Subgroups — C₃×#13276 #10631883

Order 768.

398 conjugacy classes of subgroups.

The subgroup lattice has 192 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×#500	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	4	yes	maximal
5	#13276	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
6	C₂×C₃×C₄⋊Q₈	—	2	1	no	—
7	C₂²×C₃×C₁₆	C₂²	1	2	yes	—
8	C₃×#500	—	2	2	no	—
9	C₃×#500	C₄	1	4	yes	complemented
10	C₂×C₃×#107	—	2	1	no	—
11	C₂×C₃×#107	C₂²	1	2	yes	—
12	C₂×C₃×#107	C₂²	1	2	yes	—
13	C₂×C₃×C₄×C₈	C₂²	1	1	yes	characteristic
14	C₃×C₄×C₄⋊C₄	—	2	1	no	—
15	C₃×C₄⋊C₁₆	—	2	2	no	—
16	C₃×#340	—	2	1	no	—
17	C₂×#500	C₂×C₃	1	1	yes	characteristic
18	C₂×C₄⋊C₁₆	C₂×C₃	1	1	yes	characteristic
19	C₄×#107	C₂×C₃	1	1	yes	characteristic
20	#1380	C₂×C₃	1	4	yes	—
21	C₂²×C₃×Q₈	—	4	1	no	—
22	C₂²×C₃×C₈	D₄	1	1	yes	characteristic
23	C₂²×C₃×C₈	C₂³	1	1	yes	characteristic
24	C₃×C₄⋊Q₈	—	2	4	no	—
25	C₃×C₄⋊Q₈	—	4	2	no	—
26	C₂×C₃×C₄⋊C₄	—	2	2	no	—
27	C₂×C₃×C₄⋊C₄	—	2	2	no	—
28	C₂×C₃×C₄⋊C₄	—	4	1	no	—
29	C₂×C₃×C₄²	—	4	1	no	—
30	C₂×C₃×C₄²	D₄	1	1	yes	characteristic
31	C₂×C₃×C₁₆	—	2	2	no	—
32	C₂×C₃×C₁₆	—	2	4	no	—
33	C₃×#107	—	2	4	no	—
34	C₃×#107	—	2	2	no	—
35	C₃×#107	—	2	2	no	—
36	C₃×#107	C₂×C₄	1	4	yes	—
37	C₃×#107	Q₈	1	4	yes	complemented
38	C₃×C₄×C₈	C₂×C₄	1	2	yes	—
39	C₃×C₄×C₈	D₄	1	2	yes	—
40	C₃×#95	—	4	1	no	—
41	C₂×C₄⋊Q₈	—	2	1	no	—
42	C₂²×C₁₆	C₂²×C₃	1	2	yes	—
43	#500	—	2	2	no	—
44	#500	C₃×C₄	1	4	yes	complemented
45	C₂×#107	—	2	1	no	—
46	C₂×#107	C₂²×C₃	1	2	yes	—
47	C₂×#107	C₂²×C₃	1	2	yes	—
48	C₂×C₄×C₈	C₂²×C₃	1	1	yes	characteristic
49	C₄×C₄⋊C₄	—	2	1	no	—
50	C₄⋊C₁₆	—	2	2	no	—
51	#340	—	2	1	no	—
52	C₂×C₃×Q₈	—	4	4	no	—
53	C₂×C₃×Q₈	—	8	4	no	—
54	C₂²×C₃×C₄	—	4	2	no	—
55	C₂²×C₃×C₄	—	4	2	no	—
56	C₂²×C₃×C₄	D₈	1	1	yes	characteristic
57	C₂²×C₃×C₄	D₈	1	1	yes	characteristic
58	C₂²×C₃×C₄	C₂×D₄	1	1	yes	characteristic
59	C₂×C₃×C₈	—	2	2	no	—
60	C₂×C₃×C₈	C₂²⋊C₄	1	2	yes	—
61	C₂×C₃×C₈	C₂²×C₄	1	1	yes	characteristic
62	C₂×C₃×C₈	C₂×D₄	1	1	yes	characteristic
63	C₂×C₃×C₈	C₂×Q₈	1	2	yes	—
64	C₂×C₃×C₈	#41	1	2	yes	—
65	C₃×C₄⋊C₄	—	2	4	no	—
66	C₃×C₄⋊C₄	—	2	4	no	—
67	C₃×C₄⋊C₄	—	4	4	no	—
68	C₃×C₄⋊C₄	—	4	2	no	—
69	C₃×C₄⋊C₄	—	4	2	no	—
70	C₃×C₄²	—	8	2	no	—
71	C₃×C₄²	C₂²⋊C₄	1	2	yes	—
72	C₃×C₄²	SD₁₆	1	2	yes	—
73	C₃×C₁₆	—	4	2	no	—
74	C₂²×Q₈	—	4	1	no	—
75	C₂²×C₈	C₃×D₄	1	1	yes	characteristic
76	C₂²×C₈	C₂³×C₃	1	1	yes	characteristic
77	C₄⋊Q₈	—	2	4	no	—
78	C₄⋊Q₈	—	4	2	no	—
79	C₂×C₄⋊C₄	—	2	2	no	—
80	C₂×C₄⋊C₄	—	2	2	no	—
81	C₂×C₄⋊C₄	—	4	1	no	—
82	C₂×C₄²	—	4	1	no	—
83	C₂×C₄²	C₃×D₄	1	1	yes	characteristic
84	C₂×C₁₆	—	2	2	no	—
85	C₂×C₁₆	—	2	4	no	—
86	#107	—	2	4	no	—
87	#107	—	2	2	no	—
88	#107	—	2	2	no	—
89	#107	C₂×C₃×C₄	1	4	yes	—
90	#107	C₃×Q₈	1	4	yes	complemented
91	C₄×C₈	C₂×C₃×C₄	1	2	yes	—
92	C₄×C₈	C₃×D₄	1	2	yes	—
93	#95	—	4	1	no	—
94	C₂³×C₃	C₂×D₈	1	1	yes	characteristic
95	C₃×Q₈	—	8	8	no	—
96	C₂×C₃×C₄	—	2	2	no	—
97	C₂×C₃×C₄	—	4	4	no	—
98	C₂×C₃×C₄	—	4	4	no	—
99	C₂×C₃×C₄	—	8	2	no	—
100	C₂×C₃×C₄	—	8	2	no	—
101	C₂×C₃×C₄	—	8	4	no	—
102	C₂×C₃×C₄	#102	1	2	yes	—
103	C₂×C₃×C₄	#102	1	2	yes	—
104	C₂×C₃×C₄	SD₃₂	1	2	yes	—
105	C₂×C₃×C₄	Q₃₂	1	2	yes	—
106	C₂×C₃×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
107	C₂×C₃×C₄	C₂²⋊Q₈	1	2	yes	—
108	C₂×C₃×C₄	C₂×SD₁₆	1	1	yes	characteristic
109	C₂×C₃×C₄	#137	1	2	yes	—
110	C₃×C₈	—	2	2	no	—
111	C₃×C₈	C₄×Q₈	1	2	yes	—
112	C₃×C₈	#124	1	1	yes	characteristic
113	C₃×C₈	C₄⋊Q₈	1	1	yes	characteristic
114	C₂×Q₈	—	4	4	no	—
115	C₂×Q₈	—	8	4	no	—
116	C₂²×C₄	—	4	2	no	—
117	C₂²×C₄	—	4	2	no	—
118	C₂²×C₄	C₃×D₈	1	1	yes	characteristic
119	C₂²×C₄	C₃×D₈	1	1	yes	characteristic
120	C₂²×C₄	C₂×C₃×D₄	1	1	yes	characteristic
121	C₂×C₈	—	2	2	no	—
122	C₂×C₈	C₃×C₂²⋊C₄	1	2	yes	—
123	C₂×C₈	C₂²×C₃×C₄	1	1	yes	characteristic
124	C₂×C₈	C₂×C₃×D₄	1	1	yes	characteristic
125	C₂×C₈	C₂×C₃×Q₈	1	2	yes	—
126	C₂×C₈	C₃×#41	1	2	yes	—
127	C₄⋊C₄	—	2	4	no	—
128	C₄⋊C₄	—	2	4	no	—
129	C₄⋊C₄	—	4	4	no	—
130	C₄⋊C₄	—	4	2	no	—
131	C₄⋊C₄	—	4	2	no	—
132	C₄²	—	8	2	no	—
133	C₄²	C₃×C₂²⋊C₄	1	2	yes	—
134	C₄²	C₃×SD₁₆	1	2	yes	—
135	C₁₆	—	4	2	no	—
136	C₂²×C₃	C₂×#102	1	1	yes	characteristic
137	C₂²×C₃	#474	1	2	yes	—
138	C₂²×C₃	C₂×SD₃₂	1	1	yes	characteristic
139	C₂²×C₃	C₂×Q₃₂	1	1	yes	characteristic
140	C₂²×C₃	#509	1	2	yes	—
141	C₃×C₄	—	2	2	no	—
142	C₃×C₄	—	8	4	no	—
143	C₃×C₄	—	8	4	no	—
144	C₃×C₄	#358	1	4	yes	—
145	C₃×C₄	#391	1	1	yes	characteristic
146	C₃×C₄	#419	1	1	yes	characteristic
147	C₃×C₄	#475	1	2	yes	—
148	C₂³	C₂×C₃×D₈	1	1	yes	characteristic
149	Q₈	—	8	8	no	—
150	C₂×C₄	—	2	2	no	—
151	C₂×C₄	—	4	4	no	—
152	C₂×C₄	—	4	4	no	—
153	C₂×C₄	—	8	2	no	—
154	C₂×C₄	—	8	4	no	—
155	C₂×C₄	—	8	2	no	—
156	C₂×C₄	C₃×#102	1	2	yes	—
157	C₂×C₄	C₃×#102	1	2	yes	—
158	C₂×C₄	C₃×SD₃₂	1	2	yes	—
159	C₂×C₄	C₃×Q₃₂	1	2	yes	—
160	C₂×C₄	C₂×C₃×C₂²⋊C₄	1	1	yes	characteristic
161	C₂×C₄	C₃×C₂²⋊Q₈	1	2	yes	—
162	C₂×C₄	C₂×C₃×SD₁₆	1	1	yes	characteristic
163	C₂×C₄	C₃×#137	1	2	yes	—
164	C₈	—	2	2	no	—
165	C₈	C₃×C₄×Q₈	1	2	yes	—
166	C₈	C₃×#124	1	1	yes	characteristic
167	C₈	C₃×C₄⋊Q₈	1	1	yes	characteristic
168	C₂×C₃	#1988	1	1	yes	characteristic
169	C₂×C₃	C₂×#358	1	1	yes	characteristic
170	C₂×C₃	#2140	1	1	yes	characteristic
171	C₂×C₃	#2226	1	2	yes	—
172	C₂×C₃	#2227	1	2	yes	—
173	C₂²	C₂×C₃×#102	1	1	yes	characteristic
174	C₂²	C₃×#474	1	2	yes	—
175	C₂²	C₂×C₃×SD₃₂	1	1	yes	characteristic
176	C₂²	C₂×C₃×Q₃₂	1	1	yes	characteristic
177	C₂²	C₃×#509	1	2	yes	—
178	C₄	—	2	2	no	—
179	C₄	—	8	4	no	—
180	C₄	—	8	4	no	—
181	C₄	C₃×#358	1	4	yes	—
182	C₄	C₃×#391	1	1	yes	characteristic
183	C₄	C₃×#419	1	1	yes	characteristic
184	C₄	C₃×#475	1	2	yes	—
185	C₃	#13276	1	1	yes	characteristic, direct product factor, Sylow
186	C₂	C₃×#1988	1	1	yes	characteristic
187	C₂	C₂×C₃×#358	1	1	yes	characteristic
188	C₂	C₃×#2140	1	1	yes	characteristic
189	C₂	C₃×#2226	1	2	yes	—
190	C₂	C₃×#2227	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
5. #13276	C₃	direct (N × H)
9. C₃×#500	C₄	semidirect
37. C₃×#107	Q₈	semidirect
44. #500	C₃×C₄	semidirect
90. #107	C₃×Q₈	semidirect
185. C₃	#13276	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×#500	2
C₂×C₃×C₄⋊C₁₆	2
C₃×C₄×#107	2
C₃×#1380	2
#13276	3

Subgroups — C3×#13276 #10631883

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#13276 #10631883