Subgroups — C₂×3₋¹⁺²×#106 #13145035

Order 1728.

520 conjugacy classes of subgroups.

The subgroup lattice has 174 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	1	yes	maximal, characteristic
2	C₂×C₄⋊C₄×3₋¹⁺²	C₂	1	2	yes	maximal
3	3₋¹⁺²×#106	C₂	1	4	yes	maximal, direct product factor
4	C₂×C₃²×#106	C₃	1	1	yes	maximal, characteristic
5	C₂×C₉×#106	C₃	1	3	yes	maximal, complemented
6	C₂²×C₄×3₋¹⁺²	—	2	2	no	—
7	C₂²×C₄×3₋¹⁺²	C₂²	1	1	yes	characteristic
8	C₂×C₈×3₋¹⁺²	C₄	1	4	yes	complemented
9	C₂×C₈×3₋¹⁺²	C₂²	1	2	yes	—
10	C₄⋊C₄×3₋¹⁺²	—	2	2	no	—
11	C₄⋊C₄×3₋¹⁺²	C₂²	1	4	yes	—
12	C₂²×C₃²×C₈	C₂×C₃	1	1	yes	characteristic
13	C₂×C₃²×C₄⋊C₄	C₂×C₃	1	2	yes	—
14	C₃²×#106	C₂×C₃	1	4	yes	—
15	C₂²×C₈×C₉	C₂×C₃	1	3	yes	—
16	C₂×C₉×C₄⋊C₄	C₂×C₃	1	6	yes	—
17	C₉×#106	C₂×C₃	1	12	yes	complemented
18	C₂³×3₋¹⁺²	D₄	1	1	yes	characteristic
19	C₂×C₄×3₋¹⁺²	—	2	4	no	—
20	C₂×C₄×3₋¹⁺²	—	4	4	no	—
21	C₂×C₄×3₋¹⁺²	C₂×C₄	1	2	yes	—
22	C₂×C₄×3₋¹⁺²	D₄	1	1	yes	characteristic
23	C₂×C₄×3₋¹⁺²	Q₈	1	2	yes	—
24	C₂×C₄×3₋¹⁺²	C₂³	1	1	yes	characteristic
25	C₈×3₋¹⁺²	C₂×C₄	1	4	yes	complemented
26	C₂×C₃×#106	—	3	1	no	—
27	C₂×C₃×#106	C₃²	1	1	yes	characteristic
28	C₂²×C₃²×C₄	—	2	2	no	—
29	C₂²×C₃²×C₄	C₂²×C₃	1	1	yes	characteristic
30	C₂×C₃²×C₈	C₃×C₄	1	4	yes	—
31	C₂×C₃²×C₈	C₂²×C₃	1	2	yes	—
32	C₃²×C₄⋊C₄	—	2	2	no	—
33	C₃²×C₄⋊C₄	C₂²×C₃	1	4	yes	—
34	C₂²×C₄×C₉	—	2	6	no	—
35	C₂²×C₄×C₉	C₂²×C₃	1	3	yes	—
36	C₂×C₈×C₉	C₃×C₄	1	12	yes	complemented
37	C₂×C₈×C₉	C₂²×C₃	1	6	yes	—
38	C₉×C₄⋊C₄	—	2	6	no	—
39	C₉×C₄⋊C₄	C₂²×C₃	1	12	yes	—
40	C₂²×3₋¹⁺²	C₄⋊C₄	1	2	yes	—
41	C₂²×3₋¹⁺²	SD₁₆	1	4	yes	—
42	C₂²×3₋¹⁺²	C₂×D₄	1	1	yes	characteristic
43	C₄×3₋¹⁺²	—	4	4	no	—
44	C₄×3₋¹⁺²	C₄⋊C₄	1	2	yes	—
45	C₄×3₋¹⁺²	C₂²×C₄	1	1	yes	characteristic
46	C₄×3₋¹⁺²	C₂×Q₈	1	1	yes	characteristic
47	C₂²×C₃×C₈	—	3	1	no	—
48	C₂²×C₃×C₈	C₂×C₃²	1	1	yes	characteristic
49	C₂×C₃×C₄⋊C₄	—	3	2	no	—
50	C₂×C₃×C₄⋊C₄	C₂×C₃²	1	2	yes	—
51	C₃×#106	—	3	4	no	—
52	C₃×#106	C₂×C₃²	1	4	yes	—
53	C₂³×C₃²	C₃×D₄	1	1	yes	characteristic
54	C₂×C₃²×C₄	—	2	4	no	—
55	C₂×C₃²×C₄	—	4	4	no	—
56	C₂×C₃²×C₄	C₂×C₃×C₄	1	2	yes	—
57	C₂×C₃²×C₄	C₃×D₄	1	1	yes	characteristic
58	C₂×C₃²×C₄	C₃×Q₈	1	2	yes	—
59	C₂×C₃²×C₄	C₂³×C₃	1	1	yes	characteristic
60	C₂³×C₉	C₃×D₄	1	3	yes	—
61	C₃²×C₈	C₂×C₃×C₄	1	4	yes	—
62	C₂×C₄×C₉	—	2	12	no	—
63	C₂×C₄×C₉	—	4	12	no	—
64	C₂×C₄×C₉	C₂×C₃×C₄	1	6	yes	—
65	C₂×C₄×C₉	C₃×D₄	1	3	yes	—
66	C₂×C₄×C₉	C₃×Q₈	1	6	yes	—
67	C₂×C₄×C₉	C₂³×C₃	1	3	yes	—
68	C₈×C₉	C₂×C₃×C₄	1	12	yes	complemented
69	C₂×#106	3₋¹⁺²	1	1	yes	characteristic, direct product factor, Sylow
70	C₂×3₋¹⁺²	#106	1	4	yes	direct product factor
71	C₂×3₋¹⁺²	C₂×C₄⋊C₄	1	1	yes	characteristic
72	C₂×3₋¹⁺²	C₂×SD₁₆	1	2	yes	—
73	C₂²×C₃×C₄	—	2	2	no	—
74	C₂²×C₃×C₄	—	3	1	no	—
75	C₂²×C₃×C₄	—	6	2	no	—
76	C₂²×C₃×C₄	C₂²×C₃²	1	1	yes	characteristic
77	C₂×C₃×C₈	—	3	4	no	—
78	C₂×C₃×C₈	—	3	2	no	—
79	C₂×C₃×C₈	C₃²×C₄	1	4	yes	—
80	C₂×C₃×C₈	C₂²×C₃²	1	2	yes	—
81	C₃×C₄⋊C₄	—	2	2	no	—
82	C₃×C₄⋊C₄	—	3	4	no	—
83	C₃×C₄⋊C₄	—	6	2	no	—
84	C₃×C₄⋊C₄	C₂²×C₃²	1	4	yes	—
85	C₂²×C₃²	C₃×C₄⋊C₄	1	2	yes	—
86	C₂²×C₃²	C₃×SD₁₆	1	4	yes	—
87	C₂²×C₃²	C₂×C₃×D₄	1	1	yes	characteristic
88	C₃²×C₄	—	4	4	no	—
89	C₃²×C₄	C₃×C₄⋊C₄	1	2	yes	—
90	C₃²×C₄	C₂²×C₃×C₄	1	1	yes	characteristic
91	C₃²×C₄	C₂×C₃×Q₈	1	1	yes	characteristic
92	C₂²×C₉	C₃×C₄⋊C₄	1	6	yes	—
93	C₂²×C₉	C₃×SD₁₆	1	12	yes	—
94	C₂²×C₉	C₂×C₃×D₄	1	3	yes	—
95	C₄×C₉	—	4	12	no	—
96	C₄×C₉	C₃×C₄⋊C₄	1	6	yes	—
97	C₄×C₉	C₂²×C₃×C₄	1	3	yes	—
98	C₄×C₉	C₂×C₃×Q₈	1	3	yes	—
99	C₂²×C₈	C₂×3₋¹⁺²	1	1	yes	characteristic
100	C₂×C₄⋊C₄	C₂×3₋¹⁺²	1	2	yes	—
101	#106	C₂×3₋¹⁺²	1	4	yes	direct product factor
102	3₋¹⁺²	C₂×#106	1	1	yes	characteristic, direct product factor, Sylow
103	C₂³×C₃	—	3	1	no	—
104	C₂³×C₃	C₃²×D₄	1	1	yes	characteristic
105	C₂×C₃×C₄	—	2	4	no	—
106	C₂×C₃×C₄	—	3	2	no	—
107	C₂×C₃×C₄	—	3	1	no	—
108	C₂×C₃×C₄	—	3	2	no	—
109	C₂×C₃×C₄	—	3	1	no	—
110	C₂×C₃×C₄	—	4	4	no	—
111	C₂×C₃×C₄	—	6	4	no	—
112	C₂×C₃×C₄	—	12	4	no	—
113	C₂×C₃×C₄	C₂×C₃²×C₄	1	2	yes	—
114	C₂×C₃×C₄	C₃²×D₄	1	1	yes	characteristic
115	C₂×C₃×C₄	C₃²×Q₈	1	2	yes	—
116	C₂×C₃×C₄	C₂³×C₃²	1	1	yes	characteristic
117	C₃×C₈	—	3	4	no	—
118	C₃×C₈	C₂×C₃²×C₄	1	4	yes	—
119	C₂×C₃²	C₃×#106	1	4	yes	—
120	C₂×C₃²	C₂×C₃×C₄⋊C₄	1	1	yes	characteristic
121	C₂×C₃²	C₂×C₃×SD₁₆	1	2	yes	—
122	C₂×C₉	C₃×#106	1	12	yes	complemented
123	C₂×C₉	C₂×C₃×C₄⋊C₄	1	3	yes	—
124	C₂×C₉	C₂×C₃×SD₁₆	1	6	yes	—
125	C₂²×C₄	—	2	2	no	—
126	C₂²×C₄	C₂²×3₋¹⁺²	1	1	yes	characteristic
127	C₂×C₈	C₄×3₋¹⁺²	1	4	yes	complemented
128	C₂×C₈	C₂²×3₋¹⁺²	1	2	yes	—
129	C₄⋊C₄	—	2	2	no	—
130	C₄⋊C₄	C₂²×3₋¹⁺²	1	4	yes	—
131	C₂²×C₃	—	3	1	no	—
132	C₂²×C₃	—	3	2	no	—
133	C₂²×C₃	—	3	4	no	—
134	C₂²×C₃	C₃²×C₄⋊C₄	1	2	yes	—
135	C₂²×C₃	C₃²×SD₁₆	1	4	yes	—
136	C₂²×C₃	C₂×C₃²×D₄	1	1	yes	characteristic
137	C₃×C₄	—	3	1	no	—
138	C₃×C₄	—	3	1	no	—
139	C₃×C₄	—	3	2	no	—
140	C₃×C₄	—	4	4	no	—
141	C₃×C₄	—	12	4	no	—
142	C₃×C₄	C₃²×C₄⋊C₄	1	2	yes	—
143	C₃×C₄	C₂²×C₃²×C₄	1	1	yes	characteristic
144	C₃×C₄	C₂×C₃²×Q₈	1	1	yes	characteristic
145	C₃²	C₂×C₃×#106	1	1	yes	characteristic
146	C₉	C₂×C₃×#106	1	3	yes	complemented
147	C₂³	D₄×3₋¹⁺²	1	1	yes	characteristic
148	C₂×C₄	—	2	4	no	—
149	C₂×C₄	—	4	4	no	—
150	C₂×C₄	C₂×C₄×3₋¹⁺²	1	2	yes	—
151	C₂×C₄	D₄×3₋¹⁺²	1	1	yes	characteristic
152	C₂×C₄	Q₈×3₋¹⁺²	1	2	yes	—
153	C₂×C₄	C₂³×3₋¹⁺²	1	1	yes	characteristic
154	C₈	C₂×C₄×3₋¹⁺²	1	4	yes	complemented
155	C₂×C₃	—	3	1	no	—
156	C₂×C₃	—	3	4	no	—
157	C₂×C₃	—	3	2	no	—
158	C₂×C₃	C₃²×#106	1	4	yes	—
159	C₂×C₃	C₂×C₃²×C₄⋊C₄	1	1	yes	characteristic
160	C₂×C₃	C₂×C₃²×SD₁₆	1	2	yes	—
161	C₂²	C₄⋊C₄×3₋¹⁺²	1	2	yes	—
162	C₂²	SD₁₆×3₋¹⁺²	1	4	yes	—
163	C₂²	C₂×D₄×3₋¹⁺²	1	1	yes	characteristic
164	C₄	—	4	4	no	—
165	C₄	C₄⋊C₄×3₋¹⁺²	1	2	yes	—
166	C₄	C₂²×C₄×3₋¹⁺²	1	1	yes	characteristic
167	C₄	C₂×Q₈×3₋¹⁺²	1	1	yes	characteristic
168	C₃	—	3	1	no	—
169	C₃	C₂×C₃²×#106	1	1	yes	characteristic
170	C₂	3₋¹⁺²×#106	1	4	yes	direct product factor
171	C₂	C₂×C₄⋊C₄×3₋¹⁺²	1	1	yes	characteristic
172	C₂	C₂×SD₁₆×3₋¹⁺²	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₃	C₃²×D₄
commutator	C₃×C₄	C₂²×C₃²×C₄
Frattini, cosocle	C₂×C₃×C₄	C₂³×C₃²
prime radical	C₂³×C₃²	C₃×D₄
LCS	G ▷ C₃×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₃ ◁ 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×#106	C₂	direct (N × H)
5. C₂×C₉×#106	C₃	semidirect
8. C₂×C₈×3₋¹⁺²	C₄	semidirect
17. C₉×#106	C₂×C₃	semidirect
25. C₈×3₋¹⁺²	C₂×C₄	semidirect
36. C₂×C₈×C₉	C₃×C₄	semidirect
68. C₈×C₉	C₂×C₃×C₄	semidirect
69. C₂×#106	3₋¹⁺²	direct (N × H)
70. C₂×3₋¹⁺²	#106	direct (N × H)
101. #106	C₂×3₋¹⁺²	direct (N × H)
102. 3₋¹⁺²	C₂×#106	direct (N × H)
122. C₂×C₉	C₃×#106	semidirect
127. C₂×C₈	C₄×3₋¹⁺²	semidirect
146. C₉	C₂×C₃×#106	semidirect
154. C₈	C₂×C₄×3₋¹⁺²	semidirect
170. C₂	3₋¹⁺²×#106	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×C₄⋊C₄×3₋¹⁺²	2
3₋¹⁺²×#106	2
C₂×C₃²×#106	3
C₂×C₉×#106	3

Subgroups — C2×3−1+2×#106 #13145035

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×3₋¹⁺²×#106 #13145035