Finite Group Databasebeta

← C2×#8740 #190043

Subgroups — C2×#8740 #190043

Order 512.

1086 conjugacy classes of subgroups; only the 242 normal subgroups are shown.

normal subgroups only

This group has 1086 conjugacy classes of subgroups — too many to draw. Only its 242 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×#2298C211yesmaximal, characteristic
2C2×#1822C211yesmaximal, characteristic
3C2×#1820C211yesmaximal, characteristic
4#8740C218yesmaximal, direct product factor
5C2×#1284C214yesmaximal
6C22×#117C2211yescharacteristic
7C2×#407C2212yes—
8#2298—22no—
9#2298—22no—
10#2298C414yescomplemented
11#2298C2214yes—
12C2×#388—22no—
13C2×#387—22no—
14C2×#381C2212yes—
15C2×C4×C4⋊C4C412yescomplemented
16C2×C4×C22⋊C4C2212yes—
17C2×#376C411yescharacteristic, complemented
18C2×C43C411yescharacteristic, complemented
19#1822—22no—
20#1822C2214yes—
21#1820—22no—
22#1820C2214yes—
23C2×#343—22no—
24C2×#342—22no—
25#1284C22116yes—
26C24×C4C2311yescharacteristic
27C22×M16—22no—
28C2×#117—24no—
29C2×#117—24no—
30C2×#117C2×C412yes—
31C2×#117D412yes—
32C2×#117D412yes—
33C2×#117C2312yes—
34C22×C4⋊C4—22no—
35C22×C4⋊C4D412yes—
36C22×C22⋊C4—22no—
37C22×C22⋊C4—22no—
38C22×C22⋊C4D412yes—
39C22×C42—22no—
40C22×C42C2×C411yescharacteristic
41C22×C42C2×C411yescharacteristic
42C22×C42C2×C412yes—
43C22×C42C2×C412yes—
44#407—24no—
45#407—22no—
46#407C2314yes—
47C2×C22⋊C8—24no—
48#388—28no—
49#388—28no—
50#387—28no—
51#387—28no—
52#381—24no—
53#381—22no—
54#381C2314yes—
55C4×C4⋊C4—24no—
56C4×C4⋊C4C2×C418yescomplemented
57C4×C22⋊C4—22no—
58C4×C22⋊C4—22no—
59C4×C22⋊C4C2×C414yes—
60C4×C22⋊C4C2314yes—
61#376—22no—
62#376C2×C414yescomplemented
63C2×#95—22no—
64C2×#95—22no—
65C2×#95—22no—
66C2×#95—24no—
67C43—22no—
68C43C2×C414yescomplemented
69#343—28no—
70#342—28no—
71C25C2×D411yescharacteristic
72C23×C4—22no—
73C23×C4—22no—
74C23×C4—44no—
75C23×C4—42no—
76C23×C4C22⋊C412yes—
77C23×C4C22⋊C412yes—
78C23×C4C22×C411yescharacteristic
79C23×C4C22×C411yescharacteristic
80C23×C4C2×D411yescharacteristic
81C23×C4C2×D411yescharacteristic
82C23×C4C2×D411yescharacteristic
83C23×C4#4114yes—
84C23×C4#4114yes—
85C23×C4C2411yescharacteristic
86C2×M16—24no—
87C2×M16—42no—
88C2×M16—48no—
89C22×C8—42no—
90#117—28no—
91#117—28no—
92#117—28no—
93#117—28no—
94C2×C4⋊C4—24no—
95C2×C4⋊C4—24no—
96C2×C4⋊C4—28no—
97C2×C4⋊C4—48no—
98C2×C4⋊C4C22⋊C414yes—
99C2×C4⋊C4C2×D414yes—
100C2×C22⋊C4—24no—
101C2×C22⋊C4—24no—
102C2×C22⋊C4—24no—
103C2×C22⋊C4—24no—
104C2×C22⋊C4—24no—
105C2×C22⋊C4—48no—
106C2×C22⋊C4—48no—
107C2×C22⋊C4—42no—
108C2×C22⋊C4C22⋊C414yes—
109C2×C22⋊C4C2×D414yes—
110C2×C42—22no—
111C2×C42—22no—
112C2×C42—22no—
113C2×C42—22no—
114C2×C42—22no—
115C2×C42—28no—
116C2×C42—22no—
117C2×C42—44no—
118C2×C42—44no—
119C2×C42—48no—
120C2×C42C22⋊C412yes—
121C2×C42C22⋊C412yes—
122C2×C42C22×C412yes—
123C2×C42C22×C412yes—
124C2×C42C22×C414yes—
125C2×C42C22×C414yes—
126C22⋊C8—28no—
127C22⋊C8—44no—
128#95—28no—
129#95—24no—
130#95—28no—
131#95—24no—
132#95—28no—
133#95—44no—
134C24—24no—
135C24—44no—
136C24C2×C22⋊C411yescharacteristic
137C24C2×C22⋊C411yescharacteristic
138C24#12312yes—
139C24#12312yes—
140C24C22×D411yescharacteristic
141C22×C4—24no—
142C22×C4—24no—
143C22×C4—28no—
144C22×C4—24no—
145C22×C4—22no—
146C22×C4—22no—
147C22×C4—22no—
148C22×C4—22no—
149C22×C4—24no—
150C22×C4—42no—
151C22×C4—42no—
152C22×C4—42no—
153C22×C4—48no—
154C22×C4—48no—
155C22×C4—42no—
156C22×C4—44no—
157C22×C4—44no—
158C22×C4—44no—
159C22×C4—44no—
160C22×C4—44no—
161C22×C4—44no—
162C22×C4—44no—
163C22×C4—44no—
164C22×C4—84no—
165C22×C4—88no—
166C22×C4C2×C22⋊C411yescharacteristic
167C22×C4C2×C22⋊C411yescharacteristic
168C22×C4C2×C22⋊C414yes—
169C22×C4C2×C22⋊C414yes—
170C22×C4#11712yes—
171C22×C4#11712yes—
172C22×C4#11714yes—
173C22×C4#12312yes—
174C22×C4#12312yes—
175C22×C4#12314yes—
176C22×C4#12314yes—
177C22×C4C23×C411yescharacteristic
178C22×C4C22×D411yescharacteristic
179C22×C4C2×#4114yes—
180M16—48no—
181M16—84no—
182C2×C8—44no—
183C2×C8—84no—
184C4⋊C4—28no—
185C4⋊C4—28no—
186C4⋊C4—416no—
187C22⋊C4—28no—
188C22⋊C4—28no—
189C22⋊C4—48no—
190C22⋊C4—48no—
191C22⋊C4—48no—
192C22⋊C4—84no—
193C42—24no—
194C42—24no—
195C42—24no—
196C42—24no—
197C42—44no—
198C42—44no—
199C42—44no—
200C42—44no—
201C42—416no—
202C23—24no—
203C23—24no—
204C23—28no—
205C23—24no—
206C23—22no—
207C23—22no—
208C23—42no—
209C23—42no—
210C23—44no—
211C23—88no—
212C23#38112yes—
213C23#38111yescharacteristic
214C23#38111yescharacteristic
215C23#41014yes—
216C23C22×C22⋊C411yescharacteristic
217C23C2×#12312yes—
218C2×C4—24no—
219C2×C4—24no—
220C2×C4—24no—
221C2×C4—24no—
222C2×C4—22no—
223C2×C4—22no—
224C2×C4—22no—
225C2×C4—22no—
226C2×C4—28no—
227C2×C4—24no—
228C2×C4—24no—
229C2×C4—44no—
230C2×C4—44no—
231C2×C4—44no—
232C2×C4—44no—
233C2×C4—44no—
234C2×C4—416no—
235C2×C4—44no—
236C2×C4—44no—
237C2×C4—44no—
238C2×C4—44no—
239C2×C4—44no—
240C2×C4—44no—
241C2×C4—44no—
242C2×C4—44no—
243C2×C4—44no—
244C2×C4—44no—
245C2×C4—88no—
246C2×C4—84no—
247C2×C4—88no—
248C2×C4—168no—
249C2×C4#38112yes—
250C2×C4#38112yes—
251C2×C4C2×#11712yes—
252C2×C4C2×#12312yes—
253C8—84no—
254C22—28no—
255C22—22no—
256C22—22no—
257C22—22no—
258C22—22no—
259C22—24no—
260C22—24no—
261C22—48no—
262C22—82no—
263C22—82no—
264C22—84no—
265C22#182012yes—
266C22#182012yes—
267C22#182714yes—
268C22C2×#38111yescharacteristic
269C22C2×#41012yes—
270C4—22no—
271C4—22no—
272C4—22no—
273C4—22no—
274C4—48no—
275C4—44no—
276C4—44no—
277C4—84no—
278C4—164no—
279C2—22no—
280C2—22no—
281C2—42no—
282C2—42no—
283C2#874014yesdirect product factor
284C2C2×#182011yescharacteristic
285C2C2×#182712yes—
subgroupG/H
center, socleC23#381
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C24×C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #8740C2direct (N × H)
10. #2298C4semidirect
15. C2×C4×C4⋊C4C4semidirect
17. C2×#376C4semidirect
18. C2×C43C4semidirect
56. C4×C4⋊C4C2×C4semidirect
62. #376C2×C4semidirect
68. C43C2×C4semidirect
283. C2#8740direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#22982
C2×#18222
C2×#18202
#87402
C2×#12842