Subgroups — C₃²×#4898 #13162333

Order 1728.

744 conjugacy classes of subgroups.

The subgroup lattice has 342 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²×#909	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃²×#908	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃²×#204	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃²×#901	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃³×#108	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃²×#824	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃²×#823	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃²×#471	—	3	1	no	maximal
9	C₃×#4898	C₃	1	4	yes	maximal, direct product factor
10	C₃²×#235	C₂²	1	1	yes	characteristic
11	C₂×C₃²×D₁₂	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	C₂×C₃³×C₈	C₂²	1	1	yes	characteristic
16	C₃²×#208	C₂²	1	1	yes	characteristic
17	C₃²×Dic₁₂	—	2	1	no	—
18	C₃²×D₂₄	—	2	1	no	—
19	C₃²×#204	—	2	1	no	—
20	C₃²×#204	—	2	1	no	—
21	C₃²×#204	—	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	C₃×#909	C₂×C₃	1	4	yes	complemented
30	C₃×#908	C₂×C₃	1	4	yes	—
31	C₂×C₃×#204	C₂×C₃	1	4	yes	complemented
32	C₃×#901	C₂×C₃	1	4	yes	complemented
33	C₃²×#108	—	2	4	no	—
34	C₃²×#108	D₃	1	1	yes	characteristic, complemented
35	C₃²×#108	C₂×C₃	1	4	yes	complemented
36	C₃²×#101	—	3	1	no	—
37	C₃²×#100	—	3	1	no	—
38	C₃×#824	C₂×C₃	1	4	yes	complemented
39	C₃×#823	C₂×C₃	1	4	yes	complemented
40	C₂²×C₃²×D₃	—	2	1	no	—
41	C₂×C₃³×C₄	C₂³	1	1	yes	characteristic
42	C₃²×#67	—	2	1	no	—
43	C₂×C₃²×Dic₃	—	2	1	no	—
44	C₃²×D₁₂	—	2	1	no	—
45	C₃²×D₁₂	—	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	C₃³×C₈	D₄	1	1	yes	characteristic
54	C₃³×C₈	D₄	1	1	yes	characteristic
55	C₃²×C₃⋊C₈	—	2	1	no	—
56	C₃×#471	—	3	4	no	—
57	#4898	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	—	3	1	no	—
61	C₃×#235	C₂²×C₃	1	4	yes	—
62	C₂×C₃×D₁₂	C₂²×C₃	1	4	yes	—
63	C₂×C₃×Dic₆	C₂²×C₃	1	4	yes	—
64	C₃²×Q₁₆	—	6	1	no	—
65	C₃²×SD₁₆	—	6	1	no	—
66	C₃²×SD₁₆	—	6	1	no	—
67	C₃²×SD₁₆	—	6	1	no	—
68	C₃²×SD₁₆	—	6	1	no	—
69	C₃²×D₈	—	6	1	no	—
70	C₃²×M₁₆	—	2	4	no	—
71	C₃²×M₁₆	—	2	4	no	—
72	C₃²×M₁₆	—	3	1	no	—
73	C₃²×M₁₆	—	6	1	no	—
74	C₃²×M₁₆	C₂×D₃	1	1	yes	characteristic
75	C₃²×M₁₆	C₂×D₃	1	1	yes	characteristic
76	C₃²×M₁₆	C₂²×C₃	1	4	yes	—
77	C₃²×M₁₆	C₂²×C₃	1	4	yes	—
78	C₂×C₃²×C₈	—	2	4	no	—
79	C₂×C₃²×C₈	—	6	1	no	—
80	C₂×C₃²×C₈	C₂×D₃	1	1	yes	characteristic
81	C₂×C₃²×C₈	C₂²×C₃	1	4	yes	—
82	C₃×#208	C₂²×C₃	1	4	yes	—
83	C₃×Dic₁₂	—	2	4	no	—
84	C₃×D₂₄	—	2	4	no	—
85	C₃×#204	—	2	4	no	—
86	C₃×#204	—	2	4	no	—
87	C₃×#204	—	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	#41	1	1	yes	characteristic
92	C₂×C₃²×D₃	—	2	1	no	—
93	C₂×C₃²×D₃	—	4	1	no	—
94	C₂×C₃²×D₃	—	8	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	#909	C₂×C₃²	1	1	yes	characteristic, complemented
104	#908	C₂×C₃²	1	1	yes	characteristic
105	C₂×#204	C₂×C₃²	1	1	yes	characteristic, complemented
106	#901	C₂×C₃²	1	1	yes	characteristic, complemented
107	C₃×#108	—	2	4	no	—
108	C₃×#108	C₃×D₃	1	4	yes	complemented
109	C₃×#108	C₂×C₃²	1	1	yes	characteristic, complemented
110	C₃×#101	—	3	4	no	—
111	C₃×#100	—	3	4	no	—
112	#824	C₂×C₃²	1	1	yes	characteristic, complemented
113	#823	C₂×C₃²	1	1	yes	characteristic, complemented
114	C₂³×C₃²	—	6	1	no	—
115	C₂²×C₃×D₃	—	2	4	no	—
116	C₃²×Q₈	—	3	1	no	—
117	C₃²×Q₈	—	6	1	no	—
118	C₃²×Q₈	—	6	1	no	—
119	C₃²×D₄	—	3	1	no	—
120	C₃²×D₄	—	6	1	no	—
121	C₃²×D₄	—	6	1	no	—
122	C₃²×D₄	—	6	1	no	—
123	C₂×C₃²×C₄	—	2	4	no	—
124	C₂×C₃²×C₄	—	6	1	no	—
125	C₂×C₃²×C₄	—	6	1	no	—
126	C₂×C₃²×C₄	C₂²×D₃	1	1	yes	characteristic
127	C₂×C₃²×C₄	C₂³×C₃	1	4	yes	—
128	C₃×#67	—	2	4	no	—
129	C₂×C₃×Dic₃	—	2	4	no	—
130	C₃×D₁₂	—	2	4	no	—
131	C₃×D₁₂	—	2	4	no	—
132	C₃×D₁₂	C₃×D₄	1	4	yes	—
133	C₃×C₄×D₃	—	2	4	no	—
134	C₃×Dic₆	—	2	4	no	—
135	C₃×Dic₆	—	2	4	no	—
136	C₃×Dic₆	C₃×D₄	1	4	yes	—
137	C₃²×C₈	—	2	1	no	—
138	C₃²×C₈	—	2	1	no	—
139	C₃²×C₈	—	2	4	no	—
140	C₃²×C₈	—	2	4	no	—
141	C₃²×C₈	—	2	4	no	—
142	C₃²×C₈	—	2	4	no	—
143	C₃²×C₈	—	4	4	no	—
144	C₃²×C₈	—	4	4	no	—
145	C₃²×C₈	—	6	1	no	—
146	C₃²×C₈	D₁₂	1	1	yes	characteristic
147	C₃²×C₈	D₁₂	1	1	yes	characteristic
148	C₃²×C₈	C₃×D₄	1	4	yes	—
149	C₃²×C₈	C₃×D₄	1	4	yes	—
150	C₃×C₃⋊C₈	—	2	4	no	—
151	#471	—	3	1	no	Sylow
152	C₂×C₃³	—	2	1	no	—
153	C₂×C₃³	#121	1	1	yes	characteristic
154	C₃²×D₃	—	4	1	no	—
155	C₃²×D₃	—	8	1	no	—
156	C₃×#41	—	3	4	no	—
157	C₂×C₃×Q₈	—	3	4	no	—
158	C₂×C₃×D₄	—	3	4	no	—
159	#235	C₂²×C₃²	1	1	yes	characteristic
160	C₂×D₁₂	C₂²×C₃²	1	1	yes	characteristic
161	C₂×Dic₆	C₂²×C₃²	1	1	yes	characteristic
162	C₃×Q₁₆	—	6	4	no	—
163	C₃×SD₁₆	—	6	4	no	—
164	C₃×SD₁₆	—	6	4	no	—
165	C₃×SD₁₆	—	6	4	no	—
166	C₃×SD₁₆	—	6	4	no	—
167	C₃×D₈	—	6	4	no	—
168	C₃×M₁₆	—	2	4	no	—
169	C₃×M₁₆	—	2	4	no	—
170	C₃×M₁₆	—	3	4	no	—
171	C₃×M₁₆	—	6	4	no	—
172	C₃×M₁₆	C₂×C₃×D₃	1	4	yes	—
173	C₃×M₁₆	C₂×C₃×D₃	1	4	yes	—
174	C₃×M₁₆	C₂²×C₃²	1	1	yes	characteristic
175	C₃×M₁₆	C₂²×C₃²	1	1	yes	characteristic
176	C₂×C₃×C₈	—	2	4	no	—
177	C₂×C₃×C₈	—	6	4	no	—
178	C₂×C₃×C₈	C₂×C₃×D₃	1	4	yes	—
179	C₂×C₃×C₈	C₂²×C₃²	1	1	yes	characteristic
180	#208	C₂²×C₃²	1	1	yes	characteristic
181	Dic₁₂	—	2	1	no	—
182	D₂₄	—	2	1	no	—
183	#204	—	2	1	no	—
184	#204	—	2	1	no	—
185	#204	—	2	1	no	—
186	#204	—	2	1	no	—
187	ΓL(1,𝔽₂₅)	—	2	1	no	—
188	D₃×C₈	—	2	1	no	—
189	C₂²×C₃²	—	2	4	no	—
190	C₂²×C₃²	—	6	1	no	—
191	C₂²×C₃²	—	12	1	no	—
192	C₂²×C₃²	—	24	1	no	—
193	C₂²×C₃²	#239	1	1	yes	characteristic
194	C₂²×C₃²	C₃×#41	1	4	yes	—
195	C₂×C₃×D₃	—	2	4	no	—
196	C₂×C₃×D₃	—	4	4	no	—
197	C₂×C₃×D₃	—	8	4	no	—
198	C₃²×C₄	—	2	4	no	—
199	C₃²×C₄	—	2	4	no	—
200	C₃²×C₄	—	6	1	no	—
201	C₃²×C₄	—	12	1	no	—
202	C₃²×C₄	C₂×D₁₂	1	1	yes	characteristic
203	C₃²×C₄	D₃×D₄	1	1	yes	characteristic
204	C₃²×C₄	C₂×C₃×D₄	1	4	yes	—
205	C₃²×C₄	C₂×C₃×D₄	1	4	yes	—
206	C₃×Dic₃	—	2	4	no	—
207	C₃×Dic₃	—	4	4	no	—
208	#137	—	3	1	no	—
209	AGL(1,ℤ/8)	—	3	1	no	—
210	C₂×SD₁₆	—	3	1	no	—
211	#131	—	3	1	no	—
212	#108	C₃²×D₃	1	1	yes	characteristic, complemented
213	#101	—	3	1	no	—
214	#100	—	3	1	no	—
215	C₃³	#471	1	1	yes	characteristic, complemented, Sylow
216	C₂³×C₃	—	6	4	no	—
217	C₂²×D₃	—	2	1	no	—
218	C₃×Q₈	—	3	4	no	—
219	C₃×Q₈	—	6	4	no	—
220	C₃×Q₈	—	6	4	no	—
221	C₃×D₄	—	3	4	no	—
222	C₃×D₄	—	6	4	no	—
223	C₃×D₄	—	6	4	no	—
224	C₃×D₄	—	6	4	no	—
225	C₂×C₃×C₄	—	2	4	no	—
226	C₂×C₃×C₄	—	6	4	no	—
227	C₂×C₃×C₄	—	6	4	no	—
228	C₂×C₃×C₄	C₂²×C₃×D₃	1	4	yes	—
229	C₂×C₃×C₄	C₂³×C₃²	1	1	yes	characteristic
230	#67	—	2	1	no	—
231	C₂×Dic₃	—	2	1	no	—
232	D₁₂	—	2	1	no	—
233	D₁₂	—	2	1	no	—
234	D₁₂	C₃²×D₄	1	1	yes	characteristic
235	C₄×D₃	—	2	1	no	—
236	Dic₆	—	2	1	no	—
237	Dic₆	—	2	1	no	—
238	Dic₆	C₃²×D₄	1	1	yes	characteristic
239	C₃×C₈	—	2	4	no	—
240	C₃×C₈	—	2	4	no	—
241	C₃×C₈	—	2	1	no	—
242	C₃×C₈	—	2	1	no	—
243	C₃×C₈	—	2	4	no	—
244	C₃×C₈	—	2	4	no	—
245	C₃×C₈	—	4	4	no	—
246	C₃×C₈	—	4	4	no	—
247	C₃×C₈	—	6	4	no	—
248	C₃×C₈	C₃×D₁₂	1	4	yes	—
249	C₃×C₈	C₃×D₁₂	1	4	yes	—
250	C₃×C₈	C₃²×D₄	1	1	yes	characteristic
251	C₃×C₈	C₃²×D₄	1	1	yes	characteristic
252	C₃⋊C₈	—	2	1	no	—
253	C₂×C₃²	—	2	1	no	—
254	C₂×C₃²	—	2	4	no	—
255	C₂×C₃²	—	2	4	no	—
256	C₂×C₃²	—	4	4	no	—
257	C₂×C₃²	—	12	1	no	—
258	C₂×C₃²	—	24	1	no	—
259	C₂×C₃²	#895	1	1	yes	characteristic
260	C₂×C₃²	C₃×#121	1	4	yes	—
261	C₃×D₃	—	4	4	no	—
262	C₃×D₃	—	8	4	no	—
263	#41	—	3	1	no	—
264	C₂×Q₈	—	3	1	no	—
265	C₂×D₄	—	3	1	no	—
266	Q₁₆	—	6	1	no	—
267	SD₁₆	—	6	1	no	—
268	SD₁₆	—	6	1	no	—
269	SD₁₆	—	6	1	no	—
270	SD₁₆	—	6	1	no	—
271	D₈	—	6	1	no	—
272	M₁₆	—	3	1	no	—
273	M₁₆	—	6	1	no	—
274	M₁₆	C₂×C₃²×D₃	1	1	yes	characteristic
275	M₁₆	C₂×C₃²×D₃	1	1	yes	characteristic
276	C₂×C₈	—	6	1	no	—
277	C₂×C₈	C₂×C₃²×D₃	1	1	yes	characteristic
278	C₂²×C₃	—	2	4	no	—
279	C₂²×C₃	—	6	4	no	—
280	C₂²×C₃	—	12	4	no	—
281	C₂²×C₃	—	24	4	no	—
282	C₂²×C₃	C₃×#239	1	4	yes	—
283	C₂²×C₃	C₃²×#41	1	1	yes	characteristic
284	C₂×D₃	—	2	1	no	—
285	C₂×D₃	—	4	1	no	—
286	C₂×D₃	—	8	1	no	—
287	C₃×C₄	—	2	4	no	—
288	C₃×C₄	—	2	4	no	—
289	C₃×C₄	—	6	4	no	—
290	C₃×C₄	—	12	4	no	—
291	C₃×C₄	C₂×C₃×D₁₂	1	4	yes	—
292	C₃×C₄	C₃×D₃×D₄	1	4	yes	—
293	C₃×C₄	C₂×C₃²×D₄	1	1	yes	characteristic
294	C₃×C₄	C₂×C₃²×D₄	1	1	yes	characteristic
295	Dic₃	—	2	1	no	—
296	Dic₃	—	4	1	no	—
297	C₃²	—	2	4	no	—
298	C₃²	#4898	1	1	yes	characteristic, direct product factor
299	C₃²	C₃×#471	1	4	yes	complemented
300	C₂³	—	6	1	no	—
301	Q₈	—	3	1	no	—
302	Q₈	—	6	1	no	—
303	Q₈	—	6	1	no	—
304	D₄	—	3	1	no	—
305	D₄	—	6	1	no	—
306	D₄	—	6	1	no	—
307	D₄	—	6	1	no	—
308	C₂×C₄	—	6	1	no	—
309	C₂×C₄	—	6	1	no	—
310	C₂×C₄	C₂²×C₃²×D₃	1	1	yes	characteristic
311	C₈	—	2	1	no	—
312	C₈	—	2	1	no	—
313	C₈	—	6	1	no	—
314	C₈	C₃²×D₁₂	1	1	yes	characteristic
315	C₈	C₃²×D₁₂	1	1	yes	characteristic
316	C₂×C₃	—	2	1	no	—
317	C₂×C₃	—	2	4	no	—
318	C₂×C₃	—	2	4	no	—
319	C₂×C₃	—	4	4	no	—
320	C₂×C₃	—	12	4	no	—
321	C₂×C₃	—	24	4	no	—
322	C₂×C₃	C₃×#895	1	4	yes	—
323	C₂×C₃	C₃²×#121	1	1	yes	characteristic
324	D₃	—	4	1	no	—
325	D₃	—	8	1	no	—
326	C₂²	—	6	1	no	—
327	C₂²	—	12	1	no	—
328	C₂²	—	24	1	no	—
329	C₂²	C₃²×#239	1	1	yes	characteristic
330	C₄	—	6	1	no	—
331	C₄	—	12	1	no	—
332	C₄	C₂×C₃²×D₁₂	1	1	yes	characteristic
333	C₄	C₃²×D₃×D₄	1	1	yes	characteristic
334	C₃	—	2	4	no	—
335	C₃	C₃×#4898	1	4	yes	direct product factor
336	C₃	C₃²×#471	1	1	yes	characteristic, complemented
337	C₂	—	2	1	no	—
338	C₂	—	12	1	no	—
339	C₂	—	24	1	no	—
340	C₂	C₃²×#895	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₃²	#895
commutator, cosocle	C₂×C₃×C₄	C₂³×C₃²
Frattini	C₂×C₄	C₂²×C₃²×D₃
Fitting	C₃³×#108	C₂
socle	C₂×C₃³	#121
prime radical	C₃²×#908	C₂
hypercenter	C₃²×#108	D₃
hypocenter	C₃	C₃²×#471
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₃² ◁ C₂×C₃²×C₄ ◁ C₃²×#108

Complements

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

N	H ≅ G/N	type
1. C₃²×#909	C₂	semidirect
3. C₂×C₃²×#204	C₂	semidirect
4. C₃²×#901	C₂	semidirect
5. C₃³×#108	C₂	semidirect
6. C₃²×#824	C₂	semidirect
7. C₃²×#823	C₂	semidirect
9. C₃×#4898	C₃	direct (N × H)
29. C₃×#909	C₂×C₃	semidirect
31. C₂×C₃×#204	C₂×C₃	semidirect
32. C₃×#901	C₂×C₃	semidirect
34. C₃²×#108	D₃	semidirect
35. C₃²×#108	C₂×C₃	semidirect
38. C₃×#824	C₂×C₃	semidirect
39. C₃×#823	C₂×C₃	semidirect
57. #4898	C₃²	direct (N × H)
103. #909	C₂×C₃²	semidirect
105. C₂×#204	C₂×C₃²	semidirect
106. #901	C₂×C₃²	semidirect
108. C₃×#108	C₃×D₃	semidirect
109. C₃×#108	C₂×C₃²	semidirect
112. #824	C₂×C₃²	semidirect
113. #823	C₂×C₃²	semidirect
212. #108	C₃²×D₃	semidirect
215. C₃³	#471	semidirect
298. C₃²	#4898	direct (N × H)
299. C₃²	C₃×#471	semidirect
335. C₃	C₃×#4898	direct (N × H)
336. 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
C₃²×#909	2
C₃²×#908	2
C₂×C₃²×#204	2
C₃²×#901	2
C₃³×#108	2
C₃²×#824	2
C₃²×#823	2
C₃²×#471	3
C₃×#4898	3

Subgroups — C32×#4898 #13162333

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃²×#4898 #13162333