Subgroups — C₂²×C₃²×D₃×C₈ #13179113

Order 1728.

2332 conjugacy classes of subgroups.

The subgroup lattice has 232 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃²×C₄×D₃	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₃³×C₈	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×C₃²×C₃⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₃²×D₃×C₈	C₂	1	12	yes	maximal, direct product factor
5	C₂³×C₃²×C₈	—	3	1	no	maximal
6	C₂²×C₃×D₃×C₈	C₃	1	4	yes	maximal, direct product factor
7	C₂³×C₃²×D₃	C₄	1	1	yes	characteristic
8	C₂²×C₃³×C₄	C₂²	1	1	yes	characteristic
9	C₂²×C₃²×Dic₃	C₄	1	1	yes	characteristic
10	C₂×C₃²×C₄×D₃	C₄	1	6	yes	—
11	C₂×C₃²×C₄×D₃	C₂²	1	6	yes	—
12	C₂×C₃³×C₈	C₂²	1	6	yes	complemented
13	C₂×C₃²×C₃⋊C₈	C₂²	1	6	yes	complemented
14	C₃²×D₃×C₈	C₂²	1	16	yes	direct product factor
15	C₂³×C₃²×C₄	—	3	1	no	—
16	C₂²×C₃×C₄×D₃	C₂×C₃	1	4	yes	—
17	C₂²×C₃²×C₈	—	2	4	no	—
18	C₂²×C₃²×C₈	—	3	1	no	—
19	C₂²×C₃²×C₈	—	3	12	no	—
20	C₂²×C₃²×C₈	D₃	1	1	yes	characteristic, direct product factor
21	C₂²×C₃²×C₈	C₂×C₃	1	4	yes	complemented
22	C₂²×C₃×C₃⋊C₈	C₂×C₃	1	4	yes	complemented
23	C₂×C₃×D₃×C₈	C₂×C₃	1	48	yes	direct product factor
24	C₂³×C₃³	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₃²×D₃	C₈	1	8	yes	direct product factor
26	C₂²×C₃²×D₃	C₂×C₄	1	6	yes	—
27	C₂×C₃³×C₄	C₂×C₄	1	3	yes	—
28	C₂×C₃³×C₄	C₂³	1	3	yes	—
29	C₂×C₃²×Dic₃	C₂×C₄	1	6	yes	—
30	C₃²×C₄×D₃	C₂×C₄	1	12	yes	—
31	C₃²×C₄×D₃	C₂³	1	4	yes	—
32	C₃³×C₈	C₂³	1	4	yes	complemented
33	C₃²×C₃⋊C₈	C₂³	1	4	yes	complemented
34	C₂³×C₃×C₈	—	3	4	no	—
35	C₂²×D₃×C₈	C₃²	1	1	yes	characteristic, direct product factor
36	C₂⁴×C₃²	—	3	1	no	—
37	C₂³×C₃×D₃	C₃×C₄	1	4	yes	—
38	C₂²×C₃²×C₄	—	2	4	no	—
39	C₂²×C₃²×C₄	—	3	1	no	—
40	C₂²×C₃²×C₄	—	3	6	no	—
41	C₂²×C₃²×C₄	—	3	6	no	—
42	C₂²×C₃²×C₄	C₂×D₃	1	1	yes	characteristic
43	C₂²×C₃²×C₄	C₂²×C₃	1	4	yes	—
44	C₂²×C₃×Dic₃	C₃×C₄	1	4	yes	—
45	C₂×C₃×C₄×D₃	C₃×C₄	1	24	yes	—
46	C₂×C₃×C₄×D₃	C₂²×C₃	1	24	yes	—
47	C₂×C₃²×C₈	—	2	24	no	—
48	C₂×C₃²×C₈	—	3	6	no	—
49	C₂×C₃²×C₈	—	3	16	no	—
50	C₂×C₃²×C₈	C₂×D₃	1	6	yes	direct product factor
51	C₂×C₃²×C₈	C₂²×C₃	1	24	yes	complemented
52	C₂×C₃×C₃⋊C₈	C₂²×C₃	1	24	yes	complemented
53	C₃×D₃×C₈	C₂²×C₃	1	64	yes	direct product factor
54	C₂²×C₃³	C₂×C₈	1	4	yes	complemented
55	C₂²×C₃³	C₂²×C₄	1	3	yes	—
56	C₂×C₃²×D₃	C₂×C₈	1	24	yes	direct product factor
57	C₂×C₃²×D₃	C₂²×C₄	1	4	yes	—
58	C₃³×C₄	C₂²×C₄	1	3	yes	—
59	C₃³×C₄	C₂⁴	1	1	yes	characteristic
60	C₃²×Dic₃	C₂²×C₄	1	4	yes	—
61	C₂³×C₃×C₄	—	3	4	no	—
62	C₂²×C₄×D₃	C₂×C₃²	1	1	yes	characteristic
63	C₂²×C₃×C₈	—	2	4	no	—
64	C₂²×C₃×C₈	—	3	4	no	—
65	C₂²×C₃×C₈	—	3	48	no	—
66	C₂²×C₃×C₈	C₃×D₃	1	4	yes	direct product factor
67	C₂²×C₃×C₈	C₂×C₃²	1	1	yes	characteristic, complemented
68	C₂²×C₃⋊C₈	C₂×C₃²	1	1	yes	characteristic, complemented
69	C₂×D₃×C₈	C₂×C₃²	1	12	yes	direct product factor
70	C₂³×C₃²	—	2	4	no	—
71	C₂³×C₃²	—	3	8	no	—
72	C₂³×C₃²	—	3	6	no	—
73	C₂³×C₃²	C₄×D₃	1	1	yes	characteristic
74	C₂³×C₃²	C₂×C₃×C₄	1	4	yes	—
75	C₂²×C₃×D₃	C₃×C₈	1	32	yes	direct product factor
76	C₂²×C₃×D₃	C₂×C₃×C₄	1	24	yes	—
77	C₂×C₃²×C₄	—	2	12	no	—
78	C₂×C₃²×C₄	—	2	12	no	—
79	C₂×C₃²×C₄	—	3	6	no	—
80	C₂×C₃²×C₄	—	3	12	no	—
81	C₂×C₃²×C₄	—	3	4	no	—
82	C₂×C₃²×C₄	C₄×D₃	1	3	yes	—
83	C₂×C₃²×C₄	C₂×C₃×C₄	1	12	yes	—
84	C₂×C₃²×C₄	C₂²×D₃	1	3	yes	—
85	C₂×C₃²×C₄	C₂³×C₃	1	12	yes	—
86	C₂×C₃×Dic₃	C₂×C₃×C₄	1	24	yes	—
87	C₃×C₄×D₃	C₂×C₃×C₄	1	48	yes	—
88	C₃×C₄×D₃	C₂³×C₃	1	16	yes	—
89	C₃²×C₈	—	2	16	no	—
90	C₃²×C₈	—	3	4	no	—
91	C₃²×C₈	C₂²×D₃	1	4	yes	direct product factor
92	C₃²×C₈	C₂³×C₃	1	16	yes	complemented
93	C₃×C₃⋊C₈	C₂³×C₃	1	16	yes	complemented
94	C₂³×C₈	—	3	1	no	Sylow
95	C₂×C₃³	C₂²×C₈	1	6	yes	complemented
96	C₂×C₃³	C₂³×C₄	1	1	yes	characteristic
97	C₃²×D₃	C₂²×C₈	1	8	yes	direct product factor
98	C₂⁴×C₃	—	3	4	no	—
99	C₂³×D₃	C₃²×C₄	1	1	yes	characteristic
100	C₂²×C₃×C₄	—	2	4	no	—
101	C₂²×C₃×C₄	—	3	4	no	—
102	C₂²×C₃×C₄	—	3	24	no	—
103	C₂²×C₃×C₄	—	3	24	no	—
104	C₂²×C₃×C₄	C₂×C₃×D₃	1	4	yes	—
105	C₂²×C₃×C₄	C₂²×C₃²	1	1	yes	characteristic
106	C₂²×Dic₃	C₃²×C₄	1	1	yes	characteristic
107	C₂×C₄×D₃	C₃²×C₄	1	6	yes	—
108	C₂×C₄×D₃	C₂²×C₃²	1	6	yes	—
109	C₂×C₃×C₈	—	2	24	no	—
110	C₂×C₃×C₈	—	3	24	no	—
111	C₂×C₃×C₈	—	3	64	no	—
112	C₂×C₃×C₈	C₂×C₃×D₃	1	24	yes	direct product factor
113	C₂×C₃×C₈	C₂²×C₃²	1	6	yes	complemented
114	C₂×C₃⋊C₈	C₂²×C₃²	1	6	yes	complemented
115	D₃×C₈	C₂²×C₃²	1	16	yes	direct product factor
116	C₂²×C₃²	—	2	16	no	—
117	C₂²×C₃²	—	2	12	no	—
118	C₂²×C₃²	—	3	24	no	—
119	C₂²×C₃²	—	3	4	no	—
120	C₂²×C₃²	D₃×C₈	1	4	yes	direct product factor
121	C₂²×C₃²	C₂×C₃×C₈	1	16	yes	complemented
122	C₂²×C₃²	C₂×C₄×D₃	1	3	yes	—
123	C₂²×C₃²	C₂²×C₃×C₄	1	12	yes	—
124	C₂×C₃×D₃	C₂×C₃×C₈	1	96	yes	direct product factor
125	C₂×C₃×D₃	C₂²×C₃×C₄	1	16	yes	—
126	C₃²×C₄	—	2	4	no	—
127	C₃²×C₄	—	2	12	no	—
128	C₃²×C₄	—	3	4	no	—
129	C₃²×C₄	C₂×C₄×D₃	1	3	yes	—
130	C₃²×C₄	C₂²×C₃×C₄	1	12	yes	—
131	C₃²×C₄	C₂³×D₃	1	1	yes	characteristic
132	C₃²×C₄	C₂⁴×C₃	1	4	yes	—
133	C₃×Dic₃	C₂²×C₃×C₄	1	16	yes	—
134	C₂³×C₄	—	3	1	no	—
135	C₂²×C₈	—	3	1	no	—
136	C₂²×C₈	—	3	12	no	—
137	C₂²×C₈	C₃²×D₃	1	1	yes	characteristic, direct product factor
138	C₃³	C₂³×C₈	1	1	yes	characteristic, complemented, Sylow
139	C₂³×C₃	—	2	4	no	—
140	C₂³×C₃	—	3	32	no	—
141	C₂³×C₃	—	3	24	no	—
142	C₂³×C₃	C₃×C₄×D₃	1	4	yes	—
143	C₂³×C₃	C₂×C₃²×C₄	1	1	yes	characteristic
144	C₂²×D₃	C₃²×C₈	1	8	yes	direct product factor
145	C₂²×D₃	C₂×C₃²×C₄	1	6	yes	—
146	C₂×C₃×C₄	—	2	12	no	—
147	C₂×C₃×C₄	—	2	12	no	—
148	C₂×C₃×C₄	—	3	24	no	—
149	C₂×C₃×C₄	—	3	16	no	—
150	C₂×C₃×C₄	—	3	48	no	—
151	C₂×C₃×C₄	C₃×C₄×D₃	1	12	yes	—
152	C₂×C₃×C₄	C₂×C₃²×C₄	1	3	yes	—
153	C₂×C₃×C₄	C₂²×C₃×D₃	1	12	yes	—
154	C₂×C₃×C₄	C₂³×C₃²	1	3	yes	—
155	C₂×Dic₃	C₂×C₃²×C₄	1	6	yes	—
156	C₄×D₃	C₂×C₃²×C₄	1	12	yes	—
157	C₄×D₃	C₂³×C₃²	1	4	yes	—
158	C₃×C₈	—	2	16	no	—
159	C₃×C₈	—	3	16	no	—
160	C₃×C₈	C₂²×C₃×D₃	1	16	yes	direct product factor
161	C₃×C₈	C₂³×C₃²	1	4	yes	complemented
162	C₃⋊C₈	C₂³×C₃²	1	4	yes	complemented
163	C₂×C₃²	—	2	24	no	—
164	C₂×C₃²	—	2	4	no	—
165	C₂×C₃²	—	3	8	no	—
166	C₂×C₃²	C₂×D₃×C₈	1	6	yes	direct product factor
167	C₂×C₃²	C₂²×C₃×C₈	1	24	yes	complemented
168	C₂×C₃²	C₂²×C₄×D₃	1	1	yes	characteristic
169	C₂×C₃²	C₂³×C₃×C₄	1	4	yes	—
170	C₃×D₃	C₂²×C₃×C₈	1	32	yes	direct product factor
171	C₂⁴	—	3	1	no	—
172	C₂²×C₄	—	3	1	no	—
173	C₂²×C₄	—	3	6	no	—
174	C₂²×C₄	—	3	6	no	—
175	C₂²×C₄	C₂×C₃²×D₃	1	1	yes	characteristic
176	C₂×C₈	—	3	6	no	—
177	C₂×C₈	—	3	16	no	—
178	C₂×C₈	C₂×C₃²×D₃	1	6	yes	direct product factor
179	C₂²×C₃	—	2	12	no	—
180	C₂²×C₃	—	2	16	no	—
181	C₂²×C₃	—	3	96	no	—
182	C₂²×C₃	—	3	16	no	—
183	C₂²×C₃	C₃×D₃×C₈	1	16	yes	direct product factor
184	C₂²×C₃	C₂×C₃²×C₈	1	4	yes	complemented
185	C₂²×C₃	C₂×C₃×C₄×D₃	1	12	yes	—
186	C₂²×C₃	C₂²×C₃²×C₄	1	3	yes	—
187	C₂×D₃	C₂×C₃²×C₈	1	24	yes	direct product factor
188	C₂×D₃	C₂²×C₃²×C₄	1	4	yes	—
189	C₃×C₄	—	2	4	no	—
190	C₃×C₄	—	2	12	no	—
191	C₃×C₄	—	3	16	no	—
192	C₃×C₄	C₂×C₃×C₄×D₃	1	12	yes	—
193	C₃×C₄	C₂²×C₃²×C₄	1	3	yes	—
194	C₃×C₄	C₂³×C₃×D₃	1	4	yes	—
195	C₃×C₄	C₂⁴×C₃²	1	1	yes	characteristic
196	Dic₃	C₂²×C₃²×C₄	1	4	yes	—
197	C₃²	—	2	4	no	—
198	C₃²	C₂²×D₃×C₈	1	1	yes	characteristic, direct product factor
199	C₃²	C₂³×C₃×C₈	1	4	yes	complemented
200	C₂³	—	3	6	no	—
201	C₂³	—	3	8	no	—
202	C₂³	C₃²×C₄×D₃	1	1	yes	characteristic
203	C₂×C₄	—	3	6	no	—
204	C₂×C₄	—	3	4	no	—
205	C₂×C₄	—	3	12	no	—
206	C₂×C₄	C₃²×C₄×D₃	1	3	yes	—
207	C₂×C₄	C₂²×C₃²×D₃	1	3	yes	—
208	C₈	—	3	4	no	—
209	C₈	C₂²×C₃²×D₃	1	4	yes	direct product factor
210	C₂×C₃	—	2	4	no	—
211	C₂×C₃	—	2	24	no	—
212	C₂×C₃	—	3	32	no	—
213	C₂×C₃	C₂×C₃×D₃×C₈	1	24	yes	direct product factor
214	C₂×C₃	C₂²×C₃²×C₈	1	6	yes	complemented
215	C₂×C₃	C₂²×C₃×C₄×D₃	1	4	yes	—
216	C₂×C₃	C₂³×C₃²×C₄	1	1	yes	characteristic
217	D₃	C₂²×C₃²×C₈	1	8	yes	direct product factor
218	C₂²	—	3	4	no	—
219	C₂²	—	3	24	no	—
220	C₂²	C₃²×D₃×C₈	1	4	yes	direct product factor
221	C₂²	C₂×C₃²×C₄×D₃	1	3	yes	—
222	C₄	—	3	4	no	—
223	C₄	C₂×C₃²×C₄×D₃	1	3	yes	—
224	C₄	C₂³×C₃²×D₃	1	1	yes	characteristic
225	C₃	—	2	4	no	—
226	C₃	C₂²×C₃×D₃×C₈	1	4	yes	direct product factor
227	C₃	C₂³×C₃²×C₈	1	1	yes	characteristic, complemented
228	C₂	—	3	8	no	—
229	C₂	C₂×C₃²×D₃×C₈	1	6	yes	direct product factor
230	C₂	C₂²×C₃²×C₄×D₃	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₂²×C₃²×C₈	D₃
commutator, hypocenter	C₃	C₂³×C₃²×C₈
Frattini	C₄	C₂³×C₃²×D₃
Fitting	C₂²×C₃³×C₈	C₂
socle	C₂³×C₃³	C₂×C₄
prime radical	C₂³×C₃²×D₃	C₄
cosocle	C₃×C₄	C₂⁴×C₃²
LCS	G ▷ C₃
UCS	C₁ ◁ C₂²×C₃²×C₈

Complements

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

N	H ≅ G/N	type
2. C₂²×C₃³×C₈	C₂	semidirect
3. C₂²×C₃²×C₃⋊C₈	C₂	semidirect
4. C₂×C₃²×D₃×C₈	C₂	direct (N × H)
6. C₂²×C₃×D₃×C₈	C₃	direct (N × H)
12. C₂×C₃³×C₈	C₂²	semidirect
13. C₂×C₃²×C₃⋊C₈	C₂²	semidirect
14. C₃²×D₃×C₈	C₂²	direct (N × H)
20. C₂²×C₃²×C₈	D₃	direct (N × H)
21. C₂²×C₃²×C₈	C₂×C₃	semidirect
22. C₂²×C₃×C₃⋊C₈	C₂×C₃	semidirect
23. C₂×C₃×D₃×C₈	C₂×C₃	direct (N × H)
25. C₂²×C₃²×D₃	C₈	direct (N × H)
32. C₃³×C₈	C₂³	semidirect
33. C₃²×C₃⋊C₈	C₂³	semidirect
35. C₂²×D₃×C₈	C₃²	direct (N × H)
50. C₂×C₃²×C₈	C₂×D₃	direct (N × H)
51. C₂×C₃²×C₈	C₂²×C₃	semidirect
52. C₂×C₃×C₃⋊C₈	C₂²×C₃	semidirect
53. C₃×D₃×C₈	C₂²×C₃	direct (N × H)
54. C₂²×C₃³	C₂×C₈	semidirect
56. C₂×C₃²×D₃	C₂×C₈	direct (N × H)
66. C₂²×C₃×C₈	C₃×D₃	direct (N × H)
67. C₂²×C₃×C₈	C₂×C₃²	semidirect
68. C₂²×C₃⋊C₈	C₂×C₃²	semidirect
69. C₂×D₃×C₈	C₂×C₃²	direct (N × H)
75. C₂²×C₃×D₃	C₃×C₈	direct (N × H)
91. C₃²×C₈	C₂²×D₃	direct (N × H)
92. C₃²×C₈	C₂³×C₃	semidirect
93. C₃×C₃⋊C₈	C₂³×C₃	semidirect
95. C₂×C₃³	C₂²×C₈	semidirect
97. C₃²×D₃	C₂²×C₈	direct (N × H)
112. C₂×C₃×C₈	C₂×C₃×D₃	direct (N × H)
113. C₂×C₃×C₈	C₂²×C₃²	semidirect
114. C₂×C₃⋊C₈	C₂²×C₃²	semidirect
115. D₃×C₈	C₂²×C₃²	direct (N × H)
120. C₂²×C₃²	D₃×C₈	direct (N × H)
121. C₂²×C₃²	C₂×C₃×C₈	semidirect
124. C₂×C₃×D₃	C₂×C₃×C₈	direct (N × H)
137. C₂²×C₈	C₃²×D₃	direct (N × H)
138. C₃³	C₂³×C₈	semidirect
144. C₂²×D₃	C₃²×C₈	direct (N × H)
160. C₃×C₈	C₂²×C₃×D₃	direct (N × H)
161. C₃×C₈	C₂³×C₃²	semidirect
162. C₃⋊C₈	C₂³×C₃²	semidirect
166. C₂×C₃²	C₂×D₃×C₈	direct (N × H)
167. C₂×C₃²	C₂²×C₃×C₈	semidirect
170. C₃×D₃	C₂²×C₃×C₈	direct (N × H)
178. C₂×C₈	C₂×C₃²×D₃	direct (N × H)
183. C₂²×C₃	C₃×D₃×C₈	direct (N × H)
184. C₂²×C₃	C₂×C₃²×C₈	semidirect
187. C₂×D₃	C₂×C₃²×C₈	direct (N × H)
198. C₃²	C₂²×D₃×C₈	direct (N × H)
199. C₃²	C₂³×C₃×C₈	semidirect
209. C₈	C₂²×C₃²×D₃	direct (N × H)
213. C₂×C₃	C₂×C₃×D₃×C₈	direct (N × H)
214. C₂×C₃	C₂²×C₃²×C₈	semidirect
217. D₃	C₂²×C₃²×C₈	direct (N × H)
220. C₂²	C₃²×D₃×C₈	direct (N × H)
226. C₃	C₂²×C₃×D₃×C₈	direct (N × H)
227. C₃	C₂³×C₃²×C₈	semidirect
229. C₂	C₂×C₃²×D₃×C₈	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₃²×C₄×D₃	2
C₂²×C₃³×C₈	2
C₂²×C₃²×C₃⋊C₈	2
C₂×C₃²×D₃×C₈	2
C₂³×C₃²×C₈	3
C₂²×C₃×D₃×C₈	3

Subgroups — C22×C32×D3×C8 #13179113

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₃²×D₃×C₈ #13179113