Subgroups — C₃²×#4681 #13162117

Order 1728.

764 conjugacy classes of subgroups.

The subgroup lattice has 216 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²×#869	C₂	1	1	yes	maximal, characteristic
2	C₃²×#869	C₂	1	1	yes	maximal, characteristic
3	C₃³×C₄×C₈	C₂	1	1	yes	maximal, characteristic
4	C₃²×#816	C₂	1	4	yes	maximal
5	C₃²×#487	—	3	1	no	maximal
6	C₃×#4681	C₃	1	4	yes	maximal, direct product factor
7	C₂×C₃²×Dic₆	—	2	1	no	—
8	C₂×C₃²×Dic₆	C₂²	1	2	yes	—
9	C₂×C₃³×C₈	C₂²	1	2	yes	—
10	C₃³×C₄²	C₂²	1	1	yes	characteristic
11	C₃²×C₄⋊Dic₃	—	2	1	no	—
12	C₃²×C₄⋊Dic₃	—	2	1	no	—
13	C₃²×C₄⋊Dic₃	—	2	1	no	—
14	C₃²×C₄⋊Dic₃	C₂²	1	2	yes	—
15	C₃²×C₄⋊Q₈	—	3	1	no	—
16	C₃²×C₄⋊Q₈	—	3	1	no	—
17	C₃×#869	C₂×C₃	1	4	yes	—
18	C₃×#869	C₂×C₃	1	4	yes	—
19	C₃²×#103	—	3	4	no	—
20	C₃²×C₄×C₈	—	2	4	no	—
21	C₃²×C₄×C₈	D₃	1	1	yes	characteristic
22	C₃²×C₄×C₈	C₂×C₃	1	4	yes	—
23	C₃×#816	C₂×C₃	1	16	yes	—
24	C₂×C₃³×C₄	D₄	1	1	yes	characteristic
25	C₂×C₃³×C₄	D₄	1	1	yes	characteristic
26	C₂×C₃³×C₄	C₂³	1	1	yes	characteristic
27	C₂×C₃²×Dic₃	—	2	2	no	—
28	C₂×C₃²×Dic₃	—	2	2	no	—
29	C₃²×Dic₆	—	2	4	no	—
30	C₃²×Dic₆	—	4	2	no	—
31	C₃³×C₈	—	2	2	no	—
32	C₃×#487	—	3	4	no	—
33	#4681	C₃²	1	1	yes	characteristic, direct product factor
34	C₂×C₃²×Q₈	—	3	2	no	—
35	C₂×C₃²×Q₈	—	6	1	no	—
36	C₂×C₃×Dic₆	—	2	4	no	—
37	C₂×C₃×Dic₆	C₂²×C₃	1	8	yes	—
38	C₂×C₃²×C₈	—	2	8	no	—
39	C₂×C₃²×C₈	C₂×D₃	1	2	yes	—
40	C₂×C₃²×C₈	C₂²×C₃	1	8	yes	—
41	C₃²×C₄⋊C₄	—	3	2	no	—
42	C₃²×C₄⋊C₄	—	6	1	no	—
43	C₃²×C₄⋊C₄	—	6	1	no	—
44	C₃²×C₄⋊C₄	—	6	1	no	—
45	C₃²×C₄²	—	2	4	no	—
46	C₃²×C₄²	C₂×D₃	1	1	yes	characteristic
47	C₃²×C₄²	C₂²×C₃	1	4	yes	—
48	C₃×C₄⋊Dic₃	—	2	4	no	—
49	C₃×C₄⋊Dic₃	—	2	4	no	—
50	C₃×C₄⋊Dic₃	—	2	4	no	—
51	C₃×C₄⋊Dic₃	C₂²×C₃	1	8	yes	—
52	C₂²×C₃³	C₂×D₄	1	1	yes	characteristic
53	C₃³×C₄	SD₁₆	1	2	yes	—
54	C₃³×C₄	Q₁₆	1	2	yes	—
55	C₃³×C₄	#41	1	2	yes	—
56	C₃²×Dic₃	—	4	2	no	—
57	C₃²×Dic₃	—	4	2	no	—
58	C₃×C₄⋊Q₈	—	3	4	no	—
59	C₃×C₄⋊Q₈	—	3	4	no	—
60	#869	C₂×C₃²	1	1	yes	characteristic
61	#869	C₂×C₃²	1	1	yes	characteristic
62	C₃×#103	—	3	16	no	—
63	C₃×C₄×C₈	—	2	4	no	—
64	C₃×C₄×C₈	C₃×D₃	1	4	yes	—
65	C₃×C₄×C₈	C₂×C₃²	1	1	yes	characteristic
66	#816	C₂×C₃²	1	4	yes	—
67	C₃²×Q₈	—	6	4	no	—
68	C₃²×Q₈	—	12	2	no	—
69	C₂×C₃²×C₄	—	2	4	no	—
70	C₂×C₃²×C₄	—	2	4	no	—
71	C₂×C₃²×C₄	—	2	4	no	—
72	C₂×C₃²×C₄	—	6	2	no	—
73	C₂×C₃²×C₄	—	6	2	no	—
74	C₂×C₃²×C₄	D₁₂	1	1	yes	characteristic
75	C₂×C₃²×C₄	D₁₂	1	1	yes	characteristic
76	C₂×C₃²×C₄	C₃×D₄	1	4	yes	—
77	C₂×C₃²×C₄	C₃×D₄	1	4	yes	—
78	C₂×C₃²×C₄	C₂²×D₃	1	1	yes	characteristic
79	C₂×C₃²×C₄	C₂³×C₃	1	4	yes	—
80	C₂×C₃×Dic₃	—	2	8	no	—
81	C₂×C₃×Dic₃	—	2	8	no	—
82	C₃×Dic₆	—	2	16	no	—
83	C₃×Dic₆	—	4	8	no	—
84	C₃²×C₈	—	2	16	no	—
85	C₃²×C₈	—	2	2	no	—
86	C₃²×C₈	—	2	8	no	—
87	#487	—	3	1	no	Sylow
88	C₂×C₃³	#124	1	1	yes	characteristic
89	C₂×C₃³	C₂×SD₁₆	1	1	yes	characteristic
90	C₂×C₃³	C₂×Q₁₆	1	1	yes	characteristic
91	C₂×C₃×Q₈	—	3	8	no	—
92	C₂×C₃×Q₈	—	6	4	no	—
93	C₂×Dic₆	—	2	1	no	—
94	C₂×Dic₆	C₂²×C₃²	1	2	yes	—
95	C₂×C₃×C₈	—	2	8	no	—
96	C₂×C₃×C₈	C₂×C₃×D₃	1	8	yes	—
97	C₂×C₃×C₈	C₂²×C₃²	1	2	yes	—
98	C₃×C₄⋊C₄	—	3	8	no	—
99	C₃×C₄⋊C₄	—	6	4	no	—
100	C₃×C₄⋊C₄	—	6	4	no	—
101	C₃×C₄⋊C₄	—	6	4	no	—
102	C₃×C₄²	—	2	4	no	—
103	C₃×C₄²	C₂×C₃×D₃	1	4	yes	—
104	C₃×C₄²	C₂²×C₃²	1	1	yes	characteristic
105	C₄⋊Dic₃	—	2	1	no	—
106	C₄⋊Dic₃	—	2	1	no	—
107	C₄⋊Dic₃	—	2	1	no	—
108	C₄⋊Dic₃	C₂²×C₃²	1	2	yes	—
109	C₂²×C₃²	—	2	4	no	—
110	C₂²×C₃²	C₂×D₁₂	1	1	yes	characteristic
111	C₂²×C₃²	C₂×C₃×D₄	1	4	yes	—
112	C₃²×C₄	—	2	8	no	—
113	C₃²×C₄	—	2	8	no	—
114	C₃²×C₄	—	2	8	no	—
115	C₃²×C₄	—	12	2	no	—
116	C₃²×C₄	—	12	2	no	—
117	C₃²×C₄	#204	1	2	yes	—
118	C₃²×C₄	Dic₁₂	1	2	yes	—
119	C₃²×C₄	C₃×SD₁₆	1	8	yes	—
120	C₃²×C₄	C₃×Q₁₆	1	8	yes	—
121	C₃²×C₄	#235	1	2	yes	—
122	C₃²×C₄	C₃×#41	1	8	yes	—
123	C₃×Dic₃	—	4	8	no	—
124	C₃×Dic₃	—	4	8	no	—
125	C₄⋊Q₈	—	3	1	no	—
126	C₄⋊Q₈	—	3	1	no	—
127	#103	—	3	4	no	—
128	C₄×C₈	C₃²×D₃	1	1	yes	characteristic
129	C₃³	#487	1	1	yes	characteristic, complemented, Sylow
130	C₃×Q₈	—	6	16	no	—
131	C₃×Q₈	—	12	8	no	—
132	C₂×C₃×C₄	—	2	4	no	—
133	C₂×C₃×C₄	—	2	4	no	—
134	C₂×C₃×C₄	—	2	4	no	—
135	C₂×C₃×C₄	—	6	8	no	—
136	C₂×C₃×C₄	—	6	8	no	—
137	C₂×C₃×C₄	C₃×D₁₂	1	4	yes	—
138	C₂×C₃×C₄	C₃×D₁₂	1	4	yes	—
139	C₂×C₃×C₄	C₃²×D₄	1	1	yes	characteristic
140	C₂×C₃×C₄	C₃²×D₄	1	1	yes	characteristic
141	C₂×C₃×C₄	C₂²×C₃×D₃	1	4	yes	—
142	C₂×C₃×C₄	C₂³×C₃²	1	1	yes	characteristic
143	C₂×Dic₃	—	2	2	no	—
144	C₂×Dic₃	—	2	2	no	—
145	Dic₆	—	2	4	no	—
146	Dic₆	—	4	2	no	—
147	C₃×C₈	—	2	16	no	—
148	C₃×C₈	—	2	8	no	—
149	C₃×C₈	—	2	2	no	—
150	C₂×C₃²	—	2	4	no	—
151	C₂×C₃²	—	2	4	no	—
152	C₂×C₃²	—	2	4	no	—
153	C₂×C₃²	#875	1	1	yes	characteristic
154	C₂×C₃²	C₂×#204	1	1	yes	characteristic
155	C₂×C₃²	C₂×Dic₁₂	1	1	yes	characteristic
156	C₂×C₃²	C₃×#124	1	4	yes	—
157	C₂×C₃²	C₂×C₃×SD₁₆	1	4	yes	—
158	C₂×C₃²	C₂×C₃×Q₁₆	1	4	yes	—
159	C₂×Q₈	—	3	2	no	—
160	C₂×Q₈	—	6	1	no	—
161	C₂×C₈	C₂×C₃²×D₃	1	2	yes	—
162	C₄⋊C₄	—	3	2	no	—
163	C₄⋊C₄	—	6	1	no	—
164	C₄⋊C₄	—	6	1	no	—
165	C₄⋊C₄	—	6	1	no	—
166	C₄²	C₂×C₃²×D₃	1	1	yes	characteristic
167	C₂²×C₃	—	2	4	no	—
168	C₂²×C₃	C₂×C₃×D₁₂	1	4	yes	—
169	C₂²×C₃	C₂×C₃²×D₄	1	1	yes	characteristic
170	C₃×C₄	—	2	8	no	—
171	C₃×C₄	—	2	8	no	—
172	C₃×C₄	—	2	8	no	—
173	C₃×C₄	—	12	8	no	—
174	C₃×C₄	—	12	8	no	—
175	C₃×C₄	C₃×#204	1	8	yes	—
176	C₃×C₄	C₃×Dic₁₂	1	8	yes	—
177	C₃×C₄	C₃²×SD₁₆	1	2	yes	—
178	C₃×C₄	C₃²×Q₁₆	1	2	yes	—
179	C₃×C₄	C₃×#235	1	8	yes	—
180	C₃×C₄	C₃²×#41	1	2	yes	—
181	Dic₃	—	4	2	no	—
182	Dic₃	—	4	2	no	—
183	C₃²	—	2	4	no	—
184	C₃²	#4681	1	1	yes	characteristic, direct product factor
185	C₃²	C₃×#487	1	4	yes	complemented
186	Q₈	—	6	4	no	—
187	Q₈	—	12	2	no	—
188	C₂×C₄	—	6	2	no	—
189	C₂×C₄	—	6	2	no	—
190	C₂×C₄	C₃²×D₁₂	1	1	yes	characteristic
191	C₂×C₄	C₃²×D₁₂	1	1	yes	characteristic
192	C₂×C₄	C₂²×C₃²×D₃	1	1	yes	characteristic
193	C₈	—	2	2	no	—
194	C₂×C₃	—	2	4	no	—
195	C₂×C₃	—	2	4	no	—
196	C₂×C₃	—	2	4	no	—
197	C₂×C₃	C₃×#875	1	4	yes	—
198	C₂×C₃	C₂×C₃×#204	1	4	yes	—
199	C₂×C₃	C₂×C₃×Dic₁₂	1	4	yes	—
200	C₂×C₃	C₃²×#124	1	1	yes	characteristic
201	C₂×C₃	C₂×C₃²×SD₁₆	1	1	yes	characteristic
202	C₂×C₃	C₂×C₃²×Q₁₆	1	1	yes	characteristic
203	C₂²	C₂×C₃²×D₁₂	1	1	yes	characteristic
204	C₄	—	12	2	no	—
205	C₄	—	12	2	no	—
206	C₄	C₃²×#204	1	2	yes	—
207	C₄	C₃²×Dic₁₂	1	2	yes	—
208	C₄	C₃²×#235	1	2	yes	—
209	C₃	—	2	4	no	—
210	C₃	C₃×#4681	1	4	yes	direct product factor
211	C₃	C₃²×#487	1	1	yes	characteristic, complemented
212	C₂	C₃²×#875	1	1	yes	characteristic
213	C₂	C₂×C₃²×#204	1	1	yes	characteristic
214	C₂	C₂×C₃²×Dic₁₂	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₃²	C₂×D₁₂
commutator, cosocle	C₂×C₃×C₄	C₂³×C₃²
Frattini	C₂×C₄	C₂²×C₃²×D₃
Fitting	C₃³×C₄×C₈	C₂
socle, prime radical	C₂²×C₃³	C₂×D₄
hypercenter	C₃²×C₄×C₈	D₃
hypocenter	C₃	C₃²×#487
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₃² ◁ C₃²×C₄² ◁ C₃²×C₄×C₈

Complements

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

N	H ≅ G/N	type
6. C₃×#4681	C₃	direct (N × H)
33. #4681	C₃²	direct (N × H)
129. C₃³	#487	semidirect
184. C₃²	#4681	direct (N × H)
185. C₃²	C₃×#487	semidirect
210. C₃	C₃×#4681	direct (N × H)
211. C₃	C₃²×#487	semidirect

Maximal subgroups

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

subgroup	index
C₃²×#869	2
C₃³×C₄×C₈	2
C₃²×#816	2
C₃²×#487	3
C₃×#4681	3

Subgroups — C32×#4681 #13162117

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃²×#4681 #13162117