Finite Group Databasebeta

← #135463

Subgroups — #135463

Order 512.

375 conjugacy classes of subgroups; only the 116 normal subgroups are shown.

normal subgroups only

This group has 375 conjugacy classes of subgroups — too many to draw. Only its 116 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#20543C211yesmaximal, characteristic
2C2×#1322C212yesmaximal
3#7199C214yesmaximal
4C2×#431C411yescharacteristic
5C22×C4⋊C8C2211yescharacteristic
6C4×#117C411yescharacteristic
7C2×C22⋊C16C2214yes—
8C8×C4⋊C4—22no—
9C4×C4⋊C8—22no—
10C8×C22⋊C4C412yes—
11C8×C22⋊C4C2212yes—
12#1322—22no—
13#1322—22no—
14C23×C8C2×C411yescharacteristic
15C23×C8C2311yescharacteristic
16C2×#117C811yescharacteristic
17C2×#117C811yescharacteristic
18C22×C42C2×C411yescharacteristic
19C22×C16—24no—
20#431—24no—
21C2×C4⋊C8—24no—
22C2×C4⋊C8—21no—
23C2×C4⋊C8D412yes—
24C2×C4⋊C8D412yes—
25C2×C22⋊C8C2×C412yes—
26C2×C4×C8—21no—
27C2×C4×C8—22no—
28C2×C4×C8—22no—
29C4×C4⋊C4—21no—
30C4×C4⋊C4—21no—
31C4×C22⋊C4C812yes—
32C4×C22⋊C4C2×C412yes—
33#376—21no—
34C43—21no—
35C22⋊C16—24no—
36C22⋊C16—24no—
37#336—22no—
38#336—22no—
39C23×C4C2×C811yescharacteristic
40C23×C4C2×C811yescharacteristic
41C23×C4C22×C411yescharacteristic
42C22×C8—22no—
43C22×C8—22no—
44C22×C8—81no—
45C22×C8—81no—
46C22×C8C22⋊C412yes—
47C22×C8C2×D412yes—
48C22×C8#4114yes—
49#117—22no—
50#117—24no—
51#117C1614yescomplemented
52C2×C4⋊C4—21no—
53C2×C4⋊C4—21no—
54C2×C22⋊C4C2×C812yes—
55C2×C22⋊C4C2×C812yes—
56C2×C42—21no—
57C2×C42—21no—
58C2×C42—22no—
59C2×C42—22no—
60C2×C42—22no—
61C2×C42—22no—
62C2×C42C22⋊C411yescharacteristic
63C2×C42C22⋊C411yescharacteristic
64C2×C42M1611yescharacteristic
65C2×C42M1611yescharacteristic
66C2×C16—44no—
67C2×C16—84no—
68C4⋊C8—24no—
69C4⋊C8—24no—
70C4⋊C8—24no—
71C22⋊C8—22no—
72C22⋊C8M1614yes—
73C4×C8—24no—
74C4×C8—82no—
75C4×C8—82no—
76#95—22no—
77#95—22no—
78C24C22×C811yescharacteristic
79C22×C4—22no—
80C22×C4—22no—
81C22×C4—22no—
82C22×C4—22no—
83C22×C4—22no—
84C22×C4—41no—
85C22×C4—41no—
86C22×C4—81no—
87C22×C4C22⋊C811yescharacteristic
88C22×C4C22⋊C811yescharacteristic
89C22×C4C22⋊C811yescharacteristic
90C22×C4C22⋊C811yescharacteristic
91C22×C4C2×C1612yes—
92C22×C4M3212yes—
93C22×C4C2×C22⋊C411yescharacteristic
94C22×C4#11712yes—
95C22×C4C2×M1612yes—
96C22×C4C2×M1611yescharacteristic
97C2×C8—22no—
98C2×C8—24no—
99C2×C8—42no—
100C2×C8—44no—
101C2×C8—42no—
102C2×C8—42no—
103C2×C8—42no—
104C2×C8—82no—
105C2×C8#12314yes—
106C4⋊C4—24no—
107C4⋊C4—24no—
108C22⋊C4—22no—
109C22⋊C4—22no—
110C22⋊C4C2×C1614yes—
111C22⋊C4M3214yes—
112C42—24no—
113C42—24no—
114C42—21no—
115C42—21no—
116C42—21no—
117C42—21no—
118C42—22no—
119C42—22no—
120C42—42no—
121C42—42no—
122C42—82no—
123C16—84no—
124C23—41no—
125C23—41no—
126C23C2×C22⋊C811yescharacteristic
127C23#43112yes—
128C23C22×C1611yescharacteristic
129C23C2×M3211yescharacteristic
130C23C2×M3212yes—
131C2×C4—21no—
132C2×C4—21no—
133C2×C4—21no—
134C2×C4—21no—
135C2×C4—24no—
136C2×C4—24no—
137C2×C4—21no—
138C2×C4—22no—
139C2×C4—21no—
140C2×C4—22no—
141C2×C4—21no—
142C2×C4—21no—
143C2×C4—24no—
144C2×C4—24no—
145C2×C4—42no—
146C2×C4—41no—
147C2×C4—42no—
148C2×C4—42no—
149C2×C4—42no—
150C2×C4—42no—
151C2×C4—41no—
152C2×C4—41no—
153C2×C4—82no—
154C2×C4C22⋊C1612yes—
155C2×C4C22⋊C1612yes—
156C2×C4#38111yescharacteristic
157C2×C4#40711yescharacteristic
158C2×C4#43212yes—
159C8—24no—
160C8—44no—
161C8—42no—
162C22—21no—
163C22—22no—
164C22—21no—
165C22—22no—
166C22—41no—
167C22—41no—
168C22—42no—
169C22#181711yescharacteristic
170C22C2×C22⋊C1611yescharacteristic
171C22#211211yescharacteristic
172C22#216212yes—
173C22#216312yes—
174C4—21no—
175C4—22no—
176C4—22no—
177C4—22no—
178C4—44no—
179C4—44no—
180C4—41no—
181C4—41no—
182C4#182811yescharacteristic
183C4#183011yescharacteristic
184C2—21no—
185C2—21no—
186C2—22no—
187C2—22no—
188C2#873511yescharacteristic
189C2#1284411yescharacteristic
190C2#1285111yescharacteristic
subgroupG/H
centerC2×C4#381
commutatorC23C22×C16
Frattini, cosocleC23×C8C23
socleC22#1817
prime radicalC24C22×C8
LCSG ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C23×C8 ◁ G
LECSG ▷ C23×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
51. #117C16semidirect

Maximal subgroups

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

subgroupindex
#205432
C2×#13222
#71992