Subgroups — C₂×C₃×#418 #84757

Order 384.

1008 conjugacy classes of subgroups.

The subgroup lattice has 196 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₃×D₄	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₃×M₁₆	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃×#117	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×#418	C₂	1	8	yes	maximal, direct product factor
5	C₂×C₃×#102	C₂	1	4	yes	maximal, complemented
6	C₂×#418	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
7	C₂⁵×C₃	—	2	1	no	—
8	C₂²×C₃×D₄	—	2	2	no	—
9	C₂²×C₃×D₄	—	2	1	no	—
10	C₂²×C₃×D₄	—	2	4	no	—
11	C₂²×C₃×D₄	C₄	1	8	yes	complemented
12	C₂²×C₃×D₄	C₂²	1	2	yes	—
13	C₂²×C₃×D₄	C₂²	1	4	yes	—
14	C₂³×C₃×C₄	C₂²	1	1	yes	characteristic
15	C₂×C₃×M₁₆	—	2	4	no	—
16	C₂×C₃×M₁₆	C₂²	1	4	yes	complemented
17	C₂²×C₃×C₈	C₂²	1	2	yes	complemented
18	C₃×#117	—	2	2	no	—
19	C₃×#117	C₂²	1	4	yes	complemented
20	C₂×C₃×C₄⋊C₄	C₂²	1	2	yes	complemented
21	C₂×C₃×C₂²⋊C₄	—	2	1	no	—
22	C₂×C₃×C₄²	—	2	1	no	—
23	C₃×#102	C₂²	1	16	yes	complemented
24	C₂³×D₄	C₂×C₃	1	1	yes	characteristic
25	C₂²×M₁₆	C₂×C₃	1	1	yes	characteristic, complemented
26	C₂×#117	C₂×C₃	1	1	yes	characteristic, complemented
27	#418	C₂×C₃	1	8	yes	direct product factor
28	C₂×#102	C₂×C₃	1	4	yes	complemented
29	C₂⁴×C₃	—	2	8	no	—
30	C₂⁴×C₃	—	2	4	no	—
31	C₂⁴×C₃	—	2	2	no	—
32	C₂⁴×C₃	—	4	8	no	—
33	C₂⁴×C₃	D₄	1	1	yes	characteristic
34	C₂×C₃×D₄	—	2	16	no	—
35	C₂×C₃×D₄	—	2	8	no	—
36	C₂×C₃×D₄	—	2	4	no	—
37	C₂×C₃×D₄	—	2	2	no	—
38	C₂×C₃×D₄	—	2	4	no	—
39	C₂×C₃×D₄	—	2	8	no	—
40	C₂×C₃×D₄	C₂×C₄	1	8	yes	—
41	C₂×C₃×D₄	C₂×C₄	1	16	yes	complemented
42	C₂×C₃×D₄	C₂³	1	4	yes	—
43	C₂²×C₃×C₄	—	2	2	no	—
44	C₂²×C₃×C₄	C₂×C₄	1	4	yes	—
45	C₂²×C₃×C₄	D₄	1	2	yes	—
46	C₂²×C₃×C₄	D₄	1	1	yes	characteristic
47	C₂²×C₃×C₄	D₄	1	4	yes	—
48	C₂²×C₃×C₄	C₂³	1	2	yes	—
49	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
50	C₃×M₁₆	—	2	8	no	—
51	C₂×C₃×C₈	—	2	4	no	—
52	C₂×C₃×C₈	C₂³	1	4	yes	complemented
53	C₃×C₄⋊C₄	—	2	2	no	—
54	C₃×C₄⋊C₄	C₂³	1	4	yes	complemented
55	C₃×C₂²⋊C₄	—	2	4	no	—
56	C₃×C₄²	—	2	4	no	—
57	C₂⁵	—	2	1	no	—
58	C₂²×D₄	—	2	1	no	—
59	C₂²×D₄	—	2	2	no	—
60	C₂²×D₄	—	2	4	no	—
61	C₂²×D₄	C₃×C₄	1	8	yes	complemented
62	C₂²×D₄	C₂²×C₃	1	2	yes	—
63	C₂²×D₄	C₂²×C₃	1	4	yes	—
64	C₂³×C₄	C₂²×C₃	1	1	yes	characteristic
65	C₂×M₁₆	—	2	4	no	—
66	C₂×M₁₆	C₂²×C₃	1	4	yes	complemented
67	C₂²×C₈	C₂²×C₃	1	2	yes	complemented
68	#117	—	2	2	no	—
69	#117	C₂²×C₃	1	4	yes	complemented
70	C₂×C₄⋊C₄	C₂²×C₃	1	2	yes	complemented
71	C₂×C₂²⋊C₄	—	2	1	no	—
72	C₂×C₄²	—	2	1	no	—
73	#102	C₂²×C₃	1	16	yes	complemented
74	C₂³×C₃	—	2	4	no	—
75	C₂³×C₃	—	2	8	no	—
76	C₂³×C₃	—	2	16	no	—
77	C₂³×C₃	—	2	4	no	—
78	C₂³×C₃	—	4	16	no	—
79	C₂³×C₃	—	4	32	no	—
80	C₂³×C₃	—	4	8	no	—
81	C₂³×C₃	C₂²⋊C₄	1	4	yes	—
82	C₂³×C₃	C₂×D₄	1	2	yes	—
83	C₂³×C₃	C₂×D₄	1	1	yes	characteristic
84	C₃×D₄	—	2	8	no	—
85	C₃×D₄	—	2	16	no	—
86	C₃×D₄	—	2	4	no	—
87	C₃×D₄	C₂²×C₄	1	8	yes	—
88	C₂×C₃×C₄	—	2	4	no	—
89	C₂×C₃×C₄	—	4	4	no	—
90	C₂×C₃×C₄	C₂²⋊C₄	1	8	yes	—
91	C₂×C₃×C₄	C₂²⋊C₄	1	4	yes	—
92	C₂×C₃×C₄	C₂²×C₄	1	2	yes	—
93	C₂×C₃×C₄	C₂²×C₄	1	4	yes	—
94	C₂×C₃×C₄	C₂×D₄	1	2	yes	—
95	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
96	C₂×C₃×C₄	C₂×D₄	1	2	yes	—
97	C₂×C₃×C₄	C₂×D₄	1	4	yes	—
98	C₂×C₃×C₄	C₂⁴	1	1	yes	characteristic
99	C₃×C₈	—	2	4	no	—
100	C₂⁴	—	2	8	no	—
101	C₂⁴	—	2	2	no	—
102	C₂⁴	—	2	4	no	—
103	C₂⁴	—	4	8	no	—
104	C₂⁴	C₃×D₄	1	1	yes	characteristic
105	C₂×D₄	—	2	16	no	—
106	C₂×D₄	—	2	8	no	—
107	C₂×D₄	—	2	4	no	—
108	C₂×D₄	—	2	2	no	—
109	C₂×D₄	—	2	4	no	—
110	C₂×D₄	—	2	8	no	—
111	C₂×D₄	C₂×C₃×C₄	1	8	yes	—
112	C₂×D₄	C₂×C₃×C₄	1	16	yes	complemented
113	C₂×D₄	C₂³×C₃	1	4	yes	—
114	C₂²×C₄	—	2	2	no	—
115	C₂²×C₄	C₂×C₃×C₄	1	4	yes	—
116	C₂²×C₄	C₃×D₄	1	1	yes	characteristic
117	C₂²×C₄	C₃×D₄	1	2	yes	—
118	C₂²×C₄	C₃×D₄	1	4	yes	—
119	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
120	C₂²×C₄	C₂³×C₃	1	2	yes	—
121	M₁₆	—	2	8	no	—
122	C₂×C₈	—	2	4	no	—
123	C₂×C₈	C₂³×C₃	1	4	yes	complemented
124	C₄⋊C₄	—	2	2	no	—
125	C₄⋊C₄	C₂³×C₃	1	4	yes	complemented
126	C₂²⋊C₄	—	2	4	no	—
127	C₄²	—	2	4	no	—
128	C₂²×C₃	—	2	4	no	—
129	C₂²×C₃	—	2	8	no	—
130	C₂²×C₃	—	2	8	no	—
131	C₂²×C₃	—	4	32	no	—
132	C₂²×C₃	—	4	8	no	—
133	C₂²×C₃	—	4	16	no	—
134	C₂²×C₃	C₂×C₂²⋊C₄	1	4	yes	—
135	C₂²×C₃	C₂×C₂²⋊C₄	1	2	yes	—
136	C₂²×C₃	AGL(1,ℤ/8)	1	4	yes	—
137	C₂²×C₃	C₂²×D₄	1	1	yes	characteristic
138	C₃×C₄	—	4	4	no	—
139	C₃×C₄	C₂×C₂²⋊C₄	1	2	yes	—
140	C₃×C₄	C₂×C₂²⋊C₄	1	4	yes	—
141	C₃×C₄	C₂³×C₄	1	1	yes	characteristic
142	C₃×C₄	C₂²×D₄	1	1	yes	characteristic
143	C₂³	—	2	4	no	—
144	C₂³	—	2	8	no	—
145	C₂³	—	2	16	no	—
146	C₂³	—	2	4	no	—
147	C₂³	—	4	16	no	—
148	C₂³	—	4	32	no	—
149	C₂³	—	4	8	no	—
150	C₂³	C₃×C₂²⋊C₄	1	4	yes	—
151	C₂³	C₂×C₃×D₄	1	2	yes	—
152	C₂³	C₂×C₃×D₄	1	1	yes	characteristic
153	D₄	—	2	8	no	—
154	D₄	—	2	16	no	—
155	D₄	—	2	4	no	—
156	D₄	C₂²×C₃×C₄	1	8	yes	—
157	C₂×C₄	—	2	4	no	—
158	C₂×C₄	—	4	4	no	—
159	C₂×C₄	C₃×C₂²⋊C₄	1	8	yes	—
160	C₂×C₄	C₃×C₂²⋊C₄	1	4	yes	—
161	C₂×C₄	C₂²×C₃×C₄	1	2	yes	—
162	C₂×C₄	C₂²×C₃×C₄	1	4	yes	—
163	C₂×C₄	C₂×C₃×D₄	1	2	yes	—
164	C₂×C₄	C₂×C₃×D₄	1	1	yes	characteristic
165	C₂×C₄	C₂×C₃×D₄	1	2	yes	—
166	C₂×C₄	C₂×C₃×D₄	1	4	yes	—
167	C₂×C₄	C₂⁴×C₃	1	1	yes	characteristic
168	C₈	—	2	4	no	—
169	C₂×C₃	—	2	4	no	—
170	C₂×C₃	—	4	8	no	—
171	C₂×C₃	#418	1	4	yes	direct product factor
172	C₂×C₃	C₂²×C₂²⋊C₄	1	1	yes	characteristic
173	C₂×C₃	C₂×AGL(1,ℤ/8)	1	2	yes	—
174	C₂²	—	2	4	no	—
175	C₂²	—	2	8	no	—
176	C₂²	—	2	8	no	—
177	C₂²	—	4	32	no	—
178	C₂²	—	4	8	no	—
179	C₂²	—	4	16	no	—
180	C₂²	C₂×C₃×C₂²⋊C₄	1	4	yes	—
181	C₂²	C₂×C₃×C₂²⋊C₄	1	2	yes	—
182	C₂²	C₃×AGL(1,ℤ/8)	1	4	yes	—
183	C₂²	C₂²×C₃×D₄	1	1	yes	characteristic
184	C₄	—	4	4	no	—
185	C₄	C₂×C₃×C₂²⋊C₄	1	2	yes	—
186	C₄	C₂×C₃×C₂²⋊C₄	1	4	yes	—
187	C₄	C₂³×C₃×C₄	1	1	yes	characteristic
188	C₄	C₂²×C₃×D₄	1	1	yes	characteristic
189	C₃	C₂×#418	1	1	yes	characteristic, direct product factor, Sylow
190	C₂	—	2	4	no	—
191	C₂	—	4	8	no	—
192	C₂	C₃×#418	1	4	yes	direct product factor
193	C₂	C₂²×C₃×C₂²⋊C₄	1	1	yes	characteristic
194	C₂	C₂×C₃×AGL(1,ℤ/8)	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₃	C₂×D₄
commutator	C₄	C₂³×C₃×C₄
Frattini, cosocle	C₂×C₄	C₂⁴×C₃
prime radical	C₂³×C₃×D₄	C₂
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
2. C₂²×C₃×M₁₆	C₂	semidirect
3. C₂×C₃×#117	C₂	semidirect
4. C₃×#418	C₂	direct (N × H)
5. C₂×C₃×#102	C₂	semidirect
6. C₂×#418	C₃	direct (N × H)
11. C₂²×C₃×D₄	C₄	semidirect
16. C₂×C₃×M₁₆	C₂²	semidirect
17. C₂²×C₃×C₈	C₂²	semidirect
19. C₃×#117	C₂²	semidirect
20. C₂×C₃×C₄⋊C₄	C₂²	semidirect
23. C₃×#102	C₂²	semidirect
25. C₂²×M₁₆	C₂×C₃	semidirect
26. C₂×#117	C₂×C₃	semidirect
27. #418	C₂×C₃	direct (N × H)
28. C₂×#102	C₂×C₃	semidirect
41. C₂×C₃×D₄	C₂×C₄	semidirect
52. C₂×C₃×C₈	C₂³	semidirect
54. C₃×C₄⋊C₄	C₂³	semidirect
61. C₂²×D₄	C₃×C₄	semidirect
66. C₂×M₁₆	C₂²×C₃	semidirect
67. C₂²×C₈	C₂²×C₃	semidirect
69. #117	C₂²×C₃	semidirect
70. C₂×C₄⋊C₄	C₂²×C₃	semidirect
73. #102	C₂²×C₃	semidirect
112. C₂×D₄	C₂×C₃×C₄	semidirect
123. C₂×C₈	C₂³×C₃	semidirect
125. C₄⋊C₄	C₂³×C₃	semidirect
171. C₂×C₃	#418	direct (N × H)
189. C₃	C₂×#418	direct (N × H)
192. C₂	C₃×#418	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×C₃×D₄	2
C₂²×C₃×M₁₆	2
C₂×C₃×#117	2
C₃×#418	2
C₂×C₃×#102	2
C₂×#418	3

Subgroups — C2×C3×#418 #84757

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×#418 #84757