Finite Group Databasebeta

← #134253

Subgroups — #134253

Order 512.

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

normal subgroups only

This group has 404 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#20435C211yesmaximal, characteristic
2#12788C212yesmaximal
3#11443C212yesmaximal
4#7185C212yesmaximal
5C2×#424C2211yescharacteristic
6C2×#405C411yescharacteristic
7C2×#405C411yescharacteristic
8C22×C4×C8C2211yescharacteristic
9#2296C411yescharacteristic
10C2×#364—22no—
11#2112C2214yes—
12#1771—22no—
13C8×C4⋊C4—22no—
14#1729C2211yescharacteristic
15#1728C411yescharacteristic
16#1376—22no—
17#1323—24no—
18C22×M16C2×C411yescharacteristic
19C22×M16C2×C411yescharacteristic
20C23×C8C2×C411yescharacteristic
21C23×C8C2311yescharacteristic
22C2×#117C2×C411yescharacteristic
23C2×#117C2×C411yescharacteristic
24C22×C42C2×C411yescharacteristic
25C2×M32—24no—
26#424—22no—
27#424D412yes—
28#424Q812yes—
29C2×C4⋊C8—22no—
30#405—22no—
31#405—42no—
32#405C814yes—
33C2×#97—21no—
34C2×#97—21no—
35C2×C4×C8—21no—
36C2×C4×C8—21no—
37C2×C4×C8—42no—
38C2×C4×C8D412yes—
39C2×C4×C8D412yes—
40#389—22no—
41#385—22no—
42C4×C4⋊C4—22no—
43#364—44no—
44C22⋊C16—24no—
45C22⋊C16—24no—
46#336—22no—
47#336—22no—
48C23×C4—21no—
49C23×C4C22×C411yescharacteristic
50C2×M16—21no—
51C2×M16—21no—
52C2×M16—41no—
53C2×M16—41no—
54C2×M16—41no—
55C2×M16—41no—
56C2×M16C4⋊C412yes—
57C2×M16C2×C812yes—
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×C812yes—
67C22×C8M1612yes—
68C22×C8C2×D412yes—
69C22×C8C2×D411yescharacteristic
70C22×C8C2×Q811yescharacteristic
71C22×C8#4112yes—
72#117—21no—
73#117—21no—
74#117—41no—
75#117—41no—
76#117C4⋊C412yes—
77#117C2×C812yes—
78C2×C4⋊C4—24no—
79C2×C4⋊C4—21no—
80C2×C4⋊C4M1612yes—
81C2×C22⋊C4—21no—
82C2×C22⋊C4M1612yes—
83C2×C42—21no—
84C2×C42—21no—
85C2×C42—21no—
86C2×C42—41no—
87C2×C42—41no—
88C2×C42C22⋊C412yes—
89M32—44no—
90M32—44no—
91C2×C16—44no—
92C4⋊C8—24no—
93C4⋊C8—24no—
94#97—24no—
95#97—42no—
96C4×C8—24no—
97C4×C8—24no—
98C4×C8—42no—
99C4×C8—42no—
100#95—24no—
101C24C2×M1611yescharacteristic
102C22×C4—21no—
103C22×C4—21no—
104C22×C4—21no—
105C22×C4—21no—
106C22×C4—22no—
107C22×C4—22no—
108C22×C4—22no—
109C22×C4—44no—
110C22×C4—42no—
111C22×C4C2×C22⋊C411yescharacteristic
112C22×C4C2×C4⋊C411yescharacteristic
113C22×C4#11711yescharacteristic
114C22×C4C22×C811yescharacteristic
115C22×C4C2×M1611yescharacteristic
116C22×C4C2×M1611yescharacteristic
117M16—42no—
118M16—42no—
119M16—42no—
120M16—42no—
121C2×C8—22no—
122C2×C8—21no—
123C2×C8—22no—
124C2×C8—21no—
125C2×C8—24no—
126C2×C8—24no—
127C2×C8—22no—
128C2×C8—22no—
129C2×C8—41no—
130C2×C8—41no—
131C2×C8—42no—
132C2×C8—41no—
133C2×C8—41no—
134C2×C8—41no—
135C2×C8—41no—
136C2×C8C22⋊C812yes—
137C2×C8C22⋊C812yes—
138C2×C8#9912yes—
139C2×C8#10112yes—
140C2×C8#12112yes—
141C2×C8C22⋊Q812yes—
142C4⋊C4—22no—
143C4⋊C4—48no—
144C4⋊C4—42no—
145C4⋊C4C4⋊C814yes—
146C22⋊C4—22no—
147C22⋊C4—22no—
148C22⋊C4—42no—
149C42—22no—
150C42—22no—
151C42—22no—
152C42—22no—
153C42—42no—
154C42—41no—
155C42—41no—
156C16—84no—
157C23—21no—
158C23—21no—
159C23—42no—
160C23#40711yescharacteristic
161C23#42311yescharacteristic
162C23#43111yescharacteristic
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—21no—
172C2×C4—21no—
173C2×C4—24no—
174C2×C4—24no—
175C2×C4—44no—
176C2×C4—42no—
177C2×C4—48no—
178C2×C4—42no—
179C2×C4—44no—
180C2×C4—42no—
181C2×C4#38011yescharacteristic
182C2×C4C2×C22⋊C811yescharacteristic
183C2×C4C2×#9911yescharacteristic
184C2×C4#41011yescharacteristic
185C2×C4C2×#10111yescharacteristic
186C2×C4#41311yescharacteristic
187C2×C4C2×C4⋊C811yescharacteristic
188C2×C4#43211yescharacteristic
189C8—22no—
190C8—22no—
191C8—22no—
192C8—42no—
193C8—42no—
194C8#32414yes—
195C22—22no—
196C22—21no—
197C22—21no—
198C22—22no—
199C22—22no—
200C22—42no—
201C22—42no—
202C22#146511yescharacteristic
203C22#146711yescharacteristic
204C22#179711yescharacteristic
205C4—21no—
206C4—21no—
207C4—44no—
208C4—44no—
209C4—82no—
210C4C2×#32411yescharacteristic
211C4#145811yescharacteristic
212C4#179911yescharacteristic
213C4#180111yescharacteristic
214C2—21no—
215C2—21no—
216C2—41no—
217C2—41no—
218C2#855811yescharacteristic
219C2#1279212yes—
subgroupG/H
centerC2×C4#380
commutatorC22×C4C22×C8
Frattini, cosocleC23×C8C23
socleC22#1797
prime radicalC24C2×M16
LCSG ▷ C22×C4 ▷ 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
#204352
#127882
#114432
#71852