Finite Group Databasebeta

← C2×#7056 #122240

Subgroups — C2×#7056 #122240

Order 512.

634 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 634 conjugacy classes of subgroups — too many to draw. Only its 114 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×#1954C211yesmaximal, characteristic
2C2×#1322C212yesmaximal, complemented
3#7056C214yesmaximal, direct product factor
4C22×C4×D4C411yescharacteristic
5C22×C4⋊C8C2211yescharacteristic
6C22×C22⋊C8—22no—
7C22×C4×C8C411yescharacteristic
8C2×C22⋊C16—22no—
9#1954—22no—
10#1954C412yes—
11#1954C2212yes—
12#1322—22no—
13#1322C2214yescomplemented
14C23×D4C811yescharacteristic
15C24×C4—21no—
16C23×C8—21no—
17C23×C8D412yes—
18C2×C4×D4—21no—
19C2×C4×D4—21no—
20C2×C4×D4—41no—
21C2×C4×D4—41no—
22C2×C4×D4—41no—
23C2×C4×D4—41no—
24C2×C4×D4C812yes—
25C2×C4×D4C2×C412yes—
26C22×C4⋊C4C811yescharacteristic
27C22×C22⋊C4—21no—
28C22×C42C2×C411yescharacteristic
29C22×C16—42no—
30C2×C4⋊C8—41no—
31C2×C4⋊C8—41no—
32C2×C4⋊C8C2×C411yescharacteristic
33C2×C4⋊C8D412yes—
34C2×C4⋊C8C2311yescharacteristic
35C2×C22⋊C8—24no—
36C2×C22⋊C8—24no—
37C2×C22⋊C8—48no—
38C2×C4×C8—21no—
39C2×C4×C8—41no—
40C2×C4×C8—41no—
41C2×C4×C8C2×C412yes—
42C22⋊C16—24no—
43C22⋊C16—42no—
44C25—21no—
45C22×D4—21no—
46C22×D4—22no—
47C22×D4—41no—
48C22×D4—41no—
49C22×D4—42no—
50C22×D4C1614yescomplemented
51C22×D4C2×C812yes—
52C23×C4—21no—
53C23×C4—22no—
54C23×C4—22no—
55C23×C4—44no—
56C23×C4—44no—
57C23×C4—41no—
58C23×C4—41no—
59C23×C4—41no—
60C23×C4—41no—
61C23×C4C22⋊C411yescharacteristic
62C23×C4C2×C811yescharacteristic
63C23×C4M1611yescharacteristic
64C22×C8—24no—
65C22×C8—22no—
66C22×C8—22no—
67C22×C8—22no—
68C22×C8—42no—
69C22×C8—41no—
70C22×C8—41no—
71C22×C8—44no—
72C22×C8C22⋊C412yes—
73C22×C8C2×D412yes—
74C4×D4—42no—
75C4×D4—42no—
76C4×D4—42no—
77C4×D4—42no—
78C4×D4—42no—
79C4×D4—42no—
80C4×D4—42no—
81C4×D4—42no—
82C2×C4⋊C4—21no—
83C2×C4⋊C4—41no—
84C2×C4⋊C4—41no—
85C2×C4⋊C4C2×C812yes—
86C2×C22⋊C4—22no—
87C2×C22⋊C4—22no—
88C2×C22⋊C4—42no—
89C2×C22⋊C4—42no—
90C2×C42—21no—
91C2×C42—21no—
92C2×C42—21no—
93C2×C42—21no—
94C2×C42C22⋊C411yescharacteristic
95C2×C42C2×C811yescharacteristic
96C2×C42M1611yescharacteristic
97C2×C42C22×C411yescharacteristic
98C2×C16—44no—
99C2×C16—84no—
100C4⋊C8—41no—
101C4⋊C8—42no—
102C4⋊C8—41no—
103C22⋊C8—48no—
104C22⋊C8—48no—
105C4×C8—42no—
106C4×C8—42no—
107C24—24no—
108C24—22no—
109C24—44no—
110C24—44no—
111C24—41no—
112C24—41no—
113C24—42no—
114C24C22⋊C811yescharacteristic
115C2×D4—24no—
116C2×D4—22no—
117C2×D4—42no—
118C2×D4—44no—
119C2×D4—44no—
120C2×D4—42no—
121C2×D4—42no—
122C2×D4—44no—
123C2×D4—44no—
124C2×D4—42no—
125C2×D4C2×C1614yescomplemented
126C22×C4—21no—
127C22×C4—21no—
128C22×C4—22no—
129C22×C4—22no—
130C22×C4—22no—
131C22×C4—21no—
132C22×C4—21no—
133C22×C4—22no—
134C22×C4—44no—
135C22×C4—48no—
136C22×C4—44no—
137C22×C4—42no—
138C22×C4—44no—
139C22×C4—41no—
140C22×C4—41no—
141C22×C4—42no—
142C22×C4—42no—
143C22×C4—42no—
144C22×C4—42no—
145C22×C4—84no—
146C22×C4—84no—
147C22×C4—82no—
148C22×C4—82no—
149C22×C4—82no—
150C22×C4—82no—
151C22×C4C22⋊C811yescharacteristic
152C22×C4C22⋊C811yescharacteristic
153C22×C4C22⋊C811yescharacteristic
154C22×C4#9911yescharacteristic
155C22×C4#9911yescharacteristic
156C22×C4#10011yescharacteristic
157C22×C4#10011yescharacteristic
158C22×C4C2×C1612yes—
159C22×C4M3212yes—
160C22×C4C2×C22⋊C411yescharacteristic
161C22×C4C22×C811yescharacteristic
162C22×C4C2×M1611yescharacteristic
163C2×C8—24no—
164C2×C8—22no—
165C2×C8—22no—
166C2×C8—44no—
167C2×C8—44no—
168C2×C8—42no—
169C2×C8—44no—
170C2×C8—42no—
171C2×C8—42no—
172C2×C8—42no—
173C2×C8—82no—
174C4⋊C4—42no—
175C4⋊C4—42no—
176C22⋊C4—44no—
177C22⋊C4—44no—
178C42—21no—
179C42—21no—
180C42—21no—
181C42—21no—
182C42—21no—
183C42—21no—
184C42—21no—
185C42—21no—
186C16—84no—
187C23—22no—
188C23—22no—
189C23—24no—
190C23—44no—
191C23—416no—
192C23—44no—
193C23—44no—
194C23—44no—
195C23—42no—
196C23—42no—
197C23—82no—
198C23—88no—
199C23—82no—
200C23—82no—
201C23—82no—
202C23#32311yescharacteristic
203C23#32311yescharacteristic
204C23C22⋊C1612yes—
205C23#34912yes—
206C23C2×C22⋊C811yescharacteristic
207D4—44no—
208D4—48no—
209D4—44no—
210C2×C4—22no—
211C2×C4—22no—
212C2×C4—22no—
213C2×C4—22no—
214C2×C4—22no—
215C2×C4—21no—
216C2×C4—21no—
217C2×C4—21no—
218C2×C4—21no—
219C2×C4—22no—
220C2×C4—22no—
221C2×C4—21no—
222C2×C4—21no—
223C2×C4—21no—
224C2×C4—21no—
225C2×C4—21no—
226C2×C4—21no—
227C2×C4—42no—
228C2×C4—42no—
229C2×C4—41no—
230C2×C4—41no—
231C2×C4—44no—
232C2×C4—42no—
233C2×C4—84no—
234C2×C4—84no—
235C2×C4—82no—
236C2×C4—82no—
237C2×C4—84no—
238C2×C4—82no—
239C2×C4—82no—
240C2×C4#32311yescharacteristic
241C2×C4#32311yescharacteristic
242C2×C4C22⋊C1612yes—
243C2×C4#35312yes—
244C2×C4#35512yes—
245C2×C4C2×#9911yescharacteristic
246C2×C4C2×#10011yescharacteristic
247C2×C4C22×C1611yescharacteristic
248C2×C4C2×M3211yescharacteristic
249C8—44no—
250C8—42no—
251C8—42no—
252C22—24no—
253C22—24no—
254C22—21no—
255C22—21no—
256C22—21no—
257C22—21no—
258C22—48no—
259C22—44no—
260C22—88no—
261C22—84no—
262C22—88no—
263C22—82no—
264C22—82no—
265C22#131412yes—
266C22#131412yes—
267C22#131812yes—
268C22#132112yes—
269C22C2×#32311yescharacteristic
270C22C2×C22⋊C1611yescharacteristic
271C22C2×#34911yescharacteristic
272C4—22no—
273C4—21no—
274C4—21no—
275C4—21no—
276C4—21no—
277C4—41no—
278C4—41no—
279C4—82no—
280C4#131812yes—
281C4C2×#35311yescharacteristic
282C4C2×#35511yescharacteristic
283C2—21no—
284C2—22no—
285C2—21no—
286C2—84no—
287C2#705614yesdirect product factor
288C2C2×#131411yescharacteristic
289C2C2×#131811yescharacteristic
290C2C2×#132111yescharacteristic
subgroupG/H
centerC22×C4#99
commutatorC2×C4C22×C16
Frattini, cosocleC2×C4⋊C8C23
socleC23#323
prime radicalC23×D4C8
LCSG ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C23×C4 ◁ C22×C4⋊C8 ◁ G
LECSG ▷ C2×C4⋊C8 ▷ C22×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#1322C2semidirect
3. #7056C2direct (N × H)
13. #1322C22semidirect
50. C22×D4C16semidirect
125. C2×D4C2×C16semidirect
287. C2#7056direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#19542
C2×#13222
#70562