Finite Group Databasebeta

← #122831

Subgroups — #122831

Order 512.

265 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 265 conjugacy classes of subgroups — too many to draw. Only its 66 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#28678C211yesmaximal, characteristic
2#11494C211yesmaximal, characteristic
3#8617C211yesmaximal, characteristic
4#7091C212yesmaximal
5#7089C212yesmaximal
6C2×#499C411yescharacteristic
7C2×C8×Q8C2211yescharacteristic
8#2584—21no—
9#2163—21no—
10#2087C411yescharacteristic
11#2087C2211yescharacteristic
12#2086C411yescharacteristic
13#2086C2211yescharacteristic
14#1989—21no—
15#1941—21no—
16#1941—21no—
17#1801—21no—
18C4×#106C411yescharacteristic
19#1567—21no—
20#1322—21no—
21#1322C2212yes—
22#1292C2212yes—
23C2×#125—21no—
24C2×C4×Q8D411yescharacteristic
25#499—41no—
26#499—41no—
27C8×Q8—41no—
28C8×Q8—41no—
29C2×#107C2×C411yescharacteristic
30C2×#107C2×C411yescharacteristic
31C2×#106—21no—
32C2×#106—21no—
33#423—22no—
34C2×C4⋊C8C2×C411yescharacteristic
35C2×C4⋊C8D411yescharacteristic
36C2×C4⋊C8C2311yescharacteristic
37C2×#101—41no—
38C2×C4×C8C2×C411yescharacteristic
39C2×C4×C8D411yescharacteristic
40C2×C4×C8D411yescharacteristic
41#398—41no—
42#395—41no—
43#389—21no—
44#389—21no—
45#389—21no—
46#389—21no—
47#389—41no—
48#389—41no—
49C4×C4⋊C4—21no—
50C4⋊C16—41no—
51C22⋊C16—22no—
52#346—41no—
53#343—41no—
54#340—21no—
55#340—21no—
56#340—21no—
57#340—21no—
58#340C2×C411yescharacteristic
59#340C2×C411yescharacteristic
60#335—41no—
61#335—41no—
62C22×Q8D811yescharacteristic
63C2×M16—42no—
64C22×C8C22⋊C411yescharacteristic
65C22×C8C22⋊C411yescharacteristic
66C22×C8C22×C411yescharacteristic
67C22×C8C2×D411yescharacteristic
68#125—42no—
69#125—42no—
70C4×Q8—41no—
71C4×Q8—41no—
72C2×C4⋊C4—21no—
73C2×C4⋊C4—21no—
74C2×C4⋊C4—21no—
75C2×C4⋊C4—21no—
76C2×C4⋊C4—41no—
77C2×C4⋊C4—41no—
78C2×C4⋊C4—41no—
79C2×C4⋊C4—41no—
80C2×C4⋊C4—41no—
81C2×C4⋊C4—41no—
82C2×C4⋊C4D811yescharacteristic
83C2×C4⋊C4SD1611yescharacteristic
84C2×C4⋊C4SD1611yescharacteristic
85C2×C42—41no—
86C2×C42C22⋊C411yescharacteristic
87C2×C42C22⋊C411yescharacteristic
88C2×C42C2×D411yescharacteristic
89C2×C16—42no—
90#107—41no—
91#107—41no—
92#106—24no—
93#106—41no—
94#106—41no—
95C4⋊C8—41no—
96C4⋊C8—41no—
97C4⋊C8—41no—
98C4⋊C8—42no—
99C4⋊C8—42no—
100#101—82no—
101C4×C8—21no—
102C4×C8—21no—
103C4×C8—21no—
104C4×C8—21no—
105C4×C8—21no—
106C4×C8—21no—
107#95—41no—
108#95—41no—
109#95—41no—
110#95—41no—
111#95—41no—
112C2×Q8—21no—
113C2×Q8—21no—
114C2×Q8—41no—
115C2×Q8—41no—
116C22×C4—41no—
117C22×C4—41no—
118C22×C4—41no—
119C22×C4—41no—
120C22×C4#10211yescharacteristic
121C22×C4#10211yescharacteristic
122C22×C4#10211yescharacteristic
123C22×C4#10211yescharacteristic
124C22×C4C2×C22⋊C411yescharacteristic
125C22×C4C2×D811yescharacteristic
126C22×C4C2×SD1611yescharacteristic
127M16—82no—
128M16—82no—
129C2×C8—22no—
130C2×C8—21no—
131C2×C8—21no—
132C2×C8—21no—
133C2×C8—41no—
134C2×C8—41no—
135C2×C8—41no—
136C2×C8—82no—
137C2×C8#9911yescharacteristic
138C2×C8#9911yescharacteristic
139C4⋊C4—22no—
140C4⋊C4—22no—
141C4⋊C4—41no—
142C4⋊C4—41no—
143C4⋊C4—41no—
144C4⋊C4—41no—
145C4⋊C4—41no—
146C4⋊C4—41no—
147C4⋊C4—41no—
148C4⋊C4—41no—
149C4⋊C4—41no—
150C4⋊C4—82no—
151C4⋊C4—82no—
152C4⋊C4—82no—
153C4⋊C4—81no—
154C4⋊C4—81no—
155C4⋊C4—161no—
156C4⋊C4—161no—
157C42—21no—
158C42—21no—
159C42—21no—
160C42—21no—
161C42—21no—
162C42—21no—
163C42—161no—
164C16—82no—
165C23C2×#10211yescharacteristic
166Q8—42no—
167Q8—42no—
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—41no—
182C2×C4—41no—
183C2×C4—41no—
184C2×C4—42no—
185C2×C4—42no—
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—81no—
195C2×C4—81no—
196C2×C4C2×#9911yescharacteristic
197C2×C4#42111yescharacteristic
198C2×C4#50911yescharacteristic
199C2×C4#51011yescharacteristic
200C8—21no—
201C8—41no—
202C8—41no—
203C8—41no—
204C8—162no—
205C8#35612yes—
206C22—21no—
207C22—21no—
208C22#151911yescharacteristic
209C22#213911yescharacteristic
210C22#214511yescharacteristic
211C4—21no—
212C4—21no—
213C4—21no—
214C4—41no—
215C4—41no—
216C4—41no—
217C4—41no—
218C4—41no—
219C4—82no—
220C4—82no—
221C4—161no—
222C4—161no—
223C4#212311yescharacteristic
224C4C2×#35611yescharacteristic
225C2—21no—
226C2—21no—
227C2#1187211yescharacteristic
228C2#1188311yescharacteristic
229C2#1189011yescharacteristic
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
#286782
#114942
#86172
#70912
#70892