Finite Group Databasebeta

← #302758

Subgroups — #302758

Order 512.

671 conjugacy classes of subgroups; only the 168 normal subgroups are shown.

normal subgroups only

This group has 671 conjugacy classes of subgroups — too many to draw. Only its 168 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#46153C211yesmaximal, characteristic
2#33294C212yesmaximal
3#29225C211yesmaximal, characteristic
4#10586C214yesmaximal
5#9891C212yesmaximal
6#8915C214yesmaximal
7C4×#330C211yesmaximal, characteristic
8#2868C2211yescharacteristic
9#2859C2212yes—
10#2845C2212yes—
11#2726—22no—
12#2726—24no—
13#2552—22no—
14#2507—22no—
15#2507—22no—
16#2382C2212yes—
17C4×#125C2211yescharacteristic
18#2100—22no—
19#1992—24no—
20#1992C2212yes—
21#1990—22no—
22#1990—22no—
23#1988—22no—
24#1836C2218yes—
25#1836C2212yes—
26#1767C2211yescharacteristic
27C4×#97C2212yes—
28C2×#330C2212yes—
29#1295C2212yes—
30#1295C2218yes—
31C2×C4⋊Q8—21no—
32C2×C4⋊Q8—22no—
33C2×#125—24no—
34C2×#125D414yes—
35C2×#125C2311yescharacteristic
36C2×C4⋊C8—21no—
37C2×#97—22no—
38C2×#97C2312yes—
39C2×#97C2314yes—
40C2×C4×C8—22no—
41#398—24no—
42#398—24no—
43#398—24no—
44#398—24no—
45#398—24no—
46#398—24no—
47#389—24no—
48#389—24no—
49#389—22no—
50#389—22no—
51#389—24no—
52#389—22no—
53#389—22no—
54#389—22no—
55#389—22no—
56#389D412yes—
57#387—24no—
58#387—24no—
59#387—22no—
60#384—22no—
61#384—21no—
62#384—22no—
63#384D412yes—
64#384D412yes—
65#384C2312yes—
66#382C2311yescharacteristic
67#382C2314yes—
68C4×C4⋊C4—22no—
69C4×C4⋊C4—22no—
70C4×C4⋊C4D412yes—
71C43C2311yescharacteristic
72#340—24no—
73#340—28no—
74#340—28no—
75#340—24no—
76#336—21no—
77#330Q818yescomplemented
78C22×Q8—41no—
79C22×C8—24no—
80C22×C8—22no—
81C4⋊Q8—48no—
82C4⋊Q8—44no—
83#125—22no—
84#125—48no—
85#125—88no—
86#125C2×Q814yes—
87C2×C4⋊C4—22no—
88C2×C4⋊C4—22no—
89C2×C4⋊C4—22no—
90C2×C4⋊C4—24no—
91C2×C4⋊C4—22no—
92C2×C4⋊C4—22no—
93C2×C4⋊C4—22no—
94C2×C4⋊C4—24no—
95C2×C4⋊C4—24no—
96C2×C4⋊C4—24no—
97C2×C4⋊C4—24no—
98C2×C4⋊C4—24no—
99C2×C4⋊C4—24no—
100C2×C4⋊C4—22no—
101C2×C4⋊C4—41no—
102C2×C4⋊C4—42no—
103C2×C4⋊C4C2×D412yes—
104C2×C4⋊C4C2×D414yes—
105C2×C4⋊C4C2×D414yes—
106C2×C4⋊C4C2×D412yes—
107C2×C42—22no—
108C2×C42—24no—
109C2×C42C2×D412yes—
110C2×C42C2×D412yes—
111C2×C42C2×D411yescharacteristic
112C2×C42C2×D411yescharacteristic
113C2×C42C2411yescharacteristic
114C4⋊C8—42no—
115#97—24no—
116#97—24no—
117#97—44no—
118#97C2×Q818yes—
119C4×C8—44no—
120#95—24no—
121#95—24no—
122#95—24no—
123#95—22no—
124#95—22no—
125#95—22no—
126#95—24no—
127#95—24no—
128C2×Q8—84no—
129C2×Q8—82no—
130C22×C4—22no—
131C22×C4—24no—
132C22×C4—22no—
133C22×C4—24no—
134C22×C4—42no—
135C22×C4#12011yescharacteristic
136C22×C4#12012yes—
137C22×C4#12011yescharacteristic
138C22×C4C22×D411yescharacteristic
139C22×C4C22×D412yes—
140C2×C8—28no—
141C2×C8—44no—
142C2×C8—44no—
143C2×C8—42no—
144C2×C8—42no—
145C2×C8—42no—
146C4⋊C4—22no—
147C4⋊C4—28no—
148C4⋊C4—42no—
149C4⋊C4—44no—
150C4⋊C4—44no—
151C4⋊C4—44no—
152C4⋊C4—48no—
153C4⋊C4—44no—
154C4⋊C4—44no—
155C4⋊C4—48no—
156C4⋊C4—44no—
157C4⋊C4—44no—
158C4⋊C4—82no—
159C4⋊C4—84no—
160C4⋊C4—84no—
161C4⋊C4—82no—
162C4⋊C4—84no—
163C4⋊C4—84no—
164C4⋊C4—84no—
165C4⋊C4—84no—
166C4⋊C4C4⋊Q814yes—
167C42—24no—
168C42—22no—
169C42—24no—
170C42—41no—
171C42—44no—
172C42—84no—
173C42C22×Q812yes—
174C422−1+412yes—
175C23C2×#12011yescharacteristic
176Q8—164no—
177C2×C4—22no—
178C2×C4—21no—
179C2×C4—22no—
180C2×C4—22no—
181C2×C4—21no—
182C2×C4—22no—
183C2×C4—24no—
184C2×C4—22no—
185C2×C4—21no—
186C2×C4—42no—
187C2×C4—42no—
188C2×C4—44no—
189C2×C4—44no—
190C2×C4—44no—
191C2×C4—42no—
192C2×C4—42no—
193C2×C4—44no—
194C2×C4—44no—
195C2×C4—42no—
196C2×C4—44no—
197C2×C4—84no—
198C2×C4—82no—
199C2×C4#45414yes—
200C2×C4#45412yes—
201C2×C4#45514yes—
202C2×C4C2×C4⋊Q811yescharacteristic
203C2×C4#53611yescharacteristic
204C2×C4D4×Q814yes—
205C8—48no—
206C8—82no—
207C22#243411yescharacteristic
208C22C2×#45512yes—
209C22C2×#45412yes—
210C22C2×#45411yescharacteristic
211C22#302211yescharacteristic
212C4—24no—
213C4—21no—
214C4—21no—
215C4—24no—
216C4—41no—
217C4—44no—
218C4—84no—
219C4—82no—
220C4—84no—
221C4—162no—
222C4#165814yes—
223C2C2×#165811yescharacteristic
224C2#1902312yes—
225C2#1908011yescharacteristic
226C2#2582611yescharacteristic
227C2#2582912yes—
subgroupG/H
center, socle, prime radicalC23C2×#120
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
77. #330Q8semidirect

Maximal subgroups

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

subgroupindex
#461532
#332942
#292252
#105862
#98912
#89152
C4×#3302