Finite Group Databasebeta

← #12403

Subgroups — #12403

Order 256.

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

normal subgroups only

This group has 140 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#3352C211yesmaximal, characteristic
2#2255C211yesmaximal, characteristic, complemented
3#2227C211yesmaximal, characteristic, complemented
4#1693C211yesmaximal, characteristic
5#1363C211yesmaximal, characteristic, complemented
6#1334C211yesmaximal, characteristic, complemented
7#1330C211yesmaximal, characteristic
8#552—21no—
9#500C2211yescharacteristic
10#499C2211yescharacteristic
11#485—21no—
12#484—21no—
13#479—21no—
14#470—21no—
15#461—21no—
16C4×Q16C2211yescharacteristic
17#436C2211yescharacteristic
18#428—21no—
19#367—21no—
20#366—21no—
21#366—21no—
22C4⋊C16C2211yescharacteristic
23#358—21no—
24#346C2211yescharacteristic
25#334C2211yescharacteristic
26#332—21no—
27C2×Q16D411yescharacteristic
28C2×SD16—41no—
29C2×M16—21no—
30C4⋊Q8—21no—
31#126—41no—
32#125—21no—
33#125—41no—
34#123—41no—
35C22⋊Q8—21no—
36C22⋊Q8—21no—
37C4×Q8—21no—
38C4×D4D411yescharacteristic
39#117—21no—
40#117—41no—
41C2×C4⋊C4—21no—
42C2×C16—21no—
43C2×C16—21no—
44#107—21no—
45#107—21no—
46#107—21no—
47#107D411yescharacteristic
48#107D411yescharacteristic
49#107D411yescharacteristic
50#106—21no—
51#106—21no—
52C4⋊C8—21no—
53C4⋊C8D411yescharacteristic
54#103—21no—
55#103—21no—
56#103—21no—
57#103—41no—
58C22⋊C8—21no—
59C4×C8C2311yescharacteristic
60C2×Q8—21no—
61C2×Q8—41no—
62C2×D4D811yescharacteristic
63C22×C4—21no—
64C22×C4—41no—
65Q16—21no—
66Q16—21no—
67SD16—82no—
68M16—41no—
69M16—41no—
70C2×C8—21no—
71C2×C8—41no—
72C2×C8C2×D411yescharacteristic
73C2×C8C2×D411yescharacteristic
74C4⋊C4—21no—
75C4⋊C4—21no—
76C4⋊C4—21no—
77C4⋊C4—41no—
78C4⋊C4—41no—
79C4⋊C4—41no—
80C4⋊C4—41no—
81C4⋊C4—41no—
82C4⋊C4—41no—
83C4⋊C4—41no—
84C4⋊C4—41no—
85C4⋊C4D811yescharacteristic
86C22⋊C4—21no—
87C22⋊C4—41no—
88C22⋊C4—41no—
89C22⋊C4—41no—
90C42—41no—
91C42—41no—
92C42C2×D411yescharacteristic
93C16—22no—
94C16—41no—
95C23—21no—
96Q8—41no—
97Q8—41no—
98Q8—81no—
99Q8—81no—
100D4—41no—
101C2×C4—21no—
102C2×C4—41no—
103C2×C4—41no—
104C2×C4—41no—
105C2×C4—41no—
106C2×C4—41no—
107C2×C4—41no—
108C2×C4—82no—
109C2×C4—81no—
110C2×C4#12011yescharacteristic
111C2×C4C2×D811yescharacteristic
112C2×C4AGL(1,ℤ/8)11yescharacteristic
113C8—21no—
114C8—41no—
115C8—81no—
116C8#13511yescharacteristic
117C8#13711yescharacteristic
118C22—41no—
119C22—41no—
120C22—41no—
121C22#44711yescharacteristic
122C4—21no—
123C4—21no—
124C4—41no—
125C4—81no—
126C4—81no—
127C4—81no—
128C4—81no—
129C4—161no—
130C4#45211yescharacteristic
131C4#45611yescharacteristic
132C2—81no—
133C2#164711yescharacteristic
134C2#219011yescharacteristic
135C2#219511yescharacteristic
subgroupG/H
center, socleC22#447
commutator, Frattini, cosocleC4×C8C23
prime radicalC2×D4D8
LCSG ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C42 ◁ #436 ◁ G
LECSG ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #2255C2semidirect
3. #2227C2semidirect
5. #1363C2semidirect
6. #1334C2semidirect

Maximal subgroups

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

subgroupindex
#33522
#22552
#22272
#16932
#13632
#13342
#13302