Finite Group Databasebeta

← #140624

Subgroups — #140624

Order 512.

402 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 402 conjugacy classes of subgroups — too many to draw. Only its 114 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#21143C211yesmaximal, characteristic
2#12783C211yesmaximal, characteristic
3#12599C211yesmaximal, characteristic
4C2×#1323C214yesmaximal
5C2×#424C2212yes—
6C2×#423C411yescharacteristic
7C22×C4×C8C2211yescharacteristic
8#2285C411yescharacteristic
9C2×C4⋊C16—21no—
10C2×C22⋊C16C2212yes—
11C2×C22⋊C16C2212yes—
12C2×#346—21no—
13#1939—21no—
14#1939—21no—
15#1939—21no—
16#1939—21no—
17#1729C412yes—
18#1374—21no—
19#1374—21no—
20#1323—28no—
21C22×M16C2×C412yes—
22C23×C8C2×C411yescharacteristic
23C23×C8C2311yescharacteristic
24C2×#117C2×C412yes—
25C22×C42C2×C411yescharacteristic
26C22×C16—22no—
27C22×C16—22no—
28#424—24no—
29#424—24no—
30#423—42no—
31C2×C4⋊C8—22no—
32C2×C4⋊C8—22no—
33C2×C4⋊C8—21no—
34C2×C4⋊C8—21no—
35C2×C4×C8—42no—
36C2×C4×C8D411yescharacteristic
37C2×C4×C8D411yescharacteristic
38C2×C4×C8D411yescharacteristic
39C2×C4×C8D411yescharacteristic
40#384C811yescharacteristic
41#384C811yescharacteristic
42#382C812yes—
43#380—21no—
44#380—21no—
45C4×C4⋊C4—21no—
46C4×C4⋊C4—21no—
47C4⋊C16—42no—
48C22⋊C16—22no—
49C22⋊C16—24no—
50C22⋊C16D414yes—
51#346—42no—
52#336—22no—
53#336—22no—
54C23×C4M1611yescharacteristic
55C23×C4M1611yescharacteristic
56C23×C4C22×C411yescharacteristic
57C2×M16—24no—
58C2×M16—42no—
59C2×M16—42no—
60C22×C8—22no—
61C22×C8—22no—
62C22×C8—22no—
63C22×C8—41no—
64C22×C8—41no—
65C22×C8—41no—
66C22×C8C22⋊C411yescharacteristic
67C22×C8C22⋊C411yescharacteristic
68C22×C8C2×D411yescharacteristic
69C22×C8C2×D411yescharacteristic
70C22×C8C2×D412yes—
71C22×C8#4112yes—
72#117—24no—
73#117—24no—
74C2×C4⋊C4—21no—
75C2×C4⋊C4—21no—
76C2×C4⋊C4—21no—
77C2×C4⋊C4—21no—
78C2×C4⋊C4C2×C812yes—
79C2×C4⋊C4C2×C812yes—
80C2×C22⋊C4—22no—
81C2×C42—22no—
82C2×C42—22no—
83C2×C42—22no—
84C2×C42C22⋊C411yescharacteristic
85C2×C42C22⋊C411yescharacteristic
86C2×C42C2×C811yescharacteristic
87C2×C42C2×C811yescharacteristic
88C2×C16—24no—
89C2×C16—42no—
90C2×C16—42no—
91C2×C16—42no—
92C2×C16—42no—
93C2×C16—42no—
94C4⋊C8—42no—
95C4⋊C8—44no—
96C4⋊C8—42no—
97C4⋊C8—42no—
98C4⋊C8—42no—
99C4×C8—22no—
100C4×C8—22no—
101C4×C8—41no—
102C4×C8—41no—
103#95—22no—
104#95—22no—
105C24C2×M1611yescharacteristic
106C22×C4—21no—
107C22×C4—21no—
108C22×C4—21no—
109C22×C4—21no—
110C22×C4—22no—
111C22×C4—22no—
112C22×C4—22no—
113C22×C4—22no—
114C22×C4—22no—
115C22×C4—41no—
116C22×C4—41no—
117C22×C4—41no—
118C22×C4C22⋊C811yescharacteristic
119C22×C4C22⋊C811yescharacteristic
120C22×C4C22⋊C811yescharacteristic
121C22×C4C22⋊C811yescharacteristic
122C22×C4C2×C22⋊C411yescharacteristic
123C22×C4C4×D412yes—
124C22×C4C22×C811yescharacteristic
125C22×C4#13112yes—
126M16—84no—
127C2×C8—21no—
128C2×C8—21no—
129C2×C8—21no—
130C2×C8—21no—
131C2×C8—21no—
132C2×C8—21no—
133C2×C8—21no—
134C2×C8—21no—
135C2×C8—21no—
136C2×C8—42no—
137C2×C8—42no—
138C2×C8—44no—
139C2×C8—42no—
140C2×C8—42no—
141C2×C8—42no—
142C2×C8#10112yes—
143C2×C8#12112yes—
144C2×C8#12411yescharacteristic
145C2×C8#12711yescharacteristic
146C4⋊C4—24no—
147C4⋊C4—24no—
148C4⋊C4—42no—
149C4⋊C4—42no—
150C4⋊C4—42no—
151C4⋊C4—42no—
152C22⋊C4—44no—
153C42—22no—
154C42—22no—
155C42—21no—
156C42—21no—
157C42—42no—
158C42—42no—
159C42#9912yes—
160C42#10112yes—
161C16—44no—
162C16—82no—
163C23—21no—
164C23—21no—
165C23—21no—
166C23—21no—
167C23—21no—
168C23—21no—
169C23C2×C22⋊C811yescharacteristic
170C23#43612yes—
171C2×C4—21no—
172C2×C4—21no—
173C2×C4—21no—
174C2×C4—21no—
175C2×C4—21no—
176C2×C4—21no—
177C2×C4—21no—
178C2×C4—21no—
179C2×C4—22no—
180C2×C4—22no—
181C2×C4—22no—
182C2×C4—22no—
183C2×C4—21no—
184C2×C4—21no—
185C2×C4—21no—
186C2×C4—44no—
187C2×C4—44no—
188C2×C4—42no—
189C2×C4—42no—
190C2×C4—42no—
191C2×C4—42no—
192C2×C4—42no—
193C2×C4—42no—
194C2×C4#32412yes—
195C2×C4#32412yes—
196C2×C4#35012yes—
197C2×C4#39011yescharacteristic
198C2×C4#40811yescharacteristic
199C2×C4C2×#9911yescharacteristic
200C2×C4#41011yescharacteristic
201C2×C4C2×#10111yescharacteristic
202C2×C4C2×#10111yescharacteristic
203C2×C4C8×D412yes—
204C8—22no—
205C8—21no—
206C8—21no—
207C8—22no—
208C8—41no—
209C8—82no—
210C22—22no—
211C22—22no—
212C22—22no—
213C22—41no—
214C22—41no—
215C22—41no—
216C22—41no—
217C22C2×#32411yescharacteristic
218C22#146111yescharacteristic
219C22#195411yescharacteristic
220C22C2×#35011yescharacteristic
221C22#211711yescharacteristic
222C22#211711yescharacteristic
223C22#211711yescharacteristic
224C4—22no—
225C4—22no—
226C4—22no—
227C4—22no—
228C4—41no—
229C4—82no—
230C4—82no—
231C4#147011yescharacteristic
232C4#147211yescharacteristic
233C4#196411yescharacteristic
234C4#196611yescharacteristic
235C2—42no—
236C2#965011yescharacteristic
237C2#1144311yescharacteristic
238C2#1144411yescharacteristic
239C2#1149711yescharacteristic
240C2#1149811yescharacteristic
241C2#1319211yescharacteristic
242C2#1319311yescharacteristic
subgroupG/H
centerC22×C4C2×C22⋊C4
commutatorC22×C4C22×C8
Frattini, cosocleC23×C8C23
socleC23C2×C22⋊C8
prime radicalC24C2×M16
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C4 ◁ C22×C4×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
#211432
#127832
#125992
C2×#13232