Finite Group Databasebeta

← #99259

Subgroups — #99259

Order 512.

534 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 534 conjugacy classes of subgroups — too many to draw. Only its 114 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#13954C211yesmaximal, characteristic
2#8560C212yesmaximal
3#7039C214yesmaximal
4C2×#389—22no—
5C2×#382C411yescharacteristic
6C2×C4×C4⋊C4C411yescharacteristic
7C2×C4×C4⋊C4C2211yescharacteristic
8#1797C2214yes—
9C2×#324—22no—
10#1444—22no—
11C4⋊#95C412yes—
12C4⋊#95C2212yes—
13#1280—22no—
14C22×C4⋊C4—21no—
15C22×C4⋊C4—21no—
16C22×C4⋊C4—22no—
17C22×C4⋊C4—22no—
18C22×C4⋊C4D412yes—
19C22×C42—21no—
20C22×C42C2×C411yescharacteristic
21C22×C42C2×C411yescharacteristic
22C22×C42C2311yescharacteristic
23C2×C4⋊C8—24no—
24C2×C22⋊C8—24no—
25#389—44no—
26#389—44no—
27#382—42no—
28C4×C4⋊C4—24no—
29C4×C4⋊C4—42no—
30C2×#95—22no—
31C2×#95—21no—
32C2×#95—22no—
33C2×#95C2×C412yes—
34C2×#95C2×C412yes—
35C2×#95D412yes—
36#336—24no—
37#324—44no—
38C23×C4—22no—
39C23×C4—22no—
40C23×C4C22⋊C411yescharacteristic
41C23×C4C22⋊C411yescharacteristic
42C23×C4C22⋊C412yes—
43C23×C4C22×C411yescharacteristic
44C23×C4C2×D412yes—
45C22×C8—44no—
46C2×C4⋊C4—24no—
47C2×C4⋊C4—24no—
48C2×C4⋊C4—44no—
49C2×C4⋊C4—42no—
50C2×C4⋊C4—44no—
51C2×C4⋊C4—44no—
52C2×C4⋊C4—42no—
53C2×C4⋊C4—44no—
54C2×C4⋊C4—44no—
55C2×C4⋊C4—42no—
56C2×C4⋊C4—42no—
57C2×C4⋊C4—82no—
58C2×C4⋊C4—82no—
59C2×C4⋊C4D814yes—
60C2×C4⋊C4SD1614yes—
61C2×C42—22no—
62C2×C42—22no—
63C2×C42—22no—
64C2×C42—22no—
65C2×C42—42no—
66C2×C42—44no—
67C2×C42—81no—
68C2×C42—81no—
69C2×C42#4114yes—
70C4⋊C8—84no—
71C22⋊C8—44no—
72C22⋊C8—44no—
73#95—24no—
74#95—42no—
75#95—42no—
76#95—42no—
77#95—44no—
78#95—42no—
79#95—42no—
80C24C2×C22⋊C411yescharacteristic
81C22×C4—21no—
82C22×C4—21no—
83C22×C4—22no—
84C22×C4—21no—
85C22×C4—22no—
86C22×C4—21no—
87C22×C4—21no—
88C22×C4—21no—
89C22×C4—22no—
90C22×C4—24no—
91C22×C4—41no—
92C22×C4—42no—
93C22×C4—42no—
94C22×C4—41no—
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—42no—
107C22×C4—42no—
108C22×C4—42no—
109C22×C4—42no—
110C22×C4—42no—
111C22×C4#10012yes—
112C22×C4#10112yes—
113C22×C4#10214yes—
114C22×C4#10214yes—
115C22×C4#11712yes—
116C22×C4#12314yes—
117C22×C4C2×D812yes—
118C22×C4C2×SD1612yes—
119C22×C4#13714yes—
120C2×C8—84no—
121C2×C8—84no—
122C2×C8—84no—
123C4⋊C4—28no—
124C4⋊C4—28no—
125C4⋊C4—48no—
126C4⋊C4—84no—
127C4⋊C4—84no—
128C4⋊C4—88no—
129C4⋊C4—84no—
130C42—28no—
131C42—44no—
132C42—82no—
133C42—84no—
134C23—22no—
135C23—21no—
136C23—21no—
137C23#38111yescharacteristic
138C23C2×#10011yescharacteristic
139C23C2×#10111yescharacteristic
140C23C2×#10212yes—
141C23#41912yes—
142C2×C4—24no—
143C2×C4—22no—
144C2×C4—22no—
145C2×C4—24no—
146C2×C4—24no—
147C2×C4—24no—
148C2×C4—22no—
149C2×C4—22no—
150C2×C4—24no—
151C2×C4—22no—
152C2×C4—22no—
153C2×C4—42no—
154C2×C4—42no—
155C2×C4—44no—
156C2×C4—42no—
157C2×C4—42no—
158C2×C4—42no—
159C2×C4—44no—
160C2×C4—42no—
161C2×C4—42no—
162C2×C4—42no—
163C2×C4—44no—
164C2×C4—42no—
165C2×C4—44no—
166C2×C4—44no—
167C2×C4—44no—
168C2×C4—82no—
169C2×C4—82no—
170C2×C4—82no—
171C2×C4—82no—
172C2×C4—82no—
173C2×C4—82no—
174C2×C4—82no—
175C2×C4—82no—
176C2×C4—82no—
177C2×C4—82no—
178C2×C4—82no—
179C2×C4#33114yes—
180C2×C4#48014yes—
181C2×C4#48314yes—
182C8—164no—
183C22—21no—
184C22—21no—
185C22—21no—
186C22—21no—
187C22—22no—
188C22—22no—
189C22—22no—
190C22—22no—
191C22C2×#33111yescharacteristic
192C22#154612yes—
193C22#155811yescharacteristic
194C22#182111yescharacteristic
195C22#182312yes—
196C22#182712yes—
197C22#182811yescharacteristic
198C22#183011yescharacteristic
199C4—24no—
200C4—24no—
201C4—24no—
202C4—42no—
203C4—42no—
204C4—44no—
205C4—84no—
206C4—162no—
207C2—21no—
208C2—22no—
209C2—21no—
210C2#872311yescharacteristic
211C2#874311yescharacteristic
212C2#875711yescharacteristic
213C2#876012yes—
214C2#876612yes—
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
#139542
#85602
#70392