Subgroups — C₃×#33734 #10652341

Order 768.

514 conjugacy classes of subgroups.

The subgroup lattice has 228 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#3170	C₂	1	2	yes	maximal
2	C₃×#2252	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×#2247	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×#2241	C₂	1	2	yes	maximal, complemented
5	C₃×#2216	C₂	1	4	yes	maximal
6	C₃×C₄×Q₃₂	C₂	1	4	yes	maximal, complemented
7	C₂×C₃×C₄×C₁₆	C₂	1	1	yes	maximal, characteristic
8	#33734	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
9	C₃×#533	—	2	2	no	—
10	C₂×C₃×Q₃₂	C₂²	1	4	yes	—
11	C₂²×C₃×C₁₆	C₂²	1	2	yes	—
12	C₃×#500	C₂²	1	2	yes	—
13	C₃×#494	C₂²	1	2	yes	—
14	C₃×#487	—	2	2	no	—
15	C₃×#467	—	2	2	no	—
16	C₃×#467	C₂²	1	4	yes	—
17	C₃×C₄×Q₁₆	—	2	2	no	—
18	C₃×C₄×Q₁₆	C₂²	1	4	yes	—
19	C₂×C₃×C₄×C₈	C₂²	1	1	yes	characteristic
20	C₃×#366	C₂²	1	4	yes	—
21	C₃×#358	C₂²	1	8	yes	—
22	C₃×C₄×C₁₆	C₂²	1	1	yes	characteristic
23	C₃×C₄×C₁₆	C₂²	1	2	yes	—
24	C₃×C₄×C₁₆	C₂²	1	1	yes	characteristic
25	#3170	C₂×C₃	1	2	yes	—
26	#2252	C₂×C₃	1	1	yes	characteristic, complemented
27	#2247	C₂×C₃	1	1	yes	characteristic, complemented
28	#2241	C₂×C₃	1	2	yes	complemented
29	#2216	C₂×C₃	1	4	yes	—
30	C₄×Q₃₂	C₂×C₃	1	4	yes	complemented
31	C₂×C₄×C₁₆	C₂×C₃	1	1	yes	characteristic
32	C₂×C₃×Q₁₆	—	2	2	no	—
33	C₂×C₃×Q₁₆	C₂³	1	4	yes	—
34	C₂²×C₃×C₈	D₄	1	1	yes	characteristic
35	C₂²×C₃×C₈	C₂³	1	1	yes	characteristic
36	C₃×C₄⋊Q₈	—	2	2	no	—
37	C₃×C₄⋊Q₈	—	2	2	no	—
38	C₃×#125	—	4	2	no	—
39	C₃×C₂²⋊Q₈	—	2	4	no	—
40	C₃×C₂²⋊Q₈	—	4	2	no	—
41	C₃×C₄×Q₈	—	2	4	no	—
42	C₃×C₄×Q₈	—	4	2	no	—
43	C₃×#117	—	4	2	no	—
44	C₂×C₃×C₄²	D₄	1	1	yes	characteristic
45	C₃×Q₃₂	—	2	8	no	—
46	C₂×C₃×C₁₆	—	2	4	no	—
47	C₂×C₃×C₁₆	C₂³	1	2	yes	—
48	C₂×C₃×C₁₆	C₂³	1	2	yes	—
49	C₃×#107	—	2	2	no	—
50	C₃×#107	C₂³	1	4	yes	—
51	C₃×#103	—	2	4	no	—
52	C₃×#103	—	2	4	no	—
53	C₃×C₄×C₈	D₄	1	1	yes	characteristic
54	C₃×C₄×C₈	D₄	1	1	yes	characteristic
55	C₃×C₄×C₈	C₂³	1	1	yes	characteristic
56	C₃×C₄×C₈	C₂³	1	1	yes	characteristic
57	#533	—	2	2	no	—
58	C₂×Q₃₂	C₂²×C₃	1	4	yes	—
59	C₂²×C₁₆	C₂²×C₃	1	2	yes	—
60	#500	C₂²×C₃	1	2	yes	—
61	#494	C₂²×C₃	1	2	yes	—
62	#487	—	2	2	no	—
63	#467	—	2	2	no	—
64	#467	C₂²×C₃	1	4	yes	—
65	C₄×Q₁₆	—	2	2	no	—
66	C₄×Q₁₆	C₂²×C₃	1	4	yes	—
67	C₂×C₄×C₈	C₂²×C₃	1	1	yes	characteristic
68	#366	C₂²×C₃	1	4	yes	—
69	#358	C₂²×C₃	1	8	yes	—
70	C₄×C₁₆	C₂²×C₃	1	2	yes	—
71	C₄×C₁₆	C₂²×C₃	1	1	yes	characteristic
72	C₄×C₁₆	C₂²×C₃	1	1	yes	characteristic
73	C₂×C₃×Q₈	—	2	4	no	—
74	C₂×C₃×Q₈	—	4	2	no	—
75	C₂²×C₃×C₄	D₈	1	1	yes	characteristic
76	C₂²×C₃×C₄	D₈	1	1	yes	characteristic
77	C₂²×C₃×C₄	C₂×D₄	1	1	yes	characteristic
78	C₃×Q₁₆	—	2	8	no	—
79	C₃×Q₁₆	—	4	4	no	—
80	C₂×C₃×C₈	—	2	2	no	—
81	C₂×C₃×C₈	—	2	2	no	—
82	C₂×C₃×C₈	C₂×D₄	1	1	yes	characteristic
83	C₂×C₃×C₈	C₂×D₄	1	1	yes	characteristic
84	C₂×C₃×C₈	C₂×D₄	1	1	yes	characteristic
85	C₂×C₃×C₈	C₂⁴	1	1	yes	characteristic
86	C₃×C₄⋊C₄	—	2	4	no	—
87	C₃×C₄⋊C₄	—	4	2	no	—
88	C₃×C₄⋊C₄	—	4	2	no	—
89	C₃×C₄⋊C₄	—	4	4	no	—
90	C₃×C₄⋊C₄	—	4	4	no	—
91	C₃×C₄⋊C₄	—	4	2	no	—
92	C₃×C₂²⋊C₄	—	4	4	no	—
93	C₃×C₄²	—	4	4	no	—
94	C₃×C₄²	D₈	1	1	yes	characteristic
95	C₃×C₄²	D₈	1	1	yes	characteristic
96	C₃×C₄²	C₂×D₄	1	1	yes	characteristic
97	C₃×C₄²	C₂×D₄	1	1	yes	characteristic
98	C₃×C₁₆	—	2	2	no	—
99	C₃×C₁₆	#41	1	4	yes	—
100	C₂×Q₁₆	—	2	2	no	—
101	C₂×Q₁₆	C₂³×C₃	1	4	yes	—
102	C₂²×C₈	C₃×D₄	1	1	yes	characteristic
103	C₂²×C₈	C₂³×C₃	1	1	yes	characteristic
104	C₄⋊Q₈	—	2	2	no	—
105	C₄⋊Q₈	—	2	2	no	—
106	#125	—	4	2	no	—
107	C₂²⋊Q₈	—	2	4	no	—
108	C₂²⋊Q₈	—	4	2	no	—
109	C₄×Q₈	—	2	4	no	—
110	C₄×Q₈	—	4	2	no	—
111	#117	—	4	2	no	—
112	C₂×C₄²	C₃×D₄	1	1	yes	characteristic
113	Q₃₂	—	2	8	no	—
114	C₂×C₁₆	—	2	4	no	—
115	C₂×C₁₆	C₂³×C₃	1	2	yes	—
116	C₂×C₁₆	C₂³×C₃	1	2	yes	—
117	#107	—	2	2	no	—
118	#107	C₂³×C₃	1	4	yes	—
119	#103	—	2	4	no	—
120	#103	—	2	4	no	—
121	C₄×C₈	C₃×D₄	1	1	yes	characteristic
122	C₄×C₈	C₃×D₄	1	1	yes	characteristic
123	C₄×C₈	C₂³×C₃	1	1	yes	characteristic
124	C₄×C₈	C₂³×C₃	1	1	yes	characteristic
125	C₂³×C₃	C₂×D₈	1	1	yes	characteristic
126	C₃×Q₈	—	4	4	no	—
127	C₃×Q₈	—	4	4	no	—
128	C₃×Q₈	—	8	4	no	—
129	C₂×C₃×C₄	—	2	1	no	—
130	C₂×C₃×C₄	—	2	1	no	—
131	C₂×C₃×C₄	—	2	2	no	—
132	C₂×C₃×C₄	—	4	4	no	—
133	C₂×C₃×C₄	—	4	4	no	—
134	C₂×C₃×C₄	Q₃₂	1	4	yes	complemented
135	C₂×C₃×C₄	C₂×D₈	1	1	yes	characteristic
136	C₂×C₃×C₄	C₂×D₈	1	1	yes	characteristic
137	C₂×C₃×C₄	C₂×D₈	1	1	yes	characteristic
138	C₂×C₃×C₄	C₂×D₈	1	1	yes	characteristic
139	C₂×C₃×C₄	C₂×D₈	1	1	yes	characteristic
140	C₂×C₃×C₄	C₂²×D₄	1	1	yes	characteristic
141	C₃×C₈	—	2	1	no	—
142	C₃×C₈	—	2	1	no	—
143	C₃×C₈	#135	1	2	yes	—
144	C₃×C₈	C₂×#41	1	2	yes	—
145	C₂×Q₈	—	2	4	no	—
146	C₂×Q₈	—	4	2	no	—
147	C₂²×C₄	C₃×D₈	1	1	yes	characteristic
148	C₂²×C₄	C₃×D₈	1	1	yes	characteristic
149	C₂²×C₄	C₂×C₃×D₄	1	1	yes	characteristic
150	Q₁₆	—	2	8	no	—
151	Q₁₆	—	4	4	no	—
152	C₂×C₈	—	2	2	no	—
153	C₂×C₈	—	2	2	no	—
154	C₂×C₈	C₂×C₃×D₄	1	1	yes	characteristic
155	C₂×C₈	C₂×C₃×D₄	1	1	yes	characteristic
156	C₂×C₈	C₂×C₃×D₄	1	1	yes	characteristic
157	C₂×C₈	C₂⁴×C₃	1	1	yes	characteristic
158	C₄⋊C₄	—	2	4	no	—
159	C₄⋊C₄	—	4	2	no	—
160	C₄⋊C₄	—	4	2	no	—
161	C₄⋊C₄	—	4	4	no	—
162	C₄⋊C₄	—	4	4	no	—
163	C₄⋊C₄	—	4	2	no	—
164	C₂²⋊C₄	—	4	4	no	—
165	C₄²	—	4	4	no	—
166	C₄²	C₃×D₈	1	1	yes	characteristic
167	C₄²	C₃×D₈	1	1	yes	characteristic
168	C₄²	C₂×C₃×D₄	1	1	yes	characteristic
169	C₄²	C₂×C₃×D₄	1	1	yes	characteristic
170	C₁₆	—	2	2	no	—
171	C₁₆	C₃×#41	1	4	yes	—
172	C₂²×C₃	—	2	1	no	—
173	C₂²×C₃	—	2	1	no	—
174	C₂²×C₃	C₂×Q₃₂	1	2	yes	—
175	C₂²×C₃	C₂²×D₈	1	1	yes	characteristic
176	C₃×C₄	—	2	1	no	—
177	C₃×C₄	—	2	1	no	—
178	C₃×C₄	—	8	4	no	—
179	C₃×C₄	—	8	4	no	—
180	C₃×C₄	C₂×Q₃₂	1	2	yes	—
181	C₃×C₄	C₂×Q₃₂	1	2	yes	—
182	C₃×C₄	#508	1	2	yes	—
183	C₃×C₄	#532	1	1	yes	characteristic
184	C₃×C₄	C₂×#135	1	1	yes	characteristic
185	C₂³	C₂×C₃×D₈	1	1	yes	characteristic
186	Q₈	—	4	4	no	—
187	Q₈	—	4	4	no	—
188	Q₈	—	8	4	no	—
189	C₂×C₄	—	2	2	no	—
190	C₂×C₄	—	2	1	no	—
191	C₂×C₄	—	2	1	no	—
192	C₂×C₄	—	4	4	no	—
193	C₂×C₄	—	4	4	no	—
194	C₂×C₄	C₃×Q₃₂	1	4	yes	complemented
195	C₂×C₄	C₂×C₃×D₈	1	1	yes	characteristic
196	C₂×C₄	C₂×C₃×D₈	1	1	yes	characteristic
197	C₂×C₄	C₂×C₃×D₈	1	1	yes	characteristic
198	C₂×C₄	C₂×C₃×D₈	1	1	yes	characteristic
199	C₂×C₄	C₂×C₃×D₈	1	1	yes	characteristic
200	C₂×C₄	C₂²×C₃×D₄	1	1	yes	characteristic
201	C₈	—	2	1	no	—
202	C₈	—	2	1	no	—
203	C₈	C₃×#135	1	2	yes	—
204	C₈	C₂×C₃×#41	1	2	yes	—
205	C₂×C₃	—	2	2	no	—
206	C₂×C₃	#3169	1	1	yes	characteristic
207	C₂×C₃	C₂²×Q₃₂	1	1	yes	characteristic
208	C₂×C₃	C₂×#508	1	1	yes	characteristic
209	C₂²	—	2	1	no	—
210	C₂²	—	2	1	no	—
211	C₂²	C₂×C₃×Q₃₂	1	2	yes	—
212	C₂²	C₂²×C₃×D₈	1	1	yes	characteristic
213	C₄	—	2	1	no	—
214	C₄	—	2	1	no	—
215	C₄	—	8	4	no	—
216	C₄	—	8	4	no	—
217	C₄	C₂×C₃×Q₃₂	1	2	yes	—
218	C₄	C₂×C₃×Q₃₂	1	2	yes	—
219	C₄	C₃×#508	1	2	yes	—
220	C₄	C₃×#532	1	1	yes	characteristic
221	C₄	C₂×C₃×#135	1	1	yes	characteristic
222	C₃	#33734	1	1	yes	characteristic, direct product factor, Sylow
223	C₂	—	2	2	no	—
224	C₂	C₃×#3169	1	1	yes	characteristic
225	C₂	C₂²×C₃×Q₃₂	1	1	yes	characteristic
226	C₂	C₂×C₃×#508	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₃×C₄	C₂×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₃
socle	C₂²×C₃	C₂²×D₈
prime radical	C₂³×C₃	C₂×D₈
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₃×C₄ ◁ C₂×C₃×C₄² ◁ C₂×C₃×C₄×C₈ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₃×#2252	C₂	semidirect
3. C₃×#2247	C₂	semidirect
4. C₃×#2241	C₂	semidirect
6. C₃×C₄×Q₃₂	C₂	semidirect
8. #33734	C₃	direct (N × H)
26. #2252	C₂×C₃	semidirect
27. #2247	C₂×C₃	semidirect
28. #2241	C₂×C₃	semidirect
30. C₄×Q₃₂	C₂×C₃	semidirect
134. C₂×C₃×C₄	Q₃₂	semidirect
194. C₂×C₄	C₃×Q₃₂	semidirect
222. C₃	#33734	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×#3170	2
C₃×#2252	2
C₃×#2247	2
C₃×#2241	2
C₃×#2216	2
C₃×C₄×Q₃₂	2
C₂×C₃×C₄×C₁₆	2
#33734	3

Subgroups — C3×#33734 #10652341

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#33734 #10652341