Subgroups — C₂×C₃×#1371 #10630613

Order 768.

262 conjugacy classes of subgroups.

The subgroup lattice has 158 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₄×C₁₆	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×#335	C₂	1	2	yes	maximal
3	C₃×#1371	C₂	1	4	yes	maximal, direct product factor
4	C₂×#1371	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
5	C₂²×C₃×C₁₆	C₄	1	1	yes	characteristic
6	C₂²×C₃×C₁₆	C₄	1	1	yes	characteristic
7	C₂×C₃×C₄⋊C₈	—	2	2	no	—
8	C₂×C₃×C₄×C₈	C₂²	1	1	yes	characteristic
9	C₃×C₄×C₁₆	C₄	1	2	yes	—
10	C₃×C₄×C₁₆	C₂²	1	2	yes	—
11	C₃×#335	—	2	2	no	—
12	C₃×#335	C₂²	1	4	yes	—
13	C₂×C₄×C₁₆	C₂×C₃	1	1	yes	characteristic
14	C₂×#335	C₂×C₃	1	2	yes	—
15	#1371	C₂×C₃	1	4	yes	direct product factor
16	C₂²×C₃×C₈	—	4	2	no	—
17	C₂²×C₃×C₈	C₂×C₄	1	1	yes	characteristic
18	C₂²×C₃×C₈	Q₈	1	1	yes	characteristic
19	C₂×C₃×C₄²	D₄	1	1	yes	characteristic
20	C₂×C₃×C₁₆	—	2	2	no	—
21	C₂×C₃×C₁₆	C₈	1	4	yes	complemented
22	C₂×C₃×C₁₆	C₂×C₄	1	2	yes	—
23	C₂×C₃×C₁₆	C₂×C₄	1	2	yes	—
24	C₃×C₄⋊C₈	—	2	4	no	—
25	C₃×C₄⋊C₈	—	4	2	no	—
26	C₃×C₄×C₈	C₂×C₄	1	1	yes	characteristic
27	C₃×C₄×C₈	D₄	1	1	yes	characteristic
28	C₃×C₄×C₈	Q₈	1	1	yes	characteristic
29	C₃×C₄×C₈	C₂³	1	1	yes	characteristic
30	C₂²×C₁₆	C₃×C₄	1	1	yes	characteristic
31	C₂²×C₁₆	C₃×C₄	1	1	yes	characteristic
32	C₂×C₄⋊C₈	—	2	2	no	—
33	C₂×C₄×C₈	C₂²×C₃	1	1	yes	characteristic
34	C₄×C₁₆	C₃×C₄	1	2	yes	—
35	C₄×C₁₆	C₂²×C₃	1	2	yes	—
36	#335	—	2	2	no	—
37	#335	C₂²×C₃	1	4	yes	—
38	C₂²×C₃×C₄	C₄⋊C₄	1	1	yes	characteristic
39	C₂²×C₃×C₄	D₈	1	1	yes	characteristic
40	C₂²×C₃×C₄	Q₁₆	1	1	yes	characteristic
41	C₂×C₃×C₈	—	2	2	no	—
42	C₂×C₃×C₈	—	4	4	no	—
43	C₂×C₃×C₈	—	8	4	no	—
44	C₂×C₃×C₈	C₄⋊C₄	1	1	yes	characteristic
45	C₂×C₃×C₈	C₄⋊C₄	1	1	yes	characteristic
46	C₂×C₃×C₈	C₂×C₈	1	2	yes	—
47	C₂×C₃×C₈	M₁₆	1	2	yes	—
48	C₂×C₃×C₈	C₂²×C₄	1	1	yes	characteristic
49	C₂×C₃×C₈	C₂×Q₈	1	1	yes	characteristic
50	C₃×C₄²	C₄⋊C₄	1	1	yes	characteristic
51	C₃×C₄²	D₈	1	1	yes	characteristic
52	C₃×C₄²	Q₁₆	1	1	yes	characteristic
53	C₃×C₄²	C₂×D₄	1	1	yes	characteristic
54	C₃×C₁₆	—	2	2	no	—
55	C₃×C₁₆	C₂×C₈	1	4	yes	complemented
56	C₂²×C₈	—	4	2	no	—
57	C₂²×C₈	C₂×C₃×C₄	1	1	yes	characteristic
58	C₂²×C₈	C₃×Q₈	1	1	yes	characteristic
59	C₂×C₄²	C₃×D₄	1	1	yes	characteristic
60	C₂×C₁₆	—	2	2	no	—
61	C₂×C₁₆	C₃×C₈	1	4	yes	complemented
62	C₂×C₁₆	C₂×C₃×C₄	1	2	yes	—
63	C₂×C₁₆	C₂×C₃×C₄	1	2	yes	—
64	C₄⋊C₈	—	2	4	no	—
65	C₄⋊C₈	—	4	2	no	—
66	C₄×C₈	C₂×C₃×C₄	1	1	yes	characteristic
67	C₄×C₈	C₃×D₄	1	1	yes	characteristic
68	C₄×C₈	C₃×Q₈	1	1	yes	characteristic
69	C₄×C₈	C₂³×C₃	1	1	yes	characteristic
70	C₂³×C₃	#107	1	1	yes	characteristic
71	C₂×C₃×C₄	—	2	2	no	—
72	C₂×C₃×C₄	C₄⋊C₈	1	2	yes	—
73	C₂×C₃×C₄	#107	1	1	yes	characteristic
74	C₂×C₃×C₄	#107	1	1	yes	characteristic
75	C₂×C₃×C₄	#107	1	1	yes	characteristic
76	C₂×C₃×C₄	#108	1	2	yes	—
77	C₂×C₃×C₄	D₁₆	1	2	yes	—
78	C₂×C₃×C₄	Q₃₂	1	2	yes	—
79	C₂×C₃×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
80	C₂×C₃×C₄	C₂×D₈	1	1	yes	characteristic
81	C₂×C₃×C₄	C₂×Q₁₆	1	1	yes	characteristic
82	C₃×C₈	—	2	1	no	—
83	C₃×C₈	—	2	1	no	—
84	C₃×C₈	—	8	4	no	—
85	C₃×C₈	C₄⋊C₈	1	2	yes	—
86	C₃×C₈	C₂²×C₈	1	1	yes	characteristic
87	C₃×C₈	C₂×M₁₆	1	1	yes	characteristic
88	C₂²×C₄	C₃×C₄⋊C₄	1	1	yes	characteristic
89	C₂²×C₄	C₃×D₈	1	1	yes	characteristic
90	C₂²×C₄	C₃×Q₁₆	1	1	yes	characteristic
91	C₂×C₈	—	2	2	no	—
92	C₂×C₈	—	4	4	no	—
93	C₂×C₈	—	8	4	no	—
94	C₂×C₈	C₃×C₄⋊C₄	1	1	yes	characteristic
95	C₂×C₈	C₃×C₄⋊C₄	1	1	yes	characteristic
96	C₂×C₈	C₂×C₃×C₈	1	2	yes	—
97	C₂×C₈	C₃×M₁₆	1	2	yes	—
98	C₂×C₈	C₂²×C₃×C₄	1	1	yes	characteristic
99	C₂×C₈	C₂×C₃×Q₈	1	1	yes	characteristic
100	C₄²	C₃×C₄⋊C₄	1	1	yes	characteristic
101	C₄²	C₃×D₈	1	1	yes	characteristic
102	C₄²	C₃×Q₁₆	1	1	yes	characteristic
103	C₄²	C₂×C₃×D₄	1	1	yes	characteristic
104	C₁₆	—	2	2	no	—
105	C₁₆	C₂×C₃×C₈	1	4	yes	complemented
106	C₂²×C₃	#335	1	2	yes	—
107	C₂²×C₃	#366	1	2	yes	—
108	C₂²×C₃	#368	1	2	yes	—
109	C₂²×C₃	C₂×#107	1	1	yes	characteristic
110	C₃×C₄	—	2	1	no	—
111	C₃×C₄	—	2	1	no	—
112	C₃×C₄	#335	1	2	yes	—
113	C₃×C₄	#366	1	2	yes	—
114	C₃×C₄	C₂×C₄⋊C₈	1	1	yes	characteristic
115	C₃×C₄	C₂×#108	1	1	yes	characteristic
116	C₃×C₄	C₂×D₁₆	1	1	yes	characteristic
117	C₃×C₄	C₂×Q₃₂	1	1	yes	characteristic
118	C₂³	C₃×#107	1	1	yes	characteristic
119	C₂×C₄	—	2	2	no	—
120	C₂×C₄	C₃×C₄⋊C₈	1	2	yes	—
121	C₂×C₄	C₃×#107	1	1	yes	characteristic
122	C₂×C₄	C₃×#107	1	1	yes	characteristic
123	C₂×C₄	C₃×#107	1	1	yes	characteristic
124	C₂×C₄	C₃×#108	1	2	yes	—
125	C₂×C₄	C₃×D₁₆	1	2	yes	—
126	C₂×C₄	C₃×Q₃₂	1	2	yes	—
127	C₂×C₄	C₂×C₃×C₄⋊C₄	1	1	yes	characteristic
128	C₂×C₄	C₂×C₃×D₈	1	1	yes	characteristic
129	C₂×C₄	C₂×C₃×Q₁₆	1	1	yes	characteristic
130	C₈	—	2	1	no	—
131	C₈	—	2	1	no	—
132	C₈	—	8	4	no	—
133	C₈	C₃×C₄⋊C₈	1	2	yes	—
134	C₈	C₂²×C₃×C₈	1	1	yes	characteristic
135	C₈	C₂×C₃×M₁₆	1	1	yes	characteristic
136	C₂×C₃	#1371	1	4	yes	direct product factor
137	C₂×C₃	C₂×#335	1	1	yes	characteristic
138	C₂×C₃	C₂×#366	1	1	yes	characteristic
139	C₂×C₃	C₂×#368	1	1	yes	characteristic
140	C₂²	C₃×#335	1	2	yes	—
141	C₂²	C₃×#366	1	2	yes	—
142	C₂²	C₃×#368	1	2	yes	—
143	C₂²	C₂×C₃×#107	1	1	yes	characteristic
144	C₄	—	2	1	no	—
145	C₄	—	2	1	no	—
146	C₄	C₃×#335	1	2	yes	—
147	C₄	C₃×#366	1	2	yes	—
148	C₄	C₂×C₃×C₄⋊C₈	1	1	yes	characteristic
149	C₄	C₂×C₃×#108	1	1	yes	characteristic
150	C₄	C₂×C₃×D₁₆	1	1	yes	characteristic
151	C₄	C₂×C₃×Q₃₂	1	1	yes	characteristic
152	C₃	C₂×#1371	1	1	yes	characteristic, direct product factor, Sylow
153	C₂	C₃×#1371	1	4	yes	direct product factor
154	C₂	C₂×C₃×#335	1	1	yes	characteristic
155	C₂	C₂×C₃×#366	1	1	yes	characteristic
156	C₂	C₂×C₃×#368	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₃×C₄	D₈
commutator	C₈	C₂²×C₃×C₈
Frattini, cosocle	C₄×C₈	C₂³×C₃
socle, prime radical	C₂³×C₃	#107
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃×C₄ ◁ C₂×C₃×C₄² ◁ C₂×C₃×C₄×C₈ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₃×#1371	C₂	direct (N × H)
4. C₂×#1371	C₃	direct (N × H)
15. #1371	C₂×C₃	direct (N × H)
21. C₂×C₃×C₁₆	C₈	semidirect
55. C₃×C₁₆	C₂×C₈	semidirect
61. C₂×C₁₆	C₃×C₈	semidirect
105. C₁₆	C₂×C₃×C₈	semidirect
136. C₂×C₃	#1371	direct (N × H)
152. C₃	C₂×#1371	direct (N × H)
153. C₂	C₃×#1371	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×C₄×C₁₆	2
C₂×C₃×#335	2
C₃×#1371	2
C₂×#1371	3

Subgroups — C2×C3×#1371 #10630613

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×#1371 #10630613