—1272no—
183#103—1272no—
184#103C2×D12712yescomplemented
185#102—1272no—
186#102—1272no—
187C24—1271no—
188#41—2544no—
189#41—2544no—
190C2×Q8C22×D12712yes—
191C2×D4—1272no—
192C2×D4—2542no—
193C2×D4—2542no—
194C2×D4—2542no—
195C22×C4—1271no—
196C22×C4—1271no—
197C22×C4—1271no—
198C22×C4—1271no—
199C22×C4—1271no—
200C22×C4—2542no—
201C22×C4—2542no—
202C22×C4—2542no—
203C22×C4C22×D12711yescharacteristic
204Q16—22no—
205SD16—2544no—
206D8—2542no—
207C2×C8—22no—
208C2×C8—1271no—
209C2×C8—1274no—
210C2×C8—1271no—
211C2×C8—2542no—
212C2×C8—2542no—
213C2×C8—2542no—
214C2×C8—2542no—
215C2×C8C22×D12711yescharacteristic
216C2×C8C22×D12711yescharacteristic
217C4⋊C4—22no—
218C4⋊C4—1272no—
219C4⋊C4—2542no—
220C4⋊C4C22×D12712yes—
221C22⋊C4—22no—
222C22⋊C4—2542no—
223C22⋊C4—2542no—
224C22⋊C4—2542no—
225C22⋊C4—2542no—
226C42—2542no—
227C23—1271no—
228C23—1271no—
229C23—2542no—
230C23—2542no—
231C23—2542no—
232C23—2542no—
233C23D4×D12711yescharacteristic
234Q8—22no—
235Q8—22no—
236D4—5084no—
237D4—5084no—
238D4—5084no—
239D4—2542no—
240D4—2542no—
241C2×C4—21no—
242C2×C4—21no—
243C2×C4—22no—
244C2×C4—22no—
245C2×C4—1271no—
246C2×C4—1272no—
247C2×C4—1272no—
248C2×C4—1271no—
249C2×C4—5084no—
250C2×C4—2542no—
251C2×C4—2542no—
252C2×C4—2542no—
253C2×C4—2542no—
254C2×C4—2541no—
255C2×C4—2541no—
256C2×C4D4×D12711yescharacteristic
257C2×C4C23×D12711yescharacteristic
258C8—21no—
259C8—1272no—
260C8—2541no—
261C8D4×D12712yes—
262C22—21no—
263C22—21no—
264C22—1272no—
265C22—1272no—
266C22—5082no—
267C22—5082no—
268C22—5082no—
269C22—2548no—
270C22—2541no—
271C22—2541no—
272C22—2541no—
273C22—2541no—
274C22#1686123212yescomplemented
275C22C2×D4×D12711yescharacteristic
276C4—21no—
277C4—42no—
278C4—42no—
279C4—1272no—
280C4—5082no—
281C4—2541no—
282C4C2×D4×D12711yescharacteristic
283C4#41×D12711yescharacteristic
284C2—22no—
285C2—10162no—
286C2—2542no—
287C2—2542no—
288C2#121×D12711yescharacteristic
289C2#135×D12711yescharacteristic
290C2C2×#1686123211yescharacteristic
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#467×C127C2
socleC22×C127C22×D4
hypercenter#467D127
hypocenterC127#3050
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ #467

Complements

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

NH ≅ G/Ntype
1. C2×#16861232C2semidirect
2. C22×C8×D127C2semidirect
3. #22550389C2semidirect
4. #22550114C2semidirect
5. #22550113C2semidirect
6. #22550112C2semidirect
7. #22550108C2semidirect
8. #22550107C2semidirect
9. #22549913C2semidirect
10. #22549871C2semidirect
11. #467×C127C2semidirect
15. C2×#14523235C22semidirect
16. C2×D1016C22semidirect
31. #16861143C22semidirect
32. #16861140C22semidirect
33. #16861133C22semidirect
34. #16861131C22semidirect
35. #16861129C22semidirect
36. C2×Q16×C127C22semidirect
39. #107×C127C22semidirect
40. #103×C127C22semidirect
64. C8×D127D4semidirect
118. C22×C127#135semidirect
129. C2×D127#135semidirect
146. C127#3050semidirect
151. #467D127semidirect
161. C2×Q16C2×D127semidirect
181. #107C2×D127semidirect
184. #103C2×D127semidirect
274. C22#16861232semidirect

Maximal subgroups

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

subgroupindex
C2×#168612322
C22×C8×D1272
#225503892
#225501142
#225501132
#225501122
#225501082
#225501072
#225499132
#225498712
#467×C1272
#3050127