Subgroups — C₃×#33409 #10652016

Order 768.

674 conjugacy classes of subgroups.

The subgroup lattice has 196 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃×M₃₂	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×#427	C₂	1	2	yes	maximal
3	C₃×#2159	C₂	1	8	yes	maximal, complemented
4	C₃×#2158	C₂	1	4	yes	maximal, complemented
5	#33409	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
6	C₂³×C₃×C₈	C₂²	1	1	yes	characteristic
7	C₂×C₃×#117	—	2	2	no	—
8	C₂×C₃×M₃₂	C₄	1	8	yes	complemented
9	C₂×C₃×M₃₂	C₂²	1	4	yes	—
10	C₂²×C₃×C₁₆	C₂²	1	2	yes	—
11	C₃×#427	—	2	2	no	—
12	C₃×#427	—	2	2	no	—
13	C₃×#427	C₂²	1	4	yes	—
14	C₃×#427	C₂²	1	4	yes	—
15	C₂×C₃×#107	C₂²	1	4	yes	—
16	C₂×C₃×#106	—	2	2	no	—
17	C₃×#367	C₂²	1	8	yes	complemented
18	C₃×#366	C₂²	1	8	yes	complemented
19	C₂²×M₃₂	C₂×C₃	1	1	yes	characteristic
20	C₂×#427	C₂×C₃	1	2	yes	—
21	#2159	C₂×C₃	1	8	yes	complemented
22	#2158	C₂×C₃	1	4	yes	complemented
23	C₂³×C₃×C₄	D₄	1	1	yes	characteristic
24	C₂²×C₃×C₈	C₂×C₄	1	4	yes	—
25	C₂²×C₃×C₈	D₄	1	2	yes	—
26	C₂²×C₃×C₈	D₄	1	1	yes	characteristic
27	C₂²×C₃×C₈	Q₈	1	4	yes	—
28	C₂²×C₃×C₈	C₂³	1	1	yes	characteristic
29	C₂²×C₃×C₈	C₂³	1	2	yes	—
30	C₃×#117	—	2	4	no	—
31	C₃×#117	—	2	4	no	—
32	C₃×#117	—	4	2	no	—
33	C₃×#117	—	4	2	no	—
34	C₂×C₃×C₄⋊C₄	—	2	4	no	—
35	C₂×C₃×C₂²⋊C₄	—	4	2	no	—
36	C₂×C₃×C₄²	—	4	2	no	—
37	C₃×M₃₂	C₂×C₄	1	16	yes	complemented
38	C₂×C₃×C₁₆	C₂×C₄	1	8	yes	complemented
39	C₂×C₃×C₁₆	C₂³	1	4	yes	—
40	C₃×#107	—	2	4	no	—
41	C₃×#107	C₂³	1	8	yes	—
42	C₃×#106	—	2	4	no	—
43	C₃×#106	—	2	4	no	—
44	C₂³×C₈	C₂²×C₃	1	1	yes	characteristic
45	C₂×#117	—	2	2	no	—
46	C₂×M₃₂	C₃×C₄	1	8	yes	complemented
47	C₂×M₃₂	C₂²×C₃	1	4	yes	—
48	C₂²×C₁₆	C₂²×C₃	1	2	yes	—
49	#427	—	2	2	no	—
50	#427	—	2	2	no	—
51	#427	C₂²×C₃	1	4	yes	—
52	#427	C₂²×C₃	1	4	yes	—
53	C₂×#107	C₂²×C₃	1	4	yes	—
54	C₂×#106	—	2	2	no	—
55	#367	C₂²×C₃	1	8	yes	complemented
56	#366	C₂²×C₃	1	8	yes	complemented
57	C₂⁴×C₃	D₈	1	1	yes	characteristic
58	C₂²×C₃×C₄	—	4	4	no	—
59	C₂²×C₃×C₄	C₄⋊C₄	1	4	yes	—
60	C₂²×C₃×C₄	D₈	1	1	yes	characteristic
61	C₂²×C₃×C₄	D₈	1	2	yes	—
62	C₂²×C₃×C₄	Q₁₆	1	4	yes	—
63	C₂²×C₃×C₄	C₂×D₄	1	2	yes	—
64	C₂²×C₃×C₄	C₂×D₄	1	1	yes	characteristic
65	C₂×C₃×C₈	C₄⋊C₄	1	8	yes	—
66	C₂×C₃×C₈	C₄⋊C₄	1	4	yes	—
67	C₂×C₃×C₈	C₂²×C₄	1	2	yes	—
68	C₂×C₃×C₈	C₂²×C₄	1	4	yes	—
69	C₂×C₃×C₈	C₂×D₄	1	1	yes	characteristic
70	C₂×C₃×C₈	C₂×D₄	1	2	yes	—
71	C₂×C₃×C₈	C₂×Q₈	1	2	yes	—
72	C₂×C₃×C₈	C₂×Q₈	1	4	yes	—
73	C₂×C₃×C₈	C₂⁴	1	1	yes	characteristic
74	C₃×C₄⋊C₄	—	2	8	no	—
75	C₃×C₄⋊C₄	—	4	4	no	—
76	C₃×C₂²⋊C₄	—	4	4	no	—
77	C₃×C₂²⋊C₄	—	4	4	no	—
78	C₃×C₄²	—	4	4	no	—
79	C₃×C₄²	—	4	4	no	—
80	C₃×C₁₆	C₂²×C₄	1	8	yes	—
81	C₂³×C₄	C₃×D₄	1	1	yes	characteristic
82	C₂²×C₈	C₂×C₃×C₄	1	4	yes	—
83	C₂²×C₈	C₃×D₄	1	2	yes	—
84	C₂²×C₈	C₃×D₄	1	1	yes	characteristic
85	C₂²×C₈	C₃×Q₈	1	4	yes	—
86	C₂²×C₈	C₂³×C₃	1	2	yes	—
87	C₂²×C₈	C₂³×C₃	1	1	yes	characteristic
88	#117	—	2	4	no	—
89	#117	—	2	4	no	—
90	#117	—	4	2	no	—
91	#117	—	4	2	no	—
92	C₂×C₄⋊C₄	—	2	4	no	—
93	C₂×C₂²⋊C₄	—	4	2	no	—
94	C₂×C₄²	—	4	2	no	—
95	M₃₂	C₂×C₃×C₄	1	16	yes	complemented
96	C₂×C₁₆	C₂×C₃×C₄	1	8	yes	complemented
97	C₂×C₁₆	C₂³×C₃	1	4	yes	—
98	#107	—	2	4	no	—
99	#107	C₂³×C₃	1	8	yes	—
100	#106	—	2	4	no	—
101	#106	—	2	4	no	—
102	C₂³×C₃	—	4	2	no	—
103	C₂³×C₃	#107	1	4	yes	—
104	C₂³×C₃	C₂×D₈	1	1	yes	characteristic
105	C₂³×C₃	C₂×D₈	1	2	yes	—
106	C₂×C₃×C₄	—	4	8	no	—
107	C₂×C₃×C₄	—	8	8	no	—
108	C₂×C₃×C₄	#107	1	4	yes	—
109	C₂×C₃×C₄	#107	1	8	yes	—
110	C₂×C₃×C₄	C₂×C₄⋊C₄	1	4	yes	—
111	C₂×C₃×C₄	C₂×C₄⋊C₄	1	2	yes	—
112	C₂×C₃×C₄	C₂×D₈	1	2	yes	—
113	C₂×C₃×C₄	C₂×D₈	1	1	yes	characteristic
114	C₂×C₃×C₄	C₂×Q₁₆	1	4	yes	—
115	C₂×C₃×C₄	C₂×Q₁₆	1	2	yes	—
116	C₂×C₃×C₄	C₂²×D₄	1	1	yes	characteristic
117	C₃×C₈	C₂×C₄⋊C₄	1	2	yes	—
118	C₃×C₈	C₂×C₄⋊C₄	1	4	yes	—
119	C₃×C₈	C₂³×C₄	1	1	yes	characteristic
120	C₃×C₈	C₂²×Q₈	1	1	yes	characteristic
121	C₂⁴	C₃×D₈	1	1	yes	characteristic
122	C₂²×C₄	—	4	4	no	—
123	C₂²×C₄	C₃×C₄⋊C₄	1	4	yes	—
124	C₂²×C₄	C₃×D₈	1	2	yes	—
125	C₂²×C₄	C₃×D₈	1	1	yes	characteristic
126	C₂²×C₄	C₃×Q₁₆	1	4	yes	—
127	C₂²×C₄	C₂×C₃×D₄	1	2	yes	—
128	C₂²×C₄	C₂×C₃×D₄	1	1	yes	characteristic
129	C₂×C₈	C₃×C₄⋊C₄	1	8	yes	—
130	C₂×C₈	C₃×C₄⋊C₄	1	4	yes	—
131	C₂×C₈	C₂²×C₃×C₄	1	2	yes	—
132	C₂×C₈	C₂²×C₃×C₄	1	4	yes	—
133	C₂×C₈	C₂×C₃×D₄	1	1	yes	characteristic
134	C₂×C₈	C₂×C₃×D₄	1	2	yes	—
135	C₂×C₈	C₂×C₃×Q₈	1	4	yes	—
136	C₂×C₈	C₂×C₃×Q₈	1	2	yes	—
137	C₂×C₈	C₂⁴×C₃	1	1	yes	characteristic
138	C₄⋊C₄	—	2	8	no	—
139	C₄⋊C₄	—	4	4	no	—
140	C₂²⋊C₄	—	4	4	no	—
141	C₂²⋊C₄	—	4	4	no	—
142	C₄²	—	4	4	no	—
143	C₄²	—	4	4	no	—
144	C₁₆	C₂²×C₃×C₄	1	8	yes	—
145	C₂²×C₃	—	2	4	no	—
146	C₂²×C₃	—	2	2	no	—
147	C₂²×C₃	—	4	4	no	—
148	C₂²×C₃	C₂×#107	1	2	yes	—
149	C₂²×C₃	C₂×#107	1	4	yes	—
150	C₂²×C₃	C₂²×D₈	1	1	yes	characteristic
151	C₃×C₄	—	8	8	no	—
152	C₃×C₄	C₂×#107	1	4	yes	—
153	C₃×C₄	C₂×#107	1	2	yes	—
154	C₃×C₄	C₂²×C₄⋊C₄	1	1	yes	characteristic
155	C₃×C₄	C₂²×Q₁₆	1	1	yes	characteristic
156	C₂³	—	4	2	no	—
157	C₂³	C₃×#107	1	4	yes	—
158	C₂³	C₂×C₃×D₈	1	1	yes	characteristic
159	C₂³	C₂×C₃×D₈	1	2	yes	—
160	C₂×C₄	—	4	8	no	—
161	C₂×C₄	—	8	8	no	—
162	C₂×C₄	C₃×#107	1	8	yes	—
163	C₂×C₄	C₃×#107	1	4	yes	—
164	C₂×C₄	C₂×C₃×C₄⋊C₄	1	2	yes	—
165	C₂×C₄	C₂×C₃×C₄⋊C₄	1	4	yes	—
166	C₂×C₄	C₂×C₃×D₈	1	1	yes	characteristic
167	C₂×C₄	C₂×C₃×D₈	1	2	yes	—
168	C₂×C₄	C₂×C₃×Q₁₆	1	4	yes	—
169	C₂×C₄	C₂×C₃×Q₁₆	1	2	yes	—
170	C₂×C₄	C₂²×C₃×D₄	1	1	yes	characteristic
171	C₈	C₂×C₃×C₄⋊C₄	1	2	yes	—
172	C₈	C₂×C₃×C₄⋊C₄	1	4	yes	—
173	C₈	C₂³×C₃×C₄	1	1	yes	characteristic
174	C₈	C₂²×C₃×Q₈	1	1	yes	characteristic
175	C₂×C₃	—	2	4	no	—
176	C₂×C₃	—	2	2	no	—
177	C₂×C₃	C₂²×#107	1	1	yes	characteristic
178	C₂×C₃	#3414	1	2	yes	—
179	C₂²	—	2	4	no	—
180	C₂²	—	2	2	no	—
181	C₂²	—	4	4	no	—
182	C₂²	C₂×C₃×#107	1	2	yes	—
183	C₂²	C₂×C₃×#107	1	4	yes	—
184	C₂²	C₂²×C₃×D₈	1	1	yes	characteristic
185	C₄	—	8	8	no	—
186	C₄	C₂×C₃×#107	1	2	yes	—
187	C₄	C₂×C₃×#107	1	4	yes	—
188	C₄	C₂²×C₃×C₄⋊C₄	1	1	yes	characteristic
189	C₄	C₂²×C₃×Q₁₆	1	1	yes	characteristic
190	C₃	#33409	1	1	yes	characteristic, direct product factor, Sylow
191	C₂	—	2	4	no	—
192	C₂	—	2	2	no	—
193	C₂	C₂²×C₃×#107	1	1	yes	characteristic
194	C₂	C₃×#3414	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
3. C₃×#2159	C₂	semidirect
4. C₃×#2158	C₂	semidirect
5. #33409	C₃	direct (N × H)
8. C₂×C₃×M₃₂	C₄	semidirect
17. C₃×#367	C₂²	semidirect
18. C₃×#366	C₂²	semidirect
21. #2159	C₂×C₃	semidirect
22. #2158	C₂×C₃	semidirect
37. C₃×M₃₂	C₂×C₄	semidirect
38. C₂×C₃×C₁₆	C₂×C₄	semidirect
46. C₂×M₃₂	C₃×C₄	semidirect
55. #367	C₂²×C₃	semidirect
56. #366	C₂²×C₃	semidirect
95. M₃₂	C₂×C₃×C₄	semidirect
96. C₂×C₁₆	C₂×C₃×C₄	semidirect
190. C₃	#33409	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂²×C₃×M₃₂	2
C₂×C₃×#427	2
C₃×#2159	2
C₃×#2158	2
#33409	3

Subgroups — C3×#33409 #10652016

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#33409 #10652016