Finite Group Databasebeta

← #8395

Subgroups — #8395

Order 256.

391 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 391 conjugacy classes of subgroups — too many to draw. Only its 89 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#1800C211yesmaximal, characteristic
3#1549C212yesmaximal, complemented
4#1487C212yesmaximal, complemented
5#1460C211yesmaximal, characteristic
6#554—24no—
7#554—24no—
8C2×C4⋊Q8C2211yescharacteristic
9C2×#125—21no—
10C2×C22⋊Q8—22no—
11C2×C22⋊Q8C412yes—
12C2×C4×Q8C2211yescharacteristic
13C2×C4×D4—21no—
14C2×C4×D4C2211yescharacteristic
15C22×C4⋊C4C412yes—
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×C4—22no—
27C23×C4C2×C412yes—
28C2×M16—22no—
29C4⋊Q8—22no—
30C4⋊Q8D414yes—
31#125—28no—
32C22⋊Q8—216no—
33C22⋊Q8—24no—
34C22⋊Q8—24no—
35C4×Q8—22no—
36C4×Q8D414yes—
37C4×D4—28no—
38C4×D4—22no—
39C4×D4—22no—
40C2×C4⋊C4—22no—
41C2×C4⋊C4—22no—
42C2×C4⋊C4—21no—
43C2×C4⋊C4—21no—
44C2×C4⋊C4—24no—
45C2×C4⋊C4—24no—
46C2×C4⋊C4C2×C411yescharacteristic
47C2×C4⋊C4C2×C412yes—
48C2×C4⋊C4D411yescharacteristic
49C2×C22⋊C4—22no—
50C2×C22⋊C4D412yes—
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—82no—
64C2×Q8C22⋊C412yes—
65C2×D4—22no—
66C2×D4—22no—
67C2×D4SD1612yes—
68C2×D4Q1612yes—
69C22×C4—21no—
70C22×C4—21no—
71C22×C4—22no—
72C22×C4—28no—
73C22×C4—22no—
74C22×C4—22no—
75C22×C4—44no—
76C22×C4—84no—
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—28no—
87C4⋊C4—21no—
88C4⋊C4—22no—
89C4⋊C4—24no—
90C4⋊C4—24no—
91C4⋊C4—22no—
92C4⋊C4C22⋊C412yes—
93C4⋊C4C22⋊C414yes—
94C22⋊C4—28no—
95C22⋊C4—24no—
96C42—24no—
97C42C2×D412yes—
98C42C2×D412yes—
99C23—22no—
100C23—22no—
101C23—42no—
102C23—42no—
103C23#10314yes—
104C23C2×C22⋊C411yescharacteristic
105Q8—24no—
106Q8—42no—
107Q8—84no—
108D4—24no—
109D4—24no—
110C2×C4—22no—
111C2×C4—22no—
112C2×C4—22no—
113C2×C4—22no—
114C2×C4—24no—
115C2×C4—41no—
116C2×C4—44no—
117C2×C4—42no—
118C2×C4—44no—
119C2×C4—42no—
120C2×C4—44no—
121C2×C4—42no—
122C2×C4—88no—
123C2×C4C2×C22⋊C411yescharacteristic
124C2×C4C2×C22⋊C411yescharacteristic
125C2×C4#12012yes—
126C2×C4#12012yes—
127C2×C4C2×SD1611yescharacteristic
128C2×C4C2×Q1611yescharacteristic
129C2×C4#13712yes—
130C8—82no—
131C8—82no—
132C22—21no—
133C22—21no—
134C22—24no—
135C22—44no—
136C22—44no—
137C22—42no—
138C22—42no—
139C22#37911yescharacteristic
140C22C2×#10311yescharacteristic
141C22#41911yescharacteristic
142C4—22no—
143C4—21no—
144C4—42no—
145C4—44no—
146C4—42no—
147C4—42no—
148C4#41312yes—
149C4#44812yes—
150C4#45112yes—
151C2—21no—
152C2—21no—
153C2—44no—
154C2#178511yescharacteristic
155C2#178711yescharacteristic
156C2#179511yescharacteristic
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. #1549C2semidirect
4. #1487C2semidirect

Maximal subgroups

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

subgroupindex
C2×#5542
#18002
#15492
#14872
#14602