Subgroups — C₃×#5580 #10595677

Order 576.

503 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×C₄×Q₈	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃×#947	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×#946	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×#898	C₂	1	2	yes	maximal, complemented
5	C₃×#895	C₂	1	1	yes	maximal, characteristic
6	C₃×#894	C₂	1	2	yes	maximal, complemented
7	C₃×#890	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃×C₄×D₁₂	C₂	1	2	yes	maximal, complemented
9	C₃×C₄×D₁₂	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃×#872	C₂	1	2	yes	maximal, complemented
11	C₃×C₄²×D₃	C₂	1	1	yes	maximal, characteristic, complemented
12	C₃×#529	—	3	1	no	maximal
13	#5580	C₃	1	1	yes	maximal, characteristic, direct product factor
14	C₂×C₃²×Q₈	C₂²	1	1	yes	characteristic
15	C₂×C₃×D₁₂	C₂²	1	2	yes	—
16	C₂×C₃×D₁₂	C₂²	1	1	yes	characteristic
17	C₂×C₃×C₄×D₃	C₂²	1	1	yes	characteristic
18	C₂×C₃×C₄×D₃	C₂²	1	1	yes	characteristic
19	C₂×C₃×C₄×D₃	C₂²	1	1	yes	characteristic
20	C₂×C₃×C₄×D₃	C₂²	1	2	yes	—
21	C₃²×C₄⋊C₄	C₂²	1	1	yes	characteristic
22	C₃²×C₄⋊C₄	C₂²	1	2	yes	—
23	C₃²×C₄²	C₂²	1	1	yes	characteristic
24	C₃²×C₄²	C₂²	1	2	yes	—
25	C₃×(C₂×D₃)⋊C₄	C₂²	1	2	yes	—
26	C₃×(C₂×D₃)⋊C₄	C₂²	1	2	yes	—
27	C₃×(C₂×D₃)⋊C₄	C₂²	1	2	yes	—
28	C₃×(C₂×D₃)⋊C₄	C₂²	1	2	yes	—
29	C₃×(C₂×D₃)⋊C₄	C₂²	1	2	yes	—
30	C₃×C₄⋊Dic₃	C₂²	1	2	yes	—
31	C₃×C₄⋊Dic₃	C₂²	1	1	yes	characteristic
32	C₃×Dic₃⋊C₄	C₂²	1	2	yes	—
33	C₃×Dic₃⋊C₄	C₂²	1	2	yes	—
34	C₃×C₄×Dic₃	C₂²	1	1	yes	characteristic
35	C₃×C₄×Dic₃	C₂²	1	1	yes	characteristic
36	C₃×C₄×Dic₃	C₂²	1	1	yes	characteristic
37	C₃×#126	—	3	2	no	—
38	C₃×#125	—	3	1	no	—
39	C₃×#124	—	3	1	no	—
40	C₃×#123	—	3	2	no	—
41	C₃×C₂²⋊Q₈	—	3	1	no	—
42	C₃×#121	—	3	1	no	—
43	C₃×C₄×Q₈	—	2	1	no	—
44	C₃×C₄×Q₈	D₃	1	1	yes	characteristic, complemented
45	C₃×C₄×Q₈	C₂×C₃	1	1	yes	characteristic, complemented
46	C₃×C₄×D₄	—	3	1	no	—
47	C₃×C₄×D₄	—	3	2	no	—
48	C₃×#117	—	3	2	no	—
49	C₂×C₃×C₄²	—	3	1	no	—
50	#947	C₂×C₃	1	1	yes	characteristic, complemented
51	#946	C₂×C₃	1	1	yes	characteristic, complemented
52	#898	C₂×C₃	1	2	yes	complemented
53	#895	C₂×C₃	1	1	yes	characteristic
54	#894	C₂×C₃	1	2	yes	complemented
55	#890	C₂×C₃	1	1	yes	characteristic, complemented
56	C₄×D₁₂	C₂×C₃	1	2	yes	complemented
57	C₄×D₁₂	C₂×C₃	1	1	yes	characteristic, complemented
58	#872	C₂×C₃	1	2	yes	complemented
59	C₄²×D₃	C₂×C₃	1	1	yes	characteristic, complemented
60	C₂²×C₃×D₃	C₂³	1	2	yes	—
61	C₂²×C₃×D₃	C₂³	1	1	yes	characteristic
62	C₃²×Q₈	—	2	2	no	—
63	C₂×C₃²×C₄	C₂³	1	1	yes	characteristic
64	C₂×C₃²×C₄	C₂³	1	2	yes	—
65	C₂×C₃²×C₄	C₂³	1	2	yes	—
66	C₂×C₃²×C₄	C₂³	1	1	yes	characteristic
67	C₂×C₃²×C₄	C₂³	1	1	yes	characteristic
68	C₂×C₃×Dic₃	C₂³	1	1	yes	characteristic
69	C₂×C₃×Dic₃	C₂³	1	1	yes	characteristic
70	C₂×C₃×Dic₃	C₂³	1	1	yes	characteristic
71	C₂×C₃×Dic₃	C₂³	1	2	yes	—
72	C₃×D₁₂	—	2	4	no	—
73	C₃×D₁₂	—	2	2	no	—
74	C₃×C₄×D₃	—	2	2	no	—
75	C₃×C₄×D₃	—	2	2	no	—
76	C₃×C₄×D₃	—	2	2	no	—
77	C₃×C₄×D₃	—	2	4	no	—
78	#529	—	3	1	no	Sylow
79	C₂×C₃×Q₈	—	2	1	no	—
80	C₂×C₃×Q₈	C₂×D₃	1	1	yes	characteristic
81	C₂×C₃×Q₈	C₂²×C₃	1	1	yes	characteristic
82	C₂×C₃×D₄	—	3	2	no	—
83	C₂×C₃×D₄	—	3	1	no	—
84	C₂²×C₃×C₄	—	3	1	no	—
85	C₂²×C₃×C₄	—	3	1	no	—
86	C₂²×C₃×C₄	—	3	1	no	—
87	C₂²×C₃×C₄	—	3	2	no	—
88	C₂×D₁₂	C₂²×C₃	1	2	yes	—
89	C₂×D₁₂	C₂²×C₃	1	1	yes	characteristic
90	C₂×C₄×D₃	C₂²×C₃	1	1	yes	characteristic
91	C₂×C₄×D₃	C₂²×C₃	1	1	yes	characteristic
92	C₂×C₄×D₃	C₂²×C₃	1	1	yes	characteristic
93	C₂×C₄×D₃	C₂²×C₃	1	2	yes	—
94	C₃×C₄⋊C₄	—	2	1	no	—
95	C₃×C₄⋊C₄	—	2	2	no	—
96	C₃×C₄⋊C₄	—	3	2	no	—
97	C₃×C₄⋊C₄	—	3	2	no	—
98	C₃×C₄⋊C₄	—	3	1	no	—
99	C₃×C₄⋊C₄	—	3	2	no	—
100	C₃×C₄⋊C₄	C₂×D₃	1	1	yes	characteristic
101	C₃×C₄⋊C₄	C₂×D₃	1	2	yes	—
102	C₃×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
103	C₃×C₄⋊C₄	C₂²×C₃	1	2	yes	—
104	C₃×C₂²⋊C₄	—	3	2	no	—
105	C₃×C₂²⋊C₄	—	3	2	no	—
106	C₃×C₂²⋊C₄	—	3	2	no	—
107	C₃×C₂²⋊C₄	—	3	2	no	—
108	C₃×C₂²⋊C₄	—	3	2	no	—
109	C₃×C₄²	—	2	1	no	—
110	C₃×C₄²	—	2	2	no	—
111	C₃×C₄²	—	3	1	no	—
112	C₃×C₄²	—	3	1	no	—
113	C₃×C₄²	—	3	1	no	—
114	C₃×C₄²	C₂×D₃	1	1	yes	characteristic
115	C₃×C₄²	C₂×D₃	1	2	yes	—
116	C₃×C₄²	C₂²×C₃	1	1	yes	characteristic
117	C₃×C₄²	C₂²×C₃	1	2	yes	—
118	(C₂×D₃)⋊C₄	C₂²×C₃	1	2	yes	—
119	(C₂×D₃)⋊C₄	C₂²×C₃	1	2	yes	—
120	(C₂×D₃)⋊C₄	C₂²×C₃	1	2	yes	—
121	(C₂×D₃)⋊C₄	C₂²×C₃	1	2	yes	—
122	(C₂×D₃)⋊C₄	C₂²×C₃	1	2	yes	—
123	C₄⋊Dic₃	C₂²×C₃	1	2	yes	—
124	C₄⋊Dic₃	C₂²×C₃	1	1	yes	characteristic
125	Dic₃⋊C₄	C₂²×C₃	1	2	yes	—
126	Dic₃⋊C₄	C₂²×C₃	1	2	yes	—
127	C₄×Dic₃	C₂²×C₃	1	1	yes	characteristic
128	C₄×Dic₃	C₂²×C₃	1	1	yes	characteristic
129	C₄×Dic₃	C₂²×C₃	1	1	yes	characteristic
130	C₂²×C₃²	C₂⁴	1	1	yes	characteristic
131	C₂×C₃×D₃	—	2	1	no	—
132	C₂×C₃×D₃	—	2	1	no	—
133	C₂×C₃×D₃	—	2	2	no	—
134	C₂×C₃×D₃	—	2	2	no	—
135	C₂×C₃×D₃	—	2	2	no	—
136	C₂×C₃×D₃	#41	1	2	yes	—
137	C₃²×C₄	—	2	2	no	—
138	C₃²×C₄	—	2	1	no	—
139	C₃²×C₄	—	2	2	no	—
140	C₃²×C₄	#41	1	2	yes	—
141	C₃²×C₄	#41	1	2	yes	—
142	C₃×Dic₃	—	2	1	no	—
143	C₃×Dic₃	—	2	1	no	—
144	C₃×Dic₃	—	2	2	no	—
145	C₃×Dic₃	—	2	1	no	—
146	#126	—	3	2	no	—
147	#125	—	3	1	no	—
148	#124	—	3	1	no	—
149	#123	—	3	2	no	—
150	C₂²⋊Q₈	—	3	1	no	—
151	#121	—	3	1	no	—
152	C₄×Q₈	C₃×D₃	1	1	yes	characteristic, complemented
153	C₄×D₄	—	3	1	no	—
154	C₄×D₄	—	3	2	no	—
155	#117	—	3	2	no	—
156	C₂×C₄²	—	3	1	no	—
157	C₂³×C₃	—	3	1	no	—
158	C₂³×C₃	—	3	2	no	—
159	C₂²×D₃	C₂³×C₃	1	2	yes	—
160	C₂²×D₃	C₂³×C₃	1	1	yes	characteristic
161	C₃×Q₈	—	2	4	no	—
162	C₃×Q₈	—	2	2	no	—
163	C₃×Q₈	—	2	2	no	—
164	C₃×D₄	—	6	4	no	—
165	C₃×D₄	—	6	2	no	—
166	C₂×C₃×C₄	—	2	1	no	—
167	C₂×C₃×C₄	—	2	2	no	—
168	C₂×C₃×C₄	—	2	1	no	—
169	C₂×C₃×C₄	—	2	1	no	—
170	C₂×C₃×C₄	—	2	2	no	—
171	C₂×C₃×C₄	—	3	1	no	—
172	C₂×C₃×C₄	—	3	1	no	—
173	C₂×C₃×C₄	—	3	1	no	—
174	C₂×C₃×C₄	—	3	2	no	—
175	C₂×C₃×C₄	—	6	2	no	—
176	C₂×C₃×C₄	—	6	2	no	—
177	C₂×C₃×C₄	—	6	2	no	—
178	C₂×C₃×C₄	—	6	4	no	—
179	C₂×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
180	C₂×C₃×C₄	C₂²×D₃	1	2	yes	—
181	C₂×C₃×C₄	C₂²×D₃	1	2	yes	—
182	C₂×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
183	C₂×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
184	C₂×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
185	C₂×C₃×C₄	C₂³×C₃	1	2	yes	—
186	C₂×C₃×C₄	C₂³×C₃	1	2	yes	—
187	C₂×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
188	C₂×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
189	C₂×Dic₃	C₂³×C₃	1	1	yes	characteristic
190	C₂×Dic₃	C₂³×C₃	1	1	yes	characteristic
191	C₂×Dic₃	C₂³×C₃	1	1	yes	characteristic
192	C₂×Dic₃	C₂³×C₃	1	2	yes	—
193	D₁₂	—	2	4	no	—
194	D₁₂	—	2	2	no	—
195	C₄×D₃	—	2	2	no	—
196	C₄×D₃	—	2	2	no	—
197	C₄×D₃	—	2	2	no	—
198	C₄×D₃	—	2	4	no	—
199	C₂×C₃²	C₂×#41	1	1	yes	characteristic
200	C₂×C₃²	C₂×#41	1	1	yes	characteristic
201	C₂×C₃²	C₂×#41	1	1	yes	characteristic
202	C₃×D₃	—	2	2	no	—
203	C₃×D₃	—	4	2	no	—
204	C₂×Q₈	C₂×C₃×D₃	1	1	yes	characteristic
205	C₂×D₄	—	3	2	no	—
206	C₂×D₄	—	3	1	no	—
207	C₂²×C₄	—	3	1	no	—
208	C₂²×C₄	—	3	1	no	—
209	C₂²×C₄	—	3	1	no	—
210	C₂²×C₄	—	3	2	no	—
211	C₄⋊C₄	—	3	2	no	—
212	C₄⋊C₄	—	3	2	no	—
213	C₄⋊C₄	—	3	1	no	—
214	C₄⋊C₄	—	3	2	no	—
215	C₄⋊C₄	C₂×C₃×D₃	1	1	yes	characteristic
216	C₄⋊C₄	C₂×C₃×D₃	1	2	yes	—
217	C₂²⋊C₄	—	3	2	no	—
218	C₂²⋊C₄	—	3	2	no	—
219	C₂²⋊C₄	—	3	2	no	—
220	C₂²⋊C₄	—	3	2	no	—
221	C₂²⋊C₄	—	3	2	no	—
222	C₄²	—	3	1	no	—
223	C₄²	—	3	1	no	—
224	C₄²	—	3	1	no	—
225	C₄²	C₂×C₃×D₃	1	1	yes	characteristic
226	C₄²	C₂×C₃×D₃	1	2	yes	—
227	C₂²×C₃	—	2	1	no	—
228	C₂²×C₃	—	3	2	no	—
229	C₂²×C₃	—	6	1	no	—
230	C₂²×C₃	—	6	1	no	—
231	C₂²×C₃	—	6	2	no	—
232	C₂²×C₃	—	6	2	no	—
233	C₂²×C₃	—	6	2	no	—
234	C₂²×C₃	C₂³×D₃	1	1	yes	characteristic
235	C₂²×C₃	C₂⁴×C₃	1	1	yes	characteristic
236	C₂×D₃	—	2	1	no	—
237	C₂×D₃	—	2	1	no	—
238	C₂×D₃	—	2	2	no	—
239	C₂×D₃	—	2	2	no	—
240	C₂×D₃	—	2	2	no	—
241	C₂×D₃	C₃×#41	1	2	yes	—
242	C₃×C₄	—	2	4	no	—
243	C₃×C₄	—	2	2	no	—
244	C₃×C₄	—	2	1	no	—
245	C₃×C₄	—	2	2	no	—
246	C₃×C₄	—	2	1	no	—
247	C₃×C₄	—	2	2	no	—
248	C₃×C₄	—	2	2	no	—
249	C₃×C₄	—	2	2	no	—
250	C₃×C₄	—	2	2	no	—
251	C₃×C₄	—	4	1	no	—
252	C₃×C₄	—	4	2	no	—
253	C₃×C₄	—	6	1	no	—
254	C₃×C₄	—	6	1	no	—
255	C₃×C₄	—	6	2	no	—
256	C₃×C₄	—	6	1	no	—
257	C₃×C₄	#235	1	2	yes	—
258	C₃×C₄	#239	1	2	yes	—
259	C₃×C₄	C₃×#41	1	2	yes	—
260	C₃×C₄	C₃×#41	1	2	yes	—
261	Dic₃	—	2	1	no	—
262	Dic₃	—	2	1	no	—
263	Dic₃	—	2	2	no	—
264	Dic₃	—	2	1	no	—
265	C₃²	#529	1	1	yes	characteristic, complemented, Sylow
266	C₂³	—	3	1	no	—
267	C₂³	—	3	2	no	—
268	Q₈	—	2	2	no	—
269	D₄	—	6	4	no	—
270	D₄	—	6	2	no	—
271	C₂×C₄	—	3	1	no	—
272	C₂×C₄	—	3	1	no	—
273	C₂×C₄	—	3	1	no	—
274	C₂×C₄	—	3	2	no	—
275	C₂×C₄	—	6	2	no	—
276	C₂×C₄	—	6	2	no	—
277	C₂×C₄	—	6	2	no	—
278	C₂×C₄	—	6	4	no	—
279	C₂×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
280	C₂×C₄	C₂²×C₃×D₃	1	2	yes	—
281	C₂×C₄	C₂²×C₃×D₃	1	2	yes	—
282	C₂×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
283	C₂×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
284	C₂×C₃	—	2	1	no	—
285	C₂×C₃	—	2	1	no	—
286	C₂×C₃	—	2	1	no	—
287	C₂×C₃	—	6	2	no	—
288	C₂×C₃	—	12	2	no	—
289	C₂×C₃	C₂×#235	1	1	yes	characteristic
290	C₂×C₃	C₂×#239	1	1	yes	characteristic
291	C₂×C₃	D₃×#41	1	1	yes	characteristic
292	C₂×C₃	C₂×C₃×#41	1	1	yes	characteristic
293	C₂×C₃	C₂×C₃×#41	1	1	yes	characteristic
294	C₂×C₃	C₂×C₃×#41	1	1	yes	characteristic
295	D₃	—	2	2	no	—
296	D₃	—	4	2	no	—
297	C₂²	—	3	2	no	—
298	C₂²	—	6	1	no	—
299	C₂²	—	6	1	no	—
300	C₂²	—	6	2	no	—
301	C₂²	—	6	2	no	—
302	C₂²	—	6	2	no	—
303	C₂²	C₂³×C₃×D₃	1	1	yes	characteristic
304	C₄	—	2	2	no	—
305	C₄	—	2	1	no	—
306	C₄	—	2	2	no	—
307	C₄	—	6	1	no	—
308	C₄	—	6	1	no	—
309	C₄	—	6	2	no	—
310	C₄	—	6	1	no	—
311	C₄	C₃×#235	1	2	yes	—
312	C₄	C₃×#239	1	2	yes	—
313	C₃	—	2	1	no	—
314	C₃	#5580	1	1	yes	characteristic, direct product factor
315	C₃	C₃×#529	1	1	yes	characteristic, complemented
316	C₂	—	6	2	no	—
317	C₂	—	12	2	no	—
318	C₂	C₂×C₃×#235	1	1	yes	characteristic
319	C₂	C₂×C₃×#239	1	1	yes	characteristic
320	C₂	C₃×D₃×#41	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₃×C₄	C₂²×D₃
commutator, cosocle	C₂²×C₃	C₂⁴×C₃
Frattini	C₂²	C₂³×C₃×D₃
Fitting	C₃²×C₄×Q₈	C₂
socle	C₂²×C₃²	C₂⁴
prime radical	C₃×#895	C₂
hypercenter	C₃×C₄×Q₈	D₃
hypocenter	C₃	C₃×#529
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₃×C₄ ◁ C₃×C₄×Q₈

Complements

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

N	H ≅ G/N	type
1. C₃²×C₄×Q₈	C₂	semidirect
2. C₃×#947	C₂	semidirect
3. C₃×#946	C₂	semidirect
4. C₃×#898	C₂	semidirect
6. C₃×#894	C₂	semidirect
7. C₃×#890	C₂	semidirect
8. C₃×C₄×D₁₂	C₂	semidirect
9. C₃×C₄×D₁₂	C₂	semidirect
10. C₃×#872	C₂	semidirect
11. C₃×C₄²×D₃	C₂	semidirect
13. #5580	C₃	direct (N × H)
44. C₃×C₄×Q₈	D₃	semidirect
45. C₃×C₄×Q₈	C₂×C₃	semidirect
50. #947	C₂×C₃	semidirect
51. #946	C₂×C₃	semidirect
52. #898	C₂×C₃	semidirect
54. #894	C₂×C₃	semidirect
55. #890	C₂×C₃	semidirect
56. C₄×D₁₂	C₂×C₃	semidirect
57. C₄×D₁₂	C₂×C₃	semidirect
58. #872	C₂×C₃	semidirect
59. C₄²×D₃	C₂×C₃	semidirect
152. C₄×Q₈	C₃×D₃	semidirect
265. C₃²	#529	semidirect
314. C₃	#5580	direct (N × H)
315. C₃	C₃×#529	semidirect

Maximal subgroups

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

subgroup	index
C₃²×C₄×Q₈	2
C₃×#947	2
C₃×#946	2
C₃×#898	2
C₃×#895	2
C₃×#894	2
C₃×#890	2
C₃×C₄×D₁₂	2
C₃×#872	2
C₃×C₄²×D₃	2
C₃×#529	3
#5580	3

Subgroups — C3×#5580 #10595677

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#5580 #10595677