Finite Group Databasebeta

← #12412

Subgroups — #12412

Order 256.

140 conjugacy classes of subgroups; only the 41 normal subgroups are shown.

normal subgroups only

This group has 140 conjugacy classes of subgroups — too many to draw. Only its 41 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#3389C211yesmaximal, characteristic
2#2255C211yesmaximal, characteristic, complemented
3#2224C211yesmaximal, characteristic
4#1696C211yesmaximal, characteristic, complemented
5#1363C211yesmaximal, characteristic, complemented
6#1338C211yesmaximal, characteristic, complemented
7#1333C211yesmaximal, characteristic
8#554—21no—
9#501—21no—
10#500C2211yescharacteristic
11#499C2211yescharacteristic
12#478—21no—
13#478—21no—
14#477—21no—
15#474—21no—
16#446C2211yescharacteristic
17#442—21no—
18C4×D8C2211yescharacteristic
19#367—21no—
20#366—21no—
21#366—21no—
22C4⋊C16C2211yescharacteristic
23#357—21no—
24#346C2211yescharacteristic
25#334C2211yescharacteristic
26#333—21no—
27C2×D8D411yescharacteristic
28C4⋊Q8—21no—
29C4⋊Q8—21no—
30#125—21no—
31#125—21no—
32C22⋊Q8—41no—
33C22⋊Q8—41no—
34C22⋊Q8—41no—
35C4×Q8D411yescharacteristic
36C4×D4—21no—
37C4×D4—21no—
38C4×D4—21no—
39C2×C4⋊C4—41no—
40C2×C16—21no—
41C2×C16—21no—
42#107—21no—
43#107—21no—
44#107—21no—
45#107—41no—
46#107D411yescharacteristic
47#107D411yescharacteristic
48#107D411yescharacteristic
49#106—21no—
50#106—21no—
51C4⋊C8—21no—
52C4⋊C8—21no—
53C4⋊C8D411yescharacteristic
54#103—41no—
55#102—21no—
56#102—21no—
57#102—21no—
58#97—21no—
59C4×C8C2311yescharacteristic
60C2×Q8—41no—
61C2×Q8—41no—
62C2×Q8D811yescharacteristic
63C2×D4—21no—
64C22×C4—41no—
65C22×C4—41no—
66C22×C4—41no—
67D8—21no—
68D8Q1612yes—
69C2×C8—21no—
70C2×C8—41no—
71C2×C8C2×D411yescharacteristic
72C2×C8C2×D411yescharacteristic
73C4⋊C4—21no—
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⋊C4D811yescharacteristic
85C22⋊C4—41no—
86C22⋊C4—41no—
87C22⋊C4—41no—
88C42—21no—
89C42C2×D411yescharacteristic
90C16—22no—
91C16—41no—
92C23—41no—
93Q8—41no—
94Q8—161no—
95Q8—161no—
96D4—22no—
97D4—41no—
98C2×C4—21no—
99C2×C4—21no—
100C2×C4—41no—
101C2×C4—41no—
102C2×C4—41no—
103C2×C4—82no—
104C2×C4—81no—
105C2×C4—161no—
106C2×C4—161no—
107C2×C4#12011yescharacteristic
108C2×C4C2×D811yescharacteristic
109C2×C4AGL(1,ℤ/8)11yescharacteristic
110C8—21no—
111C8—41no—
112C8—81no—
113C8C2×Q1611yescharacteristic
114C8#13711yescharacteristic
115C22—42no—
116C22—81no—
117C22—81no—
118C22#44711yescharacteristic
119C4—21no—
120C4—21no—
121C4—41no—
122C4—41no—
123C4—81no—
124C4—81no—
125C4—81no—
126C4—161no—
127C4#45111yescharacteristic
128C4#45611yescharacteristic
129C2—82no—
130C2#164911yescharacteristic
131C2#219111yescharacteristic
132C2#219211yescharacteristic
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
2. #2255C2semidirect
4. #1696C2semidirect
5. #1363C2semidirect
6. #1338C2semidirect

Maximal subgroups

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

subgroupindex
#33892
#22552
#22242
#16962
#13632
#13382
#13332