Subgroups — C₃×#13281 #10631888

Order 768.

366 conjugacy classes of subgroups.

The subgroup lattice has 172 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×#499	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×C₄×C₁₆	C₂	1	1	yes	maximal, characteristic
3	C₃×#1846	C₂	1	1	yes	maximal, characteristic
4	C₃×#1380	C₂	1	4	yes	maximal
5	#13281	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
6	C₂×C₃×#125	—	2	1	no	—
7	C₂²×C₃×C₁₆	C₂²	1	2	yes	—
8	C₃×#499	—	2	2	no	—
9	C₃×#499	C₄	1	4	yes	complemented
10	C₂×C₃×#107	C₂²	1	2	yes	—
11	C₂×C₃×#107	C₂²	1	2	yes	—
12	C₂×C₃×#106	—	2	1	no	—
13	C₂×C₃×#106	—	2	1	no	—
14	C₂×C₃×C₄×C₈	C₂²	1	1	yes	characteristic
15	C₃×#382	—	2	1	no	—
16	C₃×C₄×C₁₆	—	2	2	no	—
17	C₂×#499	C₂×C₃	1	1	yes	characteristic
18	C₂×C₄×C₁₆	C₂×C₃	1	1	yes	characteristic
19	#1846	C₂×C₃	1	1	yes	characteristic
20	#1380	C₂×C₃	1	4	yes	—
21	C₂²×C₃×C₈	D₄	1	1	yes	characteristic
22	C₂²×C₃×C₈	C₂³	1	1	yes	characteristic
23	C₃×#125	—	2	4	no	—
24	C₃×#125	—	4	2	no	—
25	C₂×C₃×C₄⋊C₄	—	2	2	no	—
26	C₂×C₃×C₄⋊C₄	—	2	2	no	—
27	C₂×C₃×C₄⋊C₄	—	4	2	no	—
28	C₂×C₃×C₄²	D₄	1	1	yes	characteristic
29	C₂×C₃×C₁₆	—	2	2	no	—
30	C₂×C₃×C₁₆	—	2	2	no	—
31	C₂×C₃×C₁₆	Q₈	1	4	yes	—
32	C₃×#107	—	2	2	no	—
33	C₃×#107	—	2	4	no	—
34	C₃×#107	C₂×C₄	1	4	yes	—
35	C₃×#106	—	2	4	no	—
36	C₃×#106	—	4	2	no	—
37	C₃×C₄×C₈	C₂×C₄	1	2	yes	—
38	C₃×C₄×C₈	D₄	1	2	yes	—
39	C₃×#95	—	4	2	no	—
40	C₂×#125	—	2	1	no	—
41	C₂²×C₁₆	C₂²×C₃	1	2	yes	—
42	#499	—	2	2	no	—
43	#499	C₃×C₄	1	4	yes	complemented
44	C₂×#107	C₂²×C₃	1	2	yes	—
45	C₂×#107	C₂²×C₃	1	2	yes	—
46	C₂×#106	—	2	1	no	—
47	C₂×#106	—	2	1	no	—
48	C₂×C₄×C₈	C₂²×C₃	1	1	yes	characteristic
49	#382	—	2	1	no	—
50	C₄×C₁₆	—	2	2	no	—
51	C₂²×C₃×C₄	—	4	2	no	—
52	C₂²×C₃×C₄	—	4	2	no	—
53	C₂²×C₃×C₄	D₈	1	2	yes	—
54	C₂²×C₃×C₄	C₂×D₄	1	1	yes	characteristic
55	C₂×C₃×C₈	C₂²⋊C₄	1	2	yes	—
56	C₂×C₃×C₈	SD₁₆	1	2	yes	—
57	C₂×C₃×C₈	Q₁₆	1	2	yes	—
58	C₂×C₃×C₈	C₂²×C₄	1	1	yes	characteristic
59	C₂×C₃×C₈	C₂×D₄	1	1	yes	characteristic
60	C₂×C₃×C₈	C₂×Q₈	1	2	yes	—
61	C₂×C₃×C₈	#41	1	2	yes	—
62	C₃×C₄⋊C₄	—	2	4	no	—
63	C₃×C₄⋊C₄	—	4	8	no	—
64	C₃×C₄⋊C₄	—	4	2	no	—
65	C₃×C₄⋊C₄	—	4	2	no	—
66	C₃×C₄⋊C₄	—	4	2	no	—
67	C₃×C₄²	C₂²⋊C₄	1	2	yes	—
68	C₃×C₄²	SD₁₆	1	2	yes	—
69	C₃×C₁₆	—	2	4	no	—
70	C₂²×C₈	C₃×D₄	1	1	yes	characteristic
71	C₂²×C₈	C₂³×C₃	1	1	yes	characteristic
72	#125	—	2	4	no	—
73	#125	—	4	2	no	—
74	C₂×C₄⋊C₄	—	2	2	no	—
75	C₂×C₄⋊C₄	—	2	2	no	—
76	C₂×C₄⋊C₄	—	4	2	no	—
77	C₂×C₄²	C₃×D₄	1	1	yes	characteristic
78	C₂×C₁₆	—	2	2	no	—
79	C₂×C₁₆	—	2	2	no	—
80	C₂×C₁₆	C₃×Q₈	1	4	yes	—
81	#107	—	2	2	no	—
82	#107	—	2	4	no	—
83	#107	C₂×C₃×C₄	1	4	yes	—
84	#106	—	2	4	no	—
85	#106	—	4	2	no	—
86	C₄×C₈	C₂×C₃×C₄	1	2	yes	—
87	C₄×C₈	C₃×D₄	1	2	yes	—
88	#95	—	4	2	no	—
89	C₂³×C₃	C₂×D₈	1	1	yes	characteristic
90	C₂×C₃×C₄	—	2	4	no	—
91	C₂×C₃×C₄	—	4	4	no	—
92	C₂×C₃×C₄	—	8	4	no	—
93	C₂×C₃×C₄	—	8	2	no	—
94	C₂×C₃×C₄	—	8	4	no	—
95	C₂×C₃×C₄	#102	1	4	yes	—
96	C₂×C₃×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
97	C₂×C₃×C₄	#124	1	1	yes	characteristic
98	C₂×C₃×C₄	C₄⋊Q₈	1	1	yes	characteristic
99	C₂×C₃×C₄	C₂×SD₁₆	1	1	yes	characteristic
100	C₂×C₃×C₄	C₂×SD₁₆	1	1	yes	characteristic
101	C₂×C₃×C₄	C₂×Q₁₆	1	1	yes	characteristic
102	C₃×C₈	#103	1	4	yes	—
103	C₃×C₈	C₄×Q₈	1	2	yes	—
104	C₃×C₈	C₂²⋊Q₈	1	2	yes	—
105	C₂²×C₄	—	4	2	no	—
106	C₂²×C₄	—	4	2	no	—
107	C₂²×C₄	C₃×D₈	1	2	yes	—
108	C₂²×C₄	C₂×C₃×D₄	1	1	yes	characteristic
109	C₂×C₈	C₃×C₂²⋊C₄	1	2	yes	—
110	C₂×C₈	C₃×SD₁₆	1	2	yes	—
111	C₂×C₈	C₃×Q₁₆	1	2	yes	—
112	C₂×C₈	C₂²×C₃×C₄	1	1	yes	characteristic
113	C₂×C₈	C₂×C₃×D₄	1	1	yes	characteristic
114	C₂×C₈	C₂×C₃×Q₈	1	2	yes	—
115	C₂×C₈	C₃×#41	1	2	yes	—
116	C₄⋊C₄	—	2	4	no	—
117	C₄⋊C₄	—	4	8	no	—
118	C₄⋊C₄	—	4	2	no	—
119	C₄⋊C₄	—	4	2	no	—
120	C₄⋊C₄	—	4	2	no	—
121	C₄²	C₃×C₂²⋊C₄	1	2	yes	—
122	C₄²	C₃×SD₁₆	1	2	yes	—
123	C₁₆	—	2	4	no	—
124	C₂²×C₃	C₂×#102	1	1	yes	characteristic
125	C₂²×C₃	#486	1	1	yes	characteristic
126	C₂²×C₃	#500	1	1	yes	characteristic
127	C₂²×C₃	#508	1	4	yes	—
128	C₃×C₄	—	2	4	no	—
129	C₃×C₄	—	8	4	no	—
130	C₃×C₄	—	16	2	no	—
131	C₃×C₄	#391	1	1	yes	characteristic
132	C₃×C₄	C₂×#103	1	1	yes	characteristic
133	C₃×C₄	#487	1	1	yes	characteristic
134	C₃×C₄	#498	1	1	yes	characteristic
135	C₂³	C₂×C₃×D₈	1	1	yes	characteristic
136	C₂×C₄	—	2	4	no	—
137	C₂×C₄	—	4	4	no	—
138	C₂×C₄	—	8	4	no	—
139	C₂×C₄	—	8	4	no	—
140	C₂×C₄	—	8	2	no	—
141	C₂×C₄	C₃×#102	1	4	yes	—
142	C₂×C₄	C₂×C₃×C₂²⋊C₄	1	1	yes	characteristic
143	C₂×C₄	C₃×#124	1	1	yes	characteristic
144	C₂×C₄	C₃×C₄⋊Q₈	1	1	yes	characteristic
145	C₂×C₄	C₂×C₃×SD₁₆	1	1	yes	characteristic
146	C₂×C₄	C₂×C₃×SD₁₆	1	1	yes	characteristic
147	C₂×C₄	C₂×C₃×Q₁₆	1	1	yes	characteristic
148	C₈	C₃×#103	1	4	yes	—
149	C₈	C₃×C₄×Q₈	1	2	yes	—
150	C₈	C₃×C₂²⋊Q₈	1	2	yes	—
151	C₂×C₃	#1990	1	1	yes	characteristic
152	C₂×C₃	#2138	1	2	yes	—
153	C₂×C₃	#2242	1	2	yes	—
154	C₂×C₃	#2253	1	2	yes	—
155	C₂²	C₂×C₃×#102	1	1	yes	characteristic
156	C₂²	C₃×#486	1	1	yes	characteristic
157	C₂²	C₃×#500	1	1	yes	characteristic
158	C₂²	C₃×#508	1	4	yes	—
159	C₄	—	2	4	no	—
160	C₄	—	8	4	no	—
161	C₄	—	16	2	no	—
162	C₄	C₃×#391	1	1	yes	characteristic
163	C₄	C₂×C₃×#103	1	1	yes	characteristic
164	C₄	C₃×#487	1	1	yes	characteristic
165	C₄	C₃×#498	1	1	yes	characteristic
166	C₃	#13281	1	1	yes	characteristic, direct product factor, Sylow
167	C₂	C₃×#1990	1	1	yes	characteristic
168	C₂	C₃×#2138	1	2	yes	—
169	C₂	C₃×#2242	1	2	yes	—
170	C₂	C₃×#2253	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
5. #13281	C₃	direct (N × H)
9. C₃×#499	C₄	semidirect
43. #499	C₃×C₄	semidirect
166. C₃	#13281	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×#499	2
C₂×C₃×C₄×C₁₆	2
C₃×#1846	2
C₃×#1380	2
#13281	3

Subgroups — C3×#13281 #10631888

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#13281 #10631888