Subgroups — 3₋¹⁺²×#487 #13145221

Order 1728.

392 conjugacy classes of subgroups.

The subgroup lattice has 186 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
2	3₋¹⁺²×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic
3	3₋¹⁺²×#103	C₂	1	4	yes	maximal
4	C₄×C₈×3₋¹⁺²	C₂	1	1	yes	maximal, characteristic
5	C₃²×#487	C₃	1	1	yes	maximal, characteristic
6	C₉×#487	C₃	1	3	yes	maximal, complemented
7	C₂×Q₈×3₋¹⁺²	—	2	1	no	—
8	C₂×Q₈×3₋¹⁺²	C₂²	1	2	yes	—
9	C₂×C₈×3₋¹⁺²	C₂²	1	2	yes	—
10	C₄⋊C₄×3₋¹⁺²	—	2	1	no	—
11	C₄⋊C₄×3₋¹⁺²	—	2	1	no	—
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×C₄⋊Q₈	C₂×C₃	1	1	yes	characteristic
17	C₃²×#103	C₂×C₃	1	4	yes	—
18	C₃²×C₄×C₈	C₂×C₃	1	1	yes	characteristic
19	C₉×C₄⋊Q₈	C₂×C₃	1	3	yes	—
20	C₉×C₄⋊Q₈	C₂×C₃	1	3	yes	—
21	C₉×#103	C₂×C₃	1	12	yes	—
22	C₄×C₈×C₉	C₂×C₃	1	3	yes	—
23	Q₈×3₋¹⁺²	—	2	4	no	—
24	Q₈×3₋¹⁺²	—	4	2	no	—
25	C₂×C₄×3₋¹⁺²	—	2	2	no	—
26	C₂×C₄×3₋¹⁺²	—	2	2	no	—
27	C₂×C₄×3₋¹⁺²	D₄	1	1	yes	characteristic
28	C₂×C₄×3₋¹⁺²	D₄	1	1	yes	characteristic
29	C₂×C₄×3₋¹⁺²	C₂³	1	1	yes	characteristic
30	C₈×3₋¹⁺²	—	2	2	no	—
31	C₃×#487	—	3	1	no	—
32	C₃×#487	C₃²	1	1	yes	characteristic
33	C₂×C₃²×Q₈	—	2	1	no	—
34	C₂×C₃²×Q₈	C₂²×C₃	1	2	yes	—
35	C₂×C₃²×C₈	C₂²×C₃	1	2	yes	—
36	C₃²×C₄⋊C₄	—	2	1	no	—
37	C₃²×C₄⋊C₄	—	2	1	no	—
38	C₃²×C₄⋊C₄	—	2	1	no	—
39	C₃²×C₄⋊C₄	C₂²×C₃	1	2	yes	—
40	C₃²×C₄²	C₂²×C₃	1	1	yes	characteristic
41	C₂×Q₈×C₉	—	2	3	no	—
42	C₂×Q₈×C₉	C₂²×C₃	1	6	yes	—
43	C₂×C₈×C₉	C₂²×C₃	1	6	yes	—
44	C₉×C₄⋊C₄	—	2	3	no	—
45	C₉×C₄⋊C₄	—	2	3	no	—
46	C₉×C₄⋊C₄	—	2	3	no	—
47	C₉×C₄⋊C₄	C₂²×C₃	1	6	yes	—
48	C₄²×C₉	C₂²×C₃	1	3	yes	—
49	C₂²×3₋¹⁺²	C₂×D₄	1	1	yes	characteristic
50	C₄×3₋¹⁺²	—	4	2	no	—
51	C₄×3₋¹⁺²	—	4	2	no	—
52	C₄×3₋¹⁺²	SD₁₆	1	2	yes	—
53	C₄×3₋¹⁺²	Q₁₆	1	2	yes	—
54	C₄×3₋¹⁺²	#41	1	2	yes	—
55	C₃×C₄⋊Q₈	—	3	1	no	—
56	C₃×C₄⋊Q₈	—	3	1	no	—
57	C₃×C₄⋊Q₈	C₂×C₃²	1	1	yes	characteristic
58	C₃×C₄⋊Q₈	C₂×C₃²	1	1	yes	characteristic
59	C₃×#103	—	3	4	no	—
60	C₃×#103	C₂×C₃²	1	4	yes	—
61	C₃×C₄×C₈	—	3	1	no	—
62	C₃×C₄×C₈	C₂×C₃²	1	1	yes	characteristic
63	C₃²×Q₈	—	2	4	no	—
64	C₃²×Q₈	—	4	2	no	—
65	C₂×C₃²×C₄	—	2	2	no	—
66	C₂×C₃²×C₄	—	2	2	no	—
67	C₂×C₃²×C₄	C₃×D₄	1	1	yes	characteristic
68	C₂×C₃²×C₄	C₃×D₄	1	1	yes	characteristic
69	C₂×C₃²×C₄	C₂³×C₃	1	1	yes	characteristic
70	C₃²×C₈	—	2	2	no	—
71	Q₈×C₉	—	2	12	no	—
72	Q₈×C₉	—	4	6	no	—
73	C₂×C₄×C₉	—	2	6	no	—
74	C₂×C₄×C₉	—	2	6	no	—
75	C₂×C₄×C₉	C₃×D₄	1	3	yes	—
76	C₂×C₄×C₉	C₃×D₄	1	3	yes	—
77	C₂×C₄×C₉	C₂³×C₃	1	3	yes	—
78	C₈×C₉	—	2	6	no	—
79	#487	3₋¹⁺²	1	1	yes	characteristic, direct product factor, Sylow
80	C₂×3₋¹⁺²	#124	1	1	yes	characteristic
81	C₂×3₋¹⁺²	C₂×SD₁₆	1	1	yes	characteristic
82	C₂×3₋¹⁺²	C₂×Q₁₆	1	1	yes	characteristic
83	C₂×C₃×Q₈	—	2	1	no	—
84	C₂×C₃×Q₈	—	3	2	no	—
85	C₂×C₃×Q₈	—	6	1	no	—
86	C₂×C₃×Q₈	C₂²×C₃²	1	2	yes	—
87	C₂×C₃×C₈	—	3	2	no	—
88	C₂×C₃×C₈	C₂²×C₃²	1	2	yes	—
89	C₃×C₄⋊C₄	—	2	1	no	—
90	C₃×C₄⋊C₄	—	2	1	no	—
91	C₃×C₄⋊C₄	—	2	1	no	—
92	C₃×C₄⋊C₄	—	3	2	no	—
93	C₃×C₄⋊C₄	—	6	1	no	—
94	C₃×C₄⋊C₄	—	6	1	no	—
95	C₃×C₄⋊C₄	—	6	1	no	—
96	C₃×C₄⋊C₄	C₂²×C₃²	1	2	yes	—
97	C₃×C₄²	—	3	1	no	—
98	C₃×C₄²	C₂²×C₃²	1	1	yes	characteristic
99	C₂²×C₃²	C₂×C₃×D₄	1	1	yes	characteristic
100	C₃²×C₄	—	4	2	no	—
101	C₃²×C₄	—	4	2	no	—
102	C₃²×C₄	C₃×SD₁₆	1	2	yes	—
103	C₃²×C₄	C₃×Q₁₆	1	2	yes	—
104	C₃²×C₄	C₃×#41	1	2	yes	—
105	C₂²×C₉	C₂×C₃×D₄	1	3	yes	—
106	C₄×C₉	—	4	6	no	—
107	C₄×C₉	—	4	6	no	—
108	C₄×C₉	C₃×SD₁₆	1	6	yes	—
109	C₄×C₉	C₃×Q₁₆	1	6	yes	—
110	C₄×C₉	C₃×#41	1	6	yes	—
111	C₄⋊Q₈	C₂×3₋¹⁺²	1	1	yes	characteristic
112	C₄⋊Q₈	C₂×3₋¹⁺²	1	1	yes	characteristic
113	#103	C₂×3₋¹⁺²	1	4	yes	—
114	C₄×C₈	C₂×3₋¹⁺²	1	1	yes	characteristic
115	3₋¹⁺²	#487	1	1	yes	characteristic, direct product factor, Sylow
116	C₃×Q₈	—	2	4	no	—
117	C₃×Q₈	—	4	2	no	—
118	C₃×Q₈	—	6	4	no	—
119	C₃×Q₈	—	12	2	no	—
120	C₂×C₃×C₄	—	2	2	no	—
121	C₂×C₃×C₄	—	2	2	no	—
122	C₂×C₃×C₄	—	3	1	no	—
123	C₂×C₃×C₄	—	3	1	no	—
124	C₂×C₃×C₄	—	3	1	no	—
125	C₂×C₃×C₄	—	6	2	no	—
126	C₂×C₃×C₄	—	6	2	no	—
127	C₂×C₃×C₄	C₃²×D₄	1	1	yes	characteristic
128	C₂×C₃×C₄	C₃²×D₄	1	1	yes	characteristic
129	C₂×C₃×C₄	C₂³×C₃²	1	1	yes	characteristic
130	C₃×C₈	—	2	2	no	—
131	C₃×C₈	—	6	2	no	—
132	C₂×C₃²	C₃×#124	1	1	yes	characteristic
133	C₂×C₃²	C₂×C₃×SD₁₆	1	1	yes	characteristic
134	C₂×C₃²	C₂×C₃×Q₁₆	1	1	yes	characteristic
135	C₂×C₉	C₃×#124	1	3	yes	—
136	C₂×C₉	C₂×C₃×SD₁₆	1	3	yes	—
137	C₂×C₉	C₂×C₃×Q₁₆	1	3	yes	—
138	C₂×Q₈	—	2	1	no	—
139	C₂×Q₈	C₂²×3₋¹⁺²	1	2	yes	—
140	C₂×C₈	C₂²×3₋¹⁺²	1	2	yes	—
141	C₄⋊C₄	—	2	1	no	—
142	C₄⋊C₄	—	2	1	no	—
143	C₄⋊C₄	—	2	1	no	—
144	C₄⋊C₄	C₂²×3₋¹⁺²	1	2	yes	—
145	C₄²	C₂²×3₋¹⁺²	1	1	yes	characteristic
146	C₂²×C₃	—	3	1	no	—
147	C₂²×C₃	C₂×C₃²×D₄	1	1	yes	characteristic
148	C₃×C₄	—	3	2	no	—
149	C₃×C₄	—	3	2	no	—
150	C₃×C₄	—	3	2	no	—
151	C₃×C₄	—	4	2	no	—
152	C₃×C₄	—	4	2	no	—
153	C₃×C₄	—	12	2	no	—
154	C₃×C₄	—	12	2	no	—
155	C₃×C₄	C₃²×SD₁₆	1	2	yes	—
156	C₃×C₄	C₃²×Q₁₆	1	2	yes	—
157	C₃×C₄	C₃²×#41	1	2	yes	—
158	C₃²	C₃×#487	1	1	yes	characteristic
159	C₉	C₃×#487	1	3	yes	complemented
160	Q₈	—	2	4	no	—
161	Q₈	—	4	2	no	—
162	C₂×C₄	—	2	2	no	—
163	C₂×C₄	—	2	2	no	—
164	C₂×C₄	D₄×3₋¹⁺²	1	1	yes	characteristic
165	C₂×C₄	D₄×3₋¹⁺²	1	1	yes	characteristic
166	C₂×C₄	C₂³×3₋¹⁺²	1	1	yes	characteristic
167	C₈	—	2	2	no	—
168	C₂×C₃	—	3	1	no	—
169	C₂×C₃	—	3	1	no	—
170	C₂×C₃	—	3	1	no	—
171	C₂×C₃	C₃²×#124	1	1	yes	characteristic
172	C₂×C₃	C₂×C₃²×SD₁₆	1	1	yes	characteristic
173	C₂×C₃	C₂×C₃²×Q₁₆	1	1	yes	characteristic
174	C₂²	C₂×D₄×3₋¹⁺²	1	1	yes	characteristic
175	C₄	—	4	2	no	—
176	C₄	—	4	2	no	—
177	C₄	SD₁₆×3₋¹⁺²	1	2	yes	—
178	C₄	Q₁₆×3₋¹⁺²	1	2	yes	—
179	C₄	#41×3₋¹⁺²	1	2	yes	—
180	C₃	—	3	1	no	—
181	C₃	C₃²×#487	1	1	yes	characteristic
182	C₂	3₋¹⁺²×#124	1	1	yes	characteristic
183	C₂	C₂×SD₁₆×3₋¹⁺²	1	1	yes	characteristic
184	C₂	C₂×Q₁₆×3₋¹⁺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	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
6. C₉×#487	C₃	semidirect
79. #487	3₋¹⁺²	direct (N × H)
115. 3₋¹⁺²	#487	direct (N × H)
159. C₉	C₃×#487	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×#103	2
C₄×C₈×3₋¹⁺²	2
C₃²×#487	3
C₉×#487	3

Subgroups — 3−1+2×#487 #13145221

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — 3₋¹⁺²×#487 #13145221