Subgroups — D₃×#4494 #11797106

Order 1152.

325 conjugacy classes of subgroups.

The subgroup lattice has 296 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×D₃×#108	C₂	1	1	yes	maximal, characteristic
2	D₃×#817	C₂	1	1	yes	maximal, characteristic
3	D₃×#812	C₂	1	1	yes	maximal, characteristic
4	#10589360	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×#4494	C₂	1	1	yes	maximal, characteristic, complemented
6	#10589138	C₂	1	1	yes	maximal, characteristic, complemented
7	#10589137	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#4494	—	3	1	no	maximal
9	D₃×#365	—	3	1	no	maximal
10	C₃×D₃×M₁₆	—	2	1	no	—
11	C₂×C₃×D₃×C₈	C₂²	1	1	yes	characteristic
12	D₃×C₄⋊Dic₃	—	2	1	no	—
13	C₃²×#108	C₂²	1	1	yes	characteristic
14	#63644	C₂²	1	1	yes	characteristic
15	#63639	C₂²	1	1	yes	characteristic
16	C₃×#822	—	2	1	no	—
17	C₃×#819	C₂²	1	1	yes	characteristic
18	C₃×#817	C₂²	1	1	yes	characteristic
19	C₃×#812	C₂²	1	1	yes	characteristic
20	#63573	—	2	1	no	—
21	#63543	C₄	1	2	yes	complemented
22	D₃×C₃⋊C₁₆	C₄	1	2	yes	complemented
23	C₂×C₃×#108	—	3	1	no	—
24	C₂×#817	—	3	1	no	—
25	C₂×#812	—	3	1	no	—
26	D₃×M₃₂	—	3	1	no	—
27	D₃×#108	D₃	1	1	yes	characteristic
28	D₃×#106	—	3	1	no	—
29	C₃×#365	—	3	1	no	—
30	#4513	—	3	1	no	—
31	#4494	—	3	1	no	—
32	#4494	—	3	1	no	—
33	#4494	—	3	1	no	—
34	#4494	—	3	1	no	—
35	#4494	—	6	1	no	—
36	#4494	D₃	1	1	yes	characteristic, direct product factor
37	#4487	—	3	1	no	—
38	C₂×C₃×C₄×D₃	D₄	1	1	yes	characteristic
39	C₂×D₃×Dic₃	—	4	1	no	—
40	C₃²×M₁₆	—	2	1	no	—
41	C₂×C₃²×C₈	C₂³	1	1	yes	characteristic
42	C₄⋊(C₃⋊Dic₃)	—	2	1	no	—
43	C₃×C₄⋊Dic₃	—	2	1	no	—
44	C₃×#208	—	2	1	no	—
45	C₂×C₃×C₃⋊C₈	D₄	1	1	yes	characteristic
46	C₃×ΓL(1,𝔽₂₅)	—	4	1	no	—
47	C₃×D₃×C₈	—	4	1	no	—
48	C₃×D₃×C₈	C₂×C₄	1	1	yes	characteristic
49	C₃×D₃×C₈	C₂×C₄	1	1	yes	characteristic
50	C₃×D₃×C₈	Q₈	1	1	yes	characteristic
51	C₃×D₃×C₈	Q₈	1	1	yes	characteristic
52	Dic₃⋊Dic₃	—	4	1	no	—
53	C₃⋊(C₃⋊C₁₆)	C₂×C₄	1	2	yes	complemented
54	C₃×C₃⋊C₁₆	C₂×C₄	1	2	yes	complemented
55	C₂×#365	—	9	1	no	Sylow
56	C₂×C₃×M₁₆	—	6	1	no	—
57	C₂²×C₃×C₈	—	3	1	no	—
58	C₂×C₄⋊Dic₃	—	6	1	no	—
59	D₃×M₁₆	—	2	1	no	—
60	C₂×D₃×C₈	C₂×D₃	1	1	yes	characteristic
61	D₃×C₄⋊C₄	—	6	1	no	—
62	C₃×M₃₂	—	3	1	no	—
63	C₃×#108	—	2	1	no	—
64	C₃×#108	—	3	1	no	—
65	C₃×#108	—	6	1	no	—
66	C₃×#108	C₂×D₃	1	1	yes	characteristic
67	C₃×#108	C₂×D₃	1	1	yes	characteristic
68	C₃×#106	—	3	1	no	—
69	#822	—	2	1	no	—
70	#819	C₂×D₃	1	1	yes	characteristic
71	#817	—	3	1	no	—
72	#817	—	3	1	no	—
73	#817	—	6	1	no	—
74	#817	—	6	1	no	—
75	#817	C₂×D₃	1	1	yes	characteristic
76	#812	—	3	1	no	—
77	#812	—	3	1	no	—
78	#812	—	3	2	no	—
79	#812	—	6	1	no	—
80	#812	C₂×D₃	1	1	yes	characteristic
81	C₂×C₃⋊C₁₆	—	3	2	no	—
82	#808	—	6	1	no	—
83	#798	—	3	2	no	—
84	D₃×C₁₆	—	3	2	no	—
85	C₂²×C₃×D₃	SD₁₆	1	1	yes	characteristic
86	C₂×C₃²×C₄	C₂×D₄	1	1	yes	characteristic
87	C₂×C₃⋊Dic₃	—	4	1	no	—
88	C₂×C₃×Dic₃	—	4	1	no	—
89	C₂×C₃×Dic₃	SD₁₆	1	1	yes	characteristic
90	C₃×C₄×D₃	C₄⋊C₄	1	1	yes	characteristic
91	C₃×C₄×D₃	C₄⋊C₄	1	1	yes	characteristic
92	C₃×C₄×D₃	SD₁₆	1	1	yes	characteristic
93	C₃×C₄×D₃	SD₁₆	1	1	yes	characteristic
94	D₃×Dic₃	—	8	2	no	—
95	C₃²×C₈	—	4	1	no	—
96	C₃²×C₈	C₂²×C₄	1	1	yes	characteristic
97	C₃²×C₈	C₂×Q₈	1	1	yes	characteristic
98	C₃×C₃⋊C₈	—	4	1	no	—
99	C₃×C₃⋊C₈	C₄⋊C₄	1	1	yes	characteristic
100	C₃×C₃⋊C₈	C₄⋊C₄	1	1	yes	characteristic
101	C₂×M₃₂	—	9	1	no	—
102	C₂×#108	—	3	1	no	—
103	C₂×#106	—	9	1	no	—
104	#365	—	3	1	no	—
105	#365	—	9	1	no	—
106	#365	—	9	1	no	—
107	#365	—	9	1	no	—
108	C₂²×C₃×C₄	—	3	1	no	—
109	C₂²×Dic₃	—	12	1	no	—
110	C₂×C₄×D₃	—	12	1	no	—
111	C₂×C₄×D₃	#67	1	1	yes	characteristic
112	C₃×M₁₆	—	2	1	no	—
113	C₃×M₁₆	—	2	1	no	—
114	C₃×M₁₆	—	4	1	no	—
115	C₃×M₁₆	—	6	1	no	—
116	C₃×M₁₆	—	12	1	no	—
117	C₂×C₃×C₈	—	2	1	no	—
118	C₂×C₃×C₈	—	3	1	no	—
119	C₂×C₃×C₈	—	3	1	no	—
120	C₂×C₃×C₈	—	3	1	no	—
121	C₂×C₃×C₈	—	3	1	no	—
122	C₂×C₃×C₈	—	3	1	no	—
123	C₂×C₃×C₈	—	12	1	no	—
124	C₂×C₃×C₈	C₂²×D₃	1	1	yes	characteristic
125	C₂×C₃×C₈	C₂²×D₃	1	1	yes	characteristic
126	C₃×C₄⋊C₄	—	6	1	no	—
127	C₄⋊Dic₃	—	2	1	no	—
128	C₄⋊Dic₃	—	6	1	no	—
129	C₄⋊Dic₃	—	6	1	no	—
130	C₄⋊Dic₃	—	12	1	no	—
131	C₄⋊Dic₃	—	12	1	no	—
132	Dic₃⋊C₄	—	12	1	no	—
133	#208	—	2	1	no	—
134	C₂×C₃⋊C₈	#67	1	1	yes	characteristic
135	ΓL(1,𝔽₂₅)	—	4	1	no	—
136	D₃×C₈	—	4	1	no	—
137	D₃×C₈	Dic₆	1	1	yes	characteristic
138	D₃×C₈	Dic₆	1	1	yes	characteristic
139	D₃×C₈	C₄×D₃	1	1	yes	characteristic
140	D₃×C₈	C₄×D₃	1	1	yes	characteristic
141	C₃×C₁₆	—	3	2	no	—
142	C₃⋊C₁₆	—	3	2	no	—
143	C₃⋊C₁₆	—	3	2	no	—
144	C₃⋊C₁₆	—	6	2	no	—
145	C₃⋊C₁₆	C₄×D₃	1	2	yes	complemented
146	C₂²×C₃²	C₂×SD₁₆	1	1	yes	characteristic
147	C₂×C₃×D₃	—	2	2	no	—
148	C₂×C₃×D₃	#106	1	1	yes	characteristic
149	C₂×C₃×D₃	#106	1	1	yes	characteristic
150	C₃²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
151	C₃²×C₄	C₂×SD₁₆	1	1	yes	characteristic
152	C₃⋊Dic₃	—	8	1	no	—
153	C₃×Dic₃	—	8	1	no	—
154	C₃×Dic₃	#106	1	1	yes	characteristic
155	C₃×Dic₃	#106	1	1	yes	characteristic
156	C₂×M₁₆	—	6	1	no	—
157	C₂²×C₈	—	3	1	no	—
158	C₂×C₄⋊C₄	—	18	1	no	—
159	M₃₂	—	3	1	no	—
160	M₃₂	—	9	2	no	—
161	M₃₂	—	9	1	no	—
162	C₂×C₁₆	—	9	2	no	—
163	#108	—	3	1	no	—
164	#108	—	6	1	no	—
165	#108	D₃²	1	1	yes	characteristic
166	#106	—	3	1	no	—
167	#106	—	9	1	no	—
168	#106	—	18	1	no	—
169	C₂³×C₃	—	3	1	no	—
170	C₂²×D₃	#215	1	1	yes	characteristic
171	C₂×C₃×C₄	—	2	1	no	—
172	C₂×C₃×C₄	—	3	1	no	—
173	C₂×C₃×C₄	—	3	1	no	—
174	C₂×C₃×C₄	—	3	1	no	—
175	C₂×C₃×C₄	—	3	1	no	—
176	C₂×C₃×C₄	—	3	1	no	—
177	C₂×C₃×C₄	—	12	1	no	—
178	C₂×C₃×C₄	D₃×D₄	1	1	yes	characteristic
179	C₂×C₃×C₄	C₂×#67	1	1	yes	characteristic
180	C₂×Dic₃	—	4	1	no	—
181	C₂×Dic₃	—	12	1	no	—
182	C₂×Dic₃	—	12	1	no	—
183	C₂×Dic₃	—	24	1	no	—
184	C₂×Dic₃	—	24	2	no	—
185	C₂×Dic₃	#215	1	1	yes	characteristic
186	C₄×D₃	—	24	2	no	—
187	C₄×D₃	Dic₃⋊C₄	1	1	yes	characteristic
188	C₄×D₃	Dic₃⋊C₄	1	1	yes	characteristic
189	C₄×D₃	#214	1	1	yes	characteristic
190	C₄×D₃	#214	1	1	yes	characteristic
191	C₃×C₈	—	2	1	no	—
192	C₃×C₈	—	2	1	no	—
193	C₃×C₈	—	3	1	no	—
194	C₃×C₈	—	3	1	no	—
195	C₃×C₈	—	4	1	no	—
196	C₃×C₈	—	4	1	no	—
197	C₃×C₈	—	8	1	no	—
198	C₃×C₈	—	12	1	no	—
199	C₃×C₈	C₂×Dic₆	1	1	yes	characteristic
200	C₃×C₈	C₂×C₄×D₃	1	1	yes	characteristic
201	C₃×C₈	C₂×C₄×D₃	1	1	yes	characteristic
202	C₃×C₈	D₃×Q₈	1	1	yes	characteristic
203	C₃⋊C₈	—	4	1	no	—
204	C₃⋊C₈	Dic₃⋊C₄	1	1	yes	characteristic
205	C₃⋊C₈	Dic₃⋊C₄	1	1	yes	characteristic
206	C₂×C₃²	—	2	1	no	—
207	C₂×C₃²	C₂×#106	1	1	yes	characteristic
208	C₃×D₃	—	2	1	no	—
209	C₃×D₃	#365	1	2	yes	complemented
210	C₂²×C₄	—	3	1	no	—
211	C₂²×C₄	—	36	1	no	—
212	M₁₆	—	2	1	no	—
213	M₁₆	—	6	1	no	—
214	M₁₆	—	12	1	no	—
215	C₂×C₈	—	3	1	no	—
216	C₂×C₈	—	3	1	no	—
217	C₂×C₈	—	3	1	no	—
218	C₂×C₈	—	3	1	no	—
219	C₂×C₈	—	3	1	no	—
220	C₂×C₈	—	12	1	no	—
221	C₂×C₈	C₂×D₃²	1	1	yes	characteristic
222	C₄⋊C₄	—	6	1	no	—
223	C₄⋊C₄	—	18	1	no	—
224	C₄⋊C₄	—	36	1	no	—
225	C₁₆	—	3	2	no	—
226	C₁₆	—	9	2	no	—
227	C₂²×C₃	—	2	1	no	—
228	C₂²×C₃	—	3	1	no	—
229	C₂²×C₃	—	3	1	no	—
230	C₂²×C₃	—	6	2	no	—
231	C₂²×C₃	D₃×SD₁₆	1	1	yes	characteristic
232	C₂²×C₃	C₂×#215	1	1	yes	characteristic
233	C₂×D₃	—	2	2	no	—
234	C₂×D₃	#808	1	1	yes	characteristic
235	C₂×D₃	#808	1	1	yes	characteristic
236	C₃×C₄	—	2	1	no	—
237	C₃×C₄	—	2	1	no	—
238	C₃×C₄	—	3	1	no	—
239	C₃×C₄	—	3	1	no	—
240	C₃×C₄	—	24	1	no	—
241	C₃×C₄	D₃×C₄⋊C₄	1	1	yes	characteristic
242	C₃×C₄	D₃×SD₁₆	1	1	yes	characteristic
243	C₃×C₄	C₂×Dic₃⋊C₄	1	1	yes	characteristic
244	C₃×C₄	C₂×#214	1	1	yes	characteristic
245	Dic₃	—	8	1	no	—
246	Dic₃	—	24	1	no	—
247	Dic₃	—	24	1	no	—
248	Dic₃	—	48	1	no	—
249	Dic₃	#808	1	1	yes	characteristic
250	Dic₃	#808	1	1	yes	characteristic
251	C₃²	C₂×#365	1	1	yes	characteristic, complemented, Sylow
252	C₂³	—	3	1	no	—
253	C₂×C₄	—	3	1	no	—
254	C₂×C₄	—	3	1	no	—
255	C₂×C₄	—	3	1	no	—
256	C₂×C₄	—	3	1	no	—
257	C₂×C₄	—	3	1	no	—
258	C₂×C₄	—	12	1	no	—
259	C₂×C₄	—	36	1	no	—
260	C₂×C₄	—	72	2	no	—
261	C₂×C₄	D₃×#67	1	1	yes	characteristic
262	C₈	—	3	1	no	—
263	C₈	—	3	1	no	—
264	C₈	—	4	1	no	—
265	C₈	—	12	1	no	—
266	C₈	D₃×Dic₆	1	1	yes	characteristic
267	C₈	C₄×D₃²	1	1	yes	characteristic
268	C₂×C₃	—	2	1	no	—
269	C₂×C₃	—	2	1	no	—
270	C₂×C₃	—	2	1	no	—
271	C₂×C₃	—	3	2	no	—
272	C₂×C₃	—	4	1	no	—
273	C₂×C₃	—	6	1	no	—
274	C₂×C₃	D₃×#106	1	1	yes	characteristic
275	C₂×C₃	C₂×#808	1	1	yes	characteristic
276	D₃	—	2	1	no	—
277	D₃	#4494	1	2	yes	direct product factor
278	C₂²	—	3	1	no	—
279	C₂²	—	3	1	no	—
280	C₂²	—	6	2	no	—
281	C₂²	D₃×#215	1	1	yes	characteristic
282	C₄	—	3	1	no	—
283	C₄	—	3	1	no	—
284	C₄	—	24	1	no	—
285	C₄	—	72	1	no	—
286	C₄	D₃×Dic₃⋊C₄	1	1	yes	characteristic
287	C₄	D₃×#214	1	1	yes	characteristic
288	C₃	—	2	1	no	—
289	C₃	D₃×#365	1	1	yes	characteristic, complemented
290	C₃	C₂×#4494	1	1	yes	characteristic, complemented
291	C₂	—	2	1	no	—
292	C₂	—	3	2	no	—
293	C₂	—	6	1	no	—
294	C₂	D₃×#808	1	1	yes	characteristic

	subgroup	G/H
center	C₂	D₃×#808
commutator	C₃²×C₈	C₂²×C₄
Frattini	C₂×C₈	C₂×D₃²
Fitting	C₃²×#108	C₂²
socle	C₂×C₃²	C₂×#106
prime radical	C₂²×C₃×D₃	SD₁₆
cosocle	C₂×C₃²×C₈	C₂³
hypercenter	#108	D₃²
hypocenter	C₃²	C₂×#365
LCS	G ▷ C₃²×C₈ ▷ C₃²×C₄ ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ #108

Complements

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

N	H ≅ G/N	type
4. #10589360	C₂	semidirect
5. C₃×#4494	C₂	semidirect
6. #10589138	C₂	semidirect
7. #10589137	C₂	semidirect
21. #63543	C₄	semidirect
22. D₃×C₃⋊C₁₆	C₄	semidirect
36. #4494	D₃	direct (N × H)
53. C₃⋊(C₃⋊C₁₆)	C₂×C₄	semidirect
54. C₃×C₃⋊C₁₆	C₂×C₄	semidirect
145. C₃⋊C₁₆	C₄×D₃	semidirect
209. C₃×D₃	#365	semidirect
251. C₃²	C₂×#365	semidirect
277. D₃	#4494	direct (N × H)
289. C₃	D₃×#365	semidirect
290. C₃	C₂×#4494	semidirect

Maximal subgroups

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

subgroup	index
C₃×D₃×#108	2
D₃×#817	2
D₃×#812	2
#10589360	2
C₃×#4494	2
#10589138	2
#10589137	2
C₂×#4494	3
D₃×#365	3

Subgroups — D3×#4494 #11797106

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₃×#4494 #11797106