Subgroups — C₃×#67669 #11771121

Order 1152.

371 conjugacy classes of subgroups.

The subgroup lattice has 266 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²×C₄×M₁₆	C₂	1	1	yes	maximal, characteristic
2	C₃×#4925	C₂	1	1	yes	maximal, characteristic
3	C₃×#4924	C₂	1	1	yes	maximal, characteristic
4	C₃×#1299	—	3	1	no	maximal
5	#67669	C₃	1	1	yes	maximal, characteristic, direct product factor
6	C₂×C₃²×M₁₆	—	2	1	no	—
7	C₂×C₃²×C₄²	C₂²	1	1	yes	characteristic
8	C₂×C₃×#208	C₄	1	2	yes	—
9	C₃²×#97	C₄	1	2	yes	—
10	C₃²×C₄×C₈	—	2	1	no	—
11	C₃×#830	—	2	1	no	—
12	C₃×C₄⋊(C₃⋊C₈)	—	2	1	no	—
13	C₃×C₄⋊(C₃⋊C₈)	—	2	1	no	—
14	C₃×C₄⋊(C₃⋊C₈)	C₄	1	2	yes	—
15	C₃×C₄×C₃⋊C₈	—	2	1	no	—
16	C₃×#431	—	3	1	no	—
17	C₃×#423	—	3	1	no	—
18	C₃×C₄×M₁₆	—	2	1	no	—
19	C₃×C₄×M₁₆	D₃	1	1	yes	characteristic
20	C₃×C₄×M₁₆	C₂×C₃	1	1	yes	characteristic
21	#4925	C₂×C₃	1	1	yes	characteristic
22	#4924	C₂×C₃	1	1	yes	characteristic
23	C₂²×C₃²×C₄	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₃²×C₄	D₄	1	1	yes	characteristic
25	C₂²×C₃²×C₄	Q₈	1	1	yes	characteristic
26	C₃²×M₁₆	—	2	4	no	—
27	C₂×C₃²×C₈	—	2	2	no	—
28	C₃²×C₄²	C₂×C₄	1	1	yes	characteristic
29	C₃²×C₄²	C₂×C₄	1	1	yes	characteristic
30	C₃²×C₄²	D₄	1	1	yes	characteristic
31	C₃²×C₄²	D₄	1	1	yes	characteristic
32	C₃×#208	—	4	2	no	—
33	C₂×C₃×C₃⋊C₈	—	2	2	no	—
34	C₂×C₃×C₃⋊C₈	—	2	2	no	—
35	#1299	—	3	1	no	Sylow
36	C₂×C₃×M₁₆	—	2	2	no	—
37	C₂×C₃×M₁₆	—	2	1	no	—
38	C₂×C₃×M₁₆	—	2	1	no	—
39	C₂×C₃×M₁₆	—	3	2	no	—
40	C₂×C₃×C₄²	—	2	1	no	—
41	C₂×C₃×C₄²	C₂×D₃	1	1	yes	characteristic
42	C₂×C₃×C₄²	C₂²×C₃	1	1	yes	characteristic
43	C₂×#208	C₃×C₄	1	2	yes	—
44	C₃×C₄⋊C₈	—	3	2	no	—
45	C₃×C₄⋊C₈	—	6	1	no	—
46	C₃×C₄⋊C₈	—	6	1	no	—
47	C₃×C₂²⋊C₈	—	6	1	no	—
48	C₃×#97	—	2	2	no	—
49	C₃×#97	Dic₃	1	2	yes	—
50	C₃×#97	C₃×C₄	1	2	yes	—
51	C₃×C₄×C₈	—	2	2	no	—
52	C₃×C₄×C₈	—	2	1	no	—
53	C₃×C₄×C₈	—	2	1	no	—
54	C₃×C₄×C₈	—	6	1	no	—
55	#830	—	2	1	no	—
56	C₄⋊(C₃⋊C₈)	—	2	1	no	—
57	C₄⋊(C₃⋊C₈)	—	2	1	no	—
58	C₄⋊(C₃⋊C₈)	C₃×C₄	1	2	yes	—
59	C₄×C₃⋊C₈	—	2	1	no	—
60	C₂³×C₃²	C₄⋊C₄	1	1	yes	characteristic
61	C₂×C₃²×C₄	—	2	2	no	—
62	C₂×C₃²×C₄	—	2	1	no	—
63	C₂×C₃²×C₄	—	2	1	no	—
64	C₂×C₃²×C₄	—	2	2	no	—
65	C₂×C₃²×C₄	C₄²	1	1	yes	characteristic
66	C₂×C₃²×C₄	C₂²⋊C₄	1	1	yes	characteristic
67	C₂×C₃²×C₄	C₂²⋊C₄	1	1	yes	characteristic
68	C₂×C₃²×C₄	C₂²⋊C₄	1	1	yes	characteristic
69	C₂×C₃²×C₄	C₄⋊C₄	1	1	yes	characteristic
70	C₂×C₃²×C₄	C₄⋊C₄	1	1	yes	characteristic
71	C₃²×C₈	—	2	4	no	—
72	C₃×C₃⋊C₈	—	4	2	no	—
73	C₃×C₃⋊C₈	—	4	2	no	—
74	#431	—	3	1	no	—
75	#423	—	3	1	no	—
76	C₄×M₁₆	C₃×D₃	1	1	yes	characteristic
77	C₂²×C₃×C₄	—	2	1	no	—
78	C₂²×C₃×C₄	—	2	1	no	—
79	C₂²×C₃×C₄	—	2	1	no	—
80	C₂²×C₃×C₄	Dic₆	1	1	yes	characteristic
81	C₂²×C₃×C₄	C₄×D₃	1	1	yes	characteristic
82	C₂²×C₃×C₄	#67	1	1	yes	characteristic
83	C₂²×C₃×C₄	C₂×C₃×C₄	1	1	yes	characteristic
84	C₂²×C₃×C₄	C₃×D₄	1	1	yes	characteristic
85	C₂²×C₃×C₄	C₃×Q₈	1	1	yes	characteristic
86	C₃×M₁₆	—	2	8	no	—
87	C₃×M₁₆	—	2	4	no	—
88	C₃×M₁₆	—	2	4	no	—
89	C₃×M₁₆	—	12	2	no	—
90	C₂×C₃×C₈	—	2	4	no	—
91	C₂×C₃×C₈	—	2	2	no	—
92	C₂×C₃×C₈	—	2	2	no	—
93	C₂×C₃×C₈	—	6	2	no	—
94	C₂×C₃×C₈	—	6	2	no	—
95	C₃×C₄²	—	2	1	no	—
96	C₃×C₄²	—	2	1	no	—
97	C₃×C₄²	—	2	1	no	—
98	C₃×C₄²	—	2	1	no	—
99	C₃×C₄²	C₄×D₃	1	1	yes	characteristic
100	C₃×C₄²	D₁₂	1	1	yes	characteristic
101	C₃×C₄²	C₂×Dic₃	1	1	yes	characteristic
102	C₃×C₄²	#67	1	1	yes	characteristic
103	C₃×C₄²	C₂×C₃×C₄	1	1	yes	characteristic
104	C₃×C₄²	C₂×C₃×C₄	1	1	yes	characteristic
105	C₃×C₄²	C₃×D₄	1	1	yes	characteristic
106	C₃×C₄²	C₃×D₄	1	1	yes	characteristic
107	#208	—	4	2	no	—
108	C₂×C₃⋊C₈	—	2	2	no	—
109	C₂×C₃⋊C₈	—	2	2	no	—
110	C₂²×C₃²	—	2	1	no	—
111	C₂²×C₃²	—	2	1	no	—
112	C₂²×C₃²	—	2	1	no	—
113	C₂²×C₃²	#95	1	1	yes	characteristic
114	C₃²×C₄	—	2	1	no	—
115	C₃²×C₄	—	2	1	no	—
116	C₃²×C₄	—	2	1	no	—
117	C₃²×C₄	#100	1	1	yes	characteristic
118	C₃²×C₄	#101	1	1	yes	characteristic
119	C₃²×C₄	#104	1	2	yes	—
120	C₃²×C₄	#108	1	2	yes	—
121	C₂×M₁₆	—	2	1	no	—
122	C₂×M₁₆	—	3	2	no	—
123	C₂×C₄²	C₂×C₃×D₃	1	1	yes	characteristic
124	C₄⋊C₈	—	3	2	no	—
125	C₄⋊C₈	—	6	1	no	—
126	C₄⋊C₈	—	6	1	no	—
127	C₂²⋊C₈	—	6	1	no	—
128	#97	C₃×Dic₃	1	2	yes	—
129	C₄×C₈	—	2	1	no	—
130	C₄×C₈	—	6	1	no	—
131	C₂³×C₃	—	2	1	no	—
132	C₂³×C₃	Dic₃⋊C₄	1	1	yes	characteristic
133	C₂³×C₃	C₃×C₄⋊C₄	1	1	yes	characteristic
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	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	2	no	—
142	C₂×C₃×C₄	—	2	2	no	—
143	C₂×C₃×C₄	—	2	1	no	—
144	C₂×C₃×C₄	—	2	1	no	—
145	C₂×C₃×C₄	—	2	1	no	—
146	C₂×C₃×C₄	—	2	1	no	—
147	C₂×C₃×C₄	—	2	1	no	—
148	C₂×C₃×C₄	—	2	1	no	—
149	C₂×C₃×C₄	—	4	2	no	—
150	C₂×C₃×C₄	—	4	1	no	—
151	C₂×C₃×C₄	—	4	1	no	—
152	C₂×C₃×C₄	C₄×Dic₃	1	1	yes	characteristic
153	C₂×C₃×C₄	Dic₃⋊C₄	1	1	yes	characteristic
154	C₂×C₃×C₄	C₄⋊Dic₃	1	1	yes	characteristic
155	C₂×C₃×C₄	(C₂×D₃)⋊C₄	1	1	yes	characteristic
156	C₂×C₃×C₄	(C₂×D₃)⋊C₄	1	1	yes	characteristic
157	C₂×C₃×C₄	#217	1	1	yes	characteristic
158	C₂×C₃×C₄	C₃×C₄²	1	1	yes	characteristic
159	C₂×C₃×C₄	C₃×C₂²⋊C₄	1	1	yes	characteristic
160	C₂×C₃×C₄	C₃×C₂²⋊C₄	1	1	yes	characteristic
161	C₂×C₃×C₄	C₃×C₂²⋊C₄	1	1	yes	characteristic
162	C₂×C₃×C₄	C₃×C₄⋊C₄	1	1	yes	characteristic
163	C₂×C₃×C₄	C₃×C₄⋊C₄	1	1	yes	characteristic
164	C₃×C₈	—	2	8	no	—
165	C₃×C₈	—	2	4	no	—
166	C₃×C₈	—	2	4	no	—
167	C₃×C₈	—	12	2	no	—
168	C₃×C₈	—	12	2	no	—
169	C₃⋊C₈	—	4	2	no	—
170	C₃⋊C₈	—	4	2	no	—
171	C₂×C₃²	—	4	1	no	—
172	C₂×C₃²	#339	1	1	yes	characteristic
173	C₂×C₃²	#341	1	1	yes	characteristic
174	C₂×C₃²	#343	1	1	yes	characteristic
175	C₂²×C₄	C₃×Dic₆	1	1	yes	characteristic
176	C₂²×C₄	C₃×C₄×D₃	1	1	yes	characteristic
177	C₂²×C₄	C₃×#67	1	1	yes	characteristic
178	M₁₆	—	2	4	no	—
179	M₁₆	—	12	2	no	—
180	C₂×C₈	—	2	2	no	—
181	C₂×C₈	—	6	2	no	—
182	C₂×C₈	—	6	2	no	—
183	C₄²	C₃×C₄×D₃	1	1	yes	characteristic
184	C₄²	C₃×D₁₂	1	1	yes	characteristic
185	C₄²	C₂×C₃×Dic₃	1	1	yes	characteristic
186	C₄²	C₃×#67	1	1	yes	characteristic
187	C₂²×C₃	—	2	2	no	—
188	C₂²×C₃	—	2	1	no	—
189	C₂²×C₃	—	2	1	no	—
190	C₂²×C₃	—	2	1	no	—
191	C₂²×C₃	—	2	1	no	—
192	C₂²×C₃	—	2	1	no	—
193	C₂²×C₃	—	2	1	no	—
194	C₂²×C₃	—	2	1	no	—
195	C₂²×C₃	—	4	1	no	—
196	C₂²×C₃	—	4	1	no	—
197	C₂²×C₃	C₃⋊#95	1	1	yes	characteristic
198	C₂²×C₃	C₃×#95	1	1	yes	characteristic
199	C₃×C₄	—	2	2	no	—
200	C₃×C₄	—	2	1	no	—
201	C₃×C₄	—	2	1	no	—
202	C₃×C₄	—	2	1	no	—
203	C₃×C₄	—	2	1	no	—
204	C₃×C₄	—	2	1	no	—
205	C₃×C₄	—	2	1	no	—
206	C₃×C₄	—	2	2	no	—
207	C₃×C₄	—	2	2	no	—
208	C₃×C₄	—	2	1	no	—
209	C₃×C₄	—	2	1	no	—
210	C₃×C₄	—	4	1	no	—
211	C₃×C₄	—	4	1	no	—
212	C₃×C₄	#805	1	2	yes	—
213	C₃×C₄	#822	1	2	yes	—
214	C₃×C₄	#823	1	1	yes	characteristic
215	C₃×C₄	#824	1	1	yes	characteristic
216	C₃×C₄	C₃×#100	1	1	yes	characteristic
217	C₃×C₄	C₃×#101	1	1	yes	characteristic
218	C₃×C₄	C₃×#104	1	2	yes	—
219	C₃×C₄	C₃×#108	1	2	yes	—
220	C₃²	#1299	1	1	yes	characteristic, complemented, Sylow
221	C₂³	C₃×Dic₃⋊C₄	1	1	yes	characteristic
222	C₂×C₄	—	2	2	no	—
223	C₂×C₄	—	2	2	no	—
224	C₂×C₄	—	2	1	no	—
225	C₂×C₄	—	2	1	no	—
226	C₂×C₄	C₃×C₄×Dic₃	1	1	yes	characteristic
227	C₂×C₄	C₃×Dic₃⋊C₄	1	1	yes	characteristic
228	C₂×C₄	C₃×C₄⋊Dic₃	1	1	yes	characteristic
229	C₂×C₄	C₃×(C₂×D₃)⋊C₄	1	1	yes	characteristic
230	C₂×C₄	C₃×(C₂×D₃)⋊C₄	1	1	yes	characteristic
231	C₂×C₄	C₃×#217	1	1	yes	characteristic
232	C₈	—	2	4	no	—
233	C₈	—	12	2	no	—
234	C₈	—	12	2	no	—
235	C₂×C₃	—	2	1	no	—
236	C₂×C₃	—	2	1	no	—
237	C₂×C₃	—	2	1	no	—
238	C₂×C₃	—	4	2	no	—
239	C₂×C₃	—	4	1	no	—
240	C₂×C₃	—	4	1	no	—
241	C₂×C₃	#4523	1	1	yes	characteristic
242	C₂×C₃	#4554	1	1	yes	characteristic
243	C₂×C₃	#4558	1	1	yes	characteristic
244	C₂×C₃	C₃×#339	1	1	yes	characteristic
245	C₂×C₃	C₃×#341	1	1	yes	characteristic
246	C₂×C₃	C₃×#343	1	1	yes	characteristic
247	C₂²	—	2	1	no	—
248	C₂²	—	2	1	no	—
249	C₂²	—	2	1	no	—
250	C₂²	C₃×C₃⋊#95	1	1	yes	characteristic
251	C₄	—	2	1	no	—
252	C₄	—	2	1	no	—
253	C₄	—	2	1	no	—
254	C₄	C₃×#805	1	2	yes	—
255	C₄	C₃×#822	1	2	yes	—
256	C₄	C₃×#823	1	1	yes	characteristic
257	C₄	C₃×#824	1	1	yes	characteristic
258	C₃	—	2	1	no	—
259	C₃	#67669	1	1	yes	characteristic, direct product factor
260	C₃	C₃×#1299	1	1	yes	characteristic, complemented
261	C₂	—	4	1	no	—
262	C₂	C₃×#4523	1	1	yes	characteristic
263	C₂	C₃×#4554	1	1	yes	characteristic
264	C₂	C₃×#4558	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₃×C₄	(C₂×D₃)⋊C₄
commutator	C₂×C₃×C₄	C₃×C₄²
Frattini	C₂×C₄²	C₂×C₃×D₃
Fitting	C₃²×C₄×M₁₆	C₂
socle	C₂²×C₃²	#95
prime radical	C₂³×C₃²	C₄⋊C₄
cosocle	C₂×C₃×C₄²	C₂²×C₃
hypercenter	C₃×C₄×M₁₆	D₃
hypocenter	C₃	C₃×#1299
LCS	G ▷ C₂×C₃×C₄ ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₃×C₄ ◁ C₂×C₃×C₄² ◁ C₃×C₄×M₁₆

Complements

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

N	H ≅ G/N	type
5. #67669	C₃	direct (N × H)
220. C₃²	#1299	semidirect
259. C₃	#67669	direct (N × H)
260. C₃	C₃×#1299	semidirect

Maximal subgroups

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

subgroup	index
C₃²×C₄×M₁₆	2
C₃×#4925	2
C₃×#4924	2
C₃×#1299	3
#67669	3

Subgroups — C3×#67669 #11771121

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#67669 #11771121