Subgroups — C₃×#5132 #10592229

Order 576.

334 conjugacy classes of subgroups.

The subgroup lattice has 254 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	C₂×C₃²×M₁₆	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃×#208	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×#824	C₂	1	2	yes	maximal, complemented
5	C₃×#823	C₂	1	2	yes	maximal, complemented
6	C₃×#413	—	3	1	no	maximal
7	#5132	C₃	1	1	yes	maximal, characteristic, direct product factor
8	C₂²×C₃²×C₄	C₂²	1	1	yes	characteristic
9	C₂×C₃×#67	C₄	1	2	yes	—
10	C₃×#235	—	2	1	no	—
11	C₃×#235	—	2	1	no	—
12	C₃×#235	—	2	1	no	—
13	C₃×#235	—	2	1	no	—
14	C₂×C₃×D₁₂	C₂²	1	1	yes	characteristic
15	C₂×C₃×C₄×D₃	C₄	1	2	yes	—
16	C₂×C₃×Dic₆	C₂²	1	1	yes	characteristic
17	C₃²×M₁₆	—	2	1	no	—
18	C₃²×M₁₆	C₂²	1	2	yes	—
19	C₂×C₃²×C₈	—	2	1	no	—
20	C₃×#208	—	2	1	no	—
21	C₃×#208	C₂²	1	2	yes	—
22	C₂×C₃×C₃⋊C₈	—	2	1	no	—
23	C₂×C₃×#41	—	3	1	no	—
24	C₂×#235	C₂×C₃	1	1	yes	characteristic
25	C₂×C₃×M₁₆	—	2	1	no	—
26	C₂×C₃×M₁₆	—	3	1	no	—
27	C₂×C₃×M₁₆	D₃	1	1	yes	characteristic, complemented
28	C₂×C₃×M₁₆	C₂×C₃	1	1	yes	characteristic, complemented
29	C₂×#208	C₂×C₃	1	1	yes	characteristic, complemented
30	C₃×#101	—	3	2	no	—
31	C₃×#100	—	3	2	no	—
32	#824	C₂×C₃	1	2	yes	complemented
33	#823	C₂×C₃	1	2	yes	complemented
34	C₂³×C₃²	C₂×C₄	1	1	yes	characteristic
35	C₂²×C₃×D₃	C₂×C₄	1	2	yes	—
36	C₂×C₃²×C₄	C₂×C₄	1	1	yes	characteristic
37	C₂×C₃²×C₄	D₄	1	1	yes	characteristic
38	C₂×C₃²×C₄	D₄	1	1	yes	characteristic
39	C₂×C₃²×C₄	D₄	1	1	yes	characteristic
40	C₂×C₃²×C₄	D₄	1	1	yes	characteristic
41	C₂×C₃²×C₄	C₂³	1	1	yes	characteristic
42	C₃×#67	—	2	2	no	—
43	C₃×#67	—	2	2	no	—
44	C₂×C₃×Dic₃	C₂×C₄	1	2	yes	—
45	C₃×D₁₂	—	2	1	no	—
46	C₃×D₁₂	—	2	1	no	—
47	C₃×C₄×D₃	—	2	2	no	—
48	C₃×C₄×D₃	—	2	2	no	—
49	C₃×Dic₆	—	2	1	no	—
50	C₃×Dic₆	—	2	1	no	—
51	C₃²×C₈	—	2	2	no	—
52	C₃×C₃⋊C₈	—	2	2	no	—
53	#413	—	3	1	no	Sylow
54	C₃×#41	—	6	1	no	—
55	C₃×#41	—	6	1	no	—
56	C₃×#41	—	6	1	no	—
57	C₃×#41	—	6	1	no	—
58	C₂×C₃×Q₈	—	3	1	no	—
59	C₂×C₃×D₄	—	3	2	no	—
60	C₂×C₃×D₄	—	3	1	no	—
61	C₂²×C₃×C₄	—	2	1	no	—
62	C₂²×C₃×C₄	—	3	2	no	—
63	C₂²×C₃×C₄	C₂×D₃	1	1	yes	characteristic
64	C₂²×C₃×C₄	C₂²×C₃	1	1	yes	characteristic
65	C₂×#67	C₃×C₄	1	2	yes	—
66	#235	—	2	1	no	—
67	#235	—	2	1	no	—
68	#235	—	2	1	no	—
69	#235	—	2	1	no	—
70	C₂×D₁₂	C₂²×C₃	1	1	yes	characteristic
71	C₂×C₄×D₃	C₃×C₄	1	2	yes	—
72	C₂×Dic₆	C₂²×C₃	1	1	yes	characteristic
73	C₃×M₁₆	—	2	2	no	—
74	C₃×M₁₆	—	2	1	no	—
75	C₃×M₁₆	—	2	1	no	—
76	C₃×M₁₆	—	2	2	no	—
77	C₃×M₁₆	—	3	2	no	—
78	C₃×M₁₆	—	6	1	no	—
79	C₃×M₁₆	C₂×D₃	1	2	yes	—
80	C₃×M₁₆	C₂²×C₃	1	2	yes	—
81	C₂×C₃×C₈	—	2	2	no	—
82	C₂×C₃×C₈	—	2	1	no	—
83	C₂×C₃×C₈	—	2	1	no	—
84	C₂×C₃×C₈	—	6	1	no	—
85	#208	—	2	1	no	—
86	#208	C₂²×C₃	1	2	yes	—
87	C₂×C₃⋊C₈	—	2	1	no	—
88	C₂²×C₃²	—	4	1	no	—
89	C₂²×C₃²	C₂²⋊C₄	1	1	yes	characteristic
90	C₂²×C₃²	C₂²⋊C₄	1	1	yes	characteristic
91	C₂²×C₃²	C₂²×C₄	1	1	yes	characteristic
92	C₂×C₃×D₃	—	2	2	no	—
93	C₂×C₃×D₃	—	4	2	no	—
94	C₃²×C₄	C₂²⋊C₄	1	1	yes	characteristic
95	C₃²×C₄	C₂²⋊C₄	1	1	yes	characteristic
96	C₃²×C₄	C₂×D₄	1	1	yes	characteristic
97	C₃²×C₄	C₂×D₄	1	1	yes	characteristic
98	C₃×Dic₃	—	2	2	no	—
99	C₂×#41	—	3	1	no	—
100	C₂×M₁₆	—	3	1	no	—
101	C₂×M₁₆	C₃×D₃	1	1	yes	characteristic, complemented
102	#101	—	3	2	no	—
103	#100	—	3	2	no	—
104	C₂³×C₃	—	2	1	no	—
105	C₂³×C₃	—	3	2	no	—
106	C₂³×C₃	C₄×D₃	1	1	yes	characteristic
107	C₂³×C₃	C₂×C₃×C₄	1	1	yes	characteristic
108	C₂²×D₃	C₂×C₃×C₄	1	2	yes	—
109	C₃×Q₈	—	6	1	no	—
110	C₃×Q₈	—	6	1	no	—
111	C₃×D₄	—	6	2	no	—
112	C₃×D₄	—	6	2	no	—
113	C₃×D₄	—	6	1	no	—
114	C₃×D₄	—	6	1	no	—
115	C₂×C₃×C₄	—	2	1	no	—
116	C₂×C₃×C₄	—	2	1	no	—
117	C₂×C₃×C₄	—	2	1	no	—
118	C₂×C₃×C₄	—	2	1	no	—
119	C₂×C₃×C₄	—	2	1	no	—
120	C₂×C₃×C₄	—	2	1	no	—
121	C₂×C₃×C₄	—	3	2	no	—
122	C₂×C₃×C₄	—	6	2	no	—
123	C₂×C₃×C₄	—	6	2	no	—
124	C₂×C₃×C₄	C₄×D₃	1	1	yes	characteristic
125	C₂×C₃×C₄	D₁₂	1	1	yes	characteristic
126	C₂×C₃×C₄	D₁₂	1	1	yes	characteristic
127	C₂×C₃×C₄	#67	1	1	yes	characteristic
128	C₂×C₃×C₄	#67	1	1	yes	characteristic
129	C₂×C₃×C₄	C₂×C₃×C₄	1	1	yes	characteristic
130	C₂×C₃×C₄	C₃×D₄	1	1	yes	characteristic
131	C₂×C₃×C₄	C₃×D₄	1	1	yes	characteristic
132	C₂×C₃×C₄	C₃×D₄	1	1	yes	characteristic
133	C₂×C₃×C₄	C₃×D₄	1	1	yes	characteristic
134	C₂×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
135	C₂×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
136	#67	—	2	2	no	—
137	#67	—	2	2	no	—
138	C₂×Dic₃	C₂×C₃×C₄	1	2	yes	—
139	D₁₂	—	2	1	no	—
140	D₁₂	—	2	1	no	—
141	C₄×D₃	—	2	2	no	—
142	C₄×D₃	—	2	2	no	—
143	Dic₆	—	2	1	no	—
144	Dic₆	—	2	1	no	—
145	C₃×C₈	—	2	4	no	—
146	C₃×C₈	—	2	2	no	—
147	C₃×C₈	—	2	2	no	—
148	C₃×C₈	—	6	2	no	—
149	C₃⋊C₈	—	2	2	no	—
150	C₂×C₃²	—	2	1	no	—
151	C₂×C₃²	—	2	1	no	—
152	C₂×C₃²	—	2	1	no	—
153	C₂×C₃²	C₂×C₂²⋊C₄	1	1	yes	characteristic
154	C₃×D₃	—	4	2	no	—
155	#41	—	6	1	no	—
156	#41	—	6	1	no	—
157	#41	—	6	1	no	—
158	#41	—	6	1	no	—
159	C₂×Q₈	—	3	1	no	—
160	C₂×D₄	—	3	2	no	—
161	C₂×D₄	—	3	1	no	—
162	C₂²×C₄	—	3	2	no	—
163	C₂²×C₄	C₂×C₃×D₃	1	1	yes	characteristic
164	M₁₆	—	2	1	no	—
165	M₁₆	—	3	2	no	—
166	M₁₆	—	6	1	no	—
167	M₁₆	C₂×C₃×D₃	1	2	yes	—
168	C₂×C₈	—	2	1	no	—
169	C₂×C₈	—	6	1	no	—
170	C₂²×C₃	—	2	1	no	—
171	C₂²×C₃	—	2	1	no	—
172	C₂²×C₃	—	2	1	no	—
173	C₂²×C₃	—	4	1	no	—
174	C₂²×C₃	—	4	1	no	—
175	C₂²×C₃	—	4	2	no	—
176	C₂²×C₃	—	6	2	no	—
177	C₂²×C₃	—	12	2	no	—
178	C₂²×C₃	(C₂×D₃)⋊C₄	1	1	yes	characteristic
179	C₂²×C₃	(C₂×D₃)⋊C₄	1	1	yes	characteristic
180	C₂²×C₃	C₃×C₂²⋊C₄	1	1	yes	characteristic
181	C₂²×C₃	C₃×C₂²⋊C₄	1	1	yes	characteristic
182	C₂²×C₃	C₂×C₄×D₃	1	1	yes	characteristic
183	C₂²×C₃	C₂²×C₃×C₄	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	1	no	—
188	C₃×C₄	—	2	1	no	—
189	C₃×C₄	—	2	1	no	—
190	C₃×C₄	—	6	2	no	—
191	C₃×C₄	(C₂×D₃)⋊C₄	1	1	yes	characteristic
192	C₃×C₄	(C₂×D₃)⋊C₄	1	1	yes	characteristic
193	C₃×C₄	C₃×C₂²⋊C₄	1	1	yes	characteristic
194	C₃×C₄	C₃×C₂²⋊C₄	1	1	yes	characteristic
195	C₃×C₄	C₂×D₁₂	1	1	yes	characteristic
196	C₃×C₄	C₂×#67	1	1	yes	characteristic
197	C₃×C₄	C₂×C₃×D₄	1	1	yes	characteristic
198	C₃×C₄	C₂×C₃×D₄	1	1	yes	characteristic
199	Dic₃	—	2	2	no	—
200	C₃²	#413	1	1	yes	characteristic, complemented, Sylow
201	C₂³	—	3	2	no	—
202	C₂³	C₃×C₄×D₃	1	1	yes	characteristic
203	Q₈	—	6	1	no	—
204	Q₈	—	6	1	no	—
205	D₄	—	6	2	no	—
206	D₄	—	6	2	no	—
207	D₄	—	6	1	no	—
208	D₄	—	6	1	no	—
209	C₂×C₄	—	3	2	no	—
210	C₂×C₄	—	6	2	no	—
211	C₂×C₄	—	6	2	no	—
212	C₂×C₄	C₃×C₄×D₃	1	1	yes	characteristic
213	C₂×C₄	C₃×D₁₂	1	1	yes	characteristic
214	C₂×C₄	C₃×D₁₂	1	1	yes	characteristic
215	C₂×C₄	C₃×#67	1	1	yes	characteristic
216	C₂×C₄	C₃×#67	1	1	yes	characteristic
217	C₂×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
218	C₈	—	2	2	no	—
219	C₈	—	6	2	no	—
220	C₂×C₃	—	2	2	no	—
221	C₂×C₃	—	2	1	no	—
222	C₂×C₃	—	2	1	no	—
223	C₂×C₃	—	2	1	no	—
224	C₂×C₃	—	2	1	no	—
225	C₂×C₃	—	2	1	no	—
226	C₂×C₃	—	2	1	no	—
227	C₂×C₃	—	2	1	no	—
228	C₂×C₃	—	4	1	no	—
229	C₂×C₃	—	4	1	no	—
230	C₂×C₃	—	12	2	no	—
231	C₂×C₃	C₂×(C₂×D₃)⋊C₄	1	1	yes	characteristic
232	C₂×C₃	C₂×C₃×C₂²⋊C₄	1	1	yes	characteristic
233	D₃	—	4	2	no	—
234	C₂²	—	4	1	no	—
235	C₂²	—	6	2	no	—
236	C₂²	—	12	2	no	—
237	C₂²	C₃×(C₂×D₃)⋊C₄	1	1	yes	characteristic
238	C₂²	C₃×(C₂×D₃)⋊C₄	1	1	yes	characteristic
239	C₂²	C₂×C₃×C₄×D₃	1	1	yes	characteristic
240	C₄	—	6	2	no	—
241	C₄	C₃×(C₂×D₃)⋊C₄	1	1	yes	characteristic
242	C₄	C₃×(C₂×D₃)⋊C₄	1	1	yes	characteristic
243	C₄	C₂×C₃×D₁₂	1	1	yes	characteristic
244	C₄	C₂×C₃×#67	1	1	yes	characteristic
245	C₃	—	2	1	no	—
246	C₃	#5132	1	1	yes	characteristic, direct product factor
247	C₃	C₃×#413	1	1	yes	characteristic, complemented
248	C₂	—	2	1	no	—
249	C₂	—	2	1	no	—
250	C₂	—	2	1	no	—
251	C₂	—	12	2	no	—
252	C₂	C₂×C₃×(C₂×D₃)⋊C₄	1	1	yes	characteristic

	subgroup	G/H
center	C₃×C₄	(C₂×D₃)⋊C₄
commutator	C₂²×C₃	C₂²×C₃×C₄
Frattini	C₂×C₄	C₂²×C₃×D₃
Fitting	C₂×C₃²×M₁₆	C₂
socle	C₂×C₃²	C₂×C₂²⋊C₄
prime radical	C₂×C₃×#235	C₂
cosocle	C₂×C₃×C₄	C₂³×C₃
hypercenter	C₂×C₃×M₁₆	D₃
hypocenter	C₃	C₃×#413
LCS	G ▷ C₂²×C₃ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₃×C₄ ◁ C₂²×C₃×C₄ ◁ C₂×C₃×M₁₆

Complements

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

N	H ≅ G/N	type
2. C₂×C₃²×M₁₆	C₂	semidirect
3. C₂×C₃×#208	C₂	semidirect
4. C₃×#824	C₂	semidirect
5. C₃×#823	C₂	semidirect
7. #5132	C₃	direct (N × H)
27. C₂×C₃×M₁₆	D₃	semidirect
28. C₂×C₃×M₁₆	C₂×C₃	semidirect
29. C₂×#208	C₂×C₃	semidirect
32. #824	C₂×C₃	semidirect
33. #823	C₂×C₃	semidirect
101. C₂×M₁₆	C₃×D₃	semidirect
200. C₃²	#413	semidirect
246. C₃	#5132	direct (N × H)
247. C₃	C₃×#413	semidirect

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
C₂×C₃²×M₁₆	2
C₂×C₃×#208	2
C₃×#824	2
C₃×#823	2
C₃×#413	3
#5132	3

Subgroups — C3×#5132 #10592229

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#5132 #10592229