Subgroups — C₃×#13273 #10631880

Order 768.

334 conjugacy classes of subgroups.

The subgroup lattice has 202 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×#364	C₂	1	1	yes	maximal, characteristic
3	C₃×#1772	C₂	1	1	yes	maximal, characteristic
4	C₃×#1386	C₂	1	2	yes	maximal
5	C₃×#1383	C₂	1	2	yes	maximal
6	#13273	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
7	C₂×C₃×#125	—	2	1	no	—
8	C₂×C₃×M₃₂	C₂²	1	2	yes	—
9	C₃×#499	—	2	2	no	—
10	C₃×#499	C₄	1	4	yes	complemented
11	C₂×C₃×#108	C₂²	1	1	yes	characteristic
12	C₃×#427	C₂²	1	1	yes	characteristic
13	C₂×C₃×#107	—	2	1	no	—
14	C₂×C₃×#106	C₂²	1	2	yes	—
15	C₃×#405	—	2	1	no	—
16	C₂×C₃×C₄×C₈	C₂²	1	1	yes	characteristic
17	C₃×#364	—	2	2	no	—
18	C₃×#339	—	2	1	no	—
19	C₂×#499	C₂×C₃	1	1	yes	characteristic
20	C₂×#364	C₂×C₃	1	1	yes	characteristic
21	#1772	C₂×C₃	1	1	yes	characteristic
22	#1386	C₂×C₃	1	2	yes	—
23	#1383	C₂×C₃	1	2	yes	—
24	C₂×C₃×M₁₆	—	2	1	no	—
25	C₂²×C₃×C₈	D₄	1	1	yes	characteristic
26	C₂²×C₃×C₈	C₂³	1	1	yes	characteristic
27	C₃×#125	—	2	4	no	—
28	C₃×#125	—	4	2	no	—
29	C₃×#117	—	2	1	no	—
30	C₂×C₃×C₄⋊C₄	—	2	2	no	—
31	C₂×C₃×C₄⋊C₄	—	4	2	no	—
32	C₂×C₃×C₄²	D₄	1	1	yes	characteristic
33	C₃×M₃₂	—	2	2	no	—
34	C₃×M₃₂	—	2	2	no	—
35	C₂×C₃×C₁₆	—	2	2	no	—
36	C₃×#108	—	2	1	no	—
37	C₃×#108	Q₈	1	2	yes	—
38	C₃×#107	—	2	4	no	—
39	C₃×#107	Q₈	1	2	yes	—
40	C₃×#106	—	2	2	no	—
41	C₃×#106	—	2	1	no	—
42	C₃×#106	C₂×C₄	1	4	yes	—
43	C₃×#97	—	4	1	no	—
44	C₃×C₄×C₈	—	4	1	no	—
45	C₃×C₄×C₈	C₂×C₄	1	2	yes	—
46	C₃×C₄×C₈	D₄	1	2	yes	—
47	C₂×#125	—	2	1	no	—
48	C₂×M₃₂	C₂²×C₃	1	2	yes	—
49	#499	—	2	2	no	—
50	#499	C₃×C₄	1	4	yes	complemented
51	C₂×#108	C₂²×C₃	1	1	yes	characteristic
52	#427	C₂²×C₃	1	1	yes	characteristic
53	C₂×#107	—	2	1	no	—
54	C₂×#106	C₂²×C₃	1	2	yes	—
55	#405	—	2	1	no	—
56	C₂×C₄×C₈	C₂²×C₃	1	1	yes	characteristic
57	#364	—	2	2	no	—
58	#339	—	2	1	no	—
59	C₂²×C₃×C₄	—	2	1	no	—
60	C₂²×C₃×C₄	—	4	2	no	—
61	C₂²×C₃×C₄	C₂×D₄	1	1	yes	characteristic
62	C₃×M₁₆	—	2	2	no	—
63	C₃×M₁₆	—	4	1	no	—
64	C₂×C₃×C₈	—	4	1	no	—
65	C₂×C₃×C₈	C₂²⋊C₄	1	2	yes	—
66	C₂×C₃×C₈	SD₁₆	1	1	yes	characteristic
67	C₂×C₃×C₈	SD₁₆	1	1	yes	characteristic
68	C₂×C₃×C₈	Q₁₆	1	2	yes	—
69	C₂×C₃×C₈	C₂²×C₄	1	1	yes	characteristic
70	C₂×C₃×C₈	C₂×D₄	1	1	yes	characteristic
71	C₂×C₃×C₈	C₂×Q₈	1	1	yes	characteristic
72	C₂×C₃×C₈	C₂×Q₈	1	1	yes	characteristic
73	C₂×C₃×C₈	#41	1	1	yes	characteristic
74	C₂×C₃×C₈	#41	1	1	yes	characteristic
75	C₃×C₄⋊C₄	—	2	4	no	—
76	C₃×C₄⋊C₄	—	2	2	no	—
77	C₃×C₄⋊C₄	—	4	8	no	—
78	C₃×C₄⋊C₄	—	4	2	no	—
79	C₃×C₂²⋊C₄	—	4	1	no	—
80	C₃×C₄²	—	2	1	no	—
81	C₃×C₄²	—	4	1	no	—
82	C₃×C₄²	C₂²⋊C₄	1	2	yes	—
83	C₃×C₁₆	—	4	2	no	—
84	C₂×M₁₆	—	2	1	no	—
85	C₂²×C₈	C₃×D₄	1	1	yes	characteristic
86	C₂²×C₈	C₂³×C₃	1	1	yes	characteristic
87	#125	—	2	4	no	—
88	#125	—	4	2	no	—
89	#117	—	2	1	no	—
90	C₂×C₄⋊C₄	—	2	2	no	—
91	C₂×C₄⋊C₄	—	4	2	no	—
92	C₂×C₄²	C₃×D₄	1	1	yes	characteristic
93	M₃₂	—	2	2	no	—
94	M₃₂	—	2	2	no	—
95	C₂×C₁₆	—	2	2	no	—
96	#108	—	2	1	no	—
97	#108	C₃×Q₈	1	2	yes	—
98	#107	—	2	4	no	—
99	#107	C₃×Q₈	1	2	yes	—
100	#106	—	2	2	no	—
101	#106	—	2	1	no	—
102	#106	C₂×C₃×C₄	1	4	yes	—
103	#97	—	4	1	no	—
104	C₄×C₈	—	4	1	no	—
105	C₄×C₈	C₂×C₃×C₄	1	2	yes	—
106	C₄×C₈	C₃×D₄	1	2	yes	—
107	C₂³×C₃	AGL(1,ℤ/8)	1	1	yes	characteristic
108	C₂×C₃×C₄	—	2	2	no	—
109	C₂×C₃×C₄	—	4	2	no	—
110	C₂×C₃×C₄	—	4	4	no	—
111	C₂×C₃×C₄	—	4	2	no	—
112	C₂×C₃×C₄	—	8	4	no	—
113	C₂×C₃×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
114	C₂×C₃×C₄	C₂²⋊Q₈	1	1	yes	characteristic
115	C₂×C₃×C₄	C₂²⋊Q₈	1	1	yes	characteristic
116	C₂×C₃×C₄	C₂×SD₁₆	1	1	yes	characteristic
117	C₂×C₃×C₄	C₂×Q₁₆	1	1	yes	characteristic
118	C₂×C₃×C₄	AGL(1,ℤ/8)	1	1	yes	characteristic
119	C₃×C₈	—	4	2	no	—
120	C₃×C₈	#103	1	2	yes	—
121	C₃×C₈	#103	1	2	yes	—
122	C₃×C₈	C₄×Q₈	1	1	yes	characteristic
123	C₃×C₈	C₄×Q₈	1	1	yes	characteristic
124	C₃×C₈	#124	1	1	yes	characteristic
125	C₃×C₈	C₄⋊Q₈	1	1	yes	characteristic
126	C₂²×C₄	—	2	1	no	—
127	C₂²×C₄	—	4	2	no	—
128	C₂²×C₄	C₂×C₃×D₄	1	1	yes	characteristic
129	M₁₆	—	2	2	no	—
130	M₁₆	—	4	1	no	—
131	C₂×C₈	—	4	1	no	—
132	C₂×C₈	C₃×C₂²⋊C₄	1	2	yes	—
133	C₂×C₈	C₃×SD₁₆	1	1	yes	characteristic
134	C₂×C₈	C₃×SD₁₆	1	1	yes	characteristic
135	C₂×C₈	C₃×Q₁₆	1	2	yes	—
136	C₂×C₈	C₂²×C₃×C₄	1	1	yes	characteristic
137	C₂×C₈	C₂×C₃×D₄	1	1	yes	characteristic
138	C₂×C₈	C₂×C₃×Q₈	1	1	yes	characteristic
139	C₂×C₈	C₂×C₃×Q₈	1	1	yes	characteristic
140	C₂×C₈	C₃×#41	1	1	yes	characteristic
141	C₂×C₈	C₃×#41	1	1	yes	characteristic
142	C₄⋊C₄	—	2	4	no	—
143	C₄⋊C₄	—	2	2	no	—
144	C₄⋊C₄	—	4	8	no	—
145	C₄⋊C₄	—	4	2	no	—
146	C₂²⋊C₄	—	4	1	no	—
147	C₄²	—	2	1	no	—
148	C₄²	—	4	1	no	—
149	C₄²	C₃×C₂²⋊C₄	1	2	yes	—
150	C₁₆	—	4	2	no	—
151	C₂²×C₃	—	2	2	no	—
152	C₂²×C₃	#418	1	1	yes	characteristic
153	C₂²×C₃	#476	1	1	yes	characteristic
154	C₂²×C₃	#477	1	1	yes	characteristic
155	C₃×C₄	—	4	2	no	—
156	C₃×C₄	—	8	4	no	—
157	C₃×C₄	—	8	2	no	—
158	C₃×C₄	#391	1	1	yes	characteristic
159	C₃×C₄	C₂×#103	1	1	yes	characteristic
160	C₃×C₄	#476	1	1	yes	characteristic
161	C₃×C₄	#477	1	1	yes	characteristic
162	C₂³	C₃×AGL(1,ℤ/8)	1	1	yes	characteristic
163	C₂×C₄	—	2	2	no	—
164	C₂×C₄	—	4	2	no	—
165	C₂×C₄	—	4	4	no	—
166	C₂×C₄	—	4	2	no	—
167	C₂×C₄	—	8	4	no	—
168	C₂×C₄	C₂×C₃×C₂²⋊C₄	1	1	yes	characteristic
169	C₂×C₄	C₃×C₂²⋊Q₈	1	1	yes	characteristic
170	C₂×C₄	C₃×C₂²⋊Q₈	1	1	yes	characteristic
171	C₂×C₄	C₂×C₃×SD₁₆	1	1	yes	characteristic
172	C₂×C₄	C₂×C₃×Q₁₆	1	1	yes	characteristic
173	C₂×C₄	C₃×AGL(1,ℤ/8)	1	1	yes	characteristic
174	C₈	—	4	2	no	—
175	C₈	C₃×#103	1	2	yes	—
176	C₈	C₃×#103	1	2	yes	—
177	C₈	C₃×C₄×Q₈	1	1	yes	characteristic
178	C₈	C₃×C₄×Q₈	1	1	yes	characteristic
179	C₈	C₃×#124	1	1	yes	characteristic
180	C₈	C₃×C₄⋊Q₈	1	1	yes	characteristic
181	C₂×C₃	—	2	1	no	—
182	C₂×C₃	—	2	1	no	—
183	C₂×C₃	#1987	1	1	yes	characteristic
184	C₂×C₃	#2231	1	2	yes	—
185	C₂²	—	2	2	no	—
186	C₂²	C₃×#418	1	1	yes	characteristic
187	C₂²	C₃×#476	1	1	yes	characteristic
188	C₂²	C₃×#477	1	1	yes	characteristic
189	C₄	—	4	2	no	—
190	C₄	—	8	4	no	—
191	C₄	—	8	2	no	—
192	C₄	C₃×#391	1	1	yes	characteristic
193	C₄	C₂×C₃×#103	1	1	yes	characteristic
194	C₄	C₃×#476	1	1	yes	characteristic
195	C₄	C₃×#477	1	1	yes	characteristic
196	C₃	#13273	1	1	yes	characteristic, direct product factor, Sylow
197	C₂	—	2	1	no	—
198	C₂	—	2	1	no	—
199	C₂	C₃×#1987	1	1	yes	characteristic
200	C₂	C₃×#2231	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₃	#476
commutator	C₂×C₈	C₂²×C₃×C₄
Frattini, cosocle	C₂²×C₈	C₂³×C₃
prime radical	C₂³×C₃	AGL(1,ℤ/8)
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃ ◁ C₂²×C₃×C₄ ◁ C₂×C₃×C₄×C₈ ◁ G

Complements

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

N	H ≅ G/N	type
6. #13273	C₃	direct (N × H)
10. C₃×#499	C₄	semidirect
50. #499	C₃×C₄	semidirect
196. C₃	#13273	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×#364	2
C₃×#1772	2
C₃×#1386	2
C₃×#1383	2
#13273	3

Subgroups — C3×#13273 #10631880

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#13273 #10631880