Finite Group Databasebeta

← #124267

Subgroups — #124267

Order 512.

384 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 384 conjugacy classes of subgroups — too many to draw. Only its 118 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
3C4×C22⋊C16C211yesmaximal, characteristic
4#7118C214yesmaximal, complemented
5C2×#424—21no—
6C2×#423C411yescharacteristic
7C22×C4×C8C2211yescharacteristic
8#2285C411yescharacteristic
9C2×C4⋊C16—21no—
10C2×C22⋊C16—21no—
11C2×C22⋊C16—21no—
12C2×C4×C16—21no—
13#1939—21no—
14#1939—21no—
15#1939C412yes—
16#1939C2212yes—
17#1729—21no—
18#1374C2212yes—
19#1374C2212yes—
20C22×M16—21no—
21C23×C8D411yescharacteristic
22C23×C8D411yescharacteristic
23C2×#117—21no—
24C22×C42C2×C411yescharacteristic
25C22×C16—22no—
26C22×C16—22no—
27#424—42no—
28#424—42no—
29#423—42no—
30C2×C4⋊C8—22no—
31C2×C4⋊C8—22no—
32C2×C4⋊C8—21no—
33C2×C4⋊C8C2×C412yes—
34C2×C4×C8—42no—
35C2×C4×C8C2×C411yescharacteristic
36C2×C4×C8D411yescharacteristic
37C2×C4×C8D411yescharacteristic
38C2×C4×C8C2311yescharacteristic
39#384C811yescharacteristic
40#384C811yescharacteristic
41#382—21no—
42#380—21no—
43#380C812yes—
44C4×C4⋊C4—21no—
45C4×C4⋊C4C2×C412yes—
46C4⋊C16—42no—
47C22⋊C16—24no—
48C22⋊C16—42no—
49C4×C16—42no—
50#336—22no—
51#336—22no—
52C23×C4C22⋊C411yescharacteristic
53C23×C4C2×C811yescharacteristic
54C23×C4M1611yescharacteristic
55C2×M16—42no—
56C2×M16—42no—
57C2×M16—42no—
58C22×C8—22no—
59C22×C8—22no—
60C22×C8—22no—
61C22×C8—41no—
62C22×C8—41no—
63C22×C8—41no—
64C22×C8C22⋊C411yescharacteristic
65C22×C8C22⋊C411yescharacteristic
66C22×C8D812yes—
67C22×C8SD1612yes—
68C22×C8C2×D411yescharacteristic
69C22×C8C2×D411yescharacteristic
70#117—42no—
71#117—42no—
72C2×C4⋊C4—21no—
73C2×C4⋊C4—21no—
74C2×C4⋊C4—21no—
75C2×C4⋊C4—21no—
76C2×C4⋊C4C2×C812yes—
77C2×C4⋊C4C2×C812yes—
78C2×C22⋊C4—22no—
79C2×C42—22no—
80C2×C42—22no—
81C2×C42—22no—
82C2×C42C22⋊C411yescharacteristic
83C2×C42C2×C811yescharacteristic
84C2×C42M1611yescharacteristic
85C2×C42C22×C411yescharacteristic
86C2×C16—24no—
87C2×C16—42no—
88C2×C16—42no—
89C2×C16—42no—
90C2×C16—42no—
91C2×C16—42no—
92C4⋊C8—42no—
93C4⋊C8—44no—
94C4⋊C8—42no—
95C4⋊C8—42no—
96C4⋊C8—42no—
97C4×C8—22no—
98C4×C8—22no—
99C4×C8—41no—
100C4×C8—41no—
101#95—22no—
102#95—22no—
103C24C22⋊C811yescharacteristic
104C22×C4—21no—
105C22×C4—21no—
106C22×C4—21no—
107C22×C4—21no—
108C22×C4—22no—
109C22×C4—22no—
110C22×C4—22no—
111C22×C4—22no—
112C22×C4—22no—
113C22×C4—22no—
114C22×C4—22no—
115C22×C4—41no—
116C22×C4C22⋊C811yescharacteristic
117C22×C4C22⋊C811yescharacteristic
118C22×C4C22⋊C811yescharacteristic
119C22×C4#10212yes—
120C22×C4#10412yes—
121C22×C4C2×C22⋊C411yescharacteristic
122C22×C4C22×C811yescharacteristic
123C22×C4C2×M1611yescharacteristic
124M16—84no—
125C2×C8—21no—
126C2×C8—21no—
127C2×C8—21no—
128C2×C8—21no—
129C2×C8—21no—
130C2×C8—21no—
131C2×C8—21no—
132C2×C8—21no—
133C2×C8—21no—
134C2×C8—42no—
135C2×C8—42no—
136C2×C8—44no—
137C2×C8—42no—
138C2×C8—42no—
139C2×C8—42no—
140C2×C8#10212yes—
141C2×C8C2×D811yescharacteristic
142C2×C8C2×SD1611yescharacteristic
143C2×C8#13712yes—
144C4⋊C4—24no—
145C4⋊C4—42no—
146C4⋊C4—42no—
147C4⋊C4—42no—
148C4⋊C4—42no—
149C4⋊C4—42no—
150C22⋊C4—44no—
151C42—22no—
152C42—22no—
153C42—21no—
154C42—21no—
155C42—42no—
156C42—42no—
157C42#9912yes—
158C42#10112yes—
159C16—44no—
160C16—82no—
161C23—21no—
162C23—21no—
163C23—21no—
164C23—21no—
165C23—21no—
166C23—21no—
167C23D4⋊C812yes—
168C23C2×C22⋊C811yescharacteristic
169C2×C4—21no—
170C2×C4—21no—
171C2×C4—21no—
172C2×C4—21no—
173C2×C4—21no—
174C2×C4—21no—
175C2×C4—22no—
176C2×C4—22no—
177C2×C4—22no—
178C2×C4—22no—
179C2×C4—21no—
180C2×C4—21no—
181C2×C4—21no—
182C2×C4—44no—
183C2×C4—44no—
184C2×C4—42no—
185C2×C4—42no—
186C2×C4—42no—
187C2×C4—42no—
188C2×C4—42no—
189C2×C4—42no—
190C2×C4#32412yes—
191C2×C4#32412yes—
192C2×C4D4⋊C812yes—
193C2×C4#35012yes—
194C2×C4#35012yes—
195C2×C4C2×#9911yescharacteristic
196C2×C4C2×#10111yescharacteristic
197C2×C4C2×#10211yescharacteristic
198C2×C4#41911yescharacteristic
199C2×C4C2×#10411yescharacteristic
200C2×C4#42111yescharacteristic
201C2×C4#50412yes—
202C8—22no—
203C8—21no—
204C8—22no—
205C8—21no—
206C8—41no—
207C8—82no—
208C22—22no—
209C22—22no—
210C22—22no—
211C22—41no—
212C22—41no—
213C22—41no—
214C22—41no—
215C22C2×#32411yescharacteristic
216C22C2×D4⋊C811yescharacteristic
217C22#147911yescharacteristic
218C22#211311yescharacteristic
219C22#211511yescharacteristic
220C22C2×#35011yescharacteristic
221C22C2×#35011yescharacteristic
222C4—22no—
223C4—22no—
224C4—22no—
225C4—41no—
226C4—82no—
227C4—82no—
228C4#132314yes—
229C4#150811yescharacteristic
230C4#151911yescharacteristic
231C4#152811yescharacteristic
232C4#154611yescharacteristic
233C2—42no—
234C2#757311yescharacteristic
235C2#1144111yescharacteristic
236C2C2×#132311yescharacteristic
237C2#1158411yescharacteristic
238C2#1158611yescharacteristic
239C2#1163811yescharacteristic
240C2#1164111yescharacteristic
subgroupG/H
centerC22×C4C2×C22⋊C4
commutatorC22×C4C22×C8
Frattini, cosocleC2×C4×C8C23
socleC23C2×C22⋊C8
prime radicalC24C22⋊C8
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C4 ◁ C22×C4×C8 ◁ G
LECSG ▷ C2×C4×C8 ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #7118C2semidirect

Maximal subgroups

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

subgroupindex
#211432
#127832
C4×C22⋊C162
#71182