Subgroups — C₃×#33721 #10652328

Order 768.

794 conjugacy classes of subgroups.

The subgroup lattice has 282 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×#510	C₂	1	2	yes	maximal, complemented
2	C₂×C₃×#508	C₂	1	1	yes	maximal, characteristic
3	C₂²×C₃×Q₃₂	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×#3156	C₂	1	2	yes	maximal, complemented
5	C₃×#2251	C₂	1	4	yes	maximal, complemented
6	C₃×#2249	C₂	1	2	yes	maximal, complemented
7	C₃×#2247	C₂	1	2	yes	maximal, complemented
8	C₃×#2108	C₂	1	1	yes	maximal, characteristic, complemented
9	#33721	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
10	C₂×C₃×#137	—	2	2	no	—
11	C₂×C₃×#135	C₂²	1	2	yes	—
12	C₂²×C₃×Q₁₆	C₂²	1	2	yes	—
13	C₃×#536	—	2	2	no	—
14	C₃×#510	—	2	8	no	—
15	C₃×#508	—	2	4	no	—
16	C₂×C₃×Q₃₂	—	2	4	no	—
17	C₂×C₃×Q₃₂	C₂²	1	1	yes	characteristic
18	C₂×C₃×Q₃₂	C₂²	1	2	yes	—
19	C₂×C₃×Q₃₂	C₂²	1	4	yes	—
20	C₂×C₃×Q₃₂	C₂²	1	2	yes	—
21	C₂×C₃×SD₃₂	C₂²	1	4	yes	—
22	C₂×C₃×SD₃₂	C₂²	1	2	yes	—
23	C₂×C₃×D₁₆	C₂²	1	1	yes	characteristic
24	C₂×C₃×M₃₂	C₂²	1	1	yes	characteristic
25	C₂²×C₃×C₁₆	C₂²	1	1	yes	characteristic
26	C₃×#497	—	2	4	no	—
27	C₃×#495	C₂²	1	4	yes	—
28	C₃×#494	C₂²	1	4	yes	—
29	C₃×#405	C₂²	1	1	yes	characteristic
30	C₃×#346	C₂²	1	2	yes	—
31	C₃×C₄×C₁₆	C₂²	1	2	yes	—
32	C₂×#510	C₂×C₃	1	2	yes	complemented
33	C₂×#508	C₂×C₃	1	1	yes	characteristic
34	C₂²×Q₃₂	C₂×C₃	1	1	yes	characteristic, complemented
35	#3156	C₂×C₃	1	2	yes	complemented
36	#2251	C₂×C₃	1	4	yes	complemented
37	#2249	C₂×C₃	1	2	yes	complemented
38	#2247	C₂×C₃	1	2	yes	complemented
39	#2108	C₂×C₃	1	1	yes	characteristic, complemented
40	C₂×C₃×#41	—	2	2	no	—
41	C₂²×C₃×Q₈	—	2	2	no	—
42	C₃×#137	—	4	8	no	—
43	C₃×#135	—	2	4	no	—
44	C₃×#135	—	2	4	no	—
45	C₂×C₃×Q₁₆	—	2	4	no	—
46	C₂×C₃×Q₁₆	—	2	4	no	—
47	C₂×C₃×Q₁₆	—	2	2	no	—
48	C₂×C₃×Q₁₆	—	2	4	no	—
49	C₂×C₃×Q₁₆	C₂³	1	4	yes	—
50	C₂×C₃×Q₁₆	C₂³	1	2	yes	—
51	C₂×C₃×SD₁₆	—	2	2	no	—
52	C₂×C₃×SD₁₆	—	2	4	no	—
53	C₂×C₃×D₈	C₂³	1	2	yes	—
54	C₂×C₃×M₁₆	D₄	1	1	yes	characteristic
55	C₂²×C₃×C₈	C₂³	1	1	yes	characteristic
56	C₃×C₄⋊Q₈	—	2	4	no	—
57	C₃×#124	—	2	4	no	—
58	C₃×#123	—	4	4	no	—
59	C₃×C₂²⋊Q₈	—	4	4	no	—
60	C₃×#117	D₄	1	1	yes	characteristic
61	C₃×Q₃₂	—	2	4	no	—
62	C₃×Q₃₂	—	2	4	no	—
63	C₃×Q₃₂	—	2	2	no	—
64	C₃×Q₃₂	—	2	8	no	—
65	C₃×SD₃₂	—	2	4	no	—
66	C₃×SD₃₂	—	2	8	no	—
67	C₃×D₁₆	—	2	2	no	—
68	C₃×M₃₂	D₄	1	4	yes	—
69	C₂×C₃×C₁₆	D₄	1	4	yes	—
70	C₂×C₃×C₁₆	C₂³	1	1	yes	characteristic
71	C₂×C₃×C₁₆	C₂³	1	1	yes	characteristic
72	C₂×C₃×C₁₆	C₂³	1	2	yes	—
73	C₃×#97	D₄	1	2	yes	—
74	C₃×C₄×C₈	C₂³	1	2	yes	—
75	C₂×#137	—	2	2	no	—
76	C₂×#135	C₂²×C₃	1	2	yes	—
77	C₂²×Q₁₆	C₂²×C₃	1	2	yes	—
78	#536	—	2	2	no	—
79	#510	—	2	8	no	—
80	#508	—	2	4	no	—
81	C₂×Q₃₂	—	2	4	no	—
82	C₂×Q₃₂	C₂²×C₃	1	2	yes	—
83	C₂×Q₃₂	C₂²×C₃	1	2	yes	—
84	C₂×Q₃₂	C₂²×C₃	1	1	yes	characteristic
85	C₂×Q₃₂	C₂²×C₃	1	4	yes	—
86	C₂×SD₃₂	C₂²×C₃	1	2	yes	—
87	C₂×SD₃₂	C₂²×C₃	1	4	yes	—
88	C₂×D₁₆	C₂²×C₃	1	1	yes	characteristic
89	C₂×M₃₂	C₂²×C₃	1	1	yes	characteristic
90	C₂²×C₁₆	C₂²×C₃	1	1	yes	characteristic
91	#497	—	2	4	no	—
92	#495	C₂²×C₃	1	4	yes	—
93	#494	C₂²×C₃	1	4	yes	—
94	#405	C₂²×C₃	1	1	yes	characteristic
95	#346	C₂²×C₃	1	2	yes	—
96	C₄×C₁₆	C₂²×C₃	1	2	yes	—
97	C₃×#41	—	4	4	no	—
98	C₃×#41	—	4	4	no	—
99	C₂×C₃×Q₈	—	2	4	no	—
100	C₂×C₃×Q₈	—	2	2	no	—
101	C₂×C₃×Q₈	—	4	2	no	—
102	C₂×C₃×Q₈	—	4	4	no	—
103	C₂×C₃×Q₈	—	4	4	no	—
104	C₂×C₃×D₄	—	2	2	no	—
105	C₂×C₃×D₄	—	4	2	no	—
106	C₂²×C₃×C₄	—	4	2	no	—
107	C₂²×C₃×C₄	—	4	2	no	—
108	C₂²×C₃×C₄	C₂×D₄	1	1	yes	characteristic
109	C₃×Q₁₆	—	2	4	no	—
110	C₃×Q₁₆	—	2	2	no	—
111	C₃×Q₁₆	—	2	4	no	—
112	C₃×Q₁₆	—	2	2	no	—
113	C₃×Q₁₆	—	4	4	no	—
114	C₃×Q₁₆	—	4	8	no	—
115	C₃×SD₁₆	—	4	4	no	—
116	C₃×SD₁₆	—	4	8	no	—
117	C₃×D₈	—	2	2	no	—
118	C₃×D₈	—	2	2	no	—
119	C₃×M₁₆	D₈	1	4	yes	—
120	C₂×C₃×C₈	C₂×D₄	1	1	yes	characteristic
121	C₂×C₃×C₈	C₂×D₄	1	2	yes	—
122	C₂×C₃×C₈	C₂×D₄	1	2	yes	—
123	C₂×C₃×C₈	C₂×D₄	1	2	yes	—
124	C₂×C₃×C₈	C₂⁴	1	1	yes	characteristic
125	C₃×C₄⋊C₄	—	4	8	no	—
126	C₃×C₄⋊C₄	D₈	1	2	yes	—
127	C₃×C₂²⋊C₄	—	4	8	no	—
128	C₃×C₂²⋊C₄	D₈	1	2	yes	—
129	C₃×C₄²	C₂×D₄	1	2	yes	—
130	C₃×C₁₆	C₂×D₄	1	2	yes	—
131	C₃×C₁₆	C₂×D₄	1	2	yes	—
132	C₃×C₁₆	C₂×D₄	1	4	yes	—
133	C₂×#41	—	2	2	no	—
134	C₂²×Q₈	—	2	2	no	—
135	#137	—	4	8	no	—
136	#135	—	2	4	no	—
137	#135	—	2	4	no	—
138	C₂×Q₁₆	—	2	4	no	—
139	C₂×Q₁₆	—	2	4	no	—
140	C₂×Q₁₆	—	2	2	no	—
141	C₂×Q₁₆	—	2	4	no	—
142	C₂×Q₁₆	C₂³×C₃	1	4	yes	—
143	C₂×Q₁₆	C₂³×C₃	1	2	yes	—
144	C₂×SD₁₆	—	2	2	no	—
145	C₂×SD₁₆	—	2	4	no	—
146	C₂×D₈	C₂³×C₃	1	2	yes	—
147	C₂×M₁₆	C₃×D₄	1	1	yes	characteristic
148	C₂²×C₈	C₂³×C₃	1	1	yes	characteristic
149	C₄⋊Q₈	—	2	4	no	—
150	#124	—	2	4	no	—
151	#123	—	4	4	no	—
152	C₂²⋊Q₈	—	4	4	no	—
153	#117	C₃×D₄	1	1	yes	characteristic
154	Q₃₂	—	2	4	no	—
155	Q₃₂	—	2	4	no	—
156	Q₃₂	—	2	2	no	—
157	Q₃₂	—	2	8	no	—
158	SD₃₂	—	2	4	no	—
159	SD₃₂	—	2	8	no	—
160	D₁₆	—	2	2	no	—
161	M₃₂	C₃×D₄	1	4	yes	—
162	C₂×C₁₆	C₃×D₄	1	4	yes	—
163	C₂×C₁₆	C₂³×C₃	1	1	yes	characteristic
164	C₂×C₁₆	C₂³×C₃	1	2	yes	—
165	C₂×C₁₆	C₂³×C₃	1	1	yes	characteristic
166	#97	C₃×D₄	1	2	yes	—
167	C₄×C₈	C₂³×C₃	1	2	yes	—
168	C₂³×C₃	—	4	2	no	—
169	C₂³×C₃	C₂×D₈	1	1	yes	characteristic
170	C₃×Q₈	—	4	2	no	—
171	C₃×Q₈	—	4	4	no	—
172	C₃×Q₈	—	4	4	no	—
173	C₃×Q₈	—	4	2	no	—
174	C₃×Q₈	—	8	4	no	—
175	C₃×D₄	—	4	2	no	—
176	C₃×D₄	—	4	2	no	—
177	C₃×D₄	—	8	4	no	—
178	C₂×C₃×C₄	—	4	4	no	—
179	C₂×C₃×C₄	—	4	2	no	—
180	C₂×C₃×C₄	—	8	4	no	—
181	C₂×C₃×C₄	—	8	4	no	—
182	C₂×C₃×C₄	#127	1	2	yes	—
183	C₂×C₃×C₄	C₂×D₈	1	4	yes	—
184	C₂×C₃×C₄	C₂×D₈	1	2	yes	—
185	C₂×C₃×C₄	C₂×D₈	1	1	yes	characteristic
186	C₂×C₃×C₄	C₂²×D₄	1	1	yes	characteristic
187	C₃×C₈	#127	1	2	yes	—
188	C₃×C₈	C₂×D₈	1	4	yes	—
189	C₃×C₈	C₂²×D₄	1	1	yes	characteristic
190	C₃×C₈	C₂²×D₄	1	1	yes	characteristic
191	#41	—	4	4	no	—
192	#41	—	4	4	no	—
193	C₂×Q₈	—	2	4	no	—
194	C₂×Q₈	—	2	2	no	—
195	C₂×Q₈	—	4	2	no	—
196	C₂×Q₈	—	4	4	no	—
197	C₂×Q₈	—	4	4	no	—
198	C₂×D₄	—	2	2	no	—
199	C₂×D₄	—	4	2	no	—
200	C₂²×C₄	—	4	2	no	—
201	C₂²×C₄	—	4	2	no	—
202	C₂²×C₄	C₂×C₃×D₄	1	1	yes	characteristic
203	Q₁₆	—	2	4	no	—
204	Q₁₆	—	2	2	no	—
205	Q₁₆	—	2	4	no	—
206	Q₁₆	—	2	2	no	—
207	Q₁₆	—	4	4	no	—
208	Q₁₆	—	4	8	no	—
209	SD₁₆	—	4	4	no	—
210	SD₁₆	—	4	8	no	—
211	D₈	—	2	2	no	—
212	D₈	—	2	2	no	—
213	M₁₆	C₃×D₈	1	4	yes	—
214	C₂×C₈	C₂×C₃×D₄	1	1	yes	characteristic
215	C₂×C₈	C₂×C₃×D₄	1	2	yes	—
216	C₂×C₈	C₂×C₃×D₄	1	2	yes	—
217	C₂×C₈	C₂×C₃×D₄	1	2	yes	—
218	C₂×C₈	C₂⁴×C₃	1	1	yes	characteristic
219	C₄⋊C₄	—	4	8	no	—
220	C₄⋊C₄	C₃×D₈	1	2	yes	—
221	C₂²⋊C₄	—	4	8	no	—
222	C₂²⋊C₄	C₃×D₈	1	2	yes	—
223	C₄²	C₂×C₃×D₄	1	2	yes	—
224	C₁₆	C₂×C₃×D₄	1	2	yes	—
225	C₁₆	C₂×C₃×D₄	1	2	yes	—
226	C₁₆	C₂×C₃×D₄	1	4	yes	—
227	C₂²×C₃	—	2	2	no	—
228	C₂²×C₃	—	8	2	no	—
229	C₂²×C₃	—	8	4	no	—
230	C₂²×C₃	#493	1	2	yes	—
231	C₂²×C₃	C₂²×D₈	1	1	yes	characteristic
232	C₃×C₄	—	2	4	no	—
233	C₃×C₄	—	8	2	no	—
234	C₃×C₄	—	8	4	no	—
235	C₃×C₄	#493	1	2	yes	—
236	C₃×C₄	C₂×#127	1	1	yes	characteristic
237	C₃×C₄	C₂²×D₈	1	1	yes	characteristic
238	C₂³	—	4	2	no	—
239	C₂³	C₂×C₃×D₈	1	1	yes	characteristic
240	Q₈	—	4	4	no	—
241	Q₈	—	4	2	no	—
242	Q₈	—	4	2	no	—
243	Q₈	—	4	4	no	—
244	Q₈	—	8	4	no	—
245	D₄	—	4	2	no	—
246	D₄	—	4	2	no	—
247	D₄	—	8	4	no	—
248	C₂×C₄	—	4	4	no	—
249	C₂×C₄	—	4	2	no	—
250	C₂×C₄	—	8	4	no	—
251	C₂×C₄	—	8	4	no	—
252	C₂×C₄	C₃×#127	1	2	yes	—
253	C₂×C₄	C₂×C₃×D₈	1	4	yes	—
254	C₂×C₄	C₂×C₃×D₈	1	2	yes	—
255	C₂×C₄	C₂×C₃×D₈	1	1	yes	characteristic
256	C₂×C₄	C₂²×C₃×D₄	1	1	yes	characteristic
257	C₈	C₃×#127	1	2	yes	—
258	C₈	C₂×C₃×D₈	1	4	yes	—
259	C₈	C₂²×C₃×D₄	1	1	yes	characteristic
260	C₈	C₂²×C₃×D₄	1	1	yes	characteristic
261	C₂×C₃	—	2	2	no	—
262	C₂×C₃	—	16	2	no	—
263	C₂×C₃	C₂×#493	1	1	yes	characteristic
264	C₂×C₃	#3417	1	2	yes	—
265	C₂²	—	2	2	no	—
266	C₂²	—	8	2	no	—
267	C₂²	—	8	4	no	—
268	C₂²	C₃×#493	1	2	yes	—
269	C₂²	C₂²×C₃×D₈	1	1	yes	characteristic
270	C₄	—	2	4	no	—
271	C₄	—	8	2	no	—
272	C₄	—	8	4	no	—
273	C₄	C₃×#493	1	2	yes	—
274	C₄	C₂×C₃×#127	1	1	yes	characteristic
275	C₄	C₂²×C₃×D₈	1	1	yes	characteristic
276	C₃	#33721	1	1	yes	characteristic, direct product factor, Sylow
277	C₂	—	2	2	no	—
278	C₂	—	16	2	no	—
279	C₂	C₂×C₃×#493	1	1	yes	characteristic
280	C₂	C₃×#3417	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₃	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₃
prime radical	C₂×C₃×#508	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃ ◁ C₃×#117 ◁ C₃×#405 ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₃×#510	C₂	semidirect
3. C₂²×C₃×Q₃₂	C₂	semidirect
4. C₃×#3156	C₂	semidirect
5. C₃×#2251	C₂	semidirect
6. C₃×#2249	C₂	semidirect
7. C₃×#2247	C₂	semidirect
8. C₃×#2108	C₂	semidirect
9. #33721	C₃	direct (N × H)
32. C₂×#510	C₂×C₃	semidirect
34. C₂²×Q₃₂	C₂×C₃	semidirect
35. #3156	C₂×C₃	semidirect
36. #2251	C₂×C₃	semidirect
37. #2249	C₂×C₃	semidirect
38. #2247	C₂×C₃	semidirect
39. #2108	C₂×C₃	semidirect
276. C₃	#33721	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₃×#510	2
C₂×C₃×#508	2
C₂²×C₃×Q₃₂	2
C₃×#3156	2
C₃×#2251	2
C₃×#2249	2
C₃×#2247	2
C₃×#2108	2
#33721	3

Subgroups — C3×#33721 #10652328

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#33721 #10652328