Finite Group Databasebeta

← #32983

Subgroups — #32983

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×#484C212yesmaximal, complemented
3#2848C211yesmaximal, characteristic
4#1872C214yesmaximal, complemented
5#1852C214yesmaximal, complemented
6#1824C212yesmaximal, 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#484—28no—
18C2×#107C2214yescomplemented
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—48no—
36C23×C4D411yescharacteristic
37C23×C4D412yes—
38C23×C4C2311yescharacteristic
39C23×C4C2312yes—
40C22×C8C2314yes—
41C22⋊Q8—24no—
42C22⋊Q8—28no—
43C2×C4⋊C4—28no—
44C2×C4⋊C4—216no—
45C2×C4⋊C4—24no—
46C2×C4⋊C4—22no—
47C2×C4⋊C4—24no—
48C2×C4⋊C4—216no—
49C2×C4⋊C4—21no—
50C2×C4⋊C4—22no—
51C2×C4⋊C4D418yes—
52C2×C4⋊C4C2312yes—
53C2×C4⋊C4C2311yescharacteristic
54C2×C4⋊C4C2314yes—
55C2×C22⋊C4—21no—
56C2×C22⋊C4—22no—
57C2×C22⋊C4—22no—
58C2×C22⋊C4—21no—
59#107—28no—
60#103—28no—
61C22⋊C8—24no—
62C22⋊C8—28no—
63#95—21no—
64#95—22no—
65C24—22no—
66C24—24no—
67C24—24no—
68C24—22no—
69C24Q1614yes—
70C24C2×D411yescharacteristic
71C24C2×D412yes—
72C2×Q8—22no—
73C2×Q8—24no—
74C22×C4—232no—
75C22×C4—22no—
76C22×C4—24no—
77C22×C4—24no—
78C22×C4—24no—
79C22×C4—24no—
80C22×C4—24no—
81C22×C4—24no—
82C22×C4—24no—
83C22×C4—216no—
84C22×C4—28no—
85C22×C4—24no—
86C22×C4—21no—
87C22×C4—22no—
88C22×C4—21no—
89C22×C4—432no—
90C22×C4—48no—
91C22×C4—48no—
92C22×C4—48no—
93C22×C4C2×D411yescharacteristic
94C22×C4C2×D414yes—
95C22×C4C2×D412yes—
96C22×C4#4118yes—
97C22×C4C2411yescharacteristic
98C2×C8—24no—
99C2×C8—28no—
100C4⋊C4—216no—
101C4⋊C4—28no—
102C4⋊C4—24no—
103C4⋊C4—22no—
104C4⋊C4—24no—
105C4⋊C4—42no—
106C4⋊C4—44no—
107C4⋊C4C2×D418yes—
108C22⋊C4—42no—
109C22⋊C4—44no—
110C22⋊C4—44no—
111C22⋊C4—42no—
112C23—28no—
113C23—28no—
114C23—216no—
115C23—22no—
116C23—22no—
117C23—22no—
118C23—22no—
119C23—22no—
120C23—22no—
121C23—28no—
122C23—216no—
123C23#12012yes—
124C23#12314yes—
125C23C2×Q1612yes—
126C23C2×Q1614yes—
127C23AGL(1,ℤ/8)14yes—
128C23#13712yes—
129C23C22×D411yescharacteristic
130Q8—44no—
131C2×C4—216no—
132C2×C4—28no—
133C2×C4—22no—
134C2×C4—22no—
135C2×C4—22no—
136C2×C4—28no—
137C2×C4—216no—
138C2×C4—28no—
139C2×C4—416no—
140C2×C4—432no—
141C2×C4—48no—
142C2×C4—41no—
143C2×C4—41no—
144C2×C4—42no—
145C2×C4—42no—
146C2×C4—42no—
147C2×C4—42no—
148C2×C4—42no—
149C2×C4#12012yes—
150C2×C4#12314yes—
151C2×C4C22×D412yes—
152C2×C4C2×#4114yes—
153C8—44no—
154C22—232no—
155C22—24no—
156C22—24no—
157C22—24no—
158C22—24no—
159C22—24no—
160C22—22no—
161C22—24no—
162C22—24no—
163C22—22no—
164C22—24no—
165C22#45114yes—
166C22#48418yescomplemented
167C22C2×#12011yescharacteristic
168C22C2×#12312yes—
169C22C22×Q1611yescharacteristic
170C22C2×AGL(1,ℤ/8)12yes—
171C22C2×#13711yescharacteristic
172C4—22no—
173C4—24no—
174C4—48no—
175C4—82no—
176C4—81no—
177C4—81no—
178C4#52514yes—
179C2—24no—
180C2—28no—
181C2#284911yescharacteristic
182C2C2×#45111yescharacteristic
183C2#300311yescharacteristic
184C2C2×#48412yes—
185C2#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×#484C2semidirect
4. #1872C2semidirect
5. #1852C2semidirect
6. #1824C2semidirect
18. C2×#107C22semidirect
19. C2×#103C22semidirect
27. #340C22semidirect
166. C22#484semidirect

Maximal subgroups

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

subgroupindex
C23×C4⋊C42
C2×#4842
#28482
#18722
#18522
#18242
#17792