Finite Group Databasebeta

← #99146

Subgroups — #99146

Order 512.

848 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 848 conjugacy classes of subgroups — too many to draw. Only its 118 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#13795C211yesmaximal, characteristic
2#8299C212yesmaximal
3#7032C214yesmaximal
4C22×#117C411yescharacteristic
5C22×#117C411yescharacteristic
6C23×C4⋊C4C2211yescharacteristic
7#1779C2214yes—
8C2×#324—22no—
9C2×#324—22no—
10C4⋊#95—22no—
11C4⋊#95—22no—
12C22⋊#95C412yes—
13C22⋊#95C2212yes—
14C24×C4C2×C411yescharacteristic
15C24×C4C2×C411yescharacteristic
16C24×C4C2311yescharacteristic
17C2×#117—22no—
18C2×#117—24no—
19C2×#117—42no—
20C2×#117—42no—
21C2×#117—42no—
22C2×#117—42no—
23C2×#117C814yescomplemented
24C22×C4⋊C4—21no—
25C22×C4⋊C4—21no—
26C22×C4⋊C4—24no—
27C22×C4⋊C4—42no—
28C22×C4⋊C4—42no—
29C22×C4⋊C4D412yes—
30C22×C4⋊C4D412yes—
31C22×C22⋊C4C2×C412yes—
32C22×C22⋊C4C2×C412yes—
33C22×C42—21no—
34C22×C42—21no—
35C2×C22⋊C8—24no—
36C2×C22⋊C8—28no—
37C2×#95—24no—
38C2×#95—24no—
39#324—44no—
40#324—44no—
41C25C22×C411yescharacteristic
42C23×C4—22no—
43C23×C4—22no—
44C23×C4—22no—
45C23×C4—22no—
46C23×C4—22no—
47C23×C4—22no—
48C23×C4—22no—
49C23×C4—22no—
50C23×C4—22no—
51C23×C4—41no—
52C23×C4—41no—
53C23×C4—41no—
54C23×C4—41no—
55C23×C4—82no—
56C23×C4—82no—
57C23×C4C22⋊C411yescharacteristic
58C23×C4C22⋊C411yescharacteristic
59C23×C4C22⋊C412yes—
60C23×C4C2×C812yes—
61C23×C4M1612yes—
62C23×C4C2×D412yes—
63C23×C4#4114yes—
64C22×C8—44no—
65#117—416no—
66#117—48no—
67#117—48no—
68C2×C4⋊C4—24no—
69C2×C4⋊C4—24no—
70C2×C4⋊C4—24no—
71C2×C4⋊C4—24no—
72C2×C4⋊C4—24no—
73C2×C4⋊C4—24no—
74C2×C4⋊C4—42no—
75C2×C4⋊C4—42no—
76C2×C4⋊C4—416no—
77C2×C4⋊C4—42no—
78C2×C4⋊C4—42no—
79C2×C4⋊C4—42no—
80C2×C4⋊C4—42no—
81C2×C22⋊C4—22no—
82C2×C22⋊C4—22no—
83C2×C22⋊C4—42no—
84C2×C22⋊C4—42no—
85C2×C22⋊C4—42no—
86C2×C22⋊C4—42no—
87C2×C22⋊C4C2×C814yes—
88C2×C22⋊C4M1614yes—
89C2×C42—24no—
90C2×C42—24no—
91C2×C42—42no—
92C2×C42—42no—
93C2×C42—42no—
94C2×C42—42no—
95C22⋊C8—44no—
96C22⋊C8—44no—
97C22⋊C8—88no—
98#95—84no—
99#95—84no—
100C24—21no—
101C24—21no—
102C24—21no—
103C24—21no—
104C24—42no—
105C24—42no—
106C24C2×C22⋊C411yescharacteristic
107C24#11712yes—
108C24C22×C811yescharacteristic
109C24C2×M1611yescharacteristic
110C24C2×M1612yes—
111C22×C4—28no—
112C22×C4—28no—
113C22×C4—28no—
114C22×C4—28no—
115C22×C4—21no—
116C22×C4—21no—
117C22×C4—21no—
118C22×C4—21no—
119C22×C4—21no—
120C22×C4—21no—
121C22×C4—21no—
122C22×C4—21no—
123C22×C4—24no—
124C22×C4—24no—
125C22×C4—24no—
126C22×C4—24no—
127C22×C4—24no—
128C22×C4—24no—
129C22×C4—22no—
130C22×C4—22no—
131C22×C4—22no—
132C22×C4—24no—
133C22×C4—24no—
134C22×C4—22no—
135C22×C4—22no—
136C22×C4—44no—
137C22×C4—48no—
138C22×C4—44no—
139C22×C4—44no—
140C22×C4—44no—
141C22×C4—44no—
142C22×C4—44no—
143C22×C4—44no—
144C22×C4—42no—
145C22×C4—42no—
146C22×C4—42no—
147C22×C4—42no—
148C22×C4—44no—
149C22×C4—84no—
150C22×C4—84no—
151C22×C4—84no—
152C22×C4—84no—
153C22×C4—84no—
154C22×C4C22⋊C812yes—
155C22×C4C22⋊C812yes—
156C22×C4#9912yes—
157C22×C4#9912yes—
158C22×C4#10012yes—
159C22×C4#10112yes—
160C22×C4#12314yes—
161C2×C8—84no—
162C2×C8—88no—
163C4⋊C4—48no—
164C4⋊C4—48no—
165C4⋊C4—48no—
166C4⋊C4—48no—
167C22⋊C4—44no—
168C22⋊C4—44no—
169C22⋊C4—44no—
170C22⋊C4—44no—
171C42—48no—
172C42—48no—
173C23—22no—
174C23—22no—
175C23—22no—
176C23—22no—
177C23—24no—
178C23—24no—
179C23—21no—
180C23—21no—
181C23—21no—
182C23—21no—
183C23—22no—
184C23—22no—
185C23—42no—
186C23—42no—
187C23—44no—
188C23—42no—
189C23—42no—
190C23—42no—
191C23—42no—
192C23—44no—
193C23—44no—
194C23#38111yescharacteristic
195C23C2×C22⋊C811yescharacteristic
196C23#40711yescharacteristic
197C23C2×#9911yescharacteristic
198C23C2×#9911yescharacteristic
199C23C2×#10011yescharacteristic
200C23C2×#10111yescharacteristic
201C23#43112yes—
202C23#43212yes—
203C2×C4—216no—
204C2×C4—216no—
205C2×C4—22no—
206C2×C4—22no—
207C2×C4—22no—
208C2×C4—22no—
209C2×C4—22no—
210C2×C4—22no—
211C2×C4—22no—
212C2×C4—22no—
213C2×C4—22no—
214C2×C4—22no—
215C2×C4—22no—
216C2×C4—22no—
217C2×C4—24no—
218C2×C4—28no—
219C2×C4—48no—
220C2×C4—48no—
221C2×C4—42no—
222C2×C4—44no—
223C2×C4—44no—
224C2×C4—44no—
225C2×C4—82no—
226C2×C4—82no—
227C2×C4—88no—
228C2×C4—82no—
229C2×C4—82no—
230C2×C4—82no—
231C2×C4—82no—
232C2×C4—84no—
233C2×C4—84no—
234C2×C4—88no—
235C2×C4—82no—
236C2×C4—82no—
237C2×C4#32314yes—
238C2×C4#32414yes—
239C8—164no—
240C22—22no—
241C22—22no—
242C22—21no—
243C22—21no—
244C22—21no—
245C22—21no—
246C22—22no—
247C22—22no—
248C22—24no—
249C22—24no—
250C22—22no—
251C22—22no—
252C22—42no—
253C22—42no—
254C22—44no—
255C22—42no—
256C22—42no—
257C22—44no—
258C22—42no—
259C22—42no—
260C22—44no—
261C22C2×#32311yescharacteristic
262C22C2×#32411yescharacteristic
263C22#146211yescharacteristic
264C22#146311yescharacteristic
265C22#181711yescharacteristic
266C22#181911yescharacteristic
267C22#182111yescharacteristic
268C22#182812yes—
269C22#183012yes—
270C4—24no—
271C4—24no—
272C4—44no—
273C4—84no—
274C4—84no—
275C4—82no—
276C2—21no—
277C2—21no—
278C2—21no—
279C2—21no—
280C2—42no—
281C2—42no—
282C2#869811yescharacteristic
283C2#873512yes—
284C2#874811yescharacteristic
285C2#875311yescharacteristic
286C2#875712yes—
subgroupG/H
center, socleC23#381
commutatorC24C22×C8
Frattini, cosocleC24×C4C23
prime radicalC25C22×C4
LCSG ▷ C24 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C24×C4 ◁ G
LECSG ▷ C24×C4 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
23. C2×#117C8semidirect

Maximal subgroups

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

subgroupindex
#137952
#82992
#70322