Subgroups — C₃×#13272 #10631879

Order 768.

366 conjugacy classes of subgroups.

The subgroup lattice has 210 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×#498	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	#13272	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
7	C₂×C₃×C₄⋊Q₈	—	2	1	no	—
8	C₂×C₃×M₃₂	C₂²	1	2	yes	—
9	C₃×#498	—	2	2	no	—
10	C₃×#498	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×#106	—	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×#498	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	C₂²×C₃×Q₈	—	4	1	no	—
25	C₂×C₃×M₁₆	—	2	1	no	—
26	C₂²×C₃×C₈	D₄	1	1	yes	characteristic
27	C₂²×C₃×C₈	C₂³	1	1	yes	characteristic
28	C₃×C₄⋊Q₈	—	2	4	no	—
29	C₃×C₄⋊Q₈	—	4	2	no	—
30	C₃×#117	—	2	1	no	—
31	C₂×C₃×C₄⋊C₄	—	2	2	no	—
32	C₂×C₃×C₄⋊C₄	—	4	1	no	—
33	C₂×C₃×C₄²	D₄	1	1	yes	characteristic
34	C₃×M₃₂	—	2	2	no	—
35	C₃×M₃₂	—	2	2	no	—
36	C₂×C₃×C₁₆	—	2	2	no	—
37	C₃×#108	—	2	1	no	—
38	C₃×#108	Q₈	1	2	yes	complemented
39	C₃×#107	Q₈	1	2	yes	complemented
40	C₃×#106	—	2	4	no	—
41	C₃×#106	—	2	2	no	—
42	C₃×#106	—	2	1	no	—
43	C₃×#106	C₂×C₄	1	4	yes	—
44	C₃×#97	—	4	1	no	—
45	C₃×C₄×C₈	—	4	1	no	—
46	C₃×C₄×C₈	C₂×C₄	1	2	yes	—
47	C₃×C₄×C₈	D₄	1	2	yes	—
48	C₂×C₄⋊Q₈	—	2	1	no	—
49	C₂×M₃₂	C₂²×C₃	1	2	yes	—
50	#498	—	2	2	no	—
51	#498	C₃×C₄	1	4	yes	complemented
52	C₂×#108	C₂²×C₃	1	1	yes	characteristic
53	#427	C₂²×C₃	1	1	yes	characteristic
54	C₂×#106	—	2	1	no	—
55	C₂×#106	C₂²×C₃	1	2	yes	—
56	#405	—	2	1	no	—
57	C₂×C₄×C₈	C₂²×C₃	1	1	yes	characteristic
58	#364	—	2	2	no	—
59	#339	—	2	1	no	—
60	C₂×C₃×Q₈	—	4	4	no	—
61	C₂×C₃×Q₈	—	8	4	no	—
62	C₂²×C₃×C₄	—	2	1	no	—
63	C₂²×C₃×C₄	—	4	2	no	—
64	C₂²×C₃×C₄	C₂×D₄	1	1	yes	characteristic
65	C₃×M₁₆	—	2	2	no	—
66	C₃×M₁₆	—	4	1	no	—
67	C₂×C₃×C₈	—	4	1	no	—
68	C₂×C₃×C₈	C₂²⋊C₄	1	2	yes	—
69	C₂×C₃×C₈	SD₁₆	1	1	yes	characteristic
70	C₂×C₃×C₈	SD₁₆	1	1	yes	characteristic
71	C₂×C₃×C₈	Q₁₆	1	2	yes	—
72	C₂×C₃×C₈	C₂²×C₄	1	1	yes	characteristic
73	C₂×C₃×C₈	C₂×D₄	1	1	yes	characteristic
74	C₂×C₃×C₈	C₂×Q₈	1	1	yes	characteristic
75	C₂×C₃×C₈	C₂×Q₈	1	1	yes	characteristic
76	C₂×C₃×C₈	#41	1	1	yes	characteristic
77	C₂×C₃×C₈	#41	1	1	yes	characteristic
78	C₃×C₄⋊C₄	—	2	4	no	—
79	C₃×C₄⋊C₄	—	2	2	no	—
80	C₃×C₄⋊C₄	—	4	4	no	—
81	C₃×C₄⋊C₄	—	4	2	no	—
82	C₃×C₂²⋊C₄	—	4	1	no	—
83	C₃×C₄²	—	2	1	no	—
84	C₃×C₄²	—	4	1	no	—
85	C₃×C₄²	C₂²⋊C₄	1	2	yes	—
86	C₃×C₁₆	—	4	2	no	—
87	C₂²×Q₈	—	4	1	no	—
88	C₂×M₁₆	—	2	1	no	—
89	C₂²×C₈	C₃×D₄	1	1	yes	characteristic
90	C₂²×C₈	C₂³×C₃	1	1	yes	characteristic
91	C₄⋊Q₈	—	2	4	no	—
92	C₄⋊Q₈	—	4	2	no	—
93	#117	—	2	1	no	—
94	C₂×C₄⋊C₄	—	2	2	no	—
95	C₂×C₄⋊C₄	—	4	1	no	—
96	C₂×C₄²	C₃×D₄	1	1	yes	characteristic
97	M₃₂	—	2	2	no	—
98	M₃₂	—	2	2	no	—
99	C₂×C₁₆	—	2	2	no	—
100	#108	—	2	1	no	—
101	#108	C₃×Q₈	1	2	yes	complemented
102	#107	C₃×Q₈	1	2	yes	complemented
103	#106	—	2	4	no	—
104	#106	—	2	2	no	—
105	#106	—	2	1	no	—
106	#106	C₂×C₃×C₄	1	4	yes	—
107	#97	—	4	1	no	—
108	C₄×C₈	—	4	1	no	—
109	C₄×C₈	C₂×C₃×C₄	1	2	yes	—
110	C₄×C₈	C₃×D₄	1	2	yes	—
111	C₂³×C₃	AGL(1,ℤ/8)	1	1	yes	characteristic
112	C₃×Q₈	—	8	8	no	—
113	C₂×C₃×C₄	—	2	4	no	—
114	C₂×C₃×C₄	—	2	2	no	—
115	C₂×C₃×C₄	—	4	4	no	—
116	C₂×C₃×C₄	—	4	2	no	—
117	C₂×C₃×C₄	—	8	4	no	—
118	C₂×C₃×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
119	C₂×C₃×C₄	C₂²⋊Q₈	1	1	yes	characteristic
120	C₂×C₃×C₄	C₂²⋊Q₈	1	1	yes	characteristic
121	C₂×C₃×C₄	C₂×SD₁₆	1	1	yes	characteristic
122	C₂×C₃×C₄	C₂×Q₁₆	1	1	yes	characteristic
123	C₂×C₃×C₄	AGL(1,ℤ/8)	1	1	yes	characteristic
124	C₃×C₈	—	4	2	no	—
125	C₃×C₈	#103	1	2	yes	—
126	C₃×C₈	#103	1	2	yes	—
127	C₃×C₈	C₄×Q₈	1	1	yes	characteristic
128	C₃×C₈	C₄×Q₈	1	1	yes	characteristic
129	C₃×C₈	#124	1	1	yes	characteristic
130	C₃×C₈	C₄⋊Q₈	1	1	yes	characteristic
131	C₂×Q₈	—	4	4	no	—
132	C₂×Q₈	—	8	4	no	—
133	C₂²×C₄	—	2	1	no	—
134	C₂²×C₄	—	4	2	no	—
135	C₂²×C₄	C₂×C₃×D₄	1	1	yes	characteristic
136	M₁₆	—	2	2	no	—
137	M₁₆	—	4	1	no	—
138	C₂×C₈	—	4	1	no	—
139	C₂×C₈	C₃×C₂²⋊C₄	1	2	yes	—
140	C₂×C₈	C₃×SD₁₆	1	1	yes	characteristic
141	C₂×C₈	C₃×SD₁₆	1	1	yes	characteristic
142	C₂×C₈	C₃×Q₁₆	1	2	yes	—
143	C₂×C₈	C₂²×C₃×C₄	1	1	yes	characteristic
144	C₂×C₈	C₂×C₃×D₄	1	1	yes	characteristic
145	C₂×C₈	C₂×C₃×Q₈	1	1	yes	characteristic
146	C₂×C₈	C₂×C₃×Q₈	1	1	yes	characteristic
147	C₂×C₈	C₃×#41	1	1	yes	characteristic
148	C₂×C₈	C₃×#41	1	1	yes	characteristic
149	C₄⋊C₄	—	2	4	no	—
150	C₄⋊C₄	—	2	2	no	—
151	C₄⋊C₄	—	4	4	no	—
152	C₄⋊C₄	—	4	2	no	—
153	C₂²⋊C₄	—	4	1	no	—
154	C₄²	—	2	1	no	—
155	C₄²	—	4	1	no	—
156	C₄²	C₃×C₂²⋊C₄	1	2	yes	—
157	C₁₆	—	4	2	no	—
158	C₂²×C₃	—	2	2	no	—
159	C₂²×C₃	#418	1	1	yes	characteristic
160	C₂²×C₃	#476	1	1	yes	characteristic
161	C₂²×C₃	#477	1	1	yes	characteristic
162	C₃×C₄	—	2	4	no	—
163	C₃×C₄	—	8	4	no	—
164	C₃×C₄	—	8	2	no	—
165	C₃×C₄	#391	1	1	yes	characteristic
166	C₃×C₄	C₂×#103	1	1	yes	characteristic
167	C₃×C₄	#476	1	1	yes	characteristic
168	C₃×C₄	#477	1	1	yes	characteristic
169	C₂³	C₃×AGL(1,ℤ/8)	1	1	yes	characteristic
170	Q₈	—	8	8	no	—
171	C₂×C₄	—	2	4	no	—
172	C₂×C₄	—	2	2	no	—
173	C₂×C₄	—	4	4	no	—
174	C₂×C₄	—	4	2	no	—
175	C₂×C₄	—	8	4	no	—
176	C₂×C₄	C₂×C₃×C₂²⋊C₄	1	1	yes	characteristic
177	C₂×C₄	C₃×C₂²⋊Q₈	1	1	yes	characteristic
178	C₂×C₄	C₃×C₂²⋊Q₈	1	1	yes	characteristic
179	C₂×C₄	C₂×C₃×SD₁₆	1	1	yes	characteristic
180	C₂×C₄	C₂×C₃×Q₁₆	1	1	yes	characteristic
181	C₂×C₄	C₃×AGL(1,ℤ/8)	1	1	yes	characteristic
182	C₈	—	4	2	no	—
183	C₈	C₃×#103	1	2	yes	—
184	C₈	C₃×#103	1	2	yes	—
185	C₈	C₃×C₄×Q₈	1	1	yes	characteristic
186	C₈	C₃×C₄×Q₈	1	1	yes	characteristic
187	C₈	C₃×#124	1	1	yes	characteristic
188	C₈	C₃×C₄⋊Q₈	1	1	yes	characteristic
189	C₂×C₃	—	2	1	no	—
190	C₂×C₃	—	2	1	no	—
191	C₂×C₃	#1987	1	1	yes	characteristic
192	C₂×C₃	#2230	1	2	yes	—
193	C₂²	—	2	2	no	—
194	C₂²	C₃×#418	1	1	yes	characteristic
195	C₂²	C₃×#476	1	1	yes	characteristic
196	C₂²	C₃×#477	1	1	yes	characteristic
197	C₄	—	2	4	no	—
198	C₄	—	8	4	no	—
199	C₄	—	8	2	no	—
200	C₄	C₃×#391	1	1	yes	characteristic
201	C₄	C₂×C₃×#103	1	1	yes	characteristic
202	C₄	C₃×#476	1	1	yes	characteristic
203	C₄	C₃×#477	1	1	yes	characteristic
204	C₃	#13272	1	1	yes	characteristic, direct product factor, Sylow
205	C₂	—	2	1	no	—
206	C₂	—	2	1	no	—
207	C₂	C₃×#1987	1	1	yes	characteristic
208	C₂	C₃×#2230	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. #13272	C₃	direct (N × H)
10. C₃×#498	C₄	semidirect
38. C₃×#108	Q₈	semidirect
39. C₃×#107	Q₈	semidirect
51. #498	C₃×C₄	semidirect
101. #108	C₃×Q₈	semidirect
102. #107	C₃×Q₈	semidirect
204. C₃	#13272	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×#498	2
C₂×C₃×#364	2
C₃×#1772	2
C₃×#1386	2
C₃×#1383	2
#13272	3

Subgroups — C3×#13272 #10631879

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#13272 #10631879