Finite Group Databasebeta

← #135535

Subgroups — #135535

Order 512.

956 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 956 conjugacy classes of subgroups — too many to draw. Only its 74 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#14154C211yesmaximal, characteristic
2#8546C212yesmaximal
3#7307C214yesmaximal, complemented
4C22×#127C2211yescharacteristic
5C2×#390—24no—
6C2×C4×C4⋊C4C412yescomplemented
7#1781—24no—
8#1781C2214yes—
9#1438C412yescomplemented
10#1438C2212yes—
11C23×D4C2×C411yescharacteristic
12C23×D4D414yes—
13C23×D4C2311yescharacteristic
14C2×#127—24no—
15C2×#127—42no—
16C2×#127—42no—
17C22×C4⋊C4—22no—
18C22×C22⋊C4—24no—
19C22×C22⋊C4—24no—
20C22×C42—22no—
21C22×C42C2×C411yescharacteristic
22C2×#99—28no—
23C2×#99—44no—
24C2×#99—44no—
25#390—416no—
26#379—24no—
27C4×C4⋊C4—42no—
28C4×C4⋊C4—42no—
29C2×#95C2×C414yes—
30C25C22⋊C412yes—
31C25C2×D412yes—
32C22×D4—28no—
33C22×D4—24no—
34C22×D4—24no—
35C22×D4—22no—
36C22×D4—48no—
37C22×D4—48no—
38C22×D4—42no—
39C22×D4—44no—
40C22×D4—42no—
41C22×D4—82no—
42C22×D4—82no—
43C22×D4#4114yes—
44C23×C4—24no—
45C23×C4C22⋊C412yes—
46C23×C4C22×C411yescharacteristic
47#127—44no—
48#127—48no—
49#127—44no—
50C2×C4⋊C4—28no—
51C2×C4⋊C4—44no—
52C2×C4⋊C4—44no—
53C2×C22⋊C4—416no—
54C2×C22⋊C4—416no—
55C2×C22⋊C4—48no—
56C2×C22⋊C4—44no—
57C2×C22⋊C4—84no—
58C2×C22⋊C4—84no—
59C2×C42—22no—
60C2×C42—22no—
61C2×C42—22no—
62C2×C42—48no—
63C2×C42—42no—
64C2×C42—42no—
65#99—88no—
66#99—88no—
67#99—88no—
68#95—44no—
69C24—24no—
70C24—22no—
71C24—24no—
72C24—44no—
73C24—44no—
74C24—44no—
75C24—44no—
76C24—44no—
77C24—44no—
78C24C2×C22⋊C411yescharacteristic
79C24#12314yes—
80C2×D4—48no—
81C2×D4—44no—
82C2×D4—48no—
83C2×D4—48no—
84C2×D4—44no—
85C2×D4—432no—
86C2×D4—48no—
87C2×D4—48no—
88C2×D4—84no—
89C2×D4—84no—
90C2×D4—84no—
91C2×D4—816no—
92C2×D4—88no—
93C2×D4—88no—
94C22×C4—22no—
95C22×C4—22no—
96C22×C4—24no—
97C22×C4—24no—
98C22×C4—22no—
99C22×C4—28no—
100C22×C4—42no—
101C22×C4—48no—
102C22×C4—48no—
103C22×C4—44no—
104C22×C4—44no—
105C22×C4—84no—
106C22×C4#9912yes—
107C22×C4#9912yes—
108C22×C4#11712yes—
109C4⋊C4—416no—
110C22⋊C4—816no—
111C22⋊C4—816no—
112C22⋊C4—88no—
113C22⋊C4—88no—
114C22⋊C4—88no—
115C42—22no—
116C42—24no—
117C42—22no—
118C42—84no—
119C42—84no—
120C23—22no—
121C23—22no—
122C23—22no—
123C23—24no—
124C23—28no—
125C23—48no—
126C23—48no—
127C23—44no—
128C23—44no—
129C23—42no—
130C23—44no—
131C23—44no—
132C23—44no—
133C23—84no—
134C23—84no—
135C23—84no—
136C23—84no—
137C23—88no—
138C23—88no—
139C23—84no—
140C23—84no—
141C23—84no—
142C23—84no—
143C23#38111yescharacteristic
144C23C2×#9912yes—
145D4—816no—
146D4—816no—
147D4—88no—
148D4—88no—
149C2×C4—28no—
150C2×C4—28no—
151C2×C4—22no—
152C2×C4—22no—
153C2×C4—22no—
154C2×C4—22no—
155C2×C4—22no—
156C2×C4—24no—
157C2×C4—44no—
158C2×C4—416no—
159C2×C4—48no—
160C2×C4—88no—
161C2×C4—84no—
162C2×C4—84no—
163C2×C4—164no—
164C2×C4—168no—
165C2×C4#35314yes—
166C2×C4#35414yes—
167C22—22no—
168C22—24no—
169C22—22no—
170C22—22no—
171C22—22no—
172C22—22no—
173C22—44no—
174C22—48no—
175C22—88no—
176C22—88no—
177C22—82no—
178C22—84no—
179C22—164no—
180C22—164no—
181C22—164no—
182C22—164no—
183C22#181911yescharacteristic
184C22#182812yes—
185C22C2×#35312yes—
186C22C2×#35412yes—
187C4—24no—
188C4—24no—
189C4—42no—
190C4—88no—
191C4—324no—
192C2—22no—
193C2—22no—
194C2—22no—
195C2—84no—
196C2—162no—
197C2#874811yescharacteristic
198C2#1285612yes—
subgroupG/H
center, socleC22#1819
commutatorC23×C4C22×C4
Frattini, cosocleC23×D4C23
prime radicalC22×#127C22
LCSG ▷ C23×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C24 ◁ C22×#127 ◁ G
LECSG ▷ C23×D4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #7307C2semidirect
6. C2×C4×C4⋊C4C4semidirect
9. #1438C4semidirect

Maximal subgroups

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

subgroupindex
#141542
#85462
#73072