Finite Group Databasebeta

← #238063

Subgroups — #238063

Order 512.

900 conjugacy classes of subgroups; only the 178 normal subgroups are shown.

normal subgroups only

This group has 900 conjugacy classes of subgroups — too many to draw. Only its 178 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
1#37481C211yesmaximal, characteristic
2#20705C212yesmaximal
3#20688C211yesmaximal, characteristic
4#10389C218yesmaximal, complemented
5#8112C211yesmaximal, characteristic
6#7946C212yesmaximal
7C22×#117C2211yescharacteristic
8C2×#407C2212yes—
9#2842—22no—
10#2667—24no—
11#2629—22no—
12#2608C2214yes—
13#2441—24no—
14#2280C2211yescharacteristic
15#2280C2211yescharacteristic
16#2280C2212yes—
17#1826—24no—
18#1826C2218yes—
19#1820C2214yes—
20#1819—22no—
21#1740—22no—
22#1733—24no—
23#1306C2214yes—
24#1284C2218yes—
25C24×C4C2311yescharacteristic
26C22×M16—22no—
27C2×#125—22no—
28C2×#117—21no—
29C2×#117—21no—
30C2×#117—21no—
31C2×#117—22no—
32C2×#117—22no—
33C2×#117—22no—
34C2×#117—22no—
35C2×#117D412yes—
36C2×#117D414yes—
37C2×#117Q812yes—
38C2×#117C2312yes—
39C22×C4⋊C4Q812yes—
40C22×C22⋊C4—21no—
41C22×C42—21no—
42#407—22no—
43#407—22no—
44#407—22no—
45#407C2314yes—
46C2×C22⋊C8—22no—
47C2×C22⋊C8—22no—
48#399—28no—
49#398—28no—
50#396—28no—
51#396—28no—
52#389—24no—
53#389—28no—
54#387—28no—
55#384—22no—
56#381—22no—
57#381C2312yes—
58#381C2312yes—
59#380—22no—
60#380—22no—
61#380—22no—
62#380—22no—
63#379—22no—
64#379—21no—
65#379D412yes—
66C4×C22⋊C4—22no—
67C4×C22⋊C4—22no—
68C4×C22⋊C4D414yes—
69C4×C22⋊C4C2314yes—
70#342—28no—
71#339—216no—
72C25C2×D411yescharacteristic
73C23×C4—22no—
74C23×C4—44no—
75C23×C4C2×D411yescharacteristic
76C23×C4C2×D412yes—
77C23×C4C2×D412yes—
78C23×C4C2×Q811yescharacteristic
79C23×C4C2×Q811yescharacteristic
80C23×C4#4112yes—
81C23×C4#4112yes—
82C23×C4#4111yescharacteristic
83C23×C4#4111yescharacteristic
84C23×C4C2411yescharacteristic
85C2×M16—24no—
86C2×M16—42no—
87C2×M16—48no—
88C22×C8—42no—
89#125—84no—
90#117—24no—
91#117—24no—
92#117—24no—
93#117—24no—
94#117—41no—
95#117—41no—
96#117—42no—
97#117—42no—
98#117—41no—
99#117—41no—
100#117—48no—
101#117—44no—
102#117—44no—
103C2×C4⋊C4—22no—
104C2×C4⋊C4—22no—
105C2×C4⋊C4—22no—
106C2×C4⋊C4—22no—
107C2×C4⋊C4—24no—
108C2×C4⋊C4—22no—
109C2×C4⋊C4—28no—
110C2×C4⋊C4—28no—
111C2×C4⋊C4—28no—
112C2×C4⋊C4—22no—
113C2×C4⋊C4—41no—
114C2×C4⋊C4C2×Q814yes—
115C2×C22⋊C4—22no—
116C2×C22⋊C4—22no—
117C2×C22⋊C4—22no—
118C2×C22⋊C4—24no—
119C2×C22⋊C4—22no—
120C2×C22⋊C4—22no—
121C2×C22⋊C4—24no—
122C2×C22⋊C4—24no—
123C2×C22⋊C4—42no—
124C2×C22⋊C4—42no—
125C2×C22⋊C4—42no—
126C2×C22⋊C4C2×D414yes—
127C2×C42—22no—
128C2×C42—22no—
129C2×C42—22no—
130C2×C42—42no—
131C2×C42—42no—
132C2×C42—41no—
133C2×C42C2×D418yes—
134C22⋊C8—24no—
135C22⋊C8—24no—
136C22⋊C8—42no—
137C22⋊C8—42no—
138#95—28no—
139#95—24no—
140#95—24no—
141#95—24no—
142#95—24no—
143C24—42no—
144C24—82no—
145C24#12012yes—
146C24C22⋊Q811yescharacteristic
147C24C22⋊Q811yescharacteristic
148C24#12412yes—
149C24C22×D411yescharacteristic
150C22×C4—24no—
151C22×C4—28no—
152C22×C4—24no—
153C22×C4—28no—
154C22×C4—24no—
155C22×C4—42no—
156C22×C4—42no—
157C22×C4—44no—
158C22×C4—44no—
159C22×C4—42no—
160C22×C4—42no—
161C22×C4—44no—
162C22×C4—42no—
163C22×C4—42no—
164C22×C4—42no—
165C22×C4—42no—
166C22×C4—84no—
167C22×C4#12011yescharacteristic
168C22×C4#12011yescharacteristic
169C22×C4C22⋊Q812yes—
170C22×C4C22⋊Q812yes—
171C22×C4C22⋊Q811yescharacteristic
172C22×C4C22⋊Q811yescharacteristic
173C22×C4C22⋊Q812yes—
174C22×C4C22⋊Q812yes—
175C22×C4#12412yes—
176C22×C4#12412yes—
177C22×C4#12411yescharacteristic
178C22×C4#12411yescharacteristic
179C22×C4#12412yes—
180C22×C4#12412yes—
181C22×C4C22×D412yes—
182C22×C4C22×Q811yescharacteristic
183C22×C4C2×#4112yes—
184C22×C4C2×#4111yescharacteristic
185M16—48no—
186M16—84no—
187C2×C8—44no—
188C2×C8—84no—
189C4⋊C4—24no—
190C4⋊C4—24no—
191C4⋊C4—42no—
192C4⋊C4—42no—
193C4⋊C4—48no—
194C4⋊C4—82no—
195C4⋊C4—82no—
196C4⋊C4—88no—
197C4⋊C4—88no—
198C4⋊C4—88no—
199C22⋊C4—42no—
200C22⋊C4—42no—
201C22⋊C4—44no—
202C22⋊C4—44no—
203C22⋊C4—44no—
204C22⋊C4—84no—
205C22⋊C4—82no—
206C22⋊C4—82no—
207C22⋊C4—84no—
208C22⋊C4—84no—
209C22⋊C4—84no—
210C42—48no—
211C42—44no—
212C42—42no—
213C42—42no—
214C42—44no—
215C42—82no—
216C23—22no—
217C23—24no—
218C23—41no—
219C23—42no—
220C23—42no—
221C23—42no—
222C23—42no—
223C23—42no—
224C23—41no—
225C23—42no—
226C23—42no—
227C23—42no—
228C23—88no—
229C23#39312yes—
230C23#39312yes—
231C23C2×#12011yescharacteristic
232C23C2×C22⋊Q811yescharacteristic
233C23C2×#12411yescharacteristic
234C2×C4—24no—
235C2×C4—22no—
236C2×C4—22no—
237C2×C4—22no—
238C2×C4—24no—
239C2×C4—24no—
240C2×C4—24no—
241C2×C4—24no—
242C2×C4—22no—
243C2×C4—22no—
244C2×C4—42no—
245C2×C4—42no—
246C2×C4—42no—
247C2×C4—42no—
248C2×C4—41no—
249C2×C4—41no—
250C2×C4—42no—
251C2×C4—44no—
252C2×C4—44no—
253C2×C4—48no—
254C2×C4—48no—
255C2×C4—44no—
256C2×C4—44no—
257C2×C4—48no—
258C2×C4—84no—
259C2×C4—84no—
260C2×C4—84no—
261C2×C4—82no—
262C2×C4—84no—
263C2×C4#39311yescharacteristic
264C2×C4#39311yescharacteristic
265C2×C4#39311yescharacteristic
266C2×C4#39311yescharacteristic
267C2×C4#45814yes—
268C2×C4C2×C22⋊Q812yes—
269C2×C4C2×#12412yes—
270C8—84no—
271C22—22no—
272C22—21no—
273C22—21no—
274C22—22no—
275C22—22no—
276C22—22no—
277C22—22no—
278C22—22no—
279C22—22no—
280C22—22no—
281C22—22no—
282C22—44no—
283C22—82no—
284C22—84no—
285C22—84no—
286C22—82no—
287C22C2×#39311yescharacteristic
288C22C2×#45812yes—
289C4—22no—
290C4—22no—
291C4—21no—
292C4—21no—
293C4—84no—
294C4—88no—
295C4—84no—
296C4—162no—
297C4#203012yes—
298C4#203012yes—
299C2—22no—
300C2—41no—
301C2—41no—
302C2—42no—
303C2—42no—
304C2C2×#203011yescharacteristic
305C2#2347212yes—
subgroupG/H
center, socleC22C2×#393
commutator, Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C23×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C24×C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #10389C2semidirect

Maximal subgroups

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

subgroupindex
#374812
#207052
#206882
#103892
#81122
#79462