Finite Group Databasebeta

← #325146

Subgroups — #325146

Order 512.

558 conjugacy classes of subgroups; only the 182 normal subgroups are shown.

normal subgroups only

This group has 558 conjugacy classes of subgroups — too many to draw. Only its 182 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#28694C212yesmaximal
3C8×#125C211yesmaximal, characteristic
4#10655C214yesmaximal
5#9905C212yesmaximal
6#9616C214yesmaximal
7#8274C211yesmaximal, characteristic
8C2×#501—22no—
9#2868C2212yes—
10C4×#125C2211yescharacteristic
11#2084—22no—
12#2083—22no—
13#1993—22no—
14#1990—22no—
15#1990—22no—
16#1948C2214yes—
17#1947C2214yes—
18#1939C2212yes—
19#1916—22no—
20#1834C2212yes—
21#1834C2212yes—
22#1831C2211yescharacteristic
23C8×C4⋊C4—22no—
24C8×C4⋊C4C2212yes—
25#1767C2212yes—
26C42×C8C2211yescharacteristic
27C2×#335C2212yes—
28#1296C2218yes—
29#1277C2212yes—
30C2×C4⋊Q8—22no—
31C2×C4⋊Q8—22no—
32C2×C4⋊Q8—22no—
33C2×#125D411yescharacteristic
34#501—48no—
35C2×#107—24no—
36C2×#106—24no—
37C2×C4⋊C8—22no—
38C2×C4⋊C8C2312yes—
39C2×C4⋊C8C2314yes—
40C2×#97—22no—
41C2×C4×C8—22no—
42C2×C4×C8D412yes—
43C2×C4×C8D412yes—
44C2×C4×C8C2311yescharacteristic
45C2×C4×C8C2311yescharacteristic
46#389D412yes—
47#387—22no—
48#384—22no—
49#384—22no—
50#384—22no—
51#384C2314yes—
52#382D411yescharacteristic
53C4×C4⋊C4—22no—
54C4×C4⋊C4C2312yes—
55C43C2311yescharacteristic
56#340—24no—
57#340—22no—
58#340—22no—
59#340—24no—
60#340—24no—
61#336—22no—
62#336—22no—
63#336—22no—
64#336—22no—
65#335Q818yescomplemented
66C22×Q8—42no—
67C22×C8—24no—
68C22×C8—22no—
69C22×C8C2×D412yes—
70C22×C8C2×D411yescharacteristic
71C22×C8C2×D411yescharacteristic
72C4⋊Q8—44no—
73C4⋊Q8—44no—
74C4⋊Q8—44no—
75C4⋊Q8—44no—
76C4⋊Q8—84no—
77#125—22no—
78#125D814yes—
79C2×C4⋊C4—22no—
80C2×C4⋊C4—22no—
81C2×C4⋊C4—24no—
82C2×C4⋊C4—24no—
83C2×C4⋊C4—24no—
84C2×C4⋊C4—42no—
85C2×C4⋊C4—42no—
86C2×C4⋊C4—42no—
87C2×C4⋊C4C2×D412yes—
88C2×C4⋊C4C2×D412yes—
89C2×C42—22no—
90C2×C42C2×D412yes—
91C2×C42C2×D411yescharacteristic
92C2×C42C2×D411yescharacteristic
93C2×C42#4112yes—
94C2×C42C2411yescharacteristic
95#107—48no—
96#106—48no—
97C4⋊C8—24no—
98C4⋊C8—24no—
99C4⋊C8—44no—
100C4⋊C8C2×Q818yes—
101#97—44no—
102C4×C8—22no—
103C4×C8—42no—
104C4×C8—42no—
105C4×C8—44no—
106C4×C8C2×Q814yes—
107#95—22no—
108#95—22no—
109#95—22no—
110C2×Q8—84no—
111C2×Q8—84no—
112C2×Q8—84no—
113C22×C4—22no—
114C22×C4—22no—
115C22×C4—44no—
116C22×C4#12112yes—
117C22×C4#12112yes—
118C22×C4C22×D411yescharacteristic
119C22×C4C22×D411yescharacteristic
120C22×C4C2×#4111yescharacteristic
121C2×C8—22no—
122C2×C8—22no—
123C2×C8—22no—
124C2×C8—28no—
125C2×C8—22no—
126C2×C8—22no—
127C2×C8—44no—
128C2×C8—44no—
129C2×C8—42no—
130C2×C8—42no—
131C2×C8—42no—
132C2×C8C4⋊Q812yes—
133C2×C8C4⋊Q812yes—
134C4⋊C4—22no—
135C4⋊C4—28no—
136C4⋊C4—24no—
137C4⋊C4—44no—
138C4⋊C4—44no—
139C4⋊C4—44no—
140C4⋊C4—44no—
141C4⋊C4—44no—
142C4⋊C4—84no—
143C4⋊C4—84no—
144C4⋊C4—84no—
145C4⋊C4—84no—
146C4⋊C4C2×D814yes—
147C42—22no—
148C42—22no—
149C42—22no—
150C42—24no—
151C42—42no—
152C42—42no—
153C42C2×D812yes—
154C42#13512yes—
155C42C22×Q812yes—
156C422−1+412yes—
157C23C2×#12111yescharacteristic
158Q8—168no—
159C2×C4—22no—
160C2×C4—22no—
161C2×C4—22no—
162C2×C4—22no—
163C2×C4—22no—
164C2×C4—24no—
165C2×C4—42no—
166C2×C4—42no—
167C2×C4—42no—
168C2×C4—42no—
169C2×C4—42no—
170C2×C4—84no—
171C2×C4—84no—
172C2×C4—84no—
173C2×C4#46614yes—
174C2×C4#47114yes—
175C2×C4C2×C4⋊Q811yescharacteristic
176C2×C4#53611yescharacteristic
177C2×C4D4×Q812yes—
178C2×C4#55712yes—
179C2×C4C22×D811yescharacteristic
180C2×C4C2×#13511yescharacteristic
181C2×C4#57512yes—
182C2×C4#57512yes—
183C2×C4#57812yes—
184C8—22no—
185C8—22no—
186C8—48no—
187C8—42no—
188C8—82no—
189C22#254611yescharacteristic
190C22#304012yes—
191C22C2×#46611yescharacteristic
192C22#305911yescharacteristic
193C22C2×#47112yes—
194C4—22no—
195C4—24no—
196C4—24no—
197C4—44no—
198C4—82no—
199C4—164no—
200C4#316211yescharacteristic
201C4#316611yescharacteristic
202C4#324012yes—
203C4Q8×D812yes—
204C4#338312yes—
205C2#1927612yes—
206C2#1936512yes—
207C2#2867511yescharacteristic
208C2#2869511yescharacteristic
209C2#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
65. #335Q8semidirect

Maximal subgroups

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

subgroupindex
#332892
#286942
C8×#1252
#106552
#99052
#96162
#82742