Subgroups — C₃×#33748 #10652355

Order 768.

592 conjugacy classes of subgroups.

The subgroup lattice has 226 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#3169	C₂	1	2	yes	maximal, complemented
2	C₃×#2254	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×#2246	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×#2240	C₂	1	2	yes	maximal, complemented
5	C₃×#2218	C₂	1	4	yes	maximal, complemented
6	C₃×#2177	C₂	1	4	yes	maximal, complemented
7	C₃×C₄×M₃₂	C₂	1	1	yes	maximal, characteristic, complemented
8	#33748	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
9	C₃×#532	—	2	2	no	—
10	C₂×C₃×D₁₆	C₂²	1	4	yes	—
11	C₂×C₃×M₃₂	C₂²	1	2	yes	—
12	C₃×#500	C₂²	1	2	yes	—
13	C₃×#493	C₂²	1	2	yes	—
14	C₃×#486	—	2	2	no	—
15	C₃×#466	—	2	2	no	—
16	C₃×#466	C₂²	1	4	yes	—
17	C₃×C₄×D₈	—	2	2	no	—
18	C₃×C₄×D₈	C₂²	1	4	yes	—
19	C₂×C₃×C₄×C₈	C₂²	1	1	yes	characteristic
20	C₃×#367	C₂²	1	4	yes	—
21	C₃×#357	C₂²	1	8	yes	—
22	C₃×#346	C₂²	1	2	yes	—
23	C₃×C₄×C₁₆	C₂²	1	1	yes	characteristic
24	C₃×C₄×C₁₆	C₂²	1	1	yes	characteristic
25	#3169	C₂×C₃	1	2	yes	complemented
26	#2254	C₂×C₃	1	1	yes	characteristic, complemented
27	#2246	C₂×C₃	1	1	yes	characteristic, complemented
28	#2240	C₂×C₃	1	2	yes	complemented
29	#2218	C₂×C₃	1	4	yes	complemented
30	#2177	C₂×C₃	1	4	yes	complemented
31	C₄×M₃₂	C₂×C₃	1	1	yes	characteristic, complemented
32	C₂×C₃×#41	—	4	2	no	—
33	C₂×C₃×D₈	—	2	2	no	—
34	C₂×C₃×D₈	C₂³	1	4	yes	—
35	C₂²×C₃×C₈	D₄	1	1	yes	characteristic
36	C₂²×C₃×C₈	C₂³	1	1	yes	characteristic
37	C₃×C₄⋊Q₈	—	2	2	no	—
38	C₃×#127	—	2	2	no	—
39	C₃×#124	—	4	2	no	—
40	C₃×#121	—	2	4	no	—
41	C₃×#121	—	4	2	no	—
42	C₃×C₄×D₄	—	2	4	no	—
43	C₃×C₄×D₄	—	4	2	no	—
44	C₂×C₃×C₄²	D₄	1	1	yes	characteristic
45	C₃×D₁₆	—	2	8	no	—
46	C₃×M₃₂	—	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×#102	—	2	4	no	—
52	C₃×#102	—	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	#532	—	2	2	no	—
58	C₂×D₁₆	C₂²×C₃	1	4	yes	—
59	C₂×M₃₂	C₂²×C₃	1	2	yes	—
60	#500	C₂²×C₃	1	2	yes	—
61	#493	C₂²×C₃	1	2	yes	—
62	#486	—	2	2	no	—
63	#466	—	2	2	no	—
64	#466	C₂²×C₃	1	4	yes	—
65	C₄×D₈	—	2	2	no	—
66	C₄×D₈	C₂²×C₃	1	4	yes	—
67	C₂×C₄×C₈	C₂²×C₃	1	1	yes	characteristic
68	#367	C₂²×C₃	1	4	yes	—
69	#357	C₂²×C₃	1	8	yes	—
70	#346	C₂²×C₃	1	2	yes	—
71	C₄×C₁₆	C₂²×C₃	1	1	yes	characteristic
72	C₄×C₁₆	C₂²×C₃	1	1	yes	characteristic
73	C₃×#41	—	8	8	no	—
74	C₂×C₃×Q₈	—	4	2	no	—
75	C₂×C₃×D₄	—	2	4	no	—
76	C₂×C₃×D₄	—	4	4	no	—
77	C₂×C₃×D₄	—	4	2	no	—
78	C₂×C₃×D₄	—	4	2	no	—
79	C₂²×C₃×C₄	—	4	4	no	—
80	C₂²×C₃×C₄	D₈	1	2	yes	—
81	C₂²×C₃×C₄	C₂×D₄	1	1	yes	characteristic
82	C₃×D₈	—	2	8	no	—
83	C₃×D₈	—	4	4	no	—
84	C₂×C₃×C₈	—	2	2	no	—
85	C₂×C₃×C₈	—	2	2	no	—
86	C₂×C₃×C₈	C₂×D₄	1	1	yes	characteristic
87	C₂×C₃×C₈	C₂×D₄	1	1	yes	characteristic
88	C₂×C₃×C₈	C₂×D₄	1	1	yes	characteristic
89	C₂×C₃×C₈	C₂⁴	1	1	yes	characteristic
90	C₃×C₄⋊C₄	—	2	4	no	—
91	C₃×C₄⋊C₄	—	4	2	no	—
92	C₃×C₂²⋊C₄	—	4	4	no	—
93	C₃×C₂²⋊C₄	—	4	4	no	—
94	C₃×C₄²	D₈	1	2	yes	—
95	C₃×C₄²	C₂×D₄	1	1	yes	characteristic
96	C₃×C₄²	C₂×D₄	1	1	yes	characteristic
97	C₃×C₁₆	—	2	2	no	—
98	C₃×C₁₆	#41	1	4	yes	complemented
99	C₂×#41	—	4	2	no	—
100	C₂×D₈	—	2	2	no	—
101	C₂×D₈	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	#127	—	2	2	no	—
106	#124	—	4	2	no	—
107	#121	—	2	4	no	—
108	#121	—	4	2	no	—
109	C₄×D₄	—	2	4	no	—
110	C₄×D₄	—	4	2	no	—
111	C₂×C₄²	C₃×D₄	1	1	yes	characteristic
112	D₁₆	—	2	8	no	—
113	M₃₂	—	2	4	no	—
114	C₂×C₁₆	C₂³×C₃	1	2	yes	—
115	C₂×C₁₆	C₂³×C₃	1	2	yes	—
116	#107	—	2	2	no	—
117	#107	C₂³×C₃	1	4	yes	—
118	#102	—	2	4	no	—
119	#102	—	2	4	no	—
120	C₄×C₈	C₃×D₄	1	1	yes	characteristic
121	C₄×C₈	C₃×D₄	1	1	yes	characteristic
122	C₄×C₈	C₂³×C₃	1	1	yes	characteristic
123	C₄×C₈	C₂³×C₃	1	1	yes	characteristic
124	C₂³×C₃	—	4	4	no	—
125	C₂³×C₃	C₂×D₈	1	1	yes	characteristic
126	C₃×Q₈	—	8	4	no	—
127	C₃×D₄	—	4	4	no	—
128	C₃×D₄	—	4	4	no	—
129	C₃×D₄	—	8	8	no	—
130	C₃×D₄	—	8	4	no	—
131	C₃×D₄	—	8	4	no	—
132	C₂×C₃×C₄	—	2	1	no	—
133	C₂×C₃×C₄	—	2	1	no	—
134	C₂×C₃×C₄	—	2	4	no	—
135	C₂×C₃×C₄	—	4	4	no	—
136	C₂×C₃×C₄	—	8	8	no	—
137	C₂×C₃×C₄	C₂×D₈	1	1	yes	characteristic
138	C₂×C₃×C₄	C₂×D₈	1	2	yes	—
139	C₂×C₃×C₄	C₂×D₈	1	2	yes	—
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	#41	—	8	8	no	—
146	C₂×Q₈	—	4	2	no	—
147	C₂×D₄	—	2	4	no	—
148	C₂×D₄	—	4	4	no	—
149	C₂×D₄	—	4	2	no	—
150	C₂×D₄	—	4	2	no	—
151	C₂²×C₄	—	4	4	no	—
152	C₂²×C₄	C₃×D₈	1	2	yes	—
153	C₂²×C₄	C₂×C₃×D₄	1	1	yes	characteristic
154	D₈	—	2	8	no	—
155	D₈	—	4	4	no	—
156	C₂×C₈	—	2	2	no	—
157	C₂×C₈	—	2	2	no	—
158	C₂×C₈	C₂×C₃×D₄	1	1	yes	characteristic
159	C₂×C₈	C₂×C₃×D₄	1	1	yes	characteristic
160	C₂×C₈	C₂×C₃×D₄	1	1	yes	characteristic
161	C₂×C₈	C₂⁴×C₃	1	1	yes	characteristic
162	C₄⋊C₄	—	2	4	no	—
163	C₄⋊C₄	—	4	2	no	—
164	C₂²⋊C₄	—	4	4	no	—
165	C₂²⋊C₄	—	4	4	no	—
166	C₄²	C₃×D₈	1	2	yes	—
167	C₄²	C₂×C₃×D₄	1	1	yes	characteristic
168	C₄²	C₂×C₃×D₄	1	1	yes	characteristic
169	C₁₆	—	2	2	no	—
170	C₁₆	C₃×#41	1	4	yes	complemented
171	C₂²×C₃	—	2	2	no	—
172	C₂²×C₃	—	2	1	no	—
173	C₂²×C₃	—	8	4	no	—
174	C₂²×C₃	—	8	4	no	—
175	C₂²×C₃	—	8	4	no	—
176	C₂²×C₃	C₂²×D₈	1	1	yes	characteristic
177	C₃×C₄	—	2	2	no	—
178	C₃×C₄	—	2	1	no	—
179	C₃×C₄	—	8	4	no	—
180	C₃×C₄	#509	1	4	yes	—
181	C₃×C₄	#532	1	1	yes	characteristic
182	C₃×C₄	C₂×#135	1	1	yes	characteristic
183	C₂³	—	4	4	no	—
184	C₂³	C₂×C₃×D₈	1	1	yes	characteristic
185	Q₈	—	8	4	no	—
186	D₄	—	4	4	no	—
187	D₄	—	4	4	no	—
188	D₄	—	8	8	no	—
189	D₄	—	8	4	no	—
190	D₄	—	8	4	no	—
191	C₂×C₄	—	2	4	no	—
192	C₂×C₄	—	2	1	no	—
193	C₂×C₄	—	2	1	no	—
194	C₂×C₄	—	4	4	no	—
195	C₂×C₄	—	8	8	no	—
196	C₂×C₄	C₂×C₃×D₈	1	1	yes	characteristic
197	C₂×C₄	C₂×C₃×D₈	1	2	yes	—
198	C₂×C₄	C₂×C₃×D₈	1	2	yes	—
199	C₂×C₄	C₂²×C₃×D₄	1	1	yes	characteristic
200	C₈	—	2	1	no	—
201	C₈	—	2	1	no	—
202	C₈	C₃×#135	1	2	yes	—
203	C₈	C₂×C₃×#41	1	2	yes	—
204	C₂×C₃	—	4	1	no	—
205	C₂×C₃	—	16	4	no	—
206	C₂×C₃	#3169	1	1	yes	characteristic
207	C₂×C₃	C₂×#509	1	2	yes	—
208	C₂²	—	2	2	no	—
209	C₂²	—	2	1	no	—
210	C₂²	—	8	4	no	—
211	C₂²	—	8	4	no	—
212	C₂²	—	8	4	no	—
213	C₂²	C₂²×C₃×D₈	1	1	yes	characteristic
214	C₄	—	2	2	no	—
215	C₄	—	2	1	no	—
216	C₄	—	8	4	no	—
217	C₄	C₃×#509	1	4	yes	—
218	C₄	C₃×#532	1	1	yes	characteristic
219	C₄	C₂×C₃×#135	1	1	yes	characteristic
220	C₃	#33748	1	1	yes	characteristic, direct product factor, Sylow
221	C₂	—	4	1	no	—
222	C₂	—	16	4	no	—
223	C₂	C₃×#3169	1	1	yes	characteristic
224	C₂	C₂×C₃×#509	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×C₄² ◁ C₂×C₃×C₄×C₈ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₃×#3169	C₂	semidirect
2. C₃×#2254	C₂	semidirect
3. C₃×#2246	C₂	semidirect
4. C₃×#2240	C₂	semidirect
5. C₃×#2218	C₂	semidirect
6. C₃×#2177	C₂	semidirect
7. C₃×C₄×M₃₂	C₂	semidirect
8. #33748	C₃	direct (N × H)
25. #3169	C₂×C₃	semidirect
26. #2254	C₂×C₃	semidirect
27. #2246	C₂×C₃	semidirect
28. #2240	C₂×C₃	semidirect
29. #2218	C₂×C₃	semidirect
30. #2177	C₂×C₃	semidirect
31. C₄×M₃₂	C₂×C₃	semidirect
98. C₃×C₁₆	#41	semidirect
170. C₁₆	C₃×#41	semidirect
220. C₃	#33748	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×#3169	2
C₃×#2254	2
C₃×#2246	2
C₃×#2240	2
C₃×#2218	2
C₃×#2177	2
C₃×C₄×M₃₂	2
#33748	3

Subgroups — C3×#33748 #10652355

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#33748 #10652355