Finite Group Databasebeta

← #10296

Subgroups — #10296

Order 256.

252 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 252 conjugacy classes of subgroups — too many to draw. Only its 61 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#3230C211yesmaximal, characteristic
2#3229C211yesmaximal, characteristic
3#2862C211yesmaximal, characteristic
4#1994C211yesmaximal, characteristic
5#1549C211yesmaximal, characteristic
6#1546C211yesmaximal, characteristic
7#1276C211yesmaximal, characteristic
8#533C2211yescharacteristic
9C2×C4⋊Q8C2211yescharacteristic
10C2×C4⋊Q8C2211yescharacteristic
11#500—21no—
12#498—21no—
13#487—21no—
14#487—21no—
15#484—22no—
16#483—22no—
17#477—22no—
18#475—22no—
19#431C2211yescharacteristic
20#431C2211yescharacteristic
21#423C2211yescharacteristic
22#391—22no—
23#391—22no—
24#382—22no—
25C43C2211yescharacteristic
26#339—22no—
27#332—22no—
28#332—22no—
29C22×Q8—21no—
30C22×Q8—21no—
31C2×M16—21no—
32C4⋊Q8—21no—
33C4⋊Q8—21no—
34C4⋊Q8—21no—
35C4⋊Q8—21no—
36C4⋊Q8—21no—
37C4⋊Q8—21no—
38C4⋊Q8—21no—
39C4⋊Q8—21no—
40C4⋊Q8D411yescharacteristic
41C4⋊Q8D411yescharacteristic
42#125—21no—
43C22⋊Q8—21no—
44C22⋊Q8—21no—
45C4×Q8—21no—
46C4×Q8—21no—
47#117—21no—
48C2×C4⋊C4—21no—
49C2×C4⋊C4—21no—
50C2×C4⋊C4D412yes—
51C2×C4⋊C4D412yes—
52C2×C42—21no—
53C2×C42—21no—
54C2×C42D412yes—
55C2×C42C2311yescharacteristic
56#107—24no—
57#106—24no—
58C4⋊C8—21no—
59C4⋊C8—21no—
60C4⋊C8D412yes—
61C4⋊C8D412yes—
62#103—24no—
63#103—24no—
64C22⋊C8—21no—
65C22⋊C8—21no—
66C4×C8—21no—
67C4×C8—21no—
68#95—28no—
69C2×Q8—21no—
70C2×Q8—21no—
71C2×Q8—44no—
72C2×Q8—41no—
73C2×Q8—41no—
74C2×Q8—41no—
75C2×Q8—41no—
76C2×Q8—42no—
77C2×Q8—42no—
78C22×C4—22no—
79C22×C4—22no—
80C22×C4—22no—
81C22×C4C2×D411yescharacteristic
82C22×C4C2×D411yescharacteristic
83C22×C4C2×D411yescharacteristic
84M16—42no—
85C2×C8—22no—
86C2×C8—22no—
87C2×C8—41no—
88C4⋊C4—22no—
89C4⋊C4—22no—
90C4⋊C4—21no—
91C4⋊C4—21no—
92C4⋊C4—22no—
93C4⋊C4—22no—
94C4⋊C4—21no—
95C4⋊C4—21no—
96C4⋊C4—41no—
97C4⋊C4—41no—
98C4⋊C4—41no—
99C4⋊C4—41no—
100C4⋊C4—41no—
101C4⋊C4—41no—
102C22⋊C4—41no—
103C42—24no—
104C42—22no—
105C42—22no—
106C42—42no—
107C42—41no—
108C42C2×D411yescharacteristic
109C42C2×D411yescharacteristic
110C42C2×D411yescharacteristic
111C42#4111yescharacteristic
112C23#12011yescharacteristic
113Q8—42no—
114Q8—42no—
115Q8—82no—
116Q8—82no—
117Q8—82no—
118Q8—81no—
119Q8—81no—
120C2×C4—22no—
121C2×C4—22no—
122C2×C4—24no—
123C2×C4—42no—
124C2×C4—42no—
125C2×C4—44no—
126C2×C4—41no—
127C2×C4—41no—
128C2×C4—42no—
129C2×C4—42no—
130C2×C4—44no—
131C2×C4#12011yescharacteristic
132C2×C4#12011yescharacteristic
133C2×C4#12111yescharacteristic
134C2×C4#12111yescharacteristic
135C2×C4#12111yescharacteristic
136C2×C4#12711yescharacteristic
137C2×C4#13512yes—
138C2×C4#13712yes—
139C8—42no—
140C8—42no—
141C8—81no—
142C22—21no—
143C22—21no—
144C22#39211yescharacteristic
145C22#45212yes—
146C4—21no—
147C4—21no—
148C4—44no—
149C4—82no—
150C4—81no—
151C4—81no—
152C4—82no—
153C4#45412yes—
154C4#45612yes—
155C4#46412yes—
156C4#49511yescharacteristic
157C4#49711yescharacteristic
158C2—22no—
159C2#200311yescharacteristic
160C2#200511yescharacteristic
161C2#201511yescharacteristic
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
#32302
#32292
#28622
#19942
#15492
#15462
#12762