Finite Group Databasebeta

← #12182

Subgroups — #12182

Order 256.

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

normal subgroups only

This group has 191 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#2612C211yesmaximal, characteristic, complemented
2#2040C211yesmaximal, characteristic
3#1616C211yesmaximal, characteristic
4#1604C211yesmaximal, characteristic
5#1354C211yesmaximal, characteristic, complemented
6#1353C211yesmaximal, characteristic, complemented
7#1324C211yesmaximal, characteristic, complemented
8#537C2211yescharacteristic
9#481—21no—
10#448—21no—
11C2×#104—21no—
12#417—21no—
13#400C2211yescharacteristic
14#398—21no—
15#395C2211yescharacteristic
16#391—21no—
17#389—21no—
18#387—21no—
19#387—21no—
20#382C2211yescharacteristic
21#376—21no—
22#343C2211yescharacteristic
23#327—21no—
24#327—21no—
25#324C2211yescharacteristic
26#324C2211yescharacteristic
27C2×#41—21no—
28C22×Q8D411yescharacteristic
29C2×SD16—41no—
30C2×M16—21no—
31C22⋊Q8—41no—
32#121—21no—
33#121—21no—
34#121—41no—
35C2×C4⋊C4—21no—
36C2×C4⋊C4—21no—
37C2×C4⋊C4—21no—
38C2×C4⋊C4—21no—
39C2×C4⋊C4D411yescharacteristic
40C2×C4⋊C4D411yescharacteristic
41C2×C4⋊C4C2311yescharacteristic
42C2×C42—21no—
43C2×C42—21no—
44C2×C42D411yescharacteristic
45#106—41no—
46#104—42no—
47#103—41no—
48#103—41no—
49#102—41no—
50#102—41no—
51C22⋊C8—21no—
52C22⋊C8—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×Q8—42no—
69C2×Q8—41no—
70C2×D4—21no—
71C2×D4—21no—
72C2×D4—21no—
73C2×D4—41no—
74C22×C4—21no—
75C22×C4—21no—
76C22×C4—21no—
77C22×C4—21no—
78C22×C4—21no—
79C22×C4—21no—
80C22×C4—41no—
81C22×C4C2×D411yescharacteristic
82C22×C4C2×D411yescharacteristic
83C22×C4C2×D411yescharacteristic
84SD16—82no—
85M16—41no—
86M16—41no—
87C2×C8—41no—
88C2×C8—41no—
89C2×C8—41no—
90C4⋊C4—21no—
91C4⋊C4—21no—
92C4⋊C4—41no—
93C4⋊C4—41no—
94C4⋊C4—41no—
95C4⋊C4—42no—
96C4⋊C4—42no—
97C4⋊C4—42no—
98C4⋊C4—81no—
99C22⋊C4—41no—
100C22⋊C4—41no—
101C42—21no—
102C42—42no—
103C42—42no—
104C42SD1612yescomplemented
105C23—41no—
106C23#12011yescharacteristic
107Q8—41no—
108Q8—41no—
109Q8—42no—
110Q8—81no—
111D4—41no—
112D4—41no—
113D4—81no—
114D4—81no—
115D4—81no—
116D4—81no—
117C2×C4—21no—
118C2×C4—21no—
119C2×C4—22no—
120C2×C4—22no—
121C2×C4—22no—
122C2×C4—21no—
123C2×C4—21no—
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—41no—
139C2×C4—41no—
140C2×C4—41no—
141C2×C4—41no—
142C2×C4—81no—
143C2×C4—81no—
144C2×C4C2×SD1611yescharacteristic
145C2×C4#13711yescharacteristic
146C8—81no—
147C8—81no—
148C8—81no—
149C22—21no—
150C22—21no—
151C22—81no—
152C22—81no—
153C22—81no—
154C22#44811yescharacteristic
155C22#45411yescharacteristic
156C22B(4,𝔽2)11yescharacteristic
157C4—21no—
158C4—21no—
159C4—42no—
160C4—42no—
161C4—41no—
162C4—41no—
163C4—42no—
164C4—81no—
165C4—81no—
166C4—81no—
167C4—81no—
168C2—21no—
169C2—21no—
170C2—161no—
171C2#159911yescharacteristic
172C2#219711yescharacteristic
173C2#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. #2612C2semidirect
5. #1354C2semidirect
6. #1353C2semidirect
7. #1324C2semidirect
104. C42SD16semidirect

Maximal subgroups

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

subgroupindex
#26122
#20402
#16162
#16042
#13542
#13532
#13242