Finite Group Databasebeta

← C2×#11511 #480280

Subgroups — C2×#11511 #480280

Order 512.

1382 conjugacy classes of subgroups; only the 250 normal subgroups are shown.

normal subgroups only

This group has 1382 conjugacy classes of subgroups — too many to draw. Only its 250 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×#382C211yesmaximal, characteristic
2C2×#1463C212yesmaximal
3#11511C218yesmaximal, direct product factor
4C2×#1324C214yesmaximal, complemented
5C23×C4⋊C4C411yescharacteristic
6C23×C4⋊C4C2211yescharacteristic
7C23×C42C411yescharacteristic
8C2×#407—22no—
9C2×#382—22no—
10C2×#382—24no—
11C2×#382C414yes—
12C2×#382C412yes—
13C2×#382C2212yes—
14C2×#382C2214yes—
15C22×#95—22no—
16#1463—24no—
17#1463C2218yes—
18C2×#324—22no—
19C2×#324C2214yes—
20#1324C22116yescomplemented
21C24×C4—21no—
22C24×C4—21no—
23C24×C4C2×C411yescharacteristic
24C24×C4D412yes—
25C22×M16—42no—
26C22×C4⋊C4—22no—
27C22×C4⋊C4—21no—
28C22×C4⋊C4—41no—
29C22×C4⋊C4—41no—
30C22×C4⋊C4—42no—
31C22×C4⋊C4—42no—
32C22×C4⋊C4—41no—
33C22×C4⋊C4—41no—
34C22×C4⋊C4C2×C414yes—
35C22×C4⋊C4C2×C412yes—
36C22×C4⋊C4C2×C412yes—
37C22×C4⋊C4C2×C411yescharacteristic
38C22×C4⋊C4D412yes—
39C22×C4⋊C4D414yes—
40C22×C4⋊C4C2311yescharacteristic
41C22×C4⋊C4C2312yes—
42C22×C42—22no—
43C22×C42—21no—
44C22×C42—28no—
45C22×C42C2×C414yes—
46C22×C42C2×C412yes—
47#407—28no—
48#407—44no—
49C2×C22⋊C8—24no—
50C2×C22⋊C8—42no—
51#382—28no—
52#382—24no—
53#382—24no—
54#382—28no—
55#382C2×C414yes—
56#382C2×C418yes—
57#382C2314yes—
58C2×#95—28no—
59C2×#95—24no—
60C2×#95—24no—
61C2×#95—28no—
62#324—28no—
63#324—24no—
64#324C2318yes—
65C25C22⋊C411yescharacteristic
66C23×C4—22no—
67C23×C4—21no—
68C23×C4—21no—
69C23×C4—24no—
70C23×C4—24no—
71C23×C4—28no—
72C23×C4—22no—
73C23×C4—24no—
74C23×C4—22no—
75C23×C4—22no—
76C23×C4—24no—
77C23×C4—24no—
78C23×C4—22no—
79C23×C4—44no—
80C23×C4—44no—
81C23×C4—48no—
82C23×C4—48no—
83C23×C4—42no—
84C23×C4—41no—
85C23×C4—41no—
86C23×C4C22⋊C412yes—
87C23×C4C22⋊C411yescharacteristic
88C23×C4C22⋊C414yes—
89C23×C4C22⋊C412yes—
90C23×C4C22×C412yes—
91C23×C4C22×C411yescharacteristic
92C23×C4C2×D412yes—
93C23×C4C2×D414yes—
94C2×M16—48no—
95C2×M16—88no—
96C22×C8—44no—
97C2×C4⋊C4—22no—
98C2×C4⋊C4—24no—
99C2×C4⋊C4—42no—
100C2×C4⋊C4—42no—
101C2×C4⋊C4—41no—
102C2×C4⋊C4—41no—
103C2×C4⋊C4—44no—
104C2×C4⋊C4—42no—
105C2×C4⋊C4—44no—
106C2×C4⋊C4—42no—
107C2×C4⋊C4—42no—
108C2×C4⋊C4—41no—
109C2×C4⋊C4—42no—
110C2×C4⋊C4—41no—
111C2×C4⋊C4—44no—
112C2×C4⋊C4—44no—
113C2×C4⋊C4—42no—
114C2×C4⋊C4—42no—
115C2×C4⋊C4—44no—
116C2×C4⋊C4—44no—
117C2×C4⋊C4—42no—
118C2×C4⋊C4—42no—
119C2×C4⋊C4C22⋊C412yes—
120C2×C4⋊C4C22⋊C414yes—
121C2×C4⋊C4C22×C414yes—
122C2×C4⋊C4C22×C411yescharacteristic
123C2×C4⋊C4C22×C412yes—
124C2×C4⋊C4C2×D412yes—
125C2×C4⋊C4C2×D414yes—
126C2×C4⋊C4C2411yescharacteristic
127C2×C42—28no—
128C2×C42—216no—
129C2×C42—24no—
130C2×C42—22no—
131C2×C42—24no—
132C2×C42—216no—
133C2×C42M1618yescomplemented
134C2×C42C22×C414yes—
135C22⋊C8—28no—
136C22⋊C8—44no—
137C22⋊C8—48no—
138#95—28no—
139#95—216no—
140#95—28no—
141C24—24no—
142C24—24no—
143C24—22no—
144C24—22no—
145C24#9912yes—
146C24#9911yescharacteristic
147C24#9911yescharacteristic
148C24C2×C22⋊C412yes—
149C24C2×C22⋊C411yescharacteristic
150C22×C4—216no—
151C22×C4—28no—
152C22×C4—28no—
153C22×C4—22no—
154C22×C4—22no—
155C22×C4—22no—
156C22×C4—21no—
157C22×C4—22no—
158C22×C4—22no—
159C22×C4—21no—
160C22×C4—21no—
161C22×C4—21no—
162C22×C4—22no—
163C22×C4—24no—
164C22×C4—24no—
165C22×C4—24no—
166C22×C4—24no—
167C22×C4—216no—
168C22×C4—24no—
169C22×C4—28no—
170C22×C4—24no—
171C22×C4—24no—
172C22×C4—22no—
173C22×C4—22no—
174C22×C4—24no—
175C22×C4—22no—
176C22×C4—24no—
177C22×C4—22no—
178C22×C4—24no—
179C22×C4—432no—
180C22×C4—48no—
181C22×C4—44no—
182C22×C4—48no—
183C22×C4—44no—
184C22×C4—48no—
185C22×C4—48no—
186C22×C4—44no—
187C22×C4—44no—
188C22×C4—48no—
189C22×C4—416no—
190C22×C4—48no—
191C22×C4—44no—
192C22×C4—42no—
193C22×C4—44no—
194C22×C4—42no—
195C22×C4—84no—
196C22×C4—88no—
197C22×C4—88no—
198C22×C4—84no—
199C22×C4—84no—
200C22×C4#10114yes—
201C22×C4C2×C22⋊C411yescharacteristic
202C22×C4C2×C22⋊C412yes—
203C22×C4C2×C22⋊C412yes—
204C22×C4C2×C22⋊C414yes—
205C22×C4C2×M1614yes—
206C22×C4C23×C411yescharacteristic
207C22×C4C22×D412yes—
208M16—816no—
209C2×C8—48no—
210C2×C8—88no—
211C4⋊C4—42no—
212C4⋊C4—44no—
213C4⋊C4—42no—
214C4⋊C4—41no—
215C4⋊C4—42no—
216C4⋊C4—44no—
217C4⋊C4—41no—
218C4⋊C4—44no—
219C4⋊C4—44no—
220C4⋊C4—48no—
221C42—216no—
222C42—24no—
223C42—28no—
224C42C2×M1618yescomplemented
225C23—216no—
226C23—216no—
227C23—22no—
228C23—22no—
229C23—24no—
230C23—22no—
231C23—24no—
232C23—22no—
233C23—24no—
234C23—22no—
235C23—24no—
236C23—22no—
237C23—24no—
238C23—22no—
239C23—22no—
240C23#35414yes—
241C23#35514yes—
242C23#40712yes—
243C23C2×#9912yes—
244C23C2×#9912yes—
245C23C2×#9911yescharacteristic
246C23C2×#9911yescharacteristic
247C23C2×#10112yes—
248C23C22×C22⋊C411yescharacteristic
249C2×C4—28no—
250C2×C4—28no—
251C2×C4—28no—
252C2×C4—28no—
253C2×C4—22no—
254C2×C4—21no—
255C2×C4—22no—
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—21no—
259C2×C4—24no—
260C2×C4—24no—
261C2×C4—24no—
262C2×C4—24no—
263C2×C4—28no—
264C2×C4—22no—
265C2×C4—24no—
266C2×C4—22no—
267C2×C4—432no—
268C2×C4—44no—
269C2×C4—44no—
270C2×C4—416no—
271C2×C4—44no—
272C2×C4—48no—
273C2×C4—48no—
274C2×C4—44no—
275C2×C4—42no—
276C2×C4—44no—
277C2×C4—42no—
278C2×C4—48no—
279C2×C4—44no—
280C2×C4—82no—
281C2×C4—84no—
282C2×C4—84no—
283C2×C4—82no—
284C2×C4—88no—
285C2×C4—84no—
286C2×C4—88no—
287C2×C4—84no—
288C2×C4#40712yes—
289C2×C4C2×#10114yes—
290C2×C4C22×M1612yes—
291C8—88no—
292C22—28no—
293C22—28no—
294C22—216no—
295C22—21no—
296C22—22no—
297C22—21no—
298C22—21no—
299C22—22no—
300C22—21no—
301C22—22no—
302C22—22no—
303C22—24no—
304C22—28no—
305C22—28no—
306C22—24no—
307C22#146312yes—
308C22#146312yes—
309C22C2×#35414yes—
310C22C2×#35412yes—
311C22C2×#35514yes—
312C22C2×#35512yes—
313C22C2×#40711yescharacteristic
314C22C22×#9911yescharacteristic
315C22C22×#10111yescharacteristic
316C4—22no—
317C4—22no—
318C4—24no—
319C4—48no—
320C4—42no—
321C4—42no—
322C4—44no—
323C4—84no—
324C2—21no—
325C2—21no—
326C2—24no—
327C2—24no—
328C2—22no—
329C2#1151114yesdirect product factor
330C2C2×#146311yescharacteristic
331C2C22×#35411yescharacteristic
332C2C22×#35511yescharacteristic
subgroupG/H
center, socleC23C2×#99
commutatorC22×C4C23×C4
Frattini, cosocleC2×C4⋊C4C24
prime radicalC25C22⋊C4
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C25 ◁ C23×C4⋊C4 ◁ G
LECSG ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #11511C2direct (N × H)
4. C2×#1324C2semidirect
20. #1324C22semidirect
133. C2×C42M16semidirect
224. C42C2×M16semidirect
329. C2#11511direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#3822
C2×#14632
#115112
C2×#13242