Finite Group Databasebeta

← #9444

Subgroups — #9444

Order 256.

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

normal subgroups only

This group has 237 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#2921C211yesmaximal, characteristic
2#2499C211yesmaximal, characteristic
3#1820C211yesmaximal, characteristic
4#1511C212yesmaximal
5#1284C212yesmaximal
6C2×C22⋊Q8C2211yescharacteristic
7C2×#117C2211yescharacteristic
8#433—24no—
9#423—21no—
10#407C2212yes—
11C4×M16—21no—
12#397—21no—
13#397—21no—
14#391C411yescharacteristic, complemented
15#391C411yescharacteristic, complemented
16#387—21no—
17#387—21no—
18#384C411yescharacteristic, complemented
19#382C411yescharacteristic, complemented
20#381C2212yes—
21C4×C22⋊C4C2211yescharacteristic
22#342—21no—
23#342—21no—
24#328—24no—
25C22×Q8C2×C411yescharacteristic
26C23×C4C2311yescharacteristic
27C2×M16—22no—
28C22⋊Q8—22no—
29C22⋊Q8—22no—
30#117—22no—
31#117—22no—
32C2×C4⋊C4—21no—
33C2×C4⋊C4—21no—
34C2×C4⋊C4—21no—
35C2×C4⋊C4C2×C411yescharacteristic
36C2×C4⋊C4C2×C411yescharacteristic
37C2×C4⋊C4C2×C411yescharacteristic
38C2×C22⋊C4—21no—
39C2×C22⋊C4—21no—
40C2×C22⋊C4—22no—
41C2×C22⋊C4D411yescharacteristic
42C2×C22⋊C4D411yescharacteristic
43C2×C42C2×C411yescharacteristic
44C2×C42C2×C411yescharacteristic
45C2×C42D411yescharacteristic
46C2×C42D411yescharacteristic
47C4⋊C8—24no—
48C22⋊C8—22no—
49C22⋊C8D414yes—
50#97—22no—
51C4×C8—22no—
52#95—21no—
53#95—21no—
54#95—21no—
55#95—21no—
56#95—21no—
57#95—21no—
58#95—21no—
59#95—22no—
60#95—22no—
61C24C2×D411yescharacteristic
62C2×Q8—22no—
63C2×Q8—41no—
64C2×Q8—41no—
65C22×C4—21no—
66C22×C4—21no—
67C22×C4—21no—
68C22×C4—21no—
69C22×C4—41no—
70C22×C4—41no—
71C22×C4—42no—
72C22×C4C22⋊C411yescharacteristic
73C22×C4C22⋊C411yescharacteristic
74C22×C4C22⋊C411yescharacteristic
75C22×C4C22⋊C411yescharacteristic
76C22×C4C22×C411yescharacteristic
77C22×C4C2×D411yescharacteristic
78C22×C4C2×D412yes—
79C22×C4#4112yes—
80M16—42no—
81M16—42no—
82C2×C8—24no—
83C4⋊C4—22no—
84C4⋊C4—22no—
85C4⋊C4—22no—
86C4⋊C4—42no—
87C4⋊C4—42no—
88C4⋊C4—41no—
89C4⋊C4—41no—
90C22⋊C4—22no—
91C22⋊C4—22no—
92C22⋊C4—42no—
93C22⋊C4—42no—
94C22⋊C4—42no—
95C22⋊C4—42no—
96C42—21no—
97C42—21no—
98C42—21no—
99C42—21no—
100C42—41no—
101C42—41no—
102C23—21no—
103C23—21no—
104C23—41no—
105C23—41no—
106C23C2×C22⋊C411yescharacteristic
107C23#12112yes—
108Q8—82no—
109C2×C4—21no—
110C2×C4—21no—
111C2×C4—21no—
112C2×C4—21no—
113C2×C4—22no—
114C2×C4—21no—
115C2×C4—21no—
116C2×C4—21no—
117C2×C4—21no—
118C2×C4—42no—
119C2×C4—42no—
120C2×C4—41no—
121C2×C4—41no—
122C2×C4—41no—
123C2×C4—41no—
124C2×C4—41no—
125C2×C4—41no—
126C2×C4—41no—
127C2×C4—41no—
128C2×C4—41no—
129C2×C4—41no—
130C2×C4—41no—
131C2×C4—41no—
132C2×C4—41no—
133C2×C4—41no—
134C2×C4—41no—
135C2×C4—82no—
136C2×C4—82no—
137C2×C4—82no—
138C2×C4C4×D412yes—
139C2×C4#12011yescharacteristic
140C2×C4#12311yescharacteristic
141C8—44no—
142C22—21no—
143C22—21no—
144C22—22no—
145C22—22no—
146C22—22no—
147C22—81no—
148C22—81no—
149C22#38611yescharacteristic
150C22#41011yescharacteristic
151C22#41011yescharacteristic
152C4—21no—
153C4—22no—
154C4—21no—
155C4—42no—
156C4—41no—
157C4—41no—
158C4—41no—
159C4—41no—
160C4—81no—
161C4—81no—
162C4—81no—
163C4—162no—
164C2—21no—
165C2—21no—
166C2—42no—
167C2#188411yescharacteristic
168C2#189911yescharacteristic
169C2#190911yescharacteristic
subgroupG/H
center, socleC22#386
commutatorC22×C4C22×C4
Frattini, cosocleC23×C4C23
prime radicalC24C2×D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
14. #391C4semidirect
15. #391C4semidirect
18. #384C4semidirect
19. #382C4semidirect

Maximal subgroups

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

subgroupindex
#29212
#24992
#18202
#15112
#12842