Finite Group Databasebeta

← #30641887

Subgroups — #30641887

Order 16256.

576 conjugacy classes of subgroups; only the 213 normal subgroups are shown.

normal subgroups only

This group has 576 conjugacy classes of subgroups — too many to draw. Only its 213 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
1#22549779C212yesmaximal, complemented
2#22549769C212yesmaximal, complemented
3#22549762C212yesmaximal, complemented
4#22549756C212yesmaximal, complemented
5#22549756C212yesmaximal, complemented
6#22549751C212yesmaximal, complemented
7#22549739C211yesmaximal, characteristic, complemented
8#22549736C211yesmaximal, characteristic, complemented
9C2×#121×C127C211yesmaximal, characteristic
10C23×Dic127C2212yes—
11C2×C4⋊Dic127C2212yes—
12C2×C4⋊Dic127C2212yes—
13C2×C22⋊Dic127C2212yes—
14C2×C22⋊Dic127C2212yes—
15C2×C22⋊Dic127C2211yescharacteristic
16C2×C22⋊Dic127C2212yes—
17C2×C22⋊Dic127C2212yes—
18C2×C22⋊Dic127C2211yescharacteristic
19C2×C4×Dic127C2212yes—
20C127⋊#95C2212yes—
21C127⋊#95C2212yes—
22C127⋊#95C2212yes—
23C127⋊#95C2212yes—
24C127⋊#95C2211yescharacteristic
25C127⋊#95C2211yescharacteristic
26C22×D4×C127C2211yescharacteristic
27C22×D4×C127C2212yes—
28C23×C4×C127C2211yescharacteristic
29#121×C127C418yescomplemented
30C2×C4⋊C4×C127C2211yescharacteristic
31C2×C22⋊C4×C127C2212yes—
32C22×Dic127—42no—
33C22×Dic127—42no—
34C22×Dic127—42no—
35C22×Dic127C2312yes—
36C22×Dic127C2312yes—
37C22×Dic127C2312yes—
38C22×Dic127C2312yes—
39C4⋊Dic127—42no—
40C4⋊Dic127—42no—
41C22⋊Dic127—42no—
42C22⋊Dic127—42no—
43C22⋊Dic127—42no—
44C22⋊Dic127—41no—
45C22⋊Dic127—41no—
46C22⋊Dic127—42no—
47C4×Dic127—42no—
48C24×C127C2312yes—
49C24×C127C2311yescharacteristic
50C2×D4×C127—42no—
51C2×D4×C127—42no—
52C2×D4×C127—42no—
53C2×D4×C127C2×C414yes—
54C2×D4×C127C2×C418yescomplemented
55C22×C4×C127—42no—
56C22×C4×C127C2×C414yescomplemented
57C22×C4×C127C2311yescharacteristic
58C22×C4×C127C2312yes—
59C22×C4×C127C2311yescharacteristic
60C4⋊C4×C127C2×C414yescomplemented
61C22⋊C4×C127C2×C418yescomplemented
62C2×Dic127—22no—
63C2×Dic127—22no—
64C2×Dic127—22no—
65C2×Dic127—22no—
66C2×Dic127—22no—
67C2×Dic127—22no—
68C2×Dic127—22no—
69C2×Dic127—22no—
70C2×Dic127—22no—
71C2×Dic127—22no—
72C2×Dic127—22no—
73C2×Dic127—22no—
74C2×Dic127—48no—
75C23×C127—22no—
76C23×C127—22no—
77C23×C127—22no—
78C23×C127—22no—
79C23×C127—22no—
80C23×C127—22no—
81C23×C127—21no—
82C23×C127—22no—
83C23×C127—22no—
84C23×C127—21no—
85C23×C127C22×C414yes—
86C23×C127C22×C412yes—
87C23×C127C2411yescharacteristic
88D4×C127—44no—
89D4×C127—48no—
90C2×C4×C127—22no—
91C2×C4×C127—22no—
92C2×C4×C127—22no—
93C2×C4×C127—22no—
94C2×C4×C127—44no—
95C2×C4×C127C22×C412yes—
96C2×C4×C127C22×C414yes—
97C2×C4×C127C22×C412yes—
98C22×C127—22no—
99C22×C127—22no—
100C22×C127—24no—
101C22×C127—24no—
102C22×C127—22no—
103C22×C127—24no—
104C22×C127—42no—
105C22×C127—42no—
106C22×C127—42no—
107C22×C127—42no—
108C22×C127—42no—
109C22×C127—42no—
110C22×C127C23×C411yescharacteristic
111C22×C127H(5,𝔽2)12yes—
112C22×C127H(5,𝔽2)12yes—
113C22×C127H(5,𝔽2)11yescharacteristic
114C22×C1272−1+411yescharacteristic
115C4×C127—22no—
116C4×C127—22no—
117C4×C127—24no—
118Dic127—42no—
119Dic127—42no—
120Dic127—42no—
121Dic127—42no—
122C2×C127—42no—
123C2×C127—44no—
124C2×C127#51811yescharacteristic
125C2×C127#51811yescharacteristic
126C2×C127#52011yescharacteristic
127C2×C127#56012yes—
128C2×C127#56212yes—
129#2377—1271nomaximal, Sylow
130C127#237711yescharacteristic, complemented, Sylow
131C2×#121D12711yescharacteristic
132#390—1272no—
133#388—1272no—
134#387—1272no—
135#386—1272no—
136#386—1272no—
137#385—1272no—
138#381—1271no—
139#380—1271no—
140C22×D4C2×D12711yescharacteristic
141C22×D4C2×D12712yes—
142C23×C4—1272no—
143C23×C4C2×D12711yescharacteristic
144#121Dic12718yescomplemented
145C2×C4⋊C4—1272no—
146C2×C4⋊C4—1272no—
147C2×C4⋊C4C2×D12711yescharacteristic
148C2×C22⋊C4—1271no—
149C2×C22⋊C4—1271no—
150C2×C22⋊C4—1272no—
151C2×C22⋊C4—1272no—
152C2×C22⋊C4—1272no—
153C2×C22⋊C4—1272no—
154C2×C22⋊C4C2×D12712yes—
155C2×C42—1272no—
156#95—1272no—
157#95—1271no—
158#95—1272no—
159#95—1271no—
160#95—1272no—
161#95—1272no—
162C24C22×D12711yescharacteristic
163C24C22×D12712yes—
164C2×D4—42no—
165C2×D4—42no—
166C2×D4—42no—
167C2×D4C2×Dic12714yes—
168C2×D4C2×Dic12718yescomplemented
169C22×C4—42no—
170C22×C4—1272no—
171C22×C4—1272no—
172C22×C4—1272no—
173C22×C4—1272no—
174C22×C4—5082no—
175C22×C4—5082no—
176C22×C4—5082no—
177C22×C4C2×Dic12714yescomplemented
178C22×C4C22×D12711yescharacteristic
179C22×C4C22×D12712yes—
180C22×C4C22×D12711yescharacteristic
181C4⋊C4—5082no—
182C4⋊C4—5082no—
183C4⋊C4C2×Dic12714yescomplemented
184C22⋊C4—5082no—
185C22⋊C4—5082no—
186C22⋊C4—5082no—
187C22⋊C4—5081no—
188C22⋊C4—5082no—
189C22⋊C4—5081no—
190C22⋊C4C2×Dic12718yescomplemented
191C42—5082no—
192C23—22no—
193C23—22no—
194C23—22no—
195C23—22no—
196C23—22no—
197C23—22no—
198C23—21no—
199C23—22no—
200C23—21no—
201C23—22no—
202C23C22×Dic12714yes—
203C23C22×Dic12712yes—
204C23C23×D12711yescharacteristic
205D4—44no—
206D4—48no—
207C2×C4—22no—
208C2×C4—22no—
209C2×C4—22no—
210C2×C4—22no—
211C2×C4—44no—
212C2×C4—5088no—
213C2×C4—2542no—
214C2×C4—2542no—
215C2×C4—2542no—
216C2×C4—2542no—
217C2×C4—2542no—
218C2×C4—2542no—
219C2×C4—2542no—
220C2×C4—2542no—
221C2×C4—2542no—
222C2×C4—2542no—
223C2×C4—2542no—
224C2×C4—2542no—
225C2×C4C22×Dic12712yes—
226C2×C4C22×Dic12714yes—
227C2×C4C22×Dic12712yes—
228C22—22no—
229C22—22no—
230C22—24no—
231C22—24no—
232C22—22no—
233C22—24no—
234C22—42no—
235C22—42no—
236C22—42no—
237C22—42no—
238C22—42no—
239C22—42no—
240C22C23×Dic12711yescharacteristic
241C22#1686125412yes—
242C22#1686125412yes—
243C22#1686125511yescharacteristic
244C22#1686125611yescharacteristic
245C4—22no—
246C4—22no—
247C4—24no—
248C4—5082no—
249C4—5082no—
250C4—5082no—
251C4—5082no—
252C2—42no—
253C2—44no—
254C2#2255035111yescharacteristic
255C2#2255035111yescharacteristic
256C2#2255035911yescharacteristic
257C2#2255061412yes—
258C2#2255062112yes—
subgroupG/H
center, FrattiniC23C23×D127
commutatorC22×C127C23×C4
Fitting, prime radicalC2×#121×C127C2
socle, cosocleC23×C127C24
hypercenterC2×#121D127
hypocenterC127#2377
LCSG ▷ C22×C127 ▷ C127
UCSC1 ◁ C23 ◁ C2×#121

Complements

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

NH ≅ G/Ntype
1. #22549779C2semidirect
2. #22549769C2semidirect
3. #22549762C2semidirect
4. #22549756C2semidirect
5. #22549756C2semidirect
6. #22549751C2semidirect
7. #22549739C2semidirect
8. #22549736C2semidirect
29. #121×C127C4semidirect
54. C2×D4×C127C2×C4semidirect
56. C22×C4×C127C2×C4semidirect
60. C4⋊C4×C127C2×C4semidirect
61. C22⋊C4×C127C2×C4semidirect
130. C127#2377semidirect
144. #121Dic127semidirect
168. C2×D4C2×Dic127semidirect
177. C22×C4C2×Dic127semidirect
183. C4⋊C4C2×Dic127semidirect
190. C22⋊C4C2×Dic127semidirect

Maximal subgroups

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

subgroupindex
#225497792
#225497692
#225497622
#225497562
#225497512
#225497392
#225497362
C2×#121×C1272
#2377127