Subgroups — C₃×#20507 #10639114

Order 768.

1054 conjugacy classes of subgroups.

The subgroup lattice has 166 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₄×Q₁₆	C₂	1	3	yes	maximal
2	C₃×#2276	C₂	1	2	yes	maximal
3	C₃×C₄×#107	C₂	1	3	yes	maximal
4	C₃×#1763	C₂	1	6	yes	maximal
5	C₃×#1726	C₂	1	1	yes	maximal, characteristic
6	#20507	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
7	C₂²×C₃×Q₁₆	C₂²	1	1	yes	characteristic
8	C₂×C₃×C₄×Q₈	C₂²	1	6	yes	—
9	C₃×C₄×Q₁₆	C₄	1	24	yes	complemented
10	C₂×C₃×#107	C₂²	1	3	yes	—
11	C₂×C₃×#103	C₂²	1	6	yes	—
12	C₂×C₃×#97	C₂²	1	3	yes	—
13	C₂×C₃×C₄×C₈	C₂²	1	3	yes	—
14	C₃×C₄×C₄⋊C₄	—	2	6	no	—
15	C₃×C₄×C₄⋊C₄	C₂²	1	6	yes	—
16	C₃×#376	—	2	2	no	—
17	C₃×#376	C₂²	1	1	yes	characteristic
18	C₃×#340	C₂²	1	6	yes	—
19	C₂×C₄×Q₁₆	C₂×C₃	1	3	yes	—
20	#2276	C₂×C₃	1	2	yes	—
21	C₄×#107	C₂×C₃	1	3	yes	—
22	#1763	C₂×C₃	1	6	yes	—
23	#1726	C₂×C₃	1	1	yes	characteristic
24	C₂²×C₃×Q₈	C₂³	1	2	yes	—
25	C₂×C₃×Q₁₆	C₂×C₄	1	12	yes	—
26	C₂²×C₃×C₈	C₂³	1	3	yes	—
27	C₂²×C₃×C₈	C₂³	1	1	yes	characteristic
28	C₃×C₄×Q₈	—	2	12	no	—
29	C₃×C₄×Q₈	C₂×C₄	1	24	yes	—
30	C₂×C₃×C₄⋊C₄	—	2	6	no	—
31	C₂×C₃×C₄⋊C₄	C₂³	1	6	yes	—
32	C₂×C₃×C₄²	—	2	6	no	—
33	C₂×C₃×C₄²	—	2	6	no	—
34	C₂×C₃×C₄²	C₂³	1	3	yes	—
35	C₃×#107	C₂×C₄	1	12	yes	—
36	C₃×#103	C₂×C₄	1	24	yes	—
37	C₃×#97	—	2	6	no	—
38	C₃×C₄×C₈	C₂×C₄	1	12	yes	—
39	C₃×#95	—	2	6	no	—
40	C₃×#95	D₄	1	1	yes	characteristic
41	C₃×#95	D₄	1	3	yes	—
42	C₂²×Q₁₆	C₂²×C₃	1	1	yes	characteristic
43	C₂×C₄×Q₈	C₂²×C₃	1	6	yes	—
44	C₄×Q₁₆	C₃×C₄	1	24	yes	complemented
45	C₂×#107	C₂²×C₃	1	3	yes	—
46	C₂×#103	C₂²×C₃	1	6	yes	—
47	C₂×#97	C₂²×C₃	1	3	yes	—
48	C₂×C₄×C₈	C₂²×C₃	1	3	yes	—
49	C₄×C₄⋊C₄	—	2	6	no	—
50	C₄×C₄⋊C₄	C₂²×C₃	1	6	yes	—
51	#376	—	2	2	no	—
52	#376	C₂²×C₃	1	1	yes	characteristic
53	#340	C₂²×C₃	1	6	yes	—
54	C₂×C₃×Q₈	—	2	6	no	—
55	C₂×C₃×Q₈	C₂²×C₄	1	12	yes	—
56	C₂²×C₃×C₄	—	2	2	no	—
57	C₂²×C₃×C₄	—	2	6	no	—
58	C₂²×C₃×C₄	C₂×D₄	1	3	yes	—
59	C₂²×C₃×C₄	C₂×D₄	1	3	yes	—
60	C₂²×C₃×C₄	C₂⁴	1	1	yes	characteristic
61	C₃×Q₁₆	C₄²	1	16	yes	complemented
62	C₂×C₃×C₈	—	2	6	no	—
63	C₂×C₃×C₈	C₂²×C₄	1	6	yes	—
64	C₂×C₃×C₈	C₂²×C₄	1	6	yes	—
65	C₃×C₄⋊C₄	—	2	24	no	—
66	C₃×C₄⋊C₄	—	2	6	no	—
67	C₃×C₄⋊C₄	C₂²×C₄	1	12	yes	—
68	C₃×C₄²	—	2	24	no	—
69	C₃×C₄²	—	8	6	no	—
70	C₃×C₄²	C₂²×C₄	1	6	yes	—
71	C₃×C₄²	#41	1	6	yes	—
72	C₂²×Q₈	C₂³×C₃	1	2	yes	—
73	C₂×Q₁₆	C₂×C₃×C₄	1	12	yes	—
74	C₂²×C₈	C₂³×C₃	1	3	yes	—
75	C₂²×C₈	C₂³×C₃	1	1	yes	characteristic
76	C₄×Q₈	—	2	12	no	—
77	C₄×Q₈	C₂×C₃×C₄	1	24	yes	—
78	C₂×C₄⋊C₄	—	2	6	no	—
79	C₂×C₄⋊C₄	C₂³×C₃	1	6	yes	—
80	C₂×C₄²	—	2	6	no	—
81	C₂×C₄²	—	2	6	no	—
82	C₂×C₄²	C₂³×C₃	1	3	yes	—
83	#107	C₂×C₃×C₄	1	12	yes	—
84	#103	C₂×C₃×C₄	1	24	yes	—
85	#97	—	2	6	no	—
86	C₄×C₈	C₂×C₃×C₄	1	12	yes	—
87	#95	—	2	6	no	—
88	#95	C₃×D₄	1	1	yes	characteristic
89	#95	C₃×D₄	1	3	yes	—
90	C₂³×C₃	C₂²×D₄	1	1	yes	characteristic
91	C₃×Q₈	—	2	12	no	—
92	C₃×Q₈	C₂×C₄²	1	8	yes	—
93	C₂×C₃×C₄	—	2	12	no	—
94	C₂×C₃×C₄	—	2	3	no	—
95	C₂×C₃×C₄	—	2	6	no	—
96	C₂×C₃×C₄	—	2	3	no	—
97	C₂×C₃×C₄	—	2	12	no	—
98	C₂×C₃×C₄	—	4	6	no	—
99	C₂×C₃×C₄	—	4	6	no	—
100	C₂×C₃×C₄	C₄×D₄	1	6	yes	—
101	C₂×C₃×C₄	C₄×D₄	1	6	yes	—
102	C₂×C₃×C₄	C₂³×C₄	1	3	yes	—
103	C₂×C₃×C₄	C₂×#41	1	3	yes	—
104	C₃×C₈	—	2	6	no	—
105	C₃×C₈	C₂×C₄²	1	4	yes	—
106	C₂×Q₈	—	2	6	no	—
107	C₂×Q₈	C₂²×C₃×C₄	1	12	yes	—
108	C₂²×C₄	—	2	2	no	—
109	C₂²×C₄	—	2	6	no	—
110	C₂²×C₄	C₂×C₃×D₄	1	3	yes	—
111	C₂²×C₄	C₂×C₃×D₄	1	3	yes	—
112	C₂²×C₄	C₂⁴×C₃	1	1	yes	characteristic
113	Q₁₆	C₃×C₄²	1	16	yes	complemented
114	C₂×C₈	—	2	6	no	—
115	C₂×C₈	C₂²×C₃×C₄	1	6	yes	—
116	C₂×C₈	C₂²×C₃×C₄	1	6	yes	—
117	C₄⋊C₄	—	2	24	no	—
118	C₄⋊C₄	—	2	6	no	—
119	C₄⋊C₄	C₂²×C₃×C₄	1	12	yes	—
120	C₄²	—	2	24	no	—
121	C₄²	—	8	6	no	—
122	C₄²	C₂²×C₃×C₄	1	6	yes	—
123	C₄²	C₃×#41	1	6	yes	—
124	C₂²×C₃	C₂×C₄×D₄	1	3	yes	—
125	C₂²×C₃	#576	1	1	yes	characteristic
126	C₂²×C₃	#578	1	3	yes	—
127	C₃×C₄	—	2	6	no	—
128	C₃×C₄	—	2	6	no	—
129	C₃×C₄	—	2	8	no	—
130	C₃×C₄	—	4	12	no	—
131	C₃×C₄	C₂²×C₄²	1	1	yes	characteristic
132	C₃×C₄	C₄×#41	1	3	yes	—
133	C₂³	C₂²×C₃×D₄	1	1	yes	characteristic
134	Q₈	—	2	12	no	—
135	Q₈	C₂×C₃×C₄²	1	8	yes	—
136	C₂×C₄	—	2	3	no	—
137	C₂×C₄	—	2	6	no	—
138	C₂×C₄	—	2	3	no	—
139	C₂×C₄	—	2	12	no	—
140	C₂×C₄	—	2	12	no	—
141	C₂×C₄	—	4	6	no	—
142	C₂×C₄	—	4	6	no	—
143	C₂×C₄	C₃×C₄×D₄	1	6	yes	—
144	C₂×C₄	C₃×C₄×D₄	1	6	yes	—
145	C₂×C₄	C₂³×C₃×C₄	1	3	yes	—
146	C₂×C₄	C₂×C₃×#41	1	3	yes	—
147	C₈	—	2	6	no	—
148	C₈	C₂×C₃×C₄²	1	4	yes	—
149	C₂×C₃	C₄²×D₄	1	1	yes	characteristic
150	C₂×C₃	#2950	1	3	yes	—
151	C₂×C₃	#2951	1	3	yes	—
152	C₂²	C₂×C₃×C₄×D₄	1	3	yes	—
153	C₂²	C₃×#576	1	1	yes	characteristic
154	C₂²	C₃×#578	1	3	yes	—
155	C₄	—	2	6	no	—
156	C₄	—	2	6	no	—
157	C₄	—	2	8	no	—
158	C₄	—	4	12	no	—
159	C₄	C₂²×C₃×C₄²	1	1	yes	characteristic
160	C₄	C₃×C₄×#41	1	3	yes	—
161	C₃	#20507	1	1	yes	characteristic, direct product factor, Sylow
162	C₂	C₃×C₄²×D₄	1	1	yes	characteristic
163	C₂	C₃×#2950	1	3	yes	—
164	C₂	C₃×#2951	1	3	yes	—

	subgroup	G/H
center, socle, prime radical	C₂³×C₃	C₂²×D₄
commutator	C₄	C₂²×C₃×C₄²
Frattini, cosocle	C₂²×C₄	C₂⁴×C₃
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₃ ◁ C₃×#376 ◁ G

Complements

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

N	H ≅ G/N	type
6. #20507	C₃	direct (N × H)
9. C₃×C₄×Q₁₆	C₄	semidirect
44. C₄×Q₁₆	C₃×C₄	semidirect
61. C₃×Q₁₆	C₄²	semidirect
113. Q₁₆	C₃×C₄²	semidirect
161. C₃	#20507	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×C₄×Q₁₆	2
C₃×#2276	2
C₃×C₄×#107	2
C₃×#1763	2
C₃×#1726	2
#20507	3

Subgroups — C3×#20507 #10639114

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#20507 #10639114