Subgroups — C₃×#33401 #10652008

Order 768.

684 conjugacy classes of subgroups.

The subgroup lattice has 168 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#3407	C₂	1	2	yes	maximal
2	C₃×C₄×#131	C₂	1	1	yes	maximal, characteristic
3	C₃×#2153	C₂	1	6	yes	maximal, complemented
4	C₂×C₃×#364	C₂	1	6	yes	maximal, complemented
5	#33401	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
6	C₂×C₃×#131	C₄	1	1	yes	characteristic
7	C₂×C₃×#131	C₂²	1	1	yes	characteristic
8	C₃×C₄×#41	C₄	1	1	yes	characteristic
9	C₃×#504	—	2	8	no	—
10	C₂×C₃×M₃₂	C₂²	1	12	yes	—
11	C₃×#405	C₄	1	3	yes	—
12	C₃×#405	C₂²	1	3	yes	—
13	C₃×C₄×M₁₆	C₄	1	3	yes	—
14	C₂×C₃×C₄×C₈	C₂²	1	3	yes	—
15	C₃×#364	C₂²	1	4	yes	—
16	C₃×#364	C₂²	1	12	yes	—
17	#3407	C₂×C₃	1	2	yes	—
18	C₄×#131	C₂×C₃	1	1	yes	characteristic
19	#2153	C₂×C₃	1	6	yes	complemented
20	C₂×#364	C₂×C₃	1	6	yes	complemented
21	C₂×C₃×#41	C₂×C₄	1	1	yes	characteristic
22	C₃×#131	C₈	1	8	yes	—
23	C₃×#131	D₄	1	4	yes	—
24	C₃×#131	Q₈	1	4	yes	—
25	C₂×C₃×M₁₆	C₂×C₄	1	3	yes	—
26	C₂×C₃×M₁₆	C₂×C₄	1	3	yes	—
27	C₂²×C₃×C₈	C₂×C₄	1	3	yes	—
28	C₂²×C₃×C₈	C₂³	1	3	yes	—
29	C₃×C₄×Q₈	—	2	1	no	—
30	C₃×C₄×D₄	—	2	3	no	—
31	C₃×#117	C₂×C₄	1	3	yes	—
32	C₂×C₃×C₄²	C₂×C₄	1	3	yes	—
33	C₃×M₃₂	—	2	12	no	—
34	C₃×M₃₂	C₂³	1	8	yes	—
35	C₂×C₃×C₁₆	—	2	12	no	—
36	C₃×#97	C₂×C₄	1	6	yes	—
37	C₃×C₄×C₈	C₂×C₄	1	3	yes	—
38	C₃×C₄×C₈	C₂×C₄	1	3	yes	—
39	C₃×C₄×C₈	C₂³	1	1	yes	characteristic
40	C₃×C₄×C₈	C₂³	1	3	yes	—
41	C₂×#131	C₃×C₄	1	1	yes	characteristic
42	C₂×#131	C₂²×C₃	1	1	yes	characteristic
43	C₄×#41	C₃×C₄	1	1	yes	characteristic
44	#504	—	2	8	no	—
45	C₂×M₃₂	C₂²×C₃	1	12	yes	—
46	#405	C₃×C₄	1	3	yes	—
47	#405	C₂²×C₃	1	3	yes	—
48	C₄×M₁₆	C₃×C₄	1	3	yes	—
49	C₂×C₄×C₈	C₂²×C₃	1	3	yes	—
50	#364	C₂²×C₃	1	4	yes	—
51	#364	C₂²×C₃	1	12	yes	—
52	C₃×#41	C₄⋊C₄	1	4	yes	—
53	C₃×#41	C₂×C₈	1	4	yes	—
54	C₂×C₃×Q₈	M₁₆	1	1	yes	characteristic
55	C₂×C₃×D₄	M₁₆	1	3	yes	—
56	C₂²×C₃×C₄	—	2	3	no	—
57	C₂²×C₃×C₄	C₂²×C₄	1	3	yes	—
58	C₃×M₁₆	C₄⋊C₄	1	12	yes	—
59	C₃×M₁₆	C₂×C₈	1	12	yes	—
60	C₂×C₃×C₈	C₂×C₈	1	12	yes	—
61	C₂×C₃×C₈	C₂²×C₄	1	3	yes	—
62	C₂×C₃×C₈	C₂²×C₄	1	1	yes	characteristic
63	C₂×C₃×C₈	C₂²×C₄	1	3	yes	—
64	C₂×C₃×C₈	C₂×D₄	1	6	yes	—
65	C₂×C₃×C₈	C₂×Q₈	1	6	yes	—
66	C₂×C₃×C₈	C₂⁴	1	1	yes	characteristic
67	C₃×C₄⋊C₄	—	2	3	no	—
68	C₃×C₂²⋊C₄	—	2	3	no	—
69	C₃×C₄²	—	2	3	no	—
70	C₃×C₄²	C₂²×C₄	1	1	yes	characteristic
71	C₃×C₄²	C₂²×C₄	1	3	yes	—
72	C₃×C₁₆	—	2	8	no	—
73	C₂×#41	C₂×C₃×C₄	1	1	yes	characteristic
74	#131	C₃×C₈	1	8	yes	—
75	#131	C₃×D₄	1	4	yes	—
76	#131	C₃×Q₈	1	4	yes	—
77	C₂×M₁₆	C₂×C₃×C₄	1	3	yes	—
78	C₂×M₁₆	C₂×C₃×C₄	1	3	yes	—
79	C₂²×C₈	C₂×C₃×C₄	1	3	yes	—
80	C₂²×C₈	C₂³×C₃	1	3	yes	—
81	C₄×Q₈	—	2	1	no	—
82	C₄×D₄	—	2	3	no	—
83	#117	C₂×C₃×C₄	1	3	yes	—
84	C₂×C₄²	C₂×C₃×C₄	1	3	yes	—
85	M₃₂	—	2	12	no	—
86	M₃₂	C₂³×C₃	1	8	yes	—
87	C₂×C₁₆	—	2	12	no	—
88	#97	C₂×C₃×C₄	1	6	yes	—
89	C₄×C₈	C₂×C₃×C₄	1	3	yes	—
90	C₄×C₈	C₂×C₃×C₄	1	3	yes	—
91	C₄×C₈	C₂³×C₃	1	1	yes	characteristic
92	C₄×C₈	C₂³×C₃	1	3	yes	—
93	C₂³×C₃	C₂×M₁₆	1	3	yes	—
94	C₃×Q₈	C₄⋊C₈	1	4	yes	—
95	C₃×D₄	C₄⋊C₈	1	12	yes	—
96	C₂×C₃×C₄	—	2	1	no	—
97	C₂×C₃×C₄	—	2	3	no	—
98	C₂×C₃×C₄	—	4	6	no	—
99	C₂×C₃×C₄	C₂×C₄⋊C₄	1	6	yes	—
100	C₂×C₃×C₄	C₂²×C₈	1	6	yes	—
101	C₂×C₃×C₄	C₂×M₁₆	1	3	yes	—
102	C₂×C₃×C₄	C₂³×C₄	1	1	yes	characteristic
103	C₃×C₈	C₂×C₄⋊C₄	1	6	yes	—
104	C₃×C₈	C₂²×C₈	1	2	yes	—
105	C₃×C₈	C₂²×C₈	1	6	yes	—
106	C₃×C₈	C₂²×D₄	1	1	yes	characteristic
107	C₃×C₈	C₂²×Q₈	1	1	yes	characteristic
108	#41	C₃×C₄⋊C₄	1	4	yes	—
109	#41	C₂×C₃×C₈	1	4	yes	—
110	C₂×Q₈	C₃×M₁₆	1	1	yes	characteristic
111	C₂×D₄	C₃×M₁₆	1	3	yes	—
112	C₂²×C₄	—	2	3	no	—
113	C₂²×C₄	C₂²×C₃×C₄	1	3	yes	—
114	M₁₆	C₃×C₄⋊C₄	1	12	yes	—
115	M₁₆	C₂×C₃×C₈	1	12	yes	—
116	C₂×C₈	C₂×C₃×C₈	1	12	yes	—
117	C₂×C₈	C₂²×C₃×C₄	1	3	yes	—
118	C₂×C₈	C₂²×C₃×C₄	1	1	yes	characteristic
119	C₂×C₈	C₂²×C₃×C₄	1	3	yes	—
120	C₂×C₈	C₂×C₃×D₄	1	6	yes	—
121	C₂×C₈	C₂×C₃×Q₈	1	6	yes	—
122	C₂×C₈	C₂⁴×C₃	1	1	yes	characteristic
123	C₄⋊C₄	—	2	3	no	—
124	C₂²⋊C₄	—	2	3	no	—
125	C₄²	—	2	3	no	—
126	C₄²	C₂²×C₃×C₄	1	1	yes	characteristic
127	C₄²	C₂²×C₃×C₄	1	3	yes	—
128	C₁₆	—	2	8	no	—
129	C₂²×C₃	—	4	3	no	—
130	C₂²×C₃	C₂×C₄⋊C₈	1	6	yes	—
131	C₂²×C₃	C₂²×M₁₆	1	1	yes	characteristic
132	C₃×C₄	—	2	2	no	—
133	C₃×C₄	—	4	3	no	—
134	C₃×C₄	C₂×C₄⋊C₈	1	6	yes	—
135	C₃×C₄	C₂²×C₄⋊C₄	1	1	yes	characteristic
136	C₃×C₄	C₂³×C₈	1	1	yes	characteristic
137	C₂³	C₂×C₃×M₁₆	1	3	yes	—
138	Q₈	C₃×C₄⋊C₈	1	4	yes	—
139	D₄	C₃×C₄⋊C₈	1	12	yes	—
140	C₂×C₄	—	2	1	no	—
141	C₂×C₄	—	2	3	no	—
142	C₂×C₄	—	4	6	no	—
143	C₂×C₄	C₂×C₃×C₄⋊C₄	1	6	yes	—
144	C₂×C₄	C₂²×C₃×C₈	1	6	yes	—
145	C₂×C₄	C₂×C₃×M₁₆	1	3	yes	—
146	C₂×C₄	C₂³×C₃×C₄	1	1	yes	characteristic
147	C₈	C₂×C₃×C₄⋊C₄	1	6	yes	—
148	C₈	C₂²×C₃×C₈	1	2	yes	—
149	C₈	C₂²×C₃×C₈	1	6	yes	—
150	C₈	C₂²×C₃×D₄	1	1	yes	characteristic
151	C₈	C₂²×C₃×Q₈	1	1	yes	characteristic
152	C₂×C₃	—	2	1	no	—
153	C₂×C₃	—	2	6	no	—
154	C₂×C₃	C₂²×C₄⋊C₈	1	1	yes	characteristic
155	C₂²	—	4	3	no	—
156	C₂²	C₂×C₃×C₄⋊C₈	1	6	yes	—
157	C₂²	C₂²×C₃×M₁₆	1	1	yes	characteristic
158	C₄	—	2	2	no	—
159	C₄	—	4	3	no	—
160	C₄	C₂×C₃×C₄⋊C₈	1	6	yes	—
161	C₄	C₂²×C₃×C₄⋊C₄	1	1	yes	characteristic
162	C₄	C₂³×C₃×C₈	1	1	yes	characteristic
163	C₃	#33401	1	1	yes	characteristic, direct product factor, Sylow
164	C₂	—	2	1	no	—
165	C₂	—	2	6	no	—
166	C₂	C₂²×C₃×C₄⋊C₈	1	1	yes	characteristic

	subgroup	G/H
center	C₃×C₈	C₂²×D₄
commutator	C₄	C₂³×C₃×C₈
Frattini, cosocle	C₂×C₈	C₂⁴×C₃
socle	C₂×C₃	C₂²×C₄⋊C₈
prime radical	C₂×C₃×#41	C₂×C₄
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₃×C₈ ◁ C₂×C₃×#131 ◁ G

Complements

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

N	H ≅ G/N	type
3. C₃×#2153	C₂	semidirect
4. C₂×C₃×#364	C₂	semidirect
5. #33401	C₃	direct (N × H)
19. #2153	C₂×C₃	semidirect
20. C₂×#364	C₂×C₃	semidirect
163. C₃	#33401	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×#3407	2
C₃×C₄×#131	2
C₃×#2153	2
C₂×C₃×#364	2
#33401	3

Subgroups — C3×#33401 #10652008

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#33401 #10652008