Subgroups — C₂×C₃×#478 #84936

Order 384.

448 conjugacy classes of subgroups.

The subgroup lattice has 208 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×#125	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₃×C₄×D₄	C₂	1	1	yes	maximal, characteristic
3	C₃×#478	C₂	1	8	yes	maximal, direct product factor
4	C₂×C₃×#107	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₃×#106	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₃×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₃×#102	C₂	1	1	yes	maximal, characteristic
8	C₂×C₃×#102	C₂	1	1	yes	maximal, characteristic
9	C₂×#478	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
10	C₂²×C₃×D₄	C₂²	1	1	yes	characteristic
11	C₂³×C₃×C₄	—	2	1	no	—
12	C₂²×C₃×C₈	C₂²	1	1	yes	characteristic
13	C₂²×C₃×C₈	C₂²	1	1	yes	characteristic
14	C₃×#125	—	2	2	no	—
15	C₃×#125	C₂²	1	4	yes	complemented
16	C₃×C₄×D₄	—	2	2	no	—
17	C₃×C₄×D₄	C₂²	1	4	yes	—
18	C₂×C₃×C₄⋊C₄	—	2	1	no	—
19	C₂×C₃×C₄⋊C₄	—	2	1	no	—
20	C₂×C₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
21	C₂×C₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
22	C₂×C₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
23	C₂×C₃×C₂²⋊C₄	—	2	1	no	—
24	C₂×C₃×C₄²	C₂²	1	1	yes	characteristic
25	C₃×#107	C₂²	1	4	yes	complemented
26	C₃×#106	C₂²	1	4	yes	complemented
27	C₃×C₄⋊C₈	C₂²	1	4	yes	complemented
28	C₃×#102	C₂²	1	4	yes	—
29	C₃×#102	C₂²	1	4	yes	—
30	C₂×#125	C₂×C₃	1	1	yes	characteristic, complemented
31	C₂×C₄×D₄	C₂×C₃	1	1	yes	characteristic
32	#478	C₂×C₃	1	8	yes	direct product factor
33	C₂×#107	C₂×C₃	1	1	yes	characteristic, complemented
34	C₂×#106	C₂×C₃	1	1	yes	characteristic, complemented
35	C₂×C₄⋊C₈	C₂×C₃	1	1	yes	characteristic, complemented
36	C₂×#102	C₂×C₃	1	1	yes	characteristic
37	C₂×#102	C₂×C₃	1	1	yes	characteristic
38	C₂⁴×C₃	—	2	1	no	—
39	C₂×C₃×D₄	—	2	1	no	—
40	C₂×C₃×D₄	—	2	2	no	—
41	C₂×C₃×D₄	Q₈	1	4	yes	—
42	C₂×C₃×D₄	C₂³	1	2	yes	—
43	C₂²×C₃×C₄	—	2	1	no	—
44	C₂²×C₃×C₄	—	2	1	no	—
45	C₂²×C₃×C₄	—	2	1	no	—
46	C₂²×C₃×C₄	—	2	4	no	—
47	C₂²×C₃×C₄	—	4	4	no	—
48	C₂²×C₃×C₄	D₄	1	1	yes	characteristic
49	C₂²×C₃×C₄	D₄	1	1	yes	characteristic
50	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
51	C₂×C₃×C₈	—	2	2	no	—
52	C₂×C₃×C₈	—	2	2	no	—
53	C₂×C₃×C₈	C₂³	1	2	yes	—
54	C₂×C₃×C₈	C₂³	1	2	yes	—
55	C₃×C₄⋊C₄	—	2	4	no	—
56	C₃×C₄⋊C₄	—	2	4	no	—
57	C₃×C₄⋊C₄	—	2	1	no	—
58	C₃×C₄⋊C₄	—	2	1	no	—
59	C₃×C₄⋊C₄	—	2	1	no	—
60	C₃×C₄⋊C₄	C₂³	1	2	yes	—
61	C₃×C₄⋊C₄	C₂³	1	2	yes	—
62	C₃×C₄⋊C₄	C₂³	1	2	yes	—
63	C₃×C₂²⋊C₄	—	2	4	no	—
64	C₃×C₄²	D₄	1	2	yes	—
65	C₃×C₄²	C₂³	1	2	yes	—
66	C₂²×D₄	C₂²×C₃	1	1	yes	characteristic
67	C₂³×C₄	—	2	1	no	—
68	C₂²×C₈	C₂²×C₃	1	1	yes	characteristic
69	C₂²×C₈	C₂²×C₃	1	1	yes	characteristic
70	#125	—	2	2	no	—
71	#125	C₂²×C₃	1	4	yes	complemented
72	C₄×D₄	—	2	2	no	—
73	C₄×D₄	C₂²×C₃	1	4	yes	—
74	C₂×C₄⋊C₄	—	2	1	no	—
75	C₂×C₄⋊C₄	—	2	1	no	—
76	C₂×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
77	C₂×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
78	C₂×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
79	C₂×C₂²⋊C₄	—	2	1	no	—
80	C₂×C₄²	C₂²×C₃	1	1	yes	characteristic
81	#107	C₂²×C₃	1	4	yes	complemented
82	#106	C₂²×C₃	1	4	yes	complemented
83	C₄⋊C₈	C₂²×C₃	1	4	yes	complemented
84	#102	C₂²×C₃	1	4	yes	—
85	#102	C₂²×C₃	1	4	yes	—
86	C₂³×C₃	—	2	4	no	—
87	C₂³×C₃	—	2	2	no	—
88	C₂³×C₃	—	4	2	no	—
89	C₂³×C₃	—	4	2	no	—
90	C₂³×C₃	C₂×D₄	1	1	yes	characteristic
91	C₃×D₄	—	2	4	no	—
92	C₃×D₄	—	2	2	no	—
93	C₃×D₄	C₂×Q₈	1	4	yes	—
94	C₂×C₃×C₄	—	2	2	no	—
95	C₂×C₃×C₄	—	2	2	no	—
96	C₂×C₃×C₄	—	2	2	no	—
97	C₂×C₃×C₄	—	2	2	no	—
98	C₂×C₃×C₄	—	2	2	no	—
99	C₂×C₃×C₄	—	4	8	no	—
100	C₂×C₃×C₄	—	4	2	no	—
101	C₂×C₃×C₄	—	4	2	no	—
102	C₂×C₃×C₄	—	4	2	no	—
103	C₂×C₃×C₄	C₂×D₄	1	2	yes	—
104	C₂×C₃×C₄	C₂×D₄	1	2	yes	—
105	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
106	C₂×C₃×C₄	C₂×Q₈	1	2	yes	—
107	C₂×C₃×C₄	#41	1	2	yes	—
108	C₂×C₃×C₄	C₂⁴	1	1	yes	characteristic
109	C₃×C₈	—	2	2	no	—
110	C₃×C₈	—	2	2	no	—
111	C₂⁴	—	2	1	no	—
112	C₂×D₄	—	2	1	no	—
113	C₂×D₄	—	2	2	no	—
114	C₂×D₄	C₃×Q₈	1	4	yes	—
115	C₂×D₄	C₂³×C₃	1	2	yes	—
116	C₂²×C₄	—	2	1	no	—
117	C₂²×C₄	—	2	1	no	—
118	C₂²×C₄	—	2	1	no	—
119	C₂²×C₄	—	2	4	no	—
120	C₂²×C₄	—	4	4	no	—
121	C₂²×C₄	C₃×D₄	1	1	yes	characteristic
122	C₂²×C₄	C₃×D₄	1	1	yes	characteristic
123	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
124	C₂×C₈	—	2	2	no	—
125	C₂×C₈	—	2	2	no	—
126	C₂×C₈	C₂³×C₃	1	2	yes	—
127	C₂×C₈	C₂³×C₃	1	2	yes	—
128	C₄⋊C₄	—	2	4	no	—
129	C₄⋊C₄	—	2	4	no	—
130	C₄⋊C₄	—	2	1	no	—
131	C₄⋊C₄	—	2	1	no	—
132	C₄⋊C₄	—	2	1	no	—
133	C₄⋊C₄	C₂³×C₃	1	2	yes	—
134	C₄⋊C₄	C₂³×C₃	1	2	yes	—
135	C₄⋊C₄	C₂³×C₃	1	2	yes	—
136	C₂²⋊C₄	—	2	4	no	—
137	C₄²	C₃×D₄	1	2	yes	—
138	C₄²	C₂³×C₃	1	2	yes	—
139	C₂²×C₃	—	2	4	no	—
140	C₂²×C₃	—	4	2	no	—
141	C₂²×C₃	—	4	2	no	—
142	C₂²×C₃	—	4	8	no	—
143	C₂²×C₃	C₂²⋊Q₈	1	2	yes	—
144	C₂²×C₃	#135	1	2	yes	—
145	C₂²×C₃	AGL(1,ℤ/8)	1	2	yes	—
146	C₂²×C₃	C₂²×D₄	1	1	yes	characteristic
147	C₃×C₄	—	2	2	no	—
148	C₃×C₄	—	2	2	no	—
149	C₃×C₄	—	4	2	no	—
150	C₃×C₄	—	4	2	no	—
151	C₃×C₄	—	4	2	no	—
152	C₃×C₄	C₂²⋊Q₈	1	2	yes	—
153	C₃×C₄	C₂²×Q₈	1	1	yes	characteristic
154	C₃×C₄	C₂×#41	1	1	yes	characteristic
155	C₂³	—	2	4	no	—
156	C₂³	—	2	2	no	—
157	C₂³	—	4	2	no	—
158	C₂³	—	4	2	no	—
159	C₂³	C₂×C₃×D₄	1	1	yes	characteristic
160	D₄	—	2	4	no	—
161	D₄	—	2	2	no	—
162	D₄	C₂×C₃×Q₈	1	4	yes	—
163	C₂×C₄	—	2	2	no	—
164	C₂×C₄	—	2	2	no	—
165	C₂×C₄	—	2	2	no	—
166	C₂×C₄	—	2	2	no	—
167	C₂×C₄	—	2	2	no	—
168	C₂×C₄	—	4	8	no	—
169	C₂×C₄	—	4	2	no	—
170	C₂×C₄	—	4	2	no	—
171	C₂×C₄	—	4	2	no	—
172	C₂×C₄	C₂×C₃×D₄	1	2	yes	—
173	C₂×C₄	C₂×C₃×D₄	1	2	yes	—
174	C₂×C₄	C₂×C₃×D₄	1	1	yes	characteristic
175	C₂×C₄	C₂×C₃×Q₈	1	2	yes	—
176	C₂×C₄	C₃×#41	1	2	yes	—
177	C₂×C₄	C₂⁴×C₃	1	1	yes	characteristic
178	C₈	—	2	2	no	—
179	C₈	—	2	2	no	—
180	C₂×C₃	—	4	4	no	—
181	C₂×C₃	#478	1	4	yes	direct product factor
182	C₂×C₃	C₂×C₂²⋊Q₈	1	1	yes	characteristic
183	C₂×C₃	C₂×#135	1	1	yes	characteristic
184	C₂×C₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
185	C₂²	—	2	4	no	—
186	C₂²	—	4	2	no	—
187	C₂²	—	4	2	no	—
188	C₂²	—	4	8	no	—
189	C₂²	C₃×C₂²⋊Q₈	1	2	yes	—
190	C₂²	C₃×#135	1	2	yes	—
191	C₂²	C₃×AGL(1,ℤ/8)	1	2	yes	—
192	C₂²	C₂²×C₃×D₄	1	1	yes	characteristic
193	C₄	—	2	2	no	—
194	C₄	—	2	2	no	—
195	C₄	—	4	2	no	—
196	C₄	—	4	2	no	—
197	C₄	—	4	2	no	—
198	C₄	C₃×C₂²⋊Q₈	1	2	yes	—
199	C₄	C₂²×C₃×Q₈	1	1	yes	characteristic
200	C₄	C₂×C₃×#41	1	1	yes	characteristic
201	C₃	C₂×#478	1	1	yes	characteristic, direct product factor, Sylow
202	C₂	—	4	4	no	—
203	C₂	C₃×#478	1	4	yes	direct product factor
204	C₂	C₂×C₃×C₂²⋊Q₈	1	1	yes	characteristic
205	C₂	C₂×C₃×#135	1	1	yes	characteristic
206	C₂	C₂×C₃×AGL(1,ℤ/8)	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₃	C₂×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₃
prime radical	C₂²×C₃×D₄	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₃ ◁ C₂×C₃×C₄² ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₃×#125	C₂	semidirect
3. C₃×#478	C₂	direct (N × H)
4. C₂×C₃×#107	C₂	semidirect
5. C₂×C₃×#106	C₂	semidirect
6. C₂×C₃×C₄⋊C₈	C₂	semidirect
9. C₂×#478	C₃	direct (N × H)
15. C₃×#125	C₂²	semidirect
25. C₃×#107	C₂²	semidirect
26. C₃×#106	C₂²	semidirect
27. C₃×C₄⋊C₈	C₂²	semidirect
30. C₂×#125	C₂×C₃	semidirect
32. #478	C₂×C₃	direct (N × H)
33. C₂×#107	C₂×C₃	semidirect
34. C₂×#106	C₂×C₃	semidirect
35. C₂×C₄⋊C₈	C₂×C₃	semidirect
71. #125	C₂²×C₃	semidirect
81. #107	C₂²×C₃	semidirect
82. #106	C₂²×C₃	semidirect
83. C₄⋊C₈	C₂²×C₃	semidirect
181. C₂×C₃	#478	direct (N × H)
201. C₃	C₂×#478	direct (N × H)
203. C₂	C₃×#478	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×#125	2
C₂×C₃×C₄×D₄	2
C₃×#478	2
C₂×C₃×#107	2
C₂×C₃×#106	2
C₂×C₃×C₄⋊C₈	2
C₂×C₃×#102	2
C₂×#478	3

Subgroups — C2×C3×#478 #84936

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×#478 #84936