Finite Group Databasebeta

← #94347

Subgroups — #94347

Order 512.

233 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 233 conjugacy classes of subgroups — too many to draw. Only its 70 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×#1400C211yesmaximal, characteristic
2#12836C211yesmaximal, characteristic
3C2×#1384C211yesmaximal, characteristic
4C22×M32C2211yescharacteristic
5C2×#427—21no—
6C2×M64—21no—
7C2×#367C411yescharacteristic, complemented
8C2×#366C411yescharacteristic, complemented
9#1808—21no—
10#1400—21no—
11#1400C412yescomplemented
12#1384—21no—
13#1384C412yescomplemented
14#1380—21no—
15C23×C8D411yescharacteristic
16C2×#117—41no—
17C2×M32—41no—
18C2×M32—41no—
19C2×M32C2×C411yescharacteristic
20C2×M32C2×C411yescharacteristic
21C2×M32D411yescharacteristic
22C2×M32Q811yescharacteristic
23C22×C16C2×C411yescharacteristic
24C22×C16D411yescharacteristic
25#427—22no—
26#427—22no—
27#427—41no—
28#427—41no—
29C2×#107—21no—
30C2×#107—21no—
31C2×#107—41no—
32C2×#106—22no—
33#380—41no—
34M64—22no—
35M64—41no—
36C2×C32—41no—
37#367—41no—
38#366—41no—
39#340—41no—
40C23×C4D811yescharacteristic
41C22×C8—21no—
42C22×C8—21no—
43C22×C8—21no—
44C22×C8—21no—
45C22×C8C22⋊C411yescharacteristic
46C22×C8C22⋊C411yescharacteristic
47C22×C8C4⋊C411yescharacteristic
48C22×C8SD1611yescharacteristic
49C22×C8SD1611yescharacteristic
50C22×C8Q1611yescharacteristic
51#117—42no—
52#117—42no—
53#117—81no—
54#117—81no—
55C2×C4⋊C4—42no—
56C2×C4⋊C4—41no—
57C2×C4⋊C4—41no—
58C2×C22⋊C4—81no—
59C2×C22⋊C4—81no—
60C2×C42—81no—
61M32—41no—
62M32—41no—
63M32—41no—
64M32—41no—
65C2×C16—21no—
66C2×C16—21no—
67C2×C16—21no—
68C2×C16—21no—
69C2×C16C4211yescharacteristic
70C2×C16C22⋊C411yescharacteristic
71C2×C16C4⋊C411yescharacteristic
72C2×C16C4⋊C411yescharacteristic
73#107—42no—
74#107—42no—
75#107—41no—
76#107—41no—
77#107—41no—
78#107—41no—
79#106—24no—
80#106—42no—
81#95—81no—
82C32—42no—
83C24D1611yescharacteristic
84C22×C4—21no—
85C22×C4—21no—
86C22×C4—21no—
87C22×C4—21no—
88C22×C4—82no—
89C22×C4—81no—
90C22×C4—81no—
91C22×C4#10211yescharacteristic
92C22×C4#10211yescharacteristic
93C22×C4#10711yescharacteristic
94C22×C4SD3211yescharacteristic
95C22×C4SD3211yescharacteristic
96C22×C4Q3211yescharacteristic
97C2×C8—21no—
98C2×C8—21no—
99C2×C8—21no—
100C2×C8—21no—
101C2×C8—21no—
102C2×C8—21no—
103C2×C8—21no—
104C2×C8—21no—
105C2×C8—21no—
106C2×C8—21no—
107C2×C8#9511yescharacteristic
108C2×C8#9911yescharacteristic
109C2×C8#9911yescharacteristic
110C2×C8#10311yescharacteristic
111C2×C8#10311yescharacteristic
112C2×C8#10411yescharacteristic
113C2×C8#10411yescharacteristic
114C2×C8#10611yescharacteristic
115C4⋊C4—44no—
116C4⋊C4—82no—
117C4⋊C4—81no—
118C4⋊C4—81no—
119C4⋊C4—81no—
120C4⋊C4—81no—
121C22⋊C4—82no—
122C22⋊C4—82no—
123C22⋊C4—162no—
124C42—82no—
125C42—82no—
126C16—21no—
127C16—21no—
128C16—21no—
129C16—21no—
130C23—21no—
131C23—21no—
132C23—41no—
133C23—41no—
134C23#35711yescharacteristic
135C23#35711yescharacteristic
136C23#36611yescharacteristic
137C2×C4—21no—
138C2×C4—21no—
139C2×C4—21no—
140C2×C4—21no—
141C2×C4—21no—
142C2×C4—21no—
143C2×C4—21no—
144C2×C4—21no—
145C2×C4—21no—
146C2×C4—21no—
147C2×C4—84no—
148C2×C4—162no—
149C2×C4—162no—
150C2×C4—161no—
151C2×C4—161no—
152C2×C4—161no—
153C2×C4—161no—
154C2×C4—161no—
155C2×C4—161no—
156C2×C4#32711yescharacteristic
157C2×C4#32711yescharacteristic
158C2×C4#34011yescharacteristic
159C2×C4#35811yescharacteristic
160C2×C4#35811yescharacteristic
161C2×C4#36011yescharacteristic
162C2×C4#36011yescharacteristic
163C2×C4#36711yescharacteristic
164C8—21no—
165C8—21no—
166C8#32811yescharacteristic
167C8#32811yescharacteristic
168C8#33911yescharacteristic
169C8#34211yescharacteristic
170C22—21no—
171C22—21no—
172C22—21no—
173C22—21no—
174C22—41no—
175C22—41no—
176C22—41no—
177C22—41no—
178C22—81no—
179C22#134711yescharacteristic
180C22#134711yescharacteristic
181C22#138011yescharacteristic
182C4—21no—
183C4—21no—
184C4—164no—
185C4—321no—
186C4—321no—
187C4#129411yescharacteristic
188C4#134811yescharacteristic
189C4#134811yescharacteristic
190C4#138611yescharacteristic
191C2—21no—
192C2—21no—
193C2—41no—
194C2—41no—
195C2#725811yescharacteristic
196C2#742211yescharacteristic
197C2#742311yescharacteristic
subgroupG/H
center, socleC22#1347
commutatorC2×C16C42
Frattini, cosocleC22×M32C22
prime radicalC24D16
LCSG ▷ C2×C16 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C4 ◁ C23×C8 ◁ C22×M32 ◁ G
LECSG ▷ C22×M32 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
7. C2×#367C4semidirect
8. C2×#366C4semidirect
11. #1400C4semidirect
13. #1384C4semidirect

Maximal subgroups

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

subgroupindex
C2×#14002
#128362
C2×#13842