Finite Group Databasebeta

← D4⋊#6249 #13146446

Subgroups — D4⋊#6249 #13146446

Order 1728.

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

normal subgroups only

This group has 177 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
1D4×#1088C211yesmaximal, characteristic
2C4⋊#6249C211yesmaximal, characteristic, complemented
3C4×#6249C211yesmaximal, characteristic, complemented
4C22×#1088—21no—
5C2×#6249—21no—
6C2×#6249—21no—
7D4×#272C411yescharacteristic
8C22⋊#1088—21no—
9C4⋊#1088C411yescharacteristic
10C4×#1088C2211yescharacteristic
11C22×#272—21no—
12C2×#1088—21no—
13C2×#1088—41no—
14C2×#1088—41no—
15C2×#1088D411yescharacteristic
16C2×#1088D411yescharacteristic
17#6249—22no—
18#6249—41no—
19D4×H(3,𝔽3)C811yescharacteristic, complemented
20#6235—21no—
21C4⋊#272C811yescharacteristic, complemented
22C4×#272C2×C411yescharacteristic
23D4⋊(C3⋊C8)—91nomaximal
24C3×D3×D4—121no—
25C22×H(3,𝔽3)—21no—
26C2×#272—21no—
27C2×#272—41no—
28C2×#272—41no—
29C2×#272C22⋊C411yescharacteristic
30#1088—21no—
31#1088—41no—
32#1088D811yescharacteristic
33#1088SD1611yescharacteristic
34C4×H(3,𝔽3)C2×C811yescharacteristic
35#1084M1611yescharacteristic
36C3×C4×D4—91no—
37C4⋊(C3⋊C8)—91no—
38C4×C3⋊C8—91no—
39C22×C3×D3—122no—
40C32×D4—41no—
41C3×#67—122no—
42C3×D12—121no—
43C3×C4×D3—121no—
44D4⋊C8—271noSylow
45C2×H(3,𝔽3)—41no—
46C2×H(3,𝔽3)C22⋊C811yescharacteristic
47#272—41no—
48#272#10211yescharacteristic
49#272#10411yescharacteristic
50C2×C3×D4—91no—
51C22×C3×C4—181no—
52D3×D4—121no—
53C3×C4⋊C4—91no—
54C3×C22⋊C4—181no—
55C3×C42—91no—
56C2×C3⋊C8—181no—
57C2×C3⋊C8—181no—
58C22×C32—42no—
59C2×C3×D3—121no—
60C2×C3×D3—122no—
61C2×C3×D3—242no—
62C2×C3×D3—242no—
63C32×C4—41no—
64C3×Dic3—121no—
65C4×D4—91no—
66C4⋊C8—271no—
67C4×C8—271no—
68H(3,𝔽3)D4⋊C811yescharacteristic, complemented, Sylow
69C23×C3—181no—
70C22×D3—122no—
71C3×D4—91no—
72C3×D4—121no—
73C3×D4—181no—
74C3×D4AGL(1,𝔽9)11yescharacteristic
75C2×C3×C4—91no—
76C2×C3×C4—91no—
77C2×C3×C4—91no—
78C2×C3×C4—181no—
79C2×C3×C4—361no—
80C2×C3×C4—361no—
81#67—122no—
82D12—121no—
83C4×D3—121no—
84C3⋊C8—182no—
85C3⋊C8—361no—
86C2×C32—41no—
87C2×C32—82no—
88C3×D3—121no—
89C3×D3—121no—
90C3×D3—242no—
91C2×D4—91no—
92C22×C4—181no—
93C2×C8—541no—
94C2×C8—541no—
95C4⋊C4—91no—
96C22⋊C4—181no—
97C42—91no—
98C22×C3—21no—
99C22×C3—91no—
100C22×C3—122no—
101C22×C3—181no—
102C22×C3—361no—
103C22×C3—361no—
104C2×D3—121no—
105C2×D3—122no—
106C2×D3—242no—
107C2×D3—242no—
108C3×C4—91no—
109C3×C4—91no—
110C3×C4—91no—
111C3×C4—121no—
112C3×C4—181no—
113C3×C4—361no—
114C3×C4C2×AGL(1,𝔽9)11yescharacteristic
115Dic3—121no—
116C32—41no—
117C23—181no—
118D4—91no—
119D4—181no—
120D4#624911yescharacteristic, complemented
121C2×C4—91no—
122C2×C4—91no—
123C2×C4—91no—
124C2×C4—181no—
125C2×C4—361no—
126C2×C4—361no—
127C8—542no—
128C8—1081no—
129C2×C3—41no—
130C2×C3—91no—
131C2×C3—91no—
132C2×C3—121no—
133C2×C3—242no—
134C2×C3—361no—
135C2×C3C22⋊AGL(1,𝔽9)11yescharacteristic
136D3—121no—
137D3—121no—
138D3—242no—
139C22—21no—
140C22—91no—
141C22—181no—
142C22—361no—
143C22—361no—
144C4—91no—
145C4—91no—
146C4—91no—
147C4—181no—
148C4—361no—
149C4C2×#624911yescharacteristic
150C3—121no—
151C3D4⋊AGL(1,𝔽9)11yescharacteristic
152C2—41no—
153C2—91no—
154C2—91no—
155C2—361no—
156C2C22⋊#624911yescharacteristic
subgroupG/H
centerC2C22⋊#6249
commutatorC4×H(3,𝔽3)C2×C8
FrattiniC3×C4C2×AGL(1,𝔽9)
FittingD4×H(3,𝔽3)C8
socleC2×C3C22⋊AGL(1,𝔽9)
prime radicalD4×#272C4
cosocleC4×#1088C22
hypercenterD4#6249
hypocenterH(3,𝔽3)D4⋊C8
LCSG ▷ C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C2 ◁ C4 ◁ D4

Complements

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

NH ≅ G/Ntype
2. C4⋊#6249C2semidirect
3. C4×#6249C2semidirect
19. D4×H(3,𝔽3)C8semidirect
21. C4⋊#272C8semidirect
68. H(3,𝔽3)D4⋊C8semidirect
120. D4#6249semidirect

Maximal subgroups

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

subgroupindex
D4×#10882
C4⋊#62492
C4×#62492
D4⋊(C3⋊C8)9