Finite Group Databasebeta

← #8398

Subgroups — #8398

Order 256.

391 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 391 conjugacy classes of subgroups — too many to draw. Only its 85 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×#554C211yesmaximal, characteristic
2#1801C211yesmaximal, characteristic, complemented
3#1549C212yesmaximal, complemented
4#1486C212yesmaximal
5#1461C211yesmaximal, characteristic, complemented
6#554—24no—
7#554—24no—
8C2×C4⋊Q8C2211yescharacteristic
9C2×#125C412yes—
10C2×C22⋊Q8—21no—
11C2×C22⋊Q8—22no—
12C2×C4×Q8C2211yescharacteristic
13C2×C4×D4C412yes—
14C2×C4×D4C2211yescharacteristic
15C22×C4⋊C4—21no—
16#431C2212yes—
17#423C2212yes—
18C2×#101—21no—
19#343—21no—
20#332—24no—
21D4⋊C8—24no—
22#324—22no—
23C22×Q8—21no—
24C22×Q8D411yescharacteristic
25C22×D4C2×C411yescharacteristic
26C23×C4—21no—
27C23×C4—22no—
28C2×M16—22no—
29C4⋊Q8—22no—
30C4⋊Q8D414yes—
31#125—24no—
32#125—24no—
33C22⋊Q8—28no—
34C22⋊Q8—216no—
35C4×Q8—22no—
36C4×Q8—22no—
37C4×D4—24no—
38C4×D4—24no—
39C4×D4—22no—
40C4×D4D414yes—
41C2×C4⋊C4—28no—
42C2×C4⋊C4—22no—
43C2×C4⋊C4—22no—
44C2×C4⋊C4—21no—
45C2×C4⋊C4C2×C412yes—
46C2×C4⋊C4C2×C411yescharacteristic
47C2×C4⋊C4D411yescharacteristic
48C2×C4⋊C4D412yes—
49C2×C22⋊C4—22no—
50C2×C22⋊C4—21no—
51C2×C42C2×C412yes—
52C2×C42C2311yescharacteristic
53C4⋊C8—22no—
54C4⋊C8—22no—
55C4⋊C8—22no—
56#101—42no—
57C22⋊C8—22no—
58C4×C8—22no—
59C24—21no—
60C2×Q8—22no—
61C2×Q8—24no—
62C2×Q8—22no—
63C2×Q8—21no—
64C2×Q8—21no—
65C2×Q8—82no—
66C2×D4—24no—
67C2×D4—21no—
68C2×D4C22⋊C412yes—
69C22×C4—21no—
70C22×C4—22no—
71C22×C4—28no—
72C22×C4—24no—
73C22×C4—42no—
74C22×C4—41no—
75C22×C4—41no—
76C22×C4—84no—
77C22×C4C22⋊C412yes—
78C22×C4C22⋊C412yes—
79C22×C4C22×C411yescharacteristic
80C22×C4C2×D411yescharacteristic
81C22×C4C2×D411yescharacteristic
82M16—42no—
83M16—42no—
84C2×C8—42no—
85C2×C8—42no—
86C4⋊C4—28no—
87C4⋊C4—28no—
88C4⋊C4—21no—
89C4⋊C4—22no—
90C4⋊C4—22no—
91C4⋊C4—24no—
92C4⋊C4—24no—
93C4⋊C4C22⋊C414yes—
94C4⋊C4C22⋊C412yes—
95C22⋊C4—28no—
96C22⋊C4—24no—
97C42—22no—
98C42—22no—
99C42C2×D412yes—
100C42C2×D412yes—
101C23—24no—
102C23—22no—
103C23—42no—
104C23—41no—
105C23—41no—
106C23C2×C22⋊C411yescharacteristic
107Q8—42no—
108Q8—42no—
109Q8—84no—
110D4—24no—
111D4—22no—
112D4#10114yescomplemented
113C2×C4—22no—
114C2×C4—22no—
115C2×C4—22no—
116C2×C4—22no—
117C2×C4—24no—
118C2×C4—22no—
119C2×C4—22no—
120C2×C4—41no—
121C2×C4—44no—
122C2×C4—41no—
123C2×C4—41no—
124C2×C4—44no—
125C2×C4—44no—
126C2×C4—42no—
127C2×C4—88no—
128C2×C4C2×C22⋊C411yescharacteristic
129C2×C4C2×C22⋊C411yescharacteristic
130C2×C4#12012yes—
131C2×C4#12012yes—
132C2×C4#13512yes—
133C2×C4AGL(1,ℤ/8)11yescharacteristic
134C2×C4#13711yescharacteristic
135C8—82no—
136C8—82no—
137C22—21no—
138C22—21no—
139C22—24no—
140C22—44no—
141C22—44no—
142C22—42no—
143C22—42no—
144C22#37911yescharacteristic
145C22#41611yescharacteristic
146C22#41711yescharacteristic
147C4—22no—
148C4—21no—
149C4—44no—
150C4—42no—
151C4—42no—
152C4—42no—
153C4C2×#10112yes—
154C4#44912yes—
155C4#45212yes—
156C2—21no—
157C2—21no—
158C2—44no—
159C2#178411yescharacteristic
160C2#178811yescharacteristic
161C2#179511yescharacteristic
subgroupG/H
center, socleC22#379
commutatorC22×C4C22×C4
Frattini, cosocleC2×C42C23
prime radicalC22×D4C2×C4
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
2. #1801C2semidirect
3. #1549C2semidirect
5. #1461C2semidirect
112. D4#101semidirect

Maximal subgroups

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

subgroupindex
C2×#5542
#18012
#15492
#14862
#14612