Subgroups — C₃×#78467 #11874288

Order 1152.

1225 conjugacy classes of subgroups.

The subgroup lattice has 326 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×#904	C₂	1	4	yes	maximal, complemented
2	C₃×#5477	C₂	1	2	yes	maximal, complemented
3	C₂×C₃²×C₄×C₈	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×#4699	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×#4696	C₂	1	4	yes	maximal, complemented
6	C₃×#4695	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃×#4693	C₂	1	2	yes	maximal, complemented
8	C₃×#3147	—	3	1	no	maximal
9	#78467	C₃	1	1	yes	maximal, characteristic, direct product factor
10	C₂×C₃×#235	C₂²	1	4	yes	—
11	C₂²×C₃²×C₈	C₂²	1	2	yes	—
12	C₂×C₃²×C₄²	C₂²	1	1	yes	characteristic
13	C₃×#930	—	2	4	no	—
14	C₂×C₃×Dic₁₂	C₂²	1	4	yes	—
15	C₃×#904	—	2	16	no	—
16	C₂×C₃×D₂₄	C₂²	1	4	yes	—
17	C₂×C₃×#204	C₂²	1	8	yes	—
18	C₃×#875	C₂²	1	4	yes	—
19	C₃×#874	C₂²	1	2	yes	—
20	C₃×C₄×D₁₂	—	2	4	no	—
21	C₃×#869	C₂²	1	2	yes	—
22	C₃²×C₄×C₈	C₂²	1	1	yes	characteristic
23	C₃²×C₄×C₈	C₂²	1	1	yes	characteristic
24	C₃²×C₄×C₈	C₂²	1	2	yes	—
25	C₂×C₃×#135	—	3	4	no	—
26	C₃×#532	—	3	2	no	—
27	C₂×#904	C₂×C₃	1	4	yes	complemented
28	#5477	C₂×C₃	1	2	yes	complemented
29	C₃×#495	—	3	4	no	—
30	C₃×#494	—	3	1	no	—
31	C₃×#493	—	3	1	no	—
32	C₃×#492	—	3	2	no	—
33	C₂×C₃×C₄×C₈	—	2	1	no	—
34	C₂×C₃×C₄×C₈	D₃	1	1	yes	characteristic, complemented
35	C₂×C₃×C₄×C₈	C₂×C₃	1	1	yes	characteristic, complemented
36	#4699	C₂×C₃	1	1	yes	characteristic, complemented
37	#4696	C₂×C₃	1	4	yes	complemented
38	#4695	C₂×C₃	1	1	yes	characteristic, complemented
39	#4693	C₂×C₃	1	2	yes	complemented
40	C₂²×C₃²×C₄	D₄	1	2	yes	—
41	C₂²×C₃²×C₄	C₂³	1	1	yes	characteristic
42	C₂×C₃×#67	—	2	4	no	—
43	C₃×#235	—	2	16	no	—
44	C₂×C₃×D₁₂	—	2	4	no	—
45	C₂×C₃×D₁₂	C₂³	1	4	yes	—
46	C₂×C₃×C₄×D₃	—	2	4	no	—
47	C₂×C₃×Dic₆	C₂³	1	4	yes	—
48	C₂×C₃²×C₈	D₄	1	8	yes	complemented
49	C₂×C₃²×C₈	C₂³	1	2	yes	—
50	C₂×C₃²×C₈	C₂³	1	2	yes	—
51	C₃²×C₄²	D₄	1	2	yes	—
52	C₃²×C₄²	C₂³	1	1	yes	characteristic
53	C₃²×C₄²	C₂³	1	1	yes	characteristic
54	C₃×(C₂×D₃)⋊C₄	—	2	8	no	—
55	C₃×C₄⋊Dic₃	—	2	4	no	—
56	C₃×Dic₁₂	—	2	8	no	—
57	C₃×D₂₄	—	2	8	no	—
58	C₃×#204	—	2	16	no	—
59	#3147	—	3	1	no	Sylow
60	C₂×C₃×#41	—	3	4	no	—
61	C₂×#235	C₂²×C₃	1	4	yes	—
62	C₃×#135	—	6	16	no	—
63	C₂×C₃×Q₁₆	—	3	4	no	—
64	C₂×C₃×SD₁₆	—	3	8	no	—
65	C₂×C₃×D₈	—	3	4	no	—
66	C₂²×C₃×C₈	—	2	2	no	—
67	C₂²×C₃×C₈	C₂×D₃	1	2	yes	—
68	C₂²×C₃×C₈	C₂²×C₃	1	2	yes	—
69	C₃×C₄⋊Q₈	—	3	2	no	—
70	C₃×#127	—	3	2	no	—
71	C₃×#124	—	3	4	no	—
72	C₃×#121	—	6	4	no	—
73	C₃×C₄×D₄	—	6	4	no	—
74	C₂×C₃×C₄²	—	2	1	no	—
75	C₂×C₃×C₄²	C₂×D₃	1	1	yes	characteristic
76	C₂×C₃×C₄²	C₂²×C₃	1	1	yes	characteristic
77	#930	—	2	4	no	—
78	C₂×Dic₁₂	C₂²×C₃	1	4	yes	—
79	#904	—	2	16	no	—
80	C₂×D₂₄	C₂²×C₃	1	4	yes	—
81	C₂×#204	C₂²×C₃	1	8	yes	—
82	#875	C₂²×C₃	1	4	yes	—
83	#874	C₂²×C₃	1	2	yes	—
84	C₄×D₁₂	—	2	4	no	—
85	#869	C₂²×C₃	1	2	yes	—
86	C₃×C₄×C₈	—	2	1	no	—
87	C₃×C₄×C₈	—	2	1	no	—
88	C₃×C₄×C₈	—	2	2	no	—
89	C₃×C₄×C₈	C₂×D₃	1	1	yes	characteristic
90	C₃×C₄×C₈	C₂×D₃	1	1	yes	characteristic
91	C₃×C₄×C₈	C₂×D₃	1	2	yes	—
92	C₃×C₄×C₈	C₂²×C₃	1	1	yes	characteristic
93	C₃×C₄×C₈	C₂²×C₃	1	1	yes	characteristic
94	C₃×C₄×C₈	C₂²×C₃	1	2	yes	—
95	C₂³×C₃²	C₂×D₄	1	1	yes	characteristic
96	C₂²×C₃×D₃	—	2	4	no	—
97	C₂×C₃²×C₄	—	2	4	no	—
98	C₂×C₃²×C₄	C₂×D₄	1	2	yes	—
99	C₂×C₃²×C₄	C₂×D₄	1	4	yes	—
100	C₂×C₃²×C₄	C₂×D₄	1	2	yes	—
101	C₂×C₃²×C₄	C₂×D₄	1	1	yes	characteristic
102	C₂×C₃²×C₄	C₂⁴	1	1	yes	characteristic
103	C₃×#67	—	4	8	no	—
104	C₂×C₃×Dic₃	—	2	4	no	—
105	C₃×D₁₂	—	2	8	no	—
106	C₃×D₁₂	—	4	8	no	—
107	C₃×C₄×D₃	—	4	8	no	—
108	C₃×Dic₆	—	2	8	no	—
109	C₃²×C₈	C₂×D₄	1	4	yes	—
110	C₃²×C₈	C₂×D₄	1	4	yes	—
111	C₂×#135	—	3	4	no	—
112	#532	—	3	2	no	—
113	#495	—	3	4	no	—
114	#494	—	3	1	no	—
115	#493	—	3	1	no	—
116	#492	—	3	2	no	—
117	C₂×C₄×C₈	C₃×D₃	1	1	yes	characteristic, complemented
118	C₃×#41	—	6	16	no	—
119	C₂×C₃×Q₈	—	3	4	no	—
120	C₂×C₃×D₄	—	3	4	no	—
121	C₂×C₃×D₄	—	6	4	no	—
122	C₂×C₃×D₄	—	6	4	no	—
123	C₂²×C₃×C₄	—	2	2	no	—
124	C₂²×C₃×C₄	—	2	1	no	—
125	C₂²×C₃×C₄	—	6	4	no	—
126	C₂²×C₃×C₄	D₁₂	1	2	yes	—
127	C₂²×C₃×C₄	C₃×D₄	1	2	yes	—
128	C₂²×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
129	C₂²×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
130	C₂×#67	—	2	4	no	—
131	#235	—	2	16	no	—
132	C₂×D₁₂	—	2	4	no	—
133	C₂×D₁₂	C₂³×C₃	1	4	yes	—
134	C₂×C₄×D₃	—	2	4	no	—
135	C₂×Dic₆	C₂³×C₃	1	4	yes	—
136	C₃×Q₁₆	—	6	8	no	—
137	C₃×SD₁₆	—	6	16	no	—
138	C₃×D₈	—	6	8	no	—
139	C₂×C₃×C₈	—	2	8	no	—
140	C₂×C₃×C₈	—	2	2	no	—
141	C₂×C₃×C₈	—	2	2	no	—
142	C₂×C₃×C₈	D₁₂	1	8	yes	complemented
143	C₂×C₃×C₈	C₃×D₄	1	8	yes	complemented
144	C₂×C₃×C₈	C₂²×D₃	1	2	yes	—
145	C₂×C₃×C₈	C₂²×D₃	1	2	yes	—
146	C₂×C₃×C₈	C₂³×C₃	1	2	yes	—
147	C₂×C₃×C₈	C₂³×C₃	1	2	yes	—
148	C₃×C₄⋊C₄	—	6	4	no	—
149	C₃×C₂²⋊C₄	—	6	8	no	—
150	C₃×C₄²	—	2	2	no	—
151	C₃×C₄²	—	2	1	no	—
152	C₃×C₄²	—	2	1	no	—
153	C₃×C₄²	D₁₂	1	2	yes	—
154	C₃×C₄²	C₃×D₄	1	2	yes	—
155	C₃×C₄²	C₂²×D₃	1	1	yes	characteristic
156	C₃×C₄²	C₂²×D₃	1	1	yes	characteristic
157	C₃×C₄²	C₂³×C₃	1	1	yes	characteristic
158	C₃×C₄²	C₂³×C₃	1	1	yes	characteristic
159	(C₂×D₃)⋊C₄	—	2	8	no	—
160	C₄⋊Dic₃	—	2	4	no	—
161	Dic₁₂	—	2	8	no	—
162	D₂₄	—	2	8	no	—
163	#204	—	2	16	no	—
164	C₂²×C₃²	—	2	2	no	—
165	C₂²×C₃²	#127	1	2	yes	—
166	C₂²×C₃²	C₂²×D₄	1	1	yes	characteristic
167	C₂×C₃×D₃	—	4	4	no	—
168	C₂×C₃×D₃	—	4	8	no	—
169	C₃²×C₄	—	2	2	no	—
170	C₃²×C₄	#127	1	2	yes	—
171	C₃²×C₄	#135	1	4	yes	complemented
172	C₃²×C₄	C₂²×D₄	1	2	yes	—
173	C₃×Dic₃	—	4	4	no	—
174	C₂×#41	—	3	4	no	—
175	#135	—	6	16	no	—
176	C₂×Q₁₆	—	3	4	no	—
177	C₂×SD₁₆	—	3	8	no	—
178	C₂×D₈	—	3	4	no	—
179	C₂²×C₈	C₂×C₃×D₃	1	2	yes	—
180	C₄⋊Q₈	—	3	2	no	—
181	#127	—	3	2	no	—
182	#124	—	3	4	no	—
183	#121	—	6	4	no	—
184	C₄×D₄	—	6	4	no	—
185	C₂×C₄²	C₂×C₃×D₃	1	1	yes	characteristic
186	C₄×C₈	C₂×C₃×D₃	1	1	yes	characteristic
187	C₄×C₈	C₂×C₃×D₃	1	1	yes	characteristic
188	C₄×C₈	C₂×C₃×D₃	1	2	yes	—
189	C₂³×C₃	—	2	1	no	—
190	C₂³×C₃	—	6	4	no	—
191	C₂³×C₃	C₂×D₁₂	1	1	yes	characteristic
192	C₂³×C₃	C₂×C₃×D₄	1	1	yes	characteristic
193	C₂²×D₃	—	2	4	no	—
194	C₃×Q₈	—	6	8	no	—
195	C₃×D₄	—	6	8	no	—
196	C₃×D₄	—	12	8	no	—
197	C₃×D₄	—	12	8	no	—
198	C₂×C₃×C₄	—	2	8	no	—
199	C₂×C₃×C₄	—	2	4	no	—
200	C₂×C₃×C₄	—	2	4	no	—
201	C₂×C₃×C₄	—	2	1	no	—
202	C₂×C₃×C₄	—	2	2	no	—
203	C₂×C₃×C₄	—	2	2	no	—
204	C₂×C₃×C₄	—	2	1	no	—
205	C₂×C₃×C₄	—	2	4	no	—
206	C₂×C₃×C₄	—	6	4	no	—
207	C₂×C₃×C₄	—	12	8	no	—
208	C₂×C₃×C₄	C₂×D₁₂	1	2	yes	—
209	C₂×C₃×C₄	C₂×D₁₂	1	4	yes	—
210	C₂×C₃×C₄	C₂×D₁₂	1	2	yes	—
211	C₂×C₃×C₄	C₂×D₁₂	1	1	yes	characteristic
212	C₂×C₃×C₄	C₂×C₃×D₄	1	2	yes	—
213	C₂×C₃×C₄	C₂×C₃×D₄	1	1	yes	characteristic
214	C₂×C₃×C₄	C₂×C₃×D₄	1	4	yes	—
215	C₂×C₃×C₄	C₂×C₃×D₄	1	2	yes	—
216	C₂×C₃×C₄	C₂³×D₃	1	1	yes	characteristic
217	C₂×C₃×C₄	C₂⁴×C₃	1	1	yes	characteristic
218	#67	—	4	8	no	—
219	C₂×Dic₃	—	2	4	no	—
220	D₁₂	—	2	8	no	—
221	D₁₂	—	4	8	no	—
222	C₄×D₃	—	4	8	no	—
223	Dic₆	—	2	8	no	—
224	C₃×C₈	—	2	4	no	—
225	C₃×C₈	—	2	4	no	—
226	C₃×C₈	C₂×D₁₂	1	4	yes	—
227	C₃×C₈	C₂×D₁₂	1	4	yes	—
228	C₃×C₈	C₂×C₃×D₄	1	4	yes	—
229	C₃×C₈	C₂×C₃×D₄	1	4	yes	—
230	C₂×C₃²	—	2	2	no	—
231	C₂×C₃²	C₂×#127	1	1	yes	characteristic
232	C₂×C₃²	C₂×#135	1	2	yes	—
233	C₃×D₃	—	8	4	no	—
234	#41	—	6	16	no	—
235	C₂×Q₈	—	3	4	no	—
236	C₂×D₄	—	3	4	no	—
237	C₂×D₄	—	6	4	no	—
238	C₂×D₄	—	6	4	no	—
239	C₂²×C₄	—	6	4	no	—
240	C₂²×C₄	C₃×D₁₂	1	2	yes	—
241	C₂²×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
242	Q₁₆	—	6	8	no	—
243	SD₁₆	—	6	16	no	—
244	D₈	—	6	8	no	—
245	C₂×C₈	C₃×D₁₂	1	8	yes	complemented
246	C₂×C₈	C₂²×C₃×D₃	1	2	yes	—
247	C₂×C₈	C₂²×C₃×D₃	1	2	yes	—
248	C₄⋊C₄	—	6	4	no	—
249	C₂²⋊C₄	—	6	8	no	—
250	C₄²	C₃×D₁₂	1	2	yes	—
251	C₄²	C₂²×C₃×D₃	1	1	yes	characteristic
252	C₄²	C₂²×C₃×D₃	1	1	yes	characteristic
253	C₂²×C₃	—	2	4	no	—
254	C₂²×C₃	—	2	2	no	—
255	C₂²×C₃	—	2	2	no	—
256	C₂²×C₃	—	2	1	no	—
257	C₂²×C₃	—	2	2	no	—
258	C₂²×C₃	—	12	4	no	—
259	C₂²×C₃	—	12	8	no	—
260	C₂²×C₃	#874	1	2	yes	—
261	C₂²×C₃	C₃×#127	1	2	yes	—
262	C₂²×C₃	C₂²×D₁₂	1	1	yes	characteristic
263	C₂²×C₃	C₂²×C₃×D₄	1	1	yes	characteristic
264	C₂×D₃	—	4	8	no	—
265	C₂×D₃	—	4	4	no	—
266	C₃×C₄	—	2	4	no	—
267	C₃×C₄	—	2	2	no	—
268	C₃×C₄	—	2	2	no	—
269	C₃×C₄	—	2	4	no	—
270	C₃×C₄	—	2	2	no	—
271	C₃×C₄	—	2	2	no	—
272	C₃×C₄	—	12	4	no	—
273	C₃×C₄	#874	1	2	yes	—
274	C₃×C₄	#904	1	4	yes	complemented
275	C₃×C₄	C₃×#127	1	2	yes	—
276	C₃×C₄	C₃×#135	1	4	yes	complemented
277	C₃×C₄	C₂²×D₁₂	1	2	yes	—
278	C₃×C₄	C₂²×C₃×D₄	1	2	yes	—
279	Dic₃	—	4	4	no	—
280	C₃²	#3147	1	1	yes	characteristic, complemented, Sylow
281	C₂³	—	6	4	no	—
282	C₂³	C₂×C₃×D₁₂	1	1	yes	characteristic
283	Q₈	—	6	8	no	—
284	D₄	—	6	8	no	—
285	D₄	—	12	8	no	—
286	D₄	—	12	8	no	—
287	C₂×C₄	—	2	4	no	—
288	C₂×C₄	—	6	4	no	—
289	C₂×C₄	—	12	8	no	—
290	C₂×C₄	C₂×C₃×D₁₂	1	2	yes	—
291	C₂×C₄	C₂×C₃×D₁₂	1	2	yes	—
292	C₂×C₄	C₂×C₃×D₁₂	1	4	yes	—
293	C₂×C₄	C₂×C₃×D₁₂	1	1	yes	characteristic
294	C₂×C₄	C₂³×C₃×D₃	1	1	yes	characteristic
295	C₈	C₂×C₃×D₁₂	1	4	yes	—
296	C₈	C₂×C₃×D₁₂	1	4	yes	—
297	C₂×C₃	—	2	4	no	—
298	C₂×C₃	—	2	2	no	—
299	C₂×C₃	—	2	2	no	—
300	C₂×C₃	—	2	1	no	—
301	C₂×C₃	—	2	2	no	—
302	C₂×C₃	—	24	4	no	—
303	C₂×C₃	C₂×#874	1	1	yes	characteristic
304	C₂×C₃	C₂×#904	1	2	yes	—
305	C₂×C₃	C₂×C₃×#127	1	1	yes	characteristic
306	C₂×C₃	C₂×C₃×#135	1	2	yes	—
307	D₃	—	8	4	no	—
308	C₂²	—	2	2	no	—
309	C₂²	—	12	4	no	—
310	C₂²	—	12	8	no	—
311	C₂²	C₃×#874	1	2	yes	—
312	C₂²	C₂²×C₃×D₁₂	1	1	yes	characteristic
313	C₄	—	2	2	no	—
314	C₄	—	12	4	no	—
315	C₄	C₃×#874	1	2	yes	—
316	C₄	C₃×#904	1	4	yes	complemented
317	C₄	C₂²×C₃×D₁₂	1	2	yes	—
318	C₃	—	2	1	no	—
319	C₃	#78467	1	1	yes	characteristic, direct product factor
320	C₃	C₃×#3147	1	1	yes	characteristic, complemented
321	C₂	—	2	2	no	—
322	C₂	—	24	4	no	—
323	C₂	C₂×C₃×#874	1	1	yes	characteristic
324	C₂	C₂×C₃×#904	1	2	yes	—

	subgroup	G/H
center	C₂×C₃×C₄	C₂×D₁₂
commutator, cosocle	C₂×C₃×C₄	C₂⁴×C₃
Frattini	C₂×C₄	C₂³×C₃×D₃
Fitting	C₂×C₃²×C₄×C₈	C₂
socle	C₂²×C₃²	C₂²×D₄
hypercenter	C₂×C₃×C₄×C₈	D₃
hypocenter	C₃	C₃×#3147
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₃×C₄ ◁ C₂×C₃×C₄² ◁ C₂×C₃×C₄×C₈

Complements

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

N	H ≅ G/N	type
1. C₂×C₃×#904	C₂	semidirect
2. C₃×#5477	C₂	semidirect
3. C₂×C₃²×C₄×C₈	C₂	semidirect
4. C₃×#4699	C₂	semidirect
5. C₃×#4696	C₂	semidirect
6. C₃×#4695	C₂	semidirect
7. C₃×#4693	C₂	semidirect
9. #78467	C₃	direct (N × H)
27. C₂×#904	C₂×C₃	semidirect
28. #5477	C₂×C₃	semidirect
34. C₂×C₃×C₄×C₈	D₃	semidirect
35. C₂×C₃×C₄×C₈	C₂×C₃	semidirect
36. #4699	C₂×C₃	semidirect
37. #4696	C₂×C₃	semidirect
38. #4695	C₂×C₃	semidirect
39. #4693	C₂×C₃	semidirect
48. C₂×C₃²×C₈	D₄	semidirect
117. C₂×C₄×C₈	C₃×D₃	semidirect
142. C₂×C₃×C₈	D₁₂	semidirect
143. C₂×C₃×C₈	C₃×D₄	semidirect
171. C₃²×C₄	#135	semidirect
245. C₂×C₈	C₃×D₁₂	semidirect
274. C₃×C₄	#904	semidirect
276. C₃×C₄	C₃×#135	semidirect
280. C₃²	#3147	semidirect
316. C₄	C₃×#904	semidirect
319. C₃	#78467	direct (N × H)
320. C₃	C₃×#3147	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃×#904	2
C₃×#5477	2
C₂×C₃²×C₄×C₈	2
C₃×#4699	2
C₃×#4696	2
C₃×#4695	2
C₃×#4693	2
C₃×#3147	3
#78467	3

Subgroups — C3×#78467 #11874288

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#78467 #11874288