Subgroups — D₄×3₋¹⁺²⋊C₈ #13142751

Order 1728.

449 conjugacy classes of subgroups.

The subgroup lattice has 268 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×D₄×3₋¹⁺²	C₂	1	1	yes	maximal, characteristic
2	C₂²×3₋¹⁺²⋊C₈	C₂	1	2	yes	maximal, complemented
3	#63389⋊C₃	C₂	1	2	yes	maximal, complemented
4	C₄⋊(3₋¹⁺²⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄×3₋¹⁺²⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃×D₄×C₃⋊C₈	—	3	1	no	maximal
7	D₄×C₉⋊C₈	C₃	1	1	yes	maximal, characteristic, complemented
8	C₂×D₄×3₋¹⁺²	C₄	1	1	yes	characteristic
9	C₂²×C₄×3₋¹⁺²	C₂²	1	2	yes	—
10	C₄⋊C₄×3₋¹⁺²	C₄	1	1	yes	characteristic
11	C₂²⋊C₄×3₋¹⁺²	C₄	1	2	yes	—
12	C₄²×3₋¹⁺²	C₂²	1	1	yes	characteristic
13	C₂×3₋¹⁺²⋊C₈	—	2	4	no	—
14	C₂×3₋¹⁺²⋊C₈	C₂²	1	1	yes	characteristic
15	C₂×3₋¹⁺²⋊C₈	C₂²	1	2	yes	—
16	C₂×3₋¹⁺²⋊C₈	C₂²	1	1	yes	characteristic
17	C₃²×C₄×D₄	D₃	1	1	yes	characteristic
18	C₂²×C₃×C₃⋊C₈	—	3	2	no	—
19	C₃×#830	—	3	2	no	—
20	C₃×C₄⋊(C₃⋊C₈)	—	3	1	no	—
21	C₃×C₄×C₃⋊C₈	—	3	1	no	—
22	C₄×D₄×C₉	—	2	1	no	—
23	C₄×D₄×C₉	C₂×C₃	1	1	yes	characteristic
24	C₂²×C₉⋊C₈	C₂×C₃	1	2	yes	complemented
25	#63389	C₂×C₃	1	2	yes	complemented
26	C₄⋊(C₉⋊C₈)	C₂×C₃	1	1	yes	characteristic, complemented
27	C₄×C₉⋊C₈	C₂×C₃	1	1	yes	characteristic, complemented
28	C₂³×3₋¹⁺²	C₂×C₄	1	2	yes	—
29	D₄×3₋¹⁺²	C₈	1	4	yes	complemented
30	C₂×C₄×3₋¹⁺²	—	2	2	no	—
31	C₂×C₄×3₋¹⁺²	—	2	2	no	—
32	C₂×C₄×3₋¹⁺²	C₂×C₄	1	2	yes	—
33	C₂×C₄×3₋¹⁺²	C₂×C₄	1	1	yes	characteristic
34	C₂×C₄×3₋¹⁺²	C₂×C₄	1	1	yes	characteristic
35	C₂×C₄×3₋¹⁺²	C₂³	1	1	yes	characteristic
36	3₋¹⁺²⋊C₈	—	2	2	no	—
37	3₋¹⁺²⋊C₈	—	2	1	no	—
38	3₋¹⁺²⋊C₈	D₄	1	2	yes	direct product factor
39	C₃×C₈×D₄	—	9	1	no	—
40	D₄×C₃⋊C₈	—	3	1	no	—
41	C₂×C₃²×D₄	Dic₃	1	1	yes	characteristic
42	C₂²×C₃²×C₄	C₂×D₃	1	2	yes	—
43	C₃²×C₄⋊C₄	Dic₃	1	1	yes	characteristic
44	C₃²×C₂²⋊C₄	Dic₃	1	2	yes	—
45	C₃²×C₄²	C₂×D₃	1	1	yes	characteristic
46	C₂×C₃×C₃⋊C₈	—	3	1	no	—
47	C₂×C₃×C₃⋊C₈	—	3	2	no	—
48	C₂×C₃×C₃⋊C₈	—	3	1	no	—
49	C₂×C₃×C₃⋊C₈	—	6	4	no	—
50	C₂×D₄×C₉	—	2	1	no	—
51	C₂×D₄×C₉	C₃×C₄	1	1	yes	characteristic
52	C₂²×C₄×C₉	—	2	2	no	—
53	C₂²×C₄×C₉	C₂²×C₃	1	2	yes	—
54	C₉×C₄⋊C₄	—	2	1	no	—
55	C₉×C₄⋊C₄	C₃×C₄	1	1	yes	characteristic
56	C₉×C₂²⋊C₄	—	2	2	no	—
57	C₉×C₂²⋊C₄	C₃×C₄	1	2	yes	—
58	C₄²×C₉	—	2	1	no	—
59	C₄²×C₉	C₂²×C₃	1	1	yes	characteristic
60	C₂×C₉⋊C₈	—	2	4	no	—
61	C₂×C₉⋊C₈	C₂²×C₃	1	1	yes	characteristic
62	C₂×C₉⋊C₈	C₂²×C₃	1	2	yes	—
63	C₂×C₉⋊C₈	C₂²×C₃	1	1	yes	characteristic
64	C₂²×3₋¹⁺²	—	2	2	no	—
65	C₂²×3₋¹⁺²	—	2	2	no	—
66	C₂²×3₋¹⁺²	C₂×C₈	1	4	yes	complemented
67	C₂²×3₋¹⁺²	C₂²×C₄	1	1	yes	characteristic
68	C₄×3₋¹⁺²	—	2	2	no	—
69	C₄×3₋¹⁺²	—	2	1	no	—
70	C₄×3₋¹⁺²	C₂×C₈	1	2	yes	complemented
71	C₄×3₋¹⁺²	C₂×D₄	1	1	yes	characteristic
72	C₄×3₋¹⁺²	#41	1	1	yes	characteristic
73	C₂²×C₃×C₈	—	9	2	no	—
74	C₃×C₄×D₄	—	3	1	no	—
75	C₃×C₄×D₄	C₃×D₃	1	1	yes	characteristic
76	C₂²×C₃⋊C₈	—	3	2	no	—
77	C₃×C₄⋊C₈	—	9	1	no	—
78	C₃×C₂²⋊C₈	—	9	2	no	—
79	C₃×C₄×C₈	—	9	1	no	—
80	#830	—	3	2	no	—
81	C₄⋊(C₃⋊C₈)	—	3	1	no	—
82	C₄×C₃⋊C₈	—	3	1	no	—
83	C₂³×C₃²	C₂×Dic₃	1	2	yes	—
84	C₃²×D₄	C₃⋊C₈	1	4	yes	—
85	C₂×C₃²×C₄	—	2	2	no	—
86	C₂×C₃²×C₄	—	2	2	no	—
87	C₂×C₃²×C₄	C₂×Dic₃	1	1	yes	characteristic
88	C₂×C₃²×C₄	C₂×Dic₃	1	2	yes	—
89	C₂×C₃²×C₄	C₂×Dic₃	1	1	yes	characteristic
90	C₂×C₃²×C₄	C₂²×D₃	1	1	yes	characteristic
91	C₂³×C₉	—	2	2	no	—
92	C₂³×C₉	C₂×C₃×C₄	1	2	yes	—
93	C₃×C₃⋊C₈	—	3	2	no	—
94	C₃×C₃⋊C₈	—	6	2	no	—
95	C₃×C₃⋊C₈	—	6	1	no	—
96	D₄×C₉	—	2	4	no	—
97	D₄×C₉	C₃×C₈	1	4	yes	complemented
98	C₂×C₄×C₉	—	2	2	no	—
99	C₂×C₄×C₉	—	2	2	no	—
100	C₂×C₄×C₉	—	2	1	no	—
101	C₂×C₄×C₉	—	2	1	no	—
102	C₂×C₄×C₉	—	2	2	no	—
103	C₂×C₄×C₉	—	2	1	no	—
104	C₂×C₄×C₉	—	4	2	no	—
105	C₂×C₄×C₉	—	4	2	no	—
106	C₂×C₄×C₉	C₂×C₃×C₄	1	1	yes	characteristic
107	C₂×C₄×C₉	C₂×C₃×C₄	1	2	yes	—
108	C₂×C₄×C₉	C₂×C₃×C₄	1	1	yes	characteristic
109	C₂×C₄×C₉	C₂³×C₃	1	1	yes	characteristic
110	C₉⋊C₈	—	2	2	no	—
111	C₉⋊C₈	—	2	1	no	—
112	C₉⋊C₈	C₃×D₄	1	2	yes	complemented
113	C₈×D₄	—	9	1	no	Sylow
114	C₂×3₋¹⁺²	—	2	4	no	—
115	C₂×3₋¹⁺²	C₄×D₄	1	1	yes	characteristic
116	C₂×3₋¹⁺²	C₂²×C₈	1	1	yes	characteristic
117	C₂×3₋¹⁺²	#131	1	1	yes	characteristic
118	C₂×C₃×D₄	—	3	1	no	—
119	C₂×C₃×D₄	C₃×Dic₃	1	1	yes	characteristic
120	C₂²×C₃×C₄	—	3	2	no	—
121	C₂²×C₃×C₄	C₂×C₃×D₃	1	2	yes	—
122	C₂×C₃×C₈	—	9	1	no	—
123	C₂×C₃×C₈	—	9	1	no	—
124	C₂×C₃×C₈	—	9	2	no	—
125	C₂×C₃×C₈	—	18	4	no	—
126	C₃×C₄⋊C₄	—	3	1	no	—
127	C₃×C₄⋊C₄	C₃×Dic₃	1	1	yes	characteristic
128	C₃×C₂²⋊C₄	—	3	2	no	—
129	C₃×C₂²⋊C₄	C₃×Dic₃	1	2	yes	—
130	C₃×C₄²	—	3	1	no	—
131	C₃×C₄²	C₂×C₃×D₃	1	1	yes	characteristic
132	C₂×C₃⋊C₈	—	3	1	no	—
133	C₂×C₃⋊C₈	—	3	1	no	—
134	C₂×C₃⋊C₈	—	3	2	no	—
135	C₂×C₃⋊C₈	—	6	4	no	—
136	C₂²×C₃²	—	2	2	no	—
137	C₂²×C₃²	—	2	2	no	—
138	C₂²×C₃²	C₂×C₃⋊C₈	1	4	yes	—
139	C₂²×C₃²	C₂²×Dic₃	1	1	yes	characteristic
140	C₃²×C₄	—	2	2	no	—
141	C₃²×C₄	—	2	1	no	—
142	C₃²×C₄	C₂×C₃⋊C₈	1	2	yes	—
143	C₃²×C₄	D₃×D₄	1	1	yes	characteristic
144	C₃²×C₄	#237	1	1	yes	characteristic
145	C₂²×C₉	—	2	2	no	—
146	C₂²×C₉	—	2	2	no	—
147	C₂²×C₉	—	2	4	no	—
148	C₂²×C₉	—	2	1	no	—
149	C₂²×C₉	—	4	2	no	—
150	C₂²×C₉	—	4	2	no	—
151	C₂²×C₉	C₂×C₃×C₈	1	4	yes	complemented
152	C₂²×C₉	C₂²×C₃×C₄	1	1	yes	characteristic
153	C₄×C₉	—	2	2	no	—
154	C₄×C₉	—	2	1	no	—
155	C₄×C₉	—	2	2	no	—
156	C₄×C₉	—	2	1	no	—
157	C₄×C₉	—	2	1	no	—
158	C₄×C₉	—	4	1	no	—
159	C₄×C₉	—	4	2	no	—
160	C₄×C₉	C₂×C₃×C₈	1	2	yes	complemented
161	C₄×C₉	C₂×C₃×D₄	1	1	yes	characteristic
162	C₄×C₉	C₃×#41	1	1	yes	characteristic
163	C₂²×C₈	—	9	2	no	—
164	C₄×D₄	AGL(1,ℤ/9)	1	1	yes	characteristic
165	C₄⋊C₈	—	9	1	no	—
166	C₂²⋊C₈	—	9	2	no	—
167	C₄×C₈	—	9	1	no	—
168	3₋¹⁺²	C₈×D₄	1	1	yes	characteristic, complemented, Sylow
169	C₂³×C₃	—	3	2	no	—
170	C₂³×C₃	C₂×C₃×Dic₃	1	2	yes	—
171	C₃×D₄	—	3	4	no	—
172	C₃×D₄	C₃×C₃⋊C₈	1	4	yes	—
173	C₂×C₃×C₄	—	2	2	no	—
174	C₂×C₃×C₄	—	2	2	no	—
175	C₂×C₃×C₄	—	3	1	no	—
176	C₂×C₃×C₄	—	3	1	no	—
177	C₂×C₃×C₄	—	3	2	no	—
178	C₂×C₃×C₄	—	3	1	no	—
179	C₂×C₃×C₄	—	6	2	no	—
180	C₂×C₃×C₄	—	6	2	no	—
181	C₂×C₃×C₄	C₂×C₃×Dic₃	1	1	yes	characteristic
182	C₂×C₃×C₄	C₂×C₃×Dic₃	1	2	yes	—
183	C₂×C₃×C₄	C₂×C₃×Dic₃	1	1	yes	characteristic
184	C₂×C₃×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
185	C₃×C₈	—	9	2	no	—
186	C₃×C₈	—	18	1	no	—
187	C₃×C₈	—	18	2	no	—
188	C₃⋊C₈	—	3	2	no	—
189	C₃⋊C₈	—	6	2	no	—
190	C₃⋊C₈	—	6	1	no	—
191	C₂×C₃²	—	2	4	no	—
192	C₂×C₃²	C₂²×C₃⋊C₈	1	1	yes	characteristic
193	C₂×C₃²	D₄×Dic₃	1	1	yes	characteristic
194	C₂×C₃²	#948	1	1	yes	characteristic
195	C₂×C₉	—	2	4	no	—
196	C₂×C₉	—	2	1	no	—
197	C₂×C₉	—	2	1	no	—
198	C₂×C₉	—	2	1	no	—
199	C₂×C₉	—	4	4	no	—
200	C₂×C₉	C₃×C₄×D₄	1	1	yes	characteristic
201	C₂×C₉	C₂²×C₃×C₈	1	1	yes	characteristic
202	C₂×C₉	C₃×#131	1	1	yes	characteristic
203	C₂×D₄	3₋¹⁺²⋊C₄	1	1	yes	characteristic
204	C₂²×C₄	C₂×AGL(1,ℤ/9)	1	2	yes	—
205	C₂×C₈	—	9	1	no	—
206	C₂×C₈	—	9	1	no	—
207	C₂×C₈	—	9	2	no	—
208	C₂×C₈	—	18	4	no	—
209	C₄⋊C₄	3₋¹⁺²⋊C₄	1	1	yes	characteristic
210	C₂²⋊C₄	3₋¹⁺²⋊C₄	1	2	yes	—
211	C₄²	C₂×AGL(1,ℤ/9)	1	1	yes	characteristic
212	C₂²×C₃	—	2	2	no	—
213	C₂²×C₃	—	2	2	no	—
214	C₂²×C₃	—	3	1	no	—
215	C₂²×C₃	—	3	4	no	—
216	C₂²×C₃	—	6	2	no	—
217	C₂²×C₃	—	6	2	no	—
218	C₂²×C₃	C₂×C₃×C₃⋊C₈	1	4	yes	—
219	C₂²×C₃	C₂²×C₃×Dic₃	1	1	yes	characteristic
220	C₃×C₄	—	2	2	no	—
221	C₃×C₄	—	2	1	no	—
222	C₃×C₄	—	3	1	no	—
223	C₃×C₄	—	3	1	no	—
224	C₃×C₄	—	3	2	no	—
225	C₃×C₄	—	6	2	no	—
226	C₃×C₄	—	6	1	no	—
227	C₃×C₄	C₂×C₃×C₃⋊C₈	1	2	yes	—
228	C₃×C₄	C₃×D₃×D₄	1	1	yes	characteristic
229	C₃×C₄	C₃×#237	1	1	yes	characteristic
230	C₃²	D₄×C₃⋊C₈	1	1	yes	characteristic
231	C₉	—	2	1	no	—
232	C₉	C₃×C₈×D₄	1	1	yes	characteristic, complemented
233	C₂³	C₂×3₋¹⁺²⋊C₄	1	2	yes	—
234	D₄	3₋¹⁺²⋊C₈	1	4	yes	direct product factor
235	C₂×C₄	—	2	2	no	—
236	C₂×C₄	—	2	2	no	—
237	C₂×C₄	C₂×3₋¹⁺²⋊C₄	1	1	yes	characteristic
238	C₂×C₄	C₂×3₋¹⁺²⋊C₄	1	2	yes	—
239	C₂×C₄	C₂×3₋¹⁺²⋊C₄	1	1	yes	characteristic
240	C₂×C₄	C₂²×AGL(1,ℤ/9)	1	1	yes	characteristic
241	C₈	—	9	2	no	—
242	C₈	—	18	2	no	—
243	C₈	—	18	1	no	—
244	C₂×C₃	—	2	4	no	—
245	C₂×C₃	—	3	1	no	—
246	C₂×C₃	—	3	1	no	—
247	C₂×C₃	—	3	1	no	—
248	C₂×C₃	—	6	4	no	—
249	C₂×C₃	C₂²×C₃×C₃⋊C₈	1	1	yes	characteristic
250	C₂×C₃	C₃×D₄×Dic₃	1	1	yes	characteristic
251	C₂×C₃	C₃×#948	1	1	yes	characteristic
252	C₂²	—	2	2	no	—
253	C₂²	—	2	2	no	—
254	C₂²	C₂×3₋¹⁺²⋊C₈	1	4	yes	complemented
255	C₂²	C₂²×3₋¹⁺²⋊C₄	1	1	yes	characteristic
256	C₄	—	2	2	no	—
257	C₄	—	2	1	no	—
258	C₄	C₂×3₋¹⁺²⋊C₈	1	2	yes	complemented
259	C₄	D₄×AGL(1,ℤ/9)	1	1	yes	characteristic
260	C₄	#3699⋊C₃	1	1	yes	characteristic
261	C₃	—	3	1	no	—
262	C₃	C₃×D₄×C₃⋊C₈	1	1	yes	characteristic
263	C₂	—	2	4	no	—
264	C₂	C₂²×3₋¹⁺²⋊C₈	1	1	yes	characteristic
265	C₂	D₄×3₋¹⁺²⋊C₄	1	1	yes	characteristic
266	C₂	#11721693	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×AGL(1,ℤ/9)
commutator	C₂×C₉	C₂²×C₃×C₈
Frattini	C₂×C₃×C₄	C₂²×C₃×D₃
Fitting	C₄×D₄×3₋¹⁺²	C₂
socle	C₂²×C₃	C₂²×C₃×Dic₃
prime radical	C₂×C₃²×D₄	Dic₃
cosocle	C₂×C₄×C₉	C₂³×C₃
hypercenter	C₄×D₄	AGL(1,ℤ/9)
hypocenter	C₉	C₃×C₈×D₄
LCS	G ▷ C₂×C₉ ▷ C₉
UCS	C₁ ◁ C₂×C₄ ◁ C₄×D₄

Complements

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

N	H ≅ G/N	type
2. C₂²×3₋¹⁺²⋊C₈	C₂	semidirect
3. #63389⋊C₃	C₂	semidirect
4. C₄⋊(3₋¹⁺²⋊C₈)	C₂	semidirect
5. C₄×3₋¹⁺²⋊C₈	C₂	semidirect
7. D₄×C₉⋊C₈	C₃	semidirect
24. C₂²×C₉⋊C₈	C₂×C₃	semidirect
25. #63389	C₂×C₃	semidirect
26. C₄⋊(C₉⋊C₈)	C₂×C₃	semidirect
27. C₄×C₉⋊C₈	C₂×C₃	semidirect
29. D₄×3₋¹⁺²	C₈	semidirect
38. 3₋¹⁺²⋊C₈	D₄	direct (N × H)
66. C₂²×3₋¹⁺²	C₂×C₈	semidirect
70. C₄×3₋¹⁺²	C₂×C₈	semidirect
97. D₄×C₉	C₃×C₈	semidirect
112. C₉⋊C₈	C₃×D₄	semidirect
151. C₂²×C₉	C₂×C₃×C₈	semidirect
160. C₄×C₉	C₂×C₃×C₈	semidirect
168. 3₋¹⁺²	C₈×D₄	semidirect
232. C₉	C₃×C₈×D₄	semidirect
234. D₄	3₋¹⁺²⋊C₈	direct (N × H)
254. C₂²	C₂×3₋¹⁺²⋊C₈	semidirect
258. C₄	C₂×3₋¹⁺²⋊C₈	semidirect

Maximal subgroups

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

subgroup	index
C₄×D₄×3₋¹⁺²	2
C₂²×3₋¹⁺²⋊C₈	2
#63389⋊C₃	2
C₄⋊(3₋¹⁺²⋊C₈)	2
C₄×3₋¹⁺²⋊C₈	2
C₃×D₄×C₃⋊C₈	3
D₄×C₉⋊C₈	3

Subgroups — D4×3−1+2⋊C8 #13142751

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₄×3₋¹⁺²⋊C₈ #13142751