Finite Group Databasebeta

← #125630

Subgroups — #125630

Order 512.

280 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 280 conjugacy classes of subgroups — too many to draw. Only its 86 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#33267C211yesmaximal, characteristic
2#12879C211yesmaximal, characteristic
3#8784C211yesmaximal, characteristic
4#7120C212yesmaximal
5#7113C212yesmaximal
6#2863—21no—
7C2×C4⋊C16—21no—
8#1969C411yescharacteristic
9#1969C411yescharacteristic
10#1969C2211yescharacteristic
11#1942C411yescharacteristic
12#1942C411yescharacteristic
13#1942C2211yescharacteristic
14#1933—21no—
15#1933—21no—
16#1933—21no—
17#1842—21no—
18C42×C8C2211yescharacteristic
19#1374—21no—
20#1374C2212yes—
21#1277—21no—
22#1277C2212yes—
23C22×Q16C2×C411yescharacteristic
24C22×C16—22no—
25C2×#107C2×C411yescharacteristic
26C2×#107C2×C411yescharacteristic
27C2×#107C2×C411yescharacteristic
28C2×C4⋊C8—22no—
29C2×C4⋊C8—41no—
30C2×C4⋊C8—41no—
31C2×#103—21no—
32C2×#103—21no—
33C2×#103—21no—
34C2×C4×C8C2×C411yescharacteristic
35C2×C4×C8C2×C411yescharacteristic
36C2×C4×C8D411yescharacteristic
37C2×C4×C8D411yescharacteristic
38C2×C4×C8D411yescharacteristic
39C2×C4×C8C2311yescharacteristic
40#391—21no—
41#391—21no—
42#391—21no—
43#389—21no—
44#389—21no—
45#389—21no—
46#387—41no—
47#387—41no—
48#387—41no—
49#383—41no—
50C43D411yescharacteristic
51C4⋊C16—42no—
52#340—21no—
53#340—21no—
54#340—21no—
55C22×Q8—21no—
56C2×Q16—22no—
57C2×Q16—41no—
58C2×Q16—41no—
59C22×C8—42no—
60C22×C8C22⋊C411yescharacteristic
61C22×C8C22⋊C411yescharacteristic
62C22×C8C22×C411yescharacteristic
63C22×C8C2×D411yescharacteristic
64C2×C4⋊C4—21no—
65C2×C4⋊C4—21no—
66C2×C4⋊C4—21no—
67C2×C4⋊C4—41no—
68C2×C4⋊C4—41no—
69C2×C4⋊C4—41no—
70C2×C42C22⋊C411yescharacteristic
71C2×C42C22⋊C411yescharacteristic
72C2×C42D811yescharacteristic
73C2×C42D811yescharacteristic
74C2×C42SD1611yescharacteristic
75C2×C42SD1611yescharacteristic
76C2×C42C2×D411yescharacteristic
77C2×C16—42no—
78C2×C16—42no—
79C2×C16—42no—
80#107—22no—
81#107—41no—
82#107—41no—
83C4⋊C8—42no—
84C4⋊C8—42no—
85C4⋊C8—82no—
86C4⋊C8—82no—
87#103—42no—
88#103—42no—
89#103—41no—
90#103—41no—
91C4×C8—22no—
92C4×C8—21no—
93C4×C8—21no—
94C4×C8—41no—
95C4×C8—41no—
96C4×C8—41no—
97C4×C8—41no—
98#95—41no—
99#95—41no—
100#95—41no—
101#95—41no—
102#95—41no—
103#95—41no—
104#95—41no—
105#95—41no—
106#95—41no—
107C2×Q8—41no—
108C2×Q8—41no—
109C2×Q8—41no—
110C2×Q8—41no—
111C2×Q8—41no—
112C2×Q8—41no—
113C22×C4—41no—
114C22×C4—41no—
115C22×C4—41no—
116C22×C4—41no—
117C22×C4#10211yescharacteristic
118C22×C4#10211yescharacteristic
119C22×C4#10211yescharacteristic
120C22×C4#10211yescharacteristic
121C22×C4C2×C22⋊C411yescharacteristic
122C22×C4C2×D811yescharacteristic
123C22×C4C2×SD1611yescharacteristic
124Q16—82no—
125C2×C8—21no—
126C2×C8—21no—
127C2×C8—21no—
128C2×C8—21no—
129C2×C8—21no—
130C2×C8—21no—
131C2×C8—21no—
132C2×C8—21no—
133C2×C8—21no—
134C2×C8—21no—
135C2×C8—82no—
136C2×C8—82no—
137C2×C8—82no—
138C2×C8#13112yes—
139C2×C8AGL(1,ℤ/8)11yescharacteristic
140C2×C8#13711yescharacteristic
141C4⋊C4—41no—
142C4⋊C4—41no—
143C4⋊C4—41no—
144C4⋊C4—41no—
145C4⋊C4—41no—
146C4⋊C4—41no—
147C4⋊C4—82no—
148C4⋊C4—82no—
149C4⋊C4—81no—
150C4⋊C4—81no—
151C42—22no—
152C42—22no—
153C42—22no—
154C42—21no—
155C42—21no—
156C42—21no—
157C42—21no—
158C42—21no—
159C42—21no—
160C42—21no—
161C42—21no—
162C16—82no—
163C23C2×#10211yescharacteristic
164Q8—81no—
165Q8—81no—
166Q8—81no—
167Q8—81no—
168C2×C4—21no—
169C2×C4—21no—
170C2×C4—21no—
171C2×C4—21no—
172C2×C4—21no—
173C2×C4—21no—
174C2×C4—21no—
175C2×C4—21no—
176C2×C4—21no—
177C2×C4—21no—
178C2×C4—21no—
179C2×C4—21no—
180C2×C4—21no—
181C2×C4—21no—
182C2×C4—81no—
183C2×C4—81no—
184C2×C4—81no—
185C2×C4—81no—
186C2×C4—81no—
187C2×C4—81no—
188C2×C4—81no—
189C2×C4—81no—
190C2×C4—81no—
191C2×C4—81no—
192C2×C4—81no—
193C2×C4—81no—
194C2×C4#35912yes—
195C2×C4#36012yes—
196C2×C4#36212yes—
197C2×C4#40811yescharacteristic
198C2×C4#41011yescharacteristic
199C2×C4#41311yescharacteristic
200C2×C4#41711yescharacteristic
201C2×C4#42111yescharacteristic
202C2×C4#42111yescharacteristic
203C2×C4#50911yescharacteristic
204C2×C4#51011yescharacteristic
205C8—22no—
206C8—21no—
207C8—21no—
208C8—41no—
209C8—41no—
210C8—162no—
211C22#148011yescharacteristic
212C22#151711yescharacteristic
213C22#154711yescharacteristic
214C22#213911yescharacteristic
215C22C2×#35911yescharacteristic
216C22C2×#36011yescharacteristic
217C22C2×#36211yescharacteristic
218C4—21no—
219C4—21no—
220C4—22no—
221C4—22no—
222C4—22no—
223C4—41no—
224C4—41no—
225C4—161no—
226C4—161no—
227C4—161no—
228C4—161no—
229C4#147311yescharacteristic
230C4#149511yescharacteristic
231C4#151211yescharacteristic
232C4#153511yescharacteristic
233C2#773211yescharacteristic
234C2#1164811yescharacteristic
235C2#1165511yescharacteristic
236C2#1186011yescharacteristic
237C2#1186311yescharacteristic
238C2#1197711yescharacteristic
239C2#1198811yescharacteristic
subgroupG/H
center, socle, prime radicalC23C2×#102
commutatorC22×C8C22×C4
Frattini, cosocleC2×C4×C8C23
LCSG ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C43 ◁ C42×C8 ◁ G
LECSG ▷ C2×C4×C8 ▷ C22×C4 ▷ C2 ▷ C1

Maximal subgroups

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

subgroupindex
#332672
#128792
#87842
#71202
#71132