Finite Group Databasebeta

← C3×#87571 #11919288

Subgroups — C3×#87571 #11919288

Order 1152.

11989 conjugacy classes of subgroups.

The subgroup lattice has 250 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C32×2−1+4C211yesmaximal, characteristic, complemented
2C3×#5968C2116yesmaximal, complemented
3C3×D3×2−1+4C2116yesmaximal, complemented
4C3×#5964C2120yesmaximal, complemented
5C2×C3×#1007C2110yesmaximal, complemented
6C3×2−1+6—31nomaximal
7#87571C311yesmaximal, characteristic, direct product factor
8C32×2−1+4C22116yescomplemented
9C2×C32×#41C22110yescomplemented
10C22×C32×Q8C2215yescomplemented
11C3×#1010C22180yescomplemented
12C3×#1009C22180yescomplemented
13C3×D3×#41C221160yescomplemented
14C3×#1007C22140yescomplemented
15C3×#1007C22180yescomplemented
16C2×C3×#239C22140yescomplemented
17C2×C3×D3×Q8C22140yescomplemented
18C3×#1004C22140yescomplemented
19C2×C3×#235C22160yescomplemented
20C3×#585—316no—
21C3×#585—320no—
22C2×C3×2−1+4—21no—
23C2×C3×2−1+4—316no—
24C2×C3×2−1+4—310no—
25C2×C3×2−1+4D311yescharacteristic, complemented
26C2×C3×2−1+4C2×C311yescharacteristic, complemented
27#5968C2×C3116yescomplemented
28D3×2−1+4C2×C3116yescomplemented
29#5964C2×C3120yescomplemented
30C2×#1007C2×C3110yescomplemented
31C32×#41C23180yes—
32C2×C32×Q8C23140yes—
33C2×C32×Q8C23110yes—
34C2×C32×D4C23110yes—
35C22×C32×C4C23115yes—
36C2×C3×#67C23140yes—
37C3×#239C231160yes—
38C3×D3×Q8C231160yes—
39C3×#237C231160yes—
40C3×D3×D4C231160yes—
41C3×#235C23180yes—
42C3×#235C231240yes—
43C2×C3×D12C23160yes—
44C2×C3×C4×D3C231120yes—
45C2×C3×Dic6C23160yes—
462−1+6—31noSylow
47C3×2−1+4—216no—
48C3×2−1+4—380no—
49C3×2−1+4—380no—
50C3×2−1+4—340no—
51C3×2−1+4C2×D3116yescomplemented
52C3×2−1+4C22×C3116yescomplemented
53C3×H(5,𝔽2)—380no—
54C3×H(5,𝔽2)—340no—
55C2×C3×#41—210no—
56C2×C3×#41—3160no—
57C2×C3×#41—340no—
58C2×C3×#41—360no—
59C2×C3×#41C2×D3110yescomplemented
60C2×C3×#41C22×C3110yescomplemented
61C22×C3×Q8—25no—
62C22×C3×Q8—340no—
63C22×C3×Q8C2×D315yescomplemented
64C22×C3×Q8C22×C315yescomplemented
65#1010C22×C3180yescomplemented
66#1009C22×C3180yescomplemented
67D3×#41C22×C31160yescomplemented
68#1007C22×C3140yescomplemented
69#1007C22×C3180yescomplemented
70C2×#239C22×C3140yescomplemented
71C2×D3×Q8C22×C3140yescomplemented
72#1004C22×C3140yescomplemented
73C2×#235C22×C3160yescomplemented
74C23×C32C2415yes—
75C22×C3×D3C24140yes—
76C32×Q8C24140yes—
77C32×D4C24140yes—
78C2×C32×C4C24160yes—
79C2×C32×C4C24110yes—
80C3×#67C24116yes—
81C3×#67C24180yes—
82C2×C3×Dic3C24140yes—
83C3×D12C24180yes—
84C3×C4×D3C241160yes—
85C3×Dic6C24180yes—
86#585—316no—
87#585—320no—
88C2×2−1+4—316no—
89C2×2−1+4—310no—
90C2×2−1+4C3×D311yescharacteristic, complemented
91C3×#41—280no—
92C3×#41—3160no—
93C3×#41—3160no—
94C3×#41—3240no—
95C3×#41—380no—
96C3×#41C22×D3180yes—
97C3×#41C23×C3180yes—
98C2×C3×Q8—210no—
99C2×C3×Q8—240no—
100C2×C3×Q8—3160no—
101C2×C3×Q8—360no—
102C2×C3×Q8C22×D3140yes—
103C2×C3×Q8C22×D3110yes—
104C2×C3×Q8C23×C3140yes—
105C2×C3×Q8C23×C3110yes—
106C2×C3×D4—210no—
107C2×C3×D4—340no—
108C2×C3×D4—3160no—
109C2×C3×D4—360no—
110C2×C3×D4C22×D3110yes—
111C2×C3×D4C23×C3110yes—
112C22×C3×C4—215no—
113C22×C3×C4—3120no—
114C22×C3×C4C22×D3115yes—
115C22×C3×C4C23×C3115yes—
116C2×#67C23×C3140yes—
117#239C23×C31160yes—
118D3×Q8C23×C31160yes—
119#237C23×C31160yes—
120D3×D4C23×C31160yes—
121#235C23×C3180yes—
122#235C23×C31240yes—
123C2×D12C23×C3160yes—
124C2×C4×D3C23×C31120yes—
125C2×Dic6C23×C3160yes—
126C22×C32—45no—
127C22×C32C25110yes—
128C22×C32C2511yescharacteristic
129C2×C3×D3—440no—
130C2×C3×D3C25116yes—
131C32×C4C25120yes—
132C3×Dic3C25116yes—
1332−1+4—380no—
1342−1+4—380no—
1352−1+4—340no—
1362−1+4C2×C3×D3116yescomplemented
137H(5,𝔽2)—380no—
138H(5,𝔽2)—340no—
139C2×#41—3160no—
140C2×#41—340no—
141C2×#41—360no—
142C2×#41C2×C3×D3110yescomplemented
143C22×Q8—340no—
144C22×Q8C2×C3×D315yescomplemented
145C23×C3—25no—
146C23×C3—340no—
147C23×C3C23×D315yes—
148C23×C3C24×C315yes—
149C22×D3C24×C3140yes—
150C3×Q8—240no—
151C3×Q8—380no—
152C3×Q8C23×D3140yes—
153C3×Q8C24×C3140yes—
154C3×D4—240no—
155C3×D4—380no—
156C3×D4—316no—
157C3×D4—380no—
158C3×D4C23×D3140yes—
159C3×D4C24×C3140yes—
160C2×C3×C4—210no—
161C2×C3×C4—260no—
162C2×C3×C4—340no—
163C2×C3×C4—3160no—
164C2×C3×C4C23×D3160yes—
165C2×C3×C4C23×D3110yes—
166C2×C3×C4C24×C3160yes—
167C2×C3×C4C24×C3110yes—
168#67C24×C3116yes—
169#67C24×C3180yes—
170C2×Dic3C24×C3140yes—
171D12C24×C3180yes—
172C4×D3C24×C31160yes—
173Dic6C24×C3180yes—
174C2×C32—21no—
175C2×C32—210no—
176C2×C32C2611yescharacteristic
177C3×D3—216no—
178#41—3160no—
179#41—3160no—
180#41—3240no—
181#41—380no—
182#41C22×C3×D3180yes—
183C2×Q8—3160no—
184C2×Q8—360no—
185C2×Q8C22×C3×D3140yes—
186C2×Q8C22×C3×D3110yes—
187C2×D4—340no—
188C2×D4—3160no—
189C2×D4—360no—
190C2×D4C22×C3×D3110yes—
191C22×C4—3120no—
192C22×C4C22×C3×D3115yes—
193C22×C3—21no—
194C22×C3—210no—
195C22×C3—316no—
196C22×C3—45no—
197C22×C3—45no—
198C22×C3—410no—
199C22×C3—1240no—
200C22×C3C24×D3110yes—
201C22×C3C24×D311yescharacteristic
202C22×C3C25×C3110yes—
203C22×C3C25×C311yescharacteristic
204C2×D3—440no—
205C2×D3C25×C3116yes—
206C3×C4—220no—
207C3×C4—316no—
208C3×C4C24×D3120yes—
209C3×C4C25×C3120yes—
210Dic3C25×C3116yes—
211C322−1+611yescharacteristic, complemented, Sylow
212C23—340no—
213C23C23×C3×D315yes—
214Q8—380no—
215Q8C23×C3×D3140yes—
216D4—380no—
217D4—316no—
218D4—380no—
219D4C23×C3×D3140yes—
220C2×C4—340no—
221C2×C4—3160no—
222C2×C4C23×C3×D3160yes—
223C2×C4C23×C3×D3110yes—
224C2×C3—22no—
225C2×C3—21no—
226C2×C3—210no—
227C2×C3—210no—
228C2×C3—21no—
229C2×C3—21no—
230C2×C3—410no—
231C2×C3—616no—
232C2×C3C25×D311yescharacteristic
233C2×C3C26×C311yescharacteristic
234D3—216no—
235C22—316no—
236C22—45no—
237C22—1240no—
238C22C24×C3×D3110yes—
239C22C24×C3×D311yescharacteristic
240C4—316no—
241C4C24×C3×D3120yes—
242C3—21no—
243C3#8757111yescharacteristic, direct product factor
244C3C3×2−1+611yescharacteristic, complemented
245C2—21no—
246C2—210no—
247C2—616no—
248C2C25×C3×D311yescharacteristic
subgroupG/H
centerC2×C3C25×D3
commutator, cosocleC2×C3C26×C3
FrattiniC2C25×C3×D3
FittingC2×C32×2−1+4C2
socleC2×C32C26
hypercenterC2×C3×2−1+4D3
hypocenterC3C3×2−1+6
LCSG ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C3 ◁ C2×C3×2−1+4

Complements

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

NH ≅ G/Ntype
1. C2×C32×2−1+4C2semidirect
2. C3×#5968C2semidirect
3. C3×D3×2−1+4C2semidirect
4. C3×#5964C2semidirect
5. C2×C3×#1007C2semidirect
7. #87571C3direct (N × H)
8. C32×2−1+4C22semidirect
9. C2×C32×#41C22semidirect
10. C22×C32×Q8C22semidirect
11. C3×#1010C22semidirect
12. C3×#1009C22semidirect
13. C3×D3×#41C22semidirect
14. C3×#1007C22semidirect
15. C3×#1007C22semidirect
16. C2×C3×#239C22semidirect
17. C2×C3×D3×Q8C22semidirect
18. C3×#1004C22semidirect
19. C2×C3×#235C22semidirect
25. C2×C3×2−1+4D3semidirect
26. C2×C3×2−1+4C2×C3semidirect
27. #5968C2×C3semidirect
28. D3×2−1+4C2×C3semidirect
29. #5964C2×C3semidirect
30. C2×#1007C2×C3semidirect
51. C3×2−1+4C2×D3semidirect
52. C3×2−1+4C22×C3semidirect
59. C2×C3×#41C2×D3semidirect
60. C2×C3×#41C22×C3semidirect
63. C22×C3×Q8C2×D3semidirect
64. C22×C3×Q8C22×C3semidirect
65. #1010C22×C3semidirect
66. #1009C22×C3semidirect
67. D3×#41C22×C3semidirect
68. #1007C22×C3semidirect
69. #1007C22×C3semidirect
70. C2×#239C22×C3semidirect
71. C2×D3×Q8C22×C3semidirect
72. #1004C22×C3semidirect
73. C2×#235C22×C3semidirect
90. C2×2−1+4C3×D3semidirect
136. 2−1+4C2×C3×D3semidirect
142. C2×#41C2×C3×D3semidirect
144. C22×Q8C2×C3×D3semidirect
211. C322−1+6semidirect
243. C3#87571direct (N × H)
244. C3C3×2−1+6semidirect

Maximal subgroups

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

subgroupindex
C2×C32×2−1+42
C3×#59682
C3×D3×2−1+42
C3×#59642
C2×C3×#10072
C3×2−1+63
#875713