Finite Group Databasebeta

← #9982

Subgroups — #9982

Order 256.

361 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 361 conjugacy classes of subgroups — too many to draw. Only its 49 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#3007C212yesmaximal
2#2817C211yesmaximal, characteristic, complemented
3#1782C211yesmaximal, characteristic
4#1511C212yesmaximal, complemented
5#1306C211yesmaximal, characteristic, complemented
6C22×#41C2211yescharacteristic
7C2×#137—22no—
8C2×C22⋊Q8C2212yes—
9#452—24no—
10#451—22no—
11#450—22no—
12#417—22no—
13C2×#99—22no—
14#407C2212yes—
15#400—22no—
16#397—22no—
17#395—22no—
18#381C2211yescharacteristic
19#381C2211yescharacteristic
20#328—24no—
21C2×#41—22no—
22C2×#41—22no—
23C2×#41—44no—
24C22×Q8D412yes—
25C22×Q8D411yescharacteristic
26C22×D4—21no—
27C22×D4D411yescharacteristic
28C23×C4—21no—
29C23×C4C2311yescharacteristic
30#137—48no—
31C2×Q16—24no—
32C2×SD16—24no—
33C2×M16—22no—
34C22⋊Q8—22no—
35C22⋊Q8—22no—
36C22⋊Q8—22no—
37C22⋊Q8—22no—
38C2×C4⋊C4—22no—
39C2×C4⋊C4—22no—
40C2×C4⋊C4D412yes—
41C2×C22⋊C4—22no—
42C2×C22⋊C4—21no—
43C2×C22⋊C4D412yes—
44#103—24no—
45#102—24no—
46#99—44no—
47C22⋊C8—22no—
48C22⋊C8D414yes—
49#95—24no—
50#95—22no—
51#95—22no—
52#95—22no—
53C24—21no—
54C24C2×D411yescharacteristic
55#41—28no—
56#41—44no—
57#41—44no—
58#41—44no—
59C2×Q8—22no—
60C2×Q8—22no—
61C2×Q8—22no—
62C2×Q8—42no—
63C2×Q8—44no—
64C2×D4—24no—
65C2×D4—22no—
66C2×D4—42no—
67C2×D4—42no—
68C2×D4—41no—
69C2×D4—41no—
70C22×C4—21no—
71C22×C4—22no—
72C22×C4—22no—
73C22×C4—22no—
74C22×C4—42no—
75C22×C4—42no—
76C22×C4—42no—
77C22×C4—42no—
78C22×C4—41no—
79C22×C4C2×D412yes—
80C22×C4C2×D411yescharacteristic
81C22×C4C2×D412yes—
82C22×C4#4111yescharacteristic
83Q16—48no—
84SD16—48no—
85M16—44no—
86C2×C8—24no—
87C4⋊C4—22no—
88C4⋊C4—22no—
89C4⋊C4—42no—
90C4⋊C4—42no—
91C4⋊C4—82no—
92C22⋊C4—42no—
93C22⋊C4—42no—
94C22⋊C4—42no—
95C22⋊C4—42no—
96C23—22no—
97C23—22no—
98C23—42no—
99C23—41no—
100C23—41no—
101C23—82no—
102C23#12011yescharacteristic
103C23#12111yescharacteristic
104C23#12711yescharacteristic
105Q8—24no—
106Q8—42no—
107Q8—44no—
108Q8—82no—
109D4—24no—
110D4—42no—
111D4—44no—
112D4—44no—
113C2×C4—24no—
114C2×C4—22no—
115C2×C4—41no—
116C2×C4—41no—
117C2×C4—42no—
118C2×C4—48no—
119C2×C4—41no—
120C2×C4—41no—
121C2×C4—44no—
122C2×C4—44no—
123C2×C4—42no—
124C2×C4—42no—
125C2×C4—42no—
126C2×C4—44no—
127C2×C4—81no—
128C2×C4—82no—
129C2×C4#12012yes—
130C2×C4#12112yes—
131C8—44no—
132C22—22no—
133C22—22no—
134C22—22no—
135C22—21no—
136C22—21no—
137C22—42no—
138C22—82no—
139C22—82no—
140C22—82no—
141C22—82no—
142C22#39211yescharacteristic
143C22B(4,𝔽2)12yes—
144C4—22no—
145C4—22no—
146C4—42no—
147C4—82no—
148C4—82no—
149C4—82no—
150C4—161no—
151C2—22no—
152C2—41no—
153C2—41no—
154C2—82no—
155C2#200912yes—
156C2#202111yescharacteristic
subgroupG/H
center, socleC22#392
commutator, Frattini, cosocleC23×C4C23
prime radicalC22×#41C22
LCSG ▷ C23×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
2. #2817C2semidirect
4. #1511C2semidirect
5. #1306C2semidirect

Maximal subgroups

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

subgroupindex
#30072
#28172
#17822
#15112
#13062