Subgroups — C₃×#3164 #85028

Order 384.

340 conjugacy classes of subgroups.

The subgroup lattice has 162 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#544	C₂	1	2	yes	maximal
2	C₃×#501	C₂	1	4	yes	maximal, complemented
3	C₃×#500	C₂	1	2	yes	maximal, complemented
4	C₃×#499	C₂	1	2	yes	maximal, complemented
5	C₃×#428	C₂	1	2	yes	maximal
6	C₃×#427	C₂	1	1	yes	maximal, characteristic
7	C₂×C₃×#107	C₂	1	1	yes	maximal, characteristic
8	C₃×#405	C₂	1	1	yes	maximal, characteristic
9	#3164	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
10	C₂×C₃×M₁₆	C₂²	1	1	yes	characteristic
11	C₂²×C₃×C₈	C₂²	1	1	yes	characteristic
12	C₃×C₄⋊Q₈	—	2	2	no	—
13	C₃×C₄⋊Q₈	C₂²	1	4	yes	—
14	C₃×#125	—	2	2	no	—
15	C₃×#125	C₂²	1	4	yes	—
16	C₃×C₂²⋊Q₈	—	2	4	no	—
17	C₃×#117	C₂²	1	2	yes	—
18	C₃×#117	C₂²	1	1	yes	characteristic
19	C₂×C₃×C₄⋊C₄	C₂²	1	2	yes	—
20	C₃×#107	C₂²	1	2	yes	—
21	C₃×#107	C₂²	1	2	yes	—
22	C₃×#107	C₂²	1	1	yes	characteristic
23	C₃×#107	C₂²	1	1	yes	characteristic
24	C₃×#107	C₂²	1	4	yes	—
25	C₃×#106	C₂²	1	2	yes	—
26	C₃×#106	C₂²	1	4	yes	—
27	C₃×#97	C₂²	1	2	yes	—
28	C₃×C₄×C₈	C₂²	1	2	yes	—
29	#544	C₂×C₃	1	2	yes	—
30	#501	C₂×C₃	1	4	yes	complemented
31	#500	C₂×C₃	1	2	yes	complemented
32	#499	C₂×C₃	1	2	yes	complemented
33	#428	C₂×C₃	1	2	yes	—
34	#427	C₂×C₃	1	1	yes	characteristic
35	C₂×#107	C₂×C₃	1	1	yes	characteristic
36	#405	C₂×C₃	1	1	yes	characteristic
37	C₂×C₃×Q₈	—	2	4	no	—
38	C₂²×C₃×C₄	—	2	2	no	—
39	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
40	C₃×M₁₆	Q₈	1	4	yes	complemented
41	C₂×C₃×C₈	Q₈	1	4	yes	complemented
42	C₂×C₃×C₈	C₂³	1	1	yes	characteristic
43	C₂×C₃×C₈	C₂³	1	2	yes	—
44	C₂×C₃×C₈	C₂³	1	1	yes	characteristic
45	C₃×C₄⋊C₄	—	2	2	no	—
46	C₃×C₄⋊C₄	—	2	4	no	—
47	C₃×C₄⋊C₄	—	2	4	no	—
48	C₃×C₄⋊C₄	—	2	4	no	—
49	C₃×C₄⋊C₄	D₄	1	1	yes	characteristic
50	C₃×C₄⋊C₄	D₄	1	1	yes	characteristic
51	C₃×C₄⋊C₄	C₂³	1	2	yes	—
52	C₃×C₄⋊C₄	C₂³	1	4	yes	—
53	C₃×C₄⋊C₄	C₂³	1	2	yes	—
54	C₃×C₂²⋊C₄	—	2	2	no	—
55	C₃×C₂²⋊C₄	D₄	1	2	yes	—
56	C₃×C₄²	—	2	2	no	—
57	C₃×C₄²	C₂³	1	2	yes	—
58	C₂×M₁₆	C₂²×C₃	1	1	yes	characteristic
59	C₂²×C₈	C₂²×C₃	1	1	yes	characteristic
60	C₄⋊Q₈	—	2	2	no	—
61	C₄⋊Q₈	C₂²×C₃	1	4	yes	—
62	#125	—	2	2	no	—
63	#125	C₂²×C₃	1	4	yes	—
64	C₂²⋊Q₈	—	2	4	no	—
65	#117	C₂²×C₃	1	2	yes	—
66	#117	C₂²×C₃	1	1	yes	characteristic
67	C₂×C₄⋊C₄	C₂²×C₃	1	2	yes	—
68	#107	C₂²×C₃	1	2	yes	—
69	#107	C₂²×C₃	1	2	yes	—
70	#107	C₂²×C₃	1	1	yes	characteristic
71	#107	C₂²×C₃	1	1	yes	characteristic
72	#107	C₂²×C₃	1	4	yes	—
73	#106	C₂²×C₃	1	2	yes	—
74	#106	C₂²×C₃	1	4	yes	—
75	#97	C₂²×C₃	1	2	yes	—
76	C₄×C₈	C₂²×C₃	1	2	yes	—
77	C₂³×C₃	C₂×D₄	1	1	yes	characteristic
78	C₃×Q₈	—	8	4	no	—
79	C₂×C₃×C₄	—	2	2	no	—
80	C₂×C₃×C₄	—	2	2	no	—
81	C₂×C₃×C₄	—	2	4	no	—
82	C₂×C₃×C₄	—	8	2	no	—
83	C₂×C₃×C₄	C₂×D₄	1	2	yes	—
84	C₂×C₃×C₄	C₂×D₄	1	2	yes	—
85	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
86	C₂×C₃×C₄	C₂×Q₈	1	2	yes	—
87	C₂×C₃×C₄	C₂×Q₈	1	2	yes	—
88	C₂×C₃×C₄	C₂⁴	1	1	yes	characteristic
89	C₃×C₈	C₂×Q₈	1	2	yes	—
90	C₃×C₈	C₂×Q₈	1	2	yes	—
91	C₃×C₈	C₂×Q₈	1	4	yes	—
92	C₂×Q₈	—	2	4	no	—
93	C₂²×C₄	—	2	2	no	—
94	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
95	M₁₆	C₃×Q₈	1	4	yes	complemented
96	C₂×C₈	C₃×Q₈	1	4	yes	complemented
97	C₂×C₈	C₂³×C₃	1	1	yes	characteristic
98	C₂×C₈	C₂³×C₃	1	2	yes	—
99	C₂×C₈	C₂³×C₃	1	1	yes	characteristic
100	C₄⋊C₄	—	2	2	no	—
101	C₄⋊C₄	—	2	4	no	—
102	C₄⋊C₄	—	2	4	no	—
103	C₄⋊C₄	—	2	4	no	—
104	C₄⋊C₄	C₃×D₄	1	1	yes	characteristic
105	C₄⋊C₄	C₃×D₄	1	1	yes	characteristic
106	C₄⋊C₄	C₂³×C₃	1	2	yes	—
107	C₄⋊C₄	C₂³×C₃	1	4	yes	—
108	C₄⋊C₄	C₂³×C₃	1	2	yes	—
109	C₂²⋊C₄	—	2	2	no	—
110	C₂²⋊C₄	C₃×D₄	1	2	yes	—
111	C₄²	—	2	2	no	—
112	C₄²	C₂³×C₃	1	2	yes	—
113	C₂²×C₃	—	2	1	no	—
114	C₂²×C₃	—	2	1	no	—
115	C₂²×C₃	C₄⋊Q₈	1	2	yes	—
116	C₂²×C₃	C₂²×D₄	1	1	yes	characteristic
117	C₃×C₄	—	2	2	no	—
118	C₃×C₄	—	2	2	no	—
119	C₃×C₄	—	4	4	no	—
120	C₃×C₄	—	4	2	no	—
121	C₃×C₄	—	4	2	no	—
122	C₃×C₄	C₄⋊Q₈	1	2	yes	—
123	C₃×C₄	C₂²×Q₈	1	1	yes	characteristic
124	C₃×C₄	C₂²×Q₈	1	1	yes	characteristic
125	C₂³	C₂×C₃×D₄	1	1	yes	characteristic
126	Q₈	—	8	4	no	—
127	C₂×C₄	—	2	2	no	—
128	C₂×C₄	—	2	2	no	—
129	C₂×C₄	—	2	4	no	—
130	C₂×C₄	—	8	2	no	—
131	C₂×C₄	C₂×C₃×D₄	1	2	yes	—
132	C₂×C₄	C₂×C₃×D₄	1	2	yes	—
133	C₂×C₄	C₂×C₃×D₄	1	1	yes	characteristic
134	C₂×C₄	C₂×C₃×Q₈	1	2	yes	—
135	C₂×C₄	C₂×C₃×Q₈	1	2	yes	—
136	C₂×C₄	C₂⁴×C₃	1	1	yes	characteristic
137	C₈	C₂×C₃×Q₈	1	2	yes	—
138	C₈	C₂×C₃×Q₈	1	2	yes	—
139	C₈	C₂×C₃×Q₈	1	4	yes	—
140	C₂×C₃	—	2	2	no	—
141	C₂×C₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
142	C₂×C₃	#576	1	1	yes	characteristic
143	C₂×C₃	#578	1	1	yes	characteristic
144	C₂²	—	2	1	no	—
145	C₂²	—	2	1	no	—
146	C₂²	C₃×C₄⋊Q₈	1	2	yes	—
147	C₂²	C₂²×C₃×D₄	1	1	yes	characteristic
148	C₄	—	2	2	no	—
149	C₄	—	2	2	no	—
150	C₄	—	4	4	no	—
151	C₄	—	4	2	no	—
152	C₄	—	4	2	no	—
153	C₄	C₃×C₄⋊Q₈	1	2	yes	—
154	C₄	C₂²×C₃×Q₈	1	1	yes	characteristic
155	C₄	C₂²×C₃×Q₈	1	1	yes	characteristic
156	C₃	#3164	1	1	yes	characteristic, direct product factor, Sylow
157	C₂	—	2	2	no	—
158	C₂	C₂×C₃×C₄⋊Q₈	1	1	yes	characteristic
159	C₂	C₃×#576	1	1	yes	characteristic
160	C₂	C₃×#578	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₃
prime radical	C₂³×C₃	C₂×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃ ◁ C₃×#117 ◁ G

Complements

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

N	H ≅ G/N	type
2. C₃×#501	C₂	semidirect
3. C₃×#500	C₂	semidirect
4. C₃×#499	C₂	semidirect
9. #3164	C₃	direct (N × H)
30. #501	C₂×C₃	semidirect
31. #500	C₂×C₃	semidirect
32. #499	C₂×C₃	semidirect
40. C₃×M₁₆	Q₈	semidirect
41. C₂×C₃×C₈	Q₈	semidirect
95. M₁₆	C₃×Q₈	semidirect
96. C₂×C₈	C₃×Q₈	semidirect
156. C₃	#3164	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×#544	2
C₃×#501	2
C₃×#500	2
C₃×#499	2
C₃×#428	2
C₃×#427	2
C₂×C₃×#107	2
C₃×#405	2
#3164	3

Subgroups — C3×#3164 #85028

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#3164 #85028