Finite Group Databasebeta

← #135191

Subgroups — #135191

Order 512.

366 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 366 conjugacy classes of subgroups — too many to draw. Only its 94 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#20366C211yesmaximal, characteristic
2#8730C212yesmaximal
3#7190C212yesmaximal
4#7188C212yesmaximal
5C2×C4×M16C411yescharacteristic
6C22×C4×C8C411yescharacteristic
7C4×#117C2211yescharacteristic
8#1746—22no—
9#1744—22no—
10#1729C411yescharacteristic
11#1729C411yescharacteristic
12#1729C2211yescharacteristic
13#1729C2211yescharacteristic
14#1458C2214yes—
15#1276—22no—
16#1274—22no—
17C22×M16C2×C411yescharacteristic
18C22×M16C2×C411yescharacteristic
19C23×C8C2×C411yescharacteristic
20C23×C8C2×C411yescharacteristic
21C2×#117C2×C411yescharacteristic
22C2×#117C2311yescharacteristic
23C22×C42C2×C411yescharacteristic
24#431—24no—
25C4×M16—22no—
26C4×M16—22no—
27C2×#97—21no—
28C2×C4×C8—21no—
29C2×C4×C8—21no—
30C2×C4×C8—21no—
31C2×C4×C8—22no—
32C2×C4×C8C814yescomplemented
33C4×C4⋊C4—22no—
34C4×C22⋊C4—22no—
35#376D412yes—
36C43D412yes—
37#336—22no—
38#336—22no—
39#336—22no—
40#336—22no—
41#324—24no—
42#323—24no—
43C23×C4—21no—
44C23×C4C22×C411yescharacteristic
45C2×M16—21no—
46C2×M16—21no—
47C2×M16—41no—
48C2×M16—41no—
49C2×M16—41no—
50C2×M16—41no—
51C2×M16M1612yes—
52C2×M16M1612yes—
53C22×C8—21no—
54C22×C8—21no—
55C22×C8—21no—
56C22×C8—21no—
57C22×C8—21no—
58C22×C8—21no—
59C22×C8—41no—
60C22×C8—41no—
61C22×C8—41no—
62C22×C8—41no—
63C22×C8C2×C812yes—
64C22×C8C2×C812yes—
65#117—41no—
66#117—41no—
67#117—41no—
68#117—41no—
69C2×C4⋊C4—21no—
70C2×C4⋊C4#4112yes—
71C2×C22⋊C4—21no—
72C2×C22⋊C4#4112yes—
73C2×C42—21no—
74C2×C42—21no—
75C2×C42—21no—
76C2×C42—24no—
77C2×C42C22⋊C412yes—
78C2×C42C22⋊C411yescharacteristic
79C2×C42C22⋊C411yescharacteristic
80C2×C42C2×C812yes—
81C2×C42M1612yes—
82C2×C42C2×D412yes—
83C4⋊C8—44no—
84C22⋊C8—44no—
85#97—24no—
86C4×C8—24no—
87C4×C8—24no—
88C4×C8—24no—
89C4×C8—44no—
90#95—24no—
91C24#11711yescharacteristic
92C22×C4—21no—
93C22×C4—21no—
94C22×C4—21no—
95C22×C4—21no—
96C22×C4—21no—
97C22×C4—21no—
98C22×C4—22no—
99C22×C4—22no—
100C22×C4—22no—
101C22×C4—44no—
102C22×C4C2×C22⋊C411yescharacteristic
103C22×C4#11711yescharacteristic
104C22×C4#12314yes—
105C22×C4C22×C811yescharacteristic
106C22×C4C2×M1611yescharacteristic
107C22×C4C2×M1611yescharacteristic
108C22×C4C2×M1611yescharacteristic
109M16—42no—
110M16—42no—
111M16—42no—
112M16—42no—
113C2×C8—22no—
114C2×C8—22no—
115C2×C8—22no—
116C2×C8—22no—
117C2×C8—42no—
118C2×C8—42no—
119C2×C8—41no—
120C2×C8—41no—
121C2×C8—41no—
122C2×C8—41no—
123C2×C8—41no—
124C2×C8—41no—
125C2×C8—41no—
126C2×C8—41no—
127C2×C8—42no—
128C2×C8—42no—
129C2×C8—84no—
130C4⋊C4—42no—
131C4⋊C4—42no—
132C22⋊C4—42no—
133C22⋊C4—42no—
134C42—21no—
135C42—21no—
136C42—22no—
137C42—22no—
138C42—22no—
139C42—22no—
140C42—22no—
141C42—22no—
142C42—48no—
143C42C22⋊C812yes—
144C42C22⋊C812yes—
145C23—21no—
146C23—21no—
147C23—42no—
148C23#38111yescharacteristic
149C23#43111yescharacteristic
150C23#43211yescharacteristic
151C2×C4—24no—
152C2×C4—22no—
153C2×C4—21no—
154C2×C4—21no—
155C2×C4—21no—
156C2×C4—21no—
157C2×C4—21no—
158C2×C4—21no—
159C2×C4—21no—
160C2×C4—21no—
161C2×C4—21no—
162C2×C4—21no—
163C2×C4—22no—
164C2×C4—24no—
165C2×C4—44no—
166C2×C4—44no—
167C2×C4—42no—
168C2×C4—44no—
169C2×C4—44no—
170C2×C4—42no—
171C2×C4C2×C22⋊C811yescharacteristic
172C2×C4#40711yescharacteristic
173C2×C4#41011yescharacteristic
174C2×C4#41011yescharacteristic
175C2×C4#41311yescharacteristic
176C2×C4#41311yescharacteristic
177C2×C4#43111yescharacteristic
178C2×C4#43211yescharacteristic
179C8—42no—
180C8—42no—
181C8—42no—
182C8—42no—
183C8—164no—
184C22—22no—
185C22—21no—
186C22—21no—
187C22—41no—
188C22—41no—
189C22—42no—
190C22—42no—
191C22#181711yescharacteristic
192C22#182011yescharacteristic
193C22#182211yescharacteristic
194C4—24no—
195C4—21no—
196C4—21no—
197C4—42no—
198C4—44no—
199C4—82no—
200C4#145811yescharacteristic
201C4#145811yescharacteristic
202C4#146411yescharacteristic
203C4#146411yescharacteristic
204C2—21no—
205C2—21no—
206C2—41no—
207C2—41no—
208C2#869911yescharacteristic
209C2#1286412yes—
subgroupG/H
centerC2×C4#410
commutatorC22×C4C22×C8
Frattini, cosocleC2×#117C23
socleC22#1820
prime radicalC24#117
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ C4×#117 ◁ G
LECSG ▷ C2×#117 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
32. C2×C4×C8C8semidirect

Maximal subgroups

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

subgroupindex
#203662
#87302
#71902
#71882