Subgroups — C₃×#5825 #10595922

Order 576.

622 conjugacy classes of subgroups.

The subgroup lattice has 242 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃²×#41	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₃×Dic₆	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×#944	C₂	1	2	yes	maximal, complemented
4	C₃×#936	C₂	1	2	yes	maximal, complemented
5	C₃×#935	C₂	1	4	yes	maximal, complemented
6	C₃×#926	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃×#924	C₂	1	4	yes	maximal, complemented
8	C₃×#536	—	3	1	no	maximal
9	#5825	C₃	1	1	yes	maximal, characteristic, direct product factor
10	C₃²×#41	—	2	4	no	—
11	C₂×C₃²×Q₈	C₂²	1	1	yes	characteristic
12	C₂×C₃²×D₄	C₂²	1	2	yes	—
13	C₂×C₃²×D₄	C₂²	1	1	yes	characteristic
14	C₂²×C₃²×C₄	C₂²	1	2	yes	—
15	C₂²×C₃²×C₄	C₂²	1	1	yes	characteristic
16	C₂²×C₃×Dic₃	C₂²	1	2	yes	—
17	C₂×C₃×Dic₆	—	2	4	no	—
18	C₂×C₃×Dic₆	C₂²	1	2	yes	—
19	C₂×C₃×Dic₆	C₂²	1	2	yes	—
20	C₃×#217	C₂²	1	8	yes	—
21	C₃×#217	C₂²	1	2	yes	—
22	C₃×C₄⋊Dic₃	C₂²	1	2	yes	—
23	C₃×Dic₃⋊C₄	C₂²	1	8	yes	—
24	C₃×C₄×Dic₃	C₂²	1	2	yes	—
25	C₂×C₃×#41	—	2	1	no	—
26	C₂×C₃×#41	D₃	1	1	yes	characteristic
27	C₂×C₃×#41	C₂×C₃	1	1	yes	characteristic
28	C₂²×C₃×Q₈	—	3	1	no	—
29	C₂²×Dic₆	C₂×C₃	1	1	yes	characteristic, complemented
30	C₃×C₄⋊Q₈	—	3	2	no	—
31	C₃×#124	—	3	2	no	—
32	C₃×#123	—	3	4	no	—
33	C₃×C₂²⋊Q₈	—	3	4	no	—
34	C₃×#117	—	3	1	no	—
35	#944	C₂×C₃	1	2	yes	complemented
36	#936	C₂×C₃	1	2	yes	complemented
37	#935	C₂×C₃	1	4	yes	complemented
38	#926	C₂×C₃	1	1	yes	characteristic, complemented
39	#924	C₂×C₃	1	4	yes	complemented
40	C₂³×C₃²	C₂³	1	2	yes	—
41	C₂³×C₃²	C₂³	1	1	yes	characteristic
42	C₃²×Q₈	—	2	2	no	—
43	C₃²×D₄	—	2	4	no	—
44	C₃²×D₄	—	2	2	no	—
45	C₂×C₃²×C₄	—	2	4	no	—
46	C₂×C₃²×C₄	D₄	1	4	yes	—
47	C₂×C₃²×C₄	C₂³	1	2	yes	—
48	C₂×C₃²×C₄	C₂³	1	1	yes	characteristic
49	C₂×C₃²×C₄	C₂³	1	1	yes	characteristic
50	C₂×C₃×Dic₃	—	2	4	no	—
51	C₂×C₃×Dic₃	C₂³	1	4	yes	—
52	C₂×C₃×Dic₃	C₂³	1	4	yes	—
53	C₃×Dic₆	—	2	4	no	—
54	C₃×Dic₆	—	2	4	no	—
55	#536	—	3	1	no	Sylow
56	C₃×#41	—	2	8	no	—
57	C₃×#41	—	2	4	no	—
58	C₃×#41	—	2	4	no	—
59	C₂×C₃×Q₈	—	2	1	no	—
60	C₂×C₃×Q₈	—	3	2	no	—
61	C₂×C₃×Q₈	—	3	2	no	—
62	C₂×C₃×Q₈	—	6	4	no	—
63	C₂×C₃×Q₈	C₂×D₃	1	1	yes	characteristic
64	C₂×C₃×Q₈	C₂²×C₃	1	1	yes	characteristic
65	C₂×C₃×D₄	—	2	2	no	—
66	C₂×C₃×D₄	—	2	1	no	—
67	C₂×C₃×D₄	C₂×D₃	1	2	yes	—
68	C₂×C₃×D₄	C₂×D₃	1	1	yes	characteristic
69	C₂×C₃×D₄	C₂²×C₃	1	2	yes	—
70	C₂×C₃×D₄	C₂²×C₃	1	1	yes	characteristic
71	C₂²×C₃×C₄	—	2	2	no	—
72	C₂²×C₃×C₄	—	2	1	no	—
73	C₂²×C₃×C₄	—	3	2	no	—
74	C₂²×C₃×C₄	C₂×D₃	1	2	yes	—
75	C₂²×C₃×C₄	C₂×D₃	1	1	yes	characteristic
76	C₂²×C₃×C₄	C₂²×C₃	1	2	yes	—
77	C₂²×C₃×C₄	C₂²×C₃	1	1	yes	characteristic
78	C₂²×Dic₃	C₂²×C₃	1	2	yes	—
79	C₂×Dic₆	—	2	4	no	—
80	C₂×Dic₆	C₂²×C₃	1	2	yes	—
81	C₂×Dic₆	C₂²×C₃	1	2	yes	—
82	C₃×C₄⋊C₄	—	3	8	no	—
83	C₃×C₄⋊C₄	—	3	2	no	—
84	C₃×C₂²⋊C₄	—	3	2	no	—
85	C₃×C₂²⋊C₄	—	3	8	no	—
86	C₃×C₄²	—	3	2	no	—
87	#217	C₂²×C₃	1	8	yes	—
88	#217	C₂²×C₃	1	2	yes	—
89	C₄⋊Dic₃	C₂²×C₃	1	2	yes	—
90	Dic₃⋊C₄	C₂²×C₃	1	8	yes	—
91	C₄×Dic₃	C₂²×C₃	1	2	yes	—
92	C₂²×C₃²	—	2	2	no	—
93	C₂²×C₃²	—	2	2	no	—
94	C₂²×C₃²	—	2	4	no	—
95	C₂²×C₃²	C₂×D₄	1	2	yes	—
96	C₂²×C₃²	C₂⁴	1	1	yes	characteristic
97	C₃²×C₄	—	2	2	no	—
98	C₃²×C₄	C₂×D₄	1	2	yes	—
99	C₃²×C₄	C₂×D₄	1	2	yes	—
100	C₃×Dic₃	—	2	4	no	—
101	C₃×Dic₃	—	2	4	no	—
102	C₂×#41	C₃×D₃	1	1	yes	characteristic
103	C₂²×Q₈	—	3	1	no	—
104	C₄⋊Q₈	—	3	2	no	—
105	#124	—	3	2	no	—
106	#123	—	3	4	no	—
107	C₂²⋊Q₈	—	3	4	no	—
108	#117	—	3	1	no	—
109	C₂³×C₃	—	2	2	no	—
110	C₂³×C₃	—	2	1	no	—
111	C₂³×C₃	C₂²×D₃	1	2	yes	—
112	C₂³×C₃	C₂²×D₃	1	1	yes	characteristic
113	C₂³×C₃	C₂³×C₃	1	2	yes	—
114	C₂³×C₃	C₂³×C₃	1	1	yes	characteristic
115	C₃×Q₈	—	2	4	no	—
116	C₃×Q₈	—	2	2	no	—
117	C₃×Q₈	—	2	2	no	—
118	C₃×Q₈	—	6	4	no	—
119	C₃×Q₈	—	6	4	no	—
120	C₃×D₄	—	2	8	no	—
121	C₃×D₄	—	2	4	no	—
122	C₃×D₄	—	2	4	no	—
123	C₃×D₄	—	2	4	no	—
124	C₃×D₄	—	2	2	no	—
125	C₃×D₄	—	2	2	no	—
126	C₂×C₃×C₄	—	2	8	no	—
127	C₂×C₃×C₄	—	2	4	no	—
128	C₂×C₃×C₄	—	2	4	no	—
129	C₂×C₃×C₄	—	2	4	no	—
130	C₂×C₃×C₄	—	2	1	no	—
131	C₂×C₃×C₄	—	2	2	no	—
132	C₂×C₃×C₄	—	2	1	no	—
133	C₂×C₃×C₄	—	3	4	no	—
134	C₂×C₃×C₄	—	3	4	no	—
135	C₂×C₃×C₄	—	6	4	no	—
136	C₂×C₃×C₄	#67	1	4	yes	—
137	C₂×C₃×C₄	C₃×D₄	1	4	yes	—
138	C₂×C₃×C₄	C₂²×D₃	1	2	yes	—
139	C₂×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
140	C₂×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
141	C₂×C₃×C₄	C₂³×C₃	1	2	yes	—
142	C₂×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
143	C₂×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
144	C₂×Dic₃	—	2	4	no	—
145	C₂×Dic₃	C₂³×C₃	1	4	yes	—
146	C₂×Dic₃	C₂³×C₃	1	4	yes	—
147	Dic₆	—	2	4	no	—
148	Dic₆	—	2	4	no	—
149	C₂×C₃²	—	2	2	no	—
150	C₂×C₃²	—	4	2	no	—
151	C₂×C₃²	C₂²×D₄	1	1	yes	characteristic
152	C₂×C₃²	2₋¹⁺⁴	1	2	yes	—
153	#41	—	2	4	no	—
154	C₂×Q₈	—	3	2	no	—
155	C₂×Q₈	—	3	2	no	—
156	C₂×Q₈	—	6	4	no	—
157	C₂×Q₈	C₂×C₃×D₃	1	1	yes	characteristic
158	C₂×D₄	C₂×C₃×D₃	1	2	yes	—
159	C₂×D₄	C₂×C₃×D₃	1	1	yes	characteristic
160	C₂²×C₄	—	3	2	no	—
161	C₂²×C₄	C₂×C₃×D₃	1	2	yes	—
162	C₂²×C₄	C₂×C₃×D₃	1	1	yes	characteristic
163	C₄⋊C₄	—	3	2	no	—
164	C₄⋊C₄	—	3	8	no	—
165	C₂²⋊C₄	—	3	2	no	—
166	C₂²⋊C₄	—	3	8	no	—
167	C₄²	—	3	2	no	—
168	C₂²×C₃	—	2	2	no	—
169	C₂²×C₃	—	2	2	no	—
170	C₂²×C₃	—	2	4	no	—
171	C₂²×C₃	—	2	4	no	—
172	C₂²×C₃	—	2	2	no	—
173	C₂²×C₃	—	2	2	no	—
174	C₂²×C₃	—	2	4	no	—
175	C₂²×C₃	—	2	1	no	—
176	C₂²×C₃	—	2	2	no	—
177	C₂²×C₃	—	4	2	no	—
178	C₂²×C₃	—	4	4	no	—
179	C₂²×C₃	C₂×#67	1	2	yes	—
180	C₂²×C₃	C₂×C₃×D₄	1	2	yes	—
181	C₂²×C₃	C₂³×D₃	1	1	yes	characteristic
182	C₂²×C₃	C₂⁴×C₃	1	1	yes	characteristic
183	C₃×C₄	—	2	4	no	—
184	C₃×C₄	—	2	2	no	—
185	C₃×C₄	—	2	2	no	—
186	C₃×C₄	—	2	2	no	—
187	C₃×C₄	—	2	2	no	—
188	C₃×C₄	—	6	4	no	—
189	C₃×C₄	—	6	4	no	—
190	C₃×C₄	C₂×#67	1	2	yes	—
191	C₃×C₄	C₂×#67	1	2	yes	—
192	C₃×C₄	C₂×C₃×D₄	1	2	yes	—
193	C₃×C₄	C₂×C₃×D₄	1	2	yes	—
194	Dic₃	—	2	4	no	—
195	Dic₃	—	2	4	no	—
196	C₃²	#536	1	1	yes	characteristic, complemented, Sylow
197	C₂³	C₂²×C₃×D₃	1	2	yes	—
198	C₂³	C₂²×C₃×D₃	1	1	yes	characteristic
199	Q₈	—	2	2	no	—
200	Q₈	—	6	4	no	—
201	Q₈	—	6	4	no	—
202	D₄	—	2	4	no	—
203	D₄	—	2	2	no	—
204	C₂×C₄	—	2	4	no	—
205	C₂×C₄	—	3	4	no	—
206	C₂×C₄	—	3	4	no	—
207	C₂×C₄	—	6	4	no	—
208	C₂×C₄	C₃×#67	1	4	yes	—
209	C₂×C₄	C₂²×C₃×D₃	1	2	yes	—
210	C₂×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
211	C₂×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
212	C₂×C₃	—	2	2	no	—
213	C₂×C₃	—	2	2	no	—
214	C₂×C₃	—	2	2	no	—
215	C₂×C₃	—	2	1	no	—
216	C₂×C₃	—	4	2	no	—
217	C₂×C₃	—	4	4	no	—
218	C₂×C₃	—	4	2	no	—
219	C₂×C₃	—	4	2	no	—
220	C₂×C₃	#1010	1	2	yes	—
221	C₂×C₃	C₂²×#67	1	1	yes	characteristic
222	C₂×C₃	C₂²×C₃×D₄	1	1	yes	characteristic
223	C₂×C₃	C₃×2₋¹⁺⁴	1	2	yes	—
224	C₂²	—	2	2	no	—
225	C₂²	—	2	4	no	—
226	C₂²	—	2	2	no	—
227	C₂²	C₂×C₃×#67	1	2	yes	—
228	C₂²	C₂³×C₃×D₃	1	1	yes	characteristic
229	C₄	—	2	2	no	—
230	C₄	—	6	4	no	—
231	C₄	—	6	4	no	—
232	C₄	C₂×C₃×#67	1	2	yes	—
233	C₄	C₂×C₃×#67	1	2	yes	—
234	C₃	—	2	1	no	—
235	C₃	#5825	1	1	yes	characteristic, direct product factor
236	C₃	C₃×#536	1	1	yes	characteristic, complemented
237	C₂	—	2	2	no	—
238	C₂	—	4	2	no	—
239	C₂	C₃×#1010	1	2	yes	—
240	C₂	C₂²×C₃×#67	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₃	C₂³×D₃
commutator, cosocle	C₂²×C₃	C₂⁴×C₃
Frattini	C₂²	C₂³×C₃×D₃
Fitting, prime radical	C₂×C₃²×#41	C₂
socle	C₂²×C₃²	C₂⁴
hypercenter	C₂×C₃×#41	D₃
hypocenter	C₃	C₃×#536
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₃ ◁ C₂×C₃×#41

Complements

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

N	H ≅ G/N	type
2. C₂²×C₃×Dic₆	C₂	semidirect
3. C₃×#944	C₂	semidirect
4. C₃×#936	C₂	semidirect
5. C₃×#935	C₂	semidirect
6. C₃×#926	C₂	semidirect
7. C₃×#924	C₂	semidirect
9. #5825	C₃	direct (N × H)
29. C₂²×Dic₆	C₂×C₃	semidirect
35. #944	C₂×C₃	semidirect
36. #936	C₂×C₃	semidirect
37. #935	C₂×C₃	semidirect
38. #926	C₂×C₃	semidirect
39. #924	C₂×C₃	semidirect
196. C₃²	#536	semidirect
235. C₃	#5825	direct (N × H)
236. C₃	C₃×#536	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃²×#41	2
C₂²×C₃×Dic₆	2
C₃×#944	2
C₃×#936	2
C₃×#935	2
C₃×#926	2
C₃×#924	2
C₃×#536	3
#5825	3

Subgroups — C3×#5825 #10595922

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#5825 #10595922