Subgroups — C₃×#60113 #10678720

Order 768.

2564 conjugacy classes of subgroups.

The subgroup lattice has 180 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×#568	C₂	1	2	yes	maximal
2	C₂×C₃×#568	C₂	1	1	yes	maximal, characteristic
3	C₃×#2916	C₂	1	16	yes	maximal, complemented
4	C₂×C₃×#430	C₂	1	12	yes	maximal, complemented
5	#60113	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
6	C₂²×C₃×#41	C₂²	1	1	yes	characteristic
7	C₃×#568	—	2	8	no	—
8	C₃×#568	C₄	1	16	yes	—
9	C₃×#568	C₂²	1	16	yes	—
10	C₂×C₃×#131	—	2	8	no	—
11	C₂×C₃×#131	C₂²	1	8	yes	—
12	C₂²×C₃×M₁₆	C₂²	1	12	yes	—
13	C₂²×C₃×M₁₆	C₂²	1	6	yes	—
14	C₃×#430	C₂²	1	16	yes	—
15	C₃×#430	C₂²	1	48	yes	complemented
16	C₂×C₃×#108	C₂²	1	48	yes	complemented
17	C₂×#568	C₂×C₃	1	2	yes	—
18	C₂×#568	C₂×C₃	1	1	yes	characteristic
19	#2916	C₂×C₃	1	16	yes	complemented
20	C₂×#430	C₂×C₃	1	12	yes	complemented
21	C₂×C₃×#41	C₂×C₄	1	8	yes	—
22	C₂×C₃×#41	D₄	1	8	yes	—
23	C₂×C₃×#41	Q₈	1	4	yes	—
24	C₂×C₃×#41	C₂³	1	4	yes	—
25	C₂²×C₃×Q₈	Q₈	1	1	yes	characteristic
26	C₂²×C₃×D₄	Q₈	1	3	yes	—
27	C₂³×C₃×C₄	C₂³	1	3	yes	—
28	C₃×#131	—	2	64	no	—
29	C₃×#131	C₂×C₄	1	64	yes	—
30	C₂×C₃×M₁₆	—	2	12	no	—
31	C₂×C₃×M₁₆	—	2	24	no	—
32	C₂×C₃×M₁₆	C₂×C₄	1	48	yes	—
33	C₂×C₃×M₁₆	C₂³	1	8	yes	—
34	C₂×C₃×M₁₆	C₂³	1	48	yes	—
35	C₂×C₃×M₁₆	C₂³	1	4	yes	—
36	C₂×C₃×M₁₆	C₂³	1	12	yes	—
37	C₂²×C₃×C₈	—	2	12	no	—
38	C₂²×C₃×C₈	C₂³	1	12	yes	—
39	C₃×#108	C₂³	1	64	yes	—
40	C₂²×#41	C₂²×C₃	1	1	yes	characteristic
41	#568	—	2	8	no	—
42	#568	C₃×C₄	1	16	yes	—
43	#568	C₂²×C₃	1	16	yes	—
44	C₂×#131	—	2	8	no	—
45	C₂×#131	C₂²×C₃	1	8	yes	—
46	C₂²×M₁₆	C₂²×C₃	1	12	yes	—
47	C₂²×M₁₆	C₂²×C₃	1	6	yes	—
48	#430	C₂²×C₃	1	16	yes	—
49	#430	C₂²×C₃	1	48	yes	complemented
50	C₂×#108	C₂²×C₃	1	48	yes	complemented
51	C₂⁴×C₃	C₂×Q₈	1	3	yes	—
52	C₃×#41	C₄⋊C₄	1	32	yes	—
53	C₃×#41	C₂²×C₄	1	16	yes	—
54	C₃×#41	C₂×D₄	1	16	yes	—
55	C₂×C₃×Q₈	C₄⋊C₄	1	8	yes	—
56	C₂×C₃×Q₈	C₂×Q₈	1	4	yes	—
57	C₂×C₃×D₄	C₄⋊C₄	1	24	yes	—
58	C₂×C₃×D₄	C₂×Q₈	1	12	yes	—
59	C₂²×C₃×C₄	C₂²×C₄	1	12	yes	—
60	C₂²×C₃×C₄	C₂×D₄	1	12	yes	—
61	C₂²×C₃×C₄	C₂×Q₈	1	3	yes	—
62	C₂²×C₃×C₄	C₂×Q₈	1	6	yes	—
63	C₂²×C₃×C₄	C₂⁴	1	6	yes	—
64	C₂²×C₃×C₄	C₂⁴	1	1	yes	characteristic
65	C₃×M₁₆	—	2	8	no	—
66	C₃×M₁₆	—	2	48	no	—
67	C₃×M₁₆	C₂²×C₄	1	16	yes	—
68	C₃×M₁₆	C₂²×C₄	1	48	yes	—
69	C₃×M₁₆	C₂⁴	1	16	yes	—
70	C₂×C₃×C₈	—	2	8	no	—
71	C₂×C₃×C₈	—	2	48	no	—
72	C₂×C₃×C₈	C₂²×C₄	1	48	yes	—
73	C₂×C₃×C₈	C₂⁴	1	8	yes	—
74	C₂×#41	C₂×C₃×C₄	1	8	yes	—
75	C₂×#41	C₃×D₄	1	8	yes	—
76	C₂×#41	C₃×Q₈	1	4	yes	—
77	C₂×#41	C₂³×C₃	1	4	yes	—
78	C₂²×Q₈	C₃×Q₈	1	1	yes	characteristic
79	C₂²×D₄	C₃×Q₈	1	3	yes	—
80	C₂³×C₄	C₂³×C₃	1	3	yes	—
81	#131	—	2	64	no	—
82	#131	C₂×C₃×C₄	1	64	yes	—
83	C₂×M₁₆	—	2	12	no	—
84	C₂×M₁₆	—	2	24	no	—
85	C₂×M₁₆	C₂×C₃×C₄	1	48	yes	—
86	C₂×M₁₆	C₂³×C₃	1	8	yes	—
87	C₂×M₁₆	C₂³×C₃	1	48	yes	—
88	C₂×M₁₆	C₂³×C₃	1	4	yes	—
89	C₂×M₁₆	C₂³×C₃	1	12	yes	—
90	C₂²×C₈	—	2	12	no	—
91	C₂²×C₈	C₂³×C₃	1	12	yes	—
92	#108	C₂³×C₃	1	64	yes	—
93	C₂³×C₃	—	8	3	no	—
94	C₂³×C₃	C₂×C₄⋊C₄	1	12	yes	—
95	C₂³×C₃	C₂²×Q₈	1	1	yes	characteristic
96	C₂³×C₃	C₂²×Q₈	1	6	yes	—
97	C₃×Q₈	C₂×C₄⋊C₄	1	16	yes	—
98	C₃×D₄	C₂×C₄⋊C₄	1	48	yes	—
99	C₂×C₃×C₄	C₂×C₄⋊C₄	1	12	yes	—
100	C₂×C₃×C₄	C₂×C₄⋊C₄	1	24	yes	—
101	C₂×C₃×C₄	C₂³×C₄	1	12	yes	—
102	C₂×C₃×C₄	C₂³×C₄	1	2	yes	—
103	C₂×C₃×C₄	C₂²×D₄	1	2	yes	—
104	C₂×C₃×C₄	C₂²×D₄	1	12	yes	—
105	C₂×C₃×C₄	C₂²×Q₈	1	1	yes	characteristic
106	C₂×C₃×C₄	C₂²×Q₈	1	6	yes	—
107	C₂×C₃×C₄	C₂⁵	1	1	yes	characteristic
108	C₃×C₈	—	2	16	no	—
109	C₃×C₈	C₂³×C₄	1	16	yes	—
110	C₂⁴	C₂×C₃×Q₈	1	3	yes	—
111	#41	C₃×C₄⋊C₄	1	32	yes	—
112	#41	C₂²×C₃×C₄	1	16	yes	—
113	#41	C₂×C₃×D₄	1	16	yes	—
114	C₂×Q₈	C₃×C₄⋊C₄	1	8	yes	—
115	C₂×Q₈	C₂×C₃×Q₈	1	4	yes	—
116	C₂×D₄	C₃×C₄⋊C₄	1	24	yes	—
117	C₂×D₄	C₂×C₃×Q₈	1	12	yes	—
118	C₂²×C₄	C₂²×C₃×C₄	1	12	yes	—
119	C₂²×C₄	C₂×C₃×D₄	1	12	yes	—
120	C₂²×C₄	C₂×C₃×Q₈	1	3	yes	—
121	C₂²×C₄	C₂×C₃×Q₈	1	6	yes	—
122	C₂²×C₄	C₂⁴×C₃	1	6	yes	—
123	C₂²×C₄	C₂⁴×C₃	1	1	yes	characteristic
124	M₁₆	—	2	8	no	—
125	M₁₆	—	2	48	no	—
126	M₁₆	C₂²×C₃×C₄	1	16	yes	—
127	M₁₆	C₂²×C₃×C₄	1	48	yes	—
128	M₁₆	C₂⁴×C₃	1	16	yes	—
129	C₂×C₈	—	2	8	no	—
130	C₂×C₈	—	2	48	no	—
131	C₂×C₈	C₂²×C₃×C₄	1	48	yes	—
132	C₂×C₈	C₂⁴×C₃	1	8	yes	—
133	C₂²×C₃	—	4	1	no	—
134	C₂²×C₃	—	4	12	no	—
135	C₂²×C₃	—	4	6	no	—
136	C₂²×C₃	C₂²×C₄⋊C₄	1	2	yes	—
137	C₂²×C₃	C₂²×C₄⋊C₄	1	12	yes	—
138	C₂²×C₃	C₂³×Q₈	1	1	yes	characteristic
139	C₃×C₄	C₂²×C₄⋊C₄	1	2	yes	—
140	C₃×C₄	C₂²×C₄⋊C₄	1	12	yes	—
141	C₃×C₄	C₂⁴×C₄	1	1	yes	characteristic
142	C₃×C₄	C₂³×D₄	1	1	yes	characteristic
143	C₂³	—	8	3	no	—
144	C₂³	C₂×C₃×C₄⋊C₄	1	12	yes	—
145	C₂³	C₂²×C₃×Q₈	1	1	yes	characteristic
146	C₂³	C₂²×C₃×Q₈	1	6	yes	—
147	Q₈	C₂×C₃×C₄⋊C₄	1	16	yes	—
148	D₄	C₂×C₃×C₄⋊C₄	1	48	yes	—
149	C₂×C₄	C₂×C₃×C₄⋊C₄	1	12	yes	—
150	C₂×C₄	C₂×C₃×C₄⋊C₄	1	24	yes	—
151	C₂×C₄	C₂³×C₃×C₄	1	12	yes	—
152	C₂×C₄	C₂³×C₃×C₄	1	2	yes	—
153	C₂×C₄	C₂²×C₃×D₄	1	2	yes	—
154	C₂×C₄	C₂²×C₃×D₄	1	12	yes	—
155	C₂×C₄	C₂²×C₃×Q₈	1	1	yes	characteristic
156	C₂×C₄	C₂²×C₃×Q₈	1	6	yes	—
157	C₂×C₄	C₂⁵×C₃	1	1	yes	characteristic
158	C₈	—	2	16	no	—
159	C₈	C₂³×C₃×C₄	1	16	yes	—
160	C₂×C₃	—	2	2	no	—
161	C₂×C₃	—	2	1	no	—
162	C₂×C₃	—	2	12	no	—
163	C₂×C₃	C₂³×C₄⋊C₄	1	1	yes	characteristic
164	C₂²	—	4	1	no	—
165	C₂²	—	4	12	no	—
166	C₂²	—	4	6	no	—
167	C₂²	C₂²×C₃×C₄⋊C₄	1	2	yes	—
168	C₂²	C₂²×C₃×C₄⋊C₄	1	12	yes	—
169	C₂²	C₂³×C₃×Q₈	1	1	yes	characteristic
170	C₄	C₂²×C₃×C₄⋊C₄	1	2	yes	—
171	C₄	C₂²×C₃×C₄⋊C₄	1	12	yes	—
172	C₄	C₂⁴×C₃×C₄	1	1	yes	characteristic
173	C₄	C₂³×C₃×D₄	1	1	yes	characteristic
174	C₃	#60113	1	1	yes	characteristic, direct product factor, Sylow
175	C₂	—	2	1	no	—
176	C₂	—	2	2	no	—
177	C₂	—	2	12	no	—
178	C₂	C₂³×C₃×C₄⋊C₄	1	1	yes	characteristic

	subgroup	G/H
center	C₃×C₄	C₂³×D₄
commutator	C₄	C₂⁴×C₃×C₄
Frattini, cosocle	C₂×C₄	C₂⁵×C₃
socle	C₂×C₃	C₂³×C₄⋊C₄
prime radical	C₂²×C₃×#41	C₂²
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₃×C₄ ◁ C₂²×C₃×#41 ◁ G

Complements

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

N	H ≅ G/N	type
3. C₃×#2916	C₂	semidirect
4. C₂×C₃×#430	C₂	semidirect
5. #60113	C₃	direct (N × H)
15. C₃×#430	C₂²	semidirect
16. C₂×C₃×#108	C₂²	semidirect
19. #2916	C₂×C₃	semidirect
20. C₂×#430	C₂×C₃	semidirect
49. #430	C₂²×C₃	semidirect
50. C₂×#108	C₂²×C₃	semidirect
174. C₃	#60113	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₃×#568	2
C₃×#2916	2
C₂×C₃×#430	2
#60113	3

Subgroups — C3×#60113 #10678720

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#60113 #10678720