Subgroups — C₃×#69552 #11790725

Order 1152.

693 conjugacy classes of subgroups.

The subgroup lattice has 268 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×#869	C₂	1	1	yes	maximal, characteristic
2	C₃²×#382	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃×#810	C₂	1	4	yes	maximal
4	C₂×C₃×#803	C₂	1	1	yes	maximal, characteristic
5	C₃×#1960	—	3	1	no	maximal
6	#69552	C₃	1	1	yes	maximal, characteristic, direct product factor
7	C₂²×C₃×Dic₆	C₂²	1	2	yes	—
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	C₂²×C₃×C₃⋊C₈	C₂²	1	2	yes	—
12	C₃×#869	—	2	2	no	—
13	C₃×#869	C₄	1	4	yes	complemented
14	C₃²×#95	—	2	2	no	—
15	C₃×#810	—	2	8	no	—
16	C₃×#803	—	2	2	no	—
17	C₂×C₃×C₄⋊Q₈	—	3	1	no	—
18	C₂×#869	C₂×C₃	1	1	yes	characteristic
19	C₂×C₃×#103	—	3	4	no	—
20	C₂×C₃×#97	—	3	1	no	—
21	C₃×#382	—	2	1	no	—
22	C₃×#382	D₃	1	1	yes	characteristic
23	C₃×#382	C₂×C₃	1	1	yes	characteristic
24	C₂×#810	C₂×C₃	1	4	yes	—
25	C₂×#803	C₂×C₃	1	1	yes	characteristic
26	C₂²×C₃²×C₄	—	2	2	no	—
27	C₂²×C₃²×C₄	D₄	1	2	yes	—
28	C₂²×C₃²×C₄	C₂³	1	1	yes	characteristic
29	C₂²×C₃×Dic₃	—	2	2	no	—
30	C₂×C₃×Dic₆	—	2	4	no	—
31	C₂×C₃×Dic₆	—	2	4	no	—
32	C₂×C₃×Dic₆	—	2	2	no	—
33	C₂×C₃×Dic₆	C₂×C₄	1	4	yes	—
34	C₃²×C₄⋊C₄	—	2	4	no	—
35	C₃²×C₄²	C₂×C₄	1	2	yes	—
36	C₃²×C₄²	D₄	1	2	yes	—
37	C₃×C₄⋊Dic₃	—	2	8	no	—
38	C₂×C₃×C₃⋊C₈	—	2	2	no	—
39	C₂×C₃×C₃⋊C₈	—	2	2	no	—
40	C₂×C₃×C₃⋊C₈	D₄	1	4	yes	—
41	#1960	—	3	1	no	Sylow
42	C₂²×C₃×Q₈	—	3	2	no	—
43	C₂²×Dic₆	C₂²×C₃	1	2	yes	—
44	C₂²×C₃×C₈	—	3	2	no	—
45	C₃×C₄⋊Q₈	—	3	4	no	—
46	C₃×C₄⋊Q₈	—	6	2	no	—
47	C₂×C₃×C₄⋊C₄	—	2	2	no	—
48	C₂×C₃×C₄⋊C₄	—	6	2	no	—
49	C₂×C₃×C₄⋊C₄	C₂×D₃	1	2	yes	—
50	C₂×C₃×C₄⋊C₄	C₂²×C₃	1	2	yes	—
51	C₂×C₃×C₄²	—	2	1	no	—
52	C₂×C₃×C₄²	C₂×D₃	1	1	yes	characteristic
53	C₂×C₃×C₄²	C₂²×C₃	1	1	yes	characteristic
54	C₂×C₄⋊Dic₃	—	2	2	no	—
55	C₂²×C₃⋊C₈	C₂²×C₃	1	2	yes	—
56	#869	—	2	2	no	—
57	#869	C₃×C₄	1	4	yes	complemented
58	C₃×#103	—	6	8	no	—
59	C₃×#97	—	6	2	no	—
60	C₃×#95	—	2	4	no	—
61	C₃×#95	—	2	2	no	—
62	C₃×#95	—	2	2	no	—
63	#810	—	2	8	no	—
64	#803	—	2	2	no	—
65	C₂³×C₃²	C₂×D₄	1	1	yes	characteristic
66	C₂×C₃²×C₄	—	2	4	no	—
67	C₂×C₃²×C₄	—	4	2	no	—
68	C₂×C₃²×C₄	—	4	4	no	—
69	C₂×C₃²×C₄	C₂²⋊C₄	1	4	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	C₂×C₃×Dic₃	—	2	4	no	—
75	C₂×C₃×Dic₃	—	4	2	no	—
76	C₂×C₃×Dic₃	—	4	2	no	—
77	C₃×Dic₆	—	2	8	no	—
78	C₃×Dic₆	—	4	4	no	—
79	C₃×C₃⋊C₈	—	2	4	no	—
80	C₂×C₄⋊Q₈	—	3	1	no	—
81	C₂×#103	—	3	4	no	—
82	C₂×#97	—	3	1	no	—
83	#382	C₃×D₃	1	1	yes	characteristic
84	C₂×C₃×Q₈	—	3	4	no	—
85	C₂×C₃×Q₈	—	6	2	no	—
86	C₂×C₃×Q₈	—	6	4	no	—
87	C₂×C₃×Q₈	—	6	4	no	—
88	C₂²×C₃×C₄	—	2	4	no	—
89	C₂²×C₃×C₄	—	2	2	no	—
90	C₂²×C₃×C₄	—	2	2	no	—
91	C₂²×C₃×C₄	—	2	2	no	—
92	C₂²×C₃×C₄	—	2	1	no	—
93	C₂²×C₃×C₄	—	6	2	no	—
94	C₂²×C₃×C₄	#67	1	2	yes	—
95	C₂²×C₃×C₄	C₃×D₄	1	2	yes	—
96	C₂²×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
97	C₂²×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
98	C₂²×Dic₃	—	2	2	no	—
99	C₂×Dic₆	—	2	4	no	—
100	C₂×Dic₆	—	2	4	no	—
101	C₂×Dic₆	—	2	2	no	—
102	C₂×Dic₆	C₂×C₃×C₄	1	4	yes	—
103	C₂×C₃×C₈	—	3	4	no	—
104	C₂×C₃×C₈	—	6	2	no	—
105	C₂×C₃×C₈	—	6	2	no	—
106	C₃×C₄⋊C₄	—	2	4	no	—
107	C₃×C₄⋊C₄	—	2	4	no	—
108	C₃×C₄⋊C₄	—	4	4	no	—
109	C₃×C₄⋊C₄	—	6	8	no	—
110	C₃×C₄²	—	2	2	no	—
111	C₃×C₄²	—	2	2	no	—
112	C₃×C₄²	C₄×D₃	1	2	yes	—
113	C₃×C₄²	D₁₂	1	2	yes	—
114	C₃×C₄²	C₂×C₃×C₄	1	2	yes	—
115	C₃×C₄²	C₃×D₄	1	2	yes	—
116	C₄⋊Dic₃	—	2	8	no	—
117	C₂×C₃⋊C₈	—	2	2	no	—
118	C₂×C₃⋊C₈	—	2	2	no	—
119	C₂×C₃⋊C₈	C₃×D₄	1	4	yes	—
120	C₂²×C₃²	C₂×C₂²⋊C₄	1	1	yes	characteristic
121	C₂²×C₃²	#124	1	1	yes	characteristic
122	C₂²×C₃²	#127	1	1	yes	characteristic
123	C₂²×C₃²	#137	1	4	yes	—
124	C₃²×C₄	—	2	4	no	—
125	C₃²×C₄	—	8	2	no	—
126	C₃²×C₄	C₄×D₄	1	2	yes	—
127	C₃²×C₄	#121	1	2	yes	—
128	C₃×Dic₃	—	4	4	no	—
129	C₂²×Q₈	—	3	2	no	—
130	C₂²×C₈	—	3	2	no	—
131	C₄⋊Q₈	—	3	4	no	—
132	C₄⋊Q₈	—	6	2	no	—
133	C₂×C₄⋊C₄	—	6	2	no	—
134	C₂×C₄⋊C₄	C₂×C₃×D₃	1	2	yes	—
135	C₂×C₄²	C₂×C₃×D₃	1	1	yes	characteristic
136	#103	—	6	8	no	—
137	#97	—	6	2	no	—
138	#95	—	2	2	no	—
139	C₂³×C₃	—	2	1	no	—
140	C₂³×C₃	C₂×#67	1	1	yes	characteristic
141	C₂³×C₃	C₂×C₃×D₄	1	1	yes	characteristic
142	C₃×Q₈	—	6	8	no	—
143	C₃×Q₈	—	12	4	no	—
144	C₂×C₃×C₄	—	2	4	no	—
145	C₂×C₃×C₄	—	2	4	no	—
146	C₂×C₃×C₄	—	2	4	no	—
147	C₂×C₃×C₄	—	2	1	no	—
148	C₂×C₃×C₄	—	2	1	no	—
149	C₂×C₃×C₄	—	2	2	no	—
150	C₂×C₃×C₄	—	2	2	no	—
151	C₂×C₃×C₄	—	4	4	no	—
152	C₂×C₃×C₄	—	4	4	no	—
153	C₂×C₃×C₄	—	4	8	no	—
154	C₂×C₃×C₄	—	4	2	no	—
155	C₂×C₃×C₄	—	4	4	no	—
156	C₂×C₃×C₄	—	4	2	no	—
157	C₂×C₃×C₄	—	4	4	no	—
158	C₂×C₃×C₄	—	6	4	no	—
159	C₂×C₃×C₄	—	12	2	no	—
160	C₂×C₃×C₄	—	12	2	no	—
161	C₂×C₃×C₄	(C₂×D₃)⋊C₄	1	4	yes	—
162	C₂×C₃×C₄	C₃×C₂²⋊C₄	1	4	yes	—
163	C₂×C₃×C₄	C₂×C₄×D₃	1	1	yes	characteristic
164	C₂×C₃×C₄	C₂×D₁₂	1	1	yes	characteristic
165	C₂×C₃×C₄	D₃×D₄	1	2	yes	—
166	C₂×C₃×C₄	#239	1	2	yes	—
167	C₂×C₃×C₄	C₂²×C₃×C₄	1	1	yes	characteristic
168	C₂×C₃×C₄	C₂×C₃×D₄	1	1	yes	characteristic
169	C₂×C₃×C₄	C₂×C₃×D₄	1	2	yes	—
170	C₂×C₃×C₄	C₃×#41	1	2	yes	—
171	C₂×Dic₃	—	2	4	no	—
172	C₂×Dic₃	—	4	2	no	—
173	C₂×Dic₃	—	4	2	no	—
174	Dic₆	—	2	8	no	—
175	Dic₆	—	4	4	no	—
176	C₃×C₈	—	6	4	no	—
177	C₃⋊C₈	—	2	4	no	—
178	C₂×C₃²	#390	1	1	yes	characteristic
179	C₂×C₃²	#419	1	2	yes	—
180	C₂×C₃²	#491	1	2	yes	—
181	C₂×C₃²	#497	1	2	yes	—
182	C₂×Q₈	—	3	4	no	—
183	C₂×Q₈	—	6	2	no	—
184	C₂×Q₈	—	6	4	no	—
185	C₂×Q₈	—	6	4	no	—
186	C₂²×C₄	—	2	2	no	—
187	C₂²×C₄	—	6	2	no	—
188	C₂²×C₄	C₃×#67	1	2	yes	—
189	C₂²×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
190	C₂×C₈	—	3	4	no	—
191	C₂×C₈	—	6	2	no	—
192	C₂×C₈	—	6	2	no	—
193	C₄⋊C₄	—	2	4	no	—
194	C₄⋊C₄	—	6	8	no	—
195	C₄²	C₃×C₄×D₃	1	2	yes	—
196	C₄²	C₃×D₁₂	1	2	yes	—
197	C₂²×C₃	—	2	1	no	—
198	C₂²×C₃	—	2	1	no	—
199	C₂²×C₃	—	2	1	no	—
200	C₂²×C₃	—	2	4	no	—
201	C₂²×C₃	C₂×(C₂×D₃)⋊C₄	1	1	yes	characteristic
202	C₂²×C₃	#940	1	1	yes	characteristic
203	C₂²×C₃	#947	1	1	yes	characteristic
204	C₂²×C₃	#951	1	4	yes	—
205	C₂²×C₃	C₂×C₃×C₂²⋊C₄	1	1	yes	characteristic
206	C₂²×C₃	C₃×#124	1	1	yes	characteristic
207	C₂²×C₃	C₃×#127	1	1	yes	characteristic
208	C₂²×C₃	C₃×#137	1	4	yes	—
209	C₃×C₄	—	2	4	no	—
210	C₃×C₄	—	2	4	no	—
211	C₃×C₄	—	2	2	no	—
212	C₃×C₄	—	2	2	no	—
213	C₃×C₄	—	4	4	no	—
214	C₃×C₄	—	8	4	no	—
215	C₃×C₄	—	8	2	no	—
216	C₃×C₄	—	8	2	no	—
217	C₃×C₄	—	12	4	no	—
218	C₃×C₄	#893	1	2	yes	—
219	C₃×C₄	#895	1	2	yes	—
220	C₃×C₄	C₃×C₄×D₄	1	2	yes	—
221	C₃×C₄	C₃×#121	1	2	yes	—
222	Dic₃	—	4	4	no	—
223	C₃²	#1960	1	1	yes	characteristic, complemented, Sylow
224	C₂³	C₂×C₃×#67	1	1	yes	characteristic
225	Q₈	—	6	8	no	—
226	Q₈	—	12	4	no	—
227	C₂×C₄	—	2	4	no	—
228	C₂×C₄	—	4	2	no	—
229	C₂×C₄	—	4	4	no	—
230	C₂×C₄	—	6	4	no	—
231	C₂×C₄	—	12	2	no	—
232	C₂×C₄	—	12	2	no	—
233	C₂×C₄	C₃×(C₂×D₃)⋊C₄	1	4	yes	—
234	C₂×C₄	C₂×C₃×C₄×D₃	1	1	yes	characteristic
235	C₂×C₄	C₂×C₃×D₁₂	1	1	yes	characteristic
236	C₂×C₄	C₃×D₃×D₄	1	2	yes	—
237	C₂×C₄	C₃×#239	1	2	yes	—
238	C₈	—	6	4	no	—
239	C₂×C₃	—	2	1	no	—
240	C₂×C₃	—	2	2	no	—
241	C₂×C₃	—	2	2	no	—
242	C₂×C₃	—	2	2	no	—
243	C₂×C₃	#4988	1	1	yes	characteristic
244	C₂×C₃	#5004	1	2	yes	—
245	C₂×C₃	#5060	1	2	yes	—
246	C₂×C₃	#5069	1	2	yes	—
247	C₂×C₃	C₃×#390	1	1	yes	characteristic
248	C₂×C₃	C₃×#419	1	2	yes	—
249	C₂×C₃	C₃×#491	1	2	yes	—
250	C₂×C₃	C₃×#497	1	2	yes	—
251	C₂²	C₂×C₃×(C₂×D₃)⋊C₄	1	1	yes	characteristic
252	C₂²	C₃×#940	1	1	yes	characteristic
253	C₂²	C₃×#947	1	1	yes	characteristic
254	C₂²	C₃×#951	1	4	yes	—
255	C₄	—	2	4	no	—
256	C₄	—	8	2	no	—
257	C₄	—	12	4	no	—
258	C₄	C₃×#893	1	2	yes	—
259	C₄	C₃×#895	1	2	yes	—
260	C₃	—	2	1	no	—
261	C₃	#69552	1	1	yes	characteristic, direct product factor
262	C₃	C₃×#1960	1	1	yes	characteristic, complemented
263	C₂	C₃×#4988	1	1	yes	characteristic
264	C₂	C₃×#5004	1	2	yes	—
265	C₂	C₃×#5060	1	2	yes	—
266	C₂	C₃×#5069	1	2	yes	—

	subgroup	G/H
center	C₂³×C₃	C₂×#67
commutator	C₂×C₃×C₄	C₂²×C₃×C₄
Frattini	C₂²×C₄	C₂²×C₃×D₃
Fitting	C₃²×#382	C₂
socle, prime radical	C₂³×C₃²	C₂×D₄
cosocle	C₂²×C₃×C₄	C₂³×C₃
hypercenter	C₃×#382	D₃
hypocenter	C₃	C₃×#1960
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂³×C₃ ◁ C₂×C₃×C₄² ◁ C₃×#382

Complements

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

N	H ≅ G/N	type
6. #69552	C₃	direct (N × H)
13. C₃×#869	C₄	semidirect
57. #869	C₃×C₄	semidirect
223. C₃²	#1960	semidirect
261. C₃	#69552	direct (N × H)
262. C₃	C₃×#1960	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃×#869	2
C₃²×#382	2
C₂×C₃×#810	2
C₂×C₃×#803	2
C₃×#1960	3
#69552	3

Subgroups — C3×#69552 #11790725

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#69552 #11790725