Subgroups — C₂×C₃×D₄² #87346

Order 384.

2536 conjugacy classes of subgroups.

The subgroup lattice has 176 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₃×D₄	C₂	1	4	yes	maximal, complemented
2	C₃×D₄²	C₂	1	16	yes	maximal, direct product factor
3	C₂×C₃×#127	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₃×#121	C₂	1	4	yes	maximal, complemented
5	C₂×C₃×#120	C₂	1	4	yes	maximal, complemented
6	C₂×C₃×C₄×D₄	C₂	1	2	yes	maximal, complemented
7	C₂×D₄²	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
8	C₂⁵×C₃	C₂²	1	4	yes	complemented
9	C₂²×C₃×D₄	—	2	32	no	—
10	C₂²×C₃×D₄	C₂²	1	2	yes	—
11	C₂²×C₃×D₄	C₂²	1	8	yes	complemented
12	C₂²×C₃×D₄	C₂²	1	4	yes	complemented
13	C₂²×C₃×D₄	C₂²	1	2	yes	complemented
14	C₂²×C₃×D₄	C₂²	1	32	yes	complemented
15	C₂³×C₃×C₄	C₂²	1	4	yes	complemented
16	C₃×#127	C₂²	1	8	yes	complemented
17	C₃×#121	C₂²	1	32	yes	complemented
18	C₃×#120	C₂²	1	32	yes	complemented
19	C₃×C₄×D₄	C₂²	1	16	yes	complemented
20	C₂×C₃×C₄⋊C₄	C₂²	1	2	yes	complemented
21	C₂×C₃×C₂²⋊C₄	C₂²	1	4	yes	complemented
22	C₂×C₃×C₂²⋊C₄	C₂²	1	4	yes	complemented
23	C₂×C₃×C₄²	C₂²	1	1	yes	characteristic, complemented
24	C₂³×D₄	C₂×C₃	1	4	yes	complemented
25	D₄²	C₂×C₃	1	16	yes	direct product factor
26	C₂×#127	C₂×C₃	1	1	yes	characteristic, complemented
27	C₂×#121	C₂×C₃	1	4	yes	complemented
28	C₂×#120	C₂×C₃	1	4	yes	complemented
29	C₂×C₄×D₄	C₂×C₃	1	2	yes	complemented
30	C₂⁴×C₃	—	2	32	no	—
31	C₂⁴×C₃	—	4	8	no	—
32	C₂⁴×C₃	C₂³	1	4	yes	—
33	C₂⁴×C₃	C₂³	1	4	yes	—
34	C₂⁴×C₃	C₂³	1	16	yes	complemented
35	C₂×C₃×D₄	—	2	32	no	—
36	C₂×C₃×D₄	—	2	128	no	—
37	C₂×C₃×D₄	—	2	16	no	—
38	C₂×C₃×D₄	—	4	4	no	—
39	C₂×C₃×D₄	D₄	1	16	yes	direct product factor
40	C₂×C₃×D₄	C₂³	1	8	yes	—
41	C₂×C₃×D₄	C₂³	1	32	yes	complemented
42	C₂×C₃×D₄	C₂³	1	16	yes	complemented
43	C₂×C₃×D₄	C₂³	1	8	yes	complemented
44	C₂²×C₃×C₄	—	2	16	no	—
45	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
46	C₂²×C₃×C₄	C₂³	1	4	yes	—
47	C₂²×C₃×C₄	C₂³	1	2	yes	—
48	C₂²×C₃×C₄	C₂³	1	16	yes	complemented
49	C₃×C₄⋊C₄	C₂³	1	8	yes	complemented
50	C₃×C₂²⋊C₄	C₂³	1	16	yes	complemented
51	C₃×C₂²⋊C₄	C₂³	1	16	yes	complemented
52	C₃×C₄²	C₂³	1	4	yes	complemented
53	C₂⁵	C₂²×C₃	1	4	yes	complemented
54	C₂²×D₄	—	2	32	no	—
55	C₂²×D₄	C₂²×C₃	1	2	yes	—
56	C₂²×D₄	C₂²×C₃	1	8	yes	complemented
57	C₂²×D₄	C₂²×C₃	1	4	yes	complemented
58	C₂²×D₄	C₂²×C₃	1	2	yes	complemented
59	C₂²×D₄	C₂²×C₃	1	32	yes	complemented
60	C₂³×C₄	C₂²×C₃	1	4	yes	complemented
61	#127	C₂²×C₃	1	8	yes	complemented
62	#121	C₂²×C₃	1	32	yes	complemented
63	#120	C₂²×C₃	1	32	yes	complemented
64	C₄×D₄	C₂²×C₃	1	16	yes	complemented
65	C₂×C₄⋊C₄	C₂²×C₃	1	2	yes	complemented
66	C₂×C₂²⋊C₄	C₂²×C₃	1	4	yes	complemented
67	C₂×C₂²⋊C₄	C₂²×C₃	1	4	yes	complemented
68	C₂×C₄²	C₂²×C₃	1	1	yes	characteristic, complemented
69	C₂³×C₃	—	2	8	no	—
70	C₂³×C₃	—	2	8	no	—
71	C₂³×C₃	—	2	16	no	—
72	C₂³×C₃	—	2	64	no	—
73	C₂³×C₃	—	2	8	no	—
74	C₂³×C₃	—	4	16	no	—
75	C₂³×C₃	—	4	64	no	—
76	C₂³×C₃	C₂×D₄	1	16	yes	complemented
77	C₂³×C₃	C₂⁴	1	8	yes	—
78	C₂³×C₃	C₂⁴	1	8	yes	—
79	C₂³×C₃	C₂⁴	1	1	yes	characteristic
80	C₃×D₄	—	2	64	no	—
81	C₃×D₄	—	2	32	no	—
82	C₃×D₄	—	4	8	no	—
83	C₃×D₄	C₂×D₄	1	32	yes	direct product factor
84	C₂×C₃×C₄	—	2	8	no	—
85	C₂×C₃×C₄	—	2	2	no	—
86	C₂×C₃×C₄	—	2	32	no	—
87	C₂×C₃×C₄	C₂×D₄	1	8	yes	complemented
88	C₂×C₃×C₄	C₂⁴	1	2	yes	—
89	C₂×C₃×C₄	C₂⁴	1	8	yes	—
90	C₂×C₃×C₄	C₂⁴	1	4	yes	—
91	C₂⁴	—	2	32	no	—
92	C₂⁴	—	4	8	no	—
93	C₂⁴	C₂³×C₃	1	4	yes	—
94	C₂⁴	C₂³×C₃	1	4	yes	—
95	C₂⁴	C₂³×C₃	1	16	yes	complemented
96	C₂×D₄	—	2	32	no	—
97	C₂×D₄	—	2	128	no	—
98	C₂×D₄	—	2	16	no	—
99	C₂×D₄	—	4	4	no	—
100	C₂×D₄	C₃×D₄	1	16	yes	direct product factor
101	C₂×D₄	C₂³×C₃	1	8	yes	—
102	C₂×D₄	C₂³×C₃	1	32	yes	complemented
103	C₂×D₄	C₂³×C₃	1	16	yes	complemented
104	C₂×D₄	C₂³×C₃	1	8	yes	complemented
105	C₂²×C₄	—	2	16	no	—
106	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
107	C₂²×C₄	C₂³×C₃	1	4	yes	—
108	C₂²×C₄	C₂³×C₃	1	2	yes	—
109	C₂²×C₄	C₂³×C₃	1	16	yes	complemented
110	C₄⋊C₄	C₂³×C₃	1	8	yes	complemented
111	C₂²⋊C₄	C₂³×C₃	1	16	yes	complemented
112	C₂²⋊C₄	C₂³×C₃	1	16	yes	complemented
113	C₄²	C₂³×C₃	1	4	yes	complemented
114	C₂²×C₃	—	2	32	no	—
115	C₂²×C₃	—	2	8	no	—
116	C₂²×C₃	—	2	8	no	—
117	C₂²×C₃	—	2	16	no	—
118	C₂²×C₃	—	2	8	no	—
119	C₂²×C₃	—	4	64	no	—
120	C₂²×C₃	—	4	16	no	—
121	C₂²×C₃	C₂²×D₄	1	4	yes	—
122	C₂²×C₃	C₂²×D₄	1	16	yes	complemented
123	C₂²×C₃	H(5,𝔽₂)	1	2	yes	—
124	C₂²×C₃	C₂⁵	1	1	yes	characteristic
125	C₃×C₄	—	2	8	no	—
126	C₃×C₄	—	2	2	no	—
127	C₃×C₄	C₂²×D₄	1	8	yes	complemented
128	C₂³	—	2	8	no	—
129	C₂³	—	2	8	no	—
130	C₂³	—	2	16	no	—
131	C₂³	—	2	64	no	—
132	C₂³	—	2	8	no	—
133	C₂³	—	4	16	no	—
134	C₂³	—	4	64	no	—
135	C₂³	C₂×C₃×D₄	1	16	yes	complemented
136	C₂³	C₂⁴×C₃	1	8	yes	—
137	C₂³	C₂⁴×C₃	1	8	yes	—
138	C₂³	C₂⁴×C₃	1	1	yes	characteristic
139	D₄	—	2	64	no	—
140	D₄	—	2	32	no	—
141	D₄	—	4	8	no	—
142	D₄	C₂×C₃×D₄	1	32	yes	direct product factor
143	C₂×C₄	—	2	8	no	—
144	C₂×C₄	—	2	2	no	—
145	C₂×C₄	—	2	32	no	—
146	C₂×C₄	C₂×C₃×D₄	1	8	yes	complemented
147	C₂×C₄	C₂⁴×C₃	1	2	yes	—
148	C₂×C₄	C₂⁴×C₃	1	8	yes	—
149	C₂×C₄	C₂⁴×C₃	1	4	yes	—
150	C₂×C₃	—	2	16	no	—
151	C₂×C₃	—	4	8	no	—
152	C₂×C₃	D₄²	1	4	yes	direct product factor
153	C₂×C₃	C₂³×D₄	1	2	yes	—
154	C₂×C₃	C₂×H(5,𝔽₂)	1	1	yes	characteristic
155	C₂²	—	2	8	no	—
156	C₂²	—	2	8	no	—
157	C₂²	—	2	32	no	—
158	C₂²	—	2	16	no	—
159	C₂²	—	2	8	no	—
160	C₂²	—	4	64	no	—
161	C₂²	—	4	16	no	—
162	C₂²	C₂²×C₃×D₄	1	4	yes	—
163	C₂²	C₂²×C₃×D₄	1	16	yes	complemented
164	C₂²	C₃×H(5,𝔽₂)	1	2	yes	—
165	C₂²	C₂⁵×C₃	1	1	yes	characteristic
166	C₄	—	2	8	no	—
167	C₄	—	2	2	no	—
168	C₄	C₂²×C₃×D₄	1	8	yes	complemented
169	C₃	C₂×D₄²	1	1	yes	characteristic, direct product factor, Sylow
170	C₂	—	2	16	no	—
171	C₂	—	4	8	no	—
172	C₂	C₃×D₄²	1	4	yes	direct product factor
173	C₂	C₂³×C₃×D₄	1	2	yes	—
174	C₂	C₂×C₃×H(5,𝔽₂)	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
1. C₂³×C₃×D₄	C₂	semidirect
2. C₃×D₄²	C₂	direct (N × H)
3. C₂×C₃×#127	C₂	semidirect
4. C₂×C₃×#121	C₂	semidirect
5. C₂×C₃×#120	C₂	semidirect
6. C₂×C₃×C₄×D₄	C₂	semidirect
7. C₂×D₄²	C₃	direct (N × H)
8. C₂⁵×C₃	C₂²	semidirect
11. C₂²×C₃×D₄	C₂²	semidirect
12. C₂²×C₃×D₄	C₂²	semidirect
13. C₂²×C₃×D₄	C₂²	semidirect
14. C₂²×C₃×D₄	C₂²	semidirect
15. C₂³×C₃×C₄	C₂²	semidirect
16. C₃×#127	C₂²	semidirect
17. C₃×#121	C₂²	semidirect
18. C₃×#120	C₂²	semidirect
19. C₃×C₄×D₄	C₂²	semidirect
20. C₂×C₃×C₄⋊C₄	C₂²	semidirect
21. C₂×C₃×C₂²⋊C₄	C₂²	semidirect
22. C₂×C₃×C₂²⋊C₄	C₂²	semidirect
23. C₂×C₃×C₄²	C₂²	semidirect
24. C₂³×D₄	C₂×C₃	semidirect
25. D₄²	C₂×C₃	direct (N × H)
26. C₂×#127	C₂×C₃	semidirect
27. C₂×#121	C₂×C₃	semidirect
28. C₂×#120	C₂×C₃	semidirect
29. C₂×C₄×D₄	C₂×C₃	semidirect
34. C₂⁴×C₃	C₂³	semidirect
39. C₂×C₃×D₄	D₄	direct (N × H)
41. C₂×C₃×D₄	C₂³	semidirect
42. C₂×C₃×D₄	C₂³	semidirect
43. C₂×C₃×D₄	C₂³	semidirect
48. C₂²×C₃×C₄	C₂³	semidirect
49. C₃×C₄⋊C₄	C₂³	semidirect
50. C₃×C₂²⋊C₄	C₂³	semidirect
51. C₃×C₂²⋊C₄	C₂³	semidirect
52. C₃×C₄²	C₂³	semidirect
53. C₂⁵	C₂²×C₃	semidirect
56. C₂²×D₄	C₂²×C₃	semidirect
57. C₂²×D₄	C₂²×C₃	semidirect
58. C₂²×D₄	C₂²×C₃	semidirect
59. C₂²×D₄	C₂²×C₃	semidirect
60. C₂³×C₄	C₂²×C₃	semidirect
61. #127	C₂²×C₃	semidirect
62. #121	C₂²×C₃	semidirect
63. #120	C₂²×C₃	semidirect
64. C₄×D₄	C₂²×C₃	semidirect
65. C₂×C₄⋊C₄	C₂²×C₃	semidirect
66. C₂×C₂²⋊C₄	C₂²×C₃	semidirect
67. C₂×C₂²⋊C₄	C₂²×C₃	semidirect
68. C₂×C₄²	C₂²×C₃	semidirect
76. C₂³×C₃	C₂×D₄	semidirect
83. C₃×D₄	C₂×D₄	direct (N × H)
87. C₂×C₃×C₄	C₂×D₄	semidirect
95. C₂⁴	C₂³×C₃	semidirect
100. C₂×D₄	C₃×D₄	direct (N × H)
102. C₂×D₄	C₂³×C₃	semidirect
103. C₂×D₄	C₂³×C₃	semidirect
104. C₂×D₄	C₂³×C₃	semidirect
109. C₂²×C₄	C₂³×C₃	semidirect
110. C₄⋊C₄	C₂³×C₃	semidirect
111. C₂²⋊C₄	C₂³×C₃	semidirect
112. C₂²⋊C₄	C₂³×C₃	semidirect
113. C₄²	C₂³×C₃	semidirect
122. C₂²×C₃	C₂²×D₄	semidirect
127. C₃×C₄	C₂²×D₄	semidirect
135. C₂³	C₂×C₃×D₄	semidirect
142. D₄	C₂×C₃×D₄	direct (N × H)
146. C₂×C₄	C₂×C₃×D₄	semidirect
152. C₂×C₃	D₄²	direct (N × H)
163. C₂²	C₂²×C₃×D₄	semidirect
168. C₄	C₂²×C₃×D₄	semidirect
169. C₃	C₂×D₄²	direct (N × H)
172. C₂	C₃×D₄²	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×C₃×D₄	2
C₃×D₄²	2
C₂×C₃×#127	2
C₂×C₃×#121	2
C₂×C₃×#120	2
C₂×C₃×C₄×D₄	2
C₂×D₄²	3

Subgroups — C2×C3×D42 #87346

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×D₄² #87346