Subgroups — #69864

Order 384.

128 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₄×M₁₆	C₂	1	1	yes	maximal, characteristic
2	#4553	C₂	1	2	yes	maximal
3	#4466	C₂	1	2	yes	maximal, complemented
4	#4463	C₂	1	2	yes	maximal, complemented
5	#2165	—	3	1	no	maximal, Sylow
6	C₂×C₃×M₁₆	C₄	1	1	yes	characteristic
7	C₂×C₃×M₁₆	C₂²	1	1	yes	characteristic
8	C₂×C₃×C₄²	C₄	1	1	yes	characteristic
9	C₃×#97	C₄	1	1	yes	characteristic
10	C₃×#97	C₂²	1	1	yes	characteristic
11	C₃×C₄×C₈	C₄	1	1	yes	characteristic
12	C₃×C₄×C₈	C₂²	1	1	yes	characteristic
13	#812	C₂²	1	4	yes	—
14	C₄×M₁₆	D₃	1	1	yes	characteristic
15	#364	—	3	2	no	—
16	#349	—	3	2	no	—
17	#347	—	3	2	no	—
18	C₂²×C₃×C₄	—	2	1	no	—
19	C₂²×C₃×C₄	C₂×C₄	1	1	yes	characteristic
20	C₃×M₁₆	—	2	2	no	—
21	C₃×M₁₆	—	2	2	no	—
22	C₂×C₃×C₈	C₂×C₄	1	1	yes	characteristic
23	C₂×C₃×C₈	C₂×C₄	1	1	yes	characteristic
24	C₂×C₃×C₈	C₂×C₄	1	1	yes	characteristic
25	C₂×C₃×C₈	C₂³	1	1	yes	characteristic
26	C₃×C₄²	—	2	1	no	—
27	C₃×C₄²	C₂×C₄	1	1	yes	characteristic
28	C₃×C₄²	C₂×C₄	1	1	yes	characteristic
29	C₃⋊C₁₆	—	2	4	no	—
30	C₂×M₁₆	Dic₃	1	1	yes	characteristic
31	C₂×M₁₆	C₂×D₃	1	1	yes	characteristic
32	C₂×C₄²	Dic₃	1	1	yes	characteristic
33	M₃₂	—	3	4	no	—
34	#97	Dic₃	1	1	yes	characteristic
35	#97	C₂×D₃	1	1	yes	characteristic
36	C₄×C₈	Dic₃	1	1	yes	characteristic
37	C₄×C₈	C₂×D₃	1	1	yes	characteristic
38	C₂³×C₃	M₁₆	1	1	yes	characteristic
39	C₂×C₃×C₄	—	2	1	no	—
40	C₂×C₃×C₄	—	2	1	no	—
41	C₂×C₃×C₄	—	2	1	no	—
42	C₂×C₃×C₄	—	2	1	no	—
43	C₂×C₃×C₄	—	4	2	no	—
44	C₂×C₃×C₄	M₁₆	1	1	yes	characteristic
45	C₂×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
46	C₃×C₈	—	2	1	no	—
47	C₃×C₈	—	2	1	no	—
48	C₃×C₈	M₁₆	1	2	yes	—
49	C₃×C₈	#41	1	2	yes	—
50	C₂²×C₄	—	2	1	no	—
51	C₂²×C₄	C₂×Dic₃	1	1	yes	characteristic
52	M₁₆	—	2	2	no	—
53	M₁₆	—	2	2	no	—
54	C₂×C₈	C₂×Dic₃	1	1	yes	characteristic
55	C₂×C₈	C₂×Dic₃	1	1	yes	characteristic
56	C₂×C₈	C₂×Dic₃	1	1	yes	characteristic
57	C₂×C₈	C₂²×D₃	1	1	yes	characteristic
58	C₄²	—	2	1	no	—
59	C₄²	C₂×Dic₃	1	1	yes	characteristic
60	C₄²	C₂×Dic₃	1	1	yes	characteristic
61	C₁₆	—	6	4	no	—
62	C₂²×C₃	—	2	1	no	—
63	C₂²×C₃	—	4	1	no	—
64	C₂²×C₃	C₂×M₁₆	1	1	yes	characteristic
65	C₃×C₄	—	2	2	no	—
66	C₃×C₄	—	2	1	no	—
67	C₃×C₄	—	4	1	no	—
68	C₃×C₄	#117	1	1	yes	characteristic
69	C₃×C₄	C₂×M₁₆	1	1	yes	characteristic
70	C₂³	#208	1	1	yes	characteristic
71	C₂×C₄	—	2	1	no	—
72	C₂×C₄	—	2	1	no	—
73	C₂×C₄	—	2	1	no	—
74	C₂×C₄	—	2	1	no	—
75	C₂×C₄	—	4	2	no	—
76	C₂×C₄	#208	1	1	yes	characteristic
77	C₂×C₄	C₂²×Dic₃	1	1	yes	characteristic
78	C₈	—	2	1	no	—
79	C₈	—	2	1	no	—
80	C₈	#208	1	2	yes	—
81	C₈	#235	1	2	yes	—
82	C₂×C₃	—	2	1	no	—
83	C₂×C₃	—	4	1	no	—
84	C₂×C₃	#432	1	1	yes	characteristic
85	C₂²	—	2	1	no	—
86	C₂²	—	4	1	no	—
87	C₂²	C₂×#208	1	1	yes	characteristic
88	C₄	—	2	2	no	—
89	C₄	—	2	1	no	—
90	C₄	—	4	1	no	—
91	C₄	C₂×#208	1	1	yes	characteristic
92	C₄	#926	1	1	yes	characteristic
93	C₃	#2165	1	1	yes	characteristic, complemented, Sylow
94	C₂	—	2	1	no	—
95	C₂	—	4	1	no	—
96	C₂	#4926	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#926
commutator	C₂×C₃×C₄	C₂²×C₄
Frattini	C₂×C₈	C₂²×D₃
Fitting	C₃×C₄×M₁₆	C₂
socle	C₂×C₃	#432
prime radical	C₂³×C₃	M₁₆
cosocle	C₂×C₃×C₈	C₂³
hypercenter	C₄×M₁₆	D₃
hypocenter	C₃	#2165
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₄ ◁ #97 ◁ C₄×M₁₆

Complements

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

N	H ≅ G/N	type
3. #4466	C₂	semidirect
4. #4463	C₂	semidirect
93. C₃	#2165	semidirect

Maximal subgroups

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

subgroup	index
C₃×C₄×M₁₆	2
#4553	2
#4466	2
#4463	2
#2165	3