Finite Group Databasebeta

← Q8⋊#6249 #13146448

Subgroups — Q8⋊#6249 #13146448

Order 1728.

143 conjugacy classes of subgroups; only the 29 normal subgroups are shown.

normal subgroups only

This group has 143 conjugacy classes of subgroups — too many to draw. Only its 29 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
1Q8×#1088C211yesmaximal, characteristic
2C4⋊#6249C211yesmaximal, characteristic
3C4×#6249C211yesmaximal, characteristic
4C2×#6249—21no—
5C2×#6249—21no—
6Q8×#272C411yescharacteristic
7C4⋊#1088—21no—
8C4⋊#1088C411yescharacteristic
9C4×#1088—21no—
10C4×#1088C2211yescharacteristic
11C2×#1088—21no—
12C2×#1088D411yescharacteristic
13C2×#1088D411yescharacteristic
14#6249—22no—
15#6249—41no—
16Q8×H(3,𝔽3)C811yescharacteristic, complemented
17C4×#272—21no—
18C4×#272C2×C411yescharacteristic
19#6229—21no—
20#6229C811yescharacteristic, complemented
21#4535—91nomaximal
22C3×D3×Q8—121no—
23C2×#272C22⋊C411yescharacteristic
24#1088—21no—
25#1088—41no—
26#1088SD1611yescharacteristic
27#1088Q1611yescharacteristic
28C4×H(3,𝔽3)—21no—
29C4×H(3,𝔽3)C2×C811yescharacteristic
30#1084—21no—
31#1084M1611yescharacteristic
32C3×C4×Q8—91no—
33C4⋊(C3⋊C8)—91no—
34C4×C3⋊C8—91no—
35C32×Q8—41no—
36C3×C4×D3—121no—
37C3×C4×D3—122no—
38C3×Dic6—121no—
39C3×Dic6—122no—
40Q8⋊C8—271noSylow
41C2×H(3,𝔽3)C22⋊C811yescharacteristic
42#272#10311yescharacteristic
43#272#10411yescharacteristic
44C2×C3×Q8—91no—
45D3×Q8—121no—
46C3×C4⋊C4—91no—
47C3×C4⋊C4—181no—
48C3×C42—91no—
49C3×C42—181no—
50C2×C3⋊C8—181no—
51C2×C3⋊C8—181no—
52C2×C3×D3—121no—
53C32×C4—42no—
54C32×C4—41no—
55C3×Dic3—121no—
56C3×Dic3—122no—
57C4×Q8—91no—
58C4⋊C8—271no—
59C4×C8—271no—
60H(3,𝔽3)Q8⋊C811yescharacteristic, complemented, Sylow
61C3×Q8—91no—
62C3×Q8—121no—
63C3×Q8—181no—
64C3×Q8AGL(1,𝔽9)11yescharacteristic
65C2×C3×C4—91no—
66C2×C3×C4—91no—
67C2×C3×C4—91no—
68C2×C3×C4—181no—
69C2×C3×C4—181no—
70C4×D3—121no—
71C4×D3—122no—
72Dic6—121no—
73Dic6—122no—
74C3⋊C8—182no—
75C3⋊C8—361no—
76C2×C32—41no—
77C3×D3—121no—
78C3×D3—121no—
79C2×Q8—91no—
80C2×C8—541no—
81C2×C8—541no—
82C4⋊C4—91no—
83C4⋊C4—181no—
84C42—91no—
85C42—181no—
86C22×C3—91no—
87C2×D3—121no—
88C3×C4—21no—
89C3×C4—91no—
90C3×C4—91no—
91C3×C4—91no—
92C3×C4—122no—
93C3×C4—121no—
94C3×C4—181no—
95C3×C4—181no—
96C3×C4—361no—
97C3×C4C2×AGL(1,𝔽9)11yescharacteristic
98Dic3—121no—
99Dic3—122no—
100C32—41no—
101Q8—91no—
102Q8—181no—
103Q8#624911yescharacteristic, complemented
104C2×C4—91no—
105C2×C4—91no—
106C2×C4—91no—
107C2×C4—181no—
108C2×C4—181no—
109C8—542no—
110C8—1081no—
111C2×C3—91no—
112C2×C3—91no—
113C2×C3—121no—
114C2×C3C22⋊AGL(1,𝔽9)11yescharacteristic
115D3—121no—
116D3—121no—
117C22—91no—
118C4—21no—
119C4—91no—
120C4—91no—
121C4—91no—
122C4—181no—
123C4—181no—
124C4—361no—
125C4C2×#624911yescharacteristic
126C3—121no—
127C3Q8⋊AGL(1,𝔽9)11yescharacteristic
128C2—91no—
129C2—91no—
130C2C22⋊#624911yescharacteristic
subgroupG/H
centerC2C22⋊#6249
commutatorC4×H(3,𝔽3)C2×C8
FrattiniC3×C4C2×AGL(1,𝔽9)
FittingQ8×H(3,𝔽3)C8
socleC2×C3C22⋊AGL(1,𝔽9)
prime radicalC2×#272C22⋊C4
cosocleC4×#1088C22
hypercenterQ8#6249
hypocenterH(3,𝔽3)Q8⋊C8
LCSG ▷ C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C2 ◁ C4 ◁ Q8

Complements

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

NH ≅ G/Ntype
16. Q8×H(3,𝔽3)C8semidirect
20. #6229C8semidirect
60. H(3,𝔽3)Q8⋊C8semidirect
103. Q8#6249semidirect

Maximal subgroups

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

subgroupindex
Q8×#10882
C4⋊#62492
C4×#62492
#45359