Finite Group Databasebeta

← #13048749

Subgroups — #13048749

Order 1296.

473 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 473 conjugacy classes of subgroups — too many to draw. Only its 63 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
1C3×C4×#272C211yesmaximal, characteristic
2#10619956C212yesmaximal, complemented
3#10619903C212yesmaximal
4#10619900C212yesmaximal, complemented
5D3×#634—32nomaximal
6#88651—31nomaximal
7C2×C3×#272C2211yescharacteristic
8C3×C4×H(3,𝔽3)C2211yescharacteristic
9C3×#1084C2211yescharacteristic
10C3⋊((C3⋊Dic3)⋊C3)C2214yes—
11#6308D312yescomplemented
12C3×#634—32no—
13C32×C4×D3—32no—
14C32×C4×D3—61no—
15#6293—34no—
16D3×C3⋊Dic3—34no—
17C4×#272D311yescharacteristic
18#634⋊C3—32no—
19#634⋊C3—62no—
20#6199—32no—
21#6196—32no—
22C2×C3×H(3,𝔽3)C2311yescharacteristic
23C3×#272Q812yescomplemented
24C2×#634—91no—
25D3×Dic6—92no—
26D3×Dic6—96no—
27C2×C32×D3—32no—
28C2×C32×D3—61no—
29C33×C4—21no—
30C33×C4C2×D312yes—
31C3⋊(C3⋊Dic3)C2×D314yes—
32C3×C3⋊Dic3—34no—
33C32×Dic3—32no—
34C32×Dic3—61no—
35C2×#272C2×D311yescharacteristic
36C4×H(3,𝔽3)—22no—
37C4×H(3,𝔽3)—41no—
38C4×H(3,𝔽3)C2×D311yescharacteristic
39#1084C2×D311yescharacteristic
40(C3⋊Dic3)⋊C3—34no—
41(C3⋊Dic3)⋊C3—64no—
42C3×H(3,𝔽3)C2×Q811yescharacteristic, complemented, Sylow
43C2×C32×C4—91no—
44C2×C3⋊Dic3—92no—
45#634—32no—
46#634—32no—
47#634—62no—
48#634—96no—
49#634—92no—
50C3×C4×D3—36no—
51C3×C4×D3—32no—
52C3×C4×D3—61no—
53C3×C4×D3—63no—
54C3×Dic6—96no—
55C3×Dic6—92no—
56#627—94no—
57#627—912no—
58D3×Dic3—94no—
59D3×Dic3—912no—
60C2×C33—21no—
61C2×C33C22×D312yes—
62C32×D3—34no—
63C32×D3—62no—
64C2×H(3,𝔽3)—22no—
65C2×H(3,𝔽3)—41no—
66C2×H(3,𝔽3)C22×D311yescharacteristic
67#272Dic612yescomplemented
68D3×Q8—272no—
69C2×Dic6—273no—
70C2×Dic6—271no—
71C22×C32—91no—
72C2×C3×D3—36no—
73C2×C3×D3—32no—
74C2×C3×D3—61no—
75C2×C3×D3—63no—
76C32×C4—22no—
77C32×C4—21no—
78C32×C4—36no—
79C32×C4—41no—
80C32×C4—63no—
81C32×C4—91no—
82C32×C4D3212yes—
83C32×C4D3211yescharacteristic
84C3⋊Dic3—34no—
85C3⋊Dic3—34no—
86C3⋊Dic3—64no—
87C3⋊Dic3—912no—
88C3×Dic3—36no—
89C3×Dic3—32no—
90C3×Dic3—61no—
91C3×Dic3—63no—
92C3×Dic3—912no—
93C3×Dic3—94no—
94C33—21no—
95C33D3×Q812yescomplemented
96H(3,𝔽3)—22no—
97H(3,𝔽3)—41no—
98H(3,𝔽3)C2×Dic611yescharacteristic, complemented
99C3×Q8—272no—
100C2×C3×C4—93no—
101C2×C3×C4—91no—
102C2×Dic3—276no—
103C2×Dic3—272no—
104C4×D3—32no—
105C4×D3—61no—
106C4×D3—274no—
107Dic6—96no—
108Dic6—92no—
109Dic6—272no—
110Dic6—274no—
111Dic6—276no—
112Dic6—272no—
113Dic6—542no—
114C2×C32—22no—
115C2×C32—21no—
116C2×C32—36no—
117C2×C32—41no—
118C2×C32—63no—
119C2×C32—92no—
120C2×C32C2×D3212yes—
121C2×C32C2×D3211yescharacteristic
122C3×D3—312no—
123C3×D3—34no—
124C3×D3—62no—
125C3×D3—66no—
126C2×Q8—811noSylow
127C22×C3—93no—
128C22×C3—91no—
129C2×D3—32no—
130C2×D3—61no—
131C3×C4—32no—
132C3×C4—62no—
133C3×C4—61no—
134C3×C4—93no—
135C3×C4—91no—
136C3×C4—121no—
137C3×C4—274no—
138C3×C4((C3⋊D3)⋊C3)⋊C213yes—
139C3×C4#111311yescharacteristic
140Dic3—32no—
141Dic3—61no—
142Dic3—912no—
143Dic3—94no—
144Dic3—274no—
145Dic3—544no—
146C32—21no—
147C32—22no—
148C32—36no—
149C32—41no—
150C32—63no—
151C32D3×Dic612yescomplemented
152C32#379711yescharacteristic
153Q8—272no—
154Q8—812no—
155C2×C4—91no—
156C2×C4—812no—
157C2×C3—32no—
158C2×C3—61no—
159C2×C3—62no—
160C2×C3—96no—
161C2×C3—92no—
162C2×C3—121no—
163C2×C3C2×((C3⋊D3)⋊C3)⋊C213yes—
164C2×C3C2×#111311yescharacteristic
165D3—34no—
166D3—62no—
167C22—91no—
168C4—91no—
169C4—274no—
170C4#6643211yescharacteristic
171C3—32no—
172C3—61no—
173C3—62no—
174C3—121no—
175C3#8865113yescomplemented
176C3#8904011yescharacteristic
177C2—92no—
178C2C2×#6643211yescharacteristic
subgroupG/H
centerC2C2×#66432
commutator, cosocleC2×C3×H(3,𝔽3)C23
FrattiniC2×C3C2×#1113
FittingC3×C4×H(3,𝔽3)C22
socleC2×C32C2×D32
prime radicalC2×C3×#272C22
hypercenterC4#66432
hypocenterC3×H(3,𝔽3)C2×Q8
LCSG ▷ C2×C3×H(3,𝔽3) ▷ C3×H(3,𝔽3)
UCSC1 ◁ C2 ◁ C4

Complements

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

NH ≅ G/Ntype
2. #10619956C2semidirect
4. #10619900C2semidirect
11. #6308D3semidirect
23. C3×#272Q8semidirect
42. C3×H(3,𝔽3)C2×Q8semidirect
67. #272Dic6semidirect
95. C33D3×Q8semidirect
98. H(3,𝔽3)C2×Dic6semidirect
151. C32D3×Dic6semidirect
175. C3#88651semidirect

Maximal subgroups

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

subgroupindex
C3×C4×#2722
#106199562
#106199032
#106199002
D3×#6343
#886513