Finite Group Databasebeta

← #94969

Subgroups — #94969

Order 512.

1378 conjugacy classes of subgroups; only the 194 normal subgroups are shown.

normal subgroups only

This group has 1378 conjugacy classes of subgroups — too many to draw. Only its 194 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#20715C212yesmaximal
2C2×C4×#95C211yesmaximal, characteristic
3#7017C214yesmaximal, complemented
4C23×C42C412yes—
5C23×C42C2211yescharacteristic
6C2×#432—22no—
7C22×#95—22no—
8#1817C2214yes—
9#1797—22no—
10#1797C414yescomplemented
11#1779—22no—
12#1749—22no—
13#1749C414yescomplemented
14C4×#95—22no—
15C4×#95C412yes—
16C4×#95C2212yes—
17C24×C4—22no—
18C24×C4C2×C411yescharacteristic
19C24×C4D412yes—
20C22×C42—22no—
21C22×C42—28no—
22C22×C42—44no—
23C22×C42—44no—
24C22×C42—44no—
25C22×C42C2×C414yes—
26C22×C42C2×C411yescharacteristic
27C22×C42C2×C412yes—
28C22×C42C2×C412yes—
29C22×C42D412yes—
30C22×C42D412yes—
31C22×C42Q812yes—
32C22×C42C2311yescharacteristic
33#432—48no—
34C2×C4⋊C8—24no—
35C2×C22⋊C8—24no—
36C2×C22⋊C8—24no—
37C2×C22⋊C8—24no—
38C2×#97—24no—
39C2×#95—216no—
40C2×#95—24no—
41C2×#95—24no—
42#336—24no—
43#336C2×C418yescomplemented
44C25C22⋊C411yescharacteristic
45C23×C4—22no—
46C23×C4—22no—
47C23×C4—24no—
48C23×C4—24no—
49C23×C4—216no—
50C23×C4—28no—
51C23×C4—24no—
52C23×C4—28no—
53C23×C4—44no—
54C23×C4—44no—
55C23×C4—48no—
56C23×C4—44no—
57C23×C4—44no—
58C23×C4C4212yes—
59C23×C4C22⋊C412yes—
60C23×C4C22⋊C412yes—
61C23×C4C22⋊C411yescharacteristic
62C23×C4C22⋊C412yes—
63C23×C4C4⋊C412yes—
64C23×C4C4⋊C412yes—
65C23×C4C4⋊C412yes—
66C23×C4C22×C411yescharacteristic
67C23×C4C2×D412yes—
68C22×C8—28no—
69C2×C42—28no—
70C2×C42—28no—
71C2×C42—24no—
72C2×C42—44no—
73C2×C42—44no—
74C2×C42—44no—
75C2×C42—44no—
76C2×C42—42no—
77C2×C42—42no—
78C2×C42—44no—
79C2×C42—44no—
80C2×C42—432no—
81C2×C42—44no—
82C2×C42—44no—
83C2×C42C4211yescharacteristic
84C2×C42C4211yescharacteristic
85C2×C42C22⋊C412yes—
86C2×C42C22⋊C412yes—
87C2×C42C4⋊C412yes—
88C2×C42C4⋊C412yes—
89C2×C42C4⋊C412yes—
90C2×C42M1618yes—
91C2×C42C22×C412yes—
92C2×C42C2×D411yescharacteristic
93C2×C42C2×Q811yescharacteristic
94C4⋊C8—48no—
95C22⋊C8—48no—
96C22⋊C8—48no—
97C22⋊C8—48no—
98#97—48no—
99#95—216no—
100#95—216no—
101C24—24no—
102C24—24no—
103C24—22no—
104C24—22no—
105C24#9512yes—
106C24#10012yes—
107C24#10112yes—
108C24C2×C22⋊C411yescharacteristic
109C22×C4—216no—
110C22×C4—216no—
111C22×C4—24no—
112C22×C4—24no—
113C22×C4—22no—
114C22×C4—22no—
115C22×C4—21no—
116C22×C4—21no—
117C22×C4—21no—
118C22×C4—21no—
119C22×C4—24no—
120C22×C4—24no—
121C22×C4—24no—
122C22×C4—24no—
123C22×C4—216no—
124C22×C4—232no—
125C22×C4—24no—
126C22×C4—24no—
127C22×C4—24no—
128C22×C4—28no—
129C22×C4—24no—
130C22×C4—24no—
131C22×C4—24no—
132C22×C4—24no—
133C22×C4—24no—
134C22×C4—24no—
135C22×C4—42no—
136C22×C4—42no—
137C22×C4—42no—
138C22×C4—42no—
139C22×C4—42no—
140C22×C4—42no—
141C22×C4—42no—
142C22×C4—42no—
143C22×C4—416no—
144C22×C4—432no—
145C22×C4—432no—
146C22×C4—416no—
147C22×C4—432no—
148C22×C4—48no—
149C22×C4—44no—
150C22×C4—44no—
151C22×C4#9512yes—
152C22×C4#9512yes—
153C22×C4#9512yes—
154C22×C4#9914yes—
155C22×C4#10418yes—
156C22×C4C2×C4211yescharacteristic
157C22×C4C2×C22⋊C412yes—
158C22×C4C2×C4⋊C412yes—
159C22×C4C2×C4⋊C411yescharacteristic
160C22×C4C2×M1614yes—
161C2×C8—48no—
162C2×C8—48no—
163C2×C8—48no—
164C42—24no—
165C42—24no—
166C42—24no—
167C42—24no—
168C42—41no—
169C42—41no—
170C42—42no—
171C42—42no—
172C42—41no—
173C42—41no—
174C42—48no—
175C42—416no—
176C42—48no—
177C23—28no—
178C23—216no—
179C23—28no—
180C23—28no—
181C23—28no—
182C23—24no—
183C23—24no—
184C23—22no—
185C23—22no—
186C23—22no—
187C23—22no—
188C23—22no—
189C23—22no—
190C23H(3,ℤ/4)14yes—
191C23#33914yes—
192C23#34314yes—
193C23C2×#9511yescharacteristic
194C23#40712yes—
195C23C2×#9912yes—
196C23C2×#10011yescharacteristic
197C23C2×#10111yescharacteristic
198C2×C4—216no—
199C2×C4—216no—
200C2×C4—24no—
201C2×C4—24no—
202C2×C4—24no—
203C2×C4—24no—
204C2×C4—22no—
205C2×C4—22no—
206C2×C4—22no—
207C2×C4—22no—
208C2×C4—22no—
209C2×C4—21no—
210C2×C4—21no—
211C2×C4—22no—
212C2×C4—21no—
213C2×C4—21no—
214C2×C4—216no—
215C2×C4—22no—
216C2×C4—22no—
217C2×C4—22no—
218C2×C4—22no—
219C2×C4—22no—
220C2×C4—22no—
221C2×C4—48no—
222C2×C4—48no—
223C2×C4—48no—
224C2×C4—48no—
225C2×C4—48no—
226C2×C4—464no—
227C2×C4—416no—
228C2×C4—416no—
229C2×C4—48no—
230C2×C4—48no—
231C2×C4—48no—
232C2×C4#33914yes—
233C2×C4#34214yes—
234C2×C4C4×M1612yes—
235C2×C4C2×#10414yes—
236C2×C4#42312yes—
237C8—88no—
238C22—216no—
239C22—216no—
240C22—24no—
241C22—24no—
242C22—24no—
243C22—22no—
244C22—24no—
245C22—22no—
246C22—22no—
247C22—24no—
248C22—22no—
249C22—24no—
250C22—24no—
251C22#128518yes—
252C22#146212yes—
253C22#146312yes—
254C22#172811yescharacteristic
255C22C2×H(3,ℤ/4)12yes—
256C22C2×#33912yes—
257C22C2×#34211yescharacteristic
258C22C2×#34311yescharacteristic
259C4—24no—
260C4—24no—
261C4—24no—
262C4—22no—
263C4—22no—
264C4—416no—
265C4—42no—
266C4—42no—
267C2—22no—
268C2—28no—
269C2—22no—
270C2#785511yescharacteristic
271C2#786711yescharacteristic
272C2#790911yescharacteristic
273C2#794212yes—
274C2C2×#128512yes—
subgroupG/H
center, socleC23C2×#95
commutatorC22×C4C2×C42
Frattini, cosocleC22×C42C23
prime radicalC25C22⋊C4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C42 ◁ G
LECSG ▷ C22×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #7017C2semidirect
10. #1797C4semidirect
13. #1749C4semidirect
43. #336C2×C4semidirect

Maximal subgroups

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

subgroupindex
#207152
C2×C4×#952
#70172