Finite Group Databasebeta

← #13463

Subgroups — #13463

Order 256.

242 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 242 conjugacy classes of subgroups — too many to draw. Only its 49 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#2804C211yesmaximal, characteristic
2#1867C212yesmaximal
3#1856C211yesmaximal, characteristic
4#1393C211yesmaximal, characteristic
5#1390C212yesmaximal
6#518C2211yescharacteristic
7#428—21no—
8C2×#100C2211yescharacteristic
9C2×#99C2212yes—
10#400—21no—
11#393—22no—
12#391—21no—
13#390C2211yescharacteristic
14#386C2212yes—
15#385—22no—
16#385—22no—
17#342—22no—
18#339—21no—
19C22×Q8—21no—
20C22×D4C2311yescharacteristic
21C23×C4D412yes—
22C2×M16—21no—
23C4×D4—41no—
24C4×D4—41no—
25#117—21no—
26C2×C4⋊C4—21no—
27C2×C4⋊C4—21no—
28C2×C4⋊C4—42no—
29C2×C4⋊C4D411yescharacteristic
30C2×C22⋊C4—22no—
31C2×C22⋊C4—22no—
32C2×C22⋊C4—22no—
33C2×C22⋊C4—22no—
34C2×C22⋊C4D412yes—
35C2×C42D411yescharacteristic
36#107—41no—
37#106—41no—
38#100—21no—
39#100Q812yescomplemented
40#99—22no—
41#99Q814yescomplemented
42#95—22no—
43#95—22no—
44#95—21no—
45#95—22no—
46#95—42no—
47C24C2×D412yes—
48C2×Q8—42no—
49C2×Q8—81no—
50C2×Q8—81no—
51C2×D4—41no—
52C2×D4—41no—
53C2×D4C2×Q811yescharacteristic
54C2×D4C2×Q812yes—
55C2×D4#4111yescharacteristic
56C22×C4—21no—
57C22×C4—22no—
58C22×C4—21no—
59C22×C4—22no—
60C22×C4—22no—
61C22×C4—22no—
62C22×C4—42no—
63C22×C4—42no—
64C22×C4—82no—
65C22×C4C2×D411yescharacteristic
66M16—22no—
67M16—41no—
68C2×C8—41no—
69C4⋊C4—21no—
70C4⋊C4—21no—
71C4⋊C4—41no—
72C4⋊C4—81no—
73C4⋊C4—81no—
74C4⋊C4—84no—
75C22⋊C4—24no—
76C22⋊C4—44no—
77C22⋊C4—42no—
78C22⋊C4—42no—
79C22⋊C4—41no—
80C22⋊C4—42no—
81C22⋊C4—82no—
82C42—41no—
83C42—41no—
84C23—22no—
85C23—22no—
86C23—42no—
87C23—42no—
88C23#12011yescharacteristic
89C23C22⋊Q812yes—
90C23C4⋊Q812yes—
91Q8—84no—
92D4—41no—
93D4—42no—
94D4—41no—
95C2×C4—21no—
96C2×C4—21no—
97C2×C4—22no—
98C2×C4—41no—
99C2×C4—41no—
100C2×C4—44no—
101C2×C4—42no—
102C2×C4—42no—
103C2×C4—42no—
104C2×C4—42no—
105C2×C4—42no—
106C2×C4—44no—
107C2×C4—41no—
108C2×C4—41no—
109C2×C4—41no—
110C2×C4—42no—
111C2×C4—42no—
112C2×C4—82no—
113C2×C4—82no—
114C2×C4—82no—
115C2×C4—82no—
116C2×C4—82no—
117C2×C4C22⋊Q811yescharacteristic
118C2×C4C4⋊Q811yescharacteristic
119C8—42no—
120C22—21no—
121C22—21no—
122C22—22no—
123C22—22no—
124C22—42no—
125C22—42no—
126C22—42no—
127C22—42no—
128C22—82no—
129C22#39511yescharacteristic
130C22B(4,𝔽2)11yescharacteristic
131C22#45811yescharacteristic
132C4—21no—
133C4—21no—
134C4—42no—
135C4—82no—
136C4—84no—
137C4—81no—
138C4—81no—
139C4—82no—
140C4—81no—
141C2—21no—
142C2—21no—
143C2—42no—
144C2—42no—
145C2#206211yescharacteristic
146C2#219811yescharacteristic
147C2#219911yescharacteristic
subgroupG/H
center, socleC22B(4,𝔽2)
commutator, Frattini, prime radical, cosocleC22×D4C23
LCSG ▷ C22×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
39. #100Q8semidirect
41. #99Q8semidirect

Maximal subgroups

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

subgroupindex
#28042
#18672
#18562
#13932
#13902