Finite Group Databasebeta

← #11721845

Subgroups — #11721845

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×D4×H(3,𝔽3)C211yesmaximal, characteristic
2C22×#1084C211yesmaximal, characteristic, complemented
3#88552C212yesmaximal, complemented
4#88552C211yesmaximal, characteristic, complemented
5#88540C212yesmaximal, complemented
6C3×#935—34nomaximal
7C23×H(3,𝔽3)C2211yescharacteristic
8C23×H(3,𝔽3)C2211yescharacteristic
9D4×H(3,𝔽3)—22no—
10C2×C4×H(3,𝔽3)C2211yescharacteristic
11C2×#1084—22no—
12C2×#1084C2212yes—
13C2×#1084C2212yes—
14C2×C32×D4D314yes—
15C22×C3×Dic3—34no—
16C3×#217—34no—
17C3×#217—38no—
18C3×Dic3⋊C4—38no—
19C22×H(3,𝔽3)—22no—
20C22×H(3,𝔽3)—21no—
21C22×H(3,𝔽3)—22no—
22C22×H(3,𝔽3)D412yes—
23C22×H(3,𝔽3)C2311yescharacteristic
24C4×H(3,𝔽3)—21no—
25#1084—22no—
26#1084—22no—
27C3×#123—91no—
28#935—34no—
29C23×C32C2×D314yes—
30C23×C32C2×D314yes—
31C32×D4—28no—
32C2×C32×C4C2×D314yes—
33C2×C3×Dic3—38no—
34C2×C3×Dic3—38no—
35C2×C3×Dic3—68no—
36C2×H(3,𝔽3)—22no—
37C2×H(3,𝔽3)—41no—
38C2×H(3,𝔽3)C2×D411yescharacteristic
39C2×H(3,𝔽3)#4112yes—
40C2×C3×D4—34no—
41C2×C3×D4C3⋊D311yescharacteristic
42C22×C3×C4—91no—
43C22×Dic3—34no—
44C3×C4⋊C4—92no—
45C3×C22⋊C4—91no—
46C3×C22⋊C4—92no—
47#217—34no—
48#217—38no—
49Dic3⋊C4—38no—
50C22×C32—28no—
51C22×C32—28no—
52C22×C32—24no—
53C22×C32#6718yes—
54C22×C32C22×D314yes—
55C32×C4—24no—
56C3×Dic3—68no—
57C3×Dic3—68no—
58#123—91noSylow
59H(3,𝔽3)#12311yescharacteristic, complemented, Sylow
60C23×C3—34no—
61C23×C3—34no—
62C23×C3C2×C3⋊D311yescharacteristic
63C23×C3C2×C3⋊D311yescharacteristic
64C3×D4—22no—
65C3×D4—68no—
66C2×C3×C4—34no—
67C2×C3×C4—92no—
68C2×C3×C4—92no—
69C2×C3×C4—182no—
70C2×C3×C4C2×C3⋊D311yescharacteristic
71C2×Dic3—38no—
72C2×Dic3—38no—
73C2×Dic3—68no—
74C2×C32—28no—
75C2×C32—44no—
76C2×C32#23718yes—
77C2×C32C2×#6714yes—
78C2×D4#27211yescharacteristic
79C22×C4—91no—
80C4⋊C4—92no—
81C22⋊C4—91no—
82C22⋊C4—92no—
83C22×C3—22no—
84C22×C3—22no—
85C22×C3—21no—
86C22×C3—38no—
87C22×C3—34no—
88C22×C3—68no—
89C22×C3—64no—
90C22×C3—68no—
91C22×C3#63812yes—
92C22×C3C22×C3⋊D311yescharacteristic
93C3×C4—21no—
94C3×C4—64no—
95C3×C4—182no—
96C3×C4—182no—
97Dic3—68no—
98Dic3—68no—
99C32#93514yescomplemented
100C23C2×#27211yescharacteristic
101C23C2×#27211yescharacteristic
102D4—22no—
103C2×C4—92no—
104C2×C4—92no—
105C2×C4—182no—
106C2×C4C2×#27211yescharacteristic
107C2×C3—22no—
108C2×C3—38no—
109C2×C3—34no—
110C2×C3—41no—
111C2×C3—68no—
112C2×C3—124no—
113C2×C3#383012yes—
114C2×C3C2×#63811yescharacteristic
115C22—22no—
116C22—22no—
117C22—21no—
118C22#623512yes—
119C22C22×#27211yescharacteristic
120C4—21no—
121C4—182no—
122C4—182no—
123C3—34no—
124C3#6414411yescharacteristic
125C2—22no—
126C2—41no—
127C2#8874412yes—
128C2C2×#623511yescharacteristic
subgroupG/H
center, Frattini, socleC22×C3C22×C3⋊D3
commutator, cosocleC22×H(3,𝔽3)C23
Fitting, prime radicalC2×D4×H(3,𝔽3)C2
hypercenterC2×C3×D4C3⋊D3
hypocenterH(3,𝔽3)#123
LCSG ▷ C22×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22×C3 ◁ C2×C3×D4

Complements

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

NH ≅ G/Ntype
2. C22×#1084C2semidirect
3. #88552C2semidirect
4. #88552C2semidirect
5. #88540C2semidirect
59. H(3,𝔽3)#123semidirect
99. C32#935semidirect

Maximal subgroups

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

subgroupindex
C2×D4×H(3,𝔽3)2
C22×#10842
#885522
#885402
C3×#9353