Subgroups — #69547

Order 384.

216 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#382	C₂	1	1	yes	maximal, characteristic
2	#4936	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄×C₃⋊C₈	C₂	1	1	yes	maximal, characteristic
4	#4529	C₂	1	4	yes	maximal
5	#1835	—	3	1	no	maximal, Sylow
6	C₂×C₃×C₄⋊C₄	C₂²	1	2	yes	—
7	C₂×C₃×C₄²	C₂²	1	1	yes	characteristic
8	C₂×C₄⋊Dic₃	C₂²	1	2	yes	—
9	C₂²×C₃⋊C₈	C₂²	1	2	yes	—
10	C₃×#95	—	2	2	no	—
11	C₃⋊#95	—	2	2	no	—
12	C₄×C₃⋊C₈	C₄	1	4	yes	complemented
13	C₂×C₄×C₈	—	3	1	no	—
14	#382	—	3	1	no	—
15	#382	D₃	1	1	yes	characteristic
16	#340	—	3	4	no	—
17	C₂²×C₃×C₄	—	2	2	no	—
18	C₂²×C₃×C₄	D₄	1	2	yes	—
19	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
20	C₂²×Dic₃	—	2	2	no	—
21	C₃×C₄⋊C₄	—	2	4	no	—
22	C₃×C₄²	C₂×C₄	1	2	yes	—
23	C₃×C₄²	Q₈	1	2	yes	—
24	C₄⋊Dic₃	—	2	4	no	—
25	C₂×C₃⋊C₈	—	2	2	no	—
26	C₂×C₃⋊C₈	—	2	2	no	—
27	C₂×C₃⋊C₈	C₂×C₄	1	4	yes	—
28	C₂²×C₈	—	3	2	no	—
29	C₂×C₄⋊C₄	—	3	2	no	—
30	C₂×C₄⋊C₄	C₂×D₃	1	2	yes	—
31	C₂×C₄²	C₂×D₃	1	1	yes	characteristic
32	C₄×C₈	—	3	4	no	—
33	#95	—	2	2	no	—
34	#95	—	6	2	no	—
35	C₂³×C₃	C₂×D₄	1	1	yes	characteristic
36	C₂×C₃×C₄	—	2	4	no	—
37	C₂×C₃×C₄	—	4	2	no	—
38	C₂×C₃×C₄	—	4	4	no	—
39	C₂×C₃×C₄	C₄⋊C₄	1	4	yes	—
40	C₂×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
41	C₂×C₃×C₄	C₂×Q₈	1	1	yes	characteristic
42	C₂×C₃×C₄	#41	1	2	yes	—
43	C₂×C₃×C₄	#41	1	2	yes	—
44	C₂×Dic₃	—	4	2	no	—
45	C₂×Dic₃	—	4	4	no	—
46	C₃⋊C₈	—	2	4	no	—
47	C₂²×C₄	—	2	2	no	—
48	C₂²×C₄	—	6	2	no	—
49	C₂²×C₄	#67	1	2	yes	—
50	C₂²×C₄	C₂²×D₃	1	1	yes	characteristic
51	C₂×C₈	—	3	4	no	—
52	C₂×C₈	—	6	2	no	—
53	C₂×C₈	—	6	2	no	—
54	C₄⋊C₄	—	2	4	no	—
55	C₄⋊C₄	—	6	4	no	—
56	C₄²	Dic₆	1	2	yes	—
57	C₄²	C₄×D₃	1	2	yes	—
58	C₂²×C₃	C₂×C₄⋊C₄	1	1	yes	characteristic
59	C₂²×C₃	#124	1	1	yes	characteristic
60	C₂²×C₃	#124	1	1	yes	characteristic
61	C₂²×C₃	#135	1	4	yes	—
62	C₃×C₄	—	2	4	no	—
63	C₃×C₄	—	8	2	no	—
64	C₃×C₄	#117	1	2	yes	—
65	C₃×C₄	#125	1	2	yes	—
66	Dic₃	—	8	2	no	—
67	C₂³	C₂×#67	1	1	yes	characteristic
68	C₂×C₄	—	2	4	no	—
69	C₂×C₄	—	4	2	no	—
70	C₂×C₄	—	4	4	no	—
71	C₂×C₄	—	12	2	no	—
72	C₂×C₄	—	12	4	no	—
73	C₂×C₄	Dic₃⋊C₄	1	4	yes	—
74	C₂×C₄	C₂×Dic₆	1	1	yes	characteristic
75	C₂×C₄	C₂×C₄×D₃	1	1	yes	characteristic
76	C₂×C₄	#237	1	2	yes	—
77	C₂×C₄	#239	1	2	yes	—
78	C₈	—	6	4	no	—
79	C₂×C₃	#382	1	1	yes	characteristic
80	C₂×C₃	#427	1	2	yes	—
81	C₂×C₃	#488	1	2	yes	—
82	C₂×C₃	#488	1	2	yes	—
83	C₂²	C₂×Dic₃⋊C₄	1	1	yes	characteristic
84	C₂²	#936	1	1	yes	characteristic
85	C₂²	#947	1	1	yes	characteristic
86	C₂²	#950	1	4	yes	—
87	C₄	—	2	4	no	—
88	C₄	—	8	2	no	—
89	C₄	—	24	2	no	—
90	C₄	#890	1	2	yes	—
91	C₄	#892	1	2	yes	—
92	C₃	#1835	1	1	yes	characteristic, complemented, Sylow
93	C₂	#4977	1	1	yes	characteristic
94	C₂	#4998	1	2	yes	—
95	C₂	#5056	1	2	yes	—
96	C₂	#5068	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×#67
commutator	C₂×C₃×C₄	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₃
Fitting	C₃×#382	C₂
socle, prime radical	C₂³×C₃	C₂×D₄
cosocle	C₂²×C₃×C₄	C₂³
hypercenter	#382	D₃
hypocenter	C₃	#1835
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #382

Complements

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

N	H ≅ G/N	type
12. C₄×C₃⋊C₈	C₄	semidirect
92. C₃	#1835	semidirect

Maximal subgroups

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

subgroup	index
C₃×#382	2
#4936	2
C₂×C₄×C₃⋊C₈	2
#4529	2
#1835	3