Finite Group Databasebeta

← #11721800

Subgroups — #11721800

Order 864.

374 conjugacy classes of subgroups; only the 62 normal subgroups are shown.

normal subgroups only

This group has 374 conjugacy classes of subgroups — too many to draw. Only its 62 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#88748C211yesmaximal, characteristic, complemented
2D4×#272C211yesmaximal, characteristic, complemented
3SD16×H(3,𝔽3)C211yesmaximal, characteristic, complemented
4#88549C211yesmaximal, characteristic, complemented
5#88545C211yesmaximal, characteristic, complemented
6#88529C211yesmaximal, characteristic, complemented
7#88527C211yesmaximal, characteristic, complemented
8C3×#914—34nomaximal
9C22×#272—21no—
10Q8×H(3,𝔽3)C2211yescharacteristic, complemented
11D4×H(3,𝔽3)C2211yescharacteristic
12#6235—21no—
13C4⋊#272—21no—
14C4⋊#272C2211yescharacteristic
15C4⋊#272C2211yescharacteristic, complemented
16C4×#272—21no—
17C4×#272C2211yescharacteristic
18C8×H(3,𝔽3)C2211yescharacteristic, complemented
19#6180C2211yescharacteristic, complemented
20C3×#239—34no—
21C3×D3×D4—34no—
22C32×SD16D314yescomplemented
23C3×#215—34no—
24C3×#213—34no—
25C3×D24—34no—
26C3×ΓL(1,𝔽25)—34no—
27C22×H(3,𝔽3)—21no—
28C2×#272—21no—
29C2×#272—21no—
30C2×#272—42no—
31C2×#272D411yescharacteristic
32C4×H(3,𝔽3)—21no—
33C4×H(3,𝔽3)C2311yescharacteristic
34#1084D411yescharacteristic
35C3×AGL(1,ℤ/8)—91no—
36#914—34no—
37C22×C3×D3—64no—
38C32×Q8C2×D314yescomplemented
39C32×D4C2×D314yes—
40C3×#67—64no—
41C3×D12—34no—
42C3×D12—34no—
43C3×D12—64no—
44C3×C4×D3—34no—
45C3×C4×D3—64no—
46C32×C8C2×D314yescomplemented
47C3×C3⋊C8—34no—
48C2×H(3,𝔽3)—41no—
49C2×H(3,𝔽3)C2×D411yescharacteristic
50#272—21no—
51#272—41no—
52#272—41no—
53C3×#41—91no—
54C2×C3×D4—91no—
55#239—34no—
56D3×D4—34no—
57C3×SD16—34no—
58C3×SD16—91no—
59C3×SD16C3⋊D311yescharacteristic
60C3×D8—91no—
61C3×D8—91no—
62C3×M16—91no—
63#215—34no—
64#213—34no—
65D24—34no—
66ΓL(1,𝔽25)—34no—
67C22×C32—24no—
68C2×C3×D3—34no—
69C2×C3×D3—64no—
70C2×C3×D3—64no—
71C2×C3×D3—128no—
72C32×C4—24no—
73C32×C4C22×D314yes—
74C3×Dic3—34no—
75AGL(1,ℤ/8)—91noSylow
76H(3,𝔽3)AGL(1,ℤ/8)11yescharacteristic, complemented, Sylow
77C23×C3—181no—
78C22×D3—64no—
79C3×Q8—34no—
80C3×Q8C2×C3⋊D311yescharacteristic
81C3×D4—34no—
82C3×D4—91no—
83C3×D4—91no—
84C3×D4—181no—
85C3×D4—181no—
86C3×D4C2×C3⋊D311yescharacteristic
87C2×C3×C4—91no—
88C2×C3×C4—181no—
89#67—64no—
90D12—34no—
91D12—34no—
92D12—64no—
93C4×D3—34no—
94C4×D3—64no—
95C3×C8—34no—
96C3×C8—91no—
97C3×C8C2×C3⋊D311yescharacteristic
98C3⋊C8—34no—
99C2×C32—44no—
100C2×C32D3×D414yes—
101C3×D3—64no—
102C3×D3—124no—
103C3×D3—124no—
104#41—91no—
105C2×D4—91no—
106SD16—91no—
107SD16#27211yescharacteristic, complemented
108D8—91no—
109D8—91no—
110M16—91no—
111C22×C3—21no—
112C22×C3—64no—
113C22×C3—91no—
114C22×C3—181no—
115C22×C3—181no—
116C22×C3—362no—
117C2×D3—34no—
118C2×D3—64no—
119C2×D3—64no—
120C2×D3—128no—
121C3×C4—21no—
122C3×C4—34no—
123C3×C4—64no—
124C3×C4—91no—
125C3×C4C22×C3⋊D311yescharacteristic
126Dic3—34no—
127C32#91414yescomplemented
128C23—181no—
129Q8C2×#27211yescharacteristic, complemented
130D4—91no—
131D4—91no—
132D4—181no—
133D4—181no—
134D4C2×#27211yescharacteristic
135C2×C4—91no—
136C2×C4—181no—
137C8—91no—
138C8C2×#27211yescharacteristic, complemented
139C2×C3—34no—
140C2×C3—41no—
141C2×C3—124no—
142C2×C3—181no—
143C2×C3—361no—
144C2×C3—361no—
145C2×C3D4×C3⋊D311yescharacteristic
146D3—64no—
147D3—124no—
148D3—124no—
149C22—21no—
150C22—91no—
151C22—181no—
152C22—181no—
153C22—362no—
154C4—21no—
155C4—91no—
156C4C22×#27211yescharacteristic
157C3—34no—
158C3#6412311yescharacteristic
159C2—41no—
160C2—181no—
161C2—361no—
162C2—361no—
163C2D4×#27211yescharacteristic
subgroupG/H
center, socleC2×C3D4×C3⋊D3
commutator, cosocleC4×H(3,𝔽3)C23
FrattiniC3×C4C22×C3⋊D3
FittingSD16×H(3,𝔽3)C2
hypercenterC3×SD16C3⋊D3
hypocenterH(3,𝔽3)AGL(1,ℤ/8)
LCSG ▷ C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C2×C3 ◁ C3×C4 ◁ C3×SD16

Complements

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

NH ≅ G/Ntype
1. #88748C2semidirect
2. D4×#272C2semidirect
3. SD16×H(3,𝔽3)C2semidirect
4. #88549C2semidirect
5. #88545C2semidirect
6. #88529C2semidirect
7. #88527C2semidirect
10. Q8×H(3,𝔽3)C22semidirect
15. C4⋊#272C22semidirect
18. C8×H(3,𝔽3)C22semidirect
19. #6180C22semidirect
22. C32×SD16D3semidirect
38. C32×Q8C2×D3semidirect
46. C32×C8C2×D3semidirect
76. H(3,𝔽3)AGL(1,ℤ/8)semidirect
107. SD16#272semidirect
127. C32#914semidirect
129. Q8C2×#272semidirect
138. C8C2×#272semidirect

Maximal subgroups

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

subgroupindex
#887482
D4×#2722
SD16×H(3,𝔽3)2
#885492
#885452
#885292
#885272
C3×#9143