Finite Group Databasebeta

← #325150

Subgroups — #325150

Order 512.

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

normal subgroups only

This group has 538 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#29213C212yesmaximal
3C8×#125C211yesmaximal, characteristic
4#10656C214yesmaximal
5#9903C212yesmaximal
6#9616C214yesmaximal
7#8274C211yesmaximal, 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
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
34C2×#107—24no—
35C2×#106—24no—
36C2×C4⋊C8—22no—
37C2×C4⋊C8C2312yes—
38C2×C4⋊C8C2314yes—
39C2×#97—22no—
40C2×C4×C8—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#335Q818yescomplemented
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—22no—
76#125Q1614yes—
77C2×C4⋊C4—22no—
78C2×C4⋊C4—22no—
79C2×C4⋊C4—24no—
80C2×C4⋊C4—24no—
81C2×C4⋊C4—24no—
82C2×C4⋊C4—42no—
83C2×C4⋊C4—42no—
84C2×C4⋊C4—42no—
85C2×C4⋊C4C2×D412yes—
86C2×C4⋊C4C2×D412yes—
87C2×C42—22no—
88C2×C42C2×D411yescharacteristic
89C2×C42C2×D412yes—
90C2×C42C2×D411yescharacteristic
91C2×C42#4112yes—
92C2×C42C2411yescharacteristic
93#107—84no—
94#106—84no—
95C4⋊C8—24no—
96C4⋊C8—24no—
97C4⋊C8—44no—
98C4⋊C8C2×Q818yes—
99#97—44no—
100C4×C8—22no—
101C4×C8—42no—
102C4×C8—42no—
103C4×C8—44no—
104C4×C8C2×Q814yes—
105#95—22no—
106#95—22no—
107#95—22no—
108C2×Q8—84no—
109C2×Q8—84no—
110C2×Q8—84no—
111C22×C4—22no—
112C22×C4—22no—
113C22×C4—44no—
114C22×C4#12112yes—
115C22×C4#12112yes—
116C22×C4C22×D411yescharacteristic
117C22×C4C22×D411yescharacteristic
118C22×C4C2×#4111yescharacteristic
119C2×C8—22no—
120C2×C8—22no—
121C2×C8—22no—
122C2×C8—28no—
123C2×C8—22no—
124C2×C8—22no—
125C2×C8—44no—
126C2×C8—44no—
127C2×C8—42no—
128C2×C8—42no—
129C2×C8—42no—
130C2×C8C4⋊Q812yes—
131C2×C8C4⋊Q812yes—
132C4⋊C4—22no—
133C4⋊C4—28no—
134C4⋊C4—24no—
135C4⋊C4—44no—
136C4⋊C4—44no—
137C4⋊C4—44no—
138C4⋊C4—44no—
139C4⋊C4—44no—
140C4⋊C4—84no—
141C4⋊C4—84no—
142C4⋊C4—84no—
143C4⋊C4—84no—
144C4⋊C4C2×Q1614yes—
145C42—22no—
146C42—22no—
147C42—22no—
148C42—24no—
149C42—42no—
150C42—42no—
151C42C2×Q1612yes—
152C42#13512yes—
153C42C22×Q812yes—
154C422−1+412yes—
155C23C2×#12111yescharacteristic
156Q8—168no—
157C2×C4—22no—
158C2×C4—22no—
159C2×C4—22no—
160C2×C4—22no—
161C2×C4—22no—
162C2×C4—24no—
163C2×C4—42no—
164C2×C4—42no—
165C2×C4—42no—
166C2×C4—42no—
167C2×C4—42no—
168C2×C4—84no—
169C2×C4—84no—
170C2×C4—84no—
171C2×C4#46714yes—
172C2×C4#47114yes—
173C2×C4C2×C4⋊Q811yescharacteristic
174C2×C4#53611yescharacteristic
175C2×C4D4×Q812yes—
176C2×C4#55712yes—
177C2×C4C22×Q1611yescharacteristic
178C2×C4C2×#13511yescharacteristic
179C2×C4#57512yes—
180C2×C4#57512yes—
181C2×C4#57612yes—
182C8—22no—
183C8—22no—
184C8—48no—
185C8—42no—
186C8—82no—
187C22#254611yescharacteristic
188C22#304012yes—
189C22C2×#46711yescharacteristic
190C22#305811yescharacteristic
191C22C2×#47112yes—
192C4—22no—
193C4—24no—
194C4—24no—
195C4—44no—
196C4—82no—
197C4—164no—
198C4#316211yescharacteristic
199C4#316611yescharacteristic
200C4#324012yes—
201C4#338012yes—
202C4Q8×Q1612yes—
203C2#1927712yes—
204C2#1936412yes—
205C2#2867611yescharacteristic
206C2#2869511yescharacteristic
207C2#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. #335Q8semidirect

Maximal subgroups

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

subgroupindex
#332892
#292132
C8×#1252
#106562
#99032
#96162
#82742