Finite Group Databasebeta

← #97029

Subgroups — #97029

Order 512.

732 conjugacy classes of subgroups; only the 130 normal subgroups are shown.

normal subgroups only

This group has 732 conjugacy classes of subgroups — too many to draw. Only its 130 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
1C2×#2330C211yesmaximal, characteristic
2#8699C211yesmaximal, characteristic
3#8699C211yesmaximal, characteristic
4#7593C214yesmaximal
5C22×#125C411yescharacteristic
6#2330—28no—
7C2×#389—22no—
8C2×#389C2212yes—
9C2×#387—22no—
10C2×#384C411yescharacteristic
11C2×C4×C4⋊C4C412yes—
12C2×#376C2211yescharacteristic
13#1817C2212yes—
14#1817C2212yes—
15#1280—22no—
16#1280—22no—
17#1271—28no—
18C2×#125—24no—
19C2×#125—24no—
20C2×#125—22no—
21C2×#125C814yescomplemented
22C22×C4⋊C4—22no—
23C22×C4⋊C4—22no—
24C22×C4⋊C4—22no—
25C22×C4⋊C4C2×C412yes—
26C22×C4⋊C4C2×C412yes—
27C22×C42—22no—
28C22×C42C2×C411yescharacteristic
29C22×C42C2×C411yescharacteristic
30C22×C42C2311yescharacteristic
31C2×C22⋊C8—22no—
32C2×C22⋊C8—22no—
33#389—216no—
34#389—24no—
35#389D418yes—
36#387—216no—
37#384—24no—
38C4×C4⋊C4—24no—
39C4×C4⋊C4—24no—
40#376—24no—
41C2×#95—22no—
42C2×#95—22no—
43C2×#95D412yes—
44C2×#95D412yes—
45#336—24no—
46#336—24no—
47C23×C4—22no—
48C23×C4—22no—
49C23×C4C22⋊C412yes—
50C23×C4C22⋊C412yes—
51C23×C4C22×C411yescharacteristic
52C23×C4C2×D411yescharacteristic
53C23×C4C2×D411yescharacteristic
54C22×C8—42no—
55C22×C8—42no—
56#125—216no—
57#125—216no—
58C2×C4⋊C4—28no—
59C2×C4⋊C4—28no—
60C2×C4⋊C4—28no—
61C2×C4⋊C4—22no—
62C2×C4⋊C4—22no—
63C2×C4⋊C4—22no—
64C2×C4⋊C4—22no—
65C2×C4⋊C4—28no—
66C2×C4⋊C4—28no—
67C2×C4⋊C4—44no—
68C2×C4⋊C4—84no—
69C2×C4⋊C4—84no—
70C2×C4⋊C4C22⋊C414yes—
71C2×C4⋊C4C22⋊C414yes—
72C2×C4⋊C4C2×C814yes—
73C2×C4⋊C4M1614yes—
74C2×C42—24no—
75C2×C42—22no—
76C2×C42—22no—
77C2×C42—22no—
78C2×C42—28no—
79C2×C42—42no—
80C2×C42—84no—
81C2×C42C2×C812yes—
82C2×C42M1612yes—
83C2×C42C2×D414yes—
84C22⋊C8—44no—
85C22⋊C8—44no—
86#95—28no—
87#95—28no—
88#95—24no—
89#95—24no—
90C24C2×C22⋊C411yescharacteristic
91C22×C4—22no—
92C22×C4—22no—
93C22×C4—22no—
94C22×C4—22no—
95C22×C4—24no—
96C22×C4—24no—
97C22×C4—22no—
98C22×C4—24no—
99C22×C4—22no—
100C22×C4—22no—
101C22×C4—22no—
102C22×C4—24no—
103C22×C4—42no—
104C22×C4—42no—
105C22×C4—44no—
106C22×C4—44no—
107C22×C4—42no—
108C22×C4—42no—
109C22×C4—42no—
110C22×C4—42no—
111C22×C4—44no—
112C22×C4—44no—
113C22×C4—44no—
114C22×C4C22⋊C814yes—
115C22×C4#10114yes—
116C22×C4C2×C22⋊C412yes—
117C22×C4#12012yes—
118C22×C4#12012yes—
119C22×C4C22×C811yescharacteristic
120C22×C4C2×M1612yes—
121C22×C4C2×M1611yescharacteristic
122C2×C8—82no—
123C2×C8—82no—
124C2×C8—84no—
125C2×C8—84no—
126C4⋊C4—232no—
127C4⋊C4—24no—
128C4⋊C4—24no—
129C4⋊C4—24no—
130C4⋊C4—44no—
131C4⋊C4—44no—
132C4⋊C4—88no—
133C4⋊C4—88no—
134C4⋊C4C22⋊C818yes—
135C42—28no—
136C42—24no—
137C42—24no—
138C42—44no—
139C42—88no—
140C23—22no—
141C23—22no—
142C23#37911yescharacteristic
143C23C2×C22⋊C811yescharacteristic
144C23#40711yescharacteristic
145C23#41012yes—
146C23C2×#10112yes—
147C2×C4—22no—
148C2×C4—22no—
149C2×C4—24no—
150C2×C4—22no—
151C2×C4—22no—
152C2×C4—24no—
153C2×C4—24no—
154C2×C4—24no—
155C2×C4—28no—
156C2×C4—22no—
157C2×C4—22no—
158C2×C4—24no—
159C2×C4—44no—
160C2×C4—44no—
161C2×C4—48no—
162C2×C4—48no—
163C2×C4—42no—
164C2×C4—44no—
165C2×C4—42no—
166C2×C4—42no—
167C2×C4—42no—
168C2×C4—44no—
169C2×C4—44no—
170C2×C4—44no—
171C2×C4—44no—
172C2×C4—44no—
173C2×C4—44no—
174C2×C4—82no—
175C2×C4—82no—
176C2×C4—88no—
177C2×C4C2×C22⋊C812yes—
178C2×C4#40712yes—
179C8—162no—
180C8—162no—
181C22—22no—
182C22—22no—
183C22—22no—
184C22—22no—
185C22—22no—
186C22—22no—
187C22#146112yes—
188C22#146712yes—
189C22#177911yescharacteristic
190C22#178211yescharacteristic
191C22#178411yescharacteristic
192C22#179512yes—
193C22#179612yes—
194C4—22no—
195C4—22no—
196C4—42no—
197C4—42no—
198C4—44no—
199C4—44no—
200C4—48no—
201C4—84no—
202C2—22no—
203C2—22no—
204C2#830311yescharacteristic
205C2#832412yes—
206C2#851811yescharacteristic
207C2#852111yescharacteristic
208C2#852512yes—
subgroupG/H
center, socleC23#379
commutatorC22×C4C22×C8
Frattini, cosocleC22×C42C23
prime radicalC24C2×C22⋊C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
21. C2×#125C8semidirect

Maximal subgroups

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

subgroupindex
C2×#23302
#86992
#75932