Subgroups — 3₋¹⁺²×#462 #13145146

Order 1728.

432 conjugacy classes of subgroups.

The subgroup lattice has 234 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Q₁₆×3₋¹⁺²	C₂	1	2	yes	maximal
2	3₋¹⁺²×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic
3	C₄×Q₈×3₋¹⁺²	C₂	1	1	yes	maximal, characteristic
4	3₋¹⁺²×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic
5	3₋¹⁺²×#103	C₂	1	2	yes	maximal
6	C₃²×#462	C₃	1	1	yes	maximal, characteristic
7	C₉×#462	C₃	1	3	yes	maximal, complemented
8	C₂×Q₈×3₋¹⁺²	C₂²	1	2	yes	—
9	C₂×Q₈×3₋¹⁺²	C₂²	1	1	yes	characteristic
10	Q₁₆×3₋¹⁺²	—	2	4	no	—
11	C₂×C₈×3₋¹⁺²	C₂²	1	2	yes	—
12	C₄⋊C₄×3₋¹⁺²	—	2	1	no	—
13	C₄⋊C₄×3₋¹⁺²	—	2	1	no	—
14	C₄⋊C₄×3₋¹⁺²	—	2	1	no	—
15	C₄⋊C₄×3₋¹⁺²	C₂²	1	1	yes	characteristic
16	C₄²×3₋¹⁺²	—	2	1	no	—
17	C₄²×3₋¹⁺²	C₂²	1	1	yes	characteristic
18	C₂×C₃²×Q₁₆	C₂×C₃	1	2	yes	—
19	C₃²×C₄⋊Q₈	C₂×C₃	1	1	yes	characteristic
20	C₃²×C₄×Q₈	C₂×C₃	1	1	yes	characteristic
21	C₃²×C₄⋊C₈	C₂×C₃	1	1	yes	characteristic
22	C₃²×#103	C₂×C₃	1	2	yes	—
23	C₂×C₉×Q₁₆	C₂×C₃	1	6	yes	—
24	C₉×C₄⋊Q₈	C₂×C₃	1	3	yes	—
25	C₄×Q₈×C₉	C₂×C₃	1	3	yes	—
26	C₉×C₄⋊C₈	C₂×C₃	1	3	yes	—
27	C₉×#103	C₂×C₃	1	6	yes	—
28	Q₈×3₋¹⁺²	—	2	1	no	—
29	Q₈×3₋¹⁺²	—	2	2	no	—
30	Q₈×3₋¹⁺²	—	2	2	no	—
31	Q₈×3₋¹⁺²	D₄	1	2	yes	—
32	C₂×C₄×3₋¹⁺²	—	2	1	no	—
33	C₂×C₄×3₋¹⁺²	—	2	2	no	—
34	C₂×C₄×3₋¹⁺²	—	2	1	no	—
35	C₂×C₄×3₋¹⁺²	D₄	1	1	yes	characteristic
36	C₂×C₄×3₋¹⁺²	D₄	1	1	yes	characteristic
37	C₂×C₄×3₋¹⁺²	C₂³	1	1	yes	characteristic
38	C₈×3₋¹⁺²	—	2	2	no	—
39	C₃×#462	—	3	1	no	—
40	C₃×#462	C₃²	1	1	yes	characteristic
41	C₂×C₃²×Q₈	C₂²×C₃	1	2	yes	—
42	C₂×C₃²×Q₈	C₂²×C₃	1	1	yes	characteristic
43	C₃²×Q₁₆	—	2	4	no	—
44	C₂×C₃²×C₈	C₂²×C₃	1	2	yes	—
45	C₃²×C₄⋊C₄	—	2	1	no	—
46	C₃²×C₄⋊C₄	—	2	1	no	—
47	C₃²×C₄⋊C₄	—	2	1	no	—
48	C₃²×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
49	C₃²×C₄²	—	2	1	no	—
50	C₃²×C₄²	C₂²×C₃	1	1	yes	characteristic
51	C₂×Q₈×C₉	C₂²×C₃	1	6	yes	—
52	C₂×Q₈×C₉	C₂²×C₃	1	3	yes	—
53	C₉×Q₁₆	—	2	12	no	—
54	C₂×C₈×C₉	C₂²×C₃	1	6	yes	—
55	C₉×C₄⋊C₄	—	2	3	no	—
56	C₉×C₄⋊C₄	—	2	3	no	—
57	C₉×C₄⋊C₄	—	2	3	no	—
58	C₉×C₄⋊C₄	C₂²×C₃	1	3	yes	—
59	C₄²×C₉	—	2	3	no	—
60	C₄²×C₉	C₂²×C₃	1	3	yes	—
61	C₂²×3₋¹⁺²	C₂×D₄	1	1	yes	characteristic
62	C₄×3₋¹⁺²	—	2	2	no	—
63	C₄×3₋¹⁺²	—	2	1	no	—
64	C₄×3₋¹⁺²	—	4	2	no	—
65	C₄×3₋¹⁺²	—	4	1	no	—
66	C₄×3₋¹⁺²	Q₁₆	1	2	yes	complemented
67	C₄×3₋¹⁺²	C₂×D₄	1	1	yes	characteristic
68	C₄×3₋¹⁺²	#41	1	1	yes	characteristic
69	C₂×C₃×Q₁₆	—	3	2	no	—
70	C₂×C₃×Q₁₆	C₂×C₃²	1	2	yes	—
71	C₃×C₄⋊Q₈	—	3	1	no	—
72	C₃×C₄⋊Q₈	C₂×C₃²	1	1	yes	characteristic
73	C₃×C₄×Q₈	—	3	1	no	—
74	C₃×C₄×Q₈	C₂×C₃²	1	1	yes	characteristic
75	C₃×C₄⋊C₈	—	3	1	no	—
76	C₃×C₄⋊C₈	C₂×C₃²	1	1	yes	characteristic
77	C₃×#103	—	3	2	no	—
78	C₃×#103	C₂×C₃²	1	2	yes	—
79	C₃²×Q₈	—	2	1	no	—
80	C₃²×Q₈	—	2	2	no	—
81	C₃²×Q₈	—	2	2	no	—
82	C₃²×Q₈	C₃×D₄	1	2	yes	—
83	C₂×C₃²×C₄	—	2	1	no	—
84	C₂×C₃²×C₄	—	2	1	no	—
85	C₂×C₃²×C₄	—	2	2	no	—
86	C₂×C₃²×C₄	C₃×D₄	1	1	yes	characteristic
87	C₂×C₃²×C₄	C₃×D₄	1	1	yes	characteristic
88	C₂×C₃²×C₄	C₂³×C₃	1	1	yes	characteristic
89	C₃²×C₈	—	2	2	no	—
90	Q₈×C₉	—	2	3	no	—
91	Q₈×C₉	—	2	6	no	—
92	Q₈×C₉	—	2	6	no	—
93	Q₈×C₉	C₃×D₄	1	6	yes	—
94	C₂×C₄×C₉	—	2	3	no	—
95	C₂×C₄×C₉	—	2	3	no	—
96	C₂×C₄×C₉	—	2	6	no	—
97	C₂×C₄×C₉	C₃×D₄	1	3	yes	—
98	C₂×C₄×C₉	C₃×D₄	1	3	yes	—
99	C₂×C₄×C₉	C₂³×C₃	1	3	yes	—
100	C₈×C₉	—	2	6	no	—
101	#462	3₋¹⁺²	1	1	yes	characteristic, direct product factor, Sylow
102	C₂×3₋¹⁺²	#121	1	1	yes	characteristic
103	C₂×3₋¹⁺²	C₂×Q₁₆	1	1	yes	characteristic
104	C₂×3₋¹⁺²	#137	1	1	yes	characteristic
105	C₂×C₃×Q₈	—	3	1	no	—
106	C₂×C₃×Q₈	—	3	2	no	—
107	C₂×C₃×Q₈	C₂²×C₃²	1	2	yes	—
108	C₂×C₃×Q₈	C₂²×C₃²	1	1	yes	characteristic
109	C₃×Q₁₆	—	2	4	no	—
110	C₃×Q₁₆	—	6	4	no	—
111	C₂×C₃×C₈	—	3	2	no	—
112	C₂×C₃×C₈	C₂²×C₃²	1	2	yes	—
113	C₃×C₄⋊C₄	—	2	1	no	—
114	C₃×C₄⋊C₄	—	2	1	no	—
115	C₃×C₄⋊C₄	—	2	1	no	—
116	C₃×C₄⋊C₄	—	3	1	no	—
117	C₃×C₄⋊C₄	—	6	1	no	—
118	C₃×C₄⋊C₄	—	6	1	no	—
119	C₃×C₄⋊C₄	—	6	1	no	—
120	C₃×C₄⋊C₄	C₂²×C₃²	1	1	yes	characteristic
121	C₃×C₄²	—	2	1	no	—
122	C₃×C₄²	—	3	1	no	—
123	C₃×C₄²	—	6	1	no	—
124	C₃×C₄²	C₂²×C₃²	1	1	yes	characteristic
125	C₂²×C₃²	C₂×C₃×D₄	1	1	yes	characteristic
126	C₃²×C₄	—	2	2	no	—
127	C₃²×C₄	—	2	1	no	—
128	C₃²×C₄	—	4	2	no	—
129	C₃²×C₄	—	4	1	no	—
130	C₃²×C₄	C₃×Q₁₆	1	2	yes	—
131	C₃²×C₄	C₂×C₃×D₄	1	1	yes	characteristic
132	C₃²×C₄	C₃×#41	1	1	yes	characteristic
133	C₂²×C₉	C₂×C₃×D₄	1	3	yes	—
134	C₄×C₉	—	2	6	no	—
135	C₄×C₉	—	2	3	no	—
136	C₄×C₉	—	4	6	no	—
137	C₄×C₉	—	4	3	no	—
138	C₄×C₉	C₃×Q₁₆	1	6	yes	complemented
139	C₄×C₉	C₂×C₃×D₄	1	3	yes	—
140	C₄×C₉	C₃×#41	1	3	yes	—
141	C₂×Q₁₆	C₂×3₋¹⁺²	1	2	yes	—
142	C₄⋊Q₈	C₂×3₋¹⁺²	1	1	yes	characteristic
143	C₄×Q₈	C₂×3₋¹⁺²	1	1	yes	characteristic
144	C₄⋊C₈	C₂×3₋¹⁺²	1	1	yes	characteristic
145	#103	C₂×3₋¹⁺²	1	2	yes	—
146	3₋¹⁺²	#462	1	1	yes	characteristic, direct product factor, Sylow
147	C₃×Q₈	—	2	1	no	—
148	C₃×Q₈	—	2	2	no	—
149	C₃×Q₈	—	2	2	no	—
150	C₃×Q₈	—	3	2	no	—
151	C₃×Q₈	—	6	1	no	—
152	C₃×Q₈	—	6	2	no	—
153	C₃×Q₈	—	6	2	no	—
154	C₃×Q₈	C₃²×D₄	1	2	yes	—
155	C₂×C₃×C₄	—	2	1	no	—
156	C₂×C₃×C₄	—	2	1	no	—
157	C₂×C₃×C₄	—	2	2	no	—
158	C₂×C₃×C₄	—	3	1	no	—
159	C₂×C₃×C₄	—	3	1	no	—
160	C₂×C₃×C₄	—	3	1	no	—
161	C₂×C₃×C₄	—	6	1	no	—
162	C₂×C₃×C₄	—	6	1	no	—
163	C₂×C₃×C₄	—	6	2	no	—
164	C₂×C₃×C₄	C₃²×D₄	1	1	yes	characteristic
165	C₂×C₃×C₄	C₃²×D₄	1	1	yes	characteristic
166	C₂×C₃×C₄	C₂³×C₃²	1	1	yes	characteristic
167	C₃×C₈	—	2	2	no	—
168	C₃×C₈	—	6	2	no	—
169	C₂×C₃²	C₃×#121	1	1	yes	characteristic
170	C₂×C₃²	C₂×C₃×Q₁₆	1	1	yes	characteristic
171	C₂×C₃²	C₃×#137	1	1	yes	characteristic
172	C₂×C₉	C₃×#121	1	3	yes	—
173	C₂×C₉	C₂×C₃×Q₁₆	1	3	yes	—
174	C₂×C₉	C₃×#137	1	3	yes	—
175	C₂×Q₈	C₂²×3₋¹⁺²	1	2	yes	—
176	C₂×Q₈	C₂²×3₋¹⁺²	1	1	yes	characteristic
177	Q₁₆	—	2	4	no	—
178	C₂×C₈	C₂²×3₋¹⁺²	1	2	yes	—
179	C₄⋊C₄	—	2	1	no	—
180	C₄⋊C₄	—	2	1	no	—
181	C₄⋊C₄	—	2	1	no	—
182	C₄⋊C₄	C₂²×3₋¹⁺²	1	1	yes	characteristic
183	C₄²	—	2	1	no	—
184	C₄²	C₂²×3₋¹⁺²	1	1	yes	characteristic
185	C₂²×C₃	—	3	1	no	—
186	C₂²×C₃	C₂×C₃²×D₄	1	1	yes	characteristic
187	C₃×C₄	—	2	2	no	—
188	C₃×C₄	—	2	1	no	—
189	C₃×C₄	—	3	2	no	—
190	C₃×C₄	—	3	1	no	—
191	C₃×C₄	—	3	1	no	—
192	C₃×C₄	—	4	2	no	—
193	C₃×C₄	—	4	1	no	—
194	C₃×C₄	—	6	1	no	—
195	C₃×C₄	—	6	2	no	—
196	C₃×C₄	—	12	2	no	—
197	C₃×C₄	—	12	1	no	—
198	C₃×C₄	C₃²×Q₁₆	1	2	yes	—
199	C₃×C₄	C₂×C₃²×D₄	1	1	yes	characteristic
200	C₃×C₄	C₃²×#41	1	1	yes	characteristic
201	C₃²	C₃×#462	1	1	yes	characteristic
202	C₉	C₃×#462	1	3	yes	complemented
203	Q₈	—	2	1	no	—
204	Q₈	—	2	2	no	—
205	Q₈	—	2	2	no	—
206	Q₈	D₄×3₋¹⁺²	1	2	yes	—
207	C₂×C₄	—	2	1	no	—
208	C₂×C₄	—	2	1	no	—
209	C₂×C₄	—	2	2	no	—
210	C₂×C₄	D₄×3₋¹⁺²	1	1	yes	characteristic
211	C₂×C₄	D₄×3₋¹⁺²	1	1	yes	characteristic
212	C₂×C₄	C₂³×3₋¹⁺²	1	1	yes	characteristic
213	C₈	—	2	2	no	—
214	C₂×C₃	—	3	1	no	—
215	C₂×C₃	—	3	1	no	—
216	C₂×C₃	—	3	1	no	—
217	C₂×C₃	C₃²×#121	1	1	yes	characteristic
218	C₂×C₃	C₂×C₃²×Q₁₆	1	1	yes	characteristic
219	C₂×C₃	C₃²×#137	1	1	yes	characteristic
220	C₂²	C₂×D₄×3₋¹⁺²	1	1	yes	characteristic
221	C₄	—	2	2	no	—
222	C₄	—	2	1	no	—
223	C₄	—	4	2	no	—
224	C₄	—	4	1	no	—
225	C₄	Q₁₆×3₋¹⁺²	1	2	yes	complemented
226	C₄	C₂×D₄×3₋¹⁺²	1	1	yes	characteristic
227	C₄	#41×3₋¹⁺²	1	1	yes	characteristic
228	C₃	—	3	1	no	—
229	C₃	C₃²×#462	1	1	yes	characteristic
230	C₂	3₋¹⁺²×#121	1	1	yes	characteristic
231	C₂	C₂×Q₁₆×3₋¹⁺²	1	1	yes	characteristic
232	C₂	3₋¹⁺²×#137	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃	C₂×C₃²×D₄
commutator, Frattini, cosocle	C₂×C₃×C₄	C₂³×C₃²
prime radical	C₂²×C₃²	C₂×C₃×D₄
LCS	G ▷ C₂×C₃×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃ ◁ C₄²×3₋¹⁺² ◁ G

Complements

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

N	H ≅ G/N	type
7. C₉×#462	C₃	semidirect
66. C₄×3₋¹⁺²	Q₁₆	semidirect
101. #462	3₋¹⁺²	direct (N × H)
138. C₄×C₉	C₃×Q₁₆	semidirect
146. 3₋¹⁺²	#462	direct (N × H)
202. C₉	C₃×#462	semidirect
225. C₄	Q₁₆×3₋¹⁺²	semidirect

Maximal subgroups

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

subgroup	index
C₂×Q₁₆×3₋¹⁺²	2
3₋¹⁺²×C₄⋊Q₈	2
C₄×Q₈×3₋¹⁺²	2
3₋¹⁺²×C₄⋊C₈	2
3₋¹⁺²×#103	2
C₃²×#462	3
C₉×#462	3

Subgroups — 3−1+2×#462 #13145146

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — 3₋¹⁺²×#462 #13145146