Finite Group Databasebeta

← #8389

Subgroups — #8389

Order 256.

369 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 369 conjugacy classes of subgroups — too many to draw. Only its 81 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
1C2×#556C211yesmaximal, characteristic
2#1800C211yesmaximal, characteristic
3#1546C212yesmaximal, complemented
4#1487C212yesmaximal, complemented
5#1460C211yesmaximal, characteristic
6#556—24no—
7#556—24no—
8C2×C4⋊Q8C2211yescharacteristic
9C2×#126—22no—
10C2×#124—21no—
11C2×C22⋊Q8C412yes—
12C2×C4×Q8—21no—
13C2×C4×Q8C2211yescharacteristic
14C2×C4×D4C2211yescharacteristic
15C2×#117C412yes—
16#431C2212yes—
17#423C2212yes—
18C2×#100—21no—
19#343—21no—
20#332—24no—
21Q8⋊C8—24no—
22#323—22no—
23C22×Q8—21no—
24C22×Q8C2×C411yescharacteristic
25C22×D4D411yescharacteristic
26C23×C4C2×C412yes—
27C2×M16—22no—
28C4⋊Q8—22no—
29C4⋊Q8D414yes—
30#126—216no—
31#124—28no—
32C22⋊Q8—24no—
33C22⋊Q8—24no—
34C4×Q8—28no—
35C4×Q8—22no—
36C4×Q8D414yes—
37C4×D4—22no—
38C4×D4—22no—
39#117—24no—
40#117—24no—
41C2×C4⋊C4—22no—
42C2×C4⋊C4—21no—
43C2×C4⋊C4—21no—
44C2×C4⋊C4C2×C411yescharacteristic
45C2×C4⋊C4C2×C412yes—
46C2×C4⋊C4D411yescharacteristic
47C2×C22⋊C4—22no—
48C2×C22⋊C4—22no—
49C2×C22⋊C4D412yes—
50C2×C42—22no—
51C2×C42—21no—
52C2×C42C2311yescharacteristic
53C4⋊C8—22no—
54C4⋊C8—22no—
55C4⋊C8—22no—
56#100—42no—
57C22⋊C8—22no—
58C4×C8—22no—
59C24C22⋊C412yes—
60C2×Q8—21no—
61C2×Q8—24no—
62C2×Q8—42no—
63C2×Q8—42no—
64C2×Q8—41no—
65C2×Q8—41no—
66C2×Q8C22⋊C412yes—
67C2×D4—22no—
68C2×D4—22no—
69C2×D4—21no—
70C2×D4—21no—
71C22×C4—21no—
72C22×C4—21no—
73C22×C4—22no—
74C22×C4—22no—
75C22×C4—22no—
76C22×C4—44no—
77C22×C4C22⋊C412yes—
78C22×C4C22×C411yescharacteristic
79C22×C4C2×D411yescharacteristic
80C22×C4C2×D411yescharacteristic
81M16—42no—
82M16—42no—
83C2×C8—42no—
84C2×C8—42no—
85C4⋊C4—28no—
86C4⋊C4—21no—
87C4⋊C4—22no—
88C4⋊C4—24no—
89C4⋊C4—24no—
90C4⋊C4—22no—
91C4⋊C4C22⋊C412yes—
92C4⋊C4C22⋊C414yes—
93C22⋊C4—28no—
94C22⋊C4—28no—
95C22⋊C4—24no—
96C42—28no—
97C42—24no—
98C42C2×D412yes—
99C42C2×D412yes—
100C23—22no—
101C23—22no—
102C23—22no—
103C23—42no—
104C23—42no—
105C23C2×C22⋊C411yescharacteristic
106Q8—24no—
107Q8—42no—
108Q8—44no—
109Q8—44no—
110D4—42no—
111D4—42no—
112C2×C4—22no—
113C2×C4—22no—
114C2×C4—22no—
115C2×C4—22no—
116C2×C4—24no—
117C2×C4—41no—
118C2×C4—44no—
119C2×C4—42no—
120C2×C4—42no—
121C2×C4—42no—
122C2×C4—42no—
123C2×C4—44no—
124C2×C4—42no—
125C2×C4C2×C22⋊C411yescharacteristic
126C2×C4C2×C22⋊C411yescharacteristic
127C2×C4#12012yes—
128C2×C4#12012yes—
129C2×C4#13512yes—
130C2×C4AGL(1,ℤ/8)11yescharacteristic
131C2×C4#13711yescharacteristic
132C8—82no—
133C8—82no—
134C22—21no—
135C22—21no—
136C22—42no—
137C22—42no—
138C22—42no—
139C22—82no—
140C22—82no—
141C22#37911yescharacteristic
142C22#41611yescharacteristic
143C22#41711yescharacteristic
144C4—22no—
145C4—21no—
146C4—42no—
147C4—42no—
148C4—42no—
149C4—44no—
150C4#41312yes—
151C4#44912yes—
152C4#45212yes—
153C2—21no—
154C2—21no—
155C2—82no—
156C2#178511yescharacteristic
157C2#178811yescharacteristic
158C2#179611yescharacteristic
subgroupG/H
center, socleC22#379
commutatorC22×C4C22×C4
Frattini, cosocleC2×C42C23
prime radicalC22×D4D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #1546C2semidirect
4. #1487C2semidirect

Maximal subgroups

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

subgroupindex
C2×#5562
#18002
#15462
#14872
#14602