Subgroups — C₂³×D₃×SD₁₆ #11708963

Order 768.

13310 conjugacy classes of subgroups.

The subgroup lattice has 304 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×D₃×Q₈	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂³×D₃×D₄	C₂	1	1	yes	maximal, characteristic
3	C₂³×C₃×SD₁₆	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂³×#215	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂³×#214	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×D₃×SD₁₆	C₂	1	56	yes	maximal, direct product factor
7	C₂³×#204	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂³×D₃×C₈	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂⁴×SD₁₆	—	3	1	no	maximal, Sylow
10	C₂⁵×D₃	—	2	1	no	—
11	C₂³×C₃×Q₈	C₂²	1	1	yes	characteristic, complemented
12	C₂³×C₃×D₄	C₂²	1	1	yes	characteristic
13	C₂³×#67	—	2	1	no	—
14	C₂²×D₃×Q₈	—	2	14	no	—
15	C₂²×D₃×Q₈	C₂²	1	28	yes	complemented
16	C₂²×D₃×D₄	—	2	14	no	—
17	C₂²×D₃×D₄	C₂²	1	28	yes	—
18	C₂³×D₁₂	C₂²	1	1	yes	characteristic
19	C₂³×C₄×D₃	—	2	1	no	—
20	C₂³×C₄×D₃	C₂²	1	1	yes	characteristic
21	C₂³×Dic₆	—	2	1	no	—
22	C₂³×Dic₆	C₂²	1	1	yes	characteristic, complemented
23	C₂²×C₃×SD₁₆	C₂²	1	28	yes	complemented
24	C₂³×C₃×C₈	C₂²	1	1	yes	characteristic, complemented
25	C₂²×#215	C₂²	1	28	yes	complemented
26	C₂²×#214	C₂²	1	28	yes	complemented
27	C₂³×C₃⋊C₈	C₂²	1	1	yes	characteristic, complemented
28	C₂×D₃×SD₁₆	C₂²	1	448	yes	direct product factor
29	C₂²×#204	C₂²	1	28	yes	complemented
30	C₂²×D₃×C₈	C₂²	1	28	yes	complemented
31	C₂⁴×Q₈	—	3	1	no	—
32	C₂⁴×D₄	—	3	1	no	—
33	C₂³×SD₁₆	—	3	1	no	—
34	C₂³×SD₁₆	—	3	1	no	—
35	C₂³×SD₁₆	—	3	56	no	—
36	C₂³×SD₁₆	—	3	1	no	—
37	C₂³×SD₁₆	D₃	1	1	yes	characteristic, direct product factor
38	C₂⁴×C₈	—	3	1	no	—
39	C₂⁵×C₃	—	2	1	no	—
40	C₂⁴×D₃	—	2	1	no	—
41	C₂⁴×D₃	—	2	28	no	—
42	C₂⁴×D₃	—	4	16	no	—
43	C₂⁴×D₃	D₄	1	1	yes	characteristic
44	C₂²×C₃×Q₈	—	2	7	no	—
45	C₂²×C₃×Q₈	C₂³	1	14	yes	complemented
46	C₂²×C₃×D₄	—	2	7	no	—
47	C₂²×C₃×D₄	C₂³	1	14	yes	—
48	C₂³×C₃×C₄	—	2	1	no	—
49	C₂³×C₃×C₄	C₂³	1	1	yes	characteristic
50	C₂²×#67	—	2	28	no	—
51	C₂³×Dic₃	—	2	1	no	—
52	C₂³×Dic₃	D₄	1	1	yes	characteristic
53	C₂×D₃×Q₈	—	2	168	no	—
54	C₂×D₃×Q₈	C₂³	1	112	yes	complemented
55	C₂×D₃×D₄	—	2	168	no	—
56	C₂×D₃×D₄	C₂³	1	112	yes	—
57	C₂²×D₁₂	—	2	7	no	—
58	C₂²×D₁₂	C₂³	1	14	yes	—
59	C₂²×C₄×D₃	—	2	28	no	—
60	C₂²×C₄×D₃	D₄	1	14	yes	—
61	C₂²×C₄×D₃	C₂³	1	14	yes	—
62	C₂²×Dic₆	—	2	7	no	—
63	C₂²×Dic₆	—	2	28	no	—
64	C₂²×Dic₆	C₂³	1	14	yes	complemented
65	C₂×C₃×SD₁₆	C₂³	1	112	yes	complemented
66	C₂²×C₃×C₈	C₂³	1	14	yes	complemented
67	C₂×#215	C₂³	1	112	yes	complemented
68	C₂×#214	C₂³	1	112	yes	complemented
69	C₂²×C₃⋊C₈	C₂³	1	14	yes	complemented
70	D₃×SD₁₆	C₂³	1	512	yes	direct product factor
71	C₂×#204	C₂³	1	112	yes	complemented
72	C₂×D₃×C₈	C₂³	1	112	yes	complemented
73	C₂⁶	—	6	1	no	—
74	C₂³×Q₈	—	3	28	no	—
75	C₂³×Q₈	—	3	1	no	—
76	C₂³×Q₈	—	6	1	no	—
77	C₂³×Q₈	—	6	14	no	—
78	C₂³×Q₈	C₂×D₃	1	1	yes	characteristic, complemented
79	C₂³×D₄	—	3	28	no	—
80	C₂³×D₄	—	3	1	no	—
81	C₂³×D₄	—	6	14	no	—
82	C₂³×D₄	—	6	1	no	—
83	C₂³×D₄	C₂×D₃	1	1	yes	characteristic
84	C₂⁴×C₄	—	3	1	no	—
85	C₂⁴×C₄	—	6	1	no	—
86	C₂²×SD₁₆	—	3	28	no	—
87	C₂²×SD₁₆	—	3	28	no	—
88	C₂²×SD₁₆	—	3	448	no	—
89	C₂²×SD₁₆	—	3	28	no	—
90	C₂²×SD₁₆	C₂×D₃	1	28	yes	direct product factor
91	C₂³×C₈	—	3	1	no	—
92	C₂³×C₈	—	3	28	no	—
93	C₂³×C₈	C₂×D₃	1	1	yes	characteristic, complemented
94	C₂⁴×C₃	—	2	14	no	—
95	C₂⁴×C₃	—	4	8	no	—
96	C₂⁴×C₃	C₂×D₄	1	1	yes	characteristic
97	C₂³×D₃	—	2	14	no	—
98	C₂³×D₃	—	2	112	no	—
99	C₂³×D₃	—	4	8	no	—
100	C₂³×D₃	—	4	224	no	—
101	C₂³×D₃	SD₁₆	1	16	yes	direct product factor
102	C₂³×D₃	C₂×D₄	1	14	yes	—
103	C₂×C₃×Q₈	—	2	42	no	—
104	C₂×C₃×Q₈	C₂⁴	1	28	yes	complemented
105	C₂×C₃×D₄	—	2	42	no	—
106	C₂×C₃×D₄	C₂⁴	1	28	yes	—
107	C₂²×C₃×C₄	—	2	14	no	—
108	C₂²×C₃×C₄	C₂×D₄	1	7	yes	—
109	C₂²×C₃×C₄	C₂⁴	1	7	yes	—
110	C₂×#67	—	2	112	no	—
111	C₂²×Dic₃	—	2	14	no	—
112	C₂²×Dic₃	C₂×D₄	1	14	yes	—
113	D₃×Q₈	—	2	224	no	—
114	D₃×Q₈	C₂⁴	1	64	yes	complemented
115	D₃×D₄	—	2	224	no	—
116	D₃×D₄	C₂⁴	1	64	yes	—
117	C₂×D₁₂	—	2	42	no	—
118	C₂×D₁₂	C₂⁴	1	28	yes	—
119	C₂×C₄×D₃	—	2	112	no	—
120	C₂×C₄×D₃	C₂×D₄	1	84	yes	—
121	C₂×C₄×D₃	C₂⁴	1	28	yes	—
122	C₂×Dic₆	—	2	42	no	—
123	C₂×Dic₆	—	2	112	no	—
124	C₂×Dic₆	C₂⁴	1	28	yes	complemented
125	C₃×SD₁₆	C₂⁴	1	64	yes	complemented
126	C₂×C₃×C₈	C₂⁴	1	28	yes	complemented
127	#215	C₂⁴	1	64	yes	complemented
128	#214	C₂⁴	1	64	yes	complemented
129	C₂×C₃⋊C₈	C₂⁴	1	28	yes	complemented
130	#204	C₂⁴	1	64	yes	complemented
131	D₃×C₈	C₂⁴	1	64	yes	complemented
132	C₂⁵	—	2	1	no	—
133	C₂⁵	—	3	1	no	—
134	C₂⁵	—	6	28	no	—
135	C₂⁵	—	6	1	no	—
136	C₂⁵	—	12	16	no	—
137	C₂²×Q₈	—	2	7	no	—
138	C₂²×Q₈	—	3	112	no	—
139	C₂²×Q₈	—	3	14	no	—
140	C₂²×Q₈	—	6	28	no	—
141	C₂²×Q₈	—	6	168	no	—
142	C₂²×Q₈	—	6	7	no	—
143	C₂²×Q₈	C₂²×D₃	1	14	yes	complemented
144	C₂²×D₄	—	2	7	no	—
145	C₂²×D₄	—	3	112	no	—
146	C₂²×D₄	—	3	14	no	—
147	C₂²×D₄	—	6	168	no	—
148	C₂²×D₄	—	6	28	no	—
149	C₂²×D₄	—	6	7	no	—
150	C₂²×D₄	C₂²×D₃	1	14	yes	—
151	C₂³×C₄	—	2	1	no	—
152	C₂³×C₄	—	3	1	no	—
153	C₂³×C₄	—	3	14	no	—
154	C₂³×C₄	—	3	14	no	—
155	C₂³×C₄	—	6	28	no	—
156	C₂³×C₄	—	6	1	no	—
157	C₂³×C₄	C₂²×D₃	1	1	yes	characteristic
158	C₂×SD₁₆	—	3	112	no	—
159	C₂×SD₁₆	—	3	112	no	—
160	C₂×SD₁₆	—	3	512	no	—
161	C₂×SD₁₆	—	3	112	no	—
162	C₂×SD₁₆	C₂²×D₃	1	112	yes	direct product factor
163	C₂²×C₈	—	3	14	no	—
164	C₂²×C₈	—	3	112	no	—
165	C₂²×C₈	C₂²×D₃	1	14	yes	complemented
166	C₂³×C₃	—	2	28	no	—
167	C₂³×C₃	—	4	56	no	—
168	C₂³×C₃	C₂×SD₁₆	1	8	yes	complemented
169	C₂³×C₃	C₂²×D₄	1	7	yes	—
170	C₂²×D₃	—	2	28	no	—
171	C₂²×D₃	—	2	64	no	—
172	C₂²×D₃	—	4	56	no	—
173	C₂²×D₃	—	4	448	no	—
174	C₂²×D₃	C₂×SD₁₆	1	112	yes	direct product factor
175	C₂²×D₃	C₂²×D₄	1	28	yes	—
176	C₃×Q₈	—	2	28	no	—
177	C₃×Q₈	C₂⁵	1	8	yes	complemented
178	C₃×D₄	—	2	28	no	—
179	C₃×D₄	C₂⁵	1	8	yes	—
180	C₂×C₃×C₄	—	2	28	no	—
181	C₂×C₃×C₄	C₂²×D₄	1	21	yes	—
182	C₂×C₃×C₄	C₂⁵	1	7	yes	—
183	#67	—	2	64	no	—
184	C₂×Dic₃	—	2	28	no	—
185	C₂×Dic₃	C₂²×D₄	1	28	yes	—
186	D₁₂	—	2	28	no	—
187	D₁₂	C₂⁵	1	8	yes	—
188	C₄×D₃	—	2	64	no	—
189	C₄×D₃	C₂²×D₄	1	56	yes	—
190	C₄×D₃	C₂⁵	1	8	yes	—
191	Dic₆	—	2	28	no	—
192	Dic₆	—	2	64	no	—
193	Dic₆	C₂⁵	1	8	yes	complemented
194	C₃×C₈	C₂⁵	1	8	yes	complemented
195	C₃⋊C₈	C₂⁵	1	8	yes	complemented
196	C₂⁴	—	2	14	no	—
197	C₂⁴	—	3	16	no	—
198	C₂⁴	—	3	14	no	—
199	C₂⁴	—	4	8	no	—
200	C₂⁴	—	6	112	no	—
201	C₂⁴	—	6	14	no	—
202	C₂⁴	—	12	224	no	—
203	C₂⁴	—	12	8	no	—
204	C₂⁴	D₃×D₄	1	1	yes	characteristic
205	C₂×Q₈	—	2	42	no	—
206	C₂×Q₈	—	3	64	no	—
207	C₂×Q₈	—	3	28	no	—
208	C₂×Q₈	—	6	112	no	—
209	C₂×Q₈	—	6	224	no	—
210	C₂×Q₈	—	6	42	no	—
211	C₂×Q₈	C₂³×D₃	1	28	yes	complemented
212	C₂×D₄	—	2	42	no	—
213	C₂×D₄	—	3	64	no	—
214	C₂×D₄	—	3	28	no	—
215	C₂×D₄	—	6	224	no	—
216	C₂×D₄	—	6	112	no	—
217	C₂×D₄	—	6	42	no	—
218	C₂×D₄	C₂³×D₃	1	28	yes	—
219	C₂²×C₄	—	2	14	no	—
220	C₂²×C₄	—	3	14	no	—
221	C₂²×C₄	—	3	84	no	—
222	C₂²×C₄	—	3	28	no	—
223	C₂²×C₄	—	6	112	no	—
224	C₂²×C₄	—	6	14	no	—
225	C₂²×C₄	D₃×D₄	1	7	yes	—
226	C₂²×C₄	C₂³×D₃	1	7	yes	—
227	SD₁₆	—	3	64	no	—
228	SD₁₆	—	3	64	no	—
229	SD₁₆	—	3	64	no	—
230	SD₁₆	C₂³×D₃	1	64	yes	direct product factor
231	C₂×C₈	—	3	28	no	—
232	C₂×C₈	—	3	64	no	—
233	C₂×C₈	C₂³×D₃	1	28	yes	complemented
234	C₂²×C₃	—	2	8	no	—
235	C₂²×C₃	—	4	56	no	—
236	C₂²×C₃	C₂²×SD₁₆	1	28	yes	complemented
237	C₂²×C₃	C₂³×D₄	1	7	yes	—
238	C₂×D₃	—	2	8	no	—
239	C₂×D₃	—	4	56	no	—
240	C₂×D₃	—	4	128	no	—
241	C₂×D₃	C₂²×SD₁₆	1	112	yes	direct product factor
242	C₂×D₃	C₂³×D₄	1	8	yes	—
243	C₃×C₄	—	2	8	no	—
244	C₃×C₄	C₂³×D₄	1	7	yes	—
245	C₃×C₄	C₂⁶	1	1	yes	characteristic
246	Dic₃	—	2	8	no	—
247	Dic₃	C₂³×D₄	1	8	yes	—
248	C₂³	—	2	28	no	—
249	C₂³	—	3	28	no	—
250	C₂³	—	3	112	no	—
251	C₂³	—	4	56	no	—
252	C₂³	—	6	64	no	—
253	C₂³	—	6	28	no	—
254	C₂³	—	12	448	no	—
255	C₂³	—	12	56	no	—
256	C₂³	D₃×SD₁₆	1	8	yes	direct product factor
257	C₂³	C₂×D₃×D₄	1	7	yes	—
258	Q₈	—	2	28	no	—
259	Q₈	—	3	8	no	—
260	Q₈	—	6	64	no	—
261	Q₈	—	6	28	no	—
262	Q₈	C₂⁴×D₃	1	8	yes	complemented
263	D₄	—	2	28	no	—
264	D₄	—	3	8	no	—
265	D₄	—	6	64	no	—
266	D₄	—	6	28	no	—
267	D₄	C₂⁴×D₃	1	8	yes	—
268	C₂×C₄	—	2	28	no	—
269	C₂×C₄	—	3	28	no	—
270	C₂×C₄	—	3	56	no	—
271	C₂×C₄	—	3	8	no	—
272	C₂×C₄	—	6	64	no	—
273	C₂×C₄	—	6	28	no	—
274	C₂×C₄	C₂×D₃×D₄	1	21	yes	—
275	C₂×C₄	C₂⁴×D₃	1	7	yes	—
276	C₈	—	3	8	no	—
277	C₈	C₂⁴×D₃	1	8	yes	complemented
278	C₂×C₃	—	4	8	no	—
279	C₂×C₃	C₂³×SD₁₆	1	14	yes	complemented
280	C₂×C₃	C₂⁴×D₄	1	1	yes	characteristic
281	D₃	—	4	8	no	—
282	D₃	C₂³×SD₁₆	1	16	yes	direct product factor
283	C₂²	—	2	8	no	—
284	C₂²	—	3	8	no	—
285	C₂²	—	3	112	no	—
286	C₂²	—	4	56	no	—
287	C₂²	—	6	8	no	—
288	C₂²	—	12	128	no	—
289	C₂²	—	12	56	no	—
290	C₂²	C₂×D₃×SD₁₆	1	28	yes	direct product factor
291	C₂²	C₂²×D₃×D₄	1	7	yes	—
292	C₄	—	2	8	no	—
293	C₄	—	3	8	no	—
294	C₄	—	6	8	no	—
295	C₄	C₂²×D₃×D₄	1	7	yes	—
296	C₄	C₂⁵×D₃	1	1	yes	characteristic
297	C₃	C₂⁴×SD₁₆	1	1	yes	characteristic, complemented, Sylow
298	C₂	—	3	16	no	—
299	C₂	—	4	8	no	—
300	C₂	—	12	8	no	—
301	C₂	C₂²×D₃×SD₁₆	1	14	yes	direct product factor
302	C₂	C₂³×D₃×D₄	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁴	D₃×D₄
commutator, cosocle	C₃×C₄	C₂⁶
Frattini	C₄	C₂⁵×D₃
Fitting	C₂³×C₃×SD₁₆	C₂
socle	C₂⁴×C₃	C₂×D₄
prime radical	C₂³×D₃×D₄	C₂
hypercenter	C₂³×SD₁₆	D₃
hypocenter	C₃	C₂⁴×SD₁₆
LCS	G ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂⁴ ◁ C₂³×C₄ ◁ C₂³×SD₁₆

Complements

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

N	H ≅ G/N	type
1. C₂³×D₃×Q₈	C₂	semidirect
3. C₂³×C₃×SD₁₆	C₂	semidirect
4. C₂³×#215	C₂	semidirect
5. C₂³×#214	C₂	semidirect
6. C₂²×D₃×SD₁₆	C₂	direct (N × H)
7. C₂³×#204	C₂	semidirect
8. C₂³×D₃×C₈	C₂	semidirect
11. C₂³×C₃×Q₈	C₂²	semidirect
15. C₂²×D₃×Q₈	C₂²	semidirect
22. C₂³×Dic₆	C₂²	semidirect
23. C₂²×C₃×SD₁₆	C₂²	semidirect
24. C₂³×C₃×C₈	C₂²	semidirect
25. C₂²×#215	C₂²	semidirect
26. C₂²×#214	C₂²	semidirect
27. C₂³×C₃⋊C₈	C₂²	semidirect
28. C₂×D₃×SD₁₆	C₂²	direct (N × H)
29. C₂²×#204	C₂²	semidirect
30. C₂²×D₃×C₈	C₂²	semidirect
37. C₂³×SD₁₆	D₃	direct (N × H)
45. C₂²×C₃×Q₈	C₂³	semidirect
54. C₂×D₃×Q₈	C₂³	semidirect
64. C₂²×Dic₆	C₂³	semidirect
65. C₂×C₃×SD₁₆	C₂³	semidirect
66. C₂²×C₃×C₈	C₂³	semidirect
67. C₂×#215	C₂³	semidirect
68. C₂×#214	C₂³	semidirect
69. C₂²×C₃⋊C₈	C₂³	semidirect
70. D₃×SD₁₆	C₂³	direct (N × H)
71. C₂×#204	C₂³	semidirect
72. C₂×D₃×C₈	C₂³	semidirect
78. C₂³×Q₈	C₂×D₃	semidirect
90. C₂²×SD₁₆	C₂×D₃	direct (N × H)
93. C₂³×C₈	C₂×D₃	semidirect
101. C₂³×D₃	SD₁₆	direct (N × H)
104. C₂×C₃×Q₈	C₂⁴	semidirect
114. D₃×Q₈	C₂⁴	semidirect
124. C₂×Dic₆	C₂⁴	semidirect
125. C₃×SD₁₆	C₂⁴	semidirect
126. C₂×C₃×C₈	C₂⁴	semidirect
127. #215	C₂⁴	semidirect
128. #214	C₂⁴	semidirect
129. C₂×C₃⋊C₈	C₂⁴	semidirect
130. #204	C₂⁴	semidirect
131. D₃×C₈	C₂⁴	semidirect
143. C₂²×Q₈	C₂²×D₃	semidirect
162. C₂×SD₁₆	C₂²×D₃	direct (N × H)
165. C₂²×C₈	C₂²×D₃	semidirect
168. C₂³×C₃	C₂×SD₁₆	semidirect
174. C₂²×D₃	C₂×SD₁₆	direct (N × H)
177. C₃×Q₈	C₂⁵	semidirect
193. Dic₆	C₂⁵	semidirect
194. C₃×C₈	C₂⁵	semidirect
195. C₃⋊C₈	C₂⁵	semidirect
211. C₂×Q₈	C₂³×D₃	semidirect
230. SD₁₆	C₂³×D₃	direct (N × H)
233. C₂×C₈	C₂³×D₃	semidirect
236. C₂²×C₃	C₂²×SD₁₆	semidirect
241. C₂×D₃	C₂²×SD₁₆	direct (N × H)
256. C₂³	D₃×SD₁₆	direct (N × H)
262. Q₈	C₂⁴×D₃	semidirect
277. C₈	C₂⁴×D₃	semidirect
279. C₂×C₃	C₂³×SD₁₆	semidirect
282. D₃	C₂³×SD₁₆	direct (N × H)
290. C₂²	C₂×D₃×SD₁₆	direct (N × H)
297. C₃	C₂⁴×SD₁₆	semidirect
301. C₂	C₂²×D₃×SD₁₆	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×D₃×Q₈	2
C₂³×D₃×D₄	2
C₂³×C₃×SD₁₆	2
C₂³×#215	2
C₂³×#214	2
C₂²×D₃×SD₁₆	2
C₂³×#204	2
C₂³×D₃×C₈	2
C₂⁴×SD₁₆	3

Subgroups — C23×D3×SD16 #11708963

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×D₃×SD₁₆ #11708963