Subgroups — C₃×#5634 #10595731

Order 576.

521 conjugacy classes of subgroups.

The subgroup lattice has 310 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²×C₂²⋊Q₈	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃×#946	C₂	1	2	yes	maximal, complemented
3	C₂×C₃×(C₂×D₃)⋊C₄	C₂	1	1	yes	maximal, characteristic
4	C₃×#924	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×#897	C₂	1	2	yes	maximal, complemented
6	C₃×#896	C₂	1	2	yes	maximal, complemented
7	C₃×#896	C₂	1	2	yes	maximal, complemented
8	C₃×D₃×C₂²⋊C₄	C₂	1	2	yes	maximal
9	C₃×#878	C₂	1	2	yes	maximal, complemented
10	C₃×#543	—	3	1	no	maximal
11	#5634	C₃	1	1	yes	maximal, characteristic, direct product factor
12	C₂³×C₃×D₃	C₂²	1	1	yes	characteristic
13	C₂×C₃²×Q₈	C₂²	1	1	yes	characteristic, complemented
14	C₂²×C₃²×C₄	C₂²	1	1	yes	characteristic
15	C₂²×C₃×Dic₃	C₂²	1	1	yes	characteristic
16	C₂×C₃×C₄×D₃	C₂²	1	2	yes	—
17	C₂×C₃×C₄×D₃	C₂²	1	2	yes	—
18	C₂×C₃×Dic₆	C₂²	1	2	yes	complemented
19	C₂×C₃×Dic₆	C₂²	1	1	yes	characteristic, complemented
20	C₃²×C₄⋊C₄	C₂²	1	1	yes	characteristic, complemented
21	C₃²×C₄⋊C₄	C₂²	1	2	yes	complemented
22	C₃²×C₂²⋊C₄	C₂²	1	2	yes	—
23	C₃×#217	C₂²	1	2	yes	—
24	C₃×(C₂×D₃)⋊C₄	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×C₄⋊Dic₃	C₂²	1	2	yes	complemented
29	C₃×C₄⋊Dic₃	C₂²	1	1	yes	characteristic, complemented
30	C₃×Dic₃⋊C₄	C₂²	1	2	yes	complemented
31	C₃×Dic₃⋊C₄	C₂²	1	2	yes	complemented
32	C₃×Dic₃⋊C₄	C₂²	1	2	yes	complemented
33	C₃×C₂²⋊Q₈	—	2	1	no	—
34	C₃×C₂²⋊Q₈	—	3	2	no	—
35	C₃×C₂²⋊Q₈	—	3	1	no	—
36	C₃×C₂²⋊Q₈	—	3	2	no	—
37	C₃×C₂²⋊Q₈	—	3	2	no	—
38	C₃×C₂²⋊Q₈	—	3	2	no	—
39	C₃×C₂²⋊Q₈	—	3	2	no	—
40	C₃×C₂²⋊Q₈	D₃	1	1	yes	characteristic, complemented
41	C₃×C₂²⋊Q₈	C₂×C₃	1	1	yes	characteristic, complemented
42	C₂×C₃×C₂²⋊C₄	—	3	1	no	—
43	C₂×C₃×C₂²⋊C₄	—	3	2	no	—
44	#946	C₂×C₃	1	2	yes	complemented
45	C₂×(C₂×D₃)⋊C₄	C₂×C₃	1	1	yes	characteristic
46	#924	C₂×C₃	1	1	yes	characteristic, complemented
47	#897	C₂×C₃	1	2	yes	complemented
48	#896	C₂×C₃	1	2	yes	complemented
49	#896	C₂×C₃	1	2	yes	complemented
50	D₃×C₂²⋊C₄	C₂×C₃	1	2	yes	—
51	#878	C₂×C₃	1	2	yes	complemented
52	C₂³×C₃²	C₂³	1	1	yes	characteristic
53	C₂²×C₃×D₃	—	4	1	no	—
54	C₂²×C₃×D₃	—	4	1	no	—
55	C₂²×C₃×D₃	Q₈	1	4	yes	complemented
56	C₂²×C₃×D₃	C₂³	1	2	yes	—
57	C₃²×Q₈	—	4	1	no	—
58	C₂×C₃²×C₄	—	4	1	no	—
59	C₂×C₃²×C₄	C₂³	1	1	yes	characteristic
60	C₂×C₃²×C₄	C₂³	1	2	yes	—
61	C₂×C₃²×C₄	C₂³	1	2	yes	—
62	C₂×C₃²×C₄	C₂³	1	1	yes	characteristic
63	C₂×C₃×Dic₃	—	4	1	no	—
64	C₂×C₃×Dic₃	C₂³	1	2	yes	—
65	C₂×C₃×Dic₃	C₂³	1	2	yes	—
66	C₂×C₃×Dic₃	C₂³	1	2	yes	—
67	C₃×C₄×D₃	—	4	2	no	—
68	C₃×C₄×D₃	—	4	2	no	—
69	C₃×Dic₆	—	4	1	no	—
70	C₃×Dic₆	—	4	2	no	—
71	#543	—	3	1	no	Sylow
72	C₂⁴×C₃	—	3	1	no	—
73	C₂³×D₃	C₂²×C₃	1	1	yes	characteristic
74	C₂×C₃×Q₈	—	2	1	no	—
75	C₂×C₃×Q₈	—	3	1	no	—
76	C₂×C₃×Q₈	—	3	2	no	—
77	C₂×C₃×Q₈	C₂×D₃	1	1	yes	characteristic, complemented
78	C₂×C₃×Q₈	C₂²×C₃	1	1	yes	characteristic, complemented
79	C₂²×C₃×C₄	—	2	1	no	—
80	C₂²×C₃×C₄	—	3	1	no	—
81	C₂²×C₃×C₄	—	3	2	no	—
82	C₂²×C₃×C₄	—	3	2	no	—
83	C₂²×C₃×C₄	C₂×D₃	1	1	yes	characteristic
84	C₂²×C₃×C₄	C₂²×C₃	1	1	yes	characteristic
85	C₂²×Dic₃	C₂²×C₃	1	1	yes	characteristic
86	C₂×C₄×D₃	C₂²×C₃	1	2	yes	—
87	C₂×C₄×D₃	C₂²×C₃	1	2	yes	—
88	C₂×Dic₆	C₂²×C₃	1	2	yes	complemented
89	C₂×Dic₆	C₂²×C₃	1	1	yes	characteristic, complemented
90	C₃×C₄⋊C₄	—	2	1	no	—
91	C₃×C₄⋊C₄	—	2	2	no	—
92	C₃×C₄⋊C₄	—	3	2	no	—
93	C₃×C₄⋊C₄	—	3	2	no	—
94	C₃×C₄⋊C₄	—	3	2	no	—
95	C₃×C₄⋊C₄	—	3	1	no	—
96	C₃×C₄⋊C₄	—	3	2	no	—
97	C₃×C₄⋊C₄	C₂×D₃	1	1	yes	characteristic, complemented
98	C₃×C₄⋊C₄	C₂×D₃	1	2	yes	complemented
99	C₃×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic, complemented
100	C₃×C₄⋊C₄	C₂²×C₃	1	2	yes	complemented
101	C₃×C₂²⋊C₄	—	2	2	no	—
102	C₃×C₂²⋊C₄	—	3	2	no	—
103	C₃×C₂²⋊C₄	—	3	2	no	—
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	C₂×D₃	1	2	yes	—
108	C₃×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
109	#217	C₂²×C₃	1	2	yes	—
110	(C₂×D₃)⋊C₄	C₂²×C₃	1	2	yes	—
111	(C₂×D₃)⋊C₄	C₂²×C₃	1	2	yes	—
112	(C₂×D₃)⋊C₄	C₂²×C₃	1	2	yes	—
113	(C₂×D₃)⋊C₄	C₂²×C₃	1	2	yes	—
114	C₄⋊Dic₃	C₂²×C₃	1	2	yes	complemented
115	C₄⋊Dic₃	C₂²×C₃	1	1	yes	characteristic, complemented
116	Dic₃⋊C₄	C₂²×C₃	1	2	yes	complemented
117	Dic₃⋊C₄	C₂²×C₃	1	2	yes	complemented
118	Dic₃⋊C₄	C₂²×C₃	1	2	yes	complemented
119	C₂²×C₃²	—	2	1	no	—
120	C₂²×C₃²	—	2	1	no	—
121	C₂²×C₃²	C₂×Q₈	1	2	yes	—
122	C₂²×C₃²	C₂⁴	1	1	yes	characteristic
123	C₂×C₃×D₃	—	2	2	no	—
124	C₂×C₃×D₃	—	2	2	no	—
125	C₂×C₃×D₃	—	4	4	no	—
126	C₂×C₃×D₃	C₂×Q₈	1	4	yes	—
127	C₃²×C₄	—	2	1	no	—
128	C₃²×C₄	—	2	2	no	—
129	C₃²×C₄	—	2	1	no	—
130	C₃²×C₄	—	2	2	no	—
131	C₃×Dic₃	—	2	2	no	—
132	C₃×Dic₃	—	2	2	no	—
133	C₃×Dic₃	—	2	2	no	—
134	C₂²⋊Q₈	—	3	2	no	—
135	C₂²⋊Q₈	—	3	1	no	—
136	C₂²⋊Q₈	—	3	2	no	—
137	C₂²⋊Q₈	—	3	2	no	—
138	C₂²⋊Q₈	—	3	2	no	—
139	C₂²⋊Q₈	—	3	2	no	—
140	C₂²⋊Q₈	C₃×D₃	1	1	yes	characteristic, complemented
141	C₂×C₂²⋊C₄	—	3	1	no	—
142	C₂×C₂²⋊C₄	—	3	2	no	—
143	C₂³×C₃	—	2	1	no	—
144	C₂³×C₃	—	3	2	no	—
145	C₂³×C₃	—	3	4	no	—
146	C₂³×C₃	—	12	1	no	—
147	C₂³×C₃	—	12	1	no	—
148	C₂³×C₃	C₂²×D₃	1	1	yes	characteristic
149	C₂³×C₃	C₂³×C₃	1	1	yes	characteristic
150	C₂²×D₃	—	4	1	no	—
151	C₂²×D₃	—	4	1	no	—
152	C₂²×D₃	C₃×Q₈	1	4	yes	complemented
153	C₂²×D₃	C₂³×C₃	1	2	yes	—
154	C₃×Q₈	—	4	1	no	—
155	C₃×Q₈	—	4	1	no	—
156	C₃×Q₈	—	4	2	no	—
157	C₃×Q₈	—	12	1	no	—
158	C₃×Q₈	—	12	2	no	—
159	C₂×C₃×C₄	—	2	1	no	—
160	C₂×C₃×C₄	—	2	2	no	—
161	C₂×C₃×C₄	—	2	2	no	—
162	C₂×C₃×C₄	—	2	1	no	—
163	C₂×C₃×C₄	—	3	2	no	—
164	C₂×C₃×C₄	—	3	2	no	—
165	C₂×C₃×C₄	—	3	2	no	—
166	C₂×C₃×C₄	—	4	1	no	—
167	C₂×C₃×C₄	—	4	1	no	—
168	C₂×C₃×C₄	—	4	2	no	—
169	C₂×C₃×C₄	—	12	2	no	—
170	C₂×C₃×C₄	—	12	2	no	—
171	C₂×C₃×C₄	—	12	1	no	—
172	C₂×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
173	C₂×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
174	C₂×C₃×C₄	C₂²×D₃	1	2	yes	—
175	C₂×C₃×C₄	C₂²×D₃	1	2	yes	—
176	C₂×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
177	C₂×C₃×C₄	C₂³×C₃	1	2	yes	—
178	C₂×C₃×C₄	C₂³×C₃	1	2	yes	—
179	C₂×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
180	C₂×Dic₃	—	4	1	no	—
181	C₂×Dic₃	C₂³×C₃	1	2	yes	—
182	C₂×Dic₃	C₂³×C₃	1	2	yes	—
183	C₂×Dic₃	C₂³×C₃	1	2	yes	—
184	C₄×D₃	—	4	2	no	—
185	C₄×D₃	—	4	2	no	—
186	Dic₆	—	4	1	no	—
187	Dic₆	—	4	2	no	—
188	C₂×C₃²	—	2	2	no	—
189	C₂×C₃²	C₂²×Q₈	1	1	yes	characteristic
190	C₂×C₃²	H(5,𝔽₂)	1	1	yes	characteristic
191	C₂×C₃²	H(5,𝔽₂)	1	1	yes	characteristic
192	C₃×D₃	—	2	4	no	—
193	C₂⁴	—	3	1	no	—
194	C₂×Q₈	—	3	1	no	—
195	C₂×Q₈	—	3	2	no	—
196	C₂×Q₈	C₂×C₃×D₃	1	1	yes	characteristic, complemented
197	C₂²×C₄	—	3	1	no	—
198	C₂²×C₄	—	3	2	no	—
199	C₂²×C₄	—	3	2	no	—
200	C₂²×C₄	C₂×C₃×D₃	1	1	yes	characteristic
201	C₄⋊C₄	—	3	2	no	—
202	C₄⋊C₄	—	3	2	no	—
203	C₄⋊C₄	—	3	2	no	—
204	C₄⋊C₄	—	3	1	no	—
205	C₄⋊C₄	—	3	2	no	—
206	C₄⋊C₄	C₂×C₃×D₃	1	1	yes	characteristic, complemented
207	C₄⋊C₄	C₂×C₃×D₃	1	2	yes	complemented
208	C₂²⋊C₄	—	3	2	no	—
209	C₂²⋊C₄	—	3	2	no	—
210	C₂²⋊C₄	—	3	2	no	—
211	C₂²⋊C₄	—	3	2	no	—
212	C₂²⋊C₄	—	3	2	no	—
213	C₂²⋊C₄	C₂×C₃×D₃	1	2	yes	—
214	C₂²×C₃	—	2	1	no	—
215	C₂²×C₃	—	2	1	no	—
216	C₂²×C₃	—	2	1	no	—
217	C₂²×C₃	—	2	1	no	—
218	C₂²×C₃	—	2	1	no	—
219	C₂²×C₃	—	2	2	no	—
220	C₂²×C₃	—	3	4	no	—
221	C₂²×C₃	—	4	1	no	—
222	C₂²×C₃	—	4	1	no	—
223	C₂²×C₃	—	6	2	no	—
224	C₂²×C₃	—	6	2	no	—
225	C₂²×C₃	—	12	4	no	—
226	C₂²×C₃	D₃×Q₈	1	2	yes	—
227	C₂²×C₃	C₂×C₃×Q₈	1	2	yes	—
228	C₂²×C₃	C₂³×D₃	1	1	yes	characteristic
229	C₂²×C₃	C₂⁴×C₃	1	1	yes	characteristic
230	C₂×D₃	—	2	2	no	—
231	C₂×D₃	—	2	2	no	—
232	C₂×D₃	—	4	4	no	—
233	C₂×D₃	C₂×C₃×Q₈	1	4	yes	—
234	C₃×C₄	—	2	2	no	—
235	C₃×C₄	—	2	1	no	—
236	C₃×C₄	—	2	1	no	—
237	C₃×C₄	—	2	2	no	—
238	C₃×C₄	—	2	2	no	—
239	C₃×C₄	—	2	2	no	—
240	C₃×C₄	—	2	1	no	—
241	C₃×C₄	—	2	2	no	—
242	C₃×C₄	—	2	1	no	—
243	C₃×C₄	—	4	1	no	—
244	C₃×C₄	—	4	2	no	—
245	C₃×C₄	—	4	2	no	—
246	C₃×C₄	—	6	2	no	—
247	C₃×C₄	—	6	2	no	—
248	C₃×C₄	—	6	2	no	—
249	Dic₃	—	2	2	no	—
250	Dic₃	—	2	2	no	—
251	Dic₃	—	2	2	no	—
252	C₃²	#543	1	1	yes	characteristic, complemented, Sylow
253	C₂³	—	3	2	no	—
254	C₂³	—	3	4	no	—
255	C₂³	—	12	1	no	—
256	C₂³	—	12	1	no	—
257	C₂³	C₂²×C₃×D₃	1	1	yes	characteristic
258	Q₈	—	4	1	no	—
259	Q₈	—	12	1	no	—
260	Q₈	—	12	2	no	—
261	C₂×C₄	—	3	2	no	—
262	C₂×C₄	—	3	2	no	—
263	C₂×C₄	—	3	2	no	—
264	C₂×C₄	—	4	1	no	—
265	C₂×C₄	—	12	2	no	—
266	C₂×C₄	—	12	2	no	—
267	C₂×C₄	—	12	1	no	—
268	C₂×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
269	C₂×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
270	C₂×C₄	C₂²×C₃×D₃	1	2	yes	—
271	C₂×C₄	C₂²×C₃×D₃	1	2	yes	—
272	C₂×C₃	—	2	2	no	—
273	C₂×C₃	—	2	2	no	—
274	C₂×C₃	—	2	1	no	—
275	C₂×C₃	—	2	1	no	—
276	C₂×C₃	—	2	1	no	—
277	C₂×C₃	—	4	2	no	—
278	C₂×C₃	—	6	4	no	—
279	C₂×C₃	#1004	1	1	yes	characteristic
280	C₂×C₃	C₂×D₃×Q₈	1	1	yes	characteristic
281	C₂×C₃	#1009	1	1	yes	characteristic
282	C₂×C₃	C₂²×C₃×Q₈	1	1	yes	characteristic
283	C₂×C₃	C₃×H(5,𝔽₂)	1	1	yes	characteristic
284	C₂×C₃	C₃×H(5,𝔽₂)	1	1	yes	characteristic
285	D₃	—	2	4	no	—
286	C₂²	—	2	1	no	—
287	C₂²	—	2	1	no	—
288	C₂²	—	3	4	no	—
289	C₂²	—	6	2	no	—
290	C₂²	—	6	2	no	—
291	C₂²	—	12	4	no	—
292	C₂²	C₃×D₃×Q₈	1	2	yes	—
293	C₂²	C₂³×C₃×D₃	1	1	yes	characteristic
294	C₄	—	2	1	no	—
295	C₄	—	2	1	no	—
296	C₄	—	2	2	no	—
297	C₄	—	2	2	no	—
298	C₄	—	6	2	no	—
299	C₄	—	6	2	no	—
300	C₄	—	6	2	no	—
301	C₃	—	2	1	no	—
302	C₃	#5634	1	1	yes	characteristic, direct product factor
303	C₃	C₃×#543	1	1	yes	characteristic, complemented
304	C₂	—	2	2	no	—
305	C₂	—	6	4	no	—
306	C₂	C₃×#1004	1	1	yes	characteristic
307	C₂	C₂×C₃×D₃×Q₈	1	1	yes	characteristic
308	C₂	C₃×#1009	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×C₂²⋊Q₈	C₂
socle	C₂²×C₃²	C₂⁴
prime radical	C₂³×C₃×D₃	C₂²
hypercenter	C₃×C₂²⋊Q₈	D₃
hypocenter	C₃	C₃×#543
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₃ ◁ C₃×C₂²⋊Q₈

Complements

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

N	H ≅ G/N	type
1. C₃²×C₂²⋊Q₈	C₂	semidirect
2. C₃×#946	C₂	semidirect
4. C₃×#924	C₂	semidirect
5. C₃×#897	C₂	semidirect
6. C₃×#896	C₂	semidirect
7. C₃×#896	C₂	semidirect
9. C₃×#878	C₂	semidirect
11. #5634	C₃	direct (N × H)
13. C₂×C₃²×Q₈	C₂²	semidirect
18. C₂×C₃×Dic₆	C₂²	semidirect
19. C₂×C₃×Dic₆	C₂²	semidirect
20. C₃²×C₄⋊C₄	C₂²	semidirect
21. C₃²×C₄⋊C₄	C₂²	semidirect
28. C₃×C₄⋊Dic₃	C₂²	semidirect
29. C₃×C₄⋊Dic₃	C₂²	semidirect
30. C₃×Dic₃⋊C₄	C₂²	semidirect
31. C₃×Dic₃⋊C₄	C₂²	semidirect
32. C₃×Dic₃⋊C₄	C₂²	semidirect
40. C₃×C₂²⋊Q₈	D₃	semidirect
41. C₃×C₂²⋊Q₈	C₂×C₃	semidirect
44. #946	C₂×C₃	semidirect
46. #924	C₂×C₃	semidirect
47. #897	C₂×C₃	semidirect
48. #896	C₂×C₃	semidirect
49. #896	C₂×C₃	semidirect
51. #878	C₂×C₃	semidirect
55. C₂²×C₃×D₃	Q₈	semidirect
77. C₂×C₃×Q₈	C₂×D₃	semidirect
78. C₂×C₃×Q₈	C₂²×C₃	semidirect
88. C₂×Dic₆	C₂²×C₃	semidirect
89. C₂×Dic₆	C₂²×C₃	semidirect
97. C₃×C₄⋊C₄	C₂×D₃	semidirect
98. C₃×C₄⋊C₄	C₂×D₃	semidirect
99. C₃×C₄⋊C₄	C₂²×C₃	semidirect
100. C₃×C₄⋊C₄	C₂²×C₃	semidirect
114. C₄⋊Dic₃	C₂²×C₃	semidirect
115. C₄⋊Dic₃	C₂²×C₃	semidirect
116. Dic₃⋊C₄	C₂²×C₃	semidirect
117. Dic₃⋊C₄	C₂²×C₃	semidirect
118. Dic₃⋊C₄	C₂²×C₃	semidirect
140. C₂²⋊Q₈	C₃×D₃	semidirect
152. C₂²×D₃	C₃×Q₈	semidirect
196. C₂×Q₈	C₂×C₃×D₃	semidirect
206. C₄⋊C₄	C₂×C₃×D₃	semidirect
207. C₄⋊C₄	C₂×C₃×D₃	semidirect
252. C₃²	#543	semidirect
302. C₃	#5634	direct (N × H)
303. C₃	C₃×#543	semidirect

Maximal subgroups

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

subgroup	index
C₃²×C₂²⋊Q₈	2
C₃×#946	2
C₂×C₃×(C₂×D₃)⋊C₄	2
C₃×#924	2
C₃×#897	2
C₃×#896	2
C₃×D₃×C₂²⋊C₄	2
C₃×#878	2
C₃×#543	3
#5634	3

Subgroups — C3×#5634 #10595731

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#5634 #10595731