Subgroups — 3₋¹⁺²×#427 #13145041

Order 1728.

456 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	C₂²×C₈×3₋¹⁺²	C₂	1	1	yes	maximal, characteristic
2	3₋¹⁺²×#117	C₂	1	2	yes	maximal
3	3₋¹⁺²×#107	C₂	1	2	yes	maximal, complemented
4	3₋¹⁺²×#106	C₂	1	2	yes	maximal, complemented
5	C₃²×#427	C₃	1	1	yes	maximal, characteristic
6	C₉×#427	C₃	1	3	yes	maximal, complemented
7	C₂²×C₄×3₋¹⁺²	C₂²	1	1	yes	characteristic
8	C₂×C₈×3₋¹⁺²	C₄	1	4	yes	complemented
9	C₂×C₈×3₋¹⁺²	C₂²	1	1	yes	characteristic
10	C₂×C₈×3₋¹⁺²	C₂²	1	1	yes	characteristic
11	C₄⋊C₄×3₋¹⁺²	C₂²	1	4	yes	—
12	C₂²⋊C₄×3₋¹⁺²	—	2	2	no	—
13	C₄²×3₋¹⁺²	—	2	2	no	—
14	C₂²×C₃²×C₈	C₂×C₃	1	1	yes	characteristic
15	C₃²×#117	C₂×C₃	1	2	yes	—
16	C₃²×#107	C₂×C₃	1	2	yes	—
17	C₃²×#106	C₂×C₃	1	2	yes	—
18	C₂²×C₈×C₉	C₂×C₃	1	3	yes	—
19	C₉×#117	C₂×C₃	1	6	yes	—
20	C₉×#107	C₂×C₃	1	6	yes	complemented
21	C₉×#106	C₂×C₃	1	6	yes	complemented
22	C₂³×3₋¹⁺²	D₄	1	1	yes	characteristic
23	C₂×C₄×3₋¹⁺²	—	2	4	no	—
24	C₂×C₄×3₋¹⁺²	C₂×C₄	1	2	yes	—
25	C₂×C₄×3₋¹⁺²	D₄	1	1	yes	characteristic
26	C₂×C₄×3₋¹⁺²	Q₈	1	2	yes	—
27	C₂×C₄×3₋¹⁺²	C₂³	1	1	yes	characteristic
28	C₈×3₋¹⁺²	C₂×C₄	1	2	yes	—
29	C₈×3₋¹⁺²	C₂×C₄	1	2	yes	—
30	C₃×#427	—	3	1	no	—
31	C₃×#427	C₃²	1	1	yes	characteristic
32	C₂²×C₃²×C₄	C₂²×C₃	1	1	yes	characteristic
33	C₂×C₃²×C₈	C₃×C₄	1	4	yes	—
34	C₂×C₃²×C₈	C₂²×C₃	1	1	yes	characteristic
35	C₂×C₃²×C₈	C₂²×C₃	1	1	yes	characteristic
36	C₃²×C₄⋊C₄	C₂²×C₃	1	4	yes	—
37	C₃²×C₂²⋊C₄	—	2	2	no	—
38	C₃²×C₄²	—	2	2	no	—
39	C₂²×C₄×C₉	C₂²×C₃	1	3	yes	—
40	C₂×C₈×C₉	C₃×C₄	1	12	yes	complemented
41	C₂×C₈×C₉	C₂²×C₃	1	3	yes	—
42	C₂×C₈×C₉	C₂²×C₃	1	3	yes	—
43	C₉×C₄⋊C₄	C₂²×C₃	1	12	yes	—
44	C₉×C₂²⋊C₄	—	2	6	no	—
45	C₄²×C₉	—	2	6	no	—
46	C₂²×3₋¹⁺²	—	2	2	no	—
47	C₂²×3₋¹⁺²	C₄⋊C₄	1	2	yes	—
48	C₂²×3₋¹⁺²	C₂×D₄	1	1	yes	characteristic
49	C₄×3₋¹⁺²	—	4	4	no	—
50	C₄×3₋¹⁺²	C₄⋊C₄	1	2	yes	—
51	C₄×3₋¹⁺²	C₂²×C₄	1	1	yes	characteristic
52	C₄×3₋¹⁺²	C₂×Q₈	1	1	yes	characteristic
53	C₂²×C₃×C₈	—	3	1	no	—
54	C₂²×C₃×C₈	C₂×C₃²	1	1	yes	characteristic
55	C₃×#117	—	3	2	no	—
56	C₃×#117	C₂×C₃²	1	2	yes	—
57	C₃×#107	—	3	2	no	—
58	C₃×#107	C₂×C₃²	1	2	yes	—
59	C₃×#106	—	3	2	no	—
60	C₃×#106	C₂×C₃²	1	2	yes	—
61	C₂³×C₃²	C₃×D₄	1	1	yes	characteristic
62	C₂×C₃²×C₄	—	2	4	no	—
63	C₂×C₃²×C₄	C₂×C₃×C₄	1	2	yes	—
64	C₂×C₃²×C₄	C₃×D₄	1	1	yes	characteristic
65	C₂×C₃²×C₄	C₃×Q₈	1	2	yes	—
66	C₂×C₃²×C₄	C₂³×C₃	1	1	yes	characteristic
67	C₂³×C₉	C₃×D₄	1	3	yes	—
68	C₃²×C₈	C₂×C₃×C₄	1	2	yes	—
69	C₃²×C₈	C₂×C₃×C₄	1	2	yes	—
70	C₂×C₄×C₉	—	2	12	no	—
71	C₂×C₄×C₉	C₂×C₃×C₄	1	6	yes	—
72	C₂×C₄×C₉	C₃×D₄	1	3	yes	—
73	C₂×C₄×C₉	C₃×Q₈	1	6	yes	—
74	C₂×C₄×C₉	C₂³×C₃	1	3	yes	—
75	C₈×C₉	C₂×C₃×C₄	1	6	yes	—
76	C₈×C₉	C₂×C₃×C₄	1	6	yes	—
77	#427	3₋¹⁺²	1	1	yes	characteristic, direct product factor, Sylow
78	C₂×3₋¹⁺²	—	2	2	no	—
79	C₂×3₋¹⁺²	C₂×C₄⋊C₄	1	1	yes	characteristic
80	C₂×3₋¹⁺²	#135	1	2	yes	—
81	C₂²×C₃×C₄	—	3	1	no	—
82	C₂²×C₃×C₄	C₂²×C₃²	1	1	yes	characteristic
83	C₂×C₃×C₈	—	3	4	no	—
84	C₂×C₃×C₈	—	3	1	no	—
85	C₂×C₃×C₈	—	3	1	no	—
86	C₂×C₃×C₈	C₃²×C₄	1	4	yes	—
87	C₂×C₃×C₈	C₂²×C₃²	1	1	yes	characteristic
88	C₂×C₃×C₈	C₂²×C₃²	1	1	yes	characteristic
89	C₃×C₄⋊C₄	—	3	4	no	—
90	C₃×C₄⋊C₄	C₂²×C₃²	1	4	yes	—
91	C₃×C₂²⋊C₄	—	2	2	no	—
92	C₃×C₂²⋊C₄	—	6	2	no	—
93	C₃×C₄²	—	2	2	no	—
94	C₃×C₄²	—	6	2	no	—
95	C₂²×C₃²	—	2	2	no	—
96	C₂²×C₃²	C₃×C₄⋊C₄	1	2	yes	—
97	C₂²×C₃²	C₂×C₃×D₄	1	1	yes	characteristic
98	C₃²×C₄	—	4	4	no	—
99	C₃²×C₄	C₃×C₄⋊C₄	1	2	yes	—
100	C₃²×C₄	C₂²×C₃×C₄	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	C₂×C₃×D₄	1	3	yes	—
105	C₄×C₉	—	4	12	no	—
106	C₄×C₉	C₃×C₄⋊C₄	1	6	yes	—
107	C₄×C₉	C₂²×C₃×C₄	1	3	yes	—
108	C₄×C₉	C₂×C₃×Q₈	1	3	yes	—
109	C₂²×C₈	C₂×3₋¹⁺²	1	1	yes	characteristic
110	#117	C₂×3₋¹⁺²	1	2	yes	—
111	#107	C₂×3₋¹⁺²	1	2	yes	complemented
112	#106	C₂×3₋¹⁺²	1	2	yes	complemented
113	3₋¹⁺²	#427	1	1	yes	characteristic, direct product factor, Sylow
114	C₂³×C₃	—	3	1	no	—
115	C₂³×C₃	C₃²×D₄	1	1	yes	characteristic
116	C₂×C₃×C₄	—	2	4	no	—
117	C₂×C₃×C₄	—	3	2	no	—
118	C₂×C₃×C₄	—	3	1	no	—
119	C₂×C₃×C₄	—	3	2	no	—
120	C₂×C₃×C₄	—	3	1	no	—
121	C₂×C₃×C₄	—	6	4	no	—
122	C₂×C₃×C₄	C₂×C₃²×C₄	1	2	yes	—
123	C₂×C₃×C₄	C₃²×D₄	1	1	yes	characteristic
124	C₂×C₃×C₄	C₃²×Q₈	1	2	yes	—
125	C₂×C₃×C₄	C₂³×C₃²	1	1	yes	characteristic
126	C₃×C₈	—	3	2	no	—
127	C₃×C₈	—	3	2	no	—
128	C₃×C₈	C₂×C₃²×C₄	1	2	yes	—
129	C₃×C₈	C₂×C₃²×C₄	1	2	yes	—
130	C₂×C₃²	—	2	2	no	—
131	C₂×C₃²	C₂×C₃×C₄⋊C₄	1	1	yes	characteristic
132	C₂×C₃²	C₃×#135	1	2	yes	—
133	C₂×C₉	—	2	6	no	—
134	C₂×C₉	C₂×C₃×C₄⋊C₄	1	3	yes	—
135	C₂×C₉	C₃×#135	1	6	yes	—
136	C₂²×C₄	C₂²×3₋¹⁺²	1	1	yes	characteristic
137	C₂×C₈	C₄×3₋¹⁺²	1	4	yes	complemented
138	C₂×C₈	C₂²×3₋¹⁺²	1	1	yes	characteristic
139	C₂×C₈	C₂²×3₋¹⁺²	1	1	yes	characteristic
140	C₄⋊C₄	C₂²×3₋¹⁺²	1	4	yes	—
141	C₂²⋊C₄	—	2	2	no	—
142	C₄²	—	2	2	no	—
143	C₂²×C₃	—	2	2	no	—
144	C₂²×C₃	—	3	1	no	—
145	C₂²×C₃	—	3	2	no	—
146	C₂²×C₃	—	6	2	no	—
147	C₂²×C₃	C₃²×C₄⋊C₄	1	2	yes	—
148	C₂²×C₃	C₂×C₃²×D₄	1	1	yes	characteristic
149	C₃×C₄	—	3	1	no	—
150	C₃×C₄	—	3	1	no	—
151	C₃×C₄	—	3	2	no	—
152	C₃×C₄	—	4	4	no	—
153	C₃×C₄	—	12	4	no	—
154	C₃×C₄	C₃²×C₄⋊C₄	1	2	yes	—
155	C₃×C₄	C₂²×C₃²×C₄	1	1	yes	characteristic
156	C₃×C₄	C₂×C₃²×Q₈	1	1	yes	characteristic
157	C₃²	C₃×#427	1	1	yes	characteristic
158	C₉	C₃×#427	1	3	yes	complemented
159	C₂³	D₄×3₋¹⁺²	1	1	yes	characteristic
160	C₂×C₄	—	2	4	no	—
161	C₂×C₄	C₂×C₄×3₋¹⁺²	1	2	yes	—
162	C₂×C₄	D₄×3₋¹⁺²	1	1	yes	characteristic
163	C₂×C₄	Q₈×3₋¹⁺²	1	2	yes	—
164	C₂×C₄	C₂³×3₋¹⁺²	1	1	yes	characteristic
165	C₈	C₂×C₄×3₋¹⁺²	1	2	yes	—
166	C₈	C₂×C₄×3₋¹⁺²	1	2	yes	—
167	C₂×C₃	—	2	2	no	—
168	C₂×C₃	—	3	1	no	—
169	C₂×C₃	—	3	2	no	—
170	C₂×C₃	—	6	2	no	—
171	C₂×C₃	C₂×C₃²×C₄⋊C₄	1	1	yes	characteristic
172	C₂×C₃	C₃²×#135	1	2	yes	—
173	C₂²	—	2	2	no	—
174	C₂²	C₄⋊C₄×3₋¹⁺²	1	2	yes	—
175	C₂²	C₂×D₄×3₋¹⁺²	1	1	yes	characteristic
176	C₄	—	4	4	no	—
177	C₄	C₄⋊C₄×3₋¹⁺²	1	2	yes	—
178	C₄	C₂²×C₄×3₋¹⁺²	1	1	yes	characteristic
179	C₄	C₂×Q₈×3₋¹⁺²	1	1	yes	characteristic
180	C₃	—	3	1	no	—
181	C₃	C₃²×#427	1	1	yes	characteristic
182	C₂	—	2	2	no	—
183	C₂	C₂×C₄⋊C₄×3₋¹⁺²	1	1	yes	characteristic
184	C₂	3₋¹⁺²×#135	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
3. 3₋¹⁺²×#107	C₂	semidirect
4. 3₋¹⁺²×#106	C₂	semidirect
6. C₉×#427	C₃	semidirect
8. C₂×C₈×3₋¹⁺²	C₄	semidirect
20. C₉×#107	C₂×C₃	semidirect
21. C₉×#106	C₂×C₃	semidirect
40. C₂×C₈×C₉	C₃×C₄	semidirect
77. #427	3₋¹⁺²	direct (N × H)
111. #107	C₂×3₋¹⁺²	semidirect
112. #106	C₂×3₋¹⁺²	semidirect
113. 3₋¹⁺²	#427	direct (N × H)
137. C₂×C₈	C₄×3₋¹⁺²	semidirect
158. C₉	C₃×#427	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₈×3₋¹⁺²	2
3₋¹⁺²×#117	2
3₋¹⁺²×#107	2
3₋¹⁺²×#106	2
C₃²×#427	3
C₉×#427	3

Subgroups — 3−1+2×#427 #13145041

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — 3₋¹⁺²×#427 #13145041