Finite Group Databasebeta

← #135551

Subgroups — #135551

Order 512.

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

normal subgroups only

This group has 608 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#14288C211yesmaximal, characteristic
2#8552C212yesmaximal
3#7308C214yesmaximal, complemented
4C22×#124C2211yescharacteristic
5C2×#383C411yescharacteristic
6C2×C4×C4⋊C4C411yescharacteristic
7#1784C2214yes—
8#1462—22no—
9#1462—22no—
10#1446C2212yes—
11#1439—22no—
12#1438C412yes—
13#1437—22no—
14C23×Q8C2311yescharacteristic
15C23×D4C2×C411yescharacteristic
16C2×#124—24no—
17C2×#124—42no—
18C2×#124—42no—
19C22×C4⋊C4—21no—
20C22×C22⋊C4—24no—
21C22×C22⋊C4D412yes—
22C22×C22⋊C4D412yes—
23C22×C42—21no—
24C22×C42C2×C411yescharacteristic
25C2×#101—28no—
26#407—24no—
27#383—42no—
28C4×C4⋊C4—42no—
29C2×#95—21no—
30C2×#95—21no—
31C2×#95—24no—
32C2×#95C2×C412yes—
33C2×#95C2×C412yes—
34#323—44no—
35#323—44no—
36C25C22⋊C412yes—
37C22×Q8—41no—
38C22×Q8—41no—
39C22×Q8—82no—
40C22×Q8#4114yes—
41C22×D4—22no—
42C22×D4—41no—
43C22×D4—41no—
44C22×D4—82no—
45C23×C4—22no—
46C23×C4—22no—
47C23×C4C22⋊C411yescharacteristic
48C23×C4C22⋊C411yescharacteristic
49C23×C4C22×C411yescharacteristic
50C23×C4C2×D412yes—
51C2×M16—44no—
52#124—416no—
53C2×C4⋊C4—44no—
54C2×C4⋊C4—42no—
55C2×C22⋊C4—24no—
56C2×C22⋊C4—24no—
57C2×C22⋊C4—24no—
58C2×C22⋊C4—24no—
59C2×C22⋊C4—42no—
60C2×C22⋊C4—42no—
61C2×C22⋊C4—48no—
62C2×C22⋊C4—84no—
63C2×C22⋊C4—84no—
64C2×C42—22no—
65C2×C42—22no—
66C2×C42—22no—
67C2×C42—42no—
68C2×C42—82no—
69#101—88no—
70C22⋊C8—44no—
71C22⋊C8—44no—
72#95—42no—
73#95—42no—
74#95—42no—
75#95—42no—
76#95—84no—
77C24—22no—
78C24—22no—
79C24—22no—
80C24—44no—
81C24—42no—
82C24—42no—
83C24—44no—
84C24C2×C22⋊C411yescharacteristic
85C2×Q8—48no—
86C2×Q8—44no—
87C2×Q8—84no—
88C2×Q8—84no—
89C2×D4—44no—
90C2×D4—84no—
91C2×D4—84no—
92C2×D4—84no—
93C22×C4—21no—
94C22×C4—21no—
95C22×C4—21no—
96C22×C4—21no—
97C22×C4—21no—
98C22×C4—21no—
99C22×C4—22no—
100C22×C4—22no—
101C22×C4—24no—
102C22×C4—42no—
103C22×C4—44no—
104C22×C4—41no—
105C22×C4—42no—
106C22×C4—41no—
107C22×C4—44no—
108C22×C4—44no—
109C22×C4—42no—
110C22×C4—42no—
111C22×C4—44no—
112C22×C4—44no—
113C22×C4—42no—
114C22×C4—42no—
115C22×C4—42no—
116C22×C4#9912yes—
117C22×C4#9912yes—
118C22×C4#11712yes—
119C22×C4#12314yes—
120M16—88no—
121C2×C8—84no—
122C4⋊C4—48no—
123C22⋊C4—48no—
124C22⋊C4—48no—
125C22⋊C4—816no—
126C42—28no—
127C42—84no—
128C23—22no—
129C23—22no—
130C23—21no—
131C23—21no—
132C23—44no—
133C23—44no—
134C23—42no—
135C23—44no—
136C23—44no—
137C23—42no—
138C23—42no—
139C23—84no—
140C23—84no—
141C23—88no—
142C23—82no—
143C23—82no—
144C23—82no—
145C23—82no—
146C23#38111yescharacteristic
147C23C2×#9911yescharacteristic
148C23C2×#9911yescharacteristic
149Q8—88no—
150D4—88no—
151C2×C4—22no—
152C2×C4—22no—
153C2×C4—22no—
154C2×C4—24no—
155C2×C4—28no—
156C2×C4—24no—
157C2×C4—42no—
158C2×C4—42no—
159C2×C4—42no—
160C2×C4—42no—
161C2×C4—42no—
162C2×C4—41no—
163C2×C4—41no—
164C2×C4—42no—
165C2×C4—42no—
166C2×C4—48no—
167C2×C4—44no—
168C2×C4—82no—
169C2×C4—82no—
170C2×C4—84no—
171C2×C4—82no—
172C2×C4—82no—
173C2×C4—84no—
174C2×C4—84no—
175C2×C4—82no—
176C2×C4—82no—
177C2×C4—82no—
178C8—164no—
179C22—24no—
180C22—24no—
181C22—21no—
182C22—21no—
183C22—21no—
184C22—21no—
185C22—21no—
186C22—21no—
187C22—88no—
188C22—82no—
189C22—82no—
190C22—82no—
191C22—162no—
192C22—164no—
193C22—162no—
194C22#181911yescharacteristic
195C22#182812yes—
196C22#213212yes—
197C22#213411yescharacteristic
198C22#213511yescharacteristic
199C4—44no—
200C4—42no—
201C4—42no—
202C4—42no—
203C4—84no—
204C4—162no—
205C2—21no—
206C2—22no—
207C2—22no—
208C2—21no—
209C2—162no—
210C2#874911yescharacteristic
211C2#1286111yescharacteristic
212C2#1286311yescharacteristic
subgroupG/H
center, socleC22#1819
commutatorC23×C4C22×C4
Frattini, cosocleC23×Q8C23
prime radicalC23×D4C2×C4
LCSG ▷ C23×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C24 ◁ C22×#124 ◁ G
LECSG ▷ C23×Q8 ▷ C23 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
3. #7308C2semidirect

Maximal subgroups

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

subgroupindex
#142882
#85522
#73082