Subgroups — C₃×#33083 #10651690

Order 768.

732 conjugacy classes of subgroups.

The subgroup lattice has 224 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#2863	C₂	1	1	yes	maximal, characteristic
2	C₃×C₄²×Q₈	C₂	1	1	yes	maximal, characteristic
3	C₃×#1928	C₂	1	3	yes	maximal
4	C₃×#1921	C₂	1	3	yes	maximal
5	C₃×#1878	C₂	1	3	yes	maximal
6	C₃×#1863	C₂	1	3	yes	maximal
7	C₃×#1839	C₂	1	1	yes	maximal, characteristic
8	#33083	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
9	C₂²×C₃×Q₁₆	C₂²	1	1	yes	characteristic
10	C₂×C₃×C₄×Q₈	C₂²	1	3	yes	—
11	C₂×C₃×#107	C₂²	1	3	yes	—
12	C₂×C₃×#103	C₂²	1	3	yes	—
13	C₂×C₃×#103	C₂²	1	3	yes	—
14	C₃×#391	C₂²	1	3	yes	—
15	C₃×#389	C₂²	1	3	yes	—
16	C₃×#387	—	2	3	no	—
17	C₃×#383	—	2	1	no	—
18	C₃×C₄×C₄⋊C₄	—	2	3	no	—
19	C₃×C₄×C₄⋊C₄	C₂²	1	3	yes	—
20	C₃×C₄³	—	2	1	no	—
21	C₃×C₄³	C₂²	1	1	yes	characteristic
22	C₃×#340	C₂²	1	3	yes	—
23	C₃×#340	C₂²	1	3	yes	—
24	C₃×#336	C₂²	1	3	yes	—
25	C₃×#336	C₂²	1	3	yes	—
26	#2863	C₂×C₃	1	1	yes	characteristic
27	C₄²×Q₈	C₂×C₃	1	1	yes	characteristic
28	#1928	C₂×C₃	1	3	yes	—
29	#1921	C₂×C₃	1	3	yes	—
30	#1878	C₂×C₃	1	3	yes	—
31	#1863	C₂×C₃	1	3	yes	—
32	#1839	C₂×C₃	1	1	yes	characteristic
33	C₂²×C₃×Q₈	C₂³	1	1	yes	characteristic
34	C₂²×C₃×Q₈	C₂³	1	1	yes	characteristic
35	C₂×C₃×Q₁₆	—	2	6	no	—
36	C₂²×C₃×C₈	C₂³	1	1	yes	characteristic
37	C₂²×C₃×C₈	C₂³	1	3	yes	—
38	C₃×C₄×Q₈	—	2	6	no	—
39	C₃×C₄×Q₈	—	2	6	no	—
40	C₂×C₃×C₄⋊C₄	—	2	3	no	—
41	C₂×C₃×C₄⋊C₄	—	2	3	no	—
42	C₂×C₃×C₄⋊C₄	C₂³	1	3	yes	—
43	C₂×C₃×C₄⋊C₄	C₂³	1	3	yes	—
44	C₂×C₃×C₄²	—	2	3	no	—
45	C₂×C₃×C₄²	—	2	3	no	—
46	C₂×C₃×C₄²	D₄	1	3	yes	—
47	C₂×C₃×C₄²	D₄	1	1	yes	characteristic
48	C₂×C₃×C₄²	C₂³	1	3	yes	—
49	C₃×#107	—	2	6	no	—
50	C₃×#103	—	2	6	no	—
51	C₃×#103	—	2	6	no	—
52	C₃×#95	—	2	3	no	—
53	C₃×#95	—	2	3	no	—
54	C₃×#95	—	2	3	no	—
55	C₃×#95	—	2	3	no	—
56	C₂²×Q₁₆	C₂²×C₃	1	1	yes	characteristic
57	C₂×C₄×Q₈	C₂²×C₃	1	3	yes	—
58	C₂×#107	C₂²×C₃	1	3	yes	—
59	C₂×#103	C₂²×C₃	1	3	yes	—
60	C₂×#103	C₂²×C₃	1	3	yes	—
61	#391	C₂²×C₃	1	3	yes	—
62	#389	C₂²×C₃	1	3	yes	—
63	#387	—	2	3	no	—
64	#383	—	2	1	no	—
65	C₄×C₄⋊C₄	—	2	3	no	—
66	C₄×C₄⋊C₄	C₂²×C₃	1	3	yes	—
67	C₄³	—	2	1	no	—
68	C₄³	C₂²×C₃	1	1	yes	characteristic
69	#340	C₂²×C₃	1	3	yes	—
70	#340	C₂²×C₃	1	3	yes	—
71	#336	C₂²×C₃	1	3	yes	—
72	#336	C₂²×C₃	1	3	yes	—
73	C₂×C₃×Q₈	—	2	3	no	—
74	C₂×C₃×Q₈	—	2	3	no	—
75	C₂×C₃×Q₈	—	2	3	no	—
76	C₂×C₃×Q₈	#41	1	6	yes	—
77	C₂²×C₃×C₄	—	2	1	no	—
78	C₂²×C₃×C₄	—	2	3	no	—
79	C₂²×C₃×C₄	—	2	1	no	—
80	C₂²×C₃×C₄	—	2	3	no	—
81	C₂²×C₃×C₄	C₂×D₄	1	3	yes	—
82	C₂²×C₃×C₄	C₂×D₄	1	3	yes	—
83	C₂²×C₃×C₄	C₂⁴	1	1	yes	characteristic
84	C₃×Q₁₆	—	4	4	no	—
85	C₂×C₃×C₈	—	2	3	no	—
86	C₂×C₃×C₈	—	2	3	no	—
87	C₂×C₃×C₈	—	2	3	no	—
88	C₂×C₃×C₈	—	2	3	no	—
89	C₃×C₄⋊C₄	—	2	12	no	—
90	C₃×C₄⋊C₄	—	2	3	no	—
91	C₃×C₄⋊C₄	—	2	3	no	—
92	C₃×C₄⋊C₄	—	2	3	no	—
93	C₃×C₄⋊C₄	—	4	6	no	—
94	C₃×C₄⋊C₄	#41	1	6	yes	—
95	C₃×C₄²	—	2	12	no	—
96	C₃×C₄²	—	2	3	no	—
97	C₃×C₄²	—	2	3	no	—
98	C₃×C₄²	—	4	3	no	—
99	C₃×C₄²	—	4	6	no	—
100	C₃×C₄²	Q₁₆	1	4	yes	complemented
101	C₂²×Q₈	C₂³×C₃	1	1	yes	characteristic
102	C₂²×Q₈	C₂³×C₃	1	1	yes	characteristic
103	C₂×Q₁₆	—	2	6	no	—
104	C₂²×C₈	C₂³×C₃	1	1	yes	characteristic
105	C₂²×C₈	C₂³×C₃	1	3	yes	—
106	C₄×Q₈	—	2	6	no	—
107	C₄×Q₈	—	2	6	no	—
108	C₂×C₄⋊C₄	—	2	3	no	—
109	C₂×C₄⋊C₄	—	2	3	no	—
110	C₂×C₄⋊C₄	C₂³×C₃	1	3	yes	—
111	C₂×C₄⋊C₄	C₂³×C₃	1	3	yes	—
112	C₂×C₄²	—	2	3	no	—
113	C₂×C₄²	—	2	3	no	—
114	C₂×C₄²	C₃×D₄	1	3	yes	—
115	C₂×C₄²	C₃×D₄	1	1	yes	characteristic
116	C₂×C₄²	C₂³×C₃	1	3	yes	—
117	#107	—	2	6	no	—
118	#103	—	2	6	no	—
119	#103	—	2	6	no	—
120	#95	—	2	3	no	—
121	#95	—	2	3	no	—
122	#95	—	2	3	no	—
123	#95	—	2	3	no	—
124	C₂³×C₃	C₂²×D₄	1	1	yes	characteristic
125	C₃×Q₈	—	2	6	no	—
126	C₃×Q₈	—	4	1	no	—
127	C₃×Q₈	—	4	3	no	—
128	C₃×Q₈	#126	1	4	yes	—
129	C₂×C₃×C₄	—	2	3	no	—
130	C₂×C₃×C₄	—	2	6	no	—
131	C₂×C₃×C₄	—	2	3	no	—
132	C₂×C₃×C₄	—	2	3	no	—
133	C₂×C₃×C₄	—	2	6	no	—
134	C₂×C₃×C₄	—	2	3	no	—
135	C₂×C₃×C₄	—	2	3	no	—
136	C₂×C₃×C₄	—	4	6	no	—
137	C₂×C₃×C₄	—	4	3	no	—
138	C₂×C₃×C₄	—	4	3	no	—
139	C₂×C₃×C₄	—	4	6	no	—
140	C₂×C₃×C₄	C₂×Q₁₆	1	6	yes	—
141	C₂×C₃×C₄	C₂×#41	1	3	yes	—
142	C₂×C₃×C₄	C₂×#41	1	3	yes	—
143	C₃×C₈	—	4	1	no	—
144	C₃×C₈	—	4	3	no	—
145	C₂×Q₈	—	2	3	no	—
146	C₂×Q₈	—	2	3	no	—
147	C₂×Q₈	—	2	3	no	—
148	C₂×Q₈	C₃×#41	1	6	yes	—
149	C₂²×C₄	—	2	1	no	—
150	C₂²×C₄	—	2	3	no	—
151	C₂²×C₄	—	2	1	no	—
152	C₂²×C₄	—	2	3	no	—
153	C₂²×C₄	C₂×C₃×D₄	1	3	yes	—
154	C₂²×C₄	C₂×C₃×D₄	1	3	yes	—
155	C₂²×C₄	C₂⁴×C₃	1	1	yes	characteristic
156	Q₁₆	—	4	4	no	—
157	C₂×C₈	—	2	3	no	—
158	C₂×C₈	—	2	3	no	—
159	C₂×C₈	—	2	3	no	—
160	C₂×C₈	—	2	3	no	—
161	C₄⋊C₄	—	2	12	no	—
162	C₄⋊C₄	—	2	3	no	—
163	C₄⋊C₄	—	2	3	no	—
164	C₄⋊C₄	—	2	3	no	—
165	C₄⋊C₄	—	4	6	no	—
166	C₄⋊C₄	C₃×#41	1	6	yes	—
167	C₄²	—	2	12	no	—
168	C₄²	—	2	3	no	—
169	C₄²	—	2	3	no	—
170	C₄²	—	4	3	no	—
171	C₄²	—	4	6	no	—
172	C₄²	C₃×Q₁₆	1	4	yes	complemented
173	C₂²×C₃	#525	1	3	yes	—
174	C₂²×C₃	C₂²×Q₁₆	1	1	yes	characteristic
175	C₂²×C₃	#575	1	3	yes	—
176	C₃×C₄	—	2	6	no	—
177	C₃×C₄	—	2	4	no	—
178	C₃×C₄	—	4	3	no	—
179	C₃×C₄	—	4	6	no	—
180	C₃×C₄	—	8	1	no	—
181	C₃×C₄	—	8	3	no	—
182	C₃×C₄	C₂×#126	1	1	yes	characteristic
183	C₃×C₄	#529	1	3	yes	—
184	C₂³	C₂²×C₃×D₄	1	1	yes	characteristic
185	Q₈	—	2	6	no	—
186	Q₈	—	4	1	no	—
187	Q₈	—	4	3	no	—
188	Q₈	C₃×#126	1	4	yes	—
189	C₂×C₄	—	2	3	no	—
190	C₂×C₄	—	2	3	no	—
191	C₂×C₄	—	2	6	no	—
192	C₂×C₄	—	2	3	no	—
193	C₂×C₄	—	2	3	no	—
194	C₂×C₄	—	2	3	no	—
195	C₂×C₄	—	2	6	no	—
196	C₂×C₄	—	4	6	no	—
197	C₂×C₄	—	4	6	no	—
198	C₂×C₄	—	4	3	no	—
199	C₂×C₄	—	4	3	no	—
200	C₂×C₄	C₂×C₃×Q₁₆	1	6	yes	—
201	C₂×C₄	C₂×C₃×#41	1	3	yes	—
202	C₂×C₄	C₂×C₃×#41	1	3	yes	—
203	C₈	—	4	1	no	—
204	C₈	—	4	3	no	—
205	C₂×C₃	#2852	1	1	yes	characteristic
206	C₂×C₃	#3104	1	3	yes	—
207	C₂×C₃	#3111	1	3	yes	—
208	C₂²	C₃×#525	1	3	yes	—
209	C₂²	C₂²×C₃×Q₁₆	1	1	yes	characteristic
210	C₂²	C₃×#575	1	3	yes	—
211	C₄	—	2	6	no	—
212	C₄	—	2	4	no	—
213	C₄	—	4	3	no	—
214	C₄	—	4	6	no	—
215	C₄	—	8	1	no	—
216	C₄	—	8	3	no	—
217	C₄	C₂×C₃×#126	1	1	yes	characteristic
218	C₄	C₃×#529	1	3	yes	—
219	C₃	#33083	1	1	yes	characteristic, direct product factor, Sylow
220	C₂	C₃×#2852	1	1	yes	characteristic
221	C₂	C₃×#3104	1	3	yes	—
222	C₂	C₃×#3111	1	3	yes	—

	subgroup	G/H
center, socle, prime radical	C₂³×C₃	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₃
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₃ ◁ C₃×C₄³ ◁ G

Complements

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

N	H ≅ G/N	type
8. #33083	C₃	direct (N × H)
100. C₃×C₄²	Q₁₆	semidirect
172. C₄²	C₃×Q₁₆	semidirect
219. C₃	#33083	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×#2863	2
C₃×C₄²×Q₈	2
C₃×#1928	2
C₃×#1921	2
C₃×#1878	2
C₃×#1863	2
C₃×#1839	2
#33083	3

Subgroups — C3×#33083 #10651690

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#33083 #10651690