Finite Group Databasebeta

← #125000

Subgroups — #125000

Order 512.

440 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 440 conjugacy classes of subgroups — too many to draw. Only its 78 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#14164C211yesmaximal, characteristic
2#8699C212yesmaximal
3#7122C214yesmaximal
4C2×#383C411yescharacteristic
5C2×C4×C4⋊C4C411yescharacteristic
6C2×#376C2211yescharacteristic
7#1817—22no—
8#1444C412yes—
9#1442C2212yes—
10#1436—22no—
11#1436—22no—
12#1280—22no—
13#1280C2214yes—
14C22×C4⋊C4—22no—
15C22×C4⋊C4C2×C412yes—
16C22×C42—21no—
17C22×C42C2×C411yescharacteristic
18C22×C42D412yes—
19C2×C22⋊C8—24no—
20#383—24no—
21C4×C4⋊C4—42no—
22#376—42no—
23C2×#95—21no—
24C2×#95—21no—
25C2×#95—21no—
26C2×#95—22no—
27C2×#95—22no—
28C2×#95—22no—
29C2×#95C2×C412yes—
30C2×#95C2×C411yescharacteristic
31C2×#95D412yes—
32C2×#95C2311yescharacteristic
33#336—44no—
34C23×C4—22no—
35C23×C4—22no—
36C23×C4C22⋊C412yes—
37C23×C4C22⋊C411yescharacteristic
38C23×C4C22⋊C411yescharacteristic
39C23×C4C22×C411yescharacteristic
40C23×C4C2×D412yes—
41C22×C8—44no—
42C2×C4⋊C4—42no—
43C2×C4⋊C4—44no—
44C2×C4⋊C4—44no—
45C2×C4⋊C4—44no—
46C2×C4⋊C4—82no—
47C2×C42—22no—
48C2×C42—22no—
49C2×C42—22no—
50C2×C42—24no—
51C2×C42—24no—
52C2×C42—42no—
53C2×C42—42no—
54C2×C42—81no—
55C2×C42—81no—
56C22⋊C8—44no—
57C22⋊C8—44no—
58#95—24no—
59#95—24no—
60#95—22no—
61#95—41no—
62#95—42no—
63#95—41no—
64#95—44no—
65#95—42no—
66#95—44no—
67#95—82no—
68#95M1614yes—
69C24C2×C22⋊C411yescharacteristic
70C22×C4—21no—
71C22×C4—21no—
72C22×C4—21no—
73C22×C4—21no—
74C22×C4—22no—
75C22×C4—21no—
76C22×C4—21no—
77C22×C4—22no—
78C22×C4—22no—
79C22×C4—22no—
80C22×C4—24no—
81C22×C4—24no—
82C22×C4—22no—
83C22×C4—22no—
84C22×C4—22no—
85C22×C4—41no—
86C22×C4—42no—
87C22×C4—41no—
88C22×C4—42no—
89C22×C4—42no—
90C22×C4—42no—
91C22×C4—42no—
92C22×C4—42no—
93C22×C4—42no—
94C22×C4—42no—
95C22×C4—42no—
96C22×C4—42no—
97C22×C4—42no—
98C22×C4—42no—
99C22×C4—42no—
100C22×C4—42no—
101C22×C4—42no—
102C22×C4—42no—
103C22×C4—42no—
104C22×C4#9912yes—
105C22×C4#10112yes—
106C22×C4C2×M1612yes—
107C22×C4#13514yes—
108C2×C8—84no—
109C2×C8—84no—
110C2×C8—84no—
111C4⋊C4—44no—
112C4⋊C4—44no—
113C4⋊C4—88no—
114C42—28no—
115C42—48no—
116C42—84no—
117C23—22no—
118C23—21no—
119C23—21no—
120C23#40711yescharacteristic
121C23C2×#9911yescharacteristic
122C23C2×#10111yescharacteristic
123C23#41612yes—
124C23#42112yes—
125C2×C4—22no—
126C2×C4—22no—
127C2×C4—22no—
128C2×C4—24no—
129C2×C4—24no—
130C2×C4—42no—
131C2×C4—42no—
132C2×C4—41no—
133C2×C4—41no—
134C2×C4—42no—
135C2×C4—42no—
136C2×C4—42no—
137C2×C4—41no—
138C2×C4—41no—
139C2×C4—42no—
140C2×C4—42no—
141C2×C4—42no—
142C2×C4—44no—
143C2×C4—44no—
144C2×C4—44no—
145C2×C4—44no—
146C2×C4—44no—
147C2×C4—44no—
148C2×C4—44no—
149C2×C4—42no—
150C2×C4—42no—
151C2×C4—44no—
152C2×C4—42no—
153C2×C4—82no—
154C2×C4—82no—
155C2×C4—82no—
156C2×C4—82no—
157C2×C4—82no—
158C2×C4—82no—
159C2×C4—82no—
160C2×C4—82no—
161C2×C4—82no—
162C2×C4—82no—
163C8—164no—
164C22—22no—
165C22—22no—
166C22—22no—
167C22—22no—
168C22—21no—
169C22—21no—
170C22—21no—
171C22—21no—
172C22#146311yescharacteristic
173C22#148812yes—
174C22#151612yes—
175C22#153711yescharacteristic
176C22#154411yescharacteristic
177C22#212012yes—
178C22#212911yescharacteristic
179C22#213411yescharacteristic
180C4—44no—
181C4—42no—
182C4—42no—
183C4—42no—
184C4—84no—
185C4—82no—
186C4—84no—
187C2—22no—
188C2—21no—
189C2—21no—
190C2#764611yescharacteristic
191C2#1151511yescharacteristic
192C2#1151811yescharacteristic
193C2#1184212yes—
194C2#1184712yes—
subgroupG/H
center, socleC23C2×#99
commutatorC23×C4C22×C4
Frattini, cosocleC2×#95C23
prime radicalC24C2×C22⋊C4
LCSG ▷ C23×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#376 ◁ G
LECSG ▷ C2×#95 ▷ C24 ▷ C2 ▷ C1

Maximal subgroups

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

subgroupindex
#141642
#86992
#71222