Finite Group Databasebeta

← #136803

Subgroups — #136803

Order 512.

340 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 340 conjugacy classes of subgroups — too many to draw. Only its 90 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#33497C211yesmaximal, characteristic, complemented
2#21639C211yesmaximal, characteristic
3#11634C212yesmaximal
4#11494C211yesmaximal, characteristic, complemented
5#7150C212yesmaximal, complemented
6C2×C8×Q8C2211yescharacteristic
7C2×#436C2211yescharacteristic
8#2398C411yescharacteristic
9#2183—24no—
10#2163—22no—
11#2163C2212yes—
12#2162C2212yes—
13#1985C411yescharacteristic
14#1955C411yescharacteristic
15#1923—22no—
16#1832C411yescharacteristic
17#1753—22no—
18#1726C2211yescharacteristic
19#1330—24no—
20#1322—22no—
21C22×M16—21no—
22C2×C4×Q8C2×C411yescharacteristic
23C2×C4×D4C2×C411yescharacteristic
24C8×Q8—22no—
25C8×Q8—22no—
26#436—22no—
27#436—22no—
28C2×C4⋊C8C2×C411yescharacteristic
29C2×C4⋊C8C2×C411yescharacteristic
30C2×C4⋊C8D412yes—
31C2×C22⋊C8—22no—
32C2×C22⋊C8—21no—
33C2×#97—21no—
34C2×#97C2×C411yescharacteristic
35C2×C4×C8—21no—
36C2×C4×C8D412yes—
37C2×C4×C8C2311yescharacteristic
38#400C812yes—
39#397—21no—
40#394C812yes—
41#393—21no—
42C4×C22⋊C4—22no—
43#376C2×C411yescharacteristic
44C4⋊C16—24no—
45C4⋊C16—22no—
46C4⋊C16D414yes—
47C22⋊C16—22no—
48C22⋊C16—22no—
49#346—24no—
50#346—22no—
51C4×C16—22no—
52#336—22no—
53#336—22no—
54C22×Q8M1611yescharacteristic
55C22×D4C2×C811yescharacteristic
56C23×C4—21no—
57C2×M16—44no—
58C2×M16—41no—
59C2×M16—41no—
60C22×C8—21no—
61C22×C8—21no—
62C22×C8C22⋊C412yes—
63C22×C8C2×D411yescharacteristic
64C22×C8C2×D411yescharacteristic
65C4×Q8—22no—
66C4×Q8—22no—
67C4×D4—22no—
68C4×D4—22no—
69C2×C4⋊C4C2×C811yescharacteristic
70C2×C4⋊C4C2×C812yes—
71C2×C4⋊C4M1611yescharacteristic
72C2×C22⋊C4—22no—
73C2×C22⋊C4—21no—
74C2×C22⋊C4—22no—
75C2×C42—21no—
76C2×C42C22⋊C412yes—
77C2×C42C22×C411yescharacteristic
78C2×C16—24no—
79C2×C16—42no—
80C4⋊C8—24no—
81C4⋊C8—22no—
82C4⋊C8—22no—
83C22⋊C8—44no—
84C22⋊C8—42no—
85#97—22no—
86#97—42no—
87C4×C8—22no—
88C4×C8—22no—
89C4×C8—42no—
90C4×C8C2×D412yes—
91C4×C8#4112yes—
92#95—22no—
93#95—22no—
94#95—21no—
95#95C2×C812yes—
96C24—21no—
97C2×Q8—22no—
98C2×Q8—42no—
99C2×D4—22no—
100C2×D4—81no—
101C22×C4—21no—
102C22×C4—22no—
103C22×C4—42no—
104C22×C4—42no—
105C22×C4—41no—
106C22×C4—41no—
107C22×C4C22⋊C812yes—
108C22×C4C22⋊C812yes—
109C22×C4C2×C22⋊C411yescharacteristic
110C22×C4C22×C811yescharacteristic
111C22×C4C2×M1611yescharacteristic
112M16—82no—
113M16—82no—
114C2×C8—22no—
115C2×C8—22no—
116C2×C8—22no—
117C2×C8—41no—
118C2×C8—41no—
119C2×C8—41no—
120C2×C8—42no—
121C2×C8—42no—
122C2×C8—41no—
123C2×C8#12011yescharacteristic
124C2×C8#12112yes—
125C2×C8#12311yescharacteristic
126C4⋊C4—24no—
127C4⋊C4—22no—
128C4⋊C4—22no—
129C22⋊C4—44no—
130C22⋊C4—42no—
131C22⋊C4—82no—
132C42—22no—
133C42—22no—
134C42—41no—
135C42—41no—
136C42C4×D412yes—
137C42#13112yes—
138C16—44no—
139C16—82no—
140C23—41no—
141C23—42no—
142C23—41no—
143C23—42no—
144C23—41no—
145C23C2×C22⋊C811yescharacteristic
146Q8—44no—
147D4—82no—
148C2×C4—22no—
149C2×C4—21no—
150C2×C4—21no—
151C2×C4—22no—
152C2×C4—22no—
153C2×C4—22no—
154C2×C4—22no—
155C2×C4—41no—
156C2×C4—42no—
157C2×C4—42no—
158C2×C4—42no—
159C2×C4—42no—
160C2×C4—42no—
161C2×C4—41no—
162C2×C4—82no—
163C2×C4—81no—
164C2×C4—81no—
165C2×C4#38611yescharacteristic
166C2×C4#40811yescharacteristic
167C2×C4C8×D412yes—
168C2×C4#43512yes—
169C8—22no—
170C8—21no—
171C8—42no—
172C8—42no—
173C8—81no—
174C8#45412yes—
175C22—21no—
176C22—21no—
177C22—82no—
178C22—82no—
179C22—81no—
180C22—81no—
181C22—81no—
182C22#186911yescharacteristic
183C22#211712yes—
184C4—21no—
185C4—22no—
186C4—22no—
187C4—42no—
188C4—42no—
189C4—82no—
190C4—81no—
191C4#188411yescharacteristic
192C4#188811yescharacteristic
193C2—22no—
194C2—161no—
195C2#896011yescharacteristic
196C2#1298411yescharacteristic
197C2#1298911yescharacteristic
subgroupG/H
centerC2×C4#386
commutatorC22×C4C22×C8
Frattini, cosocleC2×C4×C8C23
socleC22#1869
prime radicalC22×D4C2×C8
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C2×C4×C8 ◁ G
LECSG ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #33497C2semidirect
4. #11494C2semidirect
5. #7150C2semidirect

Maximal subgroups

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

subgroupindex
#334972
#216392
#116342
#114942
#71502