Subgroups — C₂×C₃³×AGL(1,ℤ/8) #13180028

Order 1728.

4172 conjugacy classes of subgroups.

The subgroup lattice has 244 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃³×#41	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×C₃³×D₄	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃³×AGL(1,ℤ/8)	C₂	1	8	yes	maximal, direct product factor
4	C₂×C₃³×SD₁₆	C₂	1	2	yes	maximal, complemented
5	C₂×C₃³×D₈	C₂	1	2	yes	maximal, complemented
6	C₂×C₃³×M₁₆	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₃²×AGL(1,ℤ/8)	C₃	1	13	yes	maximal, direct product factor
8	C₂⁴×C₃³	—	2	1	no	—
9	C₃³×#41	—	2	2	no	—
10	C₃³×#41	C₂²	1	4	yes	complemented
11	C₂×C₃³×Q₈	C₂²	1	1	yes	characteristic, complemented
12	C₂×C₃³×D₄	—	2	1	no	—
13	C₂×C₃³×D₄	—	2	2	no	—
14	C₂×C₃³×D₄	—	2	1	no	—
15	C₂×C₃³×D₄	C₂²	1	2	yes	—
16	C₂×C₃³×D₄	C₂²	1	4	yes	complemented
17	C₂×C₃³×D₄	C₂²	1	1	yes	characteristic, complemented
18	C₂²×C₃³×C₄	—	2	1	no	—
19	C₂²×C₃³×C₄	C₂²	1	1	yes	characteristic
20	C₃³×SD₁₆	C₂²	1	8	yes	complemented
21	C₃³×D₈	C₂²	1	8	yes	complemented
22	C₃³×M₁₆	C₂²	1	4	yes	complemented
23	C₂×C₃³×C₈	C₂²	1	2	yes	complemented
24	C₂×C₃²×#41	C₂×C₃	1	13	yes	complemented
25	C₂²×C₃²×D₄	C₂×C₃	1	13	yes	complemented
26	C₃²×AGL(1,ℤ/8)	C₂×C₃	1	104	yes	direct product factor
27	C₂×C₃²×SD₁₆	C₂×C₃	1	26	yes	complemented
28	C₂×C₃²×D₈	C₂×C₃	1	26	yes	complemented
29	C₂×C₃²×M₁₆	C₂×C₃	1	13	yes	complemented
30	C₂³×C₃³	—	2	4	no	—
31	C₂³×C₃³	—	2	2	no	—
32	C₂³×C₃³	—	2	1	no	—
33	C₂³×C₃³	—	4	4	no	—
34	C₂³×C₃³	D₄	1	1	yes	characteristic
35	C₃³×Q₈	—	2	1	no	—
36	C₃³×Q₈	C₂³	1	2	yes	complemented
37	C₃³×D₄	—	2	4	no	—
38	C₃³×D₄	—	2	2	no	—
39	C₃³×D₄	—	2	4	no	—
40	C₃³×D₄	—	2	1	no	—
41	C₃³×D₄	C₂³	1	4	yes	—
42	C₃³×D₄	C₂³	1	2	yes	complemented
43	C₂×C₃³×C₄	—	2	1	no	—
44	C₂×C₃³×C₄	—	2	4	no	—
45	C₂×C₃³×C₄	D₄	1	2	yes	—
46	C₂×C₃³×C₄	D₄	1	1	yes	characteristic
47	C₂×C₃³×C₄	C₂³	1	2	yes	—
48	C₂×C₃³×C₄	C₂³	1	1	yes	characteristic
49	C₃³×C₈	C₂³	1	4	yes	complemented
50	C₂×C₃×AGL(1,ℤ/8)	C₃²	1	13	yes	direct product factor
51	C₂⁴×C₃²	—	2	13	no	—
52	C₃²×#41	—	2	26	no	—
53	C₃²×#41	C₂²×C₃	1	52	yes	complemented
54	C₂×C₃²×Q₈	C₂²×C₃	1	13	yes	complemented
55	C₂×C₃²×D₄	—	2	13	no	—
56	C₂×C₃²×D₄	—	2	13	no	—
57	C₂×C₃²×D₄	—	2	26	no	—
58	C₂×C₃²×D₄	C₂²×C₃	1	26	yes	—
59	C₂×C₃²×D₄	C₂²×C₃	1	52	yes	complemented
60	C₂×C₃²×D₄	C₂²×C₃	1	13	yes	complemented
61	C₂²×C₃²×C₄	—	2	13	no	—
62	C₂²×C₃²×C₄	C₂²×C₃	1	13	yes	—
63	C₃²×SD₁₆	C₂²×C₃	1	104	yes	complemented
64	C₃²×D₈	C₂²×C₃	1	104	yes	complemented
65	C₃²×M₁₆	C₂²×C₃	1	52	yes	complemented
66	C₂×C₃²×C₈	C₂²×C₃	1	26	yes	complemented
67	C₂²×C₃³	—	2	2	no	—
68	C₂²×C₃³	—	2	2	no	—
69	C₂²×C₃³	—	2	4	no	—
70	C₂²×C₃³	—	4	8	no	—
71	C₂²×C₃³	—	4	2	no	—
72	C₂²×C₃³	—	4	4	no	—
73	C₂²×C₃³	C₂×D₄	1	2	yes	—
74	C₂²×C₃³	C₂×D₄	1	1	yes	characteristic
75	C₃³×C₄	—	2	2	no	—
76	C₃³×C₄	C₂×D₄	1	2	yes	—
77	C₃³×C₄	C₂×D₄	1	1	yes	characteristic
78	C₃³×C₄	C₂⁴	1	1	yes	characteristic
79	C₂×C₃×#41	C₂×C₃²	1	13	yes	complemented
80	C₂²×C₃×D₄	C₂×C₃²	1	13	yes	complemented
81	C₃×AGL(1,ℤ/8)	C₂×C₃²	1	104	yes	direct product factor
82	C₂×C₃×SD₁₆	C₂×C₃²	1	26	yes	complemented
83	C₂×C₃×D₈	C₂×C₃²	1	26	yes	complemented
84	C₂×C₃×M₁₆	C₂×C₃²	1	13	yes	complemented
85	C₂³×C₃²	—	2	26	no	—
86	C₂³×C₃²	—	2	52	no	—
87	C₂³×C₃²	—	2	13	no	—
88	C₂³×C₃²	—	4	52	no	—
89	C₂³×C₃²	C₃×D₄	1	13	yes	—
90	C₃²×Q₈	—	2	13	no	—
91	C₃²×Q₈	C₂³×C₃	1	26	yes	complemented
92	C₃²×D₄	—	2	52	no	—
93	C₃²×D₄	—	2	26	no	—
94	C₃²×D₄	—	2	52	no	—
95	C₃²×D₄	—	2	13	no	—
96	C₃²×D₄	C₂³×C₃	1	52	yes	—
97	C₃²×D₄	C₂³×C₃	1	26	yes	complemented
98	C₂×C₃²×C₄	—	2	13	no	—
99	C₂×C₃²×C₄	—	2	52	no	—
100	C₂×C₃²×C₄	C₃×D₄	1	13	yes	—
101	C₂×C₃²×C₄	C₃×D₄	1	26	yes	—
102	C₂×C₃²×C₄	C₂³×C₃	1	13	yes	—
103	C₂×C₃²×C₄	C₂³×C₃	1	26	yes	—
104	C₃²×C₈	C₂³×C₃	1	52	yes	complemented
105	C₂×AGL(1,ℤ/8)	C₃³	1	1	yes	characteristic, direct product factor, Sylow
106	C₂×C₃³	—	2	2	no	—
107	C₂×C₃³	—	4	2	no	—
108	C₂×C₃³	—	4	4	no	—
109	C₂×C₃³	AGL(1,ℤ/8)	1	2	yes	direct product factor
110	C₂×C₃³	C₂²×D₄	1	1	yes	characteristic
111	C₂⁴×C₃	—	2	13	no	—
112	C₃×#41	—	2	26	no	—
113	C₃×#41	C₂²×C₃²	1	52	yes	complemented
114	C₂×C₃×Q₈	C₂²×C₃²	1	13	yes	complemented
115	C₂×C₃×D₄	—	2	13	no	—
116	C₂×C₃×D₄	—	2	13	no	—
117	C₂×C₃×D₄	—	2	26	no	—
118	C₂×C₃×D₄	C₂²×C₃²	1	26	yes	—
119	C₂×C₃×D₄	C₂²×C₃²	1	52	yes	complemented
120	C₂×C₃×D₄	C₂²×C₃²	1	13	yes	complemented
121	C₂²×C₃×C₄	—	2	13	no	—
122	C₂²×C₃×C₄	C₂²×C₃²	1	13	yes	—
123	C₃×SD₁₆	C₂²×C₃²	1	104	yes	complemented
124	C₃×D₈	C₂²×C₃²	1	104	yes	complemented
125	C₃×M₁₆	C₂²×C₃²	1	52	yes	complemented
126	C₂×C₃×C₈	C₂²×C₃²	1	26	yes	complemented
127	C₂²×C₃²	—	2	26	no	—
128	C₂²×C₃²	—	2	26	no	—
129	C₂²×C₃²	—	2	52	no	—
130	C₂²×C₃²	—	4	104	no	—
131	C₂²×C₃²	—	4	26	no	—
132	C₂²×C₃²	—	4	52	no	—
133	C₂²×C₃²	C₂×C₃×D₄	1	26	yes	—
134	C₂²×C₃²	C₂×C₃×D₄	1	13	yes	—
135	C₃²×C₄	—	2	26	no	—
136	C₃²×C₄	C₂×C₃×D₄	1	26	yes	—
137	C₃²×C₄	C₂×C₃×D₄	1	13	yes	—
138	C₃²×C₄	C₂⁴×C₃	1	13	yes	—
139	C₂×#41	C₂×C₃³	1	1	yes	characteristic, complemented
140	C₂²×D₄	C₂×C₃³	1	1	yes	characteristic, complemented
141	AGL(1,ℤ/8)	C₂×C₃³	1	8	yes	direct product factor
142	C₂×SD₁₆	C₂×C₃³	1	2	yes	complemented
143	C₂×D₈	C₂×C₃³	1	2	yes	complemented
144	C₂×M₁₆	C₂×C₃³	1	1	yes	characteristic, complemented
145	C₃³	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic, direct product factor, Sylow
146	C₂³×C₃	—	2	13	no	—
147	C₂³×C₃	—	2	26	no	—
148	C₂³×C₃	—	2	52	no	—
149	C₂³×C₃	—	4	52	no	—
150	C₂³×C₃	C₃²×D₄	1	13	yes	—
151	C₃×Q₈	—	2	13	no	—
152	C₃×Q₈	C₂³×C₃²	1	26	yes	complemented
153	C₃×D₄	—	2	52	no	—
154	C₃×D₄	—	2	26	no	—
155	C₃×D₄	—	2	52	no	—
156	C₃×D₄	—	2	13	no	—
157	C₃×D₄	C₂³×C₃²	1	52	yes	—
158	C₃×D₄	C₂³×C₃²	1	26	yes	complemented
159	C₂×C₃×C₄	—	2	13	no	—
160	C₂×C₃×C₄	—	2	52	no	—
161	C₂×C₃×C₄	C₃²×D₄	1	13	yes	—
162	C₂×C₃×C₄	C₃²×D₄	1	26	yes	—
163	C₂×C₃×C₄	C₂³×C₃²	1	13	yes	—
164	C₂×C₃×C₄	C₂³×C₃²	1	26	yes	—
165	C₃×C₈	C₂³×C₃²	1	52	yes	complemented
166	C₂×C₃²	—	2	26	no	—
167	C₂×C₃²	—	4	26	no	—
168	C₂×C₃²	—	4	52	no	—
169	C₂×C₃²	C₃×AGL(1,ℤ/8)	1	26	yes	direct product factor
170	C₂×C₃²	C₂²×C₃×D₄	1	13	yes	—
171	C₂⁴	—	2	1	no	—
172	#41	—	2	2	no	—
173	#41	C₂²×C₃³	1	4	yes	complemented
174	C₂×Q₈	C₂²×C₃³	1	1	yes	characteristic, complemented
175	C₂×D₄	—	2	1	no	—
176	C₂×D₄	—	2	1	no	—
177	C₂×D₄	—	2	2	no	—
178	C₂×D₄	C₂²×C₃³	1	2	yes	—
179	C₂×D₄	C₂²×C₃³	1	4	yes	complemented
180	C₂×D₄	C₂²×C₃³	1	1	yes	characteristic, complemented
181	C₂²×C₄	—	2	1	no	—
182	C₂²×C₄	C₂²×C₃³	1	1	yes	characteristic
183	SD₁₆	C₂²×C₃³	1	8	yes	complemented
184	D₈	C₂²×C₃³	1	8	yes	complemented
185	M₁₆	C₂²×C₃³	1	4	yes	complemented
186	C₂×C₈	C₂²×C₃³	1	2	yes	complemented
187	C₂²×C₃	—	2	26	no	—
188	C₂²×C₃	—	2	26	no	—
189	C₂²×C₃	—	2	52	no	—
190	C₂²×C₃	—	4	104	no	—
191	C₂²×C₃	—	4	52	no	—
192	C₂²×C₃	—	4	26	no	—
193	C₂²×C₃	C₂×C₃²×D₄	1	26	yes	—
194	C₂²×C₃	C₂×C₃²×D₄	1	13	yes	—
195	C₃×C₄	—	2	26	no	—
196	C₃×C₄	C₂×C₃²×D₄	1	26	yes	—
197	C₃×C₄	C₂×C₃²×D₄	1	13	yes	—
198	C₃×C₄	C₂⁴×C₃²	1	13	yes	—
199	C₃²	C₂×C₃×AGL(1,ℤ/8)	1	13	yes	direct product factor
200	C₂³	—	2	1	no	—
201	C₂³	—	2	2	no	—
202	C₂³	—	2	4	no	—
203	C₂³	—	4	4	no	—
204	C₂³	C₃³×D₄	1	1	yes	characteristic
205	Q₈	—	2	1	no	—
206	Q₈	C₂³×C₃³	1	2	yes	complemented
207	D₄	—	2	4	no	—
208	D₄	—	2	2	no	—
209	D₄	—	2	4	no	—
210	D₄	—	2	1	no	—
211	D₄	C₂³×C₃³	1	4	yes	—
212	D₄	C₂³×C₃³	1	2	yes	complemented
213	C₂×C₄	—	2	1	no	—
214	C₂×C₄	—	2	4	no	—
215	C₂×C₄	C₃³×D₄	1	1	yes	characteristic
216	C₂×C₄	C₃³×D₄	1	2	yes	—
217	C₂×C₄	C₂³×C₃³	1	1	yes	characteristic
218	C₂×C₄	C₂³×C₃³	1	2	yes	—
219	C₈	C₂³×C₃³	1	4	yes	complemented
220	C₂×C₃	—	2	26	no	—
221	C₂×C₃	—	4	52	no	—
222	C₂×C₃	—	4	26	no	—
223	C₂×C₃	C₃²×AGL(1,ℤ/8)	1	26	yes	direct product factor
224	C₂×C₃	C₂²×C₃²×D₄	1	13	yes	—
225	C₂²	—	2	2	no	—
226	C₂²	—	2	2	no	—
227	C₂²	—	2	4	no	—
228	C₂²	—	4	8	no	—
229	C₂²	—	4	4	no	—
230	C₂²	—	4	2	no	—
231	C₂²	C₂×C₃³×D₄	1	2	yes	—
232	C₂²	C₂×C₃³×D₄	1	1	yes	characteristic
233	C₄	—	2	2	no	—
234	C₄	C₂×C₃³×D₄	1	2	yes	—
235	C₄	C₂×C₃³×D₄	1	1	yes	characteristic
236	C₄	C₂⁴×C₃³	1	1	yes	characteristic
237	C₃	C₂×C₃²×AGL(1,ℤ/8)	1	13	yes	direct product factor
238	C₂	—	2	2	no	—
239	C₂	—	4	4	no	—
240	C₂	—	4	2	no	—
241	C₂	C₃³×AGL(1,ℤ/8)	1	2	yes	direct product factor
242	C₂	C₂²×C₃³×D₄	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃³	C₂×D₄
commutator, Frattini, cosocle	C₄	C₂⁴×C₃³
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃³ ◁ C₂²×C₃³×C₄ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₃³×#41	C₂	semidirect
2. C₂²×C₃³×D₄	C₂	semidirect
3. C₃³×AGL(1,ℤ/8)	C₂	direct (N × H)
4. C₂×C₃³×SD₁₆	C₂	semidirect
5. C₂×C₃³×D₈	C₂	semidirect
6. C₂×C₃³×M₁₆	C₂	semidirect
7. C₂×C₃²×AGL(1,ℤ/8)	C₃	direct (N × H)
10. C₃³×#41	C₂²	semidirect
11. C₂×C₃³×Q₈	C₂²	semidirect
16. C₂×C₃³×D₄	C₂²	semidirect
17. C₂×C₃³×D₄	C₂²	semidirect
20. C₃³×SD₁₆	C₂²	semidirect
21. C₃³×D₈	C₂²	semidirect
22. C₃³×M₁₆	C₂²	semidirect
23. C₂×C₃³×C₈	C₂²	semidirect
24. C₂×C₃²×#41	C₂×C₃	semidirect
25. C₂²×C₃²×D₄	C₂×C₃	semidirect
26. C₃²×AGL(1,ℤ/8)	C₂×C₃	direct (N × H)
27. C₂×C₃²×SD₁₆	C₂×C₃	semidirect
28. C₂×C₃²×D₈	C₂×C₃	semidirect
29. C₂×C₃²×M₁₆	C₂×C₃	semidirect
36. C₃³×Q₈	C₂³	semidirect
42. C₃³×D₄	C₂³	semidirect
49. C₃³×C₈	C₂³	semidirect
50. C₂×C₃×AGL(1,ℤ/8)	C₃²	direct (N × H)
53. C₃²×#41	C₂²×C₃	semidirect
54. C₂×C₃²×Q₈	C₂²×C₃	semidirect
59. C₂×C₃²×D₄	C₂²×C₃	semidirect
60. C₂×C₃²×D₄	C₂²×C₃	semidirect
63. C₃²×SD₁₆	C₂²×C₃	semidirect
64. C₃²×D₈	C₂²×C₃	semidirect
65. C₃²×M₁₆	C₂²×C₃	semidirect
66. C₂×C₃²×C₈	C₂²×C₃	semidirect
79. C₂×C₃×#41	C₂×C₃²	semidirect
80. C₂²×C₃×D₄	C₂×C₃²	semidirect
81. C₃×AGL(1,ℤ/8)	C₂×C₃²	direct (N × H)
82. C₂×C₃×SD₁₆	C₂×C₃²	semidirect
83. C₂×C₃×D₈	C₂×C₃²	semidirect
84. C₂×C₃×M₁₆	C₂×C₃²	semidirect
91. C₃²×Q₈	C₂³×C₃	semidirect
97. C₃²×D₄	C₂³×C₃	semidirect
104. C₃²×C₈	C₂³×C₃	semidirect
105. C₂×AGL(1,ℤ/8)	C₃³	direct (N × H)
109. C₂×C₃³	AGL(1,ℤ/8)	direct (N × H)
113. C₃×#41	C₂²×C₃²	semidirect
114. C₂×C₃×Q₈	C₂²×C₃²	semidirect
119. C₂×C₃×D₄	C₂²×C₃²	semidirect
120. C₂×C₃×D₄	C₂²×C₃²	semidirect
123. C₃×SD₁₆	C₂²×C₃²	semidirect
124. C₃×D₈	C₂²×C₃²	semidirect
125. C₃×M₁₆	C₂²×C₃²	semidirect
126. C₂×C₃×C₈	C₂²×C₃²	semidirect
139. C₂×#41	C₂×C₃³	semidirect
140. C₂²×D₄	C₂×C₃³	semidirect
141. AGL(1,ℤ/8)	C₂×C₃³	direct (N × H)
142. C₂×SD₁₆	C₂×C₃³	semidirect
143. C₂×D₈	C₂×C₃³	semidirect
144. C₂×M₁₆	C₂×C₃³	semidirect
145. C₃³	C₂×AGL(1,ℤ/8)	direct (N × H)
152. C₃×Q₈	C₂³×C₃²	semidirect
158. C₃×D₄	C₂³×C₃²	semidirect
165. C₃×C₈	C₂³×C₃²	semidirect
169. C₂×C₃²	C₃×AGL(1,ℤ/8)	direct (N × H)
173. #41	C₂²×C₃³	semidirect
174. C₂×Q₈	C₂²×C₃³	semidirect
179. C₂×D₄	C₂²×C₃³	semidirect
180. C₂×D₄	C₂²×C₃³	semidirect
183. SD₁₆	C₂²×C₃³	semidirect
184. D₈	C₂²×C₃³	semidirect
185. M₁₆	C₂²×C₃³	semidirect
186. C₂×C₈	C₂²×C₃³	semidirect
199. C₃²	C₂×C₃×AGL(1,ℤ/8)	direct (N × H)
206. Q₈	C₂³×C₃³	semidirect
212. D₄	C₂³×C₃³	semidirect
219. C₈	C₂³×C₃³	semidirect
223. C₂×C₃	C₃²×AGL(1,ℤ/8)	direct (N × H)
237. C₃	C₂×C₃²×AGL(1,ℤ/8)	direct (N × H)
241. C₂	C₃³×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×C₃³×#41	2
C₂²×C₃³×D₄	2
C₃³×AGL(1,ℤ/8)	2
C₂×C₃³×SD₁₆	2
C₂×C₃³×D₈	2
C₂×C₃³×M₁₆	2
C₂×C₃²×AGL(1,ℤ/8)	3

Subgroups — C2×C33×AGL(1,ℤ/8) #13180028

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃³×AGL(1,ℤ/8) #13180028