Finite Group Databasebeta

← #135220

Subgroups — #135220

Order 512.

370 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 370 conjugacy classes of subgroups — too many to draw. Only its 98 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#20438C211yesmaximal, characteristic
2#12846C212yesmaximal
3#7185C212yesmaximal
4#7185C212yesmaximal
5C2×#423C411yescharacteristic
6C2×#405C411yescharacteristic
7C2×#405C411yescharacteristic
8C22×#97C2211yescharacteristic
9#2285C411yescharacteristic
10#2112C2214yes—
11#1771—22no—
12#1771—22no—
13#1729C2211yescharacteristic
14#1729C2211yescharacteristic
15#1377—22no—
16#1377—22no—
17C22×M16C2×C411yescharacteristic
18C22×M16C2×C411yescharacteristic
19C23×C8C2×C411yescharacteristic
20C23×C8C2311yescharacteristic
21C2×#117C2×C411yescharacteristic
22C2×#117C2×C411yescharacteristic
23C22×C42C2×C411yescharacteristic
24C2×M32—24no—
25#423—22no—
26#423—22no—
27C2×C4⋊C8—22no—
28#405—42no—
29#405—42no—
30C2×#97—22no—
31C2×#97—22no—
32C2×#97—21no—
33C2×#97—21no—
34C2×#97D412yes—
35C2×#97D412yes—
36C2×C4×C8—21no—
37C2×C4×C8—21no—
38#384—21no—
39#382—21no—
40#380C814yes—
41C4×C4⋊C4—22no—
42C22⋊C16—24no—
43C22⋊C16—24no—
44#336—22no—
45#336—22no—
46C23×C4C2×C812yes—
47C23×C4C22×C411yescharacteristic
48C2×M16—21no—
49C2×M16—21no—
50C2×M16—41no—
51C2×M16—41no—
52C2×M16—41no—
53C2×M16—41no—
54C2×M16M1612yes—
55C2×M16M1612yes—
56C22×C8—21no—
57C22×C8—21no—
58C22×C8—22no—
59C22×C8—22no—
60C22×C8—21no—
61C22×C8—21no—
62C22×C8—41no—
63C22×C8—41no—
64C22×C8C22⋊C411yescharacteristic
65C22×C8C22⋊C411yescharacteristic
66C22×C8C2×D412yes—
67C22×C8#4112yes—
68C22×C8#4112yes—
69#117—41no—
70#117—41no—
71#117—41no—
72#117—41no—
73C2×C4⋊C4—21no—
74C2×C4⋊C4—21no—
75C2×C4⋊C4—24no—
76C2×C22⋊C4C2×C812yes—
77C2×C22⋊C4C2×C812yes—
78C2×C42—21no—
79C2×C42—21no—
80C2×C42—21no—
81C2×C42—21no—
82C2×C42—21no—
83C2×C42—21no—
84C2×C42C22⋊C412yes—
85C2×C42M1611yescharacteristic
86C2×C42M1611yescharacteristic
87M32—44no—
88M32—44no—
89C2×C16—44no—
90C4⋊C8—28no—
91#97—24no—
92#97—24no—
93#97—42no—
94#97—42no—
95C4×C8—42no—
96C4×C8—42no—
97#95—24no—
98C24C22×C811yescharacteristic
99C22×C4—21no—
100C22×C4—21no—
101C22×C4—21no—
102C22×C4—21no—
103C22×C4—22no—
104C22×C4—22no—
105C22×C4—22no—
106C22×C4—22no—
107C22×C4—82no—
108C22×C4C22⋊C812yes—
109C22×C4C22⋊C812yes—
110C22×C4C2×C22⋊C411yescharacteristic
111C22×C4#11711yescharacteristic
112C22×C4#11711yescharacteristic
113C22×C4C2×M1611yescharacteristic
114C22×C4C2×M1611yescharacteristic
115C22×C4C2×M1611yescharacteristic
116M16—42no—
117M16—42no—
118M16—42no—
119M16—42no—
120C2×C8—24no—
121C2×C8—24no—
122C2×C8—21no—
123C2×C8—21no—
124C2×C8—22no—
125C2×C8—22no—
126C2×C8—21no—
127C2×C8—21no—
128C2×C8—22no—
129C2×C8—22no—
130C2×C8—42no—
131C2×C8—42no—
132C2×C8—41no—
133C2×C8—41no—
134C2×C8—41no—
135C2×C8—41no—
136C2×C8#10012yes—
137C2×C8#10112yes—
138C2×C8#12314yes—
139C4⋊C4—42no—
140C4⋊C4—42no—
141C4⋊C4—84no—
142C22⋊C4—42no—
143C22⋊C4—42no—
144C42—22no—
145C42—22no—
146C42—21no—
147C42—21no—
148C42—21no—
149C42—21no—
150C42—41no—
151C42—41no—
152C42—41no—
153C42—41no—
154C16—84no—
155C23—21no—
156C23—21no—
157C23—42no—
158C23C2×C22⋊C811yescharacteristic
159C23#43111yescharacteristic
160C23#43111yescharacteristic
161C2×C4—21no—
162C2×C4—21no—
163C2×C4—21no—
164C2×C4—21no—
165C2×C4—21no—
166C2×C4—21no—
167C2×C4—21no—
168C2×C4—21no—
169C2×C4—21no—
170C2×C4—21no—
171C2×C4—22no—
172C2×C4—22no—
173C2×C4—42no—
174C2×C4—42no—
175C2×C4—44no—
176C2×C4—44no—
177C2×C4—42no—
178C2×C4—42no—
179C2×C4—82no—
180C2×C4—82no—
181C2×C4#38111yescharacteristic
182C2×C4#40711yescharacteristic
183C2×C4#41011yescharacteristic
184C2×C4#41011yescharacteristic
185C2×C4C2×#10011yescharacteristic
186C2×C4C2×#10111yescharacteristic
187C2×C4#43211yescharacteristic
188C2×C4#43211yescharacteristic
189C8—24no—
190C8—22no—
191C8—22no—
192C8—42no—
193C8—42no—
194C22—22no—
195C22—22no—
196C22—22no—
197C22—21no—
198C22—21no—
199C22—42no—
200C22—42no—
201C22#146011yescharacteristic
202C22#146111yescharacteristic
203C22#181711yescharacteristic
204C4—21no—
205C4—21no—
206C4—42no—
207C4—42no—
208C4—82no—
209C4—82no—
210C4#146611yescharacteristic
211C4#146711yescharacteristic
212C4#182011yescharacteristic
213C4#182111yescharacteristic
214C2—21no—
215C2—21no—
216C2—41no—
217C2—41no—
218C2#870111yescharacteristic
219C2#1285112yes—
subgroupG/H
centerC2×C4#381
commutator, prime radicalC24C22×C8
Frattini, cosocleC23×C8C23
socleC22#1817
LCSG ▷ C24 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C23×C8 ◁ G
LECSG ▷ C23×C8 ▷ C2×C4 ▷ C2 ▷ C1

Maximal subgroups

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

subgroupindex
#204382
#128462
#71852