Subgroups — C₃×#68569 #11776886

Order 1152.

395 conjugacy classes of subgroups.

The subgroup lattice has 295 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#4837	C₂	1	2	yes	maximal
2	C₃×C₄×D₃×C₈	C₂	1	1	yes	maximal, characteristic
3	C₃²×#335	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×#4479	C₂	1	2	yes	maximal, complemented
5	C₃×#4458	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃×#1567	—	3	1	no	maximal
7	#68569	C₃	1	1	yes	maximal, characteristic, direct product factor
8	C₂×C₃×ΓL(1,𝔽₂₅)	—	2	2	no	—
9	C₂×C₃×D₃×C₈	C₄	1	1	yes	characteristic
10	C₂×C₃×D₃×C₈	C₄	1	1	yes	characteristic
11	C₃×C₄²×D₃	C₂²	1	1	yes	characteristic
12	C₃²×C₄⋊C₈	C₂²	1	2	yes	—
13	C₃²×C₄×C₈	C₂²	1	1	yes	characteristic
14	C₃×Dic₃⋊C₈	—	2	2	no	—
15	C₃×C₈×Dic₃	C₄	1	1	yes	characteristic
16	C₃×C₈×Dic₃	C₄	1	1	yes	characteristic
17	C₃×C₄⋊(C₃⋊C₈)	C₂²	1	2	yes	—
18	C₃×C₄×C₃⋊C₈	C₂²	1	1	yes	characteristic
19	C₃×#423	—	3	2	no	—
20	C₂×C₃×C₄×C₈	—	3	1	no	—
21	#4837	C₂×C₃	1	2	yes	—
22	C₄×D₃×C₈	C₂×C₃	1	1	yes	characteristic
23	C₃×#335	—	2	1	no	—
24	C₃×#335	—	3	2	no	—
25	C₃×#335	—	3	1	no	—
26	C₃×#335	D₃	1	1	yes	characteristic, complemented
27	C₃×#335	C₂×C₃	1	1	yes	characteristic, complemented
28	#4479	C₂×C₃	1	2	yes	complemented
29	#4458	C₂×C₃	1	1	yes	characteristic, complemented
30	C₂×C₃×C₄×D₃	C₂×C₄	1	1	yes	characteristic
31	C₂×C₃×C₄×D₃	D₄	1	1	yes	characteristic
32	C₂×C₃×C₄×D₃	Q₈	1	1	yes	characteristic
33	C₂×C₃²×C₈	—	2	2	no	—
34	C₂×C₃²×C₈	C₂×C₄	1	1	yes	characteristic
35	C₂×C₃²×C₈	C₂×C₄	1	1	yes	characteristic
36	C₃²×C₄²	C₂³	1	1	yes	characteristic
37	C₃×C₄×Dic₃	C₂×C₄	1	1	yes	characteristic
38	C₃×C₄×Dic₃	D₄	1	1	yes	characteristic
39	C₃×C₄×Dic₃	Q₈	1	1	yes	characteristic
40	C₂×C₃×C₃⋊C₈	—	2	2	no	—
41	C₂×C₃×C₃⋊C₈	C₂×C₄	1	1	yes	characteristic
42	C₂×C₃×C₃⋊C₈	C₂×C₄	1	1	yes	characteristic
43	C₃×ΓL(1,𝔽₂₅)	—	4	4	no	—
44	C₃×D₃×C₈	—	2	2	no	—
45	C₃×D₃×C₈	—	2	2	no	—
46	#1567	—	3	1	no	Sylow
47	C₂×C₃×M₁₆	—	6	2	no	—
48	C₂²×C₃×C₈	—	3	1	no	—
49	C₂²×C₃×C₈	—	3	1	no	—
50	C₂×C₃×C₄²	—	3	1	no	—
51	C₂×ΓL(1,𝔽₂₅)	—	2	2	no	—
52	C₂×D₃×C₈	C₃×C₄	1	1	yes	characteristic
53	C₂×D₃×C₈	C₃×C₄	1	1	yes	characteristic
54	C₄²×D₃	C₂²×C₃	1	1	yes	characteristic
55	C₃×C₄⋊C₈	—	2	2	no	—
56	C₃×C₄⋊C₈	—	3	2	no	—
57	C₃×C₄⋊C₈	—	6	2	no	—
58	C₃×C₄⋊C₈	C₂×D₃	1	2	yes	—
59	C₃×C₄⋊C₈	C₂²×C₃	1	2	yes	—
60	C₃×C₄×C₈	—	2	1	no	—
61	C₃×C₄×C₈	—	3	1	no	—
62	C₃×C₄×C₈	—	3	1	no	—
63	C₃×C₄×C₈	—	3	1	no	—
64	C₃×C₄×C₈	C₂×D₃	1	1	yes	characteristic
65	C₃×C₄×C₈	C₂²×C₃	1	1	yes	characteristic
66	Dic₃⋊C₈	—	2	2	no	—
67	C₈×Dic₃	C₃×C₄	1	1	yes	characteristic
68	C₈×Dic₃	C₃×C₄	1	1	yes	characteristic
69	C₄⋊(C₃⋊C₈)	C₂²×C₃	1	2	yes	—
70	C₄×C₃⋊C₈	C₂²×C₃	1	1	yes	characteristic
71	C₂²×C₃×D₃	C₄⋊C₄	1	1	yes	characteristic
72	C₂×C₃²×C₄	C₂²×C₄	1	1	yes	characteristic
73	C₂×C₃²×C₄	C₂×D₄	1	1	yes	characteristic
74	C₂×C₃²×C₄	C₂×Q₈	1	1	yes	characteristic
75	C₂×C₃×Dic₃	C₄⋊C₄	1	1	yes	characteristic
76	C₂×C₃×Dic₃	C₄⋊C₄	1	1	yes	characteristic
77	C₂×C₃×Dic₃	C₄⋊C₄	1	1	yes	characteristic
78	C₃×C₄×D₃	—	2	1	no	—
79	C₃×C₄×D₃	—	2	1	no	—
80	C₃×C₄×D₃	—	2	2	no	—
81	C₃×C₄×D₃	D₈	1	2	yes	—
82	C₃×C₄×D₃	Q₁₆	1	2	yes	—
83	C₃²×C₈	—	2	1	no	—
84	C₃²×C₈	—	4	2	no	—
85	C₃²×C₈	M₁₆	1	2	yes	complemented
86	C₃×C₃⋊C₈	—	2	1	no	—
87	C₃×C₃⋊C₈	—	4	2	no	—
88	C₃×C₃⋊C₈	M₁₆	1	2	yes	complemented
89	#423	—	3	2	no	—
90	C₂×C₄×C₈	—	3	1	no	—
91	#335	—	3	2	no	—
92	#335	—	3	1	no	—
93	#335	C₃×D₃	1	1	yes	characteristic, complemented
94	C₂²×C₃×C₄	—	3	1	no	—
95	C₂²×C₃×C₄	—	3	1	no	—
96	C₂²×C₃×C₄	—	3	1	no	—
97	C₂×C₄×D₃	C₂×C₃×C₄	1	1	yes	characteristic
98	C₂×C₄×D₃	C₃×D₄	1	1	yes	characteristic
99	C₂×C₄×D₃	C₃×Q₈	1	1	yes	characteristic
100	C₃×M₁₆	—	12	4	no	—
101	C₂×C₃×C₈	—	2	2	no	—
102	C₂×C₃×C₈	—	2	2	no	—
103	C₂×C₃×C₈	—	2	1	no	—
104	C₂×C₃×C₈	—	2	1	no	—
105	C₂×C₃×C₈	—	3	1	no	—
106	C₂×C₃×C₈	—	3	1	no	—
107	C₂×C₃×C₈	—	4	2	no	—
108	C₂×C₃×C₈	—	6	2	no	—
109	C₂×C₃×C₈	—	6	2	no	—
110	C₂×C₃×C₈	—	6	2	no	—
111	C₂×C₃×C₈	C₄×D₃	1	1	yes	characteristic
112	C₂×C₃×C₈	C₄×D₃	1	1	yes	characteristic
113	C₂×C₃×C₈	C₂×C₃×C₄	1	1	yes	characteristic
114	C₂×C₃×C₈	C₂×C₃×C₄	1	1	yes	characteristic
115	C₃×C₄²	—	2	1	no	—
116	C₃×C₄²	—	3	1	no	—
117	C₃×C₄²	—	3	1	no	—
118	C₃×C₄²	—	3	1	no	—
119	C₃×C₄²	C₂²×D₃	1	1	yes	characteristic
120	C₃×C₄²	C₂³×C₃	1	1	yes	characteristic
121	C₄×Dic₃	C₂×C₃×C₄	1	1	yes	characteristic
122	C₄×Dic₃	C₃×D₄	1	1	yes	characteristic
123	C₄×Dic₃	C₃×Q₈	1	1	yes	characteristic
124	C₂×C₃⋊C₈	—	2	2	no	—
125	C₂×C₃⋊C₈	C₂×C₃×C₄	1	1	yes	characteristic
126	C₂×C₃⋊C₈	C₂×C₃×C₄	1	1	yes	characteristic
127	ΓL(1,𝔽₂₅)	—	4	4	no	—
128	D₃×C₈	—	2	2	no	—
129	D₃×C₈	—	2	2	no	—
130	C₂²×C₃²	C₂×C₄⋊C₄	1	1	yes	characteristic
131	C₂×C₃×D₃	—	2	1	no	—
132	C₂×C₃×D₃	—	2	1	no	—
133	C₂×C₃×D₃	#107	1	2	yes	—
134	C₃²×C₄	—	2	1	no	—
135	C₃²×C₄	C₂×M₁₆	1	1	yes	characteristic
136	C₃²×C₄	C₂×M₁₆	1	1	yes	characteristic
137	C₃²×C₄	C₂×D₈	1	1	yes	characteristic
138	C₃²×C₄	C₂×Q₁₆	1	1	yes	characteristic
139	C₃×Dic₃	—	2	1	no	—
140	C₃×Dic₃	#107	1	2	yes	—
141	C₃×Dic₃	#108	1	2	yes	—
142	C₂×M₁₆	—	6	2	no	—
143	C₂²×C₈	—	3	1	no	—
144	C₂²×C₈	—	3	1	no	—
145	C₂×C₄²	—	3	1	no	—
146	C₄⋊C₈	—	3	2	no	—
147	C₄⋊C₈	—	6	2	no	—
148	C₄⋊C₈	C₂×C₃×D₃	1	2	yes	—
149	C₄×C₈	—	3	1	no	—
150	C₄×C₈	—	3	1	no	—
151	C₄×C₈	—	3	1	no	—
152	C₄×C₈	C₂×C₃×D₃	1	1	yes	characteristic
153	C₂³×C₃	—	3	1	no	—
154	C₂²×D₃	C₃×C₄⋊C₄	1	1	yes	characteristic
155	C₂×C₃×C₄	—	2	1	no	—
156	C₂×C₃×C₄	—	2	1	no	—
157	C₂×C₃×C₄	—	2	1	no	—
158	C₂×C₃×C₄	—	3	1	no	—
159	C₂×C₃×C₄	—	3	1	no	—
160	C₂×C₃×C₄	—	3	2	no	—
161	C₂×C₃×C₄	—	3	2	no	—
162	C₂×C₃×C₄	—	3	1	no	—
163	C₂×C₃×C₄	—	6	2	no	—
164	C₂×C₃×C₄	—	6	1	no	—
165	C₂×C₃×C₄	—	6	1	no	—
166	C₂×C₃×C₄	C₂×C₄×D₃	1	1	yes	characteristic
167	C₂×C₃×C₄	D₃×D₄	1	1	yes	characteristic
168	C₂×C₃×C₄	D₃×Q₈	1	1	yes	characteristic
169	C₂×C₃×C₄	C₂²×C₃×C₄	1	1	yes	characteristic
170	C₂×C₃×C₄	C₂×C₃×D₄	1	1	yes	characteristic
171	C₂×C₃×C₄	C₂×C₃×Q₈	1	1	yes	characteristic
172	C₂×Dic₃	C₃×C₄⋊C₄	1	1	yes	characteristic
173	C₂×Dic₃	C₃×C₄⋊C₄	1	1	yes	characteristic
174	C₂×Dic₃	C₃×C₄⋊C₄	1	1	yes	characteristic
175	C₄×D₃	—	2	1	no	—
176	C₄×D₃	—	2	1	no	—
177	C₄×D₃	—	2	2	no	—
178	C₄×D₃	C₃×D₈	1	2	yes	—
179	C₄×D₃	C₃×Q₁₆	1	2	yes	—
180	C₃×C₈	—	2	1	no	—
181	C₃×C₈	—	2	1	no	—
182	C₃×C₈	—	2	2	no	—
183	C₃×C₈	—	3	2	no	—
184	C₃×C₈	—	4	1	no	—
185	C₃×C₈	—	4	2	no	—
186	C₃×C₈	—	4	2	no	—
187	C₃×C₈	—	6	1	no	—
188	C₃×C₈	—	8	2	no	—
189	C₃×C₈	—	12	2	no	—
190	C₃×C₈	ΓL(1,𝔽₂₅)	1	2	yes	complemented
191	C₃×C₈	C₃×M₁₆	1	2	yes	complemented
192	C₃⋊C₈	—	2	1	no	—
193	C₃⋊C₈	—	4	2	no	—
194	C₃⋊C₈	C₃×M₁₆	1	2	yes	complemented
195	C₂×C₃²	#423	1	1	yes	characteristic
196	C₂×C₃²	C₂×#107	1	1	yes	characteristic
197	C₂×C₃²	C₂×#108	1	1	yes	characteristic
198	C₃×D₃	—	2	2	no	—
199	C₂²×C₄	—	3	1	no	—
200	C₂²×C₄	—	3	1	no	—
201	C₂²×C₄	—	3	1	no	—
202	M₁₆	—	12	4	no	—
203	C₂×C₈	—	2	2	no	—
204	C₂×C₈	—	3	1	no	—
205	C₂×C₈	—	3	1	no	—
206	C₂×C₈	—	6	2	no	—
207	C₂×C₈	—	6	2	no	—
208	C₂×C₈	—	6	2	no	—
209	C₂×C₈	C₃×C₄×D₃	1	1	yes	characteristic
210	C₂×C₈	C₃×C₄×D₃	1	1	yes	characteristic
211	C₄²	—	3	1	no	—
212	C₄²	—	3	1	no	—
213	C₄²	—	3	1	no	—
214	C₄²	C₂²×C₃×D₃	1	1	yes	characteristic
215	C₂²×C₃	—	2	1	no	—
216	C₂²×C₃	—	3	2	no	—
217	C₂²×C₃	—	6	1	no	—
218	C₂²×C₃	—	6	1	no	—
219	C₂²×C₃	D₃×C₄⋊C₄	1	1	yes	characteristic
220	C₂²×C₃	C₂×C₃×C₄⋊C₄	1	1	yes	characteristic
221	C₂×D₃	—	2	1	no	—
222	C₂×D₃	—	2	1	no	—
223	C₂×D₃	C₃×#107	1	2	yes	—
224	C₃×C₄	—	2	1	no	—
225	C₃×C₄	—	2	1	no	—
226	C₃×C₄	—	2	1	no	—
227	C₃×C₄	—	2	1	no	—
228	C₃×C₄	—	2	1	no	—
229	C₃×C₄	—	2	1	no	—
230	C₃×C₄	—	3	2	no	—
231	C₃×C₄	—	3	2	no	—
232	C₃×C₄	—	4	1	no	—
233	C₃×C₄	—	6	1	no	—
234	C₃×C₄	C₂×ΓL(1,𝔽₂₅)	1	1	yes	characteristic
235	C₃×C₄	D₃×M₁₆	1	1	yes	characteristic
236	C₃×C₄	D₃×D₈	1	1	yes	characteristic
237	C₃×C₄	D₃×Q₁₆	1	1	yes	characteristic
238	C₃×C₄	C₂×C₃×M₁₆	1	1	yes	characteristic
239	C₃×C₄	C₂×C₃×M₁₆	1	1	yes	characteristic
240	C₃×C₄	C₂×C₃×D₈	1	1	yes	characteristic
241	C₃×C₄	C₂×C₃×Q₁₆	1	1	yes	characteristic
242	Dic₃	—	2	1	no	—
243	Dic₃	C₃×#107	1	2	yes	—
244	Dic₃	C₃×#108	1	2	yes	—
245	C₃²	#1567	1	1	yes	characteristic, complemented, Sylow
246	C₂³	—	3	1	no	—
247	C₂×C₄	—	3	1	no	—
248	C₂×C₄	—	3	1	no	—
249	C₂×C₄	—	3	2	no	—
250	C₂×C₄	—	3	2	no	—
251	C₂×C₄	—	3	1	no	—
252	C₂×C₄	—	6	2	no	—
253	C₂×C₄	—	6	1	no	—
254	C₂×C₄	—	6	1	no	—
255	C₂×C₄	C₂×C₃×C₄×D₃	1	1	yes	characteristic
256	C₂×C₄	C₃×D₃×D₄	1	1	yes	characteristic
257	C₂×C₄	C₃×D₃×Q₈	1	1	yes	characteristic
258	C₈	—	2	1	no	—
259	C₈	—	3	2	no	—
260	C₈	—	4	2	no	—
261	C₈	—	6	1	no	—
262	C₈	—	12	2	no	—
263	C₈	C₃×ΓL(1,𝔽₂₅)	1	2	yes	complemented
264	C₂×C₃	—	2	1	no	—
265	C₂×C₃	—	2	1	no	—
266	C₂×C₃	—	2	1	no	—
267	C₂×C₃	—	6	2	no	—
268	C₂×C₃	#4837	1	1	yes	characteristic
269	C₂×C₃	D₃×#107	1	1	yes	characteristic
270	C₂×C₃	D₃×#108	1	1	yes	characteristic
271	C₂×C₃	C₃×#423	1	1	yes	characteristic
272	C₂×C₃	C₂×C₃×#107	1	1	yes	characteristic
273	C₂×C₃	C₂×C₃×#108	1	1	yes	characteristic
274	D₃	—	2	2	no	—
275	C₂²	—	3	2	no	—
276	C₂²	—	6	1	no	—
277	C₂²	—	6	1	no	—
278	C₂²	C₃×D₃×C₄⋊C₄	1	1	yes	characteristic
279	C₄	—	2	1	no	—
280	C₄	—	3	2	no	—
281	C₄	—	3	2	no	—
282	C₄	—	6	1	no	—
283	C₄	C₂×C₃×ΓL(1,𝔽₂₅)	1	1	yes	characteristic
284	C₄	C₃×D₃×M₁₆	1	1	yes	characteristic
285	C₄	C₃×D₃×D₈	1	1	yes	characteristic
286	C₄	C₃×D₃×Q₁₆	1	1	yes	characteristic
287	C₃	—	2	1	no	—
288	C₃	#68569	1	1	yes	characteristic, direct product factor
289	C₃	C₃×#1567	1	1	yes	characteristic, complemented
290	C₂	—	6	2	no	—
291	C₂	C₃×#4837	1	1	yes	characteristic
292	C₂	C₃×D₃×#107	1	1	yes	characteristic
293	C₂	C₃×D₃×#108	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₃×C₄	D₃×D₄
commutator	C₂×C₃×C₄	C₂²×C₃×C₄
Frattini	C₄²	C₂²×C₃×D₃
Fitting	C₃²×#335	C₂
socle	C₂²×C₃²	C₂×C₄⋊C₄
prime radical	C₂²×C₃×D₃	C₄⋊C₄
cosocle	C₃×C₄²	C₂³×C₃
hypercenter	C₃×#335	D₃
hypocenter	C₃	C₃×#1567
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₃×C₄ ◁ C₃×C₄² ◁ C₃×#335

Complements

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

N	H ≅ G/N	type
3. C₃²×#335	C₂	semidirect
4. C₃×#4479	C₂	semidirect
5. C₃×#4458	C₂	semidirect
7. #68569	C₃	direct (N × H)
26. C₃×#335	D₃	semidirect
27. C₃×#335	C₂×C₃	semidirect
28. #4479	C₂×C₃	semidirect
29. #4458	C₂×C₃	semidirect
85. C₃²×C₈	M₁₆	semidirect
88. C₃×C₃⋊C₈	M₁₆	semidirect
93. #335	C₃×D₃	semidirect
190. C₃×C₈	ΓL(1,𝔽₂₅)	semidirect
191. C₃×C₈	C₃×M₁₆	semidirect
194. C₃⋊C₈	C₃×M₁₆	semidirect
245. C₃²	#1567	semidirect
263. C₈	C₃×ΓL(1,𝔽₂₅)	semidirect
288. C₃	#68569	direct (N × H)
289. C₃	C₃×#1567	semidirect

Maximal subgroups

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

subgroup	index
C₃×#4837	2
C₃×C₄×D₃×C₈	2
C₃²×#335	2
C₃×#4479	2
C₃×#4458	2
C₃×#1567	3
#68569	3

Subgroups — C3×#68569 #11776886

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#68569 #11776886