Subgroups — C₃×#67726 #11772465

Order 1152.

337 conjugacy classes of subgroups.

The subgroup lattice has 269 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²×#427	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×#817	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃×#812	C₂	1	1	yes	maximal, characteristic
4	C₃×#1386	—	3	1	no	maximal
5	#67726	C₃	1	1	yes	maximal, characteristic, direct product factor
6	C₂²×C₃²×C₈	C₂²	1	1	yes	characteristic
7	C₃²×#117	—	2	1	no	—
8	C₂×C₃×C₄⋊Dic₃	—	2	1	no	—
9	C₃²×#107	C₄	1	2	yes	complemented
10	C₃²×#106	—	2	1	no	—
11	C₃×#817	—	2	1	no	—
12	C₃×#817	C₄	1	2	yes	complemented
13	C₃×#812	—	2	1	no	—
14	C₃×#812	C₄	1	2	yes	complemented
15	C₂×C₃×C₃⋊C₁₆	—	2	1	no	—
16	C₂×C₃×M₃₂	—	3	1	no	—
17	C₃×#427	—	2	1	no	—
18	C₃×#427	D₃	1	1	yes	characteristic
19	C₃×#427	C₂×C₃	1	1	yes	characteristic
20	C₂×C₃×#106	—	3	1	no	—
21	C₂×#817	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	C₂²×C₃×Dic₃	—	4	1	no	—
25	C₂×C₃²×C₈	C₂×C₄	1	1	yes	characteristic
26	C₂×C₃²×C₈	C₂×C₄	1	1	yes	characteristic
27	C₂×C₃²×C₈	C₂×C₄	1	1	yes	characteristic
28	C₂×C₃²×C₈	D₄	1	1	yes	characteristic
29	C₂×C₃²×C₈	D₄	1	1	yes	characteristic
30	C₂×C₃²×C₈	Q₈	1	1	yes	characteristic
31	C₃²×C₄⋊C₄	—	2	2	no	—
32	C₃²×C₂²⋊C₄	—	4	1	no	—
33	C₃²×C₄²	—	4	1	no	—
34	C₃×C₄⋊Dic₃	—	2	2	no	—
35	C₃×C₄⋊Dic₃	—	4	1	no	—
36	C₃×C₃⋊C₁₆	—	2	2	no	—
37	#1386	—	3	1	no	Sylow
38	C₂²×C₃×C₈	—	2	1	no	—
39	C₂²×C₃×C₈	C₂×D₃	1	1	yes	characteristic
40	C₂²×C₃×C₈	C₂²×C₃	1	1	yes	characteristic
41	C₃×#117	—	2	2	no	—
42	C₃×#117	—	2	1	no	—
43	C₃×#117	—	2	1	no	—
44	C₂×C₃×C₄⋊C₄	—	6	1	no	—
45	C₂×C₄⋊Dic₃	—	2	1	no	—
46	C₃×M₃₂	—	3	2	no	—
47	C₃×M₃₂	—	6	1	no	—
48	C₂×C₃×C₁₆	—	6	1	no	—
49	C₃×#107	—	2	2	no	—
50	C₃×#107	Dic₃	1	2	yes	complemented
51	C₃×#107	C₃×C₄	1	2	yes	complemented
52	C₃×#106	—	2	2	no	—
53	C₃×#106	—	2	1	no	—
54	C₃×#106	—	2	1	no	—
55	C₃×#106	—	3	2	no	—
56	C₃×#106	—	6	1	no	—
57	#817	—	2	1	no	—
58	#817	C₃×C₄	1	2	yes	complemented
59	#812	—	2	1	no	—
60	#812	C₃×C₄	1	2	yes	complemented
61	C₂×C₃⋊C₁₆	—	2	1	no	—
62	C₂³×C₃²	D₈	1	1	yes	characteristic
63	C₂×C₃²×C₄	—	4	2	no	—
64	C₂×C₃²×C₄	C₂²⋊C₄	1	1	yes	characteristic
65	C₂×C₃²×C₄	C₂²⋊C₄	1	1	yes	characteristic
66	C₂×C₃²×C₄	C₄⋊C₄	1	1	yes	characteristic
67	C₂×C₃²×C₄	SD₁₆	1	1	yes	characteristic
68	C₂×C₃²×C₄	SD₁₆	1	1	yes	characteristic
69	C₂×C₃²×C₄	Q₁₆	1	1	yes	characteristic
70	C₂×C₃×Dic₃	—	4	2	no	—
71	C₂×C₃×Dic₃	—	8	2	no	—
72	C₃²×C₈	C₄²	1	1	yes	characteristic
73	C₃²×C₈	C₂²⋊C₄	1	1	yes	characteristic
74	C₃²×C₈	C₄⋊C₄	1	1	yes	characteristic
75	C₃²×C₈	C₄⋊C₄	1	1	yes	characteristic
76	C₂×M₃₂	—	3	1	no	—
77	#427	C₃×D₃	1	1	yes	characteristic
78	C₂×#106	—	3	1	no	—
79	C₂²×C₃×C₄	—	2	1	no	—
80	C₂²×C₃×C₄	—	12	1	no	—
81	C₂²×C₃×C₄	#67	1	1	yes	characteristic
82	C₂²×C₃×C₄	C₃×D₄	1	1	yes	characteristic
83	C₂²×Dic₃	—	4	1	no	—
84	C₂×C₃×C₈	—	2	1	no	—
85	C₂×C₃×C₈	—	2	1	no	—
86	C₂×C₃×C₈	—	2	1	no	—
87	C₂×C₃×C₈	—	2	1	no	—
88	C₂×C₃×C₈	—	2	1	no	—
89	C₂×C₃×C₈	—	2	1	no	—
90	C₂×C₃×C₈	Dic₆	1	1	yes	characteristic
91	C₂×C₃×C₈	C₄×D₃	1	1	yes	characteristic
92	C₂×C₃×C₈	C₄×D₃	1	1	yes	characteristic
93	C₂×C₃×C₈	D₁₂	1	1	yes	characteristic
94	C₂×C₃×C₈	C₂×Dic₃	1	1	yes	characteristic
95	C₂×C₃×C₈	#67	1	1	yes	characteristic
96	C₂×C₃×C₈	C₂×C₃×C₄	1	1	yes	characteristic
97	C₂×C₃×C₈	C₂×C₃×C₄	1	1	yes	characteristic
98	C₂×C₃×C₈	C₂×C₃×C₄	1	1	yes	characteristic
99	C₂×C₃×C₈	C₃×D₄	1	1	yes	characteristic
100	C₂×C₃×C₈	C₃×D₄	1	1	yes	characteristic
101	C₂×C₃×C₈	C₃×Q₈	1	1	yes	characteristic
102	C₃×C₄⋊C₄	—	2	4	no	—
103	C₃×C₄⋊C₄	—	2	2	no	—
104	C₃×C₄⋊C₄	—	2	2	no	—
105	C₃×C₄⋊C₄	—	6	2	no	—
106	C₃×C₄⋊C₄	—	12	1	no	—
107	C₃×C₂²⋊C₄	—	4	2	no	—
108	C₃×C₂²⋊C₄	—	4	1	no	—
109	C₃×C₂²⋊C₄	—	4	1	no	—
110	C₃×C₄²	—	4	2	no	—
111	C₃×C₄²	—	4	1	no	—
112	C₃×C₄²	—	4	1	no	—
113	C₄⋊Dic₃	—	2	2	no	—
114	C₄⋊Dic₃	—	4	1	no	—
115	C₃×C₁₆	—	6	2	no	—
116	C₃⋊C₁₆	—	2	2	no	—
117	C₂²×C₃²	—	2	2	no	—
118	C₂²×C₃²	#102	1	1	yes	characteristic
119	C₂²×C₃²	#102	1	1	yes	characteristic
120	C₂²×C₃²	#107	1	1	yes	characteristic
121	C₃²×C₄	—	8	2	no	—
122	C₃²×C₄	#95	1	1	yes	characteristic
123	C₃²×C₄	#103	1	1	yes	characteristic
124	C₃²×C₄	#103	1	1	yes	characteristic
125	C₃²×C₄	#106	1	1	yes	characteristic
126	C₃×Dic₃	—	8	2	no	—
127	C₂²×C₈	C₂×C₃×D₃	1	1	yes	characteristic
128	#117	—	2	1	no	—
129	C₂×C₄⋊C₄	—	6	1	no	—
130	M₃₂	—	3	2	no	—
131	M₃₂	—	6	1	no	—
132	C₂×C₁₆	—	6	1	no	—
133	#107	C₃×Dic₃	1	2	yes	complemented
134	#106	—	2	1	no	—
135	#106	—	3	2	no	—
136	#106	—	6	1	no	—
137	C₂³×C₃	—	2	1	no	—
138	C₂³×C₃	#213	1	1	yes	characteristic
139	C₂³×C₃	C₃×D₈	1	1	yes	characteristic
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	—	2	1	no	—
144	C₂×C₃×C₄	—	2	1	no	—
145	C₂×C₃×C₄	—	2	1	no	—
146	C₂×C₃×C₄	—	4	4	no	—
147	C₂×C₃×C₄	—	4	2	no	—
148	C₂×C₃×C₄	—	4	2	no	—
149	C₂×C₃×C₄	—	12	2	no	—
150	C₂×C₃×C₄	—	24	2	no	—
151	C₂×C₃×C₄	Dic₃⋊C₄	1	1	yes	characteristic
152	C₂×C₃×C₄	(C₂×D₃)⋊C₄	1	1	yes	characteristic
153	C₂×C₃×C₄	#214	1	1	yes	characteristic
154	C₂×C₃×C₄	#215	1	1	yes	characteristic
155	C₂×C₃×C₄	#216	1	1	yes	characteristic
156	C₂×C₃×C₄	#217	1	1	yes	characteristic
157	C₂×C₃×C₄	C₃×C₂²⋊C₄	1	1	yes	characteristic
158	C₂×C₃×C₄	C₃×C₂²⋊C₄	1	1	yes	characteristic
159	C₂×C₃×C₄	C₃×C₄⋊C₄	1	1	yes	characteristic
160	C₂×C₃×C₄	C₃×SD₁₆	1	1	yes	characteristic
161	C₂×C₃×C₄	C₃×SD₁₆	1	1	yes	characteristic
162	C₂×C₃×C₄	C₃×Q₁₆	1	1	yes	characteristic
163	C₂×Dic₃	—	4	2	no	—
164	C₂×Dic₃	—	8	2	no	—
165	C₃×C₈	—	2	1	no	—
166	C₃×C₈	—	2	1	no	—
167	C₃×C₈	—	2	1	no	—
168	C₃×C₈	—	2	1	no	—
169	C₃×C₈	C₄×Dic₃	1	1	yes	characteristic
170	C₃×C₈	Dic₃⋊C₄	1	1	yes	characteristic
171	C₃×C₈	C₄⋊Dic₃	1	1	yes	characteristic
172	C₃×C₈	(C₂×D₃)⋊C₄	1	1	yes	characteristic
173	C₃×C₈	C₃×C₄²	1	1	yes	characteristic
174	C₃×C₈	C₃×C₂²⋊C₄	1	1	yes	characteristic
175	C₃×C₈	C₃×C₄⋊C₄	1	1	yes	characteristic
176	C₃×C₈	C₃×C₄⋊C₄	1	1	yes	characteristic
177	C₂×C₃²	—	2	1	no	—
178	C₂×C₃²	—	2	1	no	—
179	C₂×C₃²	#340	1	1	yes	characteristic
180	C₂×C₃²	#360	1	2	yes	—
181	C₂²×C₄	—	12	1	no	—
182	C₂²×C₄	C₃×#67	1	1	yes	characteristic
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	C₄⋊C₄	—	2	2	no	—
190	C₄⋊C₄	—	6	2	no	—
191	C₄⋊C₄	—	12	1	no	—
192	C₂²⋊C₄	—	4	1	no	—
193	C₄²	—	4	1	no	—
194	C₁₆	—	6	2	no	—
195	C₂²×C₃	—	2	2	no	—
196	C₂²×C₃	—	2	2	no	—
197	C₂²×C₃	—	2	1	no	—
198	C₂²×C₃	—	2	1	no	—
199	C₂²×C₃	—	2	1	no	—
200	C₂²×C₃	—	4	2	no	—
201	C₂²×C₃	#807	1	1	yes	characteristic
202	C₂²×C₃	#809	1	1	yes	characteristic
203	C₂²×C₃	#832	1	1	yes	characteristic
204	C₂²×C₃	C₃×#102	1	1	yes	characteristic
205	C₂²×C₃	C₃×#102	1	1	yes	characteristic
206	C₂²×C₃	C₃×#107	1	1	yes	characteristic
207	C₃×C₄	—	2	1	no	—
208	C₃×C₄	—	2	1	no	—
209	C₃×C₄	—	2	1	no	—
210	C₃×C₄	—	2	1	no	—
211	C₃×C₄	—	8	4	no	—
212	C₃×C₄	—	8	2	no	—
213	C₃×C₄	—	8	2	no	—
214	C₃×C₄	—	24	2	no	—
215	C₃×C₄	#808	1	1	yes	characteristic
216	C₃×C₄	#810	1	1	yes	characteristic
217	C₃×C₄	C₃⋊#95	1	1	yes	characteristic
218	C₃×C₄	#835	1	1	yes	characteristic
219	C₃×C₄	C₃×#95	1	1	yes	characteristic
220	C₃×C₄	C₃×#103	1	1	yes	characteristic
221	C₃×C₄	C₃×#103	1	1	yes	characteristic
222	C₃×C₄	C₃×#106	1	1	yes	characteristic
223	Dic₃	—	8	2	no	—
224	C₃²	#1386	1	1	yes	characteristic, complemented, Sylow
225	C₂³	C₃×#213	1	1	yes	characteristic
226	C₂×C₄	—	4	2	no	—
227	C₂×C₄	—	12	2	no	—
228	C₂×C₄	—	24	2	no	—
229	C₂×C₄	C₃×Dic₃⋊C₄	1	1	yes	characteristic
230	C₂×C₄	C₃×(C₂×D₃)⋊C₄	1	1	yes	characteristic
231	C₂×C₄	C₃×#214	1	1	yes	characteristic
232	C₂×C₄	C₃×#215	1	1	yes	characteristic
233	C₂×C₄	C₃×#216	1	1	yes	characteristic
234	C₂×C₄	C₃×#217	1	1	yes	characteristic
235	C₈	C₃×C₄×Dic₃	1	1	yes	characteristic
236	C₈	C₃×Dic₃⋊C₄	1	1	yes	characteristic
237	C₈	C₃×C₄⋊Dic₃	1	1	yes	characteristic
238	C₈	C₃×(C₂×D₃)⋊C₄	1	1	yes	characteristic
239	C₂×C₃	—	2	1	no	—
240	C₂×C₃	—	2	1	no	—
241	C₂×C₃	—	2	1	no	—
242	C₂×C₃	—	2	1	no	—
243	C₂×C₃	—	2	1	no	—
244	C₂×C₃	—	2	2	no	—
245	C₂×C₃	—	4	1	no	—
246	C₂×C₃	—	4	1	no	—
247	C₂×C₃	#4529	1	1	yes	characteristic
248	C₂×C₃	#4566	1	2	yes	—
249	C₂×C₃	C₃×#340	1	1	yes	characteristic
250	C₂×C₃	C₃×#360	1	2	yes	—
251	C₂²	—	2	2	no	—
252	C₂²	C₃×#807	1	1	yes	characteristic
253	C₂²	C₃×#809	1	1	yes	characteristic
254	C₂²	C₃×#832	1	1	yes	characteristic
255	C₄	—	8	2	no	—
256	C₄	—	24	2	no	—
257	C₄	C₃×#808	1	1	yes	characteristic
258	C₄	C₃×#810	1	1	yes	characteristic
259	C₄	C₃×C₃⋊#95	1	1	yes	characteristic
260	C₄	C₃×#835	1	1	yes	characteristic
261	C₃	—	2	1	no	—
262	C₃	#67726	1	1	yes	characteristic, direct product factor
263	C₃	C₃×#1386	1	1	yes	characteristic, complemented
264	C₂	—	2	1	no	—
265	C₂	—	2	1	no	—
266	C₂	C₃×#4529	1	1	yes	characteristic
267	C₂	C₃×#4566	1	2	yes	—

	subgroup	G/H
center	C₂²×C₃	#832
commutator	C₃×C₈	C₃×C₄²
Frattini	C₂²×C₈	C₂×C₃×D₃
Fitting	C₃²×#427	C₂
socle	C₂²×C₃²	#102
prime radical	C₂³×C₃²	D₈
cosocle	C₂²×C₃×C₈	C₂²×C₃
hypercenter	C₃×#427	D₃
hypocenter	C₃	C₃×#1386
LCS	G ▷ C₃×C₈ ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₃ ◁ C₂²×C₃×C₄ ◁ C₂²×C₃×C₈ ◁ C₃×#427

Complements

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

N	H ≅ G/N	type
5. #67726	C₃	direct (N × H)
9. C₃²×#107	C₄	semidirect
12. C₃×#817	C₄	semidirect
14. C₃×#812	C₄	semidirect
50. C₃×#107	Dic₃	semidirect
51. C₃×#107	C₃×C₄	semidirect
58. #817	C₃×C₄	semidirect
60. #812	C₃×C₄	semidirect
133. #107	C₃×Dic₃	semidirect
224. C₃²	#1386	semidirect
262. C₃	#67726	direct (N × H)
263. C₃	C₃×#1386	semidirect

Maximal subgroups

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

subgroup	index
C₃²×#427	2
C₂×C₃×#817	2
C₂×C₃×#812	2
C₃×#1386	3
#67726	3

Subgroups — C3×#67726 #11772465

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#67726 #11772465