Subgroups — C₃×#69577 #11786226

Order 1152.

653 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	C₃²×#384	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×#807	C₂	1	4	yes	maximal
3	C₃×#4935	C₂	1	1	yes	maximal, characteristic
4	C₂×C₃×C₄×C₃⋊C₈	C₂	1	1	yes	maximal, characteristic
5	C₃×#1845	—	3	1	no	maximal
6	#69577	C₃	1	1	yes	maximal, characteristic, direct product factor
7	C₂×C₃²×C₄⋊C₄	—	2	2	no	—
8	C₂×C₃²×C₄⋊C₄	C₂²	1	2	yes	—
9	C₂×C₃²×C₄²	C₂²	1	1	yes	characteristic
10	C₂×C₃×C₄⋊Dic₃	—	2	2	no	—
11	C₂×C₃×C₄⋊Dic₃	C₂²	1	2	yes	—
12	C₂²×C₃×C₃⋊C₈	C₂²	1	2	yes	—
13	C₃×#807	—	2	8	no	—
14	C₃×C₄×C₃⋊C₈	C₄	1	4	yes	complemented
15	C₂×C₃×#107	—	3	4	no	—
16	C₂×C₃×C₄×C₈	—	3	1	no	—
17	C₃×#384	—	2	1	no	—
18	C₃×#384	—	3	1	no	—
19	C₃×#384	D₃	1	1	yes	characteristic
20	C₃×#384	C₂×C₃	1	1	yes	characteristic
21	C₂×#807	C₂×C₃	1	4	yes	—
22	#4935	C₂×C₃	1	1	yes	characteristic
23	C₂×C₄×C₃⋊C₈	C₂×C₃	1	1	yes	characteristic
24	C₂²×C₃²×C₄	—	2	2	no	—
25	C₂²×C₃²×C₄	D₄	1	2	yes	—
26	C₂²×C₃²×C₄	C₂³	1	1	yes	characteristic
27	C₂²×C₃×Dic₃	—	2	2	no	—
28	C₃²×C₄⋊C₄	—	2	4	no	—
29	C₃²×C₄⋊C₄	—	4	4	no	—
30	C₃²×C₄²	C₂×C₄	1	2	yes	—
31	C₃²×C₄²	Q₈	1	2	yes	—
32	C₃×C₄⋊Dic₃	—	2	4	no	—
33	C₃×C₄⋊Dic₃	—	4	4	no	—
34	C₂×C₃×C₃⋊C₈	C₂×C₄	1	4	yes	—
35	C₂×C₃×C₃⋊C₈	D₄	1	4	yes	—
36	C₂×C₃×C₃⋊C₈	Q₈	1	4	yes	—
37	#1845	—	3	1	no	Sylow
38	C₂²×C₃×C₈	—	3	2	no	—
39	C₂×C₃×C₄⋊C₄	—	2	4	no	—
40	C₂×C₃×C₄⋊C₄	—	2	2	no	—
41	C₂×C₃×C₄⋊C₄	—	2	2	no	—
42	C₂×C₃×C₄⋊C₄	—	2	2	no	—
43	C₂×C₃×C₄⋊C₄	—	3	2	no	—
44	C₂×C₃×C₄⋊C₄	—	6	2	no	—
45	C₂×C₃×C₄⋊C₄	C₂×D₃	1	2	yes	—
46	C₂×C₃×C₄⋊C₄	C₂²×C₃	1	2	yes	—
47	C₂×C₃×C₄²	—	2	1	no	—
48	C₂×C₃×C₄²	C₂×D₃	1	1	yes	characteristic
49	C₂×C₃×C₄²	C₂²×C₃	1	1	yes	characteristic
50	C₂×C₄⋊Dic₃	—	2	2	no	—
51	C₂×C₄⋊Dic₃	C₂²×C₃	1	2	yes	—
52	C₂²×C₃⋊C₈	C₂²×C₃	1	2	yes	—
53	C₃×#107	—	6	8	no	—
54	C₃×C₄×C₈	—	3	4	no	—
55	#807	—	2	8	no	—
56	C₄×C₃⋊C₈	C₃×C₄	1	4	yes	complemented
57	C₂³×C₃²	C₂×D₄	1	1	yes	characteristic
58	C₂×C₃²×C₄	—	4	2	no	—
59	C₂×C₃²×C₄	—	4	4	no	—
60	C₂×C₃²×C₄	C₄⋊C₄	1	4	yes	—
61	C₂×C₃²×C₄	D₈	1	4	yes	—
62	C₂×C₃²×C₄	Q₁₆	1	4	yes	—
63	C₂×C₃²×C₄	C₂²×C₄	1	1	yes	characteristic
64	C₂×C₃²×C₄	C₂×D₄	1	2	yes	—
65	C₂×C₃²×C₄	C₂×Q₈	1	1	yes	characteristic
66	C₂×C₃²×C₄	C₂×Q₈	1	2	yes	—
67	C₂×C₃×Dic₃	—	4	2	no	—
68	C₂×C₃×Dic₃	—	4	4	no	—
69	C₃×C₃⋊C₈	C₄⋊C₄	1	8	yes	complemented
70	C₂×#107	—	3	4	no	—
71	C₂×C₄×C₈	—	3	1	no	—
72	#384	—	3	1	no	—
73	#384	C₃×D₃	1	1	yes	characteristic
74	C₂²×C₃×C₄	—	2	4	no	—
75	C₂²×C₃×C₄	—	2	2	no	—
76	C₂²×C₃×C₄	—	2	2	no	—
77	C₂²×C₃×C₄	—	2	2	no	—
78	C₂²×C₃×C₄	—	2	1	no	—
79	C₂²×C₃×C₄	—	6	2	no	—
80	C₂²×C₃×C₄	#67	1	2	yes	—
81	C₂²×C₃×C₄	C₃×D₄	1	2	yes	—
82	C₂²×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
83	C₂²×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
84	C₂²×Dic₃	—	2	2	no	—
85	C₂×C₃×C₈	—	3	4	no	—
86	C₂×C₃×C₈	—	3	4	no	—
87	C₂×C₃×C₈	—	3	4	no	—
88	C₃×C₄⋊C₄	—	2	4	no	—
89	C₃×C₄⋊C₄	—	2	4	no	—
90	C₃×C₄⋊C₄	—	4	4	no	—
91	C₃×C₄⋊C₄	—	4	8	no	—
92	C₃×C₄⋊C₄	—	4	4	no	—
93	C₃×C₄⋊C₄	—	4	4	no	—
94	C₃×C₄⋊C₄	—	6	4	no	—
95	C₃×C₄⋊C₄	—	12	4	no	—
96	C₃×C₄²	—	2	2	no	—
97	C₃×C₄²	—	2	2	no	—
98	C₃×C₄²	Dic₆	1	2	yes	—
99	C₃×C₄²	C₄×D₃	1	2	yes	—
100	C₃×C₄²	C₂×C₃×C₄	1	2	yes	—
101	C₃×C₄²	C₃×Q₈	1	2	yes	—
102	C₄⋊Dic₃	—	2	4	no	—
103	C₄⋊Dic₃	—	4	4	no	—
104	C₂×C₃⋊C₈	C₂×C₃×C₄	1	4	yes	—
105	C₂×C₃⋊C₈	C₃×D₄	1	4	yes	—
106	C₂×C₃⋊C₈	C₃×Q₈	1	4	yes	—
107	C₂²×C₃²	C₂×C₄⋊C₄	1	1	yes	characteristic
108	C₂²×C₃²	#127	1	1	yes	characteristic
109	C₂²×C₃²	C₄⋊Q₈	1	1	yes	characteristic
110	C₂²×C₃²	C₂×D₈	1	2	yes	—
111	C₂²×C₃²	C₂×Q₁₆	1	2	yes	—
112	C₃²×C₄	—	8	2	no	—
113	C₃²×C₄	#107	1	8	yes	complemented
114	C₃²×C₄	C₂×C₄⋊C₄	1	2	yes	—
115	C₃²×C₄	C₄⋊Q₈	1	2	yes	—
116	C₃×Dic₃	—	8	2	no	—
117	C₂²×C₈	—	3	2	no	—
118	C₂×C₄⋊C₄	—	2	2	no	—
119	C₂×C₄⋊C₄	—	3	2	no	—
120	C₂×C₄⋊C₄	—	6	2	no	—
121	C₂×C₄⋊C₄	C₂×C₃×D₃	1	2	yes	—
122	C₂×C₄²	C₂×C₃×D₃	1	1	yes	characteristic
123	#107	—	6	8	no	—
124	C₄×C₈	—	3	4	no	—
125	C₂³×C₃	—	2	1	no	—
126	C₂³×C₃	C₂×#67	1	1	yes	characteristic
127	C₂³×C₃	C₂×C₃×D₄	1	1	yes	characteristic
128	C₂×C₃×C₄	—	2	4	no	—
129	C₂×C₃×C₄	—	2	4	no	—
130	C₂×C₃×C₄	—	2	4	no	—
131	C₂×C₃×C₄	—	2	1	no	—
132	C₂×C₃×C₄	—	2	1	no	—
133	C₂×C₃×C₄	—	2	2	no	—
134	C₂×C₃×C₄	—	2	2	no	—
135	C₂×C₃×C₄	—	4	8	no	—
136	C₂×C₃×C₄	—	4	4	no	—
137	C₂×C₃×C₄	—	4	4	no	—
138	C₂×C₃×C₄	—	4	2	no	—
139	C₂×C₃×C₄	—	4	2	no	—
140	C₂×C₃×C₄	—	4	4	no	—
141	C₂×C₃×C₄	—	12	4	no	—
142	C₂×C₃×C₄	—	12	2	no	—
143	C₂×C₃×C₄	Dic₃⋊C₄	1	4	yes	—
144	C₂×C₃×C₄	#213	1	4	yes	—
145	C₂×C₃×C₄	#216	1	4	yes	—
146	C₂×C₃×C₄	C₃×C₄⋊C₄	1	4	yes	—
147	C₂×C₃×C₄	C₃×D₈	1	4	yes	—
148	C₂×C₃×C₄	C₃×Q₁₆	1	4	yes	—
149	C₂×C₃×C₄	C₂×Dic₆	1	1	yes	characteristic
150	C₂×C₃×C₄	C₂×C₄×D₃	1	1	yes	characteristic
151	C₂×C₃×C₄	D₃×D₄	1	2	yes	—
152	C₂×C₃×C₄	D₃×Q₈	1	2	yes	—
153	C₂×C₃×C₄	C₂²×C₃×C₄	1	1	yes	characteristic
154	C₂×C₃×C₄	C₂×C₃×D₄	1	2	yes	—
155	C₂×C₃×C₄	C₂×C₃×Q₈	1	1	yes	characteristic
156	C₂×C₃×C₄	C₂×C₃×Q₈	1	2	yes	—
157	C₂×Dic₃	—	4	2	no	—
158	C₂×Dic₃	—	4	4	no	—
159	C₃×C₈	—	3	8	no	—
160	C₃⋊C₈	C₃×C₄⋊C₄	1	8	yes	complemented
161	C₂×C₃²	#384	1	1	yes	characteristic
162	C₂×C₃²	C₂×#107	1	2	yes	—
163	C₂×C₃²	#493	1	1	yes	characteristic
164	C₂×C₃²	#494	1	1	yes	characteristic
165	C₂×C₃²	#500	1	2	yes	—
166	C₂²×C₄	—	2	2	no	—
167	C₂²×C₄	—	6	2	no	—
168	C₂²×C₄	C₃×#67	1	2	yes	—
169	C₂²×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
170	C₂×C₈	—	3	4	no	—
171	C₂×C₈	—	3	4	no	—
172	C₂×C₈	—	3	4	no	—
173	C₄⋊C₄	—	2	4	no	—
174	C₄⋊C₄	—	4	4	no	—
175	C₄⋊C₄	—	6	4	no	—
176	C₄⋊C₄	—	12	4	no	—
177	C₄²	C₃×Dic₆	1	2	yes	—
178	C₄²	C₃×C₄×D₃	1	2	yes	—
179	C₂²×C₃	—	2	1	no	—
180	C₂²×C₃	—	2	2	no	—
181	C₂²×C₃	—	2	1	no	—
182	C₂²×C₃	—	2	2	no	—
183	C₂²×C₃	—	2	1	no	—
184	C₂²×C₃	C₂×Dic₃⋊C₄	1	1	yes	characteristic
185	C₂²×C₃	C₂×#213	1	2	yes	—
186	C₂²×C₃	#940	1	1	yes	characteristic
187	C₂²×C₃	C₂×#216	1	2	yes	—
188	C₂²×C₃	#944	1	1	yes	characteristic
189	C₂²×C₃	C₂×C₃×C₄⋊C₄	1	1	yes	characteristic
190	C₂²×C₃	C₃×#127	1	1	yes	characteristic
191	C₂²×C₃	C₃×C₄⋊Q₈	1	1	yes	characteristic
192	C₂²×C₃	C₂×C₃×D₈	1	2	yes	—
193	C₂²×C₃	C₂×C₃×Q₁₆	1	2	yes	—
194	C₃×C₄	—	2	2	no	—
195	C₃×C₄	—	2	2	no	—
196	C₃×C₄	—	2	8	no	—
197	C₃×C₄	—	8	4	no	—
198	C₃×C₄	—	8	2	no	—
199	C₃×C₄	—	8	2	no	—
200	C₃×C₄	—	24	2	no	—
201	C₃×C₄	#807	1	8	yes	complemented
202	C₃×C₄	C₃×#107	1	8	yes	complemented
203	C₃×C₄	#888	1	2	yes	—
204	C₃×C₄	D₃×C₄⋊C₄	1	2	yes	—
205	C₃×C₄	C₂×C₃×C₄⋊C₄	1	2	yes	—
206	C₃×C₄	C₃×C₄⋊Q₈	1	2	yes	—
207	Dic₃	—	8	2	no	—
208	C₃²	#1845	1	1	yes	characteristic, complemented, Sylow
209	C₂³	C₂×C₃×#67	1	1	yes	characteristic
210	C₂×C₄	—	4	4	no	—
211	C₂×C₄	—	4	2	no	—
212	C₂×C₄	—	12	4	no	—
213	C₂×C₄	—	12	2	no	—
214	C₂×C₄	C₃×Dic₃⋊C₄	1	4	yes	—
215	C₂×C₄	C₃×#213	1	4	yes	—
216	C₂×C₄	C₃×#216	1	4	yes	—
217	C₂×C₄	C₂×C₃×Dic₆	1	1	yes	characteristic
218	C₂×C₄	C₂×C₃×C₄×D₃	1	1	yes	characteristic
219	C₂×C₄	C₃×D₃×D₄	1	2	yes	—
220	C₂×C₄	C₃×D₃×Q₈	1	2	yes	—
221	C₈	—	3	8	no	—
222	C₂×C₃	—	2	2	no	—
223	C₂×C₃	—	2	1	no	—
224	C₂×C₃	—	2	1	no	—
225	C₂×C₃	—	2	2	no	—
226	C₂×C₃	—	2	1	no	—
227	C₂×C₃	C₂×#807	1	2	yes	—
228	C₂×C₃	#4975	1	1	yes	characteristic
229	C₂×C₃	#5073	1	1	yes	characteristic
230	C₂×C₃	#5078	1	2	yes	—
231	C₂×C₃	#5092	1	1	yes	characteristic
232	C₂×C₃	C₃×#384	1	1	yes	characteristic
233	C₂×C₃	C₂×C₃×#107	1	2	yes	—
234	C₂×C₃	C₃×#493	1	1	yes	characteristic
235	C₂×C₃	C₃×#494	1	1	yes	characteristic
236	C₂×C₃	C₃×#500	1	2	yes	—
237	C₂²	C₂×C₃×Dic₃⋊C₄	1	1	yes	characteristic
238	C₂²	C₂×C₃×#213	1	2	yes	—
239	C₂²	C₃×#940	1	1	yes	characteristic
240	C₂²	C₂×C₃×#216	1	2	yes	—
241	C₂²	C₃×#944	1	1	yes	characteristic
242	C₄	—	8	2	no	—
243	C₄	—	24	2	no	—
244	C₄	C₃×#807	1	8	yes	complemented
245	C₄	C₃×#888	1	2	yes	—
246	C₄	C₃×D₃×C₄⋊C₄	1	2	yes	—
247	C₃	—	2	1	no	—
248	C₃	#69577	1	1	yes	characteristic, direct product factor
249	C₃	C₃×#1845	1	1	yes	characteristic, complemented
250	C₂	C₂×C₃×#807	1	2	yes	—
251	C₂	C₃×#4975	1	1	yes	characteristic
252	C₂	C₃×#5073	1	1	yes	characteristic
253	C₂	C₃×#5078	1	2	yes	—
254	C₂	C₃×#5092	1	1	yes	characteristic

	subgroup	G/H
center	C₂³×C₃	C₂×#67
commutator	C₂×C₃×C₄	C₂²×C₃×C₄
Frattini	C₂²×C₄	C₂²×C₃×D₃
Fitting	C₃²×#384	C₂
socle, prime radical	C₂³×C₃²	C₂×D₄
cosocle	C₂²×C₃×C₄	C₂³×C₃
hypercenter	C₃×#384	D₃
hypocenter	C₃	C₃×#1845
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂³×C₃ ◁ C₂×C₃×C₄² ◁ C₃×#384

Complements

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

N	H ≅ G/N	type
6. #69577	C₃	direct (N × H)
14. C₃×C₄×C₃⋊C₈	C₄	semidirect
56. C₄×C₃⋊C₈	C₃×C₄	semidirect
69. C₃×C₃⋊C₈	C₄⋊C₄	semidirect
113. C₃²×C₄	#107	semidirect
160. C₃⋊C₈	C₃×C₄⋊C₄	semidirect
201. C₃×C₄	#807	semidirect
202. C₃×C₄	C₃×#107	semidirect
208. C₃²	#1845	semidirect
244. C₄	C₃×#807	semidirect
248. C₃	#69577	direct (N × H)
249. C₃	C₃×#1845	semidirect

Maximal subgroups

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

subgroup	index
C₃²×#384	2
C₂×C₃×#807	2
C₃×#4935	2
C₂×C₃×C₄×C₃⋊C₈	2
C₃×#1845	3
#69577	3

Subgroups — C3×#69577 #11786226

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#69577 #11786226