Finite Group Databasebeta

← #12271

Subgroups — #12271

Order 256.

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

normal subgroups only

This group has 198 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#2572C211yesmaximal, characteristic
2#2028C211yesmaximal, characteristic, complemented
3#1617C211yesmaximal, characteristic, complemented
4#1599C211yesmaximal, characteristic
5#1353C211yesmaximal, characteristic, complemented
6#1345C211yesmaximal, characteristic
7#1319C211yesmaximal, characteristic, complemented
8#544C2211yescharacteristic
9#484—21no—
10#450—21no—
11#419—21no—
12#400C2211yescharacteristic
13#394C2211yescharacteristic
14#393—21no—
15#391C2211yescharacteristic
16#388—21no—
17#386—21no—
18#383—21no—
19#381—21no—
20C4×C22⋊C4—21no—
21#343C2211yescharacteristic
22#339—21no—
23#328—21no—
24#328—21no—
25#324C2211yescharacteristic
26#324C2211yescharacteristic
27C22×Q8D411yescharacteristic
28C22×D4D411yescharacteristic
29C23×C4—21no—
30C2×SD16—41no—
31C2×M16—21no—
32C4⋊Q8—41no—
33#125—41no—
34C22⋊Q8—21no—
35C22⋊Q8—21no—
36#117—21no—
37C2×C4⋊C4D411yescharacteristic
38C2×C4⋊C4C2311yescharacteristic
39C2×C22⋊C4—21no—
40C2×C22⋊C4—21no—
41C2×C22⋊C4—21no—
42C2×C22⋊C4—21no—
43C2×C22⋊C4—21no—
44C2×C42—21no—
45C2×C42D411yescharacteristic
46#107—41no—
47#103—41no—
48#103—41no—
49#103—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#95D411yescharacteristic
61#95D411yescharacteristic
62C24—21no—
63C2×Q8—21no—
64C2×Q8—21no—
65C2×Q8—21no—
66C2×Q8—21no—
67C2×Q8—41no—
68C2×Q8—41no—
69C2×D4—41no—
70C2×D4—42no—
71C22×C4—21no—
72C22×C4—21no—
73C22×C4—21no—
74C22×C4—21no—
75C22×C4—21no—
76C22×C4—42no—
77C22×C4—42no—
78C22×C4—81no—
79C22×C4C2×D411yescharacteristic
80C22×C4C2×D411yescharacteristic
81C22×C4C2×D411yescharacteristic
82SD16—82no—
83M16—41no—
84M16—41no—
85C2×C8—41no—
86C2×C8—41no—
87C2×C8—41no—
88C4⋊C4—21no—
89C4⋊C4—21no—
90C4⋊C4—21no—
91C4⋊C4—21no—
92C4⋊C4—41no—
93C4⋊C4—41no—
94C4⋊C4—41no—
95C4⋊C4—41no—
96C22⋊C4—42no—
97C22⋊C4—42no—
98C22⋊C4—41no—
99C22⋊C4—41no—
100C22⋊C4—41no—
101C22⋊C4—81no—
102C22⋊C4—81no—
103C42—41no—
104C42—42no—
105C42—41no—
106C23—22no—
107C23—41no—
108C23—41no—
109C23—41no—
110C23—41no—
111C23—41no—
112C23—41no—
113C23#12011yescharacteristic
114Q8—42no—
115Q8—81no—
116Q8—81no—
117Q8—81no—
118D4—42no—
119D4—81no—
120C2×C4—21no—
121C2×C4—21no—
122C2×C4—22no—
123C2×C4—21no—
124C2×C4—21no—
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—41no—
143C2×C4—84no—
144C2×C4#13511yescharacteristic
145C2×C4#13711yescharacteristic
146C8—81no—
147C8—81no—
148C8—81no—
149C22—21no—
150C22—21no—
151C22—42no—
152C22—42no—
153C22—42no—
154C22—81no—
155C22—81no—
156C22—81no—
157C22—81no—
158C22#45211yescharacteristic
159C22#45511yescharacteristic
160C22B(4,𝔽2)11yescharacteristic
161C4—21no—
162C4—21no—
163C4—42no—
164C4—41no—
165C4—41no—
166C4—81no—
167C4—81no—
168C4—81no—
169C4—81no—
170C4—81no—
171C4—81no—
172C2—21no—
173C2—21no—
174C2—82no—
175C2#160611yescharacteristic
176C2#219711yescharacteristic
177C2#220411yescharacteristic
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. #2028C2semidirect
3. #1617C2semidirect
5. #1353C2semidirect
7. #1319C2semidirect

Maximal subgroups

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

subgroupindex
#25722
#20282
#16172
#15992
#13532
#13452
#13192