Finite Group Databasebeta

← C2×#87574 #11709084

Subgroups — C2×#87574 #11709084

Order 768.

60570 conjugacy classes of subgroups.

The subgroup lattice has 244 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#585C211yesmaximal, characteristic, complemented
2#87574C2164yesmaximal, direct product factor
3C2×#5968C2112yesmaximal, complemented
4C2×D3×H(5,𝔽2)C2120yesmaximal, complemented
5C2×#5964C2115yesmaximal, complemented
6C22×#1009C2115yesmaximal, complemented
7C2×H(7,𝔽2)—31nomaximal, Sylow
8C3×#585C22132yescomplemented
9C2×C3×2−1+4C2216yescomplemented
10C2×C3×H(5,𝔽2)C22110yescomplemented
11C22×C3×#41C22115yescomplemented
12#5968C221384yescomplemented
13D3×H(5,𝔽2)C221640yescomplemented
14#5964C221480yescomplemented
15C2×#1010C22120yescomplemented
16C2×#1009C22160yescomplemented
17C2×#1009C221120yescomplemented
18C2×#1009C221480yescomplemented
19C2×D3×#41C221120yescomplemented
20C2×#1007C22130yescomplemented
21C22×#239C22130yescomplemented
22C2×#1004C22190yescomplemented
23C22×D3×D4C22190yescomplemented
24C22×#235C22145yescomplemented
25C23×D12C22115yescomplemented
26H(7,𝔽2)—364no—
27C2×#585—312no—
28C2×#585—315no—
29C2×#585D311yescharacteristic, complemented
30C22×H(5,𝔽2)—320no—
31C22×H(5,𝔽2)—315no—
32C24×D3C23130yescomplemented
33C3×2−1+4C23196yescomplemented
34C3×H(5,𝔽2)C231160yescomplemented
35C2×C3×#41C23160yescomplemented
36C2×C3×#41C23120yescomplemented
37C2×C3×#41C231240yescomplemented
38C22×C3×Q8C23115yescomplemented
39C22×C3×D4C23145yescomplemented
40C23×C3×C4C23115yescomplemented
41C22×#67C23190yescomplemented
42#1010C231320yescomplemented
43#1009C231960yescomplemented
44#1009C2311920yescomplemented
45D3×#41C2311920yescomplemented
46#1007C231480yescomplemented
47C2×#239C231120yescomplemented
48C2×#239C231480yescomplemented
49C2×D3×Q8C23140yescomplemented
50#1004C2311440yescomplemented
51C2×#237C231120yescomplemented
52C2×D3×D4C231360yescomplemented
53C2×D3×D4C2311440yescomplemented
54C2×#235C23160yescomplemented
55C2×#235C231180yescomplemented
56C2×#235C231720yescomplemented
57C22×D12C23145yescomplemented
58C22×D12C23190yescomplemented
59C22×D12C231240yescomplemented
60C22×C4×D3C23190yescomplemented
61C22×Dic6C23115yescomplemented
62#585—3384no—
63#585—3480no—
64#585C2×D3132yescomplemented
65C2×2−1+4—320no—
66C2×2−1+4—330no—
67C2×2−1+4C2×D316yescomplemented
68C2×H(5,𝔽2)—3640no—
69C2×H(5,𝔽2)—3120no—
70C2×H(5,𝔽2)—360no—
71C2×H(5,𝔽2)—3480no—
72C2×H(5,𝔽2)—390no—
73C2×H(5,𝔽2)C2×D3110yescomplemented
74C22×#41—3120no—
75C22×#41—330no—
76C22×#41—345no—
77C22×#41C2×D3115yescomplemented
78C23×D4—390no—
79C23×D4—315no—
80C24×C3C24115yes—
81C23×D3—860no—
82C23×D3C24190yes—
83C23×D3C241240yescomplemented
84C3×#41C241480yescomplemented
85C3×#41C241160yescomplemented
86C2×C3×Q8C24120yes—
87C2×C3×Q8C241120yescomplemented
88C2×C3×D4C24160yes—
89C2×C3×D4C241360yescomplemented
90C22×C3×C4C24115yes—
91C22×C3×C4C24145yes—
92C22×C3×C4C241120yescomplemented
93C2×#67C241120yes—
94C2×#67C241720yescomplemented
95C22×Dic3C24130yes—
96#239C241960yescomplemented
97D3×Q8C241320yescomplemented
98#237C241960yescomplemented
99D3×D4C2412880yescomplemented
100#235C241480yescomplemented
101#235C2411440yescomplemented
102C2×D12C24110yes—
103C2×D12C24190yes—
104C2×D12C241360yescomplemented
105C2×D12C241720yescomplemented
106C2×C4×D3C241120yes—
107C2×C4×D3C241720yescomplemented
108C2×Dic6C2416yes—
109C2×Dic6C24130yes—
110C2×Dic6C241120yescomplemented
111C25—330no—
1122−1+4—3320no—
1132−1+4—3480no—
1142−1+4C22×D3196yescomplemented
115H(5,𝔽2)—31920no—
116H(5,𝔽2)—3960no—
117H(5,𝔽2)—31440no—
118H(5,𝔽2)C22×D31160yescomplemented
119C2×#41—31920no—
120C2×#41—3120no—
121C2×#41—3480no—
122C2×#41—3120no—
123C2×#41—3180no—
124C2×#41—360no—
125C2×#41—3720no—
126C2×#41C22×D3160yescomplemented
127C2×#41C22×D3120yescomplemented
128C2×#41C22×D31240yescomplemented
129C22×Q8—340no—
130C22×Q8—315no—
131C22×Q8C22×D3115yescomplemented
132C22×D4—390no—
133C22×D4—3360no—
134C22×D4—31440no—
135C22×D4—390no—
136C22×D4—345no—
137C22×D4—3240no—
138C22×D4C22×D3145yescomplemented
139C23×C4—390no—
140C23×C4C22×D3115yescomplemented
141C23×C3—430no—
142C23×C3C25115yes—
143C23×C3C25160yes—
144C22×D3—4180no—
145C22×D3—8240no—
146C22×D3C25120yes—
147C22×D3C251360yes—
148C3×Q8C25180yes—
149C3×D4C251240yes—
150C2×C3×C4C2511yescharacteristic
151C2×C3×C4C25115yes—
152C2×C3×C4C25160yes—
153C2×C3×C4C251180yes—
154#67C251480yes—
155C2×Dic3C25112yes—
156C2×Dic3C251120yes—
157D12C25140yes—
158D12C251360yes—
159C4×D3C251480yes—
160Dic6C25124yes—
161Dic6C251120yes—
162C24—390no—
163C24—3240no—
164C24—2460no—
165C24C23×D3115yes—
166#41—3960no—
167#41—3960no—
168#41—31440no—
169#41—3480no—
170#41C23×D31480yescomplemented
171#41C23×D31160yescomplemented
172C2×Q8—3320no—
173C2×Q8—330no—
174C2×Q8—36no—
175C2×Q8—3120no—
176C2×Q8C23×D3120yes—
177C2×Q8C23×D31120yescomplemented
178C2×D4—3120no—
179C2×D4—3720no—
180C2×D4—32880no—
181C2×D4—390no—
182C2×D4—310no—
183C2×D4—3720no—
184C2×D4—3360no—
185C2×D4C23×D3160yes—
186C2×D4C23×D31360yescomplemented
187C22×C4—330no—
188C22×C4—3120no—
189C22×C4—3720no—
190C22×C4C23×D3145yes—
191C22×C4C23×D3115yes—
192C22×C4C23×D31120yescomplemented
193C22×C3—230no—
194C22×C3—460no—
195C22×C3C26130yes—
196C22×C3C2611yescharacteristic
197C2×D3—240no—
198C2×D3—4360no—
199C2×D3C26140yes—
200C3×C4C2612yes—
201C3×C4C26130yes—
202Dic3C26124yes—
203C23—320no—
204C23—3360no—
205C23—430no—
206C23—12180no—
207C23—24240no—
208C23C24×D3115yes—
209C23C24×D3160yes—
210Q8—3120no—
211Q8—324no—
212Q8C24×D3180yes—
213D4—3480no—
214D4—3360no—
215D4—340no—
216D4C24×D31240yes—
217C2×C4—3120no—
218C2×C4—312no—
219C2×C4—3480no—
220C2×C4C24×D31180yes—
221C2×C4C24×D3160yes—
222C2×C4C24×D3115yes—
223C2×C4C24×D311yescharacteristic
224C2×C3—230no—
225C2×C3H(7,𝔽2)12yescomplemented
226C2×C3C2711yescharacteristic
227D3—240no—
228C22—230no—
229C22—340no—
230C22—460no—
231C22—640no—
232C22—12360no—
233C22C25×D3130yes—
234C22C25×D311yescharacteristic
235C4—324no—
236C4C25×D3130yes—
237C4C25×D312yes—
238C3C2×H(7,𝔽2)11yescharacteristic, complemented, Sylow
239C2—230no—
240C2—640no—
241C2#8757412yesdirect product factor
242C2C26×D311yescharacteristic
subgroupG/H
centerC22C25×D3
commutator, cosocleC2×C3C27
FrattiniC2C26×D3
FittingC2×C3×#585C2
socleC22×C3C26
hypercenterC2×#585D3
hypocenterC3C2×H(7,𝔽2)
LCSG ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×#585

Complements

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

NH ≅ G/Ntype
1. C2×C3×#585C2semidirect
2. #87574C2direct (N × H)
3. C2×#5968C2semidirect
4. C2×D3×H(5,𝔽2)C2semidirect
5. C2×#5964C2semidirect
6. C22×#1009C2semidirect
8. C3×#585C22semidirect
9. C2×C3×2−1+4C22semidirect
10. C2×C3×H(5,𝔽2)C22semidirect
11. C22×C3×#41C22semidirect
12. #5968C22semidirect
13. D3×H(5,𝔽2)C22semidirect
14. #5964C22semidirect
15. C2×#1010C22semidirect
16. C2×#1009C22semidirect
17. C2×#1009C22semidirect
18. C2×#1009C22semidirect
19. C2×D3×#41C22semidirect
20. C2×#1007C22semidirect
21. C22×#239C22semidirect
22. C2×#1004C22semidirect
23. C22×D3×D4C22semidirect
24. C22×#235C22semidirect
25. C23×D12C22semidirect
29. C2×#585D3semidirect
32. C24×D3C23semidirect
33. C3×2−1+4C23semidirect
34. C3×H(5,𝔽2)C23semidirect
35. C2×C3×#41C23semidirect
36. C2×C3×#41C23semidirect
37. C2×C3×#41C23semidirect
38. C22×C3×Q8C23semidirect
39. C22×C3×D4C23semidirect
40. C23×C3×C4C23semidirect
41. C22×#67C23semidirect
42. #1010C23semidirect
43. #1009C23semidirect
44. #1009C23semidirect
45. D3×#41C23semidirect
46. #1007C23semidirect
47. C2×#239C23semidirect
48. C2×#239C23semidirect
49. C2×D3×Q8C23semidirect
50. #1004C23semidirect
51. C2×#237C23semidirect
52. C2×D3×D4C23semidirect
53. C2×D3×D4C23semidirect
54. C2×#235C23semidirect
55. C2×#235C23semidirect
56. C2×#235C23semidirect
57. C22×D12C23semidirect
58. C22×D12C23semidirect
59. C22×D12C23semidirect
60. C22×C4×D3C23semidirect
61. C22×Dic6C23semidirect
64. #585C2×D3semidirect
67. C2×2−1+4C2×D3semidirect
73. C2×H(5,𝔽2)C2×D3semidirect
77. C22×#41C2×D3semidirect
83. C23×D3C24semidirect
84. C3×#41C24semidirect
85. C3×#41C24semidirect
87. C2×C3×Q8C24semidirect
89. C2×C3×D4C24semidirect
92. C22×C3×C4C24semidirect
94. C2×#67C24semidirect
96. #239C24semidirect
97. D3×Q8C24semidirect
98. #237C24semidirect
99. D3×D4C24semidirect
100. #235C24semidirect
101. #235C24semidirect
104. C2×D12C24semidirect
105. C2×D12C24semidirect
107. C2×C4×D3C24semidirect
110. C2×Dic6C24semidirect
114. 2−1+4C22×D3semidirect
118. H(5,𝔽2)C22×D3semidirect
126. C2×#41C22×D3semidirect
127. C2×#41C22×D3semidirect
128. C2×#41C22×D3semidirect
131. C22×Q8C22×D3semidirect
138. C22×D4C22×D3semidirect
140. C23×C4C22×D3semidirect
170. #41C23×D3semidirect
171. #41C23×D3semidirect
177. C2×Q8C23×D3semidirect
186. C2×D4C23×D3semidirect
192. C22×C4C23×D3semidirect
225. C2×C3H(7,𝔽2)semidirect
238. C3C2×H(7,𝔽2)semidirect
241. C2#87574direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#5852
#875742
C2×#59682
C2×D3×H(5,𝔽2)2
C2×#59642
C22×#10092
C2×H(7,𝔽2)3