Finite Group Databasebeta

← D₃×#585 #87573

Subgroups — D₃×#585 #87573

Order 384.

5956 conjugacy classes of subgroups; only the 3203 normal subgroups are shown.

normal subgroups only

This group has 5956 conjugacy classes of subgroups — too many to draw. Only its 3203 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×#585	C₂	1	1	yes	maximal, characteristic, complemented
2	#5968	C₂	1	6	yes	maximal, complemented
3	D₃×2₋¹⁺⁴	C₂	1	6	yes	maximal, complemented
4	#5966	C₂	1	10	yes	maximal, complemented
5	D₃×H(5,𝔽₂)	C₂	1	10	yes	maximal, complemented
6	#5964	C₂	1	15	yes	maximal, complemented
7	C₂×D₃×#41	C₂	1	15	yes	maximal, complemented
8	C₂×#585	—	3	1	no	maximal, Sylow
9	C₃×2₋¹⁺⁴	C₂²	1	6	yes	complemented
10	C₃×H(5,𝔽₂)	C₂²	1	10	yes	complemented
11	C₂×C₃×#41	C₂²	1	15	yes	complemented
12	#1010	C₂²	1	60	yes	complemented
13	#1009	C₂²	1	60	yes	complemented
14	D₃×#41	C₂²	1	20	yes	complemented
15	D₃×#41	C₂²	1	60	yes	complemented
16	D₃×#41	C₂²	1	60	yes	complemented
17	D₃×#41	C₂²	1	60	yes	complemented
18	#1007	C₂²	1	30	yes	complemented
19	C₂×#239	C₂²	1	15	yes	complemented
20	C₂×D₃×Q₈	C₂²	1	15	yes	complemented
21	#1004	C₂²	1	90	yes	complemented
22	C₂×#237	C₂²	1	45	yes	complemented
23	C₂×D₃×D₄	C₂²	1	45	yes	complemented
24	C₂×#235	C₂²	1	45	yes	complemented
25	C₂²×C₄×D₃	C₂²	1	15	yes	complemented
26	#585	—	3	6	no	—
27	#585	—	3	10	no	—
28	#585	—	3	15	no	—
29	#585	D₃	1	1	yes	characteristic, direct product factor
30	C₂×2₋¹⁺⁴	—	3	6	no	—
31	C₂×H(5,𝔽₂)	—	3	10	no	—
32	C₂²×#41	—	3	15	no	—
33	C₂³×D₃	C₂³	1	15	yes	—
34	C₃×#41	C₂³	1	20	yes	complemented
35	C₃×#41	C₂³	1	60	yes	complemented
36	C₂×C₃×Q₈	C₂³	1	15	yes	complemented
37	C₂×C₃×D₄	C₂³	1	45	yes	complemented
38	C₂²×C₃×C₄	C₂³	1	15	yes	complemented
39	C₂×#67	C₂³	1	90	yes	complemented
40	C₂²×Dic₃	C₂³	1	15	yes	complemented
41	#239	C₂³	1	20	yes	complemented
42	#239	C₂³	1	60	yes	complemented
43	D₃×Q₈	C₂³	1	20	yes	—
44	D₃×Q₈	C₂³	1	60	yes	complemented
45	#237	C₂³	1	60	yes	complemented
46	#237	C₂³	1	180	yes	complemented
47	D₃×D₄	C₂³	1	60	yes	—
48	D₃×D₄	C₂³	1	180	yes	complemented
49	#235	C₂³	1	60	yes	complemented
50	#235	C₂³	1	180	yes	complemented
51	C₂×D₁₂	C₂³	1	45	yes	complemented
52	C₂×C₄×D₃	C₂³	1	15	yes	—
53	C₂×C₄×D₃	C₂³	1	45	yes	—
54	C₂×C₄×D₃	C₂³	1	45	yes	complemented
55	C₂×C₄×D₃	C₂³	1	45	yes	complemented
56	C₂×Dic₆	C₂³	1	45	yes	complemented
57	2₋¹⁺⁴	—	3	60	no	—
58	2₋¹⁺⁴	—	3	30	no	—
59	2₋¹⁺⁴	C₂×D₃	1	6	yes	complemented
60	H(5,𝔽₂)	—	3	60	no	—
61	H(5,𝔽₂)	—	3	90	no	—
62	H(5,𝔽₂)	C₂×D₃	1	10	yes	complemented
63	C₂×#41	—	3	20	no	—
64	C₂×#41	—	3	60	no	—
65	C₂×#41	—	3	60	no	—
66	C₂×#41	—	3	60	no	—
67	C₂×#41	—	3	15	no	—
68	C₂×#41	—	3	45	no	—
69	C₂×#41	—	3	45	no	—
70	C₂×#41	C₂×D₃	1	15	yes	complemented
71	C₂²×Q₈	—	3	15	no	—
72	C₂²×D₄	—	3	45	no	—
73	C₂³×C₄	—	3	15	no	—
74	C₂³×C₃	C₂⁴	1	15	yes	—
75	C₂²×D₃	—	4	30	no	—
76	C₂²×D₃	C₂⁴	1	15	yes	—
77	C₂²×D₃	C₂⁴	1	45	yes	—
78	C₃×Q₈	C₂⁴	1	20	yes	—
79	C₃×D₄	C₂⁴	1	60	yes	—
80	C₂×C₃×C₄	C₂⁴	1	45	yes	—
81	C₂×C₃×C₄	C₂⁴	1	15	yes	—
82	#67	C₂⁴	1	120	yes	—
83	C₂×Dic₃	C₂⁴	1	45	yes	—
84	C₂×Dic₃	C₂⁴	1	15	yes	—
85	D₁₂	C₂⁴	1	60	yes	—
86	C₄×D₃	C₂⁴	1	1	yes	characteristic
87	C₄×D₃	C₂⁴	1	15	yes	—
88	C₄×D₃	C₂⁴	1	15	yes	—
89	C₄×D₃	C₂⁴	1	15	yes	—
90	C₄×D₃	C₂⁴	1	90	yes	—
91	Dic₆	C₂⁴	1	60	yes	—
92	C₂⁴	—	3	15	no	—
93	#41	—	3	20	no	—
94	#41	—	3	60	no	—
95	#41	—	3	60	no	—
96	#41	—	3	180	no	—
97	#41	—	3	60	no	—
98	#41	—	3	180	no	—
99	#41	C₂²×D₃	1	20	yes	complemented
100	#41	C₂²×D₃	1	60	yes	complemented
101	C₂×Q₈	—	3	20	no	—
102	C₂×Q₈	—	3	60	no	—
103	C₂×Q₈	—	3	45	no	—
104	C₂×Q₈	C₂²×D₃	1	15	yes	complemented
105	C₂×D₄	—	3	90	no	—
106	C₂×D₄	—	3	60	no	—
107	C₂×D₄	—	3	180	no	—
108	C₂×D₄	—	3	45	no	—
109	C₂×D₄	C₂²×D₃	1	45	yes	complemented
110	C₂²×C₄	—	3	15	no	—
111	C₂²×C₄	—	3	15	no	—
112	C₂²×C₄	—	3	45	no	—
113	C₂²×C₄	—	3	45	no	—
114	C₂²×C₄	—	3	45	no	—
115	C₂²×C₄	C₂²×D₃	1	15	yes	complemented
116	C₂²×C₃	—	4	15	no	—
117	C₂²×C₃	C₂⁵	1	15	yes	—
118	C₂×D₃	—	2	30	no	—
119	C₂×D₃	—	4	45	no	—
120	C₂×D₃	C₂⁵	1	15	yes	—
121	C₂×D₃	C₂⁵	1	1	yes	characteristic
122	C₃×C₄	C₂⁵	1	15	yes	—
123	C₃×C₄	C₂⁵	1	1	yes	characteristic
124	Dic₃	C₂⁵	1	15	yes	—
125	Dic₃	C₂⁵	1	1	yes	characteristic
126	C₂³	—	3	15	no	—
127	C₂³	—	3	45	no	—
128	C₂³	—	12	30	no	—
129	C₂³	C₂³×D₃	1	15	yes	—
130	Q₈	—	3	60	no	—
131	Q₈	C₂³×D₃	1	20	yes	—
132	D₄	—	3	120	no	—
133	D₄	—	3	60	no	—
134	D₄	C₂³×D₃	1	60	yes	—
135	C₂×C₄	—	3	15	no	—
136	C₂×C₄	—	3	45	no	—
137	C₂×C₄	—	3	15	no	—
138	C₂×C₄	—	3	15	no	—
139	C₂×C₄	—	3	90	no	—
140	C₂×C₄	—	3	1	no	—
141	C₂×C₄	—	3	15	no	—
142	C₂×C₄	C₂³×D₃	1	45	yes	—
143	C₂×C₄	C₂³×D₃	1	15	yes	—
144	C₂×C₃	—	2	15	no	—
145	C₂×C₃	C₂⁶	1	1	yes	characteristic
146	D₃	—	2	15	no	—
147	D₃	#585	1	2	yes	direct product factor
148	C₂²	—	3	1	no	—
149	C₂²	—	3	15	no	—
150	C₂²	—	4	15	no	—
151	C₂²	—	6	30	no	—
152	C₂²	—	12	45	no	—
153	C₂²	C₂⁴×D₃	1	15	yes	—
154	C₄	—	3	1	no	—
155	C₄	—	3	15	no	—
156	C₄	C₂⁴×D₃	1	15	yes	—
157	C₄	C₂⁴×D₃	1	1	yes	characteristic
158	C₃	C₂×#585	1	1	yes	characteristic, complemented, Sylow
159	C₂	—	2	15	no	—
160	C₂	—	3	2	no	—
161	C₂	—	6	15	no	—
162	C₂	C₂⁵×D₃	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂⁴×D₃
commutator, socle, cosocle	C₂×C₃	C₂⁶
Frattini	C₂	C₂⁵×D₃
Fitting	C₃×#585	C₂
hypercenter	#585	D₃
hypocenter	C₃	C₂×#585
LCS	G ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₄ ◁ #585

Complements

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

N	H ≅ G/N	type
1. C₃×#585	C₂	semidirect
2. #5968	C₂	semidirect
3. D₃×2₋¹⁺⁴	C₂	semidirect
4. #5966	C₂	semidirect
5. D₃×H(5,𝔽₂)	C₂	semidirect
6. #5964	C₂	semidirect
7. C₂×D₃×#41	C₂	semidirect
9. C₃×2₋¹⁺⁴	C₂²	semidirect
10. C₃×H(5,𝔽₂)	C₂²	semidirect
11. C₂×C₃×#41	C₂²	semidirect
12. #1010	C₂²	semidirect
13. #1009	C₂²	semidirect
14. D₃×#41	C₂²	semidirect
15. D₃×#41	C₂²	semidirect
16. D₃×#41	C₂²	semidirect
17. D₃×#41	C₂²	semidirect
18. #1007	C₂²	semidirect
19. C₂×#239	C₂²	semidirect
20. C₂×D₃×Q₈	C₂²	semidirect
21. #1004	C₂²	semidirect
22. C₂×#237	C₂²	semidirect
23. C₂×D₃×D₄	C₂²	semidirect
24. C₂×#235	C₂²	semidirect
25. C₂²×C₄×D₃	C₂²	semidirect
29. #585	D₃	direct (N × H)
34. C₃×#41	C₂³	semidirect
35. C₃×#41	C₂³	semidirect
36. C₂×C₃×Q₈	C₂³	semidirect
37. C₂×C₃×D₄	C₂³	semidirect
38. C₂²×C₃×C₄	C₂³	semidirect
39. C₂×#67	C₂³	semidirect
40. C₂²×Dic₃	C₂³	semidirect
41. #239	C₂³	semidirect
42. #239	C₂³	semidirect
44. D₃×Q₈	C₂³	semidirect
45. #237	C₂³	semidirect
46. #237	C₂³	semidirect
48. D₃×D₄	C₂³	semidirect
49. #235	C₂³	semidirect
50. #235	C₂³	semidirect
51. C₂×D₁₂	C₂³	semidirect
54. C₂×C₄×D₃	C₂³	semidirect
55. C₂×C₄×D₃	C₂³	semidirect
56. C₂×Dic₆	C₂³	semidirect
59. 2₋¹⁺⁴	C₂×D₃	semidirect
62. H(5,𝔽₂)	C₂×D₃	semidirect
70. C₂×#41	C₂×D₃	semidirect
99. #41	C₂²×D₃	semidirect
100. #41	C₂²×D₃	semidirect
104. C₂×Q₈	C₂²×D₃	semidirect
109. C₂×D₄	C₂²×D₃	semidirect
115. C₂²×C₄	C₂²×D₃	semidirect
147. D₃	#585	direct (N × H)
158. C₃	C₂×#585	semidirect

Maximal subgroups

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

subgroup	index
C₃×#585	2
#5968	2
D₃×2₋¹⁺⁴	2
#5966	2
D₃×H(5,𝔽₂)	2
#5964	2
C₂×D₃×#41	2
C₂×#585	3