Subgroups — Dic₃×#798 #11795862

Order 1152.

419 conjugacy classes of subgroups.

The subgroup lattice has 269 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×#798	C₂	1	1	yes	maximal, characteristic
2	D₃×C₈×Dic₃	C₂	1	1	yes	maximal, characteristic
3	C₂×#63543	C₂	1	1	yes	maximal, characteristic
4	#10589368	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×Dic₃×C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
6	#10589104	C₂	1	1	yes	maximal, characteristic, complemented
7	Dic₃×C₃⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
8	Dic₃×M₃₂	—	3	1	no	maximal
9	C₄×#798	—	3	1	no	maximal
10	C₂×C₃×D₃×C₈	C₂²	1	1	yes	characteristic
11	C₄×D₃×Dic₃	C₄	1	1	yes	characteristic
12	C₂×D₃×C₃⋊C₈	C₄	1	1	yes	characteristic
13	C₂×C₃²×C₁₆	C₂²	1	1	yes	characteristic
14	C₈×C₃⋊Dic₃	C₂²	1	1	yes	characteristic
15	C₂×C₃⋊(C₃⋊C₁₆)	C₂²	1	1	yes	characteristic
16	C₃×C₈×Dic₃	C₂²	1	1	yes	characteristic
17	C₂×C₃×C₃⋊C₁₆	C₂²	1	1	yes	characteristic
18	C₂×C₃×C₃⋊C₁₆	C₂²	1	1	yes	characteristic
19	C₃×#798	C₄	1	4	yes	complemented
20	#63553	C₄	1	1	yes	characteristic
21	Dic₃×C₃⋊C₈	C₄	1	1	yes	characteristic
22	#63543	C₄	1	4	yes	complemented
23	C₂×C₃×M₃₂	—	3	1	no	—
24	C₂×C₈×Dic₃	—	3	1	no	—
25	C₂×#812	—	3	1	no	—
26	C₂×#798	—	3	1	no	—
27	C₂×#798	D₃	1	1	yes	characteristic
28	C₄×D₃×C₈	—	3	1	no	—
29	C₃×C₄×C₁₆	—	3	1	no	—
30	#4502	—	3	1	no	—
31	#4502	—	3	1	no	—
32	#4502	—	3	1	no	—
33	#4502	—	3	1	no	—
34	#4502	—	6	1	no	—
35	Dic₃×C₁₆	—	3	1	no	—
36	Dic₃×C₁₆	D₃	1	1	yes	characteristic, complemented
37	C₄×C₃⋊C₁₆	—	3	1	no	—
38	C₂×C₃×C₄×D₃	C₂×C₄	1	1	yes	characteristic
39	C₂×D₃×Dic₃	C₈	1	2	yes	—
40	C₂×C₃²×C₈	C₂³	1	1	yes	characteristic
41	C₄×C₃⋊Dic₃	C₂×C₄	1	1	yes	characteristic
42	C₂×C₃⋊(C₃⋊C₈)	C₂×C₄	1	1	yes	characteristic
43	C₃×C₄×Dic₃	C₂×C₄	1	1	yes	characteristic
44	C₂×C₃×C₃⋊C₈	C₂×C₄	1	1	yes	characteristic
45	C₂×C₃×C₃⋊C₈	C₂×C₄	1	1	yes	characteristic
46	C₃×D₃×C₈	C₂×C₄	1	2	yes	—
47	C₃×D₃×C₈	C₂×C₄	1	2	yes	—
48	Dic₃²	C₈	1	2	yes	—
49	D₃×C₃⋊C₈	C₈	1	4	yes	—
50	C₃²×C₁₆	C₂×C₄	1	2	yes	complemented
51	C₃⋊(C₃⋊C₁₆)	C₂×C₄	1	2	yes	complemented
52	C₃×C₃⋊C₁₆	C₂×C₄	1	2	yes	complemented
53	C₃×C₃⋊C₁₆	C₂×C₄	1	2	yes	complemented
54	C₄×M₃₂	—	9	1	no	Sylow
55	C₂²×C₃×C₈	—	3	1	no	—
56	C₂×C₄×Dic₃	—	3	1	no	—
57	C₂²×C₃⋊C₈	—	3	1	no	—
58	C₂×D₃×C₈	—	3	1	no	—
59	C₂×D₃×C₈	C₂×D₃	1	1	yes	characteristic
60	C₄²×D₃	—	3	1	no	—
61	C₃×M₃₂	—	3	4	no	—
62	C₂×C₃×C₁₆	—	2	1	no	—
63	C₂×C₃×C₁₆	—	3	1	no	—
64	C₂×C₃×C₁₆	—	3	1	no	—
65	C₂×C₃×C₁₆	C₂×D₃	1	1	yes	characteristic
66	C₂×C₃×C₁₆	C₂×D₃	1	1	yes	characteristic
67	C₃×C₄×C₈	—	3	1	no	—
68	C₈×Dic₃	—	3	1	no	—
69	C₈×Dic₃	—	3	1	no	—
70	C₈×Dic₃	—	3	1	no	—
71	C₈×Dic₃	—	3	1	no	—
72	C₈×Dic₃	—	3	1	no	—
73	C₈×Dic₃	—	6	1	no	—
74	C₈×Dic₃	C₂×D₃	1	1	yes	characteristic
75	#812	—	3	4	no	—
76	C₂×C₃⋊C₁₆	—	3	1	no	—
77	C₂×C₃⋊C₁₆	—	3	1	no	—
78	C₂×C₃⋊C₁₆	—	6	1	no	—
79	C₂×C₃⋊C₁₆	C₂×D₃	1	1	yes	characteristic
80	C₂×C₃⋊C₁₆	C₂×D₃	1	1	yes	characteristic
81	C₄×C₃⋊C₈	—	3	1	no	—
82	#798	—	3	4	no	—
83	#798	Dic₃	1	4	yes	direct product factor
84	C₂²×C₃×D₃	C₂×C₈	1	1	yes	characteristic
85	C₂×C₃²×C₄	C₂²×C₄	1	1	yes	characteristic
86	C₂×C₃⋊Dic₃	C₂×C₈	1	2	yes	—
87	C₂×C₃×Dic₃	C₂×C₈	1	1	yes	characteristic
88	C₂×C₃×Dic₃	C₂×C₈	1	2	yes	—
89	C₃×C₄×D₃	C₄²	1	2	yes	—
90	C₃×C₄×D₃	C₂×C₈	1	2	yes	—
91	D₃×Dic₃	—	2	4	no	—
92	C₃²×C₈	C₂²×C₄	1	1	yes	characteristic
93	C₃²×C₈	C₂²×C₄	1	1	yes	characteristic
94	C₃⋊(C₃⋊C₈)	C₂×C₈	1	2	yes	—
95	C₃×C₃⋊C₈	C₄²	1	2	yes	—
96	C₃×C₃⋊C₈	C₂×C₈	1	2	yes	—
97	C₂×M₃₂	—	3	1	no	—
98	C₂×M₃₂	—	9	1	no	—
99	C₂×C₄×C₈	—	9	1	no	—
100	#346	—	9	1	no	—
101	#346	—	9	1	no	—
102	C₄×C₁₆	—	3	1	no	—
103	C₄×C₁₆	—	9	1	no	—
104	C₂²×C₃×C₄	—	3	1	no	—
105	C₂²×Dic₃	—	3	2	no	—
106	C₂×C₄×D₃	—	3	2	no	—
107	C₂×C₄×D₃	C₄×D₃	1	1	yes	characteristic
108	C₂×C₃×C₈	—	2	1	no	—
109	C₂×C₃×C₈	—	3	1	no	—
110	C₂×C₃×C₈	—	3	1	no	—
111	C₂×C₃×C₈	—	3	2	no	—
112	C₂×C₃×C₈	—	3	2	no	—
113	C₂×C₃×C₈	C₂²×D₃	1	1	yes	characteristic
114	C₂×C₃×C₈	C₂²×D₃	1	1	yes	characteristic
115	C₃×C₄²	—	3	1	no	—
116	C₄×Dic₃	—	3	1	no	—
117	C₄×Dic₃	—	3	1	no	—
118	C₄×Dic₃	—	3	2	no	—
119	C₄×Dic₃	—	3	2	no	—
120	C₄×Dic₃	—	6	1	no	—
121	C₄×Dic₃	C₄×D₃	1	1	yes	characteristic
122	C₂×C₃⋊C₈	—	3	1	no	—
123	C₂×C₃⋊C₈	—	3	1	no	—
124	C₂×C₃⋊C₈	—	3	4	no	—
125	C₂×C₃⋊C₈	—	6	1	no	—
126	C₂×C₃⋊C₈	C₄×D₃	1	1	yes	characteristic
127	C₂×C₃⋊C₈	C₄×D₃	1	1	yes	characteristic
128	D₃×C₈	—	3	4	no	—
129	D₃×C₈	C₄×D₃	1	2	yes	—
130	D₃×C₈	C₂×Dic₃	1	2	yes	—
131	C₃×C₁₆	—	2	2	no	—
132	C₃×C₁₆	—	3	2	no	—
133	C₃×C₁₆	—	3	2	no	—
134	C₃×C₁₆	C₄×D₃	1	2	yes	complemented
135	C₃×C₁₆	C₂×Dic₃	1	2	yes	complemented
136	C₃⋊C₁₆	—	3	2	no	—
137	C₃⋊C₁₆	—	3	2	no	—
138	C₃⋊C₁₆	—	6	2	no	—
139	C₃⋊C₁₆	C₄×D₃	1	2	yes	complemented
140	C₃⋊C₁₆	C₂×Dic₃	1	2	yes	complemented
141	C₂²×C₃²	C₂²×C₈	1	1	yes	characteristic
142	C₂×C₃×D₃	—	2	2	no	—
143	C₂×C₃×D₃	C₄×C₈	1	2	yes	—
144	C₃²×C₄	C₂×C₄²	1	1	yes	characteristic
145	C₃²×C₄	C₂²×C₈	1	1	yes	characteristic
146	C₃⋊Dic₃	—	2	2	no	—
147	C₃×Dic₃	C₄×C₈	1	2	yes	—
148	C₃×Dic₃	M₃₂	1	4	yes	complemented
149	C₂²×C₈	—	3	1	no	—
150	C₂²×C₈	—	9	1	no	—
151	C₂×C₄²	—	9	1	no	—
152	M₃₂	—	3	4	no	—
153	M₃₂	—	9	4	no	—
154	C₂×C₁₆	—	3	1	no	—
155	C₂×C₁₆	—	3	1	no	—
156	C₂×C₁₆	—	9	1	no	—
157	C₂×C₁₆	D₃²	1	1	yes	characteristic
158	C₄×C₈	—	3	1	no	—
159	C₄×C₈	—	9	1	no	—
160	C₄×C₈	—	9	1	no	—
161	C₄×C₈	—	9	1	no	—
162	C₂³×C₃	—	3	1	no	—
163	C₂²×D₃	D₃×C₈	1	1	yes	characteristic
164	C₂×C₃×C₄	—	2	1	no	—
165	C₂×C₃×C₄	—	3	1	no	—
166	C₂×C₃×C₄	—	3	2	no	—
167	C₂×C₃×C₄	—	3	2	no	—
168	C₂×C₃×C₄	—	3	2	no	—
169	C₂×C₃×C₄	C₂×C₄×D₃	1	1	yes	characteristic
170	C₂×C₃×C₄	C₂×C₄×D₃	1	1	yes	characteristic
171	C₂×Dic₃	—	3	2	no	—
172	C₂×Dic₃	—	3	2	no	—
173	C₂×Dic₃	—	6	2	no	—
174	C₂×Dic₃	—	6	4	no	—
175	C₂×Dic₃	D₃×C₈	1	2	yes	—
176	C₂×Dic₃	D₃×C₈	1	1	yes	characteristic
177	C₄×D₃	—	6	4	no	—
178	C₄×D₃	D₃×C₈	1	2	yes	—
179	C₄×D₃	C₄×Dic₃	1	2	yes	—
180	C₃×C₈	—	2	1	no	—
181	C₃×C₈	—	2	1	no	—
182	C₃×C₈	—	3	2	no	—
183	C₃×C₈	—	3	2	no	—
184	C₃×C₈	C₂×C₄×D₃	1	1	yes	characteristic
185	C₃×C₈	C₂×C₄×D₃	1	1	yes	characteristic
186	C₃×C₈	C₂×C₄×D₃	1	1	yes	characteristic
187	C₃×C₈	C₂²×Dic₃	1	1	yes	characteristic
188	C₃⋊C₈	—	3	2	no	—
189	C₃⋊C₈	—	3	2	no	—
190	C₃⋊C₈	—	6	2	no	—
191	C₃⋊C₈	D₃×C₈	1	2	yes	—
192	C₃⋊C₈	C₄×Dic₃	1	2	yes	—
193	C₂×C₃²	C₂×C₄×C₈	1	1	yes	characteristic
194	C₂×C₃²	C₂×M₃₂	1	2	yes	—
195	C₃×D₃	—	2	2	no	—
196	C₂²×C₄	—	3	1	no	—
197	C₂²×C₄	—	9	2	no	—
198	C₂×C₈	—	3	1	no	—
199	C₂×C₈	—	3	1	no	—
200	C₂×C₈	—	3	2	no	—
201	C₂×C₈	—	3	2	no	—
202	C₂×C₈	—	9	4	no	—
203	C₂×C₈	—	9	1	no	—
204	C₂×C₈	C₂×D₃²	1	1	yes	characteristic
205	C₄²	—	3	1	no	—
206	C₄²	—	9	2	no	—
207	C₄²	—	9	1	no	—
208	C₁₆	—	3	2	no	—
209	C₁₆	—	3	2	no	—
210	C₁₆	—	9	2	no	—
211	C₁₆	D₃×Dic₃	1	2	yes	complemented
212	C₂²×C₃	—	2	1	no	—
213	C₂²×C₃	—	3	2	no	—
214	C₂²×C₃	—	6	2	no	—
215	C₂²×C₃	C₂×D₃×C₈	1	1	yes	characteristic
216	C₂²×C₃	C₂×D₃×C₈	1	1	yes	characteristic
217	C₂×D₃	—	2	2	no	—
218	C₂×D₃	C₈×Dic₃	1	2	yes	—
219	C₃×C₄	—	2	1	no	—
220	C₃×C₄	—	2	1	no	—
221	C₃×C₄	—	3	4	no	—
222	C₃×C₄	—	3	2	no	—
223	C₃×C₄	C₄²×D₃	1	1	yes	characteristic
224	C₃×C₄	C₂×D₃×C₈	1	1	yes	characteristic
225	C₃×C₄	C₂×D₃×C₈	1	1	yes	characteristic
226	C₃×C₄	C₂×C₄×Dic₃	1	1	yes	characteristic
227	Dic₃	—	6	2	no	—
228	Dic₃	—	6	2	no	—
229	Dic₃	—	12	2	no	—
230	Dic₃	#798	1	4	yes	direct product factor
231	Dic₃	C₈×Dic₃	1	2	yes	—
232	C₃²	C₄×M₃₂	1	1	yes	characteristic, complemented, Sylow
233	C₂³	—	3	1	no	—
234	C₂×C₄	—	3	2	no	—
235	C₂×C₄	—	3	1	no	—
236	C₂×C₄	—	3	2	no	—
237	C₂×C₄	—	3	2	no	—
238	C₂×C₄	—	9	2	no	—
239	C₂×C₄	—	18	4	no	—
240	C₂×C₄	C₄×D₃²	1	1	yes	characteristic
241	C₈	—	3	2	no	—
242	C₈	—	3	2	no	—
243	C₈	—	9	2	no	—
244	C₈	C₄×D₃²	1	1	yes	characteristic
245	C₈	C₂×D₃×Dic₃	1	1	yes	characteristic
246	C₂×C₃	—	2	2	no	—
247	C₂×C₃	—	2	1	no	—
248	C₂×C₃	—	6	2	no	—
249	C₂×C₃	C₄×D₃×C₈	1	1	yes	characteristic
250	C₂×C₃	C₂×#798	1	2	yes	—
251	C₂×C₃	D₃×M₃₂	1	2	yes	—
252	C₂×C₃	C₂×C₈×Dic₃	1	1	yes	characteristic
253	D₃	—	2	2	no	—
254	C₂²	—	3	2	no	—
255	C₂²	—	6	2	no	—
256	C₂²	D₃²×C₈	1	1	yes	characteristic
257	C₄	—	3	4	no	—
258	C₄	—	3	2	no	—
259	C₄	—	18	2	no	—
260	C₄	D₃²×C₈	1	1	yes	characteristic
261	C₄	C₄×D₃×Dic₃	1	1	yes	characteristic
262	C₃	—	2	1	no	—
263	C₃	C₄×#798	1	1	yes	characteristic, complemented
264	C₃	Dic₃×M₃₂	1	1	yes	characteristic, complemented
265	C₂	—	6	2	no	—
266	C₂	D₃×#798	1	2	yes	—
267	C₂	D₃×C₈×Dic₃	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₈	C₂×D₃²
commutator	C₂×C₃²	C₂×C₄×C₈
Fitting	C₂×C₃²×C₁₆	C₂²
socle	C₂²×C₃²	C₂²×C₈
prime radical	C₂²×C₃×D₃	C₂×C₈
cosocle	C₂×C₃²×C₈	C₂³
hypercenter	C₂×C₁₆	D₃²
hypocenter	C₃²	C₄×M₃₂
LCS	G ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₁₆

Complements

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

N	H ≅ G/N	type
4. #10589368	C₂	semidirect
5. C₃×Dic₃×C₁₆	C₂	semidirect
6. #10589104	C₂	semidirect
7. Dic₃×C₃⋊C₁₆	C₂	semidirect
19. C₃×#798	C₄	semidirect
22. #63543	C₄	semidirect
36. Dic₃×C₁₆	D₃	semidirect
50. C₃²×C₁₆	C₂×C₄	semidirect
51. C₃⋊(C₃⋊C₁₆)	C₂×C₄	semidirect
52. C₃×C₃⋊C₁₆	C₂×C₄	semidirect
53. C₃×C₃⋊C₁₆	C₂×C₄	semidirect
83. #798	Dic₃	direct (N × H)
134. C₃×C₁₆	C₄×D₃	semidirect
135. C₃×C₁₆	C₂×Dic₃	semidirect
139. C₃⋊C₁₆	C₄×D₃	semidirect
140. C₃⋊C₁₆	C₂×Dic₃	semidirect
148. C₃×Dic₃	M₃₂	semidirect
211. C₁₆	D₃×Dic₃	semidirect
230. Dic₃	#798	direct (N × H)
232. C₃²	C₄×M₃₂	semidirect
263. C₃	C₄×#798	semidirect
264. C₃	Dic₃×M₃₂	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃×#798	2
D₃×C₈×Dic₃	2
C₂×#63543	2
#10589368	2
C₃×Dic₃×C₁₆	2
#10589104	2
Dic₃×C₃⋊C₁₆	2
Dic₃×M₃₂	3
C₄×#798	3

Subgroups — Dic3×#798 #11795862

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Dic₃×#798 #11795862