Finite Group Databasebeta

← #7123

Subgroups — #7123

Order 256.

206 conjugacy classes of subgroups; only the 45 normal subgroups are shown.

normal subgroups only

This group has 206 conjugacy classes of subgroups — too many to draw. Only its 45 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#1444C211yesmaximal, characteristic
2#1280C211yesmaximal, characteristic
3#1280C211yesmaximal, characteristic
4C22×C4⋊C4—21no—
5C22×C4⋊C4C411yescharacteristic
6C2×C22⋊C8—21no—
7C2×C22⋊C8—21no—
8C2×#95—21no—
9C2×#95C411yescharacteristic
10C2×#95C2211yescharacteristic
11C23×C4—21no—
12C23×C4—21no—
13C23×C4C2×C411yescharacteristic
14C23×C4D411yescharacteristic
15C23×C4D411yescharacteristic
16C22×C8—41no—
17C22×C8—41no—
18C2×C4⋊C4—41no—
19C2×C4⋊C4—42no—
20C2×C4⋊C4—42no—
21C2×C4⋊C4—41no—
22C2×C4⋊C4—41no—
23C2×C4⋊C4—42no—
24C22⋊C8—22no—
25C22⋊C8—22no—
26C22⋊C8—41no—
27C22⋊C8—41no—
28#95—21no—
29#95—41no—
30#95—42no—
31#95C812yescomplemented
32C24C22⋊C411yescharacteristic
33C22×C4—22no—
34C22×C4—22no—
35C22×C4—22no—
36C22×C4—21no—
37C22×C4—21no—
38C22×C4—21no—
39C22×C4—21no—
40C22×C4—21no—
41C22×C4—21no—
42C22×C4—22no—
43C22×C4—41no—
44C22×C4—41no—
45C22×C4—41no—
46C22×C4—41no—
47C22×C4—41no—
48C22×C4—41no—
49C22×C4—41no—
50C22×C4—41no—
51C22×C4—41no—
52C22×C4—41no—
53C22×C4—41no—
54C22×C4—41no—
55C22×C4—41no—
56C22×C4—41no—
57C22×C4—41no—
58C22×C4—41no—
59C22×C4C2×C811yescharacteristic
60C22×C4M1611yescharacteristic
61C22×C4D811yescharacteristic
62C22×C4SD1611yescharacteristic
63C22×C4SD1611yescharacteristic
64C22×C4Q1611yescharacteristic
65C2×C8—42no—
66C2×C8—42no—
67C2×C8—81no—
68C2×C8—81no—
69C2×C8—81no—
70C2×C8—81no—
71C4⋊C4—42no—
72C4⋊C4—44no—
73C4⋊C4—44no—
74C4⋊C4—42no—
75C23—21no—
76C23—21no—
77C23—21no—
78C23—21no—
79C23C22⋊C811yescharacteristic
80C23#9911yescharacteristic
81C23#10111yescharacteristic
82C23#10211yescharacteristic
83C23#10311yescharacteristic
84C23#10411yescharacteristic
85C23#10411yescharacteristic
86C2×C4—24no—
87C2×C4—21no—
88C2×C4—21no—
89C2×C4—21no—
90C2×C4—21no—
91C2×C4—21no—
92C2×C4—21no—
93C2×C4—42no—
94C2×C4—42no—
95C2×C4—42no—
96C2×C4—42no—
97C2×C4—42no—
98C2×C4—42no—
99C2×C4—42no—
100C2×C4—41no—
101C2×C4—41no—
102C2×C4—41no—
103C2×C4—41no—
104C2×C4—41no—
105C2×C4—41no—
106C2×C4—41no—
107C2×C4—41no—
108C2×C4—41no—
109C2×C4—41no—
110C2×C4—41no—
111C2×C4—41no—
112C2×C4—42no—
113C2×C4—42no—
114C2×C4—42no—
115C2×C4—81no—
116C2×C4—81no—
117C2×C4—81no—
118C2×C4—81no—
119C2×C4—81no—
120C2×C4—81no—
121C8—82no—
122C8—82no—
123C22—21no—
124C22—21no—
125C22—21no—
126C22—21no—
127C22—21no—
128C22—21no—
129C22—21no—
130C22—21no—
131C22—21no—
132C22—21no—
133C22—21no—
134C22—21no—
135C22#32411yescharacteristic
136C22D4⋊C811yescharacteristic
137C22Q8⋊C811yescharacteristic
138C22#32711yescharacteristic
139C22#32811yescharacteristic
140C22#33011yescharacteristic
141C22#33211yescharacteristic
142C22AΣL(1,𝔽16)11yescharacteristic
143C22#35211yescharacteristic
144C22#35311yescharacteristic
145C22#35611yescharacteristic
146C4—44no—
147C4—41no—
148C4—41no—
149C4—41no—
150C4—41no—
151C4—41no—
152C4—41no—
153C4—82no—
154C2—21no—
155C2—21no—
156C2—21no—
157C2—21no—
158C2#127111yescharacteristic
159C2#132511yescharacteristic
160C2#132611yescharacteristic
161C2#134311yescharacteristic
162C2#134611yescharacteristic
163C2#135211yescharacteristic
164C2#135411yescharacteristic
subgroupG/H
center, socleC23#99
commutatorC22×C4C2×C8
Frattini, cosocleC2×#95C22
prime radicalC24C22⋊C4
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C24 ◁ C2×#95 ◁ G
LECSG ▷ C2×#95 ▷ C24 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
31. #95C8semidirect

Maximal subgroups

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

subgroupindex
#14442
#12802