Subgroups — C₃×#33741 #10652348

Order 768.

484 conjugacy classes of subgroups.

The subgroup lattice has 230 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#3177	C₂	1	2	yes	maximal
2	C₃×#2253	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×#2247	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×#2245	C₂	1	2	yes	maximal, complemented
5	C₃×#2219	C₂	1	2	yes	maximal
6	C₃×#2216	C₂	1	2	yes	maximal
7	C₃×#2176	C₂	1	2	yes	maximal, complemented
8	C₃×C₄×Q₃₂	C₂	1	2	yes	maximal, complemented
9	C₃×#2108	C₂	1	1	yes	maximal, characteristic
10	#33741	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
11	C₃×#541	—	2	2	no	—
12	C₂×C₃×Q₃₂	C₂²	1	2	yes	—
13	C₂×C₃×Q₃₂	C₂²	1	2	yes	—
14	C₂×C₃×M₃₂	C₂²	1	1	yes	characteristic
15	C₂²×C₃×C₁₆	C₂²	1	1	yes	characteristic
16	C₃×#499	C₂²	1	2	yes	—
17	C₃×#494	C₂²	1	2	yes	—
18	C₃×#491	—	2	2	no	—
19	C₃×#470	—	2	2	no	—
20	C₃×#467	C₂²	1	4	yes	—
21	C₃×#441	—	2	2	no	—
22	C₃×C₄×Q₁₆	C₂²	1	4	yes	—
23	C₃×#405	C₂²	1	1	yes	characteristic
24	C₃×#367	C₂²	1	2	yes	—
25	C₃×#366	C₂²	1	2	yes	—
26	C₃×#358	C₂²	1	4	yes	—
27	C₃×#358	C₂²	1	4	yes	—
28	C₃×#346	C₂²	1	1	yes	characteristic
29	C₃×#346	C₂²	1	1	yes	characteristic
30	C₃×C₄×C₁₆	C₂²	1	1	yes	characteristic
31	C₃×C₄×C₁₆	C₂²	1	1	yes	characteristic
32	#3177	C₂×C₃	1	2	yes	—
33	#2253	C₂×C₃	1	1	yes	characteristic, complemented
34	#2247	C₂×C₃	1	1	yes	characteristic, complemented
35	#2245	C₂×C₃	1	2	yes	complemented
36	#2219	C₂×C₃	1	2	yes	—
37	#2216	C₂×C₃	1	2	yes	—
38	#2176	C₂×C₃	1	2	yes	complemented
39	C₄×Q₃₂	C₂×C₃	1	2	yes	complemented
40	#2108	C₂×C₃	1	1	yes	characteristic
41	C₂×C₃×Q₁₆	—	2	2	no	—
42	C₂×C₃×Q₁₆	C₂³	1	4	yes	—
43	C₂×C₃×M₁₆	D₄	1	1	yes	characteristic
44	C₂²×C₃×C₈	C₂³	1	1	yes	characteristic
45	C₃×C₄⋊Q₈	—	2	2	no	—
46	C₃×#126	—	4	4	no	—
47	C₃×#125	—	2	2	no	—
48	C₃×#125	—	4	4	no	—
49	C₃×C₂²⋊Q₈	—	2	4	no	—
50	C₃×C₄×Q₈	—	2	4	no	—
51	C₃×#117	D₄	1	1	yes	characteristic
52	C₃×Q₃₂	—	2	4	no	—
53	C₃×Q₃₂	—	2	4	no	—
54	C₃×M₃₂	—	2	2	no	—
55	C₂×C₃×C₁₆	—	2	2	no	—
56	C₂×C₃×C₁₆	C₂³	1	1	yes	characteristic
57	C₂×C₃×C₁₆	C₂³	1	1	yes	characteristic
58	C₂×C₃×C₁₆	C₂³	1	1	yes	characteristic
59	C₂×C₃×C₁₆	C₂³	1	1	yes	characteristic
60	C₃×#107	C₂³	1	4	yes	—
61	C₃×#106	—	2	2	no	—
62	C₃×#103	—	2	4	no	—
63	C₃×#103	—	2	4	no	—
64	C₃×#97	D₄	1	1	yes	characteristic
65	C₃×#97	D₄	1	1	yes	characteristic
66	C₃×C₄×C₈	C₂³	1	1	yes	characteristic
67	C₃×C₄×C₈	C₂³	1	1	yes	characteristic
68	#541	—	2	2	no	—
69	C₂×Q₃₂	C₂²×C₃	1	2	yes	—
70	C₂×Q₃₂	C₂²×C₃	1	2	yes	—
71	C₂×M₃₂	C₂²×C₃	1	1	yes	characteristic
72	C₂²×C₁₆	C₂²×C₃	1	1	yes	characteristic
73	#499	C₂²×C₃	1	2	yes	—
74	#494	C₂²×C₃	1	2	yes	—
75	#491	—	2	2	no	—
76	#470	—	2	2	no	—
77	#467	C₂²×C₃	1	4	yes	—
78	#441	—	2	2	no	—
79	C₄×Q₁₆	C₂²×C₃	1	4	yes	—
80	#405	C₂²×C₃	1	1	yes	characteristic
81	#367	C₂²×C₃	1	2	yes	—
82	#366	C₂²×C₃	1	2	yes	—
83	#358	C₂²×C₃	1	4	yes	—
84	#358	C₂²×C₃	1	4	yes	—
85	#346	C₂²×C₃	1	1	yes	characteristic
86	#346	C₂²×C₃	1	1	yes	characteristic
87	C₄×C₁₆	C₂²×C₃	1	1	yes	characteristic
88	C₄×C₁₆	C₂²×C₃	1	1	yes	characteristic
89	C₂×C₃×Q₈	—	2	4	no	—
90	C₂²×C₃×C₄	C₂×D₄	1	1	yes	characteristic
91	C₃×Q₁₆	—	2	4	no	—
92	C₃×Q₁₆	—	2	4	no	—
93	C₃×Q₁₆	—	4	4	no	—
94	C₃×M₁₆	—	2	2	no	—
95	C₂×C₃×C₈	—	2	1	no	—
96	C₂×C₃×C₈	—	2	1	no	—
97	C₂×C₃×C₈	C₂×D₄	1	1	yes	characteristic
98	C₂×C₃×C₈	C₂×D₄	1	1	yes	characteristic
99	C₂×C₃×C₈	C₂×D₄	1	1	yes	characteristic
100	C₂×C₃×C₈	C₂⁴	1	1	yes	characteristic
101	C₃×C₄⋊C₄	—	2	4	no	—
102	C₃×C₄⋊C₄	—	4	4	no	—
103	C₃×C₄⋊C₄	—	4	4	no	—
104	C₃×C₄⋊C₄	—	4	4	no	—
105	C₃×C₄⋊C₄	—	4	4	no	—
106	C₃×C₄⋊C₄	D₈	1	2	yes	—
107	C₃×C₂²⋊C₄	—	4	4	no	—
108	C₃×C₂²⋊C₄	D₈	1	2	yes	—
109	C₃×C₄²	—	4	4	no	—
110	C₃×C₄²	C₂×D₄	1	1	yes	characteristic
111	C₃×C₄²	C₂×D₄	1	1	yes	characteristic
112	C₃×C₁₆	—	2	1	no	—
113	C₃×C₁₆	—	2	1	no	—
114	C₃×C₁₆	#41	1	2	yes	—
115	C₃×C₁₆	#41	1	2	yes	—
116	C₂×Q₁₆	—	2	2	no	—
117	C₂×Q₁₆	C₂³×C₃	1	4	yes	—
118	C₂×M₁₆	C₃×D₄	1	1	yes	characteristic
119	C₂²×C₈	C₂³×C₃	1	1	yes	characteristic
120	C₄⋊Q₈	—	2	2	no	—
121	#126	—	4	4	no	—
122	#125	—	2	2	no	—
123	#125	—	4	4	no	—
124	C₂²⋊Q₈	—	2	4	no	—
125	C₄×Q₈	—	2	4	no	—
126	#117	C₃×D₄	1	1	yes	characteristic
127	Q₃₂	—	2	4	no	—
128	Q₃₂	—	2	4	no	—
129	M₃₂	—	2	2	no	—
130	C₂×C₁₆	—	2	2	no	—
131	C₂×C₁₆	C₂³×C₃	1	1	yes	characteristic
132	C₂×C₁₆	C₂³×C₃	1	1	yes	characteristic
133	C₂×C₁₆	C₂³×C₃	1	1	yes	characteristic
134	C₂×C₁₆	C₂³×C₃	1	1	yes	characteristic
135	#107	C₂³×C₃	1	4	yes	—
136	#106	—	2	2	no	—
137	#103	—	2	4	no	—
138	#103	—	2	4	no	—
139	#97	C₃×D₄	1	1	yes	characteristic
140	#97	C₃×D₄	1	1	yes	characteristic
141	C₄×C₈	C₂³×C₃	1	1	yes	characteristic
142	C₄×C₈	C₂³×C₃	1	1	yes	characteristic
143	C₂³×C₃	C₂×D₈	1	1	yes	characteristic
144	C₃×Q₈	—	4	4	no	—
145	C₃×Q₈	—	4	4	no	—
146	C₂×C₃×C₄	—	2	1	no	—
147	C₂×C₃×C₄	—	2	1	no	—
148	C₂×C₃×C₄	—	4	4	no	—
149	C₂×C₃×C₄	—	4	4	no	—
150	C₂×C₃×C₄	C₂×D₈	1	2	yes	—
151	C₂×C₃×C₄	C₂×D₈	1	2	yes	—
152	C₂×C₃×C₄	C₂×D₈	1	1	yes	characteristic
153	C₂×C₃×C₄	C₂²×D₄	1	1	yes	characteristic
154	C₃×C₈	—	2	1	no	—
155	C₃×C₈	—	2	1	no	—
156	C₃×C₈	#135	1	2	yes	—
157	C₃×C₈	C₂×#41	1	1	yes	characteristic
158	C₃×C₈	C₂×#41	1	1	yes	characteristic
159	C₂×Q₈	—	2	4	no	—
160	C₂²×C₄	C₂×C₃×D₄	1	1	yes	characteristic
161	Q₁₆	—	2	4	no	—
162	Q₁₆	—	2	4	no	—
163	Q₁₆	—	4	4	no	—
164	M₁₆	—	2	2	no	—
165	C₂×C₈	—	2	1	no	—
166	C₂×C₈	—	2	1	no	—
167	C₂×C₈	C₂×C₃×D₄	1	1	yes	characteristic
168	C₂×C₈	C₂×C₃×D₄	1	1	yes	characteristic
169	C₂×C₈	C₂×C₃×D₄	1	1	yes	characteristic
170	C₂×C₈	C₂⁴×C₃	1	1	yes	characteristic
171	C₄⋊C₄	—	2	4	no	—
172	C₄⋊C₄	—	4	4	no	—
173	C₄⋊C₄	—	4	4	no	—
174	C₄⋊C₄	—	4	4	no	—
175	C₄⋊C₄	—	4	4	no	—
176	C₄⋊C₄	C₃×D₈	1	2	yes	—
177	C₂²⋊C₄	—	4	4	no	—
178	C₂²⋊C₄	C₃×D₈	1	2	yes	—
179	C₄²	—	4	4	no	—
180	C₄²	C₂×C₃×D₄	1	1	yes	characteristic
181	C₄²	C₂×C₃×D₄	1	1	yes	characteristic
182	C₁₆	—	2	1	no	—
183	C₁₆	—	2	1	no	—
184	C₁₆	C₃×#41	1	2	yes	—
185	C₁₆	C₃×#41	1	2	yes	—
186	C₂²×C₃	—	2	2	no	—
187	C₂²×C₃	—	2	1	no	—
188	C₂²×C₃	C₂²×D₈	1	1	yes	characteristic
189	C₃×C₄	—	2	2	no	—
190	C₃×C₄	—	2	2	no	—
191	C₃×C₄	—	2	1	no	—
192	C₃×C₄	—	8	4	no	—
193	C₃×C₄	—	8	4	no	—
194	C₃×C₄	#532	1	1	yes	characteristic
195	C₃×C₄	C₂×#135	1	1	yes	characteristic
196	C₂³	C₂×C₃×D₈	1	1	yes	characteristic
197	Q₈	—	4	4	no	—
198	Q₈	—	4	4	no	—
199	C₂×C₄	—	2	1	no	—
200	C₂×C₄	—	2	1	no	—
201	C₂×C₄	—	4	4	no	—
202	C₂×C₄	—	4	4	no	—
203	C₂×C₄	C₂×C₃×D₈	1	2	yes	—
204	C₂×C₄	C₂×C₃×D₈	1	2	yes	—
205	C₂×C₄	C₂×C₃×D₈	1	1	yes	characteristic
206	C₂×C₄	C₂²×C₃×D₄	1	1	yes	characteristic
207	C₈	—	2	1	no	—
208	C₈	—	2	1	no	—
209	C₈	C₃×#135	1	2	yes	—
210	C₈	C₂×C₃×#41	1	1	yes	characteristic
211	C₈	C₂×C₃×#41	1	1	yes	characteristic
212	C₂×C₃	—	4	1	no	—
213	C₂×C₃	#3169	1	1	yes	characteristic
214	C₂×C₃	#3417	1	2	yes	—
215	C₂²	—	2	2	no	—
216	C₂²	—	2	1	no	—
217	C₂²	C₂²×C₃×D₈	1	1	yes	characteristic
218	C₄	—	2	2	no	—
219	C₄	—	2	2	no	—
220	C₄	—	2	1	no	—
221	C₄	—	8	4	no	—
222	C₄	—	8	4	no	—
223	C₄	C₃×#532	1	1	yes	characteristic
224	C₄	C₂×C₃×#135	1	1	yes	characteristic
225	C₃	#33741	1	1	yes	characteristic, direct product factor, Sylow
226	C₂	—	4	1	no	—
227	C₂	C₃×#3169	1	1	yes	characteristic
228	C₂	C₃×#3417	1	2	yes	—

	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×#2253	C₂	semidirect
3. C₃×#2247	C₂	semidirect
4. C₃×#2245	C₂	semidirect
7. C₃×#2176	C₂	semidirect
8. C₃×C₄×Q₃₂	C₂	semidirect
10. #33741	C₃	direct (N × H)
33. #2253	C₂×C₃	semidirect
34. #2247	C₂×C₃	semidirect
35. #2245	C₂×C₃	semidirect
38. #2176	C₂×C₃	semidirect
39. C₄×Q₃₂	C₂×C₃	semidirect
225. C₃	#33741	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×#3177	2
C₃×#2253	2
C₃×#2247	2
C₃×#2245	2
C₃×#2219	2
C₃×#2216	2
C₃×#2176	2
C₃×C₄×Q₃₂	2
C₃×#2108	2
#33741	3

Subgroups — C3×#33741 #10652348

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#33741 #10652348