Subgroups — C₃×#5559 #10595656

Order 576.

519 conjugacy classes of subgroups.

The subgroup lattice has 330 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²×C₄×D₄	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃×#940	C₂	1	1	yes	maximal, characteristic
3	C₃×#936	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×#929	C₂	1	2	yes	maximal, complemented
5	C₃×C₄×#67	C₂	1	2	yes	maximal, complemented
6	C₃×#893	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃×#887	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃×#885	C₂	1	2	yes	maximal, complemented
9	C₃×#883	C₂	1	2	yes	maximal, complemented
10	C₃×#875	C₂	1	1	yes	maximal, characteristic, complemented
11	C₃×C₄×Dic₆	C₂	1	1	yes	maximal, characteristic, complemented
12	C₃×#559	—	3	1	no	maximal
13	#5559	C₃	1	1	yes	maximal, characteristic, direct product factor
14	C₂×C₃²×D₄	C₂²	1	1	yes	characteristic
15	C₂²×C₃²×C₄	C₂²	1	2	yes	—
16	C₂×C₃×#67	C₂²	1	2	yes	—
17	C₂×C₃×#67	C₂²	1	2	yes	—
18	C₂×C₃×D₁₂	C₂²	1	1	yes	characteristic
19	C₂×C₃×C₄×D₃	C₂²	1	2	yes	—
20	C₂×C₃×Dic₆	C₂²	1	1	yes	characteristic
21	C₂×C₃×Dic₆	C₂²	1	1	yes	characteristic
22	C₃²×C₄⋊C₄	C₂²	1	1	yes	characteristic
23	C₃²×C₂²⋊C₄	C₂²	1	2	yes	—
24	C₃²×C₄²	C₂²	1	1	yes	characteristic
25	C₃×#217	C₂²	1	2	yes	—
26	C₃×#217	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	1	yes	characteristic
31	C₃×Dic₃⋊C₄	C₂²	1	2	yes	—
32	C₃×Dic₃⋊C₄	C₂²	1	1	yes	characteristic
33	C₃×Dic₃⋊C₄	C₂²	1	1	yes	characteristic
34	C₃×C₄×Dic₃	C₂²	1	2	yes	—
35	C₃×C₄×Dic₃	C₂²	1	1	yes	characteristic
36	C₃×C₄×Dic₃	C₂²	1	1	yes	characteristic
37	C₃×#127	—	3	1	no	—
38	C₃×#124	—	3	1	no	—
39	C₃×#124	—	3	2	no	—
40	C₃×#124	—	3	1	no	—
41	C₃×#123	—	3	2	no	—
42	C₃×#123	—	3	2	no	—
43	C₃×C₄×Q₈	—	3	1	no	—
44	C₃×C₄×Q₈	—	3	1	no	—
45	C₃×C₄×D₄	—	2	1	no	—
46	C₃×C₄×D₄	—	3	2	no	—
47	C₃×C₄×D₄	—	3	1	no	—
48	C₃×C₄×D₄	D₃	1	1	yes	characteristic, complemented
49	C₃×C₄×D₄	C₂×C₃	1	1	yes	characteristic, complemented
50	#940	C₂×C₃	1	1	yes	characteristic
51	#936	C₂×C₃	1	1	yes	characteristic, complemented
52	#929	C₂×C₃	1	2	yes	complemented
53	C₄×#67	C₂×C₃	1	2	yes	complemented
54	#893	C₂×C₃	1	1	yes	characteristic, complemented
55	#887	C₂×C₃	1	1	yes	characteristic, complemented
56	#885	C₂×C₃	1	2	yes	complemented
57	#883	C₂×C₃	1	2	yes	complemented
58	#875	C₂×C₃	1	1	yes	characteristic, complemented
59	C₄×Dic₆	C₂×C₃	1	1	yes	characteristic, complemented
60	C₂³×C₃²	C₂³	1	2	yes	—
61	C₂²×C₃×D₃	C₂³	1	2	yes	—
62	C₃²×D₄	—	2	2	no	—
63	C₂×C₃²×C₄	—	4	2	no	—
64	C₂×C₃²×C₄	C₂³	1	2	yes	—
65	C₂×C₃²×C₄	C₂³	1	1	yes	characteristic
66	C₂×C₃²×C₄	C₂³	1	1	yes	characteristic
67	C₂×C₃²×C₄	C₂³	1	1	yes	characteristic
68	C₃×#67	—	2	4	no	—
69	C₃×#67	—	4	2	no	—
70	C₂×C₃×Dic₃	C₂³	1	1	yes	characteristic
71	C₂×C₃×Dic₃	C₂³	1	1	yes	characteristic
72	C₂×C₃×Dic₃	C₂³	1	2	yes	—
73	C₂×C₃×Dic₃	C₂³	1	1	yes	characteristic
74	C₂×C₃×Dic₃	C₂³	1	1	yes	characteristic
75	C₃×D₁₂	—	2	2	no	—
76	C₃×C₄×D₃	—	4	2	no	—
77	C₃×Dic₆	—	2	2	no	—
78	C₃×Dic₆	—	2	2	no	—
79	#559	—	3	1	no	Sylow
80	C₂×C₃×Q₈	—	3	1	no	—
81	C₂×C₃×Q₈	—	3	1	no	—
82	C₂×C₃×D₄	—	2	1	no	—
83	C₂×C₃×D₄	—	3	2	no	—
84	C₂×C₃×D₄	—	3	2	no	—
85	C₂×C₃×D₄	—	3	1	no	—
86	C₂×C₃×D₄	C₂×D₃	1	1	yes	characteristic
87	C₂×C₃×D₄	C₂²×C₃	1	1	yes	characteristic
88	C₂²×C₃×C₄	—	2	2	no	—
89	C₂²×C₃×C₄	—	3	2	no	—
90	C₂²×C₃×C₄	C₂×D₃	1	2	yes	—
91	C₂²×C₃×C₄	C₂²×C₃	1	2	yes	—
92	C₂×#67	C₂²×C₃	1	2	yes	—
93	C₂×#67	C₂²×C₃	1	2	yes	—
94	C₂×D₁₂	C₂²×C₃	1	1	yes	characteristic
95	C₂×C₄×D₃	C₂²×C₃	1	2	yes	—
96	C₂×Dic₆	C₂²×C₃	1	1	yes	characteristic
97	C₂×Dic₆	C₂²×C₃	1	1	yes	characteristic
98	C₃×C₄⋊C₄	—	2	1	no	—
99	C₃×C₄⋊C₄	—	3	1	no	—
100	C₃×C₄⋊C₄	—	3	2	no	—
101	C₃×C₄⋊C₄	—	3	1	no	—
102	C₃×C₄⋊C₄	—	3	1	no	—
103	C₃×C₄⋊C₄	C₂×D₃	1	1	yes	characteristic
104	C₃×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
105	C₃×C₂²⋊C₄	—	2	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×C₂²⋊C₄	—	3	2	no	—
110	C₃×C₂²⋊C₄	—	3	2	no	—
111	C₃×C₂²⋊C₄	C₂×D₃	1	2	yes	—
112	C₃×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
113	C₃×C₄²	—	2	1	no	—
114	C₃×C₄²	—	3	2	no	—
115	C₃×C₄²	—	3	1	no	—
116	C₃×C₄²	—	3	1	no	—
117	C₃×C₄²	C₂×D₃	1	1	yes	characteristic
118	C₃×C₄²	C₂²×C₃	1	1	yes	characteristic
119	#217	C₂²×C₃	1	2	yes	—
120	#217	C₂²×C₃	1	2	yes	—
121	(C₂×D₃)⋊C₄	C₂²×C₃	1	2	yes	—
122	(C₂×D₃)⋊C₄	C₂²×C₃	1	2	yes	—
123	(C₂×D₃)⋊C₄	C₂²×C₃	1	2	yes	—
124	C₄⋊Dic₃	C₂²×C₃	1	1	yes	characteristic
125	Dic₃⋊C₄	C₂²×C₃	1	1	yes	characteristic
126	Dic₃⋊C₄	C₂²×C₃	1	1	yes	characteristic
127	Dic₃⋊C₄	C₂²×C₃	1	2	yes	—
128	C₄×Dic₃	C₂²×C₃	1	1	yes	characteristic
129	C₄×Dic₃	C₂²×C₃	1	1	yes	characteristic
130	C₄×Dic₃	C₂²×C₃	1	2	yes	—
131	C₂²×C₃²	—	2	2	no	—
132	C₂²×C₃²	—	2	2	no	—
133	C₂²×C₃²	—	2	2	no	—
134	C₂²×C₃²	C₂⁴	1	1	yes	characteristic
135	C₂×C₃×D₃	—	2	2	no	—
136	C₂×C₃×D₃	—	2	2	no	—
137	C₂×C₃×D₃	—	2	2	no	—
138	C₃²×C₄	—	2	1	no	—
139	C₃²×C₄	—	2	1	no	—
140	C₃²×C₄	—	2	2	no	—
141	C₃²×C₄	#41	1	2	yes	—
142	C₃×Dic₃	—	2	2	no	—
143	C₃×Dic₃	—	2	1	no	—
144	C₃×Dic₃	—	2	1	no	—
145	C₃×Dic₃	—	2	1	no	—
146	C₃×Dic₃	#41	1	2	yes	—
147	#127	—	3	1	no	—
148	#124	—	3	1	no	—
149	#124	—	3	2	no	—
150	#124	—	3	1	no	—
151	#123	—	3	2	no	—
152	#123	—	3	2	no	—
153	C₄×Q₈	—	3	1	no	—
154	C₄×Q₈	—	3	1	no	—
155	C₄×D₄	—	3	2	no	—
156	C₄×D₄	—	3	1	no	—
157	C₄×D₄	C₃×D₃	1	1	yes	characteristic, complemented
158	C₂³×C₃	—	2	2	no	—
159	C₂³×C₃	—	3	2	no	—
160	C₂³×C₃	C₂²×D₃	1	2	yes	—
161	C₂³×C₃	C₂³×C₃	1	2	yes	—
162	C₂²×D₃	C₂³×C₃	1	2	yes	—
163	C₃×Q₈	—	6	2	no	—
164	C₃×Q₈	—	6	2	no	—
165	C₃×D₄	—	2	4	no	—
166	C₃×D₄	—	2	2	no	—
167	C₃×D₄	—	2	2	no	—
168	C₃×D₄	—	6	4	no	—
169	C₃×D₄	—	6	2	no	—
170	C₃×D₄	—	12	2	no	—
171	C₂×C₃×C₄	—	2	1	no	—
172	C₂×C₃×C₄	—	2	1	no	—
173	C₂×C₃×C₄	—	2	1	no	—
174	C₂×C₃×C₄	—	2	2	no	—
175	C₂×C₃×C₄	—	3	1	no	—
176	C₂×C₃×C₄	—	3	2	no	—
177	C₂×C₃×C₄	—	3	1	no	—
178	C₂×C₃×C₄	—	3	1	no	—
179	C₂×C₃×C₄	—	3	1	no	—
180	C₂×C₃×C₄	—	4	2	no	—
181	C₂×C₃×C₄	—	4	2	no	—
182	C₂×C₃×C₄	—	4	4	no	—
183	C₂×C₃×C₄	—	12	2	no	—
184	C₂×C₃×C₄	C₂²×D₃	1	2	yes	—
185	C₂×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
186	C₂×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
187	C₂×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
188	C₂×C₃×C₄	C₂³×C₃	1	2	yes	—
189	C₂×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
190	C₂×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
191	C₂×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
192	#67	—	2	4	no	—
193	#67	—	4	2	no	—
194	C₂×Dic₃	C₂³×C₃	1	1	yes	characteristic
195	C₂×Dic₃	C₂³×C₃	1	1	yes	characteristic
196	C₂×Dic₃	C₂³×C₃	1	1	yes	characteristic
197	C₂×Dic₃	C₂³×C₃	1	2	yes	—
198	C₂×Dic₃	C₂³×C₃	1	1	yes	characteristic
199	D₁₂	—	2	2	no	—
200	C₄×D₃	—	4	2	no	—
201	Dic₆	—	2	2	no	—
202	Dic₆	—	2	2	no	—
203	C₂×C₃²	—	4	2	no	—
204	C₂×C₃²	C₂×#41	1	1	yes	characteristic
205	C₂×C₃²	C₂×#41	1	1	yes	characteristic
206	C₂×C₃²	H(5,𝔽₂)	1	1	yes	characteristic
207	C₃×D₃	—	4	2	no	—
208	C₂×Q₈	—	3	1	no	—
209	C₂×Q₈	—	3	1	no	—
210	C₂×D₄	—	3	2	no	—
211	C₂×D₄	—	3	2	no	—
212	C₂×D₄	—	3	1	no	—
213	C₂×D₄	C₂×C₃×D₃	1	1	yes	characteristic
214	C₂²×C₄	—	3	2	no	—
215	C₂²×C₄	C₂×C₃×D₃	1	2	yes	—
216	C₄⋊C₄	—	3	1	no	—
217	C₄⋊C₄	—	3	2	no	—
218	C₄⋊C₄	—	3	1	no	—
219	C₄⋊C₄	—	3	1	no	—
220	C₄⋊C₄	C₂×C₃×D₃	1	1	yes	characteristic
221	C₂²⋊C₄	—	3	2	no	—
222	C₂²⋊C₄	—	3	2	no	—
223	C₂²⋊C₄	—	3	2	no	—
224	C₂²⋊C₄	—	3	2	no	—
225	C₂²⋊C₄	—	3	2	no	—
226	C₂²⋊C₄	C₂×C₃×D₃	1	2	yes	—
227	C₄²	—	3	2	no	—
228	C₄²	—	3	1	no	—
229	C₄²	—	3	1	no	—
230	C₄²	C₂×C₃×D₃	1	1	yes	characteristic
231	C₂²×C₃	—	2	4	no	—
232	C₂²×C₃	—	2	2	no	—
233	C₂²×C₃	—	2	2	no	—
234	C₂²×C₃	—	2	2	no	—
235	C₂²×C₃	—	2	2	no	—
236	C₂²×C₃	—	2	2	no	—
237	C₂²×C₃	—	2	2	no	—
238	C₂²×C₃	—	2	1	no	—
239	C₂²×C₃	—	4	2	no	—
240	C₂²×C₃	—	4	2	no	—
241	C₂²×C₃	—	6	2	no	—
242	C₂²×C₃	—	6	2	no	—
243	C₂²×C₃	—	6	2	no	—
244	C₂²×C₃	C₂³×D₃	1	1	yes	characteristic
245	C₂²×C₃	C₂⁴×C₃	1	1	yes	characteristic
246	C₂×D₃	—	2	2	no	—
247	C₂×D₃	—	2	2	no	—
248	C₂×D₃	—	2	2	no	—
249	C₃×C₄	—	2	2	no	—
250	C₃×C₄	—	2	1	no	—
251	C₃×C₄	—	2	2	no	—
252	C₃×C₄	—	2	1	no	—
253	C₃×C₄	—	2	1	no	—
254	C₃×C₄	—	2	2	no	—
255	C₃×C₄	—	2	1	no	—
256	C₃×C₄	—	2	2	no	—
257	C₃×C₄	—	3	2	no	—
258	C₃×C₄	—	4	2	no	—
259	C₃×C₄	—	4	1	no	—
260	C₃×C₄	—	6	2	no	—
261	C₃×C₄	—	6	1	no	—
262	C₃×C₄	—	6	1	no	—
263	C₃×C₄	—	6	1	no	—
264	C₃×C₄	#235	1	2	yes	—
265	C₃×C₄	C₃×#41	1	2	yes	—
266	Dic₃	—	2	2	no	—
267	Dic₃	—	2	1	no	—
268	Dic₃	—	2	1	no	—
269	Dic₃	—	2	1	no	—
270	Dic₃	C₃×#41	1	2	yes	—
271	C₃²	#559	1	1	yes	characteristic, complemented, Sylow
272	C₂³	—	3	2	no	—
273	C₂³	C₂²×C₃×D₃	1	2	yes	—
274	Q₈	—	6	2	no	—
275	Q₈	—	6	2	no	—
276	D₄	—	2	2	no	—
277	D₄	—	6	4	no	—
278	D₄	—	6	2	no	—
279	D₄	—	12	2	no	—
280	C₂×C₄	—	3	1	no	—
281	C₂×C₄	—	3	2	no	—
282	C₂×C₄	—	3	1	no	—
283	C₂×C₄	—	3	1	no	—
284	C₂×C₄	—	3	1	no	—
285	C₂×C₄	—	4	2	no	—
286	C₂×C₄	—	12	2	no	—
287	C₂×C₄	C₂²×C₃×D₃	1	2	yes	—
288	C₂×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
289	C₂×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
290	C₂×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
291	C₂×C₃	—	2	1	no	—
292	C₂×C₃	—	2	1	no	—
293	C₂×C₃	—	2	1	no	—
294	C₂×C₃	—	4	4	no	—
295	C₂×C₃	—	4	2	no	—
296	C₂×C₃	—	4	2	no	—
297	C₂×C₃	—	12	2	no	—
298	C₂×C₃	C₂×#235	1	1	yes	characteristic
299	C₂×C₃	#1004	1	1	yes	characteristic
300	C₂×C₃	D₃×#41	1	1	yes	characteristic
301	C₂×C₃	C₂×C₃×#41	1	1	yes	characteristic
302	C₂×C₃	C₂×C₃×#41	1	1	yes	characteristic
303	C₂×C₃	C₃×H(5,𝔽₂)	1	1	yes	characteristic
304	D₃	—	4	2	no	—
305	C₂²	—	2	2	no	—
306	C₂²	—	2	2	no	—
307	C₂²	—	2	2	no	—
308	C₂²	—	6	2	no	—
309	C₂²	—	6	2	no	—
310	C₂²	—	6	2	no	—
311	C₂²	C₂³×C₃×D₃	1	1	yes	characteristic
312	C₄	—	2	1	no	—
313	C₄	—	2	2	no	—
314	C₄	—	2	1	no	—
315	C₄	—	3	2	no	—
316	C₄	—	6	2	no	—
317	C₄	—	6	1	no	—
318	C₄	—	6	1	no	—
319	C₄	—	6	1	no	—
320	C₄	C₃×#235	1	2	yes	—
321	C₃	—	2	1	no	—
322	C₃	#5559	1	1	yes	characteristic, direct product factor
323	C₃	C₃×#559	1	1	yes	characteristic, complemented
324	C₂	—	4	2	no	—
325	C₂	—	12	2	no	—
326	C₂	C₂×C₃×#235	1	1	yes	characteristic
327	C₂	C₃×#1004	1	1	yes	characteristic
328	C₂	C₃×D₃×#41	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₃	C₂³×D₃
commutator, cosocle	C₂²×C₃	C₂⁴×C₃
Frattini	C₂²	C₂³×C₃×D₃
Fitting	C₃²×C₄×D₄	C₂
socle	C₂²×C₃²	C₂⁴
prime radical	C₃×#940	C₂
hypercenter	C₃×C₄×D₄	D₃
hypocenter	C₃	C₃×#559
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₃ ◁ C₃×C₄×D₄

Complements

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

N	H ≅ G/N	type
1. C₃²×C₄×D₄	C₂	semidirect
3. C₃×#936	C₂	semidirect
4. C₃×#929	C₂	semidirect
5. C₃×C₄×#67	C₂	semidirect
6. C₃×#893	C₂	semidirect
7. C₃×#887	C₂	semidirect
8. C₃×#885	C₂	semidirect
9. C₃×#883	C₂	semidirect
10. C₃×#875	C₂	semidirect
11. C₃×C₄×Dic₆	C₂	semidirect
13. #5559	C₃	direct (N × H)
48. C₃×C₄×D₄	D₃	semidirect
49. C₃×C₄×D₄	C₂×C₃	semidirect
51. #936	C₂×C₃	semidirect
52. #929	C₂×C₃	semidirect
53. C₄×#67	C₂×C₃	semidirect
54. #893	C₂×C₃	semidirect
55. #887	C₂×C₃	semidirect
56. #885	C₂×C₃	semidirect
57. #883	C₂×C₃	semidirect
58. #875	C₂×C₃	semidirect
59. C₄×Dic₆	C₂×C₃	semidirect
157. C₄×D₄	C₃×D₃	semidirect
271. C₃²	#559	semidirect
322. C₃	#5559	direct (N × H)
323. C₃	C₃×#559	semidirect

Maximal subgroups

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

subgroup	index
C₃²×C₄×D₄	2
C₃×#940	2
C₃×#936	2
C₃×#929	2
C₃×C₄×#67	2
C₃×#893	2
C₃×#887	2
C₃×#885	2
C₃×#883	2
C₃×#875	2
C₃×C₄×Dic₆	2
C₃×#559	3
#5559	3

Subgroups — C3×#5559 #10595656

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#5559 #10595656