Subgroups — C₂³×C₃×#135 #10681687

Order 768.

11422 conjugacy classes of subgroups.

The subgroup lattice has 206 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₃×#41	C₂	1	2	yes	maximal, complemented
2	C₂²×C₃×#135	C₂	1	56	yes	maximal, direct product factor
3	C₂³×C₃×Q₁₆	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂³×C₃×SD₁₆	C₂	1	2	yes	maximal, complemented
5	C₂³×C₃×D₈	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂⁴×C₃×C₈	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂³×#135	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
8	C₂²×C₃×#41	—	2	28	no	—
9	C₂²×C₃×#41	C₂²	1	56	yes	complemented
10	C₂³×C₃×Q₈	C₂²	1	2	yes	—
11	C₂³×C₃×D₄	—	2	2	no	—
12	C₂³×C₃×D₄	C₂²	1	2	yes	—
13	C₂⁴×C₃×C₄	—	2	2	no	—
14	C₂⁴×C₃×C₄	C₂²	1	1	yes	characteristic
15	C₂×C₃×#135	C₂²	1	448	yes	direct product factor
16	C₂²×C₃×Q₁₆	C₂²	1	28	yes	complemented
17	C₂²×C₃×SD₁₆	C₂²	1	56	yes	complemented
18	C₂²×C₃×D₈	C₂²	1	28	yes	complemented
19	C₂³×C₃×C₈	C₂²	1	1	yes	characteristic
20	C₂³×C₃×C₈	C₂²	1	1	yes	characteristic
21	C₂³×C₃×C₈	C₂²	1	28	yes	complemented
22	C₂³×#41	C₂×C₃	1	2	yes	complemented
23	C₂²×#135	C₂×C₃	1	56	yes	direct product factor
24	C₂³×Q₁₆	C₂×C₃	1	1	yes	characteristic, complemented
25	C₂³×SD₁₆	C₂×C₃	1	2	yes	complemented
26	C₂³×D₈	C₂×C₃	1	1	yes	characteristic, complemented
27	C₂⁴×C₈	C₂×C₃	1	1	yes	characteristic, complemented
28	C₂⁵×C₃	—	2	2	no	—
29	C₂⁵×C₃	D₄	1	1	yes	characteristic
30	C₂×C₃×#41	—	2	336	no	—
31	C₂×C₃×#41	C₂³	1	224	yes	complemented
32	C₂²×C₃×Q₈	—	2	14	no	—
33	C₂²×C₃×Q₈	C₂³	1	28	yes	—
34	C₂²×C₃×D₄	—	2	56	no	—
35	C₂²×C₃×D₄	—	2	14	no	—
36	C₂²×C₃×D₄	C₂³	1	28	yes	—
37	C₂³×C₃×C₄	—	2	2	no	—
38	C₂³×C₃×C₄	—	2	56	no	—
39	C₂³×C₃×C₄	D₄	1	14	yes	—
40	C₂³×C₃×C₄	D₄	1	1	yes	characteristic
41	C₂³×C₃×C₄	C₂³	1	1	yes	characteristic
42	C₂³×C₃×C₄	C₂³	1	14	yes	—
43	C₃×#135	C₂³	1	512	yes	direct product factor
44	C₂×C₃×Q₁₆	C₂³	1	112	yes	complemented
45	C₂×C₃×SD₁₆	C₂³	1	224	yes	complemented
46	C₂×C₃×D₈	C₂³	1	112	yes	complemented
47	C₂²×C₃×C₈	C₂³	1	14	yes	—
48	C₂²×C₃×C₈	C₂³	1	14	yes	—
49	C₂²×C₃×C₈	C₂³	1	112	yes	complemented
50	C₂²×#41	—	2	28	no	—
51	C₂²×#41	C₂²×C₃	1	56	yes	complemented
52	C₂³×Q₈	C₂²×C₃	1	2	yes	—
53	C₂³×D₄	—	2	2	no	—
54	C₂³×D₄	C₂²×C₃	1	2	yes	—
55	C₂⁴×C₄	—	2	2	no	—
56	C₂⁴×C₄	C₂²×C₃	1	1	yes	characteristic
57	C₂×#135	C₂²×C₃	1	448	yes	direct product factor
58	C₂²×Q₁₆	C₂²×C₃	1	28	yes	complemented
59	C₂²×SD₁₆	C₂²×C₃	1	56	yes	complemented
60	C₂²×D₈	C₂²×C₃	1	28	yes	complemented
61	C₂³×C₈	C₂²×C₃	1	1	yes	characteristic
62	C₂³×C₈	C₂²×C₃	1	1	yes	characteristic
63	C₂³×C₈	C₂²×C₃	1	28	yes	complemented
64	C₂⁴×C₃	—	2	8	no	—
65	C₂⁴×C₃	—	2	28	no	—
66	C₂⁴×C₃	—	4	16	no	—
67	C₂⁴×C₃	C₂×D₄	1	14	yes	—
68	C₂⁴×C₃	C₂×D₄	1	1	yes	characteristic
69	C₃×#41	—	2	448	no	—
70	C₃×#41	C₂⁴	1	128	yes	complemented
71	C₂×C₃×Q₈	—	2	84	no	—
72	C₂×C₃×Q₈	C₂⁴	1	56	yes	—
73	C₂×C₃×D₄	—	2	224	no	—
74	C₂×C₃×D₄	—	2	84	no	—
75	C₂×C₃×D₄	C₂⁴	1	56	yes	—
76	C₂²×C₃×C₄	—	2	28	no	—
77	C₂²×C₃×C₄	—	2	224	no	—
78	C₂²×C₃×C₄	C₂×D₄	1	14	yes	—
79	C₂²×C₃×C₄	C₂×D₄	1	7	yes	—
80	C₂²×C₃×C₄	C₂×D₄	1	84	yes	—
81	C₂²×C₃×C₄	C₂⁴	1	7	yes	—
82	C₂²×C₃×C₄	C₂⁴	1	28	yes	—
83	C₃×Q₁₆	C₂⁴	1	64	yes	complemented
84	C₃×SD₁₆	C₂⁴	1	128	yes	complemented
85	C₃×D₈	C₂⁴	1	64	yes	complemented
86	C₂×C₃×C₈	C₂⁴	1	28	yes	—
87	C₂×C₃×C₈	C₂⁴	1	28	yes	—
88	C₂×C₃×C₈	C₂⁴	1	64	yes	complemented
89	C₂⁵	—	2	2	no	—
90	C₂⁵	C₃×D₄	1	1	yes	characteristic
91	C₂×#41	—	2	336	no	—
92	C₂×#41	C₂³×C₃	1	224	yes	complemented
93	C₂²×Q₈	—	2	14	no	—
94	C₂²×Q₈	C₂³×C₃	1	28	yes	—
95	C₂²×D₄	—	2	56	no	—
96	C₂²×D₄	—	2	14	no	—
97	C₂²×D₄	C₂³×C₃	1	28	yes	—
98	C₂³×C₄	—	2	2	no	—
99	C₂³×C₄	—	2	56	no	—
100	C₂³×C₄	C₃×D₄	1	1	yes	characteristic
101	C₂³×C₄	C₃×D₄	1	14	yes	—
102	C₂³×C₄	C₂³×C₃	1	1	yes	characteristic
103	C₂³×C₄	C₂³×C₃	1	14	yes	—
104	#135	C₂³×C₃	1	512	yes	direct product factor
105	C₂×Q₁₆	C₂³×C₃	1	112	yes	complemented
106	C₂×SD₁₆	C₂³×C₃	1	224	yes	complemented
107	C₂×D₈	C₂³×C₃	1	112	yes	complemented
108	C₂²×C₈	C₂³×C₃	1	14	yes	—
109	C₂²×C₈	C₂³×C₃	1	14	yes	—
110	C₂²×C₈	C₂³×C₃	1	112	yes	complemented
111	C₂³×C₃	—	2	56	no	—
112	C₂³×C₃	—	2	56	no	—
113	C₂³×C₃	—	4	112	no	—
114	C₂³×C₃	#135	1	8	yes	direct product factor
115	C₂³×C₃	C₂²×D₄	1	28	yes	—
116	C₂³×C₃	C₂²×D₄	1	7	yes	—
117	C₃×Q₈	—	2	56	no	—
118	C₃×Q₈	C₂⁵	1	16	yes	—
119	C₃×D₄	—	2	128	no	—
120	C₃×D₄	—	2	56	no	—
121	C₃×D₄	C₂⁵	1	16	yes	—
122	C₂×C₃×C₄	—	2	56	no	—
123	C₂×C₃×C₄	—	2	128	no	—
124	C₂×C₃×C₄	C₂²×D₄	1	28	yes	—
125	C₂×C₃×C₄	C₂²×D₄	1	21	yes	—
126	C₂×C₃×C₄	C₂²×D₄	1	56	yes	—
127	C₂×C₃×C₄	C₂⁵	1	7	yes	—
128	C₂×C₃×C₄	C₂⁵	1	8	yes	—
129	C₃×C₈	C₂⁵	1	8	yes	—
130	C₃×C₈	C₂⁵	1	8	yes	—
131	C₂⁴	—	2	8	no	—
132	C₂⁴	—	2	28	no	—
133	C₂⁴	—	4	16	no	—
134	C₂⁴	C₂×C₃×D₄	1	14	yes	—
135	C₂⁴	C₂×C₃×D₄	1	1	yes	characteristic
136	#41	—	2	448	no	—
137	#41	C₂⁴×C₃	1	128	yes	complemented
138	C₂×Q₈	—	2	84	no	—
139	C₂×Q₈	C₂⁴×C₃	1	56	yes	—
140	C₂×D₄	—	2	224	no	—
141	C₂×D₄	—	2	84	no	—
142	C₂×D₄	C₂⁴×C₃	1	56	yes	—
143	C₂²×C₄	—	2	28	no	—
144	C₂²×C₄	—	2	224	no	—
145	C₂²×C₄	C₂×C₃×D₄	1	14	yes	—
146	C₂²×C₄	C₂×C₃×D₄	1	7	yes	—
147	C₂²×C₄	C₂×C₃×D₄	1	84	yes	—
148	C₂²×C₄	C₂⁴×C₃	1	7	yes	—
149	C₂²×C₄	C₂⁴×C₃	1	28	yes	—
150	Q₁₆	C₂⁴×C₃	1	64	yes	complemented
151	SD₁₆	C₂⁴×C₃	1	128	yes	complemented
152	D₈	C₂⁴×C₃	1	64	yes	complemented
153	C₂×C₈	C₂⁴×C₃	1	28	yes	—
154	C₂×C₈	C₂⁴×C₃	1	28	yes	—
155	C₂×C₈	C₂⁴×C₃	1	64	yes	complemented
156	C₂²×C₃	—	2	56	no	—
157	C₂²×C₃	—	2	16	no	—
158	C₂²×C₃	—	4	112	no	—
159	C₂²×C₃	C₂×#135	1	28	yes	direct product factor
160	C₂²×C₃	C₂³×D₄	1	8	yes	—
161	C₂²×C₃	C₂³×D₄	1	7	yes	—
162	C₃×C₄	—	2	16	no	—
163	C₃×C₄	C₂³×D₄	1	8	yes	—
164	C₃×C₄	C₂³×D₄	1	7	yes	—
165	C₃×C₄	C₂⁶	1	1	yes	characteristic
166	C₂³	—	2	56	no	—
167	C₂³	—	2	56	no	—
168	C₂³	—	4	112	no	—
169	C₂³	C₃×#135	1	8	yes	direct product factor
170	C₂³	C₂²×C₃×D₄	1	28	yes	—
171	C₂³	C₂²×C₃×D₄	1	7	yes	—
172	Q₈	—	2	56	no	—
173	Q₈	C₂⁵×C₃	1	16	yes	—
174	D₄	—	2	128	no	—
175	D₄	—	2	56	no	—
176	D₄	C₂⁵×C₃	1	16	yes	—
177	C₂×C₄	—	2	56	no	—
178	C₂×C₄	—	2	128	no	—
179	C₂×C₄	C₂²×C₃×D₄	1	28	yes	—
180	C₂×C₄	C₂²×C₃×D₄	1	21	yes	—
181	C₂×C₄	C₂²×C₃×D₄	1	56	yes	—
182	C₂×C₄	C₂⁵×C₃	1	7	yes	—
183	C₂×C₄	C₂⁵×C₃	1	8	yes	—
184	C₈	C₂⁵×C₃	1	8	yes	—
185	C₈	C₂⁵×C₃	1	8	yes	—
186	C₂×C₃	—	2	8	no	—
187	C₂×C₃	—	4	16	no	—
188	C₂×C₃	C₂²×#135	1	14	yes	direct product factor
189	C₂×C₃	C₂⁴×D₄	1	1	yes	characteristic
190	C₂²	—	2	56	no	—
191	C₂²	—	2	16	no	—
192	C₂²	—	4	112	no	—
193	C₂²	C₂×C₃×#135	1	28	yes	direct product factor
194	C₂²	C₂³×C₃×D₄	1	8	yes	—
195	C₂²	C₂³×C₃×D₄	1	7	yes	—
196	C₄	—	2	16	no	—
197	C₄	C₂³×C₃×D₄	1	8	yes	—
198	C₄	C₂³×C₃×D₄	1	7	yes	—
199	C₄	C₂⁶×C₃	1	1	yes	characteristic
200	C₃	C₂³×#135	1	1	yes	characteristic, direct product factor, Sylow
201	C₂	—	2	8	no	—
202	C₂	—	4	16	no	—
203	C₂	C₂²×C₃×#135	1	14	yes	direct product factor
204	C₂	C₂⁴×C₃×D₄	1	1	yes	characteristic

	subgroup	G/H
center	C₂³×C₃×C₄	D₄
commutator, Frattini, cosocle	C₄	C₂⁶×C₃
socle	C₂⁴×C₃	C₂×D₄
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₃×C₄ ◁ C₂⁴×C₃×C₄ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂³×C₃×#41	C₂	semidirect
2. C₂²×C₃×#135	C₂	direct (N × H)
3. C₂³×C₃×Q₁₆	C₂	semidirect
4. C₂³×C₃×SD₁₆	C₂	semidirect
5. C₂³×C₃×D₈	C₂	semidirect
6. C₂⁴×C₃×C₈	C₂	semidirect
7. C₂³×#135	C₃	direct (N × H)
9. C₂²×C₃×#41	C₂²	semidirect
15. C₂×C₃×#135	C₂²	direct (N × H)
16. C₂²×C₃×Q₁₆	C₂²	semidirect
17. C₂²×C₃×SD₁₆	C₂²	semidirect
18. C₂²×C₃×D₈	C₂²	semidirect
21. C₂³×C₃×C₈	C₂²	semidirect
22. C₂³×#41	C₂×C₃	semidirect
23. C₂²×#135	C₂×C₃	direct (N × H)
24. C₂³×Q₁₆	C₂×C₃	semidirect
25. C₂³×SD₁₆	C₂×C₃	semidirect
26. C₂³×D₈	C₂×C₃	semidirect
27. C₂⁴×C₈	C₂×C₃	semidirect
31. C₂×C₃×#41	C₂³	semidirect
43. C₃×#135	C₂³	direct (N × H)
44. C₂×C₃×Q₁₆	C₂³	semidirect
45. C₂×C₃×SD₁₆	C₂³	semidirect
46. C₂×C₃×D₈	C₂³	semidirect
49. C₂²×C₃×C₈	C₂³	semidirect
51. C₂²×#41	C₂²×C₃	semidirect
57. C₂×#135	C₂²×C₃	direct (N × H)
58. C₂²×Q₁₆	C₂²×C₃	semidirect
59. C₂²×SD₁₆	C₂²×C₃	semidirect
60. C₂²×D₈	C₂²×C₃	semidirect
63. C₂³×C₈	C₂²×C₃	semidirect
70. C₃×#41	C₂⁴	semidirect
83. C₃×Q₁₆	C₂⁴	semidirect
84. C₃×SD₁₆	C₂⁴	semidirect
85. C₃×D₈	C₂⁴	semidirect
88. C₂×C₃×C₈	C₂⁴	semidirect
92. C₂×#41	C₂³×C₃	semidirect
104. #135	C₂³×C₃	direct (N × H)
105. C₂×Q₁₆	C₂³×C₃	semidirect
106. C₂×SD₁₆	C₂³×C₃	semidirect
107. C₂×D₈	C₂³×C₃	semidirect
110. C₂²×C₈	C₂³×C₃	semidirect
114. C₂³×C₃	#135	direct (N × H)
137. #41	C₂⁴×C₃	semidirect
150. Q₁₆	C₂⁴×C₃	semidirect
151. SD₁₆	C₂⁴×C₃	semidirect
152. D₈	C₂⁴×C₃	semidirect
155. C₂×C₈	C₂⁴×C₃	semidirect
159. C₂²×C₃	C₂×#135	direct (N × H)
169. C₂³	C₃×#135	direct (N × H)
188. C₂×C₃	C₂²×#135	direct (N × H)
193. C₂²	C₂×C₃×#135	direct (N × H)
200. C₃	C₂³×#135	direct (N × H)
203. C₂	C₂²×C₃×#135	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×#41	2
C₂²×C₃×#135	2
C₂³×C₃×Q₁₆	2
C₂³×C₃×SD₁₆	2
C₂³×C₃×D₈	2
C₂⁴×C₃×C₈	2
C₂³×#135	3

Subgroups — C23×C3×#135 #10681687

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×C₃×#135 #10681687