Finite Group Databasebeta

← #417076

Subgroups — #417076

Order 512.

802 conjugacy classes of subgroups; only the 234 normal subgroups are shown.

normal subgroups only

This group has 802 conjugacy classes of subgroups — too many to draw. Only its 234 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#21322C212yesmaximal
2#21300C211yesmaximal, characteristic
3C2×#1834C211yesmaximal, characteristic
4#11303C218yesmaximal
5#10857C212yesmaximal
6C4×#341C211yesmaximal, characteristic
7C2×#533C2211yescharacteristic
8C2×#423C2211yescharacteristic
9C2×C4×M16C2212yes—
10C22×C4×C8C2211yescharacteristic
11C2×#384C2212yes—
12#1992—24no—
13#1990—22no—
14#1990—22no—
15#1984—22no—
16#1984—22no—
17#1834—22no—
18#1834—22no—
19C2×#343C2218yes—
20C2×#341C2214yes—
21#1736C2218yes—
22C2×#340C2214yes—
23#1729C2214yes—
24C22×M16C2312yes—
25C22×M16C2314yes—
26C23×C8C2312yes—
27#533—24no—
28#533—24no—
29C2×C4⋊Q8D411yescharacteristic
30C2×C4⋊Q8D411yescharacteristic
31C2×#125D412yes—
32C2×C22⋊Q8—21no—
33C2×C22⋊Q8—21no—
34C2×C4×Q8—21no—
35C2×C4×Q8—21no—
36C2×#117C2312yes—
37C22×C4⋊C4—22no—
38C22×C4⋊C4—22no—
39C22×C4⋊C4C2314yes—
40C22×C42C2311yescharacteristic
41#423—24no—
42C2×C4⋊C8—21no—
43C2×C4⋊C8—21no—
44C4×M16—24no—
45C4×M16—24no—
46C2×#97—22no—
47C2×C4×C8—24no—
48C2×C4×C8—22no—
49C2×C4×C8D412yes—
50C2×C4×C8D412yes—
51#384—22no—
52#384—22no—
53#384—22no—
54#384—22no—
55#343—216no—
56#341—28no—
57#340—28no—
58#340—28no—
59#340—28no—
60#336—28no—
61C22×Q8—21no—
62C22×Q8—21no—
63C23×C4—24no—
64C23×C4#4111yescharacteristic
65C23×C4#4111yescharacteristic
66C23×C4C2411yescharacteristic
67C2×M16—24no—
68C2×M16—24no—
69C2×M16—24no—
70C2×M16—24no—
71C2×M16—24no—
72C2×M16—28no—
73C2×M16—24no—
74C2×M16#4118yes—
75C22×C8—24no—
76C22×C8—24no—
77C22×C8—22no—
78C22×C8—22no—
79C22×C8—24no—
80C22×C8—22no—
81C22×C8C2×D414yes—
82C4⋊Q8—24no—
83C4⋊Q8—24no—
84#125—28no—
85C22⋊Q8—44no—
86C22⋊Q8—44no—
87C4×Q8—42no—
88C4×Q8—42no—
89C4×Q8—42no—
90C4×Q8—42no—
91#117—24no—
92#117—24no—
93C2×C4⋊C4—22no—
94C2×C4⋊C4—22no—
95C2×C4⋊C4—21no—
96C2×C4⋊C4—21no—
97C2×C4⋊C4—24no—
98C2×C4⋊C4—24no—
99C2×C4⋊C4—28no—
100C2×C4⋊C4—28no—
101C2×C4⋊C4—42no—
102C2×C4⋊C4—42no—
103C2×C4⋊C4—42no—
104C2×C4⋊C4—42no—
105C2×C4⋊C4—84no—
106C2×C4⋊C4—84no—
107C2×C4⋊C4C2×D412yes—
108C2×C4⋊C4C2×D412yes—
109C2×C22⋊C4—22no—
110C2×C42—22no—
111C2×C42C2×D411yescharacteristic
112C2×C42C2×D411yescharacteristic
113C2×C42C2×D411yescharacteristic
114C2×C42C2×D411yescharacteristic
115C2×C42#4114yes—
116C2×C42#4114yes—
117C4⋊C8—42no—
118C4⋊C8—42no—
119#97—44no—
120C4×C8—24no—
121C4×C8—24no—
122C4×C8—44no—
123C24C2×#4111yescharacteristic
124C2×Q8—42no—
125C2×Q8—42no—
126C2×Q8—44no—
127C2×Q8—42no—
128C2×Q8—42no—
129C22×C4—24no—
130C22×C4—22no—
131C22×C4—22no—
132C22×C4—22no—
133C22×C4—22no—
134C22×C4—44no—
135C22×C4—48no—
136C22×C4—84no—
137C22×C4—84no—
138C22×C4#12111yescharacteristic
139C22×C4#12111yescharacteristic
140C22×C4#12111yescharacteristic
141C22×C4#12111yescharacteristic
142C22×C4#12614yes—
143C22×C4#12614yes—
144C22×C4C22×D411yescharacteristic
145C22×C4C22×D411yescharacteristic
146C22×C4C2×#4112yes—
147C22×C4C2×#4112yes—
148C22×C4C2×#4114yes—
149C22×C42−1+414yes—
150M16—28no—
151M16—28no—
152M16—44no—
153M16—44no—
154M16—48no—
155C2×C8—24no—
156C2×C8—24no—
157C2×C8—24no—
158C2×C8—28no—
159C2×C8—44no—
160C2×C8—44no—
161C2×C8—44no—
162C2×C8—44no—
163C2×C8—42no—
164C2×C8—44no—
165C4⋊C4—24no—
166C4⋊C4—24no—
167C4⋊C4—44no—
168C4⋊C4—44no—
169C4⋊C4—42no—
170C4⋊C4—42no—
171C4⋊C4—44no—
172C4⋊C4—44no—
173C4⋊C4—48no—
174C4⋊C4—84no—
175C4⋊C4—84no—
176C4⋊C4—84no—
177C4⋊C4—84no—
178C22⋊C4—44no—
179C42—42no—
180C42—42no—
181C42#12312yes—
182C42#12312yes—
183C42#12314yes—
184C42#12314yes—
185C42#12412yes—
186C42#12412yes—
187C23—22no—
188C23—21no—
189C23—21no—
190C23C2×#12111yescharacteristic
191C23C2×#12612yes—
192C23#55614yes—
193Q8—84no—
194Q8—84no—
195C2×C4—22no—
196C2×C4—21no—
197C2×C4—21no—
198C2×C4—28no—
199C2×C4—28no—
200C2×C4—42no—
201C2×C4—42no—
202C2×C4—44no—
203C2×C4—42no—
204C2×C4—42no—
205C2×C4—84no—
206C2×C4—84no—
207C2×C4—84no—
208C2×C4—168no—
209C2×C4#39712yes—
210C2×C4#39712yes—
211C2×C4#39712yes—
212C2×C4#39712yes—
213C2×C4#47114yes—
214C2×C4#47212yes—
215C2×C4#47312yes—
216C2×C4C2×#12312yes—
217C2×C4C2×#12311yescharacteristic
218C2×C4C2×#12411yescharacteristic
219C2×C4#52914yes—
220C2×C4#53212yes—
221C2×C4#53612yes—
222C2×C4#54114yes—
223C2×C4#54814yes—
224C2×C4#55214yes—
225C2×C4#55614yes—
226C8—48no—
227C8—44no—
228C8—82no—
229C22—22no—
230C22—22no—
231C22—24no—
232C22—24no—
233C22C2×#39711yescharacteristic
234C22#251414yes—
235C22#254512yes—
236C22C2×#47112yes—
237C22C2×#47211yescharacteristic
238C22C2×#47311yescharacteristic
239C4—22no—
240C4—22no—
241C4—82no—
242C4—82no—
243C4—164no—
244C4#207814yes—
245C4#207914yes—
246C4#242114yes—
247C4#249912yes—
248C4#264211yescharacteristic
249C4#286111yescharacteristic
250C2—24no—
251C2#1524111yescharacteristic
252C2C2×#207811yescharacteristic
253C2C2×#207911yescharacteristic
254C2#2866912yes—
255C2#2867012yes—
subgroupG/H
center, socleC23C2×#121
commutator, Frattini, cosocleC23×C4C24
prime radicalC24C2×#41
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Maximal subgroups

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

subgroupindex
#213222
#213002
C2×#18342
#113032
#108572
C4×#3412