Subgroups — C₃×#2205 #72920

Order 384.

224 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×Q₈²	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃×#456	C₂	1	3	yes	maximal
3	C₃×#356	C₂	1	3	yes	maximal, complemented
4	#2205	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
5	C₃×2₋¹⁺⁴	C₂²	1	1	yes	characteristic
6	C₃×#137	—	2	3	no	—
7	C₃×C₄⋊Q₈	—	2	3	no	—
8	C₃×C₄⋊Q₈	C₂²	1	3	yes	—
9	C₃×C₄×Q₈	—	2	3	no	—
10	C₃×#104	—	2	3	no	—
11	C₃×#101	C₂²	1	3	yes	—
12	Q₈²	C₂×C₃	1	1	yes	characteristic, complemented
13	#456	C₂×C₃	1	3	yes	—
14	#356	C₂×C₃	1	3	yes	complemented
15	C₃×#41	—	2	3	no	—
16	C₃×#41	—	4	1	no	—
17	C₂×C₃×Q₈	—	2	1	no	—
18	C₂×C₃×Q₈	—	2	1	no	—
19	C₂×C₃×Q₈	—	4	1	no	—
20	C₂×C₃×Q₈	D₄	1	3	yes	—
21	C₂×C₃×Q₈	C₂³	1	1	yes	characteristic
22	C₃×Q₁₆	—	4	3	no	—
23	C₃×SD₁₆	—	4	3	no	—
24	C₃×M₁₆	—	2	3	no	—
25	C₃×C₄⋊C₄	—	2	3	no	—
26	C₃×C₄⋊C₄	—	2	6	no	—
27	C₃×C₄²	—	2	3	no	—
28	C₃×C₄²	D₄	1	3	yes	—
29	2₋¹⁺⁴	C₂²×C₃	1	1	yes	characteristic
30	#137	—	2	3	no	—
31	C₄⋊Q₈	—	2	3	no	—
32	C₄⋊Q₈	C₂²×C₃	1	3	yes	—
33	C₄×Q₈	—	2	3	no	—
34	#104	—	2	3	no	—
35	#101	C₂²×C₃	1	3	yes	—
36	C₃×Q₈	—	2	4	no	—
37	C₃×Q₈	—	2	3	no	—
38	C₃×Q₈	—	4	1	no	—
39	C₃×Q₈	—	4	3	no	—
40	C₃×Q₈	—	8	1	no	—
41	C₃×D₄	—	2	3	no	—
42	C₃×D₄	—	4	1	no	—
43	C₂×C₃×C₄	—	2	3	no	—
44	C₂×C₃×C₄	—	2	3	no	—
45	C₂×C₃×C₄	—	4	3	no	—
46	C₂×C₃×C₄	C₂×D₄	1	3	yes	—
47	C₃×C₈	—	4	3	no	—
48	#41	—	2	3	no	—
49	#41	—	4	1	no	—
50	C₂×Q₈	—	2	1	no	—
51	C₂×Q₈	—	2	1	no	—
52	C₂×Q₈	—	4	1	no	—
53	C₂×Q₈	C₃×D₄	1	3	yes	—
54	C₂×Q₈	C₂³×C₃	1	1	yes	characteristic
55	Q₁₆	—	4	3	no	—
56	SD₁₆	—	4	3	no	—
57	M₁₆	—	2	3	no	—
58	C₄⋊C₄	—	2	3	no	—
59	C₄⋊C₄	—	2	6	no	—
60	C₄²	—	2	3	no	—
61	C₄²	C₃×D₄	1	3	yes	—
62	C₂²×C₃	—	4	1	no	—
63	C₂²×C₃	#120	1	1	yes	characteristic
64	C₃×C₄	—	2	6	no	—
65	C₃×C₄	—	2	3	no	—
66	C₃×C₄	—	4	3	no	—
67	C₃×C₄	—	4	1	no	—
68	Q₈	—	2	4	no	—
69	Q₈	—	2	3	no	—
70	Q₈	—	4	1	no	—
71	Q₈	—	4	3	no	—
72	Q₈	—	8	1	no	—
73	D₄	—	2	3	no	—
74	D₄	—	4	1	no	—
75	C₂×C₄	—	2	3	no	—
76	C₂×C₄	—	2	3	no	—
77	C₂×C₄	—	4	3	no	—
78	C₂×C₄	C₂×C₃×D₄	1	3	yes	—
79	C₈	—	4	3	no	—
80	C₂×C₃	—	2	1	no	—
81	C₂×C₃	—	8	1	no	—
82	C₂×C₃	B(4,𝔽₂)	1	1	yes	characteristic
83	C₂²	—	4	1	no	—
84	C₂²	C₃×#120	1	1	yes	characteristic
85	C₄	—	2	6	no	—
86	C₄	—	2	3	no	—
87	C₄	—	4	3	no	—
88	C₄	—	4	1	no	—
89	C₃	#2205	1	1	yes	characteristic, direct product factor, Sylow
90	C₂	—	2	1	no	—
91	C₂	—	8	1	no	—
92	C₂	C₃×B(4,𝔽₂)	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₃	B(4,𝔽₂)
commutator, Frattini, cosocle	C₂×Q₈	C₂³×C₃
prime radical	C₃×2₋¹⁺⁴	C₂²
LCS	G ▷ C₂×Q₈ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₃ ◁ C₂²×C₃ ◁ C₂×C₃×Q₈ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₃×Q₈²	C₂	semidirect
3. C₃×#356	C₂	semidirect
4. #2205	C₃	direct (N × H)
12. Q₈²	C₂×C₃	semidirect
14. #356	C₂×C₃	semidirect
89. C₃	#2205	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×Q₈²	2
C₃×#456	2
C₃×#356	2
#2205	3

Subgroups — C3×#2205 #72920

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#2205 #72920