Subgroups — C₂×C₃×#472 #84929

Order 384.

488 conjugacy classes of subgroups.

The subgroup lattice has 218 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×AGL(1,ℤ/8)	C₂	1	2	yes	maximal, complemented
2	C₂²×C₃×D₈	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃×#131	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×#472	C₂	1	8	yes	maximal, direct product factor
5	C₂×C₃×#108	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₃×#100	C₂	1	2	yes	maximal, complemented
7	C₂×#472	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
8	C₂×C₃×#41	C₂²	1	1	yes	characteristic
9	C₂²×C₃×D₄	C₂²	1	2	yes	—
10	C₃×AGL(1,ℤ/8)	—	2	4	no	—
11	C₃×AGL(1,ℤ/8)	C₂²	1	8	yes	complemented
12	C₂×C₃×SD₁₆	—	2	2	no	—
13	C₂×C₃×D₈	—	2	2	no	—
14	C₂×C₃×D₈	—	2	1	no	—
15	C₂×C₃×D₈	—	2	1	no	—
16	C₂×C₃×D₈	—	2	2	no	—
17	C₂×C₃×D₈	C₂²	1	4	yes	complemented
18	C₃×#131	—	2	2	no	—
19	C₃×#131	C₂²	1	4	yes	complemented
20	C₂×C₃×M₁₆	—	2	1	no	—
21	C₂×C₃×M₁₆	C₂²	1	1	yes	characteristic
22	C₂×C₃×M₁₆	C₂²	1	2	yes	—
23	C₂²×C₃×C₈	—	2	1	no	—
24	C₂²×C₃×C₈	C₂²	1	1	yes	characteristic
25	C₃×#108	C₂²	1	4	yes	complemented
26	C₃×#100	C₂²	1	8	yes	complemented
27	C₂×AGL(1,ℤ/8)	C₂×C₃	1	2	yes	complemented
28	C₂²×D₈	C₂×C₃	1	1	yes	characteristic, complemented
29	C₂×#131	C₂×C₃	1	1	yes	characteristic, complemented
30	#472	C₂×C₃	1	8	yes	direct product factor
31	C₂×#108	C₂×C₃	1	1	yes	characteristic, complemented
32	C₂×#100	C₂×C₃	1	2	yes	complemented
33	C₂⁴×C₃	—	2	2	no	—
34	C₃×#41	—	2	1	no	—
35	C₃×#41	—	2	1	no	—
36	C₃×#41	D₄	1	2	yes	—
37	C₃×#41	C₂³	1	2	yes	—
38	C₂×C₃×Q₈	D₄	1	1	yes	characteristic
39	C₂×C₃×D₄	—	2	1	no	—
40	C₂×C₃×D₄	—	2	2	no	—
41	C₂×C₃×D₄	—	2	2	no	—
42	C₂×C₃×D₄	—	2	2	no	—
43	C₂×C₃×D₄	—	2	2	no	—
44	C₂×C₃×D₄	—	2	2	no	—
45	C₂×C₃×D₄	D₄	1	1	yes	characteristic
46	C₂×C₃×D₄	C₂³	1	4	yes	—
47	C₂²×C₃×C₄	—	2	1	no	—
48	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
49	C₃×SD₁₆	—	2	8	no	—
50	C₃×D₈	—	2	8	no	—
51	C₃×D₈	—	2	4	no	—
52	C₃×D₈	—	2	4	no	—
53	C₃×M₁₆	—	2	1	no	—
54	C₃×M₁₆	—	2	2	no	—
55	C₃×M₁₆	—	2	4	no	—
56	C₃×M₁₆	C₂³	1	2	yes	—
57	C₃×M₁₆	C₂³	1	4	yes	—
58	C₂×C₃×C₈	—	2	1	no	—
59	C₂×C₃×C₈	—	2	2	no	—
60	C₂×C₃×C₈	—	2	4	no	—
61	C₂×C₃×C₈	D₄	1	1	yes	characteristic
62	C₂×C₃×C₈	D₄	1	1	yes	characteristic
63	C₂×C₃×C₈	D₄	1	2	yes	—
64	C₂×C₃×C₈	C₂³	1	2	yes	—
65	C₂×#41	C₂²×C₃	1	1	yes	characteristic
66	C₂²×D₄	C₂²×C₃	1	2	yes	—
67	AGL(1,ℤ/8)	—	2	4	no	—
68	AGL(1,ℤ/8)	C₂²×C₃	1	8	yes	complemented
69	C₂×SD₁₆	—	2	2	no	—
70	C₂×D₈	—	2	2	no	—
71	C₂×D₈	—	2	1	no	—
72	C₂×D₈	—	2	1	no	—
73	C₂×D₈	—	2	2	no	—
74	C₂×D₈	C₂²×C₃	1	4	yes	complemented
75	#131	—	2	2	no	—
76	#131	C₂²×C₃	1	4	yes	complemented
77	C₂×M₁₆	—	2	1	no	—
78	C₂×M₁₆	C₂²×C₃	1	1	yes	characteristic
79	C₂×M₁₆	C₂²×C₃	1	2	yes	—
80	C₂²×C₈	—	2	1	no	—
81	C₂²×C₈	C₂²×C₃	1	1	yes	characteristic
82	#108	C₂²×C₃	1	4	yes	complemented
83	#100	C₂²×C₃	1	8	yes	complemented
84	C₂³×C₃	—	2	1	no	—
85	C₂³×C₃	—	2	4	no	—
86	C₂³×C₃	—	4	2	no	—
87	C₂³×C₃	—	4	2	no	—
88	C₂³×C₃	—	8	4	no	—
89	C₂³×C₃	#41	1	1	yes	characteristic
90	C₃×Q₈	—	2	1	no	—
91	C₃×Q₈	C₂×D₄	1	2	yes	—
92	C₃×D₄	—	2	2	no	—
93	C₃×D₄	—	2	2	no	—
94	C₃×D₄	—	2	1	no	—
95	C₃×D₄	—	2	4	no	—
96	C₃×D₄	—	2	4	no	—
97	C₃×D₄	—	4	2	no	—
98	C₃×D₄	—	4	2	no	—
99	C₃×D₄	C₂×D₄	1	2	yes	—
100	C₂×C₃×C₄	—	2	1	no	—
101	C₂×C₃×C₄	—	2	2	no	—
102	C₂×C₃×C₄	—	2	2	no	—
103	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
104	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
105	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
106	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
107	C₂×C₃×C₄	#41	1	1	yes	characteristic
108	C₂×C₃×C₄	C₂⁴	1	1	yes	characteristic
109	C₃×C₈	—	2	2	no	—
110	C₃×C₈	—	2	4	no	—
111	C₃×C₈	C₂×D₄	1	2	yes	—
112	C₃×C₈	C₂×D₄	1	2	yes	—
113	C₂⁴	—	2	2	no	—
114	#41	—	2	1	no	—
115	#41	—	2	1	no	—
116	#41	C₃×D₄	1	2	yes	—
117	#41	C₂³×C₃	1	2	yes	—
118	C₂×Q₈	C₃×D₄	1	1	yes	characteristic
119	C₂×D₄	—	2	1	no	—
120	C₂×D₄	—	2	2	no	—
121	C₂×D₄	—	2	2	no	—
122	C₂×D₄	—	2	2	no	—
123	C₂×D₄	—	2	2	no	—
124	C₂×D₄	—	2	2	no	—
125	C₂×D₄	C₃×D₄	1	1	yes	characteristic
126	C₂×D₄	C₂³×C₃	1	4	yes	—
127	C₂²×C₄	—	2	1	no	—
128	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
129	SD₁₆	—	2	8	no	—
130	D₈	—	2	8	no	—
131	D₈	—	2	4	no	—
132	D₈	—	2	4	no	—
133	M₁₆	—	2	1	no	—
134	M₁₆	—	2	2	no	—
135	M₁₆	—	2	4	no	—
136	M₁₆	C₂³×C₃	1	2	yes	—
137	M₁₆	C₂³×C₃	1	4	yes	—
138	C₂×C₈	—	2	1	no	—
139	C₂×C₈	—	2	2	no	—
140	C₂×C₈	—	2	4	no	—
141	C₂×C₈	C₃×D₄	1	1	yes	characteristic
142	C₂×C₈	C₃×D₄	1	1	yes	characteristic
143	C₂×C₈	C₃×D₄	1	2	yes	—
144	C₂×C₈	C₂³×C₃	1	2	yes	—
145	C₂²×C₃	—	2	2	no	—
146	C₂²×C₃	—	2	2	no	—
147	C₂²×C₃	—	4	2	no	—
148	C₂²×C₃	—	4	4	no	—
149	C₂²×C₃	—	8	4	no	—
150	C₂²×C₃	—	8	4	no	—
151	C₂²×C₃	—	8	4	no	—
152	C₂²×C₃	#121	1	1	yes	characteristic
153	C₂²×C₃	#121	1	1	yes	characteristic
154	C₂²×C₃	C₂×#41	1	1	yes	characteristic
155	C₃×C₄	—	2	2	no	—
156	C₃×C₄	#121	1	1	yes	characteristic
157	C₃×C₄	#121	1	1	yes	characteristic
158	C₃×C₄	C₂²×D₄	1	1	yes	characteristic
159	C₃×C₄	C₂²×D₄	1	1	yes	characteristic
160	C₂³	—	2	1	no	—
161	C₂³	—	2	4	no	—
162	C₂³	—	4	2	no	—
163	C₂³	—	4	2	no	—
164	C₂³	—	8	4	no	—
165	C₂³	C₃×#41	1	1	yes	characteristic
166	Q₈	—	2	1	no	—
167	Q₈	C₂×C₃×D₄	1	2	yes	—
168	D₄	—	2	2	no	—
169	D₄	—	2	2	no	—
170	D₄	—	2	1	no	—
171	D₄	—	2	4	no	—
172	D₄	—	2	4	no	—
173	D₄	—	4	2	no	—
174	D₄	—	4	2	no	—
175	D₄	C₂×C₃×D₄	1	2	yes	—
176	C₂×C₄	—	2	1	no	—
177	C₂×C₄	—	2	2	no	—
178	C₂×C₄	—	2	2	no	—
179	C₂×C₄	C₂×C₃×D₄	1	1	yes	characteristic
180	C₂×C₄	C₂×C₃×D₄	1	1	yes	characteristic
181	C₂×C₄	C₂×C₃×D₄	1	1	yes	characteristic
182	C₂×C₄	C₂×C₃×D₄	1	1	yes	characteristic
183	C₂×C₄	C₃×#41	1	1	yes	characteristic
184	C₂×C₄	C₂⁴×C₃	1	1	yes	characteristic
185	C₈	—	2	2	no	—
186	C₈	—	2	4	no	—
187	C₈	C₂×C₃×D₄	1	2	yes	—
188	C₈	C₂×C₃×D₄	1	2	yes	—
189	C₂×C₃	—	2	1	no	—
190	C₂×C₃	—	2	1	no	—
191	C₂×C₃	—	4	2	no	—
192	C₂×C₃	—	8	4	no	—
193	C₂×C₃	#472	1	2	yes	direct product factor
194	C₂×C₃	C₂×#121	1	1	yes	characteristic
195	C₂²	—	2	2	no	—
196	C₂²	—	2	2	no	—
197	C₂²	—	4	2	no	—
198	C₂²	—	4	4	no	—
199	C₂²	—	8	4	no	—
200	C₂²	—	8	4	no	—
201	C₂²	—	8	4	no	—
202	C₂²	C₃×#121	1	1	yes	characteristic
203	C₂²	C₃×#121	1	1	yes	characteristic
204	C₂²	C₂×C₃×#41	1	1	yes	characteristic
205	C₄	—	2	2	no	—
206	C₄	C₃×#121	1	1	yes	characteristic
207	C₄	C₃×#121	1	1	yes	characteristic
208	C₄	C₂²×C₃×D₄	1	1	yes	characteristic
209	C₄	C₂²×C₃×D₄	1	1	yes	characteristic
210	C₃	C₂×#472	1	1	yes	characteristic, direct product factor, Sylow
211	C₂	—	2	1	no	—
212	C₂	—	2	1	no	—
213	C₂	—	4	2	no	—
214	C₂	—	8	4	no	—
215	C₂	C₃×#472	1	2	yes	direct product factor
216	C₂	C₂×C₃×#121	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃	#121
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₃
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃ ◁ C₂²×C₃×C₄ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₃×AGL(1,ℤ/8)	C₂	semidirect
2. C₂²×C₃×D₈	C₂	semidirect
3. C₂×C₃×#131	C₂	semidirect
4. C₃×#472	C₂	direct (N × H)
5. C₂×C₃×#108	C₂	semidirect
6. C₂×C₃×#100	C₂	semidirect
7. C₂×#472	C₃	direct (N × H)
11. C₃×AGL(1,ℤ/8)	C₂²	semidirect
17. C₂×C₃×D₈	C₂²	semidirect
19. C₃×#131	C₂²	semidirect
25. C₃×#108	C₂²	semidirect
26. C₃×#100	C₂²	semidirect
27. C₂×AGL(1,ℤ/8)	C₂×C₃	semidirect
28. C₂²×D₈	C₂×C₃	semidirect
29. C₂×#131	C₂×C₃	semidirect
30. #472	C₂×C₃	direct (N × H)
31. C₂×#108	C₂×C₃	semidirect
32. C₂×#100	C₂×C₃	semidirect
68. AGL(1,ℤ/8)	C₂²×C₃	semidirect
74. C₂×D₈	C₂²×C₃	semidirect
76. #131	C₂²×C₃	semidirect
82. #108	C₂²×C₃	semidirect
83. #100	C₂²×C₃	semidirect
193. C₂×C₃	#472	direct (N × H)
210. C₃	C₂×#472	direct (N × H)
215. C₂	C₃×#472	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×C₃×AGL(1,ℤ/8)	2
C₂²×C₃×D₈	2
C₂×C₃×#131	2
C₃×#472	2
C₂×C₃×#108	2
C₂×C₃×#100	2
C₂×#472	3

Subgroups — C2×C3×#472 #84929

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×#472 #84929