Subgroups — C₃×#60626 #10679233

Order 768.

2134 conjugacy classes of subgroups.

The subgroup lattice has 342 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#3504	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃×#3498	C₂	1	1	yes	maximal, characteristic
3	C₃×#3142	C₂	1	3	yes	maximal, complemented
4	C₃×#3138	C₂	1	3	yes	maximal, complemented
5	C₃×#3136	C₂	1	12	yes	maximal, complemented
6	C₃×#3135	C₂	1	3	yes	maximal, complemented
7	C₂×C₃×#490	C₂	1	3	yes	maximal, complemented
8	C₃×#2897	C₂	1	4	yes	maximal, complemented
9	C₃×#2875	C₂	1	1	yes	maximal, characteristic, complemented
10	#60626	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
11	C₂×C₃×H(5,𝔽₂)	C₂²	1	2	yes	—
12	C₂×C₃×#131	C₂²	1	2	yes	—
13	C₃×#557	—	2	2	no	—
14	C₃×#554	—	2	6	no	—
15	C₃×#550	—	2	2	no	—
16	C₃×D₄²	—	2	6	no	—
17	C₃×#544	C₂²	1	6	yes	complemented
18	C₃×#535	C₂²	1	6	yes	—
19	C₃×#533	C₂²	1	3	yes	complemented
20	C₃×#532	C₂²	1	3	yes	—
21	C₂×C₃×#127	C₂²	1	3	yes	—
22	C₂×C₃×#125	C₂²	1	3	yes	complemented
23	C₃×#520	C₂²	1	2	yes	—
24	C₃×C₄×#41	C₂²	1	1	yes	characteristic
25	C₃×#491	C₂²	1	1	yes	characteristic, complemented
26	C₃×#491	C₂²	1	3	yes	complemented
27	C₃×#490	C₂²	1	3	yes	complemented
28	C₃×#490	C₂²	1	12	yes	complemented
29	C₃×#490	C₂²	1	6	yes	complemented
30	C₃×#490	C₂²	1	3	yes	complemented
31	C₃×#489	C₂²	1	12	yes	complemented
32	C₃×#488	C₂²	1	12	yes	complemented
33	C₃×#486	C₂²	1	12	yes	complemented
34	C₃×#418	C₂²	1	12	yes	complemented
35	C₃×#417	C₂²	1	12	yes	complemented
36	C₃×#416	C₂²	1	12	yes	complemented
37	C₂×C₃×#102	C₂²	1	12	yes	complemented
38	C₃×#405	C₂²	1	6	yes	complemented
39	C₃×C₄×M₁₆	C₂²	1	3	yes	complemented
40	C₂×C₃×#97	C₂²	1	3	yes	complemented
41	#3504	C₂×C₃	1	1	yes	characteristic, complemented
42	#3498	C₂×C₃	1	1	yes	characteristic
43	#3142	C₂×C₃	1	3	yes	complemented
44	#3138	C₂×C₃	1	3	yes	complemented
45	#3136	C₂×C₃	1	12	yes	complemented
46	#3135	C₂×C₃	1	3	yes	complemented
47	C₂×#490	C₂×C₃	1	3	yes	complemented
48	#2897	C₂×C₃	1	4	yes	complemented
49	#2875	C₂×C₃	1	1	yes	characteristic, complemented
50	C₃×H(5,𝔽₂)	—	2	16	no	—
51	C₂×C₃×#41	—	2	2	no	—
52	C₂×C₃×#41	C₂³	1	6	yes	—
53	C₂×C₃×#41	C₂³	1	1	yes	characteristic
54	C₂²×C₃×D₄	—	2	6	no	—
55	C₂²×C₃×D₄	—	2	6	no	—
56	C₂²×C₃×D₄	C₂³	1	6	yes	—
57	C₃×#131	—	2	8	no	—
58	C₂×C₃×M₁₆	C₂³	1	6	yes	—
59	C₂²×C₃×C₈	C₂³	1	6	yes	—
60	C₃×C₄⋊Q₈	—	2	6	no	—
61	C₃×C₄⋊Q₈	C₂³	1	1	yes	characteristic
62	C₃×C₄⋊Q₈	C₂³	1	6	yes	—
63	C₃×#127	—	2	6	no	—
64	C₃×#127	C₂³	1	3	yes	—
65	C₃×#127	C₂³	1	6	yes	—
66	C₃×#125	—	2	6	no	—
67	C₃×#125	—	2	6	no	—
68	C₃×#125	C₂³	1	3	yes	—
69	C₃×#125	C₂³	1	6	yes	—
70	C₃×#125	C₂³	1	1	yes	characteristic
71	C₃×#124	C₂³	1	6	yes	—
72	C₃×C₂²⋊Q₈	—	2	6	no	—
73	C₃×C₂²⋊Q₈	—	2	12	no	—
74	C₃×#121	—	2	6	no	—
75	C₃×#121	—	2	12	no	—
76	C₃×#120	—	2	12	no	—
77	C₃×C₄×Q₈	—	2	2	no	—
78	C₃×C₄×Q₈	—	2	6	no	—
79	C₃×C₄×Q₈	D₄	1	2	yes	—
80	C₃×C₄×D₄	—	2	6	no	—
81	C₃×C₄×D₄	—	2	6	no	—
82	C₃×C₄×D₄	D₄	1	6	yes	—
83	C₃×#117	C₂³	1	6	yes	—
84	C₃×#117	C₂³	1	3	yes	—
85	C₂×C₃×C₄⋊C₄	—	2	6	no	—
86	C₂×C₃×C₄⋊C₄	C₂³	1	6	yes	—
87	C₂×C₃×C₄²	C₂³	1	3	yes	—
88	C₃×#103	C₂³	1	4	yes	—
89	C₃×#103	C₂³	1	12	yes	—
90	C₃×#102	C₂³	1	12	yes	—
91	C₃×#102	C₂³	1	24	yes	—
92	C₃×#102	C₂³	1	12	yes	—
93	C₃×#97	C₂³	1	1	yes	characteristic
94	C₃×#97	C₂³	1	3	yes	—
95	C₃×#97	C₂³	1	6	yes	—
96	C₃×C₄×C₈	C₂³	1	6	yes	—
97	C₂×H(5,𝔽₂)	C₂²×C₃	1	2	yes	—
98	C₂×#131	C₂²×C₃	1	2	yes	—
99	#557	—	2	2	no	—
100	#554	—	2	6	no	—
101	#550	—	2	2	no	—
102	D₄²	—	2	6	no	—
103	#544	C₂²×C₃	1	6	yes	complemented
104	#535	C₂²×C₃	1	6	yes	—
105	#533	C₂²×C₃	1	3	yes	complemented
106	#532	C₂²×C₃	1	3	yes	—
107	C₂×#127	C₂²×C₃	1	3	yes	—
108	C₂×#125	C₂²×C₃	1	3	yes	complemented
109	#520	C₂²×C₃	1	2	yes	—
110	C₄×#41	C₂²×C₃	1	1	yes	characteristic
111	#491	C₂²×C₃	1	1	yes	characteristic, complemented
112	#491	C₂²×C₃	1	3	yes	complemented
113	#490	C₂²×C₃	1	3	yes	complemented
114	#490	C₂²×C₃	1	12	yes	complemented
115	#490	C₂²×C₃	1	6	yes	complemented
116	#490	C₂²×C₃	1	3	yes	complemented
117	#489	C₂²×C₃	1	12	yes	complemented
118	#488	C₂²×C₃	1	12	yes	complemented
119	#486	C₂²×C₃	1	12	yes	complemented
120	#418	C₂²×C₃	1	12	yes	complemented
121	#417	C₂²×C₃	1	12	yes	complemented
122	#416	C₂²×C₃	1	12	yes	complemented
123	C₂×#102	C₂²×C₃	1	12	yes	complemented
124	#405	C₂²×C₃	1	6	yes	complemented
125	C₄×M₁₆	C₂²×C₃	1	3	yes	complemented
126	C₂×#97	C₂²×C₃	1	3	yes	complemented
127	C₂⁴×C₃	—	2	6	no	—
128	C₃×#41	—	2	24	no	—
129	C₃×#41	—	4	8	no	—
130	C₃×#41	#41	1	8	yes	—
131	C₂×C₃×Q₈	—	2	6	no	—
132	C₂×C₃×Q₈	C₂×D₄	1	1	yes	characteristic
133	C₂×C₃×Q₈	C₂⁴	1	2	yes	—
134	C₂×C₃×D₄	—	2	6	no	—
135	C₂×C₃×D₄	—	2	6	no	—
136	C₂×C₃×D₄	—	2	6	no	—
137	C₂×C₃×D₄	—	2	12	no	—
138	C₂×C₃×D₄	—	2	24	no	—
139	C₂×C₃×D₄	—	4	24	no	—
140	C₂×C₃×D₄	—	8	12	no	—
141	C₂×C₃×D₄	C₂×D₄	1	3	yes	—
142	C₂×C₃×D₄	C₂⁴	1	6	yes	—
143	C₂²×C₃×C₄	—	2	6	no	—
144	C₂²×C₃×C₄	—	2	6	no	—
145	C₂²×C₃×C₄	C₂×D₄	1	6	yes	—
146	C₂²×C₃×C₄	C₂⁴	1	3	yes	—
147	C₃×M₁₆	—	2	12	no	—
148	C₂×C₃×C₈	—	2	12	no	—
149	C₂×C₃×C₈	C₂⁴	1	6	yes	—
150	C₂×C₃×C₈	C₂⁴	1	2	yes	—
151	C₃×C₄⋊C₄	—	2	6	no	—
152	C₃×C₄⋊C₄	—	2	12	no	—
153	C₃×C₄⋊C₄	—	2	6	no	—
154	C₃×C₄⋊C₄	—	2	2	no	—
155	C₃×C₄⋊C₄	—	2	2	no	—
156	C₃×C₄⋊C₄	—	2	6	no	—
157	C₃×C₄⋊C₄	C₂×D₄	1	6	yes	—
158	C₃×C₄⋊C₄	C₂⁴	1	6	yes	—
159	C₃×C₄⋊C₄	C₂⁴	1	2	yes	—
160	C₃×C₂²⋊C₄	—	2	6	no	—
161	C₃×C₂²⋊C₄	—	2	12	no	—
162	C₃×C₂²⋊C₄	C₂×D₄	1	6	yes	—
163	C₃×C₄²	—	2	6	no	—
164	C₃×C₄²	C₂×D₄	1	6	yes	—
165	C₃×C₄²	C₂⁴	1	3	yes	—
166	C₃×C₄²	C₂⁴	1	1	yes	characteristic
167	H(5,𝔽₂)	—	2	16	no	—
168	C₂×#41	—	2	2	no	—
169	C₂×#41	C₂³×C₃	1	6	yes	—
170	C₂×#41	C₂³×C₃	1	1	yes	characteristic
171	C₂²×D₄	—	2	6	no	—
172	C₂²×D₄	—	2	6	no	—
173	C₂²×D₄	C₂³×C₃	1	6	yes	—
174	#131	—	2	8	no	—
175	C₂×M₁₆	C₂³×C₃	1	6	yes	—
176	C₂²×C₈	C₂³×C₃	1	6	yes	—
177	C₄⋊Q₈	—	2	6	no	—
178	C₄⋊Q₈	C₂³×C₃	1	1	yes	characteristic
179	C₄⋊Q₈	C₂³×C₃	1	6	yes	—
180	#127	—	2	6	no	—
181	#127	C₂³×C₃	1	3	yes	—
182	#127	C₂³×C₃	1	6	yes	—
183	#125	—	2	6	no	—
184	#125	—	2	6	no	—
185	#125	C₂³×C₃	1	3	yes	—
186	#125	C₂³×C₃	1	6	yes	—
187	#125	C₂³×C₃	1	1	yes	characteristic
188	#124	C₂³×C₃	1	6	yes	—
189	C₂²⋊Q₈	—	2	6	no	—
190	C₂²⋊Q₈	—	2	12	no	—
191	#121	—	2	6	no	—
192	#121	—	2	12	no	—
193	#120	—	2	12	no	—
194	C₄×Q₈	—	2	2	no	—
195	C₄×Q₈	—	2	6	no	—
196	C₄×Q₈	C₃×D₄	1	2	yes	—
197	C₄×D₄	—	2	6	no	—
198	C₄×D₄	—	2	6	no	—
199	C₄×D₄	C₃×D₄	1	6	yes	—
200	#117	C₂³×C₃	1	6	yes	—
201	#117	C₂³×C₃	1	3	yes	—
202	C₂×C₄⋊C₄	—	2	6	no	—
203	C₂×C₄⋊C₄	C₂³×C₃	1	6	yes	—
204	C₂×C₄²	C₂³×C₃	1	3	yes	—
205	#103	C₂³×C₃	1	4	yes	—
206	#103	C₂³×C₃	1	12	yes	—
207	#102	C₂³×C₃	1	12	yes	—
208	#102	C₂³×C₃	1	24	yes	—
209	#102	C₂³×C₃	1	12	yes	—
210	#97	C₂³×C₃	1	1	yes	characteristic
211	#97	C₂³×C₃	1	3	yes	—
212	#97	C₂³×C₃	1	6	yes	—
213	C₄×C₈	C₂³×C₃	1	6	yes	—
214	C₂³×C₃	—	2	6	no	—
215	C₂³×C₃	—	4	12	no	—
216	C₂³×C₃	—	8	12	no	—
217	C₂³×C₃	C₂²×D₄	1	3	yes	—
218	C₃×Q₈	—	2	4	no	—
219	C₃×Q₈	—	8	6	no	—
220	C₃×Q₈	#124	1	4	yes	—
221	C₃×D₄	—	2	12	no	—
222	C₃×D₄	—	4	12	no	—
223	C₃×D₄	—	4	12	no	—
224	C₃×D₄	—	4	12	no	—
225	C₃×D₄	—	8	12	no	—
226	C₃×D₄	#124	1	12	yes	—
227	C₂×C₃×C₄	—	2	2	no	—
228	C₂×C₃×C₄	—	2	6	no	—
229	C₂×C₃×C₄	—	2	2	no	—
230	C₂×C₃×C₄	—	4	6	no	—
231	C₂×C₃×C₄	—	4	12	no	—
232	C₂×C₃×C₄	—	8	6	no	—
233	C₂×C₃×C₄	C₂²×D₄	1	2	yes	—
234	C₂×C₃×C₄	C₂²×D₄	1	6	yes	—
235	C₂×C₃×C₄	C₂²×D₄	1	3	yes	—
236	C₂×C₃×C₄	C₂×#41	1	12	yes	—
237	C₂×C₃×C₄	C₂⁵	1	1	yes	characteristic
238	C₃×C₈	—	2	2	no	—
239	C₃×C₈	—	2	6	no	—
240	C₂⁴	—	2	6	no	—
241	#41	—	2	24	no	—
242	#41	—	4	8	no	—
243	#41	C₃×#41	1	8	yes	—
244	C₂×Q₈	—	2	6	no	—
245	C₂×Q₈	C₂×C₃×D₄	1	1	yes	characteristic
246	C₂×Q₈	C₂⁴×C₃	1	2	yes	—
247	C₂×D₄	—	2	6	no	—
248	C₂×D₄	—	2	6	no	—
249	C₂×D₄	—	2	6	no	—
250	C₂×D₄	—	2	12	no	—
251	C₂×D₄	—	2	24	no	—
252	C₂×D₄	—	4	24	no	—
253	C₂×D₄	—	8	12	no	—
254	C₂×D₄	C₂×C₃×D₄	1	3	yes	—
255	C₂×D₄	C₂⁴×C₃	1	6	yes	—
256	C₂²×C₄	—	2	6	no	—
257	C₂²×C₄	—	2	6	no	—
258	C₂²×C₄	C₂×C₃×D₄	1	6	yes	—
259	C₂²×C₄	C₂⁴×C₃	1	3	yes	—
260	M₁₆	—	2	12	no	—
261	C₂×C₈	—	2	12	no	—
262	C₂×C₈	C₂⁴×C₃	1	6	yes	—
263	C₂×C₈	C₂⁴×C₃	1	2	yes	—
264	C₄⋊C₄	—	2	6	no	—
265	C₄⋊C₄	—	2	12	no	—
266	C₄⋊C₄	—	2	6	no	—
267	C₄⋊C₄	—	2	2	no	—
268	C₄⋊C₄	—	2	2	no	—
269	C₄⋊C₄	—	2	6	no	—
270	C₄⋊C₄	C₂×C₃×D₄	1	6	yes	—
271	C₄⋊C₄	C₂⁴×C₃	1	6	yes	—
272	C₄⋊C₄	C₂⁴×C₃	1	2	yes	—
273	C₂²⋊C₄	—	2	6	no	—
274	C₂²⋊C₄	—	2	12	no	—
275	C₂²⋊C₄	C₂×C₃×D₄	1	6	yes	—
276	C₄²	—	2	6	no	—
277	C₄²	C₂×C₃×D₄	1	6	yes	—
278	C₄²	C₂⁴×C₃	1	3	yes	—
279	C₄²	C₂⁴×C₃	1	1	yes	characteristic
280	C₂²×C₃	—	2	6	no	—
281	C₂²×C₃	—	4	12	no	—
282	C₂²×C₃	—	4	6	no	—
283	C₂²×C₃	—	16	12	no	—
284	C₂²×C₃	C₂×#124	1	6	yes	—
285	C₂²×C₃	C₂³×D₄	1	1	yes	characteristic
286	C₃×C₄	—	2	2	no	—
287	C₃×C₄	—	2	6	no	—
288	C₃×C₄	—	4	6	no	—
289	C₃×C₄	—	4	2	no	—
290	C₃×C₄	—	4	2	no	—
291	C₃×C₄	C₂×#124	1	6	yes	—
292	C₃×C₄	C₂²×#41	1	2	yes	—
293	C₂³	—	2	6	no	—
294	C₂³	—	4	12	no	—
295	C₂³	—	8	12	no	—
296	C₂³	C₂²×C₃×D₄	1	3	yes	—
297	Q₈	—	2	4	no	—
298	Q₈	—	8	6	no	—
299	Q₈	C₃×#124	1	4	yes	—
300	D₄	—	2	12	no	—
301	D₄	—	4	12	no	—
302	D₄	—	4	12	no	—
303	D₄	—	4	12	no	—
304	D₄	—	8	12	no	—
305	D₄	C₃×#124	1	12	yes	—
306	C₂×C₄	—	2	2	no	—
307	C₂×C₄	—	2	6	no	—
308	C₂×C₄	—	2	2	no	—
309	C₂×C₄	—	4	6	no	—
310	C₂×C₄	—	4	12	no	—
311	C₂×C₄	—	8	6	no	—
312	C₂×C₄	C₂²×C₃×D₄	1	2	yes	—
313	C₂×C₄	C₂²×C₃×D₄	1	6	yes	—
314	C₂×C₄	C₂²×C₃×D₄	1	3	yes	—
315	C₂×C₄	C₂×C₃×#41	1	12	yes	—
316	C₂×C₄	C₂⁵×C₃	1	1	yes	characteristic
317	C₈	—	2	2	no	—
318	C₈	—	2	6	no	—
319	C₂×C₃	—	2	6	no	—
320	C₂×C₃	—	8	6	no	—
321	C₂×C₃	C₂²×#124	1	1	yes	characteristic
322	C₂×C₃	#3585	1	2	yes	—
323	C₂²	—	2	6	no	—
324	C₂²	—	4	6	no	—
325	C₂²	—	4	12	no	—
326	C₂²	—	16	12	no	—
327	C₂²	C₂×C₃×#124	1	6	yes	—
328	C₂²	C₂³×C₃×D₄	1	1	yes	characteristic
329	C₄	—	2	2	no	—
330	C₄	—	2	6	no	—
331	C₄	—	4	2	no	—
332	C₄	—	4	6	no	—
333	C₄	—	4	2	no	—
334	C₄	C₂×C₃×#124	1	6	yes	—
335	C₄	C₂²×C₃×#41	1	2	yes	—
336	C₃	#60626	1	1	yes	characteristic, direct product factor, Sylow
337	C₂	—	2	6	no	—
338	C₂	—	8	6	no	—
339	C₂	C₂²×C₃×#124	1	1	yes	characteristic
340	C₂	C₃×#3585	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₃	C₂³×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₃
prime radical	C₃×#3498	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃ ◁ C₃×C₄×#41 ◁ G

Complements

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

N	H ≅ G/N	type
1. C₃×#3504	C₂	semidirect
3. C₃×#3142	C₂	semidirect
4. C₃×#3138	C₂	semidirect
5. C₃×#3136	C₂	semidirect
6. C₃×#3135	C₂	semidirect
7. C₂×C₃×#490	C₂	semidirect
8. C₃×#2897	C₂	semidirect
9. C₃×#2875	C₂	semidirect
10. #60626	C₃	direct (N × H)
17. C₃×#544	C₂²	semidirect
19. C₃×#533	C₂²	semidirect
22. C₂×C₃×#125	C₂²	semidirect
25. C₃×#491	C₂²	semidirect
26. C₃×#491	C₂²	semidirect
27. C₃×#490	C₂²	semidirect
28. C₃×#490	C₂²	semidirect
29. C₃×#490	C₂²	semidirect
30. C₃×#490	C₂²	semidirect
31. C₃×#489	C₂²	semidirect
32. C₃×#488	C₂²	semidirect
33. C₃×#486	C₂²	semidirect
34. C₃×#418	C₂²	semidirect
35. C₃×#417	C₂²	semidirect
36. C₃×#416	C₂²	semidirect
37. C₂×C₃×#102	C₂²	semidirect
38. C₃×#405	C₂²	semidirect
39. C₃×C₄×M₁₆	C₂²	semidirect
40. C₂×C₃×#97	C₂²	semidirect
41. #3504	C₂×C₃	semidirect
43. #3142	C₂×C₃	semidirect
44. #3138	C₂×C₃	semidirect
45. #3136	C₂×C₃	semidirect
46. #3135	C₂×C₃	semidirect
47. C₂×#490	C₂×C₃	semidirect
48. #2897	C₂×C₃	semidirect
49. #2875	C₂×C₃	semidirect
103. #544	C₂²×C₃	semidirect
105. #533	C₂²×C₃	semidirect
108. C₂×#125	C₂²×C₃	semidirect
111. #491	C₂²×C₃	semidirect
112. #491	C₂²×C₃	semidirect
113. #490	C₂²×C₃	semidirect
114. #490	C₂²×C₃	semidirect
115. #490	C₂²×C₃	semidirect
116. #490	C₂²×C₃	semidirect
117. #489	C₂²×C₃	semidirect
118. #488	C₂²×C₃	semidirect
119. #486	C₂²×C₃	semidirect
120. #418	C₂²×C₃	semidirect
121. #417	C₂²×C₃	semidirect
122. #416	C₂²×C₃	semidirect
123. C₂×#102	C₂²×C₃	semidirect
124. #405	C₂²×C₃	semidirect
125. C₄×M₁₆	C₂²×C₃	semidirect
126. C₂×#97	C₂²×C₃	semidirect
336. C₃	#60626	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃×#3504	2
C₃×#3498	2
C₃×#3142	2
C₃×#3138	2
C₃×#3136	2
C₃×#3135	2
C₂×C₃×#490	2
C₃×#2897	2
C₃×#2875	2
#60626	3

Subgroups — C3×#60626 #10679233

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#60626 #10679233