Finite Group Databasebeta

← C2×#11724627 #13181682

Subgroups — C2×#11724627 #13181682

Order 1728.

4122 conjugacy classes of subgroups; only the 663 normal subgroups are shown.

normal subgroups only

This group has 4122 conjugacy classes of subgroups — too many to draw. Only its 663 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×C3×#3832C213yesmaximal, complemented
2#11724627C2116yesmaximal, direct product factor
3C2×C4⋊#1113C219yesmaximal, complemented
4C2×#89042C213yesmaximal, complemented
5C2×#64320—33nomaximal
6C22×#1113C2216yescomplemented
7C2×C33×Q8C2211yescharacteristic, complemented
8C3×#3832C22124yescomplemented
9C2×C3×C4⋊(C3⋊D3)C2219yescomplemented
10C2×C3×C4×C3⋊D3C2219yescomplemented
11C2×#6296C2211yescharacteristic, complemented
12C2×#6295C2219yescomplemented
13C4⋊#1113C22172yescomplemented
14#89042C22124yescomplemented
15C2×#3832D313yescomplemented
16C2×C3×#239—36no—
17C2×C3×#239—63no—
18#64320—348no—
19C2×#3802—39no—
20C2×D3×D12—318no—
21C2×#3798—39no—
22C22×C3×C3⋊D3C2319yes—
23C2×#1113—412no—
24C2×#1113C23124yescomplemented
25C33×Q8C2314yescomplemented
26C2×C33×C4C2313yes—
27C2×C3×C3⋊Dic3C2313yes—
28C3×C4⋊(C3⋊D3)C23136yescomplemented
29C3×C4×C3⋊D3C23136yescomplemented
30#6296C2314yescomplemented
31#6295C23136yescomplemented
32C2×#1009—93no—
33C22×D32—318no—
34C2×C32×Q8—23no—
35C2×C32×Q8—41no—
36C2×C32×Q8C2×D313yescomplemented
37#3832C2×D3124yescomplemented
38C2×C4⋊(C3⋊D3)C2×D319yescomplemented
39C2×C4×C3⋊D3C2×D319yescomplemented
40C3×#239—348no—
41C3×#239—624no—
42C2×C3×D12—318no—
43C2×C3×D12—69no—
44C2×C3×C4×D3—318no—
45C2×C3×C4×D3—69no—
46C2×#627—33no—
47C2×#626—318no—
48C2×#625—39no—
49#3802—372no—
50D3×D12—3144no—
51#3798—372no—
52C22×C33C2411yescharacteristic
53C2×C3×C3⋊D3—218no—
54C2×C3×C3⋊D3C24118yes—
55#1113—424no—
56C33×C4C2416yes—
57C3×C3⋊Dic3C2416yes—
58C2×C3×#41—93no—
59#1009—948no—
60C2×#239—36no—
61C2×#239—63no—
62C2×D3×D4—918no—
63C2×#235—99no—
64C22×D12—99no—
65C22×C3⋊D3C22×D319yes—
66C22×C3×D3—318no—
67C22×C3×D3—69no—
68C2×D32—372no—
69C2×D32—1236no—
70C32×Q8—212no—
71C32×Q8—44no—
72C32×Q8C22×D3112yescomplemented
73C2×C32×C4—29no—
74C2×C32×C4—43no—
75C2×C32×C4C22×D319yes—
76C2×C3⋊Dic3C22×D313yes—
77C4⋊(C3⋊D3)C22×D3136yescomplemented
78C4×C3⋊D3C22×D3136yescomplemented
79C2×C3×Dic3—36no—
80C2×C3×Dic3—63no—
81C3×D12—372no—
82C3×D12—636no—
83C3×C4×D3—372no—
84C3×C4×D3—636no—
85#627—312no—
86#626—372no—
87#625—336no—
88C2×H(5,𝔽2)—271noSylow
89C2×C33H(5,𝔽2)12yescomplemented
90C2×C33C2511yescharacteristic
91C3×C3⋊D3—218no—
92C23×D3—918no—
93C3×#41—924no—
94C2×C3×Q8—23no—
95C2×C3×Q8—41no—
96C2×C3×Q8D3213yescomplemented
97C2×C3×D4—99no—
98C22×C3×C4—99no—
99C2×#67—918no—
100#239—348no—
101#239—624no—
102D3×D4—9144no—
103#235—972no—
104C2×D12—318no—
105C2×D12—69no—
106C2×D12—972no—
107C2×D12—99no—
108C2×C4×D3—318no—
109C2×C4×D3—69no—
110C2×Dic6—93no—
111C22×C32—23no—
112C22×C32—41no—
113C22×C32C23×D313yes—
114C2×C3⋊D3—218no—
115C2×C3⋊D3C23×D3118yes—
116C2×C3×D3—336no—
117C2×C3×D3—618no—
118C2×C3×D3—636no—
119C2×C3×D3—1218no—
120D32—1272no—
121C32×C4—218no—
122C32×C4—46no—
123C32×C4C23×D3118yes—
124C3⋊Dic3C23×D316yes—
125C3×Dic3—312no—
126C3×Dic3—66no—
127H(5,𝔽2)—2716no—
128C2×#41—93no—
129C2×#41—273no—
130C22×D4—279no—
131C33C2×H(5,𝔽2)11yescharacteristic, complemented, Sylow
132C23×C3—99no—
133C22×D3—318no—
134C22×D3—69no—
135C22×D3—972no—
136C22×D3—3636no—
137C3×Q8—212no—
138C3×Q8—44no—
139C3×Q8C2×D32112yescomplemented
140C3×D4—936no—
141C2×C3×C4—29no—
142C2×C3×C4—43no—
143C2×C3×C4—93no—
144C2×C3×C4—936no—
145C2×C3×C4C2×D3219yes—
146#67—972no—
147C2×Dic3—36no—
148C2×Dic3—63no—
149D12—372no—
150D12—636no—
151D12—936no—
152C4×D3—372no—
153C4×D3—636no—
154Dic6—912no—
155C2×C32—26no—
156C2×C32—23no—
157C2×C32—42no—
158C2×C32—41no—
159C2×C32#100916yescomplemented
160C2×C32C24×D313yes—
161C3⋊D3—218no—
162C3×D3—636no—
163C3×D3—1218no—
164C24—276no—
165#41—924no—
166#41—2724no—
167C2×Q8—271no—
168C2×Q8#111311yescharacteristic, complemented
169C2×D4—99no—
170C2×D4—2772no—
171C2×D4—279no—
172C22×C4—99no—
173C22×C3—23no—
174C22×C3—41no—
175C22×C3—918no—
176C22×C3—1818no—
177C22×C3C22×D3213yes—
178C2×D3—336no—
179C2×D3—618no—
180C2×D3—636no—
181C2×D3—1218no—
182C2×D3—3672no—
183C3×C4—218no—
184C3×C4—46no—
185C3×C4—96no—
186C3×C4C22×D32118yes—
187Dic3—312no—
188Dic3—66no—
189C32—23no—
190C32—41no—
191C32C2×#100913yescomplemented
192C23—99no—
193C23—2724no—
194C23—10812no—
195Q8—274no—
196Q8C2×#111314yescomplemented
197D4—936no—
198D4—2736no—
199C2×C4—93no—
200C2×C4—936no—
201C2×C4C2×#111313yes—
202C2×C3—26no—
203C2×C3—23no—
204C2×C3—42no—
205C2×C3—41no—
206C2×C3—1818no—
207C2×C3#6432016yescomplemented
208C2×C3C23×D3213yes—
209D3—636no—
210D3—1218no—
211C22—918no—
212C22—1818no—
213C22—10824no—
214C22C22×#111311yescharacteristic
215C4—96no—
216C4C22×#111316yes—
217C3—23no—
218C3—41no—
219C3C2×#6432013yescomplemented
220C2—1818no—
221C2#1172462712yesdirect product factor
222C2C23×#111311yescharacteristic
subgroupG/H
centerC22C22×#1113
commutator, cosocleC2×C33C25
FrattiniC2C23×#1113
FittingC2×C33×Q8C22
socleC22×C33C24
hypercenterC2×Q8#1113
hypocenterC33C2×H(5,𝔽2)
LCSG ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×Q8

Complements

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

NH ≅ G/Ntype
1. C2×C3×#3832C2semidirect
2. #11724627C2direct (N × H)
3. C2×C4⋊#1113C2semidirect
4. C2×#89042C2semidirect
6. C22×#1113C22semidirect
7. C2×C33×Q8C22semidirect
8. C3×#3832C22semidirect
9. C2×C3×C4⋊(C3⋊D3)C22semidirect
10. C2×C3×C4×C3⋊D3C22semidirect
11. C2×#6296C22semidirect
12. C2×#6295C22semidirect
13. C4⋊#1113C22semidirect
14. #89042C22semidirect
15. C2×#3832D3semidirect
24. C2×#1113C23semidirect
25. C33×Q8C23semidirect
28. C3×C4⋊(C3⋊D3)C23semidirect
29. C3×C4×C3⋊D3C23semidirect
30. #6296C23semidirect
31. #6295C23semidirect
36. C2×C32×Q8C2×D3semidirect
37. #3832C2×D3semidirect
38. C2×C4⋊(C3⋊D3)C2×D3semidirect
39. C2×C4×C3⋊D3C2×D3semidirect
72. C32×Q8C22×D3semidirect
77. C4⋊(C3⋊D3)C22×D3semidirect
78. C4×C3⋊D3C22×D3semidirect
89. C2×C33H(5,𝔽2)semidirect
96. C2×C3×Q8D32semidirect
131. C33C2×H(5,𝔽2)semidirect
139. C3×Q8C2×D32semidirect
159. C2×C32#1009semidirect
168. C2×Q8#1113semidirect
191. C32C2×#1009semidirect
196. Q8C2×#1113semidirect
207. C2×C3#64320semidirect
219. C3C2×#64320semidirect
221. C2#11724627direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#38322
#117246272
C2×C4⋊#11132
C2×#890422
C2×#643203