Finite Group Databasebeta

← #11849

Subgroups — #11849

Order 256.

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

normal subgroups only

This group has 188 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#2599C211yesmaximal, characteristic
2#1822C211yesmaximal, characteristic
3#1458C211yesmaximal, characteristic
4#1354C212yesmaximal, complemented
5#1346C212yesmaximal, complemented
6C2×C4×Q8C411yescharacteristic
7C2×#117C2211yescharacteristic
8#431—21no—
9#407—21no—
10#398C2211yescharacteristic
11#397C411yescharacteristic
12#395C2211yescharacteristic
13#393C411yescharacteristic
14#388—21no—
15#387—21no—
16#387—21no—
17#385—21no—
18#383C411yescharacteristic
19#343—21no—
20#343C2212yes—
21#324C2212yes—
22#323—21no—
23C22×Q8C2×C411yescharacteristic
24C23×C4D411yescharacteristic
25C2×M16—22no—
26C4×Q8—41no—
27C4×Q8—41no—
28#117—41no—
29#117—41no—
30C2×C4⋊C4—21no—
31C2×C4⋊C4—21no—
32C2×C4⋊C4—21no—
33C2×C4⋊C4C2×C411yescharacteristic
34C2×C4⋊C4D411yescharacteristic
35C2×C4⋊C4C2311yescharacteristic
36C2×C22⋊C4—21no—
37C2×C22⋊C4—21no—
38C2×C22⋊C4C2×C411yescharacteristic
39C2×C22⋊C4D411yescharacteristic
40C2×C42—21no—
41C2×C42C2×C411yescharacteristic
42C2×C42D411yescharacteristic
43C4⋊C8—41no—
44C22⋊C8—22no—
45C22⋊C8—41no—
46C22⋊C8—41no—
47C4×C8—41no—
48#95—21no—
49#95—21no—
50#95—21no—
51#95—21no—
52#95—21no—
53#95—21no—
54#95C2×C411yescharacteristic
55#95C2×C411yescharacteristic
56C24C22⋊C411yescharacteristic
57C2×Q8—42no—
58C2×Q8—41no—
59C22×C4—21no—
60C22×C4—21no—
61C22×C4—21no—
62C22×C4—22no—
63C22×C4—21no—
64C22×C4—21no—
65C22×C4—21no—
66C22×C4—41no—
67C22×C4C22⋊C411yescharacteristic
68C22×C4C22⋊C411yescharacteristic
69C22×C4C22⋊C411yescharacteristic
70C22×C4C22×C411yescharacteristic
71C22×C4C2×D411yescharacteristic
72C22×C4C2×D411yescharacteristic
73M16—42no—
74M16—42no—
75C2×C8—42no—
76C2×C8—42no—
77C4⋊C4—41no—
78C4⋊C4—41no—
79C4⋊C4—41no—
80C4⋊C4—42no—
81C4⋊C4—41no—
82C4⋊C4—41no—
83C4⋊C4—42no—
84C22⋊C4—41no—
85C22⋊C4—41no—
86C22⋊C4—42no—
87C22⋊C4—42no—
88C42—21no—
89C42—21no—
90C42—21no—
91C42—21no—
92C42—42no—
93C23—21no—
94C23—21no—
95C23—21no—
96C23—41no—
97C23—41no—
98C23C2×C22⋊C411yescharacteristic
99Q8—42no—
100Q8—42no—
101C2×C4—21no—
102C2×C4—21no—
103C2×C4—21no—
104C2×C4—21no—
105C2×C4—22no—
106C2×C4—21no—
107C2×C4—21no—
108C2×C4—21no—
109C2×C4—21no—
110C2×C4—42no—
111C2×C4—42no—
112C2×C4—41no—
113C2×C4—41no—
114C2×C4—41no—
115C2×C4—41no—
116C2×C4—41no—
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—41no—
127C2×C4#9911yescharacteristic
128C2×C4#9911yescharacteristic
129C2×C4#13512yes—
130C8—82no—
131C8—82no—
132C22—21no—
133C22—21no—
134C22—42no—
135C22—41no—
136C22—41no—
137C22—41no—
138C22—81no—
139C22C2×#9911yescharacteristic
140C22#41611yescharacteristic
141C22#42111yescharacteristic
142C4—22no—
143C4—21no—
144C4—21no—
145C4—41no—
146C4—42no—
147C4—41no—
148C4—41no—
149C4—41no—
150C4—81no—
151C4—81no—
152C4—81no—
153C2—21no—
154C2—21no—
155C2—81no—
156C2#151611yescharacteristic
157C2#212011yescharacteristic
158C2#213511yescharacteristic
subgroupG/H
center, socleC22C2×#99
commutatorC22×C4C22×C4
Frattini, cosocleC2×C4⋊C4C23
prime radicalC24C22⋊C4
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#117 ◁ G
LECSG ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #1354C2semidirect
5. #1346C2semidirect

Maximal subgroups

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

subgroupindex
#25992
#18222
#14582
#13542
#13462