Subgroups — C₃×#71977 #11785434

Order 1152.

753 conjugacy classes of subgroups.

The subgroup lattice has 289 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	C₃×#5207	C₂	1	2	yes	maximal
3	C₂×C₃×#819	C₂	1	2	yes	maximal, complemented
4	C₃×#4552	C₂	1	2	yes	maximal, complemented
5	C₃×#1810	—	3	1	no	maximal
6	#71977	C₃	1	1	yes	maximal, characteristic, direct product factor
7	C₂³×C₃²×C₄	C₂²	1	1	yes	characteristic
8	C₂²×C₃²×C₈	—	2	4	no	—
9	C₂²×C₃²×C₈	C₄	1	4	yes	—
10	C₂²×C₃²×C₈	C₂²	1	1	yes	characteristic
11	C₂²×C₃²×C₈	C₂²	1	1	yes	characteristic
12	C₂×C₃×#208	C₂²	1	4	yes	—
13	C₃×#830	—	2	4	no	—
14	C₃×#819	—	2	4	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×#108	—	3	2	no	—
19	C₃×#407	—	3	2	no	—
20	#5207	C₂×C₃	1	2	yes	—
21	C₂×#819	C₂×C₃	1	2	yes	complemented
22	C₃×#341	—	3	2	no	—
23	#4552	C₂×C₃	1	2	yes	complemented
24	C₂⁴×C₃²	Q₈	1	1	yes	characteristic
25	C₂²×C₃²×C₄	—	2	2	no	—
26	C₂²×C₃²×C₄	—	2	2	no	—
27	C₂²×C₃²×C₄	C₂×C₄	1	2	yes	—
28	C₂²×C₃²×C₄	D₄	1	2	yes	—
29	C₂²×C₃²×C₄	Q₈	1	1	yes	characteristic
30	C₂²×C₃²×C₄	C₂³	1	1	yes	characteristic
31	C₂×C₃²×C₈	—	2	2	no	—
32	C₂×C₃²×C₈	—	2	8	no	—
33	C₂×C₃²×C₈	C₂×C₄	1	2	yes	—
34	C₂×C₃²×C₈	C₂×C₄	1	2	yes	—
35	C₂×C₃²×C₈	D₄	1	4	yes	—
36	C₃×#208	—	2	4	no	—
37	C₃×#208	—	2	4	no	—
38	C₂×C₃×C₃⋊C₈	—	2	4	no	—
39	#1810	—	3	1	no	Sylow
40	C₂³×C₃×C₄	—	2	1	no	—
41	C₂³×C₃×C₄	C₂×D₃	1	1	yes	characteristic
42	C₂³×C₃×C₄	C₂²×C₃	1	1	yes	characteristic
43	C₂×C₃×M₁₆	—	3	4	no	—
44	C₂²×C₃×C₈	—	2	8	no	—
45	C₂²×C₃×C₈	—	2	4	no	—
46	C₂²×C₃×C₈	—	2	4	no	—
47	C₂²×C₃×C₈	—	2	4	no	—
48	C₂²×C₃×C₈	—	2	1	no	—
49	C₂²×C₃×C₈	—	2	1	no	—
50	C₂²×C₃×C₈	Dic₃	1	4	yes	—
51	C₂²×C₃×C₈	C₃×C₄	1	4	yes	—
52	C₂²×C₃×C₈	C₂×D₃	1	1	yes	characteristic
53	C₂²×C₃×C₈	C₂×D₃	1	1	yes	characteristic
54	C₂²×C₃×C₈	C₂²×C₃	1	1	yes	characteristic
55	C₂²×C₃×C₈	C₂²×C₃	1	1	yes	characteristic
56	C₂×#208	C₂²×C₃	1	4	yes	—
57	C₃×#108	—	6	4	no	—
58	C₃×C₂²⋊C₈	—	6	4	no	—
59	#830	—	2	4	no	—
60	#819	—	2	4	no	—
61	C₂³×C₃²	—	2	4	no	—
62	C₂³×C₃²	—	2	2	no	—
63	C₂³×C₃²	C₄⋊C₄	1	2	yes	—
64	C₂³×C₃²	C₂×Q₈	1	1	yes	characteristic
65	C₂×C₃²×C₄	—	2	2	no	—
66	C₂×C₃²×C₄	—	2	2	no	—
67	C₂×C₃²×C₄	—	2	2	no	—
68	C₂×C₃²×C₄	—	2	4	no	—
69	C₂×C₃²×C₄	C₄⋊C₄	1	2	yes	—
70	C₂×C₃²×C₄	C₂²×C₄	1	1	yes	characteristic
71	C₂×C₃²×C₄	C₂×D₄	1	1	yes	characteristic
72	C₂×C₃²×C₄	C₂×D₄	1	2	yes	—
73	C₂×C₃²×C₄	#41	1	2	yes	—
74	C₃²×C₈	—	2	2	no	—
75	C₃²×C₈	C₂²⋊C₄	1	4	yes	—
76	C₃×C₃⋊C₈	—	4	4	no	—
77	C₂³×C₈	C₃×D₃	1	1	yes	characteristic
78	C₂×#108	—	3	2	no	—
79	#407	—	3	2	no	—
80	#341	—	3	2	no	—
81	C₂⁴×C₃	—	2	1	no	—
82	C₂⁴×C₃	Dic₆	1	1	yes	characteristic
83	C₂⁴×C₃	C₃×Q₈	1	1	yes	characteristic
84	C₂²×C₃×C₄	—	2	4	no	—
85	C₂²×C₃×C₄	—	2	4	no	—
86	C₂²×C₃×C₄	—	2	2	no	—
87	C₂²×C₃×C₄	—	2	2	no	—
88	C₂²×C₃×C₄	—	2	2	no	—
89	C₂²×C₃×C₄	—	2	2	no	—
90	C₂²×C₃×C₄	—	2	1	no	—
91	C₂²×C₃×C₄	—	2	2	no	—
92	C₂²×C₃×C₄	—	2	2	no	—
93	C₂²×C₃×C₄	—	2	1	no	—
94	C₂²×C₃×C₄	Dic₆	1	1	yes	characteristic
95	C₂²×C₃×C₄	D₁₂	1	2	yes	—
96	C₂²×C₃×C₄	C₂×Dic₃	1	2	yes	—
97	C₂²×C₃×C₄	C₂×C₃×C₄	1	2	yes	—
98	C₂²×C₃×C₄	C₃×D₄	1	2	yes	—
99	C₂²×C₃×C₄	C₃×Q₈	1	1	yes	characteristic
100	C₂²×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
101	C₂²×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
102	C₃×M₁₆	—	6	4	no	—
103	C₃×M₁₆	—	6	4	no	—
104	C₂×C₃×C₈	—	2	4	no	—
105	C₂×C₃×C₈	—	2	16	no	—
106	C₂×C₃×C₈	—	2	2	no	—
107	C₂×C₃×C₈	—	2	2	no	—
108	C₂×C₃×C₈	—	2	8	no	—
109	C₂×C₃×C₈	—	2	8	no	—
110	C₂×C₃×C₈	—	2	2	no	—
111	C₂×C₃×C₈	—	2	2	no	—
112	C₂×C₃×C₈	—	2	4	no	—
113	C₂×C₃×C₈	—	6	4	no	—
114	C₂×C₃×C₈	C₂×Dic₃	1	2	yes	—
115	C₂×C₃×C₈	C₂×Dic₃	1	2	yes	—
116	C₂×C₃×C₈	#67	1	4	yes	—
117	C₂×C₃×C₈	C₂×C₃×C₄	1	2	yes	—
118	C₂×C₃×C₈	C₂×C₃×C₄	1	2	yes	—
119	C₂×C₃×C₈	C₃×D₄	1	4	yes	—
120	#208	—	2	4	no	—
121	#208	—	2	4	no	—
122	C₂×C₃⋊C₈	—	2	4	no	—
123	C₂²×C₃²	—	2	8	no	—
124	C₂²×C₃²	—	2	2	no	—
125	C₂²×C₃²	—	2	2	no	—
126	C₂²×C₃²	—	2	2	no	—
127	C₂²×C₃²	#108	1	4	yes	complemented
128	C₂²×C₃²	C₂×C₄⋊C₄	1	1	yes	characteristic
129	C₂²×C₃²	C₂²⋊Q₈	1	2	yes	—
130	C₃²×C₄	—	2	2	no	—
131	C₃²×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
132	C₃²×C₄	#117	1	1	yes	characteristic
133	C₃²×C₄	#121	1	2	yes	—
134	C₂³×C₄	C₂×C₃×D₃	1	1	yes	characteristic
135	C₂×M₁₆	—	3	4	no	—
136	C₂²×C₈	—	2	4	no	—
137	C₂²×C₈	C₃×Dic₃	1	4	yes	—
138	C₂²×C₈	C₂×C₃×D₃	1	1	yes	characteristic
139	C₂²×C₈	C₂×C₃×D₃	1	1	yes	characteristic
140	#108	—	6	4	no	—
141	C₂²⋊C₈	—	6	4	no	—
142	C₂³×C₃	—	2	4	no	—
143	C₂³×C₃	—	2	8	no	—
144	C₂³×C₃	—	2	4	no	—
145	C₂³×C₃	—	2	4	no	—
146	C₂³×C₃	—	2	2	no	—
147	C₂³×C₃	—	2	2	no	—
148	C₂³×C₃	—	2	2	no	—
149	C₂³×C₃	—	2	1	no	—
150	C₂³×C₃	C₄⋊Dic₃	1	2	yes	—
151	C₂³×C₃	C₃×C₄⋊C₄	1	2	yes	—
152	C₂³×C₃	C₂×Dic₆	1	1	yes	characteristic
153	C₂³×C₃	C₂×C₃×Q₈	1	1	yes	characteristic
154	C₂×C₃×C₄	—	2	4	no	—
155	C₂×C₃×C₄	—	2	4	no	—
156	C₂×C₃×C₄	—	2	4	no	—
157	C₂×C₃×C₄	—	2	8	no	—
158	C₂×C₃×C₄	—	2	2	no	—
159	C₂×C₃×C₄	—	2	2	no	—
160	C₂×C₃×C₄	—	2	2	no	—
161	C₂×C₃×C₄	—	2	4	no	—
162	C₂×C₃×C₄	—	2	2	no	—
163	C₂×C₃×C₄	—	2	2	no	—
164	C₂×C₃×C₄	—	2	2	no	—
165	C₂×C₃×C₄	—	2	4	no	—
166	C₂×C₃×C₄	—	2	2	no	—
167	C₂×C₃×C₄	—	2	1	no	—
168	C₂×C₃×C₄	—	2	2	no	—
169	C₂×C₃×C₄	—	2	1	no	—
170	C₂×C₃×C₄	—	2	2	no	—
171	C₂×C₃×C₄	C₄⋊Dic₃	1	2	yes	—
172	C₂×C₃×C₄	C₃×C₄⋊C₄	1	2	yes	—
173	C₂×C₃×C₄	C₂×D₁₂	1	1	yes	characteristic
174	C₂×C₃×C₄	#235	1	2	yes	—
175	C₂×C₃×C₄	C₂²×Dic₃	1	1	yes	characteristic
176	C₂×C₃×C₄	C₂×#67	1	2	yes	—
177	C₂×C₃×C₄	C₂²×C₃×C₄	1	1	yes	characteristic
178	C₂×C₃×C₄	C₂×C₃×D₄	1	1	yes	characteristic
179	C₂×C₃×C₄	C₂×C₃×D₄	1	2	yes	—
180	C₂×C₃×C₄	C₃×#41	1	2	yes	—
181	C₃×C₈	—	2	4	no	—
182	C₃×C₈	—	2	2	no	—
183	C₃×C₈	—	2	2	no	—
184	C₃×C₈	—	2	4	no	—
185	C₃×C₈	—	12	4	no	—
186	C₃×C₈	#217	1	4	yes	—
187	C₃×C₈	C₃×C₂²⋊C₄	1	4	yes	—
188	C₃⋊C₈	—	4	4	no	—
189	C₂×C₃²	—	2	4	no	—
190	C₂×C₃²	—	2	2	no	—
191	C₂×C₃²	#380	1	1	yes	characteristic
192	C₂×C₃²	C₂×#108	1	2	yes	—
193	C₂⁴	C₃×Dic₆	1	1	yes	characteristic
194	C₂²×C₄	—	2	2	no	—
195	C₂²×C₄	—	2	2	no	—
196	C₂²×C₄	C₃×Dic₆	1	1	yes	characteristic
197	C₂²×C₄	C₃×D₁₂	1	2	yes	—
198	C₂²×C₄	C₂×C₃×Dic₃	1	2	yes	—
199	C₂²×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
200	M₁₆	—	6	4	no	—
201	M₁₆	—	6	4	no	—
202	C₂×C₈	—	2	2	no	—
203	C₂×C₈	—	2	8	no	—
204	C₂×C₈	—	6	4	no	—
205	C₂×C₈	C₂×C₃×Dic₃	1	2	yes	—
206	C₂×C₈	C₂×C₃×Dic₃	1	2	yes	—
207	C₂×C₈	C₃×#67	1	4	yes	—
208	C₂²×C₃	—	2	4	no	—
209	C₂²×C₃	—	2	16	no	—
210	C₂²×C₃	—	2	8	no	—
211	C₂²×C₃	—	2	8	no	—
212	C₂²×C₃	—	2	4	no	—
213	C₂²×C₃	—	2	4	no	—
214	C₂²×C₃	—	2	2	no	—
215	C₂²×C₃	—	2	2	no	—
216	C₂²×C₃	—	2	2	no	—
217	C₂²×C₃	—	2	2	no	—
218	C₂²×C₃	—	2	2	no	—
219	C₂²×C₃	—	2	2	no	—
220	C₂²×C₃	—	2	2	no	—
221	C₂²×C₃	—	2	1	no	—
222	C₂²×C₃	—	2	4	no	—
223	C₂²×C₃	#819	1	4	yes	complemented
224	C₂²×C₃	C₃×#108	1	4	yes	complemented
225	C₂²×C₃	#924	1	2	yes	—
226	C₂²×C₃	C₂×C₄⋊Dic₃	1	1	yes	characteristic
227	C₂²×C₃	C₂×C₃×C₄⋊C₄	1	1	yes	characteristic
228	C₂²×C₃	C₃×C₂²⋊Q₈	1	2	yes	—
229	C₃×C₄	—	2	4	no	—
230	C₃×C₄	—	2	2	no	—
231	C₃×C₄	—	2	2	no	—
232	C₃×C₄	—	2	1	no	—
233	C₃×C₄	—	2	2	no	—
234	C₃×C₄	—	2	1	no	—
235	C₃×C₄	#926	1	1	yes	characteristic
236	C₃×C₄	#930	1	2	yes	—
237	C₃×C₄	C₂×#217	1	1	yes	characteristic
238	C₃×C₄	C₂×C₃×C₂²⋊C₄	1	1	yes	characteristic
239	C₃×C₄	C₃×#117	1	1	yes	characteristic
240	C₃×C₄	C₃×#121	1	2	yes	—
241	C₃²	#1810	1	1	yes	characteristic, complemented, Sylow
242	C₂³	—	2	2	no	—
243	C₂³	—	2	4	no	—
244	C₂³	C₃×C₄⋊Dic₃	1	2	yes	—
245	C₂³	C₂×C₃×Dic₆	1	1	yes	characteristic
246	C₂×C₄	—	2	2	no	—
247	C₂×C₄	—	2	2	no	—
248	C₂×C₄	—	2	4	no	—
249	C₂×C₄	—	2	2	no	—
250	C₂×C₄	C₃×C₄⋊Dic₃	1	2	yes	—
251	C₂×C₄	C₂×C₃×D₁₂	1	1	yes	characteristic
252	C₂×C₄	C₃×#235	1	2	yes	—
253	C₂×C₄	C₂²×C₃×Dic₃	1	1	yes	characteristic
254	C₂×C₄	C₂×C₃×#67	1	2	yes	—
255	C₈	—	2	2	no	—
256	C₈	—	12	4	no	—
257	C₈	C₃×#217	1	4	yes	—
258	C₂×C₃	—	2	4	no	—
259	C₂×C₃	—	2	8	no	—
260	C₂×C₃	—	2	4	no	—
261	C₂×C₃	—	2	2	no	—
262	C₂×C₃	—	2	4	no	—
263	C₂×C₃	—	2	2	no	—
264	C₂×C₃	—	2	1	no	—
265	C₂×C₃	—	2	2	no	—
266	C₂×C₃	C₂×#819	1	2	yes	—
267	C₂×C₃	#5212	1	1	yes	characteristic
268	C₂×C₃	C₃×#380	1	1	yes	characteristic
269	C₂×C₃	C₂×C₃×#108	1	2	yes	—
270	C₂²	—	2	8	no	—
271	C₂²	—	2	2	no	—
272	C₂²	—	2	2	no	—
273	C₂²	—	2	2	no	—
274	C₂²	C₃×#819	1	4	yes	complemented
275	C₂²	C₃×#924	1	2	yes	—
276	C₂²	C₂×C₃×C₄⋊Dic₃	1	1	yes	characteristic
277	C₄	—	2	2	no	—
278	C₄	C₃×#926	1	1	yes	characteristic
279	C₄	C₃×#930	1	2	yes	—
280	C₄	C₂×C₃×#217	1	1	yes	characteristic
281	C₃	—	2	1	no	—
282	C₃	#71977	1	1	yes	characteristic, direct product factor
283	C₃	C₃×#1810	1	1	yes	characteristic, complemented
284	C₂	—	2	2	no	—
285	C₂	—	2	4	no	—
286	C₂	C₂×C₃×#819	1	2	yes	—
287	C₂	C₃×#5212	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₃×C₄	C₂×D₁₂
commutator	C₂×C₃×C₄	C₂²×C₃×C₄
Frattini	C₂²×C₄	C₂²×C₃×D₃
Fitting	C₂³×C₃²×C₈	C₂
socle	C₂²×C₃²	C₂×C₄⋊C₄
prime radical	C₂⁴×C₃²	Q₈
cosocle	C₂²×C₃×C₄	C₂³×C₃
hypercenter	C₂³×C₃×C₈	D₃
hypocenter	C₃	C₃×#1810
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×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. C₂×C₃×#819	C₂	semidirect
4. C₃×#4552	C₂	semidirect
6. #71977	C₃	direct (N × H)
21. C₂×#819	C₂×C₃	semidirect
23. #4552	C₂×C₃	semidirect
127. C₂²×C₃²	#108	semidirect
223. C₂²×C₃	#819	semidirect
224. C₂²×C₃	C₃×#108	semidirect
241. C₃²	#1810	semidirect
274. C₂²	C₃×#819	semidirect
282. C₃	#71977	direct (N × H)
283. C₃	C₃×#1810	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
C₃×#5207	2
C₂×C₃×#819	2
C₃×#4552	2
C₃×#1810	3
#71977	3

Subgroups — C3×#71977 #11785434

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#71977 #11785434