Finite Group Databasebeta

← #12128

Subgroups — #12128

Order 256.

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

normal subgroups only

This group has 204 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#2799C211yesmaximal, characteristic, complemented
2#2007C211yesmaximal, characteristic
3#1616C211yesmaximal, characteristic
4#1604C211yesmaximal, characteristic
5#1354C211yesmaximal, characteristic, complemented
6#1353C211yesmaximal, characteristic, complemented
7#1319C211yesmaximal, characteristic, complemented
8C2×#137—21no—
9#537C2211yescharacteristic
10#481—21no—
11#448—21no—
12C2×#104—21no—
13#400—21no—
14#400C2211yescharacteristic
15#398—21no—
16#395C2211yescharacteristic
17#391—21no—
18#391—21no—
19#391C2211yescharacteristic
20#387—21no—
21#387—21no—
22#343C2211yescharacteristic
23#327—21no—
24#327—21no—
25#324C2211yescharacteristic
26#324C2211yescharacteristic
27C2×#41—21no—
28C22×Q8D411yescharacteristic
29C22×Q8D411yescharacteristic
30#137—42no—
31#137—42no—
32C2×Q16—41no—
33C2×SD16—41no—
34C2×SD16—41no—
35C2×M16—21no—
36C22⋊Q8—41no—
37#121—21no—
38#121—21no—
39#121—41no—
40C2×C4⋊C4—21no—
41C2×C4⋊C4—21no—
42C2×C4⋊C4—21no—
43C2×C4⋊C4—21no—
44C2×C4⋊C4D411yescharacteristic
45C2×C4⋊C4C2311yescharacteristic
46C2×C42—21no—
47C2×C42D411yescharacteristic
48#106—41no—
49#104—42no—
50#103—41no—
51#102—41no—
52C22⋊C8—21no—
53C22⋊C8—21no—
54#95—21no—
55#95—21no—
56#95—21no—
57#95—21no—
58#95—21no—
59#95—21no—
60#95—21no—
61#95—21no—
62#95—21no—
63#95D411yescharacteristic
64#95D411yescharacteristic
65#41—41no—
66#41—41no—
67#41—41no—
68#41—41no—
69C2×Q8—21no—
70C2×Q8—21no—
71C2×Q8—42no—
72C2×Q8—41no—
73C2×Q8—41no—
74C2×Q8—41no—
75C2×Q8SD1612yescomplemented
76C2×D4—21no—
77C2×D4—21no—
78C2×D4—21no—
79C2×D4—41no—
80C22×C4—21no—
81C22×C4—21no—
82C22×C4—21no—
83C22×C4—21no—
84C22×C4—21no—
85C22×C4—21no—
86C22×C4—41no—
87C22×C4C2×D411yescharacteristic
88C22×C4C2×D411yescharacteristic
89C22×C4C2×D411yescharacteristic
90Q16—44no—
91SD16—44no—
92SD16—82no—
93M16—22no—
94M16—41no—
95C2×C8—41no—
96C2×C8—41no—
97C2×C8—41no—
98C4⋊C4—21no—
99C4⋊C4—21no—
100C4⋊C4—41no—
101C4⋊C4—41no—
102C4⋊C4—42no—
103C4⋊C4—42no—
104C4⋊C4—81no—
105C4⋊C4—81no—
106C22⋊C4—41no—
107C22⋊C4—41no—
108C42—41no—
109C42—81no—
110C23—41no—
111C23#12011yescharacteristic
112Q8—42no—
113Q8—41no—
114Q8—41no—
115Q8—42no—
116Q8—81no—
117Q8—81no—
118D4—41no—
119D4—41no—
120D4—81no—
121D4—81no—
122D4—81no—
123D4—81no—
124C2×C4—21no—
125C2×C4—21no—
126C2×C4—22no—
127C2×C4—22no—
128C2×C4—21no—
129C2×C4—21no—
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—42no—
143C2×C4—41no—
144C2×C4—41no—
145C2×C4—41no—
146C2×C4—41no—
147C2×C4—41no—
148C2×C4—81no—
149C2×C4—81no—
150C2×C4C2×SD1611yescharacteristic
151C2×C4#13711yescharacteristic
152C8—42no—
153C8—81no—
154C8—81no—
155C22—21no—
156C22—21no—
157C22—81no—
158C22—81no—
159C22—81no—
160C22#44811yescharacteristic
161C22#45311yescharacteristic
162C22B(4,𝔽2)11yescharacteristic
163C4—21no—
164C4—21no—
165C4—42no—
166C4—41no—
167C4—41no—
168C4—42no—
169C4—81no—
170C4—81no—
171C4—81no—
172C4—81no—
173C4—81no—
174C2—21no—
175C2—21no—
176C2—161no—
177C2#159611yescharacteristic
178C2#219711yescharacteristic
179C2#220311yescharacteristic
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. #2799C2semidirect
5. #1354C2semidirect
6. #1353C2semidirect
7. #1319C2semidirect
75. C2×Q8SD16semidirect

Maximal subgroups

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

subgroupindex
#27992
#20072
#16162
#16042
#13542
#13532
#13192