Finite Group Databasebeta

← #9416

Subgroups — #9416

Order 256.

190 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 190 conjugacy classes of subgroups — too many to draw. Only its 57 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#3185C211yesmaximal, characteristic
2#2987C211yesmaximal, characteristic
3#1537C211yesmaximal, characteristic
4#1515C211yesmaximal, characteristic
5#1515C211yesmaximal, characteristic
6#1291C211yesmaximal, characteristic
7#1288C211yesmaximal, characteristic
8#533C2211yescharacteristic
9#533C2211yescharacteristic
10#498C411yescharacteristic, complemented
11#494C411yescharacteristic, complemented
12#487C411yescharacteristic, complemented
13#487C411yescharacteristic, complemented
14#470—22no—
15#446—22no—
16#446—22no—
17#441—22no—
18#433—22no—
19#432C2211yescharacteristic
20#432C2211yescharacteristic
21#432C2211yescharacteristic
22C4×M16C2211yescharacteristic
23#383C2211yescharacteristic
24#330—22no—
25#328—22no—
26#328—22no—
27C2×Q16—22no—
28C2×M16—21no—
29C4⋊Q8C2×C411yescharacteristic
30C4⋊Q8C2×C411yescharacteristic
31C4⋊Q8C2×C411yescharacteristic
32C4⋊Q8C2×C411yescharacteristic
33#125—21no—
34#125—21no—
35C22⋊Q8—21no—
36C22⋊Q8—21no—
37C22⋊Q8—21no—
38C22⋊Q8D412yes—
39C4×Q8—21no—
40C4×Q8—21no—
41C4×Q8—21no—
42C4×Q8D412yes—
43#117—21no—
44#117—21no—
45C2×C42C2311yescharacteristic
46#106—22no—
47C4⋊C8—24no—
48C4⋊C8—21no—
49C4⋊C8—21no—
50C4⋊C8—21no—
51#103—22no—
52#103—22no—
53C22⋊C8—21no—
54C22⋊C8—21no—
55C22⋊C8D412yes—
56#97—22no—
57#97—21no—
58#97—21no—
59#97—21no—
60#97D412yes—
61C4×C8—22no—
62C4×C8C2×C411yescharacteristic
63C4×C8C2×C411yescharacteristic
64#95—21no—
65#95—21no—
66#95—21no—
67C2×Q8—21no—
68C2×Q8—21no—
69C2×Q8—21no—
70C2×Q8C22⋊C412yes—
71C22×C4—41no—
72C22×C4C2×D411yescharacteristic
73C22×C4C2×D411yescharacteristic
74C22×C4#4111yescharacteristic
75Q16—44no—
76M16—42no—
77C2×C8—21no—
78C2×C8—21no—
79C2×C8—22no—
80C2×C8—22no—
81C2×C8—41no—
82C4⋊C4—21no—
83C4⋊C4—21no—
84C4⋊C4—21no—
85C4⋊C4—22no—
86C4⋊C4—22no—
87C4⋊C4—21no—
88C4⋊C4—21no—
89C4⋊C4—21no—
90C4⋊C4—21no—
91C4⋊C4—41no—
92C4⋊C4—41no—
93C4⋊C4C22⋊C412yes—
94C22⋊C4—22no—
95C22⋊C4—41no—
96C42—22no—
97C42—41no—
98C42C22×C411yescharacteristic
99C42C2×D411yescharacteristic
100C42C2×D411yescharacteristic
101C42#4111yescharacteristic
102C23#12111yescharacteristic
103Q8—42no—
104Q8—42no—
105Q8—81no—
106Q8—81no—
107C2×C4—22no—
108C2×C4—22no—
109C2×C4—41no—
110C2×C4—41no—
111C2×C4—41no—
112C2×C4—41no—
113C2×C4—41no—
114C2×C4—81no—
115C2×C4—81no—
116C2×C4—81no—
117C2×C4C2×C22⋊C411yescharacteristic
118C2×C4C4×D411yescharacteristic
119C2×C4C4×D411yescharacteristic
120C2×C4#12011yescharacteristic
121C2×C4#12111yescharacteristic
122C2×C4#12311yescharacteristic
123C8—22no—
124C8—42no—
125C8—42no—
126C8—41no—
127C8—81no—
128C22—21no—
129C22—21no—
130C22—21no—
131C22#38611yescharacteristic
132C4—21no—
133C4—21no—
134C4—21no—
135C4—21no—
136C4—21no—
137C4—21no—
138C4—42no—
139C4—42no—
140C4—81no—
141C4—81no—
142C4—161no—
143C2—41no—
144C2#188411yescharacteristic
145C2#190211yescharacteristic
146C2#190511yescharacteristic
subgroupG/H
center, socleC22#386
commutatorC42C22×C4
Frattini, cosocleC2×C42C23
prime radicalC23#121
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
10. #498C4semidirect
11. #494C4semidirect
12. #487C4semidirect
13. #487C4semidirect

Maximal subgroups

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

subgroupindex
#31852
#29872
#15372
#15152
#12912
#12882