Subgroups — 3₋¹⁺²×#486 #13145218

Order 1728.

472 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	3₋¹⁺²×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic, complemented
2	3₋¹⁺²×#127	C₂	1	1	yes	maximal, characteristic
3	3₋¹⁺²×#102	C₂	1	4	yes	maximal, complemented
4	C₄×C₈×3₋¹⁺²	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃²×#486	C₃	1	1	yes	maximal, characteristic
6	C₉×#486	C₃	1	3	yes	maximal, complemented
7	C₂×Q₈×3₋¹⁺²	—	2	1	no	—
8	C₂×D₄×3₋¹⁺²	—	2	1	no	—
9	C₂×D₄×3₋¹⁺²	—	2	1	no	—
10	C₂×D₄×3₋¹⁺²	C₂²	1	2	yes	—
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	C₂²	1	2	yes	—
14	C₄²×3₋¹⁺²	C₂²	1	1	yes	characteristic
15	C₃²×C₄⋊Q₈	C₂×C₃	1	1	yes	characteristic
16	C₃²×#127	C₂×C₃	1	1	yes	characteristic
17	C₃²×#102	C₂×C₃	1	4	yes	—
18	C₃²×C₄×C₈	C₂×C₃	1	1	yes	characteristic
19	C₉×C₄⋊Q₈	C₂×C₃	1	3	yes	complemented
20	C₉×#127	C₂×C₃	1	3	yes	—
21	C₉×#102	C₂×C₃	1	12	yes	complemented
22	C₄×C₈×C₉	C₂×C₃	1	3	yes	complemented
23	C₂³×3₋¹⁺²	—	2	2	no	—
24	Q₈×3₋¹⁺²	—	4	2	no	—
25	D₄×3₋¹⁺²	—	2	4	no	—
26	D₄×3₋¹⁺²	—	4	2	no	—
27	D₄×3₋¹⁺²	—	4	2	no	—
28	C₂×C₄×3₋¹⁺²	—	2	2	no	—
29	C₂×C₄×3₋¹⁺²	D₄	1	1	yes	characteristic
30	C₂×C₄×3₋¹⁺²	D₄	1	1	yes	characteristic
31	C₂×C₄×3₋¹⁺²	C₂³	1	1	yes	characteristic
32	C₈×3₋¹⁺²	—	2	2	no	—
33	C₃×#486	—	3	1	no	—
34	C₃×#486	C₃²	1	1	yes	characteristic
35	C₂×C₃²×Q₈	—	2	1	no	—
36	C₂×C₃²×D₄	—	2	1	no	—
37	C₂×C₃²×D₄	—	2	1	no	—
38	C₂×C₃²×D₄	C₂²×C₃	1	2	yes	—
39	C₂×C₃²×C₈	C₂²×C₃	1	2	yes	—
40	C₃²×C₄⋊C₄	—	2	1	no	—
41	C₃²×C₄⋊C₄	C₂²×C₃	1	2	yes	—
42	C₃²×C₄²	C₂²×C₃	1	1	yes	characteristic
43	C₂×Q₈×C₉	—	2	3	no	—
44	C₂×D₄×C₉	—	2	3	no	—
45	C₂×D₄×C₉	—	2	3	no	—
46	C₂×D₄×C₉	C₂²×C₃	1	6	yes	—
47	C₂×C₈×C₉	C₂²×C₃	1	6	yes	—
48	C₉×C₄⋊C₄	—	2	3	no	—
49	C₉×C₄⋊C₄	C₂²×C₃	1	6	yes	—
50	C₄²×C₉	C₂²×C₃	1	3	yes	—
51	C₂²×3₋¹⁺²	—	4	2	no	—
52	C₂²×3₋¹⁺²	—	4	2	no	—
53	C₂²×3₋¹⁺²	—	4	2	no	—
54	C₂²×3₋¹⁺²	C₂×D₄	1	1	yes	characteristic
55	C₄×3₋¹⁺²	—	4	2	no	—
56	C₄×3₋¹⁺²	D₈	1	2	yes	—
57	C₄×3₋¹⁺²	SD₁₆	1	2	yes	—
58	C₄×3₋¹⁺²	#41	1	2	yes	—
59	C₃×C₄⋊Q₈	—	3	1	no	—
60	C₃×C₄⋊Q₈	C₂×C₃²	1	1	yes	characteristic
61	C₃×#127	—	3	1	no	—
62	C₃×#127	C₂×C₃²	1	1	yes	characteristic
63	C₃×#102	—	3	4	no	—
64	C₃×#102	C₂×C₃²	1	4	yes	—
65	C₃×C₄×C₈	—	3	1	no	—
66	C₃×C₄×C₈	C₂×C₃²	1	1	yes	characteristic
67	C₂³×C₃²	—	2	2	no	—
68	C₃²×Q₈	—	4	2	no	—
69	C₃²×D₄	—	2	4	no	—
70	C₃²×D₄	—	4	2	no	—
71	C₃²×D₄	—	4	2	no	—
72	C₂×C₃²×C₄	—	2	2	no	—
73	C₂×C₃²×C₄	C₃×D₄	1	1	yes	characteristic
74	C₂×C₃²×C₄	C₃×D₄	1	1	yes	characteristic
75	C₂×C₃²×C₄	C₂³×C₃	1	1	yes	characteristic
76	C₂³×C₉	—	2	6	no	—
77	C₃²×C₈	—	2	2	no	—
78	Q₈×C₉	—	4	6	no	—
79	D₄×C₉	—	2	12	no	—
80	D₄×C₉	—	4	6	no	—
81	D₄×C₉	—	4	6	no	—
82	C₂×C₄×C₉	—	2	6	no	—
83	C₂×C₄×C₉	C₃×D₄	1	3	yes	—
84	C₂×C₄×C₉	C₃×D₄	1	3	yes	—
85	C₂×C₄×C₉	C₂³×C₃	1	3	yes	—
86	C₈×C₉	—	2	6	no	—
87	#486	3₋¹⁺²	1	1	yes	characteristic, direct product factor, Sylow
88	C₂×3₋¹⁺²	—	8	2	no	—
89	C₂×3₋¹⁺²	#124	1	1	yes	characteristic
90	C₂×3₋¹⁺²	C₂×D₈	1	1	yes	characteristic
91	C₂×3₋¹⁺²	C₂×SD₁₆	1	1	yes	characteristic
92	C₂×C₃×Q₈	—	2	1	no	—
93	C₂×C₃×Q₈	—	6	1	no	—
94	C₂×C₃×D₄	—	2	1	no	—
95	C₂×C₃×D₄	—	2	1	no	—
96	C₂×C₃×D₄	—	3	2	no	—
97	C₂×C₃×D₄	—	6	1	no	—
98	C₂×C₃×D₄	—	6	1	no	—
99	C₂×C₃×D₄	C₂²×C₃²	1	2	yes	—
100	C₂×C₃×C₈	—	3	2	no	—
101	C₂×C₃×C₈	C₂²×C₃²	1	2	yes	—
102	C₃×C₄⋊C₄	—	2	1	no	—
103	C₃×C₄⋊C₄	—	3	2	no	—
104	C₃×C₄⋊C₄	—	6	1	no	—
105	C₃×C₄⋊C₄	C₂²×C₃²	1	2	yes	—
106	C₃×C₄²	—	3	1	no	—
107	C₃×C₄²	C₂²×C₃²	1	1	yes	characteristic
108	C₂²×C₃²	—	4	2	no	—
109	C₂²×C₃²	—	4	2	no	—
110	C₂²×C₃²	—	4	2	no	—
111	C₂²×C₃²	C₂×C₃×D₄	1	1	yes	characteristic
112	C₃²×C₄	—	4	2	no	—
113	C₃²×C₄	C₃×D₈	1	2	yes	—
114	C₃²×C₄	C₃×SD₁₆	1	2	yes	—
115	C₃²×C₄	C₃×#41	1	2	yes	—
116	C₂²×C₉	—	4	6	no	—
117	C₂²×C₉	—	4	6	no	—
118	C₂²×C₉	—	4	6	no	—
119	C₂²×C₉	C₂×C₃×D₄	1	3	yes	—
120	C₄×C₉	—	4	6	no	—
121	C₄×C₉	C₃×D₈	1	6	yes	—
122	C₄×C₉	C₃×SD₁₆	1	6	yes	—
123	C₄×C₉	C₃×#41	1	6	yes	—
124	C₄⋊Q₈	C₂×3₋¹⁺²	1	1	yes	characteristic, complemented
125	#127	C₂×3₋¹⁺²	1	1	yes	characteristic
126	#102	C₂×3₋¹⁺²	1	4	yes	complemented
127	C₄×C₈	C₂×3₋¹⁺²	1	1	yes	characteristic, complemented
128	3₋¹⁺²	#486	1	1	yes	characteristic, direct product factor, Sylow
129	C₂³×C₃	—	2	2	no	—
130	C₂³×C₃	—	6	2	no	—
131	C₃×Q₈	—	4	2	no	—
132	C₃×Q₈	—	12	2	no	—
133	C₃×D₄	—	2	4	no	—
134	C₃×D₄	—	4	2	no	—
135	C₃×D₄	—	4	2	no	—
136	C₃×D₄	—	6	4	no	—
137	C₃×D₄	—	12	2	no	—
138	C₃×D₄	—	12	2	no	—
139	C₂×C₃×C₄	—	2	2	no	—
140	C₂×C₃×C₄	—	3	1	no	—
141	C₂×C₃×C₄	—	3	1	no	—
142	C₂×C₃×C₄	—	3	1	no	—
143	C₂×C₃×C₄	—	6	2	no	—
144	C₂×C₃×C₄	C₃²×D₄	1	1	yes	characteristic
145	C₂×C₃×C₄	C₃²×D₄	1	1	yes	characteristic
146	C₂×C₃×C₄	C₂³×C₃²	1	1	yes	characteristic
147	C₃×C₈	—	2	2	no	—
148	C₃×C₈	—	6	2	no	—
149	C₂×C₃²	—	8	2	no	—
150	C₂×C₃²	C₃×#124	1	1	yes	characteristic
151	C₂×C₃²	C₂×C₃×D₈	1	1	yes	characteristic
152	C₂×C₃²	C₂×C₃×SD₁₆	1	1	yes	characteristic
153	C₂×C₉	—	8	6	no	—
154	C₂×C₉	C₃×#124	1	3	yes	—
155	C₂×C₉	C₂×C₃×D₈	1	3	yes	—
156	C₂×C₉	C₂×C₃×SD₁₆	1	3	yes	—
157	C₂×Q₈	—	2	1	no	—
158	C₂×D₄	—	2	1	no	—
159	C₂×D₄	—	2	1	no	—
160	C₂×D₄	C₂²×3₋¹⁺²	1	2	yes	—
161	C₂×C₈	C₂²×3₋¹⁺²	1	2	yes	—
162	C₄⋊C₄	—	2	1	no	—
163	C₄⋊C₄	C₂²×3₋¹⁺²	1	2	yes	—
164	C₄²	C₂²×3₋¹⁺²	1	1	yes	characteristic
165	C₂²×C₃	—	3	1	no	—
166	C₂²×C₃	—	4	2	no	—
167	C₂²×C₃	—	4	2	no	—
168	C₂²×C₃	—	4	2	no	—
169	C₂²×C₃	—	12	2	no	—
170	C₂²×C₃	—	12	2	no	—
171	C₂²×C₃	—	12	2	no	—
172	C₂²×C₃	C₂×C₃²×D₄	1	1	yes	characteristic
173	C₃×C₄	—	3	2	no	—
174	C₃×C₄	—	3	2	no	—
175	C₃×C₄	—	3	2	no	—
176	C₃×C₄	—	4	2	no	—
177	C₃×C₄	—	12	2	no	—
178	C₃×C₄	C₃²×D₈	1	2	yes	—
179	C₃×C₄	C₃²×SD₁₆	1	2	yes	—
180	C₃×C₄	C₃²×#41	1	2	yes	—
181	C₃²	C₃×#486	1	1	yes	characteristic
182	C₉	C₃×#486	1	3	yes	complemented
183	C₂³	—	2	2	no	—
184	Q₈	—	4	2	no	—
185	D₄	—	2	4	no	—
186	D₄	—	4	2	no	—
187	D₄	—	4	2	no	—
188	C₂×C₄	—	2	2	no	—
189	C₂×C₄	D₄×3₋¹⁺²	1	1	yes	characteristic
190	C₂×C₄	D₄×3₋¹⁺²	1	1	yes	characteristic
191	C₂×C₄	C₂³×3₋¹⁺²	1	1	yes	characteristic
192	C₈	—	2	2	no	—
193	C₂×C₃	—	3	1	no	—
194	C₂×C₃	—	3	1	no	—
195	C₂×C₃	—	3	1	no	—
196	C₂×C₃	—	8	2	no	—
197	C₂×C₃	—	24	2	no	—
198	C₂×C₃	C₃²×#124	1	1	yes	characteristic
199	C₂×C₃	C₂×C₃²×D₈	1	1	yes	characteristic
200	C₂×C₃	C₂×C₃²×SD₁₆	1	1	yes	characteristic
201	C₂²	—	4	2	no	—
202	C₂²	—	4	2	no	—
203	C₂²	—	4	2	no	—
204	C₂²	C₂×D₄×3₋¹⁺²	1	1	yes	characteristic
205	C₄	—	4	2	no	—
206	C₄	D₈×3₋¹⁺²	1	2	yes	—
207	C₄	SD₁₆×3₋¹⁺²	1	2	yes	—
208	C₄	#41×3₋¹⁺²	1	2	yes	—
209	C₃	—	3	1	no	—
210	C₃	C₃²×#486	1	1	yes	characteristic
211	C₂	—	8	2	no	—
212	C₂	3₋¹⁺²×#124	1	1	yes	characteristic
213	C₂	C₂×D₈×3₋¹⁺²	1	1	yes	characteristic
214	C₂	C₂×SD₁₆×3₋¹⁺²	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃	C₂×C₃²×D₄
commutator, Frattini, cosocle	C₂×C₃×C₄	C₂³×C₃²
prime radical	C₃²×#127	C₂×C₃
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
1. 3₋¹⁺²×C₄⋊Q₈	C₂	semidirect
3. 3₋¹⁺²×#102	C₂	semidirect
4. C₄×C₈×3₋¹⁺²	C₂	semidirect
6. C₉×#486	C₃	semidirect
19. C₉×C₄⋊Q₈	C₂×C₃	semidirect
21. C₉×#102	C₂×C₃	semidirect
22. C₄×C₈×C₉	C₂×C₃	semidirect
87. #486	3₋¹⁺²	direct (N × H)
124. C₄⋊Q₈	C₂×3₋¹⁺²	semidirect
126. #102	C₂×3₋¹⁺²	semidirect
127. C₄×C₈	C₂×3₋¹⁺²	semidirect
128. 3₋¹⁺²	#486	direct (N × H)
182. C₉	C₃×#486	semidirect

Maximal subgroups

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

subgroup	index
3₋¹⁺²×C₄⋊Q₈	2
3₋¹⁺²×#127	2
3₋¹⁺²×#102	2
C₄×C₈×3₋¹⁺²	2
C₃²×#486	3
C₉×#486	3

Subgroups — 3−1+2×#486 #13145218

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — 3₋¹⁺²×#486 #13145218