Subgroups — C₂×C₃×#518 #87309

Order 384.

1920 conjugacy classes of subgroups.

The subgroup lattice has 144 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₃×D₄	C₂	1	1	yes	maximal, characteristic
2	C₃×#518	C₂	1	16	yes	maximal, direct product factor
3	C₂×C₃×C₄×D₄	C₂	1	8	yes	maximal, complemented
4	C₂×C₃×#117	C₂	1	2	yes	maximal, complemented
5	C₂²×C₃×C₂²⋊C₄	C₂	1	4	yes	maximal, complemented
6	C₂×#518	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
7	C₂⁵×C₃	C₂²	1	2	yes	—
8	C₂²×C₃×D₄	C₄	1	16	yes	complemented
9	C₂²×C₃×D₄	C₂²	1	4	yes	—
10	C₂²×C₃×D₄	C₂²	1	8	yes	—
11	C₂³×C₃×C₄	C₂²	1	1	yes	characteristic
12	C₂³×C₃×C₄	C₂²	1	8	yes	complemented
13	C₃×C₄×D₄	C₂²	1	64	yes	complemented
14	C₃×#117	C₂²	1	16	yes	complemented
15	C₂×C₃×C₄⋊C₄	C₂²	1	4	yes	complemented
16	C₂×C₃×C₂²⋊C₄	C₂²	1	4	yes	—
17	C₂×C₃×C₂²⋊C₄	C₂²	1	32	yes	complemented
18	C₂×C₃×C₂²⋊C₄	C₂²	1	8	yes	complemented
19	C₂×C₃×C₄²	C₂²	1	4	yes	complemented
20	C₂³×D₄	C₂×C₃	1	1	yes	characteristic
21	#518	C₂×C₃	1	16	yes	direct product factor
22	C₂×C₄×D₄	C₂×C₃	1	8	yes	complemented
23	C₂×#117	C₂×C₃	1	2	yes	complemented
24	C₂²×C₂²⋊C₄	C₂×C₃	1	4	yes	complemented
25	C₂⁴×C₃	—	4	8	no	—
26	C₂⁴×C₃	C₂×C₄	1	16	yes	complemented
27	C₂⁴×C₃	C₂³	1	1	yes	characteristic
28	C₂⁴×C₃	C₂³	1	4	yes	—
29	C₂⁴×C₃	C₂³	1	8	yes	—
30	C₂×C₃×D₄	C₂×C₄	1	32	yes	complemented
31	C₂×C₃×D₄	C₂×C₄	1	64	yes	complemented
32	C₂×C₃×D₄	C₂³	1	16	yes	—
33	C₂²×C₃×C₄	—	4	16	no	—
34	C₂²×C₃×C₄	C₂×C₄	1	8	yes	complemented
35	C₂²×C₃×C₄	C₂³	1	8	yes	—
36	C₂²×C₃×C₄	C₂³	1	2	yes	—
37	C₂²×C₃×C₄	C₂³	1	4	yes	—
38	C₂²×C₃×C₄	C₂³	1	32	yes	complemented
39	C₃×C₄⋊C₄	C₂³	1	16	yes	complemented
40	C₃×C₂²⋊C₄	C₂³	1	16	yes	—
41	C₃×C₂²⋊C₄	C₂³	1	32	yes	complemented
42	C₃×C₄²	C₂³	1	16	yes	complemented
43	C₂⁵	C₂²×C₃	1	2	yes	—
44	C₂²×D₄	C₃×C₄	1	16	yes	complemented
45	C₂²×D₄	C₂²×C₃	1	4	yes	—
46	C₂²×D₄	C₂²×C₃	1	8	yes	—
47	C₂³×C₄	C₂²×C₃	1	1	yes	characteristic
48	C₂³×C₄	C₂²×C₃	1	8	yes	complemented
49	C₄×D₄	C₂²×C₃	1	64	yes	complemented
50	#117	C₂²×C₃	1	16	yes	complemented
51	C₂×C₄⋊C₄	C₂²×C₃	1	4	yes	complemented
52	C₂×C₂²⋊C₄	C₂²×C₃	1	4	yes	—
53	C₂×C₂²⋊C₄	C₂²×C₃	1	32	yes	complemented
54	C₂×C₂²⋊C₄	C₂²×C₃	1	8	yes	complemented
55	C₂×C₄²	C₂²×C₃	1	4	yes	complemented
56	C₂³×C₃	—	2	16	no	—
57	C₂³×C₃	—	2	4	no	—
58	C₂³×C₃	—	4	32	no	—
59	C₂³×C₃	—	4	16	no	—
60	C₂³×C₃	C₂²×C₄	1	4	yes	—
61	C₂³×C₃	C₂²×C₄	1	16	yes	—
62	C₂³×C₃	C₂²×C₄	1	32	yes	complemented
63	C₂³×C₃	C₂⁴	1	2	yes	—
64	C₂³×C₃	C₂⁴	1	8	yes	—
65	C₂³×C₃	C₂⁴	1	1	yes	characteristic
66	C₃×D₄	C₂²×C₄	1	64	yes	complemented
67	C₂×C₃×C₄	—	2	16	no	—
68	C₂×C₃×C₄	—	4	32	no	—
69	C₂×C₃×C₄	C₂²×C₄	1	8	yes	—
70	C₂×C₃×C₄	C₂²×C₄	1	16	yes	complemented
71	C₂×C₃×C₄	C₂⁴	1	16	yes	—
72	C₂×C₃×C₄	C₂⁴	1	4	yes	—
73	C₂⁴	—	4	8	no	—
74	C₂⁴	C₂×C₃×C₄	1	16	yes	complemented
75	C₂⁴	C₂³×C₃	1	1	yes	characteristic
76	C₂⁴	C₂³×C₃	1	4	yes	—
77	C₂⁴	C₂³×C₃	1	8	yes	—
78	C₂×D₄	C₂×C₃×C₄	1	32	yes	complemented
79	C₂×D₄	C₂×C₃×C₄	1	64	yes	complemented
80	C₂×D₄	C₂³×C₃	1	16	yes	—
81	C₂²×C₄	—	4	16	no	—
82	C₂²×C₄	C₂×C₃×C₄	1	8	yes	complemented
83	C₂²×C₄	C₂³×C₃	1	8	yes	—
84	C₂²×C₄	C₂³×C₃	1	2	yes	—
85	C₂²×C₄	C₂³×C₃	1	4	yes	—
86	C₂²×C₄	C₂³×C₃	1	32	yes	complemented
87	C₄⋊C₄	C₂³×C₃	1	16	yes	complemented
88	C₂²⋊C₄	C₂³×C₃	1	16	yes	—
89	C₂²⋊C₄	C₂³×C₃	1	32	yes	complemented
90	C₄²	C₂³×C₃	1	16	yes	complemented
91	C₂²×C₃	—	2	8	no	—
92	C₂²×C₃	—	2	32	no	—
93	C₂²×C₃	—	2	16	no	—
94	C₂²×C₃	—	2	4	no	—
95	C₂²×C₃	—	4	32	no	—
96	C₂²×C₃	C₂³×C₄	1	4	yes	—
97	C₂²×C₃	C₂³×C₄	1	16	yes	—
98	C₂²×C₃	C₂³×C₄	1	2	yes	—
99	C₂²×C₃	H(5,𝔽₂)	1	4	yes	—
100	C₂²×C₃	C₂⁵	1	1	yes	characteristic
101	C₃×C₄	—	2	16	no	—
102	C₃×C₄	C₂³×C₄	1	8	yes	—
103	C₂³	—	2	16	no	—
104	C₂³	—	2	4	no	—
105	C₂³	—	4	32	no	—
106	C₂³	—	4	16	no	—
107	C₂³	C₂²×C₃×C₄	1	4	yes	—
108	C₂³	C₂²×C₃×C₄	1	16	yes	—
109	C₂³	C₂²×C₃×C₄	1	32	yes	complemented
110	C₂³	C₂⁴×C₃	1	2	yes	—
111	C₂³	C₂⁴×C₃	1	8	yes	—
112	C₂³	C₂⁴×C₃	1	1	yes	characteristic
113	D₄	C₂²×C₃×C₄	1	64	yes	complemented
114	C₂×C₄	—	2	16	no	—
115	C₂×C₄	—	4	32	no	—
116	C₂×C₄	C₂²×C₃×C₄	1	8	yes	—
117	C₂×C₄	C₂²×C₃×C₄	1	16	yes	complemented
118	C₂×C₄	C₂⁴×C₃	1	16	yes	—
119	C₂×C₄	C₂⁴×C₃	1	4	yes	—
120	C₂×C₃	—	2	4	no	—
121	C₂×C₃	—	2	16	no	—
122	C₂×C₃	#518	1	4	yes	direct product factor
123	C₂×C₃	C₂⁴×C₄	1	1	yes	characteristic
124	C₂×C₃	C₂×H(5,𝔽₂)	1	2	yes	—
125	C₂²	—	2	16	no	—
126	C₂²	—	2	4	no	—
127	C₂²	—	2	8	no	—
128	C₂²	—	2	32	no	—
129	C₂²	—	4	32	no	—
130	C₂²	C₂³×C₃×C₄	1	4	yes	—
131	C₂²	C₂³×C₃×C₄	1	16	yes	—
132	C₂²	C₂³×C₃×C₄	1	2	yes	—
133	C₂²	C₃×H(5,𝔽₂)	1	4	yes	—
134	C₂²	C₂⁵×C₃	1	1	yes	characteristic
135	C₄	—	2	16	no	—
136	C₄	C₂³×C₃×C₄	1	8	yes	—
137	C₃	C₂×#518	1	1	yes	characteristic, direct product factor, Sylow
138	C₂	—	2	4	no	—
139	C₂	—	2	16	no	—
140	C₂	C₃×#518	1	4	yes	direct product factor
141	C₂	C₂⁴×C₃×C₄	1	1	yes	characteristic
142	C₂	C₂×C₃×H(5,𝔽₂)	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₃	C₂⁴
commutator	C₂	C₂⁴×C₃×C₄
Frattini, cosocle	C₂²	C₂⁵×C₃
prime radical	C₂³×C₃×D₄	C₂
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₃×#518	C₂	direct (N × H)
3. C₂×C₃×C₄×D₄	C₂	semidirect
4. C₂×C₃×#117	C₂	semidirect
5. C₂²×C₃×C₂²⋊C₄	C₂	semidirect
6. C₂×#518	C₃	direct (N × H)
8. C₂²×C₃×D₄	C₄	semidirect
12. C₂³×C₃×C₄	C₂²	semidirect
13. C₃×C₄×D₄	C₂²	semidirect
14. C₃×#117	C₂²	semidirect
15. C₂×C₃×C₄⋊C₄	C₂²	semidirect
17. C₂×C₃×C₂²⋊C₄	C₂²	semidirect
18. C₂×C₃×C₂²⋊C₄	C₂²	semidirect
19. C₂×C₃×C₄²	C₂²	semidirect
21. #518	C₂×C₃	direct (N × H)
22. C₂×C₄×D₄	C₂×C₃	semidirect
23. C₂×#117	C₂×C₃	semidirect
24. C₂²×C₂²⋊C₄	C₂×C₃	semidirect
26. C₂⁴×C₃	C₂×C₄	semidirect
30. C₂×C₃×D₄	C₂×C₄	semidirect
31. C₂×C₃×D₄	C₂×C₄	semidirect
34. C₂²×C₃×C₄	C₂×C₄	semidirect
38. C₂²×C₃×C₄	C₂³	semidirect
39. C₃×C₄⋊C₄	C₂³	semidirect
41. C₃×C₂²⋊C₄	C₂³	semidirect
42. C₃×C₄²	C₂³	semidirect
44. C₂²×D₄	C₃×C₄	semidirect
48. C₂³×C₄	C₂²×C₃	semidirect
49. C₄×D₄	C₂²×C₃	semidirect
50. #117	C₂²×C₃	semidirect
51. C₂×C₄⋊C₄	C₂²×C₃	semidirect
53. C₂×C₂²⋊C₄	C₂²×C₃	semidirect
54. C₂×C₂²⋊C₄	C₂²×C₃	semidirect
55. C₂×C₄²	C₂²×C₃	semidirect
62. C₂³×C₃	C₂²×C₄	semidirect
66. C₃×D₄	C₂²×C₄	semidirect
70. C₂×C₃×C₄	C₂²×C₄	semidirect
74. C₂⁴	C₂×C₃×C₄	semidirect
78. C₂×D₄	C₂×C₃×C₄	semidirect
79. C₂×D₄	C₂×C₃×C₄	semidirect
82. C₂²×C₄	C₂×C₃×C₄	semidirect
86. C₂²×C₄	C₂³×C₃	semidirect
87. C₄⋊C₄	C₂³×C₃	semidirect
89. C₂²⋊C₄	C₂³×C₃	semidirect
90. C₄²	C₂³×C₃	semidirect
109. C₂³	C₂²×C₃×C₄	semidirect
113. D₄	C₂²×C₃×C₄	semidirect
117. C₂×C₄	C₂²×C₃×C₄	semidirect
122. C₂×C₃	#518	direct (N × H)
137. C₃	C₂×#518	direct (N × H)
140. C₂	C₃×#518	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×C₃×D₄	2
C₃×#518	2
C₂×C₃×C₄×D₄	2
C₂×C₃×#117	2
C₂²×C₃×C₂²⋊C₄	2
C₂×#518	3

Subgroups — C2×C3×#518 #87309

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×#518 #87309