Subgroups — C₂×C₃³×#124 #13179981

Order 1728.

4620 conjugacy classes of subgroups.

The subgroup lattice has 168 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃³×Q₈	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×C₃³×D₄	C₂	1	1	yes	maximal, characteristic
3	C₃³×#124	C₂	1	8	yes	maximal, direct product factor
4	C₂×C₃³×C₂²⋊C₄	C₂	1	4	yes	maximal, complemented
5	C₂×C₃³×C₄²	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₃²×#124	C₃	1	13	yes	maximal, direct product factor
7	C₂⁴×C₃³	C₂²	1	2	yes	—
8	C₂×C₃³×Q₈	—	2	4	no	—
9	C₂×C₃³×Q₈	C₂²	1	4	yes	complemented
10	C₂×C₃³×D₄	—	2	4	no	—
11	C₂×C₃³×D₄	C₂²	1	4	yes	—
12	C₂²×C₃³×C₄	C₂²	1	2	yes	—
13	C₂²×C₃³×C₄	C₂²	1	2	yes	—
14	C₂²×C₃³×C₄	C₂²	1	1	yes	characteristic
15	C₃³×C₂²⋊C₄	C₂²	1	16	yes	complemented
16	C₃³×C₄²	C₂²	1	4	yes	complemented
17	C₂²×C₃²×Q₈	C₂×C₃	1	13	yes	complemented
18	C₂²×C₃²×D₄	C₂×C₃	1	13	yes	—
19	C₃²×#124	C₂×C₃	1	104	yes	direct product factor
20	C₂×C₃²×C₂²⋊C₄	C₂×C₃	1	52	yes	complemented
21	C₂×C₃²×C₄²	C₂×C₃	1	13	yes	complemented
22	C₂³×C₃³	—	2	4	no	—
23	C₂³×C₃³	—	2	8	no	—
24	C₂³×C₃³	C₂³	1	4	yes	—
25	C₂³×C₃³	C₂³	1	1	yes	characteristic
26	C₃³×Q₈	—	2	8	no	—
27	C₃³×D₄	—	2	8	no	—
28	C₂×C₃³×C₄	—	2	4	no	—
29	C₂×C₃³×C₄	—	2	4	no	—
30	C₂×C₃³×C₄	D₄	1	4	yes	—
31	C₂×C₃³×C₄	C₂³	1	4	yes	—
32	C₂×C₃³×C₄	C₂³	1	4	yes	—
33	C₂×C₃³×C₄	C₂³	1	2	yes	—
34	C₂×C₃×#124	C₃²	1	13	yes	direct product factor
35	C₂⁴×C₃²	C₂²×C₃	1	26	yes	—
36	C₂×C₃²×Q₈	—	2	52	no	—
37	C₂×C₃²×Q₈	C₂²×C₃	1	52	yes	complemented
38	C₂×C₃²×D₄	—	2	52	no	—
39	C₂×C₃²×D₄	C₂²×C₃	1	52	yes	—
40	C₂²×C₃²×C₄	C₂²×C₃	1	26	yes	—
41	C₂²×C₃²×C₄	C₂²×C₃	1	26	yes	—
42	C₂²×C₃²×C₄	C₂²×C₃	1	13	yes	—
43	C₃²×C₂²⋊C₄	C₂²×C₃	1	208	yes	complemented
44	C₃²×C₄²	C₂²×C₃	1	52	yes	complemented
45	C₂²×C₃³	—	2	4	no	—
46	C₂²×C₃³	—	2	8	no	—
47	C₂²×C₃³	—	4	8	no	—
48	C₂²×C₃³	C₂×D₄	1	2	yes	—
49	C₂²×C₃³	#41	1	4	yes	—
50	C₂²×C₃³	C₂⁴	1	1	yes	characteristic
51	C₃³×C₄	—	2	4	no	—
52	C₃³×C₄	—	2	4	no	—
53	C₃³×C₄	C₂×D₄	1	4	yes	—
54	C₂²×C₃×Q₈	C₂×C₃²	1	13	yes	complemented
55	C₂²×C₃×D₄	C₂×C₃²	1	13	yes	—
56	C₃×#124	C₂×C₃²	1	104	yes	direct product factor
57	C₂×C₃×C₂²⋊C₄	C₂×C₃²	1	52	yes	complemented
58	C₂×C₃×C₄²	C₂×C₃²	1	13	yes	complemented
59	C₂³×C₃²	—	2	52	no	—
60	C₂³×C₃²	—	2	104	no	—
61	C₂³×C₃²	C₂³×C₃	1	52	yes	—
62	C₂³×C₃²	C₂³×C₃	1	13	yes	—
63	C₃²×Q₈	—	2	104	no	—
64	C₃²×D₄	—	2	104	no	—
65	C₂×C₃²×C₄	—	2	52	no	—
66	C₂×C₃²×C₄	—	2	52	no	—
67	C₂×C₃²×C₄	C₃×D₄	1	52	yes	—
68	C₂×C₃²×C₄	C₂³×C₃	1	52	yes	—
69	C₂×C₃²×C₄	C₂³×C₃	1	52	yes	—
70	C₂×C₃²×C₄	C₂³×C₃	1	26	yes	—
71	C₂×#124	C₃³	1	1	yes	characteristic, direct product factor, Sylow
72	C₂×C₃³	—	4	4	no	—
73	C₂×C₃³	#124	1	4	yes	direct product factor
74	C₂×C₃³	C₂²×D₄	1	1	yes	characteristic
75	C₂×C₃³	C₂×#41	1	2	yes	—
76	C₂⁴×C₃	C₂²×C₃²	1	26	yes	—
77	C₂×C₃×Q₈	—	2	52	no	—
78	C₂×C₃×Q₈	C₂²×C₃²	1	52	yes	complemented
79	C₂×C₃×D₄	—	2	52	no	—
80	C₂×C₃×D₄	C₂²×C₃²	1	52	yes	—
81	C₂²×C₃×C₄	C₂²×C₃²	1	26	yes	—
82	C₂²×C₃×C₄	C₂²×C₃²	1	26	yes	—
83	C₂²×C₃×C₄	C₂²×C₃²	1	13	yes	—
84	C₃×C₂²⋊C₄	C₂²×C₃²	1	208	yes	complemented
85	C₃×C₄²	C₂²×C₃²	1	52	yes	complemented
86	C₂²×C₃²	—	2	52	no	—
87	C₂²×C₃²	—	2	104	no	—
88	C₂²×C₃²	—	4	104	no	—
89	C₂²×C₃²	C₂×C₃×D₄	1	26	yes	—
90	C₂²×C₃²	C₃×#41	1	52	yes	—
91	C₂²×C₃²	C₂⁴×C₃	1	13	yes	—
92	C₃²×C₄	—	2	52	no	—
93	C₃²×C₄	—	2	52	no	—
94	C₃²×C₄	C₂×C₃×D₄	1	52	yes	—
95	C₂²×Q₈	C₂×C₃³	1	1	yes	characteristic, complemented
96	C₂²×D₄	C₂×C₃³	1	1	yes	characteristic
97	#124	C₂×C₃³	1	8	yes	direct product factor
98	C₂×C₂²⋊C₄	C₂×C₃³	1	4	yes	complemented
99	C₂×C₄²	C₂×C₃³	1	1	yes	characteristic, complemented
100	C₃³	C₂×#124	1	1	yes	characteristic, direct product factor, Sylow
101	C₂³×C₃	—	2	52	no	—
102	C₂³×C₃	—	2	104	no	—
103	C₂³×C₃	C₂³×C₃²	1	52	yes	—
104	C₂³×C₃	C₂³×C₃²	1	13	yes	—
105	C₃×Q₈	—	2	104	no	—
106	C₃×D₄	—	2	104	no	—
107	C₂×C₃×C₄	—	2	52	no	—
108	C₂×C₃×C₄	—	2	52	no	—
109	C₂×C₃×C₄	C₃²×D₄	1	52	yes	—
110	C₂×C₃×C₄	C₂³×C₃²	1	52	yes	—
111	C₂×C₃×C₄	C₂³×C₃²	1	52	yes	—
112	C₂×C₃×C₄	C₂³×C₃²	1	26	yes	—
113	C₂×C₃²	—	4	52	no	—
114	C₂×C₃²	C₃×#124	1	52	yes	direct product factor
115	C₂×C₃²	C₂²×C₃×D₄	1	13	yes	—
116	C₂×C₃²	C₂×C₃×#41	1	26	yes	—
117	C₂⁴	C₂²×C₃³	1	2	yes	—
118	C₂×Q₈	—	2	4	no	—
119	C₂×Q₈	C₂²×C₃³	1	4	yes	complemented
120	C₂×D₄	—	2	4	no	—
121	C₂×D₄	C₂²×C₃³	1	4	yes	—
122	C₂²×C₄	C₂²×C₃³	1	2	yes	—
123	C₂²×C₄	C₂²×C₃³	1	2	yes	—
124	C₂²×C₄	C₂²×C₃³	1	1	yes	characteristic
125	C₂²⋊C₄	C₂²×C₃³	1	16	yes	complemented
126	C₄²	C₂²×C₃³	1	4	yes	complemented
127	C₂²×C₃	—	2	52	no	—
128	C₂²×C₃	—	2	104	no	—
129	C₂²×C₃	—	4	104	no	—
130	C₂²×C₃	C₂×C₃²×D₄	1	26	yes	—
131	C₂²×C₃	C₃²×#41	1	52	yes	—
132	C₂²×C₃	C₂⁴×C₃²	1	13	yes	—
133	C₃×C₄	—	2	52	no	—
134	C₃×C₄	—	2	52	no	—
135	C₃×C₄	C₂×C₃²×D₄	1	52	yes	—
136	C₃²	C₂×C₃×#124	1	13	yes	direct product factor
137	C₂³	—	2	4	no	—
138	C₂³	—	2	8	no	—
139	C₂³	C₂³×C₃³	1	4	yes	—
140	C₂³	C₂³×C₃³	1	1	yes	characteristic
141	Q₈	—	2	8	no	—
142	D₄	—	2	8	no	—
143	C₂×C₄	—	2	4	no	—
144	C₂×C₄	—	2	4	no	—
145	C₂×C₄	C₃³×D₄	1	4	yes	—
146	C₂×C₄	C₂³×C₃³	1	4	yes	—
147	C₂×C₄	C₂³×C₃³	1	4	yes	—
148	C₂×C₄	C₂³×C₃³	1	2	yes	—
149	C₂×C₃	—	4	52	no	—
150	C₂×C₃	C₃²×#124	1	52	yes	direct product factor
151	C₂×C₃	C₂²×C₃²×D₄	1	13	yes	—
152	C₂×C₃	C₂×C₃²×#41	1	26	yes	—
153	C₂²	—	2	4	no	—
154	C₂²	—	2	8	no	—
155	C₂²	—	4	8	no	—
156	C₂²	C₂×C₃³×D₄	1	2	yes	—
157	C₂²	C₃³×#41	1	4	yes	—
158	C₂²	C₂⁴×C₃³	1	1	yes	characteristic
159	C₄	—	2	4	no	—
160	C₄	—	2	4	no	—
161	C₄	C₂×C₃³×D₄	1	4	yes	—
162	C₃	C₂×C₃²×#124	1	13	yes	direct product factor
163	C₂	—	4	4	no	—
164	C₂	C₃³×#124	1	4	yes	direct product factor
165	C₂	C₂²×C₃³×D₄	1	1	yes	characteristic
166	C₂	C₂×C₃³×#41	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₃³	C₂³
commutator, Frattini, cosocle	C₂²	C₂⁴×C₃³
prime radical	C₂²×C₃³×D₄	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₃³ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂²×C₃³×Q₈	C₂	semidirect
3. C₃³×#124	C₂	direct (N × H)
4. C₂×C₃³×C₂²⋊C₄	C₂	semidirect
5. C₂×C₃³×C₄²	C₂	semidirect
6. C₂×C₃²×#124	C₃	direct (N × H)
9. C₂×C₃³×Q₈	C₂²	semidirect
15. C₃³×C₂²⋊C₄	C₂²	semidirect
16. C₃³×C₄²	C₂²	semidirect
17. C₂²×C₃²×Q₈	C₂×C₃	semidirect
19. C₃²×#124	C₂×C₃	direct (N × H)
20. C₂×C₃²×C₂²⋊C₄	C₂×C₃	semidirect
21. C₂×C₃²×C₄²	C₂×C₃	semidirect
34. C₂×C₃×#124	C₃²	direct (N × H)
37. C₂×C₃²×Q₈	C₂²×C₃	semidirect
43. C₃²×C₂²⋊C₄	C₂²×C₃	semidirect
44. C₃²×C₄²	C₂²×C₃	semidirect
54. C₂²×C₃×Q₈	C₂×C₃²	semidirect
56. C₃×#124	C₂×C₃²	direct (N × H)
57. C₂×C₃×C₂²⋊C₄	C₂×C₃²	semidirect
58. C₂×C₃×C₄²	C₂×C₃²	semidirect
71. C₂×#124	C₃³	direct (N × H)
73. C₂×C₃³	#124	direct (N × H)
78. C₂×C₃×Q₈	C₂²×C₃²	semidirect
84. C₃×C₂²⋊C₄	C₂²×C₃²	semidirect
85. C₃×C₄²	C₂²×C₃²	semidirect
95. C₂²×Q₈	C₂×C₃³	semidirect
97. #124	C₂×C₃³	direct (N × H)
98. C₂×C₂²⋊C₄	C₂×C₃³	semidirect
99. C₂×C₄²	C₂×C₃³	semidirect
100. C₃³	C₂×#124	direct (N × H)
114. C₂×C₃²	C₃×#124	direct (N × H)
119. C₂×Q₈	C₂²×C₃³	semidirect
125. C₂²⋊C₄	C₂²×C₃³	semidirect
126. C₄²	C₂²×C₃³	semidirect
136. C₃²	C₂×C₃×#124	direct (N × H)
150. C₂×C₃	C₃²×#124	direct (N × H)
162. C₃	C₂×C₃²×#124	direct (N × H)
164. C₂	C₃³×#124	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×C₃³×Q₈	2
C₂²×C₃³×D₄	2
C₃³×#124	2
C₂×C₃³×C₂²⋊C₄	2
C₂×C₃³×C₄²	2
C₂×C₃²×#124	3

Subgroups — C2×C33×#124 #13179981

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃³×#124 #13179981