Finite Group Databasebeta

← #12249

Subgroups — #12249

Order 256.

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

normal subgroups only

This group has 193 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#2823C211yesmaximal, characteristic
2#2060C211yesmaximal, characteristic, complemented
3#1611C211yesmaximal, characteristic, complemented
4#1599C211yesmaximal, characteristic
5#1346C211yesmaximal, characteristic, complemented
6#1345C211yesmaximal, characteristic
7#1325C211yesmaximal, characteristic, complemented
8#544C2211yescharacteristic
9C2×#126—21no—
10#483—21no—
11#450—21no—
12#428—21no—
13#398—21no—
14#398C2211yescharacteristic
15#396—21no—
16#396—21no—
17#396—21no—
18#394—21no—
19#394C2211yescharacteristic
20#384C2211yescharacteristic
21#343C2211yescharacteristic
22#339—21no—
23#328—21no—
24#328—21no—
25#324C2211yescharacteristic
26#324C2211yescharacteristic
27C22×D4D411yescharacteristic
28C2×SD16—41no—
29C2×M16—21no—
30C4⋊Q8—41no—
31#126—42no—
32#126—42no—
33#125—41no—
34C22⋊Q8—21no—
35C22⋊Q8—21no—
36#117—21no—
37C2×C4⋊C4—21no—
38C2×C4⋊C4—21no—
39C2×C4⋊C4—21no—
40C2×C4⋊C4—21no—
41C2×C4⋊C4—21no—
42C2×C4⋊C4—21no—
43C2×C4⋊C4D411yescharacteristic
44C2×C4⋊C4D411yescharacteristic
45C2×C4⋊C4C2311yescharacteristic
46C2×C22⋊C4—21no—
47C2×C22⋊C4—21no—
48C2×C22⋊C4—21no—
49C2×C22⋊C4—21no—
50C2×C22⋊C4—21no—
51C2×C22⋊C4—21no—
52C2×C42D411yescharacteristic
53#107—41no—
54#106—41no—
55#106—41no—
56#103—41no—
57#102—41no—
58C22⋊C8—21no—
59C22⋊C8—21no—
60#95—21no—
61#95—21no—
62#95D411yescharacteristic
63#95D411yescharacteristic
64C24—21no—
65C2×Q8—21no—
66C2×Q8—21no—
67C2×D4—41no—
68C2×D4—81no—
69C22×C4—21no—
70C22×C4—21no—
71C22×C4—21no—
72C22×C4—21no—
73C22×C4—21no—
74C22×C4C2×D411yescharacteristic
75C22×C4C2×D411yescharacteristic
76C22×C4C2×D411yescharacteristic
77SD16—82no—
78M16—22no—
79M16—41no—
80C2×C8—41no—
81C2×C8—41no—
82C2×C8—41no—
83C4⋊C4—21no—
84C4⋊C4—21no—
85C4⋊C4—21no—
86C4⋊C4—21no—
87C4⋊C4—41no—
88C4⋊C4—41no—
89C4⋊C4—41no—
90C4⋊C4—42no—
91C4⋊C4—42no—
92C4⋊C4—41no—
93C4⋊C4—41no—
94C4⋊C4—41no—
95C4⋊C4—41no—
96C4⋊C4—81no—
97C4⋊C4—81no—
98C4⋊C4—81no—
99C22⋊C4—42no—
100C22⋊C4—42no—
101C22⋊C4—41no—
102C22⋊C4—41no—
103C22⋊C4—41no—
104C22⋊C4—81no—
105C22⋊C4—81no—
106C22⋊C4—81no—
107C42—21no—
108C42—41no—
109C42SD1612yescomplemented
110C23—41no—
111C23—41no—
112C23—41no—
113C23—41no—
114C23—41no—
115C23—41no—
116C23—41no—
117C23#12011yescharacteristic
118Q8—81no—
119Q8—81no—
120D4—42no—
121D4—81no—
122C2×C4—21no—
123C2×C4—21no—
124C2×C4—22no—
125C2×C4—21no—
126C2×C4—21no—
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—41no—
139C2×C4—41no—
140C2×C4—41no—
141C2×C4—41no—
142C2×C4—41no—
143C2×C4—41no—
144C2×C4—41no—
145C2×C4C2×SD1611yescharacteristic
146C2×C4AGL(1,ℤ/8)11yescharacteristic
147C8—42no—
148C8—81no—
149C8—81no—
150C22—21no—
151C22—21no—
152C22—81no—
153C22—81no—
154C22—81no—
155C22—81no—
156C22—81no—
157C22—81no—
158C22—81no—
159C22#45011yescharacteristic
160C22#45611yescharacteristic
161C22B(4,𝔽2)11yescharacteristic
162C4—21no—
163C4—21no—
164C4—42no—
165C4—41no—
166C4—41no—
167C4—81no—
168C4—81no—
169C4—81no—
170C4—81no—
171C4—81no—
172C4—81no—
173C2—21no—
174C2—21no—
175C2—161no—
176C2#160411yescharacteristic
177C2#220011yescharacteristic
178C2#220311yescharacteristic
subgroupG/H
center, socleC22B(4,𝔽2)
commutator, Frattini, cosocleC2×C4⋊C4C23
prime radicalC22×D4D4
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
2. #2060C2semidirect
3. #1611C2semidirect
5. #1346C2semidirect
7. #1325C2semidirect
109. C42SD16semidirect

Maximal subgroups

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

subgroupindex
#28232
#20602
#16112
#15992
#13462
#13452
#13252