Subgroups — C₃×#67731 #11772451

Order 1152.

301 conjugacy classes of subgroups.

The subgroup lattice has 256 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃²×#108	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×#819	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃×#812	C₂	1	1	yes	maximal, characteristic
4	C₃×#1385	—	3	1	no	maximal
5	#67731	C₃	1	1	yes	maximal, characteristic, direct product factor
6	C₂×C₃²×M₁₆	—	2	1	no	—
7	C₂²×C₃²×C₈	C₂²	1	1	yes	characteristic
8	C₂×C₃×#208	—	2	1	no	—
9	C₃²×#108	—	2	1	no	—
10	C₃²×#108	C₄	1	2	yes	—
11	C₃×#819	—	2	1	no	—
12	C₃×#819	C₄	1	2	yes	—
13	C₃×#812	—	2	1	no	—
14	C₃×#812	—	2	1	no	—
15	C₂×C₃×C₃⋊C₁₆	C₄	1	2	yes	—
16	C₂×C₃×M₃₂	—	3	1	no	—
17	C₂×C₃×#108	—	2	1	no	—
18	C₂×C₃×#108	—	3	1	no	—
19	C₂×C₃×#108	D₃	1	1	yes	characteristic
20	C₂×C₃×#108	C₂×C₃	1	1	yes	characteristic
21	C₂×#819	C₂×C₃	1	1	yes	characteristic
22	C₂×#812	C₂×C₃	1	1	yes	characteristic
23	C₂²×C₃²×C₄	D₄	1	1	yes	characteristic
24	C₃²×M₁₆	—	2	2	no	—
25	C₃²×M₁₆	—	4	1	no	—
26	C₂×C₃²×C₈	—	4	1	no	—
27	C₂×C₃²×C₈	C₂×C₄	1	1	yes	characteristic
28	C₂×C₃²×C₈	C₂×C₄	1	1	yes	characteristic
29	C₂×C₃²×C₈	C₂×C₄	1	1	yes	characteristic
30	C₂×C₃²×C₈	D₄	1	1	yes	characteristic
31	C₂×C₃²×C₈	D₄	1	1	yes	characteristic
32	C₂×C₃²×C₈	Q₈	1	1	yes	characteristic
33	C₃×#208	—	2	2	no	—
34	C₃×#208	—	4	1	no	—
35	C₂×C₃×C₃⋊C₈	—	4	1	no	—
36	C₃×C₃⋊C₁₆	—	2	2	no	—
37	#1385	—	3	1	no	Sylow
38	C₂×C₃×M₁₆	—	2	2	no	—
39	C₂×C₃×M₁₆	—	2	1	no	—
40	C₂×C₃×M₁₆	—	2	1	no	—
41	C₂×C₃×M₁₆	—	6	1	no	—
42	C₂²×C₃×C₈	—	2	1	no	—
43	C₂²×C₃×C₈	C₂×D₃	1	1	yes	characteristic
44	C₂²×C₃×C₈	C₂²×C₃	1	1	yes	characteristic
45	C₂×#208	—	2	1	no	—
46	C₃×M₃₂	—	6	1	no	—
47	C₃×M₃₂	—	6	1	no	—
48	C₂×C₃×C₁₆	—	3	2	no	—
49	C₃×#108	—	2	2	no	—
50	C₃×#108	—	2	1	no	—
51	C₃×#108	—	2	1	no	—
52	C₃×#108	—	2	2	no	—
53	C₃×#108	—	3	2	no	—
54	C₃×#108	—	6	1	no	—
55	C₃×#108	Dic₃	1	2	yes	—
56	C₃×#108	C₃×C₄	1	2	yes	—
57	#819	—	2	1	no	—
58	#819	C₃×C₄	1	2	yes	—
59	#812	—	2	1	no	—
60	#812	—	2	1	no	—
61	C₂×C₃⋊C₁₆	C₃×C₄	1	2	yes	—
62	C₂³×C₃²	Q₁₆	1	1	yes	characteristic
63	C₂×C₃²×C₄	C₂²⋊C₄	1	1	yes	characteristic
64	C₂×C₃²×C₄	C₂²⋊C₄	1	1	yes	characteristic
65	C₂×C₃²×C₄	C₄⋊C₄	1	1	yes	characteristic
66	C₂×C₃²×C₄	D₈	1	1	yes	characteristic
67	C₂×C₃²×C₄	SD₁₆	1	1	yes	characteristic
68	C₂×C₃²×C₄	SD₁₆	1	1	yes	characteristic
69	C₃²×C₈	—	4	2	no	—
70	C₃²×C₈	C₄²	1	1	yes	characteristic
71	C₃²×C₈	C₂²⋊C₄	1	1	yes	characteristic
72	C₃²×C₈	C₄⋊C₄	1	1	yes	characteristic
73	C₃²×C₈	C₄⋊C₄	1	1	yes	characteristic
74	C₃×C₃⋊C₈	—	4	2	no	—
75	C₂×M₃₂	—	3	1	no	—
76	C₂×#108	—	3	1	no	—
77	C₂×#108	C₃×D₃	1	1	yes	characteristic
78	C₂²×C₃×C₄	—	2	1	no	—
79	C₂²×C₃×C₄	#67	1	1	yes	characteristic
80	C₂²×C₃×C₄	C₃×D₄	1	1	yes	characteristic
81	C₃×M₁₆	—	2	4	no	—
82	C₃×M₁₆	—	2	2	no	—
83	C₃×M₁₆	—	2	2	no	—
84	C₃×M₁₆	—	4	2	no	—
85	C₃×M₁₆	—	4	1	no	—
86	C₃×M₁₆	—	4	1	no	—
87	C₃×M₁₆	—	6	2	no	—
88	C₃×M₁₆	—	12	1	no	—
89	C₂×C₃×C₈	—	2	1	no	—
90	C₂×C₃×C₈	—	2	1	no	—
91	C₂×C₃×C₈	—	2	1	no	—
92	C₂×C₃×C₈	—	2	1	no	—
93	C₂×C₃×C₈	—	2	1	no	—
94	C₂×C₃×C₈	—	2	1	no	—
95	C₂×C₃×C₈	—	4	2	no	—
96	C₂×C₃×C₈	—	4	1	no	—
97	C₂×C₃×C₈	—	4	1	no	—
98	C₂×C₃×C₈	—	12	1	no	—
99	C₂×C₃×C₈	Dic₆	1	1	yes	characteristic
100	C₂×C₃×C₈	C₄×D₃	1	1	yes	characteristic
101	C₂×C₃×C₈	C₄×D₃	1	1	yes	characteristic
102	C₂×C₃×C₈	D₁₂	1	1	yes	characteristic
103	C₂×C₃×C₈	C₂×Dic₃	1	1	yes	characteristic
104	C₂×C₃×C₈	#67	1	1	yes	characteristic
105	C₂×C₃×C₈	C₂×C₃×C₄	1	1	yes	characteristic
106	C₂×C₃×C₈	C₂×C₃×C₄	1	1	yes	characteristic
107	C₂×C₃×C₈	C₂×C₃×C₄	1	1	yes	characteristic
108	C₂×C₃×C₈	C₃×D₄	1	1	yes	characteristic
109	C₂×C₃×C₈	C₃×D₄	1	1	yes	characteristic
110	C₂×C₃×C₈	C₃×Q₈	1	1	yes	characteristic
111	#208	—	2	2	no	—
112	#208	—	4	1	no	—
113	C₂×C₃⋊C₈	—	4	1	no	—
114	C₃×C₁₆	—	6	2	no	—
115	C₃⋊C₁₆	—	2	2	no	—
116	C₂²×C₃²	—	4	1	no	—
117	C₂²×C₃²	#103	1	1	yes	characteristic
118	C₂²×C₃²	#103	1	1	yes	characteristic
119	C₂²×C₃²	#107	1	1	yes	characteristic
120	C₃²×C₄	#95	1	1	yes	characteristic
121	C₃²×C₄	#102	1	1	yes	characteristic
122	C₃²×C₄	#102	1	1	yes	characteristic
123	C₃²×C₄	#106	1	1	yes	characteristic
124	C₂×M₁₆	—	2	1	no	—
125	C₂×M₁₆	—	6	1	no	—
126	C₂²×C₈	C₂×C₃×D₃	1	1	yes	characteristic
127	M₃₂	—	6	1	no	—
128	M₃₂	—	6	1	no	—
129	C₂×C₁₆	—	3	2	no	—
130	#108	—	2	1	no	—
131	#108	—	3	2	no	—
132	#108	—	6	1	no	—
133	#108	C₃×Dic₃	1	2	yes	—
134	C₂³×C₃	—	2	1	no	—
135	C₂³×C₃	#216	1	1	yes	characteristic
136	C₂³×C₃	C₃×Q₁₆	1	1	yes	characteristic
137	C₂×C₃×C₄	—	2	1	no	—
138	C₂×C₃×C₄	—	2	1	no	—
139	C₂×C₃×C₄	—	2	1	no	—
140	C₂×C₃×C₄	—	2	1	no	—
141	C₂×C₃×C₄	—	2	1	no	—
142	C₂×C₃×C₄	—	2	1	no	—
143	C₂×C₃×C₄	Dic₃⋊C₄	1	1	yes	characteristic
144	C₂×C₃×C₄	(C₂×D₃)⋊C₄	1	1	yes	characteristic
145	C₂×C₃×C₄	#213	1	1	yes	characteristic
146	C₂×C₃×C₄	#214	1	1	yes	characteristic
147	C₂×C₃×C₄	#215	1	1	yes	characteristic
148	C₂×C₃×C₄	#217	1	1	yes	characteristic
149	C₂×C₃×C₄	C₃×C₂²⋊C₄	1	1	yes	characteristic
150	C₂×C₃×C₄	C₃×C₂²⋊C₄	1	1	yes	characteristic
151	C₂×C₃×C₄	C₃×C₄⋊C₄	1	1	yes	characteristic
152	C₂×C₃×C₄	C₃×D₈	1	1	yes	characteristic
153	C₂×C₃×C₄	C₃×SD₁₆	1	1	yes	characteristic
154	C₂×C₃×C₄	C₃×SD₁₆	1	1	yes	characteristic
155	C₃×C₈	—	2	1	no	—
156	C₃×C₈	—	2	1	no	—
157	C₃×C₈	—	2	1	no	—
158	C₃×C₈	—	2	1	no	—
159	C₃×C₈	—	4	4	no	—
160	C₃×C₈	—	4	2	no	—
161	C₃×C₈	—	4	2	no	—
162	C₃×C₈	—	12	2	no	—
163	C₃×C₈	C₄×Dic₃	1	1	yes	characteristic
164	C₃×C₈	Dic₃⋊C₄	1	1	yes	characteristic
165	C₃×C₈	C₄⋊Dic₃	1	1	yes	characteristic
166	C₃×C₈	(C₂×D₃)⋊C₄	1	1	yes	characteristic
167	C₃×C₈	C₃×C₄²	1	1	yes	characteristic
168	C₃×C₈	C₃×C₂²⋊C₄	1	1	yes	characteristic
169	C₃×C₈	C₃×C₄⋊C₄	1	1	yes	characteristic
170	C₃×C₈	C₃×C₄⋊C₄	1	1	yes	characteristic
171	C₃⋊C₈	—	4	2	no	—
172	C₂×C₃²	—	2	1	no	—
173	C₂×C₃²	—	2	1	no	—
174	C₂×C₃²	—	2	1	no	—
175	C₂×C₃²	#340	1	1	yes	characteristic
176	C₂²×C₄	C₃×#67	1	1	yes	characteristic
177	M₁₆	—	2	2	no	—
178	M₁₆	—	4	1	no	—
179	M₁₆	—	6	2	no	—
180	M₁₆	—	12	1	no	—
181	C₂×C₈	—	4	1	no	—
182	C₂×C₈	—	12	1	no	—
183	C₂×C₈	C₃×Dic₆	1	1	yes	characteristic
184	C₂×C₈	C₃×C₄×D₃	1	1	yes	characteristic
185	C₂×C₈	C₃×C₄×D₃	1	1	yes	characteristic
186	C₂×C₈	C₃×D₁₂	1	1	yes	characteristic
187	C₂×C₈	C₂×C₃×Dic₃	1	1	yes	characteristic
188	C₂×C₈	C₃×#67	1	1	yes	characteristic
189	C₁₆	—	6	2	no	—
190	C₂²×C₃	—	2	1	no	—
191	C₂²×C₃	—	2	1	no	—
192	C₂²×C₃	—	2	1	no	—
193	C₂²×C₃	—	4	1	no	—
194	C₂²×C₃	—	4	1	no	—
195	C₂²×C₃	—	4	2	no	—
196	C₂²×C₃	#807	1	1	yes	characteristic
197	C₂²×C₃	#810	1	1	yes	characteristic
198	C₂²×C₃	#835	1	1	yes	characteristic
199	C₂²×C₃	C₃×#103	1	1	yes	characteristic
200	C₂²×C₃	C₃×#103	1	1	yes	characteristic
201	C₂²×C₃	C₃×#107	1	1	yes	characteristic
202	C₃×C₄	—	2	1	no	—
203	C₃×C₄	—	2	1	no	—
204	C₃×C₄	—	2	1	no	—
205	C₃×C₄	—	2	1	no	—
206	C₃×C₄	#808	1	1	yes	characteristic
207	C₃×C₄	#809	1	1	yes	characteristic
208	C₃×C₄	C₃⋊#95	1	1	yes	characteristic
209	C₃×C₄	#832	1	1	yes	characteristic
210	C₃×C₄	C₃×#95	1	1	yes	characteristic
211	C₃×C₄	C₃×#102	1	1	yes	characteristic
212	C₃×C₄	C₃×#102	1	1	yes	characteristic
213	C₃×C₄	C₃×#106	1	1	yes	characteristic
214	C₃²	#1385	1	1	yes	characteristic, complemented, Sylow
215	C₂³	C₃×#216	1	1	yes	characteristic
216	C₂×C₄	C₃×Dic₃⋊C₄	1	1	yes	characteristic
217	C₂×C₄	C₃×(C₂×D₃)⋊C₄	1	1	yes	characteristic
218	C₂×C₄	C₃×#213	1	1	yes	characteristic
219	C₂×C₄	C₃×#214	1	1	yes	characteristic
220	C₂×C₄	C₃×#215	1	1	yes	characteristic
221	C₂×C₄	C₃×#217	1	1	yes	characteristic
222	C₈	—	4	2	no	—
223	C₈	—	12	2	no	—
224	C₈	C₃×C₄×Dic₃	1	1	yes	characteristic
225	C₈	C₃×Dic₃⋊C₄	1	1	yes	characteristic
226	C₈	C₃×C₄⋊Dic₃	1	1	yes	characteristic
227	C₈	C₃×(C₂×D₃)⋊C₄	1	1	yes	characteristic
228	C₂×C₃	—	2	2	no	—
229	C₂×C₃	—	2	1	no	—
230	C₂×C₃	—	2	1	no	—
231	C₂×C₃	—	2	1	no	—
232	C₂×C₃	—	2	1	no	—
233	C₂×C₃	—	2	1	no	—
234	C₂×C₃	—	2	1	no	—
235	C₂×C₃	—	2	1	no	—
236	C₂×C₃	—	4	1	no	—
237	C₂×C₃	—	4	1	no	—
238	C₂×C₃	#4529	1	1	yes	characteristic
239	C₂×C₃	C₃×#340	1	1	yes	characteristic
240	C₂²	—	4	1	no	—
241	C₂²	C₃×#807	1	1	yes	characteristic
242	C₂²	C₃×#810	1	1	yes	characteristic
243	C₂²	C₃×#835	1	1	yes	characteristic
244	C₄	C₃×#808	1	1	yes	characteristic
245	C₄	C₃×#809	1	1	yes	characteristic
246	C₄	C₃×C₃⋊#95	1	1	yes	characteristic
247	C₄	C₃×#832	1	1	yes	characteristic
248	C₃	—	2	1	no	—
249	C₃	#67731	1	1	yes	characteristic, direct product factor
250	C₃	C₃×#1385	1	1	yes	characteristic, complemented
251	C₂	—	2	1	no	—
252	C₂	—	2	1	no	—
253	C₂	—	2	1	no	—
254	C₂	C₃×#4529	1	1	yes	characteristic

	subgroup	G/H
center	C₃×C₄	#808
commutator	C₃×C₈	C₃×C₄²
Frattini	C₂²×C₈	C₂×C₃×D₃
Fitting	C₂×C₃²×#108	C₂
socle	C₂×C₃²	#340
prime radical	C₂³×C₃²	Q₁₆
cosocle	C₂²×C₃×C₈	C₂²×C₃
hypercenter	C₂×C₃×#108	D₃
hypocenter	C₃	C₃×#1385
LCS	G ▷ C₃×C₈ ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₃×C₄ ◁ C₂²×C₃×C₄ ◁ C₂²×C₃×C₈ ◁ C₂×C₃×#108

Complements

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

N	H ≅ G/N	type
5. #67731	C₃	direct (N × H)
214. C₃²	#1385	semidirect
249. C₃	#67731	direct (N × H)
250. C₃	C₃×#1385	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃²×#108	2
C₂×C₃×#819	2
C₂×C₃×#812	2
C₃×#1385	3
#67731	3

Subgroups — C3×#67731 #11772451

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#67731 #11772451