Subgroups — C₃×#69392 #11785958

Order 1152.

649 conjugacy classes of subgroups.

The subgroup lattice has 277 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²×C₄³	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×C₄⋊(C₃⋊C₈)	C₂	1	2	yes	maximal
3	C₃×#4524	C₂	1	4	yes	maximal
4	C₃×#1831	—	3	1	no	maximal
5	#69392	C₃	1	1	yes	maximal, characteristic, direct product factor
6	C₂×C₃²×C₄²	C₄	1	2	yes	—
7	C₂×C₃²×C₄²	C₄	1	1	yes	characteristic
8	C₂×C₃²×C₄²	C₄	1	1	yes	characteristic
9	C₂×C₃²×C₄²	C₂²	1	2	yes	—
10	C₂×C₃²×C₄²	C₂²	1	1	yes	characteristic
11	C₂²×C₃×C₃⋊C₈	C₂²	1	4	yes	—
12	C₃×C₄⋊(C₃⋊C₈)	—	2	4	no	—
13	C₂×C₃×C₄⋊C₈	—	3	2	no	—
14	C₃×C₄³	—	2	1	no	—
15	C₃×C₄³	D₃	1	1	yes	characteristic
16	C₃×C₄³	C₂×C₃	1	1	yes	characteristic
17	C₂×C₄⋊(C₃⋊C₈)	C₂×C₃	1	2	yes	—
18	C₃×#336	—	3	4	no	—
19	#4524	C₂×C₃	1	4	yes	—
20	C₂²×C₃²×C₄	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₃²×C₄	C₂×C₄	1	2	yes	—
22	C₂²×C₃²×C₄	C₂×C₄	1	2	yes	—
23	C₂²×C₃²×C₄	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₃²×C₄	C₂³	1	1	yes	characteristic
25	C₃²×C₄²	—	2	4	no	—
26	C₃²×C₄²	—	2	2	no	—
27	C₃²×C₄²	—	2	4	no	—
28	C₃²×C₄²	C₈	1	4	yes	complemented
29	C₃²×C₄²	D₄	1	2	yes	—
30	C₃²×C₄²	Q₈	1	2	yes	—
31	C₂×C₃×C₃⋊C₈	—	2	4	no	—
32	C₂×C₃×C₃⋊C₈	—	2	8	no	—
33	#1831	—	3	1	no	Sylow
34	C₂²×C₃×C₈	—	3	4	no	—
35	C₂×C₃×C₄²	—	2	2	no	—
36	C₂×C₃×C₄²	—	2	1	no	—
37	C₂×C₃×C₄²	—	2	1	no	—
38	C₂×C₃×C₄²	—	2	2	no	—
39	C₂×C₃×C₄²	—	2	1	no	—
40	C₂×C₃×C₄²	Dic₃	1	2	yes	—
41	C₂×C₃×C₄²	Dic₃	1	1	yes	characteristic
42	C₂×C₃×C₄²	Dic₃	1	1	yes	characteristic
43	C₂×C₃×C₄²	C₃×C₄	1	2	yes	—
44	C₂×C₃×C₄²	C₃×C₄	1	1	yes	characteristic
45	C₂×C₃×C₄²	C₃×C₄	1	1	yes	characteristic
46	C₂×C₃×C₄²	C₂×D₃	1	2	yes	—
47	C₂×C₃×C₄²	C₂×D₃	1	1	yes	characteristic
48	C₂×C₃×C₄²	C₂²×C₃	1	2	yes	—
49	C₂×C₃×C₄²	C₂²×C₃	1	1	yes	characteristic
50	C₂²×C₃⋊C₈	C₂²×C₃	1	4	yes	—
51	C₃×C₄⋊C₈	—	6	4	no	—
52	C₄⋊(C₃⋊C₈)	—	2	4	no	—
53	C₂³×C₃²	C₂²×C₄	1	1	yes	characteristic
54	C₂×C₃²×C₄	—	2	2	no	—
55	C₂×C₃²×C₄	—	2	2	no	—
56	C₂×C₃²×C₄	—	2	2	no	—
57	C₂×C₃²×C₄	—	2	2	no	—
58	C₂×C₃²×C₄	—	2	2	no	—
59	C₂×C₃²×C₄	C₄⋊C₄	1	2	yes	—
60	C₂×C₃²×C₄	C₄⋊C₄	1	2	yes	—
61	C₂×C₃²×C₄	C₂×C₈	1	4	yes	—
62	C₂×C₃²×C₄	C₂×C₈	1	2	yes	—
63	C₂×C₃²×C₄	M₁₆	1	4	yes	—
64	C₂×C₃²×C₄	M₁₆	1	2	yes	—
65	C₂×C₃²×C₄	C₂×D₄	1	1	yes	characteristic
66	C₂×C₃²×C₄	C₂×Q₈	1	1	yes	characteristic
67	C₂×C₃²×C₄	#41	1	4	yes	—
68	C₃×C₃⋊C₈	—	4	4	no	—
69	C₂×C₄⋊C₈	—	3	2	no	—
70	C₄³	C₃×D₃	1	1	yes	characteristic
71	#336	—	3	4	no	—
72	C₂²×C₃×C₄	—	2	2	no	—
73	C₂²×C₃×C₄	—	2	2	no	—
74	C₂²×C₃×C₄	—	2	1	no	—
75	C₂²×C₃×C₄	—	2	1	no	—
76	C₂²×C₃×C₄	—	2	1	no	—
77	C₂²×C₃×C₄	C₂×Dic₃	1	2	yes	—
78	C₂²×C₃×C₄	C₂×Dic₃	1	2	yes	—
79	C₂²×C₃×C₄	C₂×Dic₃	1	1	yes	characteristic
80	C₂²×C₃×C₄	C₂×Dic₃	1	1	yes	characteristic
81	C₂²×C₃×C₄	C₂×C₃×C₄	1	1	yes	characteristic
82	C₂²×C₃×C₄	C₂×C₃×C₄	1	2	yes	—
83	C₂²×C₃×C₄	C₂×C₃×C₄	1	2	yes	—
84	C₂²×C₃×C₄	C₂×C₃×C₄	1	1	yes	characteristic
85	C₂²×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
86	C₂²×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
87	C₂×C₃×C₈	—	6	4	no	—
88	C₂×C₃×C₈	—	6	8	no	—
89	C₃×C₄²	—	2	8	no	—
90	C₃×C₄²	—	2	4	no	—
91	C₃×C₄²	—	2	8	no	—
92	C₃×C₄²	—	2	4	no	—
93	C₃×C₄²	—	2	2	no	—
94	C₃×C₄²	—	2	4	no	—
95	C₃×C₄²	—	2	2	no	—
96	C₃×C₄²	—	2	4	no	—
97	C₃×C₄²	—	2	4	no	—
98	C₃×C₄²	—	2	4	no	—
99	C₃×C₄²	—	2	2	no	—
100	C₃×C₄²	—	2	2	no	—
101	C₃×C₄²	C₃⋊C₈	1	4	yes	complemented
102	C₃×C₄²	C₃×C₈	1	4	yes	complemented
103	C₃×C₄²	Dic₆	1	2	yes	—
104	C₃×C₄²	D₁₂	1	2	yes	—
105	C₃×C₄²	C₃×D₄	1	2	yes	—
106	C₃×C₄²	C₃×Q₈	1	2	yes	—
107	C₂×C₃⋊C₈	—	2	4	no	—
108	C₂×C₃⋊C₈	—	2	8	no	—
109	C₂²×C₃²	C₂×C₄⋊C₄	1	1	yes	characteristic
110	C₂²×C₃²	#117	1	2	yes	—
111	C₂²×C₃²	C₂²×C₈	1	1	yes	characteristic
112	C₂²×C₃²	C₂×M₁₆	1	1	yes	characteristic
113	C₂²×C₃²	C₂×M₁₆	1	2	yes	—
114	C₃²×C₄	—	2	2	no	—
115	C₃²×C₄	—	2	4	no	—
116	C₃²×C₄	—	2	4	no	—
117	C₃²×C₄	C₄⋊C₈	1	4	yes	—
118	C₃²×C₄	#124	1	2	yes	—
119	C₃²×C₄	#125	1	2	yes	—
120	C₂²×C₈	—	3	4	no	—
121	C₂×C₄²	C₃×Dic₃	1	2	yes	—
122	C₂×C₄²	C₃×Dic₃	1	1	yes	characteristic
123	C₂×C₄²	C₃×Dic₃	1	1	yes	characteristic
124	C₂×C₄²	C₂×C₃×D₃	1	2	yes	—
125	C₂×C₄²	C₂×C₃×D₃	1	1	yes	characteristic
126	C₄⋊C₈	—	6	4	no	—
127	C₂³×C₃	—	2	1	no	—
128	C₂³×C₃	C₂²×Dic₃	1	1	yes	characteristic
129	C₂³×C₃	C₂²×C₃×C₄	1	1	yes	characteristic
130	C₂×C₃×C₄	—	2	4	no	—
131	C₂×C₃×C₄	—	2	4	no	—
132	C₂×C₃×C₄	—	2	4	no	—
133	C₂×C₃×C₄	—	2	4	no	—
134	C₂×C₃×C₄	—	2	4	no	—
135	C₂×C₃×C₄	—	2	2	no	—
136	C₂×C₃×C₄	—	2	2	no	—
137	C₂×C₃×C₄	—	2	2	no	—
138	C₂×C₃×C₄	—	2	2	no	—
139	C₂×C₃×C₄	—	2	2	no	—
140	C₂×C₃×C₄	—	2	2	no	—
141	C₂×C₃×C₄	—	2	2	no	—
142	C₂×C₃×C₄	—	2	2	no	—
143	C₂×C₃×C₄	—	2	2	no	—
144	C₂×C₃×C₄	—	2	2	no	—
145	C₂×C₃×C₄	—	2	4	no	—
146	C₂×C₃×C₄	—	2	2	no	—
147	C₂×C₃×C₄	—	2	4	no	—
148	C₂×C₃×C₄	—	2	2	no	—
149	C₂×C₃×C₄	—	2	2	no	—
150	C₂×C₃×C₄	—	2	2	no	—
151	C₂×C₃×C₄	—	2	1	no	—
152	C₂×C₃×C₄	—	2	1	no	—
153	C₂×C₃×C₄	—	2	4	no	—
154	C₂×C₃×C₄	C₂×C₃⋊C₈	1	4	yes	—
155	C₂×C₃×C₄	C₂×C₃⋊C₈	1	2	yes	—
156	C₂×C₃×C₄	#208	1	4	yes	—
157	C₂×C₃×C₄	#208	1	2	yes	—
158	C₂×C₃×C₄	C₄⋊Dic₃	1	2	yes	—
159	C₂×C₃×C₄	C₄⋊Dic₃	1	2	yes	—
160	C₂×C₃×C₄	C₃×C₄⋊C₄	1	2	yes	—
161	C₂×C₃×C₄	C₃×C₄⋊C₄	1	2	yes	—
162	C₂×C₃×C₄	C₂×C₃×C₈	1	4	yes	—
163	C₂×C₃×C₄	C₂×C₃×C₈	1	2	yes	—
164	C₂×C₃×C₄	C₃×M₁₆	1	4	yes	—
165	C₂×C₃×C₄	C₃×M₁₆	1	2	yes	—
166	C₂×C₃×C₄	C₂×Dic₆	1	1	yes	characteristic
167	C₂×C₃×C₄	C₂×D₁₂	1	1	yes	characteristic
168	C₂×C₃×C₄	#235	1	4	yes	—
169	C₂×C₃×C₄	C₂×C₃×D₄	1	1	yes	characteristic
170	C₂×C₃×C₄	C₂×C₃×Q₈	1	1	yes	characteristic
171	C₂×C₃×C₄	C₃×#41	1	4	yes	—
172	C₃×C₈	—	12	4	no	—
173	C₃⋊C₈	—	4	4	no	—
174	C₂×C₃²	#382	1	1	yes	characteristic
175	C₂×C₃²	C₂×C₄⋊C₈	1	1	yes	characteristic
176	C₂×C₃²	#423	1	1	yes	characteristic
177	C₂×C₃²	#431	1	2	yes	—
178	C₂×C₃²	#432	1	2	yes	—
179	C₂²×C₄	C₂×C₃×Dic₃	1	2	yes	—
180	C₂²×C₄	C₂×C₃×Dic₃	1	2	yes	—
181	C₂²×C₄	C₂×C₃×Dic₃	1	1	yes	characteristic
182	C₂²×C₄	C₂×C₃×Dic₃	1	1	yes	characteristic
183	C₂²×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
184	C₂×C₈	—	6	4	no	—
185	C₂×C₈	—	6	8	no	—
186	C₄²	—	2	4	no	—
187	C₄²	—	2	2	no	—
188	C₄²	—	2	4	no	—
189	C₄²	C₃×C₃⋊C₈	1	4	yes	complemented
190	C₄²	C₃×Dic₆	1	2	yes	—
191	C₄²	C₃×D₁₂	1	2	yes	—
192	C₂²×C₃	—	2	1	no	—
193	C₂²×C₃	—	2	1	no	—
194	C₂²×C₃	—	2	2	no	—
195	C₂²×C₃	—	2	1	no	—
196	C₂²×C₃	—	2	2	no	—
197	C₂²×C₃	C₂²×C₃⋊C₈	1	1	yes	characteristic
198	C₂²×C₃	C₂×#208	1	1	yes	characteristic
199	C₂²×C₃	C₂×#208	1	2	yes	—
200	C₂²×C₃	C₂×C₄⋊Dic₃	1	1	yes	characteristic
201	C₂²×C₃	#926	1	2	yes	—
202	C₂²×C₃	C₂×C₃×C₄⋊C₄	1	1	yes	characteristic
203	C₂²×C₃	C₃×#117	1	2	yes	—
204	C₂²×C₃	C₂²×C₃×C₈	1	1	yes	characteristic
205	C₂²×C₃	C₂×C₃×M₁₆	1	1	yes	characteristic
206	C₂²×C₃	C₂×C₃×M₁₆	1	2	yes	—
207	C₃×C₄	—	2	4	no	—
208	C₃×C₄	—	2	8	no	—
209	C₃×C₄	—	2	8	no	—
210	C₃×C₄	—	2	2	no	—
211	C₃×C₄	—	2	4	no	—
212	C₃×C₄	—	2	4	no	—
213	C₃×C₄	—	2	2	no	—
214	C₃×C₄	—	2	4	no	—
215	C₃×C₄	—	2	4	no	—
216	C₃×C₄	—	2	4	no	—
217	C₃×C₄	—	2	2	no	—
218	C₃×C₄	—	2	2	no	—
219	C₃×C₄	C₄⋊(C₃⋊C₈)	1	4	yes	—
220	C₃×C₄	C₃×C₄⋊C₈	1	4	yes	—
221	C₃×C₄	#870	1	2	yes	—
222	C₃×C₄	#875	1	2	yes	—
223	C₃×C₄	C₃×#124	1	2	yes	—
224	C₃×C₄	C₃×#125	1	2	yes	—
225	C₃²	#1831	1	1	yes	characteristic, complemented, Sylow
226	C₂³	C₂²×C₃×Dic₃	1	1	yes	characteristic
227	C₂×C₄	—	2	2	no	—
228	C₂×C₄	—	2	2	no	—
229	C₂×C₄	—	2	2	no	—
230	C₂×C₄	—	2	2	no	—
231	C₂×C₄	—	2	2	no	—
232	C₂×C₄	C₂×C₃×C₃⋊C₈	1	4	yes	—
233	C₂×C₄	C₂×C₃×C₃⋊C₈	1	2	yes	—
234	C₂×C₄	C₃×#208	1	4	yes	—
235	C₂×C₄	C₃×#208	1	2	yes	—
236	C₂×C₄	C₃×C₄⋊Dic₃	1	2	yes	—
237	C₂×C₄	C₃×C₄⋊Dic₃	1	2	yes	—
238	C₂×C₄	C₂×C₃×Dic₆	1	1	yes	characteristic
239	C₂×C₄	C₂×C₃×D₁₂	1	1	yes	characteristic
240	C₂×C₄	C₃×#235	1	4	yes	—
241	C₈	—	12	4	no	—
242	C₂×C₃	—	2	1	no	—
243	C₂×C₃	—	2	1	no	—
244	C₂×C₃	—	2	2	no	—
245	C₂×C₃	—	2	2	no	—
246	C₂×C₃	—	2	1	no	—
247	C₂×C₃	C₂×C₄⋊(C₃⋊C₈)	1	1	yes	characteristic
248	C₂×C₃	#4924	1	1	yes	characteristic
249	C₂×C₃	#4925	1	2	yes	—
250	C₂×C₃	#4926	1	2	yes	—
251	C₂×C₃	#4936	1	1	yes	characteristic
252	C₂×C₃	C₃×#382	1	1	yes	characteristic
253	C₂×C₃	C₂×C₃×C₄⋊C₈	1	1	yes	characteristic
254	C₂×C₃	C₃×#423	1	1	yes	characteristic
255	C₂×C₃	C₃×#431	1	2	yes	—
256	C₂×C₃	C₃×#432	1	2	yes	—
257	C₂²	C₂²×C₃×C₃⋊C₈	1	1	yes	characteristic
258	C₂²	C₂×C₃×#208	1	1	yes	characteristic
259	C₂²	C₂×C₃×#208	1	2	yes	—
260	C₂²	C₂×C₃×C₄⋊Dic₃	1	1	yes	characteristic
261	C₂²	C₃×#926	1	2	yes	—
262	C₄	—	2	2	no	—
263	C₄	—	2	4	no	—
264	C₄	—	2	4	no	—
265	C₄	C₃×C₄⋊(C₃⋊C₈)	1	4	yes	—
266	C₄	C₃×#870	1	2	yes	—
267	C₄	C₃×#875	1	2	yes	—
268	C₃	—	2	1	no	—
269	C₃	#69392	1	1	yes	characteristic, direct product factor
270	C₃	C₃×#1831	1	1	yes	characteristic, complemented
271	C₂	C₂×C₃×C₄⋊(C₃⋊C₈)	1	1	yes	characteristic
272	C₂	C₃×#4924	1	1	yes	characteristic
273	C₂	C₃×#4925	1	2	yes	—
274	C₂	C₃×#4926	1	2	yes	—
275	C₂	C₃×#4936	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₃×C₄	C₂²×D₃
commutator	C₂²×C₃	C₂²×C₃×C₈
Frattini	C₂²×C₄	C₂²×C₃×D₃
Fitting	C₃²×C₄³	C₂
socle, prime radical	C₂³×C₃²	C₂²×C₄
cosocle	C₂²×C₃×C₄	C₂³×C₃
hypercenter	C₃×C₄³	D₃
hypocenter	C₃	C₃×#1831
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₃×C₄ ◁ C₃×C₄³

Complements

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

N	H ≅ G/N	type
5. #69392	C₃	direct (N × H)
28. C₃²×C₄²	C₈	semidirect
101. C₃×C₄²	C₃⋊C₈	semidirect
102. C₃×C₄²	C₃×C₈	semidirect
189. C₄²	C₃×C₃⋊C₈	semidirect
225. C₃²	#1831	semidirect
269. C₃	#69392	direct (N × H)
270. C₃	C₃×#1831	semidirect

Maximal subgroups

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

subgroup	index
C₃²×C₄³	2
C₂×C₃×C₄⋊(C₃⋊C₈)	2
C₃×#4524	2
C₃×#1831	3
#69392	3

Subgroups — C3×#69392 #11785958

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#69392 #11785958