Subgroups — C₃²×#4870 #13162288

Order 1728.

792 conjugacy classes of subgroups.

The subgroup lattice has 322 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²×#893	C₂	1	1	yes	maximal, characteristic
2	C₃²×#888	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃³×#106	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃²×#821	C₂	1	1	yes	maximal, characteristic
5	C₃²×Dic₃⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃²×#809	C₂	1	1	yes	maximal, characteristic
7	C₃²×#808	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃²×#476	—	3	1	no	maximal
9	C₃×#4870	C₃	1	4	yes	maximal, direct product factor
10	C₂×C₃²×D₁₂	C₂²	1	1	yes	characteristic
11	C₂×C₃²×C₄×D₃	—	2	1	no	—
12	C₂×C₃²×Dic₆	—	2	1	no	—
13	C₂×C₃³×C₈	C₂²	1	1	yes	characteristic
14	C₃³×C₄⋊C₄	C₂²	1	1	yes	characteristic
15	C₃³×C₄⋊C₄	C₂²	1	1	yes	characteristic
16	C₃²×(C₂×D₃)⋊C₄	—	2	1	no	—
17	C₃²×C₄⋊Dic₃	C₂²	1	1	yes	characteristic
18	C₃²×Dic₃⋊C₄	—	2	1	no	—
19	C₃²×C₄×Dic₃	C₂²	1	1	yes	characteristic
20	C₂×C₃²×C₃⋊C₈	C₂²	1	1	yes	characteristic
21	C₃²×C₄⋊Q₈	—	3	1	no	—
22	C₃²×C₄×D₄	—	3	1	no	—
23	C₃×#893	C₂×C₃	1	4	yes	—
24	C₃×#888	C₂×C₃	1	4	yes	complemented
25	C₃²×#106	—	2	4	no	—
26	C₃²×#106	—	3	1	no	—
27	C₃²×#106	D₃	1	1	yes	characteristic, complemented
28	C₃²×#106	C₂×C₃	1	4	yes	complemented
29	C₃²×C₄⋊C₈	—	3	1	no	—
30	C₃²×#102	—	3	1	no	—
31	C₃²×#102	—	3	1	no	—
32	C₃×#821	C₂×C₃	1	4	yes	—
33	C₃×Dic₃⋊C₈	C₂×C₃	1	4	yes	complemented
34	C₃×#809	C₂×C₃	1	4	yes	—
35	C₃×#808	C₂×C₃	1	4	yes	complemented
36	C₂²×C₃²×D₃	—	2	1	no	—
37	C₂×C₃³×C₄	—	2	1	no	—
38	C₂×C₃³×C₄	—	2	1	no	—
39	C₂×C₃³×C₄	C₂³	1	1	yes	characteristic
40	C₂×C₃²×Dic₃	—	2	1	no	—
41	C₂×C₃²×Dic₃	D₄	1	1	yes	characteristic
42	C₂×C₃²×Dic₃	D₄	1	1	yes	characteristic
43	C₃²×D₁₂	—	2	1	no	—
44	C₃²×D₁₂	Q₈	1	2	yes	complemented
45	C₃²×C₄×D₃	—	4	2	no	—
46	C₃²×Dic₆	—	4	2	no	—
47	C₃³×C₈	—	2	1	no	—
48	C₃²×C₃⋊C₈	—	2	1	no	—
49	C₃×#476	—	3	4	no	—
50	#4870	C₃²	1	1	yes	characteristic, direct product factor
51	C₂×C₃²×Q₈	—	6	1	no	—
52	C₂×C₃²×D₄	—	3	1	no	—
53	C₂²×C₃²×C₄	—	6	1	no	—
54	C₂×C₃×D₁₂	C₂²×C₃	1	4	yes	—
55	C₂×C₃×C₄×D₃	—	2	4	no	—
56	C₂×C₃×Dic₆	—	2	4	no	—
57	C₂×C₃²×C₈	—	2	4	no	—
58	C₂×C₃²×C₈	—	3	1	no	—
59	C₂×C₃²×C₈	C₂×D₃	1	1	yes	characteristic
60	C₂×C₃²×C₈	C₂²×C₃	1	4	yes	—
61	C₃²×C₄⋊C₄	—	2	4	no	—
62	C₃²×C₄⋊C₄	—	2	4	no	—
63	C₃²×C₄⋊C₄	—	3	1	no	—
64	C₃²×C₄⋊C₄	—	6	1	no	—
65	C₃²×C₄⋊C₄	C₂×D₃	1	1	yes	characteristic
66	C₃²×C₄⋊C₄	C₂×D₃	1	1	yes	characteristic
67	C₃²×C₄⋊C₄	C₂²×C₃	1	4	yes	—
68	C₃²×C₄⋊C₄	C₂²×C₃	1	4	yes	—
69	C₃²×C₂²⋊C₄	—	6	1	no	—
70	C₃²×C₄²	—	3	1	no	—
71	C₃×(C₂×D₃)⋊C₄	—	2	4	no	—
72	C₃×C₄⋊Dic₃	C₂²×C₃	1	4	yes	—
73	C₃×Dic₃⋊C₄	—	2	4	no	—
74	C₃×C₄×Dic₃	C₂²×C₃	1	4	yes	—
75	C₂×C₃×C₃⋊C₈	C₂²×C₃	1	4	yes	—
76	C₂²×C₃³	C₂×D₄	1	1	yes	characteristic
77	C₂×C₃²×D₃	—	2	2	no	—
78	C₂×C₃²×D₃	—	4	1	no	—
79	C₂×C₃²×D₃	—	4	1	no	—
80	C₃³×C₄	—	4	1	no	—
81	C₃³×C₄	—	4	1	no	—
82	C₃³×C₄	C₂×Q₈	1	1	yes	characteristic
83	C₃³×C₄	#41	1	1	yes	characteristic
84	C₃²×Dic₃	—	2	1	no	—
85	C₃²×Dic₃	—	4	1	no	—
86	C₃²×Dic₃	SD₁₆	1	2	yes	—
87	C₃×C₄⋊Q₈	—	3	4	no	—
88	C₃×C₄×D₄	—	3	4	no	—
89	#893	C₂×C₃²	1	1	yes	characteristic
90	#888	C₂×C₃²	1	1	yes	characteristic, complemented
91	C₃×#106	—	2	4	no	—
92	C₃×#106	—	3	4	no	—
93	C₃×#106	C₃×D₃	1	4	yes	complemented
94	C₃×#106	C₂×C₃²	1	1	yes	characteristic, complemented
95	C₃×C₄⋊C₈	—	3	4	no	—
96	C₃×#102	—	3	4	no	—
97	C₃×#102	—	3	4	no	—
98	#821	C₂×C₃²	1	1	yes	characteristic
99	Dic₃⋊C₈	C₂×C₃²	1	1	yes	characteristic, complemented
100	#809	C₂×C₃²	1	1	yes	characteristic
101	#808	C₂×C₃²	1	1	yes	characteristic, complemented
102	C₂³×C₃²	—	6	1	no	—
103	C₂²×C₃×D₃	—	2	4	no	—
104	C₃²×Q₈	—	12	2	no	—
105	C₃²×D₄	—	3	2	no	—
106	C₃²×D₄	—	6	1	no	—
107	C₂×C₃²×C₄	—	2	1	no	—
108	C₂×C₃²×C₄	—	2	1	no	—
109	C₂×C₃²×C₄	—	2	4	no	—
110	C₂×C₃²×C₄	—	2	4	no	—
111	C₂×C₃²×C₄	—	2	4	no	—
112	C₂×C₃²×C₄	—	3	1	no	—
113	C₂×C₃²×C₄	—	3	1	no	—
114	C₂×C₃²×C₄	—	4	4	no	—
115	C₂×C₃²×C₄	—	4	4	no	—
116	C₂×C₃²×C₄	—	6	1	no	—
117	C₂×C₃²×C₄	—	12	2	no	—
118	C₂×C₃²×C₄	C₂²×D₃	1	1	yes	characteristic
119	C₂×C₃²×C₄	C₂³×C₃	1	4	yes	—
120	C₂×C₃×Dic₃	—	2	4	no	—
121	C₂×C₃×Dic₃	C₃×D₄	1	4	yes	—
122	C₂×C₃×Dic₃	C₃×D₄	1	4	yes	—
123	C₃×D₁₂	—	2	4	no	—
124	C₃×D₁₂	C₃×Q₈	1	8	yes	complemented
125	C₃×C₄×D₃	—	4	8	no	—
126	C₃×Dic₆	—	4	8	no	—
127	C₃²×C₈	—	2	8	no	—
128	C₃²×C₈	—	2	1	no	—
129	C₃²×C₈	—	2	4	no	—
130	C₃²×C₈	—	6	1	no	—
131	C₃×C₃⋊C₈	—	2	4	no	—
132	#476	—	3	1	no	Sylow
133	C₂×C₃³	C₂²⋊Q₈	1	1	yes	characteristic
134	C₂×C₃³	C₂×SD₁₆	1	1	yes	characteristic
135	C₂×C₃³	AGL(1,ℤ/8)	1	1	yes	characteristic
136	C₃²×D₃	—	4	2	no	—
137	C₂×C₃×Q₈	—	6	4	no	—
138	C₂×C₃×D₄	—	3	4	no	—
139	C₂²×C₃×C₄	—	6	4	no	—
140	C₂×D₁₂	C₂²×C₃²	1	1	yes	characteristic
141	C₂×C₄×D₃	—	2	1	no	—
142	C₂×Dic₆	—	2	1	no	—
143	C₂×C₃×C₈	—	2	4	no	—
144	C₂×C₃×C₈	—	3	4	no	—
145	C₂×C₃×C₈	C₂×C₃×D₃	1	4	yes	—
146	C₂×C₃×C₈	C₂²×C₃²	1	1	yes	characteristic
147	C₃×C₄⋊C₄	—	2	4	no	—
148	C₃×C₄⋊C₄	—	2	4	no	—
149	C₃×C₄⋊C₄	—	3	4	no	—
150	C₃×C₄⋊C₄	—	6	4	no	—
151	C₃×C₄⋊C₄	C₂×C₃×D₃	1	4	yes	—
152	C₃×C₄⋊C₄	C₂×C₃×D₃	1	4	yes	—
153	C₃×C₄⋊C₄	C₂²×C₃²	1	1	yes	characteristic
154	C₃×C₄⋊C₄	C₂²×C₃²	1	1	yes	characteristic
155	C₃×C₂²⋊C₄	—	6	4	no	—
156	C₃×C₄²	—	3	4	no	—
157	(C₂×D₃)⋊C₄	—	2	1	no	—
158	C₄⋊Dic₃	C₂²×C₃²	1	1	yes	characteristic
159	Dic₃⋊C₄	—	2	1	no	—
160	C₄×Dic₃	C₂²×C₃²	1	1	yes	characteristic
161	C₂×C₃⋊C₈	C₂²×C₃²	1	1	yes	characteristic
162	C₂²×C₃²	—	2	4	no	—
163	C₂²×C₃²	—	6	2	no	—
164	C₂²×C₃²	—	12	1	no	—
165	C₂²×C₃²	—	12	1	no	—
166	C₂²×C₃²	D₃×D₄	1	1	yes	characteristic
167	C₂²×C₃²	C₂×C₃×D₄	1	4	yes	—
168	C₂×C₃×D₃	—	2	8	no	—
169	C₂×C₃×D₃	—	4	4	no	—
170	C₂×C₃×D₃	—	4	4	no	—
171	C₃²×C₄	—	2	4	no	—
172	C₃²×C₄	—	2	4	no	—
173	C₃²×C₄	—	3	2	no	—
174	C₃²×C₄	—	4	1	no	—
175	C₃²×C₄	—	4	1	no	—
176	C₃²×C₄	—	4	4	no	—
177	C₃²×C₄	—	4	4	no	—
178	C₃²×C₄	—	6	1	no	—
179	C₃²×C₄	—	8	4	no	—
180	C₃²×C₄	—	8	4	no	—
181	C₃²×C₄	—	12	1	no	—
182	C₃²×C₄	#235	1	1	yes	characteristic
183	C₃²×C₄	D₃×Q₈	1	1	yes	characteristic
184	C₃²×C₄	C₂×C₃×Q₈	1	4	yes	—
185	C₃²×C₄	C₃×#41	1	4	yes	—
186	C₃×Dic₃	—	2	4	no	—
187	C₃×Dic₃	—	4	4	no	—
188	C₃×Dic₃	C₃×SD₁₆	1	8	yes	—
189	C₄⋊Q₈	—	3	1	no	—
190	C₄×D₄	—	3	1	no	—
191	#106	—	3	1	no	—
192	#106	C₃²×D₃	1	1	yes	characteristic, complemented
193	C₄⋊C₈	—	3	1	no	—
194	#102	—	3	1	no	—
195	#102	—	3	1	no	—
196	C₃³	#476	1	1	yes	characteristic, complemented, Sylow
197	C₂³×C₃	—	6	4	no	—
198	C₂²×D₃	—	2	1	no	—
199	C₃×Q₈	—	12	8	no	—
200	C₃×D₄	—	3	8	no	—
201	C₃×D₄	—	6	4	no	—
202	C₂×C₃×C₄	—	2	4	no	—
203	C₂×C₃×C₄	—	2	4	no	—
204	C₂×C₃×C₄	—	2	1	no	—
205	C₂×C₃×C₄	—	2	1	no	—
206	C₂×C₃×C₄	—	2	4	no	—
207	C₂×C₃×C₄	—	3	4	no	—
208	C₂×C₃×C₄	—	3	4	no	—
209	C₂×C₃×C₄	—	4	4	no	—
210	C₂×C₃×C₄	—	4	4	no	—
211	C₂×C₃×C₄	—	6	4	no	—
212	C₂×C₃×C₄	—	12	8	no	—
213	C₂×C₃×C₄	C₂²×C₃×D₃	1	4	yes	—
214	C₂×C₃×C₄	C₂³×C₃²	1	1	yes	characteristic
215	C₂×Dic₃	—	2	1	no	—
216	C₂×Dic₃	C₃²×D₄	1	1	yes	characteristic
217	C₂×Dic₃	C₃²×D₄	1	1	yes	characteristic
218	D₁₂	—	2	1	no	—
219	D₁₂	C₃²×Q₈	1	2	yes	complemented
220	C₄×D₃	—	4	2	no	—
221	Dic₆	—	4	2	no	—
222	C₃×C₈	—	2	8	no	—
223	C₃×C₈	—	2	4	no	—
224	C₃×C₈	—	2	1	no	—
225	C₃×C₈	—	6	4	no	—
226	C₃⋊C₈	—	2	1	no	—
227	C₂×C₃²	—	2	4	no	—
228	C₂×C₃²	—	2	4	no	—
229	C₂×C₃²	—	2	4	no	—
230	C₂×C₃²	—	12	2	no	—
231	C₂×C₃²	#896	1	1	yes	characteristic
232	C₂×C₃²	D₃×SD₁₆	1	1	yes	characteristic
233	C₂×C₃²	#914	1	1	yes	characteristic
234	C₂×C₃²	C₃×C₂²⋊Q₈	1	4	yes	—
235	C₂×C₃²	C₂×C₃×SD₁₆	1	4	yes	—
236	C₂×C₃²	C₃×AGL(1,ℤ/8)	1	4	yes	—
237	C₃×D₃	—	4	8	no	—
238	C₂×Q₈	—	6	1	no	—
239	C₂×D₄	—	3	1	no	—
240	C₂²×C₄	—	6	1	no	—
241	C₂×C₈	—	3	1	no	—
242	C₂×C₈	C₂×C₃²×D₃	1	1	yes	characteristic
243	C₄⋊C₄	—	3	1	no	—
244	C₄⋊C₄	—	6	1	no	—
245	C₄⋊C₄	C₂×C₃²×D₃	1	1	yes	characteristic
246	C₄⋊C₄	C₂×C₃²×D₃	1	1	yes	characteristic
247	C₂²⋊C₄	—	6	1	no	—
248	C₄²	—	3	1	no	—
249	C₂²×C₃	—	2	4	no	—
250	C₂²×C₃	—	6	8	no	—
251	C₂²×C₃	—	12	4	no	—
252	C₂²×C₃	—	12	4	no	—
253	C₂²×C₃	C₃×D₃×D₄	1	4	yes	—
254	C₂²×C₃	C₂×C₃²×D₄	1	1	yes	characteristic
255	C₂×D₃	—	2	2	no	—
256	C₂×D₃	—	4	1	no	—
257	C₂×D₃	—	4	1	no	—
258	C₃×C₄	—	2	4	no	—
259	C₃×C₄	—	2	4	no	—
260	C₃×C₄	—	3	8	no	—
261	C₃×C₄	—	4	4	no	—
262	C₃×C₄	—	4	4	no	—
263	C₃×C₄	—	4	1	no	—
264	C₃×C₄	—	4	1	no	—
265	C₃×C₄	—	6	4	no	—
266	C₃×C₄	—	8	4	no	—
267	C₃×C₄	—	8	4	no	—
268	C₃×C₄	—	12	4	no	—
269	C₃×C₄	C₃×#235	1	4	yes	—
270	C₃×C₄	C₃×D₃×Q₈	1	4	yes	—
271	C₃×C₄	C₂×C₃²×Q₈	1	1	yes	characteristic
272	C₃×C₄	C₃²×#41	1	1	yes	characteristic
273	Dic₃	—	2	1	no	—
274	Dic₃	—	4	1	no	—
275	Dic₃	C₃²×SD₁₆	1	2	yes	—
276	C₃²	—	2	4	no	—
277	C₃²	#4870	1	1	yes	characteristic, direct product factor
278	C₃²	C₃×#476	1	4	yes	complemented
279	C₂³	—	6	1	no	—
280	Q₈	—	12	2	no	—
281	D₄	—	3	2	no	—
282	D₄	—	6	1	no	—
283	C₂×C₄	—	2	1	no	—
284	C₂×C₄	—	2	1	no	—
285	C₂×C₄	—	3	1	no	—
286	C₂×C₄	—	3	1	no	—
287	C₂×C₄	—	6	1	no	—
288	C₂×C₄	—	12	2	no	—
289	C₂×C₄	C₂²×C₃²×D₃	1	1	yes	characteristic
290	C₈	—	2	1	no	—
291	C₈	—	6	1	no	—
292	C₂×C₃	—	2	4	no	—
293	C₂×C₃	—	2	4	no	—
294	C₂×C₃	—	2	4	no	—
295	C₂×C₃	—	12	8	no	—
296	C₂×C₃	C₃×#896	1	4	yes	—
297	C₂×C₃	C₃×D₃×SD₁₆	1	4	yes	—
298	C₂×C₃	C₃×#914	1	4	yes	—
299	C₂×C₃	C₃²×C₂²⋊Q₈	1	1	yes	characteristic
300	C₂×C₃	C₂×C₃²×SD₁₆	1	1	yes	characteristic
301	C₂×C₃	C₃²×AGL(1,ℤ/8)	1	1	yes	characteristic
302	D₃	—	4	2	no	—
303	C₂²	—	6	2	no	—
304	C₂²	—	12	1	no	—
305	C₂²	—	12	1	no	—
306	C₂²	C₃²×D₃×D₄	1	1	yes	characteristic
307	C₄	—	3	2	no	—
308	C₄	—	4	1	no	—
309	C₄	—	4	1	no	—
310	C₄	—	6	1	no	—
311	C₄	—	12	1	no	—
312	C₄	C₃²×#235	1	1	yes	characteristic
313	C₄	C₃²×D₃×Q₈	1	1	yes	characteristic
314	C₃	—	2	4	no	—
315	C₃	C₃×#4870	1	4	yes	direct product factor
316	C₃	C₃²×#476	1	1	yes	characteristic, complemented
317	C₂	—	12	2	no	—
318	C₂	C₃²×#896	1	1	yes	characteristic
319	C₂	C₃²×D₃×SD₁₆	1	1	yes	characteristic
320	C₂	C₃²×#914	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₃²	D₃×D₄
commutator, cosocle	C₂×C₃×C₄	C₂³×C₃²
Frattini	C₂×C₄	C₂²×C₃²×D₃
Fitting	C₃³×#106	C₂
socle	C₂²×C₃³	C₂×D₄
prime radical	C₂×C₃²×D₁₂	C₂²
hypercenter	C₃²×#106	D₃
hypocenter	C₃	C₃²×#476
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₃² ◁ C₂×C₃²×C₄ ◁ C₃²×#106

Complements

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

N	H ≅ G/N	type
2. C₃²×#888	C₂	semidirect
3. C₃³×#106	C₂	semidirect
5. C₃²×Dic₃⋊C₈	C₂	semidirect
7. C₃²×#808	C₂	semidirect
9. C₃×#4870	C₃	direct (N × H)
24. C₃×#888	C₂×C₃	semidirect
27. C₃²×#106	D₃	semidirect
28. C₃²×#106	C₂×C₃	semidirect
33. C₃×Dic₃⋊C₈	C₂×C₃	semidirect
35. C₃×#808	C₂×C₃	semidirect
44. C₃²×D₁₂	Q₈	semidirect
50. #4870	C₃²	direct (N × H)
90. #888	C₂×C₃²	semidirect
93. C₃×#106	C₃×D₃	semidirect
94. C₃×#106	C₂×C₃²	semidirect
99. Dic₃⋊C₈	C₂×C₃²	semidirect
101. #808	C₂×C₃²	semidirect
124. C₃×D₁₂	C₃×Q₈	semidirect
192. #106	C₃²×D₃	semidirect
196. C₃³	#476	semidirect
219. D₁₂	C₃²×Q₈	semidirect
277. C₃²	#4870	direct (N × H)
278. C₃²	C₃×#476	semidirect
315. C₃	C₃×#4870	direct (N × H)
316. C₃	C₃²×#476	semidirect

Maximal subgroups

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

subgroup	index
C₃²×#893	2
C₃²×#888	2
C₃³×#106	2
C₃²×#821	2
C₃²×Dic₃⋊C₈	2
C₃²×#809	2
C₃²×#808	2
C₃²×#476	3
C₃×#4870	3

Subgroups — C32×#4870 #13162288

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃²×#4870 #13162288