Subgroups — C₂²×C₃×#117 #87305

Order 384.

1880 conjugacy classes of subgroups.

The subgroup lattice has 120 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁴×C₃×C₄	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×#117	C₂	1	24	yes	maximal, direct product factor
3	C₂²×C₃×C₄⋊C₄	C₂	1	2	yes	maximal, complemented
4	C₂²×C₃×C₂²⋊C₄	C₂	1	2	yes	maximal
5	C₂²×C₃×C₄²	C₂	1	2	yes	maximal, complemented
6	C₂²×#117	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
7	C₂⁵×C₃	C₂²	1	1	yes	characteristic
8	C₂³×C₃×C₄	C₄	1	16	yes	complemented
9	C₂³×C₃×C₄	C₂²	1	4	yes	—
10	C₂³×C₃×C₄	C₂²	1	1	yes	characteristic
11	C₂³×C₃×C₄	C₂²	1	12	yes	—
12	C₂³×C₃×C₄	C₂²	1	1	yes	characteristic
13	C₃×#117	C₂²	1	64	yes	direct product factor
14	C₂×C₃×C₄⋊C₄	C₂²	1	24	yes	complemented
15	C₂×C₃×C₂²⋊C₄	C₂²	1	24	yes	—
16	C₂×C₃×C₄²	C₂²	1	24	yes	complemented
17	C₂⁴×C₄	C₂×C₃	1	1	yes	characteristic
18	C₂×#117	C₂×C₃	1	24	yes	direct product factor
19	C₂²×C₄⋊C₄	C₂×C₃	1	2	yes	complemented
20	C₂²×C₂²⋊C₄	C₂×C₃	1	2	yes	—
21	C₂²×C₄²	C₂×C₃	1	2	yes	complemented
22	C₂⁴×C₃	—	2	8	no	—
23	C₂⁴×C₃	C₂×C₄	1	8	yes	—
24	C₂⁴×C₃	C₂³	1	6	yes	—
25	C₂⁴×C₃	C₂³	1	1	yes	characteristic
26	C₂²×C₃×C₄	—	2	16	no	—
27	C₂²×C₃×C₄	C₂×C₄	1	8	yes	—
28	C₂²×C₃×C₄	C₂×C₄	1	8	yes	—
29	C₂²×C₃×C₄	C₂×C₄	1	96	yes	complemented
30	C₂²×C₃×C₄	C₂³	1	24	yes	—
31	C₂²×C₃×C₄	C₂³	1	6	yes	—
32	C₂²×C₃×C₄	C₂³	1	6	yes	—
33	C₂²×C₃×C₄	C₂³	1	16	yes	—
34	C₃×C₄⋊C₄	C₂³	1	32	yes	complemented
35	C₃×C₂²⋊C₄	C₂³	1	32	yes	—
36	C₃×C₄²	C₂³	1	32	yes	complemented
37	C₂⁵	C₂²×C₃	1	1	yes	characteristic
38	C₂³×C₄	C₃×C₄	1	16	yes	complemented
39	C₂³×C₄	C₂²×C₃	1	4	yes	—
40	C₂³×C₄	C₂²×C₃	1	1	yes	characteristic
41	C₂³×C₄	C₂²×C₃	1	1	yes	characteristic
42	C₂³×C₄	C₂²×C₃	1	12	yes	—
43	#117	C₂²×C₃	1	64	yes	direct product factor
44	C₂×C₄⋊C₄	C₂²×C₃	1	24	yes	complemented
45	C₂×C₂²⋊C₄	C₂²×C₃	1	24	yes	—
46	C₂×C₄²	C₂²×C₃	1	24	yes	complemented
47	C₂³×C₃	—	2	32	no	—
48	C₂³×C₃	—	2	24	no	—
49	C₂³×C₃	C₂²×C₄	1	24	yes	—
50	C₂³×C₃	C₂²×C₄	1	4	yes	—
51	C₂³×C₃	#41	1	8	yes	—
52	C₂³×C₃	C₂⁴	1	4	yes	—
53	C₂³×C₃	C₂⁴	1	3	yes	—
54	C₂×C₃×C₄	—	2	48	no	—
55	C₂×C₃×C₄	C₂²×C₄	1	24	yes	—
56	C₂×C₃×C₄	C₂²×C₄	1	24	yes	—
57	C₂×C₃×C₄	C₂²×C₄	1	64	yes	complemented
58	C₂×C₃×C₄	C₂⁴	1	16	yes	—
59	C₂×C₃×C₄	C₂⁴	1	4	yes	—
60	C₂×C₃×C₄	C₂⁴	1	4	yes	—
61	C₂⁴	—	2	8	no	—
62	C₂⁴	C₂×C₃×C₄	1	8	yes	—
63	C₂⁴	C₂³×C₃	1	6	yes	—
64	C₂⁴	C₂³×C₃	1	1	yes	characteristic
65	C₂²×C₄	—	2	16	no	—
66	C₂²×C₄	C₂×C₃×C₄	1	8	yes	—
67	C₂²×C₄	C₂×C₃×C₄	1	8	yes	—
68	C₂²×C₄	C₂×C₃×C₄	1	96	yes	complemented
69	C₂²×C₄	C₂³×C₃	1	24	yes	—
70	C₂²×C₄	C₂³×C₃	1	6	yes	—
71	C₂²×C₄	C₂³×C₃	1	6	yes	—
72	C₂²×C₄	C₂³×C₃	1	16	yes	—
73	C₄⋊C₄	C₂³×C₃	1	32	yes	complemented
74	C₂²⋊C₄	C₂³×C₃	1	32	yes	—
75	C₄²	C₂³×C₃	1	32	yes	complemented
76	C₂²×C₃	—	2	48	no	—
77	C₂²×C₃	—	2	8	no	—
78	C₂²×C₃	#117	1	16	yes	direct product factor
79	C₂²×C₃	C₂³×C₄	1	8	yes	—
80	C₂²×C₃	C₂³×C₄	1	6	yes	—
81	C₂²×C₃	C₂×#41	1	12	yes	—
82	C₂²×C₃	C₂⁵	1	1	yes	characteristic
83	C₃×C₄	—	2	16	no	—
84	C₃×C₄	C₂³×C₄	1	8	yes	—
85	C₃×C₄	C₂³×C₄	1	8	yes	—
86	C₂³	—	2	32	no	—
87	C₂³	—	2	24	no	—
88	C₂³	C₂²×C₃×C₄	1	24	yes	—
89	C₂³	C₂²×C₃×C₄	1	4	yes	—
90	C₂³	C₃×#41	1	8	yes	—
91	C₂³	C₂⁴×C₃	1	4	yes	—
92	C₂³	C₂⁴×C₃	1	3	yes	—
93	C₂×C₄	—	2	48	no	—
94	C₂×C₄	C₂²×C₃×C₄	1	24	yes	—
95	C₂×C₄	C₂²×C₃×C₄	1	24	yes	—
96	C₂×C₄	C₂²×C₃×C₄	1	64	yes	complemented
97	C₂×C₄	C₂⁴×C₃	1	16	yes	—
98	C₂×C₄	C₂⁴×C₃	1	4	yes	—
99	C₂×C₄	C₂⁴×C₃	1	4	yes	—
100	C₂×C₃	—	2	8	no	—
101	C₂×C₃	C₂×#117	1	12	yes	direct product factor
102	C₂×C₃	C₂⁴×C₄	1	1	yes	characteristic
103	C₂×C₃	C₂²×#41	1	2	yes	—
104	C₂²	—	2	8	no	—
105	C₂²	—	2	48	no	—
106	C₂²	C₃×#117	1	16	yes	direct product factor
107	C₂²	C₂³×C₃×C₄	1	8	yes	—
108	C₂²	C₂³×C₃×C₄	1	6	yes	—
109	C₂²	C₂×C₃×#41	1	12	yes	—
110	C₂²	C₂⁵×C₃	1	1	yes	characteristic
111	C₄	—	2	16	no	—
112	C₄	C₂³×C₃×C₄	1	8	yes	—
113	C₄	C₂³×C₃×C₄	1	8	yes	—
114	C₃	C₂²×#117	1	1	yes	characteristic, direct product factor, Sylow
115	C₂	—	2	8	no	—
116	C₂	C₂×C₃×#117	1	12	yes	direct product factor
117	C₂	C₂⁴×C₃×C₄	1	1	yes	characteristic
118	C₂	C₂²×C₃×#41	1	2	yes	—

	subgroup	G/H
center	C₂³×C₃×C₄	C₂²
commutator	C₂	C₂⁴×C₃×C₄
Frattini, cosocle	C₂²	C₂⁵×C₃
socle	C₂⁴×C₃	C₂³
prime radical	C₂⁵×C₃	C₂²
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₃×C₄ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₃×#117	C₂	direct (N × H)
3. C₂²×C₃×C₄⋊C₄	C₂	semidirect
5. C₂²×C₃×C₄²	C₂	semidirect
6. C₂²×#117	C₃	direct (N × H)
8. C₂³×C₃×C₄	C₄	semidirect
13. C₃×#117	C₂²	direct (N × H)
14. C₂×C₃×C₄⋊C₄	C₂²	semidirect
16. C₂×C₃×C₄²	C₂²	semidirect
18. C₂×#117	C₂×C₃	direct (N × H)
19. C₂²×C₄⋊C₄	C₂×C₃	semidirect
21. C₂²×C₄²	C₂×C₃	semidirect
29. C₂²×C₃×C₄	C₂×C₄	semidirect
34. C₃×C₄⋊C₄	C₂³	semidirect
36. C₃×C₄²	C₂³	semidirect
38. C₂³×C₄	C₃×C₄	semidirect
43. #117	C₂²×C₃	direct (N × H)
44. C₂×C₄⋊C₄	C₂²×C₃	semidirect
46. C₂×C₄²	C₂²×C₃	semidirect
57. C₂×C₃×C₄	C₂²×C₄	semidirect
68. C₂²×C₄	C₂×C₃×C₄	semidirect
73. C₄⋊C₄	C₂³×C₃	semidirect
75. C₄²	C₂³×C₃	semidirect
78. C₂²×C₃	#117	direct (N × H)
96. C₂×C₄	C₂²×C₃×C₄	semidirect
101. C₂×C₃	C₂×#117	direct (N × H)
106. C₂²	C₃×#117	direct (N × H)
114. C₃	C₂²×#117	direct (N × H)
116. C₂	C₂×C₃×#117	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂⁴×C₃×C₄	2
C₂×C₃×#117	2
C₂²×C₃×C₄⋊C₄	2
C₂²×C₃×C₂²⋊C₄	2
C₂²×C₃×C₄²	2
C₂²×#117	3

Subgroups — C22×C3×#117 #87305

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₃×#117 #87305