Finite Group Databasebeta

← C2×#15467 #609040

Subgroups — C2×#15467 #609040

Order 512.

11994 conjugacy classes of subgroups; only the 1010 normal subgroups are shown.

normal subgroups only

This group has 11994 conjugacy classes of subgroups — too many to draw. Only its 1010 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×#393C213yesmaximal, complemented
2C22×#379C213yesmaximal
3#15467C2116yesmaximal, direct product factor
4C2×#1440C216yesmaximal, complemented
5C2×#1437C213yesmaximal, complemented
6C27C2211yescharacteristic
7C24×Q8C2211yescharacteristic, complemented
8C23×C22⋊C4C2213yes—
9C23×C22⋊C4C2216yes—
10C23×C22⋊C4C2216yes—
11C23×C22⋊C4C2213yes—
12C23×C42C2211yescharacteristic, complemented
13C2×#393—224no—
14C2×#393C22124yescomplemented
15C2×#379—224no—
16C2×#379C22124yes—
17C22×#95C2213yescomplemented
18C22×#95C2216yescomplemented
19C22×#95C2212yescomplemented
20C22×#95C2213yescomplemented
21#1440C22148yescomplemented
22#1437C22124yescomplemented
23C26—212no—
24C26—224no—
25C26—42no—
26C26—46no—
27C26—44no—
28C26C2313yes—
29C26C2314yes—
30C23×Q8—212no—
31C23×Q8—46no—
32C23×Q8—42no—
33C23×Q8C2314yescomplemented
34C24×C4C2316yes—
35C24×C4C2313yes—
36C24×C4C2313yes—
37C22×C22⋊C4—224no—
38C22×C22⋊C4—212no—
39C22×C22⋊C4—224no—
40C22×C22⋊C4—224no—
41C22×C22⋊C4—224no—
42C22×C22⋊C4—224no—
43C22×C22⋊C4—46no—
44C22×C22⋊C4—46no—
45C22×C22⋊C4—412no—
46C22×C22⋊C4—412no—
47C22×C22⋊C4D4124yes—
48C22×C22⋊C4C23112yes—
49C22×C22⋊C4C23124yes—
50C22×C22⋊C4C23124yes—
51C22×C22⋊C4C23112yes—
52C22×C42—42no—
53C22×C42—44no—
54C22×C42C2314yescomplemented
55#393—296no—
56#379—296no—
57C2×#95—224no—
58C2×#95—212no—
59C2×#95—46no—
60C2×#95—44no—
61C2×#95C23124yescomplemented
62C2×#95C23112yescomplemented
63C2×#95C23112yescomplemented
64C2×#95C2318yescomplemented
65C25—212no—
66C25—212no—
67C25—212no—
68C25—26no—
69C25—26no—
70C25—212no—
71C25—212no—
72C25—296no—
73C25—224no—
74C25—248no—
75C25—44no—
76C25—412no—
77C25—48no—
78C25—448no—
79C25—496no—
80C25—448no—
81C25—496no—
82C25—448no—
83C25—496no—
84C25—416no—
85C25—416no—
86C25—448no—
87C25C2×D4112yes—
88C25#41124yes—
89C25C2416yes—
90C25C2411yescharacteristic
91C22×Q8—224no—
92C22×Q8—412no—
93C22×Q8—44no—
94C22×Q8—448no—
95C22×Q8—448no—
96C22×Q8—416no—
97C23×C4—212no—
98C23×C4—212no—
99C23×C4—26no—
100C23×C4—26no—
101C23×C4—212no—
102C23×C4—26no—
103C23×C4—26no—
104C23×C4—212no—
105C23×C4—212no—
106C23×C4—212no—
107C23×C4—212no—
108C23×C4—212no—
109C23×C4—212no—
110C23×C4—212no—
111C23×C4—212no—
112C23×C4C2×D4112yes—
113C23×C4C2×D4112yes—
114C23×C4C24112yes—
115C23×C4C2416yes—
116C23×C4C2416yes—
117C2×C22⋊C4—224no—
118C2×C22⋊C4—248no—
119C2×C22⋊C4—248no—
120C2×C22⋊C4—248no—
121C2×C22⋊C4—248no—
122C2×C22⋊C4—248no—
123C2×C22⋊C4—448no—
124C2×C22⋊C4—412no—
125C2×C22⋊C4—412no—
126C2×C22⋊C4—424no—
127C2×C22⋊C4—424no—
128C2×C22⋊C4—496no—
129C2×C22⋊C4—496no—
130C2×C22⋊C4—448no—
131C2×C22⋊C4C2×D4148yes—
132C2×C42—44no—
133C2×C42—48no—
134C2×C42—416no—
135#95—248no—
136#95—224no—
137#95—412no—
138#95—48no—
139C24—224no—
140C24—248no—
141C24—224no—
142C24—248no—
143C24—248no—
144C24—248no—
145C24—248no—
146C24—248no—
147C24—248no—
148C24—248no—
149C24—212no—
150C24—212no—
151C24—212no—
152C24—26no—
153C24—26no—
154C24—212no—
155C24—212no—
156C24—296no—
157C24—448no—
158C24—496no—
159C24—448no—
160C24—496no—
161C24—4384no—
162C24—448no—
163C24—496no—
164C24—4768no—
165C24—4128no—
166C24—4384no—
167C24—4256no—
168C24—416no—
169C24—416no—
170C24—448no—
171C24—424no—
172C24—424no—
173C24—424no—
174C24—424no—
175C24—412no—
176C24—424no—
177C24—412no—
178C24—424no—
179C24—412no—
180C24—412no—
181C24#124148yes—
182C24C22×D4112yes—
183C24C22×D416yes—
184C24C2×#41124yes—
185C24C2×#41112yes—
186C24H(5,𝔽2)12yes—
187C24H(5,𝔽2)16yes—
188C24H(5,𝔽2)14yes—
189C24C2511yescharacteristic
190C2×Q8—448no—
191C2×Q8—448no—
192C2×Q8—4128no—
193C2×Q8—416no—
194C22×C4—224no—
195C22×C4—224no—
196C22×C4—248no—
197C22×C4—248no—
198C22×C4—212no—
199C22×C4—212no—
200C22×C4—26no—
201C22×C4—26no—
202C22×C4—212no—
203C22×C4—26no—
204C22×C4—26no—
205C22×C4—212no—
206C22×C4—212no—
207C22×C4—212no—
208C22×C4—212no—
209C22×C4—212no—
210C22×C4—212no—
211C22×C4—212no—
212C22×C4—212no—
213C22×C4—424no—
214C22×C4—424no—
215C22×C4—412no—
216C22×C4—412no—
217C22×C4—412no—
218C22×C4—412no—
219C22×C4—424no—
220C22×C4—424no—
221C22×C4—424no—
222C22×C4—424no—
223C22×C4—424no—
224C22×C4—424no—
225C22×C4—424no—
226C22×C4C22×D4112yes—
227C22×C4C22×D4112yes—
228C22⋊C4—448no—
229C22⋊C4—496no—
230C22⋊C4—496no—
231C22⋊C4—448no—
232C42—416no—
233C23—296no—
234C23—212no—
235C23—224no—
236C23—212no—
237C23—224no—
238C23—224no—
239C23—224no—
240C23—224no—
241C23—224no—
242C23—224no—
243C23—224no—
244C23—2192no—
245C23—4192no—
246C23—4384no—
247C23—41024no—
248C23—464no—
249C23—4192no—
250C23—4128no—
251C23—496no—
252C23—412no—
253C23—412no—
254C23—448no—
255C23—412no—
256C23—412no—
257C23—448no—
258C23—496no—
259C23—46no—
260C23—412no—
261C23—46no—
262C23—412no—
263C23—496no—
264C23—496no—
265C23—496no—
266C23—46no—
267C23—46no—
268C23#525112yes—
269C23C2×#124124yes—
270C23C2×#124112yes—
271C23#534112yes—
272C23#538124yes—
273C23D42112yes—
274C23#546124yes—
275C23#551112yes—
276C23#551124yes—
277C23H(3,𝔽4)14yes—
278C23H(3,𝔽4)14yes—
279C23C23×D413yes—
280C23C22×#4116yes—
281C23C2×H(5,𝔽2)11yescharacteristic
282C23C2×H(5,𝔽2)13yes—
283C23C2×H(5,𝔽2)12yes—
284Q8—464no—
285C2×C4—212no—
286C2×C4—212no—
287C2×C4—224no—
288C2×C4—224no—
289C2×C4—412no—
290C2×C4—448no—
291C2×C4—412no—
292C2×C4—448no—
293C2×C4—46no—
294C2×C4—46no—
295C2×C4—46no—
296C2×C4—46no—
297C2×C4—412no—
298C2×C4—412no—
299C2×C4—412no—
300C2×C4—412no—
301C2×C4—412no—
302C2×C4—412no—
303C2×C4—412no—
304C2×C4—848no—
305C22—224no—
306C22—248no—
307C22—4256no—
308C22—424no—
309C22—424no—
310C22—424no—
311C22—4192no—
312C22—412no—
313C22—412no—
314C22—424no—
315C22—424no—
316C22#2472148yes—
317C22#2561124yes—
318C22#267918yes—
319C22#2684124yes—
320C22#2736116yes—
321C22C2×#52513yes—
322C22C22×#12413yes—
323C22C2×#53413yes—
324C22C2×#53816yes—
325C22C2×D4213yes—
326C22C2×#54616yes—
327C22C2×#55113yes—
328C22C2×#55116yes—
329C22C2×H(3,𝔽4)11yescharacteristic
330C22C2×H(3,𝔽4)11yescharacteristic
331C4—412no—
332C4—412no—
333C4—812no—
334C2—424no—
335C2#15467116yesdirect product factor
336C2C2×#247216yes—
337C2C2×#256113yes—
338C2C2×#267911yescharacteristic
339C2C2×#268413yes—
340C2C2×#273612yes—
subgroupG/H
center, socleC25C24
commutator, Frattini, cosocleC24C25
prime radicalC27C22
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C25 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C22×#393C2semidirect
3. #15467C2direct (N × H)
4. C2×#1440C2semidirect
5. C2×#1437C2semidirect
7. C24×Q8C22semidirect
12. C23×C42C22semidirect
14. C2×#393C22semidirect
17. C22×#95C22semidirect
18. C22×#95C22semidirect
19. C22×#95C22semidirect
20. C22×#95C22semidirect
21. #1440C22semidirect
22. #1437C22semidirect
33. C23×Q8C23semidirect
54. C22×C42C23semidirect
61. C2×#95C23semidirect
62. C2×#95C23semidirect
63. C2×#95C23semidirect
64. C2×#95C23semidirect
335. C2#15467direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#3932
C22×#3792
#154672
C2×#14402
C2×#14372