Subgroups — C₂×C₃×#63431 #13152272

Order 1728.

978 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	C₂²×C₃×Dic₁₈	C₂	1	2	yes	maximal
2	C₂×C₃×C₄²×C₉	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃×C₄⋊Dic₉	C₂	1	4	yes	maximal
4	C₃×#63431	C₂	1	8	yes	maximal, direct product factor
5	C₂×C₃×#869	—	3	1	no	maximal
6	C₂×#63431	C₃	1	1	yes	maximal, characteristic, direct product factor
7	C₂²×C₃×C₄×C₉	C₂²	1	1	yes	characteristic
8	C₂²×C₃×C₄×C₉	C₂²	1	2	yes	—
9	C₂²×C₃×Dic₉	C₂²	1	4	yes	—
10	C₂×C₃×Dic₁₈	—	2	8	no	—
11	C₂×C₃×Dic₁₈	C₂²	1	8	yes	—
12	C₃×C₄²×C₉	C₂²	1	4	yes	—
13	C₃×C₄⋊Dic₉	C₂²	1	16	yes	—
14	C₂²×C₃×Dic₆	—	3	2	no	—
15	C₂×C₃²×C₄²	D₃	1	1	yes	characteristic
16	C₂×C₃×C₄⋊Dic₃	—	3	4	no	—
17	C₃×#869	—	3	8	no	—
18	C₂²×Dic₁₈	C₂×C₃	1	2	yes	—
19	C₂×C₄²×C₉	—	2	1	no	—
20	C₂×C₄²×C₉	C₂×C₃	1	1	yes	characteristic
21	C₂×C₄⋊Dic₉	C₂×C₃	1	4	yes	—
22	#63431	C₂×C₃	1	8	yes	direct product factor
23	C₂³×C₃×C₉	C₂³	1	1	yes	characteristic
24	C₂×C₃×C₄×C₉	D₄	1	4	yes	—
25	C₂×C₃×C₄×C₉	Q₈	1	8	yes	complemented
26	C₂×C₃×C₄×C₉	C₂³	1	2	yes	—
27	C₂×C₃×C₄×C₉	C₂³	1	4	yes	—
28	C₂×C₃×Dic₉	—	2	8	no	—
29	C₂×C₃×Dic₉	C₂³	1	8	yes	—
30	C₃×Dic₁₈	—	2	16	no	—
31	C₂×C₃×C₄⋊Q₈	—	9	1	no	—
32	C₂×#869	—	3	1	no	—
33	C₂²×C₃²×C₄	C₂×D₃	1	2	yes	—
34	C₂²×C₃²×C₄	C₂×D₃	1	1	yes	characteristic
35	C₂²×C₃×Dic₃	—	3	4	no	—
36	C₂×C₃×Dic₆	—	3	8	no	—
37	C₂×C₃×Dic₆	—	6	8	no	—
38	C₃²×C₄²	C₂×D₃	1	4	yes	—
39	C₃×C₄⋊Dic₃	—	3	16	no	—
40	C₂²×C₄×C₉	—	2	2	no	—
41	C₂²×C₄×C₉	—	2	1	no	—
42	C₂²×C₄×C₉	C₂²×C₃	1	1	yes	characteristic
43	C₂²×C₄×C₉	C₂²×C₃	1	2	yes	—
44	C₂²×Dic₉	C₂²×C₃	1	4	yes	—
45	C₂×Dic₁₈	—	2	8	no	—
46	C₂×Dic₁₈	C₂²×C₃	1	8	yes	—
47	C₄²×C₉	—	2	4	no	—
48	C₄²×C₉	C₂²×C₃	1	4	yes	—
49	C₄⋊Dic₉	C₂²×C₃	1	16	yes	—
50	C₂²×C₃×C₉	C₂×D₄	1	2	yes	—
51	C₂²×C₃×C₉	C₂×Q₈	1	4	yes	—
52	C₂²×C₃×C₉	C₂⁴	1	1	yes	characteristic
53	C₃×C₄×C₉	C₂×D₄	1	4	yes	—
54	C₃×C₄×C₉	C₂×Q₈	1	8	yes	complemented
55	C₃×Dic₉	—	2	8	no	—
56	C₂²×C₃×Q₈	—	9	2	no	—
57	C₂²×Dic₆	—	3	2	no	—
58	C₃×C₄⋊Q₈	—	9	8	no	—
59	C₂×C₃×C₄⋊C₄	—	9	4	no	—
60	C₂×C₃×C₄²	—	2	1	no	—
61	C₂×C₃×C₄²	D₉	1	1	yes	characteristic
62	C₂×C₃×C₄²	C₃×D₃	1	1	yes	characteristic
63	C₂×C₄⋊Dic₃	—	3	4	no	—
64	#869	—	3	8	no	—
65	C₂³×C₃²	C₂²×D₃	1	1	yes	characteristic
66	C₂×C₃²×C₄	Dic₆	1	8	yes	—
67	C₂×C₃²×C₄	D₁₂	1	4	yes	—
68	C₂×C₃²×C₄	C₂²×D₃	1	4	yes	—
69	C₂×C₃²×C₄	C₂²×D₃	1	2	yes	—
70	C₂×C₃×Dic₃	—	3	8	no	—
71	C₂×C₃×Dic₃	—	6	8	no	—
72	C₃×Dic₆	—	6	16	no	—
73	C₂³×C₉	—	2	1	no	—
74	C₂³×C₉	C₂³×C₃	1	1	yes	characteristic
75	C₂×C₄×C₉	—	2	8	no	—
76	C₂×C₄×C₉	—	2	4	no	—
77	C₂×C₄×C₉	—	2	4	no	—
78	C₂×C₄×C₉	—	2	2	no	—
79	C₂×C₄×C₉	C₃×D₄	1	4	yes	—
80	C₂×C₄×C₉	C₃×Q₈	1	8	yes	complemented
81	C₂×C₄×C₉	C₂³×C₃	1	2	yes	—
82	C₂×C₄×C₉	C₂³×C₃	1	4	yes	—
83	C₂×Dic₉	—	2	8	no	—
84	C₂×Dic₉	C₂³×C₃	1	8	yes	—
85	Dic₁₈	—	2	16	no	—
86	C₂×C₄⋊Q₈	—	9	1	no	Sylow
87	C₂×C₃×C₉	C₄⋊Q₈	1	4	yes	complemented
88	C₂×C₃×C₉	C₂²×D₄	1	1	yes	characteristic
89	C₂×C₃×C₉	C₂²×Q₈	1	2	yes	—
90	C₂×C₃×Q₈	—	9	8	no	—
91	C₂×C₃×Q₈	—	18	8	no	—
92	C₂²×C₃×C₄	—	2	2	no	—
93	C₂²×C₃×C₄	—	2	1	no	—
94	C₂²×C₃×C₄	—	9	4	no	—
95	C₂²×C₃×C₄	C₂×D₉	1	2	yes	—
96	C₂²×C₃×C₄	C₂×D₉	1	1	yes	characteristic
97	C₂²×C₃×C₄	C₂×C₃×D₃	1	2	yes	—
98	C₂²×C₃×C₄	C₂×C₃×D₃	1	1	yes	characteristic
99	C₂²×Dic₃	—	3	4	no	—
100	C₂×Dic₆	—	3	8	no	—
101	C₂×Dic₆	—	6	8	no	—
102	C₃×C₄⋊C₄	—	9	16	no	—
103	C₃×C₄²	—	2	4	no	—
104	C₃×C₄²	C₂×D₉	1	4	yes	—
105	C₃×C₄²	C₂×C₃×D₃	1	4	yes	—
106	C₄⋊Dic₃	—	3	16	no	—
107	C₂²×C₃²	C₂×Dic₆	1	4	yes	—
108	C₂²×C₃²	C₂×D₁₂	1	2	yes	—
109	C₂²×C₃²	C₂³×D₃	1	1	yes	characteristic
110	C₃²×C₄	C₂×Dic₆	1	8	yes	—
111	C₃²×C₄	C₂×D₁₂	1	4	yes	—
112	C₃×Dic₃	—	6	8	no	—
113	C₂²×C₉	—	2	1	no	—
114	C₂²×C₉	—	2	4	no	—
115	C₂²×C₉	—	2	2	no	—
116	C₂²×C₉	C₂×C₃×D₄	1	2	yes	—
117	C₂²×C₉	C₂×C₃×Q₈	1	4	yes	—
118	C₂²×C₉	C₂⁴×C₃	1	1	yes	characteristic
119	C₄×C₉	—	2	8	no	—
120	C₄×C₉	—	2	4	no	—
121	C₄×C₉	C₂×C₃×D₄	1	4	yes	—
122	C₄×C₉	C₂×C₃×Q₈	1	8	yes	complemented
123	Dic₉	—	2	8	no	—
124	C₂²×Q₈	—	9	2	no	—
125	C₄⋊Q₈	—	9	8	no	—
126	C₂×C₄⋊C₄	—	9	4	no	—
127	C₂×C₄²	C₃×D₉	1	1	yes	characteristic
128	C₃×C₉	C₂×C₄⋊Q₈	1	1	yes	characteristic, complemented, Sylow
129	C₂³×C₃	—	2	1	no	—
130	C₂³×C₃	C₂²×D₉	1	1	yes	characteristic
131	C₂³×C₃	C₂²×C₃×D₃	1	1	yes	characteristic
132	C₃×Q₈	—	18	16	no	—
133	C₂×C₃×C₄	—	2	8	no	—
134	C₂×C₃×C₄	—	2	4	no	—
135	C₂×C₃×C₄	—	2	4	no	—
136	C₂×C₃×C₄	—	2	2	no	—
137	C₂×C₃×C₄	—	9	8	no	—
138	C₂×C₃×C₄	—	18	8	no	—
139	C₂×C₃×C₄	Dic₁₈	1	8	yes	complemented
140	C₂×C₃×C₄	D₃₆	1	4	yes	—
141	C₂×C₃×C₄	C₂²×D₉	1	4	yes	—
142	C₂×C₃×C₄	C₂²×D₉	1	2	yes	—
143	C₂×C₃×C₄	C₃×Dic₆	1	8	yes	—
144	C₂×C₃×C₄	C₃×D₁₂	1	4	yes	—
145	C₂×C₃×C₄	C₂²×C₃×D₃	1	4	yes	—
146	C₂×C₃×C₄	C₂²×C₃×D₃	1	2	yes	—
147	C₂×Dic₃	—	3	8	no	—
148	C₂×Dic₃	—	6	8	no	—
149	Dic₆	—	6	16	no	—
150	C₂×C₃²	#869	1	4	yes	—
151	C₂×C₃²	C₂²×Dic₆	1	2	yes	—
152	C₂×C₃²	C₂²×D₁₂	1	1	yes	characteristic
153	C₂×C₉	—	2	2	no	—
154	C₂×C₉	—	2	1	no	—
155	C₂×C₉	—	2	4	no	—
156	C₂×C₉	C₃×C₄⋊Q₈	1	4	yes	complemented
157	C₂×C₉	C₂²×C₃×D₄	1	1	yes	characteristic
158	C₂×C₉	C₂²×C₃×Q₈	1	2	yes	—
159	C₂×Q₈	—	9	8	no	—
160	C₂×Q₈	—	18	8	no	—
161	C₂²×C₄	—	9	4	no	—
162	C₂²×C₄	C₂×C₃×D₉	1	2	yes	—
163	C₂²×C₄	C₂×C₃×D₉	1	1	yes	characteristic
164	C₄⋊C₄	—	9	16	no	—
165	C₄²	C₂×C₃×D₉	1	4	yes	—
166	C₂²×C₃	—	2	1	no	—
167	C₂²×C₃	—	2	4	no	—
168	C₂²×C₃	—	2	2	no	—
169	C₂²×C₃	C₂×Dic₁₈	1	4	yes	—
170	C₂²×C₃	C₂×D₃₆	1	2	yes	—
171	C₂²×C₃	C₂³×D₉	1	1	yes	characteristic
172	C₂²×C₃	C₂×C₃×Dic₆	1	4	yes	—
173	C₂²×C₃	C₂×C₃×D₁₂	1	2	yes	—
174	C₂²×C₃	C₂³×C₃×D₃	1	1	yes	characteristic
175	C₃×C₄	—	2	8	no	—
176	C₃×C₄	—	2	4	no	—
177	C₃×C₄	—	18	8	no	—
178	C₃×C₄	C₂×Dic₁₈	1	8	yes	complemented
179	C₃×C₄	C₂×D₃₆	1	4	yes	—
180	C₃×C₄	C₂×C₃×Dic₆	1	8	yes	—
181	C₃×C₄	C₂×C₃×D₁₂	1	4	yes	—
182	Dic₃	—	6	8	no	—
183	C₃²	C₂×#869	1	1	yes	characteristic
184	C₉	—	2	1	no	—
185	C₉	C₂×C₃×C₄⋊Q₈	1	1	yes	characteristic, complemented
186	C₂³	C₂²×C₃×D₉	1	1	yes	characteristic
187	Q₈	—	18	16	no	—
188	C₂×C₄	—	9	8	no	—
189	C₂×C₄	—	18	8	no	—
190	C₂×C₄	C₃×Dic₁₈	1	8	yes	complemented
191	C₂×C₄	C₃×D₃₆	1	4	yes	—
192	C₂×C₄	C₂²×C₃×D₉	1	4	yes	—
193	C₂×C₄	C₂²×C₃×D₉	1	2	yes	—
194	C₂×C₃	—	2	2	no	—
195	C₂×C₃	—	2	1	no	—
196	C₂×C₃	—	2	4	no	—
197	C₂×C₃	#63431	1	4	yes	direct product factor
198	C₂×C₃	C₂²×Dic₁₈	1	2	yes	—
199	C₂×C₃	C₂²×D₃₆	1	1	yes	characteristic
200	C₂×C₃	C₃×#869	1	4	yes	—
201	C₂×C₃	C₂²×C₃×Dic₆	1	2	yes	—
202	C₂×C₃	C₂²×C₃×D₁₂	1	1	yes	characteristic
203	C₂²	C₂×C₃×Dic₁₈	1	4	yes	—
204	C₂²	C₂×C₃×D₃₆	1	2	yes	—
205	C₂²	C₂³×C₃×D₉	1	1	yes	characteristic
206	C₄	—	18	8	no	—
207	C₄	C₂×C₃×Dic₁₈	1	8	yes	complemented
208	C₄	C₂×C₃×D₃₆	1	4	yes	—
209	C₃	—	2	1	no	—
210	C₃	C₂×#63431	1	1	yes	characteristic, direct product factor
211	C₃	C₂×C₃×#869	1	1	yes	characteristic
212	C₂	C₃×#63431	1	4	yes	direct product factor
213	C₂	C₂²×C₃×Dic₁₈	1	2	yes	—
214	C₂	C₂²×C₃×D₃₆	1	1	yes	characteristic

	subgroup	G/H
center	C₂³×C₃	C₂²×D₉
commutator, cosocle	C₂²×C₉	C₂⁴×C₃
Frattini	C₂²×C₃	C₂³×C₃×D₃
Fitting	C₂×C₃×C₄²×C₉	C₂
socle, prime radical	C₂³×C₃²	C₂²×D₃
hypercenter	C₂×C₃×C₄²	D₉
hypocenter	C₉	C₂×C₃×C₄⋊Q₈
LCS	G ▷ C₂²×C₉ ▷ C₉
UCS	C₁ ◁ C₂³×C₃ ◁ C₂×C₃×C₄²

Complements

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

N	H ≅ G/N	type
4. C₃×#63431	C₂	direct (N × H)
6. C₂×#63431	C₃	direct (N × H)
22. #63431	C₂×C₃	direct (N × H)
25. C₂×C₃×C₄×C₉	Q₈	semidirect
54. C₃×C₄×C₉	C₂×Q₈	semidirect
80. C₂×C₄×C₉	C₃×Q₈	semidirect
87. C₂×C₃×C₉	C₄⋊Q₈	semidirect
122. C₄×C₉	C₂×C₃×Q₈	semidirect
128. C₃×C₉	C₂×C₄⋊Q₈	semidirect
139. C₂×C₃×C₄	Dic₁₈	semidirect
156. C₂×C₉	C₃×C₄⋊Q₈	semidirect
178. C₃×C₄	C₂×Dic₁₈	semidirect
185. C₉	C₂×C₃×C₄⋊Q₈	semidirect
190. C₂×C₄	C₃×Dic₁₈	semidirect
197. C₂×C₃	#63431	direct (N × H)
207. C₄	C₂×C₃×Dic₁₈	semidirect
210. C₃	C₂×#63431	direct (N × H)
212. C₂	C₃×#63431	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×C₃×Dic₁₈	2
C₂×C₃×C₄²×C₉	2
C₂×C₃×C₄⋊Dic₉	2
C₃×#63431	2
C₂×C₃×#869	3
C₂×#63431	3

Subgroups — C2×C3×#63431 #13152272

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×#63431 #13152272