Finite Group Databasebeta

← #391826

Subgroups — #391826

Order 512.

1274 conjugacy classes of subgroups; only the 242 normal subgroups are shown.

normal subgroups only

This group has 1274 conjugacy classes of subgroups — too many to draw. Only its 242 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×#2627C211yesmaximal, characteristic
2#14342C211yesmaximal, characteristic
3#14339C211yesmaximal, characteristic
4#11381C212yesmaximal
5#11353C212yesmaximal
6#11203C214yesmaximal
7#11197C214yesmaximal
8C22×C22⋊Q8C2211yescharacteristic
9C23×C4⋊C4C2211yescharacteristic
10#2627—24no—
11#2627—24no—
12C2×#400C412yescomplemented
13C2×#398C414yescomplemented
14C2×#395C412yescomplemented
15C2×#385C2212yes—
16C2×#381C2212yes—
17C2×#381C2212yes—
18C2×#381C2212yes—
19C2×#380C2212yes—
20C2×#380C2211yescharacteristic
21C2×#342C2218yes—
22C2×#342C2218yes—
23#1445—24no—
24#1441—24no—
25C22⋊#95C2212yes—
26C22⋊#95C2212yes—
27C22⋊#95C2212yes—
28C23×Q8C2×C411yescharacteristic
29C24×C4C2311yescharacteristic
30C24×C4C2311yescharacteristic
31C24×C4C2311yescharacteristic
32C2×C22⋊Q8—22no—
33C2×C22⋊Q8—22no—
34C2×C22⋊Q8—44no—
35C22×C4⋊C4—22no—
36C22×C4⋊C4—44no—
37C22×C4⋊C4C2×C412yes—
38C22×C4⋊C4C2×C411yescharacteristic
39C22×C4⋊C4C2×C414yes—
40C22×C4⋊C4C2×C412yes—
41C22×C4⋊C4C2×C412yes—
42C22×C4⋊C4C2×C412yes—
43C22×C4⋊C4D414yes—
44C22×C4⋊C4D414yes—
45C22×C22⋊C4C2314yes—
46C22×C22⋊C4C2314yes—
47C22×C22⋊C4C2312yes—
48C22×C22⋊C4C2312yes—
49#400—28no—
50#398—216no—
51#395—28no—
52#385—24no—
53#385—24no—
54#381—24no—
55#381—24no—
56#381—44no—
57#381—44no—
58#380—22no—
59#380—22no—
60#380—44no—
61C2×#95—22no—
62C2×#95—24no—
63C2×#95—24no—
64C2×#95—24no—
65C2×#95—24no—
66C2×#95—24no—
67C2×#95C2×C414yes—
68C2×#95C2×C412yes—
69C2×#95C2×C414yes—
70C2×#95C2×C414yes—
71#342—48no—
72#342—48no—
73C25C2411yescharacteristic
74C22×Q8—22no—
75C22×Q8—42no—
76C22×Q8—82no—
77C23×C4—24no—
78C23×C4—22no—
79C23×C4—24no—
80C23×C4—22no—
81C23×C4—24no—
82C23×C4—24no—
83C23×C4—24no—
84C23×C4—24no—
85C23×C4—82no—
86C23×C4—82no—
87C23×C4C22×C412yes—
88C23×C4C22×C412yes—
89C23×C4C22×C412yes—
90C23×C4C22×C414yes—
91C23×C4C22×C411yescharacteristic
92C23×C4C22×C412yes—
93C23×C4C22×C411yescharacteristic
94C23×C4C2×D412yes—
95C23×C4C2×D412yes—
96C23×C4C2×D412yes—
97C23×C4C2×D412yes—
98C23×C4C2×D412yes—
99C23×C4C2×D412yes—
100C22⋊Q8—48no—
101C22⋊Q8—48no—
102C2×C4⋊C4—24no—
103C2×C4⋊C4—22no—
104C2×C4⋊C4—24no—
105C2×C4⋊C4—28no—
106C2×C4⋊C4—44no—
107C2×C4⋊C4—44no—
108C2×C4⋊C4—44no—
109C2×C4⋊C4—48no—
110C2×C4⋊C4—44no—
111C2×C4⋊C4—44no—
112C2×C4⋊C4—42no—
113C2×C4⋊C4—44no—
114C2×C4⋊C4—84no—
115C2×C4⋊C4—82no—
116C2×C4⋊C4—84no—
117C2×C4⋊C4C22⋊C418yes—
118C2×C4⋊C4C22⋊C418yes—
119C2×C22⋊C4—22no—
120C2×C22⋊C4—22no—
121C2×C22⋊C4—22no—
122C2×C22⋊C4—22no—
123C2×C22⋊C4—28no—
124C2×C22⋊C4—24no—
125C2×C22⋊C4—24no—
126C2×C22⋊C4—24no—
127C2×C22⋊C4—28no—
128C2×C22⋊C4—28no—
129C2×C22⋊C4—28no—
130C2×C22⋊C4—28no—
131C2×C22⋊C4—28no—
132C2×C22⋊C4—24no—
133#95—24no—
134#95—28no—
135#95—28no—
136#95—28no—
137#95—44no—
138#95—48no—
139#95—48no—
140#95—84no—
141#95—84no—
142#95—84no—
143C24—42no—
144C24—42no—
145C24C23×C411yescharacteristic
146C24C22×D411yescharacteristic
147C24C22×D411yescharacteristic
148C24H(5,𝔽2)14yes—
149C24H(5,𝔽2)12yes—
150C242−1+414yes—
151C242−1+412yes—
152C2×Q8—44no—
153C2×Q8—84no—
154C2×Q8—88no—
155C22×C4—22no—
156C22×C4—22no—
157C22×C4—24no—
158C22×C4—24no—
159C22×C4—24no—
160C22×C4—22no—
161C22×C4—24no—
162C22×C4—22no—
163C22×C4—24no—
164C22×C4—22no—
165C22×C4—24no—
166C22×C4—24no—
167C22×C4—22no—
168C22×C4—24no—
169C22×C4—24no—
170C22×C4—24no—
171C22×C4—24no—
172C22×C4—24no—
173C22×C4—22no—
174C22×C4—24no—
175C22×C4—22no—
176C22×C4—24no—
177C22×C4—24no—
178C22×C4—28no—
179C22×C4—42no—
180C22×C4—44no—
181C22×C4—42no—
182C22×C4—44no—
183C22×C4—44no—
184C22×C4—44no—
185C22×C4—44no—
186C22×C4—48no—
187C22×C4—48no—
188C22×C4—44no—
189C22×C4—44no—
190C22×C4—44no—
191C22×C4—44no—
192C22×C4—48no—
193C22×C4—48no—
194C22×C4—48no—
195C22×C4—48no—
196C22×C4—44no—
197C22×C4—44no—
198C22×C4—44no—
199C22×C4—44no—
200C22×C4—44no—
201C22×C4—44no—
202C22×C4—88no—
203C22×C4—88no—
204C22×C4C2×C22⋊C414yes—
205C22×C4C2×C22⋊C414yes—
206C22×C4C2×C22⋊C412yes—
207C22×C4C2×C22⋊C412yes—
208C4⋊C4—48no—
209C4⋊C4—48no—
210C4⋊C4—48no—
211C4⋊C4—44no—
212C4⋊C4—84no—
213C4⋊C4—88no—
214C4⋊C4—88no—
215C22⋊C4—44no—
216C22⋊C4—44no—
217C22⋊C4—42no—
218C22⋊C4—42no—
219C22⋊C4—44no—
220C22⋊C4—416no—
221C22⋊C4—48no—
222C22⋊C4—48no—
223C23—22no—
224C23—22no—
225C23—22no—
226C23—22no—
227C23—22no—
228C23—22no—
229C23—44no—
230C23—44no—
231C23—44no—
232C23—48no—
233C23—44no—
234C23C22×C22⋊C411yescharacteristic
235C23#51812yes—
236C23#51912yes—
237C23#52012yes—
238C23#53412yes—
239C23#53412yes—
240C23#53511yescharacteristic
241C23#53612yes—
242C23#53611yescharacteristic
243C23#53612yes—
244C23#53712yes—
245C23#56218yes—
246C23#56318yes—
247Q8—88no—
248C2×C4—22no—
249C2×C4—22no—
250C2×C4—24no—
251C2×C4—24no—
252C2×C4—24no—
253C2×C4—24no—
254C2×C4—24no—
255C2×C4—28no—
256C2×C4—24no—
257C2×C4—44no—
258C2×C4—44no—
259C2×C4—44no—
260C2×C4—44no—
261C2×C4—44no—
262C2×C4—44no—
263C2×C4—48no—
264C2×C4—48no—
265C2×C4—42no—
266C2×C4—42no—
267C2×C4—44no—
268C2×C4—44no—
269C2×C4—44no—
270C2×C4—44no—
271C2×C4—44no—
272C2×C4—84no—
273C2×C4—84no—
274C2×C4—84no—
275C2×C4—84no—
276C2×C4—88no—
277C2×C4—84no—
278C2×C4—84no—
279C2×C4—84no—
280C2×C4—88no—
281C2×C4—88no—
282C2×C4—88no—
283C2×C4—88no—
284C2×C4—84no—
285C2×C4—84no—
286C2×C4—84no—
287C2×C4—84no—
288C2×C4—84no—
289C22—24no—
290C22—24no—
291C22—24no—
292C22—22no—
293C22—24no—
294C22—24no—
295C22—22no—
296C22—22no—
297C22—22no—
298C22—22no—
299C22—22no—
300C22—22no—
301C22—22no—
302C22—48no—
303C22—48no—
304C22#230811yescharacteristic
305C22#231011yescharacteristic
306C22#231111yescharacteristic
307C22#237814yes—
308C22#281414yes—
309C22#281612yes—
310C22#281712yes—
311C22#288312yes—
312C22#288412yes—
313C4—44no—
314C4—42no—
315C4—44no—
316C4—42no—
317C4—84no—
318C4—82no—
319C4—82no—
320C4—164no—
321C4—164no—
322C2—22no—
323C2—24no—
324C2—22no—
325C2—24no—
326C2#1466011yescharacteristic
327C2#2174911yescharacteristic
328C2#2175111yescharacteristic
329C2#2182812yes—
330C2#2187712yes—
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC24C23×C4
Frattini, prime radical, cosocleC25C24
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C4⋊C4 ◁ G
LECSG ▷ C25 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
12. C2×#400C4semidirect
13. C2×#398C4semidirect
14. C2×#395C4semidirect

Maximal subgroups

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

subgroupindex
C2×#26272
#143422
#143392
#113812
#113532
#112032
#111972