Finite Group Databasebeta

← #30648676

Subgroups — #30648676

Order 16256.

410 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 410 conjugacy classes of subgroups — too many to draw. Only its 127 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×#16861185C211yesmaximal, characteristic
2#117×D127C211yesmaximal, characteristic
3#22550216C212yesmaximal, complemented
4#22550215C212yesmaximal, complemented
5#22550210C212yesmaximal, complemented
6#22550205C212yesmaximal, complemented
7#22549919C211yesmaximal, characteristic, complemented
8#22549918C211yesmaximal, characteristic, complemented
9#22549885C211yesmaximal, characteristic, complemented
10#22549839C211yesmaximal, characteristic
11#419×C127C211yesmaximal, characteristic, complemented
12C22×C4×D127C2211yescharacteristic
13C2×#14523232C2211yescharacteristic
14#16861185—21no—
15#16861185—21no—
16#16861185C2212yes—
17#16861185C2212yes—
18#16861168C2211yescharacteristic
19#16861167C2211yescharacteristic
20#16861167C2211yescharacteristic
21#16861166—21no—
22#16861165—21no—
23C2×C4⋊Dic127C2211yescharacteristic
24C2×Dic127⋊C4—21no—
25C4⋊C4×D127C2211yescharacteristic
26C4⋊C4×D127C2211yescharacteristic
27C2×(C2×D127)⋊C4—21no—
28C22⋊C4×D127—21no—
29C42×D127—21no—
30#16861143C2212yescomplemented
31#16861142C2212yescomplemented
32#16861141C2212yescomplemented
33#16861140C2212yescomplemented
34#16861133C2212yescomplemented
35#16861133C2212yescomplemented
36(C2×D127)⋊C8C2212yes—
37(C2×D127)⋊C8C2212yes—
38C22×Q8×C127C2211yescharacteristic
39C2×M16×C127C2211yescharacteristic
40#117×C127C2211yescharacteristic
41#103×C127C2212yescomplemented
42#103×C127C2212yescomplemented
43C23×D127D411yescharacteristic
44C22×Dic127—21no—
45C22×Dic127D411yescharacteristic
46C2×C4×D127—21no—
47C2×C4×D127—21no—
48C2×C4×D127—22no—
49C2×C4×D127—22no—
50C2×C4×D127D411yescharacteristic
51C2×C4×D127D411yescharacteristic
52C2×C4×D127C2311yescharacteristic
53C2×C4×D127C2311yescharacteristic
54#14523232—22no—
55C2×C127⋊C8C2312yes—
56C4⋊Dic127—21no—
57C4⋊Dic127C2312yes—
58C4⋊Dic127C2311yescharacteristic
59C4⋊Dic127C2311yescharacteristic
60Dic127⋊C4—22no—
61Dic127⋊C4—22no—
62Dic127⋊C4—21no—
63Dic127⋊C4—21no—
64C22⋊Dic127—21no—
65(C2×D127)⋊C4—22no—
66(C2×D127)⋊C4—22no—
67(C2×D127)⋊C4—21no—
68(C2×D127)⋊C4—21no—
69C4×Dic127—21no—
70C4×Dic127—21no—
71C4×Dic127—21no—
72C2×Q8×C127—21no—
73C2×Q8×C127—22no—
74C2×Q8×C127—22no—
75C2×Q8×C127C2312yes—
76C22×C4×C127—21no—
77C22×C4×C127C2311yescharacteristic
78M16×C127—22no—
79C2×C8×C127C2312yes—
80C4⋊C4×C127C2311yescharacteristic
81C4⋊C4×C127C2311yescharacteristic
82C22⋊C4×C127—21no—
83C42×C127—21no—
84C22×D127—22no—
85C22×D127—22no—
86C22×D127—41no—
87C22×D127C2×D411yescharacteristic
88C22×D127C2×D411yescharacteristic
89C2×Dic127—22no—
90C2×Dic127—21no—
91C2×Dic127—21no—
92C2×Dic127—22no—
93C2×Dic127—42no—
94C2×Dic127C2×D411yescharacteristic
95C2×Dic127C2×D411yescharacteristic
96C4×D127—22no—
97C4×D127—22no—
98C4×D127—21no—
99C4×D127—21no—
100C4×D127—21no—
101C4×D127—21no—
102C4×D127—42no—
103C4×D127—42no—
104C127⋊C8—22no—
105C23×C127C2×D411yescharacteristic
106Q8×C127—22no—
107Q8×C127—22no—
108Q8×C127—42no—
109C2×C4×C127—21no—
110C2×C4×C127—21no—
111C2×C4×C127—22no—
112C2×C4×C127—42no—
113C2×C4×C127C2×D411yescharacteristic
114C2×C4×C127#4112yes—
115C2×C4×C127#4112yes—
116C2×C4×C127C2411yescharacteristic
117C8×C127—22no—
118C22×C127—21no—
119C22×C127—21no—
120C22×C127#12312yes—
121C22×C127C22×D411yescharacteristic
122C2×D127—21no—
123C2×D127—21no—
124C2×D127—21no—
125C2×D127—21no—
126C2×D127—21no—
127C2×D127—44no—
128C2×D127#13712yes—
129C4×C127—42no—
130C4×C127—41no—
131C4×C127—41no—
132C4×C127#12312yes—
133C4×C127C2×#4111yescharacteristic
134C4×C127C2×#4111yescharacteristic
135Dic127—21no—
136Dic127—21no—
137Dic127—41no—
138Dic127—41no—
139Dic127—42no—
140C2×C127—22no—
141C2×C127C2×#12311yescharacteristic
142C2×C127C2×#13711yescharacteristic
143C2×C127#57511yescharacteristic
144D127—22no—
145D127—41no—
146#3095—1271nomaximal, Sylow
147C127#309511yescharacteristic, complemented, Sylow
148C2×C22⋊Q8—1271no—
149C2×#117—1271no—
150#485—1272no—
151#485—1272no—
152#484—1272no—
153#483—1272no—
154#428—1271no—
155#428—1271no—
156#419—1271no—
157#419D12711yescharacteristic, complemented
158#407—1271no—
159C22×Q8C2×D12711yescharacteristic
160C23×C4—1271no—
161C2×M16—1271no—
162C2×M16C2×D12711yescharacteristic
163C22⋊Q8—1272no—
164C22⋊Q8—1272no—
165C22⋊Q8—2541no—
166C22⋊Q8—2541no—
167#117—1271no—
168#117—1271no—
169#117—1271no—
170#117—2541no—
171#117—2541no—
172#117C2×D12711yescharacteristic
173C2×C4⋊C4—1271no—
174C2×C4⋊C4—1271no—
175C2×C4⋊C4—1271no—
176C2×C4⋊C4—2541no—
177C2×C22⋊C4—2541no—
178C2×C22⋊C4—2541no—
179C2×C42—2541no—
180#107—1272no—
181#107—1272no—
182#106—1272no—
183#106—1272no—
184#103—1272no—
185#103—1272no—
186#103C2×D12712yescomplemented
187#103C2×D12712yescomplemented
188C22⋊C8—1272no—
189C22⋊C8—1272no—
190C24—1271no—
191C2×Q8—21no—
192C2×Q8—22no—
193C2×Q8—22no—
194C2×Q8C22×D12712yes—
195C22×C4—21no—
196C22×C4—1271no—
197C22×C4—1271no—
198C22×C4—1271no—
199C22×C4—1271no—
200C22×C4—1271no—
201C22×C4—2541no—
202C22×C4—2541no—
203C22×C4—2541no—
204C22×C4—2542no—
205C22×C4—2542no—
206C22×C4C22×D12711yescharacteristic
207M16—22no—
208M16—2542no—
209C2×C8—1272no—
210C2×C8C22×D12712yes—
211C4⋊C4—1271no—
212C4⋊C4—1271no—
213C4⋊C4—1272no—
214C4⋊C4—2541no—
215C4⋊C4—2541no—
216C4⋊C4—2542no—
217C4⋊C4—2542no—
218C4⋊C4—2541no—
219C4⋊C4C22×D12711yescharacteristic
220C4⋊C4C22×D12711yescharacteristic
221C22⋊C4—21no—
222C22⋊C4—2541no—
223C22⋊C4—2541no—
224C22⋊C4—2542no—
225C22⋊C4—2542no—
226C22⋊C4—2541no—
227C42—21no—
228C42—2541no—
229C42—2541no—
230C42—2541no—
231C23—1271no—
232C23—1271no—
233C23—5081no—
234C23—2542no—
235C23—2542no—
236C23D4×D12711yescharacteristic
237Q8—22no—
238Q8—22no—
239Q8—42no—
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—22no—
243C2×C4—42no—
244C2×C4—1271no—
245C2×C4—1271no—
246C2×C4—5082no—
247C2×C4—5082no—
248C2×C4—5082no—
249C2×C4—2541no—
250C2×C4—2541no—
251C2×C4—2542no—
252C2×C4—2542no—
253C2×C4—2542no—
254C2×C4—2542no—
255C2×C4—2541no—
256C2×C4—2541no—
257C2×C4—2541no—
258C2×C4—2541no—
259C2×C4D4×D12711yescharacteristic
260C2×C4#1452324712yes—
261C2×C4#1452324812yes—
262C2×C4C23×D12711yescharacteristic
263C8—22no—
264C8—2542no—
265C22—21no—
266C22—21no—
267C22—1272no—
268C22—5084no—
269C22—2541no—
270C22—2541no—
271C22—2541no—
272C22—2541no—
273C22—2541no—
274C22#1686119112yes—
275C22C2×D4×D12711yescharacteristic
276C4—42no—
277C4—41no—
278C4—41no—
279C4—5081no—
280C4—5081no—
281C4—5082no—
282C4—2541no—
283C4—2541no—
284C4#1686119112yes—
285C4C2×#1452324711yescharacteristic
286C4C2×#1452324811yescharacteristic
287C2—22no—
288C2—5081no—
289C2—2542no—
290C2C2×#1686119111yescharacteristic
291C2#137×D12711yescharacteristic
292C2#2255067511yescharacteristic
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#419×C127C2
socleC22×C127C22×D4
prime radicalC23×D127D4
hypercenter#419D127
hypocenterC127#3095
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ #419

Complements

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

NH ≅ G/Ntype
3. #22550216C2semidirect
4. #22550215C2semidirect
5. #22550210C2semidirect
6. #22550205C2semidirect
7. #22549919C2semidirect
8. #22549918C2semidirect
9. #22549885C2semidirect
11. #419×C127C2semidirect
30. #16861143C22semidirect
31. #16861142C22semidirect
32. #16861141C22semidirect
33. #16861140C22semidirect
34. #16861133C22semidirect
35. #16861133C22semidirect
41. #103×C127C22semidirect
42. #103×C127C22semidirect
147. C127#3095semidirect
157. #419D127semidirect
186. #103C2×D127semidirect
187. #103C2×D127semidirect

Maximal subgroups

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

subgroupindex
C2×#168611852
#117×D1272
#225502162
#225502152
#225502102
#225502052
#225499192
#225499182
#225498852
#225498392
#419×C1272
#3095127