Finite Group Databasebeta

← #302754

Subgroups — #302754

Order 512.

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

normal subgroups only

This group has 683 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#46007C211yesmaximal, characteristic
2#33293C212yesmaximal
3#29226C211yesmaximal, characteristic
4#10586C214yesmaximal
5#9890C212yesmaximal
6#8915C214yesmaximal
7C4×#330C211yesmaximal, characteristic
8#2868C2211yescharacteristic
9#2862C2212yes—
10#2845C2212yes—
11#2806—22no—
12#2755—24no—
13#2546—22no—
14#2451—22no—
15#2446—22no—
16#2382C2212yes—
17C4×#125C2211yescharacteristic
18#2099—22no—
19#1993—24no—
20#1992C2212yes—
21#1990—22no—
22#1990—22no—
23#1987—22no—
24#1836C2218yes—
25#1836C2212yes—
26#1767C2211yescharacteristic
27C4×#97C2212yes—
28C2×#330C2212yes—
29#1295C2212yes—
30#1295C2218yes—
31C2×C4⋊Q8—22no—
32C2×C4⋊Q8—21no—
33C2×C4⋊Q8—22no—
34C2×C4⋊Q8D412yes—
35C2×C4⋊Q8D412yes—
36C2×#125C2311yescharacteristic
37C2×C4×Q8—22no—
38C2×C4⋊C8—21no—
39C2×#97—22no—
40C2×#97C2312yes—
41C2×#97C2314yes—
42C2×C4×C8—22no—
43#400—24no—
44#398—24no—
45#398—24no—
46#398—24no—
47#398—24no—
48#395—24no—
49#391—22no—
50#391—24no—
51#391—22no—
52#391—22no—
53#391—22no—
54#391—22no—
55#389—22no—
56#389—24no—
57#389—24no—
58#389D412yes—
59#387—24no—
60#387—24no—
61#387—24no—
62#387—22no—
63#384—21no—
64#384—22no—
65#384C2312yes—
66#382D414yes—
67#382C2311yescharacteristic
68#382C2314yes—
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—22no—
79C22×Q8—22no—
80C22×Q8—41no—
81C22×C8—24no—
82C22×C8—22no—
83C4⋊Q8—44no—
84C4⋊Q8—44no—
85C4⋊Q8—48no—
86C4⋊Q8—44no—
87C4⋊Q8—84no—
88#125—22no—
89#125C2×Q814yes—
90C4×Q8—84no—
91C2×C4⋊C4—22no—
92C2×C4⋊C4—22no—
93C2×C4⋊C4—22no—
94C2×C4⋊C4—22no—
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—42no—
102C2×C4⋊C4—41no—
103C2×C4⋊C4C2×D412yes—
104C2×C4⋊C4C2×D414yes—
105C2×C4⋊C4C2×D414yes—
106C2×C4⋊C4C2×D412yes—
107C2×C42—22no—
108C2×C42—24no—
109C2×C42C2×D411yescharacteristic
110C2×C42C2×D412yes—
111C2×C42C2×D412yes—
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—24no—
126#95—24no—
127#95—22no—
128#95—24no—
129#95—24no—
130C2×Q8—44no—
131C2×Q8—44no—
132C2×Q8—44no—
133C2×Q8—44no—
134C2×Q8—82no—
135C2×Q8—82no—
136C2×Q8—82no—
137C2×Q8—84no—
138C22×C4—22no—
139C22×C4—24no—
140C22×C4—22no—
141C22×C4—24no—
142C22×C4—42no—
143C22×C4#12011yescharacteristic
144C22×C4#12012yes—
145C22×C4#12011yescharacteristic
146C22×C4C22×D411yescharacteristic
147C22×C4C22×D412yes—
148C2×C8—28no—
149C2×C8—44no—
150C2×C8—44no—
151C2×C8—42no—
152C2×C8—42no—
153C2×C8—42no—
154C4⋊C4—22no—
155C4⋊C4—28no—
156C4⋊C4—42no—
157C4⋊C4—44no—
158C4⋊C4—44no—
159C4⋊C4—44no—
160C4⋊C4—48no—
161C4⋊C4—44no—
162C4⋊C4—44no—
163C4⋊C4—82no—
164C4⋊C4—84no—
165C4⋊C4—84no—
166C4⋊C4—84no—
167C4⋊C4—84no—
168C4⋊C4—84no—
169C4⋊C4—82no—
170C4⋊C4C4⋊Q814yes—
171C42—24no—
172C42—22no—
173C42—24no—
174C42—41no—
175C42—44no—
176C42—84no—
177C42C22×Q812yes—
178C422−1+412yes—
179C23C2×#12011yescharacteristic
180Q8—88no—
181Q8—164no—
182Q8—164no—
183C2×C4—22no—
184C2×C4—22no—
185C2×C4—21no—
186C2×C4—22no—
187C2×C4—21no—
188C2×C4—22no—
189C2×C4—24no—
190C2×C4—21no—
191C2×C4—22no—
192C2×C4—42no—
193C2×C4—42no—
194C2×C4—42no—
195C2×C4—44no—
196C2×C4—44no—
197C2×C4—44no—
198C2×C4—42no—
199C2×C4—44no—
200C2×C4—44no—
201C2×C4—42no—
202C2×C4—44no—
203C2×C4—82no—
204C2×C4—84no—
205C2×C4#45414yes—
206C2×C4#45412yes—
207C2×C4#45514yes—
208C2×C4C2×C4⋊Q811yescharacteristic
209C2×C4#53611yescharacteristic
210C2×C4D4×Q814yes—
211C8—48no—
212C8—82no—
213C22#243411yescharacteristic
214C22C2×#45512yes—
215C22C2×#45411yescharacteristic
216C22C2×#45412yes—
217C22#302211yescharacteristic
218C4—24no—
219C4—21no—
220C4—21no—
221C4—24no—
222C4—41no—
223C4—44no—
224C4—84no—
225C4—82no—
226C4—84no—
227C4—162no—
228C4#166214yes—
229C2C2×#166211yescharacteristic
230C2#1902712yes—
231C2#1908311yescharacteristic
232C2#2582611yescharacteristic
233C2#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
#460072
#332932
#292262
#105862
#98902
#89152
C4×#3302