Finite Group Databasebeta

← #150887

Subgroups — #150887

Order 512.

283 conjugacy classes of subgroups; only the 60 normal subgroups are shown.

normal subgroups only

This group has 283 conjugacy classes of subgroups — too many to draw. Only its 60 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#29269C211yesmaximal, characteristic
2#12800C211yesmaximal, characteristic
3#11480C211yesmaximal, characteristic
4#7435C212yesmaximal
5#7428C212yesmaximal
6#2330C411yescharacteristic, complemented
7#2326C2211yescharacteristic
8C2×#354—21no—
9#2096C411yescharacteristic, complemented
10#2094C2211yescharacteristic
11#2026C411yescharacteristic, complemented
12#2025C2211yescharacteristic
13#1983—21no—
14#1948—21no—
15#1865—21no—
16#1864—21no—
17#1832C411yescharacteristic, complemented
18#1799—21no—
19#1461—21no—
20#1393C2212yes—
21#1325—21no—
22#1318C2212yes—
23C2×#125C2×C411yescharacteristic
24C2×#124D411yescharacteristic
25C2×C4×Q8D411yescharacteristic
26C2×C4×D4D411yescharacteristic
27#431—41no—
28C2×#106—21no—
29C2×#106—21no—
30C2×C4⋊C8—21no—
31C2×#103—21no—
32C2×#103—21no—
33C2×#102—21no—
34C2×#102—21no—
35C2×#99—22no—
36#391C2×C411yescharacteristic
37#390C2311yescharacteristic
38#389—21no—
39#389—41no—
40#388—41no—
41#387—41no—
42#385—41no—
43#384C2×C411yescharacteristic
44#384D411yescharacteristic
45#380—41no—
46C4×C4⋊C4—21no—
47#376C2×C411yescharacteristic
48#354—42no—
49#342—41no—
50#340—21no—
51#340—21no—
52#336—21no—
53#336C2×C411yescharacteristic
54#336C2×C411yescharacteristic
55#324—22no—
56#324—41no—
57C22×Q8C22⋊C411yescharacteristic
58C22×D4C2×D411yescharacteristic
59C23×C4—21no—
60C22×C8—21no—
61C22×C8—21no—
62#125—42no—
63#124—41no—
64#124SD1612yes—
65#124Q1612yes—
66C4×Q8—81no—
67C4×D4—81no—
68C2×C4⋊C4—21no—
69C2×C4⋊C4—21no—
70C2×C4⋊C4—21no—
71C2×C4⋊C4—41no—
72C2×C4⋊C4—41no—
73C2×C4⋊C4—41no—
74C2×C4⋊C4—41no—
75C2×C4⋊C4—81no—
76C2×C4⋊C4C22⋊C411yescharacteristic
77C2×C4⋊C4C2×D411yescharacteristic
78C2×C22⋊C4—21no—
79C2×C22⋊C4—41no—
80C2×C22⋊C4—42no—
81C2×C22⋊C4—41no—
82C2×C42—21no—
83C2×C42—41no—
84C2×C42C22⋊C411yescharacteristic
85C2×C42C22⋊C411yescharacteristic
86C2×C42C22×C411yescharacteristic
87#106—81no—
88#106—81no—
89C4⋊C8—42no—
90C4⋊C8—81no—
91#103—42no—
92#103—81no—
93#102—42no—
94#102—81no—
95#99—42no—
96#99—42no—
97C22⋊C8—42no—
98C4×C8—81no—
99#95—21no—
100#95—21no—
101#95—41no—
102#95—41no—
103#95—41no—
104#95—81no—
105C24—21no—
106C2×Q8—21no—
107C2×Q8—41no—
108C2×Q8—41no—
109C2×Q8#10312yes—
110C2×D4—21no—
111C2×D4—41no—
112C2×D4—41no—
113C2×D4C2×SD1611yescharacteristic
114C2×D4C2×Q1611yescharacteristic
115C22×C4—21no—
116C22×C4—21no—
117C22×C4—21no—
118C22×C4—21no—
119C22×C4—21no—
120C22×C4—41no—
121C22×C4—41no—
122C22×C4—41no—
123C22×C4—41no—
124C22×C4—82no—
125C22×C4—161no—
126C22×C4#9911yescharacteristic
127C22×C4#9911yescharacteristic
128C22×C4C2×C22⋊C411yescharacteristic
129C2×C8—41no—
130C2×C8—41no—
131C2×C8—41no—
132C2×C8—41no—
133C2×C8—41no—
134C2×C8—41no—
135C2×C8—82no—
136C4⋊C4—22no—
137C4⋊C4—41no—
138C4⋊C4—41no—
139C4⋊C4—41no—
140C4⋊C4—41no—
141C4⋊C4—41no—
142C4⋊C4—42no—
143C4⋊C4—42no—
144C4⋊C4—81no—
145C4⋊C4—161no—
146C4⋊C4—161no—
147C4⋊C4—161no—
148C4⋊C4—162no—
149C22⋊C4—42no—
150C22⋊C4—42no—
151C22⋊C4—81no—
152C22⋊C4—82no—
153C22⋊C4—82no—
154C22⋊C4—161no—
155C42—21no—
156C42—41no—
157C42—41no—
158C42—41no—
159C42—41no—
160C42—161no—
161C42#10312yes—
162C23—21no—
163C23—21no—
164C23—41no—
165C23—41no—
166C23—81no—
167C23—81no—
168C23C2×#9911yescharacteristic
169Q8—42no—
170Q8—81no—
171D4—41no—
172D4—41no—
173D4—81no—
174C2×C4—21no—
175C2×C4—21no—
176C2×C4—21no—
177C2×C4—21no—
178C2×C4—22no—
179C2×C4—22no—
180C2×C4—41no—
181C2×C4—41no—
182C2×C4—41no—
183C2×C4—41no—
184C2×C4—41no—
185C2×C4—41no—
186C2×C4—41no—
187C2×C4—41no—
188C2×C4—41no—
189C2×C4—42no—
190C2×C4—41no—
191C2×C4—41no—
192C2×C4—82no—
193C2×C4—81no—
194C2×C4—81no—
195C2×C4—81no—
196C2×C4—81no—
197C2×C4—81no—
198C2×C4—81no—
199C2×C4—81no—
200C2×C4—161no—
201C2×C4—162no—
202C2×C4—162no—
203C2×C4—162no—
204C2×C4C2×#10311yescharacteristic
205C2×C4#42111yescharacteristic
206C8—81no—
207C8—81no—
208C8—162no—
209C22—21no—
210C22—21no—
211C22—41no—
212C22—41no—
213C22—81no—
214C22—81no—
215C22—81no—
216C22—81no—
217C22—161no—
218C22#152011yescharacteristic
219C22#212011yescharacteristic
220C22#212711yescharacteristic
221C4—21no—
222C4—21no—
223C4—42no—
224C4—41no—
225C4—41no—
226C4—42no—
227C4—81no—
228C4—82no—
229C4—81no—
230C4—81no—
231C4—161no—
232C4—322no—
233C2—21no—
234C2—21no—
235C2—81no—
236C2—81no—
237C2#1189711yescharacteristic
238C2#1360611yescharacteristic
239C2#1361011yescharacteristic
subgroupG/H
center, socleC22#2127
commutatorC2×C42C22×C4
Frattini, cosocle#390C23
prime radicalC22×D4C2×D4
LCSG ▷ C2×C42 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C23 ◁ C2×C42 ◁ #2326 ◁ G
LECSG ▷ #390 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
6. #2330C4semidirect
9. #2096C4semidirect
11. #2026C4semidirect
17. #1832C4semidirect

Maximal subgroups

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

subgroupindex
#292692
#128002
#114802
#74352
#74282