Subgroups — 3₋¹⁺²×#464 #13145152

Order 1728.

448 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	C₂×Q₁₆×3₋¹⁺²	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×SD₁₆×3₋¹⁺²	C₂	1	1	yes	maximal, characteristic
3	3₋¹⁺²×#124	C₂	1	1	yes	maximal, characteristic
4	C₄×Q₈×3₋¹⁺²	C₂	1	1	yes	maximal, characteristic, complemented
5	3₋¹⁺²×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
6	3₋¹⁺²×#103	C₂	1	1	yes	maximal, characteristic, complemented
7	3₋¹⁺²×#102	C₂	1	1	yes	maximal, characteristic
8	C₃²×#464	C₃	1	1	yes	maximal, characteristic
9	C₉×#464	C₃	1	3	yes	maximal, complemented
10	C₂×Q₈×3₋¹⁺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	2	no	—
14	SD₁₆×3₋¹⁺²	—	2	2	no	—
15	C₂×C₈×3₋¹⁺²	C₂²	1	1	yes	characteristic
16	C₂×C₈×3₋¹⁺²	C₂²	1	1	yes	characteristic
17	C₄⋊C₄×3₋¹⁺²	—	2	1	no	—
18	C₄⋊C₄×3₋¹⁺²	C₂²	1	1	yes	characteristic
19	C₂²⋊C₄×3₋¹⁺²	—	2	1	no	—
20	C₂²⋊C₄×3₋¹⁺²	—	2	1	no	—
21	C₄²×3₋¹⁺²	—	2	1	no	—
22	C₄²×3₋¹⁺²	C₂²	1	1	yes	characteristic
23	C₂×C₃²×Q₁₆	C₂×C₃	1	1	yes	characteristic
24	C₂×C₃²×SD₁₆	C₂×C₃	1	1	yes	characteristic
25	C₃²×#124	C₂×C₃	1	1	yes	characteristic
26	C₃²×C₄×Q₈	C₂×C₃	1	1	yes	characteristic
27	C₃²×C₄⋊C₈	C₂×C₃	1	1	yes	characteristic
28	C₃²×#103	C₂×C₃	1	1	yes	characteristic
29	C₃²×#102	C₂×C₃	1	1	yes	characteristic
30	C₂×C₉×Q₁₆	C₂×C₃	1	3	yes	complemented
31	C₂×C₉×SD₁₆	C₂×C₃	1	3	yes	—
32	C₉×#124	C₂×C₃	1	3	yes	—
33	C₄×Q₈×C₉	C₂×C₃	1	3	yes	complemented
34	C₉×C₄⋊C₈	C₂×C₃	1	3	yes	complemented
35	C₉×#103	C₂×C₃	1	3	yes	complemented
36	C₉×#102	C₂×C₃	1	3	yes	—
37	C₂³×3₋¹⁺²	—	2	1	no	—
38	Q₈×3₋¹⁺²	—	2	1	no	—
39	Q₈×3₋¹⁺²	—	2	1	no	—
40	Q₈×3₋¹⁺²	—	2	1	no	—
41	Q₈×3₋¹⁺²	D₄	1	2	yes	—
42	D₄×3₋¹⁺²	—	2	1	no	—
43	D₄×3₋¹⁺²	—	2	1	no	—
44	C₂×C₄×3₋¹⁺²	—	2	1	no	—
45	C₂×C₄×3₋¹⁺²	—	2	1	no	—
46	C₂×C₄×3₋¹⁺²	—	2	1	no	—
47	C₂×C₄×3₋¹⁺²	D₄	1	1	yes	characteristic
48	C₂×C₄×3₋¹⁺²	D₄	1	1	yes	characteristic
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	C₃×#464	—	3	1	no	—
53	C₃×#464	C₃²	1	1	yes	characteristic
54	C₂×C₃²×Q₈	C₂²×C₃	1	1	yes	characteristic
55	C₂×C₃²×Q₈	C₂²×C₃	1	1	yes	characteristic
56	C₂×C₃²×D₄	C₂²×C₃	1	1	yes	characteristic
57	C₃²×Q₁₆	—	2	2	no	—
58	C₃²×SD₁₆	—	2	2	no	—
59	C₂×C₃²×C₈	C₂²×C₃	1	1	yes	characteristic
60	C₂×C₃²×C₈	C₂²×C₃	1	1	yes	characteristic
61	C₃²×C₄⋊C₄	—	2	1	no	—
62	C₃²×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
63	C₃²×C₂²⋊C₄	—	2	1	no	—
64	C₃²×C₂²⋊C₄	—	2	1	no	—
65	C₃²×C₄²	—	2	1	no	—
66	C₃²×C₄²	C₂²×C₃	1	1	yes	characteristic
67	C₂×Q₈×C₉	C₂²×C₃	1	3	yes	—
68	C₂×Q₈×C₉	C₂²×C₃	1	3	yes	—
69	C₂×D₄×C₉	C₂²×C₃	1	3	yes	—
70	C₉×Q₁₆	—	2	6	no	—
71	C₉×SD₁₆	—	2	6	no	—
72	C₂×C₈×C₉	C₂²×C₃	1	3	yes	—
73	C₂×C₈×C₉	C₂²×C₃	1	3	yes	—
74	C₉×C₄⋊C₄	—	2	3	no	—
75	C₉×C₄⋊C₄	C₂²×C₃	1	3	yes	—
76	C₉×C₂²⋊C₄	—	2	3	no	—
77	C₉×C₂²⋊C₄	—	2	3	no	—
78	C₄²×C₉	—	2	3	no	—
79	C₄²×C₉	C₂²×C₃	1	3	yes	—
80	C₂²×3₋¹⁺²	—	4	1	no	—
81	C₂²×3₋¹⁺²	—	4	1	no	—
82	C₂²×3₋¹⁺²	—	4	1	no	—
83	C₂²×3₋¹⁺²	C₂×D₄	1	1	yes	characteristic
84	C₄×3₋¹⁺²	—	2	2	no	—
85	C₄×3₋¹⁺²	—	2	1	no	—
86	C₄×3₋¹⁺²	—	2	1	no	—
87	C₄×3₋¹⁺²	—	4	1	no	—
88	C₄×3₋¹⁺²	—	4	1	no	—
89	C₄×3₋¹⁺²	C₂×D₄	1	1	yes	characteristic
90	C₄×3₋¹⁺²	#41	1	1	yes	characteristic
91	C₂×C₃×Q₁₆	—	3	1	no	—
92	C₂×C₃×Q₁₆	C₂×C₃²	1	1	yes	characteristic
93	C₂×C₃×SD₁₆	—	3	1	no	—
94	C₂×C₃×SD₁₆	C₂×C₃²	1	1	yes	characteristic
95	C₃×#124	—	3	1	no	—
96	C₃×#124	C₂×C₃²	1	1	yes	characteristic
97	C₃×C₄×Q₈	—	3	1	no	—
98	C₃×C₄×Q₈	C₂×C₃²	1	1	yes	characteristic
99	C₃×C₄⋊C₈	—	3	1	no	—
100	C₃×C₄⋊C₈	C₂×C₃²	1	1	yes	characteristic
101	C₃×#103	—	3	1	no	—
102	C₃×#103	C₂×C₃²	1	1	yes	characteristic
103	C₃×#102	—	3	1	no	—
104	C₃×#102	C₂×C₃²	1	1	yes	characteristic
105	C₂³×C₃²	—	2	1	no	—
106	C₃²×Q₈	—	2	1	no	—
107	C₃²×Q₈	—	2	1	no	—
108	C₃²×Q₈	—	2	1	no	—
109	C₃²×Q₈	C₃×D₄	1	2	yes	—
110	C₃²×D₄	—	2	1	no	—
111	C₃²×D₄	—	2	1	no	—
112	C₂×C₃²×C₄	—	2	1	no	—
113	C₂×C₃²×C₄	—	2	1	no	—
114	C₂×C₃²×C₄	—	2	1	no	—
115	C₂×C₃²×C₄	C₃×D₄	1	1	yes	characteristic
116	C₂×C₃²×C₄	C₃×D₄	1	1	yes	characteristic
117	C₂×C₃²×C₄	C₂³×C₃	1	1	yes	characteristic
118	C₂³×C₉	—	2	3	no	—
119	C₃²×C₈	—	2	1	no	—
120	C₃²×C₈	—	2	1	no	—
121	Q₈×C₉	—	2	3	no	—
122	Q₈×C₉	—	2	3	no	—
123	Q₈×C₉	—	2	3	no	—
124	Q₈×C₉	C₃×D₄	1	6	yes	—
125	D₄×C₉	—	2	3	no	—
126	D₄×C₉	—	2	3	no	—
127	C₂×C₄×C₉	—	2	3	no	—
128	C₂×C₄×C₉	—	2	3	no	—
129	C₂×C₄×C₉	—	2	3	no	—
130	C₂×C₄×C₉	C₃×D₄	1	3	yes	—
131	C₂×C₄×C₉	C₃×D₄	1	3	yes	—
132	C₂×C₄×C₉	C₂³×C₃	1	3	yes	—
133	C₈×C₉	—	2	3	no	—
134	C₈×C₉	—	2	3	no	—
135	#464	3₋¹⁺²	1	1	yes	characteristic, direct product factor, Sylow
136	C₂×3₋¹⁺²	—	8	1	no	—
137	C₂×3₋¹⁺²	#121	1	1	yes	characteristic
138	C₂×3₋¹⁺²	#135	1	1	yes	characteristic
139	C₂×3₋¹⁺²	#137	1	1	yes	characteristic
140	C₂×C₃×Q₈	—	3	1	no	—
141	C₂×C₃×Q₈	—	3	1	no	—
142	C₂×C₃×Q₈	C₂²×C₃²	1	1	yes	characteristic
143	C₂×C₃×Q₈	C₂²×C₃²	1	1	yes	characteristic
144	C₂×C₃×D₄	—	3	1	no	—
145	C₂×C₃×D₄	C₂²×C₃²	1	1	yes	characteristic
146	C₃×Q₁₆	—	2	2	no	—
147	C₃×Q₁₆	—	6	2	no	—
148	C₃×SD₁₆	—	2	2	no	—
149	C₃×SD₁₆	—	6	2	no	—
150	C₂×C₃×C₈	—	3	1	no	—
151	C₂×C₃×C₈	—	3	1	no	—
152	C₂×C₃×C₈	C₂²×C₃²	1	1	yes	characteristic
153	C₂×C₃×C₈	C₂²×C₃²	1	1	yes	characteristic
154	C₃×C₄⋊C₄	—	2	1	no	—
155	C₃×C₄⋊C₄	—	3	1	no	—
156	C₃×C₄⋊C₄	—	6	1	no	—
157	C₃×C₄⋊C₄	C₂²×C₃²	1	1	yes	characteristic
158	C₃×C₂²⋊C₄	—	2	1	no	—
159	C₃×C₂²⋊C₄	—	2	1	no	—
160	C₃×C₂²⋊C₄	—	6	1	no	—
161	C₃×C₂²⋊C₄	—	6	1	no	—
162	C₃×C₄²	—	2	1	no	—
163	C₃×C₄²	—	3	1	no	—
164	C₃×C₄²	—	6	1	no	—
165	C₃×C₄²	C₂²×C₃²	1	1	yes	characteristic
166	C₂²×C₃²	—	4	1	no	—
167	C₂²×C₃²	—	4	1	no	—
168	C₂²×C₃²	—	4	1	no	—
169	C₂²×C₃²	C₂×C₃×D₄	1	1	yes	characteristic
170	C₃²×C₄	—	2	2	no	—
171	C₃²×C₄	—	2	1	no	—
172	C₃²×C₄	—	2	1	no	—
173	C₃²×C₄	—	4	1	no	—
174	C₃²×C₄	—	4	1	no	—
175	C₃²×C₄	C₂×C₃×D₄	1	1	yes	characteristic
176	C₃²×C₄	C₃×#41	1	1	yes	characteristic
177	C₂²×C₉	—	4	3	no	—
178	C₂²×C₉	—	4	3	no	—
179	C₂²×C₉	—	4	3	no	—
180	C₂²×C₉	C₂×C₃×D₄	1	3	yes	—
181	C₄×C₉	—	2	6	no	—
182	C₄×C₉	—	2	3	no	—
183	C₄×C₉	—	2	3	no	—
184	C₄×C₉	—	4	3	no	—
185	C₄×C₉	—	4	3	no	—
186	C₄×C₉	C₂×C₃×D₄	1	3	yes	—
187	C₄×C₉	C₃×#41	1	3	yes	—
188	C₂×Q₁₆	C₂×3₋¹⁺²	1	1	yes	characteristic, complemented
189	C₂×SD₁₆	C₂×3₋¹⁺²	1	1	yes	characteristic
190	#124	C₂×3₋¹⁺²	1	1	yes	characteristic
191	C₄×Q₈	C₂×3₋¹⁺²	1	1	yes	characteristic, complemented
192	C₄⋊C₈	C₂×3₋¹⁺²	1	1	yes	characteristic, complemented
193	#103	C₂×3₋¹⁺²	1	1	yes	characteristic, complemented
194	#102	C₂×3₋¹⁺²	1	1	yes	characteristic
195	3₋¹⁺²	#464	1	1	yes	characteristic, direct product factor, Sylow
196	C₂³×C₃	—	2	1	no	—
197	C₂³×C₃	—	6	1	no	—
198	C₃×Q₈	—	2	1	no	—
199	C₃×Q₈	—	2	1	no	—
200	C₃×Q₈	—	2	1	no	—
201	C₃×Q₈	—	3	2	no	—
202	C₃×Q₈	—	6	1	no	—
203	C₃×Q₈	—	6	1	no	—
204	C₃×Q₈	—	6	1	no	—
205	C₃×Q₈	C₃²×D₄	1	2	yes	—
206	C₃×D₄	—	2	1	no	—
207	C₃×D₄	—	2	1	no	—
208	C₃×D₄	—	6	1	no	—
209	C₃×D₄	—	6	1	no	—
210	C₂×C₃×C₄	—	2	1	no	—
211	C₂×C₃×C₄	—	2	1	no	—
212	C₂×C₃×C₄	—	2	1	no	—
213	C₂×C₃×C₄	—	3	1	no	—
214	C₂×C₃×C₄	—	3	1	no	—
215	C₂×C₃×C₄	—	3	1	no	—
216	C₂×C₃×C₄	—	6	1	no	—
217	C₂×C₃×C₄	—	6	1	no	—
218	C₂×C₃×C₄	—	6	1	no	—
219	C₂×C₃×C₄	C₃²×D₄	1	1	yes	characteristic
220	C₂×C₃×C₄	C₃²×D₄	1	1	yes	characteristic
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	—	6	1	no	—
225	C₃×C₈	—	6	1	no	—
226	C₂×C₃²	—	8	1	no	—
227	C₂×C₃²	C₃×#121	1	1	yes	characteristic
228	C₂×C₃²	C₃×#135	1	1	yes	characteristic
229	C₂×C₃²	C₃×#137	1	1	yes	characteristic
230	C₂×C₉	—	8	3	no	—
231	C₂×C₉	C₃×#121	1	3	yes	—
232	C₂×C₉	C₃×#135	1	3	yes	—
233	C₂×C₉	C₃×#137	1	3	yes	—
234	C₂×Q₈	C₂²×3₋¹⁺²	1	1	yes	characteristic
235	C₂×Q₈	C₂²×3₋¹⁺²	1	1	yes	characteristic
236	C₂×D₄	C₂²×3₋¹⁺²	1	1	yes	characteristic
237	Q₁₆	—	2	2	no	—
238	SD₁₆	—	2	2	no	—
239	C₂×C₈	C₂²×3₋¹⁺²	1	1	yes	characteristic
240	C₂×C₈	C₂²×3₋¹⁺²	1	1	yes	characteristic
241	C₄⋊C₄	—	2	1	no	—
242	C₄⋊C₄	C₂²×3₋¹⁺²	1	1	yes	characteristic
243	C₂²⋊C₄	—	2	1	no	—
244	C₂²⋊C₄	—	2	1	no	—
245	C₄²	—	2	1	no	—
246	C₄²	C₂²×3₋¹⁺²	1	1	yes	characteristic
247	C₂²×C₃	—	3	1	no	—
248	C₂²×C₃	—	4	1	no	—
249	C₂²×C₃	—	4	1	no	—
250	C₂²×C₃	—	4	1	no	—
251	C₂²×C₃	—	12	1	no	—
252	C₂²×C₃	—	12	1	no	—
253	C₂²×C₃	—	12	1	no	—
254	C₂²×C₃	C₂×C₃²×D₄	1	1	yes	characteristic
255	C₃×C₄	—	2	2	no	—
256	C₃×C₄	—	2	1	no	—
257	C₃×C₄	—	2	1	no	—
258	C₃×C₄	—	3	1	no	—
259	C₃×C₄	—	3	1	no	—
260	C₃×C₄	—	4	1	no	—
261	C₃×C₄	—	4	1	no	—
262	C₃×C₄	—	6	1	no	—
263	C₃×C₄	—	6	1	no	—
264	C₃×C₄	—	6	2	no	—
265	C₃×C₄	—	12	1	no	—
266	C₃×C₄	—	12	1	no	—
267	C₃×C₄	C₂×C₃²×D₄	1	1	yes	characteristic
268	C₃×C₄	C₃²×#41	1	1	yes	characteristic
269	C₃²	C₃×#464	1	1	yes	characteristic
270	C₉	C₃×#464	1	3	yes	complemented
271	C₂³	—	2	1	no	—
272	Q₈	—	2	1	no	—
273	Q₈	—	2	1	no	—
274	Q₈	—	2	1	no	—
275	Q₈	D₄×3₋¹⁺²	1	2	yes	—
276	D₄	—	2	1	no	—
277	D₄	—	2	1	no	—
278	C₂×C₄	—	2	1	no	—
279	C₂×C₄	—	2	1	no	—
280	C₂×C₄	—	2	1	no	—
281	C₂×C₄	D₄×3₋¹⁺²	1	1	yes	characteristic
282	C₂×C₄	D₄×3₋¹⁺²	1	1	yes	characteristic
283	C₂×C₄	C₂³×3₋¹⁺²	1	1	yes	characteristic
284	C₈	—	2	1	no	—
285	C₈	—	2	1	no	—
286	C₂×C₃	—	3	1	no	—
287	C₂×C₃	—	3	1	no	—
288	C₂×C₃	—	3	1	no	—
289	C₂×C₃	—	8	1	no	—
290	C₂×C₃	—	24	1	no	—
291	C₂×C₃	C₃²×#121	1	1	yes	characteristic
292	C₂×C₃	C₃²×#135	1	1	yes	characteristic
293	C₂×C₃	C₃²×#137	1	1	yes	characteristic
294	C₂²	—	4	1	no	—
295	C₂²	—	4	1	no	—
296	C₂²	—	4	1	no	—
297	C₂²	C₂×D₄×3₋¹⁺²	1	1	yes	characteristic
298	C₄	—	2	2	no	—
299	C₄	—	2	1	no	—
300	C₄	—	2	1	no	—
301	C₄	—	4	1	no	—
302	C₄	—	4	1	no	—
303	C₄	C₂×D₄×3₋¹⁺²	1	1	yes	characteristic
304	C₄	#41×3₋¹⁺²	1	1	yes	characteristic
305	C₃	—	3	1	no	—
306	C₃	C₃²×#464	1	1	yes	characteristic
307	C₂	—	8	1	no	—
308	C₂	3₋¹⁺²×#121	1	1	yes	characteristic
309	C₂	3₋¹⁺²×#135	1	1	yes	characteristic
310	C₂	3₋¹⁺²×#137	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃	C₂×C₃²×D₄
commutator, Frattini, cosocle	C₂×C₃×C₄	C₂³×C₃²
prime radical	C₂×C₃²×D₄	C₂²×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×Q₁₆×3₋¹⁺²	C₂	semidirect
4. C₄×Q₈×3₋¹⁺²	C₂	semidirect
5. 3₋¹⁺²×C₄⋊C₈	C₂	semidirect
6. 3₋¹⁺²×#103	C₂	semidirect
9. C₉×#464	C₃	semidirect
30. C₂×C₉×Q₁₆	C₂×C₃	semidirect
33. C₄×Q₈×C₉	C₂×C₃	semidirect
34. C₉×C₄⋊C₈	C₂×C₃	semidirect
35. C₉×#103	C₂×C₃	semidirect
135. #464	3₋¹⁺²	direct (N × H)
188. C₂×Q₁₆	C₂×3₋¹⁺²	semidirect
191. C₄×Q₈	C₂×3₋¹⁺²	semidirect
192. C₄⋊C₈	C₂×3₋¹⁺²	semidirect
193. #103	C₂×3₋¹⁺²	semidirect
195. 3₋¹⁺²	#464	direct (N × H)
270. C₉	C₃×#464	semidirect

Maximal subgroups

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

subgroup	index
C₂×Q₁₆×3₋¹⁺²	2
C₂×SD₁₆×3₋¹⁺²	2
3₋¹⁺²×#124	2
C₄×Q₈×3₋¹⁺²	2
3₋¹⁺²×C₄⋊C₈	2
3₋¹⁺²×#103	2
3₋¹⁺²×#102	2
C₃²×#464	3
C₉×#464	3

Subgroups — 3−1+2×#464 #13145152

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — 3₋¹⁺²×#464 #13145152