Subgroups — C₂×C₄×C₈×3₋¹⁺² #13144966

Order 1728.

648 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	C₂²×C₈×3₋¹⁺²	C₂	1	2	yes	maximal
2	C₂×C₄²×3₋¹⁺²	C₂	1	1	yes	maximal, characteristic
3	C₄×C₈×3₋¹⁺²	C₂	1	4	yes	maximal, direct product factor
4	C₂×C₃²×C₄×C₈	C₃	1	1	yes	maximal, characteristic
5	C₂×C₄×C₈×C₉	C₃	1	3	yes	maximal, complemented
6	C₂²×C₄×3₋¹⁺²	C₄	1	2	yes	—
7	C₂²×C₄×3₋¹⁺²	C₂²	1	1	yes	characteristic
8	C₂×C₈×3₋¹⁺²	C₄	1	8	yes	direct product factor
9	C₂×C₈×3₋¹⁺²	C₂²	1	4	yes	—
10	C₄²×3₋¹⁺²	C₄	1	2	yes	—
11	C₄²×3₋¹⁺²	C₂²	1	2	yes	—
12	C₂²×C₃²×C₈	C₂×C₃	1	2	yes	—
13	C₂×C₃²×C₄²	C₂×C₃	1	1	yes	characteristic
14	C₃²×C₄×C₈	C₂×C₃	1	4	yes	—
15	C₂²×C₈×C₉	C₂×C₃	1	6	yes	—
16	C₂×C₄²×C₉	C₂×C₃	1	3	yes	—
17	C₄×C₈×C₉	C₂×C₃	1	12	yes	complemented
18	C₂³×3₋¹⁺²	C₂×C₄	1	1	yes	characteristic
19	C₂×C₄×3₋¹⁺²	C₈	1	8	yes	direct product factor
20	C₂×C₄×3₋¹⁺²	C₂×C₄	1	1	yes	characteristic
21	C₂×C₄×3₋¹⁺²	C₂×C₄	1	4	yes	—
22	C₂×C₄×3₋¹⁺²	C₂×C₄	1	4	yes	—
23	C₂×C₄×3₋¹⁺²	C₂³	1	1	yes	characteristic
24	C₈×3₋¹⁺²	C₂×C₄	1	8	yes	direct product factor
25	C₂×C₃×C₄×C₈	—	3	1	no	—
26	C₂×C₃×C₄×C₈	C₃²	1	1	yes	characteristic
27	C₂²×C₃²×C₄	C₃×C₄	1	2	yes	—
28	C₂²×C₃²×C₄	C₂²×C₃	1	1	yes	characteristic
29	C₂×C₃²×C₈	C₃×C₄	1	8	yes	—
30	C₂×C₃²×C₈	C₂²×C₃	1	4	yes	—
31	C₃²×C₄²	C₃×C₄	1	2	yes	—
32	C₃²×C₄²	C₂²×C₃	1	2	yes	—
33	C₂²×C₄×C₉	C₃×C₄	1	6	yes	—
34	C₂²×C₄×C₉	C₂²×C₃	1	3	yes	—
35	C₂×C₈×C₉	C₃×C₄	1	24	yes	complemented
36	C₂×C₈×C₉	C₂²×C₃	1	12	yes	—
37	C₄²×C₉	C₃×C₄	1	6	yes	—
38	C₄²×C₉	C₂²×C₃	1	6	yes	—
39	C₂²×3₋¹⁺²	C₄²	1	2	yes	—
40	C₂²×3₋¹⁺²	C₂×C₈	1	4	yes	—
41	C₂²×3₋¹⁺²	C₂²×C₄	1	1	yes	characteristic
42	C₄×3₋¹⁺²	C₄²	1	2	yes	—
43	C₄×3₋¹⁺²	C₂×C₈	1	8	yes	direct product factor
44	C₄×3₋¹⁺²	C₂²×C₄	1	2	yes	—
45	C₂²×C₃×C₈	—	3	2	no	—
46	C₂²×C₃×C₈	C₂×C₃²	1	2	yes	—
47	C₂×C₃×C₄²	—	3	1	no	—
48	C₂×C₃×C₄²	C₂×C₃²	1	1	yes	characteristic
49	C₃×C₄×C₈	—	3	4	no	—
50	C₃×C₄×C₈	C₂×C₃²	1	4	yes	—
51	C₂³×C₃²	C₂×C₃×C₄	1	1	yes	characteristic
52	C₂×C₃²×C₄	C₃×C₈	1	8	yes	—
53	C₂×C₃²×C₄	C₂×C₃×C₄	1	1	yes	characteristic
54	C₂×C₃²×C₄	C₂×C₃×C₄	1	4	yes	—
55	C₂×C₃²×C₄	C₂×C₃×C₄	1	4	yes	—
56	C₂×C₃²×C₄	C₂³×C₃	1	1	yes	characteristic
57	C₂³×C₉	C₂×C₃×C₄	1	3	yes	—
58	C₃²×C₈	C₂×C₃×C₄	1	8	yes	—
59	C₂×C₄×C₉	C₃×C₈	1	24	yes	complemented
60	C₂×C₄×C₉	C₂×C₃×C₄	1	3	yes	—
61	C₂×C₄×C₉	C₂×C₃×C₄	1	12	yes	—
62	C₂×C₄×C₉	C₂×C₃×C₄	1	12	yes	—
63	C₂×C₄×C₉	C₂³×C₃	1	3	yes	—
64	C₈×C₉	C₂×C₃×C₄	1	24	yes	complemented
65	C₂×C₄×C₈	3₋¹⁺²	1	1	yes	characteristic, direct product factor, Sylow
66	C₂×3₋¹⁺²	C₄×C₈	1	4	yes	direct product factor
67	C₂×3₋¹⁺²	C₂×C₄²	1	1	yes	characteristic
68	C₂×3₋¹⁺²	C₂²×C₈	1	2	yes	—
69	C₂²×C₃×C₄	—	3	2	no	—
70	C₂²×C₃×C₄	—	3	1	no	—
71	C₂²×C₃×C₄	C₃²×C₄	1	2	yes	—
72	C₂²×C₃×C₄	C₂²×C₃²	1	1	yes	characteristic
73	C₂×C₃×C₈	—	3	8	no	—
74	C₂×C₃×C₈	—	3	4	no	—
75	C₂×C₃×C₈	C₃²×C₄	1	8	yes	—
76	C₂×C₃×C₈	C₂²×C₃²	1	4	yes	—
77	C₃×C₄²	—	3	2	no	—
78	C₃×C₄²	—	3	2	no	—
79	C₃×C₄²	C₃²×C₄	1	2	yes	—
80	C₃×C₄²	C₂²×C₃²	1	2	yes	—
81	C₂²×C₃²	C₃×C₄²	1	2	yes	—
82	C₂²×C₃²	C₂×C₃×C₈	1	4	yes	—
83	C₂²×C₃²	C₂²×C₃×C₄	1	1	yes	characteristic
84	C₃²×C₄	C₃×C₄²	1	2	yes	—
85	C₃²×C₄	C₂×C₃×C₈	1	8	yes	—
86	C₃²×C₄	C₂²×C₃×C₄	1	2	yes	—
87	C₂²×C₉	C₃×C₄²	1	6	yes	—
88	C₂²×C₉	C₂×C₃×C₈	1	12	yes	—
89	C₂²×C₉	C₂²×C₃×C₄	1	3	yes	—
90	C₄×C₉	C₃×C₄²	1	6	yes	—
91	C₄×C₉	C₂×C₃×C₈	1	24	yes	complemented
92	C₄×C₉	C₂²×C₃×C₄	1	6	yes	—
93	C₂²×C₈	C₂×3₋¹⁺²	1	2	yes	—
94	C₂×C₄²	C₂×3₋¹⁺²	1	1	yes	characteristic
95	C₄×C₈	C₂×3₋¹⁺²	1	4	yes	direct product factor
96	3₋¹⁺²	C₂×C₄×C₈	1	1	yes	characteristic, direct product factor, Sylow
97	C₂³×C₃	—	3	1	no	—
98	C₂³×C₃	C₂×C₃²×C₄	1	1	yes	characteristic
99	C₂×C₃×C₄	—	3	8	no	—
100	C₂×C₃×C₄	—	3	1	no	—
101	C₂×C₃×C₄	—	3	4	no	—
102	C₂×C₃×C₄	—	3	4	no	—
103	C₂×C₃×C₄	—	3	1	no	—
104	C₂×C₃×C₄	C₃²×C₈	1	8	yes	—
105	C₂×C₃×C₄	C₂×C₃²×C₄	1	1	yes	characteristic
106	C₂×C₃×C₄	C₂×C₃²×C₄	1	4	yes	—
107	C₂×C₃×C₄	C₂×C₃²×C₄	1	4	yes	—
108	C₂×C₃×C₄	C₂³×C₃²	1	1	yes	characteristic
109	C₃×C₈	—	3	8	no	—
110	C₃×C₈	C₂×C₃²×C₄	1	8	yes	—
111	C₂×C₃²	C₃×C₄×C₈	1	4	yes	—
112	C₂×C₃²	C₂×C₃×C₄²	1	1	yes	characteristic
113	C₂×C₃²	C₂²×C₃×C₈	1	2	yes	—
114	C₂×C₉	C₃×C₄×C₈	1	12	yes	complemented
115	C₂×C₉	C₂×C₃×C₄²	1	3	yes	—
116	C₂×C₉	C₂²×C₃×C₈	1	6	yes	—
117	C₂²×C₄	C₄×3₋¹⁺²	1	2	yes	—
118	C₂²×C₄	C₂²×3₋¹⁺²	1	1	yes	characteristic
119	C₂×C₈	C₄×3₋¹⁺²	1	8	yes	direct product factor
120	C₂×C₈	C₂²×3₋¹⁺²	1	4	yes	—
121	C₄²	C₄×3₋¹⁺²	1	2	yes	—
122	C₄²	C₂²×3₋¹⁺²	1	2	yes	—
123	C₂²×C₃	—	3	1	no	—
124	C₂²×C₃	—	3	2	no	—
125	C₂²×C₃	—	3	4	no	—
126	C₂²×C₃	C₃²×C₄²	1	2	yes	—
127	C₂²×C₃	C₂×C₃²×C₈	1	4	yes	—
128	C₂²×C₃	C₂²×C₃²×C₄	1	1	yes	characteristic
129	C₃×C₄	—	3	2	no	—
130	C₃×C₄	—	3	2	no	—
131	C₃×C₄	—	3	8	no	—
132	C₃×C₄	C₃²×C₄²	1	2	yes	—
133	C₃×C₄	C₂×C₃²×C₈	1	8	yes	—
134	C₃×C₄	C₂²×C₃²×C₄	1	2	yes	—
135	C₃²	C₂×C₃×C₄×C₈	1	1	yes	characteristic
136	C₉	C₂×C₃×C₄×C₈	1	3	yes	complemented
137	C₂³	C₂×C₄×3₋¹⁺²	1	1	yes	characteristic
138	C₂×C₄	C₈×3₋¹⁺²	1	8	yes	direct product factor
139	C₂×C₄	C₂×C₄×3₋¹⁺²	1	4	yes	—
140	C₂×C₄	C₂×C₄×3₋¹⁺²	1	1	yes	characteristic
141	C₂×C₄	C₂×C₄×3₋¹⁺²	1	4	yes	—
142	C₂×C₄	C₂³×3₋¹⁺²	1	1	yes	characteristic
143	C₈	C₂×C₄×3₋¹⁺²	1	8	yes	direct product factor
144	C₂×C₃	—	3	4	no	—
145	C₂×C₃	—	3	1	no	—
146	C₂×C₃	—	3	2	no	—
147	C₂×C₃	C₃²×C₄×C₈	1	4	yes	—
148	C₂×C₃	C₂×C₃²×C₄²	1	1	yes	characteristic
149	C₂×C₃	C₂²×C₃²×C₈	1	2	yes	—
150	C₂²	C₄²×3₋¹⁺²	1	2	yes	—
151	C₂²	C₂×C₈×3₋¹⁺²	1	4	yes	—
152	C₂²	C₂²×C₄×3₋¹⁺²	1	1	yes	characteristic
153	C₄	C₄²×3₋¹⁺²	1	2	yes	—
154	C₄	C₂×C₈×3₋¹⁺²	1	8	yes	direct product factor
155	C₄	C₂²×C₄×3₋¹⁺²	1	2	yes	—
156	C₃	—	3	1	no	—
157	C₃	C₂×C₃²×C₄×C₈	1	1	yes	characteristic
158	C₂	C₄×C₈×3₋¹⁺²	1	4	yes	direct product factor
159	C₂	C₂×C₄²×3₋¹⁺²	1	1	yes	characteristic
160	C₂	C₂²×C₈×3₋¹⁺²	1	2	yes	—

	subgroup	G/H
center	C₂×C₃×C₄×C₈	C₃²
commutator	C₃	C₂×C₃²×C₄×C₈
Frattini, cosocle	C₂×C₃×C₄	C₂³×C₃²
socle	C₂³×C₃	C₂×C₃²×C₄
prime radical	C₂³×C₃²	C₂×C₃×C₄
LCS	G ▷ C₃ ▷ C₁
UCS	C₁ ◁ C₂×C₃×C₄×C₈ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₄×C₈×3₋¹⁺²	C₂	direct (N × H)
5. C₂×C₄×C₈×C₉	C₃	semidirect
8. C₂×C₈×3₋¹⁺²	C₄	direct (N × H)
17. C₄×C₈×C₉	C₂×C₃	semidirect
19. C₂×C₄×3₋¹⁺²	C₈	direct (N × H)
24. C₈×3₋¹⁺²	C₂×C₄	direct (N × H)
35. C₂×C₈×C₉	C₃×C₄	semidirect
43. C₄×3₋¹⁺²	C₂×C₈	direct (N × H)
59. C₂×C₄×C₉	C₃×C₈	semidirect
64. C₈×C₉	C₂×C₃×C₄	semidirect
65. C₂×C₄×C₈	3₋¹⁺²	direct (N × H)
66. C₂×3₋¹⁺²	C₄×C₈	direct (N × H)
91. C₄×C₉	C₂×C₃×C₈	semidirect
95. C₄×C₈	C₂×3₋¹⁺²	direct (N × H)
96. 3₋¹⁺²	C₂×C₄×C₈	direct (N × H)
114. C₂×C₉	C₃×C₄×C₈	semidirect
119. C₂×C₈	C₄×3₋¹⁺²	direct (N × H)
136. C₉	C₂×C₃×C₄×C₈	semidirect
138. C₂×C₄	C₈×3₋¹⁺²	direct (N × H)
143. C₈	C₂×C₄×3₋¹⁺²	direct (N × H)
154. C₄	C₂×C₈×3₋¹⁺²	direct (N × H)
158. C₂	C₄×C₈×3₋¹⁺²	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₈×3₋¹⁺²	2
C₂×C₄²×3₋¹⁺²	2
C₄×C₈×3₋¹⁺²	2
C₂×C₃²×C₄×C₈	3
C₂×C₄×C₈×C₉	3

Subgroups — C2×C4×C8×3−1+2 #13144966

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₄×C₈×3₋¹⁺² #13144966