Finite Group Databasebeta

← #305052

Subgroups — #305052

Order 512.

779 conjugacy classes of subgroups; only the 176 normal subgroups are shown.

normal subgroups only

This group has 779 conjugacy classes of subgroups — too many to draw. Only its 176 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#59869C211yesmaximal, characteristic
2#29218C212yesmaximal
3#28440C211yesmaximal, characteristic
4#10352C214yesmaximal
5#9865C212yesmaximal
6#9066C214yesmaximal
7C4×#333C211yesmaximal, characteristic
8C2×#494C2212yes—
9C2×#462—22no—
10#2868C2211yescharacteristic
11#2809—22no—
12#2571C2212yes—
13#2568—22no—
14#2546—22no—
15#2508—24no—
16#2369C2212yes—
17#2355—22no—
18C4×C4⋊Q8C2212yes—
19C4×C4⋊Q8C2211yescharacteristic
20#2100—24no—
21#2016—22no—
22#1990—22no—
23#1987C2218yes—
24#1956C2212yes—
25#1834—22no—
26C4×C4⋊C8C2212yes—
27C4×C4⋊C8C2211yescharacteristic
28C2×#333C2212yes—
29#1296C2212yes—
30#1279C2218yes—
31C22×Q16—24no—
32C2×C4⋊Q8—22no—
33C2×C4⋊Q8—21no—
34C2×C4⋊Q8—21no—
35C2×C4⋊Q8D412yes—
36C2×C4⋊Q8C2312yes—
37C2×C4⋊Q8C2311yescharacteristic
38C2×C4×Q8—22no—
39C2×C4×Q8D412yes—
40#494—28no—
41#462—48no—
42C2×C4⋊C8—21no—
43C2×C4⋊C8C2314yes—
44C2×#103—24no—
45C2×C4×C8—22no—
46C2×C4×C8C2312yes—
47#400—24no—
48#400—24no—
49#398—28no—
50#391—24no—
51#391—24no—
52#391—22no—
53#391D412yes—
54#389—24no—
55#389—24no—
56#389—22no—
57#389—22no—
58#389D414yes—
59#387—28no—
60#384—22no—
61#384—22no—
62#384D412yes—
63#384C2311yescharacteristic
64#382—24no—
65#382—24no—
66C4×C4⋊C4—22no—
67C4×C4⋊C4—24no—
68C4×C4⋊C4—22no—
69C4×C4⋊C4—22no—
70C4×C4⋊C4C2314yes—
71C43C2311yescharacteristic
72#340—28no—
73#340—28no—
74#336—22no—
75#336—21no—
76#333—24no—
77C22×Q8—22no—
78C22×Q8—22no—
79C22×Q8C2×D412yes—
80C2×Q16—48no—
81C2×Q16—48no—
82C2×Q16—84no—
83C22×C8—22no—
84C22×C8—24no—
85C4⋊Q8—24no—
86C4⋊Q8—24no—
87C4⋊Q8—24no—
88C4⋊Q8—22no—
89C4⋊Q8—24no—
90C4⋊Q8—44no—
91C4⋊Q8—82no—
92C4⋊Q8—84no—
93C4⋊Q8#4114yes—
94C4×Q8—44no—
95C4×Q8—44no—
96C4×Q8—44no—
97C2×C4⋊C4—22no—
98C2×C4⋊C4—22no—
99C2×C4⋊C4—24no—
100C2×C4⋊C4—22no—
101C2×C4⋊C4—22no—
102C2×C4⋊C4—24no—
103C2×C4⋊C4—24no—
104C2×C4⋊C4—24no—
105C2×C4⋊C4—22no—
106C2×C4⋊C4—22no—
107C2×C4⋊C4—41no—
108C2×C4⋊C4—42no—
109C2×C4⋊C4—41no—
110C2×C4⋊C4C2×D414yes—
111C2×C4⋊C4C2×D414yes—
112C2×C4⋊C4C2×D412yes—
113C2×C42—22no—
114C2×C42—24no—
115C2×C42—24no—
116C2×C42C2×D411yescharacteristic
117C2×C42C2×D412yes—
118C2×C42C2×D411yescharacteristic
119C2×C42C2×D412yes—
120C2×C42C2411yescharacteristic
121C4⋊C8—24no—
122C4⋊C8—24no—
123C4⋊C8—24no—
124#103—48no—
125C4×C8—24no—
126C4×C8—44no—
127#95—24no—
128#95—28no—
129#95—28no—
130#95—24no—
131#95—24no—
132#95—24no—
133C2×Q8—22no—
134C2×Q8—42no—
135C2×Q8—42no—
136C2×Q8—44no—
137C2×Q8—44no—
138C2×Q8—44no—
139C2×Q8—44no—
140C2×Q8—44no—
141C2×Q8—42no—
142C2×Q8—42no—
143C2×Q8#12414yes—
144C22×C4—22no—
145C22×C4—24no—
146C22×C4—22no—
147C22×C4—24no—
148C22×C4—42no—
149C22×C4#12011yescharacteristic
150C22×C4#12012yes—
151C22×C4#12011yescharacteristic
152C22×C4C22×D411yescharacteristic
153C22×C4C22×D412yes—
154Q16—816no—
155C2×C8—24no—
156C2×C8—42no—
157C2×C8—42no—
158C2×C8—44no—
159C2×C8—44no—
160C2×C8—44no—
161C4⋊C4—28no—
162C4⋊C4—24no—
163C4⋊C4—24no—
164C4⋊C4—28no—
165C4⋊C4—42no—
166C4⋊C4—44no—
167C4⋊C4—44no—
168C4⋊C4—48no—
169C4⋊C4—44no—
170C4⋊C4—48no—
171C4⋊C4—82no—
172C4⋊C4—82no—
173C4⋊C4—82no—
174C4⋊C4—84no—
175C4⋊C4—84no—
176C4⋊C4—84no—
177C4⋊C4—82no—
178C42—24no—
179C42—22no—
180C42—41no—
181C42—44no—
182C42—44no—
183C42—44no—
184C42—84no—
185C42#13514yes—
186C42#13714yes—
187C42C2×#4112yes—
188C422−1+412yes—
189C23C2×#12011yescharacteristic
190Q8—44no—
191Q8—82no—
192Q8—84no—
193Q8—84no—
194Q8—88no—
195C2×C4—22no—
196C2×C4—21no—
197C2×C4—22no—
198C2×C4—21no—
199C2×C4—24no—
200C2×C4—22no—
201C2×C4—21no—
202C2×C4—42no—
203C2×C4—42no—
204C2×C4—44no—
205C2×C4—44no—
206C2×C4—44no—
207C2×C4—44no—
208C2×C4—44no—
209C2×C4—42no—
210C2×C4—44no—
211C2×C4—44no—
212C2×C4—82no—
213C2×C4—82no—
214C2×C4—82no—
215C2×C4#45214yes—
216C2×C4#45214yes—
217C2×C4#45312yes—
218C2×C4C2×#12411yescharacteristic
219C2×C4#53611yescharacteristic
220C2×C4#54814yes—
221C2×C4C2×#13512yes—
222C2×C4C2×#13712yes—
223C2×C4#57814yes—
224C8—44no—
225C8—84no—
226C22#243611yescharacteristic
227C22C2×#45212yes—
228C22#301112yes—
229C22C2×#45311yescharacteristic
230C22#302111yescharacteristic
231C4—21no—
232C4—21no—
233C4—24no—
234C4—41no—
235C4—44no—
236C4—82no—
237C4—84no—
238C4—84no—
239C4—162no—
240C4#163414yes—
241C4#336414yes—
242C4#337414yes—
243C2C2×#163411yescharacteristic
244C2#1871911yescharacteristic
245C2#1889912yes—
246C2#2591811yescharacteristic
247C2#2593112yes—
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

Maximal subgroups

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

subgroupindex
#598692
#292182
#284402
#103522
#98652
#90662
C4×#3332