Subgroups — C₃×#3169 #85033

Order 384.

456 conjugacy classes of subgroups.

The subgroup lattice has 172 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#532	C₂	1	2	yes	maximal, complemented
2	C₃×#500	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×#493	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×#486	C₂	1	2	yes	maximal, complemented
5	C₃×#466	C₂	1	4	yes	maximal, complemented
6	C₃×C₄×D₈	C₂	1	4	yes	maximal, complemented
7	C₂×C₃×C₄×C₈	C₂	1	1	yes	maximal, characteristic, complemented
8	#3169	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
9	C₂×C₃×#41	—	2	2	no	—
10	C₂×C₃×D₈	C₂²	1	4	yes	—
11	C₂²×C₃×C₈	C₂²	1	2	yes	—
12	C₃×C₄⋊Q₈	C₂²	1	2	yes	—
13	C₃×#127	C₂²	1	2	yes	—
14	C₃×#124	—	2	2	no	—
15	C₃×#121	—	2	2	no	—
16	C₃×#121	C₂²	1	4	yes	—
17	C₃×C₄×D₄	—	2	2	no	—
18	C₃×C₄×D₄	C₂²	1	4	yes	—
19	C₂×C₃×C₄²	C₂²	1	1	yes	characteristic
20	C₃×#107	C₂²	1	4	yes	—
21	C₃×#102	C₂²	1	8	yes	—
22	C₃×C₄×C₈	C₂²	1	2	yes	—
23	C₃×C₄×C₈	C₂²	1	1	yes	characteristic
24	C₃×C₄×C₈	C₂²	1	1	yes	characteristic
25	#532	C₂×C₃	1	2	yes	complemented
26	#500	C₂×C₃	1	1	yes	characteristic, complemented
27	#493	C₂×C₃	1	1	yes	characteristic, complemented
28	#486	C₂×C₃	1	2	yes	complemented
29	#466	C₂×C₃	1	4	yes	complemented
30	C₄×D₈	C₂×C₃	1	4	yes	complemented
31	C₂×C₄×C₈	C₂×C₃	1	1	yes	characteristic, complemented
32	C₃×#41	—	4	8	no	—
33	C₂×C₃×Q₈	—	2	2	no	—
34	C₂×C₃×D₄	—	2	4	no	—
35	C₂×C₃×D₄	—	2	2	no	—
36	C₂×C₃×D₄	—	2	2	no	—
37	C₂×C₃×D₄	C₂³	1	4	yes	—
38	C₂²×C₃×C₄	—	2	4	no	—
39	C₂²×C₃×C₄	D₄	1	1	yes	characteristic
40	C₂²×C₃×C₄	D₄	1	1	yes	characteristic
41	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
42	C₃×D₈	—	2	8	no	—
43	C₂×C₃×C₈	—	2	4	no	—
44	C₂×C₃×C₈	C₂³	1	2	yes	—
45	C₂×C₃×C₈	C₂³	1	2	yes	—
46	C₃×C₄⋊C₄	—	2	2	no	—
47	C₃×C₄⋊C₄	C₂³	1	4	yes	—
48	C₃×C₂²⋊C₄	—	2	4	no	—
49	C₃×C₂²⋊C₄	—	2	4	no	—
50	C₃×C₄²	D₄	1	1	yes	characteristic
51	C₃×C₄²	D₄	1	1	yes	characteristic
52	C₃×C₄²	C₂³	1	1	yes	characteristic
53	C₃×C₄²	C₂³	1	1	yes	characteristic
54	C₂×#41	—	2	2	no	—
55	C₂×D₈	C₂²×C₃	1	4	yes	—
56	C₂²×C₈	C₂²×C₃	1	2	yes	—
57	C₄⋊Q₈	C₂²×C₃	1	2	yes	—
58	#127	C₂²×C₃	1	2	yes	—
59	#124	—	2	2	no	—
60	#121	—	2	2	no	—
61	#121	C₂²×C₃	1	4	yes	—
62	C₄×D₄	—	2	2	no	—
63	C₄×D₄	C₂²×C₃	1	4	yes	—
64	C₂×C₄²	C₂²×C₃	1	1	yes	characteristic
65	#107	C₂²×C₃	1	4	yes	—
66	#102	C₂²×C₃	1	8	yes	—
67	C₄×C₈	C₂²×C₃	1	2	yes	—
68	C₄×C₈	C₂²×C₃	1	1	yes	characteristic
69	C₄×C₈	C₂²×C₃	1	1	yes	characteristic
70	C₂³×C₃	—	2	4	no	—
71	C₂³×C₃	C₂×D₄	1	1	yes	characteristic
72	C₃×Q₈	—	4	4	no	—
73	C₃×D₄	—	2	8	no	—
74	C₃×D₄	—	4	8	no	—
75	C₃×D₄	—	4	4	no	—
76	C₃×D₄	—	4	4	no	—
77	C₂×C₃×C₄	—	2	2	no	—
78	C₂×C₃×C₄	—	2	2	no	—
79	C₂×C₃×C₄	—	2	4	no	—
80	C₂×C₃×C₄	—	4	8	no	—
81	C₂×C₃×C₄	D₈	1	4	yes	complemented
82	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
83	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
84	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
85	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
86	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
87	C₂×C₃×C₄	C₂⁴	1	1	yes	characteristic
88	C₃×C₈	—	2	2	no	—
89	C₃×C₈	#41	1	4	yes	complemented
90	#41	—	4	8	no	—
91	C₂×Q₈	—	2	2	no	—
92	C₂×D₄	—	2	4	no	—
93	C₂×D₄	—	2	2	no	—
94	C₂×D₄	—	2	2	no	—
95	C₂×D₄	C₂³×C₃	1	4	yes	—
96	C₂²×C₄	—	2	4	no	—
97	C₂²×C₄	C₃×D₄	1	1	yes	characteristic
98	C₂²×C₄	C₃×D₄	1	1	yes	characteristic
99	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
100	D₈	—	2	8	no	—
101	C₂×C₈	—	2	4	no	—
102	C₂×C₈	C₂³×C₃	1	2	yes	—
103	C₂×C₈	C₂³×C₃	1	2	yes	—
104	C₄⋊C₄	—	2	2	no	—
105	C₄⋊C₄	C₂³×C₃	1	4	yes	—
106	C₂²⋊C₄	—	2	4	no	—
107	C₂²⋊C₄	—	2	4	no	—
108	C₄²	C₃×D₄	1	1	yes	characteristic
109	C₄²	C₃×D₄	1	1	yes	characteristic
110	C₄²	C₂³×C₃	1	1	yes	characteristic
111	C₄²	C₂³×C₃	1	1	yes	characteristic
112	C₂²×C₃	—	2	1	no	—
113	C₂²×C₃	—	2	1	no	—
114	C₂²×C₃	—	4	4	no	—
115	C₂²×C₃	—	4	4	no	—
116	C₂²×C₃	—	4	4	no	—
117	C₂²×C₃	C₂×D₈	1	2	yes	—
118	C₂²×C₃	C₂²×D₄	1	1	yes	characteristic
119	C₃×C₄	—	2	1	no	—
120	C₃×C₄	—	2	1	no	—
121	C₃×C₄	—	4	4	no	—
122	C₃×C₄	C₂×D₈	1	2	yes	—
123	C₃×C₄	C₂×D₈	1	2	yes	—
124	C₃×C₄	#135	1	2	yes	—
125	C₃×C₄	C₂×#41	1	2	yes	—
126	C₂³	—	2	4	no	—
127	C₂³	C₂×C₃×D₄	1	1	yes	characteristic
128	Q₈	—	4	4	no	—
129	D₄	—	2	8	no	—
130	D₄	—	4	8	no	—
131	D₄	—	4	4	no	—
132	D₄	—	4	4	no	—
133	C₂×C₄	—	2	2	no	—
134	C₂×C₄	—	2	2	no	—
135	C₂×C₄	—	2	4	no	—
136	C₂×C₄	—	4	8	no	—
137	C₂×C₄	C₃×D₈	1	4	yes	complemented
138	C₂×C₄	C₂×C₃×D₄	1	1	yes	characteristic
139	C₂×C₄	C₂×C₃×D₄	1	1	yes	characteristic
140	C₂×C₄	C₂×C₃×D₄	1	1	yes	characteristic
141	C₂×C₄	C₂×C₃×D₄	1	1	yes	characteristic
142	C₂×C₄	C₂×C₃×D₄	1	1	yes	characteristic
143	C₂×C₄	C₂⁴×C₃	1	1	yes	characteristic
144	C₈	—	2	2	no	—
145	C₈	C₃×#41	1	4	yes	complemented
146	C₂×C₃	—	2	2	no	—
147	C₂×C₃	—	8	4	no	—
148	C₂×C₃	#532	1	1	yes	characteristic
149	C₂×C₃	C₂²×D₈	1	1	yes	characteristic
150	C₂×C₃	C₂×#135	1	1	yes	characteristic
151	C₂²	—	2	1	no	—
152	C₂²	—	2	1	no	—
153	C₂²	—	4	4	no	—
154	C₂²	—	4	4	no	—
155	C₂²	—	4	4	no	—
156	C₂²	C₂×C₃×D₈	1	2	yes	—
157	C₂²	C₂²×C₃×D₄	1	1	yes	characteristic
158	C₄	—	2	1	no	—
159	C₄	—	2	1	no	—
160	C₄	—	4	4	no	—
161	C₄	C₂×C₃×D₈	1	2	yes	—
162	C₄	C₂×C₃×D₈	1	2	yes	—
163	C₄	C₃×#135	1	2	yes	—
164	C₄	C₂×C₃×#41	1	2	yes	—
165	C₃	#3169	1	1	yes	characteristic, direct product factor, Sylow
166	C₂	—	2	2	no	—
167	C₂	—	8	4	no	—
168	C₂	C₃×#532	1	1	yes	characteristic
169	C₂	C₂²×C₃×D₈	1	1	yes	characteristic
170	C₂	C₂×C₃×#135	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
1. C₃×#532	C₂	semidirect
2. C₃×#500	C₂	semidirect
3. C₃×#493	C₂	semidirect
4. C₃×#486	C₂	semidirect
5. C₃×#466	C₂	semidirect
6. C₃×C₄×D₈	C₂	semidirect
7. C₂×C₃×C₄×C₈	C₂	semidirect
8. #3169	C₃	direct (N × H)
25. #532	C₂×C₃	semidirect
26. #500	C₂×C₃	semidirect
27. #493	C₂×C₃	semidirect
28. #486	C₂×C₃	semidirect
29. #466	C₂×C₃	semidirect
30. C₄×D₈	C₂×C₃	semidirect
31. C₂×C₄×C₈	C₂×C₃	semidirect
81. C₂×C₃×C₄	D₈	semidirect
89. C₃×C₈	#41	semidirect
137. C₂×C₄	C₃×D₈	semidirect
145. C₈	C₃×#41	semidirect
165. C₃	#3169	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×#532	2
C₃×#500	2
C₃×#493	2
C₃×#486	2
C₃×#466	2
C₃×C₄×D₈	2
C₂×C₃×C₄×C₈	2
#3169	3

Subgroups — C3×#3169 #85033

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#3169 #85033