Subgroups — 3₋¹⁺²×#424 #13145032

Order 1728.

456 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	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×C₄⋊C₈	C₂	1	4	yes	maximal, complemented
5	C₃²×#424	C₃	1	1	yes	maximal, characteristic
6	C₉×#424	C₃	1	3	yes	maximal, complemented
7	C₂²×C₄×3₋¹⁺²	C₂²	1	1	yes	characteristic
8	M₁₆×3₋¹⁺²	—	2	2	no	—
9	C₂×C₈×3₋¹⁺²	—	2	2	no	—
10	C₂×C₈×3₋¹⁺²	C₂²	1	2	yes	—
11	C₂×C₈×3₋¹⁺²	C₂²	1	2	yes	—
12	C₄⋊C₄×3₋¹⁺²	C₄	1	2	yes	—
13	C₂²⋊C₄×3₋¹⁺²	C₄	1	2	yes	—
14	C₄²×3₋¹⁺²	C₂²	1	2	yes	—
15	C₂×C₃²×M₁₆	C₂×C₃	1	1	yes	characteristic
16	C₂²×C₃²×C₈	C₂×C₃	1	1	yes	characteristic
17	C₃²×#117	C₂×C₃	1	1	yes	characteristic
18	C₃²×C₄⋊C₈	C₂×C₃	1	4	yes	—
19	C₂×C₉×M₁₆	C₂×C₃	1	3	yes	—
20	C₂²×C₈×C₉	C₂×C₃	1	3	yes	—
21	C₉×#117	C₂×C₃	1	3	yes	—
22	C₉×C₄⋊C₈	C₂×C₃	1	12	yes	complemented
23	C₂³×3₋¹⁺²	C₂×C₄	1	1	yes	characteristic
24	C₂×C₄×3₋¹⁺²	C₂×C₄	1	4	yes	—
25	C₂×C₄×3₋¹⁺²	C₂×C₄	1	1	yes	characteristic
26	C₂×C₄×3₋¹⁺²	D₄	1	2	yes	—
27	C₂×C₄×3₋¹⁺²	Q₈	1	2	yes	—
28	C₂×C₄×3₋¹⁺²	C₂³	1	1	yes	characteristic
29	C₈×3₋¹⁺²	—	2	2	no	—
30	C₈×3₋¹⁺²	—	2	2	no	—
31	C₃×#424	—	3	1	no	—
32	C₃×#424	C₃²	1	1	yes	characteristic
33	C₂²×C₃²×C₄	C₂²×C₃	1	1	yes	characteristic
34	C₃²×M₁₆	—	2	2	no	—
35	C₂×C₃²×C₈	—	2	2	no	—
36	C₂×C₃²×C₈	C₂²×C₃	1	2	yes	—
37	C₂×C₃²×C₈	C₂²×C₃	1	2	yes	—
38	C₃²×C₄⋊C₄	C₃×C₄	1	2	yes	—
39	C₃²×C₂²⋊C₄	C₃×C₄	1	2	yes	—
40	C₃²×C₄²	C₂²×C₃	1	2	yes	—
41	C₂²×C₄×C₉	C₂²×C₃	1	3	yes	—
42	C₉×M₁₆	—	2	6	no	—
43	C₂×C₈×C₉	—	2	6	no	—
44	C₂×C₈×C₉	C₂²×C₃	1	6	yes	—
45	C₂×C₈×C₉	C₂²×C₃	1	6	yes	—
46	C₉×C₄⋊C₄	C₃×C₄	1	6	yes	—
47	C₉×C₂²⋊C₄	C₃×C₄	1	6	yes	—
48	C₄²×C₉	C₂²×C₃	1	6	yes	—
49	C₂²×3₋¹⁺²	—	2	2	no	—
50	C₂²×3₋¹⁺²	C₄⋊C₄	1	2	yes	—
51	C₂²×3₋¹⁺²	C₂²×C₄	1	1	yes	characteristic
52	C₄×3₋¹⁺²	—	2	4	no	—
53	C₄×3₋¹⁺²	C₄⋊C₄	1	2	yes	—
54	C₄×3₋¹⁺²	C₂×D₄	1	1	yes	characteristic
55	C₄×3₋¹⁺²	C₂×Q₈	1	1	yes	characteristic
56	C₂×C₃×M₁₆	—	3	1	no	—
57	C₂×C₃×M₁₆	C₂×C₃²	1	1	yes	characteristic
58	C₂²×C₃×C₈	—	3	1	no	—
59	C₂²×C₃×C₈	C₂×C₃²	1	1	yes	characteristic
60	C₃×#117	—	3	1	no	—
61	C₃×#117	C₂×C₃²	1	1	yes	characteristic
62	C₃×C₄⋊C₈	—	3	4	no	—
63	C₃×C₄⋊C₈	C₂×C₃²	1	4	yes	—
64	C₂³×C₃²	C₂×C₃×C₄	1	1	yes	characteristic
65	C₂×C₃²×C₄	C₂×C₃×C₄	1	4	yes	—
66	C₂×C₃²×C₄	C₂×C₃×C₄	1	1	yes	characteristic
67	C₂×C₃²×C₄	C₃×D₄	1	2	yes	—
68	C₂×C₃²×C₄	C₃×Q₈	1	2	yes	—
69	C₂×C₃²×C₄	C₂³×C₃	1	1	yes	characteristic
70	C₂³×C₉	C₂×C₃×C₄	1	3	yes	—
71	C₃²×C₈	—	2	2	no	—
72	C₃²×C₈	—	2	2	no	—
73	C₂×C₄×C₉	C₂×C₃×C₄	1	12	yes	—
74	C₂×C₄×C₉	C₂×C₃×C₄	1	3	yes	—
75	C₂×C₄×C₉	C₃×D₄	1	6	yes	—
76	C₂×C₄×C₉	C₃×Q₈	1	6	yes	—
77	C₂×C₄×C₉	C₂³×C₃	1	3	yes	—
78	C₈×C₉	—	2	6	no	—
79	C₈×C₉	—	2	6	no	—
80	#424	3₋¹⁺²	1	1	yes	characteristic, direct product factor, Sylow
81	C₂×3₋¹⁺²	—	2	2	no	—
82	C₂×3₋¹⁺²	C₂×C₄⋊C₄	1	1	yes	characteristic
83	C₂×3₋¹⁺²	#131	1	2	yes	—
84	C₂²×C₃×C₄	—	3	1	no	—
85	C₂²×C₃×C₄	C₂²×C₃²	1	1	yes	characteristic
86	C₃×M₁₆	—	2	2	no	—
87	C₃×M₁₆	—	6	2	no	—
88	C₂×C₃×C₈	—	2	2	no	—
89	C₂×C₃×C₈	—	3	2	no	—
90	C₂×C₃×C₈	—	3	2	no	—
91	C₂×C₃×C₈	—	6	2	no	—
92	C₂×C₃×C₈	C₂²×C₃²	1	2	yes	—
93	C₂×C₃×C₈	C₂²×C₃²	1	2	yes	—
94	C₃×C₄⋊C₄	—	3	2	no	—
95	C₃×C₄⋊C₄	C₃²×C₄	1	2	yes	—
96	C₃×C₂²⋊C₄	—	3	2	no	—
97	C₃×C₂²⋊C₄	C₃²×C₄	1	2	yes	—
98	C₃×C₄²	—	3	2	no	—
99	C₃×C₄²	C₂²×C₃²	1	2	yes	—
100	C₂²×C₃²	—	2	2	no	—
101	C₂²×C₃²	C₃×C₄⋊C₄	1	2	yes	—
102	C₂²×C₃²	C₂²×C₃×C₄	1	1	yes	characteristic
103	C₃²×C₄	—	2	4	no	—
104	C₃²×C₄	C₃×C₄⋊C₄	1	2	yes	—
105	C₃²×C₄	C₂×C₃×D₄	1	1	yes	characteristic
106	C₃²×C₄	C₂×C₃×Q₈	1	1	yes	characteristic
107	C₂²×C₉	—	2	6	no	—
108	C₂²×C₉	C₃×C₄⋊C₄	1	6	yes	—
109	C₂²×C₉	C₂²×C₃×C₄	1	3	yes	—
110	C₄×C₉	—	2	12	no	—
111	C₄×C₉	C₃×C₄⋊C₄	1	6	yes	—
112	C₄×C₉	C₂×C₃×D₄	1	3	yes	—
113	C₄×C₉	C₂×C₃×Q₈	1	3	yes	—
114	C₂×M₁₆	C₂×3₋¹⁺²	1	1	yes	characteristic
115	C₂²×C₈	C₂×3₋¹⁺²	1	1	yes	characteristic
116	#117	C₂×3₋¹⁺²	1	1	yes	characteristic
117	C₄⋊C₈	C₂×3₋¹⁺²	1	4	yes	complemented
118	3₋¹⁺²	#424	1	1	yes	characteristic, direct product factor, Sylow
119	C₂³×C₃	—	3	1	no	—
120	C₂³×C₃	C₂×C₃²×C₄	1	1	yes	characteristic
121	C₂×C₃×C₄	—	3	1	no	—
122	C₂×C₃×C₄	—	3	4	no	—
123	C₂×C₃×C₄	—	3	2	no	—
124	C₂×C₃×C₄	—	3	2	no	—
125	C₂×C₃×C₄	—	3	1	no	—
126	C₂×C₃×C₄	C₂×C₃²×C₄	1	4	yes	—
127	C₂×C₃×C₄	C₂×C₃²×C₄	1	1	yes	characteristic
128	C₂×C₃×C₄	C₃²×D₄	1	2	yes	—
129	C₂×C₃×C₄	C₃²×Q₈	1	2	yes	—
130	C₂×C₃×C₄	C₂³×C₃²	1	1	yes	characteristic
131	C₃×C₈	—	2	2	no	—
132	C₃×C₈	—	2	2	no	—
133	C₃×C₈	—	6	2	no	—
134	C₃×C₈	—	6	2	no	—
135	C₂×C₃²	—	2	2	no	—
136	C₂×C₃²	C₂×C₃×C₄⋊C₄	1	1	yes	characteristic
137	C₂×C₃²	C₃×#131	1	2	yes	—
138	C₂×C₉	—	2	6	no	—
139	C₂×C₉	C₂×C₃×C₄⋊C₄	1	3	yes	—
140	C₂×C₉	C₃×#131	1	6	yes	—
141	C₂²×C₄	C₂²×3₋¹⁺²	1	1	yes	characteristic
142	M₁₆	—	2	2	no	—
143	C₂×C₈	—	2	2	no	—
144	C₂×C₈	C₂²×3₋¹⁺²	1	2	yes	—
145	C₂×C₈	C₂²×3₋¹⁺²	1	2	yes	—
146	C₄⋊C₄	C₄×3₋¹⁺²	1	2	yes	—
147	C₂²⋊C₄	C₄×3₋¹⁺²	1	2	yes	—
148	C₄²	C₂²×3₋¹⁺²	1	2	yes	—
149	C₂²×C₃	—	2	2	no	—
150	C₂²×C₃	—	3	1	no	—
151	C₂²×C₃	—	3	2	no	—
152	C₂²×C₃	—	6	2	no	—
153	C₂²×C₃	C₃²×C₄⋊C₄	1	2	yes	—
154	C₂²×C₃	C₂²×C₃²×C₄	1	1	yes	characteristic
155	C₃×C₄	—	2	4	no	—
156	C₃×C₄	—	3	1	no	—
157	C₃×C₄	—	3	1	no	—
158	C₃×C₄	—	3	2	no	—
159	C₃×C₄	—	6	4	no	—
160	C₃×C₄	C₃²×C₄⋊C₄	1	2	yes	—
161	C₃×C₄	C₂×C₃²×D₄	1	1	yes	characteristic
162	C₃×C₄	C₂×C₃²×Q₈	1	1	yes	characteristic
163	C₃²	C₃×#424	1	1	yes	characteristic
164	C₉	C₃×#424	1	3	yes	complemented
165	C₂³	C₂×C₄×3₋¹⁺²	1	1	yes	characteristic
166	C₂×C₄	C₂×C₄×3₋¹⁺²	1	4	yes	—
167	C₂×C₄	C₂×C₄×3₋¹⁺²	1	1	yes	characteristic
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	2	no	—
172	C₈	—	2	2	no	—
173	C₂×C₃	—	2	2	no	—
174	C₂×C₃	—	3	1	no	—
175	C₂×C₃	—	3	2	no	—
176	C₂×C₃	—	6	2	no	—
177	C₂×C₃	C₂×C₃²×C₄⋊C₄	1	1	yes	characteristic
178	C₂×C₃	C₃²×#131	1	2	yes	—
179	C₂²	—	2	2	no	—
180	C₂²	C₄⋊C₄×3₋¹⁺²	1	2	yes	—
181	C₂²	C₂²×C₄×3₋¹⁺²	1	1	yes	characteristic
182	C₄	—	2	4	no	—
183	C₄	C₄⋊C₄×3₋¹⁺²	1	2	yes	—
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×#424	1	1	yes	characteristic
188	C₂	—	2	2	no	—
189	C₂	C₂×C₄⋊C₄×3₋¹⁺²	1	1	yes	characteristic
190	C₂	3₋¹⁺²×#131	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
4. 3₋¹⁺²×C₄⋊C₈	C₂	semidirect
6. C₉×#424	C₃	semidirect
22. C₉×C₄⋊C₈	C₂×C₃	semidirect
80. #424	3₋¹⁺²	direct (N × H)
117. C₄⋊C₈	C₂×3₋¹⁺²	semidirect
118. 3₋¹⁺²	#424	direct (N × H)
164. C₉	C₃×#424	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×C₄⋊C₈	2
C₃²×#424	3
C₉×#424	3

Subgroups — 3−1+2×#424 #13145032

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — 3₋¹⁺²×#424 #13145032