Finite Group Databasebeta

← #12115

Subgroups — #12115

Order 256.

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

normal subgroups only

This group has 211 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#3195C211yesmaximal, characteristic, complemented
2#2191C212yesmaximal, complemented
3#2010C211yesmaximal, characteristic
4#1340C212yesmaximal, complemented
5#1320C211yesmaximal, characteristic, complemented
6#575C2211yescharacteristic
7#525—21no—
8C2×C22⋊Q8—21no—
9#510—22no—
10#483—22no—
11#470—21no—
12#470C2212yes—
13#451—22no—
14#421—21no—
15#407C2211yescharacteristic
16#360—22no—
17#349C2212yes—
18H(3,ℤ/4)C2211yescharacteristic
19C2×#41—21no—
20C22×Q8—21no—
21C23×C4D411yescharacteristic
22#137—41no—
23AGL(1,ℤ/8)—41no—
24#135—22no—
25C2×Q16—21no—
26C2×Q16D412yescomplemented
27C2×M16D411yescharacteristic
28C2×M16C2311yescharacteristic
29#123—41no—
30C22⋊Q8—22no—
31C22⋊Q8—22no—
32C22⋊Q8—41no—
33C22⋊Q8—41no—
34C22⋊Q8—41no—
35#121—41no—
36#120—41no—
37C4×D4—42no—
38#117—21no—
39C2×C4⋊C4—21no—
40C2×C4⋊C4—41no—
41C2×C22⋊C4—41no—
42Q32—42no—
43SD32—42no—
44M32—22no—
45#106—21no—
46#106D412yescomplemented
47#104—42no—
48#103—22no—
49#103—22no—
50C22⋊C8—22no—
51C24D811yescharacteristic
52#41—41no—
53#41—41no—
54#41—41no—
55#41—41no—
56C2×Q8—21no—
57C2×Q8—22no—
58C2×Q8—41no—
59C2×Q8—81no—
60C2×Q8—81no—
61C2×D4—21no—
62C2×D4—42no—
63C2×D4—41no—
64C22×C4—21no—
65C22×C4—21no—
66C22×C4—41no—
67C22×C4—41no—
68C22×C4—41no—
69C22×C4—41no—
70C22×C4—41no—
71C22×C4D811yescharacteristic
72C22×C4C2×D411yescharacteristic
73Q16—22no—
74Q16—42no—
75Q16—81no—
76SD16—42no—
77D8—22no—
78M16—41no—
79M16—41no—
80C2×C8—21no—
81C2×C8C2×D412yes—
82C4⋊C4—22no—
83C4⋊C4—41no—
84C4⋊C4—42no—
85C4⋊C4—41no—
86C4⋊C4—42no—
87C4⋊C4—42no—
88C22⋊C4—42no—
89C22⋊C4—42no—
90C22⋊C4—42no—
91C22⋊C4—42no—
92C22⋊C4—41no—
93C42—22no—
94C16—42no—
95C23—21no—
96C23—41no—
97C23—42no—
98C23—41no—
99C23C2×D811yescharacteristic
100Q8—41no—
101Q8—42no—
102Q8—41no—
103Q8—42no—
104Q8—82no—
105D4—41no—
106D4—41no—
107D4—81no—
108D4—84no—
109D4—81no—
110C2×C4—21no—
111C2×C4—21no—
112C2×C4—21no—
113C2×C4—41no—
114C2×C4—41no—
115C2×C4—41no—
116C2×C4—42no—
117C2×C4—41no—
118C2×C4—42no—
119C2×C4—42no—
120C2×C4—81no—
121C2×C4—81no—
122C2×C4—82no—
123C2×C4—82no—
124C2×C4—81no—
125C2×C4#12011yescharacteristic
126C2×C4AGL(1,ℤ/8)11yescharacteristic
127C8—22no—
128C8—41no—
129C22—22no—
130C22—21no—
131C22—41no—
132C22—41no—
133C22—42no—
134C22—41no—
135C22—81no—
136C22—82no—
137C22—81no—
138C22#44711yescharacteristic
139C4—21no—
140C4—41no—
141C4—82no—
142C4—82no—
143C4—81no—
144C4—82no—
145C4#45311yescharacteristic
146C4B(4,𝔽2)11yescharacteristic
147C2—21no—
148C2—41no—
149C2—42no—
150C2—161no—
151C2#159511yescharacteristic
subgroupG/H
center, socleC2#1595
commutator, Frattini, cosocleC2×M16C23
prime radical#2010C2
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
1. #3195C2semidirect
2. #2191C2semidirect
4. #1340C2semidirect
5. #1320C2semidirect
26. C2×Q16D4semidirect
46. #106D4semidirect

Maximal subgroups

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

subgroupindex
#31952
#21912
#20102
#13402
#13202