Subgroups — C₃×#69766 #11786252

Order 1152.

577 conjugacy classes of subgroups.

The subgroup lattice has 256 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃²×C₄×C₈	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×#818	C₂	1	2	yes	maximal
3	C₂×C₃×#817	C₂	1	2	yes	maximal
4	C₃×#4936	C₂	1	2	yes	maximal
5	C₃×#1846	—	3	1	no	maximal
6	#69766	C₃	1	1	yes	maximal, characteristic, direct product factor
7	C₂²×C₃²×C₈	C₂²	1	1	yes	characteristic
8	C₂²×C₃²×C₈	C₂²	1	1	yes	characteristic
9	C₂×C₃²×C₄²	C₂²	1	1	yes	characteristic
10	C₂×C₃×C₄⋊Dic₃	C₂²	1	4	yes	—
11	C₃²×C₄×C₈	C₄	1	4	yes	complemented
12	C₃×C₃⋊#95	—	2	4	no	—
13	C₃×#818	—	2	4	no	—
14	C₃×#817	—	2	4	no	—
15	C₂×C₃×#107	—	3	2	no	—
16	C₂×C₃×#106	—	3	2	no	—
17	C₂×C₃×C₄×C₈	—	2	1	no	—
18	C₂×C₃×C₄×C₈	D₃	1	1	yes	characteristic
19	C₂×C₃×C₄×C₈	C₂×C₃	1	1	yes	characteristic
20	C₃×#382	—	3	2	no	—
21	C₂×#818	C₂×C₃	1	2	yes	—
22	C₂×#817	C₂×C₃	1	2	yes	—
23	#4936	C₂×C₃	1	2	yes	—
24	C₂²×C₃²×C₄	D₄	1	2	yes	—
25	C₂²×C₃²×C₄	C₂³	1	1	yes	characteristic
26	C₂²×C₃×Dic₃	—	2	4	no	—
27	C₂×C₃²×C₈	C₂×C₄	1	2	yes	—
28	C₂×C₃²×C₈	C₂×C₄	1	2	yes	—
29	C₂×C₃²×C₈	D₄	1	2	yes	—
30	C₂×C₃²×C₈	D₄	1	2	yes	—
31	C₂×C₃²×C₈	Q₈	1	2	yes	—
32	C₂×C₃²×C₈	Q₈	1	2	yes	—
33	C₃²×C₄²	C₂×C₄	1	2	yes	—
34	C₃²×C₄²	Q₈	1	2	yes	—
35	C₃×C₄⋊Dic₃	—	2	4	no	—
36	C₃×C₄⋊Dic₃	—	2	4	no	—
37	#1846	—	3	1	no	Sylow
38	C₂²×C₃×C₈	—	2	1	no	—
39	C₂²×C₃×C₈	—	2	1	no	—
40	C₂²×C₃×C₈	C₂×D₃	1	1	yes	characteristic
41	C₂²×C₃×C₈	C₂×D₃	1	1	yes	characteristic
42	C₂²×C₃×C₈	C₂²×C₃	1	1	yes	characteristic
43	C₂²×C₃×C₈	C₂²×C₃	1	1	yes	characteristic
44	C₂×C₃×C₄⋊C₄	—	3	4	no	—
45	C₂×C₃×C₄²	—	2	1	no	—
46	C₂×C₃×C₄²	C₂×D₃	1	1	yes	characteristic
47	C₂×C₃×C₄²	C₂²×C₃	1	1	yes	characteristic
48	C₂×C₄⋊Dic₃	C₂²×C₃	1	4	yes	—
49	C₃×#107	—	6	4	no	—
50	C₃×#106	—	6	4	no	—
51	C₃×C₄×C₈	—	2	4	no	—
52	C₃×C₄×C₈	Dic₃	1	4	yes	complemented
53	C₃×C₄×C₈	C₃×C₄	1	4	yes	complemented
54	C₃×#95	—	6	4	no	—
55	C₃⋊#95	—	2	4	no	—
56	#818	—	2	4	no	—
57	#817	—	2	4	no	—
58	C₂³×C₃²	C₂×D₄	1	1	yes	characteristic
59	C₂×C₃²×C₄	—	2	4	no	—
60	C₂×C₃²×C₄	C₄⋊C₄	1	4	yes	—
61	C₂×C₃²×C₄	C₂²×C₄	1	1	yes	characteristic
62	C₂×C₃²×C₄	C₂×D₄	1	1	yes	characteristic
63	C₂×C₃²×C₄	C₂×D₄	1	1	yes	characteristic
64	C₂×C₃²×C₄	C₂×Q₈	1	1	yes	characteristic
65	C₂×C₃²×C₄	C₂×Q₈	1	1	yes	characteristic
66	C₂×C₃²×C₄	C₂×Q₈	1	1	yes	characteristic
67	C₂×C₃×Dic₃	—	4	4	no	—
68	C₂×C₃×Dic₃	—	4	8	no	—
69	C₃²×C₈	C₄⋊C₄	1	4	yes	—
70	C₃²×C₈	C₄⋊C₄	1	4	yes	—
71	C₂×#107	—	3	2	no	—
72	C₂×#106	—	3	2	no	—
73	C₂×C₄×C₈	C₃×D₃	1	1	yes	characteristic
74	#382	—	3	2	no	—
75	C₂²×C₃×C₄	—	2	2	no	—
76	C₂²×C₃×C₄	—	2	1	no	—
77	C₂²×C₃×C₄	—	6	4	no	—
78	C₂²×C₃×C₄	D₁₂	1	2	yes	—
79	C₂²×C₃×C₄	C₃×D₄	1	2	yes	—
80	C₂²×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
81	C₂²×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
82	C₂²×Dic₃	—	2	4	no	—
83	C₂×C₃×C₈	—	2	2	no	—
84	C₂×C₃×C₈	—	2	2	no	—
85	C₂×C₃×C₈	—	2	2	no	—
86	C₂×C₃×C₈	—	2	2	no	—
87	C₂×C₃×C₈	—	2	2	no	—
88	C₂×C₃×C₈	—	2	2	no	—
89	C₂×C₃×C₈	Dic₆	1	2	yes	—
90	C₂×C₃×C₈	Dic₆	1	2	yes	—
91	C₂×C₃×C₈	D₁₂	1	2	yes	—
92	C₂×C₃×C₈	D₁₂	1	2	yes	—
93	C₂×C₃×C₈	C₂×Dic₃	1	2	yes	—
94	C₂×C₃×C₈	C₂×Dic₃	1	2	yes	—
95	C₂×C₃×C₈	C₂×C₃×C₄	1	2	yes	—
96	C₂×C₃×C₈	C₂×C₃×C₄	1	2	yes	—
97	C₂×C₃×C₈	C₃×D₄	1	2	yes	—
98	C₂×C₃×C₈	C₃×D₄	1	2	yes	—
99	C₂×C₃×C₈	C₃×Q₈	1	2	yes	—
100	C₂×C₃×C₈	C₃×Q₈	1	2	yes	—
101	C₃×C₄⋊C₄	—	6	4	no	—
102	C₃×C₄⋊C₄	—	6	4	no	—
103	C₃×C₄²	—	2	2	no	—
104	C₃×C₄²	—	2	2	no	—
105	C₃×C₄²	Dic₆	1	2	yes	—
106	C₃×C₄²	C₂×Dic₃	1	2	yes	—
107	C₃×C₄²	C₂×C₃×C₄	1	2	yes	—
108	C₃×C₄²	C₃×Q₈	1	2	yes	—
109	C₄⋊Dic₃	—	2	4	no	—
110	C₄⋊Dic₃	—	2	4	no	—
111	C₂²×C₃²	C₂×C₄⋊C₄	1	1	yes	characteristic
112	C₂²×C₃²	#127	1	1	yes	characteristic
113	C₂²×C₃²	C₄⋊Q₈	1	1	yes	characteristic
114	C₂²×C₃²	#135	1	4	yes	—
115	C₃²×C₄	—	2	4	no	—
116	C₃²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
117	C₃²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
118	C₃²×C₄	C₄⋊Q₈	1	1	yes	characteristic
119	C₃²×C₄	C₄⋊Q₈	1	1	yes	characteristic
120	C₃×Dic₃	—	8	4	no	—
121	C₂²×C₈	C₂×C₃×D₃	1	1	yes	characteristic
122	C₂²×C₈	C₂×C₃×D₃	1	1	yes	characteristic
123	C₂×C₄⋊C₄	—	3	4	no	—
124	C₂×C₄²	C₂×C₃×D₃	1	1	yes	characteristic
125	#107	—	6	4	no	—
126	#106	—	6	4	no	—
127	C₄×C₈	C₃×Dic₃	1	4	yes	complemented
128	#95	—	6	4	no	—
129	C₂³×C₃	—	2	1	no	—
130	C₂³×C₃	C₂×D₁₂	1	1	yes	characteristic
131	C₂³×C₃	C₂×C₃×D₄	1	1	yes	characteristic
132	C₂×C₃×C₄	—	2	8	no	—
133	C₂×C₃×C₄	—	2	4	no	—
134	C₂×C₃×C₄	—	2	4	no	—
135	C₂×C₃×C₄	—	2	4	no	—
136	C₂×C₃×C₄	—	2	1	no	—
137	C₂×C₃×C₄	—	2	1	no	—
138	C₂×C₃×C₄	—	2	1	no	—
139	C₂×C₃×C₄	—	2	1	no	—
140	C₂×C₃×C₄	—	2	1	no	—
141	C₂×C₃×C₄	—	2	1	no	—
142	C₂×C₃×C₄	—	12	4	no	—
143	C₂×C₃×C₄	—	12	8	no	—
144	C₂×C₃×C₄	C₄⋊Dic₃	1	4	yes	—
145	C₂×C₃×C₄	C₃×C₄⋊C₄	1	4	yes	—
146	C₂×C₃×C₄	C₂×Dic₆	1	1	yes	characteristic
147	C₂×C₃×C₄	C₂×Dic₆	1	1	yes	characteristic
148	C₂×C₃×C₄	C₂×Dic₆	1	1	yes	characteristic
149	C₂×C₃×C₄	C₂×D₁₂	1	1	yes	characteristic
150	C₂×C₃×C₄	C₂×D₁₂	1	1	yes	characteristic
151	C₂×C₃×C₄	C₂²×Dic₃	1	1	yes	characteristic
152	C₂×C₃×C₄	C₂²×C₃×C₄	1	1	yes	characteristic
153	C₂×C₃×C₄	C₂×C₃×D₄	1	1	yes	characteristic
154	C₂×C₃×C₄	C₂×C₃×D₄	1	1	yes	characteristic
155	C₂×C₃×C₄	C₂×C₃×Q₈	1	1	yes	characteristic
156	C₂×C₃×C₄	C₂×C₃×Q₈	1	1	yes	characteristic
157	C₂×C₃×C₄	C₂×C₃×Q₈	1	1	yes	characteristic
158	C₂×Dic₃	—	4	4	no	—
159	C₂×Dic₃	—	4	8	no	—
160	C₃×C₈	—	2	4	no	—
161	C₃×C₈	—	2	4	no	—
162	C₃×C₈	C₄⋊Dic₃	1	4	yes	—
163	C₃×C₈	C₄⋊Dic₃	1	4	yes	—
164	C₃×C₈	C₃×C₄⋊C₄	1	4	yes	—
165	C₃×C₈	C₃×C₄⋊C₄	1	4	yes	—
166	C₂×C₃²	#384	1	1	yes	characteristic
167	C₂×C₃²	#427	1	2	yes	—
168	C₂×C₃²	#495	1	2	yes	—
169	C₂×C₃²	#499	1	2	yes	—
170	C₂²×C₄	—	6	4	no	—
171	C₂²×C₄	C₃×D₁₂	1	2	yes	—
172	C₂²×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
173	C₂×C₈	C₃×Dic₆	1	2	yes	—
174	C₂×C₈	C₃×Dic₆	1	2	yes	—
175	C₂×C₈	C₃×D₁₂	1	2	yes	—
176	C₂×C₈	C₃×D₁₂	1	2	yes	—
177	C₂×C₈	C₂×C₃×Dic₃	1	2	yes	—
178	C₂×C₈	C₂×C₃×Dic₃	1	2	yes	—
179	C₄⋊C₄	—	6	4	no	—
180	C₄⋊C₄	—	6	4	no	—
181	C₄²	C₃×Dic₆	1	2	yes	—
182	C₄²	C₂×C₃×Dic₃	1	2	yes	—
183	C₂²×C₃	—	2	4	no	—
184	C₂²×C₃	—	2	1	no	—
185	C₂²×C₃	—	2	1	no	—
186	C₂²×C₃	—	2	1	no	—
187	C₂²×C₃	#869	1	1	yes	characteristic
188	C₂²×C₃	#874	1	1	yes	characteristic
189	C₂²×C₃	#904	1	4	yes	—
190	C₂²×C₃	C₂×C₄⋊Dic₃	1	1	yes	characteristic
191	C₂²×C₃	C₂×C₃×C₄⋊C₄	1	1	yes	characteristic
192	C₂²×C₃	C₃×#127	1	1	yes	characteristic
193	C₂²×C₃	C₃×C₄⋊Q₈	1	1	yes	characteristic
194	C₂²×C₃	C₃×#135	1	4	yes	—
195	C₃×C₄	—	2	8	no	—
196	C₃×C₄	—	2	4	no	—
197	C₃×C₄	—	2	4	no	—
198	C₃×C₄	—	2	1	no	—
199	C₃×C₄	—	2	1	no	—
200	C₃×C₄	—	2	1	no	—
201	C₃×C₄	—	2	1	no	—
202	C₃×C₄	—	24	4	no	—
203	C₃×C₄	#869	1	1	yes	characteristic
204	C₃×C₄	#869	1	1	yes	characteristic
205	C₃×C₄	C₂×C₄⋊Dic₃	1	1	yes	characteristic
206	C₃×C₄	C₂×C₄⋊Dic₃	1	1	yes	characteristic
207	C₃×C₄	C₂×C₃×C₄⋊C₄	1	1	yes	characteristic
208	C₃×C₄	C₂×C₃×C₄⋊C₄	1	1	yes	characteristic
209	C₃×C₄	C₃×C₄⋊Q₈	1	1	yes	characteristic
210	C₃×C₄	C₃×C₄⋊Q₈	1	1	yes	characteristic
211	Dic₃	—	8	4	no	—
212	C₃²	#1846	1	1	yes	characteristic, complemented, Sylow
213	C₂³	C₂×C₃×D₁₂	1	1	yes	characteristic
214	C₂×C₄	—	2	4	no	—
215	C₂×C₄	—	12	8	no	—
216	C₂×C₄	—	12	4	no	—
217	C₂×C₄	C₃×C₄⋊Dic₃	1	4	yes	—
218	C₂×C₄	C₂×C₃×Dic₆	1	1	yes	characteristic
219	C₂×C₄	C₂×C₃×Dic₆	1	1	yes	characteristic
220	C₂×C₄	C₂×C₃×Dic₆	1	1	yes	characteristic
221	C₂×C₄	C₂×C₃×D₁₂	1	1	yes	characteristic
222	C₂×C₄	C₂×C₃×D₁₂	1	1	yes	characteristic
223	C₂×C₄	C₂²×C₃×Dic₃	1	1	yes	characteristic
224	C₈	C₃×C₄⋊Dic₃	1	4	yes	—
225	C₈	C₃×C₄⋊Dic₃	1	4	yes	—
226	C₂×C₃	—	2	1	no	—
227	C₂×C₃	—	2	2	no	—
228	C₂×C₃	—	2	2	no	—
229	C₂×C₃	—	2	2	no	—
230	C₂×C₃	#4683	1	2	yes	—
231	C₂×C₃	#4696	1	2	yes	—
232	C₂×C₃	#4935	1	1	yes	characteristic
233	C₂×C₃	#5106	1	2	yes	—
234	C₂×C₃	C₃×#384	1	1	yes	characteristic
235	C₂×C₃	C₃×#427	1	2	yes	—
236	C₂×C₃	C₃×#495	1	2	yes	—
237	C₂×C₃	C₃×#499	1	2	yes	—
238	C₂²	C₃×#869	1	1	yes	characteristic
239	C₂²	C₃×#874	1	1	yes	characteristic
240	C₂²	C₃×#904	1	4	yes	—
241	C₂²	C₂×C₃×C₄⋊Dic₃	1	1	yes	characteristic
242	C₄	—	2	4	no	—
243	C₄	—	24	4	no	—
244	C₄	C₃×#869	1	1	yes	characteristic
245	C₄	C₃×#869	1	1	yes	characteristic
246	C₄	C₂×C₃×C₄⋊Dic₃	1	1	yes	characteristic
247	C₄	C₂×C₃×C₄⋊Dic₃	1	1	yes	characteristic
248	C₃	—	2	1	no	—
249	C₃	#69766	1	1	yes	characteristic, direct product factor
250	C₃	C₃×#1846	1	1	yes	characteristic, complemented
251	C₂	C₃×#4683	1	2	yes	—
252	C₂	C₃×#4696	1	2	yes	—
253	C₂	C₃×#4935	1	1	yes	characteristic
254	C₂	C₃×#5106	1	2	yes	—

	subgroup	G/H
center	C₂³×C₃	C₂×D₁₂
commutator	C₂×C₃×C₄	C₂²×C₃×C₄
Frattini	C₂²×C₄	C₂²×C₃×D₃
Fitting	C₂×C₃²×C₄×C₈	C₂
socle, prime radical	C₂³×C₃²	C₂×D₄
cosocle	C₂²×C₃×C₄	C₂³×C₃
hypercenter	C₂×C₃×C₄×C₈	D₃
hypocenter	C₃	C₃×#1846
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂³×C₃ ◁ C₂×C₃×C₄² ◁ C₂×C₃×C₄×C₈

Complements

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

N	H ≅ G/N	type
6. #69766	C₃	direct (N × H)
11. C₃²×C₄×C₈	C₄	semidirect
52. C₃×C₄×C₈	Dic₃	semidirect
53. C₃×C₄×C₈	C₃×C₄	semidirect
127. C₄×C₈	C₃×Dic₃	semidirect
212. C₃²	#1846	semidirect
249. C₃	#69766	direct (N × H)
250. C₃	C₃×#1846	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃²×C₄×C₈	2
C₂×C₃×#818	2
C₂×C₃×#817	2
C₃×#4936	2
C₃×#1846	3
#69766	3

Subgroups — C3×#69766 #11786252

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#69766 #11786252