Finite Group Databasebeta

← #13144870

Subgroups — #13144870

Order 1728.

880 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 880 conjugacy classes of subgroups — too many to draw. Only its 123 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)×H(5,𝔽2)C211yesmaximal, characteristic
2#11721845C213yesmaximal, complemented
3#11720580C213yesmaximal, complemented
4C3×#5243—34nomaximal
5#41×H(3,𝔽3)—23no—
6C2×D4×H(3,𝔽3)—23no—
7C2×D4×H(3,𝔽3)C2213yes—
8C22×#1084C2211yescharacteristic, complemented
9#88552—23no—
10#88552C2213yescomplemented
11#88540—23no—
12C32×H(5,𝔽2)D314yes—
13C3×#935—312no—
14C3×#834—312no—
15C23×H(3,𝔽3)—21no—
16C23×H(3,𝔽3)D413yes—
17C23×H(3,𝔽3)C2311yescharacteristic
18Q8×H(3,𝔽3)—21no—
19D4×H(3,𝔽3)—23no—
20D4×H(3,𝔽3)—26no—
21C2×C4×H(3,𝔽3)—23no—
22C2×C4×H(3,𝔽3)D413yes—
23C2×#1084—23no—
24C2×#1084—23no—
25C3×#458—91no—
26#5243—34no—
27C32×#41—212no—
28C2×C32×D4—212no—
29C2×C32×D4C2×D3112yes—
30C22×C3×Dic3—34no—
31C3×#217—312no—
32C3×#217—612no—
33C3×Dic3⋊C4—612no—
34C22×H(3,𝔽3)—23no—
35C22×H(3,𝔽3)—41no—
36C22×H(3,𝔽3)—43no—
37C22×H(3,𝔽3)—81no—
38C22×H(3,𝔽3)C2×D413yes—
39C4×H(3,𝔽3)—23no—
40#1084—43no—
41#1084—41no—
42C3×H(5,𝔽2)—34no—
43C3×H(5,𝔽2)C3⋊D311yescharacteristic
44C3×#123—93no—
45#935—312no—
46C3×#99—93no—
47#834—312no—
48C23×C32—24no—
49C23×C32#67112yes—
50C23×C32C22×D314yes—
51C32×Q8—24no—
52C32×D4—212no—
53C32×D4—224no—
54C2×C32×C4—212no—
55C2×C32×C4#67112yes—
56C2×C3×Dic3—612no—
57C2×C3×Dic3—612no—
58#458—91noSylow
59C2×H(3,𝔽3)—23no—
60C2×H(3,𝔽3)—43no—
61C2×H(3,𝔽3)#12011yescharacteristic
62C3×#41—23no—
63C3×#41—612no—
64C2×C3×D4—23no—
65C2×C3×D4—312no—
66C2×C3×D4—612no—
67C2×C3×D4C2×C3⋊D313yes—
68C22×C3×C4—91no—
69C22×Dic3—34no—
70C3×C4⋊C4—183no—
71C3×C22⋊C4—93no—
72C3×C22⋊C4—183no—
73#217—312no—
74#217—612no—
75Dic3⋊C4—612no—
76C22×C32—212no—
77C22×C32—44no—
78C22×C32—412no—
79C22×C32—84no—
80C22×C32C2×#67112yes—
81C32×C4—212no—
82C3×Dic3—1212no—
83C3×Dic3—124no—
84H(5,𝔽2)#27211yescharacteristic
85#123—93no—
86#99—93no—
87H(3,𝔽3)#45811yescharacteristic, complemented, Sylow
88C23×C3—21no—
89C23×C3—312no—
90C23×C3—34no—
91C23×C3—64no—
92C23×C3#63813yes—
93C23×C3C22×C3⋊D311yescharacteristic
94C3×Q8—21no—
95C3×Q8—64no—
96C3×D4—23no—
97C3×D4—26no—
98C3×D4—624no—
99C3×D4—612no—
100C2×C3×C4—23no—
101C2×C3×C4—312no—
102C2×C3×C4—612no—
103C2×C3×C4—183no—
104C2×C3×C4—183no—
105C2×C3×C4#63813yes—
106C2×Dic3—612no—
107C2×Dic3—612no—
108C2×C32—212no—
109C2×C32—412no—
110C2×C32#95314yes—
111#41—23no—
112C2×D4—23no—
113C2×D4C2×#27213yes—
114C22×C4—91no—
115C4⋊C4—183no—
116C22⋊C4—93no—
117C22⋊C4—183no—
118C22×C3—23no—
119C22×C3—312no—
120C22×C3—41no—
121C22×C3—43no—
122C22×C3—612no—
123C22×C3—81no—
124C22×C3—124no—
125C22×C3—1212no—
126C22×C3—244no—
127C22×C3C2×#63813yes—
128C3×C4—23no—
129C3×C4—612no—
130C3×C4—363no—
131C3×C4—361no—
132Dic3—124no—
133Dic3—1212no—
134C32#524314yescomplemented
135C23—21no—
136C23#623513yes—
137C23C22×#27211yescharacteristic
138Q8—21no—
139D4—23no—
140D4—26no—
141C2×C4—23no—
142C2×C4—183no—
143C2×C4—183no—
144C2×C4#623513yes—
145C2×C3—23no—
146C2×C3—34no—
147C2×C3—43no—
148C2×C3—612no—
149C2×C3—1212no—
150C2×C3#6416211yescharacteristic
151C22—23no—
152C22—41no—
153C22—43no—
154C22—81no—
155C22C2×#623513yes—
156C4—23no—
157C4—363no—
158C4—361no—
159C3—34no—
160C3#1059294211yescharacteristic
161C2—23no—
162C2—43no—
163C2#1172187611yescharacteristic
subgroupG/H
center, socleC2×C3#64162
commutator, cosocleC23×H(3,𝔽3)C23
FrattiniC23×C3C22×C3⋊D3
Fitting, prime radicalH(3,𝔽3)×H(5,𝔽2)C2
hypercenterC3×H(5,𝔽2)C3⋊D3
hypocenterH(3,𝔽3)#458
LCSG ▷ C23×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C2×C3 ◁ C23×C3 ◁ C3×H(5,𝔽2)

Complements

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

NH ≅ G/Ntype
2. #11721845C2semidirect
3. #11720580C2semidirect
8. C22×#1084C22semidirect
10. #88552C22semidirect
87. H(3,𝔽3)#458semidirect
134. C32#5243semidirect

Maximal subgroups

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

subgroupindex
H(3,𝔽3)×H(5,𝔽2)2
#117218452
#117205802
C3×#52433