Finite Group Databasebeta

← #302745

Subgroups — #302745

Order 512.

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

normal subgroups only

This group has 743 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#33316C212yesmaximal
3#29226C211yesmaximal, characteristic
4#10585C214yesmaximal
5#9891C212yesmaximal
6#8913C214yesmaximal
7C4×#330C211yesmaximal, characteristic
8C2×#501—24no—
9#2868C2211yescharacteristic
10#2859C2212yes—
11#2845C2212yes—
12#2726—22no—
13#2726—24no—
14#2552—22no—
15#2507—22no—
16#2507—22no—
17#2382C2212yes—
18C4×#125C2211yescharacteristic
19#2099—22no—
20#1992C2212yes—
21#1987—22no—
22#1847C2218yes—
23#1845—22no—
24#1844—22no—
25#1836C2212yes—
26#1767C2211yescharacteristic
27C4×#97C2212yes—
28C2×#330C2212yes—
29#1295C2212yes—
30#1295C2218yes—
31C2×C4⋊Q8—22no—
32C2×C4⋊Q8—21no—
33C2×#125—24no—
34C2×#125D414yes—
35C2×#125C2311yescharacteristic
36#501—416no—
37C2×#107—28no—
38C2×#106—28no—
39C2×C4⋊C8—21no—
40C2×#97—22no—
41C2×#97C2312yes—
42C2×#97C2314yes—
43C2×C4×C8—22no—
44#398—24no—
45#398—24no—
46#398—24no—
47#398—24no—
48#398—24no—
49#398—24no—
50#389—24no—
51#389—24no—
52#389—22no—
53#389—22no—
54#389—24no—
55#389—22no—
56#389—22no—
57#389—22no—
58#389—22no—
59#389D412yes—
60#387—24no—
61#387—24no—
62#387—22no—
63#384—22no—
64#384—21no—
65#384—22no—
66#384D412yes—
67#384D412yes—
68#384C2312yes—
69#382C2311yescharacteristic
70#382C2314yes—
71C4×C4⋊C4—22no—
72C4×C4⋊C4—22no—
73C4×C4⋊C4D412yes—
74C43C2311yescharacteristic
75#340—24no—
76#340—24no—
77#336—21no—
78#330Q818yescomplemented
79C22×Q8—41no—
80C22×C8—24no—
81C22×C8—22no—
82C4⋊Q8—44no—
83C4⋊Q8—48no—
84#125—22no—
85#125—48no—
86#125—88no—
87#125C2×Q814yes—
88C2×C4⋊C4—22no—
89C2×C4⋊C4—22no—
90C2×C4⋊C4—22no—
91C2×C4⋊C4—24no—
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—24no—
101C2×C4⋊C4—22no—
102C2×C4⋊C4—42no—
103C2×C4⋊C4—41no—
104C2×C4⋊C4C2×D412yes—
105C2×C4⋊C4C2×D414yes—
106C2×C4⋊C4C2×D414yes—
107C2×C4⋊C4C2×D412yes—
108C2×C42—22no—
109C2×C42—24no—
110C2×C42C2×D412yes—
111C2×C42C2×D411yescharacteristic
112C2×C42C2×D412yes—
113C2×C42C2×D411yescharacteristic
114C2×C42C2411yescharacteristic
115#107—416no—
116#106—416no—
117C4⋊C8—42no—
118#97—28no—
119#97—24no—
120#97—24no—
121#97C2×Q818yes—
122C4×C8—28no—
123#95—24no—
124#95—24no—
125#95—24no—
126#95—22no—
127#95—22no—
128#95—22no—
129#95—24no—
130#95—24no—
131C2×Q8—84no—
132C2×Q8—82no—
133C22×C4—22no—
134C22×C4—24no—
135C22×C4—22no—
136C22×C4—24no—
137C22×C4—42no—
138C22×C4#12011yescharacteristic
139C22×C4#12012yes—
140C22×C4#12011yescharacteristic
141C22×C4C22×D411yescharacteristic
142C22×C4C22×D412yes—
143C2×C8—28no—
144C2×C8—28no—
145C2×C8—28no—
146C2×C8—42no—
147C2×C8—42no—
148C2×C8—42no—
149C4⋊C4—22no—
150C4⋊C4—28no—
151C4⋊C4—44no—
152C4⋊C4—44no—
153C4⋊C4—42no—
154C4⋊C4—48no—
155C4⋊C4—44no—
156C4⋊C4—48no—
157C4⋊C4—44no—
158C4⋊C4—44no—
159C4⋊C4—44no—
160C4⋊C4—44no—
161C4⋊C4—82no—
162C4⋊C4—84no—
163C4⋊C4—84no—
164C4⋊C4—84no—
165C4⋊C4—84no—
166C4⋊C4—84no—
167C4⋊C4—84no—
168C4⋊C4—82no—
169C4⋊C4C4⋊Q814yes—
170C42—24no—
171C42—22no—
172C42—24no—
173C42—41no—
174C42—44no—
175C42—84no—
176C42C22×Q812yes—
177C422−1+412yes—
178C23C2×#12011yescharacteristic
179Q8—164no—
180C2×C4—22no—
181C2×C4—21no—
182C2×C4—22no—
183C2×C4—22no—
184C2×C4—21no—
185C2×C4—22no—
186C2×C4—24no—
187C2×C4—22no—
188C2×C4—21no—
189C2×C4—42no—
190C2×C4—42no—
191C2×C4—44no—
192C2×C4—44no—
193C2×C4—44no—
194C2×C4—42no—
195C2×C4—44no—
196C2×C4—42no—
197C2×C4—44no—
198C2×C4—42no—
199C2×C4—44no—
200C2×C4—84no—
201C2×C4—82no—
202C2×C4#45314yes—
203C2×C4#45412yes—
204C2×C4#45614yes—
205C2×C4C2×C4⋊Q811yescharacteristic
206C2×C4#53611yescharacteristic
207C2×C4D4×Q814yes—
208C8—216no—
209C8—82no—
210C22#243411yescharacteristic
211C22C2×#45312yes—
212C22C2×#45411yescharacteristic
213C22C2×#45612yes—
214C22#302211yescharacteristic
215C4—24no—
216C4—21no—
217C4—21no—
218C4—24no—
219C4—41no—
220C4—44no—
221C4—84no—
222C4—82no—
223C4—84no—
224C4—162no—
225C4#166314yes—
226C2C2×#166311yescharacteristic
227C2#1901712yes—
228C2#1907511yescharacteristic
229C2#2582611yescharacteristic
230C2#2583012yes—
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
78. #330Q8semidirect

Maximal subgroups

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

subgroupindex
#461532
#333162
#292262
#105852
#98912
#89132
C4×#3302