Finite Group Databasebeta

← #99271

Subgroups — #99271

Order 512.

462 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 462 conjugacy classes of subgroups — too many to draw. Only its 94 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#13964C211yesmaximal, characteristic
2#8560C212yesmaximal
3#7039C214yesmaximal
4C2×#383C411yescharacteristic
5C2×C4×C4⋊C4C2211yescharacteristic
6C2×#376C411yescharacteristic
7#1797C2214yes—
8C2×#324—22no—
9#1441—22no—
10#1436—22no—
11C4⋊#95C2212yes—
12#1434C412yes—
13#1280—22no—
14C22×C4⋊C4—22no—
15C22×C4⋊C4D412yes—
16C22×C42—21no—
17C22×C42C2×C411yescharacteristic
18C22×C42C2×C411yescharacteristic
19C22×C42C2311yescharacteristic
20C2×C4⋊C8—24no—
21C2×C22⋊C8—24no—
22#383—42no—
23C4×C4⋊C4—24no—
24#376—42no—
25C2×#95—22no—
26C2×#95—21no—
27C2×#95—21no—
28C2×#95—21no—
29C2×#95—22no—
30C2×#95—22no—
31C2×#95C2×C412yes—
32C2×#95C2×C412yes—
33C2×#95D412yes—
34#336—24no—
35#324—44no—
36C23×C4—22no—
37C23×C4—22no—
38C23×C4C22⋊C411yescharacteristic
39C23×C4C22⋊C412yes—
40C23×C4C22⋊C411yescharacteristic
41C23×C4C22×C411yescharacteristic
42C23×C4C2×D412yes—
43C22×C8—44no—
44C2×C4⋊C4—24no—
45C2×C4⋊C4—24no—
46C2×C4⋊C4—22no—
47C2×C4⋊C4—22no—
48C2×C4⋊C4—44no—
49C2×C4⋊C4—44no—
50C2×C4⋊C4—82no—
51C2×C42—22no—
52C2×C42—22no—
53C2×C42—22no—
54C2×C42—22no—
55C2×C42—42no—
56C2×C42—44no—
57C2×C42—81no—
58C2×C42—81no—
59C2×C42#4114yes—
60C4⋊C8—84no—
61C22⋊C8—44no—
62C22⋊C8—44no—
63#95—24no—
64#95—42no—
65#95—42no—
66#95—44no—
67#95—42no—
68#95—42no—
69#95—42no—
70#95—42no—
71#95—42no—
72#95—82no—
73C24C2×C22⋊C411yescharacteristic
74C22×C4—21no—
75C22×C4—21no—
76C22×C4—21no—
77C22×C4—21no—
78C22×C4—21no—
79C22×C4—21no—
80C22×C4—22no—
81C22×C4—22no—
82C22×C4—24no—
83C22×C4—22no—
84C22×C4—22no—
85C22×C4—22no—
86C22×C4—41no—
87C22×C4—42no—
88C22×C4—42no—
89C22×C4—41no—
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—42no—
105C22×C4—42no—
106C22×C4#10012yes—
107C22×C4#10112yes—
108C22×C4#11712yes—
109C22×C4#12314yes—
110C22×C4#13514yes—
111C22×C4AGL(1,ℤ/8)12yes—
112C22×C4#13712yes—
113C2×C8—84no—
114C2×C8—84no—
115C2×C8—84no—
116C4⋊C4—44no—
117C4⋊C4—44no—
118C4⋊C4—88no—
119C42—28no—
120C42—44no—
121C42—48no—
122C42—82no—
123C23—22no—
124C23—21no—
125C23—21no—
126C23#38111yescharacteristic
127C23C2×#10011yescharacteristic
128C23C2×#10111yescharacteristic
129C23#41612yes—
130C23#41712yes—
131C2×C4—24no—
132C2×C4—24no—
133C2×C4—24no—
134C2×C4—22no—
135C2×C4—22no—
136C2×C4—22no—
137C2×C4—22no—
138C2×C4—22no—
139C2×C4—44no—
140C2×C4—42no—
141C2×C4—42no—
142C2×C4—42no—
143C2×C4—42no—
144C2×C4—42no—
145C2×C4—41no—
146C2×C4—41no—
147C2×C4—42no—
148C2×C4—42no—
149C2×C4—42no—
150C2×C4—44no—
151C2×C4—44no—
152C2×C4—44no—
153C2×C4—42no—
154C2×C4—42no—
155C2×C4—42no—
156C2×C4—82no—
157C2×C4—82no—
158C2×C4—82no—
159C2×C4—82no—
160C2×C4—82no—
161C2×C4—82no—
162C2×C4—82no—
163C2×C4—82no—
164C2×C4—82no—
165C2×C4—82no—
166C2×C4—82no—
167C2×C4—82no—
168C2×C4—82no—
169C2×C4#48214yes—
170C2×C4#48514yes—
171C8—164no—
172C22—21no—
173C22—21no—
174C22—21no—
175C22—21no—
176C22—22no—
177C22—22no—
178C22—22no—
179C22—22no—
180C22#154411yescharacteristic
181C22#155112yes—
182C22#156111yescharacteristic
183C22#182111yescharacteristic
184C22#182512yes—
185C22#182712yes—
186C22#182811yescharacteristic
187C22#183011yescharacteristic
188C4—24no—
189C4—24no—
190C4—42no—
191C4—42no—
192C4—42no—
193C4—84no—
194C4—82no—
195C4—162no—
196C2—22no—
197C2—21no—
198C2—21no—
199C2#872911yescharacteristic
200C2#874311yescharacteristic
201C2#875711yescharacteristic
202C2#876312yes—
203C2#876812yes—
subgroupG/H
center, socleC23#381
commutatorC23×C4C22×C4
Frattini, cosocleC22×C42C23
prime radicalC24C2×C22⋊C4
LCSG ▷ C23×C4 ▷ C23 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C23 ▷ C1

Maximal subgroups

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

subgroupindex
#139642
#85602
#70392