Subgroups — #2984⋊C₃ #73254

Order 384.

119 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×SL(2,𝔽₃)	C₂	1	1	yes	maximal, characteristic
2	C₈×SL(2,𝔽₃)	C₂	1	2	yes	maximal, complemented
3	#2984	C₃	1	1	yes	maximal, characteristic, complemented, Sylow
4	C₂²×SL(2,𝔽₃)	C₄	1	1	yes	characteristic
5	C₄×SL(2,𝔽₃)	C₄	1	1	yes	characteristic
6	C₄×SL(2,𝔽₃)	C₂²	1	1	yes	characteristic
7	C₃×C₂²⋊C₈	—	4	1	no	maximal
8	C₂×C₄×Q₈	C₂×C₃	1	1	yes	characteristic
9	#446	—	3	2	no	—
10	C₈×Q₈	C₂×C₃	1	2	yes	complemented
11	#432	—	3	1	no	—
12	#431	—	3	1	no	—
13	C₂²×C₃×C₄	—	4	1	no	—
14	C₂×SL(2,𝔽₃)	—	2	1	no	—
15	C₂×SL(2,𝔽₃)	C₂×C₄	1	1	yes	characteristic
16	C₂×C₃×C₈	—	4	2	no	—
17	C₂²×Q₈	C₃×C₄	1	1	yes	characteristic
18	C₄×Q₈	—	3	1	no	—
19	C₄×Q₈	—	3	1	no	—
20	C₄×Q₈	C₃×C₄	1	1	yes	characteristic
21	C₄×Q₈	C₂²×C₃	1	1	yes	characteristic
22	C₂×C₄⋊C₄	—	3	1	no	—
23	C₂×C₄²	—	3	1	no	—
24	C₄⋊C₈	—	3	2	no	—
25	C₄⋊C₈	—	3	2	no	—
26	C₂²⋊C₈	—	3	1	no	—
27	C₂²⋊C₈	A₄	1	1	yes	characteristic
28	#97	—	3	2	no	—
29	C₄×C₈	—	3	2	no	—
30	C₂³×C₃	—	4	1	no	—
31	C₂×C₃×C₄	—	4	1	no	—
32	C₂×C₃×C₄	—	4	1	no	—
33	C₂×C₃×C₄	—	8	2	no	—
34	SL(2,𝔽₃)	M₁₆	1	1	yes	characteristic
35	C₃×C₈	—	8	2	no	—
36	C₂×Q₈	—	2	1	no	—
37	C₂×Q₈	—	3	1	no	—
38	C₂×Q₈	—	6	1	no	—
39	C₂×Q₈	C₂×C₃×C₄	1	1	yes	characteristic
40	C₂²×C₄	—	3	1	no	—
41	C₂²×C₄	—	3	1	no	—
42	C₂²×C₄	C₂×A₄	1	1	yes	characteristic
43	C₂×C₈	—	3	2	no	—
44	C₂×C₈	C₂×A₄	1	2	yes	—
45	C₄⋊C₄	—	3	1	no	—
46	C₄⋊C₄	—	3	1	no	—
47	C₄⋊C₄	—	3	2	no	—
48	C₄²	—	3	1	no	—
49	C₄²	—	3	1	no	—
50	C₄²	—	3	1	no	—
51	C₄²	—	3	1	no	—
52	C₂²×C₃	—	4	2	no	—
53	C₂²×C₃	—	4	1	no	—
54	C₂²×C₃	—	8	1	no	—
55	C₂²×C₃	—	8	1	no	—
56	C₃×C₄	—	4	2	no	—
57	C₃×C₄	—	8	1	no	—
58	C₂³	C₄×A₄	1	1	yes	characteristic
59	Q₈	—	3	1	no	—
60	Q₈	—	6	1	no	—
61	Q₈	—	6	1	no	—
62	Q₈	C₃×M₁₆	1	1	yes	characteristic
63	C₂×C₄	—	2	2	no	—
64	C₂×C₄	—	3	1	no	—
65	C₂×C₄	—	3	1	no	—
66	C₂×C₄	—	3	1	no	—
67	C₂×C₄	—	3	1	no	—
68	C₂×C₄	—	6	1	no	—
69	C₂×C₄	—	6	1	no	—
70	C₂×C₄	—	6	2	no	—
71	C₂×C₄	C₄×A₄	1	1	yes	characteristic
72	C₂×C₄	C₂²×A₄	1	1	yes	characteristic
73	C₈	—	2	2	no	—
74	C₈	—	6	2	no	—
75	C₂×C₃	—	4	1	no	—
76	C₂×C₃	—	4	1	no	—
77	C₂×C₃	—	4	1	no	—
78	C₂×C₃	—	8	2	no	—
79	C₂²	—	2	1	no	—
80	C₂²	—	2	1	no	—
81	C₂²	#131⋊C₃	1	2	yes	—
82	C₂²	C₂×C₄×A₄	1	1	yes	characteristic
83	C₄	—	2	1	no	—
84	C₄	—	3	1	no	—
85	C₄	—	3	1	no	—
86	C₄	—	6	1	no	—
87	C₄	—	6	1	no	—
88	C₄	—	6	1	no	—
89	C₄	2₋¹⁺⁴⋊C₃	1	2	yes	—
90	C₃	—	4	1	no	Sylow
91	C₂	—	2	2	no	—
92	C₂	#519⋊C₃	1	1	yes	characteristic
93	C₂	A₄×M₁₆	1	1	yes	characteristic
94	C₂	C₂×#131⋊C₃	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂²×A₄
commutator	C₂×Q₈	C₂×C₃×C₄
Fitting	#2984	C₃
socle	C₂²	C₂×C₄×A₄
prime radical	C₂²×SL(2,𝔽₃)	C₄
cosocle	C₄×Q₈	C₂²×C₃
hypercenter	C₂²⋊C₈	A₄
hypocenter	Q₈	C₃×M₁₆
LCS	G ▷ C₂×Q₈ ▷ Q₈
UCS	C₁ ◁ C₂×C₄ ◁ C₂²⋊C₈

Complements

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

N	H ≅ G/N	type
2. C₈×SL(2,𝔽₃)	C₂	semidirect
3. #2984	C₃	semidirect
10. C₈×Q₈	C₂×C₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×SL(2,𝔽₃)	2
C₈×SL(2,𝔽₃)	2
#2984	3
C₃×C₂²⋊C₈	4

Subgroups — #2984⋊C3 #73254

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #2984⋊C₃ #73254