Finite Group Databasebeta

← #135227

Subgroups — #135227

Order 512.

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

normal subgroups only

This group has 382 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#20435C211yesmaximal, characteristic
2#12850C212yesmaximal
3#7194C212yesmaximal
4#7191C212yesmaximal
5C2×#424C411yescharacteristic
6C2×#405C411yescharacteristic
7C2×#405C2211yescharacteristic
8C22×C4×C8C411yescharacteristic
9#2296C411yescharacteristic
10C2×#349C2214yes—
11#1771—22no—
12C8×C4⋊C4—22no—
13#1729C2211yescharacteristic
14#1728C2211yescharacteristic
15#1376—22no—
16#1375—22no—
17C22×M16C2×C411yescharacteristic
18C22×M16C2311yescharacteristic
19C23×C8C2×C411yescharacteristic
20C23×C8C2×C411yescharacteristic
21C2×#117C2×C411yescharacteristic
22C2×#117C2×C411yescharacteristic
23C22×C42C2×C411yescharacteristic
24C2×M32—24no—
25#424—22no—
26#424C814yes—
27C2×C4⋊C8—22no—
28#405—22no—
29#405—22no—
30#405—42no—
31C2×#97—21no—
32C2×#97D412yes—
33C2×C4×C8—21no—
34C2×C4×C8—21no—
35C2×C4×C8—21no—
36C2×C4×C8—42no—
37C2×C4×C8D412yes—
38#389—22no—
39#385—22no—
40C4×C4⋊C4—22no—
41#349—24no—
42#349—24no—
43#336—22no—
44#336—22no—
45C23×C4M1612yes—
46C23×C4C22×C411yescharacteristic
47C2×M16—21no—
48C2×M16—41no—
49C2×M16—41no—
50C2×M16—41no—
51C2×M16—41no—
52C2×M16C2×C812yes—
53C2×M16#4112yes—
54C2×M16#4112yes—
55C22×C8—21no—
56C22×C8—21no—
57C22×C8—22no—
58C22×C8—22no—
59C22×C8—21no—
60C22×C8—21no—
61C22×C8—21no—
62C22×C8—41no—
63C22×C8—41no—
64C22×C8C22⋊C411yescharacteristic
65C22×C8C22⋊C411yescharacteristic
66C22×C8C2×C812yes—
67C22×C8C2×D412yes—
68#117—21no—
69#117—21no—
70#117—41no—
71#117—41no—
72#117C2×C812yes—
73#117M1612yes—
74C2×C4⋊C4—21no—
75C2×C4⋊C4—24no—
76C2×C4⋊C4M1612yes—
77C2×C22⋊C4—21no—
78C2×C22⋊C4—21no—
79C2×C42—21no—
80C2×C42—21no—
81C2×C42—21no—
82C2×C42—41no—
83C2×C42—41no—
84C2×C42C22⋊C412yes—
85M32—44no—
86M32—44no—
87C2×C16—44no—
88C4⋊C8—28no—
89#97—24no—
90#97—42no—
91C4×C8—24no—
92C4×C8—42no—
93C4×C8—42no—
94C4×C8—42no—
95#95—24no—
96C24C2×M1611yescharacteristic
97C22×C4—21no—
98C22×C4—21no—
99C22×C4—21no—
100C22×C4—21no—
101C22×C4—22no—
102C22×C4—22no—
103C22×C4—22no—
104C22×C4—44no—
105C22×C4—42no—
106C22×C4C2×C22⋊C411yescharacteristic
107C22×C4#11711yescharacteristic
108C22×C4#11711yescharacteristic
109C22×C4C22×C811yescharacteristic
110C22×C4C2×M1611yescharacteristic
111C22×C4C2×M1611yescharacteristic
112M16—42no—
113M16—42no—
114M16—42no—
115M16—42no—
116C2×C8—24no—
117C2×C8—24no—
118C2×C8—21no—
119C2×C8—21no—
120C2×C8—22no—
121C2×C8—22no—
122C2×C8—21no—
123C2×C8—21no—
124C2×C8—22no—
125C2×C8—22no—
126C2×C8—42no—
127C2×C8—41no—
128C2×C8—41no—
129C2×C8—41no—
130C2×C8—41no—
131C2×C8—42no—
132C2×C8#10012yes—
133C2×C8#10112yes—
134C2×C8#12314yes—
135C4⋊C4—22no—
136C4⋊C4—22no—
137C4⋊C4—42no—
138C4⋊C4—48no—
139C22⋊C4—22no—
140C22⋊C4—22no—
141C22⋊C4—42no—
142C42—22no—
143C42—42no—
144C42—41no—
145C42—41no—
146C42—41no—
147C42—41no—
148C42C22⋊C814yes—
149C16—84no—
150C23—21no—
151C23—21no—
152C23—42no—
153C23#40711yescharacteristic
154C23#43111yescharacteristic
155C23#43211yescharacteristic
156C2×C4—21no—
157C2×C4—21no—
158C2×C4—21no—
159C2×C4—21no—
160C2×C4—21no—
161C2×C4—21no—
162C2×C4—21no—
163C2×C4—21no—
164C2×C4—21no—
165C2×C4—21no—
166C2×C4—24no—
167C2×C4—24no—
168C2×C4—44no—
169C2×C4—42no—
170C2×C4—44no—
171C2×C4—42no—
172C2×C4—44no—
173C2×C4—44no—
174C2×C4—42no—
175C2×C4#38111yescharacteristic
176C2×C4C2×C22⋊C811yescharacteristic
177C2×C4#41011yescharacteristic
178C2×C4#41011yescharacteristic
179C2×C4C2×#10011yescharacteristic
180C2×C4C2×#10111yescharacteristic
181C2×C4#43111yescharacteristic
182C2×C4#43211yescharacteristic
183C8—24no—
184C8—22no—
185C8—22no—
186C8—42no—
187C8—42no—
188C22—22no—
189C22—21no—
190C22—21no—
191C22—41no—
192C22—41no—
193C22—42no—
194C22—42no—
195C22#146611yescharacteristic
196C22#146711yescharacteristic
197C22#181711yescharacteristic
198C4—21no—
199C4—21no—
200C4—44no—
201C4—44no—
202C4—82no—
203C4#146011yescharacteristic
204C4#146111yescharacteristic
205C4#182011yescharacteristic
206C4#182111yescharacteristic
207C2—21no—
208C2—21no—
209C2—41no—
210C2—41no—
211C2#870111yescharacteristic
212C2#1153712yes—
subgroupG/H
centerC2×C4#410
commutatorC22×C4C22×C8
Frattini, cosocleC22×M16C23
socleC22#1466
prime radicalC24C2×M16
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C22×C8 ◁ C2×#405 ◁ G
LECSG ▷ C22×M16 ▷ 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
#128502
#71942
#71912