Subgroups — C₃×#71969 #11783256

Order 1152.

1017 conjugacy classes of subgroups.

The subgroup lattice has 284 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₃²×C₈	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×#926	C₂	1	2	yes	maximal
3	C₃×#4551	C₂	1	4	yes	maximal, complemented
4	C₃×#1735	—	3	1	no	maximal
5	#71969	C₃	1	1	yes	maximal, characteristic, direct product factor
6	C₂³×C₃²×C₄	C₂²	1	1	yes	characteristic
7	C₂²×C₃²×C₈	—	2	4	no	—
8	C₂²×C₃²×C₈	C₄	1	4	yes	complemented
9	C₂²×C₃²×C₈	C₂²	1	2	yes	—
10	C₂×C₃×#217	—	2	2	no	—
11	C₃×#926	—	2	4	no	—
12	C₃×#926	C₄	1	8	yes	complemented
13	C₂×C₃×C₄⋊Dic₃	C₂²	1	4	yes	—
14	C₂×C₃×C₄×Dic₃	—	2	2	no	—
15	C₂³×C₃×C₈	—	2	1	no	—
16	C₂³×C₃×C₈	D₃	1	1	yes	characteristic
17	C₂³×C₃×C₈	C₂×C₃	1	1	yes	characteristic
18	C₂×C₃×#117	—	3	2	no	—
19	C₂×#926	C₂×C₃	1	2	yes	—
20	C₃×#340	—	3	4	no	—
21	#4551	C₂×C₃	1	4	yes	complemented
22	C₂⁴×C₃²	D₄	1	1	yes	characteristic
23	C₂²×C₃²×C₄	C₂×C₄	1	2	yes	—
24	C₂²×C₃²×C₄	C₂×C₄	1	4	yes	—
25	C₂²×C₃²×C₄	D₄	1	1	yes	characteristic
26	C₂²×C₃²×C₄	D₄	1	4	yes	—
27	C₂²×C₃²×C₄	Q₈	1	2	yes	—
28	C₂²×C₃²×C₄	C₂³	1	1	yes	characteristic
29	C₂²×C₃×Dic₃	—	2	4	no	—
30	C₂×C₃²×C₈	—	2	8	no	—
31	C₂×C₃²×C₈	—	2	4	no	—
32	C₂×C₃²×C₈	C₂×C₄	1	4	yes	—
33	C₃×#217	—	2	8	no	—
34	C₃×C₄⋊Dic₃	—	2	4	no	—
35	C₃×C₄⋊Dic₃	C₂×C₄	1	8	yes	—
36	C₃×C₄×Dic₃	—	2	8	no	—
37	#1735	—	3	1	no	Sylow
38	C₂³×C₃×C₄	—	2	1	no	—
39	C₂³×C₃×C₄	C₂×D₃	1	1	yes	characteristic
40	C₂³×C₃×C₄	C₂²×C₃	1	1	yes	characteristic
41	C₂²×C₃×C₈	—	2	8	no	—
42	C₂²×C₃×C₈	—	2	4	no	—
43	C₂²×C₃×C₈	—	2	4	no	—
44	C₂²×C₃×C₈	—	2	4	no	—
45	C₂²×C₃×C₈	—	2	2	no	—
46	C₂²×C₃×C₈	Dic₃	1	4	yes	complemented
47	C₂²×C₃×C₈	C₃×C₄	1	4	yes	complemented
48	C₂²×C₃×C₈	C₂×D₃	1	2	yes	—
49	C₂²×C₃×C₈	C₂²×C₃	1	2	yes	—
50	C₃×#117	—	3	8	no	—
51	C₃×#117	—	6	4	no	—
52	C₂×C₃×C₄⋊C₄	—	3	4	no	—
53	C₂×C₃×C₂²⋊C₄	—	6	2	no	—
54	C₂×C₃×C₄²	—	6	2	no	—
55	C₂×#217	—	2	2	no	—
56	#926	—	2	4	no	—
57	#926	C₃×C₄	1	8	yes	complemented
58	C₂×C₄⋊Dic₃	C₂²×C₃	1	4	yes	—
59	C₂×C₄×Dic₃	—	2	2	no	—
60	C₂³×C₃²	—	2	4	no	—
61	C₂³×C₃²	C₂²⋊C₄	1	4	yes	—
62	C₂³×C₃²	C₄⋊C₄	1	2	yes	—
63	C₂³×C₃²	C₂×D₄	1	1	yes	characteristic
64	C₂×C₃²×C₄	C₄²	1	4	yes	—
65	C₂×C₃²×C₄	C₂²⋊C₄	1	4	yes	—
66	C₂×C₃²×C₄	C₂²⋊C₄	1	4	yes	—
67	C₂×C₃²×C₄	C₄⋊C₄	1	8	yes	—
68	C₂×C₃²×C₄	C₄⋊C₄	1	2	yes	—
69	C₂×C₃²×C₄	C₂²×C₄	1	2	yes	—
70	C₂×C₃²×C₄	C₂²×C₄	1	1	yes	characteristic
71	C₂×C₃²×C₄	C₂×D₄	1	2	yes	—
72	C₂×C₃²×C₄	C₂×Q₈	1	1	yes	characteristic
73	C₂×C₃×Dic₃	—	2	8	no	—
74	C₂×C₃×Dic₃	—	4	8	no	—
75	C₃²×C₈	—	2	4	no	—
76	C₂³×C₈	C₃×D₃	1	1	yes	characteristic
77	C₂×#117	—	3	2	no	—
78	#340	—	3	4	no	—
79	C₂⁴×C₃	—	2	1	no	—
80	C₂⁴×C₃	D₁₂	1	1	yes	characteristic
81	C₂⁴×C₃	C₃×D₄	1	1	yes	characteristic
82	C₂²×C₃×C₄	—	2	2	no	—
83	C₂²×C₃×C₄	—	2	4	no	—
84	C₂²×C₃×C₄	—	2	1	no	—
85	C₂²×C₃×C₄	—	2	2	no	—
86	C₂²×C₃×C₄	—	2	4	no	—
87	C₂²×C₃×C₄	—	2	1	no	—
88	C₂²×C₃×C₄	—	6	4	no	—
89	C₂²×C₃×C₄	Dic₆	1	2	yes	—
90	C₂²×C₃×C₄	C₄×D₃	1	4	yes	—
91	C₂²×C₃×C₄	D₁₂	1	1	yes	characteristic
92	C₂²×C₃×C₄	C₂×Dic₃	1	2	yes	—
93	C₂²×C₃×C₄	#67	1	4	yes	—
94	C₂²×C₃×C₄	C₂×C₃×C₄	1	4	yes	—
95	C₂²×C₃×C₄	C₂×C₃×C₄	1	2	yes	—
96	C₂²×C₃×C₄	C₃×D₄	1	1	yes	characteristic
97	C₂²×C₃×C₄	C₃×D₄	1	4	yes	—
98	C₂²×C₃×C₄	C₃×Q₈	1	2	yes	—
99	C₂²×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
100	C₂²×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
101	C₂²×Dic₃	—	2	4	no	—
102	C₂×C₃×C₈	—	2	16	no	—
103	C₂×C₃×C₈	—	2	8	no	—
104	C₂×C₃×C₈	—	2	8	no	—
105	C₂×C₃×C₈	—	2	8	no	—
106	C₂×C₃×C₈	—	2	4	no	—
107	C₂×C₃×C₈	—	2	4	no	—
108	C₂×C₃×C₈	—	2	4	no	—
109	C₂×C₃×C₈	C₂×Dic₃	1	4	yes	—
110	C₂×C₃×C₈	C₂×C₃×C₄	1	4	yes	—
111	C₃×C₄⋊C₄	—	3	8	no	—
112	C₃×C₄⋊C₄	—	6	4	no	—
113	C₃×C₂²⋊C₄	—	6	8	no	—
114	C₃×C₄²	—	6	8	no	—
115	#217	—	2	8	no	—
116	C₄⋊Dic₃	—	2	4	no	—
117	C₄⋊Dic₃	C₂×C₃×C₄	1	8	yes	—
118	C₄×Dic₃	—	2	8	no	—
119	C₂²×C₃²	—	2	8	no	—
120	C₂²×C₃²	—	2	4	no	—
121	C₂²×C₃²	#95	1	4	yes	—
122	C₂²×C₃²	C₂×C₂²⋊C₄	1	2	yes	—
123	C₂²×C₃²	C₂×C₄⋊C₄	1	1	yes	characteristic
124	C₂²×C₃²	#135	1	4	yes	—
125	C₃²×C₄	#95	1	4	yes	—
126	C₃²×C₄	C₂×C₄²	1	1	yes	characteristic
127	C₃²×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
128	C₃²×C₄	C₂×C₄⋊C₄	1	2	yes	—
129	C₃×Dic₃	—	4	8	no	—
130	C₂³×C₄	C₂×C₃×D₃	1	1	yes	characteristic
131	C₂²×C₈	—	2	4	no	—
132	C₂²×C₈	C₃×Dic₃	1	4	yes	complemented
133	C₂²×C₈	C₂×C₃×D₃	1	2	yes	—
134	#117	—	3	8	no	—
135	#117	—	6	4	no	—
136	C₂×C₄⋊C₄	—	3	4	no	—
137	C₂×C₂²⋊C₄	—	6	2	no	—
138	C₂×C₄²	—	6	2	no	—
139	C₂³×C₃	—	2	8	no	—
140	C₂³×C₃	—	2	4	no	—
141	C₂³×C₃	—	2	4	no	—
142	C₂³×C₃	—	2	2	no	—
143	C₂³×C₃	—	2	4	no	—
144	C₂³×C₃	—	2	1	no	—
145	C₂³×C₃	C₄⋊Dic₃	1	2	yes	—
146	C₂³×C₃	(C₂×D₃)⋊C₄	1	4	yes	—
147	C₂³×C₃	C₃×C₂²⋊C₄	1	4	yes	—
148	C₂³×C₃	C₃×C₄⋊C₄	1	2	yes	—
149	C₂³×C₃	C₂×D₁₂	1	1	yes	characteristic
150	C₂³×C₃	C₂×C₃×D₄	1	1	yes	characteristic
151	C₂×C₃×C₄	—	2	4	no	—
152	C₂×C₃×C₄	—	2	8	no	—
153	C₂×C₃×C₄	—	2	2	no	—
154	C₂×C₃×C₄	—	2	4	no	—
155	C₂×C₃×C₄	—	2	4	no	—
156	C₂×C₃×C₄	—	2	1	no	—
157	C₂×C₃×C₄	—	2	2	no	—
158	C₂×C₃×C₄	—	2	1	no	—
159	C₂×C₃×C₄	—	2	2	no	—
160	C₂×C₃×C₄	—	6	8	no	—
161	C₂×C₃×C₄	—	12	8	no	—
162	C₂×C₃×C₄	C₄×Dic₃	1	4	yes	—
163	C₂×C₃×C₄	Dic₃⋊C₄	1	8	yes	—
164	C₂×C₃×C₄	C₄⋊Dic₃	1	2	yes	—
165	C₂×C₃×C₄	(C₂×D₃)⋊C₄	1	4	yes	—
166	C₂×C₃×C₄	#217	1	4	yes	—
167	C₂×C₃×C₄	C₃×C₄²	1	4	yes	—
168	C₂×C₃×C₄	C₃×C₂²⋊C₄	1	4	yes	—
169	C₂×C₃×C₄	C₃×C₂²⋊C₄	1	4	yes	—
170	C₂×C₃×C₄	C₃×C₄⋊C₄	1	8	yes	—
171	C₂×C₃×C₄	C₃×C₄⋊C₄	1	2	yes	—
172	C₂×C₃×C₄	C₂×Dic₆	1	1	yes	characteristic
173	C₂×C₃×C₄	C₂×C₄×D₃	1	2	yes	—
174	C₂×C₃×C₄	C₂²×Dic₃	1	1	yes	characteristic
175	C₂×C₃×C₄	C₂×#67	1	2	yes	—
176	C₂×C₃×C₄	C₂²×C₃×C₄	1	2	yes	—
177	C₂×C₃×C₄	C₂²×C₃×C₄	1	1	yes	characteristic
178	C₂×C₃×C₄	C₂×C₃×D₄	1	2	yes	—
179	C₂×C₃×C₄	C₂×C₃×Q₈	1	1	yes	characteristic
180	C₂×Dic₃	—	2	8	no	—
181	C₂×Dic₃	—	4	8	no	—
182	C₃×C₈	—	2	8	no	—
183	C₃×C₈	—	2	4	no	—
184	C₃×C₈	—	2	4	no	—
185	C₂×C₃²	—	2	4	no	—
186	C₂×C₃²	C₂×#95	1	1	yes	characteristic
187	C₂×C₃²	#416	1	4	yes	—
188	C₂×C₃²	#427	1	2	yes	—
189	C₂⁴	C₃×D₁₂	1	1	yes	characteristic
190	C₂²×C₄	—	6	4	no	—
191	C₂²×C₄	C₃×Dic₆	1	2	yes	—
192	C₂²×C₄	C₃×C₄×D₃	1	4	yes	—
193	C₂²×C₄	C₃×D₁₂	1	1	yes	characteristic
194	C₂²×C₄	C₂×C₃×Dic₃	1	2	yes	—
195	C₂²×C₄	C₃×#67	1	4	yes	—
196	C₂²×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
197	C₂×C₈	—	2	8	no	—
198	C₂×C₈	—	2	4	no	—
199	C₂×C₈	C₂×C₃×Dic₃	1	4	yes	—
200	C₄⋊C₄	—	3	8	no	—
201	C₄⋊C₄	—	6	4	no	—
202	C₂²⋊C₄	—	6	8	no	—
203	C₄²	—	6	8	no	—
204	C₂²×C₃	—	2	8	no	—
205	C₂²×C₃	—	2	16	no	—
206	C₂²×C₃	—	2	8	no	—
207	C₂²×C₃	—	2	4	no	—
208	C₂²×C₃	—	2	4	no	—
209	C₂²×C₃	—	2	8	no	—
210	C₂²×C₃	—	2	4	no	—
211	C₂²×C₃	—	2	1	no	—
212	C₂²×C₃	—	2	2	no	—
213	C₂²×C₃	—	2	4	no	—
214	C₂²×C₃	C₃⋊#95	1	4	yes	—
215	C₂²×C₃	C₃×#95	1	4	yes	—
216	C₂²×C₃	#904	1	4	yes	—
217	C₂²×C₃	C₂×C₄⋊Dic₃	1	1	yes	characteristic
218	C₂²×C₃	C₂×(C₂×D₃)⋊C₄	1	2	yes	—
219	C₂²×C₃	C₂×C₃×C₂²⋊C₄	1	2	yes	—
220	C₂²×C₃	C₂×C₃×C₄⋊C₄	1	1	yes	characteristic
221	C₂²×C₃	C₃×#135	1	4	yes	—
222	C₃×C₄	—	2	1	no	—
223	C₃×C₄	—	2	2	no	—
224	C₃×C₄	—	2	1	no	—
225	C₃×C₄	—	2	4	no	—
226	C₃×C₄	—	12	8	no	—
227	C₃×C₄	C₃⋊#95	1	4	yes	—
228	C₃×C₄	C₃×#95	1	4	yes	—
229	C₃×C₄	C₂×C₄×Dic₃	1	1	yes	characteristic
230	C₃×C₄	C₂×Dic₃⋊C₄	1	2	yes	—
231	C₃×C₄	C₂×#217	1	1	yes	characteristic
232	C₃×C₄	C₂×C₃×C₄²	1	1	yes	characteristic
233	C₃×C₄	C₂×C₃×C₂²⋊C₄	1	1	yes	characteristic
234	C₃×C₄	C₂×C₃×C₄⋊C₄	1	2	yes	—
235	Dic₃	—	4	8	no	—
236	C₃²	#1735	1	1	yes	characteristic, complemented, Sylow
237	C₂³	—	2	4	no	—
238	C₂³	C₃×C₄⋊Dic₃	1	2	yes	—
239	C₂³	C₃×(C₂×D₃)⋊C₄	1	4	yes	—
240	C₂³	C₂×C₃×D₁₂	1	1	yes	characteristic
241	C₂×C₄	—	6	8	no	—
242	C₂×C₄	—	12	8	no	—
243	C₂×C₄	C₃×C₄×Dic₃	1	4	yes	—
244	C₂×C₄	C₃×Dic₃⋊C₄	1	8	yes	—
245	C₂×C₄	C₃×C₄⋊Dic₃	1	2	yes	—
246	C₂×C₄	C₃×(C₂×D₃)⋊C₄	1	4	yes	—
247	C₂×C₄	C₃×#217	1	4	yes	—
248	C₂×C₄	C₂×C₃×Dic₆	1	1	yes	characteristic
249	C₂×C₄	C₂×C₃×C₄×D₃	1	2	yes	—
250	C₂×C₄	C₂²×C₃×Dic₃	1	1	yes	characteristic
251	C₂×C₄	C₂×C₃×#67	1	2	yes	—
252	C₈	—	2	4	no	—
253	C₂×C₃	—	2	8	no	—
254	C₂×C₃	—	2	4	no	—
255	C₂×C₃	—	2	4	no	—
256	C₂×C₃	—	2	1	no	—
257	C₂×C₃	—	2	2	no	—
258	C₂×C₃	—	2	4	no	—
259	C₂×C₃	#5106	1	2	yes	—
260	C₂×C₃	#5113	1	4	yes	—
261	C₂×C₃	C₂×C₃⋊#95	1	1	yes	characteristic
262	C₂×C₃	C₂×C₃×#95	1	1	yes	characteristic
263	C₂×C₃	C₃×#416	1	4	yes	—
264	C₂×C₃	C₃×#427	1	2	yes	—
265	C₂²	—	2	4	no	—
266	C₂²	—	2	8	no	—
267	C₂²	C₃×C₃⋊#95	1	4	yes	—
268	C₂²	C₃×#904	1	4	yes	—
269	C₂²	C₂×C₃×C₄⋊Dic₃	1	1	yes	characteristic
270	C₂²	C₂×C₃×(C₂×D₃)⋊C₄	1	2	yes	—
271	C₄	—	12	8	no	—
272	C₄	C₃×C₃⋊#95	1	4	yes	—
273	C₄	C₂×C₃×C₄×Dic₃	1	1	yes	characteristic
274	C₄	C₂×C₃×Dic₃⋊C₄	1	2	yes	—
275	C₄	C₂×C₃×#217	1	1	yes	characteristic
276	C₃	—	2	1	no	—
277	C₃	#71969	1	1	yes	characteristic, direct product factor
278	C₃	C₃×#1735	1	1	yes	characteristic, complemented
279	C₂	—	2	4	no	—
280	C₂	C₃×#5106	1	2	yes	—
281	C₂	C₃×#5113	1	4	yes	—
282	C₂	C₂×C₃×C₃⋊#95	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₃×C₄	D₁₂
commutator	C₃×C₄	C₂×C₃×C₄²
Frattini	C₂²×C₄	C₂²×C₃×D₃
Fitting	C₂³×C₃²×C₈	C₂
socle	C₂³×C₃²	C₂×D₄
prime radical	C₂⁴×C₃²	D₄
cosocle	C₂²×C₃×C₄	C₂³×C₃
hypercenter	C₂³×C₃×C₈	D₃
hypocenter	C₃	C₃×#1735
LCS	G ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₃×C₄ ◁ C₂³×C₃×C₄ ◁ C₂³×C₃×C₈

Complements

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

N	H ≅ G/N	type
3. C₃×#4551	C₂	semidirect
5. #71969	C₃	direct (N × H)
8. C₂²×C₃²×C₈	C₄	semidirect
12. C₃×#926	C₄	semidirect
21. #4551	C₂×C₃	semidirect
46. C₂²×C₃×C₈	Dic₃	semidirect
47. C₂²×C₃×C₈	C₃×C₄	semidirect
57. #926	C₃×C₄	semidirect
132. C₂²×C₈	C₃×Dic₃	semidirect
236. C₃²	#1735	semidirect
277. C₃	#71969	direct (N × H)
278. C₃	C₃×#1735	semidirect

Maximal subgroups

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

subgroup	index
C₂³×C₃²×C₈	2
C₂×C₃×#926	2
C₃×#4551	2
C₃×#1735	3
#71969	3

Subgroups — C3×#71969 #11783256

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#71969 #11783256