Finite Group Databasebeta

← #30641793

Subgroups — #30641793

Order 16256.

832 conjugacy classes of subgroups; only the 209 normal subgroups are shown.

normal subgroups only

This group has 832 conjugacy classes of subgroups — too many to draw. Only its 209 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×#16861205C211yesmaximal, characteristic
2C2×C4×C22⋊D127C212yesmaximal, complemented
3C2×(C22⋊D127)⋊C4C212yesmaximal, complemented
4#22549776C212yesmaximal, complemented
5#22549776C212yesmaximal, complemented
6#22549756C211yesmaximal, characteristic, complemented
7#22549754C212yesmaximal, complemented
8#22549727C212yesmaximal, complemented
9#386×C127C211yesmaximal, characteristic, complemented
10C22×D508C2211yescharacteristic
11C22×C22⋊D127C2212yes—
12C22×C22⋊D127C2212yes—
13C23×Dic127C2211yescharacteristic
14C22×C4×D127C2212yes—
15#16861205C418yescomplemented
16C4×C22⋊D127—28no—
17(C22⋊D127)⋊C4—28no—
18C2×Dic127⋊C4C2212yes—
19C2×Dic127⋊C4C2212yes—
20C2×C22⋊Dic127C2212yes—
21C2×C22⋊Dic127C2211yescharacteristic
22C2×(C2×D127)⋊C4C2212yes—
23C2×(C2×D127)⋊C4C2212yes—
24C2×(C2×D127)⋊C4C2212yes—
25C2×C4×Dic127C2212yes—
26C2×C4×Dic127C2212yes—
27C2×C4×Dic127C2211yescharacteristic
28C127⋊#95C2212yes—
29C22×D4×C127C2211yescharacteristic
30C23×C4×C127C2211yescharacteristic
31C2×C22⋊C4×C127C2212yes—
32C2×C22⋊C4×C127C2211yescharacteristic
33#95×C127C2212yes—
34C23×D127C2312yes—
35C2×D508—41no—
36C2×D508—41no—
37C2×D508C2×C414yescomplemented
38C2×C22⋊D127—24no—
39C2×C22⋊D127—24no—
40C2×C22⋊D127—24no—
41C2×C22⋊D127—24no—
42C2×C22⋊D127C2×C418yescomplemented
43C2×C22⋊D127C2×C418yescomplemented
44C22×Dic127—24no—
45C22×Dic127—41no—
46C22×Dic127C2312yes—
47C22×Dic127C2312yes—
48C22×Dic127C2312yes—
49C2×C4×D127—24no—
50C2×C4×D127—24no—
51C2×C4×D127—42no—
52Dic127⋊C4—24no—
53Dic127⋊C4D418yescomplemented
54C22⋊Dic127—24no—
55C22⋊Dic127—41no—
56(C2×D127)⋊C4—24no—
57(C2×D127)⋊C4—24no—
58(C2×D127)⋊C4—42no—
59C4×Dic127—24no—
60C4×Dic127—24no—
61C4×Dic127C2×C414yescomplemented
62C24×C127C2311yescharacteristic
63C24×C127C2311yescharacteristic
64C2×D4×C127—41no—
65C2×D4×C127—41no—
66C2×D4×C127C2×C414yescomplemented
67C22×C4×C127—24no—
68C22×C4×C127—41no—
69C22×C4×C127C2312yes—
70C22×C4×C127C2311yescharacteristic
71C22×C4×C127C2312yes—
72C22⋊C4×C127—24no—
73C22⋊C4×C127—41no—
74C22×D127—22no—
75C22×D127—22no—
76C22×D127—22no—
77C22×D127—22no—
78C22×D127—22no—
79C22×D127—22no—
80C22×D127C22×C414yes—
81D508—44no—
82C22⋊D127—216no—
83C22⋊D127—216no—
84C2×Dic127—22no—
85C2×Dic127—22no—
86C2×Dic127—22no—
87C2×Dic127—22no—
88C2×Dic127—22no—
89C2×Dic127—44no—
90C2×Dic127C22×C414yes—
91C2×Dic127C2×D414yes—
92C2×Dic127C2×D414yes—
93C2×Dic127#4114yes—
94C4×D127—48no—
95C23×C127—22no—
96C23×C127—22no—
97C23×C127—22no—
98C23×C127—22no—
99C23×C127—21no—
100C23×C127—21no—
101C23×C127—21no—
102C23×C127—21no—
103C23×C127C22×C412yes—
104C23×C127C22×C412yes—
105C23×C127C2411yescharacteristic
106D4×C127—44no—
107C2×C4×C127—22no—
108C2×C4×C127—22no—
109C2×C4×C127—21no—
110C2×C4×C127—21no—
111C2×C4×C127—22no—
112C2×C4×C127—22no—
113C2×C4×C127—22no—
114C2×C4×C127—44no—
115C2×C4×C127C22×C412yes—
116C2×C4×C127C2×D414yes—
117C22×C127—24no—
118C22×C127—24no—
119C22×C127—22no—
120C22×C127—22no—
121C22×C127—41no—
122C22×C127—41no—
123C22×C127—42no—
124C22×C127—42no—
125C22×C127—41no—
126C22×C127—41no—
127C22×C127C23×C411yescharacteristic
128C22×C127C22×D412yes—
129C22×C127C2×#4112yes—
130C22×C127H(5,𝔽2)11yescharacteristic
131C22×C127H(5,𝔽2)11yescharacteristic
132C2×D127—24no—
133C2×D127—24no—
134C2×D127—24no—
135C2×D127—42no—
136C2×D127—42no—
137C2×D127—42no—
138C2×D127—42no—
139C4×C127—22no—
140C4×C127—24no—
141C4×C127—42no—
142Dic127—24no—
143Dic127—42no—
144Dic127C4×D418yescomplemented
145C2×C127—42no—
146C2×C127—42no—
147C2×C127C2×C4×D412yes—
148C2×C127#51811yescharacteristic
149C2×C127D4211yescharacteristic
150C2×C127#55012yes—
151C2×C127#55911yescharacteristic
152D127—44no—
153#2366—1271nomaximal, Sylow
154C127#236611yescharacteristic, complemented, Sylow
155C2×#127—1271no—
156C2×C4×D4—1272no—
157C2×C4×D4—1272no—
158#390—1272no—
159#390—1272no—
160#386—1271no—
161#386—1272no—
162#386D12711yescharacteristic, complemented
163C4×C4⋊C4—1272no—
164C22×D4—1272no—
165C22×D4—1272no—
166C22×D4—1271no—
167C22×D4C2×D12711yescharacteristic
168C23×C4—1271no—
169C23×C4—1272no—
170C23×C4C2×D12711yescharacteristic
171#127—1278no—
172C4×D4—2548no—
173C4×D4—2548no—
174C2×C4⋊C4—1272no—
175C2×C4⋊C4—1272no—
176C2×C22⋊C4—1271no—
177C2×C22⋊C4—1272no—
178C2×C22⋊C4—1272no—
179C2×C22⋊C4—1272no—
180C2×C22⋊C4—1272no—
181C2×C22⋊C4C2×D12712yes—
182C2×C22⋊C4C2×D12711yescharacteristic
183C2×C42—1271no—
184C2×C42—1272no—
185C2×C42—1272no—
186#95—1272no—
187#95C2×D12712yes—
188C24—1272no—
189C24C22×D12711yescharacteristic
190C24C22×D12711yescharacteristic
191C2×D4—41no—
192C2×D4—41no—
193C2×D4—1278no—
194C2×D4—1278no—
195C2×D4—1274no—
196C2×D4—5081no—
197C2×D4—5081no—
198C2×D4—2544no—
199C2×D4—2544no—
200C2×D4—2544no—
201C2×D4—2544no—
202C2×D4C4×D12714yescomplemented
203C22×C4—24no—
204C22×C4—41no—
205C22×C4—1272no—
206C22×C4—1272no—
207C22×C4—1272no—
208C22×C4—5082no—
209C22×C4—5081no—
210C22×C4—2544no—
211C22×C4—2544no—
212C22×C4—2544no—
213C22×C4C22×D12712yes—
214C22×C4C22×D12711yescharacteristic
215C22×C4C22×D12712yes—
216C4⋊C4—1278no—
217C4⋊C4—2544no—
218C22⋊C4—24no—
219C22⋊C4—41no—
220C22⋊C4—5082no—
221C22⋊C4—5081no—
222C22⋊C4—2544no—
223C22⋊C4—2544no—
224C22⋊C4—2544no—
225C42—1274no—
226C42—2544no—
227C42—2544no—
228C23—22no—
229C23—22no—
230C23—22no—
231C23—22no—
232C23—21no—
233C23—21no—
234C23—21no—
235C23—21no—
236C23—1274no—
237C23—2542no—
238C23—2542no—
239C23—2542no—
240C23—2542no—
241C23—2542no—
242C23—2542no—
243C23C2×C4×D12712yes—
244C23C2×C4×D12712yes—
245C23C23×D12711yescharacteristic
246D4—44no—
247D4—5084no—
248D4—25416no—
249D4—25416no—
250C2×C4—22no—
251C2×C4—22no—
252C2×C4—21no—
253C2×C4—21no—
254C2×C4—22no—
255C2×C4—22no—
256C2×C4—22no—
257C2×C4—44no—
258C2×C4—1274no—
259C2×C4—1274no—
260C2×C4—1274no—
261C2×C4—1274no—
262C2×C4—5088no—
263C2×C4—5084no—
264C2×C4—2542no—
265C2×C4—2542no—
266C2×C4—2542no—
267C2×C4—2542no—
268C2×C4—2542no—
269C2×C4C2×C4×D12712yes—
270C2×C4D4×D12714yes—
271C22—24no—
272C22—24no—
273C22—22no—
274C22—22no—
275C22—42no—
276C22—42no—
277C22—41no—
278C22—41no—
279C22—41no—
280C22—41no—
281C22—5082no—
282C22—5082no—
283C22—5082no—
284C22—5082no—
285C22—2544no—
286C22—2544no—
287C22—2544no—
288C22C22×C4×D12711yescharacteristic
289C22C2×D4×D12712yes—
290C22#41×D12712yes—
291C22#1686125411yescharacteristic
292C22#1686125411yescharacteristic
293C4—22no—
294C4—24no—
295C4—42no—
296C4—1278no—
297C4—5082no—
298C4—2544no—
299C2—42no—
300C2—42no—
301C2—5084no—
302C2C4×D4×D12712yes—
303C2#2255034911yescharacteristic
304C2#2255050811yescharacteristic
305C2#2255054412yes—
306C2#2255061011yescharacteristic
subgroupG/H
center, FrattiniC23C23×D127
commutatorC22×C127C23×C4
Fitting#386×C127C2
socle, cosocleC23×C127C24
prime radicalC2×#16861205C2
hypercenter#386D127
hypocenterC127#2366
LCSG ▷ C22×C127 ▷ C127
UCSC1 ◁ C23 ◁ #386

Complements

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

NH ≅ G/Ntype
2. C2×C4×C22⋊D127C2semidirect
3. C2×(C22⋊D127)⋊C4C2semidirect
4. #22549776C2semidirect
5. #22549776C2semidirect
6. #22549756C2semidirect
7. #22549754C2semidirect
8. #22549727C2semidirect
9. #386×C127C2semidirect
15. #16861205C4semidirect
37. C2×D508C2×C4semidirect
42. C2×C22⋊D127C2×C4semidirect
43. C2×C22⋊D127C2×C4semidirect
53. Dic127⋊C4D4semidirect
61. C4×Dic127C2×C4semidirect
66. C2×D4×C127C2×C4semidirect
144. Dic127C4×D4semidirect
154. C127#2366semidirect
162. #386D127semidirect
202. C2×D4C4×D127semidirect

Maximal subgroups

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

subgroupindex
C2×#168612052
C2×C4×C22⋊D1272
C2×(C22⋊D127)⋊C42
#225497762
#225497562
#225497542
#225497272
#386×C1272
#2366127