Subgroups — C₃×#33729 #10652336

Order 768.

474 conjugacy classes of subgroups.

The subgroup lattice has 220 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#3164	C₂	1	2	yes	maximal
2	C₃×#2255	C₂	1	4	yes	maximal, complemented
3	C₃×#2253	C₂	1	2	yes	maximal, complemented
4	C₃×#2252	C₂	1	2	yes	maximal, complemented
5	C₃×#2159	C₂	1	2	yes	maximal
6	C₃×#2158	C₂	1	1	yes	maximal, characteristic
7	C₂×C₃×#366	C₂	1	1	yes	maximal, characteristic
8	C₃×#2108	C₂	1	1	yes	maximal, characteristic
9	#33729	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
10	C₃×#544	—	2	2	no	—
11	C₂×C₃×M₃₂	C₂²	1	1	yes	characteristic
12	C₂²×C₃×C₁₆	C₂²	1	1	yes	characteristic
13	C₃×#501	—	2	4	no	—
14	C₃×#500	C₂²	1	4	yes	—
15	C₃×#499	C₂²	1	4	yes	—
16	C₃×#428	—	2	2	no	—
17	C₃×#427	C₂²	1	2	yes	—
18	C₂×C₃×#107	C₂²	1	2	yes	—
19	C₃×#405	C₂²	1	1	yes	characteristic
20	C₃×#367	C₂²	1	4	yes	—
21	C₃×#367	C₂²	1	2	yes	—
22	C₃×#366	C₂²	1	1	yes	characteristic
23	C₃×#366	C₂²	1	2	yes	—
24	C₃×#366	C₂²	1	4	yes	—
25	C₃×#366	C₂²	1	1	yes	characteristic
26	C₃×#366	C₂²	1	2	yes	—
27	C₃×#346	C₂²	1	2	yes	—
28	C₃×C₄×C₁₆	C₂²	1	2	yes	—
29	#3164	C₂×C₃	1	2	yes	—
30	#2255	C₂×C₃	1	4	yes	complemented
31	#2253	C₂×C₃	1	2	yes	complemented
32	#2252	C₂×C₃	1	2	yes	complemented
33	#2159	C₂×C₃	1	2	yes	—
34	#2158	C₂×C₃	1	1	yes	characteristic
35	C₂×#366	C₂×C₃	1	1	yes	characteristic
36	#2108	C₂×C₃	1	1	yes	characteristic
37	C₂×C₃×M₁₆	D₄	1	1	yes	characteristic
38	C₂²×C₃×C₈	C₂³	1	1	yes	characteristic
39	C₃×C₄⋊Q₈	—	2	4	no	—
40	C₃×C₄⋊Q₈	—	4	2	no	—
41	C₃×#125	—	2	4	no	—
42	C₃×#125	—	4	2	no	—
43	C₃×C₂²⋊Q₈	—	4	4	no	—
44	C₃×#117	—	2	2	no	—
45	C₃×#117	D₄	1	1	yes	characteristic
46	C₂×C₃×C₄⋊C₄	—	2	2	no	—
47	C₃×M₃₂	Q₈	1	4	yes	complemented
48	C₂×C₃×C₁₆	Q₈	1	4	yes	complemented
49	C₂×C₃×C₁₆	C₂³	1	1	yes	characteristic
50	C₂×C₃×C₁₆	C₂³	1	1	yes	characteristic
51	C₂×C₃×C₁₆	C₂³	1	2	yes	—
52	C₃×#107	—	2	2	no	—
53	C₃×#107	—	2	4	no	—
54	C₃×#107	C₂³	1	4	yes	—
55	C₃×#107	C₂³	1	2	yes	—
56	C₃×#107	C₂³	1	2	yes	—
57	C₃×#106	—	2	2	no	—
58	C₃×#106	—	2	4	no	—
59	C₃×#97	D₄	1	2	yes	—
60	C₃×C₄×C₈	C₂³	1	2	yes	—
61	#544	—	2	2	no	—
62	C₂×M₃₂	C₂²×C₃	1	1	yes	characteristic
63	C₂²×C₁₆	C₂²×C₃	1	1	yes	characteristic
64	#501	—	2	4	no	—
65	#500	C₂²×C₃	1	4	yes	—
66	#499	C₂²×C₃	1	4	yes	—
67	#428	—	2	2	no	—
68	#427	C₂²×C₃	1	2	yes	—
69	C₂×#107	C₂²×C₃	1	2	yes	—
70	#405	C₂²×C₃	1	1	yes	characteristic
71	#367	C₂²×C₃	1	2	yes	—
72	#367	C₂²×C₃	1	4	yes	—
73	#366	C₂²×C₃	1	2	yes	—
74	#366	C₂²×C₃	1	2	yes	—
75	#366	C₂²×C₃	1	1	yes	characteristic
76	#366	C₂²×C₃	1	1	yes	characteristic
77	#366	C₂²×C₃	1	4	yes	—
78	#346	C₂²×C₃	1	2	yes	—
79	C₄×C₁₆	C₂²×C₃	1	2	yes	—
80	C₂×C₃×Q₈	—	4	4	no	—
81	C₂²×C₃×C₄	—	4	2	no	—
82	C₂²×C₃×C₄	C₂×D₄	1	1	yes	characteristic
83	C₃×M₁₆	Q₁₆	1	4	yes	—
84	C₂×C₃×C₈	C₂×D₄	1	1	yes	characteristic
85	C₂×C₃×C₈	C₂×D₄	1	2	yes	—
86	C₂×C₃×C₈	C₂×Q₈	1	2	yes	—
87	C₂×C₃×C₈	C₂×Q₈	1	2	yes	—
88	C₂×C₃×C₈	C₂⁴	1	1	yes	characteristic
89	C₃×C₄⋊C₄	—	2	2	no	—
90	C₃×C₄⋊C₄	—	2	4	no	—
91	C₃×C₄⋊C₄	—	2	2	no	—
92	C₃×C₄⋊C₄	—	4	2	no	—
93	C₃×C₄⋊C₄	—	4	4	no	—
94	C₃×C₄⋊C₄	—	4	4	no	—
95	C₃×C₄⋊C₄	—	4	4	no	—
96	C₃×C₄⋊C₄	D₈	1	1	yes	characteristic
97	C₃×C₄⋊C₄	D₈	1	1	yes	characteristic
98	C₃×C₂²⋊C₄	—	4	2	no	—
99	C₃×C₂²⋊C₄	D₈	1	2	yes	—
100	C₃×C₄²	—	4	2	no	—
101	C₃×C₄²	C₂×D₄	1	2	yes	—
102	C₃×C₁₆	C₂×Q₈	1	2	yes	—
103	C₃×C₁₆	C₂×Q₈	1	2	yes	—
104	C₃×C₁₆	C₂×Q₈	1	4	yes	—
105	C₂×M₁₆	C₃×D₄	1	1	yes	characteristic
106	C₂²×C₈	C₂³×C₃	1	1	yes	characteristic
107	C₄⋊Q₈	—	2	4	no	—
108	C₄⋊Q₈	—	4	2	no	—
109	#125	—	2	4	no	—
110	#125	—	4	2	no	—
111	C₂²⋊Q₈	—	4	4	no	—
112	#117	—	2	2	no	—
113	#117	C₃×D₄	1	1	yes	characteristic
114	C₂×C₄⋊C₄	—	2	2	no	—
115	M₃₂	C₃×Q₈	1	4	yes	complemented
116	C₂×C₁₆	C₃×Q₈	1	4	yes	complemented
117	C₂×C₁₆	C₂³×C₃	1	1	yes	characteristic
118	C₂×C₁₆	C₂³×C₃	1	2	yes	—
119	C₂×C₁₆	C₂³×C₃	1	1	yes	characteristic
120	#107	—	2	2	no	—
121	#107	—	2	4	no	—
122	#107	C₂³×C₃	1	4	yes	—
123	#107	C₂³×C₃	1	2	yes	—
124	#107	C₂³×C₃	1	2	yes	—
125	#106	—	2	2	no	—
126	#106	—	2	4	no	—
127	#97	C₃×D₄	1	2	yes	—
128	C₄×C₈	C₂³×C₃	1	2	yes	—
129	C₂³×C₃	C₂×D₈	1	1	yes	characteristic
130	C₃×Q₈	—	16	4	no	—
131	C₂×C₃×C₄	—	4	2	no	—
132	C₂×C₃×C₄	—	4	2	no	—
133	C₂×C₃×C₄	—	4	4	no	—
134	C₂×C₃×C₄	—	16	2	no	—
135	C₂×C₃×C₄	C₄⋊Q₈	1	2	yes	—
136	C₂×C₃×C₄	C₂×D₈	1	2	yes	—
137	C₂×C₃×C₄	C₂×D₈	1	2	yes	—
138	C₂×C₃×C₄	C₂×D₈	1	1	yes	characteristic
139	C₂×C₃×C₄	C₂×Q₁₆	1	2	yes	—
140	C₂×C₃×C₄	C₂²×D₄	1	1	yes	characteristic
141	C₃×C₈	C₄⋊Q₈	1	2	yes	—
142	C₃×C₈	C₂×Q₁₆	1	4	yes	—
143	C₃×C₈	C₂²×Q₈	1	1	yes	characteristic
144	C₃×C₈	C₂²×Q₈	1	1	yes	characteristic
145	C₂×Q₈	—	4	4	no	—
146	C₂²×C₄	—	4	2	no	—
147	C₂²×C₄	C₂×C₃×D₄	1	1	yes	characteristic
148	M₁₆	C₃×Q₁₆	1	4	yes	—
149	C₂×C₈	C₂×C₃×D₄	1	1	yes	characteristic
150	C₂×C₈	C₂×C₃×D₄	1	2	yes	—
151	C₂×C₈	C₂×C₃×Q₈	1	2	yes	—
152	C₂×C₈	C₂×C₃×Q₈	1	2	yes	—
153	C₂×C₈	C₂⁴×C₃	1	1	yes	characteristic
154	C₄⋊C₄	—	2	2	no	—
155	C₄⋊C₄	—	2	4	no	—
156	C₄⋊C₄	—	2	2	no	—
157	C₄⋊C₄	—	4	2	no	—
158	C₄⋊C₄	—	4	4	no	—
159	C₄⋊C₄	—	4	4	no	—
160	C₄⋊C₄	—	4	4	no	—
161	C₄⋊C₄	C₃×D₈	1	1	yes	characteristic
162	C₄⋊C₄	C₃×D₈	1	1	yes	characteristic
163	C₂²⋊C₄	—	4	2	no	—
164	C₂²⋊C₄	C₃×D₈	1	2	yes	—
165	C₄²	—	4	2	no	—
166	C₄²	C₂×C₃×D₄	1	2	yes	—
167	C₁₆	C₂×C₃×Q₈	1	2	yes	—
168	C₁₆	C₂×C₃×Q₈	1	2	yes	—
169	C₁₆	C₂×C₃×Q₈	1	4	yes	—
170	C₂²×C₃	—	2	1	no	—
171	C₂²×C₃	—	2	1	no	—
172	C₂²×C₃	#500	1	2	yes	—
173	C₂²×C₃	C₂²×D₈	1	1	yes	characteristic
174	C₃×C₄	—	2	2	no	—
175	C₃×C₄	—	2	2	no	—
176	C₃×C₄	—	8	4	no	—
177	C₃×C₄	—	8	2	no	—
178	C₃×C₄	—	8	2	no	—
179	C₃×C₄	#500	1	2	yes	—
180	C₃×C₄	C₂×C₄⋊Q₈	1	1	yes	characteristic
181	C₃×C₄	C₂²×Q₁₆	1	1	yes	characteristic
182	C₂³	C₂×C₃×D₈	1	1	yes	characteristic
183	Q₈	—	16	4	no	—
184	C₂×C₄	—	4	2	no	—
185	C₂×C₄	—	4	2	no	—
186	C₂×C₄	—	4	4	no	—
187	C₂×C₄	—	16	2	no	—
188	C₂×C₄	C₃×C₄⋊Q₈	1	2	yes	—
189	C₂×C₄	C₂×C₃×D₈	1	2	yes	—
190	C₂×C₄	C₂×C₃×D₈	1	2	yes	—
191	C₂×C₄	C₂×C₃×D₈	1	1	yes	characteristic
192	C₂×C₄	C₂×C₃×Q₁₆	1	2	yes	—
193	C₂×C₄	C₂²×C₃×D₄	1	1	yes	characteristic
194	C₈	C₃×C₄⋊Q₈	1	2	yes	—
195	C₈	C₂×C₃×Q₁₆	1	4	yes	—
196	C₈	C₂²×C₃×Q₈	1	1	yes	characteristic
197	C₈	C₂²×C₃×Q₈	1	1	yes	characteristic
198	C₂×C₃	—	2	2	no	—
199	C₂×C₃	C₂×#500	1	1	yes	characteristic
200	C₂×C₃	#3415	1	1	yes	characteristic
201	C₂×C₃	#3417	1	1	yes	characteristic
202	C₂²	—	2	1	no	—
203	C₂²	—	2	1	no	—
204	C₂²	C₃×#500	1	2	yes	—
205	C₂²	C₂²×C₃×D₈	1	1	yes	characteristic
206	C₄	—	2	2	no	—
207	C₄	—	2	2	no	—
208	C₄	—	8	4	no	—
209	C₄	—	8	2	no	—
210	C₄	—	8	2	no	—
211	C₄	C₃×#500	1	2	yes	—
212	C₄	C₂×C₃×C₄⋊Q₈	1	1	yes	characteristic
213	C₄	C₂²×C₃×Q₁₆	1	1	yes	characteristic
214	C₃	#33729	1	1	yes	characteristic, direct product factor, Sylow
215	C₂	—	2	2	no	—
216	C₂	C₂×C₃×#500	1	1	yes	characteristic
217	C₂	C₃×#3415	1	1	yes	characteristic
218	C₂	C₃×#3417	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₃
prime radical	C₂³×C₃	C₂×D₈
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
2. C₃×#2255	C₂	semidirect
3. C₃×#2253	C₂	semidirect
4. C₃×#2252	C₂	semidirect
9. #33729	C₃	direct (N × H)
30. #2255	C₂×C₃	semidirect
31. #2253	C₂×C₃	semidirect
32. #2252	C₂×C₃	semidirect
47. C₃×M₃₂	Q₈	semidirect
48. C₂×C₃×C₁₆	Q₈	semidirect
115. M₃₂	C₃×Q₈	semidirect
116. C₂×C₁₆	C₃×Q₈	semidirect
214. C₃	#33729	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×#3164	2
C₃×#2255	2
C₃×#2253	2
C₃×#2252	2
C₃×#2159	2
C₃×#2158	2
C₂×C₃×#366	2
C₃×#2108	2
#33729	3

Subgroups — C3×#33729 #10652336

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#33729 #10652336