Finite Group Databasebeta

← #7656

Subgroups — #7656

Order 256.

216 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 63 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#2861C211yesmaximal, characteristic
2#1839C211yesmaximal, characteristic, complemented
3#1744C211yesmaximal, characteristic, complemented
4#1272C212yesmaximal, complemented
5#1270C212yesmaximal
6C2×#125C411yescharacteristic
7C2×#124C411yescharacteristic
8C2×C4×C8—21no—
9#391C411yescharacteristic
10#391C2211yescharacteristic
11#390C411yescharacteristic
12#390C2211yescharacteristic
13#388—21no—
14#388—21no—
15#388—21no—
16#388—21no—
17C43C2211yescharacteristic
18#336—21no—
19#336—21no—
20#336—21no—
21#336C2212yes—
22#336C2212yes—
23C22×Q8C2×C411yescharacteristic
24C22×D4C2×C411yescharacteristic
25C22×C8—22no—
26C22×C8—22no—
27#125—42no—
28#124—42no—
29C2×C4⋊C4—21no—
30C2×C4⋊C4—21no—
31C2×C4⋊C4C2×C411yescharacteristic
32C2×C4⋊C4C2×C411yescharacteristic
33C2×C22⋊C4—21no—
34C2×C22⋊C4—21no—
35C2×C22⋊C4—21no—
36C2×C22⋊C4—21no—
37C2×C22⋊C4—21no—
38C2×C22⋊C4—21no—
39C2×C42C2×C411yescharacteristic
40C2×C42C2×C411yescharacteristic
41C2×C42D411yescharacteristic
42C2×C42D411yescharacteristic
43C2×C42D411yescharacteristic
44C2×C42D411yescharacteristic
45C2×C42C2311yescharacteristic
46C4×C8—24no—
47#95—21no—
48#95—21no—
49#95—21no—
50#95—21no—
51C24—21no—
52C2×Q8—22no—
53C2×Q8—41no—
54C2×Q8—41no—
55C2×D4—22no—
56C2×D4—41no—
57C2×D4—41no—
58C22×C4—21no—
59C22×C4—21no—
60C22×C4—21no—
61C22×C4C22⋊C411yescharacteristic
62C22×C4C22⋊C411yescharacteristic
63C22×C4C22⋊C411yescharacteristic
64C22×C4C22⋊C411yescharacteristic
65C22×C4C22×C411yescharacteristic
66C22×C4C2×D411yescharacteristic
67C22×C4C2×D411yescharacteristic
68C2×C8—24no—
69C2×C8—42no—
70C2×C8—42no—
71C2×C8—42no—
72C2×C8—42no—
73C2×C8—42no—
74C4⋊C4—41no—
75C4⋊C4—41no—
76C4⋊C4—42no—
77C4⋊C4—42no—
78C22⋊C4—42no—
79C22⋊C4—42no—
80C22⋊C4—42no—
81C22⋊C4—42no—
82C22⋊C4—42no—
83C22⋊C4—42no—
84C42—22no—
85C42—22no—
86C42—22no—
87C42—22no—
88C42—22no—
89C42—41no—
90C42—41no—
91C23—41no—
92C23—41no—
93C23—41no—
94C23—41no—
95C23—41no—
96C23—41no—
97C23—41no—
98C23C2×C22⋊C411yescharacteristic
99Q8—82no—
100D4—82no—
101C2×C4—21no—
102C2×C4—21no—
103C2×C4—21no—
104C2×C4—21no—
105C2×C4—21no—
106C2×C4—21no—
107C2×C4—21no—
108C2×C4—21no—
109C2×C4—21no—
110C2×C4—21no—
111C2×C4—21no—
112C2×C4—21no—
113C2×C4—21no—
114C2×C4—21no—
115C2×C4—21no—
116C2×C4—21no—
117C2×C4—41no—
118C2×C4—41no—
119C2×C4—41no—
120C2×C4—41no—
121C2×C4—41no—
122C2×C4—41no—
123C2×C4—41no—
124C2×C4—41no—
125C2×C4—41no—
126C2×C4#10012yes—
127C2×C4#10412yes—
128C2×C4#13112yes—
129C2×C4#13512yes—
130C2×C4#13512yes—
131C8—44no—
132C8—82no—
133C22—81no—
134C22—81no—
135C22—81no—
136C22—81no—
137C22—81no—
138C22—81no—
139C22—81no—
140C22#40811yescharacteristic
141C22#41011yescharacteristic
142C22C2×#10011yescharacteristic
143C22#41611yescharacteristic
144C22#41611yescharacteristic
145C22C2×#10411yescharacteristic
146C22#42111yescharacteristic
147C4—22no—
148C4—22no—
149C4—22no—
150C4—22no—
151C4—22no—
152C4—41no—
153C4—41no—
154C4—81no—
155C4—81no—
156C4—81no—
157C2—161no—
158C2#146911yescharacteristic
159C2#148211yescharacteristic
160C2#149411yescharacteristic
161C2#150011yescharacteristic
162C2#152111yescharacteristic
163C2#153011yescharacteristic
164C2#154311yescharacteristic
subgroupG/H
center, socleC23C2×C22⋊C4
commutatorC22×C4C22×C4
Frattini, cosocleC2×C42C23
prime radicalC22×D4C2×C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C43 ◁ G
LECSG ▷ C2×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #1839C2semidirect
3. #1744C2semidirect
4. #1272C2semidirect

Maximal subgroups

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

subgroupindex
#28612
#18392
#17442
#12722
#12702