Subgroups — C₃×D₈×C₃⋊Dic₃ #13163184

Order 1728.

1108 conjugacy classes of subgroups.

The subgroup lattice has 268 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃³×D₈	C₂	1	1	yes	maximal, characteristic
2	C₃×D₄×C₃⋊Dic₃	C₂	1	2	yes	maximal, complemented
3	C₃×#63659	C₂	1	2	yes	maximal, complemented
4	C₃×#63645	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×C₈×C₃⋊Dic₃	C₂	1	1	yes	maximal, characteristic, complemented
6	D₈×C₃⋊Dic₃	C₃	1	1	yes	maximal, characteristic, direct product factor
7	C₃×Dic₃×D₈	—	3	4	no	maximal
8	C₂×C₃³×D₄	C₂²	1	2	yes	—
9	C₂²×C₃×C₃⋊Dic₃	—	2	2	no	—
10	C₃³×D₈	C₄	1	4	yes	complemented
11	C₂×C₃³×C₈	C₂²	1	1	yes	characteristic
12	C₃×#3757	—	2	2	no	—
13	C₃×C₄⋊(C₃⋊Dic₃)	C₂²	1	2	yes	—
14	C₃×C₄×C₃⋊Dic₃	C₂²	1	1	yes	characteristic
15	C₂×C₃×C₃⋊(C₃⋊C₈)	C₂²	1	1	yes	characteristic
16	C₂×C₃²×D₈	—	2	4	no	—
17	C₂×C₃²×D₈	D₃	1	4	yes	—
18	C₂×C₃²×D₈	C₂×C₃	1	1	yes	characteristic
19	D₄×C₃⋊Dic₃	C₂×C₃	1	2	yes	complemented
20	C₃×D₄×Dic₃	—	3	8	no	—
21	#63659	C₂×C₃	1	2	yes	complemented
22	#63645	C₂×C₃	1	1	yes	characteristic, complemented
23	C₈×C₃⋊Dic₃	C₂×C₃	1	1	yes	characteristic, complemented
24	C₃×#832	—	3	8	no	—
25	C₃×#818	—	3	4	no	—
26	C₃×C₈×Dic₃	—	3	4	no	—
27	C₂³×C₃³	—	2	2	no	—
28	C₃³×D₄	—	2	2	no	—
29	C₃³×D₄	C₂×C₄	1	4	yes	complemented
30	C₂×C₃³×C₄	C₂³	1	1	yes	characteristic
31	C₂×C₃×C₃⋊Dic₃	—	2	2	no	—
32	C₂×C₃×C₃⋊Dic₃	—	4	4	no	—
33	C₂×C₃×C₃⋊Dic₃	D₄	1	1	yes	characteristic
34	C₂×C₃×C₃⋊Dic₃	D₄	1	1	yes	characteristic
35	C₃³×C₈	C₂×C₄	1	2	yes	complemented
36	C₃×C₃⋊(C₃⋊C₈)	—	2	1	no	—
37	C₃×C₄×D₈	—	9	1	no	—
38	Dic₃×D₈	—	3	4	no	—
39	C₂×C₃²×D₄	—	2	8	no	—
40	C₂×C₃²×D₄	C₂×D₃	1	8	yes	—
41	C₂×C₃²×D₄	C₂²×C₃	1	2	yes	—
42	C₂²×C₃⋊Dic₃	—	2	2	no	—
43	C₂²×C₃×Dic₃	—	6	8	no	—
44	C₃²×D₈	—	2	16	no	—
45	C₃²×D₈	Dic₃	1	16	yes	complemented
46	C₃²×D₈	C₃×C₄	1	4	yes	complemented
47	C₂×C₃²×C₈	—	2	4	no	—
48	C₂×C₃²×C₈	C₂×D₃	1	4	yes	—
49	C₂×C₃²×C₈	C₂²×C₃	1	1	yes	characteristic
50	#3757	—	2	2	no	—
51	C₄⋊(C₃⋊Dic₃)	C₂²×C₃	1	2	yes	—
52	C₄×C₃⋊Dic₃	C₂²×C₃	1	1	yes	characteristic
53	C₂×C₃⋊(C₃⋊C₈)	C₂²×C₃	1	1	yes	characteristic
54	C₃×#217	—	6	8	no	—
55	C₃×C₄⋊Dic₃	—	3	8	no	—
56	C₃×C₄×Dic₃	—	3	4	no	—
57	C₂×C₃×C₃⋊C₈	—	3	4	no	—
58	C₂²×C₃³	—	2	4	no	—
59	C₂²×C₃³	—	4	2	no	—
60	C₂²×C₃³	—	4	2	no	—
61	C₂²×C₃³	C₂×D₄	1	1	yes	characteristic
62	C₃³×C₄	C₂²×C₄	1	1	yes	characteristic
63	C₃³×C₄	#41	1	1	yes	characteristic
64	C₃×C₃⋊Dic₃	—	2	1	no	—
65	C₃×C₃⋊Dic₃	—	4	2	no	—
66	C₃×C₃⋊Dic₃	D₈	1	2	yes	direct product factor
67	C₂×C₃×D₈	—	2	4	no	—
68	C₂×C₃×D₈	C₃×D₃	1	4	yes	—
69	C₂×C₃×D₈	C₃⋊D₃	1	1	yes	characteristic
70	C₃×C₄×D₄	—	9	2	no	—
71	D₄×Dic₃	—	3	8	no	—
72	C₃×#107	—	9	1	no	—
73	C₃×#102	—	9	2	no	—
74	C₃×C₄×C₈	—	9	1	no	—
75	#832	—	3	8	no	—
76	#818	—	3	4	no	—
77	C₈×Dic₃	—	3	4	no	—
78	C₂³×C₃²	—	2	2	no	—
79	C₂³×C₃²	—	2	8	no	—
80	C₂³×C₃²	—	4	8	no	—
81	C₃²×D₄	—	2	2	no	—
82	C₃²×D₄	—	2	16	no	—
83	C₃²×D₄	—	2	8	no	—
84	C₃²×D₄	—	4	8	no	—
85	C₃²×D₄	C₂×Dic₃	1	16	yes	complemented
86	C₃²×D₄	C₂×C₃×C₄	1	4	yes	complemented
87	C₂×C₃²×C₄	—	2	4	no	—
88	C₂×C₃²×C₄	C₂²×D₃	1	4	yes	—
89	C₂×C₃²×C₄	C₂³×C₃	1	1	yes	characteristic
90	C₂×C₃⋊Dic₃	—	2	2	no	—
91	C₂×C₃⋊Dic₃	—	4	4	no	—
92	C₂×C₃⋊Dic₃	C₃×D₄	1	1	yes	characteristic
93	C₂×C₃⋊Dic₃	C₃×D₄	1	1	yes	characteristic
94	C₂×C₃×Dic₃	—	3	4	no	—
95	C₂×C₃×Dic₃	—	3	4	no	—
96	C₂×C₃×Dic₃	—	6	8	no	—
97	C₂×C₃×Dic₃	—	12	16	no	—
98	C₃²×C₈	—	2	8	no	—
99	C₃²×C₈	C₂×Dic₃	1	8	yes	complemented
100	C₃²×C₈	C₂×C₃×C₄	1	2	yes	complemented
101	C₃⋊(C₃⋊C₈)	—	2	1	no	—
102	C₃×C₃⋊C₈	—	6	4	no	—
103	C₄×D₈	—	9	1	no	Sylow
104	C₂×C₃³	—	4	4	no	—
105	C₂×C₃³	C₄×D₄	1	1	yes	characteristic
106	C₂×C₃³	C₂×D₈	1	1	yes	characteristic
107	C₂×C₃³	#135	1	1	yes	characteristic
108	C₂×C₃×D₄	—	2	8	no	—
109	C₂×C₃×D₄	C₂×C₃×D₃	1	8	yes	—
110	C₂×C₃×D₄	C₂×C₃⋊D₃	1	2	yes	—
111	C₂²×C₃×C₄	—	18	2	no	—
112	C₂²×Dic₃	—	6	8	no	—
113	C₃×D₈	—	2	16	no	—
114	C₃×D₈	C₃×Dic₃	1	16	yes	complemented
115	C₃×D₈	C₃⋊Dic₃	1	4	yes	direct product factor
116	C₂×C₃×C₈	—	2	4	no	—
117	C₂×C₃×C₈	—	9	1	no	—
118	C₂×C₃×C₈	C₂×C₃×D₃	1	4	yes	—
119	C₂×C₃×C₈	C₂×C₃⋊D₃	1	1	yes	characteristic
120	C₃×C₄⋊C₄	—	9	2	no	—
121	C₃×C₂²⋊C₄	—	18	2	no	—
122	C₃×C₄²	—	9	1	no	—
123	#217	—	6	8	no	—
124	C₄⋊Dic₃	—	3	8	no	—
125	C₄×Dic₃	—	3	4	no	—
126	C₂×C₃⋊C₈	—	3	4	no	—
127	C₂²×C₃²	—	2	4	no	—
128	C₂²×C₃²	—	2	4	no	—
129	C₂²×C₃²	—	2	16	no	—
130	C₂²×C₃²	—	4	16	no	—
131	C₂²×C₃²	—	4	2	no	—
132	C₂²×C₃²	—	4	2	no	—
133	C₂²×C₃²	—	4	8	no	—
134	C₂²×C₃²	—	4	8	no	—
135	C₂²×C₃²	—	8	8	no	—
136	C₂²×C₃²	—	8	8	no	—
137	C₂²×C₃²	D₃×D₄	1	4	yes	—
138	C₂²×C₃²	C₂×C₃×D₄	1	1	yes	characteristic
139	C₃²×C₄	—	2	4	no	—
140	C₃²×C₄	—	2	4	no	—
141	C₃²×C₄	#237	1	4	yes	—
142	C₃²×C₄	C₂²×Dic₃	1	4	yes	—
143	C₃²×C₄	C₂²×C₃×C₄	1	1	yes	characteristic
144	C₃²×C₄	C₃×#41	1	1	yes	characteristic
145	C₃⋊Dic₃	—	2	1	no	—
146	C₃⋊Dic₃	—	4	2	no	—
147	C₃⋊Dic₃	C₃×D₈	1	2	yes	direct product factor
148	C₃×Dic₃	—	3	8	no	—
149	C₃×Dic₃	—	6	4	no	—
150	C₃×Dic₃	—	12	8	no	—
151	C₂×D₈	C₃×C₃⋊D₃	1	1	yes	characteristic
152	C₄×D₄	—	9	2	no	—
153	#107	—	9	1	no	—
154	#102	—	9	2	no	—
155	C₄×C₈	—	9	1	no	—
156	C₃³	C₄×D₈	1	1	yes	characteristic, complemented, Sylow
157	C₂³×C₃	—	2	8	no	—
158	C₂³×C₃	—	2	2	no	—
159	C₂³×C₃	—	4	8	no	—
160	C₃×D₄	—	2	8	no	—
161	C₃×D₄	—	2	2	no	—
162	C₃×D₄	—	2	16	no	—
163	C₃×D₄	—	4	8	no	—
164	C₃×D₄	C₂×C₃×Dic₃	1	16	yes	complemented
165	C₃×D₄	C₂×C₃⋊Dic₃	1	4	yes	complemented
166	C₂×C₃×C₄	—	2	4	no	—
167	C₂×C₃×C₄	—	9	1	no	—
168	C₂×C₃×C₄	—	9	1	no	—
169	C₂×C₃×C₄	—	18	2	no	—
170	C₂×C₃×C₄	—	36	4	no	—
171	C₂×C₃×C₄	C₂²×C₃×D₃	1	4	yes	—
172	C₂×C₃×C₄	C₂²×C₃⋊D₃	1	1	yes	characteristic
173	C₂×Dic₃	—	3	4	no	—
174	C₂×Dic₃	—	3	4	no	—
175	C₂×Dic₃	—	6	8	no	—
176	C₂×Dic₃	—	12	16	no	—
177	C₃×C₈	—	2	8	no	—
178	C₃×C₈	—	18	1	no	—
179	C₃×C₈	C₂×C₃×Dic₃	1	8	yes	complemented
180	C₃×C₈	C₂×C₃⋊Dic₃	1	2	yes	complemented
181	C₃⋊C₈	—	6	4	no	—
182	C₂×C₃²	—	2	4	no	—
183	C₂×C₃²	—	2	4	no	—
184	C₂×C₃²	—	2	4	no	—
185	C₂×C₃²	—	4	4	no	—
186	C₂×C₃²	—	4	16	no	—
187	C₂×C₃²	—	8	16	no	—
188	C₂×C₃²	D₃×D₈	1	4	yes	—
189	C₂×C₃²	#912	1	4	yes	—
190	C₂×C₃²	D₄×Dic₃	1	4	yes	—
191	C₂×C₃²	C₃×C₄×D₄	1	1	yes	characteristic
192	C₂×C₃²	C₂×C₃×D₈	1	1	yes	characteristic
193	C₂×C₃²	C₃×#135	1	1	yes	characteristic
194	C₂×D₄	C₂×C₃×C₃⋊D₃	1	2	yes	—
195	C₂²×C₄	—	18	2	no	—
196	D₈	C₃×C₃⋊Dic₃	1	4	yes	direct product factor
197	C₂×C₈	—	9	1	no	—
198	C₂×C₈	C₂×C₃×C₃⋊D₃	1	1	yes	characteristic
199	C₄⋊C₄	—	9	2	no	—
200	C₂²⋊C₄	—	18	2	no	—
201	C₄²	—	9	1	no	—
202	C₂²×C₃	—	2	16	no	—
203	C₂²×C₃	—	2	4	no	—
204	C₂²×C₃	—	2	4	no	—
205	C₂²×C₃	—	4	16	no	—
206	C₂²×C₃	—	4	8	no	—
207	C₂²×C₃	—	4	8	no	—
208	C₂²×C₃	—	4	2	no	—
209	C₂²×C₃	—	4	2	no	—
210	C₂²×C₃	—	8	8	no	—
211	C₂²×C₃	—	8	8	no	—
212	C₂²×C₃	C₃×D₃×D₄	1	4	yes	—
213	C₂²×C₃	D₄×C₃⋊D₃	1	1	yes	characteristic
214	C₃×C₄	—	2	4	no	—
215	C₃×C₄	—	2	4	no	—
216	C₃×C₄	—	9	2	no	—
217	C₃×C₄	—	18	1	no	—
218	C₃×C₄	—	36	2	no	—
219	C₃×C₄	C₃×#237	1	4	yes	—
220	C₃×C₄	C₂²×C₃×Dic₃	1	4	yes	—
221	C₃×C₄	#3830	1	1	yes	characteristic
222	C₃×C₄	C₂²×C₃⋊Dic₃	1	1	yes	characteristic
223	Dic₃	—	3	8	no	—
224	Dic₃	—	6	4	no	—
225	Dic₃	—	12	8	no	—
226	C₃²	—	2	4	no	—
227	C₃²	Dic₃×D₈	1	4	yes	complemented
228	C₃²	C₃×C₄×D₈	1	1	yes	characteristic, complemented
229	C₂³	—	2	2	no	—
230	D₄	—	2	2	no	—
231	D₄	C₂×C₃×C₃⋊Dic₃	1	4	yes	complemented
232	C₂×C₄	—	9	1	no	—
233	C₂×C₄	—	9	1	no	—
234	C₂×C₄	—	18	2	no	—
235	C₂×C₄	—	36	4	no	—
236	C₂×C₄	C₂²×C₃×C₃⋊D₃	1	1	yes	characteristic
237	C₈	—	18	1	no	—
238	C₈	C₂×C₃×C₃⋊Dic₃	1	2	yes	complemented
239	C₂×C₃	—	2	4	no	—
240	C₂×C₃	—	2	4	no	—
241	C₂×C₃	—	2	4	no	—
242	C₂×C₃	—	4	4	no	—
243	C₂×C₃	—	4	16	no	—
244	C₂×C₃	—	8	16	no	—
245	C₂×C₃	C₃×D₃×D₈	1	4	yes	—
246	C₂×C₃	C₃×#912	1	4	yes	—
247	C₂×C₃	C₃×D₄×Dic₃	1	4	yes	—
248	C₂×C₃	D₈×C₃⋊D₃	1	1	yes	characteristic
249	C₂×C₃	#64121	1	1	yes	characteristic
250	C₂×C₃	D₄×C₃⋊Dic₃	1	1	yes	characteristic
251	C₂²	—	2	4	no	—
252	C₂²	—	4	2	no	—
253	C₂²	—	4	2	no	—
254	C₂²	C₃×D₄×C₃⋊D₃	1	1	yes	characteristic
255	C₄	—	9	2	no	—
256	C₄	—	18	1	no	—
257	C₄	—	36	2	no	—
258	C₄	C₃×#3830	1	1	yes	characteristic
259	C₄	C₂²×C₃×C₃⋊Dic₃	1	1	yes	characteristic
260	C₃	—	2	4	no	—
261	C₃	C₃×Dic₃×D₈	1	4	yes	complemented
262	C₃	D₈×C₃⋊Dic₃	1	1	yes	characteristic, direct product factor
263	C₂	—	4	4	no	—
264	C₂	C₃×D₈×C₃⋊D₃	1	1	yes	characteristic
265	C₂	C₃×#64121	1	1	yes	characteristic
266	C₂	C₃×D₄×C₃⋊Dic₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₃	D₄×C₃⋊D₃
commutator	C₃²×C₄	C₂²×C₃×C₄
Frattini	C₂×C₄	C₂²×C₃×C₃⋊D₃
Fitting, prime radical	C₂×C₃³×D₈	C₂
socle	C₂²×C₃³	C₂×D₄
cosocle	C₂×C₃²×C₄	C₂³×C₃
hypercenter	C₂×C₃×D₈	C₃⋊D₃
hypocenter	C₃²	C₃×C₄×D₈
LCS	G ▷ C₃²×C₄ ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂²×C₃ ◁ C₂×C₃×C₄ ◁ C₂×C₃×D₈

Complements

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

N	H ≅ G/N	type
2. C₃×D₄×C₃⋊Dic₃	C₂	semidirect
3. C₃×#63659	C₂	semidirect
4. C₃×#63645	C₂	semidirect
5. C₃×C₈×C₃⋊Dic₃	C₂	semidirect
6. D₈×C₃⋊Dic₃	C₃	direct (N × H)
10. C₃³×D₈	C₄	semidirect
19. D₄×C₃⋊Dic₃	C₂×C₃	semidirect
21. #63659	C₂×C₃	semidirect
22. #63645	C₂×C₃	semidirect
23. C₈×C₃⋊Dic₃	C₂×C₃	semidirect
29. C₃³×D₄	C₂×C₄	semidirect
35. C₃³×C₈	C₂×C₄	semidirect
45. C₃²×D₈	Dic₃	semidirect
46. C₃²×D₈	C₃×C₄	semidirect
66. C₃×C₃⋊Dic₃	D₈	direct (N × H)
85. C₃²×D₄	C₂×Dic₃	semidirect
86. C₃²×D₄	C₂×C₃×C₄	semidirect
99. C₃²×C₈	C₂×Dic₃	semidirect
100. C₃²×C₈	C₂×C₃×C₄	semidirect
114. C₃×D₈	C₃×Dic₃	semidirect
115. C₃×D₈	C₃⋊Dic₃	direct (N × H)
147. C₃⋊Dic₃	C₃×D₈	direct (N × H)
156. C₃³	C₄×D₈	semidirect
164. C₃×D₄	C₂×C₃×Dic₃	semidirect
165. C₃×D₄	C₂×C₃⋊Dic₃	semidirect
179. C₃×C₈	C₂×C₃×Dic₃	semidirect
180. C₃×C₈	C₂×C₃⋊Dic₃	semidirect
196. D₈	C₃×C₃⋊Dic₃	direct (N × H)
227. C₃²	Dic₃×D₈	semidirect
228. C₃²	C₃×C₄×D₈	semidirect
231. D₄	C₂×C₃×C₃⋊Dic₃	semidirect
238. C₈	C₂×C₃×C₃⋊Dic₃	semidirect
261. C₃	C₃×Dic₃×D₈	semidirect
262. C₃	D₈×C₃⋊Dic₃	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₃³×D₈	2
C₃×D₄×C₃⋊Dic₃	2
C₃×#63659	2
C₃×#63645	2
C₃×C₈×C₃⋊Dic₃	2
D₈×C₃⋊Dic₃	3
C₃×Dic₃×D₈	3

Subgroups — C3×D8×C3⋊Dic3 #13163184

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×D₈×C₃⋊Dic₃ #13163184