Finite Group Databasebeta

← #327970

Subgroups — #327970

Order 512.

688 conjugacy classes of subgroups; only the 136 normal subgroups are shown.

normal subgroups only

This group has 688 conjugacy classes of subgroups — too many to draw. Only its 136 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#37643C211yesmaximal, characteristic
2#27980C211yesmaximal, characteristic
3#27955C212yesmaximal
4#10585C214yesmaximal, complemented
5#10361C214yesmaximal, complemented
6#8722C211yesmaximal, characteristic
7#8722C212yesmaximal
8#2857C2211yescharacteristic
9#2845C2214yes—
10#2629—24no—
11#2629—22no—
12#2552—22no—
13#2552—22no—
14#2552—22no—
15#2285C2212yes—
16#2285C2211yescharacteristic
17#2084—22no—
18#2083—22no—
19#2072—22no—
20#1993C2214yes—
21#1960C2212yes—
22#1873—22no—
23#1854—24no—
24#1847C2214yes—
25#1836C2212yes—
26#1749C2212yes—
27#1749C2211yescharacteristic
28#1295C2214yes—
29#1295C2218yes—
30C2×C4⋊Q8C2311yescharacteristic
31C2×#125—22no—
32C2×#125—24no—
33C2×C22⋊Q8—21no—
34C2×#117D414yes—
35C2×#117D412yes—
36C22×C42C2311yescharacteristic
37C2×#107—24no—
38C2×#106—24no—
39C2×#103—24no—
40C2×C22⋊C8—22no—
41C2×C22⋊C8—21no—
42C2×#97C2312yes—
43C2×#97C2314yes—
44#398—28no—
45#398—28no—
46#398—28no—
47#396—28no—
48#391—21no—
49#389—24no—
50#389—24no—
51#389—24no—
52#389—22no—
53#384—22no—
54#384—24no—
55#384D412yes—
56#384D412yes—
57#384D412yes—
58#384C2311yescharacteristic
59#382C2312yes—
60#382C2314yes—
61#380—22no—
62#380—22no—
63#380—22no—
64#380—22no—
65C4×C4⋊C4—22no—
66C4×C4⋊C4—22no—
67C4×C4⋊C4—22no—
68#340—24no—
69#340—24no—
70#340—24no—
71#336—21no—
72#336—22no—
73C22×Q8—21no—
74C23×C4C2×D411yescharacteristic
75C23×C4C2×D412yes—
76C22×C8—24no—
77C22×C8—22no—
78C4⋊Q8—24no—
79#125—84no—
80#125—88no—
81C22⋊Q8—44no—
82#117—82no—
83#117—84no—
84C2×C4⋊C4—22no—
85C2×C4⋊C4—22no—
86C2×C4⋊C4—22no—
87C2×C4⋊C4—22no—
88C2×C4⋊C4—22no—
89C2×C4⋊C4—22no—
90C2×C4⋊C4—21no—
91C2×C4⋊C4—22no—
92C2×C4⋊C4—22no—
93C2×C4⋊C4—28no—
94C2×C4⋊C4—28no—
95C2×C4⋊C4—28no—
96C2×C4⋊C4—41no—
97C2×C4⋊C4C2×D414yes—
98C2×C4⋊C4C2×D414yes—
99C2×C4⋊C4C2×D414yes—
100C2×C22⋊C4—24no—
101C2×C22⋊C4—22no—
102C2×C22⋊C4—41no—
103C2×C42—24no—
104C2×C42—22no—
105C2×C42—24no—
106C2×C42C2×D412yes—
107C2×C42C2×D411yescharacteristic
108C2×C42C2411yescharacteristic
109#107—84no—
110#106—84no—
111#103—84no—
112C22⋊C8—28no—
113C22⋊C8—42no—
114#97—28no—
115#97—24no—
116#95—28no—
117#95—24no—
118#95—24no—
119#95—24no—
120#95—42no—
121C24#12011yescharacteristic
122C2×Q8—42no—
123C2×Q8—42no—
124C2×Q8—42no—
125C22×C4—24no—
126C22×C4—22no—
127C22×C4—22no—
128C22×C4—24no—
129C22×C4—24no—
130C22×C4—24no—
131C22×C4—24no—
132C22×C4—42no—
133C22×C4—41no—
134C22×C4—41no—
135C22×C4—41no—
136C22×C4#12011yescharacteristic
137C22×C4#12012yes—
138C22×C4C22×D412yes—
139C22×C4C22×D411yescharacteristic
140C2×C8—28no—
141C2×C8—44no—
142C2×C8—44no—
143C2×C8—42no—
144C2×C8—42no—
145C2×C8—42no—
146C4⋊C4—44no—
147C4⋊C4—44no—
148C4⋊C4—44no—
149C4⋊C4—44no—
150C4⋊C4—44no—
151C4⋊C4—44no—
152C4⋊C4—42no—
153C4⋊C4—42no—
154C4⋊C4—42no—
155C4⋊C4—44no—
156C4⋊C4—44no—
157C4⋊C4—44no—
158C4⋊C4—44no—
159C4⋊C4—88no—
160C4⋊C4—88no—
161C4⋊C4—88no—
162C4⋊C4—82no—
163C22⋊C4—84no—
164C22⋊C4—82no—
165C22⋊C4—82no—
166C42—24no—
167C42—22no—
168C42—44no—
169C42—48no—
170C422−1+414yes—
171C23—21no—
172C23—21no—
173C23—22no—
174C23—22no—
175C23—21no—
176C23C2×#12011yescharacteristic
177Q8—84no—
178C2×C4—21no—
179C2×C4—22no—
180C2×C4—21no—
181C2×C4—22no—
182C2×C4—21no—
183C2×C4—22no—
184C2×C4—22no—
185C2×C4—21no—
186C2×C4—44no—
187C2×C4—48no—
188C2×C4—44no—
189C2×C4—44no—
190C2×C4—44no—
191C2×C4—44no—
192C2×C4—44no—
193C2×C4—44no—
194C2×C4—44no—
195C2×C4—44no—
196C2×C4—44no—
197C2×C4—81no—
198C2×C4—81no—
199C2×C4—81no—
200C2×C4—82no—
201C2×C4—81no—
202C2×C4#45412yes—
203C2×C4#45614yes—
204C2×C4#53614yes—
205C2×C4#53612yes—
206C8—48no—
207C8—82no—
208C22—41no—
209C22—42no—
210C22—41no—
211C22—41no—
212C22—42no—
213C22#261411yescharacteristic
214C22C2×#45411yescharacteristic
215C22C2×#45612yes—
216C22#301912yes—
217C22#301911yescharacteristic
218C4—22no—
219C4—22no—
220C4—24no—
221C4—24no—
222C4—84no—
223C4—84no—
224C4—84no—
225C4—161no—
226C4—161no—
227C2—81no—
228C2#1906111yescharacteristic
229C2#1906512yes—
230C2#1908411yescharacteristic
231C2#2965111yescharacteristic
232C2#2965312yes—
subgroupG/H
center, socleC23C2×#120
commutator, Frattini, cosocleC2×C42C24
prime radicalC24#120
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
4. #10585C2semidirect
5. #10361C2semidirect

Maximal subgroups

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

subgroupindex
#376432
#279802
#279552
#105852
#103612
#87222