Finite Group Databasebeta

← #325423

Subgroups — #325423

Order 512.

668 conjugacy classes of subgroups; only the 182 normal subgroups are shown.

normal subgroups only

This group has 668 conjugacy classes of subgroups — too many to draw. Only its 182 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#33289C211yesmaximal, characteristic, complemented
2#29139C212yesmaximal
3#21681C211yesmaximal, characteristic
4#10621C214yesmaximal, complemented
5#9896C212yesmaximal, complemented
6#8656C214yesmaximal, complemented
7C4×#334C211yesmaximal, characteristic, complemented
8C2×#436—22no—
9C2×#436C2212yes—
10#2868C2212yes—
11C4×#127C2211yescharacteristic
12#2023—24no—
13#1990—22no—
14#1990—22no—
15#1957—22no—
16#1954—22no—
17#1954C2212yes—
18#1842C2211yescharacteristic
19#1834—22no—
20#1834C2212yes—
21#1834C2212yes—
22#1802C2218yes—
23C4×C4⋊C8C2212yes—
24C42×C8C2211yescharacteristic
25C2×#334C2212yes—
26#1296C2218yes—
27#1277C2212yes—
28C22×M16—22no—
29C2×C4⋊Q8—22no—
30C2×C4⋊Q8—22no—
31C2×C4⋊Q8—22no—
32C2×#127D411yescharacteristic
33C2×C4×D4—22no—
34C2×C4×D4C2312yes—
35#436—28no—
36#436—48no—
37C2×C4⋊C8—22no—
38C2×C4⋊C8C2314yes—
39C2×C4⋊C8C2312yes—
40C2×#102—28no—
41C2×C22⋊C8—24no—
42C2×C22⋊C8—22no—
43C2×C4×C8—22no—
44C2×C4×C8D412yes—
45C2×C4×C8D412yes—
46C2×C4×C8C2311yescharacteristic
47C2×C4×C8C2311yescharacteristic
48#390—22no—
49#390D412yes—
50#384—22no—
51#384—22no—
52#384—22no—
53#384D411yescharacteristic
54#384C2314yes—
55C43C2311yescharacteristic
56#340—24no—
57#340—22no—
58#340—22no—
59#340—28no—
60#336—22no—
61#334—24no—
62C22×Q8—42no—
63C22×D4—22no—
64C22×D4C2×D412yes—
65C23×C4—22no—
66C2×M16—48no—
67C2×M16—44no—
68C22×C8—22no—
69C22×C8—24no—
70C22×C8C2×D412yes—
71C22×C8C2×D411yescharacteristic
72C22×C8C2×D411yescharacteristic
73C4⋊Q8—44no—
74C4⋊Q8—44no—
75C4⋊Q8—84no—
76C4⋊Q8—84no—
77#127—24no—
78C4×D4—24no—
79C4×D4—24no—
80C4×D4—44no—
81C4×D4—44no—
82C2×C4⋊C4—22no—
83C2×C4⋊C4—24no—
84C2×C4⋊C4—24no—
85C2×C4⋊C4—24no—
86C2×C4⋊C4—42no—
87C2×C4⋊C4—42no—
88C2×C4⋊C4—42no—
89C2×C4⋊C4C2×D412yes—
90C2×C22⋊C4—24no—
91C2×C22⋊C4—22no—
92C2×C42C2×D411yescharacteristic
93C2×C42C2×D412yes—
94C2×C42C2×D411yescharacteristic
95C2×C42#4112yes—
96C2×C42C2411yescharacteristic
97C4⋊C8—24no—
98C4⋊C8—24no—
99C4⋊C8—24no—
100C4⋊C8—44no—
101#102—88no—
102C22⋊C8—48no—
103C22⋊C8—44no—
104C4×C8—24no—
105C4×C8—24no—
106C4×C8—22no—
107C4×C8—44no—
108C4×C8#4114yes—
109C24—22no—
110C2×Q8—84no—
111C2×Q8—84no—
112C2×Q8—84no—
113C2×D4—24no—
114C2×D4—24no—
115C2×D4—24no—
116C2×D4—44no—
117C2×D4—44no—
118C2×D4—44no—
119C22×C4—22no—
120C22×C4—42no—
121C22×C4—42no—
122C22×C4—44no—
123C22×C4—44no—
124C22×C4—44no—
125C22×C4#12112yes—
126C22×C4#12112yes—
127C22×C4C22×D411yescharacteristic
128C22×C4C22×D411yescharacteristic
129C22×C4C2×#4111yescharacteristic
130M16—88no—
131C2×C8—22no—
132C2×C8—22no—
133C2×C8—22no—
134C2×C8—22no—
135C2×C8—22no—
136C2×C8—42no—
137C2×C8—44no—
138C2×C8—44no—
139C2×C8—44no—
140C2×C8—44no—
141C2×C8#12412yes—
142C2×C8#12412yes—
143C4⋊C4—24no—
144C4⋊C4—44no—
145C4⋊C4—44no—
146C4⋊C4—44no—
147C4⋊C4—48no—
148C4⋊C4—84no—
149C4⋊C4—84no—
150C4⋊C4—84no—
151C4⋊C4—84no—
152C22⋊C4—44no—
153C22⋊C4—48no—
154C42—22no—
155C42—24no—
156C42#13514yes—
157C42AGL(1,ℤ/8)14yes—
158C42#13714yes—
159C42C2×#4112yes—
160C422−1+412yes—
161C23—44no—
162C23—42no—
163C23—42no—
164C23—42no—
165C23—44no—
166C23C2×#12111yescharacteristic
167Q8—168no—
168D4—48no—
169D4—88no—
170C2×C4—22no—
171C2×C4—22no—
172C2×C4—22no—
173C2×C4—44no—
174C2×C4—42no—
175C2×C4—82no—
176C2×C4—82no—
177C2×C4—84no—
178C2×C4—84no—
179C2×C4—84no—
180C2×C4—84no—
181C2×C4#46314yes—
182C2×C4#46314yes—
183C2×C4#47014yes—
184C2×C4#47114yes—
185C2×C4C2×#12411yescharacteristic
186C2×C4#53611yescharacteristic
187C2×C4#54712yes—
188C2×C4#55612yes—
189C2×C4C2×#13512yes—
190C2×C4C2×AGL(1,ℤ/8)12yes—
191C2×C4C2×#13712yes—
192C2×C4#57612yes—
193C8—22no—
194C8—22no—
195C8—42no—
196C8—84no—
197C8—82no—
198C22—82no—
199C22—82no—
200C22—82no—
201C22—84no—
202C22—84no—
203C22#254711yescharacteristic
204C22C2×#46312yes—
205C22C2×#47011yescharacteristic
206C22#305811yescharacteristic
207C22C2×#47112yes—
208C4—22no—
209C4—24no—
210C4—82no—
211C4—164no—
212C4#170318yes—
213C4#313111yescharacteristic
214C4#313811yescharacteristic
215C4#322812yes—
216C4#335714yes—
217C2—162no—
218C2C2×#170312yes—
219C2#1919812yes—
220C2#2871111yescharacteristic
221C2#2871811yescharacteristic
222C2#2873211yescharacteristic
subgroupG/H
center, socleC23C2×#121
commutator, Frattini, cosocleC2×C42C24
prime radicalC2×#127D4
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
1. #33289C2semidirect
4. #10621C2semidirect
5. #9896C2semidirect
6. #8656C2semidirect
7. C4×#334C2semidirect

Maximal subgroups

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

subgroupindex
#332892
#291392
#216812
#106212
#98962
#86562
C4×#3342