Finite Group Databasebeta

← C2×#11724684 #13181761

Subgroups — C2×#11724684 #13181761

Order 1728.

12572 conjugacy classes of subgroups; only the 2267 normal subgroups are shown.

normal subgroups only

This group has 12572 conjugacy classes of subgroups — too many to draw. Only its 2267 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×C33×#41C211yesmaximal, characteristic, complemented
2#11724684C2116yesmaximal, direct product factor
3C2×#89080C212yesmaximal, complemented
4C2×D4×C3⋊(C3⋊D3)C216yesmaximal, complemented
5C2×#89076C213yesmaximal, complemented
6C22×C4⋊(C3⋊(C3⋊D3))C213yesmaximal, complemented
7C2×#64366—313nomaximal
8C23×C3⋊(C3⋊D3)C2216yescomplemented
9C33×#41C2218yescomplemented
10C2×C33×Q8C2211yescharacteristic, complemented
11C2×C33×D4C2213yescomplemented
12C22×C33×C4C2213yescomplemented
13C2×#6312C2216yescomplemented
14#89080C22116yescomplemented
15D4×C3⋊(C3⋊D3)C22148yescomplemented
16#89076C22124yescomplemented
17C2×C4⋊(C3⋊(C3⋊D3))C2213yescomplemented
18C2×C4⋊(C3⋊(C3⋊D3))C2216yescomplemented
19C2×C4⋊(C3⋊(C3⋊D3))C22124yescomplemented
20C2×C4×C3⋊(C3⋊D3)C2216yescomplemented
21C2×#6308C2211yescharacteristic, complemented
22C2×C32×#41D3113yescomplemented
23#64366—3208no—
24C2×#3832—326no—
25C2×D4×C3⋊D3—378no—
26C2×#3828—339no—
27C22×C4⋊(C3⋊D3)—339no—
28C23×C33C2313yes—
29C22×C3⋊(C3⋊D3)—412no—
30C22×C3⋊(C3⋊D3)C2316yes—
31C22×C3⋊(C3⋊D3)C23124yescomplemented
32C33×Q8C2314yescomplemented
33C33×D4C23112yescomplemented
34C2×C33×C4C2311yescharacteristic
35C2×C33×C4C2313yes—
36C2×C33×C4C23112yescomplemented
37#6312C23124yescomplemented
38C2×C3⋊(C3⋊Dic3)C2312yes—
39C4⋊(C3⋊(C3⋊D3))C23112yescomplemented
40C4⋊(C3⋊(C3⋊D3))C23124yescomplemented
41C4×C3⋊(C3⋊D3)C23124yescomplemented
42#6308C2314yescomplemented
43C2×#1009—913no—
44C23×C3⋊D3—378no—
45C32×#41C2×D31104yescomplemented
46C2×C32×Q8C2×D3113yescomplemented
47C2×C32×D4C2×D3139yescomplemented
48C22×C32×C4C2×D3139yescomplemented
49C2×#638—378no—
50#3832—3208no—
51D4×C3⋊D3—3624no—
52#3828—3312no—
53C2×C4⋊(C3⋊D3)—339no—
54C2×C4⋊(C3⋊D3)—378no—
55C2×C4⋊(C3⋊D3)—3312no—
56C2×C4×C3⋊D3—378no—
57C2×#634—313no—
58C22×C33—26no—
59C22×C33C2416yes—
60C22×C33C2411yescharacteristic
61C2×C3⋊(C3⋊D3)—212no—
62C2×C3⋊(C3⋊D3)—424no—
63C2×C3⋊(C3⋊D3)C24112yes—
64C33×C4C2412yes—
65C33×C4C2416yes—
66C3⋊(C3⋊Dic3)C2414yes—
67C2×C3×#41C3⋊D3113yescomplemented
68#1009—9208no—
69C2×#239—926no—
70C2×D3×D4—978no—
71C2×#235—939no—
72C22×D12—939no—
73C23×C32C22×D3139yes—
74C22×C3⋊D3—378no—
75C22×C3⋊D3—3312no—
76C22×C3⋊D3—12156no—
77C32×Q8C22×D3152yescomplemented
78C32×D4C22×D31156yescomplemented
79C2×C32×C4C22×D3113yes—
80C2×C32×C4C22×D3139yes—
81C2×C32×C4C22×D31156yescomplemented
82#638—3312no—
83C2×C3⋊Dic3—326no—
84C4⋊(C3⋊D3)—3156no—
85C4⋊(C3⋊D3)—3312no—
86C4×C3⋊D3—3312no—
87#634—352no—
88C2×H(5,𝔽2)—271noSylow
89C2×C33—26no—
90C2×C33H(5,𝔽2)12yescomplemented
91C2×C33C2511yescharacteristic
92C3⋊(C3⋊D3)—212no—
93C23×D3—978no—
94C3×#41C2×C3⋊D31104yescomplemented
95C2×C3×Q8C2×C3⋊D3113yescomplemented
96C2×C3×D4C2×C3⋊D3139yescomplemented
97C22×C3×C4C2×C3⋊D3139yescomplemented
98C2×#67—978no—
99#239—9208no—
100D3×D4—9624no—
101#235—9312no—
102C2×D12—939no—
103C2×D12—978no—
104C2×D12—9312no—
105C2×C4×D3—978no—
106C2×Dic6—913no—
107C22×C32—278no—
108C22×C32C23×D3178yes—
109C22×C32C23×D3113yes—
110C2×C3⋊D3—3156no—
111C2×C3⋊D3—6156no—
112C2×C3⋊D3—12312no—
113C32×C4C23×D3126yes—
114C32×C4C23×D3178yes—
115C3⋊Dic3—352no—
116H(5,𝔽2)—2716no—
117C2×#41—272no—
118C2×#41—273no—
119C2×#41C3⋊(C3⋊D3)11yescharacteristic, complemented
120C22×D4—276no—
121C22×D4—273no—
122C33C2×H(5,𝔽2)11yescharacteristic, complemented, Sylow
123C23×C3C22×C3⋊D3139yes—
124C22×D3—978no—
125C22×D3—9312no—
126C22×D3—36156no—
127C3×Q8C22×C3⋊D3152yescomplemented
128C3×D4C22×C3⋊D31156yescomplemented
129C2×C3×C4C22×C3⋊D3113yes—
130C2×C3×C4C22×C3⋊D3139yes—
131C2×C3×C4C22×C3⋊D31156yescomplemented
132#67—9312no—
133C2×Dic3—926no—
134D12—9156no—
135D12—9312no—
136C4×D3—9312no—
137Dic6—952no—
138C2×C32—278no—
139C2×C32#1009126yescomplemented
140C2×C32C24×D3113yes—
141C3⋊D3—6156no—
142C24—276no—
143#41—2716no—
144#41—2724no—
145#41C2×C3⋊(C3⋊D3)18yescomplemented
146C2×Q8—271no—
147C2×Q8C2×C3⋊(C3⋊D3)11yescharacteristic, complemented
148C2×D4—276no—
149C2×D4—2748no—
150C2×D4—276no—
151C2×D4—273no—
152C2×D4—2724no—
153C2×D4C2×C3⋊(C3⋊D3)13yescomplemented
154C22×C4—276no—
155C22×C4C2×C3⋊(C3⋊D3)13yescomplemented
156C22×C3—278no—
157C22×C3C23×C3⋊D3178yes—
158C22×C3C23×C3⋊D3113yes—
159C2×D3—9156no—
160C2×D3—18156no—
161C2×D3—36312no—
162C3×C4C23×C3⋊D3126yes—
163C3×C4C23×C3⋊D3178yes—
164Dic3—952no—
165C32C2×#1009113yescomplemented
166C23—276no—
167C23—2724no—
168C23—10812no—
169C23C22×C3⋊(C3⋊D3)13yes—
170Q8—274no—
171Q8C22×C3⋊(C3⋊D3)14yescomplemented
172D4—2724no—
173D4—2724no—
174D4—2712no—
175D4C22×C3⋊(C3⋊D3)112yescomplemented
176C2×C4—272no—
177C2×C4—2724no—
178C2×C4C22×C3⋊(C3⋊D3)13yes—
179C2×C4C22×C3⋊(C3⋊D3)11yescharacteristic
180C2×C4C22×C3⋊(C3⋊D3)112yescomplemented
181C2×C3—278no—
182C2×C3#64366126yescomplemented
183C2×C3C24×C3⋊D3113yes—
184D3—18156no—
185C22—26no—
186C22—2712no—
187C22—5412no—
188C22—10824no—
189C22C23×C3⋊(C3⋊D3)16yes—
190C22C23×C3⋊(C3⋊D3)11yescharacteristic
191C4—274no—
192C4C23×C3⋊(C3⋊D3)16yes—
193C4C23×C3⋊(C3⋊D3)12yes—
194C3C2×#64366113yescomplemented
195C2—26no—
196C2—5412no—
197C2#1172468412yesdirect product factor
198C2C24×C3⋊(C3⋊D3)11yescharacteristic
subgroupG/H
centerC22C23×C3⋊(C3⋊D3)
commutator, cosocleC2×C33C25
FrattiniC2C24×C3⋊(C3⋊D3)
FittingC2×C33×#41C2
socleC22×C33C24
hypercenterC2×#41C3⋊(C3⋊D3)
hypocenterC33C2×H(5,𝔽2)
LCSG ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×#41

Complements

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

NH ≅ G/Ntype
1. C2×C33×#41C2semidirect
2. #11724684C2direct (N × H)
3. C2×#89080C2semidirect
4. C2×D4×C3⋊(C3⋊D3)C2semidirect
5. C2×#89076C2semidirect
6. C22×C4⋊(C3⋊(C3⋊D3))C2semidirect
8. C23×C3⋊(C3⋊D3)C22semidirect
9. C33×#41C22semidirect
10. C2×C33×Q8C22semidirect
11. C2×C33×D4C22semidirect
12. C22×C33×C4C22semidirect
13. C2×#6312C22semidirect
14. #89080C22semidirect
15. D4×C3⋊(C3⋊D3)C22semidirect
16. #89076C22semidirect
17. C2×C4⋊(C3⋊(C3⋊D3))C22semidirect
18. C2×C4⋊(C3⋊(C3⋊D3))C22semidirect
19. C2×C4⋊(C3⋊(C3⋊D3))C22semidirect
20. C2×C4×C3⋊(C3⋊D3)C22semidirect
21. C2×#6308C22semidirect
22. C2×C32×#41D3semidirect
31. C22×C3⋊(C3⋊D3)C23semidirect
32. C33×Q8C23semidirect
33. C33×D4C23semidirect
36. C2×C33×C4C23semidirect
37. #6312C23semidirect
39. C4⋊(C3⋊(C3⋊D3))C23semidirect
40. C4⋊(C3⋊(C3⋊D3))C23semidirect
41. C4×C3⋊(C3⋊D3)C23semidirect
42. #6308C23semidirect
45. C32×#41C2×D3semidirect
46. C2×C32×Q8C2×D3semidirect
47. C2×C32×D4C2×D3semidirect
48. C22×C32×C4C2×D3semidirect
67. C2×C3×#41C3⋊D3semidirect
77. C32×Q8C22×D3semidirect
78. C32×D4C22×D3semidirect
81. C2×C32×C4C22×D3semidirect
90. C2×C33H(5,𝔽2)semidirect
94. C3×#41C2×C3⋊D3semidirect
95. C2×C3×Q8C2×C3⋊D3semidirect
96. C2×C3×D4C2×C3⋊D3semidirect
97. C22×C3×C4C2×C3⋊D3semidirect
119. C2×#41C3⋊(C3⋊D3)semidirect
122. C33C2×H(5,𝔽2)semidirect
127. C3×Q8C22×C3⋊D3semidirect
128. C3×D4C22×C3⋊D3semidirect
131. C2×C3×C4C22×C3⋊D3semidirect
139. C2×C32#1009semidirect
145. #41C2×C3⋊(C3⋊D3)semidirect
147. C2×Q8C2×C3⋊(C3⋊D3)semidirect
153. C2×D4C2×C3⋊(C3⋊D3)semidirect
155. C22×C4C2×C3⋊(C3⋊D3)semidirect
165. C32C2×#1009semidirect
171. Q8C22×C3⋊(C3⋊D3)semidirect
175. D4C22×C3⋊(C3⋊D3)semidirect
180. C2×C4C22×C3⋊(C3⋊D3)semidirect
182. C2×C3#64366semidirect
194. C3C2×#64366semidirect
197. C2#11724684direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C33×#412
#117246842
C2×#890802
C2×D4×C3⋊(C3⋊D3)2
C2×#890762
C22×C4⋊(C3⋊(C3⋊D3))2
C2×#643663