Finite Group Databasebeta

← #12344

Subgroups — #12344

Order 256.

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

normal subgroups only

This group has 165 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#3334C211yesmaximal, characteristic
2#2241C211yesmaximal, characteristic, complemented
3#2208C211yesmaximal, characteristic, complemented
4#1669C211yesmaximal, characteristic, complemented
5#1364C211yesmaximal, characteristic, complemented
6#1331C211yesmaximal, characteristic, complemented
7D4⋊C16C211yesmaximal, characteristic, complemented
8#547—21no—
9C2×SD32—21no—
10#500C2211yescharacteristic
11#494C2211yescharacteristic
12#480—22no—
13#474—21no—
14#474—21no—
15#466—21no—
16C4×D8C2211yescharacteristic
17C8×D4C2211yescharacteristic
18C2×#107—21no—
19C4⋊C16C2211yescharacteristic
20#358—21no—
21#358—22no—
22C4×C16C2211yescharacteristic
23#335C2211yescharacteristic
24#332—21no—
25C2×#41—41no—
26C2×Q16—21no—
27C2×Q16—21no—
28C2×D8D411yescharacteristic
29C22×C8—21no—
30C4⋊Q8—21no—
31C4⋊Q8—21no—
32#123—42no—
33C22⋊Q8—41no—
34#121—22no—
35C4×D4—21no—
36C4×D4D411yescharacteristic
37C2×C4⋊C4—22no—
38SD32—42no—
39C2×C16—21no—
40C2×C16—21no—
41#107—22no—
42#107—21no—
43#107—41no—
44#107D411yescharacteristic
45#107D412yes—
46C4⋊C8—21no—
47C4⋊C8D411yescharacteristic
48#102—22no—
49#102—21no—
50#102—41no—
51C22⋊C8—21no—
52C4×C8C2311yescharacteristic
53#41—82no—
54#41—82no—
55C2×Q8—41no—
56C2×Q8—41no—
57C2×D4—21no—
58C2×D4—42no—
59C2×D4D811yescharacteristic
60C22×C4—21no—
61C22×C4—42no—
62C22×C4—41no—
63Q16—42no—
64Q16—41no—
65Q16—41no—
66D8—21no—
67D8—21no—
68C2×C8—21no—
69C2×C8—41no—
70C2×C8—41no—
71C2×C8—41no—
72C2×C8C2×D411yescharacteristic
73C2×C8C2×D411yescharacteristic
74C4⋊C4—21no—
75C4⋊C4—22no—
76C4⋊C4—42no—
77C4⋊C4—41no—
78C4⋊C4—41no—
79C4⋊C4—41no—
80C4⋊C4D811yescharacteristic
81C22⋊C4—21no—
82C22⋊C4—42no—
83C22⋊C4—41no—
84C42C2×D411yescharacteristic
85C16—41no—
86C16—41no—
87C23—21no—
88C23—41no—
89Q8—81no—
90Q8—81no—
91Q8—81no—
92Q8—81no—
93D4—21no—
94D4—41no—
95D4—41no—
96D4—84no—
97D4SD3212yescomplemented
98C2×C4—21no—
99C2×C4—41no—
100C2×C4—41no—
101C2×C4—41no—
102C2×C4—42no—
103C2×C4—84no—
104C2×C4—81no—
105C2×C4—81no—
106C2×C4—81no—
107C2×C4#12011yescharacteristic
108C2×C4C2×D811yescharacteristic
109C2×C4AGL(1,ℤ/8)11yescharacteristic
110C8—21no—
111C8—41no—
112C8—81no—
113C8#13511yescharacteristic
114C8AGL(1,ℤ/8)11yescharacteristic
115C22—22no—
116C22—41no—
117C22—41no—
118C22—81no—
119C22—81no—
120C22—81no—
121C22#44711yescharacteristic
122C4—21no—
123C4—41no—
124C4—81no—
125C4—82no—
126C4—161no—
127C4#44911yescharacteristic
128C4#45411yescharacteristic
129C4C2×SD3211yescharacteristic
130C4#50911yescharacteristic
131C2—42no—
132C2—161no—
133C2#162511yescharacteristic
134C2#218711yescharacteristic
135C2#219511yescharacteristic
subgroupG/H
center, socleC22#447
commutator, Frattini, cosocleC4×C8C23
prime radical#3334C2
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
2. #2241C2semidirect
3. #2208C2semidirect
4. #1669C2semidirect
5. #1364C2semidirect
6. #1331C2semidirect
7. D4⋊C16C2semidirect
97. D4SD32semidirect

Maximal subgroups

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

subgroupindex
#33342
#22412
#22082
#16692
#13642
#13312
D4⋊C162