Finite Group Databasebeta

← #511719

Subgroups — #511719

Order 512.

378 conjugacy classes of subgroups; only the 130 normal subgroups are shown.

normal subgroups only

This group has 378 conjugacy classes of subgroups — too many to draw. Only its 130 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#33257C212yesmaximal
2#13539C212yesmaximal
3#13539C212yesmaximal
4#13516C212yesmaximal
5#13515C211yesmaximal, characteristic
6#13514C211yesmaximal, characteristic
7#13507C212yesmaximal
8#13507C212yesmaximal
9#12879C211yesmaximal, characteristic
10#2864—22no—
11C2×#367C2212yes—
12C2×#366C2211yescharacteristic
13C2×#366C2211yescharacteristic
14C2×C4⋊C16C2211yescharacteristic
15C2×C4⋊C16C2211yescharacteristic
16#1942C2212yes—
17#1942C2212yes—
18#1942C2212yes—
19#1942C2212yes—
20#1941—22no—
21#1941—22no—
22#1846C2212yes—
23#1846C2212yes—
24#1835—22no—
25C42×C8C2211yescharacteristic
26#1380C2212yes—
27#1380C2212yes—
28#1380C2212yes—
29#1380C2212yes—
30#1380C2211yescharacteristic
31#1380C2211yescharacteristic
32#1380C2212yes—
33#1374C2212yes—
34#1374C2212yes—
35C22×C16C2311yescharacteristic
36C22×C16C2311yescharacteristic
37C22×C16C2312yes—
38C2×#107C2312yes—
39C2×#107C2312yes—
40C2×#107C2312yes—
41C2×#107C2312yes—
42C2×#106—22no—
43C2×#106—22no—
44C2×C4×C8D412yes—
45C2×C4×C8D411yescharacteristic
46C2×C4×C8C2312yes—
47C2×C4×C8C2311yescharacteristic
48#389—22no—
49#389—22no—
50#389—22no—
51#389—22no—
52#387—42no—
53#387—42no—
54#387—42no—
55#387—42no—
56#382—22no—
57#382—22no—
58C43D411yescharacteristic
59#367—42no—
60#366—41no—
61#366—41no—
62C4⋊C16—41no—
63C4⋊C16—41no—
64#340—22no—
65#340—22no—
66#340—22no—
67#340—22no—
68C22×C8C2×D411yescharacteristic
69C22×C8C2×D412yes—
70C22×C8C2411yescharacteristic
71C2×C4⋊C4—22no—
72C2×C4⋊C4—22no—
73C2×C4⋊C4—22no—
74C2×C4⋊C4—22no—
75C2×C4⋊C4—42no—
76C2×C4⋊C4—42no—
77C2×C4⋊C4—42no—
78C2×C4⋊C4—42no—
79C2×C42D811yescharacteristic
80C2×C42D812yes—
81C2×C42D811yescharacteristic
82C2×C42C2×D412yes—
83C2×C42C2×D411yescharacteristic
84C2×C16—21no—
85C2×C16—21no—
86C2×C16—22no—
87C2×C16—22no—
88C2×C16—22no—
89C2×C16—24no—
90#107—22no—
91#107—22no—
92#107—22no—
93#107—22no—
94#107—22no—
95#107—22no—
96#107—22no—
97#107—22no—
98#106—44no—
99#106—44no—
100C4×C8—24no—
101C4×C8—21no—
102C4×C8—22no—
103C4×C8—22no—
104C4×C8—21no—
105C4×C8—41no—
106#95—42no—
107#95—42no—
108#95—42no—
109#95—42no—
110#95—42no—
111#95—42no—
112#95—42no—
113#95—42no—
114C22×C4—42no—
115C22×C4—42no—
116C22×C4—42no—
117C22×C4—42no—
118C22×C4C2×D812yes—
119C22×C4C2×D812yes—
120C22×C4C2×D811yescharacteristic
121C22×C4C2×D811yescharacteristic
122C22×C4C22×D411yescharacteristic
123C2×C8—22no—
124C2×C8—22no—
125C2×C8—21no—
126C2×C8—22no—
127C2×C8#13714yes—
128C2×C8H(5,𝔽2)11yescharacteristic
129C2×C82−1+412yes—
130C2×C82−1+412yes—
131C2×C82−1+411yescharacteristic
132C4⋊C4—42no—
133C4⋊C4—42no—
134C4⋊C4—42no—
135C4⋊C4—42no—
136C4⋊C4—42no—
137C4⋊C4—42no—
138C4⋊C4—42no—
139C4⋊C4—42no—
140C4⋊C4—84no—
141C4⋊C4—84no—
142C4⋊C4—84no—
143C4⋊C4—84no—
144C42—22no—
145C42—21no—
146C42—22no—
147C42—21no—
148C42—22no—
149C42—41no—
150C42—42no—
151C16—41no—
152C16—41no—
153C16—42no—
154C23C22×D811yescharacteristic
155C2×C4—21no—
156C2×C4—21no—
157C2×C4—22no—
158C2×C4—22no—
159C2×C4—22no—
160C2×C4—22no—
161C2×C4—22no—
162C2×C4—22no—
163C2×C4—22no—
164C2×C4—21no—
165C2×C4—82no—
166C2×C4—82no—
167C2×C4—82no—
168C2×C4—82no—
169C2×C4—82no—
170C2×C4—82no—
171C2×C4—82no—
172C2×C4—82no—
173C2×C4—82no—
174C2×C4—82no—
175C2×C4—82no—
176C2×C4—82no—
177C2×C4#50812yes—
178C2×C4#53612yes—
179C2×C4#53711yescharacteristic
180C2×C4C2×#13712yes—
181C2×C4#57511yescharacteristic
182C8—24no—
183C8—41no—
184C8#56311yescharacteristic
185C8#56311yescharacteristic
186C8#56412yes—
187C22#322612yes—
188C22#324511yescharacteristic
189C22C2×#50811yescharacteristic
190C22#341412yes—
191C22#341411yescharacteristic
192C4—22no—
193C4—42no—
194C4—42no—
195C4—41no—
196C4—162no—
197C4—162no—
198C4—162no—
199C4—162no—
200C4#282211yescharacteristic
201C4#323612yes—
202C4#325511yescharacteristic
203C2#3284811yescharacteristic
204C2#3379912yes—
205C2#3380612yes—
206C2#3381611yescharacteristic
207C2#3382211yescharacteristic
subgroupG/H
center, socle, prime radicalC23C22×D8
commutator, Frattini, cosocleC22×C8C24
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C43 ◁ C42×C8 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Maximal subgroups

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

subgroupindex
#332572
#135392
#135162
#135152
#135142
#135072
#128792