Finite Group Databasebeta

← #150691

Subgroups — #150691

Order 512.

277 conjugacy classes of subgroups; only the 54 normal subgroups are shown.

normal subgroups only

This group has 277 conjugacy classes of subgroups — too many to draw. Only its 54 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#23826C211yesmaximal, characteristic
2#12801C211yesmaximal, characteristic
3#11531C211yesmaximal, characteristic
4#7429C212yesmaximal, complemented
5#7413C212yesmaximal, complemented
6C4×#125C411yescharacteristic
7C4×#124C2211yescharacteristic
8C2×#355—21no—
9#2101—21no—
10#2074C2211yescharacteristic
11#2073C411yescharacteristic
12#2042—21no—
13#2026C2211yescharacteristic
14#2025C411yescharacteristic
15#1801—21no—
16#1763—21no—
17#1762—21no—
18#1745C411yescharacteristic
19#1465—21no—
20#1394C2212yes—
21#1324—21no—
22#1319C2212yes—
23C2×#125C2×C411yescharacteristic
24C2×#124D411yescharacteristic
25C2×C4×Q8D411yescharacteristic
26C2×C4×D4D411yescharacteristic
27#423—41no—
28C2×#103—21no—
29C2×#103—21no—
30C2×#102—21no—
31C2×#102—21no—
32C2×#101—22no—
33#407—41no—
34C2×#97—21no—
35#391C2311yescharacteristic
36#390C2×C411yescharacteristic
37#389—21no—
38#388—41no—
39#387—41no—
40#382C2×C411yescharacteristic
41#382D411yescharacteristic
42C4×C4⋊C4—21no—
43C4×C4⋊C4—41no—
44C4×C22⋊C4—41no—
45C43C2×C411yescharacteristic
46#355—42no—
47#343—41no—
48#340—21no—
49#340—21no—
50#340—21no—
51#340—21no—
52#336—21no—
53#336C2×C411yescharacteristic
54#336C2×C411yescharacteristic
55#324—22no—
56#323—41no—
57C22×Q8C2×D411yescharacteristic
58C22×D4C22⋊C411yescharacteristic
59C23×C4—21no—
60C2×M16—42no—
61C2×M16—81no—
62C22×C8—21no—
63C22×C8—21no—
64#125—42no—
65#124—21no—
66#124—21no—
67#124—41no—
68C4×Q8—41no—
69C4×Q8—41no—
70C4×D4—41no—
71C4×D4—41no—
72C2×C4⋊C4—21no—
73C2×C4⋊C4—21no—
74C2×C4⋊C4—21no—
75C2×C4⋊C4—41no—
76C2×C4⋊C4—41no—
77C2×C4⋊C4C22⋊C411yescharacteristic
78C2×C4⋊C4C2×D411yescharacteristic
79C2×C22⋊C4—21no—
80C2×C22⋊C4—41no—
81C2×C22⋊C4—41no—
82C2×C42—21no—
83C2×C42—21no—
84C2×C42—21no—
85C2×C42—41no—
86C2×C42C22⋊C411yescharacteristic
87C2×C42C22⋊C411yescharacteristic
88C2×C42C22×C411yescharacteristic
89C4⋊C8—81no—
90C4⋊C8—81no—
91#103—42no—
92#103—41no—
93#103—41no—
94#102—42no—
95#102—42no—
96#101—42no—
97#101—42no—
98C22⋊C8—42no—
99C22⋊C8—81no—
100#97—41no—
101#97—41no—
102#95—41no—
103#95—41no—
104#95—41no—
105#95—41no—
106#95—41no—
107C24—21no—
108C2×Q8—21no—
109C2×Q8—41no—
110C2×Q8—41no—
111C2×Q8AGL(1,ℤ/8)11yescharacteristic
112C2×Q8#13711yescharacteristic
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—82no—
127C22×C4#9911yescharacteristic
128C22×C4#9911yescharacteristic
129C22×C4C2×C22⋊C411yescharacteristic
130M16—82no—
131M16—82no—
132M16—162no—
133C2×C8—41no—
134C2×C8—41no—
135C2×C8—41no—
136C2×C8—41no—
137C2×C8—41no—
138C2×C8—41no—
139C2×C8—82no—
140C2×C8—82no—
141C4⋊C4—22no—
142C4⋊C4—41no—
143C4⋊C4—41no—
144C4⋊C4—41no—
145C4⋊C4—42no—
146C4⋊C4—42no—
147C4⋊C4—41no—
148C4⋊C4—41no—
149C4⋊C4—81no—
150C4⋊C4—82no—
151C22⋊C4—42no—
152C22⋊C4—42no—
153C22⋊C4—81no—
154C22⋊C4—82no—
155C42—21no—
156C42—21no—
157C42—21no—
158C42—21no—
159C42—41no—
160C42—41no—
161C42—41no—
162C42—81no—
163C42—81no—
164C42—82no—
165C23—41no—
166C23—41no—
167C23—41no—
168C23—81no—
169C23—81no—
170C23C2×#9911yescharacteristic
171Q8—41no—
172Q8—41no—
173Q8—81no—
174D4—42no—
175D4—81no—
176C2×C4—21no—
177C2×C4—21no—
178C2×C4—21no—
179C2×C4—21no—
180C2×C4—21no—
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—42no—
191C2×C4—41no—
192C2×C4—41no—
193C2×C4—41no—
194C2×C4—82no—
195C2×C4—81no—
196C2×C4—81no—
197C2×C4—81no—
198C2×C4—81no—
199C2×C4—81no—
200C2×C4—81no—
201C2×C4—81no—
202C2×C4—161no—
203C2×C4#35411yescharacteristic
204C2×C4#35411yescharacteristic
205C2×C4#41711yescharacteristic
206C2×C4#42111yescharacteristic
207C8—81no—
208C8—81no—
209C8—162no—
210C8—162no—
211C22—21no—
212C22—21no—
213C22—82no—
214C22—81no—
215C22—81no—
216C22—81no—
217C22—161no—
218C22#151311yescharacteristic
219C22#212011yescharacteristic
220C22C2×#35411yescharacteristic
221C4—21no—
222C4—21no—
223C4—21no—
224C4—21no—
225C4—41no—
226C4—41no—
227C4—41no—
228C4—81no—
229C4—81no—
230C4—82no—
231C4—81no—
232C4—81no—
233C4—161no—
234C2—21no—
235C2—21no—
236C2—161no—
237C2#1182311yescharacteristic
238C2#1360011yescharacteristic
239C2#1360311yescharacteristic
subgroupG/H
center, socleC22C2×#354
commutatorC2×C42C22×C4
Frattini, cosocle#391C23
prime radicalC22×D4C22⋊C4
LCSG ▷ C2×C42 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C42 ◁ C4×#124 ◁ G
LECSG ▷ #391 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #7429C2semidirect
5. #7413C2semidirect

Maximal subgroups

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

subgroupindex
#238262
#128012
#115312
#74292
#74132