Finite Group Databasebeta

← #8238

Subgroups — #8238

Order 256.

267 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 267 conjugacy classes of subgroups — too many to draw. Only its 85 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#2929C211yesmaximal, characteristic
2C8×C22⋊C4C212yesmaximal, complemented
3#1501C212yesmaximal
4#1298C211yesmaximal, characteristic, complemented
5#1297C211yesmaximal, characteristic, complemented
6C2×#131C411yescharacteristic, complemented
7C23×C8C2211yescharacteristic
8#525C2211yescharacteristic
9#435—22no—
10C8×D4—22no—
11#424C411yescharacteristic, complemented
12#408C411yescharacteristic, complemented
13#407C2211yescharacteristic
14C2×C22⋊C8C2212yes—
15#405C411yescharacteristic, complemented
16C2×C4×C8—22no—
17C4×C22⋊C4C2212yes—
18#336—22no—
19#328C414yescomplemented
20#327C414yescomplemented
21C2×#41C2×C411yescharacteristic
22C23×C4C2311yescharacteristic
23#131—42no—
24C2×M16—21no—
25C2×M16C2×C411yescharacteristic
26C2×M16C2×C411yescharacteristic
27C22×C8—22no—
28C22×C8—21no—
29C22×C8—42no—
30C22×C8C2×C411yescharacteristic
31C22×C8C2×C411yescharacteristic
32C22×C8D412yes—
33C22×C8D412yes—
34#123—21no—
35C22⋊Q8C2×C412yes—
36#121C2×C412yes—
37#120—21no—
38C4×D4—22no—
39#117C2×C411yescharacteristic
40C2×C22⋊C4—22no—
41C2×C42—22no—
42C4⋊C8—22no—
43C22⋊C8—22no—
44C22⋊C8—22no—
45C22⋊C8—22no—
46C22⋊C8C2×C414yes—
47#97—21no—
48C4×C8—21no—
49C4×C8—44no—
50#95C2×C414yes—
51C24#4111yescharacteristic
52#41—41no—
53#41—41no—
54C2×Q8C4211yescharacteristic
55C2×D4—21no—
56C2×D4—22no—
57C2×D4C4211yescharacteristic
58C22×C4—24no—
59C22×C4—21no—
60C22×C4—41no—
61C22×C4—41no—
62C22×C4C22×C411yescharacteristic
63C22×C4C22×C412yes—
64C22×C4C2×D412yes—
65C22×C4#4111yescharacteristic
66M16—42no—
67M16—41no—
68M16—41no—
69C2×C8—24no—
70C2×C8—22no—
71C2×C8—21no—
72C2×C8—21no—
73C2×C8—24no—
74C2×C8—41no—
75C2×C8—42no—
76C2×C8—42no—
77C2×C8—42no—
78C2×C8—42no—
79C2×C8C22⋊C411yescharacteristic
80C2×C8C22⋊C412yes—
81C2×C8C22⋊C411yescharacteristic
82C4⋊C4—22no—
83C4⋊C4C4212yes—
84C22⋊C4—28no—
85C22⋊C4—21no—
86C22⋊C4—22no—
87C22⋊C4—22no—
88C42—21no—
89C42—42no—
90C42—42no—
91C23—24no—
92C23—21no—
93C23—41no—
94C23#11711yescharacteristic
95C23C4×D412yes—
96Q8—41no—
97D4—41no—
98D4—42no—
99D4—44no—
100C2×C4—22no—
101C2×C4—21no—
102C2×C4—22no—
103C2×C4—21no—
104C2×C4—28no—
105C2×C4—22no—
106C2×C4—21no—
107C2×C4—42no—
108C2×C4—41no—
109C2×C4—44no—
110C2×C4—44no—
111C2×C4—41no—
112C2×C4—41no—
113C2×C4C2×C4211yescharacteristic
114C2×C4C2×C22⋊C411yescharacteristic
115C2×C4C4×D412yes—
116C8—22no—
117C8—21no—
118C8—44no—
119C8—41no—
120C8—41no—
121C8#9912yes—
122C22—22no—
123C22—22no—
124C22—21no—
125C22—22no—
126C22—44no—
127C22—41no—
128C22—42no—
129C22C4×C22⋊C411yescharacteristic
130C22#44314yes—
131C4—22no—
132C4—21no—
133C4—48no—
134C4—41no—
135C4—42no—
136C4C2×#9911yescharacteristic
137C4#41011yescharacteristic
138C2—24no—
139C2—41no—
140C2—81no—
141C2C4×#9911yescharacteristic
142C2#176412yes—
subgroupG/H
centerC2×C8C22⋊C4
commutatorC2×C4C2×C42
Frattini, cosocleC23×C4C23
socleC22C4×C22⋊C4
prime radical#525C22
LCSG ▷ C2×C4 ▷ C22 ▷ C1
UCSC1 ◁ C2×C8 ◁ C23×C8 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C8×C22⋊C4C2semidirect
4. #1298C2semidirect
5. #1297C2semidirect
6. C2×#131C4semidirect
11. #424C4semidirect
12. #408C4semidirect
15. #405C4semidirect
19. #328C4semidirect
20. #327C4semidirect

Maximal subgroups

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

subgroupindex
#29292
C8×C22⋊C42
#15012
#12982
#12972