Finite Group Databasebeta

← C2×#22549784 #30641925

Subgroups — C2×#22549784 #30641925

Order 16256.

1812 conjugacy classes of subgroups; only the 285 normal subgroups are shown.

normal subgroups only

This group has 1812 conjugacy classes of subgroups — too many to draw. Only its 285 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⋊D127C212yesmaximal, complemented
2C22×C22⋊Dic127C212yesmaximal, complemented
3C22×(C2×D127)⋊C4C211yesmaximal, characteristic, complemented
4#22549784C218yesmaximal, direct product factor
5C2×C127⋊#95C211yesmaximal, characteristic, complemented
6C23×D4×C127C211yesmaximal, characteristic, complemented
7C24×D127C2211yescharacteristic
8C22×C22⋊D127—28no—
9C22×C22⋊D127—28no—
10C22×C22⋊D127—28no—
11C22×C22⋊D127C2218yescomplemented
12C23×Dic127C2211yescharacteristic
13C23×Dic127C2212yes—
14C2×C22⋊Dic127—28no—
15C2×C22⋊Dic127C2218yescomplemented
16C2×(C2×D127)⋊C4—24no—
17C2×(C2×D127)⋊C4C2214yescomplemented
18C127⋊#95C2214yescomplemented
19C25×C127C2212yes—
20C22×D4×C127—28no—
21C22×D4×C127—24no—
22C22×D4×C127C2214yescomplemented
23C23×C4×C127C2211yescharacteristic
24C23×D127—24no—
25C23×D127—24no—
26C23×D127—22no—
27C23×D127—22no—
28C23×D127D414yescomplemented
29C23×D127C2312yes—
30C2×C22⋊D127—264no—
31C2×C22⋊D127—216no—
32C2×C22⋊D127—216no—
33C2×C22⋊D127—216no—
34C22×Dic127—24no—
35C22×Dic127—24no—
36C22×Dic127—22no—
37C22×Dic127—24no—
38C22×Dic127D418yescomplemented
39C22×Dic127C2312yes—
40C22×Dic127C2314yes—
41C22⋊Dic127—216no—
42(C2×D127)⋊C4—28no—
43C24×C127—24no—
44C24×C127—24no—
45C24×C127—24no—
46C24×C127—24no—
47C24×C127—24no—
48C24×C127—24no—
49C24×C127D418yescomplemented
50C24×C127C2314yes—
51C24×C127C2311yescharacteristic
52C2×D4×C127—232no—
53C2×D4×C127—216no—
54C2×D4×C127—28no—
55C22×C4×C127—24no—
56C22×C4×C127D414yescomplemented
57C22×C4×C127C2312yes—
58C22×D127—216no—
59C22×D127—28no—
60C22×D127—22no—
61C22×D127—24no—
62C22×D127—24no—
63C22×D127—22no—
64C22×D127—44no—
65C22×D127—48no—
66C22×D127—44no—
67C22×D127C2×D414yescomplemented
68C22⋊D127—264no—
69C2×Dic127—216no—
70C2×Dic127—24no—
71C2×Dic127—24no—
72C2×Dic127—22no—
73C2×Dic127—24no—
74C2×Dic127—44no—
75C2×Dic127C2×D418yescomplemented
76C23×C127—216no—
77C23×C127—216no—
78C23×C127—216no—
79C23×C127—24no—
80C23×C127—24no—
81C23×C127—24no—
82C23×C127—24no—
83C23×C127—24no—
84C23×C127—24no—
85C23×C127—48no—
86C23×C127—48no—
87C23×C127—48no—
88C23×C127—48no—
89C23×C127C2×D412yes—
90C23×C127C2×D414yes—
91C23×C127C2×D414yes—
92C23×C127C2×D412yes—
93C23×C127C2×D418yescomplemented
94C23×C127#4112yes—
95C23×C127C2411yescharacteristic
96D4×C127—232no—
97C2×C4×C127—28no—
98C2×C4×C127—24no—
99C2×C4×C127C2×D414yescomplemented
100C22×C127—28no—
101C22×C127—28no—
102C22×C127—28no—
103C22×C127—44no—
104C22×C127—44no—
105C22×C127—44no—
106C22×C127—44no—
107C22×C127—432no—
108C22×C127#12018yes—
109C22×C127#12014yes—
110C22×C127#12114yes—
111C22×C127#12118yes—
112C22×C127#12714yes—
113C22×C127C22×D411yescharacteristic
114C22×C127C22×D412yes—
115C22×C127C22×D412yes—
116C22×C127C22×D411yescharacteristic
117C22×C127C2×#4111yescharacteristic
118C2×D127—28no—
119C2×D127—24no—
120C2×D127—42no—
121C2×D127—44no—
122C2×D127—42no—
123C2×D127—416no—
124C4×C127—24no—
125Dic127—28no—
126Dic127—42no—
127C2×C127—48no—
128C2×C127#39218yescomplemented
129C2×C127C2×#12012yes—
130C2×C127C2×#12011yescharacteristic
131C2×C127C2×#12111yescharacteristic
132C2×C127C2×#12112yes—
133C2×C127C2×#12711yescharacteristic
134D127—44no—
135C2×#392—1271nomaximal, Sylow
136C127C2×#39211yescharacteristic, complemented, Sylow
137C23×D4—1272no—
138C23×D4D12711yescharacteristic, complemented
139C22×C22⋊C4—1272no—
140C22×C22⋊C4—1271no—
141#392—1278no—
142C2×#95—1271no—
143C25—1271no—
144C25C2×D12712yes—
145C22×D4—28no—
146C22×D4—24no—
147C22×D4—1278no—
148C22×D4—2548no—
149C22×D4—2548no—
150C22×D4—2548no—
151C22×D4C2×D12714yescomplemented
152C23×C4—1272no—
153C23×C4—1271no—
154C23×C4C2×D12711yescharacteristic
155C2×C22⋊C4—1278no—
156C2×C22⋊C4—1274no—
157C2×C22⋊C4—2548no—
158C2×C22⋊C4—2544no—
159#95—1274no—
160C24—24no—
161C24—24no—
162C24—24no—
163C24—24no—
164C24—24no—
165C24—24no—
166C24—1274no—
167C24—1272no—
168C24—2544no—
169C24—2542no—
170C24—2544no—
171C24—2542no—
172C24C22⋊D12718yescomplemented
173C24C22×D12714yes—
174C24C22×D12711yescharacteristic
175C2×D4—232no—
176C2×D4—216no—
177C2×D4—28no—
178C2×D4—25416no—
179C2×D4—25416no—
180C2×D4—25464no—
181C2×D4—25416no—
182C22×C4—24no—
183C22×C4—1278no—
184C22×C4—1274no—
185C22×C4—1272no—
186C22×C4—2544no—
187C22×C4—2542no—
188C22×C4—2544no—
189C22×C4—2544no—
190C22×C4C22⋊D12714yescomplemented
191C22×C4C22×D12712yes—
192C22⋊C4—25416no—
193C22⋊C4—2548no—
194C23—24no—
195C23—24no—
196C23—24no—
197C23—24no—
198C23—24no—
199C23—24no—
200C23—216no—
201C23—216no—
202C23—216no—
203C23—48no—
204C23—48no—
205C23—48no—
206C23—48no—
207C23—1274no—
208C23—5084no—
209C23—5084no—
210C23—5088no—
211C23—2544no—
212C23—2542no—
213C23—2544no—
214C23—2542no—
215C23—25416no—
216C23—2548no—
217C23D4×D12714yes—
218C23D4×D12712yes—
219C23C2×C22⋊D12712yes—
220C23C2×C22⋊D12714yes—
221C23C2×C22⋊D12718yescomplemented
222C23#1452324712yes—
223C23C23×D12711yescharacteristic
224D4—232no—
225D4—25464no—
226C2×C4—24no—
227C2×C4—28no—
228C2×C4—1278no—
229C2×C4—5084no—
230C2×C4—2544no—
231C2×C4—2542no—
232C2×C4—2544no—
233C2×C4—2544no—
234C2×C4—25416no—
235C2×C4C2×C22⋊D12714yescomplemented
236C22—28no—
237C22—28no—
238C22—28no—
239C22—432no—
240C22—44no—
241C22—44no—
242C22—44no—
243C22—44no—
244C22—50816no—
245C22—5082no—
246C22—5084no—
247C22—5082no—
248C22—2548no—
249C22—2544no—
250C22#1686117718yes—
251C22#1686117814yes—
252C22#1686118014yes—
253C22#1686118118yes—
254C22#1686120514yes—
255C22C2×D4×D12711yescharacteristic
256C22C2×D4×D12712yes—
257C22C22×C22⋊D12712yes—
258C22C22×C22⋊D12711yescharacteristic
259C22C2×#1452324711yescharacteristic
260C4—24no—
261C4—5082no—
262C4—2548no—
263C2—48no—
264C2—5084no—
265C2#2254978418yesdirect product factor
266C2C2×#1686117712yes—
267C2C2×#1686117811yescharacteristic
268C2C2×#1686118011yescharacteristic
269C2C2×#1686118112yes—
270C2C2×#1686120511yescharacteristic
subgroupG/H
centerC24C22×D127
commutator, cosocleC23×C127C24
FrattiniC23C23×D127
FittingC23×D4×C127C2
socleC24×C127C23
hypercenterC23×D4D127
hypocenterC127C2×#392
LCSG ▷ C23×C127 ▷ C127
UCSC1 ◁ C24 ◁ C23×D4

Complements

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

NH ≅ G/Ntype
1. C23×C22⋊D127C2semidirect
2. C22×C22⋊Dic127C2semidirect
3. C22×(C2×D127)⋊C4C2semidirect
4. #22549784C2direct (N × H)
5. C2×C127⋊#95C2semidirect
6. C23×D4×C127C2semidirect
11. C22×C22⋊D127C22semidirect
15. C2×C22⋊Dic127C22semidirect
17. C2×(C2×D127)⋊C4C22semidirect
18. C127⋊#95C22semidirect
22. C22×D4×C127C22semidirect
28. C23×D127D4semidirect
38. C22×Dic127D4semidirect
49. C24×C127D4semidirect
56. C22×C4×C127D4semidirect
67. C22×D127C2×D4semidirect
75. C2×Dic127C2×D4semidirect
93. C23×C127C2×D4semidirect
99. C2×C4×C127C2×D4semidirect
128. C2×C127#392semidirect
136. C127C2×#392semidirect
138. C23×D4D127semidirect
151. C22×D4C2×D127semidirect
172. C24C22⋊D127semidirect
190. C22×C4C22⋊D127semidirect
221. C23C2×C22⋊D127semidirect
235. C2×C4C2×C22⋊D127semidirect
265. C2#22549784direct (N × H)

Maximal subgroups

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

subgroupindex
C23×C22⋊D1272
C22×C22⋊Dic1272
C22×(C2×D127)⋊C42
#225497842
C2×C127⋊#952
C23×D4×C1272
C2×#392127