Finite Group Databasebeta

← C2×#7499 #151837

Subgroups — C2×#7499 #151837

Order 512.

674 conjugacy classes of subgroups; only the 138 normal subgroups are shown.

normal subgroups only

This group has 674 conjugacy classes of subgroups — too many to draw. Only its 138 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×#356C212yesmaximal
2C2×#1956C211yesmaximal, characteristic
3#7499C214yesmaximal, direct product factor
4C22×C4⋊Q8C2211yescharacteristic
5C22×#103—22no—
6C22×#101—22no—
7C22×C4×C8C411yescharacteristic
8C2×#384C411yescharacteristic
9C2×#356—22no—
10C2×#356—24no—
11C2×#356C418yescomplemented
12C2×#356C2214yes—
13#1956—22no—
14#1956C412yes—
15#1956C2212yes—
16C23×Q8D412yes—
17C22×M16—42no—
18C23×C8—21no—
19C2×C4⋊Q8—41no—
20C2×C4⋊Q8—42no—
21C2×C4⋊Q8C2×C414yes—
22C2×C4⋊Q8C2×C411yescharacteristic
23C2×C4⋊Q8D412yes—
24C2×C4⋊Q8D412yes—
25C2×C4⋊Q8Q812yes—
26C2×C4⋊Q8C2311yescharacteristic
27C22×C4⋊C4—21no—
28C22×C4⋊C4—41no—
29C22×C4⋊C4—41no—
30C22×C42C2×C411yescharacteristic
31C2×#103—24no—
32C2×#103—24no—
33C2×#103—44no—
34C2×#103—44no—
35C2×#101—28no—
36C2×#101—24no—
37C2×#101—42no—
38C2×#101—44no—
39C2×C4×C8—21no—
40C2×C4×C8—42no—
41C2×C4×C8C2×C412yes—
42#384—21no—
43#384—41no—
44#384C2×C412yes—
45#356—28no—
46#356—24no—
47#356C2×C418yescomplemented
48C22×Q8—24no—
49C22×Q8—22no—
50C22×Q8—44no—
51C22×Q8—42no—
52C22×Q8—42no—
53C22×Q8C22⋊C414yes—
54C22×Q8C22⋊C412yes—
55C22×Q8C4⋊C414yes—
56C22×Q8C2×D412yes—
57C23×C4—21no—
58C23×C4—22no—
59C23×C4—41no—
60C23×C4C22⋊C411yescharacteristic
61C2×M16—48no—
62C2×M16—44no—
63C2×M16—82no—
64C2×M16—84no—
65C22×C8—22no—
66C22×C8—22no—
67C22×C8—22no—
68C22×C8—42no—
69C22×C8—42no—
70C22×C8—82no—
71C4⋊Q8—44no—
72C4⋊Q8—41no—
73C4⋊Q8—42no—
74C4⋊Q8—42no—
75C4⋊Q8—42no—
76C4⋊Q8—41no—
77C4⋊Q8C4212yes—
78C4⋊Q8C22⋊C414yes—
79C4⋊Q8C4⋊C414yes—
80C4⋊Q8C4⋊C412yes—
81C4⋊Q8C22×C412yes—
82C4⋊Q8C2×D411yescharacteristic
83C4⋊Q8C2×Q811yescharacteristic
84C2×C4⋊C4—22no—
85C2×C4⋊C4—41no—
86C2×C4⋊C4—44no—
87C2×C4⋊C4—42no—
88C2×C4⋊C4—42no—
89C2×C4⋊C4—42no—
90C2×C4⋊C4—41no—
91C2×C4⋊C4—42no—
92C2×C4⋊C4—42no—
93C2×C4⋊C4—42no—
94C2×C4⋊C4—42no—
95C2×C4⋊C4—41no—
96C2×C4⋊C4—84no—
97C2×C42—21no—
98C2×C42—22no—
99C2×C42C4212yes—
100C2×C42C22⋊C411yescharacteristic
101C2×C42C4⋊C412yes—
102C2×C42C22×C411yescharacteristic
103#103—48no—
104#103—48no—
105#101—28no—
106#101—48no—
107#101—44no—
108C4×C8—42no—
109C4×C8—42no—
110C24#9911yescharacteristic
111C2×Q8—24no—
112C2×Q8—24no—
113C2×Q8—22no—
114C2×Q8—22no—
115C2×Q8—42no—
116C2×Q8—42no—
117C2×Q8—42no—
118C2×Q8—44no—
119C2×Q8—44no—
120C2×Q8—44no—
121C2×Q8—44no—
122C2×Q8—42no—
123C2×Q8—82no—
124C2×Q8—82no—
125C2×Q8—84no—
126C2×Q8—84no—
127C2×Q8#9514yes—
128C2×Q8C2×C22⋊C412yes—
129C2×Q8C2×C4⋊C412yes—
130C22×C4—24no—
131C22×C4—24no—
132C22×C4—22no—
133C22×C4—22no—
134C22×C4—21no—
135C22×C4—21no—
136C22×C4—22no—
137C22×C4—21no—
138C22×C4—21no—
139C22×C4—24no—
140C22×C4—24no—
141C22×C4—22no—
142C22×C4—22no—
143C22×C4—42no—
144C22×C4—42no—
145C22×C4—44no—
146C22×C4—42no—
147C22×C4—84no—
148C22×C4—81no—
149C22×C4—81no—
150C22×C4#9512yes—
151C22×C4#9911yescharacteristic
152C22×C4#9912yes—
153C22×C4C2×C22⋊C411yescharacteristic
154M16—48no—
155M16—84no—
156M16—88no—
157C2×C8—24no—
158C2×C8—42no—
159C2×C8—44no—
160C2×C8—44no—
161C2×C8—42no—
162C2×C8—84no—
163C2×C8—88no—
164C4⋊C4—42no—
165C4⋊C4—44no—
166C4⋊C4—44no—
167C4⋊C4—42no—
168C4⋊C4—42no—
169C4⋊C4—41no—
170C4⋊C4—41no—
171C4⋊C4—44no—
172C4⋊C4—82no—
173C4⋊C4—84no—
174C4⋊C4—84no—
175C42—22no—
176C42—22no—
177C42—21no—
178C42—21no—
179C42#9512yes—
180C42C2×C4211yescharacteristic
181C42C2×C4⋊C411yescharacteristic
182C23—22no—
183C23—22no—
184C23#34212yes—
185C23#35311yescharacteristic
186C23#35311yescharacteristic
187C23#35412yes—
188C23C2×#9911yescharacteristic
189Q8—44no—
190Q8—42no—
191Q8—42no—
192Q8—82no—
193Q8—82no—
194Q8—84no—
195Q8—84no—
196C2×C4—22no—
197C2×C4—22no—
198C2×C4—22no—
199C2×C4—22no—
200C2×C4—28no—
201C2×C4—28no—
202C2×C4—21no—
203C2×C4—21no—
204C2×C4—21no—
205C2×C4—21no—
206C2×C4—22no—
207C2×C4—22no—
208C2×C4—22no—
209C2×C4—22no—
210C2×C4—22no—
211C2×C4—21no—
212C2×C4—21no—
213C2×C4—22no—
214C2×C4—24no—
215C2×C4—22no—
216C2×C4—42no—
217C2×C4—42no—
218C2×C4—42no—
219C2×C4—42no—
220C2×C4—44no—
221C2×C4—44no—
222C2×C4—44no—
223C2×C4—44no—
224C2×C4—84no—
225C2×C4—82no—
226C2×C4—162no—
227C2×C4—162no—
228C2×C4#34212yes—
229C2×C4C2×#9511yescharacteristic
230C2×C4C2×#9911yescharacteristic
231C8—44no—
232C8—88no—
233C22—21no—
234C22—21no—
235C22—21no—
236C22—21no—
237C22—24no—
238C22—24no—
239C22#139312yes—
240C22#139312yes—
241C22#141012yes—
242C22#141112yes—
243C22C2×#34211yescharacteristic
244C22C2×#35311yescharacteristic
245C22C2×#35411yescharacteristic
246C4—24no—
247C4—22no—
248C4—22no—
249C4—21no—
250C4—22no—
251C4—21no—
252C4—42no—
253C4—44no—
254C4—42no—
255C4—162no—
256C2—21no—
257C2—21no—
258C2—22no—
259C2#749914yesdirect product factor
260C2C2×#139311yescharacteristic
261C2C2×#141011yescharacteristic
262C2C2×#141111yescharacteristic
subgroupG/H
center, socleC23#353
commutatorC42C2×C42
Frattini, cosocleC2×C4⋊Q8C23
prime radicalC24#99
LCSG ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C24 ◁ C23×C4 ◁ C22×C4⋊Q8 ◁ G
LECSG ▷ C2×C4⋊Q8 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #7499C2direct (N × H)
11. C2×#356C4semidirect
47. #356C2×C4semidirect
259. C2#7499direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#3562
C2×#19562
#74992