Subgroups — C₂×C₃²×AGL(1,ℤ/8) #10596374

Order 576.

894 conjugacy classes of subgroups.

The subgroup lattice has 183 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃²×#41	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×C₃²×D₄	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃²×AGL(1,ℤ/8)	C₂	1	8	yes	maximal, direct product factor
4	C₂×C₃²×SD₁₆	C₂	1	2	yes	maximal, complemented
5	C₂×C₃²×D₈	C₂	1	2	yes	maximal, complemented
6	C₂×C₃²×M₁₆	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₃×AGL(1,ℤ/8)	C₃	1	4	yes	maximal, direct product factor
8	C₂⁴×C₃²	—	2	1	no	—
9	C₃²×#41	—	2	2	no	—
10	C₃²×#41	C₂²	1	4	yes	complemented
11	C₂×C₃²×Q₈	C₂²	1	1	yes	characteristic, complemented
12	C₂×C₃²×D₄	—	2	1	no	—
13	C₂×C₃²×D₄	—	2	2	no	—
14	C₂×C₃²×D₄	—	2	1	no	—
15	C₂×C₃²×D₄	C₂²	1	2	yes	—
16	C₂×C₃²×D₄	C₂²	1	4	yes	complemented
17	C₂×C₃²×D₄	C₂²	1	1	yes	characteristic, complemented
18	C₂²×C₃²×C₄	—	2	1	no	—
19	C₂²×C₃²×C₄	C₂²	1	1	yes	characteristic
20	C₃²×SD₁₆	C₂²	1	8	yes	complemented
21	C₃²×D₈	C₂²	1	8	yes	complemented
22	C₃²×M₁₆	C₂²	1	4	yes	complemented
23	C₂×C₃²×C₈	C₂²	1	2	yes	complemented
24	C₂×C₃×#41	C₂×C₃	1	4	yes	complemented
25	C₂²×C₃×D₄	C₂×C₃	1	4	yes	complemented
26	C₃×AGL(1,ℤ/8)	C₂×C₃	1	32	yes	direct product factor
27	C₂×C₃×SD₁₆	C₂×C₃	1	8	yes	complemented
28	C₂×C₃×D₈	C₂×C₃	1	8	yes	complemented
29	C₂×C₃×M₁₆	C₂×C₃	1	4	yes	complemented
30	C₂³×C₃²	—	2	4	no	—
31	C₂³×C₃²	—	2	2	no	—
32	C₂³×C₃²	—	2	1	no	—
33	C₂³×C₃²	—	4	4	no	—
34	C₂³×C₃²	D₄	1	1	yes	characteristic
35	C₃²×Q₈	—	2	1	no	—
36	C₃²×Q₈	C₂³	1	2	yes	complemented
37	C₃²×D₄	—	2	4	no	—
38	C₃²×D₄	—	2	2	no	—
39	C₃²×D₄	—	2	4	no	—
40	C₃²×D₄	—	2	1	no	—
41	C₃²×D₄	C₂³	1	4	yes	—
42	C₃²×D₄	C₂³	1	2	yes	complemented
43	C₂×C₃²×C₄	—	2	1	no	—
44	C₂×C₃²×C₄	—	2	4	no	—
45	C₂×C₃²×C₄	D₄	1	2	yes	—
46	C₂×C₃²×C₄	D₄	1	1	yes	characteristic
47	C₂×C₃²×C₄	C₂³	1	2	yes	—
48	C₂×C₃²×C₄	C₂³	1	1	yes	characteristic
49	C₃²×C₈	C₂³	1	4	yes	complemented
50	C₂×AGL(1,ℤ/8)	C₃²	1	1	yes	characteristic, direct product factor, Sylow
51	C₂⁴×C₃	—	2	4	no	—
52	C₃×#41	—	2	8	no	—
53	C₃×#41	C₂²×C₃	1	16	yes	complemented
54	C₂×C₃×Q₈	C₂²×C₃	1	4	yes	complemented
55	C₂×C₃×D₄	—	2	4	no	—
56	C₂×C₃×D₄	—	2	4	no	—
57	C₂×C₃×D₄	—	2	8	no	—
58	C₂×C₃×D₄	C₂²×C₃	1	8	yes	—
59	C₂×C₃×D₄	C₂²×C₃	1	16	yes	complemented
60	C₂×C₃×D₄	C₂²×C₃	1	4	yes	complemented
61	C₂²×C₃×C₄	—	2	4	no	—
62	C₂²×C₃×C₄	C₂²×C₃	1	4	yes	—
63	C₃×SD₁₆	C₂²×C₃	1	32	yes	complemented
64	C₃×D₈	C₂²×C₃	1	32	yes	complemented
65	C₃×M₁₆	C₂²×C₃	1	16	yes	complemented
66	C₂×C₃×C₈	C₂²×C₃	1	8	yes	complemented
67	C₂²×C₃²	—	2	2	no	—
68	C₂²×C₃²	—	2	2	no	—
69	C₂²×C₃²	—	2	4	no	—
70	C₂²×C₃²	—	4	8	no	—
71	C₂²×C₃²	—	4	2	no	—
72	C₂²×C₃²	—	4	4	no	—
73	C₂²×C₃²	C₂×D₄	1	2	yes	—
74	C₂²×C₃²	C₂×D₄	1	1	yes	characteristic
75	C₃²×C₄	—	2	2	no	—
76	C₃²×C₄	C₂×D₄	1	2	yes	—
77	C₃²×C₄	C₂×D₄	1	1	yes	characteristic
78	C₃²×C₄	C₂⁴	1	1	yes	characteristic
79	C₂×#41	C₂×C₃²	1	1	yes	characteristic, complemented
80	C₂²×D₄	C₂×C₃²	1	1	yes	characteristic, complemented
81	AGL(1,ℤ/8)	C₂×C₃²	1	8	yes	direct product factor
82	C₂×SD₁₆	C₂×C₃²	1	2	yes	complemented
83	C₂×D₈	C₂×C₃²	1	2	yes	complemented
84	C₂×M₁₆	C₂×C₃²	1	1	yes	characteristic, complemented
85	C₂³×C₃	—	2	4	no	—
86	C₂³×C₃	—	2	8	no	—
87	C₂³×C₃	—	2	16	no	—
88	C₂³×C₃	—	4	16	no	—
89	C₂³×C₃	C₃×D₄	1	4	yes	—
90	C₃×Q₈	—	2	4	no	—
91	C₃×Q₈	C₂³×C₃	1	8	yes	complemented
92	C₃×D₄	—	2	16	no	—
93	C₃×D₄	—	2	8	no	—
94	C₃×D₄	—	2	16	no	—
95	C₃×D₄	—	2	4	no	—
96	C₃×D₄	C₂³×C₃	1	16	yes	—
97	C₃×D₄	C₂³×C₃	1	8	yes	complemented
98	C₂×C₃×C₄	—	2	4	no	—
99	C₂×C₃×C₄	—	2	16	no	—
100	C₂×C₃×C₄	C₃×D₄	1	4	yes	—
101	C₂×C₃×C₄	C₃×D₄	1	8	yes	—
102	C₂×C₃×C₄	C₂³×C₃	1	4	yes	—
103	C₂×C₃×C₄	C₂³×C₃	1	8	yes	—
104	C₃×C₈	C₂³×C₃	1	16	yes	complemented
105	C₂×C₃²	—	2	2	no	—
106	C₂×C₃²	—	4	2	no	—
107	C₂×C₃²	—	4	4	no	—
108	C₂×C₃²	AGL(1,ℤ/8)	1	2	yes	direct product factor
109	C₂×C₃²	C₂²×D₄	1	1	yes	characteristic
110	C₂⁴	—	2	1	no	—
111	#41	—	2	2	no	—
112	#41	C₂²×C₃²	1	4	yes	complemented
113	C₂×Q₈	C₂²×C₃²	1	1	yes	characteristic, complemented
114	C₂×D₄	—	2	1	no	—
115	C₂×D₄	—	2	1	no	—
116	C₂×D₄	—	2	2	no	—
117	C₂×D₄	C₂²×C₃²	1	2	yes	—
118	C₂×D₄	C₂²×C₃²	1	4	yes	complemented
119	C₂×D₄	C₂²×C₃²	1	1	yes	characteristic, complemented
120	C₂²×C₄	—	2	1	no	—
121	C₂²×C₄	C₂²×C₃²	1	1	yes	characteristic
122	SD₁₆	C₂²×C₃²	1	8	yes	complemented
123	D₈	C₂²×C₃²	1	8	yes	complemented
124	M₁₆	C₂²×C₃²	1	4	yes	complemented
125	C₂×C₈	C₂²×C₃²	1	2	yes	complemented
126	C₂²×C₃	—	2	8	no	—
127	C₂²×C₃	—	2	8	no	—
128	C₂²×C₃	—	2	16	no	—
129	C₂²×C₃	—	4	32	no	—
130	C₂²×C₃	—	4	16	no	—
131	C₂²×C₃	—	4	8	no	—
132	C₂²×C₃	C₂×C₃×D₄	1	8	yes	—
133	C₂²×C₃	C₂×C₃×D₄	1	4	yes	—
134	C₃×C₄	—	2	8	no	—
135	C₃×C₄	C₂×C₃×D₄	1	8	yes	—
136	C₃×C₄	C₂×C₃×D₄	1	4	yes	—
137	C₃×C₄	C₂⁴×C₃	1	4	yes	—
138	C₃²	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic, direct product factor, Sylow
139	C₂³	—	2	1	no	—
140	C₂³	—	2	2	no	—
141	C₂³	—	2	4	no	—
142	C₂³	—	4	4	no	—
143	C₂³	C₃²×D₄	1	1	yes	characteristic
144	Q₈	—	2	1	no	—
145	Q₈	C₂³×C₃²	1	2	yes	complemented
146	D₄	—	2	4	no	—
147	D₄	—	2	2	no	—
148	D₄	—	2	4	no	—
149	D₄	—	2	1	no	—
150	D₄	C₂³×C₃²	1	4	yes	—
151	D₄	C₂³×C₃²	1	2	yes	complemented
152	C₂×C₄	—	2	1	no	—
153	C₂×C₄	—	2	4	no	—
154	C₂×C₄	C₃²×D₄	1	1	yes	characteristic
155	C₂×C₄	C₃²×D₄	1	2	yes	—
156	C₂×C₄	C₂³×C₃²	1	1	yes	characteristic
157	C₂×C₄	C₂³×C₃²	1	2	yes	—
158	C₈	C₂³×C₃²	1	4	yes	complemented
159	C₂×C₃	—	2	8	no	—
160	C₂×C₃	—	4	16	no	—
161	C₂×C₃	—	4	8	no	—
162	C₂×C₃	C₃×AGL(1,ℤ/8)	1	8	yes	direct product factor
163	C₂×C₃	C₂²×C₃×D₄	1	4	yes	—
164	C₂²	—	2	2	no	—
165	C₂²	—	2	2	no	—
166	C₂²	—	2	4	no	—
167	C₂²	—	4	8	no	—
168	C₂²	—	4	4	no	—
169	C₂²	—	4	2	no	—
170	C₂²	C₂×C₃²×D₄	1	2	yes	—
171	C₂²	C₂×C₃²×D₄	1	1	yes	characteristic
172	C₄	—	2	2	no	—
173	C₄	C₂×C₃²×D₄	1	2	yes	—
174	C₄	C₂×C₃²×D₄	1	1	yes	characteristic
175	C₄	C₂⁴×C₃²	1	1	yes	characteristic
176	C₃	C₂×C₃×AGL(1,ℤ/8)	1	4	yes	direct product factor
177	C₂	—	2	2	no	—
178	C₂	—	4	4	no	—
179	C₂	—	4	2	no	—
180	C₂	C₃²×AGL(1,ℤ/8)	1	2	yes	direct product factor
181	C₂	C₂²×C₃²×D₄	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃²	C₂×D₄
commutator, Frattini, cosocle	C₄	C₂⁴×C₃²
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃² ◁ C₂²×C₃²×C₄ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₃²×#41	C₂	semidirect
2. C₂²×C₃²×D₄	C₂	semidirect
3. C₃²×AGL(1,ℤ/8)	C₂	direct (N × H)
4. C₂×C₃²×SD₁₆	C₂	semidirect
5. C₂×C₃²×D₈	C₂	semidirect
6. C₂×C₃²×M₁₆	C₂	semidirect
7. C₂×C₃×AGL(1,ℤ/8)	C₃	direct (N × H)
10. C₃²×#41	C₂²	semidirect
11. C₂×C₃²×Q₈	C₂²	semidirect
16. C₂×C₃²×D₄	C₂²	semidirect
17. C₂×C₃²×D₄	C₂²	semidirect
20. C₃²×SD₁₆	C₂²	semidirect
21. C₃²×D₈	C₂²	semidirect
22. C₃²×M₁₆	C₂²	semidirect
23. C₂×C₃²×C₈	C₂²	semidirect
24. C₂×C₃×#41	C₂×C₃	semidirect
25. C₂²×C₃×D₄	C₂×C₃	semidirect
26. C₃×AGL(1,ℤ/8)	C₂×C₃	direct (N × H)
27. C₂×C₃×SD₁₆	C₂×C₃	semidirect
28. C₂×C₃×D₈	C₂×C₃	semidirect
29. C₂×C₃×M₁₆	C₂×C₃	semidirect
36. C₃²×Q₈	C₂³	semidirect
42. C₃²×D₄	C₂³	semidirect
49. C₃²×C₈	C₂³	semidirect
50. C₂×AGL(1,ℤ/8)	C₃²	direct (N × H)
53. C₃×#41	C₂²×C₃	semidirect
54. C₂×C₃×Q₈	C₂²×C₃	semidirect
59. C₂×C₃×D₄	C₂²×C₃	semidirect
60. C₂×C₃×D₄	C₂²×C₃	semidirect
63. C₃×SD₁₆	C₂²×C₃	semidirect
64. C₃×D₈	C₂²×C₃	semidirect
65. C₃×M₁₆	C₂²×C₃	semidirect
66. C₂×C₃×C₈	C₂²×C₃	semidirect
79. C₂×#41	C₂×C₃²	semidirect
80. C₂²×D₄	C₂×C₃²	semidirect
81. AGL(1,ℤ/8)	C₂×C₃²	direct (N × H)
82. C₂×SD₁₆	C₂×C₃²	semidirect
83. C₂×D₈	C₂×C₃²	semidirect
84. C₂×M₁₆	C₂×C₃²	semidirect
91. C₃×Q₈	C₂³×C₃	semidirect
97. C₃×D₄	C₂³×C₃	semidirect
104. C₃×C₈	C₂³×C₃	semidirect
108. C₂×C₃²	AGL(1,ℤ/8)	direct (N × H)
112. #41	C₂²×C₃²	semidirect
113. C₂×Q₈	C₂²×C₃²	semidirect
118. C₂×D₄	C₂²×C₃²	semidirect
119. C₂×D₄	C₂²×C₃²	semidirect
122. SD₁₆	C₂²×C₃²	semidirect
123. D₈	C₂²×C₃²	semidirect
124. M₁₆	C₂²×C₃²	semidirect
125. C₂×C₈	C₂²×C₃²	semidirect
138. C₃²	C₂×AGL(1,ℤ/8)	direct (N × H)
145. Q₈	C₂³×C₃²	semidirect
151. D₄	C₂³×C₃²	semidirect
158. C₈	C₂³×C₃²	semidirect
162. C₂×C₃	C₃×AGL(1,ℤ/8)	direct (N × H)
176. C₃	C₂×C₃×AGL(1,ℤ/8)	direct (N × H)
180. C₂	C₃²×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
C₂×C₃²×#41	2
C₂²×C₃²×D₄	2
C₃²×AGL(1,ℤ/8)	2
C₂×C₃²×SD₁₆	2
C₂×C₃²×D₈	2
C₂×C₃²×M₁₆	2
C₂×C₃×AGL(1,ℤ/8)	3

Subgroups — C2×C32×AGL(1,ℤ/8) #10596374

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃²×AGL(1,ℤ/8) #10596374