Finite Group Databasebeta

← #302736

Subgroups — #302736

Order 512.

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

normal subgroups only

This group has 755 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#33314C212yesmaximal
3#29225C211yesmaximal, characteristic
4#10585C214yesmaximal
5#9890C212yesmaximal
6#8913C214yesmaximal
7C4×#330C211yesmaximal, characteristic
8C2×#500—22no—
9C2×#498—22no—
10#2868C2211yescharacteristic
11#2862C2212yes—
12#2845C2212yes—
13#2806—22no—
14#2755—24no—
15#2546—22no—
16#2451—22no—
17#2446—22no—
18#2382C2212yes—
19C4×#125C2211yescharacteristic
20#2100—22no—
21#1992C2212yes—
22#1988—22no—
23#1847—24no—
24#1847C2218yes—
25#1836C2212yes—
26#1767C2211yescharacteristic
27C4×#97C2212yes—
28C2×#330C2212yes—
29#1295C2212yes—
30#1295C2218yes—
31C2×C4⋊Q8—22no—
32C2×C4⋊Q8—22no—
33C2×C4⋊Q8—21no—
34C2×C4⋊Q8D412yes—
35C2×C4⋊Q8D412yes—
36C2×#125C2311yescharacteristic
37C2×C4×Q8—22no—
38#500—48no—
39#498—48no—
40C2×#107—28no—
41C2×#106—28no—
42C2×C4⋊C8—21no—
43C2×#97—22no—
44C2×#97C2312yes—
45C2×#97C2314yes—
46C2×C4×C8—22no—
47#400—24no—
48#398—24no—
49#398—24no—
50#398—24no—
51#398—24no—
52#395—24no—
53#391—22no—
54#391—24no—
55#391—22no—
56#391—22no—
57#391—22no—
58#391—22no—
59#389—22no—
60#389—24no—
61#389—24no—
62#389D412yes—
63#387—24no—
64#387—24no—
65#387—24no—
66#387—22no—
67#384—21no—
68#384—22no—
69#384C2312yes—
70#382D414yes—
71#382C2311yescharacteristic
72#382C2314yes—
73C4×C4⋊C4—22no—
74C4×C4⋊C4D412yes—
75C43C2311yescharacteristic
76#340—24no—
77#340—24no—
78#336—21no—
79#330Q818yescomplemented
80C22×Q8—22no—
81C22×Q8—22no—
82C22×Q8—41no—
83C22×C8—24no—
84C22×C8—22no—
85C4⋊Q8—44no—
86C4⋊Q8—44no—
87C4⋊Q8—44no—
88C4⋊Q8—44no—
89C4⋊Q8—44no—
90C4⋊Q8—84no—
91#125—22no—
92#125C2×Q814yes—
93C4×Q8—84no—
94C2×C4⋊C4—22no—
95C2×C4⋊C4—22no—
96C2×C4⋊C4—22no—
97C2×C4⋊C4—22no—
98C2×C4⋊C4—24no—
99C2×C4⋊C4—24no—
100C2×C4⋊C4—24no—
101C2×C4⋊C4—24no—
102C2×C4⋊C4—24no—
103C2×C4⋊C4—22no—
104C2×C4⋊C4—41no—
105C2×C4⋊C4—42no—
106C2×C4⋊C4C2×D412yes—
107C2×C4⋊C4C2×D414yes—
108C2×C4⋊C4C2×D414yes—
109C2×C4⋊C4C2×D412yes—
110C2×C42—22no—
111C2×C42—24no—
112C2×C42C2×D412yes—
113C2×C42C2×D411yescharacteristic
114C2×C42C2×D412yes—
115C2×C42C2×D411yescharacteristic
116C2×C42C2411yescharacteristic
117#107—416no—
118#106—416no—
119C4⋊C8—42no—
120#97—28no—
121#97—24no—
122#97—24no—
123#97C2×Q818yes—
124C4×C8—28no—
125#95—24no—
126#95—24no—
127#95—24no—
128#95—22no—
129#95—22no—
130#95—24no—
131#95—24no—
132#95—22no—
133#95—24no—
134#95—24no—
135C2×Q8—44no—
136C2×Q8—44no—
137C2×Q8—44no—
138C2×Q8—44no—
139C2×Q8—82no—
140C2×Q8—82no—
141C2×Q8—84no—
142C2×Q8—82no—
143C22×C4—22no—
144C22×C4—24no—
145C22×C4—22no—
146C22×C4—24no—
147C22×C4—42no—
148C22×C4#12011yescharacteristic
149C22×C4#12012yes—
150C22×C4#12011yescharacteristic
151C22×C4C22×D411yescharacteristic
152C22×C4C22×D412yes—
153C2×C8—28no—
154C2×C8—28no—
155C2×C8—28no—
156C2×C8—42no—
157C2×C8—42no—
158C2×C8—42no—
159C4⋊C4—22no—
160C4⋊C4—28no—
161C4⋊C4—44no—
162C4⋊C4—44no—
163C4⋊C4—42no—
164C4⋊C4—44no—
165C4⋊C4—44no—
166C4⋊C4—44no—
167C4⋊C4—44no—
168C4⋊C4—44no—
169C4⋊C4—82no—
170C4⋊C4—84no—
171C4⋊C4—82no—
172C4⋊C4—84no—
173C4⋊C4—84no—
174C4⋊C4—84no—
175C4⋊C4—84no—
176C4⋊C4C4⋊Q814yes—
177C42—24no—
178C42—22no—
179C42—24no—
180C42—41no—
181C42—44no—
182C42—84no—
183C42C22×Q812yes—
184C422−1+412yes—
185C23C2×#12011yescharacteristic
186Q8—88no—
187Q8—164no—
188Q8—164no—
189C2×C4—22no—
190C2×C4—22no—
191C2×C4—21no—
192C2×C4—22no—
193C2×C4—21no—
194C2×C4—22no—
195C2×C4—24no—
196C2×C4—21no—
197C2×C4—22no—
198C2×C4—42no—
199C2×C4—42no—
200C2×C4—42no—
201C2×C4—44no—
202C2×C4—44no—
203C2×C4—44no—
204C2×C4—44no—
205C2×C4—42no—
206C2×C4—44no—
207C2×C4—42no—
208C2×C4—44no—
209C2×C4—84no—
210C2×C4—82no—
211C2×C4#45314yes—
212C2×C4#45412yes—
213C2×C4#45614yes—
214C2×C4C2×C4⋊Q811yescharacteristic
215C2×C4#53611yescharacteristic
216C2×C4D4×Q814yes—
217C8—216no—
218C8—82no—
219C22#243411yescharacteristic
220C22C2×#45312yes—
221C22C2×#45411yescharacteristic
222C22C2×#45612yes—
223C22#302211yescharacteristic
224C4—24no—
225C4—21no—
226C4—21no—
227C4—24no—
228C4—41no—
229C4—44no—
230C4—84no—
231C4—82no—
232C4—84no—
233C4—162no—
234C4#165614yes—
235C2C2×#165611yescharacteristic
236C2#1903312yes—
237C2#1908411yescharacteristic
238C2#2582611yescharacteristic
239C2#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
79. #330Q8semidirect

Maximal subgroups

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

subgroupindex
#460072
#333142
#292252
#105852
#98902
#89132
C4×#3302