Subgroups — C₃×#135758 #72225814

Order 1536.

604 conjugacy classes of subgroups.

The subgroup lattice has 180 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#12903	C₂	1	4	yes	maximal
2	C₃×C₈×M₃₂	C₂	1	1	yes	maximal, characteristic
3	C₃×#8822	C₂	1	1	yes	maximal, characteristic
4	C₃×#8813	C₂	1	1	yes	maximal, characteristic
5	#135758	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
6	C₂×C₃×#365	—	2	8	no	—
7	C₃×C₄×M₃₂	C₂²	1	2	yes	—
8	C₃×#1848	C₂²	1	2	yes	—
9	C₃×#1844	—	2	2	no	—
10	C₃×#1844	C₂²	1	2	yes	—
11	C₃×#1566	—	2	2	no	—
12	C₂×C₃×C₈²	C₂²	1	1	yes	characteristic
13	C₃×#1311	C₄	1	2	yes	complemented
14	C₃×C₈×C₁₆	C₄	1	2	yes	complemented
15	#12903	C₂×C₃	1	4	yes	—
16	C₈×M₃₂	C₂×C₃	1	1	yes	characteristic
17	#8822	C₂×C₃	1	1	yes	characteristic
18	#8813	C₂×C₃	1	1	yes	characteristic
19	C₂×C₃×M₃₂	D₄	1	4	yes	—
20	C₂×C₃×#108	—	2	4	no	—
21	C₂×C₃×#106	—	2	4	no	—
22	C₂×C₃×#106	—	4	4	no	—
23	C₃×#423	—	2	2	no	—
24	C₂×C₃×C₄×C₈	D₄	1	2	yes	—
25	C₂×C₃×C₄×C₈	C₂³	1	1	yes	characteristic
26	C₃×#384	—	2	2	no	—
27	C₃×#365	—	4	16	no	—
28	C₃×#346	Q₈	1	4	yes	—
29	C₃×C₄×C₁₆	C₂×C₄	1	4	yes	—
30	C₃×#334	—	4	4	no	—
31	C₃×C₈²	C₂×C₄	1	1	yes	characteristic
32	C₃×C₈²	C₂×C₄	1	1	yes	characteristic
33	C₃×C₈²	Q₈	1	2	yes	—
34	C₂×#365	—	2	8	no	—
35	C₄×M₃₂	C₂²×C₃	1	2	yes	—
36	#1848	C₂²×C₃	1	2	yes	—
37	#1844	—	2	2	no	—
38	#1844	C₂²×C₃	1	2	yes	—
39	#1566	—	2	2	no	—
40	C₂×C₈²	C₂²×C₃	1	1	yes	characteristic
41	#1311	C₃×C₄	1	2	yes	complemented
42	C₈×C₁₆	C₃×C₄	1	2	yes	complemented
43	C₂×C₃×M₁₆	—	4	2	no	—
44	C₂²×C₃×C₈	SD₁₆	1	4	yes	—
45	C₂²×C₃×C₈	C₂×D₄	1	2	yes	—
46	C₂×C₃×C₄⋊C₄	—	4	2	no	—
47	C₂×C₃×C₄⋊C₄	—	4	4	no	—
48	C₂×C₃×C₄²	C₂×D₄	1	1	yes	characteristic
49	C₃×M₃₂	SD₁₆	1	16	yes	—
50	C₂×C₃×C₁₆	C₄⋊C₄	1	8	yes	—
51	C₃×#108	—	4	8	no	—
52	C₃×#106	—	4	8	no	—
53	C₃×#106	—	8	8	no	—
54	C₃×C₄⋊C₈	—	4	4	no	—
55	C₃×C₄×C₈	C₄⋊C₄	1	2	yes	—
56	C₃×C₄×C₈	C₄⋊C₄	1	2	yes	—
57	C₃×C₄×C₈	SD₁₆	1	4	yes	—
58	C₃×C₄×C₈	C₂²×C₄	1	1	yes	characteristic
59	C₃×C₄×C₈	C₂×Q₈	1	1	yes	characteristic
60	C₃×C₄×C₈	C₂×Q₈	1	2	yes	—
61	C₂×M₃₂	C₃×D₄	1	4	yes	—
62	C₂×#108	—	2	4	no	—
63	C₂×#106	—	2	4	no	—
64	C₂×#106	—	4	4	no	—
65	#423	—	2	2	no	—
66	C₂×C₄×C₈	C₃×D₄	1	2	yes	—
67	C₂×C₄×C₈	C₂³×C₃	1	1	yes	characteristic
68	#384	—	2	2	no	—
69	#365	—	4	16	no	—
70	#346	C₃×Q₈	1	4	yes	—
71	C₄×C₁₆	C₂×C₃×C₄	1	4	yes	—
72	#334	—	4	4	no	—
73	C₈²	C₂×C₃×C₄	1	1	yes	characteristic
74	C₈²	C₂×C₃×C₄	1	1	yes	characteristic
75	C₈²	C₃×Q₈	1	2	yes	—
76	C₂²×C₃×C₄	—	8	2	no	—
77	C₂²×C₃×C₄	#127	1	1	yes	characteristic
78	C₂²×C₃×C₄	C₂×SD₁₆	1	2	yes	—
79	C₃×M₁₆	—	8	4	no	—
80	C₂×C₃×C₈	—	2	8	no	—
81	C₂×C₃×C₈	—	8	2	no	—
82	C₂×C₃×C₈	#106	1	4	yes	—
83	C₂×C₃×C₈	#106	1	4	yes	—
84	C₂×C₃×C₈	C₂×C₄⋊C₄	1	2	yes	—
85	C₂×C₃×C₈	C₄⋊Q₈	1	2	yes	—
86	C₂×C₃×C₈	C₂×SD₁₆	1	8	yes	—
87	C₃×C₄⋊C₄	—	8	8	no	—
88	C₃×C₄⋊C₄	—	8	4	no	—
89	C₃×C₄²	C₂×C₄⋊C₄	1	1	yes	characteristic
90	C₃×C₄²	C₄⋊Q₈	1	1	yes	characteristic
91	C₃×C₄²	C₂×SD₁₆	1	2	yes	—
92	C₃×C₁₆	#106	1	16	yes	complemented
93	C₂×M₁₆	—	4	2	no	—
94	C₂²×C₈	C₃×SD₁₆	1	4	yes	—
95	C₂²×C₈	C₂×C₃×D₄	1	2	yes	—
96	C₂×C₄⋊C₄	—	4	2	no	—
97	C₂×C₄⋊C₄	—	4	4	no	—
98	C₂×C₄²	C₂×C₃×D₄	1	1	yes	characteristic
99	M₃₂	C₃×SD₁₆	1	16	yes	—
100	C₂×C₁₆	C₃×C₄⋊C₄	1	8	yes	—
101	#108	—	4	8	no	—
102	#106	—	4	8	no	—
103	#106	—	8	8	no	—
104	C₄⋊C₈	—	4	4	no	—
105	C₄×C₈	C₃×C₄⋊C₄	1	2	yes	—
106	C₄×C₈	C₃×C₄⋊C₄	1	2	yes	—
107	C₄×C₈	C₃×SD₁₆	1	4	yes	—
108	C₄×C₈	C₂²×C₃×C₄	1	1	yes	characteristic
109	C₄×C₈	C₂×C₃×Q₈	1	1	yes	characteristic
110	C₄×C₈	C₂×C₃×Q₈	1	2	yes	—
111	C₂³×C₃	#492	1	1	yes	characteristic
112	C₂×C₃×C₄	—	2	4	no	—
113	C₂×C₃×C₄	—	16	4	no	—
114	C₂×C₃×C₄	—	16	2	no	—
115	C₂×C₃×C₄	#384	1	1	yes	characteristic
116	C₂×C₃×C₄	C₂×#106	1	2	yes	—
117	C₂×C₃×C₄	#492	1	4	yes	—
118	C₂×C₃×C₄	#492	1	1	yes	characteristic
119	C₂×C₃×C₄	#498	1	2	yes	—
120	C₃×C₈	—	2	4	no	—
121	C₃×C₈	—	16	2	no	—
122	C₃×C₈	#365	1	8	yes	complemented
123	C₃×C₈	C₂×#106	1	4	yes	—
124	C₃×C₈	#498	1	4	yes	—
125	C₂²×C₄	—	8	2	no	—
126	C₂²×C₄	C₃×#127	1	1	yes	characteristic
127	C₂²×C₄	C₂×C₃×SD₁₆	1	2	yes	—
128	M₁₆	—	8	4	no	—
129	C₂×C₈	—	2	8	no	—
130	C₂×C₈	—	8	2	no	—
131	C₂×C₈	C₃×#106	1	4	yes	—
132	C₂×C₈	C₃×#106	1	4	yes	—
133	C₂×C₈	C₂×C₃×C₄⋊C₄	1	2	yes	—
134	C₂×C₈	C₃×C₄⋊Q₈	1	2	yes	—
135	C₂×C₈	C₂×C₃×SD₁₆	1	8	yes	—
136	C₄⋊C₄	—	8	8	no	—
137	C₄⋊C₄	—	8	4	no	—
138	C₄²	C₂×C₃×C₄⋊C₄	1	1	yes	characteristic
139	C₄²	C₃×C₄⋊Q₈	1	1	yes	characteristic
140	C₄²	C₂×C₃×SD₁₆	1	2	yes	—
141	C₁₆	C₃×#106	1	16	yes	complemented
142	C₂²×C₃	—	2	2	no	—
143	C₂²×C₃	#1705	1	2	yes	—
144	C₂²×C₃	#1844	1	1	yes	characteristic
145	C₃×C₄	—	2	2	no	—
146	C₃×C₄	—	32	2	no	—
147	C₃×C₄	#1705	1	2	yes	—
148	C₃×C₄	#1844	1	2	yes	—
149	C₃×C₄	C₂×#365	1	4	yes	—
150	C₂³	C₃×#492	1	1	yes	characteristic
151	C₂×C₄	—	2	4	no	—
152	C₂×C₄	—	16	4	no	—
153	C₂×C₄	—	16	2	no	—
154	C₂×C₄	C₃×#384	1	1	yes	characteristic
155	C₂×C₄	C₂×C₃×#106	1	2	yes	—
156	C₂×C₄	C₃×#492	1	4	yes	—
157	C₂×C₄	C₃×#492	1	1	yes	characteristic
158	C₂×C₄	C₃×#498	1	2	yes	—
159	C₈	—	2	4	no	—
160	C₈	—	16	2	no	—
161	C₈	C₃×#365	1	8	yes	complemented
162	C₈	C₂×C₃×#106	1	4	yes	—
163	C₈	C₃×#498	1	4	yes	—
164	C₂×C₃	—	2	2	no	—
165	C₂×C₃	#8813	1	1	yes	characteristic
166	C₂×C₃	#12903	1	2	yes	—
167	C₂²	—	2	2	no	—
168	C₂²	C₃×#1705	1	2	yes	—
169	C₂²	C₃×#1844	1	1	yes	characteristic
170	C₄	—	2	2	no	—
171	C₄	—	32	2	no	—
172	C₄	C₃×#1705	1	2	yes	—
173	C₄	C₃×#1844	1	2	yes	—
174	C₄	C₂×C₃×#365	1	4	yes	—
175	C₃	#135758	1	1	yes	characteristic, direct product factor, Sylow
176	C₂	—	2	2	no	—
177	C₂	C₃×#8813	1	1	yes	characteristic
178	C₂	C₃×#12903	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₃	#1844
commutator	C₄×C₈	C₂²×C₃×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³×C₃
prime radical	C₂³×C₃	#492
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃ ◁ C₂×C₃×C₄² ◁ C₂×C₃×C₈² ◁ G

Complements

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

N	H ≅ G/N	type
5. #135758	C₃	direct (N × H)
13. C₃×#1311	C₄	semidirect
14. C₃×C₈×C₁₆	C₄	semidirect
41. #1311	C₃×C₄	semidirect
42. C₈×C₁₆	C₃×C₄	semidirect
92. C₃×C₁₆	#106	semidirect
122. C₃×C₈	#365	semidirect
141. C₁₆	C₃×#106	semidirect
161. C₈	C₃×#365	semidirect
175. C₃	#135758	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×#12903	2
C₃×C₈×M₃₂	2
C₃×#8822	2
C₃×#8813	2
#135758	3

Subgroups — C3×#135758 #72225814

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#135758 #72225814