Finite Group Databasebeta

← #12965

Subgroups — #12965

Order 256.

268 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 268 conjugacy classes of subgroups — too many to draw. Only its 59 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#2372C211yesmaximal, characteristic
2C2×#354C211yesmaximal, characteristic
3#1885C211yesmaximal, characteristic
4#1867C212yesmaximal
5#1390C212yesmaximal
6C2×#124C2211yescharacteristic
7#518C2211yescharacteristic
8#419—21no—
9C2×#100C2211yescharacteristic
10C2×#99C2212yes—
11#391—21no—
12#388—22no—
13#386C2212yes—
14#385—22no—
15C4×C22⋊C4—22no—
16#376—21no—
17#354C414yescomplemented
18#342—22no—
19#339—21no—
20C22×Q8D411yescharacteristic
21C22×D4C2311yescharacteristic
22C23×C4D412yes—
23C2×M16—21no—
24#124—22no—
25#124C2×C412yes—
26#124Q812yes—
27C4×D4—42no—
28#117—21no—
29C2×C4⋊C4—21no—
30C2×C4⋊C4—42no—
31C2×C22⋊C4—22no—
32C2×C22⋊C4—22no—
33C2×C22⋊C4—22no—
34C2×C22⋊C4—22no—
35C2×C22⋊C4D412yes—
36C2×C42—22no—
37C2×C42—21no—
38C2×C42D411yescharacteristic
39#103—42no—
40#100—22no—
41#99—22no—
42#99C2×C414yes—
43#95—21no—
44#95—21no—
45#95—22no—
46#95—22no—
47#95—42no—
48C24C2×D412yes—
49C2×Q8—21no—
50C2×Q8—41no—
51C2×Q8—41no—
52C2×Q8C4⋊C412yescomplemented
53C2×D4—41no—
54C2×D4—41no—
55C2×D4C22×C411yescharacteristic
56C2×D4C2×Q811yescharacteristic
57C2×D4#4112yes—
58C22×C4—22no—
59C22×C4—21no—
60C22×C4—21no—
61C22×C4—22no—
62C22×C4—22no—
63C22×C4—22no—
64C22×C4—44no—
65C22×C4—42no—
66C22×C4—42no—
67C22×C4C2×D411yescharacteristic
68M16—41no—
69M16—41no—
70C2×C8—41no—
71C4⋊C4—22no—
72C4⋊C4—81no—
73C4⋊C4—84no—
74C22⋊C4—24no—
75C22⋊C4—24no—
76C22⋊C4—24no—
77C22⋊C4—44no—
78C22⋊C4—42no—
79C22⋊C4—42no—
80C22⋊C4—41no—
81C22⋊C4—42no—
82C42—21no—
83C42—44no—
84C42—41no—
85C42—81no—
86C42C4⋊C412yescomplemented
87C23—22no—
88C23—22no—
89C23—42no—
90C23—42no—
91C23C4×D412yes—
92C23#12011yescharacteristic
93C23C22⋊Q812yes—
94Q8—42no—
95Q8—81no—
96D4—41no—
97D4—41no—
98D4—42no—
99C2×C4—21no—
100C2×C4—21no—
101C2×C4—24no—
102C2×C4—22no—
103C2×C4—22no—
104C2×C4—42no—
105C2×C4—42no—
106C2×C4—42no—
107C2×C4—41no—
108C2×C4—41no—
109C2×C4—44no—
110C2×C4—42no—
111C2×C4—44no—
112C2×C4—42no—
113C2×C4—42no—
114C2×C4—42no—
115C2×C4—82no—
116C2×C4—82no—
117C2×C4—82no—
118C2×C4—82no—
119C2×C4—82no—
120C2×C4C2×C4⋊C411yescharacteristic
121C2×C4#12311yescharacteristic
122C8—81no—
123C22—22no—
124C22—22no—
125C22—22no—
126C22—42no—
127C22—42no—
128C22—42no—
129C22—42no—
130C22—82no—
131C22#38511yescharacteristic
132C22B(4,𝔽2)11yescharacteristic
133C22#45811yescharacteristic
134C4—21no—
135C4—21no—
136C4—42no—
137C4—42no—
138C4—44no—
139C4—82no—
140C4—84no—
141C4—82no—
142C2—21no—
143C2—21no—
144C2—42no—
145C2—42no—
146C2#186711yescharacteristic
147C2#219712yes—
subgroupG/H
center, socleC22B(4,𝔽2)
commutatorC2×D4C22×C4
Frattini, prime radical, cosocleC22×D4C23
LCSG ▷ C2×D4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C23 ◁ C22×D4 ◁ G
LECSG ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
17. #354C4semidirect
52. C2×Q8C4⋊C4semidirect
86. C42C4⋊C4semidirect

Maximal subgroups

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

subgroupindex
#23722
C2×#3542
#18852
#18672
#13902