Subgroups — 3₋¹⁺²×#423 #13145029

Order 1728.

504 conjugacy classes of subgroups.

The subgroup lattice has 192 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	2	yes	maximal
2	C₂×C₄²×3₋¹⁺²	C₂	1	1	yes	maximal, characteristic
3	3₋¹⁺²×C₄⋊C₈	C₂	1	4	yes	maximal, complemented
4	C₃²×#423	C₃	1	1	yes	maximal, characteristic
5	C₉×#423	C₃	1	3	yes	maximal, complemented
6	C₂²×C₄×3₋¹⁺²	C₄	1	2	yes	—
7	C₂²×C₄×3₋¹⁺²	C₂²	1	1	yes	characteristic
8	M₁₆×3₋¹⁺²	—	2	4	no	—
9	C₂×C₈×3₋¹⁺²	C₂²	1	4	yes	—
10	C₄²×3₋¹⁺²	C₄	1	1	yes	characteristic
11	C₄²×3₋¹⁺²	C₄	1	1	yes	characteristic
12	C₄²×3₋¹⁺²	C₂²	1	2	yes	—
13	C₂×C₃²×M₁₆	C₂×C₃	1	2	yes	—
14	C₂×C₃²×C₄²	C₂×C₃	1	1	yes	characteristic
15	C₃²×C₄⋊C₈	C₂×C₃	1	4	yes	—
16	C₂×C₉×M₁₆	C₂×C₃	1	6	yes	—
17	C₂×C₄²×C₉	C₂×C₃	1	3	yes	—
18	C₉×C₄⋊C₈	C₂×C₃	1	12	yes	complemented
19	C₂³×3₋¹⁺²	C₂×C₄	1	1	yes	characteristic
20	C₂×C₄×3₋¹⁺²	—	2	4	no	—
21	C₂×C₄×3₋¹⁺²	C₂×C₄	1	2	yes	—
22	C₂×C₄×3₋¹⁺²	C₂×C₄	1	1	yes	characteristic
23	C₂×C₄×3₋¹⁺²	C₂×C₄	1	2	yes	—
24	C₂×C₄×3₋¹⁺²	D₄	1	2	yes	—
25	C₂×C₄×3₋¹⁺²	Q₈	1	2	yes	—
26	C₂×C₄×3₋¹⁺²	C₂³	1	1	yes	characteristic
27	C₈×3₋¹⁺²	—	2	4	no	—
28	C₃×#423	—	3	1	no	—
29	C₃×#423	C₃²	1	1	yes	characteristic
30	C₂²×C₃²×C₄	C₃×C₄	1	2	yes	—
31	C₂²×C₃²×C₄	C₂²×C₃	1	1	yes	characteristic
32	C₃²×M₁₆	—	2	4	no	—
33	C₂×C₃²×C₈	C₂²×C₃	1	4	yes	—
34	C₃²×C₄²	C₃×C₄	1	1	yes	characteristic
35	C₃²×C₄²	C₃×C₄	1	1	yes	characteristic
36	C₃²×C₄²	C₂²×C₃	1	2	yes	—
37	C₂²×C₄×C₉	C₃×C₄	1	6	yes	—
38	C₂²×C₄×C₉	C₂²×C₃	1	3	yes	—
39	C₉×M₁₆	—	2	12	no	—
40	C₂×C₈×C₉	C₂²×C₃	1	12	yes	—
41	C₄²×C₉	C₃×C₄	1	3	yes	—
42	C₄²×C₉	C₃×C₄	1	3	yes	—
43	C₄²×C₉	C₂²×C₃	1	6	yes	—
44	C₂²×3₋¹⁺²	—	2	2	no	—
45	C₂²×3₋¹⁺²	C₄⋊C₄	1	2	yes	—
46	C₂²×3₋¹⁺²	C₂²×C₄	1	1	yes	characteristic
47	C₄×3₋¹⁺²	—	2	2	no	—
48	C₄×3₋¹⁺²	C₄⋊C₄	1	2	yes	—
49	C₄×3₋¹⁺²	M₁₆	1	4	yes	complemented
50	C₄×3₋¹⁺²	C₂×D₄	1	1	yes	characteristic
51	C₄×3₋¹⁺²	C₂×Q₈	1	1	yes	characteristic
52	C₂×C₃×M₁₆	—	3	2	no	—
53	C₂×C₃×M₁₆	C₂×C₃²	1	2	yes	—
54	C₂×C₃×C₄²	—	3	1	no	—
55	C₂×C₃×C₄²	C₂×C₃²	1	1	yes	characteristic
56	C₃×C₄⋊C₈	—	3	4	no	—
57	C₃×C₄⋊C₈	C₂×C₃²	1	4	yes	—
58	C₂³×C₃²	C₂×C₃×C₄	1	1	yes	characteristic
59	C₂×C₃²×C₄	—	2	4	no	—
60	C₂×C₃²×C₄	C₂×C₃×C₄	1	2	yes	—
61	C₂×C₃²×C₄	C₂×C₃×C₄	1	1	yes	characteristic
62	C₂×C₃²×C₄	C₂×C₃×C₄	1	2	yes	—
63	C₂×C₃²×C₄	C₃×D₄	1	2	yes	—
64	C₂×C₃²×C₄	C₃×Q₈	1	2	yes	—
65	C₂×C₃²×C₄	C₂³×C₃	1	1	yes	characteristic
66	C₂³×C₉	C₂×C₃×C₄	1	3	yes	—
67	C₃²×C₈	—	2	4	no	—
68	C₂×C₄×C₉	—	2	12	no	—
69	C₂×C₄×C₉	C₂×C₃×C₄	1	6	yes	—
70	C₂×C₄×C₉	C₂×C₃×C₄	1	3	yes	—
71	C₂×C₄×C₉	C₂×C₃×C₄	1	6	yes	—
72	C₂×C₄×C₉	C₃×D₄	1	6	yes	—
73	C₂×C₄×C₉	C₃×Q₈	1	6	yes	—
74	C₂×C₄×C₉	C₂³×C₃	1	3	yes	—
75	C₈×C₉	—	2	12	no	—
76	#423	3₋¹⁺²	1	1	yes	characteristic, direct product factor, Sylow
77	C₂×3₋¹⁺²	—	2	2	no	—
78	C₂×3₋¹⁺²	C₂×C₄⋊C₄	1	1	yes	characteristic
79	C₂×3₋¹⁺²	C₂×M₁₆	1	2	yes	—
80	C₂²×C₃×C₄	—	3	2	no	—
81	C₂²×C₃×C₄	—	3	1	no	—
82	C₂²×C₃×C₄	C₃²×C₄	1	2	yes	—
83	C₂²×C₃×C₄	C₂²×C₃²	1	1	yes	characteristic
84	C₃×M₁₆	—	2	4	no	—
85	C₃×M₁₆	—	6	4	no	—
86	C₂×C₃×C₈	—	3	4	no	—
87	C₂×C₃×C₈	C₂²×C₃²	1	4	yes	—
88	C₃×C₄²	—	3	1	no	—
89	C₃×C₄²	—	3	1	no	—
90	C₃×C₄²	—	3	2	no	—
91	C₃×C₄²	C₃²×C₄	1	1	yes	characteristic
92	C₃×C₄²	C₃²×C₄	1	1	yes	characteristic
93	C₃×C₄²	C₂²×C₃²	1	2	yes	—
94	C₂²×C₃²	—	2	2	no	—
95	C₂²×C₃²	C₃×C₄⋊C₄	1	2	yes	—
96	C₂²×C₃²	C₂²×C₃×C₄	1	1	yes	characteristic
97	C₃²×C₄	—	2	2	no	—
98	C₃²×C₄	C₃×C₄⋊C₄	1	2	yes	—
99	C₃²×C₄	C₃×M₁₆	1	4	yes	—
100	C₃²×C₄	C₂×C₃×D₄	1	1	yes	characteristic
101	C₃²×C₄	C₂×C₃×Q₈	1	1	yes	characteristic
102	C₂²×C₉	—	2	6	no	—
103	C₂²×C₉	C₃×C₄⋊C₄	1	6	yes	—
104	C₂²×C₉	C₂²×C₃×C₄	1	3	yes	—
105	C₄×C₉	—	2	6	no	—
106	C₄×C₉	C₃×C₄⋊C₄	1	6	yes	—
107	C₄×C₉	C₃×M₁₆	1	12	yes	complemented
108	C₄×C₉	C₂×C₃×D₄	1	3	yes	—
109	C₄×C₉	C₂×C₃×Q₈	1	3	yes	—
110	C₂×M₁₆	C₂×3₋¹⁺²	1	2	yes	—
111	C₂×C₄²	C₂×3₋¹⁺²	1	1	yes	characteristic
112	C₄⋊C₈	C₂×3₋¹⁺²	1	4	yes	complemented
113	3₋¹⁺²	#423	1	1	yes	characteristic, direct product factor, Sylow
114	C₂³×C₃	—	3	1	no	—
115	C₂³×C₃	C₂×C₃²×C₄	1	1	yes	characteristic
116	C₂×C₃×C₄	—	2	4	no	—
117	C₂×C₃×C₄	—	3	1	no	—
118	C₂×C₃×C₄	—	3	2	no	—
119	C₂×C₃×C₄	—	3	2	no	—
120	C₂×C₃×C₄	—	3	2	no	—
121	C₂×C₃×C₄	—	3	2	no	—
122	C₂×C₃×C₄	—	3	1	no	—
123	C₂×C₃×C₄	—	6	4	no	—
124	C₂×C₃×C₄	C₂×C₃²×C₄	1	2	yes	—
125	C₂×C₃×C₄	C₂×C₃²×C₄	1	1	yes	characteristic
126	C₂×C₃×C₄	C₂×C₃²×C₄	1	2	yes	—
127	C₂×C₃×C₄	C₃²×D₄	1	2	yes	—
128	C₂×C₃×C₄	C₃²×Q₈	1	2	yes	—
129	C₂×C₃×C₄	C₂³×C₃²	1	1	yes	characteristic
130	C₃×C₈	—	2	4	no	—
131	C₃×C₈	—	6	4	no	—
132	C₂×C₃²	—	2	2	no	—
133	C₂×C₃²	C₂×C₃×C₄⋊C₄	1	1	yes	characteristic
134	C₂×C₃²	C₂×C₃×M₁₆	1	2	yes	—
135	C₂×C₉	—	2	6	no	—
136	C₂×C₉	C₂×C₃×C₄⋊C₄	1	3	yes	—
137	C₂×C₉	C₂×C₃×M₁₆	1	6	yes	—
138	C₂²×C₄	C₄×3₋¹⁺²	1	2	yes	—
139	C₂²×C₄	C₂²×3₋¹⁺²	1	1	yes	characteristic
140	M₁₆	—	2	4	no	—
141	C₂×C₈	C₂²×3₋¹⁺²	1	4	yes	—
142	C₄²	C₄×3₋¹⁺²	1	1	yes	characteristic
143	C₄²	C₄×3₋¹⁺²	1	1	yes	characteristic
144	C₄²	C₂²×3₋¹⁺²	1	2	yes	—
145	C₂²×C₃	—	2	2	no	—
146	C₂²×C₃	—	3	1	no	—
147	C₂²×C₃	—	3	2	no	—
148	C₂²×C₃	—	6	2	no	—
149	C₂²×C₃	C₃²×C₄⋊C₄	1	2	yes	—
150	C₂²×C₃	C₂²×C₃²×C₄	1	1	yes	characteristic
151	C₃×C₄	—	2	2	no	—
152	C₃×C₄	—	3	1	no	—
153	C₃×C₄	—	3	1	no	—
154	C₃×C₄	—	3	2	no	—
155	C₃×C₄	—	3	4	no	—
156	C₃×C₄	—	6	2	no	—
157	C₃×C₄	C₃²×C₄⋊C₄	1	2	yes	—
158	C₃×C₄	C₃²×M₁₆	1	4	yes	—
159	C₃×C₄	C₂×C₃²×D₄	1	1	yes	characteristic
160	C₃×C₄	C₂×C₃²×Q₈	1	1	yes	characteristic
161	C₃²	C₃×#423	1	1	yes	characteristic
162	C₉	C₃×#423	1	3	yes	complemented
163	C₂³	C₂×C₄×3₋¹⁺²	1	1	yes	characteristic
164	C₂×C₄	—	2	4	no	—
165	C₂×C₄	C₂×C₄×3₋¹⁺²	1	2	yes	—
166	C₂×C₄	C₂×C₄×3₋¹⁺²	1	1	yes	characteristic
167	C₂×C₄	C₂×C₄×3₋¹⁺²	1	2	yes	—
168	C₂×C₄	D₄×3₋¹⁺²	1	2	yes	—
169	C₂×C₄	Q₈×3₋¹⁺²	1	2	yes	—
170	C₂×C₄	C₂³×3₋¹⁺²	1	1	yes	characteristic
171	C₈	—	2	4	no	—
172	C₂×C₃	—	2	2	no	—
173	C₂×C₃	—	3	1	no	—
174	C₂×C₃	—	3	2	no	—
175	C₂×C₃	—	6	2	no	—
176	C₂×C₃	C₂×C₃²×C₄⋊C₄	1	1	yes	characteristic
177	C₂×C₃	C₂×C₃²×M₁₆	1	2	yes	—
178	C₂²	—	2	2	no	—
179	C₂²	C₄⋊C₄×3₋¹⁺²	1	2	yes	—
180	C₂²	C₂²×C₄×3₋¹⁺²	1	1	yes	characteristic
181	C₄	—	2	2	no	—
182	C₄	C₄⋊C₄×3₋¹⁺²	1	2	yes	—
183	C₄	M₁₆×3₋¹⁺²	1	4	yes	complemented
184	C₄	C₂×D₄×3₋¹⁺²	1	1	yes	characteristic
185	C₄	C₂×Q₈×3₋¹⁺²	1	1	yes	characteristic
186	C₃	—	3	1	no	—
187	C₃	C₃²×#423	1	1	yes	characteristic
188	C₂	—	2	2	no	—
189	C₂	C₂×C₄⋊C₄×3₋¹⁺²	1	1	yes	characteristic
190	C₂	C₂×M₁₆×3₋¹⁺²	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
3. 3₋¹⁺²×C₄⋊C₈	C₂	semidirect
5. C₉×#423	C₃	semidirect
18. C₉×C₄⋊C₈	C₂×C₃	semidirect
49. C₄×3₋¹⁺²	M₁₆	semidirect
76. #423	3₋¹⁺²	direct (N × H)
107. C₄×C₉	C₃×M₁₆	semidirect
112. C₄⋊C₈	C₂×3₋¹⁺²	semidirect
113. 3₋¹⁺²	#423	direct (N × H)
162. C₉	C₃×#423	semidirect
183. C₄	M₁₆×3₋¹⁺²	semidirect

Maximal subgroups

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

subgroup	index
C₂×M₁₆×3₋¹⁺²	2
C₂×C₄²×3₋¹⁺²	2
3₋¹⁺²×C₄⋊C₈	2
C₃²×#423	3
C₉×#423	3

Subgroups — 3−1+2×#423 #13145029

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — 3₋¹⁺²×#423 #13145029