Finite Group Databasebeta

← #12300

Subgroups — #12300

Order 256.

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

normal subgroups only

This group has 164 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#3197C211yesmaximal, characteristic
2#2239C211yesmaximal, characteristic, complemented
3#2238C211yesmaximal, characteristic, complemented
4#2091C211yesmaximal, characteristic, complemented
5#1342C211yesmaximal, characteristic, complemented
6#1341C211yesmaximal, characteristic, complemented
7#1320C211yesmaximal, characteristic, complemented
8#525—21no—
9#485—21no—
10#482—21no—
11#470C2211yescharacteristic
12#469C2211yescharacteristic
13#468—21no—
14#452—21no—
15#449—21no—
16#430—21no—
17#430C2211yescharacteristic
18#407—21no—
19#407C2211yescharacteristic
20#365—21no—
21#365—21no—
22#362—21no—
23#361—21no—
24#349C2211yescharacteristic
25#349C2211yescharacteristic
26#338C2211yescharacteristic
27C2×#41—21no—
28C23×C4D411yescharacteristic
29C2×Q16D411yescharacteristic
30C2×SD16—21no—
31C2×D8D411yescharacteristic
32C2×M16—21no—
33C2×M16—21no—
34C2×M16D411yescharacteristic
35C2×M16C2311yescharacteristic
36#123—41no—
37C22⋊Q8—21no—
38C22⋊Q8—21no—
39#121—21no—
40#121—21no—
41#120—41no—
42C4×D4—41no—
43C4×D4—41no—
44#117—21no—
45M32—21no—
46M32—21no—
47#108—21no—
48#108—21no—
49#108—42no—
50#107—21no—
51#106D411yescharacteristic
52#106D411yescharacteristic
53#103—21no—
54#103—21no—
55#102—21no—
56#102—21no—
57C22⋊C8—21no—
58C22⋊C8—21no—
59C22⋊C8—42no—
60C24SD1611yescharacteristic
61#41—81no—
62#41—81no—
63C2×Q8—21no—
64C2×D4—21no—
65C2×D4—41no—
66C2×D4—41no—
67C2×D4—41no—
68C22×C4—21no—
69C22×C4—21no—
70C22×C4—41no—
71C22×C4—41no—
72C22×C4—41no—
73C22×C4SD1611yescharacteristic
74C22×C4C2×D411yescharacteristic
75Q16—41no—
76SD16—81no—
77D8—41no—
78M16—41no—
79M16—41no—
80M16—41no—
81M16—41no—
82M16—41no—
83M16—41no—
84C2×C8—22no—
85C2×C8—21no—
86C2×C8—41no—
87C2×C8C2×D411yescharacteristic
88C2×C8C2×D411yescharacteristic
89C4⋊C4—21no—
90C4⋊C4—21no—
91C4⋊C4—41no—
92C4⋊C4—41no—
93C22⋊C4—41no—
94C22⋊C4—41no—
95C22⋊C4—41no—
96C22⋊C4—41no—
97C22⋊C4—41no—
98C42—41no—
99C16—41no—
100C16—41no—
101C23—21no—
102C23—41no—
103C23—41no—
104C23—81no—
105C23C2×SD1611yescharacteristic
106Q8—41no—
107Q8—41no—
108D4—41no—
109D4—41no—
110D4—82no—
111D4—82no—
112D4—82no—
113C2×C4—21no—
114C2×C4—21no—
115C2×C4—21no—
116C2×C4—41no—
117C2×C4—41no—
118C2×C4—41no—
119C2×C4—41no—
120C2×C4—41no—
121C2×C4—41no—
122C2×C4—81no—
123C2×C4—82no—
124C2×C4#12011yescharacteristic
125C2×C4AGL(1,ℤ/8)11yescharacteristic
126C8—21no—
127C8—21no—
128C8—42no—
129C8—41no—
130C8—81no—
131C22—21no—
132C22—41no—
133C22—41no—
134C22—41no—
135C22—42no—
136C22—41no—
137C22—81no—
138C22—81no—
139C22—81no—
140C22—81no—
141C22#45011yescharacteristic
142C4—21no—
143C4—41no—
144C4—81no—
145C4—81no—
146C4—81no—
147C4#45311yescharacteristic
148C4#45811yescharacteristic
149C2—21no—
150C2—41no—
151C2—81no—
152C2—161no—
153C2#161311yescharacteristic
subgroupG/H
center, socleC2#1613
commutator, Frattini, cosocleC2×M16C23
prime radical#3197C2
LCSG ▷ C2×M16 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×M16 ◁ G
LECSG ▷ C2×M16 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #2239C2semidirect
3. #2238C2semidirect
4. #2091C2semidirect
5. #1342C2semidirect
6. #1341C2semidirect
7. #1320C2semidirect

Maximal subgroups

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

subgroupindex
#31972
#22392
#22382
#20912
#13422
#13412
#13202