Subgroups — C₂×C₃×#540 #87336

Order 384.

1260 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	C₃×#540	C₂	1	16	yes	maximal, direct product factor
2	C₂×C₃×#127	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃×#125	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₃×#123	C₂	1	4	yes	maximal, complemented
5	C₂×C₃×#121	C₂	1	4	yes	maximal, complemented
6	C₂×C₃×#121	C₂	1	2	yes	maximal, complemented
7	C₂×C₃×C₄×D₄	C₂	1	2	yes	maximal, complemented
8	C₂×C₃×#117	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#540	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
10	C₂²×C₃×D₄	C₂²	1	2	yes	—
11	C₂²×C₃×D₄	C₂²	1	4	yes	—
12	C₂²×C₃×D₄	C₂²	1	4	yes	—
13	C₂³×C₃×C₄	C₂²	1	1	yes	characteristic
14	C₂³×C₃×C₄	C₂²	1	4	yes	—
15	C₃×#127	C₂²	1	8	yes	complemented
16	C₃×#125	C₂²	1	8	yes	complemented
17	C₃×#123	C₂²	1	32	yes	complemented
18	C₃×#121	C₂²	1	32	yes	complemented
19	C₃×#121	C₂²	1	16	yes	complemented
20	C₃×C₄×D₄	C₂²	1	16	yes	complemented
21	C₃×#117	C₂²	1	8	yes	complemented
22	C₂×C₃×C₄⋊C₄	C₂²	1	2	yes	—
23	C₂×C₃×C₄⋊C₄	C₂²	1	2	yes	—
24	C₂×C₃×C₄⋊C₄	C₂²	1	4	yes	—
25	C₂×C₃×C₂²⋊C₄	C₂²	1	2	yes	—
26	C₂×C₃×C₂²⋊C₄	C₂²	1	4	yes	—
27	C₂×C₃×C₂²⋊C₄	C₂²	1	4	yes	—
28	C₂×C₃×C₄²	C₂²	1	1	yes	characteristic
29	C₂×C₃×C₄²	C₂²	1	1	yes	characteristic
30	#540	C₂×C₃	1	16	yes	direct product factor
31	C₂×#127	C₂×C₃	1	1	yes	characteristic, complemented
32	C₂×#125	C₂×C₃	1	1	yes	characteristic, complemented
33	C₂×#123	C₂×C₃	1	4	yes	complemented
34	C₂×#121	C₂×C₃	1	4	yes	complemented
35	C₂×#121	C₂×C₃	1	2	yes	complemented
36	C₂×C₄×D₄	C₂×C₃	1	2	yes	complemented
37	C₂×#117	C₂×C₃	1	1	yes	characteristic, complemented
38	C₂⁴×C₃	C₂³	1	1	yes	characteristic
39	C₂⁴×C₃	C₂³	1	4	yes	—
40	C₂×C₃×D₄	—	2	8	no	—
41	C₂×C₃×D₄	—	4	8	no	—
42	C₂×C₃×D₄	—	4	8	no	—
43	C₂×C₃×D₄	C₂³	1	8	yes	—
44	C₂×C₃×D₄	C₂³	1	16	yes	—
45	C₂×C₃×D₄	C₂³	1	16	yes	—
46	C₂²×C₃×C₄	—	2	4	no	—
47	C₂²×C₃×C₄	—	4	8	no	—
48	C₂²×C₃×C₄	C₂³	1	4	yes	—
49	C₂²×C₃×C₄	C₂³	1	2	yes	—
50	C₂²×C₃×C₄	C₂³	1	2	yes	—
51	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
52	C₂²×C₃×C₄	C₂³	1	4	yes	—
53	C₂²×C₃×C₄	C₂³	1	16	yes	—
54	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
55	C₃×C₄⋊C₄	C₂³	1	8	yes	—
56	C₃×C₄⋊C₄	C₂³	1	8	yes	—
57	C₃×C₄⋊C₄	C₂³	1	16	yes	—
58	C₃×C₂²⋊C₄	C₂³	1	8	yes	—
59	C₃×C₂²⋊C₄	C₂³	1	16	yes	—
60	C₃×C₂²⋊C₄	C₂³	1	16	yes	—
61	C₃×C₄²	C₂³	1	4	yes	—
62	C₃×C₄²	C₂³	1	4	yes	—
63	C₂²×D₄	C₂²×C₃	1	2	yes	—
64	C₂²×D₄	C₂²×C₃	1	4	yes	—
65	C₂²×D₄	C₂²×C₃	1	4	yes	—
66	C₂³×C₄	C₂²×C₃	1	1	yes	characteristic
67	C₂³×C₄	C₂²×C₃	1	4	yes	—
68	#127	C₂²×C₃	1	8	yes	complemented
69	#125	C₂²×C₃	1	8	yes	complemented
70	#123	C₂²×C₃	1	32	yes	complemented
71	#121	C₂²×C₃	1	32	yes	complemented
72	#121	C₂²×C₃	1	16	yes	complemented
73	C₄×D₄	C₂²×C₃	1	16	yes	complemented
74	#117	C₂²×C₃	1	8	yes	complemented
75	C₂×C₄⋊C₄	C₂²×C₃	1	2	yes	—
76	C₂×C₄⋊C₄	C₂²×C₃	1	2	yes	—
77	C₂×C₄⋊C₄	C₂²×C₃	1	4	yes	—
78	C₂×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
79	C₂×C₂²⋊C₄	C₂²×C₃	1	4	yes	—
80	C₂×C₂²⋊C₄	C₂²×C₃	1	4	yes	—
81	C₂×C₄²	C₂²×C₃	1	1	yes	characteristic
82	C₂×C₄²	C₂²×C₃	1	1	yes	characteristic
83	C₂³×C₃	—	2	2	no	—
84	C₂³×C₃	—	2	8	no	—
85	C₂³×C₃	—	2	8	no	—
86	C₂³×C₃	—	2	4	no	—
87	C₂³×C₃	—	2	8	no	—
88	C₂³×C₃	C₂⁴	1	2	yes	—
89	C₂³×C₃	C₂⁴	1	8	yes	—
90	C₂³×C₃	C₂⁴	1	1	yes	characteristic
91	C₃×D₄	—	2	16	no	—
92	C₃×D₄	—	4	16	no	—
93	C₃×D₄	—	4	16	no	—
94	C₂×C₃×C₄	—	2	2	no	—
95	C₂×C₃×C₄	—	2	4	no	—
96	C₂×C₃×C₄	—	2	8	no	—
97	C₂×C₃×C₄	—	2	4	no	—
98	C₂×C₃×C₄	—	2	8	no	—
99	C₂×C₃×C₄	—	4	16	no	—
100	C₂×C₃×C₄	#41	1	4	yes	—
101	C₂×C₃×C₄	C₂⁴	1	4	yes	—
102	C₂×C₃×C₄	C₂⁴	1	4	yes	—
103	C₂×C₃×C₄	C₂⁴	1	2	yes	—
104	C₂×C₃×C₄	C₂⁴	1	8	yes	—
105	C₂×C₃×C₄	C₂⁴	1	2	yes	—
106	C₂⁴	C₂³×C₃	1	1	yes	characteristic
107	C₂⁴	C₂³×C₃	1	4	yes	—
108	C₂×D₄	—	2	8	no	—
109	C₂×D₄	—	4	8	no	—
110	C₂×D₄	—	4	8	no	—
111	C₂×D₄	C₂³×C₃	1	8	yes	—
112	C₂×D₄	C₂³×C₃	1	16	yes	—
113	C₂×D₄	C₂³×C₃	1	16	yes	—
114	C₂²×C₄	—	2	4	no	—
115	C₂²×C₄	—	4	8	no	—
116	C₂²×C₄	C₂³×C₃	1	2	yes	—
117	C₂²×C₄	C₂³×C₃	1	2	yes	—
118	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
119	C₂²×C₄	C₂³×C₃	1	4	yes	—
120	C₂²×C₄	C₂³×C₃	1	4	yes	—
121	C₂²×C₄	C₂³×C₃	1	16	yes	—
122	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
123	C₄⋊C₄	C₂³×C₃	1	8	yes	—
124	C₄⋊C₄	C₂³×C₃	1	8	yes	—
125	C₄⋊C₄	C₂³×C₃	1	16	yes	—
126	C₂²⋊C₄	C₂³×C₃	1	8	yes	—
127	C₂²⋊C₄	C₂³×C₃	1	16	yes	—
128	C₂²⋊C₄	C₂³×C₃	1	16	yes	—
129	C₄²	C₂³×C₃	1	4	yes	—
130	C₄²	C₂³×C₃	1	4	yes	—
131	C₂²×C₃	—	2	2	no	—
132	C₂²×C₃	—	2	8	no	—
133	C₂²×C₃	—	2	8	no	—
134	C₂²×C₃	—	2	4	no	—
135	C₂²×C₃	—	2	8	no	—
136	C₂²×C₃	—	4	4	no	—
137	C₂²×C₃	—	4	16	no	—
138	C₂²×C₃	C₂×#41	1	2	yes	—
139	C₂²×C₃	H(5,𝔽₂)	1	4	yes	—
140	C₂²×C₃	C₂⁵	1	1	yes	characteristic
141	C₃×C₄	—	2	2	no	—
142	C₃×C₄	—	2	4	no	—
143	C₃×C₄	—	2	8	no	—
144	C₃×C₄	—	2	4	no	—
145	C₃×C₄	C₂×#41	1	4	yes	—
146	C₂³	—	2	2	no	—
147	C₂³	—	2	8	no	—
148	C₂³	—	2	8	no	—
149	C₂³	—	2	4	no	—
150	C₂³	—	2	8	no	—
151	C₂³	C₂⁴×C₃	1	2	yes	—
152	C₂³	C₂⁴×C₃	1	8	yes	—
153	C₂³	C₂⁴×C₃	1	1	yes	characteristic
154	D₄	—	2	16	no	—
155	D₄	—	4	16	no	—
156	D₄	—	4	16	no	—
157	C₂×C₄	—	2	2	no	—
158	C₂×C₄	—	2	4	no	—
159	C₂×C₄	—	2	8	no	—
160	C₂×C₄	—	2	4	no	—
161	C₂×C₄	—	2	8	no	—
162	C₂×C₄	—	4	16	no	—
163	C₂×C₄	C₃×#41	1	4	yes	—
164	C₂×C₄	C₂⁴×C₃	1	4	yes	—
165	C₂×C₄	C₂⁴×C₃	1	4	yes	—
166	C₂×C₄	C₂⁴×C₃	1	2	yes	—
167	C₂×C₄	C₂⁴×C₃	1	8	yes	—
168	C₂×C₄	C₂⁴×C₃	1	2	yes	—
169	C₂×C₃	—	4	2	no	—
170	C₂×C₃	—	4	8	no	—
171	C₂×C₃	#540	1	4	yes	direct product factor
172	C₂×C₃	C₂²×#41	1	1	yes	characteristic
173	C₂×C₃	C₂×H(5,𝔽₂)	1	2	yes	—
174	C₂²	—	2	2	no	—
175	C₂²	—	2	8	no	—
176	C₂²	—	2	8	no	—
177	C₂²	—	2	4	no	—
178	C₂²	—	2	8	no	—
179	C₂²	—	4	4	no	—
180	C₂²	—	4	16	no	—
181	C₂²	C₂×C₃×#41	1	2	yes	—
182	C₂²	C₃×H(5,𝔽₂)	1	4	yes	—
183	C₂²	C₂⁵×C₃	1	1	yes	characteristic
184	C₄	—	2	2	no	—
185	C₄	—	2	4	no	—
186	C₄	—	2	8	no	—
187	C₄	—	2	4	no	—
188	C₄	C₂×C₃×#41	1	4	yes	—
189	C₃	C₂×#540	1	1	yes	characteristic, direct product factor, Sylow
190	C₂	—	4	2	no	—
191	C₂	—	4	8	no	—
192	C₂	C₃×#540	1	4	yes	direct product factor
193	C₂	C₂²×C₃×#41	1	1	yes	characteristic
194	C₂	C₂×C₃×H(5,𝔽₂)	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₃	C₂⁴
commutator, Frattini, cosocle	C₂²	C₂⁵×C₃
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₃×#540	C₂	direct (N × H)
2. C₂×C₃×#127	C₂	semidirect
3. C₂×C₃×#125	C₂	semidirect
4. C₂×C₃×#123	C₂	semidirect
5. C₂×C₃×#121	C₂	semidirect
6. C₂×C₃×#121	C₂	semidirect
7. C₂×C₃×C₄×D₄	C₂	semidirect
8. C₂×C₃×#117	C₂	semidirect
9. C₂×#540	C₃	direct (N × H)
15. C₃×#127	C₂²	semidirect
16. C₃×#125	C₂²	semidirect
17. C₃×#123	C₂²	semidirect
18. C₃×#121	C₂²	semidirect
19. C₃×#121	C₂²	semidirect
20. C₃×C₄×D₄	C₂²	semidirect
21. C₃×#117	C₂²	semidirect
30. #540	C₂×C₃	direct (N × H)
31. C₂×#127	C₂×C₃	semidirect
32. C₂×#125	C₂×C₃	semidirect
33. C₂×#123	C₂×C₃	semidirect
34. C₂×#121	C₂×C₃	semidirect
35. C₂×#121	C₂×C₃	semidirect
36. C₂×C₄×D₄	C₂×C₃	semidirect
37. C₂×#117	C₂×C₃	semidirect
68. #127	C₂²×C₃	semidirect
69. #125	C₂²×C₃	semidirect
70. #123	C₂²×C₃	semidirect
71. #121	C₂²×C₃	semidirect
72. #121	C₂²×C₃	semidirect
73. C₄×D₄	C₂²×C₃	semidirect
74. #117	C₂²×C₃	semidirect
171. C₂×C₃	#540	direct (N × H)
189. C₃	C₂×#540	direct (N × H)
192. C₂	C₃×#540	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃×#540	2
C₂×C₃×#127	2
C₂×C₃×#125	2
C₂×C₃×#123	2
C₂×C₃×#121	2
C₂×C₃×C₄×D₄	2
C₂×C₃×#117	2
C₂×#540	3

Subgroups — C2×C3×#540 #87336

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×#540 #87336