Subgroups — C₃×#79042 #11859520

Order 1152.

1031 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×#907	C₂	1	2	yes	maximal, complemented
2	C₃×#5477	C₂	1	1	yes	maximal, characteristic
3	C₃²×#423	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×#5129	C₂	1	2	yes	maximal, complemented
5	C₂×C₃×#803	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃×#4836	C₂	1	8	yes	maximal, complemented
7	C₃×#2932	—	3	1	no	maximal
8	#79042	C₃	1	1	yes	maximal, characteristic, direct product factor
9	C₂×C₃×#235	C₂²	1	2	yes	—
10	C₂×C₃²×M₁₆	C₂²	1	2	yes	—
11	C₂×C₃²×C₄²	C₂²	1	1	yes	characteristic
12	C₃×#930	C₄	1	4	yes	—
13	C₂²×C₃×C₃⋊C₈	C₂²	1	1	yes	characteristic
14	C₂²×C₃×C₃⋊C₈	C₂²	1	1	yes	characteristic
15	C₃×#907	—	2	8	no	—
16	C₂×C₃×ΓL(1,𝔽₂₅)	C₂²	1	4	yes	—
17	C₂×C₃×D₃×C₈	C₂²	1	4	yes	—
18	C₃×#875	C₄	1	2	yes	—
19	C₃×#874	C₄	1	1	yes	characteristic
20	C₃×C₄×D₁₂	C₂²	1	4	yes	—
21	C₃×#869	C₄	1	1	yes	characteristic
22	C₃²×C₄⋊C₈	C₂²	1	4	yes	—
23	C₃×(C₂×D₃)⋊C₈	C₂²	1	8	yes	—
24	C₃×#803	C₂²	1	4	yes	—
25	C₂×C₃×#131	—	3	2	no	—
26	C₃×#532	—	3	1	no	—
27	C₂×#907	C₂×C₃	1	2	yes	complemented
28	#5477	C₂×C₃	1	1	yes	characteristic
29	C₃×#435	—	3	8	no	—
30	C₃×#423	—	2	1	no	—
31	C₃×#423	D₃	1	1	yes	characteristic, complemented
32	C₃×#423	C₂×C₃	1	1	yes	characteristic, complemented
33	C₃×#408	—	3	2	no	—
34	C₂×C₃×#97	—	3	1	no	—
35	#5129	C₂×C₃	1	2	yes	complemented
36	C₂×#803	C₂×C₃	1	1	yes	characteristic, complemented
37	#4836	C₂×C₃	1	8	yes	complemented
38	C₂²×C₃²×C₄	C₂×C₄	1	2	yes	—
39	C₂²×C₃²×C₄	C₂³	1	1	yes	characteristic
40	C₂×C₃×#67	C₂×C₄	1	4	yes	—
41	C₃×#235	—	2	4	no	—
42	C₃×#235	—	2	4	no	—
43	C₂×C₃×D₁₂	C₂×C₄	1	2	yes	—
44	C₂×C₃×D₁₂	C₂×C₄	1	4	yes	—
45	C₂×C₃×C₄×D₃	C₂³	1	4	yes	—
46	C₂×C₃×Dic₆	C₂×C₄	1	2	yes	—
47	C₃²×M₁₆	—	2	4	no	—
48	C₂×C₃²×C₈	C₂³	1	4	yes	—
49	C₃²×C₄²	C₂×C₄	1	1	yes	characteristic
50	C₃²×C₄²	C₂×C₄	1	1	yes	characteristic
51	C₃²×C₄²	C₂³	1	2	yes	—
52	C₃×(C₂×D₃)⋊C₄	C₂×C₄	1	8	yes	—
53	C₃×C₄⋊Dic₃	C₂×C₄	1	4	yes	—
54	C₂×C₃×C₃⋊C₈	—	2	2	no	—
55	C₂×C₃×C₃⋊C₈	D₄	1	4	yes	complemented
56	C₂×C₃×C₃⋊C₈	C₂³	1	2	yes	—
57	C₂×C₃×C₃⋊C₈	C₂³	1	2	yes	—
58	C₃×ΓL(1,𝔽₂₅)	—	2	8	no	—
59	C₃×D₃×C₈	—	2	8	no	—
60	#2932	—	3	1	no	Sylow
61	C₂×C₃×#41	—	3	2	no	—
62	C₂×#235	C₂²×C₃	1	2	yes	—
63	C₃×#131	—	6	8	no	—
64	C₂×C₃×M₁₆	—	2	2	no	—
65	C₂×C₃×M₁₆	—	3	4	no	—
66	C₂×C₃×M₁₆	C₂×D₃	1	2	yes	—
67	C₂×C₃×M₁₆	C₂²×C₃	1	2	yes	—
68	C₂²×C₃×C₈	—	3	1	no	—
69	C₂²×C₃×C₈	—	3	1	no	—
70	C₂²×C₃×C₈	—	3	4	no	—
71	C₃×C₄⋊Q₈	—	3	1	no	—
72	C₃×#127	—	3	1	no	—
73	C₃×#124	—	3	2	no	—
74	C₃×#121	—	3	4	no	—
75	C₃×C₄×D₄	—	3	4	no	—
76	C₂×C₃×C₄²	—	2	1	no	—
77	C₂×C₃×C₄²	C₂×D₃	1	1	yes	characteristic
78	C₂×C₃×C₄²	C₂²×C₃	1	1	yes	characteristic
79	#930	C₃×C₄	1	4	yes	—
80	C₂²×C₃⋊C₈	C₂²×C₃	1	1	yes	characteristic
81	C₂²×C₃⋊C₈	C₂²×C₃	1	1	yes	characteristic
82	#907	—	2	8	no	—
83	C₂×ΓL(1,𝔽₂₅)	C₂²×C₃	1	4	yes	—
84	C₂×D₃×C₈	C₂²×C₃	1	4	yes	—
85	#875	C₃×C₄	1	2	yes	—
86	#874	C₃×C₄	1	1	yes	characteristic
87	C₄×D₁₂	C₂²×C₃	1	4	yes	—
88	#869	C₃×C₄	1	1	yes	characteristic
89	C₃×C₄⋊C₈	—	2	4	no	—
90	C₃×C₄⋊C₈	C₂×D₃	1	4	yes	—
91	C₃×C₄⋊C₈	C₂²×C₃	1	4	yes	—
92	C₃×C₂²⋊C₈	—	3	8	no	—
93	C₃×#97	—	3	4	no	—
94	(C₂×D₃)⋊C₈	C₂²×C₃	1	8	yes	—
95	#803	C₂²×C₃	1	4	yes	—
96	C₂³×C₃²	C₂²×C₄	1	1	yes	characteristic
97	C₂²×C₃×D₃	C₂²×C₄	1	4	yes	—
98	C₂×C₃²×C₄	—	4	2	no	—
99	C₂×C₃²×C₄	C₂²×C₄	1	2	yes	—
100	C₂×C₃²×C₄	C₂²×C₄	1	2	yes	—
101	C₂×C₃²×C₄	C₂²×C₄	1	1	yes	characteristic
102	C₂×C₃²×C₄	C₂×D₄	1	2	yes	—
103	C₂×C₃²×C₄	#41	1	2	yes	—
104	C₂×C₃²×C₄	C₂⁴	1	1	yes	characteristic
105	C₃×#67	—	2	8	no	—
106	C₂×C₃×Dic₃	C₂²×C₄	1	4	yes	—
107	C₃×D₁₂	—	2	4	no	—
108	C₃×D₁₂	—	4	4	no	—
109	C₃×C₄×D₃	—	2	4	no	—
110	C₃×C₄×D₃	—	2	4	no	—
111	C₃×Dic₆	—	2	4	no	—
112	C₃²×C₈	—	2	4	no	—
113	C₃×C₃⋊C₈	—	2	2	no	—
114	C₃×C₃⋊C₈	C₂×D₄	1	4	yes	—
115	C₂×#131	—	3	2	no	—
116	#532	—	3	1	no	—
117	#435	—	3	8	no	—
118	#423	C₃×D₃	1	1	yes	characteristic, complemented
119	#408	—	3	2	no	—
120	C₂×#97	—	3	1	no	—
121	C₃×#41	—	6	4	no	—
122	C₃×#41	—	6	4	no	—
123	C₂×C₃×Q₈	—	3	2	no	—
124	C₂×C₃×D₄	—	3	4	no	—
125	C₂×C₃×D₄	—	3	2	no	—
126	C₂×C₃×D₄	—	3	4	no	—
127	C₂²×C₃×C₄	—	2	2	no	—
128	C₂²×C₃×C₄	—	2	1	no	—
129	C₂²×C₃×C₄	—	3	4	no	—
130	C₂²×C₃×C₄	C₄×D₃	1	2	yes	—
131	C₂²×C₃×C₄	C₂×C₃×C₄	1	2	yes	—
132	C₂²×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
133	C₂²×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
134	C₂×#67	C₂×C₃×C₄	1	4	yes	—
135	#235	—	2	4	no	—
136	#235	—	2	4	no	—
137	C₂×D₁₂	C₂×C₃×C₄	1	2	yes	—
138	C₂×D₁₂	C₂×C₃×C₄	1	4	yes	—
139	C₂×C₄×D₃	C₂³×C₃	1	4	yes	—
140	C₂×Dic₆	C₂×C₃×C₄	1	2	yes	—
141	C₃×M₁₆	—	2	4	no	—
142	C₃×M₁₆	—	2	4	no	—
143	C₃×M₁₆	—	4	4	no	—
144	C₃×M₁₆	—	6	8	no	—
145	C₂×C₃×C₈	—	2	4	no	—
146	C₂×C₃×C₈	—	3	4	no	—
147	C₂×C₃×C₈	—	3	2	no	—
148	C₂×C₃×C₈	—	3	2	no	—
149	C₂×C₃×C₈	—	6	8	no	—
150	C₂×C₃×C₈	—	6	2	no	—
151	C₂×C₃×C₈	C₂²×D₃	1	4	yes	—
152	C₂×C₃×C₈	C₂³×C₃	1	4	yes	—
153	C₃×C₄⋊C₄	—	3	4	no	—
154	C₃×C₂²⋊C₄	—	3	8	no	—
155	C₃×C₄²	—	2	1	no	—
156	C₃×C₄²	—	2	1	no	—
157	C₃×C₄²	—	2	2	no	—
158	C₃×C₄²	C₄×D₃	1	1	yes	characteristic
159	C₃×C₄²	C₄×D₃	1	1	yes	characteristic
160	C₃×C₄²	C₂×C₃×C₄	1	1	yes	characteristic
161	C₃×C₄²	C₂×C₃×C₄	1	1	yes	characteristic
162	C₃×C₄²	C₂²×D₃	1	2	yes	—
163	C₃×C₄²	C₂³×C₃	1	2	yes	—
164	(C₂×D₃)⋊C₄	C₂×C₃×C₄	1	8	yes	—
165	C₄⋊Dic₃	C₂×C₃×C₄	1	4	yes	—
166	C₂×C₃⋊C₈	—	2	2	no	—
167	C₂×C₃⋊C₈	C₃×D₄	1	4	yes	complemented
168	C₂×C₃⋊C₈	C₂³×C₃	1	2	yes	—
169	C₂×C₃⋊C₈	C₂³×C₃	1	2	yes	—
170	ΓL(1,𝔽₂₅)	—	2	8	no	—
171	D₃×C₈	—	2	8	no	—
172	C₂²×C₃²	—	2	2	no	—
173	C₂²×C₃²	C₄×D₄	1	2	yes	—
174	C₂²×C₃²	C₂³×C₄	1	1	yes	characteristic
175	C₂×C₃×D₃	—	2	4	no	—
176	C₂×C₃×D₃	—	2	8	no	—
177	C₃²×C₄	—	2	2	no	—
178	C₃²×C₄	—	2	2	no	—
179	C₃²×C₄	C₄×D₄	1	2	yes	—
180	C₃²×C₄	C₂²×D₄	1	1	yes	characteristic
181	C₃²×C₄	C₂×#41	1	1	yes	characteristic
182	C₃×Dic₃	—	2	4	no	—
183	C₂×#41	—	3	2	no	—
184	#131	—	6	8	no	—
185	C₂×M₁₆	—	3	4	no	—
186	C₂×M₁₆	C₂×C₃×D₃	1	2	yes	—
187	C₂²×C₈	—	3	1	no	—
188	C₂²×C₈	—	3	1	no	—
189	C₂²×C₈	—	3	4	no	—
190	C₄⋊Q₈	—	3	1	no	—
191	#127	—	3	1	no	—
192	#124	—	3	2	no	—
193	#121	—	3	4	no	—
194	C₄×D₄	—	3	4	no	—
195	C₂×C₄²	C₂×C₃×D₃	1	1	yes	characteristic
196	C₄⋊C₈	C₂×C₃×D₃	1	4	yes	—
197	C₂²⋊C₈	—	3	8	no	—
198	#97	—	3	4	no	—
199	C₂³×C₃	—	2	1	no	—
200	C₂³×C₃	—	3	4	no	—
201	C₂³×C₃	C₂×C₄×D₃	1	1	yes	characteristic
202	C₂³×C₃	C₂²×C₃×C₄	1	1	yes	characteristic
203	C₂²×D₃	C₂²×C₃×C₄	1	4	yes	—
204	C₃×Q₈	—	6	4	no	—
205	C₃×D₄	—	6	8	no	—
206	C₃×D₄	—	6	4	no	—
207	C₃×D₄	—	12	4	no	—
208	C₂×C₃×C₄	—	2	1	no	—
209	C₂×C₃×C₄	—	2	2	no	—
210	C₂×C₃×C₄	—	2	1	no	—
211	C₂×C₃×C₄	—	2	2	no	—
212	C₂×C₃×C₄	—	2	2	no	—
213	C₂×C₃×C₄	—	2	2	no	—
214	C₂×C₃×C₄	—	3	4	no	—
215	C₂×C₃×C₄	—	4	2	no	—
216	C₂×C₃×C₄	—	4	2	no	—
217	C₂×C₃×C₄	—	4	4	no	—
218	C₂×C₃×C₄	—	6	4	no	—
219	C₂×C₃×C₄	—	6	4	no	—
220	C₂×C₃×C₄	C₂×C₄×D₃	1	2	yes	—
221	C₂×C₃×C₄	C₂×C₄×D₃	1	2	yes	—
222	C₂×C₃×C₄	C₂×C₄×D₃	1	1	yes	characteristic
223	C₂×C₃×C₄	D₃×D₄	1	2	yes	—
224	C₂×C₃×C₄	#239	1	2	yes	—
225	C₂×C₃×C₄	C₂²×C₃×C₄	1	2	yes	—
226	C₂×C₃×C₄	C₂²×C₃×C₄	1	2	yes	—
227	C₂×C₃×C₄	C₂²×C₃×C₄	1	1	yes	characteristic
228	C₂×C₃×C₄	C₂×C₃×D₄	1	2	yes	—
229	C₂×C₃×C₄	C₃×#41	1	2	yes	—
230	C₂×C₃×C₄	C₂³×D₃	1	1	yes	characteristic
231	C₂×C₃×C₄	C₂⁴×C₃	1	1	yes	characteristic
232	#67	—	2	8	no	—
233	C₂×Dic₃	C₂²×C₃×C₄	1	4	yes	—
234	D₁₂	—	2	4	no	—
235	D₁₂	—	4	4	no	—
236	C₄×D₃	—	2	4	no	—
237	C₄×D₃	—	2	4	no	—
238	Dic₆	—	2	4	no	—
239	C₃×C₈	—	2	4	no	—
240	C₃×C₈	—	2	4	no	—
241	C₃×C₈	—	3	4	no	—
242	C₃×C₈	—	4	4	no	—
243	C₃×C₈	—	6	2	no	—
244	C₃⋊C₈	—	2	2	no	—
245	C₃⋊C₈	C₂×C₃×D₄	1	4	yes	—
246	C₂×C₃²	—	2	2	no	—
247	C₂×C₃²	C₂×C₄×D₄	1	1	yes	characteristic
248	C₂×C₃²	#568	1	2	yes	—
249	C₃×D₃	—	4	4	no	—
250	#41	—	6	4	no	—
251	#41	—	6	4	no	—
252	C₂×Q₈	—	3	2	no	—
253	C₂×D₄	—	3	4	no	—
254	C₂×D₄	—	3	2	no	—
255	C₂×D₄	—	3	4	no	—
256	C₂²×C₄	—	3	4	no	—
257	C₂²×C₄	C₃×C₄×D₃	1	2	yes	—
258	C₂²×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
259	M₁₆	—	2	4	no	—
260	M₁₆	—	6	8	no	—
261	C₂×C₈	—	3	2	no	—
262	C₂×C₈	—	3	4	no	—
263	C₂×C₈	—	3	2	no	—
264	C₂×C₈	—	6	8	no	—
265	C₂×C₈	—	6	2	no	—
266	C₂×C₈	C₂²×C₃×D₃	1	4	yes	—
267	C₄⋊C₄	—	3	4	no	—
268	C₂²⋊C₄	—	3	8	no	—
269	C₄²	C₃×C₄×D₃	1	1	yes	characteristic
270	C₄²	C₃×C₄×D₃	1	1	yes	characteristic
271	C₄²	C₂²×C₃×D₃	1	2	yes	—
272	C₂²×C₃	—	2	2	no	—
273	C₂²×C₃	—	2	2	no	—
274	C₂²×C₃	—	2	2	no	—
275	C₂²×C₃	—	2	1	no	—
276	C₂²×C₃	—	4	2	no	—
277	C₂²×C₃	—	6	8	no	—
278	C₂²×C₃	—	6	4	no	—
279	C₂²×C₃	#893	1	2	yes	—
280	C₂²×C₃	C₃×C₄×D₄	1	2	yes	—
281	C₂²×C₃	C₂²×C₄×D₃	1	1	yes	characteristic
282	C₂²×C₃	C₂³×C₃×C₄	1	1	yes	characteristic
283	C₂×D₃	—	2	8	no	—
284	C₂×D₃	—	2	4	no	—
285	C₃×C₄	—	2	4	no	—
286	C₃×C₄	—	2	2	no	—
287	C₃×C₄	—	2	2	no	—
288	C₃×C₄	—	2	2	no	—
289	C₃×C₄	—	2	2	no	—
290	C₃×C₄	—	2	2	no	—
291	C₃×C₄	—	2	1	no	—
292	C₃×C₄	—	2	1	no	—
293	C₃×C₄	—	4	2	no	—
294	C₃×C₄	—	6	4	no	—
295	C₃×C₄	#893	1	2	yes	—
296	C₃×C₄	C₃×C₄×D₄	1	2	yes	—
297	C₃×C₄	C₂×D₃×D₄	1	1	yes	characteristic
298	C₃×C₄	C₂×#239	1	1	yes	characteristic
299	C₃×C₄	C₂²×C₃×D₄	1	1	yes	characteristic
300	C₃×C₄	C₂×C₃×#41	1	1	yes	characteristic
301	Dic₃	—	2	4	no	—
302	C₃²	#2932	1	1	yes	characteristic, complemented, Sylow
303	C₂³	—	3	4	no	—
304	C₂³	C₂×C₃×C₄×D₃	1	1	yes	characteristic
305	Q₈	—	6	4	no	—
306	D₄	—	6	8	no	—
307	D₄	—	6	4	no	—
308	D₄	—	12	4	no	—
309	C₂×C₄	—	3	4	no	—
310	C₂×C₄	—	4	2	no	—
311	C₂×C₄	—	6	4	no	—
312	C₂×C₄	—	6	4	no	—
313	C₂×C₄	C₂×C₃×C₄×D₃	1	2	yes	—
314	C₂×C₄	C₂×C₃×C₄×D₃	1	2	yes	—
315	C₂×C₄	C₂×C₃×C₄×D₃	1	1	yes	characteristic
316	C₂×C₄	C₃×D₃×D₄	1	2	yes	—
317	C₂×C₄	C₃×#239	1	2	yes	—
318	C₂×C₄	C₂³×C₃×D₃	1	1	yes	characteristic
319	C₈	—	2	4	no	—
320	C₈	—	3	4	no	—
321	C₈	—	6	2	no	—
322	C₂×C₃	—	2	2	no	—
323	C₂×C₃	—	2	2	no	—
324	C₂×C₃	—	2	2	no	—
325	C₂×C₃	—	2	1	no	—
326	C₂×C₃	—	4	2	no	—
327	C₂×C₃	—	12	4	no	—
328	C₂×C₃	C₂×#893	1	1	yes	characteristic
329	C₂×C₃	#5745	1	2	yes	—
330	C₂×C₃	C₂×C₃×C₄×D₄	1	1	yes	characteristic
331	C₂×C₃	C₃×#568	1	2	yes	—
332	D₃	—	4	4	no	—
333	C₂²	—	2	2	no	—
334	C₂²	—	6	8	no	—
335	C₂²	—	6	4	no	—
336	C₂²	C₃×#893	1	2	yes	—
337	C₂²	C₂²×C₃×C₄×D₃	1	1	yes	characteristic
338	C₄	—	2	2	no	—
339	C₄	—	2	2	no	—
340	C₄	—	6	4	no	—
341	C₄	C₃×#893	1	2	yes	—
342	C₄	C₂×C₃×D₃×D₄	1	1	yes	characteristic
343	C₄	C₂×C₃×#239	1	1	yes	characteristic
344	C₃	—	2	1	no	—
345	C₃	#79042	1	1	yes	characteristic, direct product factor
346	C₃	C₃×#2932	1	1	yes	characteristic, complemented
347	C₂	—	2	2	no	—
348	C₂	—	12	4	no	—
349	C₂	C₂×C₃×#893	1	1	yes	characteristic
350	C₂	C₃×#5745	1	2	yes	—

	subgroup	G/H
center	C₂×C₃×C₄	C₂³×D₃
commutator	C₂²×C₃	C₂³×C₃×C₄
Frattini	C₂×C₄	C₂³×C₃×D₃
Fitting	C₃²×#423	C₂
socle	C₂²×C₃²	C₂³×C₄
prime radical	C₃×#5477	C₂
cosocle	C₂×C₃×C₄	C₂⁴×C₃
hypercenter	C₃×#423	D₃
hypocenter	C₃	C₃×#2932
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₃×C₄ ◁ C₃×#423

Complements

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

N	H ≅ G/N	type
1. C₂×C₃×#907	C₂	semidirect
3. C₃²×#423	C₂	semidirect
4. C₃×#5129	C₂	semidirect
5. C₂×C₃×#803	C₂	semidirect
6. C₃×#4836	C₂	semidirect
8. #79042	C₃	direct (N × H)
27. C₂×#907	C₂×C₃	semidirect
31. C₃×#423	D₃	semidirect
32. C₃×#423	C₂×C₃	semidirect
35. #5129	C₂×C₃	semidirect
36. C₂×#803	C₂×C₃	semidirect
37. #4836	C₂×C₃	semidirect
55. C₂×C₃×C₃⋊C₈	D₄	semidirect
118. #423	C₃×D₃	semidirect
167. C₂×C₃⋊C₈	C₃×D₄	semidirect
302. C₃²	#2932	semidirect
345. C₃	#79042	direct (N × H)
346. C₃	C₃×#2932	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃×#907	2
C₃×#5477	2
C₃²×#423	2
C₃×#5129	2
C₂×C₃×#803	2
C₃×#4836	2
C₃×#2932	3
#79042	3

Subgroups — C3×#79042 #11859520

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#79042 #11859520