Finite Group Databasebeta

← #12338

Subgroups — #12338

Order 256.

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

normal subgroups only

This group has 160 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#3403C211yesmaximal, characteristic
2#2251C211yesmaximal, characteristic, complemented
3#2208C211yesmaximal, characteristic, complemented
4#1690C211yesmaximal, characteristic, complemented
5#1360C211yesmaximal, characteristic, complemented
6#1338C211yesmaximal, characteristic, complemented
7#1333C211yesmaximal, characteristic, complemented
8#559—21no—
9C2×Q32—21no—
10C2×SD32—21no—
11C2×SD32—21no—
12#496—21no—
13#495C2211yescharacteristic
14#494C2211yescharacteristic
15#475—21no—
16#464—21no—
17#463—21no—
18#459—21no—
19#446C2211yescharacteristic
20#440—21no—
21C4×D8C2211yescharacteristic
22C4⋊C16C2211yescharacteristic
23#358—21no—
24#346C2211yescharacteristic
25#334C2211yescharacteristic
26#332—21no—
27C2×Q16—21no—
28C2×Q16—21no—
29C2×Q16D411yescharacteristic
30C2×SD16—21no—
31C2×SD16—21no—
32C2×D8—21no—
33C2×D8D411yescharacteristic
34C2×D8D411yescharacteristic
35C4⋊Q8—21no—
36#127—21no—
37#124—21no—
38#124—21no—
39#124—41no—
40#123—41no—
41#123—41no—
42C4×Q8—21no—
43C4×Q8D411yescharacteristic
44C4×D4—21no—
45C4×D4—21no—
46C4×D4—41no—
47Q32—42no—
48SD32—42no—
49SD32—42no—
50C2×C16—21no—
51C2×C16—21no—
52#107D411yescharacteristic
53#106—41no—
54C4⋊C8—21no—
55C4⋊C8—21no—
56C4⋊C8D411yescharacteristic
57#103—21no—
58#103—41no—
59#102—21no—
60#102—21no—
61#97—21no—
62C4×C8C2311yescharacteristic
63C2×Q8—21no—
64C2×Q8—41no—
65C2×Q8D811yescharacteristic
66C2×D4—21no—
67C2×D4—21no—
68C2×D4—41no—
69C2×D4—41no—
70C22×C4—41no—
71C22×C4—41no—
72Q16—41no—
73Q16—41no—
74Q16—41no—
75Q16—41no—
76Q16—41no—
77SD16—41no—
78SD16—41no—
79SD16—81no—
80D8—21no—
81D8—21no—
82D8—41no—
83D8—81no—
84C2×C8—21no—
85C2×C8—41no—
86C2×C8C2×D411yescharacteristic
87C2×C8C2×D411yescharacteristic
88C4⋊C4—21no—
89C4⋊C4—21no—
90C4⋊C4—41no—
91C4⋊C4—41no—
92C4⋊C4—41no—
93C4⋊C4D811yescharacteristic
94C22⋊C4—41no—
95C22⋊C4—41no—
96C22⋊C4—41no—
97C22⋊C4—41no—
98C22⋊C4—41no—
99C22⋊C4—41no—
100C42—21no—
101C42—41no—
102C42C2×D411yescharacteristic
103C16—22no—
104C16—41no—
105C23—41no—
106C23—41no—
107Q8—41no—
108Q8—41no—
109Q8—41no—
110Q8—81no—
111Q8—81no—
112D4—41no—
113D4—41no—
114D4—41no—
115D4—41no—
116D4—82no—
117D4—161no—
118C2×C4—21no—
119C2×C4—21no—
120C2×C4—41no—
121C2×C4—41no—
122C2×C4—81no—
123C2×C4—161no—
124C2×C4—161no—
125C2×C4#12011yescharacteristic
126C2×C4C2×D811yescharacteristic
127C2×C4AGL(1,ℤ/8)11yescharacteristic
128C8—21no—
129C8—41no—
130C8—81no—
131C8#13511yescharacteristic
132C8AGL(1,ℤ/8)11yescharacteristic
133C22—81no—
134C22—81no—
135C22—81no—
136C22—81no—
137C22—81no—
138C22—81no—
139C22#44711yescharacteristic
140C4—21no—
141C4—21no—
142C4—41no—
143C4—41no—
144C4—81no—
145C4—81no—
146C4—161no—
147C4#44911yescharacteristic
148C4#45311yescharacteristic
149C2—161no—
150C2—161no—
151C2#162111yescharacteristic
152C2#219111yescharacteristic
153C2#219411yescharacteristic
subgroupG/H
center, socleC22#447
commutator, Frattini, cosocleC4×C8C23
prime radical#3403C2
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. #2251C2semidirect
3. #2208C2semidirect
4. #1690C2semidirect
5. #1360C2semidirect
6. #1338C2semidirect
7. #1333C2semidirect

Maximal subgroups

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

subgroupindex
#34032
#22512
#22082
#16902
#13602
#13382
#13332