Finite Group Databasebeta

← #13143723

Subgroups — #13143723

Order 1728.

1023 conjugacy classes of subgroups; only the 159 normal subgroups are shown.

normal subgroups only

This group has 1023 conjugacy classes of subgroups — too many to draw. Only its 159 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×#88542C213yesmaximal
2C2×#88540C213yesmaximal, complemented
3H(3,𝔽3)×#95C211yesmaximal, characteristic, complemented
4C3×#4676—34nomaximal
5C23×#272C2211yescharacteristic
6C22×C4×H(3,𝔽3)C2213yes—
7C22×#1084C2213yes—
8#88542—26no—
9#88540—26no—
10C2×C3×(C2×D3)⋊C4—312no—
11C2×C3×Dic3⋊C4—312no—
12C32×#95D314yescomplemented
13C23×H(3,𝔽3)C2311yescharacteristic
14C22×#272—23no—
15C22×#272—23no—
16C22×#272Q812yes—
17C2×C4×H(3,𝔽3)—23no—
18C2×C4×H(3,𝔽3)—23no—
19C2×C4×H(3,𝔽3)—23no—
20C2×#1084—23no—
21C2×#1084—23no—
22C2×#1084D416yes—
23C3×#396—91no—
24#4676—34no—
25C23×C3×D3—34no—
26C22×C32×C4C2×D3112yes—
27C22×C3×Dic3—312no—
28C3×(C2×D3)⋊C4—624no—
29C3×Dic3⋊C4—624no—
30C22×H(3,𝔽3)C2×D413yes—
31C22×H(3,𝔽3)C2×Q811yescharacteristic
32C22×H(3,𝔽3)#4113yes—
33C2×#272—26no—
34C2×#272—43no—
35C2×#272—41no—
36C4×H(3,𝔽3)—43no—
37#1084—26no—
38C2×C3×C4⋊C4—93no—
39C2×C3×C22⋊C4—93no—
40C2×(C2×D3)⋊C4—312no—
41C2×Dic3⋊C4—312no—
42C3×#95—34no—
43C3×#95C3⋊D311yescharacteristic
44C23×C32C22×D314yes—
45C22×C3×D3—38no—
46C22×C3×D3—612no—
47C22×C3×D3—612no—
48C2×C32×C4—212no—
49C2×C32×C4—212no—
50C2×C32×C4—212no—
51C2×C3×Dic3—324no—
52C2×C3×Dic3—612no—
53C2×C3×Dic3—612no—
54#396—91noSylow
55C2×H(3,𝔽3)#12113yes—
56C2×H(3,𝔽3)C22⋊Q813yes—
57C2×H(3,𝔽3)#12611yescharacteristic
58#272—42no—
59C24×C3—91no—
60C23×D3—34no—
61C22×C3×C4—312no—
62C22×C3×C4—93no—
63C22×C3×C4C2×C3⋊D313yes—
64C22×Dic3—312no—
65C3×C4⋊C4—186no—
66C3×C22⋊C4—186no—
67(C2×D3)⋊C4—624no—
68Dic3⋊C4—624no—
69C22×C32#235112yes—
70C22×C32D3×D4112yes—
71C22×C32D3×Q814yes—
72C2×C3×D3—624no—
73C2×C3×D3—124no—
74C2×C3×D3—1212no—
75C32×C4—412no—
76C3×Dic3—624no—
77C2×C4⋊C4—93no—
78C2×C22⋊C4—93no—
79#95#27211yescharacteristic, complemented
80H(3,𝔽3)#39611yescharacteristic, complemented, Sylow
81C23×C3—34no—
82C23×C3—92no—
83C23×C3—183no—
84C23×C3—183no—
85C23×C3C22×C3⋊D311yescharacteristic
86C22×D3—38no—
87C22×D3—612no—
88C22×D3—612no—
89C2×C3×C4—23no—
90C2×C3×C4—23no—
91C2×C3×C4—23no—
92C2×C3×C4—612no—
93C2×C3×C4—612no—
94C2×C3×C4—612no—
95C2×C3×C4—96no—
96C2×C3×C4—183no—
97C2×C3×C4—183no—
98C2×Dic3—324no—
99C2×Dic3—612no—
100C2×Dic3—612no—
101C2×C32#87614yes—
102C2×C32#884112yes—
103C2×C32#896112yes—
104C3×D3—128no—
105C24—91no—
106C22×C4—93no—
107C22×C4C2×#27213yes—
108C4⋊C4—186no—
109C22⋊C4—186no—
110C22×C3—312no—
111C22×C3—312no—
112C22×C3—34no—
113C22×C3—186no—
114C22×C3—361no—
115C22×C3—363no—
116C22×C3#382813yes—
117C22×C3D4×C3⋊D313yes—
118C22×C3Q8×C3⋊D311yescharacteristic
119C2×D3—624no—
120C2×D3—1212no—
121C2×D3—124no—
122C3×C4—43no—
123C3×C4—1212no—
124C3×C4—186no—
125Dic3—624no—
126C32#467614yescomplemented
127C23—92no—
128C23—183no—
129C23—183no—
130C23C22×#27211yescharacteristic
131C2×C4—23no—
132C2×C4—23no—
133C2×C4—23no—
134C2×C4—96no—
135C2×C4—183no—
136C2×C4—183no—
137C2×C3—312no—
138C2×C3—312no—
139C2×C3—34no—
140C2×C3—362no—
141C2×C3#6408511yescharacteristic
142C2×C3#6409313yes—
143C2×C3#6410513yes—
144D3—128no—
145C22—186no—
146C22—363no—
147C22—361no—
148C22#8874013yes—
149C22D4×#27213yes—
150C22Q8×#27211yescharacteristic
151C4—43no—
152C4—186no—
153C3—34no—
154C3#1059237511yescharacteristic
155C2—362no—
156C2#1172172211yescharacteristic
157C2#1172174013yes—
158C2#1172176413yes—
subgroupG/H
center, Frattini, socleC23×C3C22×C3⋊D3
commutator, cosocleC23×H(3,𝔽3)C23
FittingH(3,𝔽3)×#95C2
prime radicalC23×#272C22
hypercenterC3×#95C3⋊D3
hypocenterH(3,𝔽3)#396
LCSG ▷ C23×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C23×C3 ◁ C3×#95

Complements

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

NH ≅ G/Ntype
2. C2×#88540C2semidirect
3. H(3,𝔽3)×#95C2semidirect
12. C32×#95D3semidirect
79. #95#272semidirect
80. H(3,𝔽3)#396semidirect
126. C32#4676semidirect

Maximal subgroups

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

subgroupindex
C2×#885422
C2×#885402
H(3,𝔽3)×#952
C3×#46763