Finite Group Databasebeta

← #11721829

Subgroups — #11721829

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
1C22×C4×H(3,𝔽3)C211yesmaximal, characteristic, complemented
2C2×#6235C211yesmaximal, characteristic
3#88552C211yesmaximal, characteristic, complemented
4#88542C212yesmaximal, complemented
5#88540C212yesmaximal, complemented
6C3×#929—34nomaximal
7C23×H(3,𝔽3)C2211yescharacteristic
8C22×#272C2211yescharacteristic
9C2×C4×H(3,𝔽3)—22no—
10C2×C4×H(3,𝔽3)C2212yes—
11#6235—22no—
12C2×#1084C2211yescharacteristic
13C2×#1084C2212yes—
14C22×C32×C4D314yescomplemented
15C2×C3×#67—34no—
16C3×#217—34no—
17C3×(C2×D3)⋊C4—38no—
18C3×Dic3⋊C4—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—21no—
26#1084—22no—
27C3×#123—91no—
28#929—34no—
29C23×C32C2×D314yes—
30C22×C3×D3—34no—
31C2×C32×C4—28no—
32C2×C32×C4C2×D318yes—
33C3×#67—68no—
34C2×C3×Dic3—34no—
35C2×C3×Dic3—38no—
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—34no—
42C22×C3×C4C3⋊D311yescharacteristic
43C2×#67—34no—
44C3×C4⋊C4—92no—
45C3×C22⋊C4—91no—
46C3×C22⋊C4—92no—
47#217—34no—
48(C2×D3)⋊C4—38no—
49Dic3⋊C4—38no—
50C22×C32—28no—
51C22×C32#6718yes—
52C22×C32C22×D314yes—
53C2×C3×D3—68no—
54C2×C3×D3—64no—
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—22no—
66C2×C3×C4—38no—
67C2×C3×C4—68no—
68C2×C3×C4—92no—
69C2×C3×C4—91no—
70C2×C3×C4C2×C3⋊D312yes—
71#67—68no—
72C2×Dic3—34no—
73C2×Dic3—38no—
74C2×C32—28no—
75C2×C32#23518yes—
76C2×C32C2×#6714yes—
77C3×D3—124no—
78C2×D4—91no—
79C22×C4#27211yescharacteristic, complemented
80C4⋊C4—92no—
81C22⋊C4—91no—
82C22⋊C4—92no—
83C22×C3—22no—
84C22×C3—38no—
85C22×C3—34no—
86C22×C3—68no—
87C22×C3—182no—
88C22×C3—181no—
89C22×C3#63812yes—
90C22×C3C22×C3⋊D311yescharacteristic
91C2×D3—64no—
92C2×D3—68no—
93C3×C4—22no—
94C3×C4—68no—
95C3×C4—182no—
96C3×C4—181no—
97Dic3—64no—
98Dic3—68no—
99C32#92914yescomplemented
100C23—91no—
101C23C2×#27211yescharacteristic
102D4—182no—
103C2×C4—22no—
104C2×C4—92no—
105C2×C4—91no—
106C2×C4C2×#27212yes—
107C2×C3—22no—
108C2×C3—38no—
109C2×C3—34no—
110C2×C3—68no—
111C2×C3—361no—
112C2×C3#382812yes—
113C2×C3C2×#63811yescharacteristic
114D3—124no—
115C22—22no—
116C22—182no—
117C22—181no—
118C22#623512yes—
119C22C22×#27211yescharacteristic
120C4—22no—
121C4—182no—
122C4—181no—
123C3—34no—
124C3#6413811yescharacteristic
125C2—22no—
126C2—361no—
127C2#8874012yes—
128C2C2×#623511yescharacteristic
subgroupG/H
center, Frattini, socleC22×C3C22×C3⋊D3
commutator, cosocleC22×H(3,𝔽3)C23
FittingC22×C4×H(3,𝔽3)C2
prime radicalC2×#6235C2
hypercenterC22×C3×C4C3⋊D3
hypocenterH(3,𝔽3)#123
LCSG ▷ C22×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22×C3 ◁ C22×C3×C4

Complements

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

NH ≅ G/Ntype
1. C22×C4×H(3,𝔽3)C2semidirect
3. #88552C2semidirect
4. #88542C2semidirect
5. #88540C2semidirect
14. C22×C32×C4D3semidirect
59. H(3,𝔽3)#123semidirect
79. C22×C4#272semidirect
99. C32#929semidirect

Maximal subgroups

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

subgroupindex
C22×C4×H(3,𝔽3)2
C2×#62352
#885522
#885422
#885402
C3×#9293