Finite Group Databasebeta

← #97079

Subgroups — #97079

Order 512.

1728 conjugacy classes of subgroups; only the 166 normal subgroups are shown.

normal subgroups only

This group has 1728 conjugacy classes of subgroups — too many to draw. Only its 166 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
1C22×#391C211yesmaximal, characteristic
2#8698C214yesmaximal
3#8299C212yesmaximal
4C24×Q8C411yescharacteristic
5C23×C4⋊C4C2211yescharacteristic
6C23×C42C411yescharacteristic
7C2×#391—28no—
8C2×#391—24no—
9C22×#95C412yes—
10C22×#95C2212yes—
11#1779C2214yes—
12C2×#324—24no—
13#1280—28no—
14C23×Q8—24no—
15C23×Q8—22no—
16C23×Q8—24no—
17C23×Q8—22no—
18C23×Q8—44no—
19C23×Q8—44no—
20C23×Q8C814yescomplemented
21C24×C4C2×C411yescharacteristic
22C24×C4C2×C412yes—
23C24×C4C2×C412yes—
24C24×C4C2×C411yescharacteristic
25C24×C4C2311yescharacteristic
26C22×C4⋊C4—24no—
27C22×C4⋊C4—44no—
28C22×C4⋊C4D414yes—
29C22×C42—24no—
30C22×C42—22no—
31C22×C42—44no—
32C2×C22⋊C8—28no—
33C2×C22⋊C8—24no—
34#391—232no—
35C2×#95—24no—
36C2×#95—24no—
37C2×#95—24no—
38C2×#95—24no—
39C2×#95—24no—
40C2×#95D418yes—
41#324—48no—
42C25C22×C411yescharacteristic
43C22×Q8—28no—
44C22×Q8—216no—
45C22×Q8—28no—
46C22×Q8—216no—
47C22×Q8—28no—
48C22×Q8—416no—
49C22×Q8—416no—
50C22×Q8—416no—
51C22×Q8—432no—
52C22×Q8—44no—
53C22×Q8—44no—
54C22×Q8—416no—
55C22×Q8—44no—
56C22×Q8—44no—
57C23×C4—22no—
58C23×C4—22no—
59C23×C4—24no—
60C23×C4—24no—
61C23×C4—22no—
62C23×C4—22no—
63C23×C4—22no—
64C23×C4—22no—
65C23×C4—22no—
66C23×C4—22no—
67C23×C4—28no—
68C23×C4—22no—
69C23×C4—42no—
70C23×C4—42no—
71C23×C4—44no—
72C23×C4—44no—
73C23×C4—44no—
74C23×C4—44no—
75C23×C4—44no—
76C23×C4—44no—
77C23×C4C22⋊C412yes—
78C23×C4C22⋊C412yes—
79C23×C4C22⋊C414yes—
80C23×C4C22⋊C414yes—
81C23×C4C2×C812yes—
82C23×C4C2×C814yes—
83C23×C4M1612yes—
84C23×C4M1614yes—
85C23×C4C2×D414yes—
86C23×C4C2×D412yes—
87C22×C8—44no—
88C2×C4⋊C4—28no—
89C2×C4⋊C4—28no—
90C2×C4⋊C4—44no—
91C2×C4⋊C4—416no—
92C2×C42—28no—
93C2×C42—416no—
94C2×C42—44no—
95C2×C42—44no—
96C22⋊C8—48no—
97C22⋊C8—416no—
98#95—216no—
99#95—216no—
100C24—22no—
101C24—24no—
102C24—22no—
103C24—24no—
104C24C2×C22⋊C411yescharacteristic
105C24C2×C22⋊C412yes—
106C24C22×C811yescharacteristic
107C24C2×M1611yescharacteristic
108C24C2×M1612yes—
109C2×Q8—232no—
110C2×Q8—464no—
111C2×Q8—464no—
112C2×Q8—416no—
113C2×Q8—416no—
114C2×Q8—416no—
115C2×Q8—416no—
116C2×Q8—416no—
117C2×Q8—416no—
118C22×C4—28no—
119C22×C4—28no—
120C22×C4—216no—
121C22×C4—22no—
122C22×C4—24no—
123C22×C4—22no—
124C22×C4—24no—
125C22×C4—22no—
126C22×C4—22no—
127C22×C4—28no—
128C22×C4—28no—
129C22×C4—28no—
130C22×C4—28no—
131C22×C4—24no—
132C22×C4—24no—
133C22×C4—24no—
134C22×C4—24no—
135C22×C4—24no—
136C22×C4—24no—
137C22×C4—24no—
138C22×C4—24no—
139C22×C4—24no—
140C22×C4—28no—
141C22×C4—22no—
142C22×C4—24no—
143C22×C4—44no—
144C22×C4—44no—
145C22×C4—44no—
146C22×C4—44no—
147C22×C4—432no—
148C22×C4—44no—
149C22×C4—48no—
150C22×C4—48no—
151C22×C4—44no—
152C22×C4—44no—
153C22×C4—44no—
154C22×C4—44no—
155C22×C4—44no—
156C22×C4—44no—
157C22×C4—44no—
158C22×C4—44no—
159C22×C4—44no—
160C22×C4—44no—
161C22×C4—48no—
162C22×C4—48no—
163C22×C4—44no—
164C22×C4—44no—
165C22×C4—44no—
166C22×C4—44no—
167C22×C4—88no—
168C22×C4—88no—
169C22×C4C22⋊C814yes—
170C22×C4C22⋊C818yes—
171C22×C4#9914yes—
172C22×C4#10014yes—
173C22×C4#10118yes—
174C22×C4#12014yes—
175C2×C8—84no—
176C2×C8—88no—
177C4⋊C4—416no—
178C42—416no—
179C23—216no—
180C23—28no—
181C23—28no—
182C23—22no—
183C23—22no—
184C23—24no—
185C23—24no—
186C23—24no—
187C23—24no—
188C23—24no—
189C23—22no—
190C23—22no—
191C23—22no—
192C23—22no—
193C23—22no—
194C23—22no—
195C23#37911yescharacteristic
196C23C2×C22⋊C811yescharacteristic
197C23C2×C22⋊C812yes—
198C23#40711yescharacteristic
199C23#40712yes—
200C23C2×#9912yes—
201C23#41014yes—
202C23C2×#10012yes—
203C23C2×#10114yes—
204Q8—464no—
205C2×C4—216no—
206C2×C4—216no—
207C2×C4—24no—
208C2×C4—24no—
209C2×C4—24no—
210C2×C4—24no—
211C2×C4—24no—
212C2×C4—24no—
213C2×C4—28no—
214C2×C4—24no—
215C2×C4—28no—
216C2×C4—28no—
217C2×C4—416no—
218C2×C4—416no—
219C2×C4—48no—
220C2×C4—48no—
221C2×C4—48no—
222C2×C4—48no—
223C2×C4—48no—
224C2×C4—48no—
225C2×C4—44no—
226C2×C4—44no—
227C2×C4—44no—
228C2×C4—44no—
229C2×C4—48no—
230C2×C4—48no—
231C2×C4—42no—
232C2×C4—44no—
233C2×C4—84no—
234C2×C4—88no—
235C2×C4—88no—
236C2×C4—84no—
237C2×C4#32318yes—
238C8—164no—
239C22—28no—
240C22—28no—
241C22—216no—
242C22—22no—
243C22—24no—
244C22—22no—
245C22—24no—
246C22—22no—
247C22—22no—
248C22—24no—
249C22—24no—
250C22—24no—
251C22—22no—
252C22—22no—
253C22—22no—
254C22—22no—
255C22C2×#32312yes—
256C22#146114yes—
257C22#146212yes—
258C22#146714yes—
259C22#177911yescharacteristic
260C22#178111yescharacteristic
261C22#178212yes—
262C22#178311yescharacteristic
263C22#178412yes—
264C4—28no—
265C4—44no—
266C4—48no—
267C4—48no—
268C4—82no—
269C2—22no—
270C2—24no—
271C2—22no—
272C2—24no—
273C2#829811yescharacteristic
274C2#830312yes—
275C2#854411yescharacteristic
276C2#855111yescharacteristic
277C2#855512yes—
subgroupG/H
center, socleC23#379
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
20. C23×Q8C8semidirect

Maximal subgroups

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

subgroupindex
C22×#3912
#86982
#82992