Subgroups — C₃×#61818 #10680425

Order 768.

1930 conjugacy classes of subgroups.

The subgroup lattice has 296 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#3502	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₃×D₄×Q₈	C₂	1	1	yes	maximal, characteristic
3	C₃×Q₈×SD₁₆	C₂	1	8	yes	maximal, complemented
4	C₃×#3359	C₂	1	8	yes	maximal, complemented
5	C₃×#3229	C₂	1	6	yes	maximal, complemented
6	C₃×#3101	C₂	1	6	yes	maximal
7	C₃×#2982	C₂	1	1	yes	maximal, characteristic, complemented
8	#61818	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
9	C₂³×C₃×Q₈	—	2	1	no	—
10	C₃×Q₈²	C₂²	1	4	yes	complemented
11	C₃×#557	—	2	4	no	—
12	C₃×#557	C₂²	1	4	yes	complemented
13	C₃×D₄×Q₈	—	2	4	no	—
14	C₃×D₄×Q₈	C₂²	1	8	yes	—
15	C₃×#544	—	2	3	no	—
16	C₃×#533	C₂²	1	6	yes	—
17	C₂×C₃×C₄⋊Q₈	C₂²	1	3	yes	—
18	C₂×C₃×C₂²⋊Q₈	—	2	3	no	—
19	C₃×#519	—	2	1	no	—
20	C₂×C₃×C₄×Q₈	C₂²	1	1	yes	characteristic
21	C₂×C₃×C₄×D₄	C₂²	1	3	yes	—
22	C₃×#498	C₂²	1	12	yes	complemented
23	C₃×#487	C₂²	1	12	yes	complemented
24	C₃×#483	C₂²	1	12	yes	—
25	C₃×#476	C₂²	1	12	yes	—
26	C₃×#475	C₂²	1	24	yes	complemented
27	C₃×#461	C₂²	1	12	yes	—
28	C₃×#450	C₂²	1	4	yes	—
29	C₃×C₈×Q₈	C₂²	1	8	yes	complemented
30	C₃×C₄×SD₁₆	C₂²	1	24	yes	—
31	C₃×#431	C₂²	1	6	yes	—
32	#3502	C₂×C₃	1	1	yes	characteristic, complemented
33	C₂×D₄×Q₈	C₂×C₃	1	1	yes	characteristic
34	Q₈×SD₁₆	C₂×C₃	1	8	yes	complemented
35	#3359	C₂×C₃	1	8	yes	complemented
36	#3229	C₂×C₃	1	6	yes	complemented
37	#3101	C₂×C₃	1	6	yes	—
38	#2982	C₂×C₃	1	1	yes	characteristic, complemented
39	C₂²×C₃×Q₈	—	2	8	no	—
40	C₂²×C₃×Q₈	—	2	3	no	—
41	C₂²×C₃×Q₈	—	4	8	no	—
42	C₂²×C₃×Q₈	D₄	1	1	yes	characteristic
43	C₂²×C₃×D₄	C₂³	1	1	yes	characteristic
44	C₂³×C₃×C₄	—	2	3	no	—
45	C₂×C₃×SD₁₆	C₂³	1	8	yes	—
46	C₃×C₄⋊Q₈	—	2	12	no	—
47	C₃×C₄⋊Q₈	—	2	6	no	—
48	C₃×C₄⋊Q₈	C₂³	1	6	yes	—
49	C₃×C₄⋊Q₈	C₂³	1	6	yes	—
50	C₃×C₄⋊Q₈	C₂³	1	12	yes	—
51	C₃×#125	—	2	12	no	—
52	C₃×#125	—	2	6	no	—
53	C₃×C₂²⋊Q₈	—	2	6	no	—
54	C₃×C₂²⋊Q₈	—	2	24	no	—
55	C₃×C₂²⋊Q₈	C₂³	1	4	yes	—
56	C₃×C₄×Q₈	—	2	4	no	—
57	C₃×C₄×Q₈	—	2	4	no	—
58	C₃×C₄×Q₈	—	2	6	no	—
59	C₃×C₄×Q₈	D₄	1	4	yes	—
60	C₃×C₄×Q₈	C₂³	1	12	yes	—
61	C₃×C₄×Q₈	C₂³	1	4	yes	—
62	C₃×C₄×D₄	—	2	6	no	—
63	C₃×C₄×D₄	C₂³	1	12	yes	—
64	C₃×#117	—	2	6	no	—
65	C₂×C₃×C₄⋊C₄	—	2	3	no	—
66	C₂×C₃×C₄⋊C₄	D₄	1	3	yes	—
67	C₂×C₃×C₄⋊C₄	C₂³	1	3	yes	—
68	C₂×C₃×C₂²⋊C₄	—	2	3	no	—
69	C₂×C₃×C₄²	C₂³	1	3	yes	—
70	C₃×#106	C₂³	1	24	yes	—
71	C₃×C₄⋊C₈	C₂³	1	12	yes	—
72	C₃×#103	C₂³	1	24	yes	—
73	C₃×#102	C₂³	1	8	yes	—
74	C₃×C₂²⋊C₈	C₂³	1	4	yes	—
75	C₃×C₄×C₈	C₂³	1	12	yes	—
76	C₂³×Q₈	—	2	1	no	—
77	Q₈²	C₂²×C₃	1	4	yes	complemented
78	#557	—	2	4	no	—
79	#557	C₂²×C₃	1	4	yes	complemented
80	D₄×Q₈	—	2	4	no	—
81	D₄×Q₈	C₂²×C₃	1	8	yes	—
82	#544	—	2	3	no	—
83	#533	C₂²×C₃	1	6	yes	—
84	C₂×C₄⋊Q₈	C₂²×C₃	1	3	yes	—
85	C₂×C₂²⋊Q₈	—	2	3	no	—
86	#519	—	2	1	no	—
87	C₂×C₄×Q₈	C₂²×C₃	1	1	yes	characteristic
88	C₂×C₄×D₄	C₂²×C₃	1	3	yes	—
89	#498	C₂²×C₃	1	12	yes	complemented
90	#487	C₂²×C₃	1	12	yes	complemented
91	#483	C₂²×C₃	1	12	yes	—
92	#476	C₂²×C₃	1	12	yes	—
93	#475	C₂²×C₃	1	24	yes	complemented
94	#461	C₂²×C₃	1	12	yes	—
95	#450	C₂²×C₃	1	4	yes	—
96	C₈×Q₈	C₂²×C₃	1	8	yes	complemented
97	C₄×SD₁₆	C₂²×C₃	1	24	yes	—
98	#431	C₂²×C₃	1	6	yes	—
99	C₂⁴×C₃	—	2	1	no	—
100	C₂×C₃×Q₈	—	2	12	no	—
101	C₂×C₃×Q₈	—	2	6	no	—
102	C₂×C₃×Q₈	—	4	32	no	—
103	C₂×C₃×Q₈	—	4	12	no	—
104	C₂×C₃×Q₈	SD₁₆	1	8	yes	complemented
105	C₂×C₃×Q₈	C₂×D₄	1	4	yes	—
106	C₂×C₃×Q₈	C₂⁴	1	4	yes	—
107	C₂×C₃×D₄	—	2	4	no	—
108	C₂×C₃×D₄	—	2	1	no	—
109	C₂×C₃×D₄	C₂⁴	1	2	yes	—
110	C₂²×C₃×C₄	—	2	3	no	—
111	C₂²×C₃×C₄	—	2	12	no	—
112	C₂²×C₃×C₄	—	4	12	no	—
113	C₂²×C₃×C₄	C₂×D₄	1	3	yes	—
114	C₂²×C₃×C₄	C₂×D₄	1	3	yes	—
115	C₂²×C₃×C₄	C₂⁴	1	1	yes	characteristic
116	C₃×SD₁₆	—	2	16	no	—
117	C₂×C₃×C₈	C₂⁴	1	8	yes	—
118	C₃×C₄⋊C₄	—	2	12	no	—
119	C₃×C₄⋊C₄	—	2	3	no	—
120	C₃×C₄⋊C₄	—	2	6	no	—
121	C₃×C₄⋊C₄	—	2	6	no	—
122	C₃×C₄⋊C₄	—	2	4	no	—
123	C₃×C₄⋊C₄	—	2	6	no	—
124	C₃×C₄⋊C₄	—	2	12	no	—
125	C₃×C₄⋊C₄	C₂×D₄	1	12	yes	—
126	C₃×C₄⋊C₄	C₂⁴	1	4	yes	—
127	C₃×C₄⋊C₄	C₂⁴	1	6	yes	—
128	C₃×C₂²⋊C₄	—	2	4	no	—
129	C₃×C₂²⋊C₄	—	2	12	no	—
130	C₃×C₄²	—	2	12	no	—
131	C₃×C₄²	C₂×D₄	1	6	yes	—
132	C₃×C₄²	C₂⁴	1	6	yes	—
133	C₂²×Q₈	—	2	8	no	—
134	C₂²×Q₈	—	2	3	no	—
135	C₂²×Q₈	—	4	8	no	—
136	C₂²×Q₈	C₃×D₄	1	1	yes	characteristic
137	C₂²×D₄	C₂³×C₃	1	1	yes	characteristic
138	C₂³×C₄	—	2	3	no	—
139	C₂×SD₁₆	C₂³×C₃	1	8	yes	—
140	C₄⋊Q₈	—	2	12	no	—
141	C₄⋊Q₈	—	2	6	no	—
142	C₄⋊Q₈	C₂³×C₃	1	6	yes	—
143	C₄⋊Q₈	C₂³×C₃	1	12	yes	—
144	C₄⋊Q₈	C₂³×C₃	1	6	yes	—
145	#125	—	2	12	no	—
146	#125	—	2	6	no	—
147	C₂²⋊Q₈	—	2	6	no	—
148	C₂²⋊Q₈	—	2	24	no	—
149	C₂²⋊Q₈	C₂³×C₃	1	4	yes	—
150	C₄×Q₈	—	2	4	no	—
151	C₄×Q₈	—	2	4	no	—
152	C₄×Q₈	—	2	6	no	—
153	C₄×Q₈	C₃×D₄	1	4	yes	—
154	C₄×Q₈	C₂³×C₃	1	12	yes	—
155	C₄×Q₈	C₂³×C₃	1	4	yes	—
156	C₄×D₄	—	2	6	no	—
157	C₄×D₄	C₂³×C₃	1	12	yes	—
158	#117	—	2	6	no	—
159	C₂×C₄⋊C₄	—	2	3	no	—
160	C₂×C₄⋊C₄	C₃×D₄	1	3	yes	—
161	C₂×C₄⋊C₄	C₂³×C₃	1	3	yes	—
162	C₂×C₂²⋊C₄	—	2	3	no	—
163	C₂×C₄²	C₂³×C₃	1	3	yes	—
164	#106	C₂³×C₃	1	24	yes	—
165	C₄⋊C₈	C₂³×C₃	1	12	yes	—
166	#103	C₂³×C₃	1	24	yes	—
167	#102	C₂³×C₃	1	8	yes	—
168	C₂²⋊C₈	C₂³×C₃	1	4	yes	—
169	C₄×C₈	C₂³×C₃	1	12	yes	—
170	C₂³×C₃	—	2	4	no	—
171	C₂³×C₃	—	2	2	no	—
172	C₂³×C₃	—	4	2	no	—
173	C₂³×C₃	—	4	2	no	—
174	C₂³×C₃	C₂²×D₄	1	1	yes	characteristic
175	C₃×Q₈	—	2	8	no	—
176	C₃×Q₈	—	4	24	no	—
177	C₃×Q₈	—	8	6	no	—
178	C₃×Q₈	C₂×SD₁₆	1	16	yes	—
179	C₃×D₄	—	2	4	no	—
180	C₃×D₄	—	2	2	no	—
181	C₃×D₄	2₋¹⁺⁴	1	4	yes	—
182	C₂×C₃×C₄	—	2	2	no	—
183	C₂×C₃×C₄	—	2	4	no	—
184	C₂×C₃×C₄	—	2	4	no	—
185	C₂×C₃×C₄	—	2	6	no	—
186	C₂×C₃×C₄	—	4	3	no	—
187	C₂×C₃×C₄	—	4	6	no	—
188	C₂×C₃×C₄	—	4	24	no	—
189	C₂×C₃×C₄	C₂×SD₁₆	1	12	yes	—
190	C₂×C₃×C₄	C₂²×D₄	1	1	yes	characteristic
191	C₂×C₃×C₄	C₂²×D₄	1	6	yes	—
192	C₂×C₃×C₄	C₂²×D₄	1	6	yes	—
193	C₂×C₃×C₄	C₂⁵	1	1	yes	characteristic
194	C₃×C₈	—	2	8	no	—
195	C₂⁴	—	2	1	no	—
196	C₂×Q₈	—	2	12	no	—
197	C₂×Q₈	—	2	6	no	—
198	C₂×Q₈	—	4	32	no	—
199	C₂×Q₈	—	4	12	no	—
200	C₂×Q₈	C₃×SD₁₆	1	8	yes	complemented
201	C₂×Q₈	C₂×C₃×D₄	1	4	yes	—
202	C₂×Q₈	C₂⁴×C₃	1	4	yes	—
203	C₂×D₄	—	2	4	no	—
204	C₂×D₄	—	2	1	no	—
205	C₂×D₄	C₂⁴×C₃	1	2	yes	—
206	C₂²×C₄	—	2	3	no	—
207	C₂²×C₄	—	2	12	no	—
208	C₂²×C₄	—	4	12	no	—
209	C₂²×C₄	C₂×C₃×D₄	1	3	yes	—
210	C₂²×C₄	C₂×C₃×D₄	1	3	yes	—
211	C₂²×C₄	C₂⁴×C₃	1	1	yes	characteristic
212	SD₁₆	—	2	16	no	—
213	C₂×C₈	C₂⁴×C₃	1	8	yes	—
214	C₄⋊C₄	—	2	12	no	—
215	C₄⋊C₄	—	2	3	no	—
216	C₄⋊C₄	—	2	6	no	—
217	C₄⋊C₄	—	2	6	no	—
218	C₄⋊C₄	—	2	4	no	—
219	C₄⋊C₄	—	2	6	no	—
220	C₄⋊C₄	—	2	12	no	—
221	C₄⋊C₄	C₂×C₃×D₄	1	12	yes	—
222	C₄⋊C₄	C₂⁴×C₃	1	4	yes	—
223	C₄⋊C₄	C₂⁴×C₃	1	6	yes	—
224	C₂²⋊C₄	—	2	4	no	—
225	C₂²⋊C₄	—	2	12	no	—
226	C₄²	—	2	12	no	—
227	C₄²	C₂×C₃×D₄	1	6	yes	—
228	C₄²	C₂⁴×C₃	1	6	yes	—
229	C₂²×C₃	—	2	1	no	—
230	C₂²×C₃	—	2	1	no	—
231	C₂²×C₃	—	2	4	no	—
232	C₂²×C₃	—	4	8	no	—
233	C₂²×C₃	—	4	2	no	—
234	C₂²×C₃	—	4	2	no	—
235	C₂²×C₃	C₂²×SD₁₆	1	2	yes	—
236	C₂²×C₃	C₂³×D₄	1	1	yes	characteristic
237	C₃×C₄	—	2	6	no	—
238	C₃×C₄	—	2	1	no	—
239	C₃×C₄	—	4	4	no	—
240	C₃×C₄	—	4	4	no	—
241	C₃×C₄	—	4	6	no	—
242	C₃×C₄	C₂²×SD₁₆	1	12	yes	—
243	C₃×C₄	C₂×2₋¹⁺⁴	1	2	yes	—
244	C₂³	—	2	4	no	—
245	C₂³	—	2	2	no	—
246	C₂³	—	4	2	no	—
247	C₂³	—	4	2	no	—
248	C₂³	C₂²×C₃×D₄	1	1	yes	characteristic
249	Q₈	—	2	8	no	—
250	Q₈	—	4	24	no	—
251	Q₈	—	8	6	no	—
252	Q₈	C₂×C₃×SD₁₆	1	16	yes	—
253	D₄	—	2	4	no	—
254	D₄	—	2	2	no	—
255	D₄	C₃×2₋¹⁺⁴	1	4	yes	—
256	C₂×C₄	—	2	2	no	—
257	C₂×C₄	—	2	4	no	—
258	C₂×C₄	—	2	4	no	—
259	C₂×C₄	—	2	6	no	—
260	C₂×C₄	—	4	3	no	—
261	C₂×C₄	—	4	6	no	—
262	C₂×C₄	—	4	24	no	—
263	C₂×C₄	C₂×C₃×SD₁₆	1	12	yes	—
264	C₂×C₄	C₂²×C₃×D₄	1	1	yes	characteristic
265	C₂×C₄	C₂²×C₃×D₄	1	6	yes	—
266	C₂×C₄	C₂²×C₃×D₄	1	6	yes	—
267	C₂×C₄	C₂⁵×C₃	1	1	yes	characteristic
268	C₈	—	2	8	no	—
269	C₂×C₃	—	2	2	no	—
270	C₂×C₃	—	4	4	no	—
271	C₂×C₃	#3486	1	1	yes	characteristic
272	C₂×C₃	C₂³×SD₁₆	1	1	yes	characteristic
273	C₂×C₃	#3586	1	1	yes	characteristic
274	C₂²	—	2	1	no	—
275	C₂²	—	2	1	no	—
276	C₂²	—	2	4	no	—
277	C₂²	—	4	8	no	—
278	C₂²	—	4	2	no	—
279	C₂²	—	4	2	no	—
280	C₂²	C₂²×C₃×SD₁₆	1	2	yes	—
281	C₂²	C₂³×C₃×D₄	1	1	yes	characteristic
282	C₄	—	2	1	no	—
283	C₄	—	2	6	no	—
284	C₄	—	4	4	no	—
285	C₄	—	4	6	no	—
286	C₄	—	4	4	no	—
287	C₄	C₂²×C₃×SD₁₆	1	12	yes	—
288	C₄	C₂×C₃×2₋¹⁺⁴	1	2	yes	—
289	C₃	#61818	1	1	yes	characteristic, direct product factor, Sylow
290	C₂	—	2	2	no	—
291	C₂	—	4	4	no	—
292	C₂	C₃×#3486	1	1	yes	characteristic
293	C₂	C₂³×C₃×SD₁₆	1	1	yes	characteristic
294	C₂	C₃×#3586	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃	C₂³×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₃
prime radical	C₂²×C₃×D₄	C₂³
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃ ◁ C₂×C₃×C₄×Q₈ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₃×#3502	C₂	semidirect
3. C₃×Q₈×SD₁₆	C₂	semidirect
4. C₃×#3359	C₂	semidirect
5. C₃×#3229	C₂	semidirect
7. C₃×#2982	C₂	semidirect
8. #61818	C₃	direct (N × H)
10. C₃×Q₈²	C₂²	semidirect
12. C₃×#557	C₂²	semidirect
22. C₃×#498	C₂²	semidirect
23. C₃×#487	C₂²	semidirect
26. C₃×#475	C₂²	semidirect
29. C₃×C₈×Q₈	C₂²	semidirect
32. #3502	C₂×C₃	semidirect
34. Q₈×SD₁₆	C₂×C₃	semidirect
35. #3359	C₂×C₃	semidirect
36. #3229	C₂×C₃	semidirect
38. #2982	C₂×C₃	semidirect
77. Q₈²	C₂²×C₃	semidirect
79. #557	C₂²×C₃	semidirect
89. #498	C₂²×C₃	semidirect
90. #487	C₂²×C₃	semidirect
93. #475	C₂²×C₃	semidirect
96. C₈×Q₈	C₂²×C₃	semidirect
104. C₂×C₃×Q₈	SD₁₆	semidirect
200. C₂×Q₈	C₃×SD₁₆	semidirect
289. C₃	#61818	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×#3502	2
C₂×C₃×D₄×Q₈	2
C₃×Q₈×SD₁₆	2
C₃×#3359	2
C₃×#3229	2
C₃×#3101	2
C₃×#2982	2
#61818	3

Subgroups — C3×#61818 #10680425

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#61818 #10680425