Finite Group Databasebeta

← #32982

Subgroups — #32982

Order 256.

867 conjugacy classes of subgroups; only the 171 normal subgroups are shown.

normal subgroups only

This group has 867 conjugacy classes of subgroups — too many to draw. Only its 171 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
1C23×C4⋊C4C211yesmaximal, characteristic
2C2×#483C212yesmaximal, complemented
3#2848C211yesmaximal, characteristic
4#1872C214yesmaximal, complemented
5#1853C214yesmaximal, complemented
6#1823C212yesmaximal, complemented
7#1779C211yesmaximal, characteristic
8C24×C4—21no—
9C24×C4C2211yescharacteristic
10C2×C22⋊Q8C2211yescharacteristic
11C2×C22⋊Q8C2212yes—
12C22×C4⋊C4—21no—
13C22×C4⋊C4—22no—
14C22×C4⋊C4—28no—
15C22×C4⋊C4C2214yes—
16C22×C4⋊C4C2212yes—
17#483—28no—
18C2×#106C2214yescomplemented
19C2×#103C2214yescomplemented
20C2×C22⋊C8C2212yes—
21C2×C22⋊C8C2214yes—
22#385—22no—
23#385—21no—
24#380C2211yescharacteristic
25#380C2212yes—
26#379—21no—
27#340C2218yescomplemented
28C25D411yescharacteristic
29C22×Q8C2311yescharacteristic
30C23×C4—24no—
31C23×C4—28no—
32C23×C4—28no—
33C23×C4—24no—
34C23×C4—22no—
35C23×C4—44no—
36C23×C4—44no—
37C23×C4D411yescharacteristic
38C23×C4D412yes—
39C23×C4C2311yescharacteristic
40C23×C4C2312yes—
41C22×C8C2314yes—
42C22⋊Q8—24no—
43C22⋊Q8—28no—
44C2×C4⋊C4—28no—
45C2×C4⋊C4—216no—
46C2×C4⋊C4—24no—
47C2×C4⋊C4—22no—
48C2×C4⋊C4—24no—
49C2×C4⋊C4—216no—
50C2×C4⋊C4—21no—
51C2×C4⋊C4—22no—
52C2×C4⋊C4D418yes—
53C2×C4⋊C4C2312yes—
54C2×C4⋊C4C2311yescharacteristic
55C2×C4⋊C4C2314yes—
56C2×C22⋊C4—21no—
57C2×C22⋊C4—22no—
58C2×C22⋊C4—21no—
59C2×C22⋊C4—22no—
60#106—28no—
61#103—28no—
62C22⋊C8—24no—
63C22⋊C8—28no—
64#95—21no—
65#95—22no—
66C24—22no—
67C24—24no—
68C24—22no—
69C24—24no—
70C24SD1614yes—
71C24C2×D411yescharacteristic
72C24C2×D412yes—
73C2×Q8—22no—
74C2×Q8—24no—
75C22×C4—232no—
76C22×C4—22no—
77C22×C4—24no—
78C22×C4—24no—
79C22×C4—24no—
80C22×C4—24no—
81C22×C4—24no—
82C22×C4—24no—
83C22×C4—24no—
84C22×C4—216no—
85C22×C4—28no—
86C22×C4—24no—
87C22×C4—21no—
88C22×C4—22no—
89C22×C4—21no—
90C22×C4—48no—
91C22×C4—432no—
92C22×C4—48no—
93C22×C4—44no—
94C22×C4—44no—
95C22×C4C2×D411yescharacteristic
96C22×C4C2×D414yes—
97C22×C4C2×D412yes—
98C22×C4#4118yes—
99C22×C4C2411yescharacteristic
100C2×C8—24no—
101C2×C8—28no—
102C4⋊C4—216no—
103C4⋊C4—28no—
104C4⋊C4—24no—
105C4⋊C4—22no—
106C4⋊C4—24no—
107C4⋊C4—42no—
108C4⋊C4—44no—
109C4⋊C4C2×D418yes—
110C22⋊C4—42no—
111C22⋊C4—44no—
112C22⋊C4—42no—
113C22⋊C4—44no—
114C23—28no—
115C23—216no—
116C23—28no—
117C23—22no—
118C23—22no—
119C23—22no—
120C23—22no—
121C23—22no—
122C23—22no—
123C23—28no—
124C23—216no—
125C23#12012yes—
126C23#12314yes—
127C23C2×SD1612yes—
128C23C2×SD1614yes—
129C23AGL(1,ℤ/8)12yes—
130C23#13714yes—
131C23C22×D411yescharacteristic
132Q8—44no—
133C2×C4—216no—
134C2×C4—28no—
135C2×C4—22no—
136C2×C4—22no—
137C2×C4—22no—
138C2×C4—28no—
139C2×C4—216no—
140C2×C4—28no—
141C2×C4—432no—
142C2×C4—44no—
143C2×C4—44no—
144C2×C4—416no—
145C2×C4—41no—
146C2×C4—41no—
147C2×C4—42no—
148C2×C4—42no—
149C2×C4—42no—
150C2×C4—42no—
151C2×C4—42no—
152C2×C4#12012yes—
153C2×C4#12314yes—
154C2×C4C22×D412yes—
155C2×C4C2×#4114yes—
156C8—44no—
157C22—232no—
158C22—24no—
159C22—24no—
160C22—24no—
161C22—24no—
162C22—24no—
163C22—22no—
164C22—24no—
165C22—24no—
166C22—22no—
167C22—24no—
168C22#45014yes—
169C22#48318yescomplemented
170C22C2×#12011yescharacteristic
171C22C2×#12312yes—
172C22C22×SD1611yescharacteristic
173C22C2×AGL(1,ℤ/8)11yescharacteristic
174C22C2×#13712yes—
175C4—22no—
176C4—24no—
177C4—48no—
178C4—82no—
179C4—81no—
180C4—81no—
181C4#52514yes—
182C2—24no—
183C2—28no—
184C2#284911yescharacteristic
185C2C2×#45011yescharacteristic
186C2#300411yescharacteristic
187C2C2×#48312yes—
188C2#309212yes—
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC25D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C24×C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. C2×#483C2semidirect
4. #1872C2semidirect
5. #1853C2semidirect
6. #1823C2semidirect
18. C2×#106C22semidirect
19. C2×#103C22semidirect
27. #340C22semidirect
169. C22#483semidirect

Maximal subgroups

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

subgroupindex
C23×C4⋊C42
C2×#4832
#28482
#18722
#18532
#18232
#17792