Subgroups — C₃×#10589418 #13145976

Order 1728.

844 conjugacy classes of subgroups.

The subgroup lattice has 196 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃³×C₈	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×#3748	C₂	1	2	yes	maximal
3	#10589418	C₃	1	1	yes	maximal, characteristic, direct product factor
4	C₃×#4552	—	3	4	no	maximal
5	C₂²×C₃³×C₄	C₂²	1	1	yes	characteristic
6	C₂×C₃³×C₈	—	2	2	no	—
7	C₂×C₃³×C₈	C₄	1	2	yes	—
8	C₃×#3748	—	2	2	no	—
9	C₃×#3748	C₄	1	4	yes	—
10	C₂×C₃×C₃⋊(C₃⋊C₈)	—	2	2	no	—
11	C₂²×C₃²×C₈	—	2	4	no	—
12	C₂²×C₃²×C₈	D₃	1	4	yes	—
13	C₂²×C₃²×C₈	C₂×C₃	1	1	yes	characteristic
14	C₂×#3748	C₂×C₃	1	2	yes	—
15	C₂×C₃×#208	—	3	8	no	—
16	C₂³×C₃³	Q₈	1	1	yes	characteristic
17	C₂×C₃³×C₄	C₂×C₄	1	2	yes	—
18	C₂×C₃³×C₄	C₂×C₄	1	1	yes	characteristic
19	C₂×C₃³×C₄	D₄	1	1	yes	characteristic
20	C₂×C₃³×C₄	D₄	1	2	yes	—
21	C₃³×C₈	—	2	2	no	—
22	C₃×C₃⋊(C₃⋊C₈)	—	2	4	no	—
23	C₃×#341	—	9	1	no	—
24	#4552	—	3	4	no	—
25	C₂²×C₃²×C₄	—	2	4	no	—
26	C₂²×C₃²×C₄	C₂×D₃	1	4	yes	—
27	C₂²×C₃²×C₄	C₂²×C₃	1	1	yes	characteristic
28	C₂×C₃²×C₈	—	2	16	no	—
29	C₂×C₃²×C₈	—	2	8	no	—
30	C₂×C₃²×C₈	—	2	8	no	—
31	C₂×C₃²×C₈	—	2	2	no	—
32	C₂×C₃²×C₈	Dic₃	1	8	yes	—
33	C₂×C₃²×C₈	C₃×C₄	1	2	yes	—
34	#3748	—	2	2	no	—
35	#3748	C₃×C₄	1	4	yes	—
36	C₂×C₃⋊(C₃⋊C₈)	—	2	2	no	—
37	C₃×#208	—	3	16	no	—
38	C₃×#208	—	6	8	no	—
39	C₂×C₃×C₃⋊C₈	—	6	8	no	—
40	C₂²×C₃³	—	2	2	no	—
41	C₂²×C₃³	C₄⋊C₄	1	2	yes	—
42	C₂²×C₃³	C₄⋊C₄	1	1	yes	characteristic
43	C₃³×C₄	C₄²	1	1	yes	characteristic
44	C₃³×C₄	C₂²⋊C₄	1	2	yes	—
45	C₃³×C₄	C₂²⋊C₄	1	1	yes	characteristic
46	C₂×C₃×M₁₆	—	9	2	no	—
47	C₂²×C₃×C₈	—	2	4	no	—
48	C₂²×C₃×C₈	C₃×D₃	1	4	yes	—
49	C₂²×C₃×C₈	C₃⋊D₃	1	1	yes	characteristic
50	C₂×#208	—	3	8	no	—
51	C₂³×C₃²	—	2	4	no	—
52	C₂³×C₃²	Dic₆	1	4	yes	—
53	C₂³×C₃²	C₃×Q₈	1	1	yes	characteristic
54	C₂×C₃²×C₄	—	2	8	no	—
55	C₂×C₃²×C₄	—	2	4	no	—
56	C₂×C₃²×C₄	—	2	4	no	—
57	C₂×C₃²×C₄	—	2	8	no	—
58	C₂×C₃²×C₄	C₄×D₃	1	8	yes	—
59	C₂×C₃²×C₄	D₁₂	1	4	yes	—
60	C₂×C₃²×C₄	C₂×Dic₃	1	4	yes	—
61	C₂×C₃²×C₄	#67	1	8	yes	—
62	C₂×C₃²×C₄	C₂×C₃×C₄	1	2	yes	—
63	C₂×C₃²×C₄	C₂×C₃×C₄	1	1	yes	characteristic
64	C₂×C₃²×C₄	C₃×D₄	1	1	yes	characteristic
65	C₂×C₃²×C₄	C₃×D₄	1	2	yes	—
66	C₃²×C₈	—	2	16	no	—
67	C₃²×C₈	—	2	8	no	—
68	C₃²×C₈	—	2	2	no	—
69	C₃⋊(C₃⋊C₈)	—	2	4	no	—
70	C₃×C₃⋊C₈	—	6	16	no	—
71	#341	—	9	1	no	Sylow
72	C₂×C₃³	—	2	2	no	—
73	C₂×C₃³	#95	1	1	yes	characteristic
74	C₂×C₃³	#108	1	2	yes	—
75	C₂²×C₃×C₄	—	2	4	no	—
76	C₂²×C₃×C₄	C₂×C₃×D₃	1	4	yes	—
77	C₂²×C₃×C₄	C₂×C₃⋊D₃	1	1	yes	characteristic
78	C₃×M₁₆	—	9	4	no	—
79	C₃×M₁₆	—	18	2	no	—
80	C₂×C₃×C₈	—	2	16	no	—
81	C₂×C₃×C₈	—	2	8	no	—
82	C₂×C₃×C₈	—	2	2	no	—
83	C₂×C₃×C₈	—	2	8	no	—
84	C₂×C₃×C₈	—	18	2	no	—
85	C₂×C₃×C₈	C₃×Dic₃	1	8	yes	—
86	C₂×C₃×C₈	C₃⋊Dic₃	1	2	yes	—
87	#208	—	3	16	no	—
88	#208	—	6	8	no	—
89	C₂×C₃⋊C₈	—	6	8	no	—
90	C₂²×C₃²	—	2	16	no	—
91	C₂²×C₃²	—	2	8	no	—
92	C₂²×C₃²	—	2	2	no	—
93	C₂²×C₃²	—	2	8	no	—
94	C₂²×C₃²	—	2	4	no	—
95	C₂²×C₃²	Dic₃⋊C₄	1	8	yes	—
96	C₂²×C₃²	C₄⋊Dic₃	1	4	yes	—
97	C₂²×C₃²	C₃×C₄⋊C₄	1	2	yes	—
98	C₂²×C₃²	C₃×C₄⋊C₄	1	1	yes	characteristic
99	C₃²×C₄	—	2	4	no	—
100	C₃²×C₄	—	2	8	no	—
101	C₃²×C₄	—	2	4	no	—
102	C₃²×C₄	C₄×Dic₃	1	4	yes	—
103	C₃²×C₄	(C₂×D₃)⋊C₄	1	8	yes	—
104	C₃²×C₄	#217	1	4	yes	—
105	C₃²×C₄	C₃×C₄²	1	1	yes	characteristic
106	C₃²×C₄	C₃×C₂²⋊C₄	1	2	yes	—
107	C₃²×C₄	C₃×C₂²⋊C₄	1	1	yes	characteristic
108	C₂×M₁₆	—	9	2	no	—
109	C₂²×C₈	C₃×C₃⋊D₃	1	1	yes	characteristic
110	C₃³	#341	1	1	yes	characteristic, complemented, Sylow
111	C₂³×C₃	—	2	4	no	—
112	C₂³×C₃	C₃×Dic₆	1	4	yes	—
113	C₂³×C₃	#634	1	1	yes	characteristic
114	C₂×C₃×C₄	—	2	8	no	—
115	C₂×C₃×C₄	—	2	4	no	—
116	C₂×C₃×C₄	—	2	4	no	—
117	C₂×C₃×C₄	—	2	8	no	—
118	C₂×C₃×C₄	C₃×C₄×D₃	1	8	yes	—
119	C₂×C₃×C₄	C₃×D₁₂	1	4	yes	—
120	C₂×C₃×C₄	C₂×C₃×Dic₃	1	4	yes	—
121	C₂×C₃×C₄	C₃×#67	1	8	yes	—
122	C₂×C₃×C₄	C₄×C₃⋊D₃	1	2	yes	—
123	C₂×C₃×C₄	C₄⋊(C₃⋊D₃)	1	1	yes	characteristic
124	C₂×C₃×C₄	C₂×C₃⋊Dic₃	1	1	yes	characteristic
125	C₂×C₃×C₄	#638	1	2	yes	—
126	C₃×C₈	—	2	16	no	—
127	C₃×C₈	—	2	8	no	—
128	C₃×C₈	—	2	2	no	—
129	C₃×C₈	—	18	4	no	—
130	C₃⋊C₈	—	6	16	no	—
131	C₂×C₃²	—	2	16	no	—
132	C₂×C₃²	—	2	8	no	—
133	C₂×C₃²	—	2	2	no	—
134	C₂×C₃²	—	2	8	no	—
135	C₂×C₃²	—	2	4	no	—
136	C₂×C₃²	#819	1	8	yes	—
137	C₂×C₃²	C₃⋊#95	1	4	yes	—
138	C₂×C₃²	C₃×#95	1	1	yes	characteristic
139	C₂×C₃²	C₃×#108	1	2	yes	—
140	C₂²×C₄	C₂×C₃×C₃⋊D₃	1	1	yes	characteristic
141	M₁₆	—	9	4	no	—
142	M₁₆	—	18	2	no	—
143	C₂×C₈	—	2	2	no	—
144	C₂×C₈	—	18	2	no	—
145	C₂×C₈	C₃×C₃⋊Dic₃	1	2	yes	—
146	C₂²×C₃	—	2	16	no	—
147	C₂²×C₃	—	2	8	no	—
148	C₂²×C₃	—	2	2	no	—
149	C₂²×C₃	—	2	8	no	—
150	C₂²×C₃	—	2	4	no	—
151	C₂²×C₃	C₃×Dic₃⋊C₄	1	8	yes	—
152	C₂²×C₃	C₃×C₄⋊Dic₃	1	4	yes	—
153	C₂²×C₃	#3750	1	2	yes	—
154	C₂²×C₃	C₄⋊(C₃⋊Dic₃)	1	1	yes	characteristic
155	C₃×C₄	—	2	4	no	—
156	C₃×C₄	—	2	8	no	—
157	C₃×C₄	—	2	4	no	—
158	C₃×C₄	C₃×C₄×Dic₃	1	4	yes	—
159	C₃×C₄	C₃×(C₂×D₃)⋊C₄	1	8	yes	—
160	C₃×C₄	C₃×#217	1	4	yes	—
161	C₃×C₄	C₄×C₃⋊Dic₃	1	1	yes	characteristic
162	C₃×C₄	#3752	1	2	yes	—
163	C₃×C₄	#3757	1	1	yes	characteristic
164	C₃²	—	2	4	no	—
165	C₃²	#4552	1	4	yes	complemented
166	C₃²	C₃×#341	1	1	yes	characteristic, complemented
167	C₂³	C₃×#634	1	1	yes	characteristic
168	C₂×C₄	C₃×C₄×C₃⋊D₃	1	2	yes	—
169	C₂×C₄	C₃×C₄⋊(C₃⋊D₃)	1	1	yes	characteristic
170	C₂×C₄	C₂×C₃×C₃⋊Dic₃	1	1	yes	characteristic
171	C₂×C₄	C₃×#638	1	2	yes	—
172	C₈	—	2	2	no	—
173	C₈	—	18	4	no	—
174	C₂×C₃	—	2	16	no	—
175	C₂×C₃	—	2	8	no	—
176	C₂×C₃	—	2	2	no	—
177	C₂×C₃	—	2	8	no	—
178	C₂×C₃	—	2	4	no	—
179	C₂×C₃	C₃×#819	1	8	yes	—
180	C₂×C₃	C₃×C₃⋊#95	1	4	yes	—
181	C₂×C₃	#63646	1	2	yes	—
182	C₂×C₃	#63658	1	1	yes	characteristic
183	C₂²	—	2	2	no	—
184	C₂²	C₃×#3750	1	2	yes	—
185	C₂²	C₃×C₄⋊(C₃⋊Dic₃)	1	1	yes	characteristic
186	C₄	C₃×C₄×C₃⋊Dic₃	1	1	yes	characteristic
187	C₄	C₃×#3752	1	2	yes	—
188	C₄	C₃×#3757	1	1	yes	characteristic
189	C₃	—	2	4	no	—
190	C₃	C₃×#4552	1	4	yes	complemented
191	C₃	#10589418	1	1	yes	characteristic, direct product factor
192	C₂	—	2	2	no	—
193	C₂	C₃×#63646	1	2	yes	—
194	C₂	C₃×#63658	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₃×C₄	C₄⋊(C₃⋊D₃)
commutator	C₃²×C₄	C₃×C₄²
Frattini	C₂²×C₄	C₂×C₃×C₃⋊D₃
Fitting	C₂²×C₃³×C₈	C₂
socle	C₂²×C₃³	C₄⋊C₄
prime radical	C₂³×C₃³	Q₈
cosocle	C₂²×C₃²×C₄	C₂²×C₃
hypercenter	C₂²×C₃×C₈	C₃⋊D₃
hypocenter	C₃²	C₃×#341
LCS	G ▷ C₃²×C₄ ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂×C₃×C₄ ◁ C₂²×C₃×C₄ ◁ C₂²×C₃×C₈

Complements

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

N	H ≅ G/N	type
3. #10589418	C₃	direct (N × H)
110. C₃³	#341	semidirect
165. C₃²	#4552	semidirect
166. C₃²	C₃×#341	semidirect
190. C₃	C₃×#4552	semidirect
191. C₃	#10589418	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂²×C₃³×C₈	2
C₂×C₃×#3748	2
#10589418	3
C₃×#4552	3

Subgroups — C3×#10589418 #13145976

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#10589418 #13145976