Finite Group Databasebeta

← #19006

Subgroups — #19006

Order 256.

469 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 469 conjugacy classes of subgroups — too many to draw. Only its 111 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×#562C211yesmaximal, characteristic
2#3215C212yesmaximal, complemented
3#3166C211yesmaximal, characteristic, complemented
4#1661C218yesmaximal, complemented
5#1534C211yesmaximal, characteristic, complemented
6#1527C212yesmaximal, complemented
7#562—22no—
8#562—24no—
9#562C2214yes—
10#533C2211yescharacteristic
11C2×#125C2211yescharacteristic
12C2×#124C2212yes—
13C2×#123—22no—
14C2×C22⋊Q8—22no—
15C2×#121—22no—
16#501C2214yescomplemented
17#499—22no—
18#497C2214yescomplemented
19#489C2214yescomplemented
20#464—24no—
21#451—22no—
22#450—22no—
23#432C2212yes—
24#428—22no—
25C4×M16C2211yescharacteristic
26#330C2214yescomplemented
27#329C2218yescomplemented
28C22×Q8D412yes—
29C22×D4—22no—
30C22×D4D412yes—
31C23×C4—22no—
32C2×Q16—24no—
33C2×SD16—24no—
34C2×M16—21no—
35C4⋊Q8C2312yes—
36#125—22no—
37#125—21no—
38#125D412yes—
39#125C2312yes—
40#124—22no—
41#124—22no—
42#124D414yes—
43#124C2314yes—
44#123—28no—
45#123—28no—
46C22⋊Q8—28no—
47C22⋊Q8—28no—
48C22⋊Q8—22no—
49#121—28no—
50#121—28no—
51C4×Q8—22no—
52#117—21no—
53C2×C4⋊C4—22no—
54C2×C4⋊C4—22no—
55C2×C4⋊C4D412yes—
56C2×C22⋊C4—22no—
57C2×C22⋊C4—22no—
58C2×C22⋊C4—22no—
59C2×C42C2311yescharacteristic
60#107—24no—
61#106—24no—
62C4⋊C8—22no—
63#103—24no—
64#102—24no—
65C22⋊C8—22no—
66#97C2312yes—
67#97C2314yes—
68C4×C8—21no—
69C24—22no—
70C2×Q8—22no—
71C2×Q8—22no—
72C2×Q8—82no—
73C2×Q8C2×D414yes—
74C2×D4—28no—
75C2×D4—22no—
76C2×D4—82no—
77C2×D4—84no—
78C2×D4C2×D414yes—
79C22×C4—22no—
80C22×C4—22no—
81C22×C4—28no—
82C22×C4—84no—
83C22×C4C2×D411yescharacteristic
84C22×C4C2×D412yes—
85Q16—84no—
86SD16—84no—
87M16—24no—
88C2×C8—22no—
89C2×C8—24no—
90C4⋊C4—28no—
91C4⋊C4—24no—
92C4⋊C4—22no—
93C4⋊C4—24no—
94C4⋊C4—22no—
95C4⋊C4—22no—
96C4⋊C4—42no—
97C4⋊C4C2×D414yes—
98C22⋊C4—28no—
99C22⋊C4—24no—
100C22⋊C4—24no—
101C22⋊C4—24no—
102C22⋊C4—24no—
103C22⋊C4—41no—
104C42—41no—
105C42C2×D411yescharacteristic
106C42C2×D412yes—
107C42C2411yescharacteristic
108C23—24no—
109C23—44no—
110C23—44no—
111C23—44no—
112C23#12011yescharacteristic
113Q8—44no—
114Q8—44no—
115Q8—82no—
116D4—44no—
117D4—82no—
118D4—88no—
119C2×C4—22no—
120C2×C4—24no—
121C2×C4—24no—
122C2×C4—42no—
123C2×C4—44no—
124C2×C4—44no—
125C2×C4—42no—
126C2×C4—88no—
127C2×C4#12011yescharacteristic
128C2×C4#12012yes—
129C2×C4C22×D411yescharacteristic
130C2×C4C22×D412yes—
131C8—24no—
132C8—44no—
133C22—22no—
134C22—44no—
135C22—44no—
136C22—44no—
137C22—88no—
138C22#45612yes—
139C22C2×#12011yescharacteristic
140C4—21no—
141C4—21no—
142C4—22no—
143C4—22no—
144C4—44no—
145C4—44no—
146C4—82no—
147C2—22no—
148C2—84no—
149C2C2×#45611yescharacteristic
150C2#301912yes—
subgroupG/H
center, socleC22C2×#120
commutator, Frattini, cosocleC42C24
prime radicalC2×#562C2
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ G
LECSG ▷ C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #3215C2semidirect
3. #3166C2semidirect
4. #1661C2semidirect
5. #1534C2semidirect
6. #1527C2semidirect
16. #501C22semidirect
18. #497C22semidirect
19. #489C22semidirect
26. #330C22semidirect
27. #329C22semidirect

Maximal subgroups

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

subgroupindex
C2×#5622
#32152
#31662
#16612
#15342
#15272