Finite Group Databasebeta

← #12966

Subgroups — #12966

Order 256.

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

normal subgroups only

This group has 290 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#2380C211yesmaximal, characteristic
2C2×#353C211yesmaximal, characteristic
3#1887C211yesmaximal, characteristic
4#1867C212yesmaximal, complemented
5#1390C212yesmaximal, complemented
6C2×#127C2211yescharacteristic
7#518C2211yescharacteristic
8#418—21no—
9C2×#100C2211yescharacteristic
10C2×#99C2212yes—
11#386—21no—
12#386—22no—
13#386—22no—
14#386C2212yes—
15#385—22no—
16#383—21no—
17#353C414yescomplemented
18#342—22no—
19#339—21no—
20C22×D4—22no—
21C22×D4D411yescharacteristic
22C22×D4C2311yescharacteristic
23C23×C4—21no—
24C23×C4D412yes—
25C2×M16—21no—
26#127—22no—
27#127C2×C412yes—
28#127Q812yes—
29C4×D4—42no—
30#117—21no—
31C2×C4⋊C4—42no—
32C2×C22⋊C4—22no—
33C2×C22⋊C4—22no—
34C2×C22⋊C4—21no—
35C2×C22⋊C4—22no—
36C2×C22⋊C4—22no—
37C2×C22⋊C4D412yes—
38C2×C42D411yescharacteristic
39#102—42no—
40#100—22no—
41#99—22no—
42#99C2×C414yescomplemented
43#95—21no—
44#95—22no—
45#95—21no—
46#95—22no—
47#95—42no—
48C24—21no—
49C24C2×D412yes—
50C2×D4—24no—
51C2×D4—24no—
52C2×D4—21no—
53C2×D4—41no—
54C2×D4—41no—
55C2×D4—41no—
56C2×D4—41no—
57C2×D4—82no—
58C2×D4—82no—
59C2×D4C4⋊C412yescomplemented
60C2×D4C22×C411yescharacteristic
61C2×D4C2×Q811yescharacteristic
62C2×D4#4112yes—
63C22×C4—21no—
64C22×C4—22no—
65C22×C4—22no—
66C22×C4—22no—
67C22×C4—22no—
68C22×C4—42no—
69C22×C4—42no—
70C22×C4—82no—
71C22×C4—82no—
72C22×C4—81no—
73C22×C4C2×D411yescharacteristic
74M16—41no—
75M16—41no—
76C2×C8—41no—
77C4⋊C4—22no—
78C4⋊C4—84no—
79C22⋊C4—24no—
80C22⋊C4—42no—
81C22⋊C4—42no—
82C22⋊C4—41no—
83C22⋊C4—42no—
84C22⋊C4—82no—
85C22⋊C4—82no—
86C22⋊C4—81no—
87C42—21no—
88C42—41no—
89C42C4⋊C412yescomplemented
90C23—22no—
91C23—22no—
92C23—22no—
93C23—42no—
94C23—42no—
95C23—42no—
96C23—42no—
97C23—41no—
98C23—41no—
99C23C4×D412yes—
100C23#12011yescharacteristic
101C23C22⋊Q812yes—
102D4—41no—
103D4—41no—
104D4—42no—
105D4—42no—
106D4—81no—
107D4—84no—
108D4—84no—
109C2×C4—21no—
110C2×C4—21no—
111C2×C4—22no—
112C2×C4—42no—
113C2×C4—44no—
114C2×C4—42no—
115C2×C4—42no—
116C2×C4—42no—
117C2×C4—42no—
118C2×C4—44no—
119C2×C4—42no—
120C2×C4—42no—
121C2×C4—42no—
122C2×C4—82no—
123C2×C4—82no—
124C2×C4—84no—
125C2×C4—82no—
126C2×C4—82no—
127C2×C4—82no—
128C2×C4C2×C4⋊C411yescharacteristic
129C2×C4#12311yescharacteristic
130C8—81no—
131C22—22no—
132C22—22no—
133C22—22no—
134C22—42no—
135C22—42no—
136C22—44no—
137C22—42no—
138C22—42no—
139C22—42no—
140C22—82no—
141C22—81no—
142C22—81no—
143C22—82no—
144C22#38511yescharacteristic
145C22B(4,𝔽2)11yescharacteristic
146C22#45811yescharacteristic
147C4—21no—
148C4—21no—
149C4—42no—
150C4—82no—
151C4—84no—
152C4—82no—
153C4—82no—
154C2—21no—
155C2—21no—
156C2—42no—
157C2—42no—
158C2—82no—
159C2#186711yescharacteristic
160C2#220012yes—
subgroupG/H
center, socleC22B(4,𝔽2)
commutatorC2×D4C22×C4
Frattini, cosocleC22×D4C23
prime radicalC2×#127C22
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
4. #1867C2semidirect
5. #1390C2semidirect
17. #353C4semidirect
42. #99C2×C4semidirect
59. C2×D4C4⋊C4semidirect
89. C42C4⋊C4semidirect

Maximal subgroups

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

subgroupindex
#23802
C2×#3532
#18872
#18672
#13902