Finite Group Databasebeta

← #30642001

Subgroups — #30642001

Order 16256.

528 conjugacy classes of subgroups; only the 159 normal subgroups are shown.

normal subgroups only

This group has 528 conjugacy classes of subgroups — too many to draw. Only its 159 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×Q8×Dic127C211yesmaximal, characteristic, complemented
2C2×#16861167C211yesmaximal, characteristic
3#22549815C211yesmaximal, characteristic, complemented
4#22549813C211yesmaximal, characteristic
5#22549808C212yesmaximal, complemented
6#22549798C212yesmaximal
7#22549787C211yesmaximal, characteristic
8C4⋊C4×Dic127C212yesmaximal, complemented
9#22549727C211yesmaximal, characteristic, complemented
10#22549723C212yesmaximal
11#395×C127C211yesmaximal, characteristic, complemented
12C22×C4×D127C2211yescharacteristic
13Q8×Dic127—24no—
14#16861167—24no—
15C2×C4⋊Dic127C2212yes—
16C2×C4⋊Dic127C2211yescharacteristic
17C2×C4⋊Dic127C2212yes—
18C2×C4⋊Dic127C2211yescharacteristic
19C2×Dic127⋊C4C2212yes—
20C2×(C2×D127)⋊C4C2212yes—
21C2×(C2×D127)⋊C4C2211yescharacteristic
22C2×(C2×D127)⋊C4C2211yescharacteristic
23C2×(C2×D127)⋊C4C2212yes—
24C2×C4×Dic127C2211yescharacteristic
25C2×C4×Dic127C2212yes—
26C2×C4×Dic127C2212yes—
27C2×C4×Dic127C2211yescharacteristic
28C127⋊#95C2212yes—
29C127⋊#95C2211yescharacteristic
30C127⋊#95C2212yes—
31C127⋊#95C2212yes—
32C22×Q8×C127C2211yescharacteristic
33C2×C4⋊C4×C127C2212yes—
34C2×C4⋊C4×C127C2211yescharacteristic
35#95×C127C2212yes—
36#95×C127C2211yescharacteristic
37C23×D127C2311yescharacteristic
38C22×Dic127C2312yes—
39C22×Dic127C2311yescharacteristic
40C22×Dic127C2312yes—
41C22×Dic127C2311yescharacteristic
42C22×Dic127C2311yescharacteristic
43C2×C4×D127—22no—
44C2×C4×D127—22no—
45C2×C4×D127Q814yescomplemented
46C4⋊Dic127—22no—
47C4⋊Dic127—24no—
48C4⋊Dic127—22no—
49C4⋊Dic127—24no—
50Dic127⋊C4—24no—
51(C2×D127)⋊C4—22no—
52(C2×D127)⋊C4—22no—
53(C2×D127)⋊C4—24no—
54(C2×D127)⋊C4—24no—
55C4×Dic127—22no—
56C4×Dic127—24no—
57C4×Dic127—24no—
58C4×Dic127D414yes—
59C2×Q8×C127—22no—
60C2×Q8×C127—24no—
61C22×C4×C127C2311yescharacteristic
62C22×C4×C127C2311yescharacteristic
63C22×C4×C127C2311yescharacteristic
64C22×C4×C127C2312yes—
65C22×C4×C127C2312yes—
66C4⋊C4×C127—22no—
67C4⋊C4×C127—24no—
68C22×D127—21no—
69C22×D127—22no—
70C22×D127—22no—
71C22×D127—21no—
72C22×D127C2×Q812yes—
73C2×Dic127—22no—
74C2×Dic127—21no—
75C2×Dic127—22no—
76C2×Dic127—22no—
77C2×Dic127—22no—
78C2×Dic127—22no—
79C2×Dic127—22no—
80C2×Dic127—22no—
81C2×Dic127—22no—
82C2×Dic127C2×D412yes—
83C2×Dic127C2×D412yes—
84C2×Dic127C2×Q812yes—
85C2×Dic127#4112yes—
86C4×D127—28no—
87C23×C127C2411yescharacteristic
88Q8×C127—28no—
89C2×C4×C127—21no—
90C2×C4×C127—22no—
91C2×C4×C127—22no—
92C2×C4×C127—22no—
93C2×C4×C127—22no—
94C2×C4×C127—22no—
95C2×C4×C127—21no—
96C2×C4×C127—21no—
97C2×C4×C127—22no—
98C2×C4×C127C2×D412yes—
99C2×C4×C127C2×Q812yes—
100C2×C4×C127#4114yes—
101C2×C4×C127#4114yes—
102C22×C127C22×D411yescharacteristic
103C22×C127C22×Q811yescharacteristic
104C22×C127C2×#4111yescharacteristic
105C22×C127C2×#4112yes—
106C22×C127C2×#4112yes—
107C2×D127—22no—
108C2×D127—24no—
109C2×D127—41no—
110C2×D127—42no—
111C2×D127—41no—
112C4×C127—22no—
113C4×C127—22no—
114C4×C127—24no—
115C4×C127—24no—
116C4×C127—41no—
117Dic127—22no—
118Dic127—42no—
119Dic127—41no—
120Dic127—42no—
121Dic127C22⋊Q814yes—
122C2×C127C2×C22⋊Q811yescharacteristic
123C2×C127#52511yescharacteristic
124C2×C127#52912yes—
125C2×C127#53211yescharacteristic
126C2×C127#53312yes—
127D127—42no—
128#2396—1271nomaximal, Sylow
129C127#239611yescharacteristic, complemented, Sylow
130C2×C4×Q8—1271no—
131C2×#117—1271no—
132#400—1271no—
133#399—1271no—
134#398—1272no—
135#396—1272no—
136#395D12711yescharacteristic, complemented
137#393—1271no—
138C4×C4⋊C4—1272no—
139C4×C4⋊C4—1271no—
140C4×C22⋊C4—1272no—
141C22×Q8C2×D12711yescharacteristic
142C23×C4—1271no—
143C4×Q8—2544no—
144#117—2544no—
145C2×C4⋊C4—1271no—
146C2×C4⋊C4—1272no—
147C2×C4⋊C4—1271no—
148C2×C4⋊C4—1272no—
149C2×C4⋊C4—1272no—
150C2×C4⋊C4C2×D12711yescharacteristic
151C2×C4⋊C4C2×D12712yes—
152C2×C22⋊C4—1271no—
153C2×C22⋊C4—1271no—
154C2×C22⋊C4—1272no—
155C2×C22⋊C4—1272no—
156C2×C42—1271no—
157C2×C42—1271no—
158C2×C42—1272no—
159C2×C42—1272no—
160#95—1272no—
161#95—1272no—
162#95—1271no—
163#95—1272no—
164#95C2×D12712yes—
165#95C2×D12711yescharacteristic
166C24—1271no—
167C2×Q8—22no—
168C2×Q8—24no—
169C22×C4—1271no—
170C22×C4—1271no—
171C22×C4—1274no—
172C22×C4—1272no—
173C22×C4—1271no—
174C22×C4—1272no—
175C22×C4—2542no—
176C22×C4—2542no—
177C22×C4C22×D12711yescharacteristic
178C22×C4C22×D12711yescharacteristic
179C22×C4C22×D12711yescharacteristic
180C22×C4C22×D12712yes—
181C22×C4C22×D12712yes—
182C4⋊C4—22no—
183C4⋊C4—24no—
184C4⋊C4—2542no—
185C4⋊C4—2544no—
186C4⋊C4—2542no—
187C4⋊C4—2544no—
188C4⋊C4—2544no—
189C22⋊C4—2542no—
190C22⋊C4—2542no—
191C22⋊C4—2544no—
192C22⋊C4—2544no—
193C42—1274no—
194C42—2544no—
195C42—2542no—
196C42—2544no—
197C23—1272no—
198C23—2541no—
199C23—2541no—
200C23—2542no—
201C23—2542no—
202C23C23×D12711yescharacteristic
203Q8—28no—
204C2×C4—22no—
205C2×C4—21no—
206C2×C4—22no—
207C2×C4—21no—
208C2×C4—22no—
209C2×C4—22no—
210C2×C4—22no—
211C2×C4—21no—
212C2×C4—22no—
213C2×C4—1272no—
214C2×C4—1272no—
215C2×C4—1272no—
216C2×C4—1272no—
217C2×C4—2548no—
218C2×C4—2541no—
219C2×C4—2542no—
220C2×C4—2542no—
221C2×C4—2542no—
222C2×C4—2542no—
223C2×C4—2542no—
224C2×C4—2542no—
225C2×C4—2542no—
226C2×C4—2542no—
227C2×C4D4×D12712yes—
228C2×C4Q8×D12712yes—
229C2×C4#1452324714yes—
230C2×C4#1452324914yes—
231C22—5081no—
232C22—5082no—
233C22—5081no—
234C22—2542no—
235C22—2544no—
236C22C2×D4×D12711yescharacteristic
237C22C2×Q8×D12711yescharacteristic
238C22#41×D12711yescharacteristic
239C22C2×#1452324712yes—
240C22C2×#1452324912yes—
241C4—22no—
242C4—22no—
243C4—24no—
244C4—24no—
245C4—41no—
246C4—1274no—
247C4—5082no—
248C4—5081no—
249C4—5082no—
250C4—2542no—
251C2—5082no—
252C2C22⋊Q8×D12711yescharacteristic
253C2#2255038211yescharacteristic
254C2#2255041212yes—
255C2#2255042711yescharacteristic
256C2#2255043112yes—
subgroupG/H
center, FrattiniC23C23×D127
commutator, socle, cosocleC23×C127C24
Fitting#395×C127C2
prime radicalC23×D127C23
hypercenter#395D127
hypocenterC127#2396
LCSG ▷ C23×C127 ▷ C127
UCSC1 ◁ C23 ◁ #395

Complements

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

NH ≅ G/Ntype
1. C2×Q8×Dic127C2semidirect
3. #22549815C2semidirect
5. #22549808C2semidirect
8. C4⋊C4×Dic127C2semidirect
9. #22549727C2semidirect
11. #395×C127C2semidirect
45. C2×C4×D127Q8semidirect
129. C127#2396semidirect
136. #395D127semidirect

Maximal subgroups

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

subgroupindex
C2×Q8×Dic1272
C2×#168611672
#225498152
#225498132
#225498082
#225497982
#225497872
C4⋊C4×Dic1272
#225497272
#225497232
#395×C1272
#2396127