Subgroups — C₂²×C₃×AGL(1,ℤ/8) #87462

Order 384.

1720 conjugacy classes of subgroups.

The subgroup lattice has 190 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃×#41	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂³×C₃×D₄	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃×AGL(1,ℤ/8)	C₂	1	24	yes	maximal, direct product factor
4	C₂²×C₃×SD₁₆	C₂	1	2	yes	maximal, complemented
5	C₂²×C₃×D₈	C₂	1	2	yes	maximal, complemented
6	C₂²×C₃×M₁₆	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×AGL(1,ℤ/8)	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
8	C₂⁵×C₃	—	2	1	no	—
9	C₂×C₃×#41	—	2	6	no	—
10	C₂×C₃×#41	C₂²	1	12	yes	complemented
11	C₂²×C₃×Q₈	C₂²	1	1	yes	characteristic, complemented
12	C₂²×C₃×D₄	—	2	1	no	—
13	C₂²×C₃×D₄	—	2	6	no	—
14	C₂²×C₃×D₄	—	2	1	no	—
15	C₂²×C₃×D₄	C₂²	1	2	yes	—
16	C₂²×C₃×D₄	C₂²	1	12	yes	complemented
17	C₂²×C₃×D₄	C₂²	1	1	yes	characteristic, complemented
18	C₂³×C₃×C₄	—	2	1	no	—
19	C₂³×C₃×C₄	C₂²	1	1	yes	characteristic
20	C₃×AGL(1,ℤ/8)	C₂²	1	64	yes	direct product factor
21	C₂×C₃×SD₁₆	C₂²	1	24	yes	complemented
22	C₂×C₃×D₈	C₂²	1	24	yes	complemented
23	C₂×C₃×M₁₆	C₂²	1	12	yes	complemented
24	C₂²×C₃×C₈	C₂²	1	2	yes	complemented
25	C₂²×#41	C₂×C₃	1	1	yes	characteristic, complemented
26	C₂³×D₄	C₂×C₃	1	1	yes	characteristic, complemented
27	C₂×AGL(1,ℤ/8)	C₂×C₃	1	24	yes	direct product factor
28	C₂²×SD₁₆	C₂×C₃	1	2	yes	complemented
29	C₂²×D₈	C₂×C₃	1	2	yes	complemented
30	C₂²×M₁₆	C₂×C₃	1	1	yes	characteristic, complemented
31	C₂⁴×C₃	—	2	12	no	—
32	C₂⁴×C₃	—	2	2	no	—
33	C₂⁴×C₃	—	2	1	no	—
34	C₂⁴×C₃	—	4	8	no	—
35	C₂⁴×C₃	D₄	1	1	yes	characteristic
36	C₃×#41	—	2	24	no	—
37	C₃×#41	C₂³	1	16	yes	complemented
38	C₂×C₃×Q₈	—	2	3	no	—
39	C₂×C₃×Q₈	C₂³	1	6	yes	complemented
40	C₂×C₃×D₄	—	2	12	no	—
41	C₂×C₃×D₄	—	2	6	no	—
42	C₂×C₃×D₄	—	2	12	no	—
43	C₂×C₃×D₄	—	2	24	no	—
44	C₂×C₃×D₄	—	2	3	no	—
45	C₂×C₃×D₄	C₂³	1	12	yes	—
46	C₂×C₃×D₄	C₂³	1	16	yes	complemented
47	C₂×C₃×D₄	C₂³	1	6	yes	complemented
48	C₂²×C₃×C₄	—	2	1	no	—
49	C₂²×C₃×C₄	—	2	12	no	—
50	C₂²×C₃×C₄	D₄	1	6	yes	—
51	C₂²×C₃×C₄	D₄	1	1	yes	characteristic
52	C₂²×C₃×C₄	C₂³	1	6	yes	—
53	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
54	C₃×SD₁₆	C₂³	1	32	yes	complemented
55	C₃×D₈	C₂³	1	32	yes	complemented
56	C₃×M₁₆	C₂³	1	16	yes	complemented
57	C₂×C₃×C₈	C₂³	1	12	yes	complemented
58	C₂⁵	—	2	1	no	—
59	C₂×#41	—	2	6	no	—
60	C₂×#41	C₂²×C₃	1	12	yes	complemented
61	C₂²×Q₈	C₂²×C₃	1	1	yes	characteristic, complemented
62	C₂²×D₄	—	2	1	no	—
63	C₂²×D₄	—	2	1	no	—
64	C₂²×D₄	—	2	6	no	—
65	C₂²×D₄	C₂²×C₃	1	2	yes	—
66	C₂²×D₄	C₂²×C₃	1	12	yes	complemented
67	C₂²×D₄	C₂²×C₃	1	1	yes	characteristic, complemented
68	C₂³×C₄	—	2	1	no	—
69	C₂³×C₄	C₂²×C₃	1	1	yes	characteristic
70	AGL(1,ℤ/8)	C₂²×C₃	1	64	yes	direct product factor
71	C₂×SD₁₆	C₂²×C₃	1	24	yes	complemented
72	C₂×D₈	C₂²×C₃	1	24	yes	complemented
73	C₂×M₁₆	C₂²×C₃	1	12	yes	complemented
74	C₂²×C₈	C₂²×C₃	1	2	yes	complemented
75	C₂³×C₃	—	2	4	no	—
76	C₂³×C₃	—	2	6	no	—
77	C₂³×C₃	—	2	12	no	—
78	C₂³×C₃	—	2	16	no	—
79	C₂³×C₃	—	4	48	no	—
80	C₂³×C₃	—	4	4	no	—
81	C₂³×C₃	—	4	8	no	—
82	C₂³×C₃	C₂×D₄	1	6	yes	—
83	C₂³×C₃	C₂×D₄	1	1	yes	characteristic
84	C₃×Q₈	—	2	6	no	—
85	C₃×Q₈	C₂⁴	1	4	yes	complemented
86	C₃×D₄	—	2	16	no	—
87	C₃×D₄	—	2	12	no	—
88	C₃×D₄	—	2	16	no	—
89	C₃×D₄	—	2	6	no	—
90	C₃×D₄	C₂⁴	1	8	yes	—
91	C₃×D₄	C₂⁴	1	4	yes	complemented
92	C₂×C₃×C₄	—	2	6	no	—
93	C₂×C₃×C₄	—	2	16	no	—
94	C₂×C₃×C₄	C₂×D₄	1	6	yes	—
95	C₂×C₃×C₄	C₂×D₄	1	3	yes	—
96	C₂×C₃×C₄	C₂×D₄	1	12	yes	—
97	C₂×C₃×C₄	C₂⁴	1	3	yes	—
98	C₂×C₃×C₄	C₂⁴	1	4	yes	—
99	C₃×C₈	C₂⁴	1	8	yes	complemented
100	C₂⁴	—	2	1	no	—
101	C₂⁴	—	2	2	no	—
102	C₂⁴	—	2	12	no	—
103	C₂⁴	—	4	8	no	—
104	C₂⁴	C₃×D₄	1	1	yes	characteristic
105	#41	—	2	24	no	—
106	#41	C₂³×C₃	1	16	yes	complemented
107	C₂×Q₈	—	2	3	no	—
108	C₂×Q₈	C₂³×C₃	1	6	yes	complemented
109	C₂×D₄	—	2	12	no	—
110	C₂×D₄	—	2	6	no	—
111	C₂×D₄	—	2	12	no	—
112	C₂×D₄	—	2	24	no	—
113	C₂×D₄	—	2	3	no	—
114	C₂×D₄	C₂³×C₃	1	12	yes	—
115	C₂×D₄	C₂³×C₃	1	16	yes	complemented
116	C₂×D₄	C₂³×C₃	1	6	yes	complemented
117	C₂²×C₄	—	2	1	no	—
118	C₂²×C₄	—	2	12	no	—
119	C₂²×C₄	C₃×D₄	1	1	yes	characteristic
120	C₂²×C₄	C₃×D₄	1	6	yes	—
121	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
122	C₂²×C₄	C₂³×C₃	1	6	yes	—
123	SD₁₆	C₂³×C₃	1	32	yes	complemented
124	D₈	C₂³×C₃	1	32	yes	complemented
125	M₁₆	C₂³×C₃	1	16	yes	complemented
126	C₂×C₈	C₂³×C₃	1	12	yes	complemented
127	C₂²×C₃	—	2	12	no	—
128	C₂²×C₃	—	2	4	no	—
129	C₂²×C₃	—	2	8	no	—
130	C₂²×C₃	—	4	32	no	—
131	C₂²×C₃	—	4	12	no	—
132	C₂²×C₃	—	4	24	no	—
133	C₂²×C₃	AGL(1,ℤ/8)	1	4	yes	direct product factor
134	C₂²×C₃	C₂²×D₄	1	4	yes	—
135	C₂²×C₃	C₂²×D₄	1	3	yes	—
136	C₃×C₄	—	2	4	no	—
137	C₃×C₄	C₂²×D₄	1	4	yes	—
138	C₃×C₄	C₂²×D₄	1	3	yes	—
139	C₃×C₄	C₂⁵	1	1	yes	characteristic
140	C₂³	—	2	4	no	—
141	C₂³	—	2	6	no	—
142	C₂³	—	2	12	no	—
143	C₂³	—	2	16	no	—
144	C₂³	—	4	48	no	—
145	C₂³	—	4	8	no	—
146	C₂³	—	4	4	no	—
147	C₂³	C₂×C₃×D₄	1	6	yes	—
148	C₂³	C₂×C₃×D₄	1	1	yes	characteristic
149	Q₈	—	2	6	no	—
150	Q₈	C₂⁴×C₃	1	4	yes	complemented
151	D₄	—	2	16	no	—
152	D₄	—	2	12	no	—
153	D₄	—	2	16	no	—
154	D₄	—	2	6	no	—
155	D₄	C₂⁴×C₃	1	8	yes	—
156	D₄	C₂⁴×C₃	1	4	yes	complemented
157	C₂×C₄	—	2	6	no	—
158	C₂×C₄	—	2	16	no	—
159	C₂×C₄	C₂×C₃×D₄	1	6	yes	—
160	C₂×C₄	C₂×C₃×D₄	1	3	yes	—
161	C₂×C₄	C₂×C₃×D₄	1	12	yes	—
162	C₂×C₄	C₂⁴×C₃	1	3	yes	—
163	C₂×C₄	C₂⁴×C₃	1	4	yes	—
164	C₈	C₂⁴×C₃	1	8	yes	complemented
165	C₂×C₃	—	2	4	no	—
166	C₂×C₃	—	4	4	no	—
167	C₂×C₃	—	4	8	no	—
168	C₂×C₃	C₂×AGL(1,ℤ/8)	1	6	yes	direct product factor
169	C₂×C₃	C₂³×D₄	1	1	yes	characteristic
170	C₂²	—	2	12	no	—
171	C₂²	—	2	4	no	—
172	C₂²	—	2	8	no	—
173	C₂²	—	4	32	no	—
174	C₂²	—	4	24	no	—
175	C₂²	—	4	12	no	—
176	C₂²	C₃×AGL(1,ℤ/8)	1	4	yes	direct product factor
177	C₂²	C₂²×C₃×D₄	1	4	yes	—
178	C₂²	C₂²×C₃×D₄	1	3	yes	—
179	C₄	—	2	4	no	—
180	C₄	C₂²×C₃×D₄	1	4	yes	—
181	C₄	C₂²×C₃×D₄	1	3	yes	—
182	C₄	C₂⁵×C₃	1	1	yes	characteristic
183	C₃	C₂²×AGL(1,ℤ/8)	1	1	yes	characteristic, direct product factor, Sylow
184	C₂	—	2	4	no	—
185	C₂	—	4	8	no	—
186	C₂	—	4	4	no	—
187	C₂	C₂×C₃×AGL(1,ℤ/8)	1	6	yes	direct product factor
188	C₂	C₂³×C₃×D₄	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₃	C₂×D₄
commutator, Frattini, cosocle	C₄	C₂⁵×C₃
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₃ ◁ C₂³×C₃×C₄ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂²×C₃×#41	C₂	semidirect
2. C₂³×C₃×D₄	C₂	semidirect
3. C₂×C₃×AGL(1,ℤ/8)	C₂	direct (N × H)
4. C₂²×C₃×SD₁₆	C₂	semidirect
5. C₂²×C₃×D₈	C₂	semidirect
6. C₂²×C₃×M₁₆	C₂	semidirect
7. C₂²×AGL(1,ℤ/8)	C₃	direct (N × H)
10. C₂×C₃×#41	C₂²	semidirect
11. C₂²×C₃×Q₈	C₂²	semidirect
16. C₂²×C₃×D₄	C₂²	semidirect
17. C₂²×C₃×D₄	C₂²	semidirect
20. C₃×AGL(1,ℤ/8)	C₂²	direct (N × H)
21. C₂×C₃×SD₁₆	C₂²	semidirect
22. C₂×C₃×D₈	C₂²	semidirect
23. C₂×C₃×M₁₆	C₂²	semidirect
24. C₂²×C₃×C₈	C₂²	semidirect
25. C₂²×#41	C₂×C₃	semidirect
26. C₂³×D₄	C₂×C₃	semidirect
27. C₂×AGL(1,ℤ/8)	C₂×C₃	direct (N × H)
28. C₂²×SD₁₆	C₂×C₃	semidirect
29. C₂²×D₈	C₂×C₃	semidirect
30. C₂²×M₁₆	C₂×C₃	semidirect
37. C₃×#41	C₂³	semidirect
39. C₂×C₃×Q₈	C₂³	semidirect
46. C₂×C₃×D₄	C₂³	semidirect
47. C₂×C₃×D₄	C₂³	semidirect
54. C₃×SD₁₆	C₂³	semidirect
55. C₃×D₈	C₂³	semidirect
56. C₃×M₁₆	C₂³	semidirect
57. C₂×C₃×C₈	C₂³	semidirect
60. C₂×#41	C₂²×C₃	semidirect
61. C₂²×Q₈	C₂²×C₃	semidirect
66. C₂²×D₄	C₂²×C₃	semidirect
67. C₂²×D₄	C₂²×C₃	semidirect
70. AGL(1,ℤ/8)	C₂²×C₃	direct (N × H)
71. C₂×SD₁₆	C₂²×C₃	semidirect
72. C₂×D₈	C₂²×C₃	semidirect
73. C₂×M₁₆	C₂²×C₃	semidirect
74. C₂²×C₈	C₂²×C₃	semidirect
85. C₃×Q₈	C₂⁴	semidirect
91. C₃×D₄	C₂⁴	semidirect
99. C₃×C₈	C₂⁴	semidirect
106. #41	C₂³×C₃	semidirect
108. C₂×Q₈	C₂³×C₃	semidirect
115. C₂×D₄	C₂³×C₃	semidirect
116. C₂×D₄	C₂³×C₃	semidirect
123. SD₁₆	C₂³×C₃	semidirect
124. D₈	C₂³×C₃	semidirect
125. M₁₆	C₂³×C₃	semidirect
126. C₂×C₈	C₂³×C₃	semidirect
133. C₂²×C₃	AGL(1,ℤ/8)	direct (N × H)
150. Q₈	C₂⁴×C₃	semidirect
156. D₄	C₂⁴×C₃	semidirect
164. C₈	C₂⁴×C₃	semidirect
168. C₂×C₃	C₂×AGL(1,ℤ/8)	direct (N × H)
176. C₂²	C₃×AGL(1,ℤ/8)	direct (N × H)
183. C₃	C₂²×AGL(1,ℤ/8)	direct (N × H)
187. C₂	C₂×C₃×AGL(1,ℤ/8)	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₃×#41	2
C₂³×C₃×D₄	2
C₂×C₃×AGL(1,ℤ/8)	2
C₂²×C₃×SD₁₆	2
C₂²×C₃×D₈	2
C₂²×C₃×M₁₆	2
C₂²×AGL(1,ℤ/8)	3

Subgroups — C22×C3×AGL(1,ℤ/8) #87462

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₃×AGL(1,ℤ/8) #87462