Finite Group Databasebeta

← C2×#13092577 #15682604

Subgroups — C2×#13092577 #15682604

Order 2944.

760 conjugacy classes of subgroups; only the 165 normal subgroups are shown.

normal subgroups only

This group has 760 conjugacy classes of subgroups — too many to draw. Only its 165 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
1C22×#67288C211yesmaximal, characteristic
2C22×#67275C211yesmaximal, characteristic, complemented
3C22×#67274C211yesmaximal, characteristic, complemented
4#13092577C218yesmaximal, direct product factor
5C2×#10624992C211yesmaximal, characteristic, complemented
6C2×#10624991C211yesmaximal, characteristic, complemented
7C2×C22⋊(C23⋊C8)C211yesmaximal, characteristic, complemented
8C2×C23×C22⋊Q8C211yesmaximal, characteristic, complemented
9C22×Q8×C23C2211yescharacteristic
10C23×C4×C23C2211yescharacteristic
11C22×C22⋊D23—21no—
12C2×#67288—22no—
13C2×#67288—24no—
14C2×#67288—24no—
15C2×#67288C2214yes—
16C22×D92C2211yescharacteristic
17C22×C4×D23—21no—
18C22×Dic46C2211yescharacteristic
19C23×C22⋊Q8—22no—
20C23×C22⋊Q8C2214yescomplemented
21C2×C4⋊C4×C23—21no—
22C2×C4⋊C4×C23C2211yescharacteristic
23C2×C22⋊C4×C23—21no—
24C2×#67275—24no—
25C2×#67275C2214yescomplemented
26C2×#67274—24no—
27C2×#67274C2214yescomplemented
28C22×C23⋊C8C2211yescharacteristic
29C22×C23⋊C8C2211yescharacteristic
30C22⋊(C23⋊C8)C2214yescomplemented
31#10624992C2214yescomplemented
32#10624991C2214yescomplemented
33C24×C23D411yescharacteristic
34C23×D23—21no—
35C2×Q8×C23—21no—
36C2×Q8×C23—22no—
37C2×Q8×C23—22no—
38C2×Q8×C23C2312yes—
39C22×C4×C23—22no—
40C22×C4×C23—22no—
41C22×C4×C23—21no—
42C22×C4×C23—21no—
43C22×C4×C23D411yescharacteristic
44C22×C4×C23D412yes—
45C22×C4×C23C2312yes—
46C22×C4×C23C2311yescharacteristic
47C2×C22⋊D23—28no—
48C2×C22⋊D23—24no—
49C22×Dic23—21no—
50#67288—216no—
51#67288—28no—
52#67288—28no—
53C2×D92—22no—
54C2×D92—21no—
55C2×D92D414yes—
56C2×D92C2312yes—
57C2×C4×D23—28no—
58C2×C4×D23—24no—
59C2×Dic46—22no—
60C2×Dic46—21no—
61C2×Dic46D414yes—
62C2×Dic46C2312yes—
63C4⋊C4×C23—24no—
64C4⋊C4×C23—21no—
65C4⋊C4×C23C2312yes—
66C22⋊C4×C23—24no—
67#67275—28no—
68#67274—28no—
69C2×C23⋊C8—22no—
70C2×C23⋊C8—22no—
71C2×C23⋊C8C2312yes—
72C2×C23⋊C8C2312yes—
73C23×C23—22no—
74C23×C23—22no—
75C23×C23—22no—
76C23×C23C2×D412yes—
77C23×C23C2×D411yescharacteristic
78C22×D23—24no—
79C22×D23—22no—
80C22×D23—42no—
81C22×D23—42no—
82Q8×C23—22no—
83Q8×C23—22no—
84Q8×C23—42no—
85C2×C4×C23—22no—
86C2×C4×C23—24no—
87C2×C4×C23—22no—
88C2×C4×C23—22no—
89C2×C4×C23—22no—
90C2×C4×C23—22no—
91C2×C4×C23—42no—
92C2×C4×C23—42no—
93C2×C4×C23C2×D412yes—
94C2×C4×C23C2×D412yes—
95C2×C4×C23C2×D412yes—
96C2×C4×C23C2×D411yescharacteristic
97C2×C4×C23C2411yescharacteristic
98C22⋊D23—216no—
99C2×Dic23—24no—
100C2×Dic23—22no—
101D92—24no—
102D92—22no—
103D92C2×D414yes—
104C4×D23—216no—
105Dic46—24no—
106Dic46—22no—
107Dic46C2×D414yes—
108C23⋊C8—22no—
109C23⋊C8—22no—
110C2×#452—231nomaximal, Sylow
111C22×C23—22no—
112C22×C23—22no—
113C22×C23—22no—
114C22×C23—44no—
115C22×C23#12012yes—
116C22×C23#13512yes—
117C22×C23#13712yes—
118C22×C23C22×D411yescharacteristic
119C2×D23—24no—
120C2×D23—42no—
121C2×D23—42no—
122C2×D23—48no—
123C4×C23—22no—
124C4×C23—42no—
125C4×C23—42no—
126C4×C23#12012yes—
127C4×C23C22×D411yescharacteristic
128C4×C23C22×D411yescharacteristic
129Dic23—24no—
130C22×#41—231no—
131C22×Q16—231no—
132C22×SD16—231no—
133C2×C22⋊Q8D2311yescharacteristic, complemented
134#452—238no—
135C2×#103—231no—
136C2×#102—231no—
137C2×C22⋊C8—231no—
138C2×C23—42no—
139C2×C23#45214yescomplemented
140C2×C23C2×#12011yescharacteristic
141C2×C23C2×#13511yescharacteristic
142C2×C23C2×#13711yescharacteristic
143D23—44no—
144C2×#41—234no—
145C2×#41—464no—
146C2×#41—464no—
147C2×#41—462no—
148C22×Q8—231no—
149C22×Q8C2×D2311yescharacteristic
150C22×D4—231no—
151C22×D4—461no—
152C23×C4—461no—
153C23×C4C2×D2311yescharacteristic
154C2×Q16—234no—
155C2×Q16—464no—
156C2×SD16—234no—
157C2×SD16—464no—
158C22×C8—231no—
159C22×C8—231no—
160C22⋊Q8—22no—
161C22⋊Q8C2×D2314yescomplemented
162C2×C4⋊C4—21no—
163C2×C4⋊C4C2×D2311yescharacteristic
164C2×C22⋊C4—21no—
165#103—234no—
166#102—234no—
167C22⋊C8—234no—
168C23C2×#45211yescharacteristic, complemented, Sylow
169C24—461no—
170C24C22⋊D2311yescharacteristic
171#41—468no—
172#41—468no—
173#41—4616no—
174C2×Q8—21no—
175C2×Q8—22no—
176C2×Q8—22no—
177C2×Q8—234no—
178C2×Q8—232no—
179C2×Q8—462no—
180C2×Q8—461no—
181C2×Q8C22×D2312yes—
182C2×D4—234no—
183C2×D4—232no—
184C2×D4—468no—
185C2×D4—464no—
186C2×D4—462no—
187C2×D4—461no—
188C22×C4—22no—
189C22×C4—22no—
190C22×C4—21no—
191C22×C4—21no—
192C22×C4—468no—
193C22×C4—461no—
194C22×C4—464no—
195C22×C4C22⋊D2312yes—
196C22×C4C22⋊D2311yescharacteristic
197C22×C4C22×D2312yes—
198C22×C4C22×D2311yescharacteristic
199Q16—468no—
200SD16—468no—
201C2×C8—232no—
202C2×C8—232no—
203C2×C8—462no—
204C2×C8—462no—
205C4⋊C4—24no—
206C4⋊C4—21no—
207C4⋊C4C22×D2312yes—
208C22⋊C4—24no—
209C23—22no—
210C23—22no—
211C23—22no—
212C23—464no—
213C23—462no—
214C23—922no—
215C23—922no—
216C23C2×C22⋊D2312yes—
217C23C2×C22⋊D2311yescharacteristic
218Q8—22no—
219Q8—22no—
220Q8—42no—
221Q8—234no—
222Q8—464no—
223Q8—462no—
224D4—234no—
225D4—4616no—
226D4—464no—
227D4—462no—
228C2×C4—22no—
229C2×C4—24no—
230C2×C4—22no—
231C2×C4—22no—
232C2×C4—22no—
233C2×C4—22no—
234C2×C4—42no—
235C2×C4—42no—
236C2×C4—4616no—
237C2×C4—464no—
238C2×C4—462no—
239C2×C4D4×D2312yes—
240C2×C4D4×D2312yes—
241C2×C4C2×C22⋊D2312yes—
242C2×C4C2×C22⋊D2311yescharacteristic
243C2×C4C23×D2311yescharacteristic
244C8—462no—
245C8—462no—
246C22—22no—
247C22—22no—
248C22—22no—
249C22—44no—
250C22—464no—
251C22—922no—
252C22—922no—
253C22—928no—
254C22#1062510812yes—
255C22#1062511312yes—
256C22#1062512112yes—
257C22C22×C22⋊D2311yescharacteristic
258C4—22no—
259C4—42no—
260C4—42no—
261C4—464no—
262C4#1062510812yes—
263C4C2×D4×D2311yescharacteristic
264C4C2×D4×D2311yescharacteristic
265C2—42no—
266C2—924no—
267C2#1309257714yesdirect product factor
268C2C2×#1062510811yescharacteristic
269C2C2×#1062512111yescharacteristic
270C2C2×#1062511311yescharacteristic
subgroupG/H
centerC23C2×C22⋊D23
commutator, cosocleC2×C4×C23C24
FrattiniC2×C4C23×D23
FittingC2×C23×C22⋊Q8C2
socleC23×C23C2×D4
prime radicalC22×#67288C2
hypercenterC2×C22⋊Q8D23
hypocenterC23C2×#452
LCSG ▷ C2×C4×C23 ▷ C2×C23 ▷ C23
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×C22⋊Q8

Complements

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

NH ≅ G/Ntype
2. C22×#67275C2semidirect
3. C22×#67274C2semidirect
4. #13092577C2direct (N × H)
5. C2×#10624992C2semidirect
6. C2×#10624991C2semidirect
7. C2×C22⋊(C23⋊C8)C2semidirect
8. C2×C23×C22⋊Q8C2semidirect
20. C23×C22⋊Q8C22semidirect
25. C2×#67275C22semidirect
27. C2×#67274C22semidirect
30. C22⋊(C23⋊C8)C22semidirect
31. #10624992C22semidirect
32. #10624991C22semidirect
133. C2×C22⋊Q8D23semidirect
139. C2×C23#452semidirect
161. C22⋊Q8C2×D23semidirect
168. C23C2×#452semidirect
267. C2#13092577direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#672882
C22×#672752
C22×#672742
#130925772
C2×#106249922
C2×#106249912
C2×C22⋊(C23⋊C8)2
C2×C23×C22⋊Q82
C2×#45223