Finite Group Databasebeta

← C₂³×D₃×D₄ #87557

Subgroups — C₂³×D₃×D₄ #87557

Order 384.

10552 conjugacy classes of subgroups; only the 3557 normal subgroups are shown.

normal subgroups only

This group has 10552 conjugacy classes of subgroups — too many to draw. Only its 3557 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁵×D₃	C₂	1	2	yes	maximal, complemented
2	C₂³×C₃×D₄	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂³×#67	C₂	1	2	yes	maximal, complemented
4	C₂²×D₃×D₄	C₂	1	56	yes	maximal, direct product factor
5	C₂³×D₁₂	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂³×C₄×D₃	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂⁴×D₄	—	3	1	no	maximal, Sylow
8	C₂⁵×C₃	C₂²	1	2	yes	complemented
9	C₂⁴×D₃	—	2	32	no	—
10	C₂⁴×D₃	C₂²	1	1	yes	characteristic
11	C₂⁴×D₃	C₂²	1	56	yes	complemented
12	C₂⁴×D₃	C₂²	1	2	yes	complemented
13	C₂²×C₃×D₄	C₂²	1	28	yes	complemented
14	C₂³×C₃×C₄	C₂²	1	1	yes	characteristic, complemented
15	C₂²×#67	C₂²	1	56	yes	complemented
16	C₂³×Dic₃	C₂²	1	1	yes	characteristic, complemented
17	C₂×D₃×D₄	C₂²	1	448	yes	direct product factor
18	C₂²×D₁₂	C₂²	1	28	yes	complemented
19	C₂²×C₄×D₃	C₂²	1	28	yes	complemented
20	C₂⁶	—	3	2	no	—
21	C₂³×D₄	—	3	2	no	—
22	C₂³×D₄	—	3	56	no	—
23	C₂³×D₄	—	3	1	no	—
24	C₂³×D₄	D₃	1	1	yes	characteristic, direct product factor
25	C₂⁴×C₄	—	3	1	no	—
26	C₂⁴×C₃	—	2	16	no	—
27	C₂⁴×C₃	C₂³	1	1	yes	characteristic
28	C₂⁴×C₃	C₂³	1	28	yes	complemented
29	C₂³×D₃	—	2	16	no	—
30	C₂³×D₃	—	2	448	no	—
31	C₂³×D₃	D₄	1	16	yes	direct product factor
32	C₂³×D₃	C₂³	1	14	yes	—
33	C₂³×D₃	C₂³	1	224	yes	complemented
34	C₂³×D₃	C₂³	1	28	yes	complemented
35	C₂×C₃×D₄	C₂³	1	112	yes	complemented
36	C₂²×C₃×C₄	C₂³	1	14	yes	complemented
37	C₂×#67	C₂³	1	224	yes	complemented
38	C₂²×Dic₃	C₂³	1	14	yes	complemented
39	D₃×D₄	C₂³	1	512	yes	direct product factor
40	C₂×D₁₂	C₂³	1	112	yes	complemented
41	C₂×C₄×D₃	C₂³	1	112	yes	complemented
42	C₂⁵	—	3	1	no	—
43	C₂⁵	—	3	2	no	—
44	C₂⁵	—	3	56	no	—
45	C₂⁵	—	6	32	no	—
46	C₂⁵	C₂×D₃	1	2	yes	complemented
47	C₂²×D₄	—	3	56	no	—
48	C₂²×D₄	—	3	448	no	—
49	C₂²×D₄	—	3	28	no	—
50	C₂²×D₄	C₂×D₃	1	28	yes	direct product factor
51	C₂³×C₄	—	3	1	no	—
52	C₂³×C₄	—	3	28	no	—
53	C₂³×C₄	C₂×D₃	1	1	yes	characteristic, complemented
54	C₂³×C₃	—	2	112	no	—
55	C₂³×C₃	C₂×D₄	1	8	yes	complemented
56	C₂³×C₃	C₂⁴	1	7	yes	—
57	C₂³×C₃	C₂⁴	1	56	yes	complemented
58	C₂²×D₃	—	2	112	no	—
59	C₂²×D₃	—	2	896	no	—
60	C₂²×D₃	C₂×D₄	1	112	yes	direct product factor
61	C₂²×D₃	C₂⁴	1	28	yes	—
62	C₂²×D₃	C₂⁴	1	128	yes	complemented
63	C₂²×D₃	C₂⁴	1	56	yes	complemented
64	C₃×D₄	C₂⁴	1	64	yes	complemented
65	C₂×C₃×C₄	C₂⁴	1	28	yes	complemented
66	#67	C₂⁴	1	128	yes	complemented
67	C₂×Dic₃	C₂⁴	1	28	yes	complemented
68	D₁₂	C₂⁴	1	64	yes	complemented
69	C₄×D₃	C₂⁴	1	64	yes	complemented
70	C₂⁴	—	2	16	no	—
71	C₂⁴	—	3	16	no	—
72	C₂⁴	—	3	14	no	—
73	C₂⁴	—	3	28	no	—
74	C₂⁴	—	3	224	no	—
75	C₂⁴	—	6	448	no	—
76	C₂⁴	—	6	16	no	—
77	C₂⁴	C₂²×D₃	1	1	yes	characteristic
78	C₂⁴	C₂²×D₃	1	28	yes	complemented
79	C₂×D₄	—	3	224	no	—
80	C₂×D₄	—	3	512	no	—
81	C₂×D₄	—	3	112	no	—
82	C₂×D₄	C₂²×D₃	1	112	yes	direct product factor
83	C₂²×C₄	—	3	14	no	—
84	C₂²×C₄	—	3	112	no	—
85	C₂²×C₄	C₂²×D₃	1	14	yes	complemented
86	C₂²×C₃	—	2	112	no	—
87	C₂²×C₃	C₂²×D₄	1	28	yes	complemented
88	C₂²×C₃	C₂⁵	1	7	yes	—
89	C₂²×C₃	C₂⁵	1	16	yes	complemented
90	C₂×D₃	—	2	112	no	—
91	C₂×D₃	—	2	256	no	—
92	C₂×D₃	C₂²×D₄	1	112	yes	direct product factor
93	C₂×D₃	C₂⁵	1	8	yes	—
94	C₂×D₃	C₂⁵	1	16	yes	complemented
95	C₃×C₄	C₂⁵	1	8	yes	complemented
96	Dic₃	C₂⁵	1	8	yes	complemented
97	C₂³	—	2	112	no	—
98	C₂³	—	3	112	no	—
99	C₂³	—	3	28	no	—
100	C₂³	—	3	56	no	—
101	C₂³	—	3	128	no	—
102	C₂³	—	6	896	no	—
103	C₂³	—	6	112	no	—
104	C₂³	D₃×D₄	1	8	yes	direct product factor
105	C₂³	C₂³×D₃	1	7	yes	—
106	C₂³	C₂³×D₃	1	56	yes	complemented
107	D₄	—	3	128	no	—
108	D₄	—	3	64	no	—
109	D₄	C₂³×D₃	1	64	yes	direct product factor
110	C₂×C₄	—	3	28	no	—
111	C₂×C₄	—	3	64	no	—
112	C₂×C₄	C₂³×D₃	1	28	yes	complemented
113	C₂×C₃	—	2	16	no	—
114	C₂×C₃	C₂³×D₄	1	14	yes	complemented
115	C₂×C₃	C₂⁶	1	1	yes	characteristic
116	D₃	—	2	16	no	—
117	D₃	C₂³×D₄	1	16	yes	direct product factor
118	C₂²	—	2	112	no	—
119	C₂²	—	3	112	no	—
120	C₂²	—	3	8	no	—
121	C₂²	—	3	16	no	—
122	C₂²	—	6	256	no	—
123	C₂²	—	6	112	no	—
124	C₂²	C₂×D₃×D₄	1	28	yes	direct product factor
125	C₂²	C₂⁴×D₃	1	7	yes	—
126	C₂²	C₂⁴×D₃	1	16	yes	complemented
127	C₄	—	3	8	no	—
128	C₄	C₂⁴×D₃	1	8	yes	complemented
129	C₃	C₂⁴×D₄	1	1	yes	characteristic, complemented, Sylow
130	C₂	—	2	16	no	—
131	C₂	—	3	16	no	—
132	C₂	—	6	16	no	—
133	C₂	C₂²×D₃×D₄	1	14	yes	direct product factor
134	C₂	C₂⁵×D₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁴	C₂²×D₃
commutator, cosocle	C₂×C₃	C₂⁶
Frattini	C₂	C₂⁵×D₃
Fitting	C₂³×C₃×D₄	C₂
socle	C₂⁴×C₃	C₂³
hypercenter	C₂³×D₄	D₃
hypocenter	C₃	C₂⁴×D₄
LCS	G ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂⁴ ◁ C₂³×D₄

Complements

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

N	H ≅ G/N	type
1. C₂⁵×D₃	C₂	semidirect
2. C₂³×C₃×D₄	C₂	semidirect
3. C₂³×#67	C₂	semidirect
4. C₂²×D₃×D₄	C₂	direct (N × H)
5. C₂³×D₁₂	C₂	semidirect
6. C₂³×C₄×D₃	C₂	semidirect
8. C₂⁵×C₃	C₂²	semidirect
11. C₂⁴×D₃	C₂²	semidirect
12. C₂⁴×D₃	C₂²	semidirect
13. C₂²×C₃×D₄	C₂²	semidirect
14. C₂³×C₃×C₄	C₂²	semidirect
15. C₂²×#67	C₂²	semidirect
16. C₂³×Dic₃	C₂²	semidirect
17. C₂×D₃×D₄	C₂²	direct (N × H)
18. C₂²×D₁₂	C₂²	semidirect
19. C₂²×C₄×D₃	C₂²	semidirect
24. C₂³×D₄	D₃	direct (N × H)
28. C₂⁴×C₃	C₂³	semidirect
31. C₂³×D₃	D₄	direct (N × H)
33. C₂³×D₃	C₂³	semidirect
34. C₂³×D₃	C₂³	semidirect
35. C₂×C₃×D₄	C₂³	semidirect
36. C₂²×C₃×C₄	C₂³	semidirect
37. C₂×#67	C₂³	semidirect
38. C₂²×Dic₃	C₂³	semidirect
39. D₃×D₄	C₂³	direct (N × H)
40. C₂×D₁₂	C₂³	semidirect
41. C₂×C₄×D₃	C₂³	semidirect
46. C₂⁵	C₂×D₃	semidirect
50. C₂²×D₄	C₂×D₃	direct (N × H)
53. C₂³×C₄	C₂×D₃	semidirect
55. C₂³×C₃	C₂×D₄	semidirect
57. C₂³×C₃	C₂⁴	semidirect
60. C₂²×D₃	C₂×D₄	direct (N × H)
62. C₂²×D₃	C₂⁴	semidirect
63. C₂²×D₃	C₂⁴	semidirect
64. C₃×D₄	C₂⁴	semidirect
65. C₂×C₃×C₄	C₂⁴	semidirect
66. #67	C₂⁴	semidirect
67. C₂×Dic₃	C₂⁴	semidirect
68. D₁₂	C₂⁴	semidirect
69. C₄×D₃	C₂⁴	semidirect
78. C₂⁴	C₂²×D₃	semidirect
82. C₂×D₄	C₂²×D₃	direct (N × H)
85. C₂²×C₄	C₂²×D₃	semidirect
87. C₂²×C₃	C₂²×D₄	semidirect
89. C₂²×C₃	C₂⁵	semidirect
92. C₂×D₃	C₂²×D₄	direct (N × H)
94. C₂×D₃	C₂⁵	semidirect
95. C₃×C₄	C₂⁵	semidirect
96. Dic₃	C₂⁵	semidirect
104. C₂³	D₃×D₄	direct (N × H)
106. C₂³	C₂³×D₃	semidirect
109. D₄	C₂³×D₃	direct (N × H)
112. C₂×C₄	C₂³×D₃	semidirect
114. C₂×C₃	C₂³×D₄	semidirect
117. D₃	C₂³×D₄	direct (N × H)
124. C₂²	C₂×D₃×D₄	direct (N × H)
126. C₂²	C₂⁴×D₃	semidirect
128. C₄	C₂⁴×D₃	semidirect
129. C₃	C₂⁴×D₄	semidirect
133. C₂	C₂²×D₃×D₄	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂⁵×D₃	2
C₂³×C₃×D₄	2
C₂³×#67	2
C₂²×D₃×D₄	2
C₂³×D₁₂	2
C₂³×C₄×D₃	2
C₂⁴×D₄	3