Finite Group Databasebeta

← C2×C3×#5450 #11919772

Subgroups — C2×C3×#5450 #11919772

Order 1152.

736 conjugacy classes of subgroups; only the 132 normal subgroups are shown.

normal subgroups only

This group has 736 conjugacy classes of subgroups — too many to draw. Only its 132 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×#5450C214yesmaximal, direct product factor
2C2×C3×#120⋊C3C213yesmaximal, complemented
3C2×C3×H(3,𝔽4)C311yesmaximal, characteristic, complemented
4C2×#5450C313yesmaximal, direct product factor
5C2×C3×C22⋊A4C2211yescharacteristic, complemented
6C23×C3×A4—41nomaximal
7C3×#120⋊C3C2216yescomplemented
8C3×H(3,𝔽4)—34no—
9C3×H(3,𝔽4)C2×C314yescomplemented
10C2×C3×#124—33no—
11C2×C3×#120—31no—
12C2×C3×#120C2×C313yescomplemented
13#5450C2×C3112yesdirect product factor
14C2×#120⋊C3C2×C319yescomplemented
15C3×C22⋊A4C2311yescharacteristic, complemented
16C22×C3×A4—43no—
17C22×C3×A4—44no—
18C2×H(3,𝔽4)C3211yescharacteristic, complemented, Sylow
19C25×C3A411yescharacteristic, complemented
20C25×C3C22×C311yescharacteristic, complemented
21C2×C22⋊A4C22×C313yescomplemented
22C23×A4—43no—
23C22×C3×Q8—31no—
24C22×C3×D4—33no—
25C3×#124—312no—
26C3×#124—312no—
27C3×#120—34no—
28C3×#120—34no—
29C3×#120—36no—
30C3×#120C22×C316yescomplemented
31C2×C3×C22⋊C4—33no—
32C2×C3×C22⋊C4—33no—
33C2×C3×C42—31no—
34#120⋊C3C22×C3118yescomplemented
35C23×C32—161no—
36C2×C3×A4—41no—
37C2×C3×A4—46no—
38C2×C3×A4—161no—
39H(3,𝔽4)—34no—
40H(3,𝔽4)C2×C3214yescomplemented
41C2×#124—33no—
42C2×#120—31no—
43C2×#120C2×C3213yescomplemented
44C24×C3—31no—
45C24×C3—31no—
46C24×C3—121no—
47C24×C3—121no—
48C24×C3—121no—
49C24×C3—121no—
50C24×C3C2×A413yes—
51C24×C3C2×A414yescomplemented
52C24×C3C23×C311yescharacteristic, complemented
53C22⋊A4C23×C313yescomplemented
54C22×A4—49no—
55C22×A4—412no—
56C2×C3×Q8—34no—
57C2×C3×Q8—121no—
58C2×C3×Q8—121no—
59C2×C3×D4—36no—
60C2×C3×D4—36no—
61C2×C3×D4—123no—
62C2×C3×D4—123no—
63C22×C3×C4—33no—
64C3×C22⋊C4—36no—
65C3×C22⋊C4—312no—
66C3×C22⋊C4—36no—
67C3×C42—34no—
68C22×C32—164no—
69C22×C32—163no—
70C3×A4—41no—
71C3×A4—161no—
72C25C3×A411yescharacteristic, complemented
73C25C22×C3211yescharacteristic, complemented
74C22×Q8—31no—
75C22×D4—33no—
76#124—312no—
77#124—312no—
78#120—34no—
79#120—34no—
80#120—36no—
81#120C22×C3216yescomplemented
82C2×C22⋊C4—33no—
83C2×C22⋊C4—33no—
84C2×C42—31no—
85C23×C3—31no—
86C23×C3—31no—
87C23×C3—41no—
88C23×C3—41no—
89C23×C3—63no—
90C23×C3—63no—
91C23×C3—63no—
92C23×C3—63no—
93C23×C3—121no—
94C23×C3—124no—
95C23×C3—121no—
96C23×C3—124no—
97C23×C3—121no—
98C23×C3—123no—
99C23×C3—121no—
100C23×C3—123no—
101C23×C3—163no—
102C23×C3C22×A411yescharacteristic
103C23×C3C22×A416yes—
104C2×A4—418no—
105C2×A4—43no—
106C2×A4—163no—
107C3×Q8—124no—
108C3×D4—126no—
109C3×D4—126no—
110C2×C3×C4—36no—
111C2×C3×C4—63no—
112C2×C3×C4—63no—
113C2×C32—166no—
114C2×C32—161no—
115C24—31no—
116C24—31no—
117C24—121no—
118C24—121no—
119C24—121no—
120C24—121no—
121C24C2×C3×A413yes—
122C24C2×C3×A414yescomplemented
123C24C23×C3211yescharacteristic, complemented
124C2×Q8—34no—
125C2×Q8—121no—
126C2×Q8—121no—
127C2×D4—36no—
128C2×D4—36no—
129C2×D4—123no—
130C2×D4—123no—
131C22×C4—33no—
132C22⋊C4—36no—
133C22⋊C4—312no—
134C22⋊C4—36no—
135C42—34no—
136C22×C3—31no—
137C22×C3—31no—
138C22×C3—44no—
139C22×C3—43no—
140C22×C3—41no—
141C22×C3—66no—
142C22×C3—63no—
143C22×C3—63no—
144C22×C3—123no—
145C22×C3—123no—
146C22×C3—124no—
147C22×C3—121no—
148C22×C3—121no—
149C22×C3—123no—
150C22×C3—121no—
151C22×C3—1612no—
152C22×C3—169no—
153C22×C3C23×A411yescharacteristic
154A4—43no—
155A4—163no—
156C3×C4—66no—
157C32—161noSylow
158C23—31no—
159C23—31no—
160C23—41no—
161C23—41no—
162C23—63no—
163C23—63no—
164C23—63no—
165C23—63no—
166C23—121no—
167C23—124no—
168C23—121no—
169C23—124no—
170C23—121no—
171C23—123no—
172C23—121no—
173C23—123no—
174C23C22×C3×A411yescharacteristic
175C23C22×C3×A416yes—
176Q8—124no—
177D4—126no—
178D4—126no—
179C2×C4—36no—
180C2×C4—63no—
181C2×C4—63no—
182C2×C3—31no—
183C2×C3—31no—
184C2×C3—46no—
185C2×C3—121no—
186C2×C3—121no—
187C2×C3—1618no—
188C2×C3—163no—
189C2×C3#545011yescharacteristic, direct product factor
190C22—31no—
191C22—31no—
192C22—44no—
193C22—43no—
194C22—41no—
195C22—66no—
196C22—63no—
197C22—63no—
198C22—123no—
199C22—123no—
200C22—124no—
201C22—121no—
202C22—121no—
203C22—123no—
204C22—121no—
205C22C23×C3×A411yescharacteristic
206C4—66no—
207C3—163no—
208C3C2×#545011yescharacteristic, direct product factor
209C2—31no—
210C2—31no—
211C2—46no—
212C2—121no—
213C2—121no—
214C2C3×#545011yescharacteristic, direct product factor
subgroupG/H
center, hypercenterC2×C3#5450
commutator, cosocle, hypocenterC24C23×C32
FrattiniC22C23×C3×A4
FittingC2×C3×H(3,𝔽4)C3
socleC23×C3C22×A4
LCSG ▷ C24
UCSC1 ◁ C2×C3

Complements

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

NH ≅ G/Ntype
1. C3×#5450C2direct (N × H)
2. C2×C3×#120⋊C3C2semidirect
3. C2×C3×H(3,𝔽4)C3semidirect
4. C2×#5450C3direct (N × H)
5. C2×C3×C22⋊A4C22semidirect
7. C3×#120⋊C3C22semidirect
9. C3×H(3,𝔽4)C2×C3semidirect
12. C2×C3×#120C2×C3semidirect
13. #5450C2×C3direct (N × H)
14. C2×#120⋊C3C2×C3semidirect
15. C3×C22⋊A4C23semidirect
18. C2×H(3,𝔽4)C32semidirect
19. C25×C3A4semidirect
20. C25×C3C22×C3semidirect
21. C2×C22⋊A4C22×C3semidirect
30. C3×#120C22×C3semidirect
34. #120⋊C3C22×C3semidirect
40. H(3,𝔽4)C2×C32semidirect
43. C2×#120C2×C32semidirect
51. C24×C3C2×A4semidirect
52. C24×C3C23×C3semidirect
53. C22⋊A4C23×C3semidirect
72. C25C3×A4semidirect
73. C25C22×C32semidirect
81. #120C22×C32semidirect
122. C24C2×C3×A4semidirect
123. C24C23×C32semidirect
189. C2×C3#5450direct (N × H)
208. C3C2×#5450direct (N × H)
214. C2C3×#5450direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#54502
C2×C3×#120⋊C32
C2×C3×H(3,𝔽4)3
C2×#54503
C23×C3×A44