Subgroups — C₂²×C₃×C₄×D₄ #87306

Order 384.

2528 conjugacy classes of subgroups.

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

	subgroup	G/H
center	C₂³×C₃×C₄	C₂²
commutator	C₂	C₂⁴×C₃×C₄
Frattini, cosocle	C₂²	C₂⁵×C₃
socle	C₂⁴×C₃	C₂³
prime radical	C₂³×C₃×D₄	C₂
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. C₂⁴×C₃×C₄	C₂	semidirect
3. C₂×C₃×C₄×D₄	C₂	direct (N × H)
4. C₂²×C₃×C₄⋊C₄	C₂	semidirect
5. C₂²×C₃×C₂²⋊C₄	C₂	semidirect
6. C₂²×C₃×C₄²	C₂	semidirect
7. C₂²×C₄×D₄	C₃	direct (N × H)
9. C₂²×C₃×D₄	C₄	direct (N × H)
16. C₂³×C₃×C₄	C₂²	semidirect
17. C₃×C₄×D₄	C₂²	direct (N × H)
18. C₂×C₃×C₄⋊C₄	C₂²	semidirect
19. C₂×C₃×C₂²⋊C₄	C₂²	semidirect
20. C₂×C₃×C₄²	C₂²	semidirect
22. C₂⁴×C₄	C₂×C₃	semidirect
23. C₂×C₄×D₄	C₂×C₃	direct (N × H)
24. C₂²×C₄⋊C₄	C₂×C₃	semidirect
25. C₂²×C₂²⋊C₄	C₂×C₃	semidirect
26. C₂²×C₄²	C₂×C₃	semidirect
29. C₂⁴×C₃	C₂×C₄	semidirect
32. C₂×C₃×D₄	C₂×C₄	direct (N × H)
37. C₂²×C₃×C₄	C₂×C₄	semidirect
38. C₂²×C₃×C₄	D₄	direct (N × H)
43. C₂²×C₃×C₄	C₂³	semidirect
44. C₃×C₄⋊C₄	C₂³	semidirect
45. C₃×C₂²⋊C₄	C₂³	semidirect
46. C₃×C₄²	C₂³	semidirect
48. C₂²×D₄	C₃×C₄	direct (N × H)
55. C₂³×C₄	C₂²×C₃	semidirect
56. C₄×D₄	C₂²×C₃	direct (N × H)
57. C₂×C₄⋊C₄	C₂²×C₃	semidirect
58. C₂×C₂²⋊C₄	C₂²×C₃	semidirect
59. C₂×C₄²	C₂²×C₃	semidirect
64. C₂³×C₃	C₂²×C₄	semidirect
69. C₃×D₄	C₂²×C₄	direct (N × H)
73. C₂×C₃×C₄	C₂²×C₄	semidirect
74. C₂×C₃×C₄	C₂×D₄	direct (N × H)
81. C₂⁴	C₂×C₃×C₄	semidirect
84. C₂×D₄	C₂×C₃×C₄	direct (N × H)
89. C₂²×C₄	C₂×C₃×C₄	semidirect
90. C₂²×C₄	C₃×D₄	direct (N × H)
95. C₂²×C₄	C₂³×C₃	semidirect
96. C₄⋊C₄	C₂³×C₃	semidirect
97. C₂²⋊C₄	C₂³×C₃	semidirect
98. C₄²	C₂³×C₃	semidirect
102. C₂²×C₃	C₄×D₄	direct (N × H)
104. C₂²×C₃	C₂³×C₄	semidirect
110. C₃×C₄	C₂³×C₄	semidirect
111. C₃×C₄	C₂²×D₄	direct (N × H)
116. C₂³	C₂²×C₃×C₄	semidirect
121. D₄	C₂²×C₃×C₄	direct (N × H)
125. C₂×C₄	C₂²×C₃×C₄	semidirect
126. C₂×C₄	C₂×C₃×D₄	direct (N × H)
132. C₂×C₃	C₂×C₄×D₄	direct (N × H)
139. C₂²	C₃×C₄×D₄	direct (N × H)
141. C₂²	C₂³×C₃×C₄	semidirect
147. C₄	C₂³×C₃×C₄	semidirect
148. C₄	C₂²×C₃×D₄	direct (N × H)
149. C₃	C₂²×C₄×D₄	direct (N × H)
151. C₂	C₂×C₃×C₄×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
C₂³×C₃×D₄	2
C₂⁴×C₃×C₄	2
C₂×C₃×C₄×D₄	2
C₂²×C₃×C₄⋊C₄	2
C₂²×C₃×C₂²⋊C₄	2
C₂²×C₃×C₄²	2
C₂²×C₄×D₄	3

Subgroups — C22×C3×C4×D4 #87306

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₃×C₄×D₄ #87306