Finite Group Databasebeta

← #140589

Subgroups — #140589

Order 512.

351 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 351 conjugacy classes of subgroups — too many to draw. Only its 94 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#21636C211yesmaximal, characteristic
2#12901C211yesmaximal, characteristic, complemented
3#12541C211yesmaximal, characteristic, complemented
4#11611C212yesmaximal, complemented
5#11587C212yesmaximal
6C2×#446C2211yescharacteristic
7C2×#436C2211yescharacteristic
8#2398C411yescharacteristic
9C2×#364—22no—
10#2150C2212yes—
11C2×#347—22no—
12C4×M32C2212yes—
13#1985C411yescharacteristic
14#1955C411yescharacteristic
15#1924—22no—
16#1832C411yescharacteristic
17#1753—22no—
18#1726C2211yescharacteristic
19#1338—24no—
20#1330—24no—
21C22×M16—21no—
22C2×C4×Q8C2×C411yescharacteristic
23C2×C4×D4C2×C411yescharacteristic
24C2×M32—22no—
25C2×M32—22no—
26#446—22no—
27#446—22no—
28#436—22no—
29#436—22no—
30C2×C4⋊C8—21no—
31C2×C4⋊C8C2×C411yescharacteristic
32C2×C4⋊C8C2×C411yescharacteristic
33C2×C22⋊C8—22no—
34C2×C22⋊C8—21no—
35C2×#97—21no—
36C2×#97C2×C411yescharacteristic
37C2×#97D412yes—
38C2×C4×C8D412yes—
39C2×C4×C8C2311yescharacteristic
40#400—21no—
41#397—21no—
42#394—21no—
43#393—21no—
44C4×C22⋊C4—22no—
45#376C2×C411yescharacteristic
46#364—44no—
47C4⋊C16—22no—
48C4⋊C16—22no—
49#347—28no—
50#346D414yes—
51C4×C16—22no—
52#336—22no—
53#336—22no—
54C22×Q8M1611yescharacteristic
55C22×D4M1611yescharacteristic
56C23×C4—21no—
57C2×M16—41no—
58C2×M16—41no—
59C2×M16—42no—
60C2×M16—42no—
61C22×C8—21no—
62C22×C8—21no—
63C22×C8C22⋊C412yes—
64C22×C8C2×D411yescharacteristic
65C22×C8C2×D411yescharacteristic
66C4×Q8—22no—
67C4×Q8—22no—
68C4×D4—22no—
69C4×D4—22no—
70C2×C4⋊C4—21no—
71C2×C4⋊C4M1611yescharacteristic
72C2×C4⋊C4M1611yescharacteristic
73C2×C22⋊C4—21no—
74C2×C22⋊C4—22no—
75C2×C22⋊C4—22no—
76C2×C42—21no—
77C2×C42C22⋊C412yes—
78C2×C42C22×C411yescharacteristic
79M32—28no—
80M32—44no—
81C2×C16—24no—
82C2×C16—42no—
83C4⋊C8—22no—
84C4⋊C8—22no—
85C4⋊C8—42no—
86C22⋊C8—44no—
87C22⋊C8—42no—
88#97—24no—
89#97—22no—
90#97—41no—
91#97—41no—
92C4×C8—22no—
93C4×C8—22no—
94C4×C8C2×D412yes—
95C4×C8#4112yes—
96#95—22no—
97#95—22no—
98#95—21no—
99#95—21no—
100C24—21no—
101C2×Q8—22no—
102C2×Q8—81no—
103C2×D4—22no—
104C2×D4—81no—
105C22×C4—22no—
106C22×C4—21no—
107C22×C4—21no—
108C22×C4—21no—
109C22×C4—41no—
110C22×C4—41no—
111C22×C4—42no—
112C22×C4—42no—
113C22×C4C2×C22⋊C411yescharacteristic
114C22×C4C2×M1611yescharacteristic
115C22×C4C2×M1611yescharacteristic
116M16—82no—
117M16—81no—
118M16—81no—
119C2×C8—22no—
120C2×C8—22no—
121C2×C8—22no—
122C2×C8—41no—
123C2×C8—41no—
124C2×C8—42no—
125C2×C8—42no—
126C2×C8#10414yes—
127C2×C8#12112yes—
128C2×C8#12411yescharacteristic
129C2×C8#12711yescharacteristic
130C2×C8AGL(1,ℤ/8)12yes—
131C2×C8#13712yes—
132C4⋊C4—22no—
133C4⋊C4—22no—
134C4⋊C4—42no—
135C22⋊C4—42no—
136C22⋊C4—82no—
137C22⋊C4—82no—
138C42—22no—
139C42—22no—
140C42—41no—
141C42—41no—
142C42C4×D412yes—
143C42#13112yes—
144C16—28no—
145C16—82no—
146C23—41no—
147C23—42no—
148C23—41no—
149C23—41no—
150C23—42no—
151C23#40711yescharacteristic
152Q8—82no—
153D4—82no—
154C2×C4—41no—
155C2×C4—41no—
156C2×C4—41no—
157C2×C4—42no—
158C2×C4—42no—
159C2×C4—42no—
160C2×C4—41no—
161C2×C4—41no—
162C2×C4—41no—
163C2×C4—41no—
164C2×C4—42no—
165C2×C4—42no—
166C2×C4—81no—
167C2×C4—81no—
168C2×C4—82no—
169C2×C4#39011yescharacteristic
170C2×C4#40811yescharacteristic
171C2×C4#41712yes—
172C2×C4C2×#10412yes—
173C2×C4#43512yes—
174C2×C4#43512yes—
175C8—22no—
176C8—24no—
177C8—81no—
178C8—81no—
179C8#48912yes—
180C8#49611yescharacteristic
181C8#49711yescharacteristic
182C22—21no—
183C22—21no—
184C22—82no—
185C22—82no—
186C22—81no—
187C22—81no—
188C22—81no—
189C22#148912yes—
190C22#195511yescharacteristic
191C4—22no—
192C4—22no—
193C4—82no—
194C4—81no—
195C4—81no—
196C4—81no—
197C4#148312yes—
198C4#195911yescharacteristic
199C4#196211yescharacteristic
200C2—22no—
201C2—161no—
202C2#964611yescharacteristic
203C2#1320111yescharacteristic
204C2#1320111yescharacteristic
subgroupG/H
centerC2×C4#390
commutatorC2×C42C22×C4
Frattini, cosocleC2×C4×C8C23
socleC22#1955
prime radicalC22×D4M16
LCSG ▷ C2×C42 ▷ 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
2. #12901C2semidirect
3. #12541C2semidirect
4. #11611C2semidirect

Maximal subgroups

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

subgroupindex
#216362
#129012
#125412
#116112
#115872