Subgroups — C₃×#68848 #11786045

Order 1152.

489 conjugacy classes of subgroups.

The subgroup lattice has 293 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₈×Dic₃	C₂	1	1	yes	maximal, characteristic
2	C₃×#4977	C₂	1	2	yes	maximal
3	C₃²×#340	C₂	1	1	yes	maximal, characteristic
4	C₃×#4551	C₂	1	1	yes	maximal, characteristic
5	C₃×#4529	C₂	1	2	yes	maximal
6	C₃×#1835	—	3	1	no	maximal
7	#68848	C₃	1	1	yes	maximal, characteristic, direct product factor
8	C₂²×C₃²×C₈	C₂²	1	1	yes	characteristic
9	C₂×C₃²×C₄⋊C₄	C₂²	1	2	yes	—
10	C₂×C₃×C₄⋊Dic₃	C₂²	1	2	yes	—
11	C₂×C₃×C₄×Dic₃	C₂²	1	1	yes	characteristic
12	C₂²×C₃×C₃⋊C₈	C₂²	1	1	yes	characteristic
13	C₃×C₃⋊#95	—	2	2	no	—
14	C₃×C₃⋊#95	—	2	2	no	—
15	C₃×C₈×Dic₃	C₄	1	4	yes	complemented
16	C₂×C₃×C₄×C₈	—	3	1	no	—
17	C₃×#382	—	3	2	no	—
18	C₂×C₈×Dic₃	C₂×C₃	1	1	yes	characteristic
19	#4977	C₂×C₃	1	2	yes	—
20	C₃×#340	—	2	1	no	—
21	C₃×#340	—	3	1	no	—
22	C₃×#340	—	3	2	no	—
23	C₃×#340	D₃	1	1	yes	characteristic
24	C₃×#340	C₂×C₃	1	1	yes	characteristic
25	#4551	C₂×C₃	1	1	yes	characteristic
26	#4529	C₂×C₃	1	2	yes	—
27	C₂²×C₃²×C₄	—	2	2	no	—
28	C₂²×C₃²×C₄	C₂³	1	1	yes	characteristic
29	C₂²×C₃×Dic₃	—	2	2	no	—
30	C₂²×C₃×Dic₃	D₄	1	1	yes	characteristic
31	C₂²×C₃×Dic₃	D₄	1	1	yes	characteristic
32	C₂×C₃²×C₈	—	2	2	no	—
33	C₂×C₃²×C₈	C₂×C₄	1	2	yes	—
34	C₃²×C₄⋊C₄	—	2	2	no	—
35	C₃²×C₄⋊C₄	—	2	2	no	—
36	C₃×C₄⋊Dic₃	—	2	2	no	—
37	C₃×C₄⋊Dic₃	—	2	2	no	—
38	C₃×C₄×Dic₃	C₂×C₄	1	2	yes	—
39	C₃×C₄×Dic₃	Q₈	1	2	yes	—
40	C₂×C₃×C₃⋊C₈	—	2	2	no	—
41	C₂×C₃×C₃⋊C₈	C₂×C₄	1	2	yes	—
42	#1835	—	3	1	no	Sylow
43	C₂²×C₃×C₈	—	2	1	no	—
44	C₂²×C₃×C₈	—	3	1	no	—
45	C₂²×C₃×C₈	C₂×D₃	1	1	yes	characteristic
46	C₂²×C₃×C₈	C₂²×C₃	1	1	yes	characteristic
47	C₂×C₃×C₄⋊C₄	—	2	2	no	—
48	C₂×C₃×C₄⋊C₄	—	3	2	no	—
49	C₂×C₃×C₄⋊C₄	C₂×D₃	1	2	yes	—
50	C₂×C₃×C₄⋊C₄	C₂²×C₃	1	2	yes	—
51	C₂×C₃×C₄²	—	3	1	no	—
52	C₂×C₄⋊Dic₃	C₂²×C₃	1	2	yes	—
53	C₂×C₄×Dic₃	C₂²×C₃	1	1	yes	characteristic
54	C₂²×C₃⋊C₈	C₂²×C₃	1	1	yes	characteristic
55	C₃×C₄×C₈	—	3	4	no	—
56	C₃×#95	—	6	2	no	—
57	C₃×#95	—	6	2	no	—
58	C₃⋊#95	—	2	2	no	—
59	C₃⋊#95	—	2	2	no	—
60	C₈×Dic₃	C₃×C₄	1	4	yes	complemented
61	C₂³×C₃²	C₂×D₄	1	1	yes	characteristic
62	C₂×C₃²×C₄	—	4	2	no	—
63	C₂×C₃²×C₄	—	4	2	no	—
64	C₂×C₃²×C₄	—	4	2	no	—
65	C₂×C₃²×C₄	C₂²×C₄	1	1	yes	characteristic
66	C₂×C₃²×C₄	C₂×Q₈	1	1	yes	characteristic
67	C₂×C₃²×C₄	#41	1	2	yes	—
68	C₂×C₃²×C₄	#41	1	2	yes	—
69	C₂×C₃×Dic₃	—	2	1	no	—
70	C₂×C₃×Dic₃	—	2	2	no	—
71	C₂×C₃×Dic₃	—	2	1	no	—
72	C₂×C₃×Dic₃	—	4	2	no	—
73	C₂×C₃×Dic₃	—	4	2	no	—
74	C₂×C₃×Dic₃	—	4	2	no	—
75	C₂×C₃×Dic₃	C₄⋊C₄	1	2	yes	—
76	C₂×C₃×Dic₃	C₄⋊C₄	1	2	yes	—
77	C₃²×C₈	—	2	2	no	—
78	C₃×C₃⋊C₈	—	2	2	no	—
79	C₂×C₄×C₈	—	3	1	no	—
80	#382	—	3	2	no	—
81	#340	—	3	1	no	—
82	#340	—	3	2	no	—
83	#340	C₃×D₃	1	1	yes	characteristic
84	C₂²×C₃×C₄	—	2	2	no	—
85	C₂²×C₃×C₄	—	2	2	no	—
86	C₂²×C₃×C₄	—	2	1	no	—
87	C₂²×C₃×C₄	—	3	1	no	—
88	C₂²×C₃×C₄	—	3	1	no	—
89	C₂²×C₃×C₄	—	4	2	no	—
90	C₂²×C₃×C₄	—	6	2	no	—
91	C₂²×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
92	C₂²×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
93	C₂²×Dic₃	—	2	2	no	—
94	C₂²×Dic₃	C₃×D₄	1	1	yes	characteristic
95	C₂²×Dic₃	C₃×D₄	1	1	yes	characteristic
96	C₂×C₃×C₈	—	2	2	no	—
97	C₂×C₃×C₈	—	2	2	no	—
98	C₂×C₃×C₈	—	2	2	no	—
99	C₂×C₃×C₈	—	3	2	no	—
100	C₂×C₃×C₈	—	4	2	no	—
101	C₂×C₃×C₈	—	6	2	no	—
102	C₂×C₃×C₈	C₄×D₃	1	2	yes	—
103	C₂×C₃×C₈	C₂×C₃×C₄	1	2	yes	—
104	C₃×C₄⋊C₄	—	2	4	no	—
105	C₃×C₄⋊C₄	—	2	2	no	—
106	C₃×C₄⋊C₄	—	2	2	no	—
107	C₃×C₄⋊C₄	—	2	2	no	—
108	C₃×C₄⋊C₄	—	2	2	no	—
109	C₃×C₄⋊C₄	—	4	2	no	—
110	C₃×C₄⋊C₄	—	6	2	no	—
111	C₃×C₄⋊C₄	—	6	2	no	—
112	C₃×C₄²	—	3	2	no	—
113	C₃×C₄²	—	3	2	no	—
114	C₄⋊Dic₃	—	2	2	no	—
115	C₄⋊Dic₃	—	2	2	no	—
116	C₄×Dic₃	C₂×C₃×C₄	1	2	yes	—
117	C₄×Dic₃	C₃×Q₈	1	2	yes	—
118	C₂×C₃⋊C₈	—	2	2	no	—
119	C₂×C₃⋊C₈	C₂×C₃×C₄	1	2	yes	—
120	C₂²×C₃²	C₂×C₄⋊C₄	1	1	yes	characteristic
121	C₂²×C₃²	#124	1	2	yes	—
122	C₂²×C₃²	#135	1	1	yes	characteristic
123	C₂²×C₃²	#135	1	2	yes	—
124	C₂²×C₃²	#135	1	1	yes	characteristic
125	C₃²×C₄	—	8	2	no	—
126	C₃²×C₄	#117	1	1	yes	characteristic
127	C₃²×C₄	#117	1	1	yes	characteristic
128	C₃²×C₄	#125	1	2	yes	—
129	C₃×Dic₃	—	2	2	no	—
130	C₃×Dic₃	—	2	2	no	—
131	C₃×Dic₃	—	8	2	no	—
132	C₂²×C₈	—	3	1	no	—
133	C₂²×C₈	C₂×C₃×D₃	1	1	yes	characteristic
134	C₂×C₄⋊C₄	—	3	2	no	—
135	C₂×C₄⋊C₄	C₂×C₃×D₃	1	2	yes	—
136	C₂×C₄²	—	3	1	no	—
137	C₄×C₈	—	3	4	no	—
138	#95	—	6	2	no	—
139	#95	—	6	2	no	—
140	C₂³×C₃	—	2	1	no	—
141	C₂³×C₃	D₃×D₄	1	1	yes	characteristic
142	C₂³×C₃	C₂×C₃×D₄	1	1	yes	characteristic
143	C₂×C₃×C₄	—	2	1	no	—
144	C₂×C₃×C₄	—	2	2	no	—
145	C₂×C₃×C₄	—	2	2	no	—
146	C₂×C₃×C₄	—	2	1	no	—
147	C₂×C₃×C₄	—	3	2	no	—
148	C₂×C₃×C₄	—	3	2	no	—
149	C₂×C₃×C₄	—	4	4	no	—
150	C₂×C₃×C₄	—	4	2	no	—
151	C₂×C₃×C₄	—	4	2	no	—
152	C₂×C₃×C₄	—	4	2	no	—
153	C₂×C₃×C₄	—	4	2	no	—
154	C₂×C₃×C₄	—	4	2	no	—
155	C₂×C₃×C₄	—	4	2	no	—
156	C₂×C₃×C₄	—	6	1	no	—
157	C₂×C₃×C₄	—	6	2	no	—
158	C₂×C₃×C₄	—	6	1	no	—
159	C₂×C₃×C₄	—	8	2	no	—
160	C₂×C₃×C₄	—	8	2	no	—
161	C₂×C₃×C₄	—	12	2	no	—
162	C₂×C₃×C₄	—	12	2	no	—
163	C₂×C₃×C₄	—	12	2	no	—
164	C₂×C₃×C₄	C₂×C₄×D₃	1	1	yes	characteristic
165	C₂×C₃×C₄	#235	1	2	yes	—
166	C₂×C₃×C₄	#237	1	2	yes	—
167	C₂×C₃×C₄	D₃×Q₈	1	1	yes	characteristic
168	C₂×C₃×C₄	C₂²×C₃×C₄	1	1	yes	characteristic
169	C₂×C₃×C₄	C₂×C₃×Q₈	1	1	yes	characteristic
170	C₂×C₃×C₄	C₃×#41	1	2	yes	—
171	C₂×C₃×C₄	C₃×#41	1	2	yes	—
172	C₂×Dic₃	—	2	1	no	—
173	C₂×Dic₃	—	2	2	no	—
174	C₂×Dic₃	—	2	1	no	—
175	C₂×Dic₃	—	4	2	no	—
176	C₂×Dic₃	—	4	2	no	—
177	C₂×Dic₃	—	4	2	no	—
178	C₂×Dic₃	C₃×C₄⋊C₄	1	2	yes	—
179	C₂×Dic₃	C₃×C₄⋊C₄	1	2	yes	—
180	C₃×C₈	—	2	2	no	—
181	C₃×C₈	—	2	2	no	—
182	C₃×C₈	—	4	2	no	—
183	C₃×C₈	—	6	2	no	—
184	C₃⋊C₈	—	2	2	no	—
185	C₂×C₃²	#382	1	1	yes	characteristic
186	C₂×C₃²	#427	1	1	yes	characteristic
187	C₂×C₃²	#427	1	1	yes	characteristic
188	C₂×C₃²	#488	1	2	yes	—
189	C₂×C₃²	#488	1	2	yes	—
190	C₂²×C₄	—	2	2	no	—
191	C₂²×C₄	—	3	1	no	—
192	C₂²×C₄	—	3	1	no	—
193	C₂²×C₄	—	6	2	no	—
194	C₂²×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
195	C₂×C₈	—	2	2	no	—
196	C₂×C₈	—	3	2	no	—
197	C₂×C₈	—	6	2	no	—
198	C₂×C₈	C₃×C₄×D₃	1	2	yes	—
199	C₄⋊C₄	—	2	2	no	—
200	C₄⋊C₄	—	2	2	no	—
201	C₄⋊C₄	—	6	2	no	—
202	C₄⋊C₄	—	6	2	no	—
203	C₄²	—	3	2	no	—
204	C₄²	—	3	2	no	—
205	C₂²×C₃	—	2	2	no	—
206	C₂²×C₃	—	2	1	no	—
207	C₂²×C₃	—	2	1	no	—
208	C₂²×C₃	—	2	2	no	—
209	C₂²×C₃	—	2	1	no	—
210	C₂²×C₃	#885	1	2	yes	—
211	C₂²×C₃	D₃×C₄⋊C₄	1	1	yes	characteristic
212	C₂²×C₃	#912	1	1	yes	characteristic
213	C₂²×C₃	#916	1	2	yes	—
214	C₂²×C₃	#919	1	1	yes	characteristic
215	C₂²×C₃	C₂×C₃×C₄⋊C₄	1	1	yes	characteristic
216	C₂²×C₃	C₃×#124	1	2	yes	—
217	C₂²×C₃	C₃×#135	1	1	yes	characteristic
218	C₂²×C₃	C₃×#135	1	2	yes	—
219	C₂²×C₃	C₃×#135	1	1	yes	characteristic
220	C₃×C₄	—	2	1	no	—
221	C₃×C₄	—	2	2	no	—
222	C₃×C₄	—	2	1	no	—
223	C₃×C₄	—	6	2	no	—
224	C₃×C₄	—	6	2	no	—
225	C₃×C₄	—	8	4	no	—
226	C₃×C₄	—	8	2	no	—
227	C₃×C₄	—	8	2	no	—
228	C₃×C₄	—	24	2	no	—
229	C₃×C₄	#872	1	1	yes	characteristic
230	C₃×C₄	#877	1	1	yes	characteristic
231	C₃×C₄	#889	1	2	yes	—
232	C₃×C₄	C₃×#117	1	1	yes	characteristic
233	C₃×C₄	C₃×#117	1	1	yes	characteristic
234	C₃×C₄	C₃×#125	1	2	yes	—
235	Dic₃	—	2	2	no	—
236	Dic₃	—	2	2	no	—
237	Dic₃	—	8	2	no	—
238	C₃²	#1835	1	1	yes	characteristic, complemented, Sylow
239	C₂³	C₃×D₃×D₄	1	1	yes	characteristic
240	C₂×C₄	—	3	2	no	—
241	C₂×C₄	—	3	2	no	—
242	C₂×C₄	—	4	2	no	—
243	C₂×C₄	—	4	2	no	—
244	C₂×C₄	—	4	2	no	—
245	C₂×C₄	—	6	1	no	—
246	C₂×C₄	—	6	2	no	—
247	C₂×C₄	—	6	1	no	—
248	C₂×C₄	—	12	2	no	—
249	C₂×C₄	—	12	2	no	—
250	C₂×C₄	—	12	2	no	—
251	C₂×C₄	C₂×C₃×C₄×D₃	1	1	yes	characteristic
252	C₂×C₄	C₃×#235	1	2	yes	—
253	C₂×C₄	C₃×#237	1	2	yes	—
254	C₂×C₄	C₃×D₃×Q₈	1	1	yes	characteristic
255	C₈	—	2	2	no	—
256	C₈	—	6	2	no	—
257	C₂×C₃	—	2	1	no	—
258	C₂×C₃	—	2	2	no	—
259	C₂×C₃	—	2	2	no	—
260	C₂×C₃	—	2	1	no	—
261	C₂×C₃	—	2	1	no	—
262	C₂×C₃	#4650	1	1	yes	characteristic
263	C₂×C₃	#4768	1	2	yes	—
264	C₂×C₃	#4800	1	2	yes	—
265	C₂×C₃	#4860	1	1	yes	characteristic
266	C₂×C₃	#4880	1	1	yes	characteristic
267	C₂×C₃	C₃×#382	1	1	yes	characteristic
268	C₂×C₃	C₃×#427	1	1	yes	characteristic
269	C₂×C₃	C₃×#427	1	1	yes	characteristic
270	C₂×C₃	C₃×#488	1	2	yes	—
271	C₂×C₃	C₃×#488	1	2	yes	—
272	C₂²	C₃×#885	1	2	yes	—
273	C₂²	C₃×D₃×C₄⋊C₄	1	1	yes	characteristic
274	C₂²	C₃×#912	1	1	yes	characteristic
275	C₂²	C₃×#916	1	2	yes	—
276	C₂²	C₃×#919	1	1	yes	characteristic
277	C₄	—	6	2	no	—
278	C₄	—	6	2	no	—
279	C₄	—	8	2	no	—
280	C₄	—	24	2	no	—
281	C₄	C₃×#872	1	1	yes	characteristic
282	C₄	C₃×#877	1	1	yes	characteristic
283	C₄	C₃×#889	1	2	yes	—
284	C₃	—	2	1	no	—
285	C₃	#68848	1	1	yes	characteristic, direct product factor
286	C₃	C₃×#1835	1	1	yes	characteristic, complemented
287	C₂	C₃×#4650	1	1	yes	characteristic
288	C₂	C₃×#4768	1	2	yes	—
289	C₂	C₃×#4800	1	2	yes	—
290	C₂	C₃×#4860	1	1	yes	characteristic
291	C₂	C₃×#4880	1	1	yes	characteristic

	subgroup	G/H
center	C₂³×C₃	D₃×D₄
commutator	C₂×C₃×C₄	C₂²×C₃×C₄
Frattini	C₂²×C₄	C₂²×C₃×D₃
Fitting	C₃²×#340	C₂
socle, prime radical	C₂³×C₃²	C₂×D₄
cosocle	C₂²×C₃×C₄	C₂³×C₃
hypercenter	C₃×#340	D₃
hypocenter	C₃	C₃×#1835
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂³×C₃ ◁ C₂²×C₃×C₄ ◁ C₃×#340

Complements

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

N	H ≅ G/N	type
7. #68848	C₃	direct (N × H)
15. C₃×C₈×Dic₃	C₄	semidirect
60. C₈×Dic₃	C₃×C₄	semidirect
238. C₃²	#1835	semidirect
285. C₃	#68848	direct (N × H)
286. C₃	C₃×#1835	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃×C₈×Dic₃	2
C₃×#4977	2
C₃²×#340	2
C₃×#4551	2
C₃×#4529	2
C₃×#1835	3
#68848	3

Subgroups — C3×#68848 #11786045

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#68848 #11786045