Subgroups — C₂×C₃×#576 #87465

Order 384.

1512 conjugacy classes of subgroups.

The subgroup lattice has 168 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×H(5,𝔽₂)	C₂	1	2	yes	maximal, complemented
2	C₃×#576	C₂	1	16	yes	maximal, direct product factor
3	C₂×C₃×AGL(1,ℤ/8)	C₂	1	6	yes	maximal, complemented
4	C₂×C₃×#135	C₂	1	3	yes	maximal, complemented
5	C₂²×C₃×D₈	C₂	1	3	yes	maximal, complemented
6	C₂×C₃×#131	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#576	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
8	C₃×H(5,𝔽₂)	—	2	8	no	—
9	C₃×H(5,𝔽₂)	C₂²	1	16	yes	complemented
10	C₂×C₃×#41	—	2	2	no	—
11	C₂×C₃×#41	C₂²	1	1	yes	characteristic
12	C₂×C₃×#41	C₂²	1	6	yes	complemented
13	C₂²×C₃×D₄	—	2	6	no	—
14	C₂²×C₃×D₄	C₂²	1	6	yes	complemented
15	C₃×AGL(1,ℤ/8)	C₂²	1	48	yes	complemented
16	C₃×#135	C₂²	1	24	yes	complemented
17	C₂×C₃×Q₁₆	C₂²	1	1	yes	characteristic, complemented
18	C₂×C₃×SD₁₆	C₂²	1	6	yes	complemented
19	C₂×C₃×D₈	C₂²	1	3	yes	complemented
20	C₂×C₃×D₈	C₂²	1	6	yes	complemented
21	C₂×C₃×D₈	C₂²	1	24	yes	complemented
22	C₃×#131	C₂²	1	8	yes	complemented
23	C₂×C₃×M₁₆	C₂²	1	3	yes	complemented
24	C₂²×C₃×C₈	C₂²	1	3	yes	complemented
25	C₂×H(5,𝔽₂)	C₂×C₃	1	2	yes	complemented
26	#576	C₂×C₃	1	16	yes	direct product factor
27	C₂×AGL(1,ℤ/8)	C₂×C₃	1	6	yes	complemented
28	C₂×#135	C₂×C₃	1	3	yes	complemented
29	C₂²×D₈	C₂×C₃	1	3	yes	complemented
30	C₂×#131	C₂×C₃	1	1	yes	characteristic, complemented
31	C₂⁴×C₃	—	2	6	no	—
32	C₃×#41	—	2	16	no	—
33	C₃×#41	—	2	12	no	—
34	C₃×#41	D₄	1	4	yes	—
35	C₃×#41	C₂³	1	4	yes	—
36	C₃×#41	C₂³	1	24	yes	complemented
37	C₂×C₃×Q₈	D₄	1	1	yes	characteristic
38	C₂×C₃×Q₈	C₂³	1	2	yes	—
39	C₂×C₃×D₄	—	2	6	no	—
40	C₂×C₃×D₄	—	2	48	no	—
41	C₂×C₃×D₄	—	2	6	no	—
42	C₂×C₃×D₄	—	2	12	no	—
43	C₂×C₃×D₄	D₄	1	3	yes	—
44	C₂×C₃×D₄	C₂³	1	6	yes	—
45	C₂×C₃×D₄	C₂³	1	24	yes	complemented
46	C₂²×C₃×C₄	—	2	6	no	—
47	C₂²×C₃×C₄	C₂³	1	3	yes	—
48	C₃×Q₁₆	C₂³	1	4	yes	complemented
49	C₃×SD₁₆	C₂³	1	24	yes	complemented
50	C₃×D₈	C₂³	1	12	yes	complemented
51	C₃×D₈	C₂³	1	24	yes	complemented
52	C₃×M₁₆	C₂³	1	12	yes	complemented
53	C₂×C₃×C₈	C₂³	1	1	yes	characteristic
54	C₂×C₃×C₈	C₂³	1	3	yes	—
55	C₂×C₃×C₈	C₂³	1	12	yes	complemented
56	H(5,𝔽₂)	—	2	8	no	—
57	H(5,𝔽₂)	C₂²×C₃	1	16	yes	complemented
58	C₂×#41	—	2	2	no	—
59	C₂×#41	C₂²×C₃	1	1	yes	characteristic
60	C₂×#41	C₂²×C₃	1	6	yes	complemented
61	C₂²×D₄	—	2	6	no	—
62	C₂²×D₄	C₂²×C₃	1	6	yes	complemented
63	AGL(1,ℤ/8)	C₂²×C₃	1	48	yes	complemented
64	#135	C₂²×C₃	1	24	yes	complemented
65	C₂×Q₁₆	C₂²×C₃	1	1	yes	characteristic, complemented
66	C₂×SD₁₆	C₂²×C₃	1	6	yes	complemented
67	C₂×D₈	C₂²×C₃	1	3	yes	complemented
68	C₂×D₈	C₂²×C₃	1	6	yes	complemented
69	C₂×D₈	C₂²×C₃	1	24	yes	complemented
70	#131	C₂²×C₃	1	8	yes	complemented
71	C₂×M₁₆	C₂²×C₃	1	3	yes	complemented
72	C₂²×C₈	C₂²×C₃	1	3	yes	complemented
73	C₂³×C₃	—	2	6	no	—
74	C₂³×C₃	—	2	24	no	—
75	C₂³×C₃	—	8	12	no	—
76	C₂³×C₃	C₂×D₄	1	3	yes	—
77	C₃×Q₈	—	2	2	no	—
78	C₃×Q₈	C₂×D₄	1	4	yes	—
79	C₃×Q₈	C₂⁴	1	4	yes	—
80	C₃×D₄	—	2	24	no	—
81	C₃×D₄	—	2	6	no	—
82	C₃×D₄	—	2	24	no	—
83	C₃×D₄	C₂×D₄	1	12	yes	—
84	C₃×D₄	C₂⁴	1	12	yes	—
85	C₂×C₃×C₄	—	2	2	no	—
86	C₂×C₃×C₄	—	2	24	no	—
87	C₂×C₃×C₄	C₂×D₄	1	6	yes	—
88	C₂×C₃×C₄	C₂×D₄	1	3	yes	—
89	C₂×C₃×C₄	C₂⁴	1	6	yes	—
90	C₂×C₃×C₄	C₂⁴	1	1	yes	characteristic
91	C₃×C₈	C₂⁴	1	2	yes	—
92	C₃×C₈	C₂⁴	1	6	yes	—
93	C₂⁴	—	2	6	no	—
94	#41	—	2	16	no	—
95	#41	—	2	12	no	—
96	#41	C₃×D₄	1	4	yes	—
97	#41	C₂³×C₃	1	4	yes	—
98	#41	C₂³×C₃	1	24	yes	complemented
99	C₂×Q₈	C₃×D₄	1	1	yes	characteristic
100	C₂×Q₈	C₂³×C₃	1	2	yes	—
101	C₂×D₄	—	2	6	no	—
102	C₂×D₄	—	2	48	no	—
103	C₂×D₄	—	2	6	no	—
104	C₂×D₄	—	2	12	no	—
105	C₂×D₄	C₃×D₄	1	3	yes	—
106	C₂×D₄	C₂³×C₃	1	6	yes	—
107	C₂×D₄	C₂³×C₃	1	24	yes	complemented
108	C₂²×C₄	—	2	6	no	—
109	C₂²×C₄	C₂³×C₃	1	3	yes	—
110	Q₁₆	C₂³×C₃	1	4	yes	complemented
111	SD₁₆	C₂³×C₃	1	24	yes	complemented
112	D₈	C₂³×C₃	1	12	yes	complemented
113	D₈	C₂³×C₃	1	24	yes	complemented
114	M₁₆	C₂³×C₃	1	12	yes	complemented
115	C₂×C₈	C₂³×C₃	1	1	yes	characteristic
116	C₂×C₈	C₂³×C₃	1	3	yes	—
117	C₂×C₈	C₂³×C₃	1	12	yes	complemented
118	C₂²×C₃	—	2	6	no	—
119	C₂²×C₃	—	2	12	no	—
120	C₂²×C₃	—	4	12	no	—
121	C₂²×C₃	—	8	24	no	—
122	C₂²×C₃	C₂²×D₄	1	6	yes	—
123	C₂²×C₃	C₂²×D₄	1	1	yes	characteristic
124	C₃×C₄	—	2	4	no	—
125	C₃×C₄	C₂²×D₄	1	6	yes	—
126	C₃×C₄	C₂²×D₄	1	1	yes	characteristic
127	C₃×C₄	C₂⁵	1	1	yes	characteristic
128	C₂³	—	2	6	no	—
129	C₂³	—	2	24	no	—
130	C₂³	—	8	12	no	—
131	C₂³	C₂×C₃×D₄	1	3	yes	—
132	Q₈	—	2	2	no	—
133	Q₈	C₂×C₃×D₄	1	4	yes	—
134	Q₈	C₂⁴×C₃	1	4	yes	—
135	D₄	—	2	24	no	—
136	D₄	—	2	6	no	—
137	D₄	—	2	24	no	—
138	D₄	C₂×C₃×D₄	1	12	yes	—
139	D₄	C₂⁴×C₃	1	12	yes	—
140	C₂×C₄	—	2	2	no	—
141	C₂×C₄	—	2	24	no	—
142	C₂×C₄	C₂×C₃×D₄	1	3	yes	—
143	C₂×C₄	C₂×C₃×D₄	1	6	yes	—
144	C₂×C₄	C₂⁴×C₃	1	1	yes	characteristic
145	C₂×C₄	C₂⁴×C₃	1	6	yes	—
146	C₈	C₂⁴×C₃	1	2	yes	—
147	C₈	C₂⁴×C₃	1	6	yes	—
148	C₂×C₃	—	2	6	no	—
149	C₂×C₃	—	4	12	no	—
150	C₂×C₃	#576	1	2	yes	direct product factor
151	C₂×C₃	C₂³×D₄	1	1	yes	characteristic
152	C₂²	—	2	6	no	—
153	C₂²	—	2	12	no	—
154	C₂²	—	4	12	no	—
155	C₂²	—	8	24	no	—
156	C₂²	C₂²×C₃×D₄	1	6	yes	—
157	C₂²	C₂²×C₃×D₄	1	1	yes	characteristic
158	C₄	—	2	4	no	—
159	C₄	C₂²×C₃×D₄	1	6	yes	—
160	C₄	C₂²×C₃×D₄	1	1	yes	characteristic
161	C₄	C₂⁵×C₃	1	1	yes	characteristic
162	C₃	C₂×#576	1	1	yes	characteristic, direct product factor, Sylow
163	C₂	—	2	6	no	—
164	C₂	—	4	12	no	—
165	C₂	C₃×#576	1	2	yes	direct product factor
166	C₂	C₂³×C₃×D₄	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃	C₂²×D₄
commutator, Frattini, cosocle	C₄	C₂⁵×C₃
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃ ◁ C₂×C₃×#41 ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₃×H(5,𝔽₂)	C₂	semidirect
2. C₃×#576	C₂	direct (N × H)
3. C₂×C₃×AGL(1,ℤ/8)	C₂	semidirect
4. C₂×C₃×#135	C₂	semidirect
5. C₂²×C₃×D₈	C₂	semidirect
6. C₂×C₃×#131	C₂	semidirect
7. C₂×#576	C₃	direct (N × H)
9. C₃×H(5,𝔽₂)	C₂²	semidirect
12. C₂×C₃×#41	C₂²	semidirect
14. C₂²×C₃×D₄	C₂²	semidirect
15. C₃×AGL(1,ℤ/8)	C₂²	semidirect
16. C₃×#135	C₂²	semidirect
17. C₂×C₃×Q₁₆	C₂²	semidirect
18. C₂×C₃×SD₁₆	C₂²	semidirect
19. C₂×C₃×D₈	C₂²	semidirect
20. C₂×C₃×D₈	C₂²	semidirect
21. C₂×C₃×D₈	C₂²	semidirect
22. C₃×#131	C₂²	semidirect
23. C₂×C₃×M₁₆	C₂²	semidirect
24. C₂²×C₃×C₈	C₂²	semidirect
25. C₂×H(5,𝔽₂)	C₂×C₃	semidirect
26. #576	C₂×C₃	direct (N × H)
27. C₂×AGL(1,ℤ/8)	C₂×C₃	semidirect
28. C₂×#135	C₂×C₃	semidirect
29. C₂²×D₈	C₂×C₃	semidirect
30. C₂×#131	C₂×C₃	semidirect
36. C₃×#41	C₂³	semidirect
45. C₂×C₃×D₄	C₂³	semidirect
48. C₃×Q₁₆	C₂³	semidirect
49. C₃×SD₁₆	C₂³	semidirect
50. C₃×D₈	C₂³	semidirect
51. C₃×D₈	C₂³	semidirect
52. C₃×M₁₆	C₂³	semidirect
55. C₂×C₃×C₈	C₂³	semidirect
57. H(5,𝔽₂)	C₂²×C₃	semidirect
60. C₂×#41	C₂²×C₃	semidirect
62. C₂²×D₄	C₂²×C₃	semidirect
63. AGL(1,ℤ/8)	C₂²×C₃	semidirect
64. #135	C₂²×C₃	semidirect
65. C₂×Q₁₆	C₂²×C₃	semidirect
66. C₂×SD₁₆	C₂²×C₃	semidirect
67. C₂×D₈	C₂²×C₃	semidirect
68. C₂×D₈	C₂²×C₃	semidirect
69. C₂×D₈	C₂²×C₃	semidirect
70. #131	C₂²×C₃	semidirect
71. C₂×M₁₆	C₂²×C₃	semidirect
72. C₂²×C₈	C₂²×C₃	semidirect
98. #41	C₂³×C₃	semidirect
107. C₂×D₄	C₂³×C₃	semidirect
110. Q₁₆	C₂³×C₃	semidirect
111. SD₁₆	C₂³×C₃	semidirect
112. D₈	C₂³×C₃	semidirect
113. D₈	C₂³×C₃	semidirect
114. M₁₆	C₂³×C₃	semidirect
117. C₂×C₈	C₂³×C₃	semidirect
150. C₂×C₃	#576	direct (N × H)
162. C₃	C₂×#576	direct (N × H)
165. C₂	C₃×#576	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₃×H(5,𝔽₂)	2
C₃×#576	2
C₂×C₃×AGL(1,ℤ/8)	2
C₂×C₃×#135	2
C₂²×C₃×D₈	2
C₂×C₃×#131	2
C₂×#576	3

Subgroups — C2×C3×#576 #87465

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×#576 #87465