Subgroups — C₃²×#577 #10596378

Order 576.

774 conjugacy classes of subgroups.

The subgroup lattice has 162 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²×2₋¹⁺⁴	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃²×H(5,𝔽₂)	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃²×#137	C₂	1	3	yes	maximal, complemented
4	C₃²×AGL(1,ℤ/8)	C₂	1	3	yes	maximal, complemented
5	C₃²×#135	C₂	1	3	yes	maximal, complemented
6	C₂×C₃²×SD₁₆	C₂	1	3	yes	maximal, complemented
7	C₃²×#131	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃×#577	C₃	1	4	yes	maximal, direct product factor
9	C₃²×#41	—	2	1	no	—
10	C₃²×#41	—	2	3	no	—
11	C₃²×#41	C₂²	1	3	yes	—
12	C₃²×#41	C₂²	1	1	yes	characteristic
13	C₃²×#41	C₂²	1	3	yes	complemented
14	C₂×C₃²×Q₈	—	2	1	no	—
15	C₂×C₃²×Q₈	C₂²	1	3	yes	complemented
16	C₂×C₃²×D₄	—	2	3	no	—
17	C₂×C₃²×D₄	C₂²	1	3	yes	—
18	C₃²×Q₁₆	C₂²	1	3	yes	complemented
19	C₃²×SD₁₆	C₂²	1	3	yes	complemented
20	C₃²×SD₁₆	C₂²	1	6	yes	complemented
21	C₃²×SD₁₆	C₂²	1	1	yes	characteristic, complemented
22	C₃²×D₈	C₂²	1	3	yes	complemented
23	C₃²×M₁₆	C₂²	1	3	yes	complemented
24	C₂×C₃²×C₈	C₂²	1	3	yes	complemented
25	C₃×2₋¹⁺⁴	C₂×C₃	1	4	yes	complemented
26	C₃×H(5,𝔽₂)	C₂×C₃	1	4	yes	complemented
27	C₃×#137	C₂×C₃	1	12	yes	complemented
28	C₃×AGL(1,ℤ/8)	C₂×C₃	1	12	yes	complemented
29	C₃×#135	C₂×C₃	1	12	yes	complemented
30	C₂×C₃×SD₁₆	C₂×C₃	1	12	yes	complemented
31	C₃×#131	C₂×C₃	1	4	yes	complemented
32	C₂³×C₃²	—	2	3	no	—
33	C₃²×Q₈	—	2	3	no	—
34	C₃²×Q₈	D₄	1	1	yes	characteristic
35	C₃²×Q₈	C₂³	1	3	yes	—
36	C₃²×Q₈	C₂³	1	1	yes	characteristic
37	C₃²×D₄	—	2	3	no	—
38	C₃²×D₄	—	2	3	no	—
39	C₃²×D₄	—	2	3	no	—
40	C₃²×D₄	D₄	1	3	yes	—
41	C₃²×D₄	C₂³	1	1	yes	characteristic
42	C₃²×D₄	C₂³	1	3	yes	—
43	C₂×C₃²×C₄	—	2	3	no	—
44	C₂×C₃²×C₄	—	2	3	no	—
45	C₂×C₃²×C₄	—	2	3	no	—
46	C₂×C₃²×C₄	C₂³	1	3	yes	—
47	C₃²×C₈	C₂³	1	1	yes	characteristic
48	C₃²×C₈	C₂³	1	3	yes	—
49	#577	C₃²	1	1	yes	characteristic, direct product factor, Sylow
50	C₃×#41	—	2	4	no	—
51	C₃×#41	—	2	12	no	—
52	C₃×#41	C₂²×C₃	1	12	yes	—
53	C₃×#41	C₂²×C₃	1	4	yes	—
54	C₃×#41	C₂²×C₃	1	12	yes	complemented
55	C₂×C₃×Q₈	—	2	4	no	—
56	C₂×C₃×Q₈	C₂²×C₃	1	12	yes	complemented
57	C₂×C₃×D₄	—	2	12	no	—
58	C₂×C₃×D₄	C₂²×C₃	1	12	yes	—
59	C₃×Q₁₆	C₂²×C₃	1	12	yes	complemented
60	C₃×SD₁₆	C₂²×C₃	1	12	yes	complemented
61	C₃×SD₁₆	C₂²×C₃	1	24	yes	complemented
62	C₃×SD₁₆	C₂²×C₃	1	4	yes	complemented
63	C₃×D₈	C₂²×C₃	1	12	yes	complemented
64	C₃×M₁₆	C₂²×C₃	1	12	yes	complemented
65	C₂×C₃×C₈	C₂²×C₃	1	12	yes	complemented
66	C₂²×C₃²	—	2	1	no	—
67	C₂²×C₃²	—	2	3	no	—
68	C₂²×C₃²	—	8	3	no	—
69	C₂²×C₃²	C₂×D₄	1	3	yes	—
70	C₃²×C₄	—	2	1	no	—
71	C₃²×C₄	—	2	3	no	—
72	C₃²×C₄	C₂×D₄	1	3	yes	—
73	C₃²×C₄	C₂⁴	1	1	yes	characteristic
74	2₋¹⁺⁴	C₂×C₃²	1	1	yes	characteristic, complemented
75	H(5,𝔽₂)	C₂×C₃²	1	1	yes	characteristic, complemented
76	#137	C₂×C₃²	1	3	yes	complemented
77	AGL(1,ℤ/8)	C₂×C₃²	1	3	yes	complemented
78	#135	C₂×C₃²	1	3	yes	complemented
79	C₂×SD₁₆	C₂×C₃²	1	3	yes	complemented
80	#131	C₂×C₃²	1	1	yes	characteristic, complemented
81	C₂³×C₃	—	2	12	no	—
82	C₃×Q₈	—	2	12	no	—
83	C₃×Q₈	C₃×D₄	1	4	yes	—
84	C₃×Q₈	C₂³×C₃	1	12	yes	—
85	C₃×Q₈	C₂³×C₃	1	4	yes	—
86	C₃×D₄	—	2	12	no	—
87	C₃×D₄	—	2	12	no	—
88	C₃×D₄	—	2	12	no	—
89	C₃×D₄	C₃×D₄	1	12	yes	—
90	C₃×D₄	C₂³×C₃	1	4	yes	—
91	C₃×D₄	C₂³×C₃	1	12	yes	—
92	C₂×C₃×C₄	—	2	12	no	—
93	C₂×C₃×C₄	—	2	12	no	—
94	C₂×C₃×C₄	—	2	12	no	—
95	C₂×C₃×C₄	C₂³×C₃	1	12	yes	—
96	C₃×C₈	C₂³×C₃	1	4	yes	—
97	C₃×C₈	C₂³×C₃	1	12	yes	—
98	C₂×C₃²	—	2	3	no	—
99	C₂×C₃²	—	4	1	no	—
100	C₂×C₃²	—	4	3	no	—
101	C₂×C₃²	C₂²×D₄	1	1	yes	characteristic
102	#41	—	2	1	no	—
103	#41	—	2	3	no	—
104	#41	C₂²×C₃²	1	3	yes	—
105	#41	C₂²×C₃²	1	1	yes	characteristic
106	#41	C₂²×C₃²	1	3	yes	complemented
107	C₂×Q₈	—	2	1	no	—
108	C₂×Q₈	C₂²×C₃²	1	3	yes	complemented
109	C₂×D₄	—	2	3	no	—
110	C₂×D₄	C₂²×C₃²	1	3	yes	—
111	Q₁₆	C₂²×C₃²	1	3	yes	complemented
112	SD₁₆	C₂²×C₃²	1	3	yes	complemented
113	SD₁₆	C₂²×C₃²	1	6	yes	complemented
114	SD₁₆	C₂²×C₃²	1	1	yes	characteristic, complemented
115	D₈	C₂²×C₃²	1	3	yes	complemented
116	M₁₆	C₂²×C₃²	1	3	yes	complemented
117	C₂×C₈	C₂²×C₃²	1	3	yes	complemented
118	C₂²×C₃	—	2	4	no	—
119	C₂²×C₃	—	2	12	no	—
120	C₂²×C₃	—	8	12	no	—
121	C₂²×C₃	C₂×C₃×D₄	1	12	yes	—
122	C₃×C₄	—	2	4	no	—
123	C₃×C₄	—	2	12	no	—
124	C₃×C₄	C₂×C₃×D₄	1	12	yes	—
125	C₃×C₄	C₂⁴×C₃	1	4	yes	—
126	C₃²	#577	1	1	yes	characteristic, direct product factor, Sylow
127	C₂³	—	2	3	no	—
128	Q₈	—	2	3	no	—
129	Q₈	C₃²×D₄	1	1	yes	characteristic
130	Q₈	C₂³×C₃²	1	3	yes	—
131	Q₈	C₂³×C₃²	1	1	yes	characteristic
132	D₄	—	2	3	no	—
133	D₄	—	2	3	no	—
134	D₄	—	2	3	no	—
135	D₄	C₃²×D₄	1	3	yes	—
136	D₄	C₂³×C₃²	1	1	yes	characteristic
137	D₄	C₂³×C₃²	1	3	yes	—
138	C₂×C₄	—	2	3	no	—
139	C₂×C₄	—	2	3	no	—
140	C₂×C₄	—	2	3	no	—
141	C₂×C₄	C₂³×C₃²	1	3	yes	—
142	C₈	C₂³×C₃²	1	1	yes	characteristic
143	C₈	C₂³×C₃²	1	3	yes	—
144	C₂×C₃	—	2	12	no	—
145	C₂×C₃	—	4	12	no	—
146	C₂×C₃	—	4	4	no	—
147	C₂×C₃	C₂²×C₃×D₄	1	4	yes	—
148	C₂²	—	2	1	no	—
149	C₂²	—	2	3	no	—
150	C₂²	—	8	3	no	—
151	C₂²	C₂×C₃²×D₄	1	3	yes	—
152	C₄	—	2	1	no	—
153	C₄	—	2	3	no	—
154	C₄	C₂×C₃²×D₄	1	3	yes	—
155	C₄	C₂⁴×C₃²	1	1	yes	characteristic
156	C₃	C₃×#577	1	4	yes	direct product factor
157	C₂	—	2	3	no	—
158	C₂	—	4	3	no	—
159	C₂	—	4	1	no	—
160	C₂	C₂²×C₃²×D₄	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₃²	C₂²×D₄
commutator, Frattini, cosocle	C₄	C₂⁴×C₃²
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₃² ◁ C₃²×#41 ◁ G

Complements

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

N	H ≅ G/N	type
1. C₃²×2₋¹⁺⁴	C₂	semidirect
2. C₃²×H(5,𝔽₂)	C₂	semidirect
3. C₃²×#137	C₂	semidirect
4. C₃²×AGL(1,ℤ/8)	C₂	semidirect
5. C₃²×#135	C₂	semidirect
6. C₂×C₃²×SD₁₆	C₂	semidirect
7. C₃²×#131	C₂	semidirect
8. C₃×#577	C₃	direct (N × H)
13. C₃²×#41	C₂²	semidirect
15. C₂×C₃²×Q₈	C₂²	semidirect
18. C₃²×Q₁₆	C₂²	semidirect
19. C₃²×SD₁₆	C₂²	semidirect
20. C₃²×SD₁₆	C₂²	semidirect
21. C₃²×SD₁₆	C₂²	semidirect
22. C₃²×D₈	C₂²	semidirect
23. C₃²×M₁₆	C₂²	semidirect
24. C₂×C₃²×C₈	C₂²	semidirect
25. C₃×2₋¹⁺⁴	C₂×C₃	semidirect
26. C₃×H(5,𝔽₂)	C₂×C₃	semidirect
27. C₃×#137	C₂×C₃	semidirect
28. C₃×AGL(1,ℤ/8)	C₂×C₃	semidirect
29. C₃×#135	C₂×C₃	semidirect
30. C₂×C₃×SD₁₆	C₂×C₃	semidirect
31. C₃×#131	C₂×C₃	semidirect
49. #577	C₃²	direct (N × H)
54. C₃×#41	C₂²×C₃	semidirect
56. C₂×C₃×Q₈	C₂²×C₃	semidirect
59. C₃×Q₁₆	C₂²×C₃	semidirect
60. C₃×SD₁₆	C₂²×C₃	semidirect
61. C₃×SD₁₆	C₂²×C₃	semidirect
62. C₃×SD₁₆	C₂²×C₃	semidirect
63. C₃×D₈	C₂²×C₃	semidirect
64. C₃×M₁₆	C₂²×C₃	semidirect
65. C₂×C₃×C₈	C₂²×C₃	semidirect
74. 2₋¹⁺⁴	C₂×C₃²	semidirect
75. H(5,𝔽₂)	C₂×C₃²	semidirect
76. #137	C₂×C₃²	semidirect
77. AGL(1,ℤ/8)	C₂×C₃²	semidirect
78. #135	C₂×C₃²	semidirect
79. C₂×SD₁₆	C₂×C₃²	semidirect
80. #131	C₂×C₃²	semidirect
106. #41	C₂²×C₃²	semidirect
108. C₂×Q₈	C₂²×C₃²	semidirect
111. Q₁₆	C₂²×C₃²	semidirect
112. SD₁₆	C₂²×C₃²	semidirect
113. SD₁₆	C₂²×C₃²	semidirect
114. SD₁₆	C₂²×C₃²	semidirect
115. D₈	C₂²×C₃²	semidirect
116. M₁₆	C₂²×C₃²	semidirect
117. C₂×C₈	C₂²×C₃²	semidirect
126. C₃²	#577	direct (N × H)
156. C₃	C₃×#577	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃²×2₋¹⁺⁴	2
C₃²×H(5,𝔽₂)	2
C₃²×#137	2
C₃²×AGL(1,ℤ/8)	2
C₃²×#135	2
C₂×C₃²×SD₁₆	2
C₃²×#131	2
C₃×#577	3

Subgroups — C32×#577 #10596378

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃²×#577 #10596378