Finite Group Databasebeta

← #93059

Subgroups — #93059

Order 512.

243 conjugacy classes of subgroups; only the 58 normal subgroups are shown.

normal subgroups only

This group has 243 conjugacy classes of subgroups — too many to draw. Only its 58 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#11460C211yesmaximal, characteristic
2#8598C211yesmaximal, characteristic
3#7980C211yesmaximal, characteristic
4C2×#431C2211yescharacteristic
5#2283—21no—
6#2163—21no—
7#1797—21no—
8C2×#334C411yescharacteristic, complemented
9C2×#334C411yescharacteristic, complemented
10#1322C412yescomplemented
11#1314—21no—
12#1289—21no—
13#1289C412yes—
14#1277—21no—
15C22×C42D411yescharacteristic
16#431—41no—
17#431—41no—
18C2×C4⋊C8—21no—
19C2×C4⋊C8—21no—
20C2×C4⋊C8C2×C411yescharacteristic
21C2×C4⋊C8Q811yescharacteristic
22C2×C22⋊C8—41no—
23C2×C22⋊C8C2×C411yescharacteristic
24C2×C22⋊C8D411yescharacteristic
25C2×C4×C8C2×C411yescharacteristic
26C2×C4×C8D411yescharacteristic
27#383—22no—
28#382—41no—
29#381—41no—
30#380—22no—
31C4×C4⋊C4—41no—
32C4×C22⋊C4—41no—
33#376—22no—
34C4⋊C16—41no—
35C22⋊C16—22no—
36#346—41no—
37#336—41no—
38#334—41no—
39#334—41no—
40C23×C4C22⋊C411yescharacteristic
41C23×C4D811yescharacteristic
42C23×C4SD1611yescharacteristic
43C22×C8—41no—
44C22×C8—41no—
45C22×C8C4211yescharacteristic
46C22×C8C22⋊C411yescharacteristic
47C22×C8C4⋊C411yescharacteristic
48C22×C8C4⋊C411yescharacteristic
49C2×C4⋊C4—42no—
50C2×C22⋊C4—42no—
51C2×C42—21no—
52C2×C42—21no—
53C2×C42—21no—
54C2×C42—21no—
55C2×C42—42no—
56C2×C42C22⋊C411yescharacteristic
57C2×C42C4⋊C411yescharacteristic
58C2×C42SD1611yescharacteristic
59C2×C42Q1611yescharacteristic
60C2×C16—42no—
61C4⋊C8—41no—
62C4⋊C8—41no—
63C4⋊C8—41no—
64C4⋊C8—41no—
65C4⋊C8—41no—
66C4⋊C8—41no—
67C22⋊C8—41no—
68C22⋊C8—41no—
69C22⋊C8—82no—
70C4×C8—21no—
71C4×C8—21no—
72C4×C8—21no—
73C4×C8—21no—
74#95—24no—
75#95—42no—
76#95—42no—
77#95—42no—
78C24#10211yescharacteristic
79C22×C4—22no—
80C22×C4—21no—
81C22×C4—21no—
82C22×C4—21no—
83C22×C4—21no—
84C22×C4—21no—
85C22×C4—21no—
86C22×C4—41no—
87C22×C4—41no—
88C22×C4—41no—
89C22×C4—44no—
90C22×C4—81no—
91C22×C4#9511yescharacteristic
92C22×C4#10211yescharacteristic
93C22×C4#10311yescharacteristic
94C22×C4#10311yescharacteristic
95C22×C4#10611yescharacteristic
96C22×C4#10711yescharacteristic
97C2×C8—21no—
98C2×C8—21no—
99C2×C8—21no—
100C2×C8—21no—
101C2×C8—41no—
102C2×C8—41no—
103C2×C8—41no—
104C2×C8—41no—
105C2×C8—81no—
106C2×C8—81no—
107C2×C8—81no—
108C2×C8—81no—
109C2×C8—81no—
110C2×C8—81no—
111C4⋊C4—82no—
112C4⋊C4—82no—
113C22⋊C4—82no—
114C22⋊C4—82no—
115C42—21no—
116C42—21no—
117C42—21no—
118C42—21no—
119C42—21no—
120C42—21no—
121C42—84no—
122C42#9911yescharacteristic
123C42#9911yescharacteristic
124C42#10411yescharacteristic
125C42#10411yescharacteristic
126C16—82no—
127C23—21no—
128C23—21no—
129C23—41no—
130C23—41no—
131C23#34011yescharacteristic
132C23#36012yes—
133C2×C4—21no—
134C2×C4—21no—
135C2×C4—21no—
136C2×C4—21no—
137C2×C4—21no—
138C2×C4—21no—
139C2×C4—21no—
140C2×C4—21no—
141C2×C4—21no—
142C2×C4—41no—
143C2×C4—42no—
144C2×C4—42no—
145C2×C4—42no—
146C2×C4—41no—
147C2×C4—41no—
148C2×C4—41no—
149C2×C4—41no—
150C2×C4—81no—
151C2×C4—81no—
152C2×C4—84no—
153C2×C4—84no—
154C2×C4—84no—
155C2×C4#32711yescharacteristic
156C2×C4#32711yescharacteristic
157C2×C4#32811yescharacteristic
158C2×C4#32811yescharacteristic
159C2×C4#33911yescharacteristic
160C2×C4#34211yescharacteristic
161C8—41no—
162C8—41no—
163C8—41no—
164C8—41no—
165C8—161no—
166C8—161no—
167C22—21no—
168C22—21no—
169C22—22no—
170C22—42no—
171C22—41no—
172C22—41no—
173C22—81no—
174C22#129411yescharacteristic
175C22#138411yescharacteristic
176C22#138611yescharacteristic
177C4—21no—
178C4—21no—
179C4—21no—
180C4—21no—
181C4—21no—
182C4—41no—
183C4—41no—
184C4—41no—
185C4—164no—
186C4#139111yescharacteristic
187C4#139511yescharacteristic
188C2—21no—
189C2—21no—
190C2—42no—
191C2#725211yescharacteristic
192C2#725611yescharacteristic
193C2#726211yescharacteristic
subgroupG/H
center, socleC22#1294
commutatorC22×C8C42
Frattini, cosocleC2×#431C22
prime radicalC24#102
LCSG ▷ C22×C8 ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#431 ◁ G
LECSG ▷ C2×#431 ▷ C22×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
8. C2×#334C4semidirect
9. C2×#334C4semidirect
10. #1322C4semidirect

Maximal subgroups

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

subgroupindex
#114602
#85982
#79802