Finite Group Databasebeta

← #9304

Subgroups — #9304

Order 256.

218 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 218 conjugacy classes of subgroups — too many to draw. Only its 75 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#3078C211yesmaximal, characteristic
2#2982C211yesmaximal, characteristic
3#1805C211yesmaximal, characteristic
4#1518C211yesmaximal, characteristic
5#1481C211yesmaximal, characteristic
6#1301C211yesmaximal, characteristic
7#1286C211yesmaximal, characteristic
8#533C2211yescharacteristic
9C2×C4×Q8C2211yescharacteristic
10#479—22no—
11#477C412yescomplemented
12#475C412yescomplemented
13C8×Q8—22no—
14C8×Q8—22no—
15#431—22no—
16#431C2211yescharacteristic
17#431C2211yescharacteristic
18#428—21no—
19#423C2211yescharacteristic
20C2×C4⋊C8C2211yescharacteristic
21#419—21no—
22C2×#103—21no—
23#382C2211yescharacteristic
24#340—21no—
25#328—22no—
26Q8⋊C8—22no—
27C22×Q8D411yescharacteristic
28C2×M16—21no—
29C22×C8—21no—
30C4⋊Q8C2×C411yescharacteristic
31C4⋊Q8C2×C411yescharacteristic
32#125—21no—
33C22⋊Q8—21no—
34C22⋊Q8—21no—
35C4×Q8—21no—
36C4×Q8—21no—
37C4×Q8—21no—
38C4×Q8—21no—
39C4×Q8C2×C412yes—
40C4×Q8D412yes—
41#117—21no—
42C2×C4⋊C4—21no—
43C2×C4⋊C4—21no—
44C2×C4⋊C4D411yescharacteristic
45C2×C42—21no—
46C2×C42C2311yescharacteristic
47#107—22no—
48#106—22no—
49C4⋊C8—24no—
50C4⋊C8—21no—
51C4⋊C8—21no—
52C4⋊C8—21no—
53C4⋊C8—21no—
54C4⋊C8—21no—
55C4⋊C8C2×C412yes—
56#103—22no—
57#103—22no—
58#103—42no—
59C22⋊C8—21no—
60C22⋊C8D412yes—
61C4×C8—24no—
62C4×C8—21no—
63C4×C8D412yes—
64#95—21no—
65#95—21no—
66C2×Q8—21no—
67C2×Q8—21no—
68C2×Q8—21no—
69C2×Q8—21no—
70C2×Q8—21no—
71C2×Q8C22⋊C412yes—
72C2×Q8SD1612yes—
73C2×Q8Q1612yes—
74C22×C4—21no—
75C22×C4—21no—
76C22×C4—41no—
77C22×C4C2×D411yescharacteristic
78C22×C4C2×D411yescharacteristic
79C22×C4#4111yescharacteristic
80M16—41no—
81M16—41no—
82C2×C8—22no—
83C2×C8—22no—
84C2×C8—22no—
85C2×C8—41no—
86C2×C8—41no—
87C2×C8—41no—
88C4⋊C4—22no—
89C4⋊C4—21no—
90C4⋊C4—22no—
91C4⋊C4—21no—
92C4⋊C4—21no—
93C4⋊C4—21no—
94C4⋊C4—21no—
95C4⋊C4—41no—
96C4⋊C4—41no—
97C4⋊C4—41no—
98C4⋊C4—41no—
99C4⋊C4C22⋊C412yes—
100C22⋊C4—41no—
101C42—22no—
102C42—22no—
103C42—41no—
104C42C22×C411yescharacteristic
105C42C2×D411yescharacteristic
106C42C2×D411yescharacteristic
107C42#4111yescharacteristic
108C23#12111yescharacteristic
109Q8—22no—
110Q8—22no—
111Q8—22no—
112Q8—42no—
113Q8—81no—
114Q8#10314yes—
115C2×C4—22no—
116C2×C4—22no—
117C2×C4—22no—
118C2×C4—22no—
119C2×C4—41no—
120C2×C4—41no—
121C2×C4—41no—
122C2×C4—41no—
123C2×C4—81no—
124C2×C4—81no—
125C2×C4—81no—
126C2×C4—81no—
127C2×C4C2×C22⋊C411yescharacteristic
128C2×C4C4×D411yescharacteristic
129C2×C4C4×D411yescharacteristic
130C2×C4#12011yescharacteristic
131C2×C4#12111yescharacteristic
132C2×C4#12311yescharacteristic
133C2×C4C2×SD1611yescharacteristic
134C2×C4C2×Q1611yescharacteristic
135C2×C4#13511yescharacteristic
136C2×C4#13511yescharacteristic
137C8—42no—
138C8—42no—
139C8—42no—
140C8—81no—
141C22—21no—
142C22—21no—
143C22#38611yescharacteristic
144C22#46511yescharacteristic
145C22#46711yescharacteristic
146C4—21no—
147C4—21no—
148C4—24no—
149C4—21no—
150C4—21no—
151C4—42no—
152C4—81no—
153C4—81no—
154C4—161no—
155C4C2×#10311yescharacteristic
156C4#41611yescharacteristic
157C4#46511yescharacteristic
158C4#46711yescharacteristic
159C2—21no—
160C2—21no—
161C2#188111yescharacteristic
162C2#189211yescharacteristic
163C2#190511yescharacteristic
subgroupG/H
center, socleC22#386
commutatorC42C22×C4
Frattini, cosocleC2×C42C23
prime radicalC23#121
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
11. #477C4semidirect
12. #475C4semidirect

Maximal subgroups

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

subgroupindex
#30782
#29822
#18052
#15182
#14812
#13012
#12862