Finite Group Databasebeta

← #7631172

Subgroups — #7631172

Order 512.

6914 conjugacy classes of subgroups.

The subgroup lattice has 296 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3502C212yesmaximal
2#36645C216yesmaximal
3#36593C214yesmaximal, complemented
4Q8×C4⋊Q8C214yesmaximal, complemented
5#36474C218yesmaximal, complemented
6#36466C2116yesmaximal, complemented
7#35173C216yesmaximal
8#35167C2112yesmaximal
9#34182C214yesmaximal
10C2×C42×Q8C211yesmaximal, characteristic
11#3502—232no—
12C2×Q82C2218yes—
13C2×#557C22116yes—
14C2×#557C2218yes—
15C2×#544C22112yes—
16C2×#533C22112yes—
17C2×#519C2214yes—
18C22×C4×Q8C2211yescharacteristic
19C22×C4×Q8C2212yes—
20#2868C2216yes—
21#2868C2216yes—
22#2862C22112yes—
23#2859C22124yes—
24#2857C2214yes—
25#2857C22124yes—
26#2857C22112yes—
27#2659C22124yes—
28#2584C22148yes—
29#2567C22124yes—
30#2552C22148yes—
31#2551C22148yes—
32#2550C22196yes—
33#2546C22124yes—
34#2351C22132yes—
35Q8×C4⋊C4C22116yes—
36Q8×C4⋊C4C22116yes—
37#2316C22124yes—
38C4×C4⋊Q8C22124yes—
39C4×C4⋊Q8C22112yes—
40C4×C4⋊Q8C22112yes—
41#2285C22124yes—
42C42×Q8C2218yes—
43C42×Q8C2214yes—
44C42×Q8C2214yes—
45C2×C4×C4⋊C4C2216yes—
46C2×C4×C4⋊C4C2213yes—
47C2×C43C2213yes—
48C23×Q8C2311yescharacteristic
49Q82—264no—
50#557—264no—
51#557—2128no—
52#544—296no—
53#533—296no—
54C2×C4⋊Q8C2314yes—
55C2×C4⋊Q8C23112yes—
56C2×C4⋊Q8C23112yes—
57C2×C4⋊Q8C23148yes—
58C2×C4⋊Q8C23112yes—
59C2×C4⋊Q8C23124yes—
60C2×#125C23124yes—
61C2×#125C23148yes—
62C2×C22⋊Q8C2318yes—
63C2×C22⋊Q8C23124yes—
64#519—232no—
65C2×C4×Q8D4116yes—
66C2×C4×Q8Q8132yescomplemented
67C2×C4×Q8C23124yes—
68C2×C4×Q8C23124yes—
69C2×C4×Q8C23116yes—
70C2×C4×Q8C23116yes—
71C2×C4×Q8C2314yes—
72C2×C4×Q8C2314yes—
73C2×C4×Q8C2318yes—
74C2×C4×Q8C2318yes—
75C2×#117C23124yes—
76C22×C4⋊C4C2313yes—
77C22×C4⋊C4C2316yes—
78C22×C42C2311yescharacteristic
79C22×C42C2316yes—
80C22×C42C2313yes—
81C22×C42C2313yes—
82C22×C42C2316yes—
83#398C23196yes—
84#395C23196yes—
85#391C2318yes—
86#391C23124yes—
87#391C23148yes—
88#389C23116yes—
89#389C23148yes—
90#389C23148yes—
91#389C23148yes—
92#389C23196yes—
93#389C23196yes—
94#384C2314yes—
95#384C23112yes—
96#384C23124yes—
97#384C23124yes—
98#384C23124yes—
99#382C23148yes—
100#380C23148yes—
101#380C23116yes—
102C4×C4⋊C4C23148yes—
103C4×C4⋊C4C23148yes—
104C4×C4⋊C4C23112yes—
105C4×C4⋊C4C23112yes—
106C4×C4⋊C4C23124yes—
107C4×C4⋊C4C23124yes—
108C2×#95C2316yes—
109C43C23112yes—
110C43C2316yes—
111C43C2316yes—
112C22×Q8C2×D418yes—
113C22×Q8C2×Q8116yes—
114C22×Q8C2418yes—
115C22×Q8C24124yes—
116C22×Q8C2414yes—
117C23×C4C2411yescharacteristic
118C23×C4C2412yes—
119C23×C4C2413yes—
120C23×C4C2416yes—
121C23×C4C2413yes—
122C4⋊Q8—2192no—
123C4⋊Q8—248no—
124C4⋊Q8—248no—
125C4⋊Q8—216no—
126C4⋊Q8—448no—
127C4⋊Q8—424no—
128#125—2192no—
129#125—296no—
130C22⋊Q8—232no—
131C22⋊Q8—296no—
132C4×Q8—264no—
133C4×Q8—264no—
134C4×Q8—296no—
135C4×Q8—296no—
136C4×Q8C2×D4132yes—
137C4×Q8C2×D4132yes—
138C4×Q8C2×Q8164yes—
139C4×Q8C2×Q8164yes—
140#117—296no—
141C2×C4⋊C4C2×D4124yes—
142C2×C4⋊C4C2×Q8148yes—
143C2×C4⋊C4C24124yes—
144C2×C4⋊C4C24124yes—
145C2×C4⋊C4C24116yes—
146C2×C4⋊C4C2418yes—
147C2×C4⋊C4C24124yes—
148C2×C4⋊C4C24148yes—
149C2×C4⋊C4C24116yes—
150C2×C4⋊C4C24116yes—
151C2×C4⋊C4C24148yes—
152C2×C4⋊C4C24148yes—
153C2×C4⋊C4C24112yes—
154C2×C4⋊C4C24124yes—
155C2×C22⋊C4C24116yes—
156C2×C42—24no—
157C2×C42—46no—
158C2×C42—412no—
159C2×C42C2×D4124yes—
160C2×C42C2×Q8148yes—
161C2×C42C24148yes—
162C2×C42C24112yes—
163C2×C42C2416yes—
164C2×C42C2412yes—
165C2×C42C2411yescharacteristic
166C2×C42C2416yes—
167C2×C42C24112yes—
168C2×C42C2416yes—
169C2×C42C2416yes—
170C2×C42C24112yes—
171C2×C42C24112yes—
172C2×C42C2411yescharacteristic
173#95C24124yes—
174#95C24132yes—
175#95C24196yes—
176C24C2511yescharacteristic
177C2×Q8—216no—
178C2×Q8—248no—
179C2×Q8—216no—
180C2×Q8—216no—
181C2×Q8—448no—
182C2×Q8C4⋊Q8164yescomplemented
183C2×Q8C22×D4116yes—
184C2×Q8C22×Q8132yes—
185C22×C4—22no—
186C22×C4—24no—
187C22×C4—22no—
188C22×C4—24no—
189C22×C4—46no—
190C22×C4—46no—
191C22×C4—46no—
192C22×C4C22×D414yes—
193C22×C4C22×D4112yes—
194C22×C4C22×D4112yes—
195C22×C4C22×Q818yes—
196C22×C4C22×Q8124yes—
197C22×C4C22×Q8124yes—
198C22×C4C25116yes—
199C22×C4C25116yes—
200C22×C4C2516yes—
201C22×C4C2511yescharacteristic
202C22×C4C2512yes—
203C22×C4C25112yes—
204C22×C4C2516yes—
205C22×C4C2511yescharacteristic
206C22×C4C2512yes—
207C4⋊C4—248no—
208C4⋊C4—248no—
209C4⋊C4—232no—
210C4⋊C4—216no—
211C4⋊C4—248no—
212C4⋊C4—296no—
213C4⋊C4—232no—
214C4⋊C4—232no—
215C4⋊C4—448no—
216C4⋊C4—448no—
217C4⋊C4C22×D4148yes—
218C4⋊C4C22×Q8196yes—
219C22⋊C4—232no—
220C42—296no—
221C42—22no—
222C42—24no—
223C42—212no—
224C42—224no—
225C42—46no—
226C42—412no—
227C42C22×D4124yes—
228C42C22×D4124yes—
229C42C22×Q8148yes—
230C42C22×Q8148yes—
231C422−1+414yes—
232C23—21no—
233C23—21no—
234C23—22no—
235C23C23×D412yes—
236C23C23×Q814yes—
237C23C2611yescharacteristic
238Q8—432no—
239Q8—848no—
240Q8C2×C4⋊Q8164yes—
241C2×C4—21no—
242C2×C4—22no—
243C2×C4—212no—
244C2×C4—212no—
245C2×C4—212no—
246C2×C4—21no—
247C2×C4—22no—
248C2×C4—28no—
249C2×C4—216no—
250C2×C4—216no—
251C2×C4—216no—
252C2×C4—216no—
253C2×C4—232no—
254C2×C4—216no—
255C2×C4—412no—
256C2×C4—424no—
257C2×C4C2×C4⋊Q8148yes—
258C2×C4C23×D412yes—
259C2×C4C23×D4112yes—
260C2×C4C23×D4112yes—
261C2×C4C23×D412yes—
262C2×C4C23×Q814yes—
263C2×C4C23×Q8124yes—
264C2×C4C23×Q8124yes—
265C2×C4C23×Q814yes—
266C2×C4C2×2−1+412yes—
267C2×C4C2×2−1+414yes—
268C2×C4#58512yes—
269C2×C4#58514yes—
270C22—22no—
271C22—24no—
272C22—22no—
273C22—24no—
274C22C22×C4⋊Q814yes—
275C22C24×D411yescharacteristic
276C22C24×Q812yes—
277C22C22×2−1+411yescharacteristic
278C22C2×#58511yescharacteristic
279C22C2×#58512yes—
280C4—22no—
281C4—24no—
282C4—212no—
283C4—224no—
284C4—416no—
285C4—416no—
286C4C22×C4⋊Q8124yes—
287C4#349714yes—
288C4#350218yescomplemented
289C2—24no—
290C2C23×C4⋊Q811yescharacteristic
291C2C2×#349711yescharacteristic
292C2C2×#350212yes—
293C2#6281211yescharacteristic
294C2#6281312yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC23C26
prime radicalC24C25
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C23 ◁ G
LECSG ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #36593C2semidirect
4. Q8×C4⋊Q8C2semidirect
5. #36474C2semidirect
6. #36466C2semidirect
66. C2×C4×Q8Q8semidirect
182. C2×Q8C4⋊Q8semidirect
288. C4#3502semidirect

Maximal subgroups

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

subgroupindex
C2×#35022
#366452
#365932
Q8×C4⋊Q82
#364742
#364662
#351732
#351672
#341822
C2×C42×Q82