Subgroups — 3₋¹⁺²×#518 #13154538

Order 1728.

1352 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	C₂²×D₄×3₋¹⁺²	C₂	1	1	yes	maximal, characteristic
2	C₄×D₄×3₋¹⁺²	C₂	1	8	yes	maximal, complemented
3	3₋¹⁺²×#117	C₂	1	2	yes	maximal, complemented
4	C₂×C₂²⋊C₄×3₋¹⁺²	C₂	1	4	yes	maximal, complemented
5	C₃²×#518	C₃	1	1	yes	maximal, characteristic
6	C₉×#518	C₃	1	3	yes	maximal, complemented
7	C₂⁴×3₋¹⁺²	C₂²	1	2	yes	—
8	C₂×D₄×3₋¹⁺²	C₄	1	8	yes	complemented
9	C₂×D₄×3₋¹⁺²	C₂²	1	4	yes	—
10	C₂²×C₄×3₋¹⁺²	C₂²	1	1	yes	characteristic
11	C₂²×C₄×3₋¹⁺²	C₂²	1	8	yes	complemented
12	C₄⋊C₄×3₋¹⁺²	C₂²	1	4	yes	complemented
13	C₂²⋊C₄×3₋¹⁺²	C₂²	1	4	yes	—
14	C₂²⋊C₄×3₋¹⁺²	C₂²	1	8	yes	complemented
15	C₄²×3₋¹⁺²	C₂²	1	4	yes	complemented
16	C₂²×C₃²×D₄	C₂×C₃	1	1	yes	characteristic
17	C₃²×C₄×D₄	C₂×C₃	1	8	yes	—
18	C₃²×#117	C₂×C₃	1	2	yes	—
19	C₂×C₃²×C₂²⋊C₄	C₂×C₃	1	4	yes	—
20	C₂²×D₄×C₉	C₂×C₃	1	3	yes	—
21	C₄×D₄×C₉	C₂×C₃	1	24	yes	complemented
22	C₉×#117	C₂×C₃	1	6	yes	complemented
23	C₂×C₉×C₂²⋊C₄	C₂×C₃	1	12	yes	complemented
24	C₂³×3₋¹⁺²	—	4	4	no	—
25	C₂³×3₋¹⁺²	C₂×C₄	1	8	yes	complemented
26	C₂³×3₋¹⁺²	C₂³	1	1	yes	characteristic
27	C₂³×3₋¹⁺²	C₂³	1	4	yes	—
28	D₄×3₋¹⁺²	C₂×C₄	1	16	yes	complemented
29	C₂×C₄×3₋¹⁺²	—	4	8	no	—
30	C₂×C₄×3₋¹⁺²	C₂×C₄	1	4	yes	complemented
31	C₂×C₄×3₋¹⁺²	C₂³	1	8	yes	—
32	C₂×C₄×3₋¹⁺²	C₂³	1	2	yes	—
33	C₃×#518	—	3	1	no	—
34	C₃×#518	C₃²	1	1	yes	characteristic
35	C₂⁴×C₃²	C₂²×C₃	1	2	yes	—
36	C₂×C₃²×D₄	C₃×C₄	1	8	yes	—
37	C₂×C₃²×D₄	C₂²×C₃	1	4	yes	—
38	C₂²×C₃²×C₄	C₂²×C₃	1	8	yes	—
39	C₂²×C₃²×C₄	C₂²×C₃	1	1	yes	characteristic
40	C₂⁴×C₉	C₂²×C₃	1	6	yes	—
41	C₃²×C₄⋊C₄	C₂²×C₃	1	4	yes	—
42	C₃²×C₂²⋊C₄	C₂²×C₃	1	4	yes	—
43	C₃²×C₂²⋊C₄	C₂²×C₃	1	8	yes	—
44	C₃²×C₄²	C₂²×C₃	1	4	yes	—
45	C₂×D₄×C₉	C₃×C₄	1	24	yes	complemented
46	C₂×D₄×C₉	C₂²×C₃	1	12	yes	—
47	C₂²×C₄×C₉	C₂²×C₃	1	3	yes	—
48	C₂²×C₄×C₉	C₂²×C₃	1	24	yes	complemented
49	C₉×C₄⋊C₄	C₂²×C₃	1	12	yes	complemented
50	C₉×C₂²⋊C₄	C₂²×C₃	1	12	yes	—
51	C₉×C₂²⋊C₄	C₂²×C₃	1	24	yes	complemented
52	C₄²×C₉	C₂²×C₃	1	12	yes	complemented
53	C₂²×3₋¹⁺²	—	2	8	no	—
54	C₂²×3₋¹⁺²	—	2	2	no	—
55	C₂²×3₋¹⁺²	—	4	8	no	—
56	C₂²×3₋¹⁺²	C₂²×C₄	1	2	yes	—
57	C₂²×3₋¹⁺²	C₂²×C₄	1	8	yes	—
58	C₂²×3₋¹⁺²	C₂⁴	1	1	yes	characteristic
59	C₄×3₋¹⁺²	—	2	8	no	—
60	C₄×3₋¹⁺²	C₂²×C₄	1	4	yes	—
61	C₂²×C₃×D₄	—	3	1	no	—
62	C₂²×C₃×D₄	C₂×C₃²	1	1	yes	characteristic
63	C₃×C₄×D₄	—	3	8	no	—
64	C₃×C₄×D₄	C₂×C₃²	1	8	yes	—
65	C₃×#117	—	3	2	no	—
66	C₃×#117	C₂×C₃²	1	2	yes	—
67	C₂×C₃×C₂²⋊C₄	—	3	4	no	—
68	C₂×C₃×C₂²⋊C₄	C₂×C₃²	1	4	yes	—
69	C₂³×C₃²	—	4	4	no	—
70	C₂³×C₃²	C₂×C₃×C₄	1	8	yes	—
71	C₂³×C₃²	C₂³×C₃	1	1	yes	characteristic
72	C₂³×C₃²	C₂³×C₃	1	4	yes	—
73	C₃²×D₄	C₂×C₃×C₄	1	16	yes	—
74	C₂×C₃²×C₄	—	4	8	no	—
75	C₂×C₃²×C₄	C₂×C₃×C₄	1	4	yes	—
76	C₂×C₃²×C₄	C₂³×C₃	1	8	yes	—
77	C₂×C₃²×C₄	C₂³×C₃	1	2	yes	—
78	C₂³×C₉	—	4	12	no	—
79	C₂³×C₉	C₂×C₃×C₄	1	24	yes	complemented
80	C₂³×C₉	C₂³×C₃	1	3	yes	—
81	C₂³×C₉	C₂³×C₃	1	12	yes	—
82	D₄×C₉	C₂×C₃×C₄	1	48	yes	complemented
83	C₂×C₄×C₉	—	4	24	no	—
84	C₂×C₄×C₉	C₂×C₃×C₄	1	12	yes	complemented
85	C₂×C₄×C₉	C₂³×C₃	1	24	yes	—
86	C₂×C₄×C₉	C₂³×C₃	1	6	yes	—
87	#518	3₋¹⁺²	1	1	yes	characteristic, direct product factor, Sylow
88	C₂×3₋¹⁺²	—	2	2	no	—
89	C₂×3₋¹⁺²	—	2	8	no	—
90	C₂×3₋¹⁺²	C₂³×C₄	1	1	yes	characteristic
91	C₂×3₋¹⁺²	H(5,𝔽₂)	1	2	yes	—
92	C₂⁴×C₃	—	3	2	no	—
93	C₂⁴×C₃	C₂²×C₃²	1	2	yes	—
94	C₂×C₃×D₄	—	3	8	no	—
95	C₂×C₃×D₄	—	3	4	no	—
96	C₂×C₃×D₄	C₃²×C₄	1	8	yes	—
97	C₂×C₃×D₄	C₂²×C₃²	1	4	yes	—
98	C₂²×C₃×C₄	—	3	1	no	—
99	C₂²×C₃×C₄	—	3	8	no	—
100	C₂²×C₃×C₄	C₂²×C₃²	1	1	yes	characteristic
101	C₂²×C₃×C₄	C₂²×C₃²	1	8	yes	—
102	C₃×C₄⋊C₄	—	3	4	no	—
103	C₃×C₄⋊C₄	C₂²×C₃²	1	4	yes	—
104	C₃×C₂²⋊C₄	—	3	4	no	—
105	C₃×C₂²⋊C₄	—	3	8	no	—
106	C₃×C₂²⋊C₄	C₂²×C₃²	1	4	yes	—
107	C₃×C₂²⋊C₄	C₂²×C₃²	1	8	yes	—
108	C₃×C₄²	—	3	4	no	—
109	C₃×C₄²	C₂²×C₃²	1	4	yes	—
110	C₂²×C₃²	—	2	8	no	—
111	C₂²×C₃²	—	2	2	no	—
112	C₂²×C₃²	—	4	8	no	—
113	C₂²×C₃²	C₂²×C₃×C₄	1	2	yes	—
114	C₂²×C₃²	C₂²×C₃×C₄	1	8	yes	—
115	C₂²×C₃²	C₂⁴×C₃	1	1	yes	characteristic
116	C₃²×C₄	—	2	8	no	—
117	C₃²×C₄	C₂²×C₃×C₄	1	4	yes	—
118	C₂²×C₉	—	2	24	no	—
119	C₂²×C₉	—	2	6	no	—
120	C₂²×C₉	—	4	24	no	—
121	C₂²×C₉	C₂²×C₃×C₄	1	6	yes	—
122	C₂²×C₉	C₂²×C₃×C₄	1	24	yes	—
123	C₂²×C₉	C₂⁴×C₃	1	3	yes	—
124	C₄×C₉	—	2	24	no	—
125	C₄×C₉	C₂²×C₃×C₄	1	12	yes	—
126	C₂²×D₄	C₂×3₋¹⁺²	1	1	yes	characteristic
127	C₄×D₄	C₂×3₋¹⁺²	1	8	yes	complemented
128	#117	C₂×3₋¹⁺²	1	2	yes	complemented
129	C₂×C₂²⋊C₄	C₂×3₋¹⁺²	1	4	yes	complemented
130	3₋¹⁺²	#518	1	1	yes	characteristic, direct product factor, Sylow
131	C₂³×C₃	—	3	8	no	—
132	C₂³×C₃	—	3	1	no	—
133	C₂³×C₃	—	3	4	no	—
134	C₂³×C₃	—	4	4	no	—
135	C₂³×C₃	—	12	4	no	—
136	C₂³×C₃	C₂×C₃²×C₄	1	8	yes	—
137	C₂³×C₃	C₂³×C₃²	1	1	yes	characteristic
138	C₂³×C₃	C₂³×C₃²	1	4	yes	—
139	C₃×D₄	—	3	16	no	—
140	C₃×D₄	C₂×C₃²×C₄	1	16	yes	—
141	C₂×C₃×C₄	—	3	4	no	—
142	C₂×C₃×C₄	—	3	2	no	—
143	C₂×C₃×C₄	—	3	8	no	—
144	C₂×C₃×C₄	—	4	8	no	—
145	C₂×C₃×C₄	—	12	8	no	—
146	C₂×C₃×C₄	C₂×C₃²×C₄	1	4	yes	—
147	C₂×C₃×C₄	C₂³×C₃²	1	8	yes	—
148	C₂×C₃×C₄	C₂³×C₃²	1	2	yes	—
149	C₂×C₃²	—	2	2	no	—
150	C₂×C₃²	—	2	8	no	—
151	C₂×C₃²	C₂³×C₃×C₄	1	1	yes	characteristic
152	C₂×C₃²	C₃×H(5,𝔽₂)	1	2	yes	—
153	C₂×C₉	—	2	6	no	—
154	C₂×C₉	—	2	24	no	—
155	C₂×C₉	C₂³×C₃×C₄	1	3	yes	—
156	C₂×C₉	C₃×H(5,𝔽₂)	1	6	yes	—
157	C₂⁴	C₂²×3₋¹⁺²	1	2	yes	—
158	C₂×D₄	C₄×3₋¹⁺²	1	8	yes	complemented
159	C₂×D₄	C₂²×3₋¹⁺²	1	4	yes	—
160	C₂²×C₄	C₂²×3₋¹⁺²	1	1	yes	characteristic
161	C₂²×C₄	C₂²×3₋¹⁺²	1	8	yes	complemented
162	C₄⋊C₄	C₂²×3₋¹⁺²	1	4	yes	complemented
163	C₂²⋊C₄	C₂²×3₋¹⁺²	1	4	yes	—
164	C₂²⋊C₄	C₂²×3₋¹⁺²	1	8	yes	complemented
165	C₄²	C₂²×3₋¹⁺²	1	4	yes	complemented
166	C₂²×C₃	—	2	8	no	—
167	C₂²×C₃	—	2	2	no	—
168	C₂²×C₃	—	3	1	no	—
169	C₂²×C₃	—	3	2	no	—
170	C₂²×C₃	—	3	8	no	—
171	C₂²×C₃	—	4	8	no	—
172	C₂²×C₃	—	6	8	no	—
173	C₂²×C₃	—	6	2	no	—
174	C₂²×C₃	—	12	8	no	—
175	C₂²×C₃	C₂²×C₃²×C₄	1	2	yes	—
176	C₂²×C₃	C₂²×C₃²×C₄	1	8	yes	—
177	C₂²×C₃	C₂⁴×C₃²	1	1	yes	characteristic
178	C₃×C₄	—	2	8	no	—
179	C₃×C₄	—	3	4	no	—
180	C₃×C₄	—	6	8	no	—
181	C₃×C₄	C₂²×C₃²×C₄	1	4	yes	—
182	C₃²	C₃×#518	1	1	yes	characteristic
183	C₉	C₃×#518	1	3	yes	complemented
184	C₂³	—	4	4	no	—
185	C₂³	C₂×C₄×3₋¹⁺²	1	8	yes	complemented
186	C₂³	C₂³×3₋¹⁺²	1	1	yes	characteristic
187	C₂³	C₂³×3₋¹⁺²	1	4	yes	—
188	D₄	C₂×C₄×3₋¹⁺²	1	16	yes	complemented
189	C₂×C₄	—	4	8	no	—
190	C₂×C₄	C₂×C₄×3₋¹⁺²	1	4	yes	complemented
191	C₂×C₄	C₂³×3₋¹⁺²	1	8	yes	—
192	C₂×C₄	C₂³×3₋¹⁺²	1	2	yes	—
193	C₂×C₃	—	2	2	no	—
194	C₂×C₃	—	2	8	no	—
195	C₂×C₃	—	3	1	no	—
196	C₂×C₃	—	3	2	no	—
197	C₂×C₃	—	6	2	no	—
198	C₂×C₃	—	6	8	no	—
199	C₂×C₃	C₂³×C₃²×C₄	1	1	yes	characteristic
200	C₂×C₃	C₃²×H(5,𝔽₂)	1	2	yes	—
201	C₂²	—	2	8	no	—
202	C₂²	—	2	2	no	—
203	C₂²	—	4	8	no	—
204	C₂²	C₂²×C₄×3₋¹⁺²	1	2	yes	—
205	C₂²	C₂²×C₄×3₋¹⁺²	1	8	yes	—
206	C₂²	C₂⁴×3₋¹⁺²	1	1	yes	characteristic
207	C₄	—	2	8	no	—
208	C₄	C₂²×C₄×3₋¹⁺²	1	4	yes	—
209	C₃	—	3	1	no	—
210	C₃	C₃²×#518	1	1	yes	characteristic
211	C₂	—	2	2	no	—
212	C₂	—	2	8	no	—
213	C₂	C₂³×C₄×3₋¹⁺²	1	1	yes	characteristic
214	C₂	3₋¹⁺²×H(5,𝔽₂)	1	2	yes	—

	subgroup	G/H
center, Frattini, socle, cosocle	C₂²×C₃	C₂⁴×C₃²
commutator	C₂×C₃	C₂³×C₃²×C₄
prime radical	C₂²×C₃²×D₄	C₂×C₃
LCS	G ▷ C₂×C₃ ▷ C₁
UCS	C₁ ◁ C₂²×C₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₄×D₄×3₋¹⁺²	C₂	semidirect
3. 3₋¹⁺²×#117	C₂	semidirect
4. C₂×C₂²⋊C₄×3₋¹⁺²	C₂	semidirect
6. C₉×#518	C₃	semidirect
8. C₂×D₄×3₋¹⁺²	C₄	semidirect
11. C₂²×C₄×3₋¹⁺²	C₂²	semidirect
12. C₄⋊C₄×3₋¹⁺²	C₂²	semidirect
14. C₂²⋊C₄×3₋¹⁺²	C₂²	semidirect
15. C₄²×3₋¹⁺²	C₂²	semidirect
21. C₄×D₄×C₉	C₂×C₃	semidirect
22. C₉×#117	C₂×C₃	semidirect
23. C₂×C₉×C₂²⋊C₄	C₂×C₃	semidirect
25. C₂³×3₋¹⁺²	C₂×C₄	semidirect
28. D₄×3₋¹⁺²	C₂×C₄	semidirect
30. C₂×C₄×3₋¹⁺²	C₂×C₄	semidirect
45. C₂×D₄×C₉	C₃×C₄	semidirect
48. C₂²×C₄×C₉	C₂²×C₃	semidirect
49. C₉×C₄⋊C₄	C₂²×C₃	semidirect
51. C₉×C₂²⋊C₄	C₂²×C₃	semidirect
52. C₄²×C₉	C₂²×C₃	semidirect
79. C₂³×C₉	C₂×C₃×C₄	semidirect
82. D₄×C₉	C₂×C₃×C₄	semidirect
84. C₂×C₄×C₉	C₂×C₃×C₄	semidirect
87. #518	3₋¹⁺²	direct (N × H)
127. C₄×D₄	C₂×3₋¹⁺²	semidirect
128. #117	C₂×3₋¹⁺²	semidirect
129. C₂×C₂²⋊C₄	C₂×3₋¹⁺²	semidirect
130. 3₋¹⁺²	#518	direct (N × H)
158. C₂×D₄	C₄×3₋¹⁺²	semidirect
161. C₂²×C₄	C₂²×3₋¹⁺²	semidirect
162. C₄⋊C₄	C₂²×3₋¹⁺²	semidirect
164. C₂²⋊C₄	C₂²×3₋¹⁺²	semidirect
165. C₄²	C₂²×3₋¹⁺²	semidirect
183. C₉	C₃×#518	semidirect
185. C₂³	C₂×C₄×3₋¹⁺²	semidirect
188. D₄	C₂×C₄×3₋¹⁺²	semidirect
190. C₂×C₄	C₂×C₄×3₋¹⁺²	semidirect

Maximal subgroups

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

subgroup	index
C₂²×D₄×3₋¹⁺²	2
C₄×D₄×3₋¹⁺²	2
3₋¹⁺²×#117	2
C₂×C₂²⋊C₄×3₋¹⁺²	2
C₃²×#518	3
C₉×#518	3

Subgroups — 3−1+2×#518 #13154538

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — 3₋¹⁺²×#518 #13154538