Subgroups — C₃×#69276 #11797349

Order 1152.

301 conjugacy classes of subgroups.

The subgroup lattice has 292 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#4689	C₂	1	1	yes	maximal, characteristic
2	C₃²×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×(C₂×D₃)⋊C₁₆	C₂	1	1	yes	maximal, characteristic
4	C₃×(C₂×D₃)⋊C₁₆	C₂	1	1	yes	maximal, characteristic
5	C₃×#4502	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃×Dic₃×C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃×C₄⋊(C₃⋊C₁₆)	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃×#2164	—	3	1	no	maximal
9	#69276	C₃	1	1	yes	maximal, characteristic, direct product factor
10	C₂×C₃×ΓL(1,𝔽₂₅)	C₄	1	1	yes	characteristic
11	C₂×C₃×D₃×C₈	C₂²	1	1	yes	characteristic
12	C₃×#872	C₄	1	1	yes	characteristic
13	C₂×C₃²×C₁₆	C₂²	1	1	yes	characteristic
14	C₂×C₃²×C₁₆	C₂²	1	1	yes	characteristic
15	C₃²×C₄×C₈	C₂²	1	1	yes	characteristic
16	C₃×#815	C₄	1	1	yes	characteristic
17	C₃×C₈×Dic₃	C₂²	1	1	yes	characteristic
18	C₂×C₃×C₃⋊C₁₆	C₂²	1	1	yes	characteristic
19	C₂×C₃×C₃⋊C₁₆	C₂²	1	1	yes	characteristic
20	C₃×#803	C₄	1	1	yes	characteristic
21	C₃×#405	—	3	1	no	—
22	#4689	C₂×C₃	1	1	yes	characteristic
23	C₃×C₄⋊C₁₆	—	2	1	no	—
24	C₃×C₄⋊C₁₆	—	3	1	no	—
25	C₃×C₄⋊C₁₆	D₃	1	1	yes	characteristic, complemented
26	C₃×C₄⋊C₁₆	C₂×C₃	1	1	yes	characteristic, complemented
27	C₃×C₂²⋊C₁₆	—	3	1	no	—
28	C₃×C₂²⋊C₁₆	—	3	1	no	—
29	C₃×#346	—	3	1	no	—
30	C₃×C₄×C₁₆	—	3	1	no	—
31	(C₂×D₃)⋊C₁₆	C₂×C₃	1	1	yes	characteristic
32	(C₂×D₃)⋊C₁₆	C₂×C₃	1	1	yes	characteristic
33	#4502	C₂×C₃	1	1	yes	characteristic, complemented
34	Dic₃×C₁₆	C₂×C₃	1	1	yes	characteristic, complemented
35	C₄⋊(C₃⋊C₁₆)	C₂×C₃	1	1	yes	characteristic, complemented
36	C₂×C₃×C₄×D₃	C₂×C₄	1	1	yes	characteristic
37	C₂×C₃²×C₈	C₂×C₄	1	1	yes	characteristic
38	C₂×C₃²×C₈	C₂³	1	1	yes	characteristic
39	C₃²×C₄²	C₂×C₄	1	1	yes	characteristic
40	C₃×(C₂×D₃)⋊C₄	C₈	1	1	yes	characteristic
41	C₃×(C₂×D₃)⋊C₄	C₈	1	1	yes	characteristic
42	C₃×Dic₃⋊C₄	C₈	1	1	yes	characteristic
43	C₃×Dic₃⋊C₄	C₈	1	1	yes	characteristic
44	C₃×C₄×Dic₃	C₂×C₄	1	1	yes	characteristic
45	C₂×C₃×C₃⋊C₈	C₂×C₄	1	1	yes	characteristic
46	C₂×C₃×C₃⋊C₈	C₂×C₄	1	1	yes	characteristic
47	C₃×ΓL(1,𝔽₂₅)	—	2	2	no	—
48	C₃×D₃×C₈	—	2	1	no	—
49	C₃×D₃×C₈	—	2	1	no	—
50	C₃²×C₁₆	—	2	1	no	—
51	C₃²×C₁₆	—	2	1	no	—
52	C₃×C₃⋊C₁₆	—	2	1	no	—
53	C₃×C₃⋊C₁₆	—	2	1	no	—
54	#2164	—	3	1	no	Sylow
55	C₂×C₃×M₁₆	—	3	1	no	—
56	C₂²×C₃×C₈	—	3	1	no	—
57	C₃×#117	—	3	1	no	—
58	C₂×ΓL(1,𝔽₂₅)	C₃×C₄	1	1	yes	characteristic
59	C₂×D₃×C₈	C₂²×C₃	1	1	yes	characteristic
60	#872	C₃×C₄	1	1	yes	characteristic
61	C₂×C₃×C₁₆	—	2	1	no	—
62	C₂×C₃×C₁₆	—	2	1	no	—
63	C₂×C₃×C₁₆	—	3	1	no	—
64	C₂×C₃×C₁₆	—	3	1	no	—
65	C₂×C₃×C₁₆	C₂×D₃	1	1	yes	characteristic
66	C₂×C₃×C₁₆	C₂×D₃	1	1	yes	characteristic
67	C₂×C₃×C₁₆	C₂²×C₃	1	1	yes	characteristic
68	C₂×C₃×C₁₆	C₂²×C₃	1	1	yes	characteristic
69	C₃×#97	—	3	1	no	—
70	C₃×#97	—	3	1	no	—
71	C₃×C₄×C₈	—	2	1	no	—
72	C₃×C₄×C₈	—	3	1	no	—
73	C₃×C₄×C₈	C₂×D₃	1	1	yes	characteristic
74	C₃×C₄×C₈	C₂²×C₃	1	1	yes	characteristic
75	#815	C₃×C₄	1	1	yes	characteristic
76	C₈×Dic₃	C₂²×C₃	1	1	yes	characteristic
77	C₂×C₃⋊C₁₆	C₂²×C₃	1	1	yes	characteristic
78	C₂×C₃⋊C₁₆	C₂²×C₃	1	1	yes	characteristic
79	#803	C₃×C₄	1	1	yes	characteristic
80	C₂²×C₃×D₃	C₂×C₈	1	1	yes	characteristic
81	C₂×C₃²×C₄	C₂×C₈	1	1	yes	characteristic
82	C₂×C₃²×C₄	C₂×C₈	1	1	yes	characteristic
83	C₂×C₃²×C₄	C₂²×C₄	1	1	yes	characteristic
84	C₂×C₃×Dic₃	C₂×C₈	1	1	yes	characteristic
85	C₂×C₃×Dic₃	C₂×C₈	1	1	yes	characteristic
86	C₂×C₃×Dic₃	C₂×C₈	1	1	yes	characteristic
87	C₃×C₄×D₃	—	2	1	no	—
88	C₃×C₄×D₃	—	2	1	no	—
89	C₃²×C₈	—	2	1	no	—
90	C₃²×C₈	#41	1	1	yes	characteristic
91	C₃²×C₈	#41	1	1	yes	characteristic
92	C₃×C₃⋊C₈	—	2	1	no	—
93	C₃×C₃⋊C₈	M₁₆	1	2	yes	—
94	#405	—	3	1	no	—
95	C₄⋊C₁₆	—	3	1	no	—
96	C₄⋊C₁₆	C₃×D₃	1	1	yes	characteristic, complemented
97	C₂²⋊C₁₆	—	3	1	no	—
98	C₂²⋊C₁₆	—	3	1	no	—
99	#346	—	3	1	no	—
100	C₄×C₁₆	—	3	1	no	—
101	C₂²×C₃×C₄	—	3	1	no	—
102	C₂×C₄×D₃	C₂×C₃×C₄	1	1	yes	characteristic
103	C₃×M₁₆	—	6	2	no	—
104	C₂×C₃×C₈	—	2	1	no	—
105	C₂×C₃×C₈	—	2	1	no	—
106	C₂×C₃×C₈	—	3	1	no	—
107	C₂×C₃×C₈	—	3	1	no	—
108	C₂×C₃×C₈	—	6	1	no	—
109	C₂×C₃×C₈	—	6	1	no	—
110	C₂×C₃×C₈	C₄×D₃	1	1	yes	characteristic
111	C₂×C₃×C₈	C₂×C₃×C₄	1	1	yes	characteristic
112	C₂×C₃×C₈	C₂²×D₃	1	1	yes	characteristic
113	C₂×C₃×C₈	C₂³×C₃	1	1	yes	characteristic
114	C₃×C₄⋊C₄	—	3	1	no	—
115	C₃×C₄⋊C₄	—	3	1	no	—
116	C₃×C₂²⋊C₄	—	3	1	no	—
117	C₃×C₂²⋊C₄	—	3	1	no	—
118	C₃×C₄²	—	2	1	no	—
119	C₃×C₄²	—	3	1	no	—
120	C₃×C₄²	C₄×D₃	1	1	yes	characteristic
121	C₃×C₄²	C₂×C₃×C₄	1	1	yes	characteristic
122	(C₂×D₃)⋊C₄	C₃×C₈	1	1	yes	characteristic
123	(C₂×D₃)⋊C₄	C₃×C₈	1	1	yes	characteristic
124	Dic₃⋊C₄	C₃×C₈	1	1	yes	characteristic
125	Dic₃⋊C₄	C₃×C₈	1	1	yes	characteristic
126	C₄×Dic₃	C₂×C₃×C₄	1	1	yes	characteristic
127	C₂×C₃⋊C₈	C₂×C₃×C₄	1	1	yes	characteristic
128	C₂×C₃⋊C₈	C₂×C₃×C₄	1	1	yes	characteristic
129	ΓL(1,𝔽₂₅)	—	2	2	no	—
130	D₃×C₈	—	2	1	no	—
131	D₃×C₈	—	2	1	no	—
132	C₃×C₁₆	—	2	1	no	—
133	C₃×C₁₆	—	2	1	no	—
134	C₃×C₁₆	—	2	1	no	—
135	C₃×C₁₆	—	2	1	no	—
136	C₃×C₁₆	—	4	1	no	—
137	C₃×C₁₆	—	4	1	no	—
138	C₃×C₁₆	—	6	1	no	—
139	C₃×C₁₆	—	6	1	no	—
140	C₃⋊C₁₆	—	2	1	no	—
141	C₃⋊C₁₆	—	2	1	no	—
142	C₂²×C₃²	C₂²×C₈	1	1	yes	characteristic
143	C₂×C₃×D₃	—	2	1	no	—
144	C₂×C₃×D₃	—	2	1	no	—
145	C₂×C₃×D₃	—	2	1	no	—
146	C₃²×C₄	—	2	1	no	—
147	C₃²×C₄	—	2	1	no	—
148	C₃²×C₄	#117	1	1	yes	characteristic
149	C₃²×C₄	C₂×M₁₆	1	1	yes	characteristic
150	C₃×Dic₃	—	2	1	no	—
151	C₃×Dic₃	—	2	1	no	—
152	C₃×Dic₃	—	2	1	no	—
153	C₂×M₁₆	—	3	1	no	—
154	C₂²×C₈	—	3	1	no	—
155	#117	—	3	1	no	—
156	C₂×C₁₆	—	3	1	no	—
157	C₂×C₁₆	—	3	1	no	—
158	C₂×C₁₆	C₂×C₃×D₃	1	1	yes	characteristic
159	C₂×C₁₆	C₂×C₃×D₃	1	1	yes	characteristic
160	#97	—	3	1	no	—
161	#97	—	3	1	no	—
162	C₄×C₈	—	3	1	no	—
163	C₄×C₈	C₂×C₃×D₃	1	1	yes	characteristic
164	C₂³×C₃	—	3	1	no	—
165	C₂²×D₃	C₂×C₃×C₈	1	1	yes	characteristic
166	C₂×C₃×C₄	—	2	1	no	—
167	C₂×C₃×C₄	—	2	1	no	—
168	C₂×C₃×C₄	—	2	1	no	—
169	C₂×C₃×C₄	—	3	1	no	—
170	C₂×C₃×C₄	—	3	1	no	—
171	C₂×C₃×C₄	—	3	1	no	—
172	C₂×C₃×C₄	—	6	1	no	—
173	C₂×C₃×C₄	—	6	1	no	—
174	C₂×C₃×C₄	D₃×C₈	1	1	yes	characteristic
175	C₂×C₃×C₄	D₃×C₈	1	1	yes	characteristic
176	C₂×C₃×C₄	C₂×C₃×C₈	1	1	yes	characteristic
177	C₂×C₃×C₄	C₂×C₃×C₈	1	1	yes	characteristic
178	C₂×C₃×C₄	C₂×C₄×D₃	1	1	yes	characteristic
179	C₂×C₃×C₄	C₂²×C₃×C₄	1	1	yes	characteristic
180	C₂×Dic₃	C₂×C₃×C₈	1	1	yes	characteristic
181	C₂×Dic₃	C₂×C₃×C₈	1	1	yes	characteristic
182	C₂×Dic₃	C₂×C₃×C₈	1	1	yes	characteristic
183	C₄×D₃	—	2	1	no	—
184	C₄×D₃	—	2	1	no	—
185	C₃×C₈	—	2	1	no	—
186	C₃×C₈	—	2	1	no	—
187	C₃×C₈	—	2	1	no	—
188	C₃×C₈	—	2	1	no	—
189	C₃×C₈	—	3	2	no	—
190	C₃×C₈	—	4	1	no	—
191	C₃×C₈	—	6	1	no	—
192	C₃×C₈	#237	1	1	yes	characteristic
193	C₃×C₈	#239	1	1	yes	characteristic
194	C₃×C₈	C₃×#41	1	1	yes	characteristic
195	C₃×C₈	C₃×#41	1	1	yes	characteristic
196	C₃⋊C₈	—	2	1	no	—
197	C₃⋊C₈	C₃×M₁₆	1	2	yes	—
198	C₂×C₃²	#431	1	1	yes	characteristic
199	C₂×C₃²	#504	1	1	yes	characteristic
200	C₂×C₃²	#504	1	1	yes	characteristic
201	C₃×D₃	—	4	1	no	—
202	C₂²×C₄	—	3	1	no	—
203	M₁₆	—	6	2	no	—
204	C₂×C₈	—	3	1	no	—
205	C₂×C₈	—	3	1	no	—
206	C₂×C₈	—	6	1	no	—
207	C₂×C₈	—	6	1	no	—
208	C₂×C₈	C₃×C₄×D₃	1	1	yes	characteristic
209	C₂×C₈	C₂²×C₃×D₃	1	1	yes	characteristic
210	C₄⋊C₄	—	3	1	no	—
211	C₄⋊C₄	—	3	1	no	—
212	C₂²⋊C₄	—	3	1	no	—
213	C₂²⋊C₄	—	3	1	no	—
214	C₄²	—	3	1	no	—
215	C₄²	C₃×C₄×D₃	1	1	yes	characteristic
216	C₁₆	—	2	1	no	—
217	C₁₆	—	2	1	no	—
218	C₁₆	—	6	1	no	—
219	C₁₆	—	6	1	no	—
220	C₂²×C₃	—	2	1	no	—
221	C₂²×C₃	—	6	1	no	—
222	C₂²×C₃	—	6	1	no	—
223	C₂²×C₃	—	6	1	no	—
224	C₂²×C₃	C₂×D₃×C₈	1	1	yes	characteristic
225	C₂²×C₃	C₂²×C₃×C₈	1	1	yes	characteristic
226	C₂×D₃	—	2	1	no	—
227	C₂×D₃	—	2	1	no	—
228	C₂×D₃	—	2	1	no	—
229	C₃×C₄	—	2	2	no	—
230	C₃×C₄	—	2	1	no	—
231	C₃×C₄	—	2	1	no	—
232	C₃×C₄	—	2	1	no	—
233	C₃×C₄	—	2	1	no	—
234	C₃×C₄	—	2	1	no	—
235	C₃×C₄	—	2	1	no	—
236	C₃×C₄	—	4	1	no	—
237	C₃×C₄	—	6	1	no	—
238	C₃×C₄	—	6	1	no	—
239	C₃×C₄	—	6	1	no	—
240	C₃×C₄	#892	1	1	yes	characteristic
241	C₃×C₄	D₃×M₁₆	1	1	yes	characteristic
242	C₃×C₄	C₃×#117	1	1	yes	characteristic
243	C₃×C₄	C₂×C₃×M₁₆	1	1	yes	characteristic
244	Dic₃	—	2	1	no	—
245	Dic₃	—	2	1	no	—
246	Dic₃	—	2	1	no	—
247	C₃²	#2164	1	1	yes	characteristic, complemented, Sylow
248	C₂³	—	3	1	no	—
249	C₂×C₄	—	3	1	no	—
250	C₂×C₄	—	3	1	no	—
251	C₂×C₄	—	3	1	no	—
252	C₂×C₄	—	6	1	no	—
253	C₂×C₄	—	6	1	no	—
254	C₂×C₄	C₃×D₃×C₈	1	1	yes	characteristic
255	C₂×C₄	C₃×D₃×C₈	1	1	yes	characteristic
256	C₂×C₄	C₂×C₃×C₄×D₃	1	1	yes	characteristic
257	C₈	—	2	1	no	—
258	C₈	—	3	2	no	—
259	C₈	—	6	1	no	—
260	C₈	C₃×#237	1	1	yes	characteristic
261	C₈	C₃×#239	1	1	yes	characteristic
262	C₂×C₃	—	2	1	no	—
263	C₂×C₃	—	2	1	no	—
264	C₂×C₃	—	2	1	no	—
265	C₂×C₃	—	12	1	no	—
266	C₂×C₃	#4833	1	1	yes	characteristic
267	C₂×C₃	#4901	1	1	yes	characteristic
268	C₂×C₃	#4907	1	1	yes	characteristic
269	C₂×C₃	C₃×#431	1	1	yes	characteristic
270	C₂×C₃	C₃×#504	1	1	yes	characteristic
271	C₂×C₃	C₃×#504	1	1	yes	characteristic
272	D₃	—	4	1	no	—
273	C₂²	—	6	1	no	—
274	C₂²	—	6	1	no	—
275	C₂²	—	6	1	no	—
276	C₂²	C₂×C₃×D₃×C₈	1	1	yes	characteristic
277	C₄	—	2	1	no	—
278	C₄	—	2	1	no	—
279	C₄	—	6	1	no	—
280	C₄	—	6	1	no	—
281	C₄	—	6	1	no	—
282	C₄	C₃×#892	1	1	yes	characteristic
283	C₄	C₃×D₃×M₁₆	1	1	yes	characteristic
284	C₃	—	2	1	no	—
285	C₃	#69276	1	1	yes	characteristic, direct product factor
286	C₃	C₃×#2164	1	1	yes	characteristic, complemented
287	C₂	—	12	1	no	—
288	C₂	C₃×#4833	1	1	yes	characteristic
289	C₂	C₃×#4901	1	1	yes	characteristic
290	C₂	C₃×#4907	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₃×C₈	C₂²×D₃
commutator	C₂²×C₃	C₂²×C₃×C₈
Frattini	C₂×C₈	C₂²×C₃×D₃
Fitting	C₃²×C₄⋊C₁₆	C₂
socle	C₂²×C₃²	C₂²×C₈
prime radical	C₂²×C₃×D₃	C₂×C₈
cosocle	C₂×C₃×C₈	C₂³×C₃
hypercenter	C₃×C₄⋊C₁₆	D₃
hypocenter	C₃	C₃×#2164
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₃×C₈ ◁ C₃×C₄⋊C₁₆

Complements

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

N	H ≅ G/N	type
2. C₃²×C₄⋊C₁₆	C₂	semidirect
5. C₃×#4502	C₂	semidirect
6. C₃×Dic₃×C₁₆	C₂	semidirect
7. C₃×C₄⋊(C₃⋊C₁₆)	C₂	semidirect
9. #69276	C₃	direct (N × H)
25. C₃×C₄⋊C₁₆	D₃	semidirect
26. C₃×C₄⋊C₁₆	C₂×C₃	semidirect
33. #4502	C₂×C₃	semidirect
34. Dic₃×C₁₆	C₂×C₃	semidirect
35. C₄⋊(C₃⋊C₁₆)	C₂×C₃	semidirect
96. C₄⋊C₁₆	C₃×D₃	semidirect
247. C₃²	#2164	semidirect
285. C₃	#69276	direct (N × H)
286. C₃	C₃×#2164	semidirect

Maximal subgroups

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

subgroup	index
C₃×#4689	2
C₃²×C₄⋊C₁₆	2
C₃×(C₂×D₃)⋊C₁₆	2
C₃×#4502	2
C₃×Dic₃×C₁₆	2
C₃×C₄⋊(C₃⋊C₁₆)	2
C₃×#2164	3
#69276	3

Subgroups — C3×#69276 #11797349

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#69276 #11797349