Finite Group Databasebeta

← #135382

Subgroups — #135382

Order 512.

345 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 345 conjugacy classes of subgroups — too many to draw. Only its 84 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#20820C211yesmaximal, characteristic
2#12847C211yesmaximal, characteristic
3#10814C211yesmaximal, characteristic
4#7262C214yesmaximal
5C2×#405C2211yescharacteristic
6#2314—21no—
7#2112C2212yes—
8#1846C411yescharacteristic, complemented
9#1844C411yescharacteristic, complemented
10#1836—21no—
11#1809—21no—
12#1807C2212yes—
13#1776—21no—
14C4×#106C412yescomplemented
15#1740—21no—
16#1735C2212yes—
17C2×H(3,ℤ/4)—21no—
18#1377—21no—
19#1377—21no—
20C22×M16D411yescharacteristic
21C23×C8C2311yescharacteristic
22C2×#117—22no—
23C2×#117D411yescharacteristic
24C2×M32—22no—
25C2×#107—21no—
26C2×#106—21no—
27C2×#106C2×C412yes—
28C2×#106C2×C412yes—
29#405—24no—
30C2×#97D411yescharacteristic
31C2×#97D411yescharacteristic
32C2×C4×C8C2×C411yescharacteristic
33C2×C4×C8C2×C411yescharacteristic
34#388—42no—
35#387—42no—
36#384—21no—
37#382—21no—
38#380—22no—
39C4×C4⋊C4—22no—
40C22⋊C16—24no—
41#342—42no—
42#340—22no—
43#340—22no—
44#340—24no—
45H(3,ℤ/4)—24no—
46C23×C4C2×D411yescharacteristic
47C2×M16—22no—
48C2×M16—22no—
49C2×M16—41no—
50C22×C8—41no—
51C22×C8—41no—
52C22×C8C22⋊C411yescharacteristic
53C22×C8C22⋊C411yescharacteristic
54C22×C8C22×C411yescharacteristic
55C22×C8C2×D411yescharacteristic
56C22×C8#4114yes—
57#117—22no—
58#117—22no—
59#117—24no—
60#117—42no—
61#117—42no—
62#117—42no—
63C2×C4⋊C4—22no—
64C2×C4⋊C4—22no—
65C2×C4⋊C4—42no—
66C2×C4⋊C4—42no—
67C2×C4⋊C4SD1612yes—
68C2×C22⋊C4—42no—
69C2×C22⋊C4—42no—
70C2×C22⋊C4D812yes—
71C2×C42—42no—
72C2×C42—42no—
73C2×C42C22⋊C411yescharacteristic
74C2×C42C22⋊C411yescharacteristic
75M32—42no—
76M32—42no—
77C2×C16—42no—
78#107—42no—
79#106—42no—
80#106—42no—
81#106—42no—
82#97—22no—
83#97—22no—
84C4×C8—22no—
85C4×C8—22no—
86#95—42no—
87#95—42no—
88#95—42no—
89C24C2×D811yescharacteristic
90C22×C4—21no—
91C22×C4—21no—
92C22×C4—21no—
93C22×C4—21no—
94C22×C4—42no—
95C22×C4—42no—
96C22×C4—82no—
97C22×C4#10212yes—
98C22×C4#10212yes—
99C22×C4C2×C22⋊C411yescharacteristic
100C22×C4#12312yes—
101C22×C4C2×SD1611yescharacteristic
102C22×C4#13712yes—
103M16—44no—
104C2×C8—21no—
105C2×C8—21no—
106C2×C8—21no—
107C2×C8—21no—
108C2×C8—21no—
109C2×C8—21no—
110C2×C8—21no—
111C2×C8—21no—
112C2×C8—21no—
113C2×C8—44no—
114C2×C8#10412yes—
115C2×C8#11712yes—
116C2×C8#12312yes—
117C4⋊C4—24no—
118C4⋊C4—44no—
119C4⋊C4—42no—
120C4⋊C4—42no—
121C4⋊C4—42no—
122C4⋊C4—42no—
123C4⋊C4—82no—
124C4⋊C4—162no—
125C22⋊C4—24no—
126C22⋊C4—44no—
127C22⋊C4—82no—
128C22⋊C4—162no—
129C42—21no—
130C42—21no—
131C42—21no—
132C42—21no—
133C42—44no—
134C42—82no—
135C42—162no—
136C16—82no—
137C23—21no—
138C23—21no—
139C23—42no—
140C23C2×#10211yescharacteristic
141C23#48012yes—
142C2×C4—21no—
143C2×C4—21no—
144C2×C4—22no—
145C2×C4—22no—
146C2×C4—22no—
147C2×C4—22no—
148C2×C4—22no—
149C2×C4—22no—
150C2×C4—42no—
151C2×C4—42no—
152C2×C4—84no—
153C2×C4—82no—
154C2×C4—82no—
155C2×C4—82no—
156C2×C4—82no—
157C2×C4—82no—
158C2×C4—82no—
159C2×C4—164no—
160C2×C4#38111yescharacteristic
161C2×C4#41011yescharacteristic
162C2×C4#41011yescharacteristic
163C2×C4#41911yescharacteristic
164C2×C4C2×#10411yescharacteristic
165C2×C4#42111yescharacteristic
166C2×C4#48312yes—
167C8—22no—
168C8—22no—
169C8—22no—
170C8—41no—
171C22—22no—
172C22—22no—
173C22—24no—
174C22—81no—
175C22—81no—
176C22#150211yescharacteristic
177C22#151711yescharacteristic
178C22#182311yescharacteristic
179C4—22no—
180C4—42no—
181C4—42no—
182C4—162no—
183C4—162no—
184C4—164no—
185C4#150611yescharacteristic
186C4#151511yescharacteristic
187C4#182011yescharacteristic
188C4#182611yescharacteristic
189C2—21no—
190C2—21no—
191C2—42no—
192C2#871611yescharacteristic
193C2#1287312yes—
subgroupG/H
center, socleC22#1823
commutatorC22×C8C22×C4
Frattini, cosocleC23×C8C23
prime radicalC24C2×D8
LCSG ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#405 ◁ G
LECSG ▷ C23×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
8. #1846C4semidirect
9. #1844C4semidirect
14. C4×#106C4semidirect

Maximal subgroups

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

subgroupindex
#208202
#128472
#108142
#72622