Finite Group Databasebeta

← #135549

Subgroups — #135549

Order 512.

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

normal subgroups only

This group has 642 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#8544C212yesmaximal
3#7307C214yesmaximal
4C22×#124C2211yescharacteristic
5C2×#383C411yescharacteristic, complemented
6C2×C4×C4⋊C4C411yescharacteristic, complemented
7#1819—22no—
8#1819—22no—
9#1781C2214yes—
10#1446C412yescomplemented
11#1439—22no—
12#1438C2212yes—
13#1437—22no—
14C23×Q8C2×C411yescharacteristic
15C23×D4C2311yescharacteristic
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×#99—28no—
26#383—42no—
27#379—24no—
28C4×C4⋊C4—42no—
29C2×#95—21no—
30C2×#95—21no—
31C2×#95—24no—
32C2×#95C2×C412yes—
33C2×#95C2×C412yes—
34#342—44no—
35#342—44no—
36C25C2×D412yes—
37C22×Q8—22no—
38C22×Q8—41no—
39C22×Q8—41no—
40C22×Q8—82no—
41C22×D4—41no—
42C22×D4—41no—
43C22×D4—82no—
44C22×D4#4114yes—
45C23×C4—22no—
46C23×C4—22no—
47C23×C4C22⋊C412yes—
48C23×C4C22⋊C411yescharacteristic
49C23×C4C22⋊C411yescharacteristic
50C23×C4C22×C411yescharacteristic
51#124—416no—
52C2×C4⋊C4—44no—
53C2×C4⋊C4—42no—
54C2×C22⋊C4—24no—
55C2×C22⋊C4—24no—
56C2×C22⋊C4—24no—
57C2×C22⋊C4—24no—
58C2×C22⋊C4—42no—
59C2×C22⋊C4—42no—
60C2×C22⋊C4—48no—
61C2×C22⋊C4—44no—
62C2×C22⋊C4—44no—
63C2×C22⋊C4—44no—
64C2×C22⋊C4—84no—
65C2×C22⋊C4—84no—
66C2×C42—22no—
67C2×C42—22no—
68C2×C42—22no—
69C2×C42—42no—
70C2×C42—82no—
71#99—88no—
72#95—42no—
73#95—42no—
74#95—42no—
75#95—42no—
76#95—84no—
77C24—22no—
78C24—22no—
79C24—44no—
80C24—44no—
81C24—42no—
82C24—42no—
83C24C2×C22⋊C411yescharacteristic
84C24#12314yes—
85C2×Q8—44no—
86C2×Q8—84no—
87C2×Q8—84no—
88C2×Q8—84no—
89C2×D4—48no—
90C2×D4—44no—
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—24no—
100C22×C4—22no—
101C22×C4—22no—
102C22×C4—22no—
103C22×C4—42no—
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—44no—
114C22×C4—42no—
115C22×C4—42no—
116C22×C4—42no—
117C22×C4—84no—
118C22×C4#9912yes—
119C22×C4#9912yes—
120C22×C4#11712yes—
121C4⋊C4—48no—
122C22⋊C4—48no—
123C22⋊C4—48no—
124C22⋊C4—816no—
125C22⋊C4—88no—
126C22⋊C4—88no—
127C22⋊C4—84no—
128C22⋊C4—84no—
129C42—28no—
130C42—84no—
131C23—22no—
132C23—22no—
133C23—21no—
134C23—21no—
135C23—24no—
136C23—28no—
137C23—44no—
138C23—44no—
139C23—44no—
140C23—42no—
141C23—42no—
142C23—82no—
143C23—82no—
144C23—82no—
145C23—82no—
146C23—84no—
147C23—84no—
148C23—88no—
149C23#38111yescharacteristic
150C23C2×#9911yescharacteristic
151C23C2×#9911yescharacteristic
152Q8—88no—
153D4—88no—
154C2×C4—22no—
155C2×C4—22no—
156C2×C4—22no—
157C2×C4—24no—
158C2×C4—42no—
159C2×C4—42no—
160C2×C4—42no—
161C2×C4—42no—
162C2×C4—42no—
163C2×C4—41no—
164C2×C4—41no—
165C2×C4—42no—
166C2×C4—42no—
167C2×C4—48no—
168C2×C4—44no—
169C2×C4—42no—
170C2×C4—44no—
171C2×C4—84no—
172C2×C4—82no—
173C2×C4—82no—
174C2×C4—82no—
175C2×C4—82no—
176C2×C4—84no—
177C2×C4—84no—
178C2×C4—82no—
179C2×C4—82no—
180C2×C4—82no—
181C2×C4—164no—
182C2×C4—164no—
183C2×C4—164no—
184C22—24no—
185C22—24no—
186C22—21no—
187C22—21no—
188C22—21no—
189C22—21no—
190C22—21no—
191C22—21no—
192C22—44no—
193C22—44no—
194C22—44no—
195C22—88no—
196C22—162no—
197C22—164no—
198C22—162no—
199C22#181911yescharacteristic
200C22#182812yes—
201C22#212612yes—
202C22#212811yescharacteristic
203C22#212911yescharacteristic
204C4—42no—
205C4—42no—
206C4—42no—
207C4—84no—
208C4—82no—
209C4—162no—
210C4—324no—
211C2—21no—
212C2—22no—
213C2—22no—
214C2—21no—
215C2—84no—
216C2#874911yescharacteristic
217C2#1285711yescharacteristic
218C2#1285911yescharacteristic
subgroupG/H
center, socleC22#1819
commutatorC23×C4C22×C4
Frattini, prime radical, cosocleC23×D4C23
LCSG ▷ C23×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C24 ◁ C22×#124 ◁ G
LECSG ▷ C23×D4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
5. C2×#383C4semidirect
6. C2×C4×C4⋊C4C4semidirect
10. #1446C4semidirect

Maximal subgroups

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

subgroupindex
#142882
#85442
#73072