Finite Group Databasebeta

← #12272

Subgroups — #12272

Order 256.

186 conjugacy classes of subgroups; only the 35 normal subgroups are shown.

normal subgroups only

This group has 186 conjugacy classes of subgroups — too many to draw. Only its 35 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#2593C211yesmaximal, characteristic
2#2027C211yesmaximal, characteristic
3#1617C211yesmaximal, characteristic, complemented
4#1611C211yesmaximal, characteristic, complemented
5#1353C211yesmaximal, characteristic, complemented
6#1346C211yesmaximal, characteristic, complemented
7#1318C211yesmaximal, characteristic
8#544C2211yescharacteristic
9#484—21no—
10#483—21no—
11#418—21no—
12#400C2211yescharacteristic
13#398C2211yescharacteristic
14#394—21no—
15#390C2211yescharacteristic
16#388—21no—
17#388—21no—
18#388—21no—
19#388—21no—
20C4×C4⋊C4—21no—
21#343C2211yescharacteristic
22#339—21no—
23#328—21no—
24#328—21no—
25#324C2211yescharacteristic
26#324C2211yescharacteristic
27C22×D4D411yescharacteristic
28C2×M16—21no—
29C4⋊Q8—41no—
30#125—41no—
31C22⋊Q8—21no—
32C22⋊Q8—21no—
33#117—21no—
34C2×C4⋊C4—21no—
35C2×C4⋊C4—21no—
36C2×C4⋊C4D411yescharacteristic
37C2×C4⋊C4D411yescharacteristic
38C2×C4⋊C4C2311yescharacteristic
39C2×C22⋊C4—21no—
40C2×C22⋊C4—21no—
41C2×C22⋊C4—21no—
42C2×C22⋊C4—21no—
43C2×C22⋊C4—21no—
44C2×C22⋊C4—21no—
45C2×C42—21no—
46C2×C42—21no—
47C2×C42D411yescharacteristic
48#107—41no—
49#106—41no—
50#103—41no—
51#103—41no—
52#102—41no—
53#102—41no—
54C22⋊C8—21no—
55C22⋊C8—21no—
56#95—21no—
57#95—21no—
58#95—21no—
59#95—21no—
60#95D411yescharacteristic
61#95D411yescharacteristic
62C24—21no—
63C2×Q8—21no—
64C2×Q8—21no—
65C2×D4—21no—
66C2×D4—21no—
67C2×D4—41no—
68C2×D4—41no—
69C22×C4—21no—
70C22×C4—21no—
71C22×C4—21no—
72C22×C4—21no—
73C22×C4—21no—
74C22×C4C2×D411yescharacteristic
75C22×C4C2×D411yescharacteristic
76C22×C4C2×D411yescharacteristic
77M16—41no—
78M16—41no—
79C2×C8—41no—
80C2×C8—41no—
81C2×C8—41no—
82C4⋊C4—21no—
83C4⋊C4—21no—
84C4⋊C4—21no—
85C4⋊C4—21no—
86C4⋊C4—41no—
87C4⋊C4—41no—
88C4⋊C4—41no—
89C4⋊C4—42no—
90C4⋊C4—41no—
91C4⋊C4—41no—
92C4⋊C4—41no—
93C4⋊C4—41no—
94C22⋊C4—42no—
95C22⋊C4—42no—
96C22⋊C4—42no—
97C22⋊C4—42no—
98C22⋊C4—41no—
99C22⋊C4—81no—
100C22⋊C4—81no—
101C42—41no—
102C42—42no—
103C42—41no—
104C42—81no—
105C23—41no—
106C23—41no—
107C23—41no—
108C23—41no—
109C23—41no—
110C23—41no—
111C23—41no—
112C23#12011yescharacteristic
113Q8—81no—
114Q8—81no—
115D4—42no—
116D4—81no—
117C2×C4—21no—
118C2×C4—21no—
119C2×C4—22no—
120C2×C4—22no—
121C2×C4—21no—
122C2×C4—21no—
123C2×C4—41no—
124C2×C4—41no—
125C2×C4—41no—
126C2×C4—41no—
127C2×C4—41no—
128C2×C4—41no—
129C2×C4—41no—
130C2×C4—41no—
131C2×C4—41no—
132C2×C4—41no—
133C2×C4—41no—
134C2×C4—41no—
135C2×C4—41no—
136C2×C4—41no—
137C2×C4—41no—
138C2×C4—41no—
139C2×C4—41no—
140C2×C4#13511yescharacteristic
141C2×C4#13711yescharacteristic
142C8—81no—
143C8—81no—
144C8—81no—
145C22—21no—
146C22—21no—
147C22—81no—
148C22—81no—
149C22—81no—
150C22—81no—
151C22—81no—
152C22—81no—
153C22—81no—
154C22#45211yescharacteristic
155C22#45511yescharacteristic
156C22B(4,𝔽2)11yescharacteristic
157C4—21no—
158C4—21no—
159C4—42no—
160C4—42no—
161C4—41no—
162C4—41no—
163C4—81no—
164C4—81no—
165C4—81no—
166C4—81no—
167C4—81no—
168C2—21no—
169C2—21no—
170C2—161no—
171C2#160611yescharacteristic
172C2#219811yescharacteristic
173C2#220411yescharacteristic
subgroupG/H
center, socleC22B(4,𝔽2)
commutator, Frattini, cosocleC2×C4⋊C4C23
prime radicalC22×D4D4
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
3. #1617C2semidirect
4. #1611C2semidirect
5. #1353C2semidirect
6. #1346C2semidirect

Maximal subgroups

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

subgroupindex
#25932
#20272
#16172
#16112
#13532
#13462
#13182