Finite Group Databasebeta

← #124156

Subgroups — #124156

Order 512.

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

normal subgroups only

This group has 321 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#21352C211yesmaximal, characteristic
2C4×#349C212yesmaximal
3#7124C214yesmaximal, complemented
4C2×#432C411yescharacteristic
5C2×C4×M16C2211yescharacteristic
6#2283C411yescharacteristic
7C2×#349—22no—
8C4×M32—22no—
9#1940C412yes—
10#1940C2212yes—
11#1851—22no—
12#1768—22no—
13#1377C2214yes—
14C22×M16D412yes—
15C22×C42C2×C411yescharacteristic
16C2×M32—24no—
17#432—42no—
18C2×C4⋊C8—22no—
19C2×C4⋊C8—22no—
20C2×C4⋊C8C2×C412yes—
21C2×C22⋊C8—21no—
22C4×M16—42no—
23C2×#97—21no—
24C2×#97C2×C411yescharacteristic
25C2×#97C2311yescharacteristic
26C2×C4×C8D412yes—
27#383C812yes—
28#382—21no—
29#381C812yes—
30#380—21no—
31C4×C4⋊C4C2×C412yes—
32C4×C22⋊C4—21no—
33#376—21no—
34#349—28no—
35#346—42no—
36C4×C16—42no—
37#336—22no—
38#336—22no—
39C23×C4C22⋊C411yescharacteristic
40C23×C4C2×C811yescharacteristic
41C23×C4M1611yescharacteristic
42C2×M16—24no—
43C2×M16—24no—
44C2×M16—42no—
45C22×C8—22no—
46C22×C8C22⋊C412yes—
47C22×C8C2×D412yes—
48C2×C4⋊C4—22no—
49C2×C22⋊C4—22no—
50C2×C42—22no—
51C2×C42—22no—
52C2×C42—22no—
53C2×C42C22⋊C411yescharacteristic
54C2×C42C2×C811yescharacteristic
55C2×C42M1611yescharacteristic
56C2×C42C22×C411yescharacteristic
57M32—28no—
58M32—44no—
59C2×C16—44no—
60C4⋊C8—42no—
61C4⋊C8—44no—
62C4⋊C8—44no—
63C22⋊C8—42no—
64#97—41no—
65#97—41no—
66#97—42no—
67C4×C8—22no—
68C4×C8—22no—
69#95—22no—
70#95—22no—
71#95—22no—
72#95C2×C814yes—
73C24C22⋊C811yescharacteristic
74C22×C4—21no—
75C22×C4—21no—
76C22×C4—22no—
77C22×C4—22no—
78C22×C4—22no—
79C22×C4—22no—
80C22×C4—24no—
81C22×C4—81no—
82C22×C4—81no—
83C22×C4C22⋊C811yescharacteristic
84C22×C4C22⋊C811yescharacteristic
85C22×C4C22⋊C811yescharacteristic
86C22×C4C2×C22⋊C411yescharacteristic
87C22×C4C22×C811yescharacteristic
88C22×C4C2×M1611yescharacteristic
89M16—48no—
90C2×C8—22no—
91C2×C8—22no—
92C2×C8—22no—
93C2×C8—22no—
94C2×C8—42no—
95C2×C8—42no—
96C2×C8—42no—
97C2×C8#10414yes—
98C2×C8#13514yes—
99C4⋊C4—82no—
100C22⋊C4—82no—
101C42—22no—
102C42—22no—
103C42—21no—
104C42—21no—
105C42—42no—
106C42—42no—
107C42#9912yes—
108C42#10112yes—
109C16—48no—
110C23—21no—
111C23—22no—
112C23—41no—
113C23—41no—
114C23C2×C22⋊C811yescharacteristic
115C2×C4—21no—
116C2×C4—21no—
117C2×C4—21no—
118C2×C4—22no—
119C2×C4—41no—
120C2×C4—41no—
121C2×C4—41no—
122C2×C4—42no—
123C2×C4—41no—
124C2×C4—41no—
125C2×C4—41no—
126C2×C4—44no—
127C2×C4—44no—
128C2×C4—44no—
129C2×C4—82no—
130C2×C4—82no—
131C2×C4#32412yes—
132C2×C4#32412yes—
133C2×C4C2×#9911yescharacteristic
134C2×C4C2×#10111yescharacteristic
135C2×C4#41612yes—
136C2×C4C2×#10412yes—
137C8—24no—
138C8—24no—
139C8—82no—
140C22—21no—
141C22—21no—
142C22—41no—
143C22—41no—
144C22—41no—
145C22—82no—
146C22C2×#32411yescharacteristic
147C22#147612yes—
148C4—22no—
149C4—22no—
150C4—41no—
151C4—41no—
152C4—41no—
153C4—84no—
154C4#150112yes—
155C4#152611yescharacteristic
156C4#154411yescharacteristic
157C2—22no—
158C2—81no—
159C2#756411yescharacteristic
160C2#1145611yescharacteristic
161C2#1145811yescharacteristic
subgroupG/H
centerC2×C4C2×#99
commutatorC22×C4C22×C8
Frattini, cosocleC2×#97C23
socleC22C2×#324
prime radicalC24C22⋊C8
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ C2×C4×M16 ◁ G
LECSG ▷ C2×#97 ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #7124C2semidirect

Maximal subgroups

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

subgroupindex
#213522
C4×#3492
#71242