Finite Group Databasebeta

← C2×#11515 #480285

Subgroups — C2×#11515 #480285

Order 512.

1142 conjugacy classes of subgroups; only the 234 normal subgroups are shown.

normal subgroups only

This group has 1142 conjugacy classes of subgroups — too many to draw. Only its 234 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×#2317C211yesmaximal, characteristic
2C2×#1464C212yesmaximal
3#11515C218yesmaximal, direct product factor
4C2×#1324C214yesmaximal, complemented
5C22×#117C411yescharacteristic
6C22×#117C2211yescharacteristic
7C23×C4⋊C4C411yescharacteristic
8C2×#432—22no—
9#2317—24no—
10#2317C414yes—
11#2317C2214yes—
12C2×#389—22no—
13C2×#385—22no—
14C2×#384C412yes—
15C2×#382C2212yes—
16#1464—24no—
17#1464C2218yes—
18C2×#324C2214yes—
19C2×#323—22no—
20#1324C22116yescomplemented
21C24×C4—21no—
22C24×C4C2×C411yescharacteristic
23C2×#117—22no—
24C2×#117—28no—
25C2×#117—42no—
26C2×#117—42no—
27C2×#117C2×C412yes—
28C2×#117C2×C414yes—
29C2×#117D414yes—
30C2×#117C2312yes—
31C22×C4⋊C4—21no—
32C22×C4⋊C4—22no—
33C22×C4⋊C4—21no—
34C22×C4⋊C4—24no—
35C22×C4⋊C4—42no—
36C22×C4⋊C4—42no—
37C22×C4⋊C4C2×C414yes—
38C22×C4⋊C4C2×C412yes—
39C22×C4⋊C4C2×C411yescharacteristic
40C22×C4⋊C4C2311yescharacteristic
41C22×C22⋊C4—21no—
42C22×C22⋊C4D412yes—
43C22×C42C2×C412yes—
44C22×C42D412yes—
45#432—28no—
46#432—44no—
47C2×C4⋊C8—42no—
48C2×C22⋊C8—24no—
49C2×#97—42no—
50#389—28no—
51#389—28no—
52#385—28no—
53#385—28no—
54#384—24no—
55#384C2×C418yes—
56#382—24no—
57#382C2×C414yes—
58#382C2314yes—
59C2×#95—24no—
60#324—24no—
61#324C2318yes—
62#323—28no—
63C25C22⋊C411yescharacteristic
64C23×C4—24no—
65C23×C4—22no—
66C23×C4—22no—
67C23×C4—24no—
68C23×C4—22no—
69C23×C4—22no—
70C23×C4—24no—
71C23×C4—42no—
72C23×C4—42no—
73C23×C4—84no—
74C23×C4C22⋊C412yes—
75C23×C4C22⋊C412yes—
76C23×C4C22⋊C411yescharacteristic
77C23×C4C22×C412yes—
78C23×C4C22×C411yescharacteristic
79C23×C4C2×D412yes—
80C22×C8—44no—
81#117—216no—
82#117—216no—
83#117—44no—
84#117—44no—
85#117—44no—
86#117—44no—
87C2×C4⋊C4—28no—
88C2×C4⋊C4—24no—
89C2×C4⋊C4—22no—
90C2×C4⋊C4—24no—
91C2×C4⋊C4—216no—
92C2×C4⋊C4—41no—
93C2×C4⋊C4—41no—
94C2×C4⋊C4—41no—
95C2×C4⋊C4—41no—
96C2×C4⋊C4—42no—
97C2×C4⋊C4—42no—
98C2×C4⋊C4—48no—
99C2×C4⋊C4—42no—
100C2×C4⋊C4—42no—
101C2×C4⋊C4—48no—
102C2×C4⋊C4—88no—
103C2×C4⋊C4C22⋊C412yes—
104C2×C4⋊C4C22×C414yes—
105C2×C4⋊C4C22×C412yes—
106C2×C4⋊C4C22×C411yescharacteristic
107C2×C4⋊C4C2×D412yes—
108C2×C4⋊C4C2411yescharacteristic
109C2×C22⋊C4—28no—
110C2×C22⋊C4—24no—
111C2×C22⋊C4—42no—
112C2×C22⋊C4—42no—
113C2×C22⋊C4C22⋊C414yes—
114C2×C22⋊C4C2×D414yes—
115C2×C42—22no—
116C2×C42—22no—
117C2×C42—24no—
118C2×C42—22no—
119C2×C42—24no—
120C2×C42C22⋊C414yes—
121C2×C42M1618yes—
122C2×C42C22×C414yes—
123C2×C42C2×D414yes—
124C4⋊C8—48no—
125C22⋊C8—28no—
126C22⋊C8—44no—
127#97—48no—
128#95—28no—
129#95—28no—
130C24—22no—
131C24—21no—
132C24—21no—
133C24—42no—
134C24—42no—
135C24#10114yes—
136C24C2×C22⋊C411yescharacteristic
137C24C2×C22⋊C412yes—
138C22×C4—216no—
139C22×C4—216no—
140C22×C4—22no—
141C22×C4—21no—
142C22×C4—21no—
143C22×C4—24no—
144C22×C4—22no—
145C22×C4—24no—
146C22×C4—22no—
147C22×C4—24no—
148C22×C4—24no—
149C22×C4—24no—
150C22×C4—24no—
151C22×C4—24no—
152C22×C4—28no—
153C22×C4—22no—
154C22×C4—22no—
155C22×C4—22no—
156C22×C4—22no—
157C22×C4—24no—
158C22×C4—44no—
159C22×C4—44no—
160C22×C4—44no—
161C22×C4—44no—
162C22×C4—48no—
163C22×C4—48no—
164C22×C4—42no—
165C22×C4—42no—
166C22×C4—88no—
167C22×C4—88no—
168C22×C4—88no—
169C22×C4#9912yes—
170C22×C4#9912yes—
171C22×C4C2×C22⋊C411yescharacteristic
172C22×C4C2×C22⋊C412yes—
173C22×C4C2×C22⋊C412yes—
174C22×C4C2×C22⋊C414yes—
175C22×C4C2×M1614yes—
176C22×C4C23×C411yescharacteristic
177C22×C4C22×D412yes—
178C2×C8—48no—
179C2×C8—88no—
180C4⋊C4—216no—
181C4⋊C4—42no—
182C4⋊C4—42no—
183C4⋊C4—41no—
184C4⋊C4—42no—
185C4⋊C4—41no—
186C4⋊C4—44no—
187C4⋊C4—44no—
188C4⋊C4—48no—
189C4⋊C4—816no—
190C22⋊C4—216no—
191C22⋊C4—44no—
192C22⋊C4—44no—
193C42—24no—
194C42—28no—
195C42—24no—
196C42—24no—
197C42—24no—
198C42—24no—
199C42C2×M1618yes—
200C23—22no—
201C23—22no—
202C23—22no—
203C23—22no—
204C23—24no—
205C23—24no—
206C23—42no—
207C23—48no—
208C23—42no—
209C23—42no—
210C23—42no—
211C23—48no—
212C23—42no—
213C23—42no—
214C23#40712yes—
215C23C2×#9911yescharacteristic
216C23C2×#9911yescharacteristic
217C23C2×#10112yes—
218C23C2×#10114yes—
219C23C22×C22⋊C411yescharacteristic
220C2×C4—216no—
221C2×C4—216no—
222C2×C4—22no—
223C2×C4—21no—
224C2×C4—21no—
225C2×C4—21no—
226C2×C4—22no—
227C2×C4—21no—
228C2×C4—24no—
229C2×C4—24no—
230C2×C4—24no—
231C2×C4—24no—
232C2×C4—28no—
233C2×C4—22no—
234C2×C4—22no—
235C2×C4—24no—
236C2×C4—44no—
237C2×C4—44no—
238C2×C4—416no—
239C2×C4—44no—
240C2×C4—44no—
241C2×C4—44no—
242C2×C4—42no—
243C2×C4—42no—
244C2×C4—48no—
245C2×C4—44no—
246C2×C4—82no—
247C2×C4—82no—
248C2×C4—84no—
249C2×C4—816no—
250C2×C4—84no—
251C2×C4#40712yes—
252C2×C4C2×#9914yes—
253C2×C4C22×M1612yes—
254C8—88no—
255C22—21no—
256C22—21no—
257C22—21no—
258C22—21no—
259C22—24no—
260C22—24no—
261C22—24no—
262C22—44no—
263C22—44no—
264C22—48no—
265C22—42no—
266C22—42no—
267C22—48no—
268C22#146312yes—
269C22#146312yes—
270C22#212912yes—
271C22#213412yes—
272C22C2×#40711yescharacteristic
273C22C22×#9911yescharacteristic
274C22C22×#10111yescharacteristic
275C4—22no—
276C4—22no—
277C4—24no—
278C4—48no—
279C4—42no—
280C4—42no—
281C4—44no—
282C4—84no—
283C2—21no—
284C2—21no—
285C2—22no—
286C2—44no—
287C2#1151514yesdirect product factor
288C2C2×#146311yescharacteristic
289C2C2×#212911yescharacteristic
290C2C2×#213411yescharacteristic
subgroupG/H
center, socleC23C2×#99
commutatorC22×C4C23×C4
Frattini, cosocleC2×C4⋊C4C24
prime radicalC25C22⋊C4
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×#117 ◁ G
LECSG ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #11515C2direct (N × H)
4. C2×#1324C2semidirect
20. #1324C22semidirect
287. C2#11515direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#23172
C2×#14642
#115152
C2×#13242