Subgroups — C₃×#542 #5872

Order 192.

216 conjugacy classes of subgroups.

The subgroup lattice has 148 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃×#126	C₂	1	2	yes	maximal, complemented
3	C₃×#124	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×#124	C₂	1	2	yes	maximal, complemented
5	C₃×#123	C₂	1	2	yes	maximal, complemented
6	C₃×C₂²⋊Q₈	C₂	1	2	yes	maximal, complemented
7	C₃×C₂²⋊Q₈	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃×#121	C₂	1	1	yes	maximal, characteristic
9	C₃×C₄×Q₈	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃×C₄×D₄	C₂	1	1	yes	maximal, characteristic, complemented
11	C₃×#117	C₂	1	1	yes	maximal, characteristic, complemented
12	#542	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
13	C₂×C₃×Q₈	C₂²	1	1	yes	characteristic
14	C₂×C₃×Q₈	C₂²	1	2	yes	—
15	C₂×C₃×D₄	C₂²	1	1	yes	characteristic
16	C₂×C₃×D₄	C₂²	1	2	yes	—
17	C₂²×C₃×C₄	C₂²	1	1	yes	characteristic
18	C₂²×C₃×C₄	C₂²	1	2	yes	—
19	C₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
20	C₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
21	C₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
22	C₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
23	C₃×C₄⋊C₄	C₂²	1	2	yes	—
24	C₃×C₄⋊C₄	C₂²	1	2	yes	—
25	C₃×C₄⋊C₄	C₂²	1	2	yes	—
26	C₃×C₂²⋊C₄	C₂²	1	1	yes	characteristic
27	C₃×C₂²⋊C₄	C₂²	1	1	yes	characteristic
28	C₃×C₂²⋊C₄	C₂²	1	2	yes	—
29	C₃×C₂²⋊C₄	C₂²	1	2	yes	—
30	C₃×C₂²⋊C₄	C₂²	1	2	yes	—
31	C₃×C₂²⋊C₄	C₂²	1	2	yes	—
32	C₃×C₂²⋊C₄	C₂²	1	2	yes	—
33	C₃×C₄²	C₂²	1	1	yes	characteristic
34	C₃×C₄²	C₂²	1	1	yes	characteristic
35	C₃×C₄²	C₂²	1	2	yes	—
36	C₄⋊Q₈	C₂×C₃	1	1	yes	characteristic, complemented
37	#126	C₂×C₃	1	2	yes	complemented
38	#124	C₂×C₃	1	1	yes	characteristic, complemented
39	#124	C₂×C₃	1	2	yes	complemented
40	#123	C₂×C₃	1	2	yes	complemented
41	C₂²⋊Q₈	C₂×C₃	1	2	yes	complemented
42	C₂²⋊Q₈	C₂×C₃	1	1	yes	characteristic, complemented
43	#121	C₂×C₃	1	1	yes	characteristic
44	C₄×Q₈	C₂×C₃	1	1	yes	characteristic, complemented
45	C₄×D₄	C₂×C₃	1	1	yes	characteristic, complemented
46	#117	C₂×C₃	1	1	yes	characteristic, complemented
47	C₂³×C₃	C₂³	1	1	yes	characteristic
48	C₂³×C₃	C₂³	1	2	yes	—
49	C₃×Q₈	—	2	2	no	—
50	C₃×Q₈	—	4	2	no	—
51	C₃×D₄	—	2	2	no	—
52	C₃×D₄	—	4	2	no	—
53	C₂×C₃×C₄	—	2	2	no	—
54	C₂×C₃×C₄	—	4	2	no	—
55	C₂×C₃×C₄	C₂³	1	1	yes	characteristic
56	C₂×C₃×C₄	C₂³	1	1	yes	characteristic
57	C₂×C₃×C₄	C₂³	1	2	yes	—
58	C₂×C₃×C₄	C₂³	1	1	yes	characteristic
59	C₂×C₃×C₄	C₂³	1	1	yes	characteristic
60	C₂×C₃×C₄	C₂³	1	2	yes	—
61	C₂×C₃×C₄	C₂³	1	1	yes	characteristic
62	C₂×C₃×C₄	C₂³	1	2	yes	—
63	C₂×C₃×C₄	C₂³	1	1	yes	characteristic
64	C₂×Q₈	C₂²×C₃	1	1	yes	characteristic
65	C₂×Q₈	C₂²×C₃	1	2	yes	—
66	C₂×D₄	C₂²×C₃	1	1	yes	characteristic
67	C₂×D₄	C₂²×C₃	1	2	yes	—
68	C₂²×C₄	C₂²×C₃	1	1	yes	characteristic
69	C₂²×C₄	C₂²×C₃	1	2	yes	—
70	C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
71	C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
72	C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
73	C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
74	C₄⋊C₄	C₂²×C₃	1	2	yes	—
75	C₄⋊C₄	C₂²×C₃	1	2	yes	—
76	C₄⋊C₄	C₂²×C₃	1	2	yes	—
77	C₂²⋊C₄	C₂²×C₃	1	1	yes	characteristic
78	C₂²⋊C₄	C₂²×C₃	1	1	yes	characteristic
79	C₂²⋊C₄	C₂²×C₃	1	2	yes	—
80	C₂²⋊C₄	C₂²×C₃	1	2	yes	—
81	C₂²⋊C₄	C₂²×C₃	1	2	yes	—
82	C₂²⋊C₄	C₂²×C₃	1	2	yes	—
83	C₂²⋊C₄	C₂²×C₃	1	2	yes	—
84	C₄²	C₂²×C₃	1	1	yes	characteristic
85	C₄²	C₂²×C₃	1	1	yes	characteristic
86	C₄²	C₂²×C₃	1	2	yes	—
87	C₂²×C₃	—	2	1	no	—
88	C₂²×C₃	—	2	2	no	—
89	C₂²×C₃	—	2	2	no	—
90	C₂²×C₃	—	2	1	no	—
91	C₂²×C₃	—	2	1	no	—
92	C₂²×C₃	—	2	2	no	—
93	C₂²×C₃	C₂⁴	1	1	yes	characteristic
94	C₃×C₄	—	2	1	no	—
95	C₃×C₄	—	2	2	no	—
96	C₃×C₄	—	2	1	no	—
97	C₃×C₄	—	2	2	no	—
98	C₃×C₄	—	2	2	no	—
99	C₃×C₄	—	2	1	no	—
100	C₃×C₄	—	2	1	no	—
101	C₃×C₄	—	2	1	no	—
102	C₃×C₄	#41	1	2	yes	—
103	C₂³	C₂³×C₃	1	1	yes	characteristic
104	C₂³	C₂³×C₃	1	2	yes	—
105	Q₈	—	2	2	no	—
106	Q₈	—	4	2	no	—
107	D₄	—	2	2	no	—
108	D₄	—	4	2	no	—
109	C₂×C₄	—	2	2	no	—
110	C₂×C₄	—	4	2	no	—
111	C₂×C₄	C₂³×C₃	1	1	yes	characteristic
112	C₂×C₄	C₂³×C₃	1	1	yes	characteristic
113	C₂×C₄	C₂³×C₃	1	2	yes	—
114	C₂×C₄	C₂³×C₃	1	1	yes	characteristic
115	C₂×C₄	C₂³×C₃	1	1	yes	characteristic
116	C₂×C₄	C₂³×C₃	1	2	yes	—
117	C₂×C₄	C₂³×C₃	1	1	yes	characteristic
118	C₂×C₄	C₂³×C₃	1	2	yes	—
119	C₂×C₄	C₂³×C₃	1	1	yes	characteristic
120	C₂×C₃	—	4	1	no	—
121	C₂×C₃	—	4	2	no	—
122	C₂×C₃	C₂×#41	1	1	yes	characteristic
123	C₂×C₃	H(5,𝔽₂)	1	1	yes	characteristic
124	C₂×C₃	2₋¹⁺⁴	1	1	yes	characteristic
125	C₂²	—	2	1	no	—
126	C₂²	—	2	2	no	—
127	C₂²	—	2	2	no	—
128	C₂²	—	2	1	no	—
129	C₂²	—	2	1	no	—
130	C₂²	—	2	2	no	—
131	C₂²	C₂⁴×C₃	1	1	yes	characteristic
132	C₄	—	2	1	no	—
133	C₄	—	2	2	no	—
134	C₄	—	2	1	no	—
135	C₄	—	2	2	no	—
136	C₄	—	2	2	no	—
137	C₄	—	2	1	no	—
138	C₄	—	2	1	no	—
139	C₄	—	2	1	no	—
140	C₄	C₃×#41	1	2	yes	—
141	C₃	#542	1	1	yes	characteristic, direct product factor, Sylow
142	C₂	—	4	1	no	—
143	C₂	—	4	2	no	—
144	C₂	C₂×C₃×#41	1	1	yes	characteristic
145	C₂	C₃×H(5,𝔽₂)	1	1	yes	characteristic
146	C₂	C₃×2₋¹⁺⁴	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃	C₂⁴
commutator, Frattini, cosocle	C₂²	C₂⁴×C₃
prime radical	C₃×#121	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₃×C₄⋊Q₈	C₂	semidirect
2. C₃×#126	C₂	semidirect
3. C₃×#124	C₂	semidirect
4. C₃×#124	C₂	semidirect
5. C₃×#123	C₂	semidirect
6. C₃×C₂²⋊Q₈	C₂	semidirect
7. C₃×C₂²⋊Q₈	C₂	semidirect
9. C₃×C₄×Q₈	C₂	semidirect
10. C₃×C₄×D₄	C₂	semidirect
11. C₃×#117	C₂	semidirect
12. #542	C₃	direct (N × H)
36. C₄⋊Q₈	C₂×C₃	semidirect
37. #126	C₂×C₃	semidirect
38. #124	C₂×C₃	semidirect
39. #124	C₂×C₃	semidirect
40. #123	C₂×C₃	semidirect
41. C₂²⋊Q₈	C₂×C₃	semidirect
42. C₂²⋊Q₈	C₂×C₃	semidirect
44. C₄×Q₈	C₂×C₃	semidirect
45. C₄×D₄	C₂×C₃	semidirect
46. #117	C₂×C₃	semidirect
141. C₃	#542	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×C₄⋊Q₈	2
C₃×#126	2
C₃×#124	2
C₃×#123	2
C₃×C₂²⋊Q₈	2
C₃×#121	2
C₃×C₄×Q₈	2
C₃×C₄×D₄	2
C₃×#117	2
#542	3

Subgroups — C3×#542 #5872

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#542 #5872