Finite Group Databasebeta

← #324462

Subgroups — #324462

Order 512.

642 conjugacy classes of subgroups; only the 158 normal subgroups are shown.

normal subgroups only

This group has 642 conjugacy classes of subgroups — too many to draw. Only its 158 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#30230C211yesmaximal, characteristic, complemented
2#29149C211yesmaximal, characteristic
3#28509C211yesmaximal, characteristic
4#22260C211yesmaximal, characteristic
5#10481C214yesmaximal, complemented
6#9897C211yesmaximal, characteristic, complemented
7#9702C211yesmaximal, characteristic, complemented
8#9215C214yesmaximal, complemented
9#8278C211yesmaximal, characteristic, complemented
10C2×#476C2214yes—
11C2×C8×D4C2212yes—
12#2868C2211yescharacteristic
13#2859C2211yescharacteristic
14C4×#127C2211yescharacteristic
15#2100—24no—
16#2094—22no—
17#2025—22no—
18#1990C2212yes—
19#1990C2212yes—
20#1961—21no—
21#1957—21no—
22#1954C2212yes—
23#1869—24no—
24#1844—21no—
25#1836—21no—
26#1831C2211yescharacteristic
27#1802C2214yes—
28#1767C2211yescharacteristic
29C4×C4⋊C8C2211yescharacteristic
30#1744C2211yescharacteristic
31C2×#335C2212yes—
32C2×#333C2214yes—
33#1296C2214yes—
34#1277C2212yes—
35C23×C8—22no—
36C2×C4⋊Q8—22no—
37C2×C4⋊Q8C2312yes—
38C2×#127D411yescharacteristic
39C2×#125—21no—
40C2×C4×D4—22no—
41C2×C4×D4C2312yes—
42#476—28no—
43#476—44no—
44C8×D4—44no—
45C2×#106—24no—
46C2×C4⋊C8C2312yes—
47C2×C4⋊C8C2312yes—
48C2×C4⋊C8C2312yes—
49C2×#102—24no—
50C2×#102—24no—
51C2×C22⋊C8—22no—
52C2×C22⋊C8—24no—
53C2×#97—21no—
54C2×C4×C8—21no—
55C2×C4×C8C2311yescharacteristic
56C2×C4×C8C2311yescharacteristic
57#390—22no—
58#390D412yes—
59#389—22no—
60#389—22no—
61#384—21no—
62#384—21no—
63#384—22no—
64#384D411yescharacteristic
65#384C2312yes—
66#382—21no—
67C43C2311yescharacteristic
68#340—24no—
69#340—24no—
70#340—24no—
71#336—22no—
72#336—21no—
73#336—21no—
74#336D414yes—
75#335—24no—
76#333—44no—
77C22×Q8—22no—
78C22×D4—22no—
79C22×D4C2×D412yes—
80C23×C4—22no—
81C22×C8—22no—
82C22×C8—22no—
83C22×C8—22no—
84C22×C8—44no—
85C22×C8—84no—
86C22×C8C2×D412yes—
87C22×C8C2×D412yes—
88C4⋊Q8—24no—
89C4⋊Q8—24no—
90C4⋊Q8—84no—
91#127—22no—
92#127Q1614yes—
93#125—82no—
94C4×D4—24no—
95C4×D4—24no—
96C4×D4—84no—
97C2×C4⋊C4—22no—
98C2×C4⋊C4—22no—
99C2×C4⋊C4—22no—
100C2×C4⋊C4—22no—
101C2×C4⋊C4—22no—
102C2×C4⋊C4—22no—
103C2×C4⋊C4—42no—
104C2×C4⋊C4—42no—
105C2×C4⋊C4—42no—
106C2×C4⋊C4C2×D412yes—
107C2×C22⋊C4—22no—
108C2×C22⋊C4—24no—
109C2×C42C2×D412yes—
110C2×C42C2×D411yescharacteristic
111C2×C42C2×D411yescharacteristic
112C2×C42#4112yes—
113C2×C42C2411yescharacteristic
114#106—48no—
115C4⋊C8—22no—
116C4⋊C8—22no—
117C4⋊C8—22no—
118C4⋊C8—24no—
119C4⋊C8#4114yes—
120#102—48no—
121#102—84no—
122C22⋊C8—44no—
123C22⋊C8—84no—
124#97—42no—
125C4×C8—22no—
126C4×C8—22no—
127C4×C8—42no—
128#95—42no—
129C24—22no—
130C2×Q8—44no—
131C2×Q8—84no—
132C2×D4—24no—
133C2×D4—22no—
134C2×D4—24no—
135C2×D4—28no—
136C2×D4—82no—
137C2×D4—82no—
138C2×D4C2×Q1614yes—
139C22×C4—22no—
140C22×C4—42no—
141C22×C4—42no—
142C22×C4—44no—
143C22×C4—44no—
144C22×C4—42no—
145C22×C4—42no—
146C22×C4#12112yes—
147C22×C4#12112yes—
148C22×C4C22×D411yescharacteristic
149C22×C4C22×D411yescharacteristic
150C22×C4C2×#4111yescharacteristic
151C2×C8—22no—
152C2×C8—22no—
153C2×C8—24no—
154C2×C8—24no—
155C2×C8—24no—
156C2×C8—42no—
157C2×C8—42no—
158C2×C8—42no—
159C2×C8—42no—
160C2×C8—42no—
161C2×C8—44no—
162C2×C8—42no—
163C2×C8—88no—
164C4⋊C4—24no—
165C4⋊C4—42no—
166C4⋊C4—42no—
167C4⋊C4—44no—
168C4⋊C4—44no—
169C4⋊C4—44no—
170C4⋊C4—82no—
171C4⋊C4—82no—
172C4⋊C4—84no—
173C4⋊C4—84no—
174C4⋊C4—84no—
175C22⋊C4—44no—
176C22⋊C4—84no—
177C42—24no—
178C42—22no—
179C42—42no—
180C42C2×Q1612yes—
181C42#13512yes—
182C42C2×#4112yes—
183C42H(5,𝔽2)12yes—
184C23—24no—
185C23—44no—
186C23—42no—
187C23—42no—
188C23—44no—
189C23C2×#12111yescharacteristic
190Q8—164no—
191D4—28no—
192D4—44no—
193D4—88no—
194C2×C4—22no—
195C2×C4—22no—
196C2×C4—22no—
197C2×C4—22no—
198C2×C4—22no—
199C2×C4—22no—
200C2×C4—44no—
201C2×C4—42no—
202C2×C4—84no—
203C2×C4—82no—
204C2×C4—82no—
205C2×C4—82no—
206C2×C4—82no—
207C2×C4—82no—
208C2×C4—84no—
209C2×C4—82no—
210C2×C4#46714yes—
211C2×C4#53211yescharacteristic
212C2×C4#53511yescharacteristic
213C2×C4#54612yes—
214C2×C4#55912yes—
215C2×C4C22×Q1611yescharacteristic
216C2×C4C2×#13511yescharacteristic
217C2×C4#57714yes—
218C2×C4#57712yes—
219C8—42no—
220C8—42no—
221C8—44no—
222C8—82no—
223C8—82no—
224C22—44no—
225C22—48no—
226C22—84no—
227C22—82no—
228C22—82no—
229C22#254411yescharacteristic
230C22#304212yes—
231C22C2×#46711yescharacteristic
232C22#305711yescharacteristic
233C22#306812yes—
234C4—24no—
235C4—22no—
236C4—42no—
237C4—82no—
238C4—162no—
239C4—162no—
240C4#317412yes—
241C4#322212yes—
242C4#329912yes—
243C4#330312yes—
244C2—84no—
245C2#1939312yes—
246C2#1951312yes—
247C2#2859811yescharacteristic
248C2#2862211yescharacteristic
249C2#2863511yescharacteristic
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. #30230C2semidirect
5. #10481C2semidirect
6. #9897C2semidirect
7. #9702C2semidirect
8. #9215C2semidirect
9. #8278C2semidirect

Maximal subgroups

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

subgroupindex
#302302
#291492
#285092
#222602
#104812
#98972
#97022
#92152
#82782