Finite Group Databasebeta

← #13048754

Subgroups — #13048754

Order 1296.

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

normal subgroups only

This group has 645 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
1C2×#66432C212yesmaximal, complemented
2C3×C4×#272C211yesmaximal, characteristic, complemented
3C4⋊(C3⋊((C3⋊D3)⋊C3))C212yesmaximal, complemented
4#10619902C212yesmaximal, complemented
5D3×C4⋊(C3⋊D3)—32nomaximal
6#88655—31nomaximal
7C2×C3×#272C2211yescharacteristic
8C2×C3⋊((C3⋊D3)⋊C3)C2214yescomplemented
9#66432—24no—
10C3×C4×H(3,𝔽3)C2211yescharacteristic, complemented
11C3×#1084C2211yescharacteristic, complemented
12C2×D3×C3⋊D3—34no—
13C4⋊(C3⋊(C3⋊D3))D312yescomplemented
14C3×C4⋊(C3⋊D3)—32no—
15C32×C4×D3—32no—
16C32×C4×D3—61no—
17#6292—34no—
18C2×((C3⋊D3)⋊C3)⋊C2—32no—
19C4×#272D311yescharacteristic, complemented
20C4⋊((C3⋊D3)⋊C3)—32no—
21C4⋊((C3⋊D3)⋊C3)—62no—
22#6198—32no—
23C2×C3×H(3,𝔽3)C2311yescharacteristic
24C3×#272D412yescomplemented
25C3⋊((C3⋊D3)⋊C3)—24no—
26C2×C4⋊(C3⋊D3)—91no—
27D3×D12—92no—
28D3×D12—96no—
29C2×C3⋊(C3⋊D3)C2×D314yescomplemented
30C2×C3×C3⋊D3—34no—
31C2×C32×D3—32no—
32C2×C32×D3—61no—
33D3×C3⋊D3—68no—
34C33×C4—21no—
35C33×C4C2×D312yescomplemented
36C32×Dic3—32no—
37C32×Dic3—61no—
38C2×#272C2×D311yescharacteristic
39C2×(C3⋊D3)⋊C3—34no—
40C2×(C3⋊D3)⋊C3—64no—
41((C3⋊D3)⋊C3)⋊C2—64no—
42C4×H(3,𝔽3)—22no—
43C4×H(3,𝔽3)—41no—
44C4×H(3,𝔽3)C2×D311yescharacteristic, complemented
45#1084C2×D311yescharacteristic, complemented
46C3×H(3,𝔽3)C2×D411yescharacteristic, complemented, Sylow
47C22×C3⋊D3—92no—
48C2×D32—94no—
49C2×D32—912no—
50C2×C32×C4—91no—
51C4⋊(C3⋊D3)—32no—
52C4⋊(C3⋊D3)—32no—
53C4⋊(C3⋊D3)—62no—
54C4⋊(C3⋊D3)—96no—
55C4⋊(C3⋊D3)—92no—
56C3×D12—96no—
57C3×D12—92no—
58C3×C4×D3—36no—
59C3×C4×D3—32no—
60C3×C4×D3—61no—
61C3×C4×D3—63no—
62#626—94no—
63#626—912no—
64C2×C33—21no—
65C2×C33C22×D312yes—
66C3⋊(C3⋊D3)—24no—
67C3×C3⋊D3—64no—
68C32×D3—34no—
69C32×D3—62no—
70C2×H(3,𝔽3)—22no—
71C2×H(3,𝔽3)—41no—
72C2×H(3,𝔽3)C22×D311yescharacteristic
73#272D1212yescomplemented
74(C3⋊D3)⋊C3—64no—
75(C3⋊D3)⋊C3—124no—
76D3×D4—272no—
77C2×D12—273no—
78C2×D12—271no—
79C22×C32—91no—
80C2×C3⋊D3—34no—
81C2×C3⋊D3—34no—
82C2×C3⋊D3—64no—
83C2×C3⋊D3—912no—
84C2×C3⋊D3—184no—
85C2×C3×D3—36no—
86C2×C3×D3—32no—
87C2×C3×D3—61no—
88C2×C3×D3—63no—
89C2×C3×D3—912no—
90C2×C3×D3—94no—
91D32—188no—
92D32—1824no—
93C32×C4—22no—
94C32×C4—21no—
95C32×C4—36no—
96C32×C4—41no—
97C32×C4—63no—
98C32×C4—91no—
99C32×C4D3211yescharacteristic
100C32×C4D3212yescomplemented
101C3×Dic3—36no—
102C3×Dic3—32no—
103C3×Dic3—61no—
104C3×Dic3—63no—
105C33—21no—
106C33D3×D412yescomplemented
107H(3,𝔽3)—22no—
108H(3,𝔽3)—41no—
109H(3,𝔽3)C2×D1211yescharacteristic, complemented
110C22×D3—274no—
111C22×D3—276no—
112C22×D3—272no—
113C3×D4—272no—
114C2×C3×C4—93no—
115C2×C3×C4—91no—
116#67—274no—
117D12—96no—
118D12—92no—
119D12—272no—
120D12—276no—
121D12—272no—
122D12—542no—
123C4×D3—32no—
124C4×D3—61no—
125C2×C32—22no—
126C2×C32—21no—
127C2×C32—36no—
128C2×C32—41no—
129C2×C32—63no—
130C2×C32—92no—
131C2×C32C2×D3212yes—
132C2×C32C2×D3211yescharacteristic
133C3⋊D3—64no—
134C3⋊D3—64no—
135C3⋊D3—124no—
136C3⋊D3—1812no—
137C3×D3—312no—
138C3×D3—34no—
139C3×D3—62no—
140C3×D3—66no—
141C3×D3—1812no—
142C3×D3—184no—
143C2×D4—811noSylow
144C22×C3—93no—
145C22×C3—91no—
146C22×C3—274no—
147C2×D3—32no—
148C2×D3—61no—
149C2×D3—912no—
150C2×D3—94no—
151C2×D3—274no—
152C2×D3—544no—
153C2×D3—548no—
154C2×D3—5412no—
155C2×D3—544no—
156C3×C4—32no—
157C3×C4—61no—
158C3×C4—62no—
159C3×C4—93no—
160C3×C4—91no—
161C3×C4—121no—
162C3×C4((C3⋊D3)⋊C3)⋊C213yescomplemented
163C3×C4#111311yescharacteristic
164Dic3—32no—
165Dic3—61no—
166C32—21no—
167C32—22no—
168C32—36no—
169C32—41no—
170C32—63no—
171C32D3×D1212yescomplemented
172C32#380211yescharacteristic
173C23—812no—
174D4—272no—
175D4—812no—
176C2×C4—91no—
177C2×C3—32no—
178C2×C3—61no—
179C2×C3—62no—
180C2×C3—96no—
181C2×C3—92no—
182C2×C3—121no—
183C2×C3—544no—
184C2×C3C2×((C3⋊D3)⋊C3)⋊C213yes—
185C2×C3C2×#111311yescharacteristic
186D3—34no—
187D3—62no—
188D3—1812no—
189D3—184no—
190D3—544no—
191D3—1084no—
192C22—91no—
193C22—274no—
194C22—1624no—
195C4—91no—
196C4#6643211yescharacteristic, complemented
197C3—32no—
198C3—61no—
199C3—62no—
200C3—121no—
201C3#8865513yescomplemented
202C3C4⋊#111311yescharacteristic
203C2—92no—
204C2—544no—
205C2C2×#6643211yescharacteristic
subgroupG/H
centerC2C2×#66432
commutator, cosocleC2×C3×H(3,𝔽3)C23
FrattiniC2×C3C2×#1113
FittingC3×C4×H(3,𝔽3)C22
socleC2×C32C2×D32
hypercenterC4#66432
hypocenterC3×H(3,𝔽3)C2×D4
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
1. C2×#66432C2semidirect
2. C3×C4×#272C2semidirect
3. C4⋊(C3⋊((C3⋊D3)⋊C3))C2semidirect
4. #10619902C2semidirect
8. C2×C3⋊((C3⋊D3)⋊C3)C22semidirect
10. C3×C4×H(3,𝔽3)C22semidirect
11. C3×#1084C22semidirect
13. C4⋊(C3⋊(C3⋊D3))D3semidirect
19. C4×#272D3semidirect
24. C3×#272D4semidirect
29. C2×C3⋊(C3⋊D3)C2×D3semidirect
35. C33×C4C2×D3semidirect
44. C4×H(3,𝔽3)C2×D3semidirect
45. #1084C2×D3semidirect
46. C3×H(3,𝔽3)C2×D4semidirect
73. #272D12semidirect
100. C32×C4D32semidirect
106. C33D3×D4semidirect
109. H(3,𝔽3)C2×D12semidirect
162. C3×C4((C3⋊D3)⋊C3)⋊C2semidirect
171. C32D3×D12semidirect
196. C4#66432semidirect
201. C3#88655semidirect

Maximal subgroups

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

subgroupindex
C2×#664322
C3×C4×#2722
C4⋊(C3⋊((C3⋊D3)⋊C3))2
#106199022
D3×C4⋊(C3⋊D3)3
#886553