Finite Group Databasebeta

← #9280

Subgroups — #9280

Order 256.

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

normal subgroups only

This group has 191 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#3231C211yesmaximal, characteristic
2#2985C211yesmaximal, characteristic
3#1506C211yesmaximal, characteristic
4#1506C211yesmaximal, characteristic
5#1497C211yesmaximal, characteristic
6#1287C211yesmaximal, characteristic
7#1274C211yesmaximal, characteristic
8#533C2211yescharacteristic
9#533C2211yescharacteristic
10#500C411yescharacteristic, complemented
11#498C411yescharacteristic, complemented
12#487C411yescharacteristic, complemented
13#487C411yescharacteristic, complemented
14#485—22no—
15#479—22no—
16#446—22no—
17#446—21no—
18C8×Q8—21no—
19#433—22no—
20#432C2211yescharacteristic
21#432C2211yescharacteristic
22#431C2211yescharacteristic
23#431C2211yescharacteristic
24#376C2211yescharacteristic
25#328—22no—
26#328—22no—
27Q8⋊C8—21no—
28Q8⋊C8—21no—
29C4⋊Q8C2×C411yescharacteristic
30C4⋊Q8C2×C411yescharacteristic
31C4⋊Q8C2×C411yescharacteristic
32C4⋊Q8C2×C411yescharacteristic
33#125—21no—
34#125—21no—
35C22⋊Q8—21no—
36C22⋊Q8—21no—
37C22⋊Q8—21no—
38C22⋊Q8D412yes—
39C4×Q8—21no—
40C4×Q8—21no—
41C4×Q8—21no—
42C4×Q8D412yes—
43#117—21no—
44#117—21no—
45C2×C42—21no—
46C2×C42C2311yescharacteristic
47#107—22no—
48#106—22no—
49C4⋊C8—22no—
50C4⋊C8—22no—
51C4⋊C8—21no—
52C4⋊C8—21no—
53C4⋊C8—21no—
54C4⋊C8—21no—
55#103—22no—
56#103—22no—
57C22⋊C8—21no—
58C22⋊C8—21no—
59C22⋊C8—21no—
60C22⋊C8D412yes—
61#97—22no—
62#97—21no—
63#97—21no—
64C4×C8—22no—
65C4×C8C2×C411yescharacteristic
66C4×C8C2×C411yescharacteristic
67C4×C8D411yescharacteristic
68C4×C8D411yescharacteristic
69#95—21no—
70#95—21no—
71C2×Q8—21no—
72C2×Q8—21no—
73C2×Q8—21no—
74C2×Q8C22⋊C412yes—
75C22×C4—41no—
76C22×C4C2×D411yescharacteristic
77C22×C4C2×D411yescharacteristic
78C22×C4#4111yescharacteristic
79C2×C8—21no—
80C2×C8—21no—
81C2×C8—21no—
82C2×C8—22no—
83C2×C8—21no—
84C2×C8—41no—
85C4⋊C4—21no—
86C4⋊C4—21no—
87C4⋊C4—21no—
88C4⋊C4—22no—
89C4⋊C4—22no—
90C4⋊C4—21no—
91C4⋊C4—21no—
92C4⋊C4—21no—
93C4⋊C4—21no—
94C4⋊C4—41no—
95C4⋊C4—41no—
96C4⋊C4C22⋊C412yes—
97C22⋊C4—22no—
98C22⋊C4—41no—
99C42—22no—
100C42—22no—
101C42—41no—
102C42—41no—
103C42C22×C411yescharacteristic
104C42C2×D411yescharacteristic
105C42C2×D411yescharacteristic
106C42#4111yescharacteristic
107C23#12111yescharacteristic
108Q8—42no—
109Q8—41no—
110Q8—41no—
111Q8—41no—
112Q8—41no—
113Q8—81no—
114C2×C4—21no—
115C2×C4—21no—
116C2×C4—22no—
117C2×C4—22no—
118C2×C4—41no—
119C2×C4—41no—
120C2×C4—42no—
121C2×C4—41no—
122C2×C4—41no—
123C2×C4—81no—
124C2×C4—81no—
125C2×C4C2×C22⋊C411yescharacteristic
126C2×C4C4×D411yescharacteristic
127C2×C4C4×D411yescharacteristic
128C2×C4#12011yescharacteristic
129C2×C4#12111yescharacteristic
130C2×C4#12311yescharacteristic
131C8—41no—
132C8—42no—
133C8—41no—
134C8—41no—
135C8—41no—
136C8—81no—
137C22—21no—
138C22—21no—
139C22—21no—
140C22#38611yescharacteristic
141C4—21no—
142C4—21no—
143C4—21no—
144C4—21no—
145C4—21no—
146C4—42no—
147C4—42no—
148C4—82no—
149C4—81no—
150C4—81no—
151C4#44311yescharacteristic
152C4#44411yescharacteristic
153C2—41no—
154C2#187611yescharacteristic
155C2#190511yescharacteristic
156C2#190911yescharacteristic
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
10. #500C4semidirect
11. #498C4semidirect
12. #487C4semidirect
13. #487C4semidirect

Maximal subgroups

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

subgroupindex
#32312
#29852
#15062
#14972
#12872
#12742