Subgroups — C₂×C₃×#909 #10595844

Order 576.

550 conjugacy classes of subgroups.

The subgroup lattice has 248 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×#235	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₃×Dic₆	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃²×M₁₆	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×#909	C₂	1	8	yes	maximal, direct product factor
5	C₂×C₃×Dic₁₂	C₂	1	2	yes	maximal, complemented
6	C₂×C₃×#204	C₂	1	2	yes	maximal, complemented
7	C₂×C₃×#137	—	3	1	no	maximal
8	C₂×#909	C₃	1	1	yes	maximal, characteristic, direct product factor
9	C₂²×C₃²×C₄	C₂²	1	1	yes	characteristic
10	C₂×C₃×#67	—	2	1	no	—
11	C₂²×C₃×Dic₃	—	2	1	no	—
12	C₃×#235	—	2	2	no	—
13	C₃×#235	C₂²	1	4	yes	—
14	C₂×C₃×D₁₂	C₂²	1	1	yes	characteristic
15	C₂×C₃×C₄×D₃	—	2	1	no	—
16	C₂×C₃×Dic₆	—	2	2	no	—
17	C₂×C₃×Dic₆	—	2	1	no	—
18	C₂×C₃×Dic₆	C₂²	1	2	yes	—
19	C₂×C₃×Dic₆	C₂²	1	1	yes	characteristic
20	C₂×C₃×Dic₆	C₂²	1	4	yes	complemented
21	C₃²×M₁₆	C₂²	1	4	yes	complemented
22	C₂×C₃²×C₈	C₂²	1	2	yes	—
23	C₃×Dic₁₂	C₂²	1	8	yes	complemented
24	C₃×#204	C₂²	1	8	yes	complemented
25	C₂×C₃×#41	—	3	1	no	—
26	C₂²×C₃×Q₈	—	3	1	no	—
27	C₂×#235	C₂×C₃	1	1	yes	characteristic
28	C₂²×Dic₆	C₂×C₃	1	1	yes	characteristic, complemented
29	C₃×#137	—	3	8	no	—
30	C₂×C₃×Q₁₆	—	3	2	no	—
31	C₂×C₃×SD₁₆	—	3	2	no	—
32	C₂×C₃×M₁₆	—	2	1	no	—
33	C₂×C₃×M₁₆	D₃	1	1	yes	characteristic, complemented
34	C₂×C₃×M₁₆	C₂×C₃	1	1	yes	characteristic, complemented
35	#909	C₂×C₃	1	8	yes	direct product factor
36	C₂×Dic₁₂	C₂×C₃	1	2	yes	complemented
37	C₂×#204	C₂×C₃	1	2	yes	complemented
38	C₂³×C₃²	D₄	1	1	yes	characteristic
39	C₂²×C₃×D₃	—	2	1	no	—
40	C₂×C₃²×C₄	D₄	1	1	yes	characteristic
41	C₂×C₃²×C₄	D₄	1	2	yes	—
42	C₂×C₃²×C₄	C₂³	1	1	yes	characteristic
43	C₂×C₃²×C₄	C₂³	1	2	yes	—
44	C₃×#67	—	2	4	no	—
45	C₂×C₃×Dic₃	—	2	2	no	—
46	C₂×C₃×Dic₃	—	2	4	no	—
47	C₂×C₃×Dic₃	—	2	1	no	—
48	C₃×D₁₂	—	2	1	no	—
49	C₃×D₁₂	C₂³	1	2	yes	—
50	C₃×C₄×D₃	—	2	4	no	—
51	C₃×Dic₆	—	2	2	no	—
52	C₃×Dic₆	—	2	4	no	—
53	C₃×Dic₆	—	2	1	no	—
54	C₃×Dic₆	C₂³	1	4	yes	—
55	C₃×Dic₆	C₂³	1	2	yes	—
56	C₃²×C₈	C₂³	1	4	yes	—
57	C₂×#137	—	3	1	no	Sylow
58	C₃×#41	—	3	4	no	—
59	C₃×#41	—	6	2	no	—
60	C₂×C₃×Q₈	—	3	2	no	—
61	C₂×C₃×Q₈	—	3	1	no	—
62	C₂×C₃×Q₈	—	3	4	no	—
63	C₂×C₃×Q₈	—	6	2	no	—
64	C₂×C₃×Q₈	—	6	1	no	—
65	C₂×C₃×D₄	—	3	1	no	—
66	C₂×C₃×D₄	—	6	1	no	—
67	C₂²×C₃×C₄	—	2	1	no	—
68	C₂²×C₃×C₄	—	6	1	no	—
69	C₂²×C₃×C₄	—	6	1	no	—
70	C₂²×C₃×C₄	C₂×D₃	1	1	yes	characteristic
71	C₂²×C₃×C₄	C₂²×C₃	1	1	yes	characteristic
72	C₂×#67	—	2	1	no	—
73	C₂²×Dic₃	—	2	1	no	—
74	#235	—	2	2	no	—
75	#235	C₂²×C₃	1	4	yes	—
76	C₂×D₁₂	C₂²×C₃	1	1	yes	characteristic
77	C₂×C₄×D₃	—	2	1	no	—
78	C₂×Dic₆	—	2	2	no	—
79	C₂×Dic₆	—	2	1	no	—
80	C₂×Dic₆	C₂²×C₃	1	2	yes	—
81	C₂×Dic₆	C₂²×C₃	1	1	yes	characteristic
82	C₂×Dic₆	C₂²×C₃	1	4	yes	complemented
83	C₃×Q₁₆	—	3	8	no	—
84	C₃×SD₁₆	—	3	8	no	—
85	C₃×M₁₆	—	2	4	no	—
86	C₃×M₁₆	C₂×D₃	1	4	yes	complemented
87	C₃×M₁₆	C₂²×C₃	1	4	yes	complemented
88	C₂×C₃×C₈	—	2	2	no	—
89	C₂×C₃×C₈	C₂×D₃	1	2	yes	—
90	C₂×C₃×C₈	C₂²×C₃	1	2	yes	—
91	Dic₁₂	C₂²×C₃	1	8	yes	complemented
92	#204	C₂²×C₃	1	8	yes	complemented
93	C₂²×C₃²	—	2	2	no	—
94	C₂²×C₃²	C₂×D₄	1	2	yes	—
95	C₂²×C₃²	C₂×D₄	1	1	yes	characteristic
96	C₂×C₃×D₃	—	2	2	no	—
97	C₂×C₃×D₃	—	4	2	no	—
98	C₃²×C₄	C₂×D₄	1	1	yes	characteristic
99	C₃²×C₄	C₂×D₄	1	2	yes	—
100	C₃²×C₄	C₂⁴	1	1	yes	characteristic
101	C₃×Dic₃	—	2	4	no	—
102	C₃×Dic₃	—	2	2	no	—
103	C₂×#41	—	3	1	no	—
104	C₂²×Q₈	—	3	1	no	—
105	#137	—	3	8	no	—
106	C₂×Q₁₆	—	3	2	no	—
107	C₂×SD₁₆	—	3	2	no	—
108	C₂×M₁₆	C₃×D₃	1	1	yes	characteristic, complemented
109	C₂³×C₃	—	2	1	no	—
110	C₂³×C₃	—	6	1	no	—
111	C₂³×C₃	D₁₂	1	1	yes	characteristic
112	C₂³×C₃	C₃×D₄	1	1	yes	characteristic
113	C₂²×D₃	—	2	1	no	—
114	C₃×Q₈	—	3	4	no	—
115	C₃×Q₈	—	3	2	no	—
116	C₃×Q₈	—	6	4	no	—
117	C₃×Q₈	—	6	2	no	—
118	C₃×Q₈	—	6	1	no	—
119	C₃×D₄	—	3	2	no	—
120	C₃×D₄	—	6	4	no	—
121	C₃×D₄	—	6	1	no	—
122	C₂×C₃×C₄	—	2	1	no	—
123	C₂×C₃×C₄	—	2	2	no	—
124	C₂×C₃×C₄	—	2	1	no	—
125	C₂×C₃×C₄	—	2	2	no	—
126	C₂×C₃×C₄	—	6	4	no	—
127	C₂×C₃×C₄	—	6	1	no	—
128	C₂×C₃×C₄	—	6	2	no	—
129	C₂×C₃×C₄	—	6	4	no	—
130	C₂×C₃×C₄	D₁₂	1	2	yes	—
131	C₂×C₃×C₄	D₁₂	1	1	yes	characteristic
132	C₂×C₃×C₄	C₃×D₄	1	1	yes	characteristic
133	C₂×C₃×C₄	C₃×D₄	1	2	yes	—
134	C₂×C₃×C₄	C₂²×D₃	1	2	yes	—
135	C₂×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
136	C₂×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
137	C₂×C₃×C₄	C₂³×C₃	1	2	yes	—
138	#67	—	2	4	no	—
139	C₂×Dic₃	—	2	2	no	—
140	C₂×Dic₃	—	2	4	no	—
141	C₂×Dic₃	—	2	1	no	—
142	D₁₂	—	2	1	no	—
143	D₁₂	C₂³×C₃	1	2	yes	—
144	C₄×D₃	—	2	4	no	—
145	Dic₆	—	2	2	no	—
146	Dic₆	—	2	4	no	—
147	Dic₆	—	2	1	no	—
148	Dic₆	C₂³×C₃	1	4	yes	—
149	Dic₆	C₂³×C₃	1	2	yes	—
150	C₃×C₈	—	2	4	no	—
151	C₃×C₈	C₂²×D₃	1	4	yes	—
152	C₃×C₈	C₂³×C₃	1	4	yes	—
153	C₂×C₃²	—	2	2	no	—
154	C₂×C₃²	#137	1	2	yes	complemented
155	C₂×C₃²	C₂²×D₄	1	1	yes	characteristic
156	C₃×D₃	—	4	2	no	—
157	#41	—	3	4	no	—
158	#41	—	6	2	no	—
159	C₂×Q₈	—	3	2	no	—
160	C₂×Q₈	—	3	1	no	—
161	C₂×Q₈	—	3	4	no	—
162	C₂×Q₈	—	6	1	no	—
163	C₂×Q₈	—	6	2	no	—
164	C₂×D₄	—	3	1	no	—
165	C₂×D₄	—	6	1	no	—
166	C₂²×C₄	—	6	1	no	—
167	C₂²×C₄	—	6	1	no	—
168	C₂²×C₄	C₂×C₃×D₃	1	1	yes	characteristic
169	Q₁₆	—	3	8	no	—
170	SD₁₆	—	3	8	no	—
171	M₁₆	C₂×C₃×D₃	1	4	yes	complemented
172	C₂×C₈	C₂×C₃×D₃	1	2	yes	—
173	C₂²×C₃	—	2	2	no	—
174	C₂²×C₃	—	2	2	no	—
175	C₂²×C₃	—	2	1	no	—
176	C₂²×C₃	—	2	2	no	—
177	C₂²×C₃	—	4	2	no	—
178	C₂²×C₃	—	6	2	no	—
179	C₂²×C₃	—	12	2	no	—
180	C₂²×C₃	C₂×D₁₂	1	2	yes	—
181	C₂²×C₃	C₂×D₁₂	1	1	yes	characteristic
182	C₂²×C₃	C₂×C₃×D₄	1	2	yes	—
183	C₂²×C₃	C₂×C₃×D₄	1	1	yes	characteristic
184	C₂×D₃	—	2	2	no	—
185	C₂×D₃	—	4	2	no	—
186	C₃×C₄	—	2	1	no	—
187	C₃×C₄	—	2	2	no	—
188	C₃×C₄	—	2	1	no	—
189	C₃×C₄	—	6	2	no	—
190	C₃×C₄	—	6	4	no	—
191	C₃×C₄	C₂×D₁₂	1	2	yes	—
192	C₃×C₄	C₂×D₁₂	1	1	yes	characteristic
193	C₃×C₄	C₂×C₃×D₄	1	1	yes	characteristic
194	C₃×C₄	C₂×C₃×D₄	1	2	yes	—
195	C₃×C₄	C₂³×D₃	1	1	yes	characteristic
196	C₃×C₄	C₂⁴×C₃	1	1	yes	characteristic
197	Dic₃	—	2	4	no	—
198	Dic₃	—	2	2	no	—
199	C₃²	C₂×#137	1	1	yes	characteristic, complemented, Sylow
200	C₂³	—	6	1	no	—
201	C₂³	C₃×D₁₂	1	1	yes	characteristic
202	Q₈	—	3	4	no	—
203	Q₈	—	3	2	no	—
204	Q₈	—	6	4	no	—
205	Q₈	—	6	2	no	—
206	Q₈	—	6	1	no	—
207	D₄	—	3	2	no	—
208	D₄	—	6	4	no	—
209	D₄	—	6	1	no	—
210	C₂×C₄	—	6	1	no	—
211	C₂×C₄	—	6	4	no	—
212	C₂×C₄	—	6	2	no	—
213	C₂×C₄	—	6	4	no	—
214	C₂×C₄	C₃×D₁₂	1	1	yes	characteristic
215	C₂×C₄	C₃×D₁₂	1	2	yes	—
216	C₂×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
217	C₂×C₄	C₂²×C₃×D₃	1	2	yes	—
218	C₈	C₂²×C₃×D₃	1	4	yes	—
219	C₂×C₃	—	2	2	no	—
220	C₂×C₃	—	2	2	no	—
221	C₂×C₃	—	2	2	no	—
222	C₂×C₃	—	2	1	no	—
223	C₂×C₃	—	4	2	no	—
224	C₂×C₃	—	12	2	no	—
225	C₂×C₃	#909	1	2	yes	direct product factor
226	C₂×C₃	C₃×#137	1	2	yes	complemented
227	C₂×C₃	C₂²×D₁₂	1	1	yes	characteristic
228	C₂×C₃	C₂²×C₃×D₄	1	1	yes	characteristic
229	D₃	—	4	2	no	—
230	C₂²	—	2	2	no	—
231	C₂²	—	6	2	no	—
232	C₂²	—	12	2	no	—
233	C₂²	C₂×C₃×D₁₂	1	2	yes	—
234	C₂²	C₂×C₃×D₁₂	1	1	yes	characteristic
235	C₄	—	6	2	no	—
236	C₄	—	6	4	no	—
237	C₄	C₂×C₃×D₁₂	1	2	yes	—
238	C₄	C₂×C₃×D₁₂	1	1	yes	characteristic
239	C₄	C₂³×C₃×D₃	1	1	yes	characteristic
240	C₃	—	2	1	no	—
241	C₃	C₂×#909	1	1	yes	characteristic, direct product factor
242	C₃	C₂×C₃×#137	1	1	yes	characteristic, complemented
243	C₂	—	2	2	no	—
244	C₂	—	12	2	no	—
245	C₂	C₃×#909	1	2	yes	direct product factor
246	C₂	C₂²×C₃×D₁₂	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₃	C₂×D₁₂
commutator, cosocle	C₃×C₄	C₂⁴×C₃
Frattini	C₄	C₂³×C₃×D₃
Fitting	C₂×C₃²×M₁₆	C₂
socle	C₂²×C₃²	C₂×D₄
prime radical	C₂×C₃×#235	C₂
hypercenter	C₂×C₃×M₁₆	D₃
hypocenter	C₃	C₂×C₃×#137
LCS	G ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₃ ◁ C₂²×C₃×C₄ ◁ C₂×C₃×M₁₆

Complements

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

N	H ≅ G/N	type
2. C₂²×C₃×Dic₆	C₂	semidirect
3. C₂×C₃²×M₁₆	C₂	semidirect
4. C₃×#909	C₂	direct (N × H)
5. C₂×C₃×Dic₁₂	C₂	semidirect
6. C₂×C₃×#204	C₂	semidirect
8. C₂×#909	C₃	direct (N × H)
20. C₂×C₃×Dic₆	C₂²	semidirect
21. C₃²×M₁₆	C₂²	semidirect
23. C₃×Dic₁₂	C₂²	semidirect
24. C₃×#204	C₂²	semidirect
28. C₂²×Dic₆	C₂×C₃	semidirect
33. C₂×C₃×M₁₆	D₃	semidirect
34. C₂×C₃×M₁₆	C₂×C₃	semidirect
35. #909	C₂×C₃	direct (N × H)
36. C₂×Dic₁₂	C₂×C₃	semidirect
37. C₂×#204	C₂×C₃	semidirect
82. C₂×Dic₆	C₂²×C₃	semidirect
86. C₃×M₁₆	C₂×D₃	semidirect
87. C₃×M₁₆	C₂²×C₃	semidirect
91. Dic₁₂	C₂²×C₃	semidirect
92. #204	C₂²×C₃	semidirect
108. C₂×M₁₆	C₃×D₃	semidirect
154. C₂×C₃²	#137	semidirect
171. M₁₆	C₂×C₃×D₃	semidirect
199. C₃²	C₂×#137	semidirect
225. C₂×C₃	#909	direct (N × H)
226. C₂×C₃	C₃×#137	semidirect
241. C₃	C₂×#909	direct (N × H)
242. C₃	C₂×C₃×#137	semidirect
245. C₂	C₃×#909	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₃×#235	2
C₂²×C₃×Dic₆	2
C₂×C₃²×M₁₆	2
C₃×#909	2
C₂×C₃×Dic₁₂	2
C₂×C₃×#204	2
C₂×C₃×#137	3
C₂×#909	3

Subgroups — C2×C3×#909 #10595844

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×#909 #10595844