Subgroups — 3₋¹⁺²×#471 #13145173

Order 1728.

416 conjugacy classes of subgroups.

The subgroup lattice has 312 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	3₋¹⁺²×#137	C₂	1	1	yes	maximal, characteristic, complemented
2	3₋¹⁺²×AGL(1,ℤ/8)	C₂	1	1	yes	maximal, characteristic
3	C₂×SD₁₆×3₋¹⁺²	C₂	1	1	yes	maximal, characteristic, complemented
4	3₋¹⁺²×#131	C₂	1	1	yes	maximal, characteristic, complemented
5	3₋¹⁺²×#108	C₂	1	1	yes	maximal, characteristic, complemented
6	3₋¹⁺²×#101	C₂	1	1	yes	maximal, characteristic, complemented
7	3₋¹⁺²×#100	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃²×#471	C₃	1	1	yes	maximal, characteristic
9	C₉×#471	C₃	1	3	yes	maximal, complemented
10	#41×3₋¹⁺²	C₂²	1	1	yes	characteristic
11	C₂×Q₈×3₋¹⁺²	C₂²	1	1	yes	characteristic
12	C₂×D₄×3₋¹⁺²	C₂²	1	1	yes	characteristic
13	Q₁₆×3₋¹⁺²	—	2	1	no	—
14	SD₁₆×3₋¹⁺²	—	2	1	no	—
15	SD₁₆×3₋¹⁺²	—	2	1	no	—
16	SD₁₆×3₋¹⁺²	—	2	1	no	—
17	SD₁₆×3₋¹⁺²	—	2	1	no	—
18	D₈×3₋¹⁺²	—	2	1	no	—
19	M₁₆×3₋¹⁺²	—	2	1	no	—
20	M₁₆×3₋¹⁺²	C₂²	1	1	yes	characteristic
21	M₁₆×3₋¹⁺²	C₂²	1	1	yes	characteristic
22	M₁₆×3₋¹⁺²	C₂²	1	1	yes	characteristic
23	C₂×C₈×3₋¹⁺²	—	2	1	no	—
24	C₂×C₈×3₋¹⁺²	C₂²	1	1	yes	characteristic
25	C₃²×#137	C₂×C₃	1	1	yes	characteristic
26	C₃²×AGL(1,ℤ/8)	C₂×C₃	1	1	yes	characteristic
27	C₂×C₃²×SD₁₆	C₂×C₃	1	1	yes	characteristic
28	C₃²×#131	C₂×C₃	1	1	yes	characteristic
29	C₃²×#108	C₂×C₃	1	1	yes	characteristic
30	C₃²×#101	C₂×C₃	1	1	yes	characteristic
31	C₃²×#100	C₂×C₃	1	1	yes	characteristic
32	C₉×#137	C₂×C₃	1	3	yes	complemented
33	C₉×AGL(1,ℤ/8)	C₂×C₃	1	3	yes	—
34	C₂×C₉×SD₁₆	C₂×C₃	1	3	yes	complemented
35	C₉×#131	C₂×C₃	1	3	yes	complemented
36	C₉×#108	C₂×C₃	1	3	yes	complemented
37	C₉×#101	C₂×C₃	1	3	yes	complemented
38	C₉×#100	C₂×C₃	1	3	yes	complemented
39	C₂³×3₋¹⁺²	—	2	1	no	—
40	Q₈×3₋¹⁺²	—	2	1	no	—
41	Q₈×3₋¹⁺²	—	2	1	no	—
42	Q₈×3₋¹⁺²	D₄	1	1	yes	characteristic
43	D₄×3₋¹⁺²	—	2	1	no	—
44	D₄×3₋¹⁺²	—	2	1	no	—
45	D₄×3₋¹⁺²	—	2	1	no	—
46	D₄×3₋¹⁺²	D₄	1	1	yes	characteristic
47	C₂×C₄×3₋¹⁺²	—	2	1	no	—
48	C₂×C₄×3₋¹⁺²	—	2	1	no	—
49	C₂×C₄×3₋¹⁺²	C₂³	1	1	yes	characteristic
50	C₈×3₋¹⁺²	—	2	1	no	—
51	C₈×3₋¹⁺²	—	2	1	no	—
52	C₈×3₋¹⁺²	—	2	1	no	—
53	C₈×3₋¹⁺²	D₄	1	1	yes	characteristic
54	C₈×3₋¹⁺²	D₄	1	1	yes	characteristic
55	C₃×#471	—	3	1	no	—
56	C₃×#471	C₃²	1	1	yes	characteristic
57	C₃²×#41	C₂²×C₃	1	1	yes	characteristic
58	C₂×C₃²×Q₈	C₂²×C₃	1	1	yes	characteristic
59	C₂×C₃²×D₄	C₂²×C₃	1	1	yes	characteristic
60	C₃²×Q₁₆	—	2	1	no	—
61	C₃²×SD₁₆	—	2	1	no	—
62	C₃²×SD₁₆	—	2	1	no	—
63	C₃²×SD₁₆	—	2	1	no	—
64	C₃²×SD₁₆	—	2	1	no	—
65	C₃²×D₈	—	2	1	no	—
66	C₃²×M₁₆	—	2	1	no	—
67	C₃²×M₁₆	C₂²×C₃	1	1	yes	characteristic
68	C₃²×M₁₆	C₂²×C₃	1	1	yes	characteristic
69	C₃²×M₁₆	C₂²×C₃	1	1	yes	characteristic
70	C₂×C₃²×C₈	—	2	1	no	—
71	C₂×C₃²×C₈	C₂²×C₃	1	1	yes	characteristic
72	C₉×#41	C₂²×C₃	1	3	yes	—
73	C₂×Q₈×C₉	C₂²×C₃	1	3	yes	—
74	C₂×D₄×C₉	C₂²×C₃	1	3	yes	—
75	C₉×Q₁₆	—	2	3	no	—
76	C₉×SD₁₆	—	2	3	no	—
77	C₉×SD₁₆	—	2	3	no	—
78	C₉×SD₁₆	—	2	3	no	—
79	C₉×SD₁₆	—	2	3	no	—
80	C₉×D₈	—	2	3	no	—
81	C₉×M₁₆	—	2	3	no	—
82	C₉×M₁₆	C₂²×C₃	1	3	yes	—
83	C₉×M₁₆	C₂²×C₃	1	3	yes	—
84	C₉×M₁₆	C₂²×C₃	1	3	yes	—
85	C₂×C₈×C₉	—	2	3	no	—
86	C₂×C₈×C₉	C₂²×C₃	1	3	yes	—
87	C₂²×3₋¹⁺²	—	2	1	no	—
88	C₂²×3₋¹⁺²	—	4	1	no	—
89	C₂²×3₋¹⁺²	—	8	1	no	—
90	C₂²×3₋¹⁺²	#41	1	1	yes	characteristic
91	C₄×3₋¹⁺²	—	2	1	no	—
92	C₄×3₋¹⁺²	—	4	1	no	—
93	C₄×3₋¹⁺²	C₂×D₄	1	1	yes	characteristic
94	C₄×3₋¹⁺²	C₂×D₄	1	1	yes	characteristic
95	C₃×#137	—	3	1	no	—
96	C₃×#137	C₂×C₃²	1	1	yes	characteristic
97	C₃×AGL(1,ℤ/8)	—	3	1	no	—
98	C₃×AGL(1,ℤ/8)	C₂×C₃²	1	1	yes	characteristic
99	C₂×C₃×SD₁₆	—	3	1	no	—
100	C₂×C₃×SD₁₆	C₂×C₃²	1	1	yes	characteristic
101	C₃×#131	—	3	1	no	—
102	C₃×#131	C₂×C₃²	1	1	yes	characteristic
103	C₃×#108	—	3	1	no	—
104	C₃×#108	C₂×C₃²	1	1	yes	characteristic
105	C₃×#101	—	3	1	no	—
106	C₃×#101	C₂×C₃²	1	1	yes	characteristic
107	C₃×#100	—	3	1	no	—
108	C₃×#100	C₂×C₃²	1	1	yes	characteristic
109	C₂³×C₃²	—	2	1	no	—
110	C₃²×Q₈	—	2	1	no	—
111	C₃²×Q₈	—	2	1	no	—
112	C₃²×Q₈	C₃×D₄	1	1	yes	characteristic
113	C₃²×D₄	—	2	1	no	—
114	C₃²×D₄	—	2	1	no	—
115	C₃²×D₄	—	2	1	no	—
116	C₃²×D₄	C₃×D₄	1	1	yes	characteristic
117	C₂×C₃²×C₄	—	2	1	no	—
118	C₂×C₃²×C₄	—	2	1	no	—
119	C₂×C₃²×C₄	C₂³×C₃	1	1	yes	characteristic
120	C₂³×C₉	—	2	3	no	—
121	C₃²×C₈	—	2	1	no	—
122	C₃²×C₈	—	2	1	no	—
123	C₃²×C₈	—	2	1	no	—
124	C₃²×C₈	C₃×D₄	1	1	yes	characteristic
125	C₃²×C₈	C₃×D₄	1	1	yes	characteristic
126	Q₈×C₉	—	2	3	no	—
127	Q₈×C₉	—	2	3	no	—
128	Q₈×C₉	C₃×D₄	1	3	yes	—
129	D₄×C₉	—	2	3	no	—
130	D₄×C₉	—	2	3	no	—
131	D₄×C₉	—	2	3	no	—
132	D₄×C₉	C₃×D₄	1	3	yes	—
133	C₂×C₄×C₉	—	2	3	no	—
134	C₂×C₄×C₉	—	2	3	no	—
135	C₂×C₄×C₉	C₂³×C₃	1	3	yes	—
136	C₈×C₉	—	2	3	no	—
137	C₈×C₉	—	2	3	no	—
138	C₈×C₉	—	2	3	no	—
139	C₈×C₉	C₃×D₄	1	3	yes	—
140	C₈×C₉	C₃×D₄	1	3	yes	—
141	#471	3₋¹⁺²	1	1	yes	characteristic, direct product factor, Sylow
142	C₂×3₋¹⁺²	—	2	1	no	—
143	C₂×3₋¹⁺²	—	4	1	no	—
144	C₂×3₋¹⁺²	—	8	1	no	—
145	C₂×3₋¹⁺²	#121	1	1	yes	characteristic
146	C₃×#41	—	3	1	no	—
147	C₃×#41	C₂²×C₃²	1	1	yes	characteristic
148	C₂×C₃×Q₈	—	3	1	no	—
149	C₂×C₃×Q₈	C₂²×C₃²	1	1	yes	characteristic
150	C₂×C₃×D₄	—	3	1	no	—
151	C₂×C₃×D₄	C₂²×C₃²	1	1	yes	characteristic
152	C₃×Q₁₆	—	2	1	no	—
153	C₃×Q₁₆	—	6	1	no	—
154	C₃×SD₁₆	—	2	1	no	—
155	C₃×SD₁₆	—	2	1	no	—
156	C₃×SD₁₆	—	2	1	no	—
157	C₃×SD₁₆	—	2	1	no	—
158	C₃×SD₁₆	—	6	1	no	—
159	C₃×SD₁₆	—	6	1	no	—
160	C₃×SD₁₆	—	6	1	no	—
161	C₃×SD₁₆	—	6	1	no	—
162	C₃×D₈	—	2	1	no	—
163	C₃×D₈	—	6	1	no	—
164	C₃×M₁₆	—	2	1	no	—
165	C₃×M₁₆	—	3	1	no	—
166	C₃×M₁₆	—	3	1	no	—
167	C₃×M₁₆	—	3	1	no	—
168	C₃×M₁₆	—	6	1	no	—
169	C₃×M₁₆	C₂²×C₃²	1	1	yes	characteristic
170	C₃×M₁₆	C₂²×C₃²	1	1	yes	characteristic
171	C₃×M₁₆	C₂²×C₃²	1	1	yes	characteristic
172	C₂×C₃×C₈	—	2	1	no	—
173	C₂×C₃×C₈	—	3	1	no	—
174	C₂×C₃×C₈	—	6	1	no	—
175	C₂×C₃×C₈	C₂²×C₃²	1	1	yes	characteristic
176	C₂²×C₃²	—	2	1	no	—
177	C₂²×C₃²	—	4	1	no	—
178	C₂²×C₃²	—	8	1	no	—
179	C₂²×C₃²	C₃×#41	1	1	yes	characteristic
180	C₃²×C₄	—	2	1	no	—
181	C₃²×C₄	—	4	1	no	—
182	C₃²×C₄	C₂×C₃×D₄	1	1	yes	characteristic
183	C₃²×C₄	C₂×C₃×D₄	1	1	yes	characteristic
184	C₂²×C₉	—	2	3	no	—
185	C₂²×C₉	—	4	3	no	—
186	C₂²×C₉	—	8	3	no	—
187	C₂²×C₉	C₃×#41	1	3	yes	—
188	C₄×C₉	—	2	3	no	—
189	C₄×C₉	—	4	3	no	—
190	C₄×C₉	C₂×C₃×D₄	1	3	yes	—
191	C₄×C₉	C₂×C₃×D₄	1	3	yes	—
192	#137	C₂×3₋¹⁺²	1	1	yes	characteristic, complemented
193	AGL(1,ℤ/8)	C₂×3₋¹⁺²	1	1	yes	characteristic
194	C₂×SD₁₆	C₂×3₋¹⁺²	1	1	yes	characteristic, complemented
195	#131	C₂×3₋¹⁺²	1	1	yes	characteristic, complemented
196	#108	C₂×3₋¹⁺²	1	1	yes	characteristic, complemented
197	#101	C₂×3₋¹⁺²	1	1	yes	characteristic, complemented
198	#100	C₂×3₋¹⁺²	1	1	yes	characteristic, complemented
199	3₋¹⁺²	#471	1	1	yes	characteristic, direct product factor, Sylow
200	C₂³×C₃	—	2	1	no	—
201	C₂³×C₃	—	6	1	no	—
202	C₃×Q₈	—	2	1	no	—
203	C₃×Q₈	—	2	1	no	—
204	C₃×Q₈	—	3	1	no	—
205	C₃×Q₈	—	6	1	no	—
206	C₃×Q₈	—	6	1	no	—
207	C₃×Q₈	C₃²×D₄	1	1	yes	characteristic
208	C₃×D₄	—	2	1	no	—
209	C₃×D₄	—	2	1	no	—
210	C₃×D₄	—	2	1	no	—
211	C₃×D₄	—	3	1	no	—
212	C₃×D₄	—	6	1	no	—
213	C₃×D₄	—	6	1	no	—
214	C₃×D₄	—	6	1	no	—
215	C₃×D₄	C₃²×D₄	1	1	yes	characteristic
216	C₂×C₃×C₄	—	2	1	no	—
217	C₂×C₃×C₄	—	2	1	no	—
218	C₂×C₃×C₄	—	3	1	no	—
219	C₂×C₃×C₄	—	6	1	no	—
220	C₂×C₃×C₄	—	6	1	no	—
221	C₂×C₃×C₄	C₂³×C₃²	1	1	yes	characteristic
222	C₃×C₈	—	2	1	no	—
223	C₃×C₈	—	2	1	no	—
224	C₃×C₈	—	2	1	no	—
225	C₃×C₈	—	3	1	no	—
226	C₃×C₈	—	3	1	no	—
227	C₃×C₈	—	6	1	no	—
228	C₃×C₈	—	6	1	no	—
229	C₃×C₈	—	6	1	no	—
230	C₃×C₈	C₃²×D₄	1	1	yes	characteristic
231	C₃×C₈	C₃²×D₄	1	1	yes	characteristic
232	C₂×C₃²	—	2	1	no	—
233	C₂×C₃²	—	4	1	no	—
234	C₂×C₃²	—	8	1	no	—
235	C₂×C₃²	C₃×#121	1	1	yes	characteristic
236	C₂×C₉	—	2	3	no	—
237	C₂×C₉	—	4	3	no	—
238	C₂×C₉	—	8	3	no	—
239	C₂×C₉	C₃×#121	1	3	yes	—
240	#41	C₂²×3₋¹⁺²	1	1	yes	characteristic
241	C₂×Q₈	C₂²×3₋¹⁺²	1	1	yes	characteristic
242	C₂×D₄	C₂²×3₋¹⁺²	1	1	yes	characteristic
243	Q₁₆	—	2	1	no	—
244	SD₁₆	—	2	1	no	—
245	SD₁₆	—	2	1	no	—
246	SD₁₆	—	2	1	no	—
247	SD₁₆	—	2	1	no	—
248	D₈	—	2	1	no	—
249	M₁₆	—	2	1	no	—
250	M₁₆	C₂²×3₋¹⁺²	1	1	yes	characteristic
251	M₁₆	C₂²×3₋¹⁺²	1	1	yes	characteristic
252	M₁₆	C₂²×3₋¹⁺²	1	1	yes	characteristic
253	C₂×C₈	—	2	1	no	—
254	C₂×C₈	C₂²×3₋¹⁺²	1	1	yes	characteristic
255	C₂²×C₃	—	2	1	no	—
256	C₂²×C₃	—	3	1	no	—
257	C₂²×C₃	—	4	1	no	—
258	C₂²×C₃	—	6	1	no	—
259	C₂²×C₃	—	8	1	no	—
260	C₂²×C₃	—	12	1	no	—
261	C₂²×C₃	—	24	1	no	—
262	C₂²×C₃	C₃²×#41	1	1	yes	characteristic
263	C₃×C₄	—	2	1	no	—
264	C₃×C₄	—	3	1	no	—
265	C₃×C₄	—	3	1	no	—
266	C₃×C₄	—	4	1	no	—
267	C₃×C₄	—	6	1	no	—
268	C₃×C₄	—	12	1	no	—
269	C₃×C₄	C₂×C₃²×D₄	1	1	yes	characteristic
270	C₃×C₄	C₂×C₃²×D₄	1	1	yes	characteristic
271	C₃²	C₃×#471	1	1	yes	characteristic
272	C₉	C₃×#471	1	3	yes	complemented
273	C₂³	—	2	1	no	—
274	Q₈	—	2	1	no	—
275	Q₈	—	2	1	no	—
276	Q₈	D₄×3₋¹⁺²	1	1	yes	characteristic
277	D₄	—	2	1	no	—
278	D₄	—	2	1	no	—
279	D₄	—	2	1	no	—
280	D₄	D₄×3₋¹⁺²	1	1	yes	characteristic
281	C₂×C₄	—	2	1	no	—
282	C₂×C₄	—	2	1	no	—
283	C₂×C₄	C₂³×3₋¹⁺²	1	1	yes	characteristic
284	C₈	—	2	1	no	—
285	C₈	—	2	1	no	—
286	C₈	—	2	1	no	—
287	C₈	D₄×3₋¹⁺²	1	1	yes	characteristic
288	C₈	D₄×3₋¹⁺²	1	1	yes	characteristic
289	C₂×C₃	—	2	1	no	—
290	C₂×C₃	—	3	1	no	—
291	C₂×C₃	—	4	1	no	—
292	C₂×C₃	—	6	1	no	—
293	C₂×C₃	—	8	1	no	—
294	C₂×C₃	—	12	1	no	—
295	C₂×C₃	—	24	1	no	—
296	C₂×C₃	C₃²×#121	1	1	yes	characteristic
297	C₂²	—	2	1	no	—
298	C₂²	—	4	1	no	—
299	C₂²	—	8	1	no	—
300	C₂²	#41×3₋¹⁺²	1	1	yes	characteristic
301	C₄	—	2	1	no	—
302	C₄	—	4	1	no	—
303	C₄	C₂×D₄×3₋¹⁺²	1	1	yes	characteristic
304	C₄	C₂×D₄×3₋¹⁺²	1	1	yes	characteristic
305	C₃	—	3	1	no	—
306	C₃	C₃²×#471	1	1	yes	characteristic
307	C₂	—	2	1	no	—
308	C₂	—	4	1	no	—
309	C₂	—	8	1	no	—
310	C₂	3₋¹⁺²×#121	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₃	C₃²×#121
commutator, Frattini, cosocle	C₂×C₃×C₄	C₂³×C₃²
prime radical	C₃²×AGL(1,ℤ/8)	C₂×C₃
LCS	G ▷ C₂×C₃×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₃ ◁ C₂×C₄×3₋¹⁺² ◁ G

Complements

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

N	H ≅ G/N	type
1. 3₋¹⁺²×#137	C₂	semidirect
3. C₂×SD₁₆×3₋¹⁺²	C₂	semidirect
4. 3₋¹⁺²×#131	C₂	semidirect
5. 3₋¹⁺²×#108	C₂	semidirect
6. 3₋¹⁺²×#101	C₂	semidirect
7. 3₋¹⁺²×#100	C₂	semidirect
9. C₉×#471	C₃	semidirect
32. C₉×#137	C₂×C₃	semidirect
34. C₂×C₉×SD₁₆	C₂×C₃	semidirect
35. C₉×#131	C₂×C₃	semidirect
36. C₉×#108	C₂×C₃	semidirect
37. C₉×#101	C₂×C₃	semidirect
38. C₉×#100	C₂×C₃	semidirect
141. #471	3₋¹⁺²	direct (N × H)
192. #137	C₂×3₋¹⁺²	semidirect
194. C₂×SD₁₆	C₂×3₋¹⁺²	semidirect
195. #131	C₂×3₋¹⁺²	semidirect
196. #108	C₂×3₋¹⁺²	semidirect
197. #101	C₂×3₋¹⁺²	semidirect
198. #100	C₂×3₋¹⁺²	semidirect
199. 3₋¹⁺²	#471	direct (N × H)
272. C₉	C₃×#471	semidirect

Maximal subgroups

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

subgroup	index
3₋¹⁺²×#137	2
3₋¹⁺²×AGL(1,ℤ/8)	2
C₂×SD₁₆×3₋¹⁺²	2
3₋¹⁺²×#131	2
3₋¹⁺²×#108	2
3₋¹⁺²×#101	2
3₋¹⁺²×#100	2
C₃²×#471	3
C₉×#471	3

Subgroups — 3−1+2×#471 #13145173

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — 3₋¹⁺²×#471 #13145173