Finite Group Databasebeta

← #125563

Subgroups — #125563

Order 512.

256 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 256 conjugacy classes of subgroups — too many to draw. Only its 82 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#33257C211yesmaximal, characteristic
2#12894C211yesmaximal, characteristic
3#8785C211yesmaximal, characteristic
4#7121C212yesmaximal
5#7120C212yesmaximal
6#2864—21no—
7#1942C411yescharacteristic
8#1942C411yescharacteristic
9#1942C2211yescharacteristic
10#1942C2211yescharacteristic
11#1941—21no—
12#1941—21no—
13#1846C411yescharacteristic
14#1846C411yescharacteristic
15#1835—21no—
16#1831—21no—
17C42×C8C2211yescharacteristic
18#1374—21no—
19#1374—21no—
20#1374C2212yes—
21#1277—21no—
22#1277C2212yes—
23C22×C16—22no—
24C2×#107C2×C411yescharacteristic
25C2×#107C2×C411yescharacteristic
26C2×#107C2×C411yescharacteristic
27C2×#107C2×C411yescharacteristic
28C2×#106—21no—
29C2×#106—21no—
30C2×C4⋊C8—22no—
31C2×C4×C8C2×C411yescharacteristic
32C2×C4×C8C2×C411yescharacteristic
33C2×C4×C8D411yescharacteristic
34C2×C4×C8D411yescharacteristic
35C2×C4×C8D411yescharacteristic
36C2×C4×C8C2311yescharacteristic
37#389—21no—
38#389—21no—
39#389—21no—
40#389—21no—
41#387—41no—
42#387—41no—
43#387—41no—
44#387—41no—
45#382—21no—
46#382—21no—
47C43D411yescharacteristic
48#340—21no—
49#340—21no—
50#340—21no—
51#340—21no—
52#336—41no—
53#336—41no—
54C22×C8—42no—
55C22×C8C22⋊C411yescharacteristic
56C22×C8C22⋊C411yescharacteristic
57C22×C8C22×C411yescharacteristic
58C22×C8C2×D411yescharacteristic
59C2×C4⋊C4—21no—
60C2×C4⋊C4—21no—
61C2×C4⋊C4—21no—
62C2×C4⋊C4—21no—
63C2×C4⋊C4—41no—
64C2×C4⋊C4—41no—
65C2×C4⋊C4—41no—
66C2×C4⋊C4—41no—
67C2×C42C22⋊C411yescharacteristic
68C2×C42C22⋊C411yescharacteristic
69C2×C42D811yescharacteristic
70C2×C42D811yescharacteristic
71C2×C42SD1611yescharacteristic
72C2×C42SD1611yescharacteristic
73C2×C42C2×D411yescharacteristic
74C2×C16—42no—
75C2×C16—42no—
76C2×C16—42no—
77#107—22no—
78#107—22no—
79#107—41no—
80#107—41no—
81#106—42no—
82#106—42no—
83C4⋊C8—42no—
84C4⋊C8—42no—
85C4×C8—22no—
86C4×C8—22no—
87C4×C8—22no—
88C4×C8—41no—
89C4×C8—41no—
90C4×C8—41no—
91#95—41no—
92#95—41no—
93#95—41no—
94#95—41no—
95#95—41no—
96#95—41no—
97#95—41no—
98#95—41no—
99C22×C4—41no—
100C22×C4—41no—
101C22×C4—41no—
102C22×C4—41no—
103C22×C4#10211yescharacteristic
104C22×C4#10211yescharacteristic
105C22×C4#10211yescharacteristic
106C22×C4#10211yescharacteristic
107C22×C4C2×C22⋊C411yescharacteristic
108C22×C4C2×D811yescharacteristic
109C22×C4C2×SD1611yescharacteristic
110C2×C8—21no—
111C2×C8—21no—
112C2×C8—21no—
113C2×C8—21no—
114C2×C8—21no—
115C2×C8—21no—
116C2×C8—21no—
117C2×C8—21no—
118C2×C8—21no—
119C2×C8—82no—
120C2×C8—82no—
121C2×C8—82no—
122C2×C8#10112yes—
123C2×C8#13112yes—
124C2×C8#13712yes—
125C4⋊C4—41no—
126C4⋊C4—41no—
127C4⋊C4—41no—
128C4⋊C4—41no—
129C4⋊C4—41no—
130C4⋊C4—41no—
131C4⋊C4—41no—
132C4⋊C4—41no—
133C4⋊C4—82no—
134C4⋊C4—82no—
135C4⋊C4—82no—
136C4⋊C4—82no—
137C42—21no—
138C42—21no—
139C42—21no—
140C42—21no—
141C42—21no—
142C42—21no—
143C42—22no—
144C42—41no—
145C42—41no—
146C42—41no—
147C16—82no—
148C23C2×#10211yescharacteristic
149C2×C4—21no—
150C2×C4—21no—
151C2×C4—21no—
152C2×C4—21no—
153C2×C4—21no—
154C2×C4—21no—
155C2×C4—21no—
156C2×C4—21no—
157C2×C4—21no—
158C2×C4—21no—
159C2×C4—21no—
160C2×C4—21no—
161C2×C4—21no—
162C2×C4—21no—
163C2×C4—21no—
164C2×C4—21no—
165C2×C4—21no—
166C2×C4—81no—
167C2×C4—81no—
168C2×C4—81no—
169C2×C4—81no—
170C2×C4—81no—
171C2×C4—81no—
172C2×C4—81no—
173C2×C4—81no—
174C2×C4—81no—
175C2×C4—81no—
176C2×C4—81no—
177C2×C4—81no—
178C2×C4#40811yescharacteristic
179C2×C4#41011yescharacteristic
180C2×C4C2×#10111yescharacteristic
181C2×C4#41911yescharacteristic
182C2×C4#42111yescharacteristic
183C2×C4#42111yescharacteristic
184C2×C4#50812yes—
185C8—22no—
186C8—22no—
187C8—22no—
188C8—41no—
189C8—162no—
190C22#148011yescharacteristic
191C22#151711yescharacteristic
192C22#154611yescharacteristic
193C22#213811yescharacteristic
194C22#214311yescharacteristic
195C22#214511yescharacteristic
196C22#214811yescharacteristic
197C4—22no—
198C4—41no—
199C4—41no—
200C4—41no—
201C4—41no—
202C4—41no—
203C4—161no—
204C4—161no—
205C4—161no—
206C4—161no—
207C4#147011yescharacteristic
208C4#149711yescharacteristic
209C4#151511yescharacteristic
210C4#153411yescharacteristic
211C2#772311yescharacteristic
212C2#1164711yescharacteristic
213C2#1165611yescharacteristic
214C2#1185511yescharacteristic
215C2#1186611yescharacteristic
216C2#1197011yescharacteristic
217C2#1197311yescharacteristic
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
#332572
#128942
#87852
#71212
#71202