Finite Group Databasebeta

← #499703

Subgroups — #499703

Order 512.

460 conjugacy classes of subgroups; only the 170 normal subgroups are shown.

normal subgroups only

This group has 460 conjugacy classes of subgroups — too many to draw. Only its 170 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#21723C212yesmaximal
2#13506C212yesmaximal
3#13505C212yesmaximal
4#13453C212yesmaximal
5#13453C212yesmaximal
6#12674C212yesmaximal
7C4×#367C211yesmaximal, characteristic
8C4×#366C211yesmaximal, characteristic
9#12592C211yesmaximal, characteristic
10C2×#499C2212yes—
11#2330—22no—
12C2×#367C2212yes—
13C2×#367C2212yes—
14C2×#366C2212yes—
15C2×#366C2212yes—
16C2×#346C2211yescharacteristic
17C2×C4×C16C2211yescharacteristic
18#1992—22no—
19#1944—22no—
20#1942C2212yes—
21#1942C2212yes—
22#1845C2212yes—
23#1776—22no—
24C4×#107C2212yes—
25C4×#107C2212yes—
26#1726C2211yescharacteristic
27#1380C2211yescharacteristic
28#1380C2211yescharacteristic
29#1380C2211yescharacteristic
30#1380C2211yescharacteristic
31#1380C2212yes—
32#1380C2212yes—
33#1374C2212yes—
34#1374C2212yes—
35C2×#125—22no—
36C22×C16C2311yescharacteristic
37C22×C16C2311yescharacteristic
38C22×C16C2312yes—
39#499—24no—
40#499—24no—
41C2×#107—22no—
42C2×#107C2312yes—
43C2×#107C2312yes—
44C2×#107C2312yes—
45C2×#107C2312yes—
46C2×#106—22no—
47C2×#97D412yes—
48C2×#97D411yescharacteristic
49C2×C4×C8C2312yes—
50C2×C4×C8C2311yescharacteristic
51#389—22no—
52#389—22no—
53#389—42no—
54#389—42no—
55#387—42no—
56#387—42no—
57#384—22no—
58C4×C4⋊C4—22no—
59C4×C4⋊C4—22no—
60#376D411yescharacteristic
61#367—24no—
62#367—24no—
63#366—24no—
64#366—24no—
65#346Q814yes—
66C4×C16Q814yes—
67#340—22no—
68#340—22no—
69#340—22no—
70#340—22no—
71C22×C8C2×D411yescharacteristic
72C22×C8C2×D412yes—
73C22×C8C2411yescharacteristic
74#125—28no—
75#125—44no—
76C2×C4⋊C4—22no—
77C2×C4⋊C4—22no—
78C2×C4⋊C4—22no—
79C2×C4⋊C4—22no—
80C2×C4⋊C4—42no—
81C2×C4⋊C4—42no—
82C2×C4⋊C4—42no—
83C2×C4⋊C4—42no—
84C2×C4⋊C4—42no—
85C2×C4⋊C4—42no—
86C2×C42—42no—
87C2×C42—42no—
88C2×C42C2×D412yes—
89C2×C42C2×D411yescharacteristic
90C2×C16—22no—
91C2×C16—22no—
92C2×C16—22no—
93C2×C16—22no—
94C2×C16C2×Q812yes—
95C2×C16C2×Q812yes—
96C2×C16C2×Q814yes—
97#107—22no—
98#107—22no—
99#107—22no—
100#107—22no—
101#107—22no—
102#107—22no—
103#107—44no—
104#107#4114yes—
105#107#4114yes—
106#106—28no—
107#97—42no—
108#97Q1614yes—
109C4×C8—22no—
110C4×C8—22no—
111C4×C8C2×Q812yes—
112C4×C8C2×Q812yes—
113#95—42no—
114#95—42no—
115#95—42no—
116#95—42no—
117#95D811yescharacteristic
118#95D812yes—
119#95D811yescharacteristic
120C22×C4—42no—
121C22×C4—42no—
122C22×C4—42no—
123C22×C4—42no—
124C22×C4C2×D812yes—
125C22×C4C2×D812yes—
126C22×C4C2×D811yescharacteristic
127C22×C4C2×D811yescharacteristic
128C22×C4C22×D411yescharacteristic
129C2×C8—22no—
130C2×C8—22no—
131C2×C8—22no—
132C2×C8C4⋊Q812yes—
133C2×C8C2×Q1614yes—
134C2×C8C22×Q811yescharacteristic
135C2×C8C22×Q811yescharacteristic
136C2×C8C2×#4112yes—
137C2×C8C2×#4112yes—
138C4⋊C4—24no—
139C4⋊C4—24no—
140C4⋊C4—48no—
141C4⋊C4—48no—
142C4⋊C4—42no—
143C4⋊C4—42no—
144C4⋊C4—42no—
145C4⋊C4—42no—
146C4⋊C4—42no—
147C4⋊C4—42no—
148C4⋊C4—162no—
149C4⋊C4—162no—
150C4⋊C4—162no—
151C4⋊C4—162no—
152C42—22no—
153C42—22no—
154C42—162no—
155C42—162no—
156C42C4⋊Q812yes—
157C42C2×Q1612yes—
158C16—22no—
159C16—24no—
160C16—22no—
161C23C22×D811yescharacteristic
162C2×C4—21no—
163C2×C4—22no—
164C2×C4—21no—
165C2×C4—22no—
166C2×C4—22no—
167C2×C4—22no—
168C2×C4—22no—
169C2×C4—22no—
170C2×C4—22no—
171C2×C4—44no—
172C2×C4—44no—
173C2×C4—82no—
174C2×C4—82no—
175C2×C4—82no—
176C2×C4—82no—
177C2×C4—82no—
178C2×C4—82no—
179C2×C4—82no—
180C2×C4—82no—
181C2×C4—82no—
182C2×C4—82no—
183C2×C4#50012yes—
184C2×C4#50012yes—
185C2×C4#52512yes—
186C2×C4C2×C4⋊Q811yescharacteristic
187C2×C4C22×Q1611yescharacteristic
188C2×C4#57512yes—
189C8—24no—
190C8—22no—
191C8#53311yescharacteristic
192C8#53312yes—
193C8#53311yescharacteristic
194C22#310012yes—
195C22C2×#50011yescharacteristic
196C22#341511yescharacteristic
197C22#341612yes—
198C22#341711yescharacteristic
199C4—22no—
200C4—22no—
201C4—42no—
202C4—42no—
203C4—84no—
204C4—84no—
205C4—162no—
206C4—162no—
207C4#239911yescharacteristic
208C4#310412yes—
209C4#316211yescharacteristic
210C2#2383511yescharacteristic
211C2#3367912yes—
212C2#3368012yes—
213C2#3372911yescharacteristic
214C2#3373011yescharacteristic
subgroupG/H
center, socle, prime radicalC23C22×D8
commutator, Frattini, cosocleC22×C8C24
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ #376 ◁ #1726 ◁ 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
#217232
#135062
#135052
#134532
#126742
C4×#3672
C4×#3662
#125922