Subgroups — D₃×#344 #69025

Order 384.

388 conjugacy classes of subgroups.

The subgroup lattice has 308 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₃×M₁₆	C₂	1	1	yes	maximal, characteristic
2	C₂×D₃×M₁₆	C₂	1	1	yes	maximal, characteristic
3	D₃×#117	C₂	1	1	yes	maximal, characteristic
4	C₃×#344	C₂	1	1	yes	maximal, characteristic, complemented
5	#4559	C₂	1	1	yes	maximal, characteristic, complemented
6	#4556	C₂	1	1	yes	maximal, characteristic, complemented
7	#4533	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#344	—	3	1	no	maximal, Sylow
9	C₂²×C₄×D₃	C₂²	1	1	yes	characteristic
10	C₂×C₃×M₁₆	C₂²	1	1	yes	characteristic
11	C₂×C₃×M₁₆	C₂²	1	1	yes	characteristic
12	C₃×#117	C₂²	1	1	yes	characteristic
13	#926	C₂²	1	1	yes	characteristic
14	C₂×#208	C₂²	1	1	yes	characteristic
15	C₂×#208	C₂²	1	1	yes	characteristic
16	D₃×M₁₆	—	2	1	no	—
17	D₃×M₁₆	—	2	1	no	—
18	D₃×M₁₆	—	2	1	no	—
19	D₃×M₁₆	—	2	1	no	—
20	D₃×M₁₆	—	2	1	no	—
21	D₃×M₁₆	—	2	1	no	—
22	D₃×M₁₆	C₄	1	2	yes	complemented
23	D₃×M₁₆	C₄	1	2	yes	complemented
24	C₂×ΓL(1,𝔽₂₅)	—	2	1	no	—
25	C₂×ΓL(1,𝔽₂₅)	C₄	1	2	yes	complemented
26	C₂×D₃×C₈	—	2	1	no	—
27	C₂×D₃×C₈	C₄	1	2	yes	complemented
28	#892	—	2	1	no	—
29	D₃×C₄⋊C₄	—	2	1	no	—
30	D₃×C₂²⋊C₄	C₄	1	2	yes	—
31	#877	C₄	1	2	yes	—
32	#872	—	2	1	no	—
33	C₄²×D₃	—	2	1	no	—
34	C₂²×M₁₆	—	3	1	no	—
35	C₂²×M₁₆	—	3	1	no	—
36	C₂×#117	—	3	1	no	—
37	#344	—	3	1	no	—
38	#344	—	3	1	no	—
39	#344	—	3	1	no	—
40	#344	D₃	1	1	yes	characteristic, direct product factor
41	C₂³×D₃	C₂×C₄	1	1	yes	characteristic
42	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
43	C₂²×Dic₃	C₂×C₄	1	1	yes	characteristic
44	C₂×C₄×D₃	—	2	2	no	—
45	C₂×C₄×D₃	C₂×C₄	1	1	yes	characteristic
46	C₂×C₄×D₃	C₂×C₄	1	1	yes	characteristic
47	C₂×C₄×D₃	C₂×C₄	1	1	yes	characteristic
48	C₂×C₄×D₃	C₂×C₄	1	1	yes	characteristic
49	C₂×C₄×D₃	D₄	1	1	yes	characteristic
50	C₂×C₄×D₃	D₄	1	1	yes	characteristic
51	C₂×C₄×D₃	D₄	1	1	yes	characteristic
52	C₂×C₄×D₃	D₄	1	1	yes	characteristic
53	C₂×C₄×D₃	D₄	1	1	yes	characteristic
54	C₂×C₄×D₃	D₄	1	1	yes	characteristic
55	C₂×C₄×D₃	Q₈	1	1	yes	characteristic
56	C₂×C₄×D₃	Q₈	1	1	yes	characteristic
57	C₃×M₁₆	—	2	1	no	—
58	C₃×M₁₆	—	2	1	no	—
59	C₃×M₁₆	—	2	1	no	—
60	C₃×M₁₆	C₂×C₄	1	2	yes	complemented
61	C₂×C₃×C₈	—	2	1	no	—
62	C₂×C₃×C₈	C₂×C₄	1	2	yes	complemented
63	C₃×C₄⋊C₄	—	2	1	no	—
64	C₃×C₂²⋊C₄	C₂×C₄	1	2	yes	—
65	C₃×C₄²	—	2	1	no	—
66	#217	C₂×C₄	1	2	yes	—
67	(C₂×D₃)⋊C₄	—	2	2	no	—
68	C₄⋊Dic₃	—	2	1	no	—
69	Dic₃⋊C₄	—	2	2	no	—
70	C₄×Dic₃	—	2	1	no	—
71	C₄×Dic₃	—	2	2	no	—
72	#208	—	2	1	no	—
73	#208	—	2	1	no	—
74	#208	—	2	1	no	—
75	#208	C₂×C₄	1	2	yes	complemented
76	C₂×C₃⋊C₈	—	2	1	no	—
77	C₂×C₃⋊C₈	C₂×C₄	1	2	yes	complemented
78	ΓL(1,𝔽₂₅)	—	2	2	no	—
79	ΓL(1,𝔽₂₅)	—	2	2	no	—
80	ΓL(1,𝔽₂₅)	—	2	2	no	—
81	ΓL(1,𝔽₂₅)	—	2	2	no	—
82	D₃×C₈	—	2	2	no	—
83	D₃×C₈	—	2	2	no	—
84	D₃×C₈	—	2	2	no	—
85	D₃×C₈	—	2	2	no	—
86	C₂³×C₄	—	3	1	no	—
87	C₂×M₁₆	—	3	1	no	—
88	C₂×M₁₆	—	3	2	no	—
89	C₂×M₁₆	—	3	1	no	—
90	C₂×M₁₆	—	3	2	no	—
91	C₂×M₁₆	—	3	2	no	—
92	C₂×M₁₆	—	6	1	no	—
93	C₂×M₁₆	—	6	1	no	—
94	C₂×M₁₆	—	6	1	no	—
95	C₂×M₁₆	—	6	1	no	—
96	C₂×M₁₆	—	6	1	no	—
97	C₂×M₁₆	—	6	1	no	—
98	C₂×M₁₆	—	6	1	no	—
99	C₂×M₁₆	C₂×D₃	1	1	yes	characteristic
100	C₂×M₁₆	C₂×D₃	1	1	yes	characteristic
101	C₂²×C₈	—	3	2	no	—
102	C₂²×C₈	—	6	1	no	—
103	#117	—	3	1	no	—
104	#117	—	3	2	no	—
105	#117	—	6	1	no	—
106	#117	—	6	1	no	—
107	#117	C₂×D₃	1	1	yes	characteristic
108	C₂×C₄⋊C₄	—	6	1	no	—
109	C₂×C₂²⋊C₄	—	3	2	no	—
110	C₂×C₄²	—	6	1	no	—
111	C₂³×C₃	C₂²×C₄	1	1	yes	characteristic
112	C₂²×D₃	—	4	2	no	—
113	C₂²×D₃	C₄²	1	1	yes	characteristic
114	C₂²×D₃	C₄²	1	1	yes	characteristic
115	C₂²×D₃	C₂²⋊C₄	1	1	yes	characteristic
116	C₂²×D₃	C₂²⋊C₄	1	1	yes	characteristic
117	C₂²×D₃	C₄⋊C₄	1	1	yes	characteristic
118	C₂²×D₃	C₄⋊C₄	1	1	yes	characteristic
119	C₂×C₃×C₄	—	2	2	no	—
120	C₂×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
121	C₂×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
122	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
123	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
124	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
125	C₂×C₃×C₄	C₂×Q₈	1	1	yes	characteristic
126	C₂×Dic₃	—	2	2	no	—
127	C₂×Dic₃	C₄²	1	1	yes	characteristic
128	C₂×Dic₃	C₄²	1	1	yes	characteristic
129	C₂×Dic₃	C₂²⋊C₄	1	1	yes	characteristic
130	C₂×Dic₃	C₂²⋊C₄	1	1	yes	characteristic
131	C₂×Dic₃	C₄⋊C₄	1	1	yes	characteristic
132	C₂×Dic₃	C₄⋊C₄	1	1	yes	characteristic
133	C₄×D₃	—	4	4	no	—
134	C₄×D₃	C₂²⋊C₄	1	1	yes	characteristic
135	C₄×D₃	C₂²⋊C₄	1	1	yes	characteristic
136	C₄×D₃	C₂²⋊C₄	1	1	yes	characteristic
137	C₄×D₃	C₂²⋊C₄	1	1	yes	characteristic
138	C₄×D₃	C₂²⋊C₄	1	1	yes	characteristic
139	C₄×D₃	C₂²⋊C₄	1	1	yes	characteristic
140	C₄×D₃	C₂²⋊C₄	1	1	yes	characteristic
141	C₄×D₃	C₂²⋊C₄	1	1	yes	characteristic
142	C₄×D₃	C₄⋊C₄	1	1	yes	characteristic
143	C₄×D₃	C₄⋊C₄	1	1	yes	characteristic
144	C₄×D₃	C₄⋊C₄	1	1	yes	characteristic
145	C₄×D₃	C₄⋊C₄	1	1	yes	characteristic
146	C₄×D₃	C₄⋊C₄	1	1	yes	characteristic
147	C₄×D₃	C₄⋊C₄	1	1	yes	characteristic
148	C₄×D₃	C₄⋊C₄	1	1	yes	characteristic
149	C₄×D₃	C₄⋊C₄	1	1	yes	characteristic
150	C₃×C₈	—	2	2	no	—
151	C₃×C₈	—	2	2	no	—
152	C₃⋊C₈	—	2	2	no	—
153	C₃⋊C₈	—	2	2	no	—
154	C₂⁴	—	3	1	no	—
155	C₂²×C₄	—	3	1	no	—
156	C₂²×C₄	—	3	1	no	—
157	C₂²×C₄	—	3	1	no	—
158	C₂²×C₄	—	3	1	no	—
159	C₂²×C₄	—	3	1	no	—
160	C₂²×C₄	—	3	1	no	—
161	C₂²×C₄	—	3	1	no	—
162	C₂²×C₄	—	3	1	no	—
163	C₂²×C₄	—	3	1	no	—
164	C₂²×C₄	—	3	1	no	—
165	C₂²×C₄	—	3	1	no	—
166	C₂²×C₄	—	3	1	no	—
167	C₂²×C₄	—	3	1	no	—
168	C₂²×C₄	—	6	2	no	—
169	C₂²×C₄	C₂²×D₃	1	1	yes	characteristic
170	M₁₆	—	2	1	no	—
171	M₁₆	—	2	1	no	—
172	M₁₆	—	2	1	no	—
173	M₁₆	—	3	2	no	—
174	M₁₆	—	6	2	no	—
175	M₁₆	—	6	2	no	—
176	M₁₆	—	6	2	no	—
177	M₁₆	—	6	2	no	—
178	M₁₆	—	6	1	no	—
179	M₁₆	—	6	1	no	—
180	M₁₆	—	6	1	no	—
181	M₁₆	C₄×D₃	1	2	yes	complemented
182	C₂×C₈	—	2	1	no	—
183	C₂×C₈	—	3	2	no	—
184	C₂×C₈	—	6	2	no	—
185	C₂×C₈	—	6	2	no	—
186	C₂×C₈	—	6	2	no	—
187	C₂×C₈	—	6	2	no	—
188	C₂×C₈	—	6	1	no	—
189	C₂×C₈	C₄×D₃	1	2	yes	complemented
190	C₄⋊C₄	—	2	1	no	—
191	C₄⋊C₄	—	6	2	no	—
192	C₄⋊C₄	—	6	1	no	—
193	C₂²⋊C₄	—	3	2	no	—
194	C₂²⋊C₄	—	6	2	no	—
195	C₂²⋊C₄	C₄×D₃	1	2	yes	—
196	C₄²	—	2	1	no	—
197	C₄²	—	6	2	no	—
198	C₄²	—	6	1	no	—
199	C₂²×C₃	—	4	1	no	—
200	C₂²×C₃	C₂×C₄²	1	1	yes	characteristic
201	C₂²×C₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
202	C₂²×C₃	C₂×C₄⋊C₄	1	1	yes	characteristic
203	C₂×D₃	—	2	2	no	—
204	C₂×D₃	—	2	2	no	—
205	C₂×D₃	—	2	2	no	—
206	C₂×D₃	—	4	1	no	—
207	C₂×D₃	—	4	1	no	—
208	C₂×D₃	—	4	1	no	—
209	C₂×D₃	#95	1	1	yes	characteristic
210	C₂×D₃	#95	1	1	yes	characteristic
211	C₂×D₃	#95	1	1	yes	characteristic
212	C₂×D₃	#95	1	1	yes	characteristic
213	C₃×C₄	—	4	2	no	—
214	C₃×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
215	C₃×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
216	C₃×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
217	C₃×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
218	Dic₃	—	4	2	no	—
219	Dic₃	#95	1	1	yes	characteristic
220	Dic₃	#95	1	1	yes	characteristic
221	Dic₃	#95	1	1	yes	characteristic
222	Dic₃	#95	1	1	yes	characteristic
223	C₂³	—	3	1	no	—
224	C₂³	—	3	1	no	—
225	C₂³	—	3	1	no	—
226	C₂³	—	3	1	no	—
227	C₂³	—	3	1	no	—
228	C₂³	—	3	1	no	—
229	C₂³	—	12	2	no	—
230	C₂³	C₂×C₄×D₃	1	1	yes	characteristic
231	C₂×C₄	—	2	2	no	—
232	C₂×C₄	—	3	1	no	—
233	C₂×C₄	—	3	1	no	—
234	C₂×C₄	—	3	1	no	—
235	C₂×C₄	—	3	1	no	—
236	C₂×C₄	—	3	1	no	—
237	C₂×C₄	—	3	1	no	—
238	C₂×C₄	—	3	1	no	—
239	C₂×C₄	—	3	1	no	—
240	C₂×C₄	—	3	1	no	—
241	C₂×C₄	—	3	1	no	—
242	C₂×C₄	—	3	1	no	—
243	C₂×C₄	—	3	1	no	—
244	C₂×C₄	—	3	1	no	—
245	C₂×C₄	—	3	1	no	—
246	C₂×C₄	—	3	1	no	—
247	C₂×C₄	—	3	1	no	—
248	C₂×C₄	—	3	1	no	—
249	C₂×C₄	—	3	1	no	—
250	C₂×C₄	—	3	1	no	—
251	C₂×C₄	—	3	1	no	—
252	C₂×C₄	—	3	1	no	—
253	C₂×C₄	—	3	1	no	—
254	C₂×C₄	—	6	2	no	—
255	C₂×C₄	—	12	4	no	—
256	C₂×C₄	C₂×C₄×D₃	1	1	yes	characteristic
257	C₂×C₄	C₂×C₄×D₃	1	1	yes	characteristic
258	C₂×C₄	D₃×D₄	1	1	yes	characteristic
259	C₂×C₄	D₃×D₄	1	1	yes	characteristic
260	C₂×C₄	D₃×D₄	1	1	yes	characteristic
261	C₂×C₄	D₃×Q₈	1	1	yes	characteristic
262	C₈	—	2	2	no	—
263	C₈	—	2	2	no	—
264	C₈	—	6	2	no	—
265	C₈	—	6	2	no	—
266	C₂×C₃	—	2	1	no	—
267	C₂×C₃	—	2	1	no	—
268	C₂×C₃	—	2	1	no	—
269	C₂×C₃	C₂×#95	1	1	yes	characteristic
270	D₃	—	2	1	no	—
271	D₃	—	2	1	no	—
272	D₃	—	2	1	no	—
273	D₃	#344	1	2	yes	direct product factor
274	C₂²	—	3	1	no	—
275	C₂²	—	3	1	no	—
276	C₂²	—	3	1	no	—
277	C₂²	—	3	1	no	—
278	C₂²	—	4	1	no	—
279	C₂²	—	6	2	no	—
280	C₂²	—	6	2	no	—
281	C₂²	—	6	2	no	—
282	C₂²	—	12	1	no	—
283	C₂²	—	12	1	no	—
284	C₂²	—	12	1	no	—
285	C₂²	C₄²×D₃	1	1	yes	characteristic
286	C₂²	D₃×C₂²⋊C₄	1	1	yes	characteristic
287	C₂²	D₃×C₄⋊C₄	1	1	yes	characteristic
288	C₄	—	3	1	no	—
289	C₄	—	3	1	no	—
290	C₄	—	3	1	no	—
291	C₄	—	3	1	no	—
292	C₄	—	4	2	no	—
293	C₄	—	12	2	no	—
294	C₄	D₃×C₂²⋊C₄	1	1	yes	characteristic
295	C₄	D₃×C₂²⋊C₄	1	1	yes	characteristic
296	C₄	D₃×C₄⋊C₄	1	1	yes	characteristic
297	C₄	D₃×C₄⋊C₄	1	1	yes	characteristic
298	C₃	C₂×#344	1	1	yes	characteristic, complemented, Sylow
299	C₂	—	2	1	no	—
300	C₂	—	2	1	no	—
301	C₂	—	2	1	no	—
302	C₂	—	3	2	no	—
303	C₂	—	6	1	no	—
304	C₂	—	6	1	no	—
305	C₂	—	6	1	no	—
306	C₂	D₃×#95	1	1	yes	characteristic

	subgroup	G/H
center	C₄	D₃×C₂²⋊C₄
commutator	C₂²×C₃	C₂×C₄²
Frattini	C₂²×C₄	C₂²×D₃
Fitting	C₃×#344	C₂
socle	C₂×C₃	C₂×#95
prime radical	C₂³×D₃	C₂×C₄
cosocle	C₂²×C₃×C₄	C₂³
hypercenter	#344	D₃
hypocenter	C₃	C₂×#344
LCS	G ▷ C₂²×C₃ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #344

Complements

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

N	H ≅ G/N	type
4. C₃×#344	C₂	semidirect
5. #4559	C₂	semidirect
6. #4556	C₂	semidirect
7. #4533	C₂	semidirect
22. D₃×M₁₆	C₄	semidirect
23. D₃×M₁₆	C₄	semidirect
25. C₂×ΓL(1,𝔽₂₅)	C₄	semidirect
27. C₂×D₃×C₈	C₄	semidirect
40. #344	D₃	direct (N × H)
60. C₃×M₁₆	C₂×C₄	semidirect
62. C₂×C₃×C₈	C₂×C₄	semidirect
75. #208	C₂×C₄	semidirect
77. C₂×C₃⋊C₈	C₂×C₄	semidirect
181. M₁₆	C₄×D₃	semidirect
189. C₂×C₈	C₄×D₃	semidirect
273. D₃	#344	direct (N × H)
298. C₃	C₂×#344	semidirect

Maximal subgroups

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

subgroup	index
C₂×D₃×M₁₆	2
D₃×#117	2
C₃×#344	2
#4559	2
#4556	2
#4533	2
C₂×#344	3

Subgroups — D3×#344 #69025

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₃×#344 #69025