Finite Group Databasebeta

← #10101

Subgroups — #10101

Order 256.

237 conjugacy classes of subgroups; only the 51 normal subgroups are shown.

normal subgroups only

This group has 237 conjugacy classes of subgroups — too many to draw. Only its 51 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#3236C211yesmaximal, characteristic
2#3165C211yesmaximal, characteristic
3#2660C211yesmaximal, characteristic
4#1556C211yesmaximal, characteristic
5#1528C211yesmaximal, characteristic
6#1506C211yesmaximal, characteristic
7#1300C211yesmaximal, characteristic
8#533C2211yescharacteristic
9C2×C4⋊Q8C2211yescharacteristic
10C2×#125C2211yescharacteristic
11#501—21no—
12#501—22no—
13#500—21no—
14#498—21no—
15#491—21no—
16#484—21no—
17#483—21no—
18#479—22no—
19#432C2211yescharacteristic
20#432C2211yescharacteristic
21#428—22no—
22C4×M16C2211yescharacteristic
23#398—22no—
24#395—22no—
25#384—21no—
26#382—21no—
27#376C2211yescharacteristic
28#332—22no—
29#330—22no—
30#328—22no—
31C22×Q8—21no—
32C2×M16—21no—
33C4⋊Q8—21no—
34C4⋊Q8—21no—
35C4⋊Q8—21no—
36C4⋊Q8—21no—
37C4⋊Q8D411yescharacteristic
38C4⋊Q8D411yescharacteristic
39#125—21no—
40#125—21no—
41#125—21no—
42#125—21no—
43#125—21no—
44C22⋊Q8—21no—
45C22⋊Q8—21no—
46C4×Q8—21no—
47C4×Q8—21no—
48#117—21no—
49C2×C4⋊C4—21no—
50C2×C4⋊C4—21no—
51C2×C4⋊C4—21no—
52C2×C4⋊C4—24no—
53C2×C4⋊C4D411yescharacteristic
54C2×C4⋊C4D411yescharacteristic
55C2×C4⋊C4D412yes—
56C2×C42—21no—
57C2×C42C2311yescharacteristic
58#107—24no—
59#107—22no—
60#106—24no—
61#106—22no—
62C4⋊C8—21no—
63C4⋊C8D412yes—
64#103—22no—
65#103—22no—
66C22⋊C8—21no—
67C22⋊C8—21no—
68#97—21no—
69#97—21no—
70#97D412yes—
71C4×C8—21no—
72#95—24no—
73#95—21no—
74#95D412yes—
75C2×Q8—21no—
76C2×Q8—21no—
77C2×Q8—41no—
78C2×Q8—41no—
79C2×Q8—82no—
80C22×C4—22no—
81C22×C4—21no—
82C22×C4—22no—
83C22×C4—21no—
84C22×C4C2×D411yescharacteristic
85C22×C4C2×D411yescharacteristic
86C22×C4C2×D411yescharacteristic
87M16—24no—
88C2×C8—22no—
89C2×C8—22no—
90C2×C8—41no—
91C4⋊C4—21no—
92C4⋊C4—21no—
93C4⋊C4—21no—
94C4⋊C4—22no—
95C4⋊C4—21no—
96C4⋊C4—21no—
97C4⋊C4—21no—
98C4⋊C4—21no—
99C4⋊C4—22no—
100C4⋊C4—21no—
101C4⋊C4—41no—
102C4⋊C4—41no—
103C4⋊C4—41no—
104C4⋊C4—41no—
105C4⋊C4—41no—
106C4⋊C4—41no—
107C4⋊C4—41no—
108C4⋊C4—41no—
109C4⋊C4—84no—
110C22⋊C4—41no—
111C42—24no—
112C42—41no—
113C42C2×D411yescharacteristic
114C42C2×D411yescharacteristic
115C42C2×D411yescharacteristic
116C42#4111yescharacteristic
117C23#12011yescharacteristic
118Q8—42no—
119Q8—81no—
120Q8—82no—
121Q8—82no—
122C2×C4—22no—
123C2×C4—22no—
124C2×C4—24no—
125C2×C4—41no—
126C2×C4—42no—
127C2×C4—42no—
128C2×C4—42no—
129C2×C4—44no—
130C2×C4—41no—
131C2×C4—41no—
132C2×C4—41no—
133C2×C4—41no—
134C2×C4—42no—
135C2×C4—42no—
136C2×C4#12011yescharacteristic
137C2×C4#12011yescharacteristic
138C2×C4#12111yescharacteristic
139C2×C4#12111yescharacteristic
140C2×C4#12111yescharacteristic
141C2×C4#12711yescharacteristic
142C8—24no—
143C8—42no—
144C8—81no—
145C22—21no—
146C22—21no—
147C22#39211yescharacteristic
148C22#45612yes—
149C4—21no—
150C4—21no—
151C4—21no—
152C4—21no—
153C4—21no—
154C4—44no—
155C4—81no—
156C4—81no—
157C4—81no—
158C4—82no—
159C4—81no—
160C4#45612yes—
161C2—22no—
162C2#200511yescharacteristic
163C2#200511yescharacteristic
164C2#200811yescharacteristic
subgroupG/H
center, socleC22#392
commutator, Frattini, cosocleC2×C42C23
prime radicalC23#120
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Maximal subgroups

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

subgroupindex
#32362
#31652
#26602
#15562
#15282
#15062
#13002