Subgroups — C₃×#5555 #10595652

Order 576.

631 conjugacy classes of subgroups.

The subgroup lattice has 266 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×#237	C₂	1	2	yes	maximal, complemented
2	C₃²×C₄×D₄	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×#930	C₂	1	2	yes	maximal, complemented
4	C₂×C₃×C₄⋊Dic₃	C₂	1	2	yes	maximal, complemented
5	C₃×#897	C₂	1	2	yes	maximal, complemented
6	C₃×#886	C₂	1	4	yes	maximal, complemented
7	C₃×C₄×D₁₂	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃×#869	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃×#547	—	3	1	no	maximal
10	#5555	C₃	1	1	yes	maximal, characteristic, direct product factor
11	C₂×C₃²×D₄	C₂²	1	1	yes	characteristic
12	C₂²×C₃²×C₄	C₂²	1	2	yes	—
13	C₂×C₃×#67	C₂²	1	4	yes	—
14	C₂²×C₃×Dic₃	C₂²	1	4	yes	—
15	C₃×#237	—	2	8	no	—
16	C₂×C₃×D₁₂	C₂²	1	1	yes	characteristic
17	C₂×C₃×C₄×D₃	C₂²	1	2	yes	—
18	C₂×C₃×Dic₆	C₂²	1	2	yes	—
19	C₃²×C₄⋊C₄	C₂²	1	1	yes	characteristic
20	C₃²×C₂²⋊C₄	C₂²	1	2	yes	—
21	C₃²×C₄²	C₂²	1	1	yes	characteristic
22	C₃×(C₂×D₃)⋊C₄	C₂²	1	2	yes	—
23	C₃×(C₂×D₃)⋊C₄	C₂²	1	4	yes	—
24	C₃×C₄⋊Dic₃	C₂²	1	2	yes	—
25	C₃×C₄⋊Dic₃	C₂²	1	4	yes	—
26	C₃×C₄⋊Dic₃	C₂²	1	1	yes	characteristic
27	C₃×C₄⋊Dic₃	C₂²	1	2	yes	—
28	C₂×C₃×#41	—	3	2	no	—
29	C₂×#237	C₂×C₃	1	2	yes	complemented
30	C₃×C₄⋊Q₈	—	3	1	no	—
31	C₃×#123	—	3	4	no	—
32	C₃×C₂²⋊Q₈	—	3	2	no	—
33	C₃×#121	—	3	2	no	—
34	C₃×C₄×D₄	—	2	1	no	—
35	C₃×C₄×D₄	—	3	1	no	—
36	C₃×C₄×D₄	D₃	1	1	yes	characteristic, complemented
37	C₃×C₄×D₄	C₂×C₃	1	1	yes	characteristic, complemented
38	C₂×C₃×C₄⋊C₄	—	3	2	no	—
39	#930	C₂×C₃	1	2	yes	complemented
40	C₂×C₄⋊Dic₃	C₂×C₃	1	2	yes	complemented
41	#897	C₂×C₃	1	2	yes	complemented
42	#886	C₂×C₃	1	4	yes	complemented
43	C₄×D₁₂	C₂×C₃	1	1	yes	characteristic, complemented
44	#869	C₂×C₃	1	1	yes	characteristic, complemented
45	C₂³×C₃²	C₂³	1	2	yes	—
46	C₂²×C₃×D₃	C₂³	1	2	yes	—
47	C₃²×D₄	D₄	1	4	yes	complemented
48	C₂×C₃²×C₄	—	2	4	no	—
49	C₂×C₃²×C₄	C₂³	1	1	yes	characteristic
50	C₂×C₃²×C₄	C₂³	1	2	yes	—
51	C₂×C₃²×C₄	C₂³	1	1	yes	characteristic
52	C₂×C₃²×C₄	C₂³	1	1	yes	characteristic
53	C₃×#67	—	2	8	no	—
54	C₂×C₃×Dic₃	—	2	8	no	—
55	C₂×C₃×Dic₃	C₂³	1	4	yes	—
56	C₂×C₃×Dic₃	C₂³	1	2	yes	—
57	C₃×D₁₂	—	2	2	no	—
58	C₃×C₄×D₃	—	2	4	no	—
59	C₃×Dic₆	—	2	4	no	—
60	#547	—	3	1	no	Sylow
61	C₃×#41	—	6	8	no	—
62	C₂×C₃×Q₈	—	3	2	no	—
63	C₂×C₃×D₄	—	2	1	no	—
64	C₂×C₃×D₄	—	3	4	no	—
65	C₂×C₃×D₄	—	3	1	no	—
66	C₂×C₃×D₄	C₂×D₃	1	1	yes	characteristic
67	C₂×C₃×D₄	C₂²×C₃	1	1	yes	characteristic
68	C₂²×C₃×C₄	—	2	2	no	—
69	C₂²×C₃×C₄	—	3	4	no	—
70	C₂²×C₃×C₄	—	3	2	no	—
71	C₂²×C₃×C₄	C₂×D₃	1	2	yes	—
72	C₂²×C₃×C₄	C₂²×C₃	1	2	yes	—
73	C₂×#67	C₂²×C₃	1	4	yes	—
74	C₂²×Dic₃	C₂²×C₃	1	4	yes	—
75	#237	—	2	8	no	—
76	C₂×D₁₂	C₂²×C₃	1	1	yes	characteristic
77	C₂×C₄×D₃	C₂²×C₃	1	2	yes	—
78	C₂×Dic₆	C₂²×C₃	1	2	yes	—
79	C₃×C₄⋊C₄	—	2	1	no	—
80	C₃×C₄⋊C₄	—	3	1	no	—
81	C₃×C₄⋊C₄	—	3	2	no	—
82	C₃×C₄⋊C₄	—	3	4	no	—
83	C₃×C₄⋊C₄	—	3	2	no	—
84	C₃×C₄⋊C₄	C₂×D₃	1	1	yes	characteristic
85	C₃×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
86	C₃×C₂²⋊C₄	—	2	2	no	—
87	C₃×C₂²⋊C₄	—	3	2	no	—
88	C₃×C₂²⋊C₄	—	3	4	no	—
89	C₃×C₂²⋊C₄	C₂×D₃	1	2	yes	—
90	C₃×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
91	C₃×C₄²	—	2	1	no	—
92	C₃×C₄²	C₂×D₃	1	1	yes	characteristic
93	C₃×C₄²	C₂²×C₃	1	1	yes	characteristic
94	(C₂×D₃)⋊C₄	C₂²×C₃	1	2	yes	—
95	(C₂×D₃)⋊C₄	C₂²×C₃	1	4	yes	—
96	C₄⋊Dic₃	C₂²×C₃	1	2	yes	—
97	C₄⋊Dic₃	C₂²×C₃	1	2	yes	—
98	C₄⋊Dic₃	C₂²×C₃	1	1	yes	characteristic
99	C₄⋊Dic₃	C₂²×C₃	1	4	yes	—
100	C₂²×C₃²	—	2	2	no	—
101	C₂²×C₃²	—	2	2	no	—
102	C₂²×C₃²	C₂×D₄	1	4	yes	—
103	C₂²×C₃²	C₂⁴	1	1	yes	characteristic
104	C₂×C₃×D₃	—	2	2	no	—
105	C₂×C₃×D₃	—	2	2	no	—
106	C₂×C₃×D₃	—	2	2	no	—
107	C₃²×C₄	—	2	2	no	—
108	C₃²×C₄	—	2	1	no	—
109	C₃²×C₄	C₂×D₄	1	2	yes	—
110	C₃²×C₄	#41	1	2	yes	complemented
111	C₃×Dic₃	—	2	4	no	—
112	C₃×Dic₃	—	2	2	no	—
113	C₂×#41	—	3	2	no	—
114	C₄⋊Q₈	—	3	1	no	—
115	#123	—	3	4	no	—
116	C₂²⋊Q₈	—	3	2	no	—
117	#121	—	3	2	no	—
118	C₄×D₄	—	3	1	no	—
119	C₄×D₄	C₃×D₃	1	1	yes	characteristic, complemented
120	C₂×C₄⋊C₄	—	3	2	no	—
121	C₂³×C₃	—	2	2	no	—
122	C₂³×C₃	—	3	2	no	—
123	C₂³×C₃	C₂²×D₃	1	2	yes	—
124	C₂³×C₃	C₂³×C₃	1	2	yes	—
125	C₂²×D₃	C₂³×C₃	1	2	yes	—
126	C₃×Q₈	—	6	4	no	—
127	C₃×D₄	—	2	4	no	—
128	C₃×D₄	—	6	8	no	—
129	C₃×D₄	—	6	2	no	—
130	C₃×D₄	D₁₂	1	4	yes	complemented
131	C₃×D₄	C₃×D₄	1	4	yes	complemented
132	C₂×C₃×C₄	—	2	4	no	—
133	C₂×C₃×C₄	—	2	4	no	—
134	C₂×C₃×C₄	—	2	1	no	—
135	C₂×C₃×C₄	—	2	1	no	—
136	C₂×C₃×C₄	—	2	1	no	—
137	C₂×C₃×C₄	—	2	2	no	—
138	C₂×C₃×C₄	—	3	2	no	—
139	C₂×C₃×C₄	—	3	4	no	—
140	C₂×C₃×C₄	—	4	4	no	—
141	C₂×C₃×C₄	—	6	8	no	—
142	C₂×C₃×C₄	—	6	4	no	—
143	C₂×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
144	C₂×C₃×C₄	C₂²×D₃	1	2	yes	—
145	C₂×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
146	C₂×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
147	C₂×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
148	C₂×C₃×C₄	C₂³×C₃	1	2	yes	—
149	C₂×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
150	C₂×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
151	#67	—	2	8	no	—
152	C₂×Dic₃	—	2	8	no	—
153	C₂×Dic₃	C₂³×C₃	1	4	yes	—
154	C₂×Dic₃	C₂³×C₃	1	2	yes	—
155	D₁₂	—	2	2	no	—
156	C₄×D₃	—	2	4	no	—
157	Dic₆	—	2	4	no	—
158	C₂×C₃²	—	2	4	no	—
159	C₂×C₃²	C₂²×D₄	1	1	yes	characteristic
160	C₂×C₃²	C₂×#41	1	1	yes	characteristic
161	C₂×C₃²	2₋¹⁺⁴	1	1	yes	characteristic
162	C₃×D₃	—	4	2	no	—
163	#41	—	6	8	no	—
164	C₂×Q₈	—	3	2	no	—
165	C₂×D₄	—	3	4	no	—
166	C₂×D₄	—	3	1	no	—
167	C₂×D₄	C₂×C₃×D₃	1	1	yes	characteristic
168	C₂²×C₄	—	3	4	no	—
169	C₂²×C₄	—	3	2	no	—
170	C₂²×C₄	C₂×C₃×D₃	1	2	yes	—
171	C₄⋊C₄	—	3	1	no	—
172	C₄⋊C₄	—	3	2	no	—
173	C₄⋊C₄	—	3	4	no	—
174	C₄⋊C₄	—	3	2	no	—
175	C₄⋊C₄	C₂×C₃×D₃	1	1	yes	characteristic
176	C₂²⋊C₄	—	3	2	no	—
177	C₂²⋊C₄	—	3	4	no	—
178	C₂²⋊C₄	C₂×C₃×D₃	1	2	yes	—
179	C₄²	C₂×C₃×D₃	1	1	yes	characteristic
180	C₂²×C₃	—	2	2	no	—
181	C₂²×C₃	—	2	2	no	—
182	C₂²×C₃	—	2	2	no	—
183	C₂²×C₃	—	2	2	no	—
184	C₂²×C₃	—	2	1	no	—
185	C₂²×C₃	—	2	4	no	—
186	C₂²×C₃	—	4	2	no	—
187	C₂²×C₃	—	4	2	no	—
188	C₂²×C₃	—	6	2	no	—
189	C₂²×C₃	—	6	2	no	—
190	C₂²×C₃	—	6	2	no	—
191	C₂²×C₃	C₂×D₁₂	1	4	yes	—
192	C₂²×C₃	C₂×C₃×D₄	1	4	yes	—
193	C₂²×C₃	C₂³×D₃	1	1	yes	characteristic
194	C₂²×C₃	C₂⁴×C₃	1	1	yes	characteristic
195	C₂×D₃	—	2	2	no	—
196	C₂×D₃	—	2	2	no	—
197	C₂×D₃	—	2	2	no	—
198	C₃×C₄	—	2	2	no	—
199	C₃×C₄	—	2	1	no	—
200	C₃×C₄	—	2	2	no	—
201	C₃×C₄	—	2	1	no	—
202	C₃×C₄	—	2	2	no	—
203	C₃×C₄	—	2	2	no	—
204	C₃×C₄	—	4	2	no	—
205	C₃×C₄	—	4	1	no	—
206	C₃×C₄	—	6	2	no	—
207	C₃×C₄	—	6	4	no	—
208	C₃×C₄	C₂×D₁₂	1	2	yes	—
209	C₃×C₄	#237	1	2	yes	complemented
210	C₃×C₄	C₂×C₃×D₄	1	2	yes	—
211	C₃×C₄	C₃×#41	1	2	yes	complemented
212	Dic₃	—	2	4	no	—
213	Dic₃	—	2	2	no	—
214	C₃²	#547	1	1	yes	characteristic, complemented, Sylow
215	C₂³	—	3	2	no	—
216	C₂³	C₂²×C₃×D₃	1	2	yes	—
217	Q₈	—	6	4	no	—
218	D₄	—	6	8	no	—
219	D₄	—	6	2	no	—
220	D₄	C₃×D₁₂	1	4	yes	complemented
221	C₂×C₄	—	2	4	no	—
222	C₂×C₄	—	3	2	no	—
223	C₂×C₄	—	3	4	no	—
224	C₂×C₄	—	6	8	no	—
225	C₂×C₄	—	6	4	no	—
226	C₂×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
227	C₂×C₄	C₂²×C₃×D₃	1	2	yes	—
228	C₂×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
229	C₂×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
230	C₂×C₃	—	2	4	no	—
231	C₂×C₃	—	2	4	no	—
232	C₂×C₃	—	2	1	no	—
233	C₂×C₃	—	2	1	no	—
234	C₂×C₃	—	2	1	no	—
235	C₂×C₃	—	4	4	no	—
236	C₂×C₃	—	12	2	no	—
237	C₂×C₃	C₂²×D₁₂	1	1	yes	characteristic
238	C₂×C₃	C₂×#237	1	1	yes	characteristic
239	C₂×C₃	#1010	1	1	yes	characteristic
240	C₂×C₃	C₂²×C₃×D₄	1	1	yes	characteristic
241	C₂×C₃	C₂×C₃×#41	1	1	yes	characteristic
242	C₂×C₃	C₃×2₋¹⁺⁴	1	1	yes	characteristic
243	D₃	—	4	2	no	—
244	C₂²	—	2	2	no	—
245	C₂²	—	2	2	no	—
246	C₂²	—	6	2	no	—
247	C₂²	—	6	2	no	—
248	C₂²	—	6	2	no	—
249	C₂²	C₂×C₃×D₁₂	1	4	yes	—
250	C₂²	C₂³×C₃×D₃	1	1	yes	characteristic
251	C₄	—	2	2	no	—
252	C₄	—	2	1	no	—
253	C₄	—	6	2	no	—
254	C₄	—	6	4	no	—
255	C₄	C₂×C₃×D₁₂	1	2	yes	—
256	C₄	C₃×#237	1	2	yes	complemented
257	C₃	—	2	1	no	—
258	C₃	#5555	1	1	yes	characteristic, direct product factor
259	C₃	C₃×#547	1	1	yes	characteristic, complemented
260	C₂	—	2	4	no	—
261	C₂	—	12	2	no	—
262	C₂	C₂²×C₃×D₁₂	1	1	yes	characteristic
263	C₂	C₂×C₃×#237	1	1	yes	characteristic
264	C₂	C₃×#1010	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
hypercenter	C₃×C₄×D₄	D₃
hypocenter	C₃	C₃×#547
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. C₂×C₃×#237	C₂	semidirect
2. C₃²×C₄×D₄	C₂	semidirect
3. C₃×#930	C₂	semidirect
4. C₂×C₃×C₄⋊Dic₃	C₂	semidirect
5. C₃×#897	C₂	semidirect
6. C₃×#886	C₂	semidirect
7. C₃×C₄×D₁₂	C₂	semidirect
8. C₃×#869	C₂	semidirect
10. #5555	C₃	direct (N × H)
29. C₂×#237	C₂×C₃	semidirect
36. C₃×C₄×D₄	D₃	semidirect
37. C₃×C₄×D₄	C₂×C₃	semidirect
39. #930	C₂×C₃	semidirect
40. C₂×C₄⋊Dic₃	C₂×C₃	semidirect
41. #897	C₂×C₃	semidirect
42. #886	C₂×C₃	semidirect
43. C₄×D₁₂	C₂×C₃	semidirect
44. #869	C₂×C₃	semidirect
47. C₃²×D₄	D₄	semidirect
110. C₃²×C₄	#41	semidirect
119. C₄×D₄	C₃×D₃	semidirect
130. C₃×D₄	D₁₂	semidirect
131. C₃×D₄	C₃×D₄	semidirect
209. C₃×C₄	#237	semidirect
211. C₃×C₄	C₃×#41	semidirect
214. C₃²	#547	semidirect
220. D₄	C₃×D₁₂	semidirect
256. C₄	C₃×#237	semidirect
258. C₃	#5555	direct (N × H)
259. C₃	C₃×#547	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃×#237	2
C₃²×C₄×D₄	2
C₃×#930	2
C₂×C₃×C₄⋊Dic₃	2
C₃×#897	2
C₃×#886	2
C₃×C₄×D₁₂	2
C₃×#869	2
C₃×#547	3
#5555	3

Subgroups — C3×#5555 #10595652

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#5555 #10595652