Finite Group Databasebeta

← C2×#36474 #7630158

Subgroups — C2×#36474 #7630158

Order 512.

7506 conjugacy classes of subgroups.

The subgroup lattice has 320 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×Q82C211yesmaximal, characteristic
2C22×#557C211yesmaximal, characteristic
3C2×#2862C213yesmaximal
4#36474C2132yesmaximal, direct product factor
5C2×#2567C213yesmaximal
6C2×#2546C213yesmaximal
7C2×#2550C2112yesmaximal
8C2×#2351C214yesmaximal
9C2×C4×C4⋊Q8C213yesmaximal
10C2×C42×Q8C211yesmaximal, characteristic
11C2×Q82—216no—
12C2×Q82C22116yes—
13C2×#557—216no—
14C2×#557C22116yes—
15C22×C4⋊Q8C2213yes—
16C22×C4⋊Q8C2213yes—
17C22×C4⋊Q8C2216yes—
18C22×C4⋊Q8C2213yes—
19C22×#125C2216yes—
20C22×C4×Q8C2212yes—
21C22×C4×Q8C2212yes—
22C22×C4×Q8C2213yes—
23C22×C4×Q8C2213yes—
24C22×C4×Q8C2211yescharacteristic
25C22×C4×Q8C2212yes—
26#2862C22148yes—
27#2567C22148yes—
28#2550C221192yes—
29#2546C22148yes—
30C2×#398C22112yes—
31C2×#395C22112yes—
32#2351C22164yes—
33C4×C4⋊Q8C22148yes—
34C2×#391C2213yes—
35C2×#391C22112yes—
36C2×#391C2216yes—
37C2×#391C2211yescharacteristic
38C2×#389C22112yes—
39C2×#389C22112yes—
40C2×#389C22112yes—
41C2×#384C2213yes—
42C2×#382C22112yes—
43C42×Q8C22116yes—
44C2×C4×C4⋊C4C22112yes—
45C2×C4×C4⋊C4C2216yes—
46C2×C4×C4⋊C4C2213yes—
47C2×C43C2213yes—
48C23×Q8C2313yes—
49C23×Q8C2311yescharacteristic
50C23×Q8C2316yes—
51C23×Q8C2311yescharacteristic
52Q82—2128no—
53#557—2128no—
54C2×C4⋊Q8—224no—
55C2×C4⋊Q8—248no—
56C2×C4⋊Q8—224no—
57C2×C4⋊Q8—224no—
58C2×C4⋊Q8C23124yes—
59C2×C4⋊Q8C23124yes—
60C2×C4⋊Q8C23124yes—
61C2×C4⋊Q8C23148yes—
62C2×#125—248no—
63C2×#125C23148yes—
64C2×C4×Q8—216no—
65C2×C4×Q8—216no—
66C2×C4×Q8—224no—
67C2×C4×Q8—224no—
68C2×C4×Q8D4116yes—
69C2×C4×Q8Q8132yescomplemented
70C2×C4×Q8C23124yes—
71C2×C4×Q8C23124yes—
72C2×C4×Q8C23116yes—
73C2×C4×Q8C23116yes—
74C2×C4×Q8C2318yes—
75C2×C4×Q8C23116yes—
76C22×C4⋊C4C2316yes—
77C22×C4⋊C4C2316yes—
78C22×C4⋊C4C2316yes—
79C22×C4⋊C4C23112yes—
80C22×C4⋊C4C2311yescharacteristic
81C22×C4⋊C4C2316yes—
82C22×C4⋊C4C2316yes—
83C22×C4⋊C4C2313yes—
84C22×C4⋊C4C23112yes—
85C22×C4⋊C4C2313yes—
86C22×C4⋊C4C2316yes—
87C22×C42C2316yes—
88C22×C42C2316yes—
89C22×C42C2313yes—
90C22×C42C2311yescharacteristic
91C22×C42C2316yes—
92C22×C42C2313yes—
93C22×C42C2316yes—
94#398C23196yes—
95#395C23196yes—
96#391C2318yes—
97#391C23124yes—
98#391C23148yes—
99#391C23196yes—
100#389C23196yes—
101#389C23196yes—
102#389C23196yes—
103#384C23124yes—
104#382C23196yes—
105C4×C4⋊C4C23196yes—
106C4×C4⋊C4C23124yes—
107C4×C4⋊C4C23148yes—
108C2×#95C2316yes—
109C2×#95C23124yes—
110C2×#95C2314yes—
111C2×#95C23112yes—
112C43C23124yes—
113C22×Q8—24no—
114C22×Q8—212no—
115C22×Q8—224no—
116C22×Q8—212no—
117C22×Q8—24no—
118C22×Q8—212no—
119C22×Q8—424no—
120C22×Q8C2×D418yes—
121C22×Q8C2×Q8116yes—
122C22×Q8#4118yes—
123C22×Q8C24112yes—
124C22×Q8C2414yes—
125C22×Q8C24124yes—
126C22×Q8C2414yes—
127C23×C4C2412yes—
128C23×C4C2416yes—
129C23×C4C2416yes—
130C23×C4C2413yes—
131C23×C4C2416yes—
132C23×C4C2412yes—
133C23×C4C2412yes—
134C23×C4C2413yes—
135C23×C4C2411yescharacteristic
136C4⋊Q8—296no—
137C4⋊Q8—2192no—
138C4⋊Q8—296no—
139C4⋊Q8—296no—
140#125—2192no—
141C4×Q8—264no—
142C4×Q8—264no—
143C4×Q8—296no—
144C4×Q8—296no—
145C4×Q8C2×D4164yes—
146C4×Q8C2×Q81128yescomplemented
147C2×C4⋊C4—24no—
148C2×C4⋊C4—224no—
149C2×C4⋊C4—224no—
150C2×C4⋊C4—224no—
151C2×C4⋊C4—224no—
152C2×C4⋊C4—212no—
153C2×C4⋊C4—224no—
154C2×C4⋊C4—248no—
155C2×C4⋊C4—424no—
156C2×C4⋊C4C2×D4124yes—
157C2×C4⋊C4C2×Q8148yes—
158C2×C4⋊C4C2414yes—
159C2×C4⋊C4C24124yes—
160C2×C4⋊C4C24124yes—
161C2×C4⋊C4C24124yes—
162C2×C4⋊C4C24112yes—
163C2×C4⋊C4C24124yes—
164C2×C4⋊C4C24124yes—
165C2×C4⋊C4C24148yes—
166C2×C4⋊C4C24148yes—
167C2×C4⋊C4C24112yes—
168C2×C4⋊C4C24124yes—
169C2×C42—224no—
170C2×C42—224no—
171C2×C42—212no—
172C2×C42—24no—
173C2×C42—412no—
174C2×C42C2×D4124yes—
175C2×C42C2×Q8148yes—
176C2×C42C24124yes—
177C2×C42C24124yes—
178C2×C42C24112yes—
179C2×C42C2414yes—
180C2×C42C24124yes—
181C2×C42C24112yes—
182C2×C42C24124yes—
183#95C24116yes—
184#95C24124yes—
185#95C24148yes—
186#95C24196yes—
187C24C2511yescharacteristic
188C2×Q8—232no—
189C2×Q8—28no—
190C2×Q8—224no—
191C2×Q8—248no—
192C2×Q8—224no—
193C2×Q8—28no—
194C2×Q8—224no—
195C2×Q8—448no—
196C2×Q8—448no—
197C2×Q8—848no—
198C2×Q8C4⋊Q8164yescomplemented
199C2×Q8C22×D4116yes—
200C2×Q8C22×Q8132yes—
201C2×Q8C2×#41116yes—
202C22×C4—24no—
203C22×C4—212no—
204C22×C4—24no—
205C22×C4—24no—
206C22×C4—24no—
207C22×C4—212no—
208C22×C4—212no—
209C22×C4—24no—
210C22×C4—212no—
211C22×C4—212no—
212C22×C4—212no—
213C22×C4—212no—
214C22×C4—28no—
215C22×C4—224no—
216C22×C4C22×D4112yes—
217C22×C4C22×D4112yes—
218C22×C4C22×D414yes—
219C22×C4C22×Q818yes—
220C22×C4C22×Q8124yes—
221C22×C4C22×Q8124yes—
222C22×C4C2×#4118yes—
223C22×C4C2×#4114yes—
224C22×C42−1+414yes—
225C22×C4C25112yes—
226C22×C4C25112yes—
227C22×C4C2514yes—
228C22×C4C2516yes—
229C22×C4C2514yes—
230C22×C4C25112yes—
231C22×C4C2514yes—
232C22×C4C2516yes—
233C22×C4C2512yes—
234C4⋊C4—28no—
235C4⋊C4—248no—
236C4⋊C4—248no—
237C4⋊C4—248no—
238C4⋊C4—248no—
239C4⋊C4—224no—
240C4⋊C4—248no—
241C4⋊C4—296no—
242C4⋊C4—448no—
243C4⋊C4C22×D4148yes—
244C4⋊C4C22×Q8196yes—
245C42—248no—
246C42—248no—
247C42—224no—
248C42—28no—
249C42—424no—
250C42C22×D4148yes—
251C42C22×Q8196yes—
252C23C23×D412yes—
253C23C23×Q814yes—
254C23C22×#4112yes—
255C23C2×2−1+412yes—
256C23#58514yes—
257C23C2611yescharacteristic
258Q8—232no—
259Q8—448no—
260Q8—848no—
261Q8C2×C4⋊Q8164yescomplemented
262C2×C4—24no—
263C2×C4—212no—
264C2×C4—24no—
265C2×C4—24no—
266C2×C4—24no—
267C2×C4—212no—
268C2×C4—212no—
269C2×C4—216no—
270C2×C4—248no—
271C2×C4—224no—
272C2×C4—216no—
273C2×C4—24no—
274C2×C4—212no—
275C2×C4—212no—
276C2×C4—212no—
277C2×C4—212no—
278C2×C4—28no—
279C2×C4—224no—
280C2×C4—424no—
281C2×C4—48no—
282C2×C4—424no—
283C2×C4C2×C4⋊Q8148yes—
284C2×C4#548116yes—
285C2×C4C23×D4112yes—
286C2×C4C23×D4112yes—
287C2×C4C23×D414yes—
288C2×C4C23×Q818yes—
289C2×C4C23×Q8124yes—
290C2×C4C23×Q8124yes—
291C2×C4C22×#4118yes—
292C2×C4C22×#4114yes—
293C2×C4C2×2−1+414yes—
294C22C22×C4⋊Q814yes—
295C22C2×#54814yes—
296C22Q8×#4118yes—
297C22#350014yes—
298C22#350218yes—
299C22C24×D411yescharacteristic
300C22C24×Q812yes—
301C22C23×#4111yescharacteristic
302C22C22×2−1+411yescharacteristic
303C22C2×#58512yes—
304C4—28no—
305C4—212no—
306C4—28no—
307C4—224no—
308C4—44no—
309C4—412no—
310C4—412no—
311C4C22×C4⋊Q8124yes—
312C4C2×#54818yes—
313C2#3647418yesdirect product factor
314C2C23×C4⋊Q811yescharacteristic
315C2C22×#54811yescharacteristic
316C2C2×Q8×#4112yes—
317C2C2×#350212yes—
318C2C2×#350011yescharacteristic
subgroupG/H
center, socle, prime radicalC24C25
commutator, Frattini, cosocleC23C26
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #36474C2direct (N × H)
69. C2×C4×Q8Q8semidirect
146. C4×Q8C2×Q8semidirect
198. C2×Q8C4⋊Q8semidirect
261. Q8C2×C4⋊Q8semidirect
313. C2#36474direct (N × H)

Maximal subgroups

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

subgroupindex
C22×Q822
C22×#5572
C2×#28622
#364742
C2×#25672
C2×#25462
C2×#25502
C2×#23512
C2×C4×C4⋊Q82
C2×C42×Q82