Finite Group Databasebeta

← #12391

Subgroups — #12391

Order 256.

145 conjugacy classes of subgroups; only the 39 normal subgroups are shown.

normal subgroups only

This group has 145 conjugacy classes of subgroups — too many to draw. Only its 39 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#3402C211yesmaximal, characteristic
2#2243C211yesmaximal, characteristic
3#2227C211yesmaximal, characteristic, complemented
4#1694C211yesmaximal, characteristic, complemented
5#1360C211yesmaximal, characteristic
6#1338C211yesmaximal, characteristic, complemented
7#1334C211yesmaximal, characteristic, complemented
8#556—21no—
9#500C2211yescharacteristic
10#497—21no—
11#495C2211yescharacteristic
12#464—21no—
13#463—21no—
14#462—21no—
15#462—21no—
16#446C2211yescharacteristic
17#440—21no—
18C4×Q16C2211yescharacteristic
19#366—21no—
20C4⋊C16C2211yescharacteristic
21#358—21no—
22#358—21no—
23#357—21no—
24#346C2211yescharacteristic
25#334C2211yescharacteristic
26#332—21no—
27C2×Q16—21no—
28C2×Q16—41no—
29C2×Q16D411yescharacteristic
30C2×Q16D411yescharacteristic
31C2×SD16—21no—
32C2×SD16—21no—
33C2×D8D411yescharacteristic
34C4⋊Q8—21no—
35C4⋊Q8—21no—
36#126—41no—
37#126—41no—
38#124—21no—
39#124—21no—
40C22⋊Q8—41no—
41C4×Q8—21no—
42C4×Q8—21no—
43C4×Q8D411yescharacteristic
44C4×D4—21no—
45#117—41no—
46C2×C16—21no—
47C2×C16—21no—
48#107—21no—
49#107—21no—
50#107D411yescharacteristic
51C4⋊C8—21no—
52C4⋊C8—21no—
53C4⋊C8D411yescharacteristic
54#103—21no—
55#103—21no—
56#103—41no—
57#102—21no—
58#97—21no—
59C4×C8C2311yescharacteristic
60C2×Q8—21no—
61C2×Q8—21no—
62C2×Q8—41no—
63C2×Q8D811yescharacteristic
64C2×D4—21no—
65C22×C4—41no—
66Q16—21no—
67Q16—21no—
68Q16—41no—
69Q16—81no—
70Q16—82no—
71SD16—41no—
72SD16—41no—
73SD16—81no—
74D8—41no—
75C2×C8—21no—
76C2×C8—41no—
77C2×C8C2×D411yescharacteristic
78C2×C8C2×D411yescharacteristic
79C4⋊C4—21no—
80C4⋊C4—21no—
81C4⋊C4—41no—
82C4⋊C4—41no—
83C4⋊C4—41no—
84C4⋊C4—41no—
85C4⋊C4—41no—
86C4⋊C4—41no—
87C4⋊C4D811yescharacteristic
88C22⋊C4—41no—
89C22⋊C4—41no—
90C22⋊C4—41no—
91C22⋊C4—41no—
92C22⋊C4—41no—
93C42—21no—
94C42—41no—
95C42—41no—
96C42C2×D411yescharacteristic
97C16—41no—
98C16—41no—
99C23—41no—
100Q8—41no—
101Q8—41no—
102Q8—41no—
103Q8—41no—
104Q8—41no—
105Q8—81no—
106Q8—81no—
107D4—41no—
108D4—41no—
109C2×C4—21no—
110C2×C4—21no—
111C2×C4—41no—
112C2×C4—41no—
113C2×C4—41no—
114C2×C4—82no—
115C2×C4—81no—
116C2×C4#12011yescharacteristic
117C2×C4C2×D811yescharacteristic
118C2×C4AGL(1,ℤ/8)11yescharacteristic
119C8—21no—
120C8—41no—
121C8—81no—
122C8#13511yescharacteristic
123C8#13711yescharacteristic
124C22—81no—
125C22—81no—
126C22—81no—
127C22#44711yescharacteristic
128C4—21no—
129C4—21no—
130C4—41no—
131C4—41no—
132C4—81no—
133C4—81no—
134C4—81no—
135C4—161no—
136C4#45211yescharacteristic
137C4#45511yescharacteristic
138C2—161no—
139C2#164111yescharacteristic
140C2#219011yescharacteristic
141C2#219511yescharacteristic
subgroupG/H
center, socleC22#447
commutator, Frattini, cosocleC4×C8C23
prime radicalC2×D8D4
LCSG ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C42 ◁ #446 ◁ G
LECSG ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #2227C2semidirect
4. #1694C2semidirect
6. #1338C2semidirect
7. #1334C2semidirect

Maximal subgroups

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

subgroupindex
#34022
#22432
#22272
#16942
#13602
#13382
#13342