Subgroups — C₃²×#4748 #13162184

Order 1728.

808 conjugacy classes of subgroups.

The subgroup lattice has 350 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃²×#214	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃²×#932	C₂	1	1	yes	maximal, characteristic
3	C₃²×#909	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃²×#907	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃³×#100	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃²×#824	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃²×#822	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃²×#471	—	3	1	no	maximal
9	C₃×#4748	C₃	1	4	yes	maximal, direct product factor
10	C₂×C₃³×D₄	C₂²	1	1	yes	characteristic
11	C₃²×#235	C₂²	1	1	yes	characteristic
12	C₂×C₃²×Dic₆	C₂²	1	1	yes	characteristic
13	C₃³×M₁₆	C₂²	1	1	yes	characteristic
14	C₃³×M₁₆	C₂²	1	1	yes	characteristic
15	C₃²×#214	—	2	1	no	—
16	C₃²×#214	—	2	1	no	—
17	C₃²×#214	—	2	1	no	—
18	C₃²×#213	—	2	1	no	—
19	C₃²×#208	C₂²	1	1	yes	characteristic
20	C₂×C₃²×C₃⋊C₈	C₂²	1	1	yes	characteristic
21	C₃²×Dic₁₂	—	2	1	no	—
22	C₃²×#204	—	2	1	no	—
23	C₃²×ΓL(1,𝔽₂₅)	—	2	1	no	—
24	C₃²×D₃×C₈	—	2	1	no	—
25	C₃²×#137	—	3	1	no	—
26	C₃²×AGL(1,ℤ/8)	—	3	1	no	—
27	C₂×C₃²×SD₁₆	—	3	1	no	—
28	C₃²×#131	—	3	1	no	—
29	C₂×C₃×#214	C₂×C₃	1	4	yes	complemented
30	C₃×#932	C₂×C₃	1	4	yes	—
31	C₃×#909	C₂×C₃	1	4	yes	complemented
32	C₃×#907	C₂×C₃	1	4	yes	complemented
33	C₃²×#108	—	3	1	no	—
34	C₃²×#101	—	3	1	no	—
35	C₃²×#100	—	2	4	no	—
36	C₃²×#100	D₃	1	1	yes	characteristic, complemented
37	C₃²×#100	C₂×C₃	1	4	yes	complemented
38	C₃×#824	C₂×C₃	1	4	yes	complemented
39	C₃×#822	C₂×C₃	1	4	yes	complemented
40	C₂³×C₃³	—	2	1	no	—
41	C₃³×D₄	—	2	1	no	—
42	C₃³×D₄	—	2	1	no	—
43	C₂×C₃³×C₄	C₂³	1	1	yes	characteristic
44	C₃²×#67	—	2	1	no	—
45	C₂×C₃²×Dic₃	—	2	1	no	—
46	C₃²×D₁₂	D₄	1	1	yes	characteristic
47	C₃²×C₄×D₃	—	2	1	no	—
48	C₃²×Dic₆	—	2	1	no	—
49	C₃²×Dic₆	—	2	1	no	—
50	C₃²×Dic₆	D₄	1	1	yes	characteristic
51	C₃³×C₈	—	2	1	no	—
52	C₃³×C₈	—	2	1	no	—
53	C₃²×C₃⋊C₈	—	2	1	no	—
54	C₃²×C₃⋊C₈	D₄	1	1	yes	characteristic
55	C₃²×C₃⋊C₈	D₄	1	1	yes	characteristic
56	C₃×#471	—	3	4	no	—
57	#4748	C₃²	1	1	yes	characteristic, direct product factor
58	C₃²×#41	—	3	1	no	—
59	C₂×C₃²×Q₈	—	3	1	no	—
60	C₂×C₃²×D₄	—	2	4	no	—
61	C₂×C₃²×D₄	C₂×D₃	1	1	yes	characteristic
62	C₂×C₃²×D₄	C₂²×C₃	1	4	yes	—
63	C₃×#235	C₂²×C₃	1	4	yes	—
64	C₂×C₃×Dic₆	C₂²×C₃	1	4	yes	—
65	C₃²×Q₁₆	—	6	1	no	—
66	C₃²×SD₁₆	—	6	1	no	—
67	C₃²×SD₁₆	—	6	1	no	—
68	C₃²×SD₁₆	—	6	1	no	—
69	C₃²×SD₁₆	—	6	1	no	—
70	C₃²×D₈	—	6	1	no	—
71	C₃²×M₁₆	—	2	4	no	—
72	C₃²×M₁₆	—	2	4	no	—
73	C₃²×M₁₆	—	3	1	no	—
74	C₃²×M₁₆	—	6	1	no	—
75	C₃²×M₁₆	C₂×D₃	1	1	yes	characteristic
76	C₃²×M₁₆	C₂×D₃	1	1	yes	characteristic
77	C₃²×M₁₆	C₂²×C₃	1	4	yes	—
78	C₃²×M₁₆	C₂²×C₃	1	4	yes	—
79	C₂×C₃²×C₈	—	3	1	no	—
80	C₂×C₃²×C₈	—	6	1	no	—
81	C₃×#214	—	2	4	no	—
82	C₃×#214	—	2	4	no	—
83	C₃×#214	—	2	4	no	—
84	C₃×#213	—	2	4	no	—
85	C₃×#208	C₂²×C₃	1	4	yes	—
86	C₂×C₃×C₃⋊C₈	C₂²×C₃	1	4	yes	—
87	C₃×Dic₁₂	—	2	4	no	—
88	C₃×#204	—	2	4	no	—
89	C₃×ΓL(1,𝔽₂₅)	—	2	4	no	—
90	C₃×D₃×C₈	—	2	4	no	—
91	C₂²×C₃³	—	4	1	no	—
92	C₂²×C₃³	—	8	1	no	—
93	C₂²×C₃³	#41	1	1	yes	characteristic
94	C₂×C₃²×D₃	—	2	1	no	—
95	C₃³×C₄	C₂×D₄	1	1	yes	characteristic
96	C₃³×C₄	C₂×D₄	1	1	yes	characteristic
97	C₃²×Dic₃	—	2	1	no	—
98	C₃²×Dic₃	—	4	1	no	—
99	C₃×#137	—	3	4	no	—
100	C₃×AGL(1,ℤ/8)	—	3	4	no	—
101	C₂×C₃×SD₁₆	—	3	4	no	—
102	C₃×#131	—	3	4	no	—
103	C₂×#214	C₂×C₃²	1	1	yes	characteristic, complemented
104	#932	C₂×C₃²	1	1	yes	characteristic
105	#909	C₂×C₃²	1	1	yes	characteristic, complemented
106	#907	C₂×C₃²	1	1	yes	characteristic, complemented
107	C₃×#108	—	3	4	no	—
108	C₃×#101	—	3	4	no	—
109	C₃×#100	—	2	4	no	—
110	C₃×#100	C₃×D₃	1	4	yes	complemented
111	C₃×#100	C₂×C₃²	1	1	yes	characteristic, complemented
112	#824	C₂×C₃²	1	1	yes	characteristic, complemented
113	#822	C₂×C₃²	1	1	yes	characteristic, complemented
114	C₂³×C₃²	—	2	8	no	—
115	C₂³×C₃²	—	2	1	no	—
116	C₂³×C₃²	—	2	4	no	—
117	C₃²×Q₈	—	3	1	no	—
118	C₃²×Q₈	—	6	1	no	—
119	C₃²×Q₈	—	6	1	no	—
120	C₃²×D₄	—	2	1	no	—
121	C₃²×D₄	—	2	1	no	—
122	C₃²×D₄	—	2	4	no	—
123	C₃²×D₄	—	2	4	no	—
124	C₃²×D₄	—	3	1	no	—
125	C₃²×D₄	—	4	4	no	—
126	C₃²×D₄	—	4	4	no	—
127	C₃²×D₄	—	6	1	no	—
128	C₂×C₃²×C₄	—	2	4	no	—
129	C₂×C₃²×C₄	—	6	1	no	—
130	C₂×C₃²×C₄	—	6	1	no	—
131	C₂×C₃²×C₄	C₂²×D₃	1	1	yes	characteristic
132	C₂×C₃²×C₄	C₂³×C₃	1	4	yes	—
133	C₃×#67	—	2	4	no	—
134	C₂×C₃×Dic₃	—	2	4	no	—
135	C₃×D₁₂	C₃×D₄	1	4	yes	—
136	C₃×C₄×D₃	—	2	4	no	—
137	C₃×Dic₆	—	2	4	no	—
138	C₃×Dic₆	—	2	4	no	—
139	C₃×Dic₆	C₃×D₄	1	4	yes	—
140	C₃²×C₈	—	2	1	no	—
141	C₃²×C₈	—	2	1	no	—
142	C₃²×C₈	—	2	4	no	—
143	C₃²×C₈	—	2	4	no	—
144	C₃²×C₈	—	3	1	no	—
145	C₃²×C₈	—	3	1	no	—
146	C₃²×C₈	—	4	4	no	—
147	C₃²×C₈	—	4	4	no	—
148	C₃²×C₈	—	6	1	no	—
149	C₃×C₃⋊C₈	—	2	4	no	—
150	C₃×C₃⋊C₈	C₃×D₄	1	4	yes	—
151	C₃×C₃⋊C₈	C₃×D₄	1	4	yes	—
152	#471	—	3	1	no	Sylow
153	C₂×C₃³	—	2	1	no	—
154	C₂×C₃³	—	8	1	no	—
155	C₂×C₃³	#121	1	1	yes	characteristic
156	C₃²×D₃	—	4	1	no	—
157	C₃×#41	—	3	4	no	—
158	C₂×C₃×Q₈	—	3	4	no	—
159	C₂×C₃×D₄	—	2	4	no	—
160	C₂×C₃×D₄	C₂×C₃×D₃	1	4	yes	—
161	C₂×C₃×D₄	C₂²×C₃²	1	1	yes	characteristic
162	#235	C₂²×C₃²	1	1	yes	characteristic
163	C₂×Dic₆	C₂²×C₃²	1	1	yes	characteristic
164	C₃×Q₁₆	—	6	4	no	—
165	C₃×SD₁₆	—	6	4	no	—
166	C₃×SD₁₆	—	6	4	no	—
167	C₃×SD₁₆	—	6	4	no	—
168	C₃×SD₁₆	—	6	4	no	—
169	C₃×D₈	—	6	4	no	—
170	C₃×M₁₆	—	2	4	no	—
171	C₃×M₁₆	—	2	4	no	—
172	C₃×M₁₆	—	3	4	no	—
173	C₃×M₁₆	—	6	4	no	—
174	C₃×M₁₆	C₂×C₃×D₃	1	4	yes	—
175	C₃×M₁₆	C₂×C₃×D₃	1	4	yes	—
176	C₃×M₁₆	C₂²×C₃²	1	1	yes	characteristic
177	C₃×M₁₆	C₂²×C₃²	1	1	yes	characteristic
178	C₂×C₃×C₈	—	3	4	no	—
179	C₂×C₃×C₈	—	6	4	no	—
180	#214	—	2	1	no	—
181	#214	—	2	1	no	—
182	#214	—	2	1	no	—
183	#213	—	2	1	no	—
184	#208	C₂²×C₃²	1	1	yes	characteristic
185	C₂×C₃⋊C₈	C₂²×C₃²	1	1	yes	characteristic
186	Dic₁₂	—	2	1	no	—
187	#204	—	2	1	no	—
188	ΓL(1,𝔽₂₅)	—	2	1	no	—
189	D₃×C₈	—	2	1	no	—
190	C₂²×C₃²	—	2	4	no	—
191	C₂²×C₃²	—	4	8	no	—
192	C₂²×C₃²	—	4	1	no	—
193	C₂²×C₃²	—	4	4	no	—
194	C₂²×C₃²	—	6	1	no	—
195	C₂²×C₃²	—	8	1	no	—
196	C₂²×C₃²	—	8	4	no	—
197	C₂²×C₃²	—	8	8	no	—
198	C₂²×C₃²	#235	1	1	yes	characteristic
199	C₂²×C₃²	C₃×#41	1	4	yes	—
200	C₂×C₃×D₃	—	2	4	no	—
201	C₃²×C₄	—	2	4	no	—
202	C₃²×C₄	—	2	4	no	—
203	C₃²×C₄	—	6	1	no	—
204	C₃²×C₄	—	12	1	no	—
205	C₃²×C₄	D₃×D₄	1	1	yes	characteristic
206	C₃²×C₄	D₃×D₄	1	1	yes	characteristic
207	C₃²×C₄	C₂×C₃×D₄	1	4	yes	—
208	C₃²×C₄	C₂×C₃×D₄	1	4	yes	—
209	C₃×Dic₃	—	2	4	no	—
210	C₃×Dic₃	—	4	4	no	—
211	#137	—	3	1	no	—
212	AGL(1,ℤ/8)	—	3	1	no	—
213	C₂×SD₁₆	—	3	1	no	—
214	#131	—	3	1	no	—
215	#108	—	3	1	no	—
216	#101	—	3	1	no	—
217	#100	C₃²×D₃	1	1	yes	characteristic, complemented
218	C₃³	#471	1	1	yes	characteristic, complemented, Sylow
219	C₂³×C₃	—	2	8	no	—
220	C₂³×C₃	—	2	4	no	—
221	C₂³×C₃	—	2	1	no	—
222	C₃×Q₈	—	3	4	no	—
223	C₃×Q₈	—	6	4	no	—
224	C₃×Q₈	—	6	4	no	—
225	C₃×D₄	—	2	4	no	—
226	C₃×D₄	—	2	4	no	—
227	C₃×D₄	—	2	1	no	—
228	C₃×D₄	—	2	1	no	—
229	C₃×D₄	—	3	4	no	—
230	C₃×D₄	—	4	4	no	—
231	C₃×D₄	—	4	4	no	—
232	C₃×D₄	—	6	4	no	—
233	C₂×C₃×C₄	—	2	4	no	—
234	C₂×C₃×C₄	—	6	4	no	—
235	C₂×C₃×C₄	—	6	4	no	—
236	C₂×C₃×C₄	C₂²×C₃×D₃	1	4	yes	—
237	C₂×C₃×C₄	C₂³×C₃²	1	1	yes	characteristic
238	#67	—	2	1	no	—
239	C₂×Dic₃	—	2	1	no	—
240	D₁₂	C₃²×D₄	1	1	yes	characteristic
241	C₄×D₃	—	2	1	no	—
242	Dic₆	—	2	1	no	—
243	Dic₆	—	2	1	no	—
244	Dic₆	C₃²×D₄	1	1	yes	characteristic
245	C₃×C₈	—	2	4	no	—
246	C₃×C₈	—	2	4	no	—
247	C₃×C₈	—	2	1	no	—
248	C₃×C₈	—	2	1	no	—
249	C₃×C₈	—	3	4	no	—
250	C₃×C₈	—	3	4	no	—
251	C₃×C₈	—	4	4	no	—
252	C₃×C₈	—	4	4	no	—
253	C₃×C₈	—	6	4	no	—
254	C₃⋊C₈	—	2	1	no	—
255	C₃⋊C₈	C₃²×D₄	1	1	yes	characteristic
256	C₃⋊C₈	C₃²×D₄	1	1	yes	characteristic
257	C₂×C₃²	—	2	1	no	—
258	C₂×C₃²	—	2	4	no	—
259	C₂×C₃²	—	2	4	no	—
260	C₂×C₃²	—	4	4	no	—
261	C₂×C₃²	—	8	8	no	—
262	C₂×C₃²	—	8	1	no	—
263	C₂×C₃²	—	8	4	no	—
264	C₂×C₃²	—	12	1	no	—
265	C₂×C₃²	#884	1	1	yes	characteristic
266	C₂×C₃²	C₃×#121	1	4	yes	—
267	C₃×D₃	—	4	4	no	—
268	#41	—	3	1	no	—
269	C₂×Q₈	—	3	1	no	—
270	C₂×D₄	C₂×C₃²×D₃	1	1	yes	characteristic
271	Q₁₆	—	6	1	no	—
272	SD₁₆	—	6	1	no	—
273	SD₁₆	—	6	1	no	—
274	SD₁₆	—	6	1	no	—
275	SD₁₆	—	6	1	no	—
276	D₈	—	6	1	no	—
277	M₁₆	—	3	1	no	—
278	M₁₆	—	6	1	no	—
279	M₁₆	C₂×C₃²×D₃	1	1	yes	characteristic
280	M₁₆	C₂×C₃²×D₃	1	1	yes	characteristic
281	C₂×C₈	—	3	1	no	—
282	C₂×C₈	—	6	1	no	—
283	C₂²×C₃	—	2	4	no	—
284	C₂²×C₃	—	4	8	no	—
285	C₂²×C₃	—	4	4	no	—
286	C₂²×C₃	—	4	1	no	—
287	C₂²×C₃	—	6	4	no	—
288	C₂²×C₃	—	8	1	no	—
289	C₂²×C₃	—	8	4	no	—
290	C₂²×C₃	—	8	8	no	—
291	C₂²×C₃	C₃×#235	1	4	yes	—
292	C₂²×C₃	C₃²×#41	1	1	yes	characteristic
293	C₂×D₃	—	2	1	no	—
294	C₃×C₄	—	2	4	no	—
295	C₃×C₄	—	2	4	no	—
296	C₃×C₄	—	6	4	no	—
297	C₃×C₄	—	12	4	no	—
298	C₃×C₄	C₃×D₃×D₄	1	4	yes	—
299	C₃×C₄	C₃×D₃×D₄	1	4	yes	—
300	C₃×C₄	C₂×C₃²×D₄	1	1	yes	characteristic
301	C₃×C₄	C₂×C₃²×D₄	1	1	yes	characteristic
302	Dic₃	—	2	1	no	—
303	Dic₃	—	4	1	no	—
304	C₃²	—	2	4	no	—
305	C₃²	#4748	1	1	yes	characteristic, direct product factor
306	C₃²	C₃×#471	1	4	yes	complemented
307	C₂³	—	2	1	no	—
308	Q₈	—	3	1	no	—
309	Q₈	—	6	1	no	—
310	Q₈	—	6	1	no	—
311	D₄	—	2	1	no	—
312	D₄	—	2	1	no	—
313	D₄	—	3	1	no	—
314	D₄	—	6	1	no	—
315	C₂×C₄	—	6	1	no	—
316	C₂×C₄	—	6	1	no	—
317	C₂×C₄	C₂²×C₃²×D₃	1	1	yes	characteristic
318	C₈	—	2	1	no	—
319	C₈	—	2	1	no	—
320	C₈	—	3	1	no	—
321	C₈	—	3	1	no	—
322	C₈	—	6	1	no	—
323	C₂×C₃	—	2	1	no	—
324	C₂×C₃	—	2	4	no	—
325	C₂×C₃	—	2	4	no	—
326	C₂×C₃	—	4	4	no	—
327	C₂×C₃	—	8	8	no	—
328	C₂×C₃	—	8	1	no	—
329	C₂×C₃	—	8	4	no	—
330	C₂×C₃	—	12	4	no	—
331	C₂×C₃	C₃×#884	1	4	yes	—
332	C₂×C₃	C₃²×#121	1	1	yes	characteristic
333	D₃	—	4	1	no	—
334	C₂²	—	4	1	no	—
335	C₂²	—	6	1	no	—
336	C₂²	—	8	1	no	—
337	C₂²	C₃²×#235	1	1	yes	characteristic
338	C₄	—	6	1	no	—
339	C₄	—	12	1	no	—
340	C₄	C₃²×D₃×D₄	1	1	yes	characteristic
341	C₄	C₃²×D₃×D₄	1	1	yes	characteristic
342	C₃	—	2	4	no	—
343	C₃	C₃×#4748	1	4	yes	direct product factor
344	C₃	C₃²×#471	1	1	yes	characteristic, complemented
345	C₂	—	2	1	no	—
346	C₂	—	8	1	no	—
347	C₂	—	12	1	no	—
348	C₂	C₃²×#884	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₃²	#884
commutator, cosocle	C₂×C₃×C₄	C₂³×C₃²
Frattini	C₂×C₄	C₂²×C₃²×D₃
Fitting	C₃³×#100	C₂
socle	C₂×C₃³	#121
prime radical	C₃²×#932	C₂
hypercenter	C₃²×#100	D₃
hypocenter	C₃	C₃²×#471
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₃² ◁ C₂×C₃²×C₄ ◁ C₃²×#100

Complements

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

N	H ≅ G/N	type
1. C₂×C₃²×#214	C₂	semidirect
3. C₃²×#909	C₂	semidirect
4. C₃²×#907	C₂	semidirect
5. C₃³×#100	C₂	semidirect
6. C₃²×#824	C₂	semidirect
7. C₃²×#822	C₂	semidirect
9. C₃×#4748	C₃	direct (N × H)
29. C₂×C₃×#214	C₂×C₃	semidirect
31. C₃×#909	C₂×C₃	semidirect
32. C₃×#907	C₂×C₃	semidirect
36. C₃²×#100	D₃	semidirect
37. C₃²×#100	C₂×C₃	semidirect
38. C₃×#824	C₂×C₃	semidirect
39. C₃×#822	C₂×C₃	semidirect
57. #4748	C₃²	direct (N × H)
103. C₂×#214	C₂×C₃²	semidirect
105. #909	C₂×C₃²	semidirect
106. #907	C₂×C₃²	semidirect
110. C₃×#100	C₃×D₃	semidirect
111. C₃×#100	C₂×C₃²	semidirect
112. #824	C₂×C₃²	semidirect
113. #822	C₂×C₃²	semidirect
217. #100	C₃²×D₃	semidirect
218. C₃³	#471	semidirect
305. C₃²	#4748	direct (N × H)
306. C₃²	C₃×#471	semidirect
343. C₃	C₃×#4748	direct (N × H)
344. C₃	C₃²×#471	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃²×#214	2
C₃²×#932	2
C₃²×#909	2
C₃²×#907	2
C₃³×#100	2
C₃²×#824	2
C₃²×#822	2
C₃²×#471	3
C₃×#4748	3

Subgroups — C32×#4748 #13162184

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃²×#4748 #13162184