Finite Group Databasebeta

← #13144502

Subgroups — #13144502

Order 1728.

528 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 528 conjugacy classes of subgroups — too many to draw. Only its 109 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
1H(3,𝔽3)×#124C211yesmaximal, characteristic
2#11721716C211yesmaximal, characteristic, complemented
3#11720583C212yesmaximal, complemented
4#11720578C212yesmaximal
5C4×#6180C211yesmaximal, characteristic, complemented
6C3×#5056—34nomaximal
7C2×Q8×H(3,𝔽3)C2211yescharacteristic
8C2×D4×H(3,𝔽3)C2211yescharacteristic
9C22⋊C4×H(3,𝔽3)—22no—
10C42×H(3,𝔽3)C2211yescharacteristic
11C4⋊#1084C2212yes—
12#88540—22no—
13C2×#6180C2212yes—
14C32×#124D314yes—
15C3×#870—34no—
16C3×#835—38no—
17C3×#832—38no—
18C3×C4×C3⋊C8—34no—
19C23×H(3,𝔽3)—21no—
20Q8×H(3,𝔽3)—22no—
21D4×H(3,𝔽3)—22no—
22C2×C4×H(3,𝔽3)—21no—
23C2×C4×H(3,𝔽3)D412yes—
24C2×C4×H(3,𝔽3)C2311yescharacteristic
25C2×#1084—22no—
26#6180—22no—
27C3×#488—91no—
28#5056—34no—
29C2×C32×Q8C2×D314yes—
30C2×C32×D4C2×D314yes—
31C32×C22⋊C4—28no—
32C32×C42C2×D314yes—
33C3×C4⋊Dic3—38no—
34C3×Dic3⋊C4—68no—
35C2×C3×C3⋊C8—38no—
36C22×H(3,𝔽3)—41no—
37C22×H(3,𝔽3)—42no—
38C22×H(3,𝔽3)C2×D411yescharacteristic
39C4×H(3,𝔽3)—22no—
40C4×H(3,𝔽3)—41no—
41C4×H(3,𝔽3)#4112yes—
42#1084—42no—
43C3×#125—91no—
44C3×#124—34no—
45C3×#124C3⋊D311yescharacteristic
46#870—34no—
47C3×#103—92no—
48C3×#102—92no—
49C3×C4×C8—91no—
50#835—38no—
51#832—38no—
52C4×C3⋊C8—34no—
53C23×C32—24no—
54C32×Q8—28no—
55C32×D4—28no—
56C2×C32×C4—24no—
57C2×C32×C4#6718yes—
58C2×C32×C4C22×D314yes—
59C2×C3×Dic3—68no—
60C3×C3⋊C8—68no—
61#488—91noSylow
62C2×H(3,𝔽3)—81no—
63C2×H(3,𝔽3)#12411yescharacteristic
64C2×H(3,𝔽3)#13512yes—
65C2×C3×Q8—34no—
66C2×C3×Q8C2×C3⋊D311yescharacteristic
67C2×C3×D4—34no—
68C2×C3×D4C2×C3⋊D311yescharacteristic
69C2×C3×C8—92no—
70C3×C4⋊C4—92no—
71C3×C4⋊C4—182no—
72C3×C22⋊C4—22no—
73C3×C22⋊C4—68no—
74C3×C42—34no—
75C3×C42C2×C3⋊D311yescharacteristic
76C4⋊Dic3—38no—
77Dic3⋊C4—68no—
78C2×C3⋊C8—38no—
79C22×C32—44no—
80C22×C32—48no—
81C22×C32C2×#6714yes—
82C32×C4—28no—
83C32×C4—44no—
84C32×C4#23718yes—
85C3×Dic3—128no—
86#125—91no—
87#124#27211yescharacteristic
88#103—92no—
89#102—92no—
90C4×C8—91no—
91H(3,𝔽3)#48811yescharacteristic, complemented, Sylow
92C23×C3—21no—
93C23×C3—64no—
94C3×Q8—22no—
95C3×Q8—68no—
96C3×D4—22no—
97C3×D4—68no—
98C2×C3×C4—21no—
99C2×C3×C4—38no—
100C2×C3×C4—34no—
101C2×C3×C4—64no—
102C2×C3×C4—182no—
103C2×C3×C4#63812yes—
104C2×C3×C4C22×C3⋊D311yescharacteristic
105C2×Dic3—68no—
106C3×C8—182no—
107C3⋊C8—68no—
108C2×C32—84no—
109C2×C32#93614yes—
110C2×C32#95018yes—
111C2×Q8C2×#27211yescharacteristic
112C2×D4C2×#27211yescharacteristic
113C2×C8—92no—
114C4⋊C4—92no—
115C4⋊C4—182no—
116C22⋊C4—22no—
117C42C2×#27211yescharacteristic
118C22×C3—34no—
119C22×C3—41no—
120C22×C3—42no—
121C22×C3—124no—
122C22×C3—128no—
123C22×C3C2×#63811yescharacteristic
124C3×C4—22no—
125C3×C4—38no—
126C3×C4—41no—
127C3×C4—68no—
128C3×C4—124no—
129C3×C4—362no—
130C3×C4#383012yes—
131Dic3—128no—
132C32#505614yescomplemented
133C23—21no—
134Q8—22no—
135D4—22no—
136C2×C4—21no—
137C2×C4—182no—
138C2×C4#623512yes—
139C2×C4C22×#27211yescharacteristic
140C8—182no—
141C2×C3—34no—
142C2×C3—38no—
143C2×C3—81no—
144C2×C3—244no—
145C2×C3#6414511yescharacteristic
146C2×C3#6415912yes—
147C22—41no—
148C22—42no—
149C22C2×#623511yescharacteristic
150C4—22no—
151C4—41no—
152C4—362no—
153C4#8874412yes—
154C3—34no—
155C3#1059275511yescharacteristic
156C2—81no—
157C2#1172184611yescharacteristic
158C2#1172187112yes—
subgroupG/H
center, socleC22×C3C2×#638
commutator, cosocleC2×C4×H(3,𝔽3)C23
FrattiniC2×C3×C4C22×C3⋊D3
FittingH(3,𝔽3)×#124C2
prime radicalC2×D4×H(3,𝔽3)C22
hypercenterC3×#124C3⋊D3
hypocenterH(3,𝔽3)#488
LCSG ▷ C2×C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22×C3 ◁ C3×C42 ◁ C3×#124

Complements

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

NH ≅ G/Ntype
2. #11721716C2semidirect
3. #11720583C2semidirect
5. C4×#6180C2semidirect
91. H(3,𝔽3)#488semidirect
132. C32#5056semidirect

Maximal subgroups

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

subgroupindex
H(3,𝔽3)×#1242
#117217162
#117205832
#117205782
C4×#61802
C3×#50563