Finite Group Databasebeta

← #11721742

Subgroups — #11721742

Order 864.

429 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 429 conjugacy classes of subgroups — too many to draw. Only its 88 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
1C2×#6235C211yesmaximal, characteristic
2C22×#1084C211yesmaximal, characteristic, complemented
3C22⋊C4×H(3,𝔽3)C211yesmaximal, characteristic, complemented
4#88542C212yesmaximal, complemented
5C4⋊#1084C212yesmaximal, complemented
6C3×#886—34nomaximal
7C23×H(3,𝔽3)C2211yescharacteristic
8C22×#272C2211yescharacteristic
9C2×C4×H(3,𝔽3)C2212yes—
10#6235—22no—
11C2×#1084—22no—
12C2×#1084C2212yes—
13C2×#1084C2211yescharacteristic
14C2×C3×#67—34no—
15C22×C3×Dic3—34no—
16C32×C22⋊C4D314yescomplemented
17C3×(C2×D3)⋊C4—38no—
18C3×C4⋊Dic3—38no—
19C22×H(3,𝔽3)—22no—
20C22×H(3,𝔽3)D412yes—
21C22×H(3,𝔽3)C2311yescharacteristic
22C2×#272—21no—
23C2×#272—22no—
24C4×H(3,𝔽3)—22no—
25#1084—22no—
26#1084—21no—
27C3×#123—91no—
28#886—34no—
29C23×C32C2×D314yes—
30C22×C3×D3—34no—
31C2×C32×C4C2×D318yes—
32C3×#67—68no—
33C2×C3×Dic3—34no—
34C2×C3×Dic3—38no—
35C2×C3×Dic3—68no—
36C2×H(3,𝔽3)—22no—
37C2×H(3,𝔽3)C2×D411yescharacteristic
38C2×H(3,𝔽3)#4112yes—
39#272—41no—
40C2×C3×D4—91no—
41C22×C3×C4—91no—
42C2×#67—34no—
43C22×Dic3—34no—
44C3×C4⋊C4—92no—
45C3×C22⋊C4—34no—
46C3×C22⋊C4—92no—
47C3×C22⋊C4C3⋊D311yescharacteristic
48(C2×D3)⋊C4—38no—
49C4⋊Dic3—38no—
50C22×C32—28no—
51C22×C32D1218yes—
52C22×C32C22×D314yes—
53C2×C3×D3—64no—
54C2×C3×D3—68no—
55C32×C4—28no—
56C3×Dic3—68no—
57C3×Dic3—64no—
58#123—91noSylow
59H(3,𝔽3)#12311yescharacteristic, complemented, Sylow
60C23×C3—34no—
61C23×C3—91no—
62C23×C3C2×C3⋊D311yescharacteristic
63C22×D3—34no—
64C3×D4—182no—
65C2×C3×C4—38no—
66C2×C3×C4—92no—
67C2×C3×C4—91no—
68C2×C3×C4—182no—
69C2×C3×C4C2×C3⋊D312yes—
70#67—68no—
71C2×Dic3—34no—
72C2×Dic3—38no—
73C2×Dic3—68no—
74C2×C32—28no—
75C2×C32C2×D1214yes—
76C2×C32#23718yes—
77C3×D3—124no—
78C2×D4—91no—
79C22×C4—91no—
80C4⋊C4—92no—
81C22⋊C4—92no—
82C22⋊C4#27211yescharacteristic, complemented
83C22×C3—22no—
84C22×C3—38no—
85C22×C3—34no—
86C22×C3—68no—
87C22×C3—181no—
88C22×C3—182no—
89C22×C3C4⋊(C3⋊D3)12yes—
90C22×C3C22×C3⋊D311yescharacteristic
91C2×D3—64no—
92C2×D3—68no—
93C3×C4—22no—
94C3×C4—68no—
95C3×C4—182no—
96C3×C4—181no—
97Dic3—68no—
98Dic3—64no—
99C32#88614yescomplemented
100C23—91no—
101C23C2×#27211yescharacteristic
102D4—182no—
103C2×C4—92no—
104C2×C4—91no—
105C2×C4—182no—
106C2×C4C2×#27212yes—
107C2×C3—22no—
108C2×C3—34no—
109C2×C3—38no—
110C2×C3—68no—
111C2×C3—361no—
112C2×C3C2×C4⋊(C3⋊D3)11yescharacteristic
113C2×C3#383012yes—
114D3—124no—
115C22—22no—
116C22—181no—
117C22—182no—
118C22C4⋊#27212yes—
119C22C22×#27211yescharacteristic
120C4—22no—
121C4—182no—
122C4—181no—
123C3—34no—
124C3#6409511yescharacteristic
125C2—22no—
126C2—361no—
127C2C2×C4⋊#27211yescharacteristic
128C2#8874412yes—
subgroupG/H
center, Frattini, socleC22×C3C22×C3⋊D3
commutator, cosocleC22×H(3,𝔽3)C23
FittingC22⋊C4×H(3,𝔽3)C2
prime radicalC2×#6235C2
hypercenterC3×C22⋊C4C3⋊D3
hypocenterH(3,𝔽3)#123
LCSG ▷ C22×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22×C3 ◁ C3×C22⋊C4

Complements

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

NH ≅ G/Ntype
2. C22×#1084C2semidirect
3. C22⋊C4×H(3,𝔽3)C2semidirect
4. #88542C2semidirect
5. C4⋊#1084C2semidirect
16. C32×C22⋊C4D3semidirect
59. H(3,𝔽3)#123semidirect
82. C22⋊C4#272semidirect
99. C32#886semidirect

Maximal subgroups

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

subgroupindex
C2×#62352
C22×#10842
C22⋊C4×H(3,𝔽3)2
#885422
C4⋊#10842
C3×#8863