Subgroups — C₃×C₈×#634 #13162716

Order 1728.

788 conjugacy classes of subgroups.

The subgroup lattice has 240 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₄×#634	C₂	1	1	yes	maximal, characteristic
2	C₃³×C₄×C₈	C₂	1	1	yes	maximal, characteristic
3	C₃×#63641	C₂	1	2	yes	maximal
4	C₃×C₈×C₃⋊Dic₃	C₂	1	2	yes	maximal
5	C₃×C₄⋊(C₃⋊(C₃⋊C₈))	C₂	1	1	yes	maximal, characteristic
6	C₈×#634	C₃	1	1	yes	maximal, characteristic, direct product factor
7	C₃×C₈×Dic₆	—	3	4	no	maximal
8	C₂×C₃×#634	C₄	1	1	yes	characteristic
9	C₂×C₃³×C₈	C₂²	1	1	yes	characteristic
10	C₂×C₃³×C₈	C₂²	1	1	yes	characteristic
11	C₃³×C₄²	C₂²	1	1	yes	characteristic
12	C₃×C₄⋊(C₃⋊Dic₃)	C₄	1	1	yes	characteristic
13	C₃×#3750	C₄	1	2	yes	—
14	C₃×C₄×C₃⋊Dic₃	C₂²	1	2	yes	—
15	C₂×C₃×C₃⋊(C₃⋊C₈)	C₂²	1	2	yes	—
16	C₄×#634	C₂×C₃	1	1	yes	characteristic
17	C₃×C₄×Dic₆	—	3	4	no	—
18	C₃²×C₄×C₈	—	2	4	no	—
19	C₃²×C₄×C₈	D₃	1	4	yes	—
20	C₃²×C₄×C₈	C₂×C₃	1	1	yes	characteristic
21	#63641	C₂×C₃	1	2	yes	—
22	C₈×C₃⋊Dic₃	C₂×C₃	1	2	yes	—
23	C₄⋊(C₃⋊(C₃⋊C₈))	C₂×C₃	1	1	yes	characteristic
24	C₃×Dic₃⋊C₈	—	3	8	no	—
25	C₃×C₈×Dic₃	—	3	8	no	—
26	C₃×C₄⋊(C₃⋊C₈)	—	3	4	no	—
27	C₂×C₃³×C₄	C₂×C₄	1	1	yes	characteristic
28	C₂×C₃³×C₄	C₂×C₄	1	1	yes	characteristic
29	C₂×C₃³×C₄	C₂³	1	1	yes	characteristic
30	C₂×C₃×C₃⋊Dic₃	C₂×C₄	1	2	yes	—
31	C₂×C₃×C₃⋊Dic₃	C₂×C₄	1	2	yes	—
32	C₃×#634	C₈	1	4	yes	direct product factor
33	C₃³×C₈	—	2	1	no	—
34	C₃³×C₈	Q₈	1	2	yes	complemented
35	C₃×C₃⋊(C₃⋊C₈)	—	2	2	no	—
36	C₃×C₈×Q₈	—	9	1	no	—
37	C₈×Dic₆	—	3	4	no	—
38	C₂×#634	C₃×C₄	1	1	yes	characteristic
39	C₂×C₃×Dic₆	—	3	4	no	—
40	C₂×C₃²×C₈	—	2	4	no	—
41	C₂×C₃²×C₈	—	2	4	no	—
42	C₂×C₃²×C₈	C₂×D₃	1	4	yes	—
43	C₂×C₃²×C₈	C₂×D₃	1	4	yes	—
44	C₂×C₃²×C₈	C₂²×C₃	1	1	yes	characteristic
45	C₂×C₃²×C₈	C₂²×C₃	1	1	yes	characteristic
46	C₃²×C₄²	—	2	4	no	—
47	C₃²×C₄²	C₂×D₃	1	4	yes	—
48	C₃²×C₄²	C₂²×C₃	1	1	yes	characteristic
49	C₄⋊(C₃⋊Dic₃)	C₃×C₄	1	1	yes	characteristic
50	#3750	C₃×C₄	1	2	yes	—
51	C₄×C₃⋊Dic₃	C₂²×C₃	1	2	yes	—
52	C₂×C₃⋊(C₃⋊C₈)	C₂²×C₃	1	2	yes	—
53	C₃×C₄⋊Dic₃	—	3	4	no	—
54	C₃×Dic₃⋊C₄	—	3	8	no	—
55	C₃×C₄×Dic₃	—	3	8	no	—
56	C₂×C₃×C₃⋊C₈	—	3	8	no	—
57	C₂²×C₃³	C₂²×C₄	1	1	yes	characteristic
58	C₃³×C₄	—	2	1	no	—
59	C₃³×C₄	C₂×C₈	1	2	yes	—
60	C₃³×C₄	C₂×Q₈	1	1	yes	characteristic
61	C₃³×C₄	#41	1	1	yes	characteristic
62	C₃×C₃⋊Dic₃	—	2	2	no	—
63	C₃×C₃⋊Dic₃	C₂×C₈	1	4	yes	—
64	C₃×C₄×Q₈	—	9	1	no	—
65	C₄×Dic₆	—	3	4	no	—
66	C₃×C₄⋊C₈	—	9	2	no	—
67	C₃×C₄⋊C₈	—	9	1	no	—
68	C₃×C₄×C₈	—	2	4	no	—
69	C₃×C₄×C₈	—	9	2	no	—
70	C₃×C₄×C₈	C₃×D₃	1	4	yes	—
71	C₃×C₄×C₈	C₃⋊D₃	1	1	yes	characteristic
72	Dic₃⋊C₈	—	3	8	no	—
73	C₈×Dic₃	—	3	8	no	—
74	C₄⋊(C₃⋊C₈)	—	3	4	no	—
75	C₂×C₃²×C₄	—	2	4	no	—
76	C₂×C₃²×C₄	—	2	4	no	—
77	C₂×C₃²×C₄	—	2	4	no	—
78	C₂×C₃²×C₄	C₄×D₃	1	4	yes	—
79	C₂×C₃²×C₄	C₄×D₃	1	4	yes	—
80	C₂×C₃²×C₄	C₂×C₃×C₄	1	1	yes	characteristic
81	C₂×C₃²×C₄	C₂×C₃×C₄	1	1	yes	characteristic
82	C₂×C₃²×C₄	C₂²×D₃	1	4	yes	—
83	C₂×C₃²×C₄	C₂³×C₃	1	1	yes	characteristic
84	C₂×C₃⋊Dic₃	C₂×C₃×C₄	1	2	yes	—
85	C₂×C₃⋊Dic₃	C₂×C₃×C₄	1	2	yes	—
86	#634	C₃×C₈	1	4	yes	direct product factor
87	C₂×C₃×Dic₃	—	3	8	no	—
88	C₂×C₃×Dic₃	—	3	8	no	—
89	C₃×Dic₆	—	3	16	no	—
90	C₃²×C₈	—	2	8	no	—
91	C₃²×C₈	—	2	4	no	—
92	C₃²×C₈	—	2	1	no	—
93	C₃²×C₈	—	2	8	no	—
94	C₃²×C₈	Dic₆	1	8	yes	complemented
95	C₃²×C₈	C₃×Q₈	1	2	yes	complemented
96	C₃⋊(C₃⋊C₈)	—	2	2	no	—
97	C₃×C₃⋊C₈	—	6	8	no	—
98	C₈×Q₈	—	9	1	no	Sylow
99	C₂×C₃³	C₄×Q₈	1	1	yes	characteristic
100	C₂×C₃³	C₂²×C₈	1	1	yes	characteristic
101	C₂×C₃³	#131	1	1	yes	characteristic
102	C₂×C₃×Q₈	—	9	1	no	—
103	C₂×Dic₆	—	3	4	no	—
104	C₂×C₃×C₈	—	2	4	no	—
105	C₂×C₃×C₈	—	2	4	no	—
106	C₂×C₃×C₈	—	9	2	no	—
107	C₂×C₃×C₈	C₂×C₃×D₃	1	4	yes	—
108	C₂×C₃×C₈	C₂×C₃×D₃	1	4	yes	—
109	C₂×C₃×C₈	C₂×C₃⋊D₃	1	1	yes	characteristic
110	C₂×C₃×C₈	C₂×C₃⋊D₃	1	1	yes	characteristic
111	C₃×C₄⋊C₄	—	9	2	no	—
112	C₃×C₄⋊C₄	—	9	1	no	—
113	C₃×C₄²	—	2	4	no	—
114	C₃×C₄²	—	9	2	no	—
115	C₃×C₄²	C₂×C₃×D₃	1	4	yes	—
116	C₃×C₄²	C₂×C₃⋊D₃	1	1	yes	characteristic
117	C₄⋊Dic₃	—	3	4	no	—
118	Dic₃⋊C₄	—	3	8	no	—
119	C₄×Dic₃	—	3	8	no	—
120	C₂×C₃⋊C₈	—	3	8	no	—
121	C₂²×C₃²	—	2	4	no	—
122	C₂²×C₃²	C₂×C₄×D₃	1	4	yes	—
123	C₂²×C₃²	C₂²×C₃×C₄	1	1	yes	characteristic
124	C₃²×C₄	—	2	8	no	—
125	C₃²×C₄	—	2	4	no	—
126	C₃²×C₄	—	2	1	no	—
127	C₃²×C₄	—	2	8	no	—
128	C₃²×C₄	—	2	4	no	—
129	C₃²×C₄	—	2	4	no	—
130	C₃²×C₄	D₃×C₈	1	8	yes	—
131	C₃²×C₄	C₂×C₃×C₈	1	2	yes	—
132	C₃²×C₄	C₂×Dic₆	1	4	yes	—
133	C₃²×C₄	#235	1	4	yes	—
134	C₃²×C₄	C₂×C₃×Q₈	1	1	yes	characteristic
135	C₃²×C₄	C₃×#41	1	1	yes	characteristic
136	C₃⋊Dic₃	—	2	2	no	—
137	C₃⋊Dic₃	C₂×C₃×C₈	1	4	yes	—
138	C₃×Dic₃	—	3	16	no	—
139	C₃×Dic₃	—	6	8	no	—
140	C₄×Q₈	—	9	1	no	—
141	C₄⋊C₈	—	9	2	no	—
142	C₄⋊C₈	—	9	1	no	—
143	C₄×C₈	—	9	2	no	—
144	C₄×C₈	C₃×C₃⋊D₃	1	1	yes	characteristic
145	C₃³	C₈×Q₈	1	1	yes	characteristic, complemented, Sylow
146	C₃×Q₈	—	9	4	no	—
147	C₂×C₃×C₄	—	2	4	no	—
148	C₂×C₃×C₄	—	2	4	no	—
149	C₂×C₃×C₄	—	2	4	no	—
150	C₂×C₃×C₄	—	9	2	no	—
151	C₂×C₃×C₄	—	9	2	no	—
152	C₂×C₃×C₄	C₃×C₄×D₃	1	4	yes	—
153	C₂×C₃×C₄	C₃×C₄×D₃	1	4	yes	—
154	C₂×C₃×C₄	C₄×C₃⋊D₃	1	1	yes	characteristic
155	C₂×C₃×C₄	C₄×C₃⋊D₃	1	1	yes	characteristic
156	C₂×C₃×C₄	C₂²×C₃×D₃	1	4	yes	—
157	C₂×C₃×C₄	C₂²×C₃⋊D₃	1	1	yes	characteristic
158	C₂×Dic₃	—	3	8	no	—
159	C₂×Dic₃	—	3	8	no	—
160	Dic₆	—	3	16	no	—
161	C₃×C₈	—	2	8	no	—
162	C₃×C₈	—	2	4	no	—
163	C₃×C₈	—	2	1	no	—
164	C₃×C₈	—	2	8	no	—
165	C₃×C₈	—	18	2	no	—
166	C₃×C₈	C₃×Dic₆	1	8	yes	complemented
167	C₃×C₈	#634	1	2	yes	direct product factor
168	C₃⋊C₈	—	6	8	no	—
169	C₂×C₃²	—	2	4	no	—
170	C₂×C₃²	—	2	4	no	—
171	C₂×C₃²	—	2	4	no	—
172	C₂×C₃²	C₄×Dic₆	1	4	yes	—
173	C₂×C₃²	C₂×D₃×C₈	1	4	yes	—
174	C₂×C₃²	#901	1	4	yes	—
175	C₂×C₃²	C₃×C₄×Q₈	1	1	yes	characteristic
176	C₂×C₃²	C₂²×C₃×C₈	1	1	yes	characteristic
177	C₂×C₃²	C₃×#131	1	1	yes	characteristic
178	C₂×Q₈	—	9	1	no	—
179	C₂×C₈	—	9	2	no	—
180	C₂×C₈	C₂×C₃×C₃⋊D₃	1	1	yes	characteristic
181	C₂×C₈	C₂×C₃×C₃⋊D₃	1	1	yes	characteristic
182	C₄⋊C₄	—	9	2	no	—
183	C₄⋊C₄	—	9	1	no	—
184	C₄²	—	9	2	no	—
185	C₄²	C₂×C₃×C₃⋊D₃	1	1	yes	characteristic
186	C₂²×C₃	—	2	4	no	—
187	C₂²×C₃	C₂×C₃×C₄×D₃	1	4	yes	—
188	C₂²×C₃	C₂×C₄×C₃⋊D₃	1	1	yes	characteristic
189	C₃×C₄	—	2	8	no	—
190	C₃×C₄	—	2	4	no	—
191	C₃×C₄	—	2	1	no	—
192	C₃×C₄	—	2	8	no	—
193	C₃×C₄	—	2	4	no	—
194	C₃×C₄	—	2	4	no	—
195	C₃×C₄	—	9	4	no	—
196	C₃×C₄	—	18	2	no	—
197	C₃×C₄	C₃×D₃×C₈	1	8	yes	—
198	C₃×C₄	C₈×C₃⋊D₃	1	2	yes	—
199	C₃×C₄	C₂×C₃×Dic₆	1	4	yes	—
200	C₃×C₄	C₃×#235	1	4	yes	—
201	C₃×C₄	C₂×#634	1	1	yes	characteristic
202	C₃×C₄	#3828	1	1	yes	characteristic
203	Dic₃	—	3	16	no	—
204	Dic₃	—	6	8	no	—
205	C₃²	—	2	4	no	—
206	C₃²	C₈×Dic₆	1	4	yes	complemented
207	C₃²	C₃×C₈×Q₈	1	1	yes	characteristic, complemented
208	Q₈	—	9	4	no	—
209	C₂×C₄	—	9	2	no	—
210	C₂×C₄	—	9	2	no	—
211	C₂×C₄	C₃×C₄×C₃⋊D₃	1	1	yes	characteristic
212	C₂×C₄	C₃×C₄×C₃⋊D₃	1	1	yes	characteristic
213	C₂×C₄	C₂²×C₃×C₃⋊D₃	1	1	yes	characteristic
214	C₈	—	2	1	no	—
215	C₈	—	18	2	no	—
216	C₈	C₃×#634	1	2	yes	direct product factor
217	C₂×C₃	—	2	4	no	—
218	C₂×C₃	—	2	4	no	—
219	C₂×C₃	—	2	4	no	—
220	C₂×C₃	C₃×C₄×Dic₆	1	4	yes	—
221	C₂×C₃	C₂×C₃×D₃×C₈	1	4	yes	—
222	C₂×C₃	C₃×#901	1	4	yes	—
223	C₂×C₃	C₄×#634	1	1	yes	characteristic
224	C₂×C₃	C₂×C₈×C₃⋊D₃	1	1	yes	characteristic
225	C₂×C₃	#64110	1	1	yes	characteristic
226	C₂²	C₂×C₃×C₄×C₃⋊D₃	1	1	yes	characteristic
227	C₄	—	2	1	no	—
228	C₄	—	9	4	no	—
229	C₄	—	18	2	no	—
230	C₄	C₃×C₈×C₃⋊D₃	1	2	yes	—
231	C₄	C₂×C₃×#634	1	1	yes	characteristic
232	C₄	C₃×#3828	1	1	yes	characteristic
233	C₃	—	2	4	no	—
234	C₃	C₃×C₈×Dic₆	1	4	yes	complemented
235	C₃	C₈×#634	1	1	yes	characteristic, direct product factor
236	C₂	C₃×C₄×#634	1	1	yes	characteristic
237	C₂	C₂×C₃×C₈×C₃⋊D₃	1	1	yes	characteristic
238	C₂	C₃×#64110	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₃×C₈	C₂×C₃⋊D₃
commutator	C₂×C₃²	C₂²×C₃×C₈
Frattini	C₂×C₄	C₂²×C₃×C₃⋊D₃
Fitting	C₃³×C₄×C₈	C₂
socle, prime radical	C₂²×C₃³	C₂²×C₄
cosocle	C₂×C₃²×C₄	C₂³×C₃
hypercenter	C₃×C₄×C₈	C₃⋊D₃
hypocenter	C₃²	C₃×C₈×Q₈
LCS	G ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂×C₃×C₈ ◁ C₃×C₄×C₈

Complements

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

N	H ≅ G/N	type
6. C₈×#634	C₃	direct (N × H)
32. C₃×#634	C₈	direct (N × H)
34. C₃³×C₈	Q₈	semidirect
86. #634	C₃×C₈	direct (N × H)
94. C₃²×C₈	Dic₆	semidirect
95. C₃²×C₈	C₃×Q₈	semidirect
145. C₃³	C₈×Q₈	semidirect
166. C₃×C₈	C₃×Dic₆	semidirect
167. C₃×C₈	#634	direct (N × H)
206. C₃²	C₈×Dic₆	semidirect
207. C₃²	C₃×C₈×Q₈	semidirect
216. C₈	C₃×#634	direct (N × H)
234. C₃	C₃×C₈×Dic₆	semidirect
235. C₃	C₈×#634	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×C₄×#634	2
C₃³×C₄×C₈	2
C₃×#63641	2
C₃×C₈×C₃⋊Dic₃	2
C₃×C₄⋊(C₃⋊(C₃⋊C₈))	2
C₈×#634	3
C₃×C₈×Dic₆	3

Subgroups — C3×C8×#634 #13162716

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₈×#634 #13162716