Subgroups — C₃×#33765 #10652372

Order 768.

616 conjugacy classes of subgroups.

The subgroup lattice has 264 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#3058	C₂	1	2	yes	maximal, complemented
2	C₃×#2234	C₂	1	4	yes	maximal, complemented
3	C₃×#2218	C₂	1	2	yes	maximal, complemented
4	C₃×#2215	C₂	1	2	yes	maximal, complemented
5	C₃×#2184	C₂	1	4	yes	maximal, complemented
6	C₃×#2113	C₂	1	1	yes	maximal, characteristic, complemented
7	#33765	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
8	C₂²×C₃×D₈	C₂²	1	2	yes	complemented
9	C₂×C₃×#131	C₂²	1	1	yes	characteristic
10	C₃×#535	—	2	2	no	—
11	C₂×C₃×D₁₆	C₂²	1	2	yes	—
12	C₂×C₃×D₁₆	C₂²	1	2	yes	—
13	C₂×C₃×M₃₂	C₂²	1	1	yes	characteristic, complemented
14	C₂²×C₃×C₁₆	C₂²	1	1	yes	characteristic, complemented
15	C₃×#469	—	2	2	no	—
16	C₃×#469	—	2	2	no	—
17	C₃×#466	C₂²	1	4	yes	—
18	C₃×#466	C₂²	1	4	yes	complemented
19	C₃×#427	C₂²	1	2	yes	complemented
20	C₃×#418	—	2	2	no	—
21	C₃×#367	C₂²	1	2	yes	complemented
22	C₃×#366	C₂²	1	2	yes	complemented
23	C₃×#357	C₂²	1	4	yes	complemented
24	C₃×#357	C₂²	1	4	yes	complemented
25	C₃×C₂²⋊C₁₆	C₂²	1	4	yes	complemented
26	#3058	C₂×C₃	1	2	yes	complemented
27	#2234	C₂×C₃	1	4	yes	complemented
28	#2218	C₂×C₃	1	2	yes	complemented
29	#2215	C₂×C₃	1	2	yes	complemented
30	#2184	C₂×C₃	1	4	yes	complemented
31	#2113	C₂×C₃	1	1	yes	characteristic, complemented
32	C₂×C₃×#41	D₄	1	1	yes	characteristic
33	C₂²×C₃×D₄	—	2	2	no	—
34	C₂×C₃×D₈	—	2	2	no	—
35	C₂×C₃×D₈	—	4	2	no	—
36	C₂×C₃×D₈	—	4	2	no	—
37	C₂×C₃×D₈	C₂³	1	4	yes	—
38	C₃×#131	—	4	1	no	—
39	C₃×#131	—	4	1	no	—
40	C₂×C₃×M₁₆	D₄	1	2	yes	—
41	C₂×C₃×M₁₆	D₄	1	1	yes	characteristic
42	C₂²×C₃×C₈	C₂³	1	1	yes	characteristic
43	C₂²×C₃×C₈	C₂³	1	2	yes	—
44	C₃×#127	—	4	2	no	—
45	C₃×#124	—	4	2	no	—
46	C₃×#121	—	2	4	no	—
47	C₃×#121	—	2	4	no	—
48	C₃×#120	—	4	2	no	—
49	C₃×#120	—	4	2	no	—
50	C₃×#117	—	2	2	no	—
51	C₃×D₁₆	—	4	2	no	—
52	C₃×D₁₆	—	4	2	no	—
53	C₃×M₃₂	—	4	1	no	—
54	C₂×C₃×C₁₆	—	4	1	no	—
55	C₂×C₃×C₁₆	C₂³	1	2	yes	—
56	C₂×C₃×C₁₆	C₂³	1	2	yes	—
57	C₃×#107	C₂³	1	4	yes	—
58	C₃×#106	—	2	2	no	—
59	C₃×#102	—	2	4	no	—
60	C₃×#102	—	2	4	no	—
61	C₂²×D₈	C₂²×C₃	1	2	yes	complemented
62	C₂×#131	C₂²×C₃	1	1	yes	characteristic
63	#535	—	2	2	no	—
64	C₂×D₁₆	C₂²×C₃	1	2	yes	—
65	C₂×D₁₆	C₂²×C₃	1	2	yes	—
66	C₂×M₃₂	C₂²×C₃	1	1	yes	characteristic, complemented
67	C₂²×C₁₆	C₂²×C₃	1	1	yes	characteristic, complemented
68	#469	—	2	2	no	—
69	#469	—	2	2	no	—
70	#466	C₂²×C₃	1	4	yes	—
71	#466	C₂²×C₃	1	4	yes	complemented
72	#427	C₂²×C₃	1	2	yes	complemented
73	#418	—	2	2	no	—
74	#367	C₂²×C₃	1	2	yes	complemented
75	#366	C₂²×C₃	1	2	yes	complemented
76	#357	C₂²×C₃	1	4	yes	complemented
77	#357	C₂²×C₃	1	4	yes	complemented
78	C₂²⋊C₁₆	C₂²×C₃	1	4	yes	complemented
79	C₂⁴×C₃	—	4	2	no	—
80	C₃×#41	—	4	1	no	—
81	C₃×#41	—	4	1	no	—
82	C₂×C₃×Q₈	D₈	1	1	yes	characteristic
83	C₂×C₃×D₄	—	2	4	no	—
84	C₂×C₃×D₄	—	4	4	no	—
85	C₂×C₃×D₄	—	4	4	no	—
86	C₂×C₃×D₄	—	4	4	no	—
87	C₂×C₃×D₄	—	4	4	no	—
88	C₂×C₃×D₄	—	4	2	no	—
89	C₂×C₃×D₄	D₈	1	1	yes	characteristic
90	C₂×C₃×D₄	D₈	1	2	yes	—
91	C₂²×C₃×C₄	C₂×D₄	1	2	yes	—
92	C₂²×C₃×C₄	C₂×D₄	1	1	yes	characteristic
93	C₃×D₈	—	2	4	no	—
94	C₃×D₈	—	2	4	no	—
95	C₃×D₈	—	4	4	no	—
96	C₃×M₁₆	—	4	2	no	—
97	C₃×M₁₆	—	4	1	no	—
98	C₂×C₃×C₈	—	2	1	no	—
99	C₂×C₃×C₈	—	2	1	no	—
100	C₂×C₃×C₈	—	2	2	no	—
101	C₂×C₃×C₈	—	2	2	no	—
102	C₂×C₃×C₈	C₂×D₄	1	1	yes	characteristic
103	C₂×C₃×C₈	C₂×D₄	1	2	yes	—
104	C₂×C₃×C₈	C₂⁴	1	1	yes	characteristic
105	C₃×C₄⋊C₄	—	2	4	no	—
106	C₃×C₂²⋊C₄	—	4	2	no	—
107	C₃×C₂²⋊C₄	—	4	4	no	—
108	C₃×C₂²⋊C₄	—	4	4	no	—
109	C₃×C₄²	—	4	2	no	—
110	C₃×C₁₆	—	2	2	no	—
111	C₃×C₁₆	—	2	2	no	—
112	C₂×#41	C₃×D₄	1	1	yes	characteristic
113	C₂²×D₄	—	2	2	no	—
114	C₂×D₈	—	2	2	no	—
115	C₂×D₈	—	4	2	no	—
116	C₂×D₈	—	4	2	no	—
117	C₂×D₈	C₂³×C₃	1	4	yes	—
118	#131	—	4	1	no	—
119	#131	—	4	1	no	—
120	C₂×M₁₆	C₃×D₄	1	2	yes	—
121	C₂×M₁₆	C₃×D₄	1	1	yes	characteristic
122	C₂²×C₈	C₂³×C₃	1	2	yes	—
123	C₂²×C₈	C₂³×C₃	1	1	yes	characteristic
124	#127	—	4	2	no	—
125	#124	—	4	2	no	—
126	#121	—	2	4	no	—
127	#121	—	2	4	no	—
128	#120	—	4	2	no	—
129	#120	—	4	2	no	—
130	#117	—	2	2	no	—
131	D₁₆	—	4	2	no	—
132	D₁₆	—	4	2	no	—
133	M₃₂	—	4	1	no	—
134	C₂×C₁₆	—	4	1	no	—
135	C₂×C₁₆	C₂³×C₃	1	2	yes	—
136	C₂×C₁₆	C₂³×C₃	1	2	yes	—
137	#107	C₂³×C₃	1	4	yes	—
138	#106	—	2	2	no	—
139	#102	—	2	4	no	—
140	#102	—	2	4	no	—
141	C₂³×C₃	—	4	4	no	—
142	C₂³×C₃	—	8	4	no	—
143	C₂³×C₃	—	16	4	no	—
144	C₂³×C₃	C₂×D₈	1	2	yes	—
145	C₂³×C₃	C₂×D₈	1	1	yes	characteristic
146	C₃×Q₈	—	4	1	no	—
147	C₃×D₄	—	4	2	no	—
148	C₃×D₄	—	4	1	no	—
149	C₃×D₄	—	4	4	no	—
150	C₃×D₄	—	4	4	no	—
151	C₃×D₄	—	8	4	no	—
152	C₃×D₄	—	16	4	no	—
153	C₃×D₄	—	16	4	no	—
154	C₂×C₃×C₄	—	2	2	no	—
155	C₂×C₃×C₄	—	2	1	no	—
156	C₂×C₃×C₄	—	2	2	no	—
157	C₂×C₃×C₄	—	2	1	no	—
158	C₂×C₃×C₄	—	4	4	no	—
159	C₂×C₃×C₄	C₂×D₈	1	2	yes	—
160	C₂×C₃×C₄	C₂×D₈	1	1	yes	characteristic
161	C₂×C₃×C₄	C₂²×D₄	1	1	yes	characteristic
162	C₃×C₈	—	2	1	no	—
163	C₃×C₈	—	2	2	no	—
164	C₃×C₈	H(5,𝔽₂)	1	1	yes	characteristic
165	C₃×C₈	H(5,𝔽₂)	1	1	yes	characteristic
166	C₂⁴	—	4	2	no	—
167	#41	—	4	1	no	—
168	#41	—	4	1	no	—
169	C₂×Q₈	C₃×D₈	1	1	yes	characteristic
170	C₂×D₄	—	2	4	no	—
171	C₂×D₄	—	4	4	no	—
172	C₂×D₄	—	4	4	no	—
173	C₂×D₄	—	4	4	no	—
174	C₂×D₄	—	4	4	no	—
175	C₂×D₄	—	4	2	no	—
176	C₂×D₄	C₃×D₈	1	1	yes	characteristic
177	C₂×D₄	C₃×D₈	1	2	yes	—
178	C₂²×C₄	C₂×C₃×D₄	1	2	yes	—
179	C₂²×C₄	C₂×C₃×D₄	1	1	yes	characteristic
180	D₈	—	2	4	no	—
181	D₈	—	2	4	no	—
182	D₈	—	4	4	no	—
183	M₁₆	—	4	1	no	—
184	M₁₆	—	4	2	no	—
185	C₂×C₈	—	2	1	no	—
186	C₂×C₈	—	2	1	no	—
187	C₂×C₈	—	2	2	no	—
188	C₂×C₈	—	2	2	no	—
189	C₂×C₈	C₂×C₃×D₄	1	1	yes	characteristic
190	C₂×C₈	C₂×C₃×D₄	1	2	yes	—
191	C₂×C₈	C₂⁴×C₃	1	1	yes	characteristic
192	C₄⋊C₄	—	2	4	no	—
193	C₂²⋊C₄	—	4	2	no	—
194	C₂²⋊C₄	—	4	4	no	—
195	C₂²⋊C₄	—	4	4	no	—
196	C₄²	—	4	2	no	—
197	C₁₆	—	2	2	no	—
198	C₁₆	—	2	2	no	—
199	C₂²×C₃	—	2	2	no	—
200	C₂²×C₃	—	2	4	no	—
201	C₂²×C₃	—	2	2	no	—
202	C₂²×C₃	—	2	1	no	—
203	C₂²×C₃	—	8	4	no	—
204	C₂²×C₃	—	8	4	no	—
205	C₂²×C₃	—	8	4	no	—
206	C₂²×C₃	—	16	8	no	—
207	C₂²×C₃	C₂²×D₈	1	1	yes	characteristic
208	C₃×C₄	—	2	2	no	—
209	C₃×C₄	—	2	1	no	—
210	C₃×C₄	—	8	4	no	—
211	C₃×C₄	#534	1	1	yes	characteristic
212	C₃×C₄	#575	1	1	yes	characteristic
213	C₂³	—	4	4	no	—
214	C₂³	—	8	4	no	—
215	C₂³	—	16	4	no	—
216	C₂³	C₂×C₃×D₈	1	2	yes	—
217	C₂³	C₂×C₃×D₈	1	1	yes	characteristic
218	Q₈	—	4	1	no	—
219	D₄	—	4	2	no	—
220	D₄	—	4	1	no	—
221	D₄	—	4	4	no	—
222	D₄	—	4	4	no	—
223	D₄	—	8	4	no	—
224	D₄	—	16	4	no	—
225	D₄	—	16	4	no	—
226	C₂×C₄	—	2	2	no	—
227	C₂×C₄	—	2	1	no	—
228	C₂×C₄	—	2	2	no	—
229	C₂×C₄	—	2	1	no	—
230	C₂×C₄	—	4	4	no	—
231	C₂×C₄	C₂×C₃×D₈	1	2	yes	—
232	C₂×C₄	C₂×C₃×D₈	1	1	yes	characteristic
233	C₂×C₄	C₂²×C₃×D₄	1	1	yes	characteristic
234	C₈	—	2	2	no	—
235	C₈	—	2	1	no	—
236	C₈	C₃×H(5,𝔽₂)	1	1	yes	characteristic
237	C₈	C₃×H(5,𝔽₂)	1	1	yes	characteristic
238	C₂×C₃	—	4	1	no	—
239	C₂×C₃	—	4	2	no	—
240	C₂×C₃	—	16	4	no	—
241	C₂×C₃	#3186	1	1	yes	characteristic
242	C₂×C₃	#3415	1	2	yes	—
243	C₂²	—	2	2	no	—
244	C₂²	—	2	4	no	—
245	C₂²	—	2	2	no	—
246	C₂²	—	2	1	no	—
247	C₂²	—	8	4	no	—
248	C₂²	—	8	4	no	—
249	C₂²	—	8	4	no	—
250	C₂²	—	16	8	no	—
251	C₂²	C₂²×C₃×D₈	1	1	yes	characteristic
252	C₄	—	2	2	no	—
253	C₄	—	2	1	no	—
254	C₄	—	8	4	no	—
255	C₄	C₃×#534	1	1	yes	characteristic
256	C₄	C₃×#575	1	1	yes	characteristic
257	C₃	#33765	1	1	yes	characteristic, direct product factor, Sylow
258	C₂	—	4	2	no	—
259	C₂	—	4	1	no	—
260	C₂	—	16	4	no	—
261	C₂	C₃×#3186	1	1	yes	characteristic
262	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 ◁ C₂×C₃×#41 ◁ C₂×C₃×#131 ◁ G

Complements

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

N	H ≅ G/N	type
1. C₃×#3058	C₂	semidirect
2. C₃×#2234	C₂	semidirect
3. C₃×#2218	C₂	semidirect
4. C₃×#2215	C₂	semidirect
5. C₃×#2184	C₂	semidirect
6. C₃×#2113	C₂	semidirect
7. #33765	C₃	direct (N × H)
8. C₂²×C₃×D₈	C₂²	semidirect
13. C₂×C₃×M₃₂	C₂²	semidirect
14. C₂²×C₃×C₁₆	C₂²	semidirect
18. C₃×#466	C₂²	semidirect
19. C₃×#427	C₂²	semidirect
21. C₃×#367	C₂²	semidirect
22. C₃×#366	C₂²	semidirect
23. C₃×#357	C₂²	semidirect
24. C₃×#357	C₂²	semidirect
25. C₃×C₂²⋊C₁₆	C₂²	semidirect
26. #3058	C₂×C₃	semidirect
27. #2234	C₂×C₃	semidirect
28. #2218	C₂×C₃	semidirect
29. #2215	C₂×C₃	semidirect
30. #2184	C₂×C₃	semidirect
31. #2113	C₂×C₃	semidirect
61. C₂²×D₈	C₂²×C₃	semidirect
66. C₂×M₃₂	C₂²×C₃	semidirect
67. C₂²×C₁₆	C₂²×C₃	semidirect
71. #466	C₂²×C₃	semidirect
72. #427	C₂²×C₃	semidirect
74. #367	C₂²×C₃	semidirect
75. #366	C₂²×C₃	semidirect
76. #357	C₂²×C₃	semidirect
77. #357	C₂²×C₃	semidirect
78. C₂²⋊C₁₆	C₂²×C₃	semidirect
257. C₃	#33765	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×#3058	2
C₃×#2234	2
C₃×#2218	2
C₃×#2215	2
C₃×#2184	2
C₃×#2113	2
#33765	3

Subgroups — C3×#33765 #10652372

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#33765 #10652372