Finite Group Databasebeta

← C2×#7646 #160940

Subgroups — C2×#7646 #160940

Order 512.

1342 conjugacy classes of subgroups; only the 290 normal subgroups are shown.

normal subgroups only

This group has 1342 conjugacy classes of subgroups — too many to draw. Only its 290 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×#2296C211yesmaximal, characteristic
2C2×#1817C212yesmaximal
3#7646C218yesmaximal, direct product factor
4C2×#1271C214yesmaximal, complemented
5C22×#117C411yescharacteristic
6C22×#117C411yescharacteristic
7C23×C42C2211yescharacteristic
8C22×C22⋊C8—22no—
9#2296—24no—
10#2296C414yes—
11#2296C2214yes—
12C2×#389C412yes—
13C2×#389C2212yes—
14C2×#385—22no—
15C2×C4×C4⋊C4—22no—
16#1817—24no—
17#1817C2218yes—
18C2×#336—22no—
19C2×#336C2214yes—
20#1271C22116yescomplemented
21C24×C4C2×C411yescharacteristic
22C24×C4D412yes—
23C23×C8—24no—
24C2×#117—22no—
25C2×#117—22no—
26C2×#117—28no—
27C2×#117—44no—
28C2×#117C2×C414yes—
29C2×#117C2×C414yes—
30C22×C4⋊C4—22no—
31C22×C4⋊C4—22no—
32C22×C4⋊C4C2×C412yes—
33C22×C4⋊C4C2×C412yes—
34C22×C22⋊C4—21no—
35C22×C22⋊C4—21no—
36C22×C42—21no—
37C22×C42—21no—
38C22×C42—44no—
39C22×C42C2×C411yescharacteristic
40C22×C42C2×C412yes—
41C22×C42D414yes—
42C22×C42D412yes—
43C22×C42C2312yes—
44C22×C42C2311yescharacteristic
45C2×C22⋊C8—28no—
46C2×C22⋊C8—48no—
47#389—24no—
48#389—22no—
49#389—22no—
50#389C2×C418yes—
51#389C2×C414yes—
52#389C2314yes—
53#385—28no—
54#385—28no—
55C4×C4⋊C4—28no—
56C4×C4⋊C4—28no—
57C2×#95—22no—
58C2×#95—22no—
59#336—24no—
60#336—28no—
61#336C2318yes—
62C25C22⋊C411yescharacteristic
63C23×C4—24no—
64C23×C4—24no—
65C23×C4—24no—
66C23×C4—28no—
67C23×C4—22no—
68C23×C4—28no—
69C23×C4—22no—
70C23×C4—44no—
71C23×C4C22⋊C412yes—
72C23×C4C22⋊C412yes—
73C23×C4C22⋊C414yes—
74C23×C4C22⋊C411yescharacteristic
75C23×C4C22×C412yes—
76C23×C4C22×C411yescharacteristic
77C23×C4C2×D414yes—
78C23×C4C2×D412yes—
79C22×C8—28no—
80C22×C8—48no—
81C22×C8—48no—
82C22×C8—48no—
83#117—216no—
84#117—216no—
85#117—48no—
86#117—48no—
87C2×C4⋊C4—22no—
88C2×C4⋊C4—24no—
89C2×C4⋊C4—28no—
90C2×C4⋊C4—24no—
91C2×C4⋊C4—22no—
92C2×C4⋊C4—24no—
93C2×C4⋊C4—44no—
94C2×C4⋊C4—48no—
95C2×C4⋊C4—48no—
96C2×C4⋊C4M1618yes—
97C2×C4⋊C4C22×C414yes—
98C2×C4⋊C4C22×C414yes—
99C2×C22⋊C4—28no—
100C2×C22⋊C4—24no—
101C2×C22⋊C4—24no—
102C2×C22⋊C4—44no—
103C2×C42—28no—
104C2×C42—24no—
105C2×C42—24no—
106C2×C42—42no—
107C2×C42—44no—
108C2×C42—44no—
109C2×C42—48no—
110C2×C42—42no—
111C2×C42—44no—
112C2×C42—44no—
113C2×C42C22⋊C414yes—
114C2×C42C22⋊C412yes—
115C2×C42C22×C412yes—
116C2×C42C22×C411yescharacteristic
117C2×C42C2×D414yes—
118C2×C42C2×D412yes—
119C2×C42C2411yescharacteristic
120C22⋊C8—416no—
121#95—28no—
122#95—24no—
123#95—24no—
124C24—22no—
125C24—22no—
126C24—24no—
127C24—24no—
128C24#9914yes—
129C24C2×C22⋊C412yes—
130C24C2×C22⋊C411yescharacteristic
131C22×C4—232no—
132C22×C4—22no—
133C22×C4—22no—
134C22×C4—22no—
135C22×C4—24no—
136C22×C4—24no—
137C22×C4—24no—
138C22×C4—22no—
139C22×C4—24no—
140C22×C4—24no—
141C22×C4—24no—
142C22×C4—24no—
143C22×C4—24no—
144C22×C4—24no—
145C22×C4—28no—
146C22×C4—28no—
147C22×C4—28no—
148C22×C4—24no—
149C22×C4—24no—
150C22×C4—24no—
151C22×C4—28no—
152C22×C4—28no—
153C22×C4—24no—
154C22×C4—44no—
155C22×C4—44no—
156C22×C4—432no—
157C22×C4—44no—
158C22×C4—44no—
159C22×C4—44no—
160C22×C4—44no—
161C22×C4—44no—
162C22×C4#10114yes—
163C22×C4C2×C22⋊C414yes—
164C22×C4C2×C22⋊C411yescharacteristic
165C22×C4C2×C22⋊C412yes—
166C22×C4C2×C22⋊C412yes—
167C22×C4C2×M1614yes—
168C22×C4#13518yes—
169C22×C4C23×C411yescharacteristic
170C22×C4C22×D412yes—
171C2×C8—48no—
172C2×C8—48no—
173C2×C8—48no—
174C2×C8—816no—
175C4⋊C4—24no—
176C4⋊C4—28no—
177C4⋊C4—44no—
178C4⋊C4—416no—
179C4⋊C4—44no—
180C4⋊C4—48no—
181C4⋊C4—48no—
182C4⋊C4C2×M1618yes—
183C22⋊C4—216no—
184C22⋊C4—48no—
185C42—216no—
186C42—44no—
187C42—42no—
188C42—41no—
189C42—42no—
190C42—44no—
191C42—42no—
192C42—41no—
193C42—48no—
194C23—22no—
195C23—22no—
196C23—24no—
197C23—24no—
198C23—28no—
199C23—216no—
200C23—44no—
201C23—44no—
202C23—48no—
203C23#40712yes—
204C23C2×#9914yes—
205C23C2×#9912yes—
206C23C2×#10112yes—
207C23#41614yes—
208C23#42114yes—
209C23C22×C22⋊C411yescharacteristic
210C2×C4—216no—
211C2×C4—216no—
212C2×C4—22no—
213C2×C4—22no—
214C2×C4—21no—
215C2×C4—21no—
216C2×C4—22no—
217C2×C4—22no—
218C2×C4—21no—
219C2×C4—21no—
220C2×C4—22no—
221C2×C4—22no—
222C2×C4—28no—
223C2×C4—28no—
224C2×C4—216no—
225C2×C4—28no—
226C2×C4—24no—
227C2×C4—24no—
228C2×C4—24no—
229C2×C4—416no—
230C2×C4—416no—
231C2×C4—44no—
232C2×C4—44no—
233C2×C4—416no—
234C2×C4—48no—
235C2×C4—44no—
236C2×C4—44no—
237C2×C4—44no—
238C2×C4—88no—
239C2×C4#40712yes—
240C2×C4C2×#10114yes—
241C2×C4#41614yes—
242C2×C4C22×M1612yes—
243C2×C4C2×#13514yes—
244C8—88no—
245C22—28no—
246C22—24no—
247C22—44no—
248C22—42no—
249C22—416no—
250C22—42no—
251C22#146314yes—
252C22#148818yes—
253C22#151618yes—
254C22#153714yes—
255C22#154414yes—
256C22C2×#40711yescharacteristic
257C22C22×#9911yescharacteristic
258C22C22×#10111yescharacteristic
259C22C2×#41612yes—
260C22C2×#42112yes—
261C4—24no—
262C4—24no—
263C4—22no—
264C4—22no—
265C4—24no—
266C4—48no—
267C4—44no—
268C4—84no—
269C2—44no—
270C2#764618yesdirect product factor
271C2C2×#146311yescharacteristic
272C2C2×#148812yes—
273C2C2×#151612yes—
274C2C2×#153711yescharacteristic
275C2C2×#154411yescharacteristic
subgroupG/H
center, socleC24C2×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC25C22⋊C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C24 ◁ C23×C42 ◁ G
LECSG ▷ C2×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #7646C2direct (N × H)
4. C2×#1271C2semidirect
20. #1271C22semidirect
270. C2#7646direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#22962
C2×#18172
#76462
C2×#12712