Finite Group Databasebeta

← #325228

Subgroups — #325228

Order 512.

526 conjugacy classes of subgroups; only the 174 normal subgroups are shown.

normal subgroups only

This group has 526 conjugacy classes of subgroups — too many to draw. Only its 174 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#33289C211yesmaximal, characteristic
2#29213C212yesmaximal
3#21673C211yesmaximal, characteristic
4#10655C214yesmaximal
5#9903C212yesmaximal
6#9622C214yesmaximal
7#8273C211yesmaximal, characteristic
8#2868C2212yes—
9C4×#125C2211yescharacteristic
10#2058—22no—
11#2057—22no—
12#1992—22no—
13#1990—22no—
14#1990—22no—
15#1948C2214yes—
16#1947C2214yes—
17#1939C2212yes—
18#1916—22no—
19#1847—22no—
20#1834C2212yes—
21#1834C2212yes—
22#1831C2211yescharacteristic
23#1771—22no—
24#1770C2212yes—
25#1767C2212yes—
26C42×C8C2211yescharacteristic
27C2×#334C2212yes—
28#1296C2218yes—
29#1277C2212yes—
30C2×C4⋊Q8—22no—
31C2×C4⋊Q8—22no—
32C2×C4⋊Q8—22no—
33C2×#125D411yescharacteristic
34C2×#107—24no—
35C2×#106—24no—
36C2×C4⋊C8—22no—
37C2×C4⋊C8C2312yes—
38C2×C4⋊C8C2314yes—
39C2×#97—22no—
40C2×#97—22no—
41C2×C4×C8D412yes—
42C2×C4×C8D412yes—
43C2×C4×C8C2311yescharacteristic
44C2×C4×C8C2311yescharacteristic
45#389D412yes—
46#387—22no—
47#384—22no—
48#384—22no—
49#384—22no—
50#384C2314yes—
51#382D411yescharacteristic
52C4×C4⋊C4—22no—
53C4×C4⋊C4C2312yes—
54C43C2311yescharacteristic
55#340—24no—
56#340—22no—
57#340—22no—
58#340—24no—
59#340—24no—
60#336—22no—
61#336—22no—
62#336—22no—
63#336—22no—
64#334Q818yescomplemented
65C22×Q8—42no—
66C22×C8—24no—
67C22×C8—22no—
68C22×C8C2×D412yes—
69C22×C8C2×D411yescharacteristic
70C22×C8C2×D411yescharacteristic
71C4⋊Q8—44no—
72C4⋊Q8—44no—
73C4⋊Q8—84no—
74C4⋊Q8—84no—
75#125—24no—
76C2×C4⋊C4—22no—
77C2×C4⋊C4—22no—
78C2×C4⋊C4—24no—
79C2×C4⋊C4—24no—
80C2×C4⋊C4—24no—
81C2×C4⋊C4—42no—
82C2×C4⋊C4—42no—
83C2×C4⋊C4—42no—
84C2×C4⋊C4C2×D412yes—
85C2×C4⋊C4C2×D412yes—
86C2×C42—22no—
87C2×C42C2×D411yescharacteristic
88C2×C42C2×D412yes—
89C2×C42C2×D411yescharacteristic
90C2×C42#4112yes—
91C2×C42C2411yescharacteristic
92#107—84no—
93#106—84no—
94C4⋊C8—24no—
95C4⋊C8—24no—
96C4⋊C8—44no—
97C4⋊C8C2×Q818yes—
98#97—44no—
99#97—44no—
100C4×C8—22no—
101C4×C8—42no—
102C4×C8—42no—
103C4×C8C2×Q814yes—
104#95—22no—
105#95—22no—
106#95—22no—
107C2×Q8—84no—
108C2×Q8—84no—
109C2×Q8—84no—
110C22×C4—22no—
111C22×C4—22no—
112C22×C4—44no—
113C22×C4#12112yes—
114C22×C4#12112yes—
115C22×C4C22×D411yescharacteristic
116C22×C4C22×D411yescharacteristic
117C22×C4C2×#4111yescharacteristic
118C2×C8—22no—
119C2×C8—22no—
120C2×C8—22no—
121C2×C8—28no—
122C2×C8—22no—
123C2×C8—22no—
124C2×C8—44no—
125C2×C8—44no—
126C2×C8—42no—
127C2×C8—42no—
128C2×C8—42no—
129C2×C8C4⋊Q812yes—
130C2×C8C4⋊Q812yes—
131C4⋊C4—24no—
132C4⋊C4—24no—
133C4⋊C4—44no—
134C4⋊C4—44no—
135C4⋊C4—44no—
136C4⋊C4—44no—
137C4⋊C4—44no—
138C4⋊C4—44no—
139C4⋊C4—84no—
140C4⋊C4—84no—
141C4⋊C4—84no—
142C4⋊C4—84no—
143C42—22no—
144C42—22no—
145C42—22no—
146C42—24no—
147C42—42no—
148C42—42no—
149C42#13714yes—
150C42C22×Q812yes—
151C422−1+412yes—
152C23C2×#12111yescharacteristic
153Q8—168no—
154C2×C4—22no—
155C2×C4—22no—
156C2×C4—22no—
157C2×C4—22no—
158C2×C4—22no—
159C2×C4—42no—
160C2×C4—42no—
161C2×C4—42no—
162C2×C4—42no—
163C2×C4—42no—
164C2×C4—42no—
165C2×C4—84no—
166C2×C4—84no—
167C2×C4—84no—
168C2×C4#47014yes—
169C2×C4#47114yes—
170C2×C4C2×C4⋊Q811yescharacteristic
171C2×C4#53611yescharacteristic
172C2×C4D4×Q812yes—
173C2×C4#55712yes—
174C2×C4C2×#13712yes—
175C2×C4#57512yes—
176C2×C4#57512yes—
177C2×C4#57612yes—
178C8—22no—
179C8—22no—
180C8—48no—
181C8—42no—
182C8—82no—
183C22#254611yescharacteristic
184C22#304012yes—
185C22C2×#47011yescharacteristic
186C22#305811yescharacteristic
187C22C2×#47112yes—
188C4—22no—
189C4—24no—
190C4—24no—
191C4—82no—
192C4—82no—
193C4—164no—
194C4#316211yescharacteristic
195C4#316611yescharacteristic
196C4#324012yes—
197C4#338514yes—
198C2#1930212yes—
199C2#1936412yes—
200C2#2868211yescharacteristic
201C2#2869511yescharacteristic
202C2#2870011yescharacteristic
subgroupG/H
center, socle, prime radicalC23C2×#121
commutator, Frattini, cosocleC2×C42C24
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C43 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
64. #334Q8semidirect

Maximal subgroups

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

subgroupindex
#332892
#292132
#216732
#106552
#99032
#96222
#82732