Finite Group Databasebeta

← #13580

Subgroups — #13580

Order 256.

201 conjugacy classes of subgroups; only the 45 normal subgroups are shown.

normal subgroups only

This group has 201 conjugacy classes of subgroups — too many to draw. Only its 45 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#3152C211yesmaximal, characteristic
2#2132C211yesmaximal, characteristic
3#2126C211yesmaximal, characteristic
4#1407C214yesmaximal, complemented
5C2×#137—21no—
6C2×#137—21no—
7#532C2211yescharacteristic
8C2×C4⋊Q8C411yescharacteristic, complemented
9#497C412yescomplemented
10#497C2212yes—
11#494—21no—
12#492—21no—
13#413—21no—
14#410—21no—
15C4×M16C411yescharacteristic, complemented
16#356—21no—
17#355C2212yes—
18#354C2212yes—
19#353—21no—
20C2×#41D411yescharacteristic
21C2×#41D411yescharacteristic
22C22×Q8—21no—
23#137—42no—
24#137—42no—
25#137—42no—
26#137—42no—
27C2×Q16—22no—
28C2×Q16—22no—
29C2×SD16—22no—
30C2×SD16—22no—
31C2×M16—21no—
32C2×M16—41no—
33C4⋊Q8—21no—
34C4⋊Q8—41no—
35C4⋊Q8C2×C412yes—
36C4⋊Q8D411yescharacteristic
37#127D411yescharacteristic
38#124C2×C411yescharacteristic
39#124C2311yescharacteristic
40#121—41no—
41C4×D4—41no—
42#117—41no—
43C2×C4⋊C4—41no—
44C2×C42C2×C411yescharacteristic
45#101—22no—
46#100—41no—
47#99—22no—
48#99—41no—
49#97C2×C412yes—
50C4×C8—21no—
51#41—41no—
52#41—41no—
53#41—41no—
54#41—41no—
55C2×Q8—22no—
56C2×Q8—41no—
57C2×Q8—41no—
58C2×Q8—41no—
59C2×Q8—41no—
60C2×Q8—41no—
61C2×Q8C22⋊C411yescharacteristic
62C2×Q8C2×D411yescharacteristic
63C2×D4—21no—
64C2×D4—21no—
65C2×D4—41no—
66C2×D4C22⋊C411yescharacteristic
67C2×D4C2×D411yescharacteristic
68C22×C4—21no—
69C22×C4—21no—
70C22×C4—21no—
71C22×C4—41no—
72C22×C4C22⋊C411yescharacteristic
73Q16—44no—
74Q16—44no—
75SD16—44no—
76SD16—44no—
77M16—22no—
78M16—22no—
79M16—42no—
80M16—81no—
81C2×C8—22no—
82C2×C8—81no—
83C4⋊C4—41no—
84C4⋊C4—42no—
85C4⋊C4—42no—
86C4⋊C4—81no—
87C22⋊C4—41no—
88C22⋊C4—41no—
89C22⋊C4—42no—
90C42—21no—
91C42—81no—
92C42C22⋊C411yescharacteristic
93C42C22×C411yescharacteristic
94C23—21no—
95C23—21no—
96C23#9911yescharacteristic
97Q8—41no—
98Q8—41no—
99Q8—42no—
100Q8—42no—
101Q8—81no—
102Q8—81no—
103D4—41no—
104D4—41no—
105D4—82no—
106D4—81no—
107D4—81no—
108C2×C4—21no—
109C2×C4—21no—
110C2×C4—21no—
111C2×C4—21no—
112C2×C4—21no—
113C2×C4—21no—
114C2×C4—42no—
115C2×C4—41no—
116C2×C4—41no—
117C2×C4—41no—
118C2×C4—41no—
119C2×C4—42no—
120C2×C4—81no—
121C2×C4—81no—
122C2×C4—82no—
123C2×C4#9911yescharacteristic
124C2×C4C2×C22⋊C411yescharacteristic
125C8—24no—
126C8—82no—
127C22—21no—
128C22—41no—
129C22—41no—
130C22—41no—
131C22—81no—
132C22—81no—
133C22C2×#9911yescharacteristic
134C4—22no—
135C4—21no—
136C4—41no—
137C4—41no—
138C4—41no—
139C4—82no—
140C4—162no—
141C4#35311yescharacteristic
142C4#35311yescharacteristic
143C2—21no—
144C2—41no—
145C2—81no—
146C2—81no—
147C2C2×#35311yescharacteristic
subgroupG/H
center, socleC2C2×#353
commutatorC42C22×C4
Frattini, cosocle#124C23
prime radical#532C22
LCSG ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2 ◁ C2×C4 ◁ C22×C4 ◁ #532 ◁ G
LECSG ▷ #124 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #1407C2semidirect
8. C2×C4⋊Q8C4semidirect
9. #497C4semidirect
15. C4×M16C4semidirect

Maximal subgroups

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

subgroupindex
#31522
#21322
#21262
#14072