Subgroups — C₃×#13277 #10631884

Order 768.

366 conjugacy classes of subgroups.

The subgroup lattice has 212 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×C₄×#107	C₂	1	1	yes	maximal, characteristic
4	C₃×#1380	C₂	1	2	yes	maximal
5	C₃×#1380	C₂	1	2	yes	maximal
6	#13277	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
7	C₂×C₃×#125	—	2	1	no	—
8	C₂²×C₃×C₁₆	C₂²	1	2	yes	—
9	C₃×#499	—	2	2	no	—
10	C₃×#499	C₄	1	4	yes	complemented
11	C₂×C₃×#107	C₂²	1	1	yes	characteristic
12	C₂×C₃×#107	C₂²	1	1	yes	characteristic
13	C₂×C₃×#107	C₂²	1	2	yes	—
14	C₂×C₃×#106	—	2	1	no	—
15	C₂×C₃×C₄×C₈	C₂²	1	1	yes	characteristic
16	C₃×C₄×C₄⋊C₄	—	2	1	no	—
17	C₃×C₄⋊C₁₆	—	2	2	no	—
18	C₃×#340	—	2	1	no	—
19	C₂×#499	C₂×C₃	1	1	yes	characteristic
20	C₂×C₄⋊C₁₆	C₂×C₃	1	1	yes	characteristic
21	C₄×#107	C₂×C₃	1	1	yes	characteristic
22	#1380	C₂×C₃	1	2	yes	—
23	#1380	C₂×C₃	1	2	yes	—
24	C₂²×C₃×C₈	D₄	1	1	yes	characteristic
25	C₂²×C₃×C₈	C₂³	1	1	yes	characteristic
26	C₃×#125	—	2	4	no	—
27	C₃×#125	—	4	2	no	—
28	C₂×C₃×C₄⋊C₄	—	2	2	no	—
29	C₂×C₃×C₄⋊C₄	—	2	1	no	—
30	C₂×C₃×C₄⋊C₄	—	2	1	no	—
31	C₂×C₃×C₄⋊C₄	—	4	1	no	—
32	C₂×C₃×C₄⋊C₄	—	4	1	no	—
33	C₂×C₃×C₄²	—	4	1	no	—
34	C₂×C₃×C₄²	D₄	1	1	yes	characteristic
35	C₂×C₃×C₁₆	—	2	2	no	—
36	C₂×C₃×C₁₆	—	2	4	no	—
37	C₃×#107	—	2	1	no	—
38	C₃×#107	—	2	1	no	—
39	C₃×#107	—	2	2	no	—
40	C₃×#107	C₂×C₄	1	2	yes	—
41	C₃×#107	C₂×C₄	1	2	yes	—
42	C₃×#107	Q₈	1	4	yes	—
43	C₃×#106	—	2	4	no	—
44	C₃×C₄×C₈	C₂×C₄	1	2	yes	—
45	C₃×C₄×C₈	D₄	1	2	yes	—
46	C₃×#95	—	4	1	no	—
47	C₂×#125	—	2	1	no	—
48	C₂²×C₁₆	C₂²×C₃	1	2	yes	—
49	#499	—	2	2	no	—
50	#499	C₃×C₄	1	4	yes	complemented
51	C₂×#107	C₂²×C₃	1	1	yes	characteristic
52	C₂×#107	C₂²×C₃	1	1	yes	characteristic
53	C₂×#107	C₂²×C₃	1	2	yes	—
54	C₂×#106	—	2	1	no	—
55	C₂×C₄×C₈	C₂²×C₃	1	1	yes	characteristic
56	C₄×C₄⋊C₄	—	2	1	no	—
57	C₄⋊C₁₆	—	2	2	no	—
58	#340	—	2	1	no	—
59	C₂²×C₃×C₄	—	4	2	no	—
60	C₂²×C₃×C₄	—	4	1	no	—
61	C₂²×C₃×C₄	—	4	1	no	—
62	C₂²×C₃×C₄	D₈	1	1	yes	characteristic
63	C₂²×C₃×C₄	D₈	1	1	yes	characteristic
64	C₂²×C₃×C₄	C₂×D₄	1	1	yes	characteristic
65	C₂×C₃×C₈	—	2	2	no	—
66	C₂×C₃×C₈	C₂²⋊C₄	1	2	yes	—
67	C₂×C₃×C₈	C₂²×C₄	1	1	yes	characteristic
68	C₂×C₃×C₈	C₂×D₄	1	1	yes	characteristic
69	C₂×C₃×C₈	C₂×Q₈	1	2	yes	—
70	C₂×C₃×C₈	#41	1	2	yes	—
71	C₃×C₄⋊C₄	—	2	2	no	—
72	C₃×C₄⋊C₄	—	2	2	no	—
73	C₃×C₄⋊C₄	—	2	4	no	—
74	C₃×C₄⋊C₄	—	4	4	no	—
75	C₃×C₄⋊C₄	—	4	4	no	—
76	C₃×C₄⋊C₄	—	4	1	no	—
77	C₃×C₄⋊C₄	—	4	1	no	—
78	C₃×C₄⋊C₄	—	4	2	no	—
79	C₃×C₄²	—	8	2	no	—
80	C₃×C₄²	C₂²⋊C₄	1	2	yes	—
81	C₃×C₄²	SD₁₆	1	2	yes	—
82	C₃×C₁₆	—	4	2	no	—
83	C₂²×C₈	C₃×D₄	1	1	yes	characteristic
84	C₂²×C₈	C₂³×C₃	1	1	yes	characteristic
85	#125	—	2	4	no	—
86	#125	—	4	2	no	—
87	C₂×C₄⋊C₄	—	2	1	no	—
88	C₂×C₄⋊C₄	—	2	1	no	—
89	C₂×C₄⋊C₄	—	2	2	no	—
90	C₂×C₄⋊C₄	—	4	1	no	—
91	C₂×C₄⋊C₄	—	4	1	no	—
92	C₂×C₄²	—	4	1	no	—
93	C₂×C₄²	C₃×D₄	1	1	yes	characteristic
94	C₂×C₁₆	—	2	2	no	—
95	C₂×C₁₆	—	2	4	no	—
96	#107	—	2	1	no	—
97	#107	—	2	1	no	—
98	#107	—	2	2	no	—
99	#107	C₂×C₃×C₄	1	2	yes	—
100	#107	C₂×C₃×C₄	1	2	yes	—
101	#107	C₃×Q₈	1	4	yes	—
102	#106	—	2	4	no	—
103	C₄×C₈	C₂×C₃×C₄	1	2	yes	—
104	C₄×C₈	C₃×D₄	1	2	yes	—
105	#95	—	4	1	no	—
106	C₂³×C₃	C₂×D₈	1	1	yes	characteristic
107	C₂×C₃×C₄	—	2	2	no	—
108	C₂×C₃×C₄	—	2	1	no	—
109	C₂×C₃×C₄	—	2	1	no	—
110	C₂×C₃×C₄	—	4	2	no	—
111	C₂×C₃×C₄	—	4	2	no	—
112	C₂×C₃×C₄	—	4	4	no	—
113	C₂×C₃×C₄	—	8	2	no	—
114	C₂×C₃×C₄	—	8	2	no	—
115	C₂×C₃×C₄	—	8	2	no	—
116	C₂×C₃×C₄	—	8	1	no	—
117	C₂×C₃×C₄	—	8	1	no	—
118	C₂×C₃×C₄	#102	1	2	yes	—
119	C₂×C₃×C₄	#102	1	2	yes	—
120	C₂×C₃×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
121	C₂×C₃×C₄	C₂²⋊Q₈	1	2	yes	—
122	C₂×C₃×C₄	C₂×SD₁₆	1	1	yes	characteristic
123	C₂×C₃×C₄	#137	1	2	yes	—
124	C₃×C₈	—	2	2	no	—
125	C₃×C₈	C₄×Q₈	1	2	yes	—
126	C₃×C₈	#124	1	1	yes	characteristic
127	C₃×C₈	C₄⋊Q₈	1	1	yes	characteristic
128	C₂²×C₄	—	4	2	no	—
129	C₂²×C₄	—	4	1	no	—
130	C₂²×C₄	—	4	1	no	—
131	C₂²×C₄	C₃×D₈	1	1	yes	characteristic
132	C₂²×C₄	C₃×D₈	1	1	yes	characteristic
133	C₂²×C₄	C₂×C₃×D₄	1	1	yes	characteristic
134	C₂×C₈	—	2	2	no	—
135	C₂×C₈	C₃×C₂²⋊C₄	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	2	yes	—
139	C₂×C₈	C₃×#41	1	2	yes	—
140	C₄⋊C₄	—	2	2	no	—
141	C₄⋊C₄	—	2	2	no	—
142	C₄⋊C₄	—	2	4	no	—
143	C₄⋊C₄	—	4	4	no	—
144	C₄⋊C₄	—	4	4	no	—
145	C₄⋊C₄	—	4	1	no	—
146	C₄⋊C₄	—	4	1	no	—
147	C₄⋊C₄	—	4	2	no	—
148	C₄²	—	8	2	no	—
149	C₄²	C₃×C₂²⋊C₄	1	2	yes	—
150	C₄²	C₃×SD₁₆	1	2	yes	—
151	C₁₆	—	4	2	no	—
152	C₂²×C₃	C₂×#102	1	1	yes	characteristic
153	C₂²×C₃	#474	1	2	yes	—
154	C₂²×C₃	#508	1	2	yes	—
155	C₂²×C₃	#509	1	1	yes	characteristic
156	C₂²×C₃	#510	1	1	yes	characteristic
157	C₃×C₄	—	2	2	no	—
158	C₃×C₄	—	2	2	no	—
159	C₃×C₄	—	8	2	no	—
160	C₃×C₄	—	8	2	no	—
161	C₃×C₄	—	8	4	no	—
162	C₃×C₄	#391	1	1	yes	characteristic
163	C₃×C₄	#419	1	1	yes	characteristic
164	C₃×C₄	#475	1	2	yes	—
165	C₂³	C₂×C₃×D₈	1	1	yes	characteristic
166	C₂×C₄	—	2	2	no	—
167	C₂×C₄	—	2	1	no	—
168	C₂×C₄	—	2	1	no	—
169	C₂×C₄	—	4	2	no	—
170	C₂×C₄	—	4	2	no	—
171	C₂×C₄	—	4	4	no	—
172	C₂×C₄	—	8	2	no	—
173	C₂×C₄	—	8	2	no	—
174	C₂×C₄	—	8	2	no	—
175	C₂×C₄	—	8	1	no	—
176	C₂×C₄	—	8	1	no	—
177	C₂×C₄	C₃×#102	1	2	yes	—
178	C₂×C₄	C₃×#102	1	2	yes	—
179	C₂×C₄	C₂×C₃×C₂²⋊C₄	1	1	yes	characteristic
180	C₂×C₄	C₃×C₂²⋊Q₈	1	2	yes	—
181	C₂×C₄	C₂×C₃×SD₁₆	1	1	yes	characteristic
182	C₂×C₄	C₃×#137	1	2	yes	—
183	C₈	—	2	2	no	—
184	C₈	C₃×C₄×Q₈	1	2	yes	—
185	C₈	C₃×#124	1	1	yes	characteristic
186	C₈	C₃×C₄⋊Q₈	1	1	yes	characteristic
187	C₂×C₃	#1988	1	1	yes	characteristic
188	C₂×C₃	#2138	1	1	yes	characteristic
189	C₂×C₃	#2139	1	1	yes	characteristic
190	C₂×C₃	#2228	1	2	yes	—
191	C₂×C₃	#2229	1	2	yes	—
192	C₂²	C₂×C₃×#102	1	1	yes	characteristic
193	C₂²	C₃×#474	1	2	yes	—
194	C₂²	C₃×#508	1	2	yes	—
195	C₂²	C₃×#509	1	1	yes	characteristic
196	C₂²	C₃×#510	1	1	yes	characteristic
197	C₄	—	2	2	no	—
198	C₄	—	2	2	no	—
199	C₄	—	8	2	no	—
200	C₄	—	8	2	no	—
201	C₄	—	8	4	no	—
202	C₄	C₃×#391	1	1	yes	characteristic
203	C₄	C₃×#419	1	1	yes	characteristic
204	C₄	C₃×#475	1	2	yes	—
205	C₃	#13277	1	1	yes	characteristic, direct product factor, Sylow
206	C₂	C₃×#1988	1	1	yes	characteristic
207	C₂	C₃×#2138	1	1	yes	characteristic
208	C₂	C₃×#2139	1	1	yes	characteristic
209	C₂	C₃×#2228	1	2	yes	—
210	C₂	C₃×#2229	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
6. #13277	C₃	direct (N × H)
10. C₃×#499	C₄	semidirect
50. #499	C₃×C₄	semidirect
205. C₃	#13277	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×C₄×#107	2
C₃×#1380	2
#13277	3

Subgroups — C3×#13277 #10631884

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#13277 #10631884