Finite Group Databasebeta

← #149880

Subgroups — #149880

Order 512.

272 conjugacy classes of subgroups; only the 56 normal subgroups are shown.

normal subgroups only

This group has 272 conjugacy classes of subgroups — too many to draw. Only its 56 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#22227C211yesmaximal, characteristic
2#11535C211yesmaximal, characteristic
3#11534C211yesmaximal, characteristic
4#7371C212yesmaximal, complemented
5#7358C212yesmaximal, complemented
6C2×#488C411yescharacteristic
7#2330C411yescharacteristic
8#2326C2211yescharacteristic
9#1992C2211yescharacteristic
10#1959C411yescharacteristic
11#1927—21no—
12#1926—21no—
13#1865—21no—
14#1864—21no—
15#1844C2211yescharacteristic
16#1746C411yescharacteristic
17#1467—21no—
18#1465—21no—
19#1325C2212yes—
20#1325C2212yes—
21#1319—21no—
22#1318—21no—
23C2×#125C2×C411yescharacteristic
24C2×#124C2×C411yescharacteristic
25C2×C4×Q8D411yescharacteristic
26C2×C4×D4D411yescharacteristic
27#488—22no—
28#488—22no—
29#432—41no—
30C2×#106—21no—
31C2×#106—21no—
32C2×#103—21no—
33C2×#103—21no—
34C2×#102—21no—
35C2×#102—21no—
36#407—41no—
37C2×#97C2×C411yescharacteristic
38C2×C4×C8C2×C411yescharacteristic
39#391D411yescharacteristic
40#390D411yescharacteristic
41#389—21no—
42#389—41no—
43#388—41no—
44#387—41no—
45#385—41no—
46#384C2×C411yescharacteristic
47#384C2311yescharacteristic
48C4×C4⋊C4—21no—
49#376C2×C411yescharacteristic
50#340—21no—
51#340—21no—
52#336—21no—
53#336—21no—
54#324—22no—
55#324—22no—
56#324—41no—
57#323—41no—
58C22×Q8C22⋊C411yescharacteristic
59C22×D4C22⋊C411yescharacteristic
60C23×C4—21no—
61C2×M16—81no—
62C22×C8—21no—
63C22×C8—21no—
64#125—21no—
65#125—41no—
66#125M1612yescomplemented
67#124—21no—
68#124—21no—
69#124—41no—
70C4×Q8—81no—
71C4×D4—81no—
72C2×C4⋊C4—21no—
73C2×C4⋊C4—21no—
74C2×C4⋊C4—21no—
75C2×C4⋊C4—41no—
76C2×C4⋊C4—41no—
77C2×C4⋊C4—41no—
78C2×C4⋊C4—41no—
79C2×C4⋊C4C2×D411yescharacteristic
80C2×C4⋊C4C2×D411yescharacteristic
81C2×C22⋊C4—21no—
82C2×C22⋊C4—41no—
83C2×C22⋊C4—41no—
84C2×C42—21no—
85C2×C42—41no—
86C2×C42C22⋊C411yescharacteristic
87C2×C42C22⋊C411yescharacteristic
88C2×C42C22×C411yescharacteristic
89#106—81no—
90#106—81no—
91C4⋊C8—81no—
92#103—24no—
93#103—81no—
94#102—24no—
95#102—81no—
96C22⋊C8—42no—
97C22⋊C8—42no—
98C22⋊C8—81no—
99#97—41no—
100#97—81no—
101C4×C8—21no—
102C4×C8M1612yescomplemented
103#95—21no—
104#95—21no—
105#95—41no—
106#95—41no—
107#95—41no—
108C24—21no—
109C2×Q8—21no—
110C2×Q8—21no—
111C2×Q8—41no—
112C2×Q8—41no—
113C2×D4—21no—
114C2×D4—21no—
115C2×D4—41no—
116C2×D4—41no—
117C22×C4—21no—
118C22×C4—21no—
119C22×C4—21no—
120C22×C4—21no—
121C22×C4—21no—
122C22×C4—41no—
123C22×C4—41no—
124C22×C4—41no—
125C22×C4—41no—
126C22×C4—161no—
127C22×C4#9911yescharacteristic
128C22×C4#9911yescharacteristic
129C22×C4C2×C22⋊C411yescharacteristic
130M16—162no—
131C2×C8—22no—
132C2×C8—41no—
133C2×C8—41no—
134C2×C8—41no—
135C2×C8—41no—
136C2×C8—41no—
137C2×C8—82no—
138C2×C8—82no—
139C4⋊C4—22no—
140C4⋊C4—41no—
141C4⋊C4—41no—
142C4⋊C4—42no—
143C4⋊C4—42no—
144C4⋊C4—41no—
145C4⋊C4—41no—
146C4⋊C4—41no—
147C4⋊C4—81no—
148C4⋊C4—161no—
149C4⋊C4—161no—
150C4⋊C4—161no—
151C22⋊C4—42no—
152C22⋊C4—42no—
153C22⋊C4—81no—
154C22⋊C4—161no—
155C42—21no—
156C42—41no—
157C42—41no—
158C42—41no—
159C42—41no—
160C42—161no—
161C42C2×M1611yescharacteristic
162C42C2×M1611yescharacteristic
163C23—41no—
164C23—41no—
165C23—41no—
166C23—81no—
167C23—81no—
168C23C2×#9911yescharacteristic
169Q8—42no—
170Q8—81no—
171D4—42no—
172D4—81no—
173C2×C4—21no—
174C2×C4—21no—
175C2×C4—21no—
176C2×C4—21no—
177C2×C4—21no—
178C2×C4—21no—
179C2×C4—41no—
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—41no—
190C2×C4—41no—
191C2×C4—42no—
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#40711yescharacteristic
202C2×C4#41311yescharacteristic
203C8—42no—
204C8—81no—
205C8—162no—
206C8—162no—
207C22—21no—
208C22—21no—
209C22—82no—
210C22—81no—
211C22—81no—
212C22—81no—
213C22—161no—
214C22#146511yescharacteristic
215C22#212711yescharacteristic
216C22#213511yescharacteristic
217C4—21no—
218C4—21no—
219C4—41no—
220C4—41no—
221C4—41no—
222C4—41no—
223C4—81no—
224C4—82no—
225C4—81no—
226C4—81no—
227C4—81no—
228C4—161no—
229C2—21no—
230C2—21no—
231C2—161no—
232C2#1153511yescharacteristic
233C2#1361011yescharacteristic
234C2#1361111yescharacteristic
subgroupG/H
center, socleC22#2127
commutatorC2×C42C22×C4
Frattini, cosocle#384C23
prime radicalC22×D4C22⋊C4
LCSG ▷ C2×C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C23 ◁ C2×C42 ◁ #2326 ◁ G
LECSG ▷ #384 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #7371C2semidirect
5. #7358C2semidirect
66. #125M16semidirect
102. C4×C8M16semidirect

Maximal subgroups

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

subgroupindex
#222272
#115352
#115342
#73712
#73582