Finite Group Databasebeta

← #12222

Subgroups — #12222

Order 256.

173 conjugacy classes of subgroups; only the 37 normal subgroups are shown.

normal subgroups only

This group has 173 conjugacy classes of subgroups — too many to draw. Only its 37 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#2799C211yesmaximal, characteristic
2#2028C211yesmaximal, characteristic
3#1617C211yesmaximal, characteristic
4#1605C211yesmaximal, characteristic
5#1354C211yesmaximal, characteristic
6#1353C211yesmaximal, characteristic
7#1319C211yesmaximal, characteristic
8#544C2211yescharacteristic
9#484—21no—
10#451—21no—
11#419—21no—
12#400—21no—
13#400C2211yescharacteristic
14#398—21no—
15#395C2211yescharacteristic
16#391—21no—
17#391—21no—
18#391C2211yescharacteristic
19#387—21no—
20#387—21no—
21#343C2211yescharacteristic
22#339—21no—
23#328—21no—
24#328—21no—
25#324C2211yescharacteristic
26#324C2211yescharacteristic
27C22×Q8D411yescharacteristic
28C22×Q8D411yescharacteristic
29C2×Q16—41no—
30C2×M16—21no—
31C4⋊Q8—41no—
32#125—41no—
33C22⋊Q8—21no—
34C22⋊Q8—21no—
35#117—21no—
36C2×C4⋊C4—21no—
37C2×C4⋊C4—21no—
38C2×C4⋊C4—21no—
39C2×C4⋊C4—21no—
40C2×C4⋊C4D411yescharacteristic
41C2×C4⋊C4C2311yescharacteristic
42C2×C42—21no—
43C2×C42D411yescharacteristic
44#107—41no—
45#103—41no—
46#103—41no—
47#103—41no—
48#103—41no—
49C22⋊C8—21no—
50C22⋊C8—21no—
51#95—21no—
52#95—21no—
53#95—21no—
54#95—21no—
55#95—21no—
56#95—21no—
57#95—21no—
58#95—21no—
59#95—21no—
60#95D411yescharacteristic
61#95D411yescharacteristic
62C2×Q8—21no—
63C2×Q8—21no—
64C2×Q8—21no—
65C2×Q8—42no—
66C2×Q8—41no—
67C2×Q8—41no—
68C2×Q8—41no—
69C2×Q8Q1612yescomplemented
70C22×C4—21no—
71C22×C4—21no—
72C22×C4—21no—
73C22×C4—21no—
74C22×C4—21no—
75C22×C4—21no—
76C22×C4C2×D411yescharacteristic
77C22×C4C2×D411yescharacteristic
78C22×C4C2×D411yescharacteristic
79Q16—82no—
80M16—41no—
81M16—41no—
82C2×C8—41no—
83C2×C8—41no—
84C2×C8—41no—
85C4⋊C4—21no—
86C4⋊C4—21no—
87C4⋊C4—21no—
88C4⋊C4—21no—
89C4⋊C4—41no—
90C4⋊C4—41no—
91C4⋊C4—42no—
92C4⋊C4—42no—
93C4⋊C4—41no—
94C4⋊C4—41no—
95C4⋊C4—81no—
96C4⋊C4—81no—
97C22⋊C4—41no—
98C42—41no—
99C42—41no—
100C42—81no—
101C23#12011yescharacteristic
102Q8—42no—
103Q8—42no—
104Q8—81no—
105Q8—81no—
106Q8—81no—
107Q8—81no—
108C2×C4—21no—
109C2×C4—21no—
110C2×C4—22no—
111C2×C4—22no—
112C2×C4—21no—
113C2×C4—21no—
114C2×C4—41no—
115C2×C4—41no—
116C2×C4—41no—
117C2×C4—41no—
118C2×C4—41no—
119C2×C4—41no—
120C2×C4—41no—
121C2×C4—41no—
122C2×C4—41no—
123C2×C4—41no—
124C2×C4—41no—
125C2×C4—41no—
126C2×C4—42no—
127C2×C4—41no—
128C2×C4—41no—
129C2×C4—41no—
130C2×C4—41no—
131C2×C4—41no—
132C2×C4—41no—
133C2×C4C2×Q1611yescharacteristic
134C2×C4#13711yescharacteristic
135C8—81no—
136C8—81no—
137C8—81no—
138C22—21no—
139C22—21no—
140C22#45111yescharacteristic
141C22#45511yescharacteristic
142C22B(4,𝔽2)11yescharacteristic
143C4—21no—
144C4—21no—
145C4—42no—
146C4—41no—
147C4—41no—
148C4—42no—
149C4—81no—
150C4—81no—
151C4—81no—
152C4—81no—
153C4—81no—
154C4—81no—
155C2—21no—
156C2—21no—
157C2#160211yescharacteristic
158C2#219711yescharacteristic
159C2#220311yescharacteristic
subgroupG/H
center, socleC22B(4,𝔽2)
commutator, Frattini, cosocleC2×C4⋊C4C23
prime radicalC23#120
LCSG ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ G
LECSG ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
69. C2×Q8Q16semidirect

Maximal subgroups

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

subgroupindex
#27992
#20282
#16172
#16052
#13542
#13532
#13192