Subgroups — 3₋¹⁺²×#466 #13145158

Order 1728.

536 conjugacy classes of subgroups.

The subgroup lattice has 264 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₈×3₋¹⁺²	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×C₈×3₋¹⁺²	C₂	1	1	yes	maximal, characteristic, complemented
3	3₋¹⁺²×#121	C₂	1	2	yes	maximal, complemented
4	3₋¹⁺²×#107	C₂	1	1	yes	maximal, characteristic, complemented
5	3₋¹⁺²×#102	C₂	1	2	yes	maximal, complemented
6	C₃²×#466	C₃	1	1	yes	maximal, characteristic
7	C₉×#466	C₃	1	3	yes	maximal, complemented
8	C₂×D₄×3₋¹⁺²	—	2	2	no	—
9	C₂×D₄×3₋¹⁺²	C₂²	1	2	yes	—
10	C₂²×C₄×3₋¹⁺²	C₂²	1	1	yes	characteristic
11	D₈×3₋¹⁺²	—	2	2	no	—
12	C₂×C₈×3₋¹⁺²	—	2	2	no	—
13	C₂×C₈×3₋¹⁺²	C₂²	1	1	yes	characteristic
14	C₂×C₈×3₋¹⁺²	C₂²	1	1	yes	characteristic
15	C₄⋊C₄×3₋¹⁺²	C₂²	1	2	yes	—
16	C₂²⋊C₄×3₋¹⁺²	—	2	2	no	—
17	C₂×C₃²×D₈	C₂×C₃	1	1	yes	characteristic
18	C₂²×C₃²×C₈	C₂×C₃	1	1	yes	characteristic
19	C₃²×#121	C₂×C₃	1	2	yes	—
20	C₃²×#107	C₂×C₃	1	1	yes	characteristic
21	C₃²×#102	C₂×C₃	1	2	yes	—
22	C₂×C₉×D₈	C₂×C₃	1	3	yes	complemented
23	C₂²×C₈×C₉	C₂×C₃	1	3	yes	complemented
24	C₉×#121	C₂×C₃	1	6	yes	complemented
25	C₉×#107	C₂×C₃	1	3	yes	complemented
26	C₉×#102	C₂×C₃	1	6	yes	complemented
27	C₂³×3₋¹⁺²	—	2	2	no	—
28	C₂³×3₋¹⁺²	D₄	1	1	yes	characteristic
29	D₄×3₋¹⁺²	—	2	2	no	—
30	D₄×3₋¹⁺²	—	2	2	no	—
31	D₄×3₋¹⁺²	—	4	4	no	—
32	C₂×C₄×3₋¹⁺²	—	2	1	no	—
33	C₂×C₄×3₋¹⁺²	—	2	1	no	—
34	C₂×C₄×3₋¹⁺²	—	2	2	no	—
35	C₂×C₄×3₋¹⁺²	D₄	1	1	yes	characteristic
36	C₂×C₄×3₋¹⁺²	C₂³	1	1	yes	characteristic
37	C₈×3₋¹⁺²	—	2	1	no	—
38	C₈×3₋¹⁺²	D₄	1	2	yes	complemented
39	C₃×#466	—	3	1	no	—
40	C₃×#466	C₃²	1	1	yes	characteristic
41	C₂×C₃²×D₄	—	2	2	no	—
42	C₂×C₃²×D₄	C₂²×C₃	1	2	yes	—
43	C₂²×C₃²×C₄	C₂²×C₃	1	1	yes	characteristic
44	C₃²×D₈	—	2	2	no	—
45	C₂×C₃²×C₈	—	2	2	no	—
46	C₂×C₃²×C₈	C₂²×C₃	1	1	yes	characteristic
47	C₂×C₃²×C₈	C₂²×C₃	1	1	yes	characteristic
48	C₃²×C₄⋊C₄	C₂²×C₃	1	2	yes	—
49	C₃²×C₂²⋊C₄	—	2	2	no	—
50	C₂×D₄×C₉	—	2	6	no	—
51	C₂×D₄×C₉	C₂²×C₃	1	6	yes	—
52	C₂²×C₄×C₉	C₂²×C₃	1	3	yes	—
53	C₉×D₈	—	2	6	no	—
54	C₂×C₈×C₉	—	2	6	no	—
55	C₂×C₈×C₉	C₂²×C₃	1	3	yes	—
56	C₂×C₈×C₉	C₂²×C₃	1	3	yes	—
57	C₉×C₄⋊C₄	C₂²×C₃	1	6	yes	—
58	C₉×C₂²⋊C₄	—	2	6	no	—
59	C₂²×3₋¹⁺²	—	2	1	no	—
60	C₂²×3₋¹⁺²	—	2	1	no	—
61	C₂²×3₋¹⁺²	—	4	2	no	—
62	C₂²×3₋¹⁺²	—	4	2	no	—
63	C₂²×3₋¹⁺²	—	4	2	no	—
64	C₂²×3₋¹⁺²	D₈	1	2	yes	complemented
65	C₂²×3₋¹⁺²	C₂×D₄	1	1	yes	characteristic
66	C₄×3₋¹⁺²	—	2	1	no	—
67	C₄×3₋¹⁺²	—	4	2	no	—
68	C₄×3₋¹⁺²	C₂×D₄	1	1	yes	characteristic
69	C₄×3₋¹⁺²	#41	1	1	yes	characteristic
70	C₂×C₃×D₈	—	3	1	no	—
71	C₂×C₃×D₈	C₂×C₃²	1	1	yes	characteristic
72	C₂²×C₃×C₈	—	3	1	no	—
73	C₂²×C₃×C₈	C₂×C₃²	1	1	yes	characteristic
74	C₃×#121	—	3	2	no	—
75	C₃×#121	C₂×C₃²	1	2	yes	—
76	C₃×#107	—	3	1	no	—
77	C₃×#107	C₂×C₃²	1	1	yes	characteristic
78	C₃×#102	—	3	2	no	—
79	C₃×#102	C₂×C₃²	1	2	yes	—
80	C₂³×C₃²	—	2	2	no	—
81	C₂³×C₃²	C₃×D₄	1	1	yes	characteristic
82	C₃²×D₄	—	2	2	no	—
83	C₃²×D₄	—	2	2	no	—
84	C₃²×D₄	—	4	4	no	—
85	C₂×C₃²×C₄	—	2	1	no	—
86	C₂×C₃²×C₄	—	2	1	no	—
87	C₂×C₃²×C₄	—	2	2	no	—
88	C₂×C₃²×C₄	C₃×D₄	1	1	yes	characteristic
89	C₂×C₃²×C₄	C₂³×C₃	1	1	yes	characteristic
90	C₂³×C₉	—	2	6	no	—
91	C₂³×C₉	C₃×D₄	1	3	yes	—
92	C₃²×C₈	—	2	1	no	—
93	C₃²×C₈	C₃×D₄	1	2	yes	—
94	D₄×C₉	—	2	6	no	—
95	D₄×C₉	—	2	6	no	—
96	D₄×C₉	—	4	12	no	—
97	C₂×C₄×C₉	—	2	3	no	—
98	C₂×C₄×C₉	—	2	3	no	—
99	C₂×C₄×C₉	—	2	6	no	—
100	C₂×C₄×C₉	C₃×D₄	1	3	yes	—
101	C₂×C₄×C₉	C₂³×C₃	1	3	yes	—
102	C₈×C₉	—	2	3	no	—
103	C₈×C₉	C₃×D₄	1	6	yes	complemented
104	#466	3₋¹⁺²	1	1	yes	characteristic, direct product factor, Sylow
105	C₂×3₋¹⁺²	—	2	2	no	—
106	C₂×3₋¹⁺²	—	8	2	no	—
107	C₂×3₋¹⁺²	#121	1	1	yes	characteristic
108	C₂×3₋¹⁺²	C₂×D₈	1	1	yes	characteristic
109	C₂×3₋¹⁺²	#135	1	1	yes	characteristic
110	C₂×C₃×D₄	—	2	2	no	—
111	C₂×C₃×D₄	—	3	2	no	—
112	C₂×C₃×D₄	—	6	2	no	—
113	C₂×C₃×D₄	C₂²×C₃²	1	2	yes	—
114	C₂²×C₃×C₄	—	3	1	no	—
115	C₂²×C₃×C₄	C₂²×C₃²	1	1	yes	characteristic
116	C₃×D₈	—	2	2	no	—
117	C₃×D₈	—	6	2	no	—
118	C₂×C₃×C₈	—	2	2	no	—
119	C₂×C₃×C₈	—	3	1	no	—
120	C₂×C₃×C₈	—	3	1	no	—
121	C₂×C₃×C₈	—	6	2	no	—
122	C₂×C₃×C₈	C₂²×C₃²	1	1	yes	characteristic
123	C₂×C₃×C₈	C₂²×C₃²	1	1	yes	characteristic
124	C₃×C₄⋊C₄	—	3	2	no	—
125	C₃×C₄⋊C₄	C₂²×C₃²	1	2	yes	—
126	C₃×C₂²⋊C₄	—	2	2	no	—
127	C₃×C₂²⋊C₄	—	6	2	no	—
128	C₂²×C₃²	—	2	1	no	—
129	C₂²×C₃²	—	2	1	no	—
130	C₂²×C₃²	—	4	2	no	—
131	C₂²×C₃²	—	4	2	no	—
132	C₂²×C₃²	—	4	2	no	—
133	C₂²×C₃²	C₃×D₈	1	2	yes	—
134	C₂²×C₃²	C₂×C₃×D₄	1	1	yes	characteristic
135	C₃²×C₄	—	2	1	no	—
136	C₃²×C₄	—	4	2	no	—
137	C₃²×C₄	C₂×C₃×D₄	1	1	yes	characteristic
138	C₃²×C₄	C₃×#41	1	1	yes	characteristic
139	C₂²×C₉	—	2	3	no	—
140	C₂²×C₉	—	2	3	no	—
141	C₂²×C₉	—	4	6	no	—
142	C₂²×C₉	—	4	6	no	—
143	C₂²×C₉	—	4	6	no	—
144	C₂²×C₉	C₃×D₈	1	6	yes	complemented
145	C₂²×C₉	C₂×C₃×D₄	1	3	yes	—
146	C₄×C₉	—	2	3	no	—
147	C₄×C₉	—	4	6	no	—
148	C₄×C₉	C₂×C₃×D₄	1	3	yes	—
149	C₄×C₉	C₃×#41	1	3	yes	—
150	C₂×D₈	C₂×3₋¹⁺²	1	1	yes	characteristic, complemented
151	C₂²×C₈	C₂×3₋¹⁺²	1	1	yes	characteristic, complemented
152	#121	C₂×3₋¹⁺²	1	2	yes	complemented
153	#107	C₂×3₋¹⁺²	1	1	yes	characteristic, complemented
154	#102	C₂×3₋¹⁺²	1	2	yes	complemented
155	3₋¹⁺²	#466	1	1	yes	characteristic, direct product factor, Sylow
156	C₂³×C₃	—	2	2	no	—
157	C₂³×C₃	—	3	1	no	—
158	C₂³×C₃	—	6	2	no	—
159	C₂³×C₃	C₃²×D₄	1	1	yes	characteristic
160	C₃×D₄	—	2	2	no	—
161	C₃×D₄	—	2	2	no	—
162	C₃×D₄	—	4	4	no	—
163	C₃×D₄	—	6	2	no	—
164	C₃×D₄	—	6	2	no	—
165	C₃×D₄	—	12	4	no	—
166	C₂×C₃×C₄	—	2	1	no	—
167	C₂×C₃×C₄	—	2	1	no	—
168	C₂×C₃×C₄	—	2	2	no	—
169	C₂×C₃×C₄	—	3	1	no	—
170	C₂×C₃×C₄	—	3	1	no	—
171	C₂×C₃×C₄	—	6	1	no	—
172	C₂×C₃×C₄	—	6	1	no	—
173	C₂×C₃×C₄	—	6	2	no	—
174	C₂×C₃×C₄	C₃²×D₄	1	1	yes	characteristic
175	C₂×C₃×C₄	C₂³×C₃²	1	1	yes	characteristic
176	C₃×C₈	—	2	1	no	—
177	C₃×C₈	—	3	2	no	—
178	C₃×C₈	—	6	1	no	—
179	C₃×C₈	C₃²×D₄	1	2	yes	—
180	C₂×C₃²	—	2	2	no	—
181	C₂×C₃²	—	8	2	no	—
182	C₂×C₃²	C₃×#121	1	1	yes	characteristic
183	C₂×C₃²	C₂×C₃×D₈	1	1	yes	characteristic
184	C₂×C₃²	C₃×#135	1	1	yes	characteristic
185	C₂×C₉	—	2	6	no	—
186	C₂×C₉	—	8	6	no	—
187	C₂×C₉	C₃×#121	1	3	yes	—
188	C₂×C₉	C₂×C₃×D₈	1	3	yes	—
189	C₂×C₉	C₃×#135	1	3	yes	—
190	C₂×D₄	—	2	2	no	—
191	C₂×D₄	C₂²×3₋¹⁺²	1	2	yes	—
192	C₂²×C₄	C₂²×3₋¹⁺²	1	1	yes	characteristic
193	D₈	—	2	2	no	—
194	C₂×C₈	—	2	2	no	—
195	C₂×C₈	C₂²×3₋¹⁺²	1	1	yes	characteristic
196	C₂×C₈	C₂²×3₋¹⁺²	1	1	yes	characteristic
197	C₄⋊C₄	C₂²×3₋¹⁺²	1	2	yes	—
198	C₂²⋊C₄	—	2	2	no	—
199	C₂²×C₃	—	2	1	no	—
200	C₂²×C₃	—	2	1	no	—
201	C₂²×C₃	—	3	1	no	—
202	C₂²×C₃	—	3	2	no	—
203	C₂²×C₃	—	4	2	no	—
204	C₂²×C₃	—	4	2	no	—
205	C₂²×C₃	—	4	2	no	—
206	C₂²×C₃	—	6	1	no	—
207	C₂²×C₃	—	6	1	no	—
208	C₂²×C₃	—	12	2	no	—
209	C₂²×C₃	—	12	2	no	—
210	C₂²×C₃	—	12	2	no	—
211	C₂²×C₃	C₃²×D₈	1	2	yes	—
212	C₂²×C₃	C₂×C₃²×D₄	1	1	yes	characteristic
213	C₃×C₄	—	2	1	no	—
214	C₃×C₄	—	3	1	no	—
215	C₃×C₄	—	3	1	no	—
216	C₃×C₄	—	4	2	no	—
217	C₃×C₄	—	6	1	no	—
218	C₃×C₄	—	12	2	no	—
219	C₃×C₄	C₂×C₃²×D₄	1	1	yes	characteristic
220	C₃×C₄	C₃²×#41	1	1	yes	characteristic
221	C₃²	C₃×#466	1	1	yes	characteristic
222	C₉	C₃×#466	1	3	yes	complemented
223	C₂³	—	2	2	no	—
224	C₂³	D₄×3₋¹⁺²	1	1	yes	characteristic
225	D₄	—	2	2	no	—
226	D₄	—	2	2	no	—
227	D₄	—	4	4	no	—
228	C₂×C₄	—	2	1	no	—
229	C₂×C₄	—	2	1	no	—
230	C₂×C₄	—	2	2	no	—
231	C₂×C₄	D₄×3₋¹⁺²	1	1	yes	characteristic
232	C₂×C₄	C₂³×3₋¹⁺²	1	1	yes	characteristic
233	C₈	—	2	1	no	—
234	C₈	D₄×3₋¹⁺²	1	2	yes	complemented
235	C₂×C₃	—	2	2	no	—
236	C₂×C₃	—	3	1	no	—
237	C₂×C₃	—	3	1	no	—
238	C₂×C₃	—	3	1	no	—
239	C₂×C₃	—	6	2	no	—
240	C₂×C₃	—	8	2	no	—
241	C₂×C₃	—	24	2	no	—
242	C₂×C₃	C₃²×#121	1	1	yes	characteristic
243	C₂×C₃	C₂×C₃²×D₈	1	1	yes	characteristic
244	C₂×C₃	C₃²×#135	1	1	yes	characteristic
245	C₂²	—	2	1	no	—
246	C₂²	—	2	1	no	—
247	C₂²	—	4	2	no	—
248	C₂²	—	4	2	no	—
249	C₂²	—	4	2	no	—
250	C₂²	D₈×3₋¹⁺²	1	2	yes	complemented
251	C₂²	C₂×D₄×3₋¹⁺²	1	1	yes	characteristic
252	C₄	—	2	1	no	—
253	C₄	—	4	2	no	—
254	C₄	C₂×D₄×3₋¹⁺²	1	1	yes	characteristic
255	C₄	#41×3₋¹⁺²	1	1	yes	characteristic
256	C₃	—	3	1	no	—
257	C₃	C₃²×#466	1	1	yes	characteristic
258	C₂	—	2	2	no	—
259	C₂	—	8	2	no	—
260	C₂	3₋¹⁺²×#121	1	1	yes	characteristic
261	C₂	C₂×D₈×3₋¹⁺²	1	1	yes	characteristic
262	C₂	3₋¹⁺²×#135	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃	C₂×C₃²×D₄
commutator, Frattini, cosocle	C₂×C₃×C₄	C₂³×C₃²
prime radical	C₃²×#466	C₃
LCS	G ▷ C₂×C₃×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃ ◁ C₂²×C₄×3₋¹⁺² ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×D₈×3₋¹⁺²	C₂	semidirect
2. C₂²×C₈×3₋¹⁺²	C₂	semidirect
3. 3₋¹⁺²×#121	C₂	semidirect
4. 3₋¹⁺²×#107	C₂	semidirect
5. 3₋¹⁺²×#102	C₂	semidirect
7. C₉×#466	C₃	semidirect
22. C₂×C₉×D₈	C₂×C₃	semidirect
23. C₂²×C₈×C₉	C₂×C₃	semidirect
24. C₉×#121	C₂×C₃	semidirect
25. C₉×#107	C₂×C₃	semidirect
26. C₉×#102	C₂×C₃	semidirect
38. C₈×3₋¹⁺²	D₄	semidirect
64. C₂²×3₋¹⁺²	D₈	semidirect
103. C₈×C₉	C₃×D₄	semidirect
104. #466	3₋¹⁺²	direct (N × H)
144. C₂²×C₉	C₃×D₈	semidirect
150. C₂×D₈	C₂×3₋¹⁺²	semidirect
151. C₂²×C₈	C₂×3₋¹⁺²	semidirect
152. #121	C₂×3₋¹⁺²	semidirect
153. #107	C₂×3₋¹⁺²	semidirect
154. #102	C₂×3₋¹⁺²	semidirect
155. 3₋¹⁺²	#466	direct (N × H)
222. C₉	C₃×#466	semidirect
234. C₈	D₄×3₋¹⁺²	semidirect
250. C₂²	D₈×3₋¹⁺²	semidirect

Maximal subgroups

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

subgroup	index
C₂×D₈×3₋¹⁺²	2
C₂²×C₈×3₋¹⁺²	2
3₋¹⁺²×#121	2
3₋¹⁺²×#107	2
3₋¹⁺²×#102	2
C₃²×#466	3
C₉×#466	3

Subgroups — 3−1+2×#466 #13145158

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — 3₋¹⁺²×#466 #13145158