Finite Group Databasebeta

← #29639

Subgroups — #29639

Order 256.

646 conjugacy classes of subgroups; only the 147 normal subgroups are shown.

normal subgroups only

This group has 646 conjugacy classes of subgroups — too many to draw. Only its 147 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
1C22×C4⋊Q8C211yesmaximal, characteristic, complemented
2C2×#481C212yesmaximal
3#2854C211yesmaximal, characteristic
4#2058C214yesmaximal, complemented
5#2023C214yesmaximal, complemented
6#1824C212yesmaximal, complemented
7#1817C211yesmaximal, characteristic, complemented
8C23×Q8—21no—
9C2×C4⋊Q8—28no—
10C2×C4⋊Q8—22no—
11C2×C4⋊Q8C2214yes—
12C2×#123—21no—
13C2×#121C2211yescharacteristic
14C22×C4⋊C4—21no—
15C22×C4⋊C4C2212yes—
16C22×C42C2211yescharacteristic
17#481—28no—
18C2×#106C2214yes—
19C2×#102C2214yes—
20C2×C22⋊C8C2212yes—
21#390C2212yes—
22#389C2212yes—
23#388—22no—
24#386—21no—
25#380C2211yescharacteristic
26#340C2218yes—
27#336C2214yes—
28C22×Q8—28no—
29C22×Q8—24no—
30C22×Q8—48no—
31C22×D4—21no—
32C22×D4C2311yescharacteristic
33C23×C4—22no—
34C23×C4D411yescharacteristic
35C23×C4D411yescharacteristic
36C23×C4C2311yescharacteristic
37C22×C8C2314yes—
38C4⋊Q8—216no—
39C4⋊Q8—216no—
40#123—44no—
41#121—24no—
42C2×C4⋊C4—28no—
43C2×C4⋊C4—24no—
44C2×C4⋊C4—22no—
45C2×C4⋊C4—24no—
46C2×C4⋊C4—22no—
47C2×C4⋊C4D418yes—
48C2×C4⋊C4C2311yescharacteristic
49C2×C4⋊C4C2312yes—
50C2×C4⋊C4C2314yes—
51C2×C22⋊C4—21no—
52C2×C22⋊C4—21no—
53C2×C22⋊C4—22no—
54C2×C22⋊C4—22no—
55C2×C42—24no—
56C2×C42D412yes—
57C2×C42C2312yes—
58#106—28no—
59#102—28no—
60C22⋊C8—24no—
61#95—21no—
62#95—22no—
63C24—21no—
64C24C2×D411yescharacteristic
65C2×Q8—216no—
66C2×Q8—432no—
67C2×Q8—48no—
68C2×Q8—48no—
69C2×D4—22no—
70C2×D4—24no—
71C2×D4—42no—
72C2×D4—42no—
73C2×D4—42no—
74C22×C4—22no—
75C22×C4—22no—
76C22×C4—22no—
77C22×C4—24no—
78C22×C4—24no—
79C22×C4—28no—
80C22×C4—24no—
81C22×C4—22no—
82C22×C4—21no—
83C22×C4—48no—
84C22×C4SD1614yes—
85C22×C4C2×D411yescharacteristic
86C22×C4C2×D414yes—
87C22×C4C2×D412yes—
88C22×C4C2×D412yes—
89C22×C4C2411yescharacteristic
90C2×C8—24no—
91C2×C8—28no—
92C4⋊C4—216no—
93C4⋊C4—28no—
94C4⋊C4—24no—
95C4⋊C4—22no—
96C4⋊C4—24no—
97C4⋊C4—44no—
98C4⋊C4C2×D418yes—
99C22⋊C4—42no—
100C22⋊C4—42no—
101C22⋊C4—44no—
102C22⋊C4—44no—
103C42—24no—
104C42—24no—
105C23—22no—
106C23—22no—
107C23—41no—
108C23—42no—
109C23—41no—
110C23—42no—
111C23—41no—
112C23#12012yes—
113C23C2×SD1612yes—
114C23#13712yes—
115C23C22×D411yescharacteristic
116Q8—432no—
117D4—44no—
118D4—84no—
119C2×C4—28no—
120C2×C4—22no—
121C2×C4—22no—
122C2×C4—22no—
123C2×C4—22no—
124C2×C4—22no—
125C2×C4—28no—
126C2×C4—28no—
127C2×C4—28no—
128C2×C4—416no—
129C2×C4—48no—
130C2×C4—41no—
131C2×C4—42no—
132C2×C4—42no—
133C2×C4—41no—
134C2×C4—42no—
135C2×C4—41no—
136C2×C4#12012yes—
137C2×C4C2×SD1614yes—
138C2×C4AGL(1,ℤ/8)14yes—
139C2×C4C22×D412yes—
140C2×C42−1+414yes—
141C8—44no—
142C22—22no—
143C22—24no—
144C22—24no—
145C22—22no—
146C22—81no—
147C22—82no—
148C22—82no—
149C22—81no—
150C22—81no—
151C22#44814yes—
152C22C2×#12011yescharacteristic
153C22#53612yes—
154C22C22×SD1611yescharacteristic
155C22C2×AGL(1,ℤ/8)12yes—
156C22C2×#13711yescharacteristic
157C4—22no—
158C4—24no—
159C4—24no—
160C4—48no—
161C4—82no—
162C4—81no—
163C4#53614yes—
164C2—24no—
165C2—161no—
166C2#261411yescharacteristic
167C2C2×#44811yescharacteristic
168C2#300311yescharacteristic
169C2#322712yes—
170C2#323912yes—
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC2×#121C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C22×C4⋊Q8C2semidirect
4. #2058C2semidirect
5. #2023C2semidirect
6. #1824C2semidirect
7. #1817C2semidirect

Maximal subgroups

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

subgroupindex
C22×C4⋊Q82
C2×#4812
#28542
#20582
#20232
#18242
#18172