Finite Group Databasebeta

← #93005

Subgroups — #93005

Order 512.

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

normal subgroups only

This group has 239 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#13190C211yesmaximal, characteristic
2#11466C211yesmaximal, characteristic
3#8160C211yesmaximal, characteristic
4C2×C8×D4C2211yescharacteristic
5C2×C4⋊C16C411yescharacteristic, complemented
6#2112—21no—
7C2×C22⋊C16—21no—
8C2×C22⋊C16—21no—
9C2×#346C411yescharacteristic, complemented
10C4×#99C411yescharacteristic
11#1752—21no—
12#1459C411yescharacteristic
13#1322C412yescomplemented
14#1314—21no—
15#1282—21no—
16C23×C8D411yescharacteristic
17C23×C8D411yescharacteristic
18C2×C4×D4C2×C411yescharacteristic
19C2×M32—41no—
20C22×C16—21no—
21C22×C16—21no—
22C8×D4—41no—
23C8×D4—41no—
24C2×C4⋊C8C2×C411yescharacteristic
25C2×#99C812yes—
26C2×C22⋊C8—21no—
27C2×C22⋊C8D411yescharacteristic
28C2×C22⋊C8Q811yescharacteristic
29C2×#97—41no—
30C2×C4×C8C2×C411yescharacteristic
31C4×C22⋊C4—21no—
32C4⋊C16—41no—
33C22⋊C16—22no—
34C22⋊C16—42no—
35C22⋊C16—42no—
36C22⋊C16—41no—
37#346—41no—
38#342—21no—
39#336—41no—
40#324C812yes—
41#323—21no—
42C22×D4C2×C811yescharacteristic
43C23×C4C22⋊C411yescharacteristic
44C23×C4C4⋊C411yescharacteristic
45C22×C8—21no—
46C22×C8—21no—
47C22×C8—21no—
48C22×C8—21no—
49C22×C8—41no—
50C22×C8—41no—
51C22×C8—41no—
52C22×C8—41no—
53C22×C8—41no—
54C22×C8C22⋊C411yescharacteristic
55C22×C8C22⋊C411yescharacteristic
56C22×C8C4⋊C411yescharacteristic
57C22×C8C4⋊C411yescharacteristic
58C4×D4—41no—
59C4×D4—41no—
60C2×C4⋊C4C2×C811yescharacteristic
61C2×C22⋊C4—22no—
62C2×C22⋊C4M1611yescharacteristic
63C2×C22⋊C4M1611yescharacteristic
64C2×C42—41no—
65C2×C42C4211yescharacteristic
66M32—81no—
67M32—81no—
68C2×C16—22no—
69C2×C16—41no—
70C2×C16—41no—
71C2×C16—42no—
72C2×C16—41no—
73C2×C16—41no—
74C2×C16—41no—
75C4⋊C8—41no—
76#99—42no—
77C22⋊C8—22no—
78C22⋊C8—41no—
79C22⋊C8—41no—
80C22⋊C8—41no—
81#97—81no—
82#97—81no—
83C4×C8—21no—
84C4×C8—21no—
85#95—41no—
86C24C22⋊C811yescharacteristic
87C24C4⋊C811yescharacteristic
88C2×D4—22no—
89C2×D4—41no—
90C2×D4—41no—
91C22×C4—21no—
92C22×C4—21no—
93C22×C4—21no—
94C22×C4—21no—
95C22×C4—41no—
96C22×C4—41no—
97C22×C4—41no—
98C22×C4—41no—
99C22×C4—42no—
100C22×C4#9511yescharacteristic
101C22×C4C4×C811yescharacteristic
102C22×C4#9711yescharacteristic
103C22×C4C22⋊C811yescharacteristic
104C22×C4C4⋊C811yescharacteristic
105C2×C8—21no—
106C2×C8—21no—
107C2×C8—21no—
108C2×C8—21no—
109C2×C8—21no—
110C2×C8—41no—
111C2×C8—41no—
112C2×C8—41no—
113C2×C8—41no—
114C2×C8—41no—
115C2×C8—41no—
116C2×C8—41no—
117C2×C8—42no—
118C2×C8—81no—
119C2×C8—81no—
120C2×C8—81no—
121C2×C8—81no—
122C2×C8#9911yescharacteristic
123C2×C8#9911yescharacteristic
124C4⋊C4—41no—
125C22⋊C4—41no—
126C22⋊C4—41no—
127C22⋊C4—42no—
128C22⋊C4—42no—
129C42—21no—
130C42—21no—
131C42—81no—
132C42—81no—
133C16—42no—
134C16—82no—
135C16—81no—
136C23—21no—
137C23—21no—
138C23—21no—
139C23—21no—
140C23—21no—
141C23—41no—
142C23—41no—
143C23—41no—
144C23—41no—
145C23#33611yescharacteristic
146C23#34912yes—
147D4—44no—
148C2×C4—21no—
149C2×C4—21no—
150C2×C4—21no—
151C2×C4—21no—
152C2×C4—21no—
153C2×C4—21no—
154C2×C4—21no—
155C2×C4—21no—
156C2×C4—41no—
157C2×C4—41no—
158C2×C4—41no—
159C2×C4—41no—
160C2×C4—41no—
161C2×C4—41no—
162C2×C4—41no—
163C2×C4—81no—
164C2×C4—81no—
165C2×C4—82no—
166C2×C4—82no—
167C2×C4—82no—
168C2×C4#34211yescharacteristic
169C2×C4#34411yescharacteristic
170C8—21no—
171C8—41no—
172C8—41no—
173C8—41no—
174C8—82no—
175C8#35311yescharacteristic
176C8#35411yescharacteristic
177C22—21no—
178C22—21no—
179C22—22no—
180C22—42no—
181C22—42no—
182C22—41no—
183C22—41no—
184C22—41no—
185C22—41no—
186C22—41no—
187C22—81no—
188C22—81no—
189C22#128211yescharacteristic
190C22#137711yescharacteristic
191C22#137811yescharacteristic
192C4—21no—
193C4—41no—
194C4—41no—
195C4—41no—
196C4—41no—
197C4—162no—
198C4#139311yescharacteristic
199C4#139511yescharacteristic
200C2—21no—
201C2—21no—
202C2—42no—
203C2—81no—
204C2#720111yescharacteristic
205C2#720511yescharacteristic
206C2#720811yescharacteristic
subgroupG/H
centerC2×C4#342
commutatorC22×C4C4×C8
Frattini, cosocleC2×C8×D4C22
socleC22#1282
prime radicalC22×D4C2×C8
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C22×C8 ◁ C2×C8×D4 ◁ G
LECSG ▷ C2×C8×D4 ▷ C22×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
5. C2×C4⋊C16C4semidirect
9. C2×#346C4semidirect
13. #1322C4semidirect

Maximal subgroups

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

subgroupindex
#131902
#114662
#81602