Finite Group Databasebeta

← #12147

Subgroups — #12147

Order 256.

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

normal subgroups only

This group has 177 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#2769C211yesmaximal, characteristic, complemented
2#2040C211yesmaximal, characteristic
3#1616C211yesmaximal, characteristic
4#1610C211yesmaximal, characteristic
5#1353C211yesmaximal, characteristic, complemented
6#1346C211yesmaximal, characteristic, complemented
7#1324C211yesmaximal, characteristic, complemented
8#537C2211yescharacteristic
9#481—21no—
10#480—21no—
11C2×#104—21no—
12#417—21no—
13#400—21no—
14#400C2211yescharacteristic
15#398—21no—
16#398C2211yescharacteristic
17#389—21no—
18#387—21no—
19#387—21no—
20#383—21no—
21#382C2211yescharacteristic
22#343C2211yescharacteristic
23#327—21no—
24#327—21no—
25#324C2211yescharacteristic
26#324C2211yescharacteristic
27C2×#41—21no—
28C2×M16—21no—
29C22⋊Q8—41no—
30#121—21no—
31#121—21no—
32#121—41no—
33C2×C4⋊C4—21no—
34C2×C4⋊C4—21no—
35C2×C4⋊C4—21no—
36C2×C4⋊C4—21no—
37C2×C4⋊C4D411yescharacteristic
38C2×C4⋊C4D411yescharacteristic
39C2×C4⋊C4D411yescharacteristic
40C2×C4⋊C4C2311yescharacteristic
41C2×C42—21no—
42C2×C42D411yescharacteristic
43#107—41no—
44#106—41no—
45#104—42no—
46#103—41no—
47#102—41no—
48#102—41no—
49#102—41no—
50C22⋊C8—21no—
51C22⋊C8—21no—
52#95—21no—
53#95—21no—
54#95—21no—
55#95—21no—
56#95—21no—
57#95—21no—
58#95—21no—
59#95—21no—
60#95—21no—
61#95D411yescharacteristic
62#95D411yescharacteristic
63#41—41no—
64#41—41no—
65#41—41no—
66#41—41no—
67C2×Q8—21no—
68C2×D4—21no—
69C2×D4—21no—
70C2×D4—21no—
71C2×D4—41no—
72C22×C4—21no—
73C22×C4—21no—
74C22×C4—21no—
75C22×C4—21no—
76C22×C4—21no—
77C22×C4—21no—
78C22×C4—41no—
79C22×C4C2×D411yescharacteristic
80C22×C4C2×D411yescharacteristic
81C22×C4C2×D411yescharacteristic
82M16—41no—
83M16—41no—
84C2×C8—41no—
85C2×C8—41no—
86C2×C8—41no—
87C4⋊C4—21no—
88C4⋊C4—21no—
89C4⋊C4—41no—
90C4⋊C4—41no—
91C4⋊C4—41no—
92C4⋊C4—41no—
93C4⋊C4—42no—
94C4⋊C4—42no—
95C4⋊C4—41no—
96C4⋊C4—41no—
97C4⋊C4—81no—
98C22⋊C4—41no—
99C22⋊C4—41no—
100C42—21no—
101C42—21no—
102C42—42no—
103C23—41no—
104C23#12011yescharacteristic
105Q8—41no—
106Q8—41no—
107D4—41no—
108D4—41no—
109D4—81no—
110D4—81no—
111D4—81no—
112D4—81no—
113C2×C4—21no—
114C2×C4—21no—
115C2×C4—22no—
116C2×C4—21no—
117C2×C4—21no—
118C2×C4—41no—
119C2×C4—41no—
120C2×C4—41no—
121C2×C4—41no—
122C2×C4—41no—
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—81no—
139C2×C4—81no—
140C2×C4#13511yescharacteristic
141C2×C4AGL(1,ℤ/8)11yescharacteristic
142C8—81no—
143C8—81no—
144C8—81no—
145C22—21no—
146C22—21no—
147C22—81no—
148C22—81no—
149C22—81no—
150C22#44911yescharacteristic
151C22#45411yescharacteristic
152C22B(4,𝔽2)11yescharacteristic
153C4—21no—
154C4—21no—
155C4—42no—
156C4—41no—
157C4—41no—
158C4—81no—
159C4—81no—
160C4—81no—
161C4—81no—
162C4—81no—
163C4—81no—
164C2—21no—
165C2—21no—
166C2—161no—
167C2#159711yescharacteristic
168C2#220111yescharacteristic
169C2#220411yescharacteristic
subgroupG/H
center, socleC22B(4,𝔽2)
commutator, Frattini, cosocleC2×C4⋊C4C23
prime radical#537C22
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
1. #2769C2semidirect
5. #1353C2semidirect
6. #1346C2semidirect
7. #1324C2semidirect

Maximal subgroups

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

subgroupindex
#27692
#20402
#16162
#16102
#13532
#13462
#13242