Subgroups — C₃×#33410 #10652017

Order 768.

658 conjugacy classes of subgroups.

The subgroup lattice has 160 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×#504	C₂	1	1	yes	maximal, characteristic
2	C₃×#2913	C₂	1	2	yes	maximal
3	C₃×#2159	C₂	1	6	yes	maximal, complemented
4	C₃×#2158	C₂	1	3	yes	maximal, complemented
5	C₂×C₃×#366	C₂	1	3	yes	maximal, complemented
6	#33410	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
7	C₂×C₃×#131	C₂²	1	1	yes	characteristic
8	C₃×#520	—	2	2	no	—
9	C₃×#504	C₄	1	8	yes	complemented
10	C₂×C₃×M₃₂	C₂²	1	3	yes	—
11	C₂²×C₃×C₁₆	C₂²	1	3	yes	—
12	C₃×#428	—	2	6	no	—
13	C₃×#427	C₂²	1	6	yes	—
14	C₂×C₃×#107	C₂²	1	6	yes	—
15	C₃×#367	C₂²	1	6	yes	—
16	C₃×#366	C₂²	1	3	yes	—
17	C₃×#366	C₂²	1	6	yes	—
18	C₃×#366	C₂²	1	1	yes	characteristic
19	C₂×#504	C₂×C₃	1	1	yes	characteristic
20	#2913	C₂×C₃	1	2	yes	—
21	#2159	C₂×C₃	1	6	yes	complemented
22	#2158	C₂×C₃	1	3	yes	complemented
23	C₂×#366	C₂×C₃	1	3	yes	complemented
24	C₂×C₃×#41	D₄	1	1	yes	characteristic
25	C₃×#131	C₂×C₄	1	4	yes	—
26	C₃×#131	Q₈	1	4	yes	—
27	C₂×C₃×M₁₆	D₄	1	3	yes	—
28	C₂²×C₃×C₈	C₂³	1	3	yes	—
29	C₃×C₄×Q₈	—	4	2	no	—
30	C₃×C₄×D₄	—	4	6	no	—
31	C₃×#117	—	2	6	no	—
32	C₂×C₃×C₄⋊C₄	—	2	6	no	—
33	C₃×M₃₂	C₂×C₄	1	12	yes	complemented
34	C₂×C₃×C₁₆	C₂×C₄	1	12	yes	complemented
35	C₂×C₃×C₁₆	C₂³	1	1	yes	characteristic
36	C₂×C₃×C₁₆	C₂³	1	3	yes	—
37	C₃×#107	—	2	6	no	—
38	C₃×#107	C₂³	1	6	yes	—
39	C₃×#107	C₂³	1	2	yes	—
40	C₃×#106	—	2	6	no	—
41	C₂×#131	C₂²×C₃	1	1	yes	characteristic
42	#520	—	2	2	no	—
43	#504	C₃×C₄	1	8	yes	complemented
44	C₂×M₃₂	C₂²×C₃	1	3	yes	—
45	C₂²×C₁₆	C₂²×C₃	1	3	yes	—
46	#428	—	2	6	no	—
47	#427	C₂²×C₃	1	6	yes	—
48	C₂×#107	C₂²×C₃	1	6	yes	—
49	#367	C₂²×C₃	1	6	yes	—
50	#366	C₂²×C₃	1	3	yes	—
51	#366	C₂²×C₃	1	6	yes	—
52	#366	C₂²×C₃	1	1	yes	characteristic
53	C₃×#41	C₄⋊C₄	1	4	yes	—
54	C₃×#41	Q₁₆	1	4	yes	—
55	C₂×C₃×Q₈	D₈	1	1	yes	characteristic
56	C₂×C₃×D₄	D₈	1	3	yes	—
57	C₂²×C₃×C₄	—	4	6	no	—
58	C₂²×C₃×C₄	C₂×D₄	1	3	yes	—
59	C₃×M₁₆	C₄⋊C₄	1	12	yes	—
60	C₂×C₃×C₈	C₂²×C₄	1	6	yes	—
61	C₂×C₃×C₈	C₂×D₄	1	3	yes	—
62	C₂×C₃×C₈	C₂×Q₈	1	6	yes	—
63	C₂×C₃×C₈	C₂⁴	1	1	yes	characteristic
64	C₃×C₄⋊C₄	—	2	2	no	—
65	C₃×C₄⋊C₄	—	2	6	no	—
66	C₃×C₄⋊C₄	—	4	6	no	—
67	C₃×C₂²⋊C₄	—	4	6	no	—
68	C₃×C₄²	—	4	6	no	—
69	C₃×C₁₆	C₂²×C₄	1	2	yes	—
70	C₃×C₁₆	C₂²×C₄	1	6	yes	—
71	C₂×#41	C₃×D₄	1	1	yes	characteristic
72	#131	C₂×C₃×C₄	1	4	yes	—
73	#131	C₃×Q₈	1	4	yes	—
74	C₂×M₁₆	C₃×D₄	1	3	yes	—
75	C₂²×C₈	C₂³×C₃	1	3	yes	—
76	C₄×Q₈	—	4	2	no	—
77	C₄×D₄	—	4	6	no	—
78	#117	—	2	6	no	—
79	C₂×C₄⋊C₄	—	2	6	no	—
80	M₃₂	C₂×C₃×C₄	1	12	yes	complemented
81	C₂×C₁₆	C₂×C₃×C₄	1	12	yes	complemented
82	C₂×C₁₆	C₂³×C₃	1	1	yes	characteristic
83	C₂×C₁₆	C₂³×C₃	1	3	yes	—
84	#107	—	2	6	no	—
85	#107	C₂³×C₃	1	6	yes	—
86	#107	C₂³×C₃	1	2	yes	—
87	#106	—	2	6	no	—
88	C₂³×C₃	C₂×D₈	1	3	yes	—
89	C₃×Q₈	#107	1	4	yes	—
90	C₃×D₄	#107	1	12	yes	—
91	C₂×C₃×C₄	—	4	2	no	—
92	C₂×C₃×C₄	—	4	6	no	—
93	C₂×C₃×C₄	—	16	6	no	—
94	C₂×C₃×C₄	C₂×C₄⋊C₄	1	6	yes	—
95	C₂×C₃×C₄	C₂×D₈	1	3	yes	—
96	C₂×C₃×C₄	C₂×Q₁₆	1	6	yes	—
97	C₂×C₃×C₄	C₂²×D₄	1	1	yes	characteristic
98	C₃×C₈	C₂×C₄⋊C₄	1	6	yes	—
99	C₃×C₈	C₂³×C₄	1	1	yes	characteristic
100	C₃×C₈	C₂²×Q₈	1	1	yes	characteristic
101	#41	C₃×C₄⋊C₄	1	4	yes	—
102	#41	C₃×Q₁₆	1	4	yes	—
103	C₂×Q₈	C₃×D₈	1	1	yes	characteristic
104	C₂×D₄	C₃×D₈	1	3	yes	—
105	C₂²×C₄	—	4	6	no	—
106	C₂²×C₄	C₂×C₃×D₄	1	3	yes	—
107	M₁₆	C₃×C₄⋊C₄	1	12	yes	—
108	C₂×C₈	C₂²×C₃×C₄	1	6	yes	—
109	C₂×C₈	C₂×C₃×D₄	1	3	yes	—
110	C₂×C₈	C₂×C₃×Q₈	1	6	yes	—
111	C₂×C₈	C₂⁴×C₃	1	1	yes	characteristic
112	C₄⋊C₄	—	2	2	no	—
113	C₄⋊C₄	—	2	6	no	—
114	C₄⋊C₄	—	4	6	no	—
115	C₂²⋊C₄	—	4	6	no	—
116	C₄²	—	4	6	no	—
117	C₁₆	C₂²×C₃×C₄	1	2	yes	—
118	C₁₆	C₂²×C₃×C₄	1	6	yes	—
119	C₂²×C₃	—	2	3	no	—
120	C₂²×C₃	—	2	3	no	—
121	C₂²×C₃	C₂×#107	1	6	yes	—
122	C₂²×C₃	C₂²×D₈	1	1	yes	characteristic
123	C₃×C₄	—	8	6	no	—
124	C₃×C₄	—	8	2	no	—
125	C₃×C₄	C₂×#107	1	6	yes	—
126	C₃×C₄	C₂²×C₄⋊C₄	1	1	yes	characteristic
127	C₃×C₄	C₂²×Q₁₆	1	1	yes	characteristic
128	C₂³	C₂×C₃×D₈	1	3	yes	—
129	Q₈	C₃×#107	1	4	yes	—
130	D₄	C₃×#107	1	12	yes	—
131	C₂×C₄	—	4	2	no	—
132	C₂×C₄	—	4	6	no	—
133	C₂×C₄	—	16	6	no	—
134	C₂×C₄	C₂×C₃×C₄⋊C₄	1	6	yes	—
135	C₂×C₄	C₂×C₃×D₈	1	3	yes	—
136	C₂×C₄	C₂×C₃×Q₁₆	1	6	yes	—
137	C₂×C₄	C₂²×C₃×D₄	1	1	yes	characteristic
138	C₈	C₂×C₃×C₄⋊C₄	1	6	yes	—
139	C₈	C₂³×C₃×C₄	1	1	yes	characteristic
140	C₈	C₂²×C₃×Q₈	1	1	yes	characteristic
141	C₂×C₃	—	2	6	no	—
142	C₂×C₃	C₂²×#107	1	1	yes	characteristic
143	C₂×C₃	#3415	1	1	yes	characteristic
144	C₂×C₃	#3417	1	1	yes	characteristic
145	C₂²	—	2	3	no	—
146	C₂²	—	2	3	no	—
147	C₂²	C₂×C₃×#107	1	6	yes	—
148	C₂²	C₂²×C₃×D₈	1	1	yes	characteristic
149	C₄	—	8	6	no	—
150	C₄	—	8	2	no	—
151	C₄	C₂×C₃×#107	1	6	yes	—
152	C₄	C₂²×C₃×C₄⋊C₄	1	1	yes	characteristic
153	C₄	C₂²×C₃×Q₁₆	1	1	yes	characteristic
154	C₃	#33410	1	1	yes	characteristic, direct product factor, Sylow
155	C₂	—	2	6	no	—
156	C₂	C₂²×C₃×#107	1	1	yes	characteristic
157	C₂	C₃×#3415	1	1	yes	characteristic
158	C₂	C₃×#3417	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃	C₂²×D₈
commutator	C₈	C₂³×C₃×C₄
Frattini, cosocle	C₂×C₈	C₂⁴×C₃
prime radical	C₂×C₃×#41	D₄
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃ ◁ C₂×C₃×#41 ◁ C₂×C₃×#131 ◁ 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. C₂×C₃×#366	C₂	semidirect
6. #33410	C₃	direct (N × H)
9. C₃×#504	C₄	semidirect
21. #2159	C₂×C₃	semidirect
22. #2158	C₂×C₃	semidirect
23. C₂×#366	C₂×C₃	semidirect
33. C₃×M₃₂	C₂×C₄	semidirect
34. C₂×C₃×C₁₆	C₂×C₄	semidirect
43. #504	C₃×C₄	semidirect
80. M₃₂	C₂×C₃×C₄	semidirect
81. C₂×C₁₆	C₂×C₃×C₄	semidirect
154. C₃	#33410	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₃×#504	2
C₃×#2913	2
C₃×#2159	2
C₃×#2158	2
C₂×C₃×#366	2
#33410	3

Subgroups — C3×#33410 #10652017

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#33410 #10652017