Finite Group Databasebeta

← #13135775

Subgroups — #13135775

Order 1728.

374 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 374 conjugacy classes of subgroups — too many to draw. Only its 61 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#11721845C211yesmaximal, characteristic
2H(3,𝔽3)×#99C211yesmaximal, characteristic
3#11720579C211yesmaximal, characteristic
4C3×#4472—34nomaximal
5C2×D4×H(3,𝔽3)C2211yescharacteristic
6C22×#1084C411yescharacteristic, complemented
7C22⋊C4×H(3,𝔽3)—21no—
8#88552—21no—
9#88552C411yescharacteristic, complemented
10#88540—21no—
11#88538—21no—
12C3×#935—34no—
13C32×#99D314yes—
14C3×#833—34no—
15C23×H(3,𝔽3)C2×C411yescharacteristic
16C23×H(3,𝔽3)D411yescharacteristic
17D4×H(3,𝔽3)—41no—
18C2×C4×H(3,𝔽3)—41no—
19C2×C4×H(3,𝔽3)D411yescharacteristic
20C2×#1084—21no—
21C2×#1084—21no—
22C2×#1084—41no—
23#6180—41no—
24C3×#352—91no—
25#4472—34no—
26C2×C32×D4C2×D314yes—
27C22×C3×Dic3—34no—
28C32×C22⋊C4—24no—
29C3×#217—34no—
30C3×#217—64no—
31C3×Dic3⋊C4—64no—
32C3×#208—64no—
33C22×H(3,𝔽3)—21no—
34C22×H(3,𝔽3)—21no—
35C22×H(3,𝔽3)—41no—
36C22×H(3,𝔽3)—41no—
37C22×H(3,𝔽3)C22⋊C411yescharacteristic
38C4×H(3,𝔽3)—21no—
39C4×H(3,𝔽3)—81no—
40#1084—41no—
41#1084—41no—
42C3×#123—91no—
43#935—34no—
44C3×#100—91no—
45C3×#99—34no—
46C3×#99C3⋊D311yescharacteristic
47#833—34no—
48C23×C32C4×D314yes—
49C23×C32#6714yes—
50C32×D4—44no—
51C2×C32×C4—44no—
52C2×C32×C4D1214yes—
53C2×C3×Dic3—64no—
54C2×C3×Dic3—64no—
55C2×C3×Dic3—124no—
56C3×C3⋊C8—124no—
57#352—91noSylow
58C2×H(3,𝔽3)—21no—
59C2×H(3,𝔽3)—41no—
60C2×H(3,𝔽3)—41no—
61C2×H(3,𝔽3)#9911yescharacteristic
62C2×C3×D4—34no—
63C2×C3×D4C2×C3⋊D311yescharacteristic
64C22×C3×C4—91no—
65C22×Dic3—34no—
66C3×M16—181no—
67C3×C4⋊C4—181no—
68C3×C22⋊C4—21no—
69C3×C22⋊C4—64no—
70C3×C22⋊C4—91no—
71C3×C22⋊C4—181no—
72#217—34no—
73#217—64no—
74Dic3⋊C4—64no—
75#208—64no—
76C22×C32—24no—
77C22×C32—24no—
78C22×C32—44no—
79C22×C32—44no—
80C22×C32(C2×D3)⋊C414yes—
81C32×C4—24no—
82C32×C4—84no—
83C3×Dic3—124no—
84C3×Dic3—124no—
85#123—91no—
86#100—91no—
87#99#27211yescharacteristic
88H(3,𝔽3)#35211yescharacteristic, complemented, Sylow
89C23×C3—34no—
90C23×C3—34no—
91C23×C3C4×C3⋊D311yescharacteristic
92C23×C3#63811yescharacteristic
93C3×D4—41no—
94C3×D4—124no—
95C2×C3×C4—34no—
96C2×C3×C4—41no—
97C2×C3×C4—124no—
98C2×C3×C4—181no—
99C2×C3×C4—181no—
100C2×C3×C4—361no—
101C2×C3×C4C4⋊(C3⋊D3)11yescharacteristic
102C2×Dic3—64no—
103C2×Dic3—64no—
104C2×Dic3—124no—
105C3×C8—361no—
106C3⋊C8—124no—
107C2×C32—24no—
108C2×C32—44no—
109C2×C32—44no—
110C2×C32#80614yes—
111C2×D4C2×#27211yescharacteristic
112C22×C4—91no—
113M16—181no—
114C4⋊C4—181no—
115C22⋊C4—21no—
116C22⋊C4—91no—
117C22⋊C4—181no—
118C22×C3—21no—
119C22×C3—21no—
120C22×C3—34no—
121C22×C3—41no—
122C22×C3—41no—
123C22×C3—64no—
124C22×C3—64no—
125C22×C3—124no—
126C22×C3—124no—
127C22×C3#375211yescharacteristic
128C3×C4—21no—
129C3×C4—64no—
130C3×C4—81no—
131C3×C4—244no—
132C3×C4—361no—
133C3×C4—361no—
134Dic3—124no—
135Dic3—124no—
136C32#447214yescomplemented
137C23C4×#27211yescharacteristic
138C23#623511yescharacteristic
139D4—41no—
140C2×C4—41no—
141C2×C4—181no—
142C2×C4—181no—
143C2×C4—361no—
144C2×C4C4⋊#27211yescharacteristic
145C8—361no—
146C2×C3—21no—
147C2×C3—34no—
148C2×C3—41no—
149C2×C3—41no—
150C2×C3—64no—
151C2×C3—124no—
152C2×C3—124no—
153C2×C3#6363311yescharacteristic
154C22—21no—
155C22—21no—
156C22—41no—
157C22—41no—
158C22#8854211yescharacteristic
159C4—21no—
160C4—81no—
161C4—361no—
162C4—361no—
163C3—34no—
164C3#1058933811yescharacteristic
165C2—21no—
166C2—41no—
167C2—41no—
168C2#1172052411yescharacteristic
subgroupG/H
center, socleC2×C3#63633
commutatorC23×H(3,𝔽3)C2×C4
FrattiniC2×C3×D4C2×C3⋊D3
FittingH(3,𝔽3)×#99C2
prime radical, cosocleC2×D4×H(3,𝔽3)C22
hypercenterC3×#99C3⋊D3
hypocenterH(3,𝔽3)#352
LCSG ▷ C23×H(3,𝔽3) ▷ C22×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C2×C3 ◁ C22×C3 ◁ C2×C3×D4 ◁ C3×#99

Complements

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

NH ≅ G/Ntype
6. C22×#1084C4semidirect
9. #88552C4semidirect
88. H(3,𝔽3)#352semidirect
136. C32#4472semidirect

Maximal subgroups

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

subgroupindex
#117218452
H(3,𝔽3)×#992
#117205792
C3×#44723