Finite Group Databasebeta

← #30648675

Subgroups — #30648675

Order 16256.

410 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 410 conjugacy classes of subgroups — too many to draw. Only its 121 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
1C22⋊Q8×D127C211yesmaximal, characteristic
2C2×#16861167C211yesmaximal, characteristic
3#22550219C212yesmaximal, complemented
4#22550215C212yesmaximal, complemented
5#22550211C211yesmaximal, characteristic, complemented
6#22550207C211yesmaximal, characteristic, complemented
7#22550204C211yesmaximal, characteristic, complemented
8#22549917C212yesmaximal, complemented
9#22549883C212yesmaximal, complemented
10#22549838C211yesmaximal, characteristic
11#484×C127C211yesmaximal, characteristic, complemented
12C2×Q8×D127C2211yescharacteristic
13C22×C4×D127C2211yescharacteristic
14C2×#14523231C2212yes—
15#16861189C2211yescharacteristic
16#16861188C2211yescharacteristic
17#16861187—21no—
18#16861186—21no—
19#16861185C2211yescharacteristic
20#16861167—21no—
21#16861167—21no—
22#16861167C2212yes—
23#16861167C2212yes—
24C2×C4⋊Dic127C2211yescharacteristic
25C4⋊C4×D127—21no—
26C4⋊C4×D127C2211yescharacteristic
27C2×(C2×D127)⋊C4—21no—
28C22⋊C4×D127—21no—
29C2×C4×Dic127—21no—
30#16861142C2212yescomplemented
31#16861141C2212yescomplemented
32#16861140C2212yescomplemented
33#16861134C2212yescomplemented
34#16861133C2212yescomplemented
35#16861132C2212yescomplemented
36C22⋊(C127⋊C8)C2211yescharacteristic
37(C2×D127)⋊C8C2212yes—
38C22⋊Q8×C127C2211yescharacteristic
39C2×C4⋊C4×C127C2211yescharacteristic
40#107×C127C2212yescomplemented
41#103×C127C2212yescomplemented
42C22⋊C8×C127C2211yescharacteristic
43C23×D127D411yescharacteristic
44C2×Dic254—21no—
45C2×Dic254C2311yescharacteristic
46Q8×D127—24no—
47C22×Dic127—21no—
48C22×Dic127D411yescharacteristic
49C2×C4×D127—24no—
50C2×C4×D127—21no—
51C2×C4×D127—21no—
52C2×C4×D127D411yescharacteristic
53C2×C4×D127D411yescharacteristic
54C2×C4×D127C2311yescharacteristic
55C2×C4×D127C2311yescharacteristic
56#14523231—24no—
57C2×C127⋊C8C2312yes—
58C4⋊Dic127—21no—
59C4⋊Dic127—21no—
60C4⋊Dic127C2311yescharacteristic
61C4⋊Dic127C2312yes—
62Dic127⋊C4—21no—
63Dic127⋊C4—21no—
64Dic127⋊C4—21no—
65C22⋊Dic127—21no—
66(C2×D127)⋊C4—21no—
67(C2×D127)⋊C4—21no—
68(C2×D127)⋊C4—22no—
69(C2×D127)⋊C4—22no—
70C4×Dic127—22no—
71C4×Dic127—22no—
72C2×Q8×C127C2311yescharacteristic
73C22×C4×C127—21no—
74C22×C4×C127C2311yescharacteristic
75C2×C8×C127C2312yes—
76C4⋊C4×C127—21no—
77C4⋊C4×C127—21no—
78C4⋊C4×C127C2311yescharacteristic
79C4⋊C4×C127C2312yes—
80C22⋊C4×C127—21no—
81C22×D127—22no—
82C22×D127—22no—
83C22×D127—41no—
84C22×D127C2×D411yescharacteristic
85C22×D127C2×D411yescharacteristic
86Dic254—22no—
87Dic254—42no—
88C2×Dic127—22no—
89C2×Dic127—22no—
90C2×Dic127—21no—
91C2×Dic127—21no—
92C2×Dic127—42no—
93C2×Dic127C2×D411yescharacteristic
94C2×Dic127C2×D411yescharacteristic
95C4×D127—22no—
96C4×D127—22no—
97C4×D127—22no—
98C4×D127—42no—
99C4×D127—42no—
100C4×D127#4114yes—
101C127⋊C8—22no—
102C23×C127C2×D411yescharacteristic
103Q8×C127—22no—
104C2×C4×C127—22no—
105C2×C4×C127—21no—
106C2×C4×C127—21no—
107C2×C4×C127—22no—
108C2×C4×C127—42no—
109C2×C4×C127C2×D411yescharacteristic
110C2×C4×C127C2411yescharacteristic
111C8×C127—22no—
112C22×C127—21no—
113C22×C127—21no—
114C22×C127#13712yes—
115C22×C127C22×D411yescharacteristic
116C2×D127—21no—
117C2×D127—21no—
118C2×D127—21no—
119C2×D127—21no—
120C2×D127—21no—
121C2×D127—44no—
122C2×D127#12312yes—
123C4×C127—21no—
124C4×C127—41no—
125C4×C127—42no—
126C4×C127—41no—
127C4×C127C2×#4112yes—
128Dic127—21no—
129Dic127—41no—
130Dic127—42no—
131Dic127—41no—
132Dic127#12312yes—
133C2×C127—22no—
134C2×C127C2×#12311yescharacteristic
135C2×C127C2×#13711yescharacteristic
136C2×C127#57511yescharacteristic
137D127—22no—
138D127—41no—
139#3095—1271nomaximal, Sylow
140C127#309511yescharacteristic, complemented, Sylow
141C2×C22⋊Q8—1271no—
142C2×#117—1271no—
143#485—1272no—
144#485—1272no—
145#484—1271no—
146#484D12711yescharacteristic, complemented
147#483—1271no—
148#483—1271no—
149#428—1272no—
150#419—1272no—
151#407—1271no—
152C22×Q8—1271no—
153C23×C4—1271no—
154C2×M16—1272no—
155C22⋊Q8—1271no—
156C22⋊Q8—1271no—
157C22⋊Q8—1271no—
158C22⋊Q8—2541no—
159C22⋊Q8—2541no—
160C22⋊Q8C2×D12711yescharacteristic
161#117—1272no—
162#117—1272no—
163#117—2541no—
164#117—2541no—
165C2×C4⋊C4—1271no—
166C2×C4⋊C4—1271no—
167C2×C4⋊C4—2541no—
168C2×C4⋊C4C2×D12711yescharacteristic
169C2×C22⋊C4—2541no—
170C2×C22⋊C4—2541no—
171C2×C42—2541no—
172#107—1272no—
173#107C2×D12712yescomplemented
174#106—1272no—
175#106—1272no—
176#103—1272no—
177#103—1272no—
178#103—1272no—
179#103C2×D12712yescomplemented
180C22⋊C8—1271no—
181C22⋊C8—1272no—
182C22⋊C8C2×D12711yescharacteristic
183C24—1271no—
184C2×Q8—1271no—
185C2×Q8—2541no—
186C2×Q8—2544no—
187C2×Q8C22×D12711yescharacteristic
188C22×C4—21no—
189C22×C4—1271no—
190C22×C4—1271no—
191C22×C4—1271no—
192C22×C4—1271no—
193C22×C4—1271no—
194C22×C4—2544no—
195C22×C4—2541no—
196C22×C4—2541no—
197C22×C4—2541no—
198C22×C4C22×D12711yescharacteristic
199M16—2544no—
200C2×C8—1272no—
201C2×C8C22×D12712yes—
202C4⋊C4—21no—
203C4⋊C4—21no—
204C4⋊C4—1272no—
205C4⋊C4—1271no—
206C4⋊C4—2541no—
207C4⋊C4—2541no—
208C4⋊C4—2541no—
209C4⋊C4—2541no—
210C4⋊C4—2541no—
211C4⋊C4C22×D12712yes—
212C4⋊C4C22×D12711yescharacteristic
213C22⋊C4—21no—
214C22⋊C4—2542no—
215C22⋊C4—2542no—
216C22⋊C4—2541no—
217C22⋊C4—2541no—
218C22⋊C4—2541no—
219C42—2542no—
220C42—2542no—
221C23—1271no—
222C23—1271no—
223C23—5081no—
224C23—2542no—
225C23—2542no—
226C23D4×D12711yescharacteristic
227Q8—22no—
228Q8—5082no—
229Q8—2542no—
230C2×C4—22no—
231C2×C4—22no—
232C2×C4—21no—
233C2×C4—21no—
234C2×C4—42no—
235C2×C4—1271no—
236C2×C4—1274no—
237C2×C4—1271no—
238C2×C4—5082no—
239C2×C4—5082no—
240C2×C4—5082no—
241C2×C4—2542no—
242C2×C4—2542no—
243C2×C4—2542no—
244C2×C4—2542no—
245C2×C4—2541no—
246C2×C4—2541no—
247C2×C4—2542no—
248C2×C4D4×D12711yescharacteristic
249C2×C4C23×D12711yescharacteristic
250C8—22no—
251C8—2542no—
252C22—21no—
253C22—21no—
254C22—1272no—
255C22—5084no—
256C22—2541no—
257C22—2541no—
258C22—2541no—
259C22—2541no—
260C22—2541no—
261C22#1686124112yes—
262C22C2×D4×D12711yescharacteristic
263C4—21no—
264C4—41no—
265C4—42no—
266C4—41no—
267C4—1272no—
268C4—5081no—
269C4—5082no—
270C4—5081no—
271C4—2541no—
272C4#41×D12712yes—
273C2—22no—
274C2—5081no—
275C2—2542no—
276C2#123×D12711yescharacteristic
277C2C2×#1686124111yescharacteristic
278C2#2255067211yescharacteristic
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#484×C127C2
socleC22×C127C22×D4
prime radicalC23×D127D4
hypercenter#484D127
hypocenterC127#3095
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ #484

Complements

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

NH ≅ G/Ntype
3. #22550219C2semidirect
4. #22550215C2semidirect
5. #22550211C2semidirect
6. #22550207C2semidirect
7. #22550204C2semidirect
8. #22549917C2semidirect
9. #22549883C2semidirect
11. #484×C127C2semidirect
30. #16861142C22semidirect
31. #16861141C22semidirect
32. #16861140C22semidirect
33. #16861134C22semidirect
34. #16861133C22semidirect
35. #16861132C22semidirect
40. #107×C127C22semidirect
41. #103×C127C22semidirect
140. C127#3095semidirect
146. #484D127semidirect
173. #107C2×D127semidirect
179. #103C2×D127semidirect

Maximal subgroups

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

subgroupindex
C22⋊Q8×D1272
C2×#168611672
#225502192
#225502152
#225502112
#225502072
#225502042
#225499172
#225498832
#225498382
#484×C1272
#3095127