Finite Group Databasebeta

← #124577

Subgroups — #124577

Order 512.

372 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 372 conjugacy classes of subgroups — too many to draw. Only its 118 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#21349C211yesmaximal, characteristic
2#12783C212yesmaximal
3#7118C214yesmaximal, complemented
4C2×#431C411yescharacteristic
5C22×C4×C8C2211yescharacteristic
6C4×#117C411yescharacteristic
7C2×C4⋊C16—22no—
8C2×C22⋊C16—22no—
9#1939C412yes—
10#1939C2212yes—
11#1850—22no—
12C4×C4⋊C8—22no—
13#1374C2214yes—
14C23×C8D412yes—
15C2×#117C811yescharacteristic
16C2×#117C811yescharacteristic
17C22×C42C2×C411yescharacteristic
18C22×C16—24no—
19#431—42no—
20C2×C4⋊C8—22no—
21C2×C4⋊C8—22no—
22C2×C4⋊C8C2×C412yes—
23C2×C22⋊C8—21no—
24C2×C4×C8—21no—
25C2×C4×C8—42no—
26C2×C4×C8C2×C411yescharacteristic
27C2×C4×C8D412yes—
28C2×C4×C8C2311yescharacteristic
29C4×C4⋊C4C812yes—
30C4×C4⋊C4C2×C412yes—
31C4×C22⋊C4—21no—
32C4×C22⋊C4—21no—
33#376—21no—
34C43—21no—
35C4⋊C16—44no—
36C22⋊C16—44no—
37#336—22no—
38#336—22no—
39C23×C4C22⋊C411yescharacteristic
40C23×C4C2×C811yescharacteristic
41C23×C4M1611yescharacteristic
42C22×C8—22no—
43C22×C8—22no—
44C22×C8—22no—
45C22×C8—42no—
46C22×C8—42no—
47C22×C8C22⋊C412yes—
48C22×C8C2×D412yes—
49#117—22no—
50#117—42no—
51#117C1614yescomplemented
52C2×C4⋊C4C2×C812yes—
53C2×C4⋊C4C2×C812yes—
54C2×C22⋊C4—21no—
55C2×C22⋊C4—21no—
56C2×C42—21no—
57C2×C42—22no—
58C2×C42—22no—
59C2×C42—21no—
60C2×C42—22no—
61C2×C42—22no—
62C2×C42C22⋊C411yescharacteristic
63C2×C42C2×C811yescharacteristic
64C2×C42M1611yescharacteristic
65C2×C42C22×C411yescharacteristic
66C2×C16—44no—
67C2×C16—44no—
68C2×C16—44no—
69C4⋊C8—42no—
70C4⋊C8—44no—
71C4⋊C8—42no—
72C4⋊C8—42no—
73C22⋊C8—42no—
74C4×C8—24no—
75C4×C8—41no—
76C4×C8—41no—
77C4×C8—42no—
78#95—22no—
79#95—22no—
80C24C22⋊C811yescharacteristic
81C22×C4—22no—
82C22×C4—22no—
83C22×C4—22no—
84C22×C4—22no—
85C22×C4—21no—
86C22×C4—21no—
87C22×C4—22no—
88C22×C4—22no—
89C22×C4—22no—
90C22×C4—42no—
91C22×C4C22⋊C811yescharacteristic
92C22×C4C22⋊C811yescharacteristic
93C22×C4C22⋊C811yescharacteristic
94C22×C4C2×C1612yes—
95C22×C4M3212yes—
96C22×C4C2×C22⋊C411yescharacteristic
97C22×C4C22×C811yescharacteristic
98C22×C4C2×M1611yescharacteristic
99C2×C8—22no—
100C2×C8—22no—
101C2×C8—22no—
102C2×C8—22no—
103C2×C8—22no—
104C2×C8—44no—
105C2×C8—44no—
106C2×C8—42no—
107C2×C8—42no—
108C2×C8—42no—
109C2×C8AGL(1,ℤ/8)12yes—
110C2×C8#13712yes—
111C4⋊C4—22no—
112C4⋊C4—42no—
113C4⋊C4C2×C1614yes—
114C22⋊C4—24no—
115C22⋊C4—42no—
116C42—22no—
117C42—22no—
118C42—24no—
119C42—21no—
120C42—21no—
121C42—44no—
122C42—42no—
123C42—42no—
124C42—42no—
125C42#9912yes—
126C42#10112yes—
127C16—84no—
128C23—21no—
129C23—22no—
130C23—21no—
131C23—22no—
132C23C22⋊C1612yes—
133C23C2×C22⋊C811yescharacteristic
134C2×C4—21no—
135C2×C4—22no—
136C2×C4—21no—
137C2×C4—24no—
138C2×C4—24no—
139C2×C4—24no—
140C2×C4—22no—
141C2×C4—21no—
142C2×C4—21no—
143C2×C4—22no—
144C2×C4—44no—
145C2×C4—42no—
146C2×C4—42no—
147C2×C4—41no—
148C2×C4—41no—
149C2×C4—42no—
150C2×C4—42no—
151C2×C4—42no—
152C2×C4—42no—
153C2×C4#32412yes—
154C2×C4#32412yes—
155C2×C4C22⋊C1612yes—
156C2×C4#34912yes—
157C2×C4C2×#9911yescharacteristic
158C2×C4C2×#10111yescharacteristic
159C2×C4#41712yes—
160C2×C4#42112yes—
161C2×C4C22×C1611yescharacteristic
162C2×C4C2×M3211yescharacteristic
163C8—22no—
164C8—22no—
165C8—42no—
166C8—82no—
167C22—24no—
168C22—22no—
169C22—42no—
170C22—41no—
171C22—41no—
172C22C2×#32411yescharacteristic
173C22#147712yes—
174C22C2×C22⋊C1611yescharacteristic
175C22C2×#34911yescharacteristic
176C22#211712yes—
177C4—21no—
178C4—21no—
179C4—22no—
180C4—44no—
181C4—41no—
182C4—41no—
183C4—82no—
184C4#132214yes—
185C4#151312yes—
186C4#153711yescharacteristic
187C4#156111yescharacteristic
188C2—42no—
189C2#759811yescharacteristic
190C2C2×#132211yescharacteristic
191C2#1144311yescharacteristic
192C2#1162112yes—
193C2#1162812yes—
subgroupG/H
centerC22×C4C2×C22⋊C4
commutatorC2×C4C22×C16
Frattini, cosocleC2×C4×C8C23
socleC23C2×C22⋊C8
prime radicalC24C22⋊C8
LCSG ▷ C2×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C4 ◁ C22×C4×C8 ◁ G
LECSG ▷ C2×C4×C8 ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #7118C2semidirect
51. #117C16semidirect

Maximal subgroups

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

subgroupindex
#213492
#127832
#71182