Finite Group Databasebeta

← #325445

Subgroups — #325445

Order 512.

624 conjugacy classes of subgroups; only the 154 normal subgroups are shown.

normal subgroups only

This group has 624 conjugacy classes of subgroups — too many to draw. Only its 154 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#33256C211yesmaximal, characteristic, complemented
2#29130C212yesmaximal
3#21681C211yesmaximal, characteristic
4#10619C214yesmaximal, complemented
5#9906C212yesmaximal, complemented
6#8656C212yesmaximal, complemented
7#8656C212yesmaximal, complemented
8#8273C211yesmaximal, characteristic, complemented
9C2×#436—22no—
10C2×#436C2212yes—
11#2859C2212yes—
12C4×#127C2211yescharacteristic
13#2100—22no—
14#2041—22no—
15#1954—22no—
16#1954C2212yes—
17#1941—22no—
18#1941—22no—
19#1936—24no—
20#1844C2212yes—
21#1842C2211yescharacteristic
22#1834C2211yescharacteristic
23#1834C2211yescharacteristic
24#1802C2214yes—
25#1802C2214yes—
26#1767C2212yes—
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×#106—22no—
40C2×#106—22no—
41C2×C4⋊C8—22no—
42C2×C4⋊C8C2314yes—
43C2×C4⋊C8C2312yes—
44C2×#102—24no—
45C2×#102—24no—
46C2×C22⋊C8—24no—
47C2×C22⋊C8—22no—
48C2×#97—22no—
49C2×C4×C8D412yes—
50C2×C4×C8D412yes—
51C2×C4×C8C2311yescharacteristic
52C2×C4×C8C2311yescharacteristic
53#390—22no—
54#390D412yes—
55#389—22no—
56#389—22no—
57#389—22no—
58#389—22no—
59#384D411yescharacteristic
60#384C2312yes—
61#384C2312yes—
62#382—22no—
63C43C2311yescharacteristic
64#340—21no—
65#340—21no—
66#340—21no—
67#340—21no—
68#340—24no—
69#340—24no—
70#336—22no—
71#334—24no—
72C22×D4—22no—
73C22×D4C2×D412yes—
74C23×C4—22no—
75C2×M16—48no—
76C2×M16—44no—
77C22×C8—22no—
78C22×C8—24no—
79C22×C8C2×D411yescharacteristic
80C22×C8C2×D411yescharacteristic
81C22×C8C2×D412yes—
82#127—42no—
83#125—84no—
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×D412yes—
101C2×C42C2×D411yescharacteristic
102C2×C42C2×D411yescharacteristic
103C2×C42#4112yes—
104C2×C42C2411yescharacteristic
105#106—44no—
106#106—44no—
107C4⋊C8—24no—
108C4⋊C8—24no—
109C4⋊C8—24no—
110C4⋊C8—44no—
111#102—48no—
112#102—84no—
113C22⋊C8—44no—
114C22⋊C8—48no—
115#97—82no—
116C4×C8—24no—
117C4×C8—22no—
118C4×C8—42no—
119C4×C8#4114yes—
120#95—42no—
121#95—42no—
122C24—22no—
123C2×D4—24no—
124C2×D4—24no—
125C2×D4—24no—
126C2×D4—44no—
127C2×D4—44no—
128C2×D4—44no—
129C22×C4—22no—
130C22×C4—42no—
131C22×C4—42no—
132C22×C4—44no—
133C22×C4—44no—
134C22×C4—42no—
135C22×C4—42no—
136C22×C4#12112yes—
137C22×C4#12112yes—
138C22×C4C22×D411yescharacteristic
139C22×C4C22×D411yescharacteristic
140C22×C4C2×#4111yescharacteristic
141M16—88no—
142C2×C8—22no—
143C2×C8—22no—
144C2×C8—22no—
145C2×C8—22no—
146C2×C8—42no—
147C2×C8—44no—
148C2×C8—44no—
149C2×C8—44no—
150C2×C8—44no—
151C2×C8#12412yes—
152C2×C8#12412yes—
153C2×C8AGL(1,ℤ/8)14yes—
154C4⋊C4—24no—
155C4⋊C4—44no—
156C4⋊C4—42no—
157C4⋊C4—42no—
158C4⋊C4—44no—
159C4⋊C4—42no—
160C4⋊C4—42no—
161C4⋊C4—42no—
162C4⋊C4—42no—
163C4⋊C4—82no—
164C4⋊C4—82no—
165C4⋊C4—84no—
166C4⋊C4—84no—
167C22⋊C4—48no—
168C22⋊C4—44no—
169C42—22no—
170C42—22no—
171C42—22no—
172C42—22no—
173C42—24no—
174C42C2×#4112yes—
175C422−1+412yes—
176C23—44no—
177C23—42no—
178C23—42no—
179C23—42no—
180C23—44no—
181C23C2×#12111yescharacteristic
182D4—48no—
183D4—88no—
184C2×C4—22no—
185C2×C4—22no—
186C2×C4—22no—
187C2×C4—22no—
188C2×C4—22no—
189C2×C4—22no—
190C2×C4—44no—
191C2×C4—42no—
192C2×C4—82no—
193C2×C4—82no—
194C2×C4—84no—
195C2×C4—82no—
196C2×C4—82no—
197C2×C4—82no—
198C2×C4—82no—
199C2×C4—82no—
200C2×C4—82no—
201C2×C4#47114yes—
202C2×C4C2×#12411yescharacteristic
203C2×C4#53611yescharacteristic
204C2×C4#54712yes—
205C2×C4#55612yes—
206C2×C4C2×AGL(1,ℤ/8)12yes—
207C2×C4#57512yes—
208C2×C4#57512yes—
209C2×C4#57712yes—
210C8—22no—
211C8—22no—
212C8—24no—
213C8—84no—
214C8—82no—
215C22—82no—
216C22—82no—
217C22—82no—
218C22—84no—
219C22—84no—
220C22#254711yescharacteristic
221C22#303612yes—
222C22#305411yescharacteristic
223C22#306011yescharacteristic
224C22C2×#47112yes—
225C4—22no—
226C4—24no—
227C4—42no—
228C4—82no—
229C4—162no—
230C4—162no—
231C4#313511yescharacteristic
232C4#313811yescharacteristic
233C4#323412yes—
234C4#335114yes—
235C2—162no—
236C2#1928212yes—
237C2#1933112yes—
238C2#2871011yescharacteristic
239C2#2872211yescharacteristic
240C2#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. #33256C2semidirect
4. #10619C2semidirect
5. #9906C2semidirect
6. #8656C2semidirect
7. #8656C2semidirect
8. #8273C2semidirect

Maximal subgroups

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

subgroupindex
#332562
#291302
#216812
#106192
#99062
#86562
#82732