Finite Group Databasebeta

← #12775

Subgroups — #12775

Order 256.

370 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 370 conjugacy classes of subgroups — too many to draw. Only its 71 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×#563C211yesmaximal, characteristic
2C2×#356C212yesmaximal, complemented
3C2×#355C211yesmaximal, characteristic
4#1888C211yesmaximal, characteristic
5#1886C212yesmaximal, complemented
6#563—24no—
7#563—22no—
8#563C414yescomplemented
9C2×C4⋊Q8C2212yes—
10C2×#126—22no—
11C2×#125—21no—
12C2×#124C2211yescharacteristic
13C2×#123—21no—
14C2×C22⋊Q8—22no—
15#519C2211yescharacteristic
16#419—22no—
17#418—21no—
18C2×#101C2212yes—
19C2×#101C2211yescharacteristic
20#356—24no—
21#355—22no—
22#339—22no—
23#339—21no—
24C22×Q8D412yes—
25C22×Q8C2311yescharacteristic
26C22×D4D411yescharacteristic
27C23×C4—21no—
28C2×M16—21no—
29C2×M16—22no—
30C4⋊Q8—22no—
31C4⋊Q8—22no—
32C4⋊Q8C2×C414yes—
33C4⋊Q8D414yes—
34#126—28no—
35#126—28no—
36#125—24no—
37#125—24no—
38#124—22no—
39#124C2×C412yes—
40#124D412yes—
41#123—24no—
42#123—24no—
43C22⋊Q8—28no—
44C22⋊Q8—28no—
45C4×Q8—42no—
46#117—21no—
47#117—22no—
48C2×C4⋊C4—21no—
49C2×C4⋊C4—22no—
50C2×C4⋊C4—21no—
51C2×C4⋊C4—22no—
52C2×C4⋊C4—22no—
53C2×C22⋊C4—22no—
54C2×C22⋊C4—21no—
55C2×C22⋊C4—22no—
56C2×C42D411yescharacteristic
57C2×C42D412yes—
58#103—44no—
59#102—42no—
60#101—22no—
61#101—22no—
62#101—22no—
63C24—21no—
64C2×Q8—22no—
65C2×Q8—41no—
66C2×Q8—41no—
67C2×Q8—42no—
68C2×Q8—42no—
69C2×Q8C22⋊C414yes—
70C2×Q8C22×C411yescharacteristic
71C2×Q8C2×D412yes—
72C2×Q8C2×D411yescharacteristic
73C2×D4—21no—
74C2×D4—41no—
75C2×D4—41no—
76C2×D4C22⋊C412yes—
77C22×C4—21no—
78C22×C4—22no—
79C22×C4—22no—
80C22×C4—24no—
81C22×C4—81no—
82C22×C4—81no—
83C22×C4C2×D412yes—
84C22×C4C2×D411yescharacteristic
85M16—41no—
86M16—42no—
87M16—42no—
88M16—41no—
89C2×C8—41no—
90C2×C8—42no—
91C4⋊C4—24no—
92C4⋊C4—28no—
93C4⋊C4—24no—
94C4⋊C4—24no—
95C4⋊C4—24no—
96C4⋊C4—24no—
97C4⋊C4—24no—
98C4⋊C4—22no—
99C4⋊C4—24no—
100C22⋊C4—28no—
101C22⋊C4—24no—
102C22⋊C4—24no—
103C22⋊C4—24no—
104C22⋊C4—41no—
105C42—21no—
106C42—22no—
107C42—41no—
108C42—42no—
109C42C22⋊C412yes—
110C42C22⋊C414yes—
111C23—22no—
112C23—44no—
113C23—41no—
114C23—41no—
115C23#12011yescharacteristic
116Q8—41no—
117Q8—42no—
118Q8—41no—
119Q8—44no—
120Q8—82no—
121D4—42no—
122D4—81no—
123C2×C4—21no—
124C2×C4—22no—
125C2×C4—21no—
126C2×C4—22no—
127C2×C4—22no—
128C2×C4—24no—
129C2×C4—24no—
130C2×C4—42no—
131C2×C4—44no—
132C2×C4—42no—
133C2×C4—42no—
134C2×C4—42no—
135C2×C4—84no—
136C2×C4C2×C22⋊C412yes—
137C2×C4C2×C22⋊C411yescharacteristic
138C2×C4#12012yes—
139C2×C4#12011yescharacteristic
140C8—81no—
141C8—82no—
142C22—22no—
143C22—44no—
144C22—42no—
145C22—82no—
146C22—82no—
147C22#37911yescharacteristic
148C22B(4,𝔽2)11yescharacteristic
149C22#45811yescharacteristic
150C4—21no—
151C4—22no—
152C4—21no—
153C4—22no—
154C4—42no—
155C4—44no—
156C4—44no—
157C4—82no—
158C2—21no—
159C2—21no—
160C2—82no—
161C2#179011yescharacteristic
162C2#220312yes—
subgroupG/H
center, socleC22B(4,𝔽2)
commutatorC2×Q8C22×C4
Frattini, cosocleC22×Q8C23
prime radicalC22×D4D4
LCSG ▷ C2×Q8 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C23 ◁ C22×Q8 ◁ G
LECSG ▷ C22×Q8 ▷ C22 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#356C2semidirect
5. #1886C2semidirect
8. #563C4semidirect

Maximal subgroups

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

subgroupindex
C2×#5632
C2×#3562
C2×#3552
#18882
#18862