Subgroups — 3₋¹⁺²×#430 #13145050

Order 1728.

408 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	C₂×M₁₆×3₋¹⁺²	C₂	1	2	yes	maximal
2	C₂×M₁₆×3₋¹⁺²	C₂	1	1	yes	maximal, characteristic
3	3₋¹⁺²×#108	C₂	1	4	yes	maximal, complemented
4	C₃²×#430	C₃	1	1	yes	maximal, characteristic
5	C₉×#430	C₃	1	3	yes	maximal, complemented
6	C₂²×C₄×3₋¹⁺²	C₂²	1	1	yes	characteristic
7	M₁₆×3₋¹⁺²	—	2	2	no	—
8	M₁₆×3₋¹⁺²	C₄	1	4	yes	—
9	M₁₆×3₋¹⁺²	C₂²	1	4	yes	—
10	C₂×C₈×3₋¹⁺²	—	2	2	no	—
11	C₂×C₈×3₋¹⁺²	C₂²	1	2	yes	—
12	C₂×C₃²×M₁₆	C₂×C₃	1	2	yes	—
13	C₂×C₃²×M₁₆	C₂×C₃	1	1	yes	characteristic
14	C₃²×#108	C₂×C₃	1	4	yes	—
15	C₂×C₉×M₁₆	C₂×C₃	1	6	yes	—
16	C₂×C₉×M₁₆	C₂×C₃	1	3	yes	—
17	C₉×#108	C₂×C₃	1	12	yes	complemented
18	C₂³×3₋¹⁺²	Q₈	1	1	yes	characteristic
19	C₂×C₄×3₋¹⁺²	C₂×C₄	1	2	yes	—
20	C₂×C₄×3₋¹⁺²	D₄	1	2	yes	—
21	C₂×C₄×3₋¹⁺²	Q₈	1	1	yes	characteristic
22	C₂×C₄×3₋¹⁺²	C₂³	1	1	yes	characteristic
23	C₈×3₋¹⁺²	—	2	4	no	—
24	C₈×3₋¹⁺²	C₂×C₄	1	4	yes	—
25	C₃×#430	—	3	1	no	—
26	C₃×#430	C₃²	1	1	yes	characteristic
27	C₂²×C₃²×C₄	C₂²×C₃	1	1	yes	characteristic
28	C₃²×M₁₆	—	2	2	no	—
29	C₃²×M₁₆	C₃×C₄	1	4	yes	—
30	C₃²×M₁₆	C₂²×C₃	1	4	yes	—
31	C₂×C₃²×C₈	—	2	2	no	—
32	C₂×C₃²×C₈	C₂²×C₃	1	2	yes	—
33	C₂²×C₄×C₉	C₂²×C₃	1	3	yes	—
34	C₉×M₁₆	—	2	6	no	—
35	C₉×M₁₆	C₃×C₄	1	12	yes	—
36	C₉×M₁₆	C₂²×C₃	1	12	yes	—
37	C₂×C₈×C₉	—	2	6	no	—
38	C₂×C₈×C₉	C₂²×C₃	1	6	yes	—
39	C₂²×3₋¹⁺²	—	4	1	no	—
40	C₂²×3₋¹⁺²	C₄⋊C₄	1	2	yes	—
41	C₂²×3₋¹⁺²	C₂×Q₈	1	1	yes	characteristic
42	C₄×3₋¹⁺²	C₄⋊C₄	1	2	yes	—
43	C₄×3₋¹⁺²	C₂²×C₄	1	1	yes	characteristic
44	C₄×3₋¹⁺²	C₂×D₄	1	1	yes	characteristic
45	C₂×C₃×M₁₆	—	3	1	no	—
46	C₂×C₃×M₁₆	—	3	2	no	—
47	C₂×C₃×M₁₆	C₂×C₃²	1	2	yes	—
48	C₂×C₃×M₁₆	C₂×C₃²	1	1	yes	characteristic
49	C₃×#108	—	3	4	no	—
50	C₃×#108	C₂×C₃²	1	4	yes	—
51	C₂³×C₃²	C₃×Q₈	1	1	yes	characteristic
52	C₂×C₃²×C₄	C₂×C₃×C₄	1	2	yes	—
53	C₂×C₃²×C₄	C₃×D₄	1	2	yes	—
54	C₂×C₃²×C₄	C₃×Q₈	1	1	yes	characteristic
55	C₂×C₃²×C₄	C₂³×C₃	1	1	yes	characteristic
56	C₂³×C₉	C₃×Q₈	1	3	yes	—
57	C₃²×C₈	—	2	4	no	—
58	C₃²×C₈	C₂×C₃×C₄	1	4	yes	—
59	C₂×C₄×C₉	C₂×C₃×C₄	1	6	yes	—
60	C₂×C₄×C₉	C₃×D₄	1	6	yes	—
61	C₂×C₄×C₉	C₃×Q₈	1	3	yes	—
62	C₂×C₄×C₉	C₂³×C₃	1	3	yes	—
63	C₈×C₉	—	2	12	no	—
64	C₈×C₉	C₂×C₃×C₄	1	12	yes	—
65	#430	3₋¹⁺²	1	1	yes	characteristic, direct product factor, Sylow
66	C₂×3₋¹⁺²	—	2	2	no	—
67	C₂×3₋¹⁺²	—	2	1	no	—
68	C₂×3₋¹⁺²	C₂×C₄⋊C₄	1	1	yes	characteristic
69	C₂²×C₃×C₄	—	3	1	no	—
70	C₂²×C₃×C₄	C₂²×C₃²	1	1	yes	characteristic
71	C₃×M₁₆	—	2	2	no	—
72	C₃×M₁₆	—	3	4	no	—
73	C₃×M₁₆	—	3	4	no	—
74	C₃×M₁₆	—	6	2	no	—
75	C₃×M₁₆	C₃²×C₄	1	4	yes	—
76	C₃×M₁₆	C₂²×C₃²	1	4	yes	—
77	C₂×C₃×C₈	—	2	2	no	—
78	C₂×C₃×C₈	—	3	2	no	—
79	C₂×C₃×C₈	—	6	2	no	—
80	C₂×C₃×C₈	C₂²×C₃²	1	2	yes	—
81	C₂²×C₃²	—	4	1	no	—
82	C₂²×C₃²	C₃×C₄⋊C₄	1	2	yes	—
83	C₂²×C₃²	C₂×C₃×Q₈	1	1	yes	characteristic
84	C₃²×C₄	C₃×C₄⋊C₄	1	2	yes	—
85	C₃²×C₄	C₂²×C₃×C₄	1	1	yes	characteristic
86	C₃²×C₄	C₂×C₃×D₄	1	1	yes	characteristic
87	C₂²×C₉	—	4	3	no	—
88	C₂²×C₉	C₃×C₄⋊C₄	1	6	yes	—
89	C₂²×C₉	C₂×C₃×Q₈	1	3	yes	—
90	C₄×C₉	C₃×C₄⋊C₄	1	6	yes	—
91	C₄×C₉	C₂²×C₃×C₄	1	3	yes	—
92	C₄×C₉	C₂×C₃×D₄	1	3	yes	—
93	C₂×M₁₆	C₂×3₋¹⁺²	1	2	yes	—
94	C₂×M₁₆	C₂×3₋¹⁺²	1	1	yes	characteristic
95	#108	C₂×3₋¹⁺²	1	4	yes	complemented
96	3₋¹⁺²	#430	1	1	yes	characteristic, direct product factor, Sylow
97	C₂³×C₃	—	3	1	no	—
98	C₂³×C₃	C₃²×Q₈	1	1	yes	characteristic
99	C₂×C₃×C₄	—	3	2	no	—
100	C₂×C₃×C₄	—	3	2	no	—
101	C₂×C₃×C₄	—	3	1	no	—
102	C₂×C₃×C₄	—	3	1	no	—
103	C₂×C₃×C₄	C₂×C₃²×C₄	1	2	yes	—
104	C₂×C₃×C₄	C₃²×D₄	1	2	yes	—
105	C₂×C₃×C₄	C₃²×Q₈	1	1	yes	characteristic
106	C₂×C₃×C₄	C₂³×C₃²	1	1	yes	characteristic
107	C₃×C₈	—	2	4	no	—
108	C₃×C₈	—	3	4	no	—
109	C₃×C₈	—	6	4	no	—
110	C₃×C₈	C₂×C₃²×C₄	1	4	yes	—
111	C₂×C₃²	—	2	2	no	—
112	C₂×C₃²	—	2	1	no	—
113	C₂×C₃²	C₂×C₃×C₄⋊C₄	1	1	yes	characteristic
114	C₂×C₉	—	2	6	no	—
115	C₂×C₉	—	2	3	no	—
116	C₂×C₉	C₂×C₃×C₄⋊C₄	1	3	yes	—
117	C₂²×C₄	C₂²×3₋¹⁺²	1	1	yes	characteristic
118	M₁₆	—	2	2	no	—
119	M₁₆	C₄×3₋¹⁺²	1	4	yes	—
120	M₁₆	C₂²×3₋¹⁺²	1	4	yes	—
121	C₂×C₈	—	2	2	no	—
122	C₂×C₈	C₂²×3₋¹⁺²	1	2	yes	—
123	C₂²×C₃	—	3	1	no	—
124	C₂²×C₃	—	3	2	no	—
125	C₂²×C₃	—	4	1	no	—
126	C₂²×C₃	—	12	1	no	—
127	C₂²×C₃	C₃²×C₄⋊C₄	1	2	yes	—
128	C₂²×C₃	C₂×C₃²×Q₈	1	1	yes	characteristic
129	C₃×C₄	—	3	1	no	—
130	C₃×C₄	—	3	1	no	—
131	C₃×C₄	—	3	2	no	—
132	C₃×C₄	C₃²×C₄⋊C₄	1	2	yes	—
133	C₃×C₄	C₂²×C₃²×C₄	1	1	yes	characteristic
134	C₃×C₄	C₂×C₃²×D₄	1	1	yes	characteristic
135	C₃²	C₃×#430	1	1	yes	characteristic
136	C₉	C₃×#430	1	3	yes	complemented
137	C₂³	Q₈×3₋¹⁺²	1	1	yes	characteristic
138	C₂×C₄	C₂×C₄×3₋¹⁺²	1	2	yes	—
139	C₂×C₄	D₄×3₋¹⁺²	1	2	yes	—
140	C₂×C₄	Q₈×3₋¹⁺²	1	1	yes	characteristic
141	C₂×C₄	C₂³×3₋¹⁺²	1	1	yes	characteristic
142	C₈	—	2	4	no	—
143	C₈	C₂×C₄×3₋¹⁺²	1	4	yes	—
144	C₂×C₃	—	2	2	no	—
145	C₂×C₃	—	2	1	no	—
146	C₂×C₃	—	3	1	no	—
147	C₂×C₃	—	6	1	no	—
148	C₂×C₃	—	6	2	no	—
149	C₂×C₃	C₂×C₃²×C₄⋊C₄	1	1	yes	characteristic
150	C₂²	—	4	1	no	—
151	C₂²	C₄⋊C₄×3₋¹⁺²	1	2	yes	—
152	C₂²	C₂×Q₈×3₋¹⁺²	1	1	yes	characteristic
153	C₄	C₄⋊C₄×3₋¹⁺²	1	2	yes	—
154	C₄	C₂²×C₄×3₋¹⁺²	1	1	yes	characteristic
155	C₄	C₂×D₄×3₋¹⁺²	1	1	yes	characteristic
156	C₃	—	3	1	no	—
157	C₃	C₃²×#430	1	1	yes	characteristic
158	C₂	—	2	1	no	—
159	C₂	—	2	2	no	—
160	C₂	C₂×C₄⋊C₄×3₋¹⁺²	1	1	yes	characteristic

	subgroup	G/H
center	C₃×C₄	C₂×C₃²×D₄
commutator	C₃×C₄	C₂²×C₃²×C₄
Frattini, cosocle	C₂×C₃×C₄	C₂³×C₃²
socle	C₂×C₃	C₂×C₃²×C₄⋊C₄
prime radical	C₂³×C₃²	C₃×Q₈
LCS	G ▷ C₃×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₃×C₄ ◁ C₂²×C₄×3₋¹⁺² ◁ G

Complements

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

N	H ≅ G/N	type
3. 3₋¹⁺²×#108	C₂	semidirect
5. C₉×#430	C₃	semidirect
17. C₉×#108	C₂×C₃	semidirect
65. #430	3₋¹⁺²	direct (N × H)
95. #108	C₂×3₋¹⁺²	semidirect
96. 3₋¹⁺²	#430	direct (N × H)
136. C₉	C₃×#430	semidirect

Maximal subgroups

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

subgroup	index
C₂×M₁₆×3₋¹⁺²	2
3₋¹⁺²×#108	2
C₃²×#430	3
C₉×#430	3

Subgroups — 3−1+2×#430 #13145050

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — 3₋¹⁺²×#430 #13145050