Subgroups — #67544

Order 384.

108 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#5058	C₂	1	1	yes	maximal, characteristic
2	C₃×#354	C₂	1	1	yes	maximal, characteristic, complemented
3	#4542	C₂	1	1	yes	maximal, characteristic, complemented
4	#1406	—	3	1	no	maximal, Sylow
5	C₃×#124	C₂²	1	1	yes	characteristic
6	C₂×#215	—	2	1	no	—
7	C₂×#213	—	2	1	no	—
8	#874	C₄	1	1	yes	characteristic, complemented
9	C₃×#99	—	2	1	no	—
10	#836	—	2	1	no	—
11	#803	C₄	1	1	yes	characteristic, complemented
12	#496	—	3	1	no	—
13	#355	—	3	1	no	—
14	#354	D₃	1	1	yes	characteristic, complemented
15	C₂×C₃×Q₈	D₄	1	1	yes	characteristic
16	C₂×C₃×D₄	D₄	1	1	yes	characteristic
17	C₂×D₁₂	—	2	1	no	—
18	C₂×D₁₂	—	4	1	no	—
19	C₃×C₂²⋊C₄	—	4	1	no	—
20	C₃×C₂²⋊C₄	—	4	1	no	—
21	C₃×C₄²	C₂×C₄	1	1	yes	characteristic
22	#215	—	4	2	no	—
23	#213	—	4	2	no	—
24	#208	—	4	1	no	—
25	C₂×C₃⋊C₈	—	2	1	no	—
26	C₂×SD₁₆	—	6	1	no	—
27	C₂×D₈	—	6	1	no	—
28	#127	—	3	1	no	—
29	#124	C₂×D₃	1	1	yes	characteristic
30	#101	—	6	1	no	—
31	#99	—	2	1	no	—
32	#97	—	3	1	no	—
33	C₂³×C₃	—	2	1	no	—
34	C₂²×D₃	—	4	1	no	—
35	C₃×Q₈	—	4	1	no	—
36	C₃×D₄	—	4	1	no	—
37	C₂×C₃×C₄	—	2	1	no	—
38	C₂×C₃×C₄	—	2	1	no	—
39	C₂×C₃×C₄	—	8	1	no	—
40	C₂×C₃×C₄	C₂²⋊C₄	1	1	yes	characteristic
41	D₁₂	—	4	1	no	—
42	D₁₂	—	4	1	no	—
43	D₁₂	—	8	2	no	—
44	C₃⋊C₈	—	2	2	no	—
45	C₃⋊C₈	—	8	1	no	—
46	C₂×Q₈	D₁₂	1	1	yes	characteristic
47	C₂×D₄	—	6	1	no	—
48	C₂×D₄	—	12	1	no	—
49	C₂×D₄	#67	1	1	yes	characteristic
50	SD₁₆	—	12	2	no	—
51	D₈	—	12	2	no	—
52	M₁₆	—	12	1	no	—
53	C₂×C₈	—	6	1	no	—
54	C₂²⋊C₄	—	4	1	no	—
55	C₂²⋊C₄	—	4	1	no	—
56	C₄²	C₄×D₃	1	1	yes	characteristic
57	C₂²×C₃	—	4	1	no	—
58	C₂²×C₃	—	8	1	no	—
59	C₂²×C₃	#99	1	1	yes	characteristic
60	C₂×D₃	—	8	2	no	—
61	C₂×D₃	—	8	1	no	—
62	C₃×C₄	—	2	1	no	—
63	C₃×C₄	—	4	1	no	—
64	C₃×C₄	—	4	1	no	—
65	C₃×C₄	—	16	1	no	—
66	C₂³	—	2	1	no	—
67	C₂³	—	12	1	no	—
68	Q₈	—	4	1	no	—
69	D₄	—	4	1	no	—
70	D₄	—	12	1	no	—
71	D₄	—	12	1	no	—
72	D₄	—	24	2	no	—
73	C₂×C₄	—	2	1	no	—
74	C₂×C₄	—	2	1	no	—
75	C₂×C₄	—	8	1	no	—
76	C₂×C₄	(C₂×D₃)⋊C₄	1	1	yes	characteristic
77	C₈	—	6	2	no	—
78	C₈	—	24	1	no	—
79	C₂×C₃	—	2	1	no	—
80	C₂×C₃	—	8	1	no	—
81	C₂×C₃	#353	1	1	yes	characteristic
82	D₃	—	16	1	no	—
83	C₂²	—	4	1	no	—
84	C₂²	—	8	1	no	—
85	C₂²	—	24	2	no	—
86	C₂²	—	24	1	no	—
87	C₂²	#806	1	1	yes	characteristic
88	C₄	—	2	1	no	—
89	C₄	—	4	1	no	—
90	C₄	—	4	1	no	—
91	C₄	—	16	1	no	—
92	C₃	#1406	1	1	yes	characteristic, complemented, Sylow
93	C₂	—	2	1	no	—
94	C₂	—	8	1	no	—
95	C₂	—	48	1	no	—
96	C₂	#4474	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#4474
commutator	C₃×C₄²	C₂×C₄
Frattini	#124	C₂×D₃
Fitting	C₃×#354	C₂
socle	C₂×C₃	#353
prime radical	#5058	C₂
cosocle	C₃×#124	C₂²
hypercenter	#354	D₃
hypocenter	C₃	#1406
LCS	G ▷ C₃×C₄² ▷ C₂×C₃×C₄ ▷ C₂²×C₃ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂×C₄ ◁ #124 ◁ #354

Complements

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

N	H ≅ G/N	type
2. C₃×#354	C₂	semidirect
3. #4542	C₂	semidirect
8. #874	C₄	semidirect
11. #803	C₄	semidirect
14. #354	D₃	semidirect
92. C₃	#1406	semidirect

Maximal subgroups

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

subgroup	index
#5058	2
C₃×#354	2
#4542	2
#1406	3