Finite Group Databasebeta

← #122741

Subgroups — #122741

Order 512.

267 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has 267 conjugacy classes of subgroups — too many to draw. Only its 64 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#28190C211yesmaximal, characteristic
2#11493C211yesmaximal, characteristic
3#8621C211yesmaximal, characteristic
4#7083C212yesmaximal
5#7082C212yesmaximal
6C2×C8×Q8C2211yescharacteristic
7C2×C4×Q16C411yescharacteristic
8#2581—21no—
9#2162—21no—
10#2086C411yescharacteristic
11#2086C2211yescharacteristic
12#2056C411yescharacteristic
13#2056C2211yescharacteristic
14#1933—21no—
15#1933—21no—
16#1835C411yescharacteristic
17#1805—21no—
18#1568—21no—
19#1467—21no—
20#1322—21no—
21#1322C2212yes—
22#1290C2212yes—
23C22×Q16C2×C411yescharacteristic
24C2×C4×Q8—21no—
25C2×C4×Q8D411yescharacteristic
26C8×Q8—41no—
27C8×Q8—41no—
28C4×Q16—41no—
29C4×Q16—41no—
30#432—22no—
31C2×#107C2×C411yescharacteristic
32C2×C4⋊C8C2×C411yescharacteristic
33C2×C4⋊C8D411yescharacteristic
34C2×C4⋊C8C2311yescharacteristic
35C2×#103—21no—
36C2×#103—21no—
37C2×#103—21no—
38C2×C4×C8C2×C411yescharacteristic
39C2×C4×C8D411yescharacteristic
40C2×C4×C8D411yescharacteristic
41#400—41no—
42#395—41no—
43#391—21no—
44#391—21no—
45#389—21no—
46#389—21no—
47#387—41no—
48#387—41no—
49#382—21no—
50C4⋊C16—41no—
51C22⋊C16—22no—
52C4×C16—41no—
53#340—21no—
54#340—21no—
55#340—21no—
56#340C2×C411yescharacteristic
57#340C2×C411yescharacteristic
58#335—41no—
59#334—41no—
60#324—41no—
61#324—41no—
62C22×Q8—21no—
63C22×Q8SD1611yescharacteristic
64C2×Q16—41no—
65C2×Q16—42no—
66C22×C8C22⋊C411yescharacteristic
67C22×C8C22⋊C411yescharacteristic
68C22×C8C22×C411yescharacteristic
69C22×C8C2×D411yescharacteristic
70C4×Q8—41no—
71C4×Q8—41no—
72C4×Q8—42no—
73C4×Q8—42no—
74C2×C4⋊C4—21no—
75C2×C4⋊C4—21no—
76C2×C4⋊C4—21no—
77C2×C4⋊C4—41no—
78C2×C4⋊C4—41no—
79C2×C4⋊C4—41no—
80C2×C4⋊C4D811yescharacteristic
81C2×C4⋊C4D811yescharacteristic
82C2×C4⋊C4SD1611yescharacteristic
83C2×C42—41no—
84C2×C42C22⋊C411yescharacteristic
85C2×C42C22⋊C411yescharacteristic
86C2×C42C2×D411yescharacteristic
87C2×C16—42no—
88#107—41no—
89C4⋊C8—41no—
90C4⋊C8—41no—
91C4⋊C8—41no—
92C4⋊C8—42no—
93#103—42no—
94#103—42no—
95#103—41no—
96#103—41no—
97C22⋊C8—42no—
98#97—42no—
99C4×C8—21no—
100C4×C8—21no—
101C4×C8—21no—
102C4×C8—21no—
103C4×C8—21no—
104C4×C8—21no—
105#95—41no—
106#95—41no—
107#95—41no—
108#95—41no—
109#95—41no—
110#95—41no—
111#95—41no—
112#95—41no—
113C2×Q8—22no—
114C2×Q8—22no—
115C2×Q8—41no—
116C2×Q8—41no—
117C2×Q8—41no—
118C2×Q8—41no—
119C2×Q8—41no—
120C2×Q8—41no—
121C2×Q8—41no—
122C22×C4—41no—
123C22×C4—41no—
124C22×C4—41no—
125C22×C4—41no—
126C22×C4#10211yescharacteristic
127C22×C4#10211yescharacteristic
128C22×C4#10211yescharacteristic
129C22×C4#10211yescharacteristic
130C22×C4C2×C22⋊C411yescharacteristic
131C22×C4C2×D811yescharacteristic
132C22×C4C2×SD1611yescharacteristic
133Q16—42no—
134Q16—42no—
135C2×C8—21no—
136C2×C8—21no—
137C2×C8—21no—
138C2×C8—41no—
139C2×C8—41no—
140C2×C8—41no—
141C2×C8—41no—
142C2×C8—82no—
143C2×C8#9911yescharacteristic
144C2×C8#9911yescharacteristic
145C4⋊C4—41no—
146C4⋊C4—41no—
147C4⋊C4—41no—
148C4⋊C4—41no—
149C4⋊C4—41no—
150C4⋊C4—41no—
151C4⋊C4—41no—
152C4⋊C4—41no—
153C4⋊C4—41no—
154C4⋊C4—82no—
155C4⋊C4—82no—
156C4⋊C4—82no—
157C42—21no—
158C42—21no—
159C42—21no—
160C42—21no—
161C42—21no—
162C42—21no—
163C42—82no—
164C16—82no—
165C23C2×#10211yescharacteristic
166Q8—44no—
167Q8—42no—
168Q8—42no—
169Q8—81no—
170Q8—81no—
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—21no—
183C2×C4—21no—
184C2×C4—41no—
185C2×C4—41no—
186C2×C4—41no—
187C2×C4—42no—
188C2×C4—81no—
189C2×C4—81no—
190C2×C4—81no—
191C2×C4—81no—
192C2×C4—81no—
193C2×C4—81no—
194C2×C4—81no—
195C2×C4—81no—
196C2×C4—81no—
197C2×C4—81no—
198C2×C4—81no—
199C2×C4#35912yes—
200C2×C4C2×#9911yescharacteristic
201C2×C4#42111yescharacteristic
202C8—21no—
203C8—21no—
204C8—41no—
205C8—41no—
206C8—41no—
207C8—162no—
208C22—21no—
209C22—21no—
210C22#151911yescharacteristic
211C22C2×#35911yescharacteristic
212C22#214811yescharacteristic
213C4—21no—
214C4—22no—
215C4—41no—
216C4—41no—
217C4—41no—
218C4—41no—
219C4—41no—
220C4—82no—
221C4—161no—
222C4—161no—
223C4—161no—
224C4#212011yescharacteristic
225C4#213311yescharacteristic
226C2—21no—
227C2—21no—
228C2#1186911yescharacteristic
229C2#1187911yescharacteristic
230C2#1189311yescharacteristic
subgroupG/H
center, socleC22#1519
commutatorC22×C8C22×C4
Frattini, cosocleC2×C4⋊C8C23
prime radicalC23C2×#102
LCSG ▷ C22×C8 ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C8×Q8 ◁ G
LECSG ▷ C2×C4⋊C8 ▷ C22×C4 ▷ C22 ▷ C1

Maximal subgroups

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

subgroupindex
#281902
#114932
#86212
#70832
#70822