Subgroups — 3₋¹⁺²×#405 #13144975

Order 1728.

520 conjugacy classes of subgroups.

The subgroup lattice has 210 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×M₁₆×3₋¹⁺²	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₈×3₋¹⁺²	C₂	1	1	yes	maximal, characteristic
3	3₋¹⁺²×#117	C₂	1	1	yes	maximal, characteristic
4	3₋¹⁺²×#97	C₂	1	2	yes	maximal, complemented
5	C₄×C₈×3₋¹⁺²	C₂	1	2	yes	maximal, complemented
6	C₃²×#405	C₃	1	1	yes	maximal, characteristic
7	C₉×#405	C₃	1	3	yes	maximal, complemented
8	C₂²×C₄×3₋¹⁺²	C₂²	1	1	yes	characteristic
9	M₁₆×3₋¹⁺²	C₄	1	4	yes	complemented
10	C₂×C₈×3₋¹⁺²	C₄	1	4	yes	complemented
11	C₂×C₈×3₋¹⁺²	C₂²	1	1	yes	characteristic
12	C₂×C₈×3₋¹⁺²	C₂²	1	2	yes	—
13	C₂×C₈×3₋¹⁺²	C₂²	1	1	yes	characteristic
14	C₄⋊C₄×3₋¹⁺²	C₄	1	2	yes	—
15	C₂²⋊C₄×3₋¹⁺²	C₄	1	2	yes	—
16	C₄²×3₋¹⁺²	C₂²	1	2	yes	—
17	C₂×C₃²×M₁₆	C₂×C₃	1	1	yes	characteristic
18	C₂²×C₃²×C₈	C₂×C₃	1	1	yes	characteristic
19	C₃²×#117	C₂×C₃	1	1	yes	characteristic
20	C₃²×#97	C₂×C₃	1	2	yes	—
21	C₃²×C₄×C₈	C₂×C₃	1	2	yes	—
22	C₂×C₉×M₁₆	C₂×C₃	1	3	yes	—
23	C₂²×C₈×C₉	C₂×C₃	1	3	yes	—
24	C₉×#117	C₂×C₃	1	3	yes	—
25	C₉×#97	C₂×C₃	1	6	yes	complemented
26	C₄×C₈×C₉	C₂×C₃	1	6	yes	complemented
27	C₂³×3₋¹⁺²	C₂×C₄	1	1	yes	characteristic
28	C₂×C₄×3₋¹⁺²	C₂×C₄	1	4	yes	—
29	C₂×C₄×3₋¹⁺²	C₂×C₄	1	1	yes	characteristic
30	C₂×C₄×3₋¹⁺²	C₂×C₄	1	2	yes	—
31	C₂×C₄×3₋¹⁺²	C₂×C₄	1	2	yes	—
32	C₂×C₄×3₋¹⁺²	C₂³	1	1	yes	characteristic
33	C₈×3₋¹⁺²	C₂×C₄	1	2	yes	—
34	C₈×3₋¹⁺²	C₂×C₄	1	2	yes	—
35	C₈×3₋¹⁺²	C₂×C₄	1	4	yes	—
36	C₃×#405	—	3	1	no	—
37	C₃×#405	C₃²	1	1	yes	characteristic
38	C₂²×C₃²×C₄	C₂²×C₃	1	1	yes	characteristic
39	C₃²×M₁₆	C₃×C₄	1	4	yes	—
40	C₂×C₃²×C₈	C₃×C₄	1	4	yes	—
41	C₂×C₃²×C₈	C₂²×C₃	1	2	yes	—
42	C₂×C₃²×C₈	C₂²×C₃	1	1	yes	characteristic
43	C₂×C₃²×C₈	C₂²×C₃	1	1	yes	characteristic
44	C₃²×C₄⋊C₄	C₃×C₄	1	2	yes	—
45	C₃²×C₂²⋊C₄	C₃×C₄	1	2	yes	—
46	C₃²×C₄²	C₂²×C₃	1	2	yes	—
47	C₂²×C₄×C₉	C₂²×C₃	1	3	yes	—
48	C₉×M₁₆	C₃×C₄	1	12	yes	complemented
49	C₂×C₈×C₉	C₃×C₄	1	12	yes	complemented
50	C₂×C₈×C₉	C₂²×C₃	1	3	yes	—
51	C₂×C₈×C₉	C₂²×C₃	1	6	yes	—
52	C₂×C₈×C₉	C₂²×C₃	1	3	yes	—
53	C₉×C₄⋊C₄	C₃×C₄	1	6	yes	—
54	C₉×C₂²⋊C₄	C₃×C₄	1	6	yes	—
55	C₄²×C₉	C₂²×C₃	1	6	yes	—
56	C₂²×3₋¹⁺²	—	2	2	no	—
57	C₂²×3₋¹⁺²	C₄²	1	2	yes	—
58	C₂²×3₋¹⁺²	C₂²×C₄	1	1	yes	characteristic
59	C₄×3₋¹⁺²	—	2	4	no	—
60	C₄×3₋¹⁺²	C₄²	1	2	yes	—
61	C₄×3₋¹⁺²	C₂²×C₄	1	1	yes	characteristic
62	C₄×3₋¹⁺²	C₂²×C₄	1	1	yes	characteristic
63	C₂×C₃×M₁₆	—	3	1	no	—
64	C₂×C₃×M₁₆	C₂×C₃²	1	1	yes	characteristic
65	C₂²×C₃×C₈	—	3	1	no	—
66	C₂²×C₃×C₈	C₂×C₃²	1	1	yes	characteristic
67	C₃×#117	—	3	1	no	—
68	C₃×#117	C₂×C₃²	1	1	yes	characteristic
69	C₃×#97	—	3	2	no	—
70	C₃×#97	C₂×C₃²	1	2	yes	—
71	C₃×C₄×C₈	—	3	2	no	—
72	C₃×C₄×C₈	C₂×C₃²	1	2	yes	—
73	C₂³×C₃²	C₂×C₃×C₄	1	1	yes	characteristic
74	C₂×C₃²×C₄	C₂×C₃×C₄	1	4	yes	—
75	C₂×C₃²×C₄	C₂×C₃×C₄	1	1	yes	characteristic
76	C₂×C₃²×C₄	C₂×C₃×C₄	1	2	yes	—
77	C₂×C₃²×C₄	C₂×C₃×C₄	1	2	yes	—
78	C₂×C₃²×C₄	C₂³×C₃	1	1	yes	characteristic
79	C₂³×C₉	C₂×C₃×C₄	1	3	yes	—
80	C₃²×C₈	C₂×C₃×C₄	1	2	yes	—
81	C₃²×C₈	C₂×C₃×C₄	1	2	yes	—
82	C₃²×C₈	C₂×C₃×C₄	1	4	yes	—
83	C₂×C₄×C₉	C₂×C₃×C₄	1	12	yes	—
84	C₂×C₄×C₉	C₂×C₃×C₄	1	3	yes	—
85	C₂×C₄×C₉	C₂×C₃×C₄	1	6	yes	—
86	C₂×C₄×C₉	C₂×C₃×C₄	1	6	yes	—
87	C₂×C₄×C₉	C₂³×C₃	1	3	yes	—
88	C₈×C₉	C₂×C₃×C₄	1	6	yes	—
89	C₈×C₉	C₂×C₃×C₄	1	6	yes	—
90	C₈×C₉	C₂×C₃×C₄	1	12	yes	—
91	#405	3₋¹⁺²	1	1	yes	characteristic, direct product factor, Sylow
92	C₂×3₋¹⁺²	—	2	2	no	—
93	C₂×3₋¹⁺²	C₂×C₄²	1	1	yes	characteristic
94	C₂×3₋¹⁺²	#131	1	2	yes	—
95	C₂²×C₃×C₄	—	3	1	no	—
96	C₂²×C₃×C₄	C₂²×C₃²	1	1	yes	characteristic
97	C₃×M₁₆	—	3	4	no	—
98	C₃×M₁₆	C₃²×C₄	1	4	yes	—
99	C₂×C₃×C₈	—	3	4	no	—
100	C₂×C₃×C₈	—	3	1	no	—
101	C₂×C₃×C₈	—	3	2	no	—
102	C₂×C₃×C₈	—	3	1	no	—
103	C₂×C₃×C₈	C₃²×C₄	1	4	yes	—
104	C₂×C₃×C₈	C₂²×C₃²	1	2	yes	—
105	C₂×C₃×C₈	C₂²×C₃²	1	1	yes	characteristic
106	C₂×C₃×C₈	C₂²×C₃²	1	1	yes	characteristic
107	C₃×C₄⋊C₄	—	3	2	no	—
108	C₃×C₄⋊C₄	C₃²×C₄	1	2	yes	—
109	C₃×C₂²⋊C₄	—	3	2	no	—
110	C₃×C₂²⋊C₄	C₃²×C₄	1	2	yes	—
111	C₃×C₄²	—	3	2	no	—
112	C₃×C₄²	C₂²×C₃²	1	2	yes	—
113	C₂²×C₃²	—	2	2	no	—
114	C₂²×C₃²	C₃×C₄²	1	2	yes	—
115	C₂²×C₃²	C₂²×C₃×C₄	1	1	yes	characteristic
116	C₃²×C₄	—	2	4	no	—
117	C₃²×C₄	C₃×C₄²	1	2	yes	—
118	C₃²×C₄	C₂²×C₃×C₄	1	1	yes	characteristic
119	C₃²×C₄	C₂²×C₃×C₄	1	1	yes	characteristic
120	C₂²×C₉	—	2	6	no	—
121	C₂²×C₉	C₃×C₄²	1	6	yes	—
122	C₂²×C₉	C₂²×C₃×C₄	1	3	yes	—
123	C₄×C₉	—	2	12	no	—
124	C₄×C₉	C₃×C₄²	1	6	yes	—
125	C₄×C₉	C₂²×C₃×C₄	1	3	yes	—
126	C₄×C₉	C₂²×C₃×C₄	1	3	yes	—
127	C₂×M₁₆	C₂×3₋¹⁺²	1	1	yes	characteristic
128	C₂²×C₈	C₂×3₋¹⁺²	1	1	yes	characteristic
129	#117	C₂×3₋¹⁺²	1	1	yes	characteristic
130	#97	C₂×3₋¹⁺²	1	2	yes	complemented
131	C₄×C₈	C₂×3₋¹⁺²	1	2	yes	complemented
132	3₋¹⁺²	#405	1	1	yes	characteristic, direct product factor, Sylow
133	C₂³×C₃	—	3	1	no	—
134	C₂³×C₃	C₂×C₃²×C₄	1	1	yes	characteristic
135	C₂×C₃×C₄	—	3	1	no	—
136	C₂×C₃×C₄	—	3	2	no	—
137	C₂×C₃×C₄	—	3	2	no	—
138	C₂×C₃×C₄	—	3	4	no	—
139	C₂×C₃×C₄	—	3	1	no	—
140	C₂×C₃×C₄	C₂×C₃²×C₄	1	4	yes	—
141	C₂×C₃×C₄	C₂×C₃²×C₄	1	1	yes	characteristic
142	C₂×C₃×C₄	C₂×C₃²×C₄	1	2	yes	—
143	C₂×C₃×C₄	C₂×C₃²×C₄	1	2	yes	—
144	C₂×C₃×C₄	C₂³×C₃²	1	1	yes	characteristic
145	C₃×C₈	—	3	2	no	—
146	C₃×C₈	—	3	2	no	—
147	C₃×C₈	—	3	4	no	—
148	C₃×C₈	C₂×C₃²×C₄	1	2	yes	—
149	C₃×C₈	C₂×C₃²×C₄	1	2	yes	—
150	C₃×C₈	C₂×C₃²×C₄	1	4	yes	—
151	C₂×C₃²	—	2	2	no	—
152	C₂×C₃²	C₂×C₃×C₄²	1	1	yes	characteristic
153	C₂×C₃²	C₃×#131	1	2	yes	—
154	C₂×C₉	—	2	6	no	—
155	C₂×C₉	C₂×C₃×C₄²	1	3	yes	—
156	C₂×C₉	C₃×#131	1	6	yes	—
157	C₂²×C₄	C₂²×3₋¹⁺²	1	1	yes	characteristic
158	M₁₆	C₄×3₋¹⁺²	1	4	yes	complemented
159	C₂×C₈	C₄×3₋¹⁺²	1	4	yes	complemented
160	C₂×C₈	C₂²×3₋¹⁺²	1	1	yes	characteristic
161	C₂×C₈	C₂²×3₋¹⁺²	1	2	yes	—
162	C₂×C₈	C₂²×3₋¹⁺²	1	1	yes	characteristic
163	C₄⋊C₄	C₄×3₋¹⁺²	1	2	yes	—
164	C₂²⋊C₄	C₄×3₋¹⁺²	1	2	yes	—
165	C₄²	C₂²×3₋¹⁺²	1	2	yes	—
166	C₂²×C₃	—	2	2	no	—
167	C₂²×C₃	—	3	1	no	—
168	C₂²×C₃	—	3	2	no	—
169	C₂²×C₃	—	6	2	no	—
170	C₂²×C₃	C₃²×C₄²	1	2	yes	—
171	C₂²×C₃	C₂²×C₃²×C₄	1	1	yes	characteristic
172	C₃×C₄	—	2	4	no	—
173	C₃×C₄	—	3	1	no	—
174	C₃×C₄	—	3	1	no	—
175	C₃×C₄	—	3	2	no	—
176	C₃×C₄	—	6	4	no	—
177	C₃×C₄	C₃²×C₄²	1	2	yes	—
178	C₃×C₄	C₂²×C₃²×C₄	1	1	yes	characteristic
179	C₃×C₄	C₂²×C₃²×C₄	1	1	yes	characteristic
180	C₃²	C₃×#405	1	1	yes	characteristic
181	C₉	C₃×#405	1	3	yes	complemented
182	C₂³	C₂×C₄×3₋¹⁺²	1	1	yes	characteristic
183	C₂×C₄	C₂×C₄×3₋¹⁺²	1	4	yes	—
184	C₂×C₄	C₂×C₄×3₋¹⁺²	1	2	yes	—
185	C₂×C₄	C₂×C₄×3₋¹⁺²	1	2	yes	—
186	C₂×C₄	C₂×C₄×3₋¹⁺²	1	1	yes	characteristic
187	C₂×C₄	C₂³×3₋¹⁺²	1	1	yes	characteristic
188	C₈	C₂×C₄×3₋¹⁺²	1	2	yes	—
189	C₈	C₂×C₄×3₋¹⁺²	1	2	yes	—
190	C₈	C₂×C₄×3₋¹⁺²	1	4	yes	—
191	C₂×C₃	—	2	2	no	—
192	C₂×C₃	—	3	1	no	—
193	C₂×C₃	—	3	2	no	—
194	C₂×C₃	—	6	2	no	—
195	C₂×C₃	C₂×C₃²×C₄²	1	1	yes	characteristic
196	C₂×C₃	C₃²×#131	1	2	yes	—
197	C₂²	—	2	2	no	—
198	C₂²	C₄²×3₋¹⁺²	1	2	yes	—
199	C₂²	C₂²×C₄×3₋¹⁺²	1	1	yes	characteristic
200	C₄	—	2	4	no	—
201	C₄	C₄²×3₋¹⁺²	1	2	yes	—
202	C₄	C₂²×C₄×3₋¹⁺²	1	1	yes	characteristic
203	C₄	C₂²×C₄×3₋¹⁺²	1	1	yes	characteristic
204	C₃	—	3	1	no	—
205	C₃	C₃²×#405	1	1	yes	characteristic
206	C₂	—	2	2	no	—
207	C₂	C₂×C₄²×3₋¹⁺²	1	1	yes	characteristic
208	C₂	3₋¹⁺²×#131	1	2	yes	—

	subgroup	G/H
center	C₂×C₃×C₈	C₂²×C₃²
commutator	C₂×C₃	C₂×C₃²×C₄²
Frattini, cosocle	C₂×C₃×C₄	C₂³×C₃²
socle	C₂²×C₃	C₂²×C₃²×C₄
prime radical	C₂³×C₃²	C₂×C₃×C₄
LCS	G ▷ C₂×C₃ ▷ C₁
UCS	C₁ ◁ C₂×C₃×C₈ ◁ G

Complements

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

N	H ≅ G/N	type
4. 3₋¹⁺²×#97	C₂	semidirect
5. C₄×C₈×3₋¹⁺²	C₂	semidirect
7. C₉×#405	C₃	semidirect
9. M₁₆×3₋¹⁺²	C₄	semidirect
10. C₂×C₈×3₋¹⁺²	C₄	semidirect
25. C₉×#97	C₂×C₃	semidirect
26. C₄×C₈×C₉	C₂×C₃	semidirect
48. C₉×M₁₆	C₃×C₄	semidirect
49. C₂×C₈×C₉	C₃×C₄	semidirect
91. #405	3₋¹⁺²	direct (N × H)
130. #97	C₂×3₋¹⁺²	semidirect
131. C₄×C₈	C₂×3₋¹⁺²	semidirect
132. 3₋¹⁺²	#405	direct (N × H)
158. M₁₆	C₄×3₋¹⁺²	semidirect
159. C₂×C₈	C₄×3₋¹⁺²	semidirect
181. C₉	C₃×#405	semidirect

Maximal subgroups

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

subgroup	index
C₂×M₁₆×3₋¹⁺²	2
C₂²×C₈×3₋¹⁺²	2
3₋¹⁺²×#117	2
3₋¹⁺²×#97	2
C₄×C₈×3₋¹⁺²	2
C₃²×#405	3
C₉×#405	3

Subgroups — 3−1+2×#405 #13144975

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — 3₋¹⁺²×#405 #13144975