Finite Group Databasebeta

← #13135776

Subgroups — #13135776

Order 1728.

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

normal subgroups only

This group has 385 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#11721846C211yesmaximal, characteristic
2H(3,𝔽3)×#99C211yesmaximal, characteristic
3#11720580C211yesmaximal, characteristic
4C3×#4473—34nomaximal
5C2×D4×H(3,𝔽3)C2211yescharacteristic
6C2×#6229C411yescharacteristic, complemented
7C22⋊C4×H(3,𝔽3)—21no—
8#88552—21no—
9#88552—21no—
10#88552—21no—
11C4×#1084C411yescharacteristic, complemented
12C3×#936—34no—
13C32×#99D314yes—
14C3×#834—34no—
15C23×H(3,𝔽3)D411yescharacteristic
16C23×H(3,𝔽3)D411yescharacteristic
17D4×H(3,𝔽3)—41no—
18C2×C4×H(3,𝔽3)—41no—
19C2×C4×H(3,𝔽3)C2×C411yescharacteristic
20C2×#1084—21no—
21C2×#1084—21no—
22C2×#1084—41no—
23#6229—41no—
24C3×#354—91no—
25#4473—34no—
26C2×C32×D4C2×D314yes—
27C2×C3×Dic6—34no—
28C32×C22⋊C4—24no—
29C3×#217—64no—
30C3×#217—64no—
31C3×#217—64no—
32C3×C4×Dic3—34no—
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—
42#1084—81no—
43C3×#124—91no—
44#936—34no—
45C3×#99—34no—
46C3×#99—91no—
47C3×#99C3⋊D311yescharacteristic
48#834—34no—
49C23×C32D1214yes—
50C23×C32#6714yes—
51C32×D4—44no—
52C2×C32×C4—44no—
53C2×C32×C4C4×D314yes—
54C2×C3×Dic3—64no—
55C2×C3×Dic3—64no—
56C2×C3×Dic3—124no—
57C3×Dic6—124no—
58#354—91noSylow
59C2×H(3,𝔽3)—21no—
60C2×H(3,𝔽3)—41no—
61C2×H(3,𝔽3)—41no—
62C2×H(3,𝔽3)#9911yescharacteristic
63C2×C3×Q8—91no—
64C2×C3×D4—34no—
65C2×C3×D4C2×C3⋊D311yescharacteristic
66C2×Dic6—34no—
67C3×C22⋊C4—21no—
68C3×C22⋊C4—64no—
69C3×C22⋊C4—181no—
70C3×C22⋊C4—181no—
71C3×C22⋊C4—181no—
72C3×C42—91no—
73#217—64no—
74#217—64no—
75#217—64no—
76C4×Dic3—34no—
77C22×C32—24no—
78C22×C32—24no—
79C22×C32—44no—
80C22×C32—44no—
81C22×C32(C2×D3)⋊C414yes—
82C32×C4—24no—
83C32×C4—84no—
84C3×Dic3—124no—
85C3×Dic3—124no—
86C3×Dic3—244no—
87#124—91no—
88#99—91no—
89#99#27211yescharacteristic
90H(3,𝔽3)#35411yescharacteristic, complemented, Sylow
91C23×C3—34no—
92C23×C3—34no—
93C23×C3C4⋊(C3⋊D3)11yescharacteristic
94C23×C3#63811yescharacteristic
95C3×Q8—361no—
96C3×D4—41no—
97C3×D4—124no—
98C2×C3×C4—34no—
99C2×C3×C4—41no—
100C2×C3×C4—124no—
101C2×C3×C4—181no—
102C2×C3×C4—181no—
103C2×C3×C4—361no—
104C2×C3×C4C4×C3⋊D311yescharacteristic
105C2×Dic3—64no—
106C2×Dic3—64no—
107C2×Dic3—124no—
108Dic6—124no—
109C2×C32—24no—
110C2×C32—44no—
111C2×C32—44no—
112C2×C32#80614yes—
113C2×Q8—91no—
114C2×D4C2×#27211yescharacteristic
115C22⋊C4—21no—
116C22⋊C4—181no—
117C22⋊C4—181no—
118C22⋊C4—181no—
119C42—91no—
120C22×C3—21no—
121C22×C3—21no—
122C22×C3—34no—
123C22×C3—41no—
124C22×C3—41no—
125C22×C3—64no—
126C22×C3—64no—
127C22×C3—124no—
128C22×C3—124no—
129C22×C3#375211yescharacteristic
130C3×C4—21no—
131C3×C4—64no—
132C3×C4—81no—
133C3×C4—244no—
134C3×C4—361no—
135C3×C4—361no—
136C3×C4—721no—
137Dic3—124no—
138Dic3—124no—
139Dic3—244no—
140C32#447314yescomplemented
141C23C4⋊#27211yescharacteristic
142C23#623511yescharacteristic
143Q8—361no—
144D4—41no—
145C2×C4—41no—
146C2×C4—181no—
147C2×C4—181no—
148C2×C4—361no—
149C2×C4C4×#27211yescharacteristic
150C2×C3—21no—
151C2×C3—34no—
152C2×C3—41no—
153C2×C3—41no—
154C2×C3—64no—
155C2×C3—124no—
156C2×C3—124no—
157C2×C3#6363311yescharacteristic
158C22—21no—
159C22—21no—
160C22—41no—
161C22—41no—
162C22#8854211yescharacteristic
163C4—21no—
164C4—81no—
165C4—361no—
166C4—361no—
167C4—721no—
168C3—34no—
169C3#1058933911yescharacteristic
170C2—21no—
171C2—41no—
172C2—41no—
173C2#1172052411yescharacteristic
subgroupG/H
center, socleC2×C3#63633
commutatorC2×C4×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)#354
LCSG ▷ C2×C4×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. C2×#6229C4semidirect
11. C4×#1084C4semidirect
90. H(3,𝔽3)#354semidirect
140. C32#4473semidirect

Maximal subgroups

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

subgroupindex
#117218462
H(3,𝔽3)×#992
#117205802
C3×#44733