Subgroups — SD₁₆×3₋¹⁺²⋊C₄ #13143379

Order 1728.

400 conjugacy classes of subgroups.

The subgroup lattice has 325 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×SD₁₆×3₋¹⁺²	C₂	1	1	yes	maximal, characteristic
2	Q₈×3₋¹⁺²⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
3	D₄×3₋¹⁺²⋊C₄	C₂	1	1	yes	maximal, characteristic
4	#63395⋊C₃	C₂	1	1	yes	maximal, characteristic, complemented
5	#63392⋊C₃	C₂	1	1	yes	maximal, characteristic
6	#63377⋊C₃	C₂	1	1	yes	maximal, characteristic, complemented
7	C₈×3₋¹⁺²⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃×Dic₃×SD₁₆	—	3	1	no	maximal
9	SD₁₆×Dic₉	C₃	1	1	yes	maximal, characteristic, complemented
10	C₂×Q₈×3₋¹⁺²	C₂²	1	1	yes	characteristic
11	C₂×D₄×3₋¹⁺²	C₂²	1	1	yes	characteristic
12	C₂²×3₋¹⁺²⋊C₄	—	2	1	no	—
13	SD₁₆×3₋¹⁺²	C₄	1	4	yes	complemented
14	C₂×C₈×3₋¹⁺²	C₂²	1	1	yes	characteristic
15	#3676⋊C₃	—	2	1	no	—
16	C₄⋊(3₋¹⁺²⋊C₄)	—	2	1	no	—
17	C₄⋊(3₋¹⁺²⋊C₄)	C₂²	1	1	yes	characteristic
18	C₄⋊(3₋¹⁺²⋊C₄)	C₂²	1	1	yes	characteristic
19	C₄×3₋¹⁺²⋊C₄	—	2	1	no	—
20	C₄×3₋¹⁺²⋊C₄	C₂²	1	1	yes	characteristic
21	C₂×3₋¹⁺²⋊C₈	C₂²	1	1	yes	characteristic
22	C₂×C₃²×SD₁₆	D₃	1	1	yes	characteristic
23	C₃×Q₈×Dic₃	—	3	1	no	—
24	C₃×D₄×Dic₃	—	3	1	no	—
25	C₃×#835	—	3	1	no	—
26	C₃×#832	—	3	1	no	—
27	C₃×#817	—	3	1	no	—
28	C₃×C₈×Dic₃	—	3	1	no	—
29	C₂×C₉×SD₁₆	—	2	1	no	—
30	C₂×C₉×SD₁₆	C₂×C₃	1	1	yes	characteristic
31	Q₈×Dic₉	C₂×C₃	1	1	yes	characteristic, complemented
32	D₄×Dic₉	C₂×C₃	1	1	yes	characteristic
33	#63395	C₂×C₃	1	1	yes	characteristic, complemented
34	#63392	C₂×C₃	1	1	yes	characteristic
35	#63377	C₂×C₃	1	1	yes	characteristic, complemented
36	C₈×Dic₉	C₂×C₃	1	1	yes	characteristic, complemented
37	C₂³×3₋¹⁺²	—	2	1	no	—
38	Q₈×3₋¹⁺²	—	2	1	no	—
39	Q₈×3₋¹⁺²	C₂×C₄	1	2	yes	complemented
40	D₄×3₋¹⁺²	—	2	1	no	—
41	D₄×3₋¹⁺²	C₂×C₄	1	2	yes	—
42	C₂×C₄×3₋¹⁺²	—	2	1	no	—
43	C₂×C₄×3₋¹⁺²	C₂³	1	1	yes	characteristic
44	C₂×3₋¹⁺²⋊C₄	—	2	1	no	—
45	C₂×3₋¹⁺²⋊C₄	—	2	1	no	—
46	C₂×3₋¹⁺²⋊C₄	—	4	2	no	—
47	C₂×3₋¹⁺²⋊C₄	D₄	1	1	yes	characteristic
48	C₂×3₋¹⁺²⋊C₄	D₄	1	1	yes	characteristic
49	C₈×3₋¹⁺²	C₂×C₄	1	2	yes	complemented
50	3₋¹⁺²⋊C₈	—	2	1	no	—
51	C₃×C₄×SD₁₆	—	9	1	no	—
52	Dic₃×SD₁₆	—	3	1	no	—
53	C₂×C₃²×Q₈	C₂×D₃	1	1	yes	characteristic
54	C₂×C₃²×D₄	C₂×D₃	1	1	yes	characteristic
55	C₂²×C₃×Dic₃	—	6	1	no	—
56	C₃²×SD₁₆	Dic₃	1	4	yes	—
57	C₂×C₃²×C₈	C₂×D₃	1	1	yes	characteristic
58	C₃×#217	—	6	1	no	—
59	C₃×C₄⋊Dic₃	—	3	1	no	—
60	C₃×C₄⋊Dic₃	—	3	1	no	—
61	C₃×C₄⋊Dic₃	—	6	1	no	—
62	C₃×C₄×Dic₃	—	3	1	no	—
63	C₃×C₄×Dic₃	—	6	1	no	—
64	C₂×C₃×C₃⋊C₈	—	3	1	no	—
65	C₂×Q₈×C₉	—	2	1	no	—
66	C₂×Q₈×C₉	C₂²×C₃	1	1	yes	characteristic
67	C₂×D₄×C₉	—	2	1	no	—
68	C₂×D₄×C₉	C₂²×C₃	1	1	yes	characteristic
69	C₂²×Dic₉	—	2	1	no	—
70	C₉×SD₁₆	—	2	4	no	—
71	C₉×SD₁₆	C₃×C₄	1	4	yes	complemented
72	C₂×C₈×C₉	—	2	1	no	—
73	C₂×C₈×C₉	C₂²×C₃	1	1	yes	characteristic
74	#3676	—	2	1	no	—
75	C₄⋊Dic₉	—	2	1	no	—
76	C₄⋊Dic₉	C₂²×C₃	1	1	yes	characteristic
77	C₄⋊Dic₉	C₂²×C₃	1	1	yes	characteristic
78	C₄×Dic₉	—	2	1	no	—
79	C₄×Dic₉	C₂²×C₃	1	1	yes	characteristic
80	C₂×C₉⋊C₈	C₂²×C₃	1	1	yes	characteristic
81	C₂²×3₋¹⁺²	—	2	2	no	—
82	C₂²×3₋¹⁺²	—	4	1	no	—
83	C₂²×3₋¹⁺²	—	4	1	no	—
84	C₂²×3₋¹⁺²	C₂×D₄	1	1	yes	characteristic
85	C₄×3₋¹⁺²	—	2	2	no	—
86	C₄×3₋¹⁺²	C₂²×C₄	1	1	yes	characteristic
87	C₄×3₋¹⁺²	#41	1	1	yes	characteristic
88	3₋¹⁺²⋊C₄	—	2	1	no	—
89	3₋¹⁺²⋊C₄	—	4	1	no	—
90	3₋¹⁺²⋊C₄	—	4	1	no	—
91	3₋¹⁺²⋊C₄	SD₁₆	1	2	yes	direct product factor
92	C₂×C₃×SD₁₆	—	3	1	no	—
93	C₂×C₃×SD₁₆	C₃×D₃	1	1	yes	characteristic
94	C₃×C₄×Q₈	—	9	1	no	—
95	C₃×C₄×D₄	—	9	1	no	—
96	Q₈×Dic₃	—	3	1	no	—
97	D₄×Dic₃	—	3	1	no	—
98	C₃×#106	—	9	1	no	—
99	C₃×#103	—	9	1	no	—
100	C₃×#102	—	9	1	no	—
101	C₃×C₄×C₈	—	9	1	no	—
102	#835	—	3	1	no	—
103	#832	—	3	1	no	—
104	#817	—	3	1	no	—
105	C₈×Dic₃	—	3	1	no	—
106	C₂³×C₃²	—	2	1	no	—
107	C₃²×Q₈	—	2	1	no	—
108	C₃²×Q₈	C₂×Dic₃	1	2	yes	—
109	C₃²×D₄	—	2	1	no	—
110	C₃²×D₄	C₂×Dic₃	1	2	yes	—
111	C₂×C₃²×C₄	—	2	1	no	—
112	C₂×C₃²×C₄	C₂²×D₃	1	1	yes	characteristic
113	C₂×C₃×Dic₃	—	3	1	no	—
114	C₂×C₃×Dic₃	—	3	1	no	—
115	C₂×C₃×Dic₃	—	6	1	no	—
116	C₂×C₃×Dic₃	—	6	1	no	—
117	C₂×C₃×Dic₃	—	12	2	no	—
118	C₂³×C₉	—	2	1	no	—
119	C₂³×C₉	—	4	1	no	—
120	C₃²×C₈	C₂×Dic₃	1	2	yes	—
121	C₃×C₃⋊C₈	—	6	1	no	—
122	Q₈×C₉	—	2	1	no	—
123	Q₈×C₉	—	2	2	no	—
124	Q₈×C₉	—	4	1	no	—
125	Q₈×C₉	C₂×C₃×C₄	1	2	yes	complemented
126	D₄×C₉	—	2	1	no	—
127	D₄×C₉	—	2	2	no	—
128	D₄×C₉	—	4	1	no	—
129	D₄×C₉	C₂×C₃×C₄	1	2	yes	—
130	C₂×C₄×C₉	—	2	1	no	—
131	C₂×C₄×C₉	—	2	1	no	—
132	C₂×C₄×C₉	—	4	1	no	—
133	C₂×C₄×C₉	C₂³×C₃	1	1	yes	characteristic
134	C₂×Dic₉	—	2	1	no	—
135	C₂×Dic₉	—	2	1	no	—
136	C₂×Dic₉	—	4	2	no	—
137	C₂×Dic₉	C₃×D₄	1	1	yes	characteristic
138	C₂×Dic₉	C₃×D₄	1	1	yes	characteristic
139	C₈×C₉	—	2	2	no	—
140	C₈×C₉	C₂×C₃×C₄	1	2	yes	complemented
141	C₉⋊C₈	—	2	1	no	—
142	C₄×SD₁₆	—	9	1	no	Sylow
143	C₂×3₋¹⁺²	—	4	2	no	—
144	C₂×3₋¹⁺²	C₄×D₄	1	1	yes	characteristic
145	C₂×3₋¹⁺²	C₂×SD₁₆	1	1	yes	characteristic
146	C₂×3₋¹⁺²	#135	1	1	yes	characteristic
147	C₂×C₃×Q₈	—	3	1	no	—
148	C₂×C₃×Q₈	C₂×C₃×D₃	1	1	yes	characteristic
149	C₂×C₃×D₄	—	3	1	no	—
150	C₂×C₃×D₄	C₂×C₃×D₃	1	1	yes	characteristic
151	C₂²×C₃×C₄	—	18	1	no	—
152	C₂²×Dic₃	—	6	1	no	—
153	C₃×SD₁₆	—	3	4	no	—
154	C₃×SD₁₆	C₃×Dic₃	1	4	yes	—
155	C₂×C₃×C₈	—	3	1	no	—
156	C₂×C₃×C₈	—	9	1	no	—
157	C₂×C₃×C₈	C₂×C₃×D₃	1	1	yes	characteristic
158	C₃×C₄⋊C₄	—	9	1	no	—
159	C₃×C₄⋊C₄	—	9	1	no	—
160	C₃×C₄⋊C₄	—	18	1	no	—
161	C₃×C₂²⋊C₄	—	18	1	no	—
162	C₃×C₄²	—	9	1	no	—
163	C₃×C₄²	—	18	1	no	—
164	#217	—	6	1	no	—
165	C₄⋊Dic₃	—	3	1	no	—
166	C₄⋊Dic₃	—	3	1	no	—
167	C₄⋊Dic₃	—	6	1	no	—
168	C₄×Dic₃	—	3	1	no	—
169	C₄×Dic₃	—	6	1	no	—
170	C₂×C₃⋊C₈	—	3	1	no	—
171	C₂²×C₃²	—	2	2	no	—
172	C₂²×C₃²	—	4	1	no	—
173	C₂²×C₃²	—	4	1	no	—
174	C₂²×C₃²	D₃×D₄	1	1	yes	characteristic
175	C₃²×C₄	—	2	2	no	—
176	C₃²×C₄	#237	1	1	yes	characteristic
177	C₃²×C₄	C₂²×Dic₃	1	1	yes	characteristic
178	C₃×Dic₃	—	3	2	no	—
179	C₃×Dic₃	—	6	1	no	—
180	C₃×Dic₃	—	12	1	no	—
181	C₃×Dic₃	—	12	1	no	—
182	C₂²×C₉	—	2	2	no	—
183	C₂²×C₉	—	2	1	no	—
184	C₂²×C₉	—	4	2	no	—
185	C₂²×C₉	—	4	1	no	—
186	C₂²×C₉	—	4	1	no	—
187	C₂²×C₉	—	8	1	no	—
188	C₂²×C₉	—	8	1	no	—
189	C₂²×C₉	C₂×C₃×D₄	1	1	yes	characteristic
190	C₄×C₉	—	2	2	no	—
191	C₄×C₉	—	2	1	no	—
192	C₄×C₉	—	2	1	no	—
193	C₄×C₉	—	4	2	no	—
194	C₄×C₉	C₂²×C₃×C₄	1	1	yes	characteristic
195	C₄×C₉	C₃×#41	1	1	yes	characteristic
196	Dic₉	—	2	1	no	—
197	Dic₉	—	4	1	no	—
198	Dic₉	—	4	1	no	—
199	Dic₉	C₃×SD₁₆	1	2	yes	complemented
200	C₂×SD₁₆	AGL(1,ℤ/9)	1	1	yes	characteristic
201	C₄×Q₈	—	9	1	no	—
202	C₄×D₄	—	9	1	no	—
203	#106	—	9	1	no	—
204	#103	—	9	1	no	—
205	#102	—	9	1	no	—
206	C₄×C₈	—	9	1	no	—
207	3₋¹⁺²	C₄×SD₁₆	1	1	yes	characteristic, complemented, Sylow
208	C₂³×C₃	—	2	1	no	—
209	C₂³×C₃	—	6	1	no	—
210	C₃×Q₈	—	2	1	no	—
211	C₃×Q₈	—	3	2	no	—
212	C₃×Q₈	—	6	1	no	—
213	C₃×Q₈	C₂×C₃×Dic₃	1	2	yes	—
214	C₃×D₄	—	2	1	no	—
215	C₃×D₄	—	3	2	no	—
216	C₃×D₄	—	6	1	no	—
217	C₃×D₄	C₂×C₃×Dic₃	1	2	yes	—
218	C₂×C₃×C₄	—	2	1	no	—
219	C₂×C₃×C₄	—	3	1	no	—
220	C₂×C₃×C₄	—	6	1	no	—
221	C₂×C₃×C₄	—	9	1	no	—
222	C₂×C₃×C₄	—	9	1	no	—
223	C₂×C₃×C₄	—	18	1	no	—
224	C₂×C₃×C₄	—	18	1	no	—
225	C₂×C₃×C₄	—	36	2	no	—
226	C₂×C₃×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
227	C₂×Dic₃	—	3	1	no	—
228	C₂×Dic₃	—	3	1	no	—
229	C₂×Dic₃	—	6	1	no	—
230	C₂×Dic₃	—	6	1	no	—
231	C₂×Dic₃	—	12	2	no	—
232	C₃×C₈	—	3	2	no	—
233	C₃×C₈	—	18	1	no	—
234	C₃×C₈	C₂×C₃×Dic₃	1	2	yes	—
235	C₃⋊C₈	—	6	1	no	—
236	C₂×C₃²	—	4	2	no	—
237	C₂×C₃²	D₃×SD₁₆	1	1	yes	characteristic
238	C₂×C₃²	#916	1	1	yes	characteristic
239	C₂×C₃²	D₄×Dic₃	1	1	yes	characteristic
240	C₂×C₉	—	2	1	no	—
241	C₂×C₉	—	2	1	no	—
242	C₂×C₉	—	2	1	no	—
243	C₂×C₉	—	4	2	no	—
244	C₂×C₉	—	8	2	no	—
245	C₂×C₉	C₃×C₄×D₄	1	1	yes	characteristic
246	C₂×C₉	C₂×C₃×SD₁₆	1	1	yes	characteristic
247	C₂×C₉	C₃×#135	1	1	yes	characteristic
248	C₂×Q₈	C₂×AGL(1,ℤ/9)	1	1	yes	characteristic
249	C₂×D₄	C₂×AGL(1,ℤ/9)	1	1	yes	characteristic
250	C₂²×C₄	—	18	1	no	—
251	SD₁₆	3₋¹⁺²⋊C₄	1	4	yes	direct product factor
252	C₂×C₈	—	9	1	no	—
253	C₂×C₈	C₂×AGL(1,ℤ/9)	1	1	yes	characteristic
254	C₄⋊C₄	—	9	1	no	—
255	C₄⋊C₄	—	9	1	no	—
256	C₄⋊C₄	—	18	1	no	—
257	C₂²⋊C₄	—	18	1	no	—
258	C₄²	—	9	1	no	—
259	C₄²	—	18	1	no	—
260	C₂²×C₃	—	2	2	no	—
261	C₂²×C₃	—	3	1	no	—
262	C₂²×C₃	—	4	1	no	—
263	C₂²×C₃	—	4	1	no	—
264	C₂²×C₃	—	6	2	no	—
265	C₂²×C₃	—	12	1	no	—
266	C₂²×C₃	—	12	1	no	—
267	C₂²×C₃	C₃×D₃×D₄	1	1	yes	characteristic
268	C₃×C₄	—	2	2	no	—
269	C₃×C₄	—	3	1	no	—
270	C₃×C₄	—	3	1	no	—
271	C₃×C₄	—	6	2	no	—
272	C₃×C₄	—	9	2	no	—
273	C₃×C₄	—	18	1	no	—
274	C₃×C₄	—	36	1	no	—
275	C₃×C₄	—	36	1	no	—
276	C₃×C₄	C₃×#237	1	1	yes	characteristic
277	C₃×C₄	C₂²×C₃×Dic₃	1	1	yes	characteristic
278	Dic₃	—	3	2	no	—
279	Dic₃	—	6	1	no	—
280	Dic₃	—	12	1	no	—
281	Dic₃	—	12	1	no	—
282	C₃²	Dic₃×SD₁₆	1	1	yes	characteristic
283	C₉	—	2	1	no	—
284	C₉	C₃×C₄×SD₁₆	1	1	yes	characteristic, complemented
285	C₂³	—	2	1	no	—
286	Q₈	—	2	1	no	—
287	Q₈	C₂×3₋¹⁺²⋊C₄	1	2	yes	complemented
288	D₄	—	2	1	no	—
289	D₄	C₂×3₋¹⁺²⋊C₄	1	2	yes	—
290	C₂×C₄	—	2	1	no	—
291	C₂×C₄	—	9	1	no	—
292	C₂×C₄	—	9	1	no	—
293	C₂×C₄	—	18	1	no	—
294	C₂×C₄	—	18	1	no	—
295	C₂×C₄	—	36	2	no	—
296	C₂×C₄	C₂²×AGL(1,ℤ/9)	1	1	yes	characteristic
297	C₈	—	18	1	no	—
298	C₈	C₂×3₋¹⁺²⋊C₄	1	2	yes	complemented
299	C₂×C₃	—	3	1	no	—
300	C₂×C₃	—	3	1	no	—
301	C₂×C₃	—	3	1	no	—
302	C₂×C₃	—	4	2	no	—
303	C₂×C₃	—	12	2	no	—
304	C₂×C₃	C₃×D₃×SD₁₆	1	1	yes	characteristic
305	C₂×C₃	C₃×#916	1	1	yes	characteristic
306	C₂×C₃	C₃×D₄×Dic₃	1	1	yes	characteristic
307	C₂²	—	2	2	no	—
308	C₂²	—	4	1	no	—
309	C₂²	—	4	1	no	—
310	C₂²	D₄×AGL(1,ℤ/9)	1	1	yes	characteristic
311	C₄	—	2	2	no	—
312	C₄	—	9	2	no	—
313	C₄	—	18	1	no	—
314	C₄	—	36	1	no	—
315	C₄	—	36	1	no	—
316	C₄	#3699⋊C₃	1	1	yes	characteristic
317	C₄	C₂²×3₋¹⁺²⋊C₄	1	1	yes	characteristic
318	C₃	—	3	1	no	—
319	C₃	C₃×Dic₃×SD₁₆	1	1	yes	characteristic
320	C₂	—	4	2	no	—
321	C₂	SD₁₆×AGL(1,ℤ/9)	1	1	yes	characteristic
322	C₂	#63478⋊C₃	1	1	yes	characteristic
323	C₂	D₄×3₋¹⁺²⋊C₄	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₄×AGL(1,ℤ/9)
commutator	C₄×C₉	C₂²×C₃×C₄
Frattini	C₂×C₃×C₄	C₂²×C₃×D₃
Fitting	C₂×SD₁₆×3₋¹⁺²	C₂
socle	C₂²×C₃	C₃×D₃×D₄
prime radical	C₂×C₃²×D₄	C₂×D₃
cosocle	C₂×C₄×C₉	C₂³×C₃
hypercenter	C₂×SD₁₆	AGL(1,ℤ/9)
hypocenter	C₉	C₃×C₄×SD₁₆
LCS	G ▷ C₄×C₉ ▷ C₂×C₉ ▷ C₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×SD₁₆

Complements

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

N	H ≅ G/N	type
2. Q₈×3₋¹⁺²⋊C₄	C₂	semidirect
4. #63395⋊C₃	C₂	semidirect
6. #63377⋊C₃	C₂	semidirect
7. C₈×3₋¹⁺²⋊C₄	C₂	semidirect
9. SD₁₆×Dic₉	C₃	semidirect
13. SD₁₆×3₋¹⁺²	C₄	semidirect
31. Q₈×Dic₉	C₂×C₃	semidirect
33. #63395	C₂×C₃	semidirect
35. #63377	C₂×C₃	semidirect
36. C₈×Dic₉	C₂×C₃	semidirect
39. Q₈×3₋¹⁺²	C₂×C₄	semidirect
49. C₈×3₋¹⁺²	C₂×C₄	semidirect
71. C₉×SD₁₆	C₃×C₄	semidirect
91. 3₋¹⁺²⋊C₄	SD₁₆	direct (N × H)
125. Q₈×C₉	C₂×C₃×C₄	semidirect
140. C₈×C₉	C₂×C₃×C₄	semidirect
199. Dic₉	C₃×SD₁₆	semidirect
207. 3₋¹⁺²	C₄×SD₁₆	semidirect
251. SD₁₆	3₋¹⁺²⋊C₄	direct (N × H)
284. C₉	C₃×C₄×SD₁₆	semidirect
287. Q₈	C₂×3₋¹⁺²⋊C₄	semidirect
298. C₈	C₂×3₋¹⁺²⋊C₄	semidirect

Maximal subgroups

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

subgroup	index
C₂×SD₁₆×3₋¹⁺²	2
Q₈×3₋¹⁺²⋊C₄	2
D₄×3₋¹⁺²⋊C₄	2
#63395⋊C₃	2
#63392⋊C₃	2
#63377⋊C₃	2
C₈×3₋¹⁺²⋊C₄	2
C₃×Dic₃×SD₁₆	3
SD₁₆×Dic₉	3

Subgroups — SD16×3−1+2⋊C4 #13143379

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — SD₁₆×3₋¹⁺²⋊C₄ #13143379