Subgroups — C₃×#3176 #85040

Order 384.

386 conjugacy classes of subgroups.

The subgroup lattice has 170 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#540	C₂	1	2	yes	maximal, complemented
2	C₃×#499	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×#493	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×#490	C₂	1	2	yes	maximal, complemented
5	C₃×#469	C₂	1	2	yes	maximal, complemented
6	C₃×#466	C₂	1	2	yes	maximal, complemented
7	C₃×#442	C₂	1	2	yes	maximal, complemented
8	C₃×C₄×D₈	C₂	1	2	yes	maximal, complemented
9	C₃×#405	C₂	1	1	yes	maximal, characteristic, complemented
10	#3176	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
11	C₂×C₃×D₈	C₂²	1	2	yes	—
12	C₂×C₃×D₈	C₂²	1	2	yes	—
13	C₂×C₃×M₁₆	C₂²	1	1	yes	characteristic
14	C₂²×C₃×C₈	C₂²	1	1	yes	characteristic
15	C₃×#127	C₂²	1	2	yes	—
16	C₃×#125	C₂²	1	2	yes	—
17	C₃×#123	—	2	4	no	—
18	C₃×#121	—	2	4	no	—
19	C₃×#121	C₂²	1	4	yes	—
20	C₃×C₄×D₄	C₂²	1	4	yes	—
21	C₃×#117	C₂²	1	1	yes	characteristic
22	C₃×#107	C₂²	1	2	yes	—
23	C₃×#106	C₂²	1	2	yes	—
24	C₃×#102	C₂²	1	4	yes	—
25	C₃×#102	C₂²	1	4	yes	—
26	C₃×#97	C₂²	1	1	yes	characteristic
27	C₃×#97	C₂²	1	1	yes	characteristic
28	C₃×C₄×C₈	C₂²	1	1	yes	characteristic
29	C₃×C₄×C₈	C₂²	1	1	yes	characteristic
30	#540	C₂×C₃	1	2	yes	complemented
31	#499	C₂×C₃	1	1	yes	characteristic, complemented
32	#493	C₂×C₃	1	1	yes	characteristic, complemented
33	#490	C₂×C₃	1	2	yes	complemented
34	#469	C₂×C₃	1	2	yes	complemented
35	#466	C₂×C₃	1	2	yes	complemented
36	#442	C₂×C₃	1	2	yes	complemented
37	C₄×D₈	C₂×C₃	1	2	yes	complemented
38	#405	C₂×C₃	1	1	yes	characteristic, complemented
39	C₂×C₃×D₄	—	2	4	no	—
40	C₂×C₃×D₄	—	2	4	no	—
41	C₂×C₃×D₄	C₂³	1	4	yes	—
42	C₂²×C₃×C₄	—	2	4	no	—
43	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
44	C₃×D₈	—	2	4	no	—
45	C₃×D₈	—	2	4	no	—
46	C₃×M₁₆	—	2	2	no	—
47	C₂×C₃×C₈	—	2	2	no	—
48	C₂×C₃×C₈	C₂³	1	1	yes	characteristic
49	C₂×C₃×C₈	C₂³	1	1	yes	characteristic
50	C₂×C₃×C₈	C₂³	1	1	yes	characteristic
51	C₂×C₃×C₈	C₂³	1	1	yes	characteristic
52	C₃×C₄⋊C₄	—	2	4	no	—
53	C₃×C₄⋊C₄	D₄	1	2	yes	—
54	C₃×C₄⋊C₄	C₂³	1	4	yes	—
55	C₃×C₂²⋊C₄	—	2	4	no	—
56	C₃×C₂²⋊C₄	—	2	4	no	—
57	C₃×C₂²⋊C₄	D₄	1	2	yes	—
58	C₃×C₄²	C₂³	1	1	yes	characteristic
59	C₃×C₄²	C₂³	1	1	yes	characteristic
60	C₂×D₈	C₂²×C₃	1	2	yes	—
61	C₂×D₈	C₂²×C₃	1	2	yes	—
62	C₂×M₁₆	C₂²×C₃	1	1	yes	characteristic
63	C₂²×C₈	C₂²×C₃	1	1	yes	characteristic
64	#127	C₂²×C₃	1	2	yes	—
65	#125	C₂²×C₃	1	2	yes	—
66	#123	—	2	4	no	—
67	#121	—	2	4	no	—
68	#121	C₂²×C₃	1	4	yes	—
69	C₄×D₄	C₂²×C₃	1	4	yes	—
70	#117	C₂²×C₃	1	1	yes	characteristic
71	#107	C₂²×C₃	1	2	yes	—
72	#106	C₂²×C₃	1	2	yes	—
73	#102	C₂²×C₃	1	4	yes	—
74	#102	C₂²×C₃	1	4	yes	—
75	#97	C₂²×C₃	1	1	yes	characteristic
76	#97	C₂²×C₃	1	1	yes	characteristic
77	C₄×C₈	C₂²×C₃	1	1	yes	characteristic
78	C₄×C₈	C₂²×C₃	1	1	yes	characteristic
79	C₂³×C₃	—	2	4	no	—
80	C₂³×C₃	C₂×D₄	1	1	yes	characteristic
81	C₃×D₄	—	2	4	no	—
82	C₃×D₄	—	2	4	no	—
83	C₃×D₄	—	8	4	no	—
84	C₃×D₄	—	8	4	no	—
85	C₂×C₃×C₄	—	2	1	no	—
86	C₂×C₃×C₄	—	2	1	no	—
87	C₂×C₃×C₄	—	2	4	no	—
88	C₂×C₃×C₄	—	8	4	no	—
89	C₂×C₃×C₄	C₂×D₄	1	2	yes	—
90	C₂×C₃×C₄	C₂×D₄	1	2	yes	—
91	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
92	C₂×C₃×C₄	C₂⁴	1	1	yes	characteristic
93	C₃×C₈	—	2	1	no	—
94	C₃×C₈	—	2	1	no	—
95	C₃×C₈	#41	1	2	yes	—
96	C₃×C₈	#41	1	2	yes	—
97	C₂×D₄	—	2	4	no	—
98	C₂×D₄	—	2	4	no	—
99	C₂×D₄	C₂³×C₃	1	4	yes	—
100	C₂²×C₄	—	2	4	no	—
101	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
102	D₈	—	2	4	no	—
103	D₈	—	2	4	no	—
104	M₁₆	—	2	2	no	—
105	C₂×C₈	—	2	2	no	—
106	C₂×C₈	C₂³×C₃	1	1	yes	characteristic
107	C₂×C₈	C₂³×C₃	1	1	yes	characteristic
108	C₂×C₈	C₂³×C₃	1	1	yes	characteristic
109	C₂×C₈	C₂³×C₃	1	1	yes	characteristic
110	C₄⋊C₄	—	2	4	no	—
111	C₄⋊C₄	C₃×D₄	1	2	yes	—
112	C₄⋊C₄	C₂³×C₃	1	4	yes	—
113	C₂²⋊C₄	—	2	4	no	—
114	C₂²⋊C₄	—	2	4	no	—
115	C₂²⋊C₄	C₃×D₄	1	2	yes	—
116	C₄²	C₂³×C₃	1	1	yes	characteristic
117	C₄²	C₂³×C₃	1	1	yes	characteristic
118	C₂²×C₃	—	2	1	no	—
119	C₂²×C₃	—	2	2	no	—
120	C₂²×C₃	—	4	4	no	—
121	C₂²×C₃	—	4	4	no	—
122	C₂²×C₃	—	4	4	no	—
123	C₂²×C₃	C₂²×D₄	1	1	yes	characteristic
124	C₃×C₄	—	2	1	no	—
125	C₃×C₄	—	2	2	no	—
126	C₃×C₄	—	2	2	no	—
127	C₃×C₄	—	4	4	no	—
128	C₃×C₄	C₂×#41	1	1	yes	characteristic
129	C₃×C₄	C₂×#41	1	1	yes	characteristic
130	C₂³	—	2	4	no	—
131	C₂³	C₂×C₃×D₄	1	1	yes	characteristic
132	D₄	—	2	4	no	—
133	D₄	—	2	4	no	—
134	D₄	—	8	4	no	—
135	D₄	—	8	4	no	—
136	C₂×C₄	—	2	1	no	—
137	C₂×C₄	—	2	1	no	—
138	C₂×C₄	—	2	4	no	—
139	C₂×C₄	—	8	4	no	—
140	C₂×C₄	C₂×C₃×D₄	1	2	yes	—
141	C₂×C₄	C₂×C₃×D₄	1	2	yes	—
142	C₂×C₄	C₂×C₃×D₄	1	1	yes	characteristic
143	C₂×C₄	C₂⁴×C₃	1	1	yes	characteristic
144	C₈	—	2	1	no	—
145	C₈	—	2	1	no	—
146	C₈	C₃×#41	1	2	yes	—
147	C₈	C₃×#41	1	2	yes	—
148	C₂×C₃	—	4	1	no	—
149	C₂×C₃	—	8	4	no	—
150	C₂×C₃	#532	1	1	yes	characteristic
151	C₂×C₃	#576	1	2	yes	—
152	C₂²	—	2	2	no	—
153	C₂²	—	2	1	no	—
154	C₂²	—	4	4	no	—
155	C₂²	—	4	4	no	—
156	C₂²	—	4	4	no	—
157	C₂²	C₂²×C₃×D₄	1	1	yes	characteristic
158	C₄	—	2	2	no	—
159	C₄	—	2	2	no	—
160	C₄	—	2	1	no	—
161	C₄	—	4	4	no	—
162	C₄	C₂×C₃×#41	1	1	yes	characteristic
163	C₄	C₂×C₃×#41	1	1	yes	characteristic
164	C₃	#3176	1	1	yes	characteristic, direct product factor, Sylow
165	C₂	—	4	1	no	—
166	C₂	—	8	4	no	—
167	C₂	C₃×#532	1	1	yes	characteristic
168	C₂	C₃×#576	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₃	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₃
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
1. C₃×#540	C₂	semidirect
2. C₃×#499	C₂	semidirect
3. C₃×#493	C₂	semidirect
4. C₃×#490	C₂	semidirect
5. C₃×#469	C₂	semidirect
6. C₃×#466	C₂	semidirect
7. C₃×#442	C₂	semidirect
8. C₃×C₄×D₈	C₂	semidirect
9. C₃×#405	C₂	semidirect
10. #3176	C₃	direct (N × H)
30. #540	C₂×C₃	semidirect
31. #499	C₂×C₃	semidirect
32. #493	C₂×C₃	semidirect
33. #490	C₂×C₃	semidirect
34. #469	C₂×C₃	semidirect
35. #466	C₂×C₃	semidirect
36. #442	C₂×C₃	semidirect
37. C₄×D₈	C₂×C₃	semidirect
38. #405	C₂×C₃	semidirect
164. C₃	#3176	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×#540	2
C₃×#499	2
C₃×#493	2
C₃×#490	2
C₃×#469	2
C₃×#466	2
C₃×#442	2
C₃×C₄×D₈	2
C₃×#405	2
#3176	3

Subgroups — C3×#3176 #85040

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#3176 #85040