Finite Group Databasebeta

← #125035

Subgroups — #125035

Order 512.

265 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 265 conjugacy classes of subgroups — too many to draw. Only its 66 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#21660C211yesmaximal, characteristic
2#8801C212yesmaximal, complemented
3#7134C212yesmaximal, complemented
4#7106C212yesmaximal
5#3131C411yescharacteristic
6#2960C411yescharacteristic
7#2931C411yescharacteristic
8#1994C2211yescharacteristic
9#1962C2211yescharacteristic
10#1938—22no—
11#1837C411yescharacteristic
12#1765—22no—
13C4×#97C2211yescharacteristic
14#1281—22no—
15#1276C2214yes—
16#1275—22no—
17C2×#131—21no—
18#533C2×C411yescharacteristic
19#532C2×C411yescharacteristic
20#488—22no—
21#487—21no—
22#486—21no—
23#446—41no—
24C8×Q8—41no—
25#436—41no—
26C8×D4—41no—
27#431—24no—
28#424—21no—
29#423C2×C411yescharacteristic
30#423C2×C411yescharacteristic
31C2×#104—22no—
32#416—22no—
33#408—21no—
34#405—21no—
35C2×#97D412yes—
36C2×#97D412yes—
37C2×C4×C8C2×C411yescharacteristic
38C2×C4×C8C2×C411yescharacteristic
39C43C2311yescharacteristic
40#341—22no—
41#339—22no—
42C2×#41—21no—
43#131—44no—
44C2×M16—21no—
45C2×M16—21no—
46C2×M16—41no—
47C22×C8—41no—
48C22×C8C22⋊C412yes—
49C22×C8C22⋊C411yescharacteristic
50C22×C8C22⋊C411yescharacteristic
51C4⋊Q8—21no—
52C4⋊Q8M1611yescharacteristic
53#127M1611yescharacteristic
54#125M1612yes—
55#124—21no—
56C22⋊Q8—41no—
57#121—41no—
58C4×Q8—41no—
59C4×D4—41no—
60#117—21no—
61C2×C42—41no—
62C2×C42C22×C411yescharacteristic
63C2×C42C2×D412yes—
64C4⋊C8—41no—
65C4⋊C8—41no—
66C4⋊C8—44no—
67C4⋊C8—42no—
68#104—44no—
69#103—24no—
70#102—24no—
71C22⋊C8—42no—
72C22⋊C8—44no—
73#97—24no—
74#97—42no—
75#97—41no—
76C4×C8—21no—
77C4×C8—21no—
78C4×C8—21no—
79C4×C8—21no—
80C4×C8—41no—
81C4×C8—44no—
82#41—42no—
83#41—42no—
84C2×Q8—21no—
85C2×Q8—41no—
86C2×D4—21no—
87C2×D4—41no—
88C2×D4—41no—
89C22×C4—21no—
90C22×C4—22no—
91C22×C4—41no—
92C22×C4C2×C22⋊C411yescharacteristic
93M16—42no—
94M16—42no—
95M16—82no—
96C2×C8—22no—
97C2×C8—22no—
98C2×C8—21no—
99C2×C8—22no—
100C2×C8—22no—
101C2×C8—22no—
102C2×C8—41no—
103C2×C8—41no—
104C2×C8—82no—
105C2×C8—84no—
106C2×C8#10011yescharacteristic
107C2×C8#10011yescharacteristic
108C4⋊C4—22no—
109C4⋊C4—41no—
110C4⋊C4—42no—
111C4⋊C4—41no—
112C22⋊C4—42no—
113C22⋊C4—41no—
114C42—21no—
115C42—22no—
116C42—41no—
117C42—82no—
118C42C2×M1611yescharacteristic
119C42C2×M1611yescharacteristic
120C42#13514yes—
121C23—41no—
122C23#41011yescharacteristic
123Q8—42no—
124Q8—82no—
125D4—42no—
126D4—82no—
127D4—82no—
128C2×C4—21no—
129C2×C4—21no—
130C2×C4—24no—
131C2×C4—42no—
132C2×C4—42no—
133C2×C4—41no—
134C2×C4—82no—
135C2×C4—81no—
136C2×C4—81no—
137C2×C4#40711yescharacteristic
138C2×C4C2×#10011yescharacteristic
139C2×C4#41612yes—
140C2×C4#42112yes—
141C8—21no—
142C8—21no—
143C8—41no—
144C8—42no—
145C8—81no—
146C8—81no—
147C8—164no—
148C22—41no—
149C22—82no—
150C22—81no—
151C22#146611yescharacteristic
152C22#152112yes—
153C4—22no—
154C4—44no—
155C4—41no—
156C4—82no—
157C4—81no—
158C4#148812yes—
159C4#153311yescharacteristic
160C4#154311yescharacteristic
161C2—21no—
162C2—22no—
163C2—161no—
164C2#764911yescharacteristic
subgroupG/H
centerC4#1543
commutatorC2×C42C22×C4
Frattini, cosocleC43C23
socleC2#7649
prime radical#532C2×C4
LCSG ▷ C2×C42 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C4 ◁ C22×C8 ◁ C4×#97 ◁ G
LECSG ▷ C43 ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #8801C2semidirect
3. #7134C2semidirect

Maximal subgroups

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

subgroupindex
#216602
#88012
#71342
#71062