Finite Group Databasebeta

← #12400

Subgroups — #12400

Order 256.

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

normal subgroups only

This group has 143 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#3355C211yesmaximal, characteristic
2#2255C211yesmaximal, characteristic, complemented
3#2224C211yesmaximal, characteristic, complemented
4#1695C211yesmaximal, characteristic, complemented
5#1363C211yesmaximal, characteristic, complemented
6#1333C211yesmaximal, characteristic, complemented
7#1330C211yesmaximal, characteristic, complemented
8#553—21no—
9#500C2211yescharacteristic
10#499C2211yescharacteristic
11#482—21no—
12#480—21no—
13#478—21no—
14#476—21no—
15#469—21no—
16C4×D8C2211yescharacteristic
17#436C2211yescharacteristic
18#428—21no—
19#367—21no—
20#366—21no—
21#366—21no—
22C4⋊C16C2211yescharacteristic
23#357—21no—
24#346C2211yescharacteristic
25#334C2211yescharacteristic
26#333—21no—
27C2×D8D411yescharacteristic
28C2×M16—21no—
29C4⋊Q8—21no—
30#126—41no—
31#125—21no—
32#123—41no—
33#121—21no—
34#121—21no—
35#121—41no—
36C4×D4—21no—
37C4×D4—41no—
38C4×D4D411yescharacteristic
39#117—21no—
40C2×C4⋊C4—21no—
41C2×C16—21no—
42C2×C16—21no—
43#107—21no—
44#107—21no—
45#107—21no—
46#107D411yescharacteristic
47#107D411yescharacteristic
48#107D411yescharacteristic
49#106—21no—
50#106—21no—
51#106—41no—
52C4⋊C8—21no—
53C4⋊C8D411yescharacteristic
54#102—21no—
55#102—21no—
56#102—21no—
57#102—41no—
58C22⋊C8—21no—
59C4×C8C2311yescharacteristic
60C2×Q8—41no—
61C2×D4—21no—
62C2×D4—41no—
63C2×D4—41no—
64C2×D4D811yescharacteristic
65C22×C4—21no—
66C22×C4—41no—
67C22×C4—41no—
68D8—21no—
69D8—21no—
70M16—41no—
71M16—41no—
72C2×C8—21no—
73C2×C8—41no—
74C2×C8C2×D411yescharacteristic
75C2×C8C2×D411yescharacteristic
76C4⋊C4—21no—
77C4⋊C4—21no—
78C4⋊C4—21no—
79C4⋊C4—41no—
80C4⋊C4—41no—
81C4⋊C4—41no—
82C4⋊C4—41no—
83C4⋊C4—41no—
84C4⋊C4D811yescharacteristic
85C22⋊C4—21no—
86C22⋊C4—41no—
87C22⋊C4—41no—
88C22⋊C4—41no—
89C22⋊C4—41no—
90C42—41no—
91C42C2×D411yescharacteristic
92C16—22no—
93C16—41no—
94C23—21no—
95C23—41no—
96Q8—161no—
97D4—41no—
98D4—41no—
99D4—41no—
100D4—82no—
101D4—161no—
102C2×C4—21no—
103C2×C4—41no—
104C2×C4—41no—
105C2×C4—41no—
106C2×C4—41no—
107C2×C4—41no—
108C2×C4—81no—
109C2×C4—161no—
110C2×C4—161no—
111C2×C4#12011yescharacteristic
112C2×C4C2×D811yescharacteristic
113C2×C4AGL(1,ℤ/8)11yescharacteristic
114C8—21no—
115C8—41no—
116C8—81no—
117C8#13511yescharacteristic
118C8#13711yescharacteristic
119C22—41no—
120C22—41no—
121C22—41no—
122C22—81no—
123C22—81no—
124C22—81no—
125C22#44711yescharacteristic
126C4—21no—
127C4—21no—
128C4—41no—
129C4—81no—
130C4—81no—
131C4—81no—
132C4—161no—
133C4#45211yescharacteristic
134C4#45611yescharacteristic
135C2—81no—
136C2—161no—
137C2#164711yescharacteristic
138C2#219111yescharacteristic
139C2#219211yescharacteristic
subgroupG/H
center, socleC22#447
commutator, Frattini, cosocleC4×C8C23
prime radical#3355C2
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. #2224C2semidirect
4. #1695C2semidirect
5. #1363C2semidirect
6. #1333C2semidirect
7. #1330C2semidirect

Maximal subgroups

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

subgroupindex
#33552
#22552
#22242
#16952
#13632
#13332
#13302