Finite Group Databasebeta

← D₃×2₋¹⁺⁶ #11709085

Subgroups — D₃×2₋¹⁺⁶ #11709085

Order 768.

59186 conjugacy classes of subgroups; only the 32041 normal subgroups are shown.

normal subgroups only

This group has 59186 conjugacy classes of subgroups — too many to draw. Only its 32041 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×2₋¹⁺⁶	C₂	1	1	yes	maximal, characteristic, complemented
2	D₃×#585	C₂	1	36	yes	maximal, complemented
3	#87572	C₂	1	36	yes	maximal, complemented
4	#87571	C₂	1	27	yes	maximal, complemented
5	C₂×D₃×2₋¹⁺⁴	C₂	1	27	yes	maximal, complemented
6	C₂×2₋¹⁺⁶	—	3	1	no	maximal, Sylow
7	C₃×#585	C₂²	1	36	yes	complemented
8	C₂×C₃×2₋¹⁺⁴	C₂²	1	27	yes	complemented
9	#5968	C₂²	1	216	yes	complemented
10	D₃×2₋¹⁺⁴	C₂²	1	216	yes	complemented
11	D₃×2₋¹⁺⁴	C₂²	1	432	yes	complemented
12	#5966	C₂²	1	360	yes	complemented
13	D₃×H(5,𝔽₂)	C₂²	1	120	yes	complemented
14	#5964	C₂²	1	540	yes	complemented
15	C₂×#1010	C₂²	1	270	yes	complemented
16	C₂×D₃×#41	C₂²	1	270	yes	complemented
17	C₂×#1007	C₂²	1	135	yes	complemented
18	C₂²×D₃×Q₈	C₂²	1	45	yes	complemented
19	2₋¹⁺⁶	—	3	36	no	—
20	2₋¹⁺⁶	—	3	27	no	—
21	2₋¹⁺⁶	D₃	1	1	yes	characteristic, direct product factor
22	C₂×#585	—	3	36	no	—
23	C₂²×2₋¹⁺⁴	—	3	27	no	—
24	C₃×2₋¹⁺⁴	C₂³	1	216	yes	complemented
25	C₃×H(5,𝔽₂)	C₂³	1	120	yes	complemented
26	C₂×C₃×#41	C₂³	1	270	yes	complemented
27	C₂²×C₃×Q₈	C₂³	1	45	yes	complemented
28	#1010	C₂³	1	2160	yes	complemented
29	#1009	C₂³	1	720	yes	complemented
30	D₃×#41	C₂³	1	720	yes	—
31	D₃×#41	C₂³	1	2160	yes	complemented
32	#1007	C₂³	1	1080	yes	complemented
33	C₂×#239	C₂³	1	270	yes	complemented
34	C₂×D₃×Q₈	C₂³	1	270	yes	—
35	C₂×D₃×Q₈	C₂³	1	540	yes	complemented
36	#1004	C₂³	1	1080	yes	complemented
37	C₂×#237	C₂³	1	810	yes	complemented
38	C₂×D₃×D₄	C₂³	1	270	yes	—
39	C₂×#235	C₂³	1	810	yes	complemented
40	C₂²×C₄×D₃	C₂³	1	135	yes	—
41	C₂²×Dic₆	C₂³	1	135	yes	complemented
42	#585	—	3	216	no	—
43	#585	—	3	360	no	—
44	#585	—	3	540	no	—
45	#585	C₂×D₃	1	36	yes	complemented
46	C₂×2₋¹⁺⁴	—	3	216	no	—
47	C₂×2₋¹⁺⁴	—	3	432	no	—
48	C₂×2₋¹⁺⁴	—	3	270	no	—
49	C₂×2₋¹⁺⁴	—	3	135	no	—
50	C₂×2₋¹⁺⁴	C₂×D₃	1	27	yes	complemented
51	C₂×H(5,𝔽₂)	—	3	120	no	—
52	C₂²×#41	—	3	270	no	—
53	C₂³×Q₈	—	3	45	no	—
54	C₂³×D₃	C₂⁴	1	45	yes	—
55	C₃×#41	C₂⁴	1	720	yes	—
56	C₂×C₃×Q₈	C₂⁴	1	270	yes	—
57	C₂×C₃×D₄	C₂⁴	1	270	yes	—
58	C₂²×C₃×C₄	C₂⁴	1	135	yes	—
59	C₂×#67	C₂⁴	1	540	yes	—
60	C₂²×Dic₃	C₂⁴	1	135	yes	—
61	#239	C₂⁴	1	720	yes	—
62	D₃×Q₈	C₂⁴	1	120	yes	—
63	D₃×Q₈	C₂⁴	1	1080	yes	—
64	#237	C₂⁴	1	2160	yes	—
65	D₃×D₄	C₂⁴	1	216	yes	—
66	D₃×D₄	C₂⁴	1	1080	yes	—
67	#235	C₂⁴	1	2160	yes	—
68	C₂×D₁₂	C₂⁴	1	270	yes	—
69	C₂×C₄×D₃	C₂⁴	1	270	yes	—
70	C₂×C₄×D₃	C₂⁴	1	810	yes	—
71	C₂×Dic₆	C₂⁴	1	810	yes	—
72	2₋¹⁺⁴	—	3	2160	no	—
73	2₋¹⁺⁴	—	3	1080	no	—
74	2₋¹⁺⁴	C₂²×D₃	1	216	yes	complemented
75	H(5,𝔽₂)	—	3	720	no	—
76	H(5,𝔽₂)	—	3	1080	no	—
77	H(5,𝔽₂)	C₂²×D₃	1	120	yes	complemented
78	C₂×#41	—	3	720	no	—
79	C₂×#41	—	3	2160	no	—
80	C₂×#41	—	3	270	no	—
81	C₂×#41	—	3	810	no	—
82	C₂×#41	—	3	810	no	—
83	C₂×#41	C₂²×D₃	1	270	yes	complemented
84	C₂²×Q₈	—	3	270	no	—
85	C₂²×Q₈	—	3	540	no	—
86	C₂²×Q₈	—	3	135	no	—
87	C₂²×Q₈	C₂²×D₃	1	45	yes	complemented
88	C₂²×D₄	—	3	270	no	—
89	C₂³×C₄	—	3	135	no	—
90	C₂³×C₃	C₂⁵	1	45	yes	—
91	C₂²×D₃	—	4	90	no	—
92	C₂²×D₃	C₂⁵	1	27	yes	—
93	C₂²×D₃	C₂⁵	1	135	yes	—
94	C₃×Q₈	C₂⁵	1	120	yes	—
95	C₃×D₄	C₂⁵	1	216	yes	—
96	C₂×C₃×C₄	C₂⁵	1	270	yes	—
97	#67	C₂⁵	1	432	yes	—
98	C₂×Dic₃	C₂⁵	1	270	yes	—
99	D₁₂	C₂⁵	1	216	yes	—
100	C₄×D₃	C₂⁵	1	36	yes	—
101	C₄×D₃	C₂⁵	1	540	yes	—
102	Dic₆	C₂⁵	1	360	yes	—
103	C₂⁴	—	3	45	no	—
104	#41	—	3	720	no	—
105	#41	—	3	2160	no	—
106	#41	—	3	2160	no	—
107	#41	C₂³×D₃	1	720	yes	—
108	C₂×Q₈	—	3	120	no	—
109	C₂×Q₈	—	3	1080	no	—
110	C₂×Q₈	—	3	810	no	—
111	C₂×Q₈	C₂³×D₃	1	270	yes	—
112	C₂×D₄	—	3	540	no	—
113	C₂×D₄	—	3	216	no	—
114	C₂×D₄	—	3	1080	no	—
115	C₂×D₄	—	3	270	no	—
116	C₂×D₄	C₂³×D₃	1	270	yes	—
117	C₂²×C₄	—	3	135	no	—
118	C₂²×C₄	—	3	270	no	—
119	C₂²×C₄	—	3	810	no	—
120	C₂²×C₄	C₂³×D₃	1	135	yes	—
121	C₂²×C₃	—	4	45	no	—
122	C₂²×C₃	C₂⁶	1	27	yes	—
123	C₂×D₃	—	2	54	no	—
124	C₂×D₃	—	4	135	no	—
125	C₂×D₃	C₂⁶	1	27	yes	—
126	C₂×D₃	C₂⁶	1	1	yes	characteristic
127	C₃×C₄	C₂⁶	1	36	yes	—
128	Dic₃	C₂⁶	1	36	yes	—
129	C₂³	—	3	27	no	—
130	C₂³	—	3	135	no	—
131	C₂³	—	12	90	no	—
132	C₂³	C₂⁴×D₃	1	45	yes	—
133	Q₈	—	3	360	no	—
134	Q₈	C₂⁴×D₃	1	120	yes	—
135	D₄	—	3	432	no	—
136	D₄	—	3	216	no	—
137	D₄	C₂⁴×D₃	1	216	yes	—
138	C₂×C₄	—	3	270	no	—
139	C₂×C₄	—	3	36	no	—
140	C₂×C₄	—	3	540	no	—
141	C₂×C₄	C₂⁴×D₃	1	270	yes	—
142	C₂×C₃	—	2	27	no	—
143	C₂×C₃	C₂⁷	1	1	yes	characteristic
144	D₃	—	2	27	no	—
145	D₃	2₋¹⁺⁶	1	2	yes	direct product factor
146	C₂²	—	3	1	no	—
147	C₂²	—	3	27	no	—
148	C₂²	—	4	45	no	—
149	C₂²	—	6	54	no	—
150	C₂²	—	12	135	no	—
151	C₂²	C₂⁵×D₃	1	27	yes	—
152	C₄	—	3	36	no	—
153	C₄	C₂⁵×D₃	1	36	yes	—
154	C₃	C₂×2₋¹⁺⁶	1	1	yes	characteristic, complemented, Sylow
155	C₂	—	2	27	no	—
156	C₂	—	3	2	no	—
157	C₂	—	6	27	no	—
158	C₂	C₂⁶×D₃	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂	C₂⁶×D₃
commutator, socle, cosocle	C₂×C₃	C₂⁷
Fitting	C₃×2₋¹⁺⁶	C₂
hypercenter	2₋¹⁺⁶	D₃
hypocenter	C₃	C₂×2₋¹⁺⁶
LCS	G ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂ ◁ 2₋¹⁺⁶

Complements

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

N	H ≅ G/N	type
1. C₃×2₋¹⁺⁶	C₂	semidirect
2. D₃×#585	C₂	semidirect
3. #87572	C₂	semidirect
4. #87571	C₂	semidirect
5. C₂×D₃×2₋¹⁺⁴	C₂	semidirect
7. C₃×#585	C₂²	semidirect
8. C₂×C₃×2₋¹⁺⁴	C₂²	semidirect
9. #5968	C₂²	semidirect
10. D₃×2₋¹⁺⁴	C₂²	semidirect
11. D₃×2₋¹⁺⁴	C₂²	semidirect
12. #5966	C₂²	semidirect
13. D₃×H(5,𝔽₂)	C₂²	semidirect
14. #5964	C₂²	semidirect
15. C₂×#1010	C₂²	semidirect
16. C₂×D₃×#41	C₂²	semidirect
17. C₂×#1007	C₂²	semidirect
18. C₂²×D₃×Q₈	C₂²	semidirect
21. 2₋¹⁺⁶	D₃	direct (N × H)
24. C₃×2₋¹⁺⁴	C₂³	semidirect
25. C₃×H(5,𝔽₂)	C₂³	semidirect
26. C₂×C₃×#41	C₂³	semidirect
27. C₂²×C₃×Q₈	C₂³	semidirect
28. #1010	C₂³	semidirect
29. #1009	C₂³	semidirect
31. D₃×#41	C₂³	semidirect
32. #1007	C₂³	semidirect
33. C₂×#239	C₂³	semidirect
35. C₂×D₃×Q₈	C₂³	semidirect
36. #1004	C₂³	semidirect
37. C₂×#237	C₂³	semidirect
39. C₂×#235	C₂³	semidirect
41. C₂²×Dic₆	C₂³	semidirect
45. #585	C₂×D₃	semidirect
50. C₂×2₋¹⁺⁴	C₂×D₃	semidirect
74. 2₋¹⁺⁴	C₂²×D₃	semidirect
77. H(5,𝔽₂)	C₂²×D₃	semidirect
83. C₂×#41	C₂²×D₃	semidirect
87. C₂²×Q₈	C₂²×D₃	semidirect
145. D₃	2₋¹⁺⁶	direct (N × H)
154. C₃	C₂×2₋¹⁺⁶	semidirect

Maximal subgroups

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

subgroup	index
C₃×2₋¹⁺⁶	2
D₃×#585	2
#87572	2
#87571	2
C₂×D₃×2₋¹⁺⁴	2
C₂×2₋¹⁺⁶	3