Finite Group Databasebeta

← #13610

Subgroups — #13610

Order 256.

168 conjugacy classes of subgroups; only the 43 normal subgroups are shown.

normal subgroups only

This group has 168 conjugacy classes of subgroups — too many to draw. Only its 43 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
1#3222C211yesmaximal, characteristic
2#2128C211yesmaximal, characteristic
3#2127C211yesmaximal, characteristic
4#1412C212yesmaximal, complemented
5#1409C212yesmaximal, complemented
6#541C411yescharacteristic, complemented
7#535C2211yescharacteristic
8#497C411yescharacteristic, complemented
9#492C2211yescharacteristic
10#490C2211yescharacteristic
11#488C411yescharacteristic, complemented
12#464—21no—
13#460—21no—
14#452—21no—
15#450—21no—
16#433C411yescharacteristic, complemented
17#410—21no—
18C2×#99—21no—
19#354—21no—
20#353—21no—
21#353C2212yes—
22#353C2212yes—
23C2×#41D411yescharacteristic
24C22×D4D411yescharacteristic
25C2×Q16—21no—
26C2×SD16—21no—
27C2×SD16—21no—
28C2×SD16—21no—
29C4⋊Q8C2×C411yescharacteristic, complemented
30#127D411yescharacteristic
31#127C2311yescharacteristic
32#126—41no—
33#125—41no—
34#125C2×C411yescharacteristic, complemented
35#124C2×C411yescharacteristic
36#124D411yescharacteristic
37C22⋊Q8—21no—
38#121—41no—
39#120—41no—
40C4×Q8—21no—
41#117—41no—
42#117C2×C411yescharacteristic
43C2×C22⋊C4—41no—
44C4⋊C8—21no—
45#103—21no—
46#102—21no—
47#102—21no—
48#102—21no—
49#99—22no—
50#99—22no—
51#99—41no—
52#99—41no—
53C22⋊C8—21no—
54#97C2×C411yescharacteristic, complemented
55C4×C8C2×C411yescharacteristic, complemented
56C24—21no—
57#41—81no—
58C2×Q8—21no—
59C2×Q8C22⋊C411yescharacteristic
60C2×D4—21no—
61C2×D4—21no—
62C2×D4—41no—
63C2×D4—41no—
64C2×D4—81no—
65C2×D4C22⋊C411yescharacteristic
66C2×D4C2×D411yescharacteristic
67C2×D4C2×D411yescharacteristic
68C22×C4—21no—
69C22×C4—81no—
70C22×C4C22⋊C411yescharacteristic
71Q16—81no—
72SD16—81no—
73SD16—81no—
74SD16—81no—
75C2×C8—21no—
76C2×C8—21no—
77C4⋊C4—21no—
78C4⋊C4—21no—
79C4⋊C4—41no—
80C4⋊C4—41no—
81C4⋊C4—41no—
82C4⋊C4—41no—
83C4⋊C4—81no—
84C22⋊C4—21no—
85C22⋊C4—41no—
86C22⋊C4—41no—
87C22⋊C4—41no—
88C22⋊C4—42no—
89C22⋊C4—42no—
90C22⋊C4—81no—
91C42—41no—
92C42—81no—
93C42C22⋊C411yescharacteristic
94C42C22×C411yescharacteristic
95C23—21no—
96C23—21no—
97C23—21no—
98C23—41no—
99C23—41no—
100C23—161no—
101C23#9911yescharacteristic
102Q8—41no—
103Q8—41no—
104Q8—41no—
105D4—41no—
106D4—41no—
107D4—41no—
108D4—81no—
109D4—81no—
110D4—161no—
111D4—161no—
112C2×C4—21no—
113C2×C4—21no—
114C2×C4—21no—
115C2×C4—41no—
116C2×C4—41no—
117C2×C4—41no—
118C2×C4—41no—
119C2×C4—41no—
120C2×C4—82no—
121C2×C4—82no—
122C2×C4—162no—
123C2×C4#9911yescharacteristic
124C2×C4C2×C22⋊C411yescharacteristic
125C8—41no—
126C8—41no—
127C22—21no—
128C22—41no—
129C22—41no—
130C22—41no—
131C22—41no—
132C22—81no—
133C22—81no—
134C22—81no—
135C22—82no—
136C22—161no—
137C22C2×#9911yescharacteristic
138C4—21no—
139C4—21no—
140C4—41no—
141C4—41no—
142C4—41no—
143C4—81no—
144C4—81no—
145C4—162no—
146C4—162no—
147C2—21no—
148C2—41no—
149C2—81no—
150C2—81no—
151C2—81no—
152C2#212711yescharacteristic
subgroupG/H
center, socleC2#2127
commutatorC42C22×C4
Frattini, cosocle#127C23
prime radical#535C22
LCSG ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2 ◁ C22 ◁ C22×C4 ◁ #535 ◁ G
LECSG ▷ #127 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #1412C2semidirect
5. #1409C2semidirect
6. #541C4semidirect
8. #497C4semidirect
11. #488C4semidirect
16. #433C4semidirect
29. C4⋊Q8C2×C4semidirect
34. #125C2×C4semidirect
54. #97C2×C4semidirect
55. C4×C8C2×C4semidirect

Maximal subgroups

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

subgroupindex
#32222
#21282
#21272
#14122
#14092