Finite Group Databasebeta

← #11709048

Subgroups — #11709048

Order 768.

10894 conjugacy classes of subgroups; only the 3815 normal subgroups are shown.

normal subgroups only

This group has 10894 conjugacy classes of subgroups — too many to draw. Only its 3815 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
1#87574C212yesmaximal, complemented
2C3×#3573C211yesmaximal, characteristic, complemented
3#87167C2130yesmaximal, complemented
4#87163C2115yesmaximal, complemented
5C2×#5752C2115yesmaximal, complemented
6#63091—31nomaximal, Sylow
7C3×#585C2211yescharacteristic
8#5968—212no—
9D3×H(5,𝔽2)—220no—
10#5964C22130yescomplemented
11C2×#1009C22130yescomplemented
12C3×#568C22115yescomplemented
13C2×C3×#131C22115yescomplemented
14#5753C22120yescomplemented
15#5752C22160yescomplemented
16#5752C221120yescomplemented
17#5751C221120yescomplemented
18#5748C22190yescomplemented
19C2×#908C22190yescomplemented
20C2×#904C22145yescomplemented
21C22×D24C22115yescomplemented
22H(7,𝔽2)—32no—
23#3585—330no—
24#3584—315no—
25C2×#576—315no—
26#3573D311yescharacteristic, complemented
27C3×2−1+4D416yes—
28C3×H(5,𝔽2)D4110yes—
29C2×C3×#41C23115yes—
30#1010C23140yescomplemented
31#1009—2120no—
32#1009C231120yescomplemented
33D3×#41—2120no—
34#1007—230no—
35C2×#239—230no—
36#1004—290no—
37C2×D3×D4—290no—
38C2×#235C23190yescomplemented
39C22×D12C23130yescomplemented
40C3×#131C23120yescomplemented
41C3×#131C23160yescomplemented
42C2×C3×M16C23145yescomplemented
43C22×C3×C8C23115yescomplemented
44#909C231120yescomplemented
45#908C231360yescomplemented
46C2×Dic12C23115yescomplemented
47#904C23160yescomplemented
48#904C231180yescomplemented
49C2×D24C23145yescomplemented
50C2×D24C23190yescomplemented
51C2×#204C23190yescomplemented
52#585—330no—
53#585—612no—
54#585C2×D311yescharacteristic
55C2×H(5,𝔽2)—330no—
56C2×H(5,𝔽2)—620no—
57#578—320no—
58#577—3120no—
59#576—360no—
60#576—3120no—
61#575—390no—
62C2×AGL(1,ℤ/8)—390no—
63C2×#135—345no—
64C22×D8—315no—
65#568C2×D3115yescomplemented
66C2×#131C2×D3115yescomplemented
67C23×D3—230no—
68C3×#41C2×D4160yes—
69C3×#41C24120yes—
70C2×C3×Q8C2×D4115yes—
71C2×C3×D4C2×D4145yes—
72C22×C3×C4C24115yes—
73C2×#67—290no—
74#239—2120no—
75D3×Q8—240no—
76#237—2120no—
77D3×D4—2360no—
78#235—2180no—
79#235C241120yes—
80C2×D12—290no—
81C2×D12C24190yes—
82C2×C4×D3—290no—
83C2×Dic6C24130yes—
84C3×M16C24160yes—
85C2×C3×C8C24145yes—
86C2×C3×C8C24115yes—
87Dic12C2416yes—
88Dic12C24130yes—
89D24C24110yes—
90D24C24190yes—
91#204C241120yes—
922−1+4—340no—
932−1+4—630no—
942−1+4D1216yes—
95H(5,𝔽2)—3120no—
96H(5,𝔽2)—6120no—
97H(5,𝔽2)—690no—
98H(5,𝔽2)D12110yes—
99C2×#41—390no—
100C2×#41—6120no—
101C2×#41—630no—
102C2×#41C22×D3115yes—
103C22×D4—330no—
104C22×D4—690no—
105#137—3120no—
106AGL(1,ℤ/8)—3360no—
107#135—360no—
108#135—3180no—
109C2×Q16—315no—
110C2×SD16—390no—
111C2×D8—345no—
112C2×D8—390no—
113#131C22×D3120yescomplemented
114#131C22×D3160yescomplemented
115C2×M16C22×D3145yescomplemented
116C22×C8C22×D3115yescomplemented
117C23×C3C22×D4115yes—
118C22×D3—290no—
119C22×D3—1630no—
120C3×Q8C22×D4120yes—
121C3×D4C22×D4160yes—
122C2×C3×C4C22×D4145yes—
123C2×C3×C4C25115yes—
124#67—2120no—
125C2×Dic3—230no—
126D12—290no—
127D12C25120yes—
128C4×D3—2120no—
129Dic6—230no—
130Dic6C25112yes—
131C3×C8C25115yes—
132C3×C8C2511yescharacteristic
133C24—630no—
134#41—3120no—
135#41—6120no—
136#41—6120no—
137#41—6180no—
138#41C2×D12160yes—
139#41C23×D3120yes—
140C2×Q8—330no—
141C2×Q8—640no—
142C2×Q8C2×D12115yes—
143C2×D4—390no—
144C2×D4—690no—
145C2×D4—6360no—
146C2×D4—690no—
147C2×D4C2×D12145yes—
148C22×C4—690no—
149C22×C4C23×D3115yes—
150Q16—36no—
151Q16—330no—
152SD16—3120no—
153D8—310no—
154D8—390no—
155M16C23×D3160yes—
156C2×C8C23×D3115yes—
157C2×C8C23×D3145yes—
158C22×C3—415no—
159C22×C3C23×D4115yes—
160C2×D3—220no—
161C2×D3—890no—
162C3×C4C23×D4115yes—
163C3×C4C2611yescharacteristic
164Dic3—212no—
165C23—690no—
166C23—4830no—
167C23C22×D12115yes—
168Q8—312no—
169Q8—630no—
170Q8C22×D12120yes—
171D4—320no—
172D4—6120no—
173D4—690no—
174D4C22×D12160yes—
175C2×C4—630no—
176C2×C4—6120no—
177C2×C4C22×D12145yes—
178C2×C4C24×D3115yes—
179C8C24×D311yescharacteristic
180C8C24×D3115yes—
181C2×C3—215no—
182C2×C3C24×D411yescharacteristic
183D3—420no—
184C22—415no—
185C22—620no—
186C22—2490no—
187C22C23×D12115yes—
188C4—612no—
189C4C23×D12115yes—
190C4C25×D311yescharacteristic
191C3#6309111yescharacteristic, complemented, Sylow
192C2—215no—
193C2—1220no—
194C2C24×D1211yescharacteristic
subgroupG/H
centerC2C24×D12
commutator, cosocleC3×C4C26
FrattiniC4C25×D3
FittingC3×#3573C2
socleC2×C3C24×D4
hypercenter#3573D3
hypocenterC3#63091
LCSG ▷ C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2 ◁ #585 ◁ #3573

Complements

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

NH ≅ G/Ntype
1. #87574C2semidirect
2. C3×#3573C2semidirect
3. #87167C2semidirect
4. #87163C2semidirect
5. C2×#5752C2semidirect
10. #5964C22semidirect
11. C2×#1009C22semidirect
12. C3×#568C22semidirect
13. C2×C3×#131C22semidirect
14. #5753C22semidirect
15. #5752C22semidirect
16. #5752C22semidirect
17. #5751C22semidirect
18. #5748C22semidirect
19. C2×#908C22semidirect
20. C2×#904C22semidirect
21. C22×D24C22semidirect
26. #3573D3semidirect
30. #1010C23semidirect
32. #1009C23semidirect
38. C2×#235C23semidirect
39. C22×D12C23semidirect
40. C3×#131C23semidirect
41. C3×#131C23semidirect
42. C2×C3×M16C23semidirect
43. C22×C3×C8C23semidirect
44. #909C23semidirect
45. #908C23semidirect
46. C2×Dic12C23semidirect
47. #904C23semidirect
48. #904C23semidirect
49. C2×D24C23semidirect
50. C2×D24C23semidirect
51. C2×#204C23semidirect
65. #568C2×D3semidirect
66. C2×#131C2×D3semidirect
113. #131C22×D3semidirect
114. #131C22×D3semidirect
115. C2×M16C22×D3semidirect
116. C22×C8C22×D3semidirect
191. C3#63091semidirect

Maximal subgroups

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

subgroupindex
#875742
C3×#35732
#871672
#871632
C2×#57522
#630913