Subgroups — C₃×#13269 #10631876

Order 768.

308 conjugacy classes of subgroups.

The subgroup lattice has 204 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#2961	C₂	1	1	yes	maximal, characteristic
2	C₃×#2153	C₂	1	1	yes	maximal, characteristic
3	C₃×#2110	C₂	1	1	yes	maximal, characteristic
4	C₃×#1378	C₂	1	2	yes	maximal
5	C₃×#1375	C₂	1	2	yes	maximal
6	#13269	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
7	C₃×#533	C₄	1	1	yes	characteristic
8	C₂×C₃×M₃₂	C₂²	1	2	yes	—
9	C₂×C₃×M₃₂	C₂²	1	2	yes	—
10	C₃×#446	—	2	2	no	—
11	C₃×#446	—	2	2	no	—
12	C₃×#424	C₄	1	1	yes	characteristic
13	C₃×#424	C₄	1	1	yes	characteristic
14	C₃×#423	C₄	1	1	yes	characteristic
15	C₃×#405	C₂²	1	1	yes	characteristic
16	C₃×#405	C₂²	1	1	yes	characteristic
17	C₂×C₃×#97	C₂²	1	1	yes	characteristic
18	C₃×#364	—	2	2	no	—
19	C₃×#347	—	2	2	no	—
20	#2961	C₂×C₃	1	1	yes	characteristic
21	#2153	C₂×C₃	1	1	yes	characteristic
22	#2110	C₂×C₃	1	1	yes	characteristic
23	#1378	C₂×C₃	1	2	yes	—
24	#1375	C₂×C₃	1	2	yes	—
25	C₂×C₃×M₁₆	C₂×C₄	1	1	yes	characteristic
26	C₂×C₃×M₁₆	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₃×C₈	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₃×C₈	C₂³	1	1	yes	characteristic
29	C₃×C₄⋊Q₈	—	2	1	no	—
30	C₃×#125	—	2	1	no	—
31	C₃×C₂²⋊Q₈	—	2	2	no	—
32	C₃×C₄×Q₈	—	2	2	no	—
33	C₃×#117	C₂×C₄	1	1	yes	characteristic
34	C₃×#117	C₂×C₄	1	1	yes	characteristic
35	C₂×C₃×C₄²	C₂×C₄	1	1	yes	characteristic
36	C₃×M₃₂	—	2	2	no	—
37	C₃×M₃₂	—	2	2	no	—
38	C₃×M₃₂	—	2	2	no	—
39	C₃×M₃₂	—	2	2	no	—
40	C₂×C₃×C₁₆	—	2	2	no	—
41	C₂×C₃×C₁₆	Q₈	1	4	yes	complemented
42	C₃×C₄⋊C₈	—	2	2	no	—
43	C₃×C₄⋊C₈	—	2	2	no	—
44	C₃×C₄⋊C₈	—	2	2	no	—
45	C₃×#97	—	2	1	no	—
46	C₃×#97	—	2	1	no	—
47	C₃×#97	—	2	1	no	—
48	C₃×#97	D₄	1	2	yes	—
49	C₃×C₄×C₈	—	2	1	no	—
50	C₃×C₄×C₈	D₄	1	2	yes	—
51	#533	C₃×C₄	1	1	yes	characteristic
52	C₂×M₃₂	C₂²×C₃	1	2	yes	—
53	C₂×M₃₂	C₂²×C₃	1	2	yes	—
54	#446	—	2	2	no	—
55	#446	—	2	2	no	—
56	#424	C₃×C₄	1	1	yes	characteristic
57	#424	C₃×C₄	1	1	yes	characteristic
58	#423	C₃×C₄	1	1	yes	characteristic
59	#405	C₂²×C₃	1	1	yes	characteristic
60	#405	C₂²×C₃	1	1	yes	characteristic
61	C₂×#97	C₂²×C₃	1	1	yes	characteristic
62	#364	—	2	2	no	—
63	#347	—	2	2	no	—
64	C₂×C₃×Q₈	—	2	2	no	—
65	C₂²×C₃×C₄	—	2	1	no	—
66	C₂²×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
67	C₃×M₁₆	—	2	2	no	—
68	C₃×M₁₆	—	2	2	no	—
69	C₂×C₃×C₈	—	2	1	no	—
70	C₂×C₃×C₈	—	2	1	no	—
71	C₂×C₃×C₈	—	2	2	no	—
72	C₂×C₃×C₈	C₂²⋊C₄	1	2	yes	—
73	C₂×C₃×C₈	C₂×D₄	1	1	yes	characteristic
74	C₂×C₃×C₈	C₂×D₄	1	1	yes	characteristic
75	C₂×C₃×C₈	C₂×Q₈	1	2	yes	—
76	C₂×C₃×C₈	#41	1	2	yes	—
77	C₃×C₄⋊C₄	—	2	4	no	—
78	C₃×C₄⋊C₄	—	2	2	no	—
79	C₃×C₄⋊C₄	—	2	1	no	—
80	C₃×C₄⋊C₄	—	2	1	no	—
81	C₃×C₂²⋊C₄	—	2	1	no	—
82	C₃×C₂²⋊C₄	M₁₆	1	2	yes	—
83	C₃×C₄²	—	2	1	no	—
84	C₃×C₄²	—	2	1	no	—
85	C₃×C₄²	C₂²⋊C₄	1	2	yes	—
86	C₃×C₄²	M₁₆	1	1	yes	characteristic
87	C₃×C₄²	M₁₆	1	1	yes	characteristic
88	C₃×C₁₆	—	2	4	no	—
89	C₃×C₁₆	—	4	2	no	—
90	C₂×M₁₆	C₂×C₃×C₄	1	1	yes	characteristic
91	C₂×M₁₆	C₂×C₃×C₄	1	1	yes	characteristic
92	C₂²×C₈	C₂×C₃×C₄	1	1	yes	characteristic
93	C₂²×C₈	C₂³×C₃	1	1	yes	characteristic
94	C₄⋊Q₈	—	2	1	no	—
95	#125	—	2	1	no	—
96	C₂²⋊Q₈	—	2	2	no	—
97	C₄×Q₈	—	2	2	no	—
98	#117	C₂×C₃×C₄	1	1	yes	characteristic
99	#117	C₂×C₃×C₄	1	1	yes	characteristic
100	C₂×C₄²	C₂×C₃×C₄	1	1	yes	characteristic
101	M₃₂	—	2	2	no	—
102	M₃₂	—	2	2	no	—
103	M₃₂	—	2	2	no	—
104	M₃₂	—	2	2	no	—
105	C₂×C₁₆	—	2	2	no	—
106	C₂×C₁₆	C₃×Q₈	1	4	yes	complemented
107	C₄⋊C₈	—	2	2	no	—
108	C₄⋊C₈	—	2	2	no	—
109	C₄⋊C₈	—	2	2	no	—
110	#97	—	2	1	no	—
111	#97	—	2	1	no	—
112	#97	—	2	1	no	—
113	#97	C₃×D₄	1	2	yes	—
114	C₄×C₈	—	2	1	no	—
115	C₄×C₈	C₃×D₄	1	2	yes	—
116	C₂³×C₃	C₂×M₁₆	1	1	yes	characteristic
117	C₃×Q₈	—	8	2	no	—
118	C₂×C₃×C₄	—	2	1	no	—
119	C₂×C₃×C₄	—	2	1	no	—
120	C₂×C₃×C₄	—	2	2	no	—
121	C₂×C₃×C₄	—	2	2	no	—
122	C₂×C₃×C₄	—	8	1	no	—
123	C₂×C₃×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
124	C₂×C₃×C₄	C₄×Q₈	1	2	yes	—
125	C₂×C₃×C₄	C₂×M₁₆	1	1	yes	characteristic
126	C₂×C₃×C₄	#131	1	2	yes	—
127	C₃×C₈	—	2	2	no	—
128	C₃×C₈	—	4	1	no	—
129	C₃×C₈	—	4	1	no	—
130	C₃×C₈	C₂²⋊Q₈	1	2	yes	—
131	C₃×C₈	#124	1	1	yes	characteristic
132	C₃×C₈	C₄⋊Q₈	1	1	yes	characteristic
133	C₂×Q₈	—	2	2	no	—
134	C₂²×C₄	—	2	1	no	—
135	C₂²×C₄	C₂²×C₃×C₄	1	1	yes	characteristic
136	M₁₆	—	2	2	no	—
137	M₁₆	—	2	2	no	—
138	C₂×C₈	—	2	1	no	—
139	C₂×C₈	—	2	1	no	—
140	C₂×C₈	—	2	2	no	—
141	C₂×C₈	C₃×C₂²⋊C₄	1	2	yes	—
142	C₂×C₈	C₂×C₃×D₄	1	1	yes	characteristic
143	C₂×C₈	C₂×C₃×D₄	1	1	yes	characteristic
144	C₂×C₈	C₂×C₃×Q₈	1	2	yes	—
145	C₂×C₈	C₃×#41	1	2	yes	—
146	C₄⋊C₄	—	2	4	no	—
147	C₄⋊C₄	—	2	2	no	—
148	C₄⋊C₄	—	2	1	no	—
149	C₄⋊C₄	—	2	1	no	—
150	C₂²⋊C₄	—	2	1	no	—
151	C₂²⋊C₄	C₃×M₁₆	1	2	yes	—
152	C₄²	—	2	1	no	—
153	C₄²	—	2	1	no	—
154	C₄²	C₃×C₂²⋊C₄	1	2	yes	—
155	C₄²	C₃×M₁₆	1	1	yes	characteristic
156	C₄²	C₃×M₁₆	1	1	yes	characteristic
157	C₁₆	—	2	4	no	—
158	C₁₆	—	4	2	no	—
159	C₂²×C₃	—	4	1	no	—
160	C₂²×C₃	#407	1	1	yes	characteristic
161	C₂²×C₃	#446	1	2	yes	—
162	C₃×C₄	—	4	1	no	—
163	C₃×C₄	—	4	1	no	—
164	C₃×C₄	—	4	2	no	—
165	C₃×C₄	—	4	2	no	—
166	C₃×C₄	#391	1	1	yes	characteristic
167	C₃×C₄	#408	1	1	yes	characteristic
168	C₃×C₄	#446	1	2	yes	—
169	C₂³	C₂×C₃×M₁₆	1	1	yes	characteristic
170	Q₈	—	8	2	no	—
171	C₂×C₄	—	2	1	no	—
172	C₂×C₄	—	2	1	no	—
173	C₂×C₄	—	2	2	no	—
174	C₂×C₄	—	2	2	no	—
175	C₂×C₄	—	8	1	no	—
176	C₂×C₄	C₂×C₃×C₂²⋊C₄	1	1	yes	characteristic
177	C₂×C₄	C₃×C₄×Q₈	1	2	yes	—
178	C₂×C₄	C₂×C₃×M₁₆	1	1	yes	characteristic
179	C₂×C₄	C₃×#131	1	2	yes	—
180	C₈	—	2	2	no	—
181	C₈	—	4	1	no	—
182	C₈	—	4	1	no	—
183	C₈	C₃×C₂²⋊Q₈	1	2	yes	—
184	C₈	C₃×#124	1	1	yes	characteristic
185	C₈	C₃×C₄⋊Q₈	1	1	yes	characteristic
186	C₂×C₃	—	2	1	no	—
187	C₂×C₃	—	2	2	no	—
188	C₂×C₃	#1985	1	1	yes	characteristic
189	C₂²	—	4	1	no	—
190	C₂²	C₃×#407	1	1	yes	characteristic
191	C₂²	C₃×#446	1	2	yes	—
192	C₄	—	4	1	no	—
193	C₄	—	4	1	no	—
194	C₄	—	4	2	no	—
195	C₄	—	4	2	no	—
196	C₄	C₃×#391	1	1	yes	characteristic
197	C₄	C₃×#408	1	1	yes	characteristic
198	C₄	C₃×#446	1	2	yes	—
199	C₃	#13269	1	1	yes	characteristic, direct product factor, Sylow
200	C₂	—	2	1	no	—
201	C₂	—	2	2	no	—
202	C₂	C₃×#1985	1	1	yes	characteristic

	subgroup	G/H
center	C₃×C₄	#391
commutator	C₂²×C₄	C₂²×C₃×C₄
Frattini, cosocle	C₂²×C₈	C₂³×C₃
socle	C₂×C₃	#1985
prime radical	C₂³×C₃	C₂×M₁₆
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₃×C₄ ◁ C₂²×C₃×C₈ ◁ G

Complements

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

N	H ≅ G/N	type
6. #13269	C₃	direct (N × H)
41. C₂×C₃×C₁₆	Q₈	semidirect
106. C₂×C₁₆	C₃×Q₈	semidirect
199. C₃	#13269	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃×#2961	2
C₃×#2153	2
C₃×#2110	2
C₃×#1378	2
C₃×#1375	2
#13269	3

Subgroups — C3×#13269 #10631876

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#13269 #10631876