Subgroups — C₃×#33720 #10652327

Order 768.

858 conjugacy classes of subgroups.

The subgroup lattice has 292 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×#509	C₂	1	2	yes	maximal, complemented
2	C₂×C₃×#508	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×C₃×D₁₆	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×#3155	C₂	1	2	yes	maximal, complemented
5	C₃×#2250	C₂	1	4	yes	maximal, complemented
6	C₃×#2249	C₂	1	2	yes	maximal, complemented
7	C₃×#2246	C₂	1	2	yes	maximal, complemented
8	C₃×#2108	C₂	1	1	yes	maximal, characteristic, complemented
9	#33720	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
10	C₂×C₃×AGL(1,ℤ/8)	—	2	2	no	—
11	C₂×C₃×#135	C₂²	1	2	yes	—
12	C₂²×C₃×D₈	C₂²	1	2	yes	—
13	C₃×#535	—	2	2	no	—
14	C₃×#509	—	2	8	no	—
15	C₃×#508	—	2	4	no	—
16	C₂×C₃×Q₃₂	C₂²	1	1	yes	characteristic, complemented
17	C₂×C₃×SD₃₂	C₂²	1	4	yes	complemented
18	C₂×C₃×SD₃₂	C₂²	1	2	yes	complemented
19	C₂×C₃×D₁₆	—	2	4	no	—
20	C₂×C₃×D₁₆	C₂²	1	2	yes	—
21	C₂×C₃×D₁₆	C₂²	1	2	yes	—
22	C₂×C₃×D₁₆	C₂²	1	1	yes	characteristic, complemented
23	C₂×C₃×D₁₆	C₂²	1	4	yes	complemented
24	C₂×C₃×M₃₂	C₂²	1	1	yes	characteristic
25	C₂²×C₃×C₁₆	C₂²	1	1	yes	characteristic
26	C₃×#496	—	2	4	no	—
27	C₃×#495	C₂²	1	4	yes	complemented
28	C₃×#493	C₂²	1	4	yes	complemented
29	C₃×#405	C₂²	1	1	yes	characteristic
30	C₃×#346	C₂²	1	2	yes	complemented
31	C₃×C₄×C₁₆	C₂²	1	2	yes	complemented
32	C₂×#509	C₂×C₃	1	2	yes	complemented
33	C₂×#508	C₂×C₃	1	1	yes	characteristic, complemented
34	C₂²×D₁₆	C₂×C₃	1	1	yes	characteristic, complemented
35	#3155	C₂×C₃	1	2	yes	complemented
36	#2250	C₂×C₃	1	4	yes	complemented
37	#2249	C₂×C₃	1	2	yes	complemented
38	#2246	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×D₄	—	2	2	no	—
42	C₃×AGL(1,ℤ/8)	—	4	8	no	—
43	C₃×#135	—	2	4	no	—
44	C₃×#135	—	2	4	no	—
45	C₂×C₃×Q₁₆	C₂³	1	2	yes	—
46	C₂×C₃×SD₁₆	—	2	2	no	—
47	C₂×C₃×SD₁₆	—	2	4	no	—
48	C₂×C₃×D₈	—	2	4	no	—
49	C₂×C₃×D₈	—	2	4	no	—
50	C₂×C₃×D₈	—	2	2	no	—
51	C₂×C₃×D₈	—	2	4	no	—
52	C₂×C₃×D₈	C₂³	1	4	yes	—
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×#127	—	2	4	no	—
57	C₃×#124	—	2	4	no	—
58	C₃×#121	—	4	4	no	—
59	C₃×#120	—	4	4	no	—
60	C₃×#117	D₄	1	1	yes	characteristic
61	C₃×Q₃₂	—	2	2	no	—
62	C₃×SD₃₂	—	2	4	no	—
63	C₃×SD₃₂	—	2	8	no	—
64	C₃×D₁₆	—	2	4	no	—
65	C₃×D₁₆	—	2	4	no	—
66	C₃×D₁₆	—	2	2	no	—
67	C₃×D₁₆	—	2	8	no	—
68	C₃×M₃₂	D₄	1	4	yes	complemented
69	C₂×C₃×C₁₆	D₄	1	4	yes	complemented
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×AGL(1,ℤ/8)	—	2	2	no	—
76	C₂×#135	C₂²×C₃	1	2	yes	—
77	C₂²×D₈	C₂²×C₃	1	2	yes	—
78	#535	—	2	2	no	—
79	#509	—	2	8	no	—
80	#508	—	2	4	no	—
81	C₂×Q₃₂	C₂²×C₃	1	1	yes	characteristic, complemented
82	C₂×SD₃₂	C₂²×C₃	1	2	yes	complemented
83	C₂×SD₃₂	C₂²×C₃	1	4	yes	complemented
84	C₂×D₁₆	—	2	4	no	—
85	C₂×D₁₆	C₂²×C₃	1	2	yes	—
86	C₂×D₁₆	C₂²×C₃	1	2	yes	—
87	C₂×D₁₆	C₂²×C₃	1	1	yes	characteristic, complemented
88	C₂×D₁₆	C₂²×C₃	1	4	yes	complemented
89	C₂×M₃₂	C₂²×C₃	1	1	yes	characteristic
90	C₂²×C₁₆	C₂²×C₃	1	1	yes	characteristic
91	#496	—	2	4	no	—
92	#495	C₂²×C₃	1	4	yes	complemented
93	#493	C₂²×C₃	1	4	yes	complemented
94	#405	C₂²×C₃	1	1	yes	characteristic
95	#346	C₂²×C₃	1	2	yes	complemented
96	C₄×C₁₆	C₂²×C₃	1	2	yes	complemented
97	C₂⁴×C₃	—	4	2	no	—
98	C₃×#41	—	4	4	no	—
99	C₃×#41	—	4	4	no	—
100	C₂×C₃×Q₈	—	2	2	no	—
101	C₂×C₃×D₄	—	2	4	no	—
102	C₂×C₃×D₄	—	2	2	no	—
103	C₂×C₃×D₄	—	4	2	no	—
104	C₂×C₃×D₄	—	4	4	no	—
105	C₂×C₃×D₄	—	4	4	no	—
106	C₂×C₃×D₄	—	4	2	no	—
107	C₂×C₃×D₄	—	4	8	no	—
108	C₂²×C₃×C₄	—	4	2	no	—
109	C₂²×C₃×C₄	C₂×D₄	1	1	yes	characteristic
110	C₃×Q₁₆	—	2	2	no	—
111	C₃×Q₁₆	—	2	2	no	—
112	C₃×SD₁₆	—	4	4	no	—
113	C₃×SD₁₆	—	4	8	no	—
114	C₃×D₈	—	2	4	no	—
115	C₃×D₈	—	2	2	no	—
116	C₃×D₈	—	2	4	no	—
117	C₃×D₈	—	2	2	no	—
118	C₃×D₈	—	4	4	no	—
119	C₃×D₈	—	4	8	no	—
120	C₃×M₁₆	D₈	1	4	yes	—
121	C₂×C₃×C₈	C₂×D₄	1	1	yes	characteristic
122	C₂×C₃×C₈	C₂×D₄	1	2	yes	—
123	C₂×C₃×C₈	C₂×D₄	1	2	yes	—
124	C₂×C₃×C₈	C₂×D₄	1	2	yes	—
125	C₂×C₃×C₈	C₂⁴	1	1	yes	characteristic
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×D₄	—	2	2	no	—
135	AGL(1,ℤ/8)	—	4	8	no	—
136	#135	—	2	4	no	—
137	#135	—	2	4	no	—
138	C₂×Q₁₆	C₂³×C₃	1	2	yes	—
139	C₂×SD₁₆	—	2	2	no	—
140	C₂×SD₁₆	—	2	4	no	—
141	C₂×D₈	—	2	4	no	—
142	C₂×D₈	—	2	4	no	—
143	C₂×D₈	—	2	2	no	—
144	C₂×D₈	—	2	4	no	—
145	C₂×D₈	C₂³×C₃	1	4	yes	—
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	#127	—	2	4	no	—
150	#124	—	2	4	no	—
151	#121	—	4	4	no	—
152	#120	—	4	4	no	—
153	#117	C₃×D₄	1	1	yes	characteristic
154	Q₃₂	—	2	2	no	—
155	SD₃₂	—	2	4	no	—
156	SD₃₂	—	2	8	no	—
157	D₁₆	—	2	4	no	—
158	D₁₆	—	2	4	no	—
159	D₁₆	—	2	2	no	—
160	D₁₆	—	2	8	no	—
161	M₃₂	C₃×D₄	1	4	yes	complemented
162	C₂×C₁₆	C₃×D₄	1	4	yes	complemented
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	4	no	—
169	C₂³×C₃	—	4	2	no	—
170	C₂³×C₃	—	8	4	no	—
171	C₂³×C₃	—	16	4	no	—
172	C₂³×C₃	C₂×D₈	1	1	yes	characteristic
173	C₃×Q₈	—	4	2	no	—
174	C₃×Q₈	—	4	2	no	—
175	C₃×D₄	—	4	4	no	—
176	C₃×D₄	—	4	2	no	—
177	C₃×D₄	—	4	2	no	—
178	C₃×D₄	—	4	4	no	—
179	C₃×D₄	—	8	4	no	—
180	C₃×D₄	—	8	4	no	—
181	C₃×D₄	—	16	8	no	—
182	C₂×C₃×C₄	—	4	2	no	—
183	C₂×C₃×C₄	—	8	4	no	—
184	C₂×C₃×C₄	#127	1	2	yes	—
185	C₂×C₃×C₄	C₂×D₈	1	4	yes	—
186	C₂×C₃×C₄	C₂×D₈	1	2	yes	—
187	C₂×C₃×C₄	C₂×D₈	1	1	yes	characteristic
188	C₂×C₃×C₄	C₂²×D₄	1	1	yes	characteristic
189	C₃×C₈	#127	1	2	yes	—
190	C₃×C₈	C₂×D₈	1	4	yes	—
191	C₃×C₈	C₂²×D₄	1	1	yes	characteristic
192	C₃×C₈	C₂²×D₄	1	1	yes	characteristic
193	C₂⁴	—	4	2	no	—
194	#41	—	4	4	no	—
195	#41	—	4	4	no	—
196	C₂×Q₈	—	2	2	no	—
197	C₂×D₄	—	2	4	no	—
198	C₂×D₄	—	2	2	no	—
199	C₂×D₄	—	4	2	no	—
200	C₂×D₄	—	4	4	no	—
201	C₂×D₄	—	4	4	no	—
202	C₂×D₄	—	4	2	no	—
203	C₂×D₄	—	4	8	no	—
204	C₂²×C₄	—	4	2	no	—
205	C₂²×C₄	C₂×C₃×D₄	1	1	yes	characteristic
206	Q₁₆	—	2	2	no	—
207	Q₁₆	—	2	2	no	—
208	SD₁₆	—	4	4	no	—
209	SD₁₆	—	4	8	no	—
210	D₈	—	2	4	no	—
211	D₈	—	2	2	no	—
212	D₈	—	2	4	no	—
213	D₈	—	2	2	no	—
214	D₈	—	4	4	no	—
215	D₈	—	4	8	no	—
216	M₁₆	C₃×D₈	1	4	yes	—
217	C₂×C₈	C₂×C₃×D₄	1	1	yes	characteristic
218	C₂×C₈	C₂×C₃×D₄	1	2	yes	—
219	C₂×C₈	C₂×C₃×D₄	1	2	yes	—
220	C₂×C₈	C₂×C₃×D₄	1	2	yes	—
221	C₂×C₈	C₂⁴×C₃	1	1	yes	characteristic
222	C₄⋊C₄	C₃×D₈	1	2	yes	—
223	C₂²⋊C₄	—	4	8	no	—
224	C₂²⋊C₄	C₃×D₈	1	2	yes	—
225	C₄²	C₂×C₃×D₄	1	2	yes	—
226	C₁₆	C₂×C₃×D₄	1	2	yes	—
227	C₁₆	C₂×C₃×D₄	1	2	yes	—
228	C₁₆	C₂×C₃×D₄	1	4	yes	—
229	C₂²×C₃	—	2	2	no	—
230	C₂²×C₃	—	8	2	no	—
231	C₂²×C₃	—	8	4	no	—
232	C₂²×C₃	—	8	4	no	—
233	C₂²×C₃	—	8	8	no	—
234	C₂²×C₃	—	16	8	no	—
235	C₂²×C₃	#493	1	2	yes	—
236	C₂²×C₃	C₂²×D₈	1	1	yes	characteristic
237	C₃×C₄	—	2	4	no	—
238	C₃×C₄	—	8	2	no	—
239	C₃×C₄	#493	1	2	yes	—
240	C₃×C₄	C₂×#127	1	1	yes	characteristic
241	C₃×C₄	C₂²×D₈	1	1	yes	characteristic
242	C₂³	—	4	4	no	—
243	C₂³	—	4	2	no	—
244	C₂³	—	8	4	no	—
245	C₂³	—	16	4	no	—
246	C₂³	C₂×C₃×D₈	1	1	yes	characteristic
247	Q₈	—	4	2	no	—
248	Q₈	—	4	2	no	—
249	D₄	—	4	2	no	—
250	D₄	—	4	4	no	—
251	D₄	—	4	4	no	—
252	D₄	—	4	2	no	—
253	D₄	—	8	4	no	—
254	D₄	—	8	4	no	—
255	D₄	—	16	8	no	—
256	C₂×C₄	—	4	2	no	—
257	C₂×C₄	—	8	4	no	—
258	C₂×C₄	C₃×#127	1	2	yes	—
259	C₂×C₄	C₂×C₃×D₈	1	4	yes	—
260	C₂×C₄	C₂×C₃×D₈	1	2	yes	—
261	C₂×C₄	C₂×C₃×D₈	1	1	yes	characteristic
262	C₂×C₄	C₂²×C₃×D₄	1	1	yes	characteristic
263	C₈	C₃×#127	1	2	yes	—
264	C₈	C₂×C₃×D₈	1	4	yes	—
265	C₈	C₂²×C₃×D₄	1	1	yes	characteristic
266	C₈	C₂²×C₃×D₄	1	1	yes	characteristic
267	C₂×C₃	—	2	2	no	—
268	C₂×C₃	—	16	2	no	—
269	C₂×C₃	—	16	4	no	—
270	C₂×C₃	C₂×#493	1	1	yes	characteristic
271	C₂×C₃	#3415	1	2	yes	—
272	C₂²	—	2	2	no	—
273	C₂²	—	8	2	no	—
274	C₂²	—	8	4	no	—
275	C₂²	—	8	4	no	—
276	C₂²	—	8	8	no	—
277	C₂²	—	16	8	no	—
278	C₂²	C₃×#493	1	2	yes	—
279	C₂²	C₂²×C₃×D₈	1	1	yes	characteristic
280	C₄	—	2	4	no	—
281	C₄	—	8	2	no	—
282	C₄	C₃×#493	1	2	yes	—
283	C₄	C₂×C₃×#127	1	1	yes	characteristic
284	C₄	C₂²×C₃×D₈	1	1	yes	characteristic
285	C₃	#33720	1	1	yes	characteristic, direct product factor, Sylow
286	C₂	—	2	2	no	—
287	C₂	—	16	4	no	—
288	C₂	—	16	2	no	—
289	C₂	C₂×C₃×#493	1	1	yes	characteristic
290	C₂	C₃×#3415	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₃	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₃
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×#509	C₂	semidirect
2. C₂×C₃×#508	C₂	semidirect
3. C₂²×C₃×D₁₆	C₂	semidirect
4. C₃×#3155	C₂	semidirect
5. C₃×#2250	C₂	semidirect
6. C₃×#2249	C₂	semidirect
7. C₃×#2246	C₂	semidirect
8. C₃×#2108	C₂	semidirect
9. #33720	C₃	direct (N × H)
16. C₂×C₃×Q₃₂	C₂²	semidirect
17. C₂×C₃×SD₃₂	C₂²	semidirect
18. C₂×C₃×SD₃₂	C₂²	semidirect
22. C₂×C₃×D₁₆	C₂²	semidirect
23. C₂×C₃×D₁₆	C₂²	semidirect
27. C₃×#495	C₂²	semidirect
28. C₃×#493	C₂²	semidirect
30. C₃×#346	C₂²	semidirect
31. C₃×C₄×C₁₆	C₂²	semidirect
32. C₂×#509	C₂×C₃	semidirect
33. C₂×#508	C₂×C₃	semidirect
34. C₂²×D₁₆	C₂×C₃	semidirect
35. #3155	C₂×C₃	semidirect
36. #2250	C₂×C₃	semidirect
37. #2249	C₂×C₃	semidirect
38. #2246	C₂×C₃	semidirect
39. #2108	C₂×C₃	semidirect
68. C₃×M₃₂	D₄	semidirect
69. C₂×C₃×C₁₆	D₄	semidirect
81. C₂×Q₃₂	C₂²×C₃	semidirect
82. C₂×SD₃₂	C₂²×C₃	semidirect
83. C₂×SD₃₂	C₂²×C₃	semidirect
87. C₂×D₁₆	C₂²×C₃	semidirect
88. C₂×D₁₆	C₂²×C₃	semidirect
92. #495	C₂²×C₃	semidirect
93. #493	C₂²×C₃	semidirect
95. #346	C₂²×C₃	semidirect
96. C₄×C₁₆	C₂²×C₃	semidirect
161. M₃₂	C₃×D₄	semidirect
162. C₂×C₁₆	C₃×D₄	semidirect
285. C₃	#33720	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×#509	2
C₂×C₃×#508	2
C₂²×C₃×D₁₆	2
C₃×#3155	2
C₃×#2250	2
C₃×#2249	2
C₃×#2246	2
C₃×#2108	2
#33720	3

Subgroups — C3×#33720 #10652327

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#33720 #10652327