Finite Group Databasebeta

← #30639567

Subgroups — #30639567

Order 16256.

408 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 408 conjugacy classes of subgroups — too many to draw. Only its 101 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×#16861124C214yesmaximal, complemented
2#22549736C211yesmaximal, characteristic, complemented
3C4×C22⋊Dic127C211yesmaximal, characteristic, complemented
4C2×#121×C127C211yesmaximal, characteristic
5C2×C4⋊Dic127—21no—
6C2×C22⋊Dic127C2212yes—
7C2×C22⋊Dic127C2212yes—
8C2×C4×Dic127—21no—
9#16861124—28no—
10C127⋊#95—21no—
11C127⋊#95—21no—
12C22×D4×C127C411yescharacteristic, complemented
13C22×D4×C127C2212yes—
14C23×C4×C127C2211yescharacteristic
15#121×C127—21no—
16#121×C127—21no—
17#121×C127—21no—
18#121×C127—21no—
19C2×C4⋊C4×C127C411yescharacteristic, complemented
20C2×C22⋊C4×C127C412yescomplemented
21C22×Dic127—22no—
22C22×Dic127—22no—
23C4⋊Dic127—42no—
24C22⋊Dic127—24no—
25C22⋊Dic127—22no—
26C22⋊Dic127D414yescomplemented
27C4×Dic127—42no—
28C24×C127C2×C412yes—
29C24×C127C2311yescharacteristic
30C2×D4×C127—22no—
31C2×D4×C127—22no—
32C2×D4×C127—24no—
33C2×D4×C127—24no—
34C2×D4×C127—21no—
35C2×D4×C127—21no—
36C2×D4×C127—42no—
37C22×C4×C127—42no—
38C22×C4×C127C2×C412yes—
39C22×C4×C127C2×C411yescharacteristic
40C22×C4×C127C2×C411yescharacteristic
41C22×C4×C127D411yescharacteristic
42C22×C4×C127D411yescharacteristic
43C22×C4×C127D411yescharacteristic
44C22×C4×C127D411yescharacteristic
45C4⋊C4×C127—21no—
46C4⋊C4×C127—21no—
47C22⋊C4×C127—22no—
48C22⋊C4×C127—22no—
49C2×Dic127—24no—
50C2×Dic127—42no—
51C2×Dic127—42no—
52C2×Dic127—42no—
53C2×Dic127—42no—
54C2×Dic127—42no—
55C23×C127—22no—
56C23×C127—24no—
57C23×C127—41no—
58C23×C127—41no—
59C23×C127—42no—
60C23×C127—42no—
61C23×C127C22×C411yescharacteristic
62C23×C127C2×D412yes—
63C23×C127C2×D411yescharacteristic
64C23×C127C2×D411yescharacteristic
65C23×C127#4112yes—
66D4×C127—42no—
67D4×C127—44no—
68D4×C127—42no—
69D4×C127—42no—
70D4×C127—42no—
71C2×C4×C127—22no—
72C2×C4×C127—22no—
73C2×C4×C127—22no—
74C2×C4×C127—22no—
75C2×C4×C127—24no—
76C2×C4×C127—41no—
77C2×C4×C127—41no—
78C2×C4×C127—41no—
79C2×C4×C127C22⋊C411yescharacteristic
80C2×C4×C127C22⋊C411yescharacteristic
81C2×C4×C127C22⋊C411yescharacteristic
82C2×C4×C127C22⋊C411yescharacteristic
83C22×C127—21no—
84C22×C127—21no—
85C22×C127—21no—
86C22×C127—21no—
87C22×C127—21no—
88C22×C127—21no—
89C22×C127—42no—
90C22×C127—42no—
91C22×C127—42no—
92C22×C127—42no—
93C22×C127—42no—
94C22×C127—84no—
95C22×C127C2×C22⋊C411yescharacteristic
96C22×C127C4×D412yes—
97C22×C127#12112yes—
98C22×C127#12411yescharacteristic
99C22×C127#12711yescharacteristic
100C4×C127—21no—
101C4×C127—21no—
102C4×C127—21no—
103C4×C127—42no—
104C4×C127#9912yescomplemented
105Dic127—44no—
106Dic127—82no—
107C2×C127—22no—
108C2×C127—21no—
109C2×C127—22no—
110C2×C127—21no—
111C2×C127—82no—
112C2×C127#39011yescharacteristic
113C2×C127C2×#9911yescharacteristic
114C2×C127#41011yescharacteristic
115#1964—1271nomaximal, Sylow
116C127#196411yescharacteristic, complemented, Sylow
117C2×#121D12711yescharacteristic
118C2×#99—1274no—
119#380—1271no—
120C4×C22⋊C4—1271no—
121C22×D4Dic12711yescharacteristic, complemented
122C22×D4C2×D12712yes—
123C23×C4C2×D12711yescharacteristic
124#121—21no—
125#121—21no—
126#121—21no—
127#121—21no—
128C2×C4⋊C4—2541no—
129C2×C4⋊C4Dic12711yescharacteristic, complemented
130C2×C22⋊C4—1272no—
131C2×C22⋊C4—1272no—
132C2×C22⋊C4Dic12712yescomplemented
133C2×C42—2541no—
134#99—2548no—
135#95—2541no—
136#95—2541no—
137C24C2×Dic12712yes—
138C24C22×D12711yescharacteristic
139C2×D4—24no—
140C2×D4—22no—
141C2×D4—22no—
142C2×D4—24no—
143C2×D4—21no—
144C2×D4—21no—
145C2×D4—42no—
146C22×C4—42no—
147C22×C4—2542no—
148C22×C4—2542no—
149C22×C4C2×Dic12712yes—
150C22×C4C2×Dic12711yescharacteristic
151C22×C4C2×Dic12711yescharacteristic
152C22×C4C22⋊D12711yescharacteristic
153C22×C4C22⋊D12711yescharacteristic
154C22×C4C22⋊D12711yescharacteristic
155C22×C4C22⋊D12711yescharacteristic
156C4⋊C4—21no—
157C4⋊C4—21no—
158C4⋊C4—5082no—
159C22⋊C4—22no—
160C22⋊C4—22no—
161C22⋊C4—1274no—
162C22⋊C4—2544no—
163C22⋊C4—2542no—
164C42—5082no—
165C23—22no—
166C23—24no—
167C23—41no—
168C23—41no—
169C23—42no—
170C23—42no—
171C23C22×Dic12711yescharacteristic
172C23D4×D12712yes—
173C23C2×C22⋊D12711yescharacteristic
174C23C2×C22⋊D12711yescharacteristic
175C23#1452324712yes—
176D4—42no—
177D4—42no—
178D4—42no—
179D4—44no—
180D4—42no—
181C2×C4—22no—
182C2×C4—22no—
183C2×C4—22no—
184C2×C4—22no—
185C2×C4—24no—
186C2×C4—41no—
187C2×C4—41no—
188C2×C4—41no—
189C2×C4—5082no—
190C2×C4—5082no—
191C2×C4—5082no—
192C2×C4—5082no—
193C2×C4—5082no—
194C2×C4—2544no—
195C2×C4C22⋊Dic12711yescharacteristic
196C2×C4C22⋊Dic12711yescharacteristic
197C2×C4C22⋊Dic12711yescharacteristic
198C2×C4C22⋊Dic12711yescharacteristic
199C22—21no—
200C22—21no—
201C22—21no—
202C22—21no—
203C22—21no—
204C22—21no—
205C22—42no—
206C22—42no—
207C22—42no—
208C22—42no—
209C22—42no—
210C22—84no—
211C22C2×C22⋊Dic12711yescharacteristic
212C22D4×Dic12712yes—
213C22#1686118012yes—
214C22#1686119611yescharacteristic
215C22#1686120511yescharacteristic
216C4—21no—
217C4—21no—
218C4—21no—
219C4—42no—
220C4—10162no—
221C4—5084no—
222C4#1686112412yescomplemented
223C2—21no—
224C2—22no—
225C2—22no—
226C2—21no—
227C2—82no—
228C2#2254977911yescharacteristic
229C2C2×#1686112411yescharacteristic
230C2#2254985011yescharacteristic
subgroupG/H
centerC22C2×C22⋊Dic127
commutatorC23×C127C22×C4
FrattiniC24C22×D127
Fitting, prime radicalC2×#121×C127C2
socleC22×C127C2×C22⋊C4
cosocleC24×C127C23
hypercenterC2×#121D127
hypocenterC127#1964
LCSG ▷ C23×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#121

Complements

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

NH ≅ G/Ntype
1. C2×#16861124C2semidirect
2. #22549736C2semidirect
3. C4×C22⋊Dic127C2semidirect
12. C22×D4×C127C4semidirect
19. C2×C4⋊C4×C127C4semidirect
20. C2×C22⋊C4×C127C4semidirect
26. C22⋊Dic127D4semidirect
104. C4×C127#99semidirect
116. C127#1964semidirect
121. C22×D4Dic127semidirect
129. C2×C4⋊C4Dic127semidirect
132. C2×C22⋊C4Dic127semidirect
222. C4#16861124semidirect

Maximal subgroups

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

subgroupindex
C2×#168611242
#225497362
C4×C22⋊Dic1272
C2×#121×C1272
#1964127