Subgroups — C₃×#13265 #10631872

Order 768.

338 conjugacy classes of subgroups.

The subgroup lattice has 228 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#2960	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic
3	C₃×C₄×M₃₂	C₂	1	1	yes	maximal, characteristic
4	C₃×#1379	C₂	1	4	yes	maximal
5	#13265	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
6	C₃×#533	C₄	1	1	yes	characteristic
7	C₂×C₃×M₃₂	C₂²	1	2	yes	—
8	C₂²×C₃×C₁₆	C₂²	1	2	yes	—
9	C₃×#446	—	2	1	no	—
10	C₃×#446	—	2	1	no	—
11	C₃×C₈×Q₈	—	2	1	no	—
12	C₃×C₈×Q₈	—	2	1	no	—
13	C₃×#424	C₄	1	2	yes	—
14	C₃×#423	C₄	1	1	yes	characteristic
15	C₃×#405	C₂²	1	2	yes	—
16	C₂×C₃×C₄×C₈	C₂²	1	1	yes	characteristic
17	C₃×C₄⋊C₁₆	—	2	1	no	—
18	C₃×C₄⋊C₁₆	—	2	1	no	—
19	C₃×#346	—	2	1	no	—
20	C₃×C₄×C₁₆	—	2	1	no	—
21	#2960	C₂×C₃	1	1	yes	characteristic
22	C₂×C₄⋊C₁₆	C₂×C₃	1	1	yes	characteristic
23	C₄×M₃₂	C₂×C₃	1	1	yes	characteristic
24	#1379	C₂×C₃	1	4	yes	—
25	C₂×C₃×M₁₆	C₂×C₄	1	2	yes	—
26	C₂²×C₃×C₈	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₃×C₈	C₂³	1	1	yes	characteristic
28	C₃×C₄⋊Q₈	C₈	1	1	yes	characteristic
29	C₃×C₄⋊Q₈	C₈	1	1	yes	characteristic
30	C₃×#125	C₈	1	2	yes	—
31	C₃×C₂²⋊Q₈	—	2	1	no	—
32	C₃×C₂²⋊Q₈	—	2	1	no	—
33	C₃×C₄×Q₈	—	2	1	no	—
34	C₃×C₄×Q₈	—	2	1	no	—
35	C₃×#117	C₂×C₄	1	2	yes	—
36	C₂×C₃×C₄²	C₂×C₄	1	1	yes	characteristic
37	C₃×M₃₂	—	2	2	no	—
38	C₃×M₃₂	Q₈	1	4	yes	complemented
39	C₂×C₃×C₁₆	—	2	2	no	—
40	C₂×C₃×C₁₆	—	2	2	no	—
41	C₂×C₃×C₁₆	—	2	4	no	—
42	C₃×C₄⋊C₈	—	2	2	no	—
43	C₃×C₄⋊C₈	—	2	2	no	—
44	C₃×C₄⋊C₈	—	2	1	no	—
45	C₃×C₄⋊C₈	—	2	1	no	—
46	C₃×#97	—	2	2	no	—
47	C₃×C₄×C₈	—	2	2	no	—
48	C₃×C₄×C₈	D₄	1	1	yes	characteristic
49	C₃×C₄×C₈	D₄	1	1	yes	characteristic
50	C₃×C₄×C₈	D₄	1	1	yes	characteristic
51	C₃×C₄×C₈	D₄	1	1	yes	characteristic
52	#533	C₃×C₄	1	1	yes	characteristic
53	C₂×M₃₂	C₂²×C₃	1	2	yes	—
54	C₂²×C₁₆	C₂²×C₃	1	2	yes	—
55	#446	—	2	1	no	—
56	#446	—	2	1	no	—
57	C₈×Q₈	—	2	1	no	—
58	C₈×Q₈	—	2	1	no	—
59	#424	C₃×C₄	1	2	yes	—
60	#423	C₃×C₄	1	1	yes	characteristic
61	#405	C₂²×C₃	1	2	yes	—
62	C₂×C₄×C₈	C₂²×C₃	1	1	yes	characteristic
63	C₄⋊C₁₆	—	2	1	no	—
64	C₄⋊C₁₆	—	2	1	no	—
65	#346	—	2	1	no	—
66	C₄×C₁₆	—	2	1	no	—
67	C₂×C₃×Q₈	—	2	1	no	—
68	C₂×C₃×Q₈	—	2	1	no	—
69	C₂²×C₃×C₄	M₁₆	1	1	yes	characteristic
70	C₂²×C₃×C₄	M₁₆	1	1	yes	characteristic
71	C₂²×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
72	C₃×M₁₆	—	2	4	no	—
73	C₂×C₃×C₈	—	2	2	no	—
74	C₂×C₃×C₈	—	2	2	no	—
75	C₂×C₃×C₈	C₂²⋊C₄	1	1	yes	characteristic
76	C₂×C₃×C₈	C₂²⋊C₄	1	1	yes	characteristic
77	C₂×C₃×C₈	C₂×D₄	1	1	yes	characteristic
78	C₂×C₃×C₈	C₂×D₄	1	1	yes	characteristic
79	C₂×C₃×C₈	C₂×Q₈	1	2	yes	—
80	C₂×C₃×C₈	#41	1	2	yes	—
81	C₃×C₄⋊C₄	—	2	2	no	—
82	C₃×C₄⋊C₄	—	2	2	no	—
83	C₃×C₄⋊C₄	—	2	1	no	—
84	C₃×C₄⋊C₄	—	2	1	no	—
85	C₃×C₄⋊C₄	C₂×C₈	1	2	yes	—
86	C₃×C₄⋊C₄	C₂×C₈	1	2	yes	—
87	C₃×C₂²⋊C₄	—	2	2	no	—
88	C₃×C₄²	—	2	2	no	—
89	C₃×C₄²	C₂²⋊C₄	1	1	yes	characteristic
90	C₃×C₄²	C₂²⋊C₄	1	1	yes	characteristic
91	C₃×C₄²	C₂×C₈	1	1	yes	characteristic
92	C₃×C₄²	C₂×C₈	1	1	yes	characteristic
93	C₃×C₁₆	—	2	4	no	—
94	C₃×C₁₆	—	4	2	no	—
95	C₂×M₁₆	C₂×C₃×C₄	1	2	yes	—
96	C₂²×C₈	C₂×C₃×C₄	1	1	yes	characteristic
97	C₂²×C₈	C₂³×C₃	1	1	yes	characteristic
98	C₄⋊Q₈	C₃×C₈	1	1	yes	characteristic
99	C₄⋊Q₈	C₃×C₈	1	1	yes	characteristic
100	#125	C₃×C₈	1	2	yes	—
101	C₂²⋊Q₈	—	2	1	no	—
102	C₂²⋊Q₈	—	2	1	no	—
103	C₄×Q₈	—	2	1	no	—
104	C₄×Q₈	—	2	1	no	—
105	#117	C₂×C₃×C₄	1	2	yes	—
106	C₂×C₄²	C₂×C₃×C₄	1	1	yes	characteristic
107	M₃₂	—	2	2	no	—
108	M₃₂	C₃×Q₈	1	4	yes	complemented
109	C₂×C₁₆	—	2	2	no	—
110	C₂×C₁₆	—	2	2	no	—
111	C₂×C₁₆	—	2	4	no	—
112	C₄⋊C₈	—	2	2	no	—
113	C₄⋊C₈	—	2	2	no	—
114	C₄⋊C₈	—	2	1	no	—
115	C₄⋊C₈	—	2	1	no	—
116	#97	—	2	2	no	—
117	C₄×C₈	—	2	2	no	—
118	C₄×C₈	C₃×D₄	1	1	yes	characteristic
119	C₄×C₈	C₃×D₄	1	1	yes	characteristic
120	C₄×C₈	C₃×D₄	1	1	yes	characteristic
121	C₄×C₈	C₃×D₄	1	1	yes	characteristic
122	C₂³×C₃	C₂×M₁₆	1	1	yes	characteristic
123	C₃×Q₈	—	4	2	no	—
124	C₃×Q₈	—	4	2	no	—
125	C₂×C₃×C₄	—	2	2	no	—
126	C₂×C₃×C₄	—	2	2	no	—
127	C₂×C₃×C₄	—	2	2	no	—
128	C₂×C₃×C₄	—	4	1	no	—
129	C₂×C₃×C₄	C₂²⋊C₈	1	1	yes	characteristic
130	C₂×C₃×C₄	C₂²⋊C₈	1	1	yes	characteristic
131	C₂×C₃×C₄	C₂²⋊C₈	1	1	yes	characteristic
132	C₂×C₃×C₄	C₂²⋊C₈	1	1	yes	characteristic
133	C₂×C₃×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
134	C₂×C₃×C₄	C₄×Q₈	1	2	yes	—
135	C₂×C₃×C₄	C₂²×C₈	1	1	yes	characteristic
136	C₂×C₃×C₄	#131	1	2	yes	—
137	C₃×C₈	—	2	1	no	—
138	C₃×C₈	—	2	1	no	—
139	C₃×C₈	—	4	2	no	—
140	C₃×C₈	C₂²⋊Q₈	1	2	yes	—
141	C₃×C₈	#124	1	1	yes	characteristic
142	C₃×C₈	C₄⋊Q₈	1	1	yes	characteristic
143	C₂×Q₈	—	2	1	no	—
144	C₂×Q₈	—	2	1	no	—
145	C₂²×C₄	C₃×M₁₆	1	1	yes	characteristic
146	C₂²×C₄	C₃×M₁₆	1	1	yes	characteristic
147	C₂²×C₄	C₂²×C₃×C₄	1	1	yes	characteristic
148	M₁₆	—	2	4	no	—
149	C₂×C₈	—	2	2	no	—
150	C₂×C₈	—	2	2	no	—
151	C₂×C₈	C₃×C₂²⋊C₄	1	1	yes	characteristic
152	C₂×C₈	C₃×C₂²⋊C₄	1	1	yes	characteristic
153	C₂×C₈	C₂×C₃×D₄	1	1	yes	characteristic
154	C₂×C₈	C₂×C₃×D₄	1	1	yes	characteristic
155	C₂×C₈	C₂×C₃×Q₈	1	2	yes	—
156	C₂×C₈	C₃×#41	1	2	yes	—
157	C₄⋊C₄	—	2	2	no	—
158	C₄⋊C₄	—	2	2	no	—
159	C₄⋊C₄	—	2	1	no	—
160	C₄⋊C₄	—	2	1	no	—
161	C₄⋊C₄	C₂×C₃×C₈	1	2	yes	—
162	C₄⋊C₄	C₂×C₃×C₈	1	2	yes	—
163	C₂²⋊C₄	—	2	2	no	—
164	C₄²	—	2	2	no	—
165	C₄²	C₃×C₂²⋊C₄	1	1	yes	characteristic
166	C₄²	C₃×C₂²⋊C₄	1	1	yes	characteristic
167	C₄²	C₂×C₃×C₈	1	1	yes	characteristic
168	C₄²	C₂×C₃×C₈	1	1	yes	characteristic
169	C₁₆	—	2	4	no	—
170	C₁₆	—	4	2	no	—
171	C₂²×C₃	—	2	1	no	—
172	C₂²×C₃	—	2	1	no	—
173	C₂²×C₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
174	C₂²×C₃	#446	1	2	yes	—
175	C₃×C₄	—	2	1	no	—
176	C₃×C₄	—	2	1	no	—
177	C₃×C₄	—	2	1	no	—
178	C₃×C₄	—	4	2	no	—
179	C₃×C₄	—	4	2	no	—
180	C₃×C₄	#350	1	2	yes	—
181	C₃×C₄	#391	1	1	yes	characteristic
182	C₃×C₄	#408	1	1	yes	characteristic
183	C₃×C₄	C₈×Q₈	1	2	yes	—
184	C₂³	C₂×C₃×M₁₆	1	1	yes	characteristic
185	Q₈	—	4	2	no	—
186	Q₈	—	4	2	no	—
187	C₂×C₄	—	2	2	no	—
188	C₂×C₄	—	2	2	no	—
189	C₂×C₄	—	2	2	no	—
190	C₂×C₄	—	4	1	no	—
191	C₂×C₄	C₃×C₂²⋊C₈	1	1	yes	characteristic
192	C₂×C₄	C₃×C₂²⋊C₈	1	1	yes	characteristic
193	C₂×C₄	C₃×C₂²⋊C₈	1	1	yes	characteristic
194	C₂×C₄	C₃×C₂²⋊C₈	1	1	yes	characteristic
195	C₂×C₄	C₂×C₃×C₂²⋊C₄	1	1	yes	characteristic
196	C₂×C₄	C₃×C₄×Q₈	1	2	yes	—
197	C₂×C₄	C₂²×C₃×C₈	1	1	yes	characteristic
198	C₂×C₄	C₃×#131	1	2	yes	—
199	C₈	—	2	1	no	—
200	C₈	—	2	1	no	—
201	C₈	—	4	2	no	—
202	C₈	C₃×C₂²⋊Q₈	1	2	yes	—
203	C₈	C₃×#124	1	1	yes	characteristic
204	C₈	C₃×C₄⋊Q₈	1	1	yes	characteristic
205	C₂×C₃	—	2	2	no	—
206	C₂×C₃	#1984	1	1	yes	characteristic
207	C₂×C₃	C₂×#350	1	1	yes	characteristic
208	C₂×C₃	#2117	1	1	yes	characteristic
209	C₂²	—	2	1	no	—
210	C₂²	—	2	1	no	—
211	C₂²	C₂×C₃×C₂²⋊C₈	1	1	yes	characteristic
212	C₂²	C₃×#446	1	2	yes	—
213	C₄	—	2	1	no	—
214	C₄	—	2	1	no	—
215	C₄	—	2	1	no	—
216	C₄	—	4	2	no	—
217	C₄	—	4	2	no	—
218	C₄	C₃×#350	1	2	yes	—
219	C₄	C₃×#391	1	1	yes	characteristic
220	C₄	C₃×#408	1	1	yes	characteristic
221	C₄	C₃×C₈×Q₈	1	2	yes	—
222	C₃	#13265	1	1	yes	characteristic, direct product factor, Sylow
223	C₂	—	2	2	no	—
224	C₂	C₃×#1984	1	1	yes	characteristic
225	C₂	C₂×C₃×#350	1	1	yes	characteristic
226	C₂	C₃×#2117	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₃×C₈	C₂×D₄
commutator	C₂×C₄	C₂²×C₃×C₈
Frattini, cosocle	C₂²×C₈	C₂³×C₃
socle	C₂²×C₃	C₂×C₂²⋊C₈
prime radical	C₂³×C₃	C₂×M₁₆
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₃×C₈ ◁ C₂×C₃×C₄×C₈ ◁ G

Complements

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

N	H ≅ G/N	type
5. #13265	C₃	direct (N × H)
38. C₃×M₃₂	Q₈	semidirect
108. M₃₂	C₃×Q₈	semidirect
222. C₃	#13265	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×#2960	2
C₂×C₃×C₄⋊C₁₆	2
C₃×C₄×M₃₂	2
C₃×#1379	2
#13265	3

Subgroups — C3×#13265 #10631872

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#13265 #10631872