Subgroups — #68098

Order 384.

198 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#4699	C₂	1	1	yes	maximal, characteristic, complemented
2	#4694	C₂	1	2	yes	maximal
3	#4693	C₂	1	1	yes	maximal, characteristic
4	#4682	C₂	1	2	yes	maximal, complemented
5	C₃×#322	C₂	1	1	yes	maximal, characteristic, complemented
6	#1722	—	3	1	no	maximal, Sylow
7	C₂×Dic₁₂	—	2	2	no	—
8	C₂×#204	—	2	2	no	—
9	#874	C₂²	1	1	yes	characteristic
10	#869	C₂²	1	1	yes	characteristic
11	#869	C₂²	1	2	yes	—
12	C₃×C₄×C₈	C₂²	1	2	yes	—
13	C₃×C₄×C₈	C₂²	1	1	yes	characteristic
14	#821	—	2	4	no	—
15	#818	—	2	4	no	—
16	#500	—	3	2	no	—
17	#494	—	3	1	no	—
18	#492	—	3	1	no	—
19	#486	—	3	2	no	—
20	#322	D₃	1	1	yes	characteristic, complemented
21	C₂×D₁₂	—	2	2	no	—
22	C₂×D₁₂	—	2	1	no	—
23	C₂×Dic₆	—	2	1	no	—
24	C₂×Dic₆	—	2	2	no	—
25	C₂×C₃×C₈	D₄	1	4	yes	—
26	C₂×C₃×C₈	D₄	1	2	yes	—
27	C₃×C₄²	C₂³	1	1	yes	characteristic
28	C₄⋊Dic₃	—	2	2	no	—
29	C₄⋊Dic₃	—	2	2	no	—
30	C₄⋊Dic₃	—	2	2	no	—
31	Dic₁₂	—	4	4	no	—
32	#204	—	4	4	no	—
33	C₂×Q₁₆	—	6	2	no	—
34	C₂×SD₁₆	—	6	2	no	—
35	C₄⋊Q₈	—	3	2	no	—
36	C₄⋊Q₈	—	3	1	no	—
37	#127	—	3	1	no	—
38	#107	—	6	4	no	—
39	#102	—	6	4	no	—
40	C₄×C₈	C₂×D₃	1	2	yes	—
41	C₄×C₈	C₂×D₃	1	1	yes	characteristic
42	C₂²×D₃	—	4	1	no	—
43	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
44	C₂×C₃×C₄	C₂×D₄	1	2	yes	—
45	C₂×Dic₃	—	4	1	no	—
46	C₂×Dic₃	—	4	2	no	—
47	D₁₂	—	4	2	no	—
48	D₁₂	—	4	4	no	—
49	Dic₆	—	4	2	no	—
50	Dic₆	—	4	4	no	—
51	C₃×C₈	—	2	4	no	—
52	C₃×C₈	D₈	1	4	yes	—
53	C₂×Q₈	—	6	1	no	—
54	C₂×Q₈	—	6	2	no	—
55	C₂×D₄	—	6	2	no	—
56	C₂×D₄	—	6	1	no	—
57	Q₁₆	—	12	4	no	—
58	SD₁₆	—	12	4	no	—
59	C₂×C₈	D₁₂	1	4	yes	—
60	C₂×C₈	D₁₂	1	2	yes	—
61	C₄⋊C₄	—	6	2	no	—
62	C₄⋊C₄	—	6	2	no	—
63	C₄⋊C₄	—	6	2	no	—
64	C₄²	C₂²×D₃	1	1	yes	characteristic
65	C₂²×C₃	#127	1	1	yes	characteristic
66	C₂×D₃	—	8	1	no	—
67	C₂×D₃	—	8	2	no	—
68	C₃×C₄	C₂×D₈	1	2	yes	—
69	C₃×C₄	#137	1	4	yes	—
70	Dic₃	—	8	1	no	—
71	Dic₃	—	8	2	no	—
72	C₂³	—	12	1	no	—
73	Q₈	—	12	2	no	—
74	Q₈	—	12	4	no	—
75	D₄	—	12	2	no	—
76	D₄	—	12	4	no	—
77	C₂×C₄	—	12	2	no	—
78	C₂×C₄	—	12	1	no	—
79	C₂×C₄	C₂×D₁₂	1	1	yes	characteristic
80	C₂×C₄	C₂×D₁₂	1	2	yes	—
81	C₈	—	2	4	no	—
82	C₈	D₂₄	1	4	yes	—
83	C₂×C₃	#493	1	1	yes	characteristic
84	C₂×C₃	#497	1	2	yes	—
85	D₃	—	16	1	no	—
86	C₂²	—	24	1	no	—
87	C₂²	—	24	2	no	—
88	C₂²	#874	1	1	yes	characteristic
89	C₄	—	24	1	no	—
90	C₄	—	24	2	no	—
91	C₄	C₂×D₂₄	1	2	yes	—
92	C₄	#909	1	4	yes	—
93	C₃	#1722	1	1	yes	characteristic, complemented, Sylow
94	C₂	—	48	1	no	—
95	C₂	#4695	1	1	yes	characteristic
96	C₂	#4715	1	2	yes	—

	subgroup	G/H
center	C₂²	#874
commutator, cosocle	C₃×C₄²	C₂³
Frattini	C₄²	C₂²×D₃
Fitting	C₃×#322	C₂
socle	C₂²×C₃	#127
prime radical	#874	C₂²
hypercenter	#322	D₃
hypocenter	C₃	#1722
LCS	G ▷ C₃×C₄² ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #322

Complements

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

N	H ≅ G/N	type
1. #4699	C₂	semidirect
4. #4682	C₂	semidirect
5. C₃×#322	C₂	semidirect
20. #322	D₃	semidirect
93. C₃	#1722	semidirect

Maximal subgroups

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

subgroup	index
#4699	2
#4694	2
#4693	2
#4682	2
C₃×#322	2
#1722	3