Finite Group Databasebeta

← #12293

Subgroups — #12293

Order 256.

158 conjugacy classes of subgroups; only the 39 normal subgroups are shown.

normal subgroups only

This group has 158 conjugacy classes of subgroups — too many to draw. Only its 39 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#3250C211yesmaximal, characteristic
2#2236C211yesmaximal, characteristic, complemented
3#2233C211yesmaximal, characteristic
4#2087C211yesmaximal, characteristic, complemented
5#1348C211yesmaximal, characteristic, complemented
6#1347C211yesmaximal, characteristic
7#1322C211yesmaximal, characteristic, complemented
8#529—21no—
9#479—21no—
10#478—21no—
11#467C2211yescharacteristic
12#466C2211yescharacteristic
13#465—21no—
14#464—21no—
15#463—21no—
16C2×#107C2211yescharacteristic
17C2×C4⋊C8C2211yescharacteristic
18#389—21no—
19#367—21no—
20#366—21no—
21#358—21no—
22#357—21no—
23C22⋊C16C2211yescharacteristic
24C22⋊C16C2211yescharacteristic
25#340—21no—
26#340C2211yescharacteristic
27C2×Q16D411yescharacteristic
28C2×SD16—21no—
29C2×D8D411yescharacteristic
30C22×C8D411yescharacteristic
31C22×C8C2311yescharacteristic
32#126—41no—
33#125—21no—
34#124—21no—
35#123—41no—
36C22⋊Q8—21no—
37#121—21no—
38C4×Q8—21no—
39C4×D4—21no—
40C4×D4—41no—
41#117—41no—
42C2×C4⋊C4—21no—
43C2×C4⋊C4—21no—
44C2×C4⋊C4—41no—
45C2×C42D411yescharacteristic
46C2×C16—21no—
47C2×C16—21no—
48#107—21no—
49#107—21no—
50#107D411yescharacteristic
51#107D411yescharacteristic
52#106—21no—
53C4⋊C8—21no—
54C4⋊C8—21no—
55#103—21no—
56#103—21no—
57#102—21no—
58#102—21no—
59#95—41no—
60C2×Q8—21no—
61C2×D4—21no—
62C2×D4—41no—
63C22×C4—41no—
64C22×C4—41no—
65C22×C4—41no—
66C22×C4D811yescharacteristic
67C22×C4D811yescharacteristic
68C22×C4C2×D411yescharacteristic
69Q16—41no—
70SD16—81no—
71D8—41no—
72C2×C8—21no—
73C2×C8—41no—
74C2×C8—41no—
75C2×C8C2×D411yescharacteristic
76C2×C8C2×D411yescharacteristic
77C4⋊C4—22no—
78C4⋊C4—21no—
79C4⋊C4—21no—
80C4⋊C4—41no—
81C4⋊C4—41no—
82C4⋊C4—41no—
83C4⋊C4—41no—
84C4⋊C4—41no—
85C4⋊C4—41no—
86C4⋊C4—41no—
87C4⋊C4—82no—
88C22⋊C4—41no—
89C22⋊C4—41no—
90C22⋊C4—41no—
91C22⋊C4—41no—
92C22⋊C4—41no—
93C42—21no—
94C42—21no—
95C42—41no—
96C16—41no—
97C16—41no—
98C23—41no—
99C23C2×D811yescharacteristic
100Q8—41no—
101Q8—41no—
102D4—41no—
103D4—41no—
104D4—81no—
105D4—81no—
106C2×C4—21no—
107C2×C4—21no—
108C2×C4—21no—
109C2×C4—21no—
110C2×C4—21no—
111C2×C4—21no—
112C2×C4—41no—
113C2×C4—42no—
114C2×C4—41no—
115C2×C4—41no—
116C2×C4—41no—
117C2×C4—82no—
118C2×C4—81no—
119C2×C4—81no—
120C2×C4—81no—
121C2×C4—81no—
122C2×C4—81no—
123C2×C4#12011yescharacteristic
124C2×C4AGL(1,ℤ/8)11yescharacteristic
125C8—21no—
126C8—21no—
127C8—41no—
128C22—21no—
129C22—21no—
130C22—81no—
131C22—81no—
132C22—81no—
133C22#44711yescharacteristic
134C22#50811yescharacteristic
135C22#51011yescharacteristic
136C4—21no—
137C4—41no—
138C4—41no—
139C4—82no—
140C4—81no—
141C4—81no—
142C4—81no—
143C4—161no—
144C4#45611yescharacteristic
145C4#45811yescharacteristic
146C2—21no—
147C2—21no—
148C2—161no—
149C2#161011yescharacteristic
150C2#218911yescharacteristic
151C2#219011yescharacteristic
subgroupG/H
center, socleC22#447
commutator, Frattini, cosocleC22×C8C23
prime radical#466C22
LCSG ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4⋊C8 ◁ G
LECSG ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #2236C2semidirect
4. #2087C2semidirect
5. #1348C2semidirect
7. #1322C2semidirect

Maximal subgroups

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

subgroupindex
#32502
#22362
#22332
#20872
#13482
#13472
#13222