Finite Group Databasebeta

← #18999

Subgroups — #18999

Order 256.

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

normal subgroups only

This group has 425 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×#563C211yesmaximal, characteristic
2#3236C212yesmaximal, complemented
3#3140C211yesmaximal, characteristic
4#1664C218yesmaximal, complemented
5#1533C211yesmaximal, characteristic
6#1528C212yesmaximal, complemented
7#563—24no—
8#563—22no—
9#563C2214yes—
10#533C2211yescharacteristic
11C2×C4⋊Q8—21no—
12C2×#126—22no—
13C2×#125C2212yes—
14C2×#124C2211yescharacteristic
15C2×#123—21no—
16C2×C22⋊Q8—22no—
17#501C2214yescomplemented
18#491C2214yescomplemented
19#489C2214yes—
20#488—22no—
21#484—22no—
22#483—22no—
23#479—24no—
24#432C2212yes—
25#419—21no—
26#418—21no—
27C4×M16C2211yescharacteristic
28#330C2218yescomplemented
29#329C2214yes—
30C22×Q8—21no—
31C22×Q8D411yescharacteristic
32C22×D4D411yescharacteristic
33C23×C4—21no—
34C2×M16—21no—
35C4⋊Q8—24no—
36C4⋊Q8—24no—
37C4⋊Q8C2312yes—
38#126—28no—
39#126—28no—
40#125—22no—
41#125—22no—
42#125—21no—
43#125D414yes—
44#125C2314yes—
45#124—22no—
46#124D412yes—
47#124C2312yes—
48#123—24no—
49#123—24no—
50C22⋊Q8—28no—
51C22⋊Q8—28no—
52C22⋊Q8—22no—
53C4×Q8—22no—
54#117—21no—
55C2×C4⋊C4—22no—
56C2×C4⋊C4—22no—
57C2×C4⋊C4—22no—
58C2×C4⋊C4D412yes—
59C2×C4⋊C4D412yes—
60C2×C22⋊C4—22no—
61C2×C22⋊C4—21no—
62C2×C22⋊C4—22no—
63C2×C42—21no—
64C2×C42C2311yescharacteristic
65#107—24no—
66#106—24no—
67C4⋊C8—22no—
68#103—24no—
69#103—24no—
70#103—24no—
71#102—24no—
72C22⋊C8—22no—
73#97C2314yes—
74#97C2312yes—
75C4×C8—21no—
76C24—21no—
77C2×Q8—24no—
78C2×Q8—21no—
79C2×Q8—22no—
80C2×Q8—42no—
81C2×Q8—82no—
82C2×Q8C2×D412yes—
83C2×D4—21no—
84C2×D4—42no—
85C2×D4C2×D412yes—
86C22×C4—22no—
87C22×C4—21no—
88C22×C4—22no—
89C22×C4—24no—
90C22×C4—82no—
91C22×C4C2×D411yescharacteristic
92C22×C4C2×D412yes—
93M16—42no—
94C2×C8—22no—
95C2×C8—24no—
96C4⋊C4—28no—
97C4⋊C4—24no—
98C4⋊C4—24no—
99C4⋊C4—22no—
100C4⋊C4—22no—
101C4⋊C4—24no—
102C4⋊C4—24no—
103C4⋊C4—22no—
104C4⋊C4—22no—
105C4⋊C4—42no—
106C4⋊C4C2×D414yes—
107C4⋊C4C2×D414yes—
108C22⋊C4—28no—
109C22⋊C4—24no—
110C22⋊C4—24no—
111C22⋊C4—24no—
112C22⋊C4—41no—
113C42—24no—
114C42—41no—
115C42C2×D411yescharacteristic
116C42C2×D412yes—
117C42C2411yescharacteristic
118C23—22no—
119C23—44no—
120C23—42no—
121C23#12011yescharacteristic
122Q8—42no—
123Q8—44no—
124Q8—81no—
125Q8—84no—
126D4—42no—
127D4—81no—
128C2×C4—22no—
129C2×C4—24no—
130C2×C4—22no—
131C2×C4—24no—
132C2×C4—42no—
133C2×C4—44no—
134C2×C4—44no—
135C2×C4—42no—
136C2×C4—42no—
137C2×C4—84no—
138C2×C4#12011yescharacteristic
139C2×C4#12012yes—
140C2×C4C22×D412yes—
141C2×C4C22×D411yescharacteristic
142C8—44no—
143C8—42no—
144C22—22no—
145C22—42no—
146C22—44no—
147C22—84no—
148C22#45512yes—
149C22C2×#12011yescharacteristic
150C4—21no—
151C4—21no—
152C4—22no—
153C4—22no—
154C4—44no—
155C4—42no—
156C4—44no—
157C4—82no—
158C2—22no—
159C2—82no—
160C2C2×#45511yescharacteristic
161C2#302012yes—
subgroupG/H
center, socleC22C2×#120
commutator, Frattini, cosocleC42C24
prime radicalC22×D4D4
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. #3236C2semidirect
4. #1664C2semidirect
6. #1528C2semidirect
17. #501C22semidirect
18. #491C22semidirect
28. #330C22semidirect

Maximal subgroups

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

subgroupindex
C2×#5632
#32362
#31402
#16642
#15332
#15282