Finite Group Databasebeta

← #325417

Subgroups — #325417

Order 512.

624 conjugacy classes of subgroups; only the 162 normal subgroups are shown.

normal subgroups only

This group has 624 conjugacy classes of subgroups — too many to draw. Only its 162 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#33291C211yesmaximal, characteristic, complemented
2#29121C212yesmaximal
3#21681C211yesmaximal, characteristic
4#10625C214yesmaximal, complemented
5#9900C212yesmaximal, complemented
6#8656C212yesmaximal, complemented
7#8656C212yesmaximal, complemented
8C4×#334C211yesmaximal, characteristic, complemented
9C2×#436—22no—
10C2×#436C2212yes—
11#2859C2212yes—
12C4×#127C2211yescharacteristic
13#2099—22no—
14#2041—22no—
15#1991—22no—
16#1991—22no—
17#1954—22no—
18#1954C2212yes—
19#1934—24no—
20#1842C2211yescharacteristic
21#1834C2212yes—
22#1834C2211yescharacteristic
23#1834C2211yescharacteristic
24#1802C2214yes—
25#1802C2214yes—
26C4×C4⋊C8C2212yes—
27C42×C8C2211yescharacteristic
28C2×#334C2212yes—
29#1296C2214yes—
30#1296C2214yes—
31#1277C2212yes—
32C22×M16—22no—
33C2×#127D411yescharacteristic
34C2×#125—22no—
35C2×C4×D4—22no—
36C2×C4×D4C2312yes—
37#436—28no—
38#436—48no—
39C2×C4⋊C8—22no—
40C2×C4⋊C8C2314yes—
41C2×C4⋊C8C2312yes—
42C2×#102—24no—
43C2×#102—24no—
44C2×C22⋊C8—24no—
45C2×C22⋊C8—22no—
46C2×C4×C8—22no—
47C2×C4×C8D412yes—
48C2×C4×C8D412yes—
49C2×C4×C8C2311yescharacteristic
50C2×C4×C8C2311yescharacteristic
51#390—22no—
52#390D412yes—
53#389—22no—
54#389—22no—
55#389—22no—
56#389—22no—
57#384D411yescharacteristic
58#384C2312yes—
59#384C2312yes—
60#382—22no—
61C43C2311yescharacteristic
62#340—24no—
63#340—21no—
64#340—21no—
65#340—21no—
66#340—21no—
67#340—24no—
68#340—24no—
69#336—22no—
70#334—24no—
71C22×D4—22no—
72C22×D4C2×D412yes—
73C23×C4—22no—
74C2×M16—48no—
75C2×M16—44no—
76C22×C8—22no—
77C22×C8—24no—
78C22×C8C2×D412yes—
79C22×C8C2×D411yescharacteristic
80C22×C8C2×D411yescharacteristic
81#127—42no—
82#125—44no—
83#125—44no—
84C4×D4—24no—
85C4×D4—24no—
86C4×D4—44no—
87C4×D4—44no—
88C2×C4⋊C4—22no—
89C2×C4⋊C4—22no—
90C2×C4⋊C4—22no—
91C2×C4⋊C4—22no—
92C2×C4⋊C4—22no—
93C2×C4⋊C4—22no—
94C2×C4⋊C4—22no—
95C2×C4⋊C4—42no—
96C2×C4⋊C4—42no—
97C2×C4⋊C4C2×D412yes—
98C2×C22⋊C4—24no—
99C2×C22⋊C4—22no—
100C2×C42C2×D411yescharacteristic
101C2×C42C2×D411yescharacteristic
102C2×C42C2×D412yes—
103C2×C42#4112yes—
104C2×C42C2411yescharacteristic
105C4⋊C8—24no—
106C4⋊C8—24no—
107C4⋊C8—24no—
108C4⋊C8—44no—
109#102—48no—
110#102—84no—
111C22⋊C8—48no—
112C22⋊C8—44no—
113C4×C8—24no—
114C4×C8—24no—
115C4×C8—22no—
116C4×C8—82no—
117C4×C8#4114yes—
118#95—42no—
119#95—42no—
120C24—22no—
121C2×D4—24no—
122C2×D4—24no—
123C2×D4—24no—
124C2×D4—44no—
125C2×D4—44no—
126C2×D4—44no—
127C22×C4—22no—
128C22×C4—42no—
129C22×C4—42no—
130C22×C4—44no—
131C22×C4—44no—
132C22×C4—42no—
133C22×C4—42no—
134C22×C4#12112yes—
135C22×C4#12112yes—
136C22×C4C22×D411yescharacteristic
137C22×C4C22×D411yescharacteristic
138C22×C4C2×#4111yescharacteristic
139M16—88no—
140C2×C8—22no—
141C2×C8—22no—
142C2×C8—22no—
143C2×C8—22no—
144C2×C8—22no—
145C2×C8—42no—
146C2×C8—44no—
147C2×C8—44no—
148C2×C8—44no—
149C2×C8—44no—
150C2×C8#12412yes—
151C2×C8#12412yes—
152C4⋊C4—24no—
153C4⋊C4—44no—
154C4⋊C4—42no—
155C4⋊C4—42no—
156C4⋊C4—42no—
157C4⋊C4—42no—
158C4⋊C4—44no—
159C4⋊C4—42no—
160C4⋊C4—42no—
161C4⋊C4—82no—
162C4⋊C4—82no—
163C4⋊C4—84no—
164C4⋊C4—84no—
165C22⋊C4—44no—
166C22⋊C4—48no—
167C42—22no—
168C42—22no—
169C42—42no—
170C42#13514yes—
171C42AGL(1,ℤ/8)14yes—
172C42C2×#4112yes—
173C422−1+412yes—
174C23—44no—
175C23—42no—
176C23—42no—
177C23—42no—
178C23—44no—
179C23C2×#12111yescharacteristic
180D4—48no—
181D4—88no—
182C2×C4—22no—
183C2×C4—22no—
184C2×C4—22no—
185C2×C4—22no—
186C2×C4—22no—
187C2×C4—44no—
188C2×C4—42no—
189C2×C4—82no—
190C2×C4—82no—
191C2×C4—84no—
192C2×C4—82no—
193C2×C4—82no—
194C2×C4—82no—
195C2×C4—82no—
196C2×C4—82no—
197C2×C4—82no—
198C2×C4#46314yes—
199C2×C4#46314yes—
200C2×C4C2×#12411yescharacteristic
201C2×C4#53611yescharacteristic
202C2×C4#54712yes—
203C2×C4#55612yes—
204C2×C4C2×#13512yes—
205C2×C4C2×AGL(1,ℤ/8)12yes—
206C2×C4#57512yes—
207C2×C4#57712yes—
208C8—22no—
209C8—22no—
210C8—42no—
211C8—84no—
212C8—82no—
213C22—84no—
214C22—82no—
215C22—82no—
216C22—82no—
217C22—84no—
218C22#254711yescharacteristic
219C22C2×#46312yes—
220C22#305411yescharacteristic
221C22#306011yescharacteristic
222C22#306712yes—
223C4—24no—
224C4—22no—
225C4—24no—
226C4—82no—
227C4—162no—
228C4—162no—
229C4#313111yescharacteristic
230C4#313811yescharacteristic
231C4#322812yes—
232C4#335114yes—
233C2—162no—
234C2#1914212yes—
235C2#1920012yes—
236C2#2871011yescharacteristic
237C2#2871811yescharacteristic
238C2#2873311yescharacteristic
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. #33291C2semidirect
4. #10625C2semidirect
5. #9900C2semidirect
6. #8656C2semidirect
7. #8656C2semidirect
8. C4×#334C2semidirect

Maximal subgroups

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

subgroupindex
#332912
#291212
#216812
#106252
#99002
#86562
C4×#3342