Finite Group Databasebeta

← #243203

Subgroups — #243203

Order 512.

634 conjugacy classes of subgroups; only the 202 normal subgroups are shown.

normal subgroups only

This group has 634 conjugacy classes of subgroups — too many to draw. Only its 202 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#21987C212yesmaximal
2#21351C211yesmaximal, characteristic
3C2×#1834C211yesmaximal, characteristic
4#10656C214yesmaximal, complemented
5#10655C214yesmaximal, complemented
6#8040C212yesmaximal
7#7875C211yesmaximal, characteristic
8C2×#432C2211yescharacteristic
9C2×#432C2212yes—
10C22×C4×C8C2211yescharacteristic
11C2×#384C2212yes—
12#2283C2211yescharacteristic
13#1948C2214yes—
14#1947C2214yes—
15#1939C2212yes—
16#1939C2212yes—
17#1853—22no—
18#1852—22no—
19#1850—22no—
20#1847—22no—
21#1836—22no—
22#1834—22no—
23#1834C2212yes—
24#1834C2212yes—
25#1768—22no—
26C2×#340—22no—
27#1296C2218yes—
28#1277C2212yes—
29#1277C2212yes—
30C23×C8D412yes—
31C22×C4⋊C4—22no—
32C22×C4⋊C4—22no—
33C22×C4⋊C4—22no—
34C22×C42C2311yescharacteristic
35#432—24no—
36#432—24no—
37#432Q818yes—
38C2×#107—24no—
39C2×#106—24no—
40C2×C4⋊C8—22no—
41C2×C4⋊C8—22no—
42C2×C4⋊C8C2312yes—
43C2×C4⋊C8C2314yes—
44C2×C22⋊C8—22no—
45C2×C22⋊C8—21no—
46C2×#97—22no—
47C2×#97—21no—
48C2×C4×C8—24no—
49C2×C4×C8D412yes—
50C2×C4×C8C2311yescharacteristic
51C2×C4×C8C2311yescharacteristic
52#384—22no—
53#384—22no—
54#384—22no—
55#384C2314yes—
56#383—21no—
57#382D411yescharacteristic
58#382D411yescharacteristic
59#381—21no—
60#380D412yes—
61C4×C4⋊C4C2312yes—
62C4×C22⋊C4—21no—
63#376—21no—
64#340—24no—
65#340—24no—
66#340—24no—
67#340—24no—
68#336—22no—
69#336—22no—
70C23×C4—42no—
71C23×C4C2×D411yescharacteristic
72C23×C4C2×D411yescharacteristic
73C23×C4#4111yescharacteristic
74C22×C8—24no—
75C22×C8—24no—
76C22×C8—22no—
77C22×C8—42no—
78C22×C8—42no—
79C22×C8C2×D412yes—
80C22×C8C2×D412yes—
81C2×C4⋊C4—24no—
82C2×C4⋊C4—24no—
83C2×C4⋊C4—24no—
84C2×C4⋊C4—44no—
85C2×C4⋊C4—44no—
86C2×C4⋊C4—42no—
87C2×C4⋊C4—42no—
88C2×C4⋊C4—42no—
89C2×C4⋊C4—84no—
90C2×C4⋊C4—84no—
91C2×C4⋊C4C2×D412yes—
92C2×C4⋊C4C2×D412yes—
93C2×C22⋊C4—22no—
94C2×C42—22no—
95C2×C42C2×D411yescharacteristic
96C2×C42C2×D411yescharacteristic
97C2×C42C2×Q814yes—
98C2×C42#4114yes—
99C2×C42#4111yescharacteristic
100C2×C42C2411yescharacteristic
101#107—84no—
102#106—84no—
103C4⋊C8—24no—
104C4⋊C8—24no—
105C4⋊C8—44no—
106C4⋊C8—44no—
107C4⋊C8C2×Q818yes—
108C22⋊C8—28no—
109C22⋊C8—42no—
110#97—28no—
111#97—42no—
112C4×C8—24no—
113C4×C8—22no—
114C4×C8—22no—
115#95—22no—
116#95—21no—
117#95—21no—
118#95—21no—
119#95—21no—
120#95—22no—
121C24#12111yescharacteristic
122C22×C4—24no—
123C22×C4—22no—
124C22×C4—22no—
125C22×C4—22no—
126C22×C4—22no—
127C22×C4—22no—
128C22×C4—22no—
129C22×C4—44no—
130C22×C4—84no—
131C22×C4—84no—
132C22×C4—84no—
133C22×C4#12111yescharacteristic
134C22×C4#12111yescharacteristic
135C22×C4#12111yescharacteristic
136C22×C4C22⋊Q814yes—
137C22×C4#12312yes—
138C22×C4#12514yes—
139C22×C4C4⋊Q812yes—
140C22×C4C22×D411yescharacteristic
141C22×C4C22×D411yescharacteristic
142C22×C4C2×#4111yescharacteristic
143C2×C8—28no—
144C2×C8—22no—
145C2×C8—22no—
146C2×C8—22no—
147C2×C8—22no—
148C2×C8—22no—
149C2×C8—22no—
150C2×C8—48no—
151C2×C8—44no—
152C2×C8—44no—
153C2×C8—42no—
154C2×C8—42no—
155C2×C8—42no—
156C4⋊C4—24no—
157C4⋊C4—24no—
158C4⋊C4—44no—
159C4⋊C4—44no—
160C4⋊C4—44no—
161C4⋊C4—44no—
162C4⋊C4—84no—
163C4⋊C4—84no—
164C4⋊C4—84no—
165C4⋊C4—84no—
166C22⋊C4—44no—
167C42—42no—
168C42—42no—
169C42C22⋊Q814yes—
170C42#12312yes—
171C42#12514yes—
172C42C4⋊Q812yes—
173C42C22×Q812yes—
174C42C2×#4112yes—
175C23—22no—
176C23—21no—
177C23—21no—
178C23#39812yes—
179C23#47114yes—
180C23C2×#12111yescharacteristic
181C2×C4—28no—
182C2×C4—22no—
183C2×C4—21no—
184C2×C4—21no—
185C2×C4—22no—
186C2×C4—22no—
187C2×C4—28no—
188C2×C4—24no—
189C2×C4—22no—
190C2×C4—22no—
191C2×C4—42no—
192C2×C4—42no—
193C2×C4—42no—
194C2×C4—42no—
195C2×C4—42no—
196C2×C4—42no—
197C2×C4—84no—
198C2×C4—84no—
199C2×C4—84no—
200C2×C4—168no—
201C2×C4#39812yes—
202C2×C4#39812yes—
203C2×C4#39812yes—
204C2×C4C2×C22⋊Q812yes—
205C2×C4C2×#12311yescharacteristic
206C2×C4C2×#12512yes—
207C2×C4C2×C4⋊Q811yescharacteristic
208C2×C4#57512yes—
209C2×C4#57512yes—
210C2×C4#57612yes—
211C2×C4#57812yes—
212C8—48no—
213C8—42no—
214C8—42no—
215C8—82no—
216C22—22no—
217C22—22no—
218C22—24no—
219C22—24no—
220C22#209014yes—
221C22C2×#39811yescharacteristic
222C22#304012yes—
223C22#305811yescharacteristic
224C22#305911yescharacteristic
225C22C2×#47112yes—
226C4—22no—
227C4—22no—
228C4—22no—
229C4—22no—
230C4—82no—
231C4—82no—
232C4—164no—
233C4#308314yes—
234C4#309312yes—
235C4#316611yescharacteristic
236C4#316611yescharacteristic
237C2—24no—
238C2#1936412yes—
239C2#1936512yes—
240C2#2365911yescharacteristic
241C2#2366711yescharacteristic
242C2C2×#209011yescharacteristic
subgroupG/H
center, socleC23C2×#121
commutator, Frattini, cosocleC2×C42C24
prime radicalC24#121
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. #10656C2semidirect
5. #10655C2semidirect

Maximal subgroups

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

subgroupindex
#219872
#213512
C2×#18342
#106562
#106552
#80402
#78752