Subgroups — C₃×#78499 #11874427

Order 1152.

1201 conjugacy classes of subgroups.

The subgroup lattice has 283 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×#97	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×#4715	C₂	1	4	yes	maximal, complemented
5	C₃×#4712	C₂	1	4	yes	maximal, complemented
6	C₃×#3150	—	3	1	no	maximal
7	#78499	C₃	1	1	yes	maximal, characteristic, direct product factor
8	C₂×C₃×#235	C₂²	1	4	yes	—
9	C₂²×C₃²×C₈	C₂²	1	2	yes	—
10	C₂×C₃²×C₄²	C₂²	1	1	yes	characteristic
11	C₃×#930	—	2	4	no	—
12	C₂×C₃×Dic₁₂	C₂²	1	4	yes	—
13	C₃×#904	—	2	16	no	—
14	C₂×C₃×D₂₄	C₂²	1	4	yes	—
15	C₂×C₃×#204	C₂²	1	8	yes	—
16	C₃×#875	C₂²	1	4	yes	—
17	C₃×#874	C₂²	1	2	yes	—
18	C₃×C₄×D₁₂	—	2	4	no	—
19	C₃×#869	C₂²	1	2	yes	—
20	C₃²×#97	C₂²	1	4	yes	—
21	C₂×C₃×#135	—	3	4	no	—
22	C₃×#532	—	3	2	no	—
23	C₂×#904	C₂×C₃	1	4	yes	complemented
24	#5477	C₂×C₃	1	2	yes	complemented
25	C₃×#497	—	3	4	no	—
26	C₃×#496	—	3	4	no	—
27	C₂×C₃×#97	—	2	1	no	—
28	C₂×C₃×#97	D₃	1	1	yes	characteristic, complemented
29	C₂×C₃×#97	C₂×C₃	1	1	yes	characteristic, complemented
30	#4715	C₂×C₃	1	4	yes	complemented
31	#4712	C₂×C₃	1	4	yes	complemented
32	C₂²×C₃²×C₄	D₄	1	2	yes	—
33	C₂²×C₃²×C₄	C₂³	1	1	yes	characteristic
34	C₂×C₃×#67	—	2	4	no	—
35	C₃×#235	—	2	16	no	—
36	C₂×C₃×D₁₂	—	2	4	no	—
37	C₂×C₃×D₁₂	C₂³	1	4	yes	—
38	C₂×C₃×C₄×D₃	—	2	4	no	—
39	C₂×C₃×Dic₆	C₂³	1	4	yes	—
40	C₂×C₃²×C₈	D₄	1	8	yes	complemented
41	C₂×C₃²×C₈	C₂³	1	4	yes	—
42	C₃²×C₄²	D₄	1	2	yes	—
43	C₃²×C₄²	C₂³	1	2	yes	—
44	C₃×(C₂×D₃)⋊C₄	—	2	8	no	—
45	C₃×C₄⋊Dic₃	—	2	4	no	—
46	C₃×Dic₁₂	—	2	8	no	—
47	C₃×D₂₄	—	2	8	no	—
48	C₃×#204	—	2	16	no	—
49	#3150	—	3	1	no	Sylow
50	C₂×C₃×#41	—	3	4	no	—
51	C₂×#235	C₂²×C₃	1	4	yes	—
52	C₃×#135	—	6	16	no	—
53	C₂×C₃×Q₁₆	—	3	4	no	—
54	C₂×C₃×SD₁₆	—	3	8	no	—
55	C₂×C₃×D₈	—	3	4	no	—
56	C₂²×C₃×C₈	—	2	2	no	—
57	C₂²×C₃×C₈	C₂×D₃	1	2	yes	—
58	C₂²×C₃×C₈	C₂²×C₃	1	2	yes	—
59	C₃×C₄⋊Q₈	—	3	2	no	—
60	C₃×#127	—	3	2	no	—
61	C₃×#124	—	3	4	no	—
62	C₃×#121	—	6	4	no	—
63	C₃×C₄×D₄	—	6	4	no	—
64	C₂×C₃×C₄²	—	2	1	no	—
65	C₂×C₃×C₄²	C₂×D₃	1	1	yes	characteristic
66	C₂×C₃×C₄²	C₂²×C₃	1	1	yes	characteristic
67	#930	—	2	4	no	—
68	C₂×Dic₁₂	C₂²×C₃	1	4	yes	—
69	#904	—	2	16	no	—
70	C₂×D₂₄	C₂²×C₃	1	4	yes	—
71	C₂×#204	C₂²×C₃	1	8	yes	—
72	#875	C₂²×C₃	1	4	yes	—
73	#874	C₂²×C₃	1	2	yes	—
74	C₄×D₁₂	—	2	4	no	—
75	#869	C₂²×C₃	1	2	yes	—
76	C₃×#97	—	2	4	no	—
77	C₃×#97	C₂×D₃	1	4	yes	—
78	C₃×#97	C₂²×C₃	1	4	yes	—
79	C₂³×C₃²	C₂×D₄	1	1	yes	characteristic
80	C₂²×C₃×D₃	—	2	4	no	—
81	C₂×C₃²×C₄	—	4	2	no	—
82	C₂×C₃²×C₄	C₂×D₄	1	4	yes	—
83	C₂×C₃²×C₄	C₂×D₄	1	4	yes	—
84	C₂×C₃²×C₄	C₂×D₄	1	1	yes	characteristic
85	C₂×C₃²×C₄	C₂⁴	1	1	yes	characteristic
86	C₃×#67	—	4	8	no	—
87	C₂×C₃×Dic₃	—	2	4	no	—
88	C₃×D₁₂	—	2	8	no	—
89	C₃×D₁₂	—	4	8	no	—
90	C₃×C₄×D₃	—	4	8	no	—
91	C₃×Dic₆	—	2	8	no	—
92	C₃²×C₈	C₂×D₄	1	8	yes	—
93	C₂×#135	—	3	4	no	—
94	#532	—	3	2	no	—
95	#497	—	3	4	no	—
96	#496	—	3	4	no	—
97	C₂×#97	C₃×D₃	1	1	yes	characteristic, complemented
98	C₃×#41	—	6	16	no	—
99	C₂×C₃×Q₈	—	3	4	no	—
100	C₂×C₃×D₄	—	3	4	no	—
101	C₂×C₃×D₄	—	6	4	no	—
102	C₂×C₃×D₄	—	6	4	no	—
103	C₂²×C₃×C₄	—	2	2	no	—
104	C₂²×C₃×C₄	—	2	1	no	—
105	C₂²×C₃×C₄	—	6	4	no	—
106	C₂²×C₃×C₄	D₁₂	1	2	yes	—
107	C₂²×C₃×C₄	C₃×D₄	1	2	yes	—
108	C₂²×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
109	C₂²×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
110	C₂×#67	—	2	4	no	—
111	#235	—	2	16	no	—
112	C₂×D₁₂	—	2	4	no	—
113	C₂×D₁₂	C₂³×C₃	1	4	yes	—
114	C₂×C₄×D₃	—	2	4	no	—
115	C₂×Dic₆	C₂³×C₃	1	4	yes	—
116	C₃×Q₁₆	—	6	8	no	—
117	C₃×SD₁₆	—	6	16	no	—
118	C₃×D₈	—	6	8	no	—
119	C₂×C₃×C₈	—	2	8	no	—
120	C₂×C₃×C₈	—	2	4	no	—
121	C₂×C₃×C₈	D₁₂	1	8	yes	complemented
122	C₂×C₃×C₈	C₃×D₄	1	8	yes	complemented
123	C₂×C₃×C₈	C₂²×D₃	1	4	yes	—
124	C₂×C₃×C₈	C₂³×C₃	1	4	yes	—
125	C₃×C₄⋊C₄	—	6	4	no	—
126	C₃×C₂²⋊C₄	—	6	8	no	—
127	C₃×C₄²	—	2	2	no	—
128	C₃×C₄²	—	2	2	no	—
129	C₃×C₄²	D₁₂	1	2	yes	—
130	C₃×C₄²	C₃×D₄	1	2	yes	—
131	C₃×C₄²	C₂²×D₃	1	2	yes	—
132	C₃×C₄²	C₂³×C₃	1	2	yes	—
133	(C₂×D₃)⋊C₄	—	2	8	no	—
134	C₄⋊Dic₃	—	2	4	no	—
135	Dic₁₂	—	2	8	no	—
136	D₂₄	—	2	8	no	—
137	#204	—	2	16	no	—
138	C₂²×C₃²	—	2	2	no	—
139	C₂²×C₃²	#127	1	2	yes	—
140	C₂²×C₃²	C₂²×D₄	1	1	yes	characteristic
141	C₂×C₃×D₃	—	4	4	no	—
142	C₂×C₃×D₃	—	4	8	no	—
143	C₃²×C₄	—	2	4	no	—
144	C₃²×C₄	#127	1	2	yes	—
145	C₃²×C₄	C₂²×D₄	1	2	yes	—
146	C₃×Dic₃	—	4	4	no	—
147	C₂×#41	—	3	4	no	—
148	#135	—	6	16	no	—
149	C₂×Q₁₆	—	3	4	no	—
150	C₂×SD₁₆	—	3	8	no	—
151	C₂×D₈	—	3	4	no	—
152	C₂²×C₈	C₂×C₃×D₃	1	2	yes	—
153	C₄⋊Q₈	—	3	2	no	—
154	#127	—	3	2	no	—
155	#124	—	3	4	no	—
156	#121	—	6	4	no	—
157	C₄×D₄	—	6	4	no	—
158	C₂×C₄²	C₂×C₃×D₃	1	1	yes	characteristic
159	#97	C₂×C₃×D₃	1	4	yes	—
160	C₂³×C₃	—	2	1	no	—
161	C₂³×C₃	—	6	4	no	—
162	C₂³×C₃	C₂×D₁₂	1	1	yes	characteristic
163	C₂³×C₃	C₂×C₃×D₄	1	1	yes	characteristic
164	C₂²×D₃	—	2	4	no	—
165	C₃×Q₈	—	6	8	no	—
166	C₃×D₄	—	6	8	no	—
167	C₃×D₄	—	12	8	no	—
168	C₃×D₄	—	12	8	no	—
169	C₂×C₃×C₄	—	2	1	no	—
170	C₂×C₃×C₄	—	2	1	no	—
171	C₂×C₃×C₄	—	2	4	no	—
172	C₂×C₃×C₄	—	2	4	no	—
173	C₂×C₃×C₄	—	4	2	no	—
174	C₂×C₃×C₄	—	4	2	no	—
175	C₂×C₃×C₄	—	4	4	no	—
176	C₂×C₃×C₄	—	6	4	no	—
177	C₂×C₃×C₄	—	12	8	no	—
178	C₂×C₃×C₄	C₂×D₁₂	1	4	yes	—
179	C₂×C₃×C₄	C₂×D₁₂	1	4	yes	—
180	C₂×C₃×C₄	C₂×D₁₂	1	1	yes	characteristic
181	C₂×C₃×C₄	C₂×C₃×D₄	1	4	yes	—
182	C₂×C₃×C₄	C₂×C₃×D₄	1	1	yes	characteristic
183	C₂×C₃×C₄	C₂×C₃×D₄	1	4	yes	—
184	C₂×C₃×C₄	C₂³×D₃	1	1	yes	characteristic
185	C₂×C₃×C₄	C₂⁴×C₃	1	1	yes	characteristic
186	#67	—	4	8	no	—
187	C₂×Dic₃	—	2	4	no	—
188	D₁₂	—	2	8	no	—
189	D₁₂	—	4	8	no	—
190	C₄×D₃	—	4	8	no	—
191	Dic₆	—	2	8	no	—
192	C₃×C₈	—	2	8	no	—
193	C₃×C₈	C₂×D₁₂	1	8	yes	—
194	C₃×C₈	C₂×C₃×D₄	1	8	yes	—
195	C₂×C₃²	—	2	2	no	—
196	C₂×C₃²	C₂×#127	1	1	yes	characteristic
197	C₂×C₃²	#575	1	2	yes	—
198	C₃×D₃	—	8	4	no	—
199	#41	—	6	16	no	—
200	C₂×Q₈	—	3	4	no	—
201	C₂×D₄	—	3	4	no	—
202	C₂×D₄	—	6	4	no	—
203	C₂×D₄	—	6	4	no	—
204	C₂²×C₄	—	6	4	no	—
205	C₂²×C₄	C₃×D₁₂	1	2	yes	—
206	C₂²×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
207	Q₁₆	—	6	8	no	—
208	SD₁₆	—	6	16	no	—
209	D₈	—	6	8	no	—
210	C₂×C₈	C₃×D₁₂	1	8	yes	complemented
211	C₂×C₈	C₂²×C₃×D₃	1	4	yes	—
212	C₄⋊C₄	—	6	4	no	—
213	C₂²⋊C₄	—	6	8	no	—
214	C₄²	C₃×D₁₂	1	2	yes	—
215	C₄²	C₂²×C₃×D₃	1	2	yes	—
216	C₂²×C₃	—	2	4	no	—
217	C₂²×C₃	—	2	2	no	—
218	C₂²×C₃	—	2	2	no	—
219	C₂²×C₃	—	2	1	no	—
220	C₂²×C₃	—	2	2	no	—
221	C₂²×C₃	—	12	4	no	—
222	C₂²×C₃	—	12	8	no	—
223	C₂²×C₃	#874	1	2	yes	—
224	C₂²×C₃	C₃×#127	1	2	yes	—
225	C₂²×C₃	C₂²×D₁₂	1	1	yes	characteristic
226	C₂²×C₃	C₂²×C₃×D₄	1	1	yes	characteristic
227	C₂×D₃	—	4	8	no	—
228	C₂×D₃	—	4	4	no	—
229	C₃×C₄	—	2	4	no	—
230	C₃×C₄	—	2	4	no	—
231	C₃×C₄	—	2	2	no	—
232	C₃×C₄	—	2	2	no	—
233	C₃×C₄	—	4	4	no	—
234	C₃×C₄	—	12	4	no	—
235	C₃×C₄	#874	1	2	yes	—
236	C₃×C₄	C₃×#127	1	2	yes	—
237	C₃×C₄	C₂²×D₁₂	1	2	yes	—
238	C₃×C₄	C₂²×C₃×D₄	1	2	yes	—
239	Dic₃	—	4	4	no	—
240	C₃²	#3150	1	1	yes	characteristic, complemented, Sylow
241	C₂³	—	6	4	no	—
242	C₂³	C₂×C₃×D₁₂	1	1	yes	characteristic
243	Q₈	—	6	8	no	—
244	D₄	—	6	8	no	—
245	D₄	—	12	8	no	—
246	D₄	—	12	8	no	—
247	C₂×C₄	—	4	2	no	—
248	C₂×C₄	—	6	4	no	—
249	C₂×C₄	—	12	8	no	—
250	C₂×C₄	C₂×C₃×D₁₂	1	4	yes	—
251	C₂×C₄	C₂×C₃×D₁₂	1	4	yes	—
252	C₂×C₄	C₂×C₃×D₁₂	1	1	yes	characteristic
253	C₂×C₄	C₂³×C₃×D₃	1	1	yes	characteristic
254	C₈	C₂×C₃×D₁₂	1	8	yes	—
255	C₂×C₃	—	2	4	no	—
256	C₂×C₃	—	2	2	no	—
257	C₂×C₃	—	2	2	no	—
258	C₂×C₃	—	2	2	no	—
259	C₂×C₃	—	2	1	no	—
260	C₂×C₃	—	24	4	no	—
261	C₂×C₃	C₂×#874	1	1	yes	characteristic
262	C₂×C₃	#5748	1	2	yes	—
263	C₂×C₃	C₂×C₃×#127	1	1	yes	characteristic
264	C₂×C₃	C₃×#575	1	2	yes	—
265	D₃	—	8	4	no	—
266	C₂²	—	2	2	no	—
267	C₂²	—	12	8	no	—
268	C₂²	—	12	4	no	—
269	C₂²	C₃×#874	1	2	yes	—
270	C₂²	C₂²×C₃×D₁₂	1	1	yes	characteristic
271	C₄	—	2	4	no	—
272	C₄	—	12	4	no	—
273	C₄	C₃×#874	1	2	yes	—
274	C₄	C₂²×C₃×D₁₂	1	2	yes	—
275	C₃	—	2	1	no	—
276	C₃	#78499	1	1	yes	characteristic, direct product factor
277	C₃	C₃×#3150	1	1	yes	characteristic, complemented
278	C₂	—	2	2	no	—
279	C₂	—	24	4	no	—
280	C₂	C₂×C₃×#874	1	1	yes	characteristic
281	C₂	C₃×#5748	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×#97	C₂
socle	C₂²×C₃²	C₂²×D₄
hypercenter	C₂×C₃×#97	D₃
hypocenter	C₃	C₃×#3150
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₃×C₄ ◁ C₂×C₃×C₄² ◁ C₂×C₃×#97

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×#97	C₂	semidirect
4. C₃×#4715	C₂	semidirect
5. C₃×#4712	C₂	semidirect
7. #78499	C₃	direct (N × H)
23. C₂×#904	C₂×C₃	semidirect
24. #5477	C₂×C₃	semidirect
28. C₂×C₃×#97	D₃	semidirect
29. C₂×C₃×#97	C₂×C₃	semidirect
30. #4715	C₂×C₃	semidirect
31. #4712	C₂×C₃	semidirect
40. C₂×C₃²×C₈	D₄	semidirect
97. C₂×#97	C₃×D₃	semidirect
121. C₂×C₃×C₈	D₁₂	semidirect
122. C₂×C₃×C₈	C₃×D₄	semidirect
210. C₂×C₈	C₃×D₁₂	semidirect
240. C₃²	#3150	semidirect
276. C₃	#78499	direct (N × H)
277. C₃	C₃×#3150	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×#97	2
C₃×#4715	2
C₃×#4712	2
C₃×#3150	3
#78499	3

Subgroups — C3×#78499 #11874427

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#78499 #11874427