Finite Group Databasebeta

← #12169

Subgroups — #12169

Order 256.

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

normal subgroups only

This group has 171 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#3262C211yesmaximal, characteristic
2#2188C211yesmaximal, characteristic, complemented
3#2185C211yesmaximal, characteristic
4#2047C211yesmaximal, characteristic, complemented
5#1350C211yesmaximal, characteristic, complemented
6#1349C211yesmaximal, characteristic
7#1323C211yesmaximal, characteristic, complemented
8C22×Q16C2211yescharacteristic
9#542—21no—
10C2×Q32—21no—
11C2×SD32—21no—
12#477—21no—
13#474—21no—
14#468—21no—
15#467C2211yescharacteristic
16#466C2211yescharacteristic
17#464—21no—
18#463—21no—
19#424C2211yescharacteristic
20#419—21no—
21C2×#101—21no—
22#358—21no—
23#358—21no—
24C22⋊C16C2211yescharacteristic
25C22⋊C16C2211yescharacteristic
26#341C2211yescharacteristic
27C22×Q8—21no—
28C2×Q16—21no—
29C2×Q16—21no—
30C2×Q16—42no—
31C2×Q16D411yescharacteristic
32C2×SD16—21no—
33C2×D8D411yescharacteristic
34C2×M16—21no—
35C2×M16D411yescharacteristic
36C22×C8C2311yescharacteristic
37C4⋊Q8—21no—
38#126—41no—
39#124—21no—
40#124—41no—
41#123—41no—
42C22⋊Q8—21no—
43C22⋊Q8—41no—
44#121—21no—
45C4×Q8—21no—
46C4×D4—21no—
47#117D411yescharacteristic
48Q32—42no—
49SD32—42no—
50C2×C16—21no—
51C2×C16—21no—
52#107—21no—
53#107D411yescharacteristic
54#107D411yescharacteristic
55C4⋊C8—21no—
56C4⋊C8—21no—
57#103—21no—
58#103—21no—
59#103—41no—
60#103—41no—
61#102—21no—
62#102—21no—
63#101—42no—
64C2×Q8—22no—
65C2×Q8—21no—
66C2×Q8—41no—
67C2×Q8—41no—
68C2×Q8—41no—
69C2×Q8—41no—
70C2×Q8—41no—
71C2×Q8—41no—
72C2×D4—21no—
73C2×D4—41no—
74C22×C4—41no—
75C22×C4—41no—
76C22×C4C2×D411yescharacteristic
77Q16—22no—
78Q16—22no—
79Q16—41no—
80Q16—41no—
81Q16—41no—
82SD16—81no—
83D8—41no—
84M16—41no—
85M16—41no—
86M16—41no—
87C2×C8—21no—
88C2×C8—41no—
89C2×C8—41no—
90C2×C8C2×D411yescharacteristic
91C2×C8C2×D411yescharacteristic
92C4⋊C4—21no—
93C4⋊C4—21no—
94C4⋊C4—41no—
95C4⋊C4—41no—
96C4⋊C4—41no—
97C4⋊C4D811yescharacteristic
98C4⋊C4D811yescharacteristic
99C22⋊C4—21no—
100C22⋊C4—41no—
101C22⋊C4—41no—
102C22⋊C4—41no—
103C22⋊C4—41no—
104C22⋊C4—41no—
105C42—21no—
106C42—41no—
107C16—41no—
108C16—41no—
109C23—41no—
110C23AGL(1,ℤ/8)11yescharacteristic
111Q8—42no—
112Q8—41no—
113Q8—41no—
114Q8—42no—
115Q8—81no—
116Q8—81no—
117Q8—161no—
118D4—41no—
119D4—41no—
120D4—161no—
121C2×C4—21no—
122C2×C4—21no—
123C2×C4—21no—
124C2×C4—21no—
125C2×C4—42no—
126C2×C4—41no—
127C2×C4—41no—
128C2×C4—41no—
129C2×C4—81no—
130C2×C4—81no—
131C2×C4—161no—
132C2×C4#12011yescharacteristic
133C2×C4C2×D811yescharacteristic
134C8—21no—
135C8—21no—
136C8—41no—
137C8—81no—
138C22—21no—
139C22—21no—
140C22—21no—
141C22—81no—
142C22—81no—
143C22—81no—
144C22#44711yescharacteristic
145C4—21no—
146C4—41no—
147C4—41no—
148C4—82no—
149C4—81no—
150C4—81no—
151C4—81no—
152C4#45411yescharacteristic
153C4B(4,𝔽2)11yescharacteristic
154C2—41no—
155C2—161no—
156C2#159811yescharacteristic
157C2#219411yescharacteristic
158C2#219511yescharacteristic
subgroupG/H
center, socleC22#447
commutator, Frattini, cosocleC22×C8C23
prime radical#466C22
LCSG ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C4 ◁ #424 ◁ G
LECSG ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #2188C2semidirect
4. #2047C2semidirect
5. #1350C2semidirect
7. #1323C2semidirect

Maximal subgroups

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

subgroupindex
#32622
#21882
#21852
#20472
#13502
#13492
#13232