Finite Group Databasebeta

← #30638644

Subgroups — #30638644

Order 16256.

880 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 880 conjugacy classes of subgroups — too many to draw. Only its 109 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
1C23×C22⋊D127C211yesmaximal, characteristic
2C2×#16861123C214yesmaximal, complemented
3#22549732C211yesmaximal, characteristic, complemented
4#379×C127C211yesmaximal, characteristic, complemented
5C24×D127C411yescharacteristic, complemented
6C22×C22⋊D127—28no—
7C22×C22⋊D127—22no—
8C22×C22⋊D127—22no—
9C22×C22⋊D127C412yescomplemented
10C22×C22⋊D127C2212yes—
11C23×Dic127C411yescharacteristic, complemented
12C2×C22⋊Dic127—22no—
13C2×C22⋊Dic127C2212yescomplemented
14#16861123—28no—
15C25×C127C2211yescharacteristic
16C2×C22⋊C4×C127—22no—
17C2×C22⋊C4×C127C2212yescomplemented
18C23×D127—24no—
19C23×D127—22no—
20C23×D127—44no—
21C23×D127C2×C412yes—
22C2×C22⋊D127—216no—
23C2×C22⋊D127—216no—
24C2×C22⋊D127—24no—
25C2×C22⋊D127—24no—
26C2×C22⋊D127—24no—
27C2×C22⋊D127—24no—
28C2×C22⋊D127—22no—
29C2×C22⋊D127—22no—
30C2×C22⋊D127D418yes—
31C22×Dic127—24no—
32C22×Dic127—22no—
33C22×Dic127—22no—
34C22×Dic127C2×C412yes—
35C22⋊Dic127—24no—
36C22⋊Dic127—44no—
37C24×C127—24no—
38C24×C127—44no—
39C24×C127C2×C411yescharacteristic
40C24×C127C2×C411yescharacteristic
41C24×C127D412yes—
42C24×C127D412yes—
43C24×C127C2311yescharacteristic
44C22×C4×C127—22no—
45C22⋊C4×C127—24no—
46C22⋊C4×C127—44no—
47C22×D127—28no—
48C22×D127—22no—
49C22×D127—22no—
50C22×D127—416no—
51C22×D127—44no—
52C22×D127—44no—
53C22×D127—44no—
54C22×D127C22⋊C414yescomplemented
55C22⋊D127—216no—
56C22⋊D127—28no—
57C22⋊D127—28no—
58C2×Dic127—28no—
59C2×Dic127—22no—
60C2×Dic127—22no—
61C2×Dic127—44no—
62C2×Dic127—42no—
63C2×Dic127C22⋊C414yescomplemented
64C23×C127—24no—
65C23×C127—22no—
66C23×C127—24no—
67C23×C127—24no—
68C23×C127—22no—
69C23×C127—24no—
70C23×C127—24no—
71C23×C127—42no—
72C23×C127—42no—
73C23×C127—42no—
74C23×C127—42no—
75C23×C127—416no—
76C23×C127C22⋊C412yes—
77C23×C127C22⋊C412yes—
78C23×C127C22×C411yescharacteristic
79C23×C127C2×D414yes—
80C23×C127C2×D411yescharacteristic
81C23×C127C2×D411yescharacteristic
82C2×C4×C127—44no—
83C2×C4×C127—42no—
84C22×C127—28no—
85C22×C127—22no—
86C22×C127—22no—
87C22×C127—22no—
88C22×C127—22no—
89C22×C127—22no—
90C22×C127—22no—
91C22×C127—22no—
92C22×C127—22no—
93C22×C127—44no—
94C22×C127—48no—
95C22×C127—48no—
96C22×C127—44no—
97C22×C127#9914yescomplemented
98C22×C127C2×C22⋊C412yes—
99C22×C127C2×C22⋊C411yescharacteristic
100C22×C127#12012yes—
101C22×C127#12012yes—
102C2×D127—24no—
103C2×D127—416no—
104C2×D127—44no—
105C2×D127—44no—
106C2×D127—44no—
107C4×C127—82no—
108Dic127—24no—
109Dic127—82no—
110C2×C127—22no—
111C2×C127—22no—
112C2×C127—24no—
113C2×C127—22no—
114C2×C127—42no—
115C2×C127#37911yescharacteristic
116C2×C127C2×#9912yes—
117D127—44no—
118#1781—1271nomaximal, Sylow
119C127#178111yescharacteristic, complemented, Sylow
120C23×D4—1271no—
121C2×#99—1274no—
122#379—1271no—
123#379D12711yescharacteristic, complemented
124C25—1271no—
125C25C2×D12711yescharacteristic
126C22×D4—1272no—
127C22×D4—1272no—
128C22×D4—2548no—
129C22×D4—2542no—
130C22×D4—2542no—
131C23×C4—1271no—
132C2×C22⋊C4—22no—
133C2×C22⋊C4—1272no—
134C2×C22⋊C4—2542no—
135C2×C22⋊C4C2×D12712yescomplemented
136#99—2548no—
137C24—24no—
138C24—44no—
139C24—1272no—
140C24—5084no—
141C24—2544no—
142C24—2542no—
143C24C4×D12711yescharacteristic
144C24C4×D12711yescharacteristic
145C24C22⋊D12712yes—
146C24D50812yes—
147C24C22×D12711yescharacteristic
148C2×D4—1278no—
149C2×D4—2544no—
150C2×D4—2544no—
151C2×D4—25416no—
152C2×D4—25416no—
153C2×D4—2544no—
154C2×D4—2544no—
155C2×D4—2542no—
156C2×D4—2542no—
157C22×C4—22no—
158C22×C4—1272no—
159C22×C4—2544no—
160C22×C4—2542no—
161C22×C4—2542no—
162C22⋊C4—24no—
163C22⋊C4—44no—
164C22⋊C4—5084no—
165C22⋊C4—2544no—
166C23—24no—
167C23—22no—
168C23—24no—
169C23—24no—
170C23—22no—
171C23—24no—
172C23—24no—
173C23—42no—
174C23—42no—
175C23—42no—
176C23—42no—
177C23—416no—
178C23—1274no—
179C23—5084no—
180C23—50816no—
181C23—5084no—
182C23—5084no—
183C23—2548no—
184C23—2542no—
185C23—2542no—
186C23(C2×D127)⋊C412yes—
187C23(C2×D127)⋊C412yes—
188C23C2×C4×D12711yescharacteristic
189C23D4×D12714yes—
190C23C2×C22⋊D12711yescharacteristic
191C23C2×D50811yescharacteristic
192D4—25416no—
193D4—2548no—
194D4—2548no—
195C2×C4—44no—
196C2×C4—42no—
197C2×C4—1274no—
198C2×C4—5084no—
199C2×C4—5082no—
200C2×C4—2548no—
201C2×C4—2542no—
202C2×C4—2542no—
203C22—28no—
204C22—22no—
205C22—22no—
206C22—22no—
207C22—22no—
208C22—22no—
209C22—22no—
210C22—22no—
211C22—22no—
212C22—44no—
213C22—48no—
214C22—48no—
215C22—44no—
216C22—5084no—
217C22—50816no—
218C22—5084no—
219C22—5084no—
220C22—2544no—
221C22#1686112314yescomplemented
222C22C22⋊C4×D12712yes—
223C22C2×(C2×D127)⋊C411yescharacteristic
224C22#1686117712yes—
225C22#1686117912yes—
226C4—82no—
227C4—10162no—
228C4—2544no—
229C2—22no—
230C2—22no—
231C2—24no—
232C2—22no—
233C2—42no—
234C2—5084no—
235C2#2254973111yescharacteristic
236C2C2×#1686112312yes—
subgroupG/H
centerC22C2×(C2×D127)⋊C4
commutatorC23×C127C22×C4
FrattiniC24C22×D127
Fitting#379×C127C2
socleC22×C127C2×C22⋊C4
prime radicalC23×C22⋊D127C2
cosocleC24×C127C23
hypercenter#379D127
hypocenterC127#1781
LCSG ▷ C23×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C25 ◁ #379

Complements

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

NH ≅ G/Ntype
2. C2×#16861123C2semidirect
3. #22549732C2semidirect
4. #379×C127C2semidirect
5. C24×D127C4semidirect
9. C22×C22⋊D127C4semidirect
11. C23×Dic127C4semidirect
13. C2×C22⋊Dic127C22semidirect
17. C2×C22⋊C4×C127C22semidirect
54. C22×D127C22⋊C4semidirect
63. C2×Dic127C22⋊C4semidirect
97. C22×C127#99semidirect
119. C127#1781semidirect
123. #379D127semidirect
135. C2×C22⋊C4C2×D127semidirect
221. C22#16861123semidirect

Maximal subgroups

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

subgroupindex
C23×C22⋊D1272
C2×#168611232
#225497322
#379×C1272
#1781127