Subgroups — 3₋¹⁺²×#460 #13145140

Order 1728.

512 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×SD₁₆×3₋¹⁺²	C₂	1	2	yes	maximal, complemented
2	3₋¹⁺²×#127	C₂	1	1	yes	maximal, characteristic
3	C₄×Q₈×3₋¹⁺²	C₂	1	1	yes	maximal, characteristic, complemented
4	3₋¹⁺²×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
5	3₋¹⁺²×#102	C₂	1	2	yes	maximal, complemented
6	C₃²×#460	C₃	1	1	yes	maximal, characteristic
7	C₉×#460	C₃	1	3	yes	maximal, complemented
8	C₂×Q₈×3₋¹⁺²	C₂²	1	1	yes	characteristic
9	C₂×D₄×3₋¹⁺²	—	2	1	no	—
10	C₂×D₄×3₋¹⁺²	—	2	1	no	—
11	C₂×D₄×3₋¹⁺²	C₂²	1	2	yes	—
12	SD₁₆×3₋¹⁺²	—	2	4	no	—
13	C₂×C₈×3₋¹⁺²	C₂²	1	2	yes	—
14	C₄⋊C₄×3₋¹⁺²	—	2	1	no	—
15	C₄⋊C₄×3₋¹⁺²	C₂²	1	1	yes	characteristic
16	C₄²×3₋¹⁺²	—	2	1	no	—
17	C₄²×3₋¹⁺²	C₂²	1	1	yes	characteristic
18	C₂×C₃²×SD₁₆	C₂×C₃	1	2	yes	—
19	C₃²×#127	C₂×C₃	1	1	yes	characteristic
20	C₃²×C₄×Q₈	C₂×C₃	1	1	yes	characteristic
21	C₃²×C₄⋊C₈	C₂×C₃	1	1	yes	characteristic
22	C₃²×#102	C₂×C₃	1	2	yes	—
23	C₂×C₉×SD₁₆	C₂×C₃	1	6	yes	complemented
24	C₉×#127	C₂×C₃	1	3	yes	—
25	C₄×Q₈×C₉	C₂×C₃	1	3	yes	complemented
26	C₉×C₄⋊C₈	C₂×C₃	1	3	yes	complemented
27	C₉×#102	C₂×C₃	1	6	yes	complemented
28	C₂³×3₋¹⁺²	—	2	2	no	—
29	Q₈×3₋¹⁺²	—	2	1	no	—
30	Q₈×3₋¹⁺²	D₄	1	2	yes	complemented
31	D₄×3₋¹⁺²	—	2	2	no	—
32	D₄×3₋¹⁺²	—	2	2	no	—
33	D₄×3₋¹⁺²	—	4	2	no	—
34	D₄×3₋¹⁺²	—	4	2	no	—
35	C₂×C₄×3₋¹⁺²	—	2	1	no	—
36	C₂×C₄×3₋¹⁺²	—	2	1	no	—
37	C₂×C₄×3₋¹⁺²	D₄	1	1	yes	characteristic
38	C₂×C₄×3₋¹⁺²	D₄	1	1	yes	characteristic
39	C₂×C₄×3₋¹⁺²	C₂³	1	1	yes	characteristic
40	C₈×3₋¹⁺²	—	2	2	no	—
41	C₃×#460	—	3	1	no	—
42	C₃×#460	C₃²	1	1	yes	characteristic
43	C₂×C₃²×Q₈	C₂²×C₃	1	1	yes	characteristic
44	C₂×C₃²×D₄	—	2	1	no	—
45	C₂×C₃²×D₄	—	2	1	no	—
46	C₂×C₃²×D₄	C₂²×C₃	1	2	yes	—
47	C₃²×SD₁₆	—	2	4	no	—
48	C₂×C₃²×C₈	C₂²×C₃	1	2	yes	—
49	C₃²×C₄⋊C₄	—	2	1	no	—
50	C₃²×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
51	C₃²×C₄²	—	2	1	no	—
52	C₃²×C₄²	C₂²×C₃	1	1	yes	characteristic
53	C₂×Q₈×C₉	C₂²×C₃	1	3	yes	—
54	C₂×D₄×C₉	—	2	3	no	—
55	C₂×D₄×C₉	—	2	3	no	—
56	C₂×D₄×C₉	C₂²×C₃	1	6	yes	—
57	C₉×SD₁₆	—	2	12	no	—
58	C₂×C₈×C₉	C₂²×C₃	1	6	yes	—
59	C₉×C₄⋊C₄	—	2	3	no	—
60	C₉×C₄⋊C₄	C₂²×C₃	1	3	yes	—
61	C₄²×C₉	—	2	3	no	—
62	C₄²×C₉	C₂²×C₃	1	3	yes	—
63	C₂²×3₋¹⁺²	—	4	2	no	—
64	C₂²×3₋¹⁺²	—	4	2	no	—
65	C₂²×3₋¹⁺²	—	4	2	no	—
66	C₂²×3₋¹⁺²	C₂×D₄	1	1	yes	characteristic
67	C₄×3₋¹⁺²	—	2	2	no	—
68	C₄×3₋¹⁺²	—	2	1	no	—
69	C₄×3₋¹⁺²	—	4	1	no	—
70	C₄×3₋¹⁺²	SD₁₆	1	2	yes	complemented
71	C₄×3₋¹⁺²	C₂×D₄	1	1	yes	characteristic
72	C₄×3₋¹⁺²	#41	1	1	yes	characteristic
73	C₂×C₃×SD₁₆	—	3	2	no	—
74	C₂×C₃×SD₁₆	C₂×C₃²	1	2	yes	—
75	C₃×#127	—	3	1	no	—
76	C₃×#127	C₂×C₃²	1	1	yes	characteristic
77	C₃×C₄×Q₈	—	3	1	no	—
78	C₃×C₄×Q₈	C₂×C₃²	1	1	yes	characteristic
79	C₃×C₄⋊C₈	—	3	1	no	—
80	C₃×C₄⋊C₈	C₂×C₃²	1	1	yes	characteristic
81	C₃×#102	—	3	2	no	—
82	C₃×#102	C₂×C₃²	1	2	yes	—
83	C₂³×C₃²	—	2	2	no	—
84	C₃²×Q₈	—	2	1	no	—
85	C₃²×Q₈	C₃×D₄	1	2	yes	—
86	C₃²×D₄	—	2	2	no	—
87	C₃²×D₄	—	2	2	no	—
88	C₃²×D₄	—	4	2	no	—
89	C₃²×D₄	—	4	2	no	—
90	C₂×C₃²×C₄	—	2	1	no	—
91	C₂×C₃²×C₄	—	2	1	no	—
92	C₂×C₃²×C₄	C₃×D₄	1	1	yes	characteristic
93	C₂×C₃²×C₄	C₃×D₄	1	1	yes	characteristic
94	C₂×C₃²×C₄	C₂³×C₃	1	1	yes	characteristic
95	C₂³×C₉	—	2	6	no	—
96	C₃²×C₈	—	2	2	no	—
97	Q₈×C₉	—	2	3	no	—
98	Q₈×C₉	C₃×D₄	1	6	yes	complemented
99	D₄×C₉	—	2	6	no	—
100	D₄×C₉	—	2	6	no	—
101	D₄×C₉	—	4	6	no	—
102	D₄×C₉	—	4	6	no	—
103	C₂×C₄×C₉	—	2	3	no	—
104	C₂×C₄×C₉	—	2	3	no	—
105	C₂×C₄×C₉	C₃×D₄	1	3	yes	—
106	C₂×C₄×C₉	C₃×D₄	1	3	yes	—
107	C₂×C₄×C₉	C₂³×C₃	1	3	yes	—
108	C₈×C₉	—	2	6	no	—
109	#460	3₋¹⁺²	1	1	yes	characteristic, direct product factor, Sylow
110	C₂×3₋¹⁺²	—	8	2	no	—
111	C₂×3₋¹⁺²	#121	1	1	yes	characteristic
112	C₂×3₋¹⁺²	C₂×SD₁₆	1	1	yes	characteristic
113	C₂×3₋¹⁺²	AGL(1,ℤ/8)	1	1	yes	characteristic
114	C₂×C₃×Q₈	—	3	1	no	—
115	C₂×C₃×Q₈	C₂²×C₃²	1	1	yes	characteristic
116	C₂×C₃×D₄	—	2	1	no	—
117	C₂×C₃×D₄	—	2	1	no	—
118	C₂×C₃×D₄	—	3	2	no	—
119	C₂×C₃×D₄	—	6	1	no	—
120	C₂×C₃×D₄	—	6	1	no	—
121	C₂×C₃×D₄	C₂²×C₃²	1	2	yes	—
122	C₃×SD₁₆	—	2	4	no	—
123	C₃×SD₁₆	—	6	4	no	—
124	C₂×C₃×C₈	—	3	2	no	—
125	C₂×C₃×C₈	C₂²×C₃²	1	2	yes	—
126	C₃×C₄⋊C₄	—	2	1	no	—
127	C₃×C₄⋊C₄	—	3	1	no	—
128	C₃×C₄⋊C₄	—	6	1	no	—
129	C₃×C₄⋊C₄	C₂²×C₃²	1	1	yes	characteristic
130	C₃×C₄²	—	2	1	no	—
131	C₃×C₄²	—	3	1	no	—
132	C₃×C₄²	—	6	1	no	—
133	C₃×C₄²	C₂²×C₃²	1	1	yes	characteristic
134	C₂²×C₃²	—	4	2	no	—
135	C₂²×C₃²	—	4	2	no	—
136	C₂²×C₃²	—	4	2	no	—
137	C₂²×C₃²	C₂×C₃×D₄	1	1	yes	characteristic
138	C₃²×C₄	—	2	2	no	—
139	C₃²×C₄	—	2	1	no	—
140	C₃²×C₄	—	4	1	no	—
141	C₃²×C₄	C₃×SD₁₆	1	2	yes	—
142	C₃²×C₄	C₂×C₃×D₄	1	1	yes	characteristic
143	C₃²×C₄	C₃×#41	1	1	yes	characteristic
144	C₂²×C₉	—	4	6	no	—
145	C₂²×C₉	—	4	6	no	—
146	C₂²×C₉	—	4	6	no	—
147	C₂²×C₉	C₂×C₃×D₄	1	3	yes	—
148	C₄×C₉	—	2	6	no	—
149	C₄×C₉	—	2	3	no	—
150	C₄×C₉	—	4	3	no	—
151	C₄×C₉	C₃×SD₁₆	1	6	yes	complemented
152	C₄×C₉	C₂×C₃×D₄	1	3	yes	—
153	C₄×C₉	C₃×#41	1	3	yes	—
154	C₂×SD₁₆	C₂×3₋¹⁺²	1	2	yes	complemented
155	#127	C₂×3₋¹⁺²	1	1	yes	characteristic
156	C₄×Q₈	C₂×3₋¹⁺²	1	1	yes	characteristic, complemented
157	C₄⋊C₈	C₂×3₋¹⁺²	1	1	yes	characteristic, complemented
158	#102	C₂×3₋¹⁺²	1	2	yes	complemented
159	3₋¹⁺²	#460	1	1	yes	characteristic, direct product factor, Sylow
160	C₂³×C₃	—	2	2	no	—
161	C₂³×C₃	—	6	2	no	—
162	C₃×Q₈	—	2	1	no	—
163	C₃×Q₈	—	3	2	no	—
164	C₃×Q₈	—	6	1	no	—
165	C₃×Q₈	C₃²×D₄	1	2	yes	—
166	C₃×D₄	—	2	2	no	—
167	C₃×D₄	—	2	2	no	—
168	C₃×D₄	—	4	2	no	—
169	C₃×D₄	—	4	2	no	—
170	C₃×D₄	—	6	2	no	—
171	C₃×D₄	—	6	2	no	—
172	C₃×D₄	—	12	2	no	—
173	C₃×D₄	—	12	2	no	—
174	C₂×C₃×C₄	—	2	1	no	—
175	C₂×C₃×C₄	—	2	1	no	—
176	C₂×C₃×C₄	—	3	1	no	—
177	C₂×C₃×C₄	—	3	1	no	—
178	C₂×C₃×C₄	—	3	1	no	—
179	C₂×C₃×C₄	—	6	1	no	—
180	C₂×C₃×C₄	—	6	1	no	—
181	C₂×C₃×C₄	C₃²×D₄	1	1	yes	characteristic
182	C₂×C₃×C₄	C₃²×D₄	1	1	yes	characteristic
183	C₂×C₃×C₄	C₂³×C₃²	1	1	yes	characteristic
184	C₃×C₈	—	2	2	no	—
185	C₃×C₈	—	6	2	no	—
186	C₂×C₃²	—	8	2	no	—
187	C₂×C₃²	C₃×#121	1	1	yes	characteristic
188	C₂×C₃²	C₂×C₃×SD₁₆	1	1	yes	characteristic
189	C₂×C₃²	C₃×AGL(1,ℤ/8)	1	1	yes	characteristic
190	C₂×C₉	—	8	6	no	—
191	C₂×C₉	C₃×#121	1	3	yes	—
192	C₂×C₉	C₂×C₃×SD₁₆	1	3	yes	—
193	C₂×C₉	C₃×AGL(1,ℤ/8)	1	3	yes	—
194	C₂×Q₈	C₂²×3₋¹⁺²	1	1	yes	characteristic
195	C₂×D₄	—	2	1	no	—
196	C₂×D₄	—	2	1	no	—
197	C₂×D₄	C₂²×3₋¹⁺²	1	2	yes	—
198	SD₁₆	—	2	4	no	—
199	C₂×C₈	C₂²×3₋¹⁺²	1	2	yes	—
200	C₄⋊C₄	—	2	1	no	—
201	C₄⋊C₄	C₂²×3₋¹⁺²	1	1	yes	characteristic
202	C₄²	—	2	1	no	—
203	C₄²	C₂²×3₋¹⁺²	1	1	yes	characteristic
204	C₂²×C₃	—	3	1	no	—
205	C₂²×C₃	—	4	2	no	—
206	C₂²×C₃	—	4	2	no	—
207	C₂²×C₃	—	4	2	no	—
208	C₂²×C₃	—	12	2	no	—
209	C₂²×C₃	—	12	2	no	—
210	C₂²×C₃	—	12	2	no	—
211	C₂²×C₃	C₂×C₃²×D₄	1	1	yes	characteristic
212	C₃×C₄	—	2	2	no	—
213	C₃×C₄	—	2	1	no	—
214	C₃×C₄	—	3	1	no	—
215	C₃×C₄	—	3	1	no	—
216	C₃×C₄	—	3	2	no	—
217	C₃×C₄	—	4	1	no	—
218	C₃×C₄	—	6	1	no	—
219	C₃×C₄	—	6	2	no	—
220	C₃×C₄	—	12	1	no	—
221	C₃×C₄	C₃²×SD₁₆	1	2	yes	—
222	C₃×C₄	C₂×C₃²×D₄	1	1	yes	characteristic
223	C₃×C₄	C₃²×#41	1	1	yes	characteristic
224	C₃²	C₃×#460	1	1	yes	characteristic
225	C₉	C₃×#460	1	3	yes	complemented
226	C₂³	—	2	2	no	—
227	Q₈	—	2	1	no	—
228	Q₈	D₄×3₋¹⁺²	1	2	yes	complemented
229	D₄	—	2	2	no	—
230	D₄	—	2	2	no	—
231	D₄	—	4	2	no	—
232	D₄	—	4	2	no	—
233	C₂×C₄	—	2	1	no	—
234	C₂×C₄	—	2	1	no	—
235	C₂×C₄	D₄×3₋¹⁺²	1	1	yes	characteristic
236	C₂×C₄	D₄×3₋¹⁺²	1	1	yes	characteristic
237	C₂×C₄	C₂³×3₋¹⁺²	1	1	yes	characteristic
238	C₈	—	2	2	no	—
239	C₂×C₃	—	3	1	no	—
240	C₂×C₃	—	3	1	no	—
241	C₂×C₃	—	3	1	no	—
242	C₂×C₃	—	8	2	no	—
243	C₂×C₃	—	24	2	no	—
244	C₂×C₃	C₃²×#121	1	1	yes	characteristic
245	C₂×C₃	C₂×C₃²×SD₁₆	1	1	yes	characteristic
246	C₂×C₃	C₃²×AGL(1,ℤ/8)	1	1	yes	characteristic
247	C₂²	—	4	2	no	—
248	C₂²	—	4	2	no	—
249	C₂²	—	4	2	no	—
250	C₂²	C₂×D₄×3₋¹⁺²	1	1	yes	characteristic
251	C₄	—	2	2	no	—
252	C₄	—	2	1	no	—
253	C₄	—	4	1	no	—
254	C₄	SD₁₆×3₋¹⁺²	1	2	yes	complemented
255	C₄	C₂×D₄×3₋¹⁺²	1	1	yes	characteristic
256	C₄	#41×3₋¹⁺²	1	1	yes	characteristic
257	C₃	—	3	1	no	—
258	C₃	C₃²×#460	1	1	yes	characteristic
259	C₂	—	8	2	no	—
260	C₂	3₋¹⁺²×#121	1	1	yes	characteristic
261	C₂	C₂×SD₁₆×3₋¹⁺²	1	1	yes	characteristic
262	C₂	3₋¹⁺²×AGL(1,ℤ/8)	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×#127	C₂×C₃
LCS	G ▷ C₂×C₃×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃ ◁ C₄²×3₋¹⁺² ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×SD₁₆×3₋¹⁺²	C₂	semidirect
3. C₄×Q₈×3₋¹⁺²	C₂	semidirect
4. 3₋¹⁺²×C₄⋊C₈	C₂	semidirect
5. 3₋¹⁺²×#102	C₂	semidirect
7. C₉×#460	C₃	semidirect
23. C₂×C₉×SD₁₆	C₂×C₃	semidirect
25. C₄×Q₈×C₉	C₂×C₃	semidirect
26. C₉×C₄⋊C₈	C₂×C₃	semidirect
27. C₉×#102	C₂×C₃	semidirect
30. Q₈×3₋¹⁺²	D₄	semidirect
70. C₄×3₋¹⁺²	SD₁₆	semidirect
98. Q₈×C₉	C₃×D₄	semidirect
109. #460	3₋¹⁺²	direct (N × H)
151. C₄×C₉	C₃×SD₁₆	semidirect
154. C₂×SD₁₆	C₂×3₋¹⁺²	semidirect
156. C₄×Q₈	C₂×3₋¹⁺²	semidirect
157. C₄⋊C₈	C₂×3₋¹⁺²	semidirect
158. #102	C₂×3₋¹⁺²	semidirect
159. 3₋¹⁺²	#460	direct (N × H)
225. C₉	C₃×#460	semidirect
228. Q₈	D₄×3₋¹⁺²	semidirect
254. C₄	SD₁₆×3₋¹⁺²	semidirect

Maximal subgroups

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

subgroup	index
C₂×SD₁₆×3₋¹⁺²	2
3₋¹⁺²×#127	2
C₄×Q₈×3₋¹⁺²	2
3₋¹⁺²×C₄⋊C₈	2
3₋¹⁺²×#102	2
C₃²×#460	3
C₉×#460	3

Subgroups — 3−1+2×#460 #13145140

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — 3₋¹⁺²×#460 #13145140