Finite Group Databasebeta

← #12319

Subgroups — #12319

Order 256.

195 conjugacy classes of subgroups; only the 43 normal subgroups are shown.

normal subgroups only

This group has 195 conjugacy classes of subgroups — too many to draw. Only its 43 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#3283C211yesmaximal, characteristic, complemented
2#2248C211yesmaximal, characteristic, complemented
3#2206C211yesmaximal, characteristic, complemented
4#1667C211yesmaximal, characteristic, complemented
5#1362C211yesmaximal, characteristic, complemented
6#1331C211yesmaximal, characteristic, complemented
7D4⋊C16C211yesmaximal, characteristic, complemented
8C2×#135—21no—
9#547—21no—
10C2×SD32—22no—
11C2×D16—21no—
12#494C2211yescharacteristic
13#493C2211yescharacteristic
14#467—21no—
15#461—21no—
16#459—21no—
17#452—22no—
18C4×D8C2211yescharacteristic
19C8×D4C2211yescharacteristic
20C4⋊C16C2211yescharacteristic
21#357—21no—
22C4×C16C2211yescharacteristic
23#335C2211yescharacteristic
24#331—21no—
25C2×#41—22no—
26#135—42no—
27#135—42no—
28C2×Q16—21no—
29C2×Q16D412yes—
30C2×SD16—22no—
31C2×D8—21no—
32C2×D8—21no—
33C2×D8—41no—
34C2×D8D411yescharacteristic
35C22×C8—21no—
36C4⋊Q8—21no—
37#127—21no—
38#123—42no—
39C22⋊Q8—22no—
40#121—41no—
41C4×D4—21no—
42C4×D4D411yescharacteristic
43C2×C4⋊C4—41no—
44SD32—44no—
45D16—42no—
46C2×C16—21no—
47C2×C16—21no—
48#107D411yescharacteristic
49C4⋊C8—21no—
50C4⋊C8D411yescharacteristic
51#103—22no—
52#102—21no—
53#102—41no—
54C22⋊C8—21no—
55C4×C8C2311yescharacteristic
56#41—44no—
57#41—82no—
58C2×Q8—22no—
59C2×D4—21no—
60C2×D4—42no—
61C2×D4—41no—
62C2×D4—41no—
63C2×D4—41no—
64C2×D4D811yescharacteristic
65C22×C4—21no—
66C22×C4—42no—
67C22×C4—41no—
68Q16—42no—
69Q16—41no—
70Q16—41no—
71SD16—44no—
72D8—21no—
73D8—42no—
74D8—41no—
75D8—41no—
76D8—82no—
77D8D812yescomplemented
78C2×C8—21no—
79C2×C8—41no—
80C2×C8—41no—
81C2×C8—41no—
82C2×C8C2×D411yescharacteristic
83C2×C8C2×D411yescharacteristic
84C4⋊C4—21no—
85C4⋊C4—42no—
86C4⋊C4—41no—
87C4⋊C4—41no—
88C4⋊C4D811yescharacteristic
89C22⋊C4—21no—
90C22⋊C4—42no—
91C22⋊C4—41no—
92C42C2×D411yescharacteristic
93C16—22no—
94C16—41no—
95C23—21no—
96C23—41no—
97C23—81no—
98Q8—42no—
99Q8—42no—
100D4—22no—
101D4—21no—
102D4—21no—
103D4—41no—
104D4—84no—
105D4—81no—
106D4—81no—
107D4—81no—
108D4—81no—
109D4—161no—
110C2×C4—21no—
111C2×C4—41no—
112C2×C4—41no—
113C2×C4—41no—
114C2×C4—42no—
115C2×C4—84no—
116C2×C4—81no—
117C2×C4—81no—
118C2×C4#12011yescharacteristic
119C2×C4C2×D811yescharacteristic
120C2×C4AGL(1,ℤ/8)11yescharacteristic
121C8—21no—
122C8—41no—
123C8—81no—
124C8C2×D811yescharacteristic
125C8AGL(1,ℤ/8)11yescharacteristic
126C22—41no—
127C22—42no—
128C22—41no—
129C22—41no—
130C22—81no—
131C22—81no—
132C22—161no—
133C22—161no—
134C22—161no—
135C22#44711yescharacteristic
136C4—21no—
137C4—41no—
138C4—82no—
139C4—81no—
140C4#44711yescharacteristic
141C4#45311yescharacteristic
142C4#50811yescharacteristic
143C4#50911yescharacteristic
144C2—81no—
145C2—82no—
146C2—321no—
147C2#161911yescharacteristic
148C2#218611yescharacteristic
149C2#219311yescharacteristic
subgroupG/H
center, socleC22#447
commutator, Frattini, cosocleC4×C8C23
LCSG ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C42 ◁ C8×D4 ◁ G
LECSG ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #3283C2semidirect
2. #2248C2semidirect
3. #2206C2semidirect
4. #1667C2semidirect
5. #1362C2semidirect
6. #1331C2semidirect
7. D4⋊C16C2semidirect
77. D8D8semidirect

Maximal subgroups

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

subgroupindex
#32832
#22482
#22062
#16672
#13622
#13312
D4⋊C162