Subgroups — D₄×C₃⋊(C₉⋊C₈) #13144398

Order 1728.

670 conjugacy classes of subgroups.

The subgroup lattice has 240 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₄×D₄×C₉	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₃⋊(C₉⋊C₈)	C₂	1	2	yes	maximal, complemented
3	#11720571	C₂	1	2	yes	maximal, complemented
4	C₄⋊(C₃⋊(C₉⋊C₈))	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄×C₃⋊(C₉⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
6	D₄×C₃⋊(C₃⋊C₈)	—	3	1	no	maximal
7	D₄×C₉⋊C₈	—	3	3	no	maximal
8	C₂×C₃×D₄×C₉	C₄	1	1	yes	characteristic
9	C₂²×C₃×C₄×C₉	C₂²	1	2	yes	—
10	C₃×C₉×C₄⋊C₄	C₄	1	1	yes	characteristic
11	C₃×C₉×C₂²⋊C₄	C₄	1	2	yes	—
12	C₃×C₄²×C₉	C₂²	1	1	yes	characteristic
13	C₂×C₃⋊(C₉⋊C₈)	—	2	4	no	—
14	C₂×C₃⋊(C₉⋊C₈)	C₂²	1	1	yes	characteristic
15	C₂×C₃⋊(C₉⋊C₈)	C₂²	1	2	yes	—
16	C₂×C₃⋊(C₉⋊C₈)	C₂²	1	1	yes	characteristic
17	C₃²×C₄×D₄	D₃	1	1	yes	characteristic
18	C₂²×C₃⋊(C₃⋊C₈)	—	3	2	no	—
19	#63657	—	3	2	no	—
20	C₄⋊(C₃⋊(C₃⋊C₈))	—	3	1	no	—
21	C₄×C₃⋊(C₃⋊C₈)	—	3	1	no	—
22	C₄×D₄×C₉	D₃	1	3	yes	—
23	C₂²×C₉⋊C₈	—	3	6	no	—
24	#63389	—	3	6	no	—
25	C₄⋊(C₉⋊C₈)	—	3	3	no	—
26	C₄×C₉⋊C₈	—	3	3	no	—
27	C₂³×C₃×C₉	C₂×C₄	1	2	yes	—
28	C₃×D₄×C₉	C₈	1	4	yes	complemented
29	C₂×C₃×C₄×C₉	—	2	2	no	—
30	C₂×C₃×C₄×C₉	—	2	2	no	—
31	C₂×C₃×C₄×C₉	C₂×C₄	1	2	yes	—
32	C₂×C₃×C₄×C₉	C₂×C₄	1	1	yes	characteristic
33	C₂×C₃×C₄×C₉	C₂×C₄	1	1	yes	characteristic
34	C₂×C₃×C₄×C₉	C₂³	1	1	yes	characteristic
35	C₃⋊(C₉⋊C₈)	—	2	2	no	—
36	C₃⋊(C₉⋊C₈)	—	2	1	no	—
37	C₃⋊(C₉⋊C₈)	D₄	1	2	yes	direct product factor
38	D₄×C₃⋊C₈	—	9	3	no	—
39	D₄×C₃⋊C₈	—	9	1	no	—
40	C₂×C₃²×D₄	Dic₃	1	1	yes	characteristic
41	C₂²×C₃²×C₄	C₂×D₃	1	2	yes	—
42	C₃²×C₄⋊C₄	Dic₃	1	1	yes	characteristic
43	C₃²×C₂²⋊C₄	Dic₃	1	2	yes	—
44	C₃²×C₄²	C₂×D₃	1	1	yes	characteristic
45	C₂×C₃⋊(C₃⋊C₈)	—	3	1	no	—
46	C₂×C₃⋊(C₃⋊C₈)	—	3	2	no	—
47	C₂×C₃⋊(C₃⋊C₈)	—	3	1	no	—
48	C₂×C₃⋊(C₃⋊C₈)	—	6	4	no	—
49	C₂×D₄×C₉	Dic₃	1	3	yes	—
50	C₂²×C₄×C₉	C₂×D₃	1	6	yes	—
51	C₉×C₄⋊C₄	Dic₃	1	3	yes	—
52	C₉×C₂²⋊C₄	Dic₃	1	6	yes	—
53	C₄²×C₉	C₂×D₃	1	3	yes	—
54	C₂×C₉⋊C₈	—	3	3	no	—
55	C₂×C₉⋊C₈	—	3	3	no	—
56	C₂×C₉⋊C₈	—	3	6	no	—
57	C₂×C₉⋊C₈	—	6	12	no	—
58	C₂²×C₃×C₉	—	2	2	no	—
59	C₂²×C₃×C₉	—	2	2	no	—
60	C₂²×C₃×C₉	C₂×C₈	1	4	yes	complemented
61	C₂²×C₃×C₉	C₂²×C₄	1	1	yes	characteristic
62	C₃×C₄×C₉	—	2	2	no	—
63	C₃×C₄×C₉	—	2	1	no	—
64	C₃×C₄×C₉	C₂×C₈	1	2	yes	complemented
65	C₃×C₄×C₉	C₂×D₄	1	1	yes	characteristic
66	C₃×C₄×C₉	#41	1	1	yes	characteristic
67	C₃×C₄×D₄	D₉	1	3	yes	—
68	C₃×C₄×D₄	C₃⋊D₃	1	1	yes	characteristic
69	C₂²×C₃⋊C₈	—	9	6	no	—
70	C₂²×C₃⋊C₈	—	9	2	no	—
71	#830	—	9	6	no	—
72	#830	—	9	2	no	—
73	C₄⋊(C₃⋊C₈)	—	9	3	no	—
74	C₄⋊(C₃⋊C₈)	—	9	1	no	—
75	C₄×C₃⋊C₈	—	9	3	no	—
76	C₄×C₃⋊C₈	—	9	1	no	—
77	C₂³×C₃²	C₂×Dic₃	1	2	yes	—
78	C₃²×D₄	C₃⋊C₈	1	4	yes	—
79	C₂×C₃²×C₄	—	2	2	no	—
80	C₂×C₃²×C₄	—	2	2	no	—
81	C₂×C₃²×C₄	C₂×Dic₃	1	1	yes	characteristic
82	C₂×C₃²×C₄	C₂×Dic₃	1	2	yes	—
83	C₂×C₃²×C₄	C₂×Dic₃	1	1	yes	characteristic
84	C₂×C₃²×C₄	C₂²×D₃	1	1	yes	characteristic
85	C₂³×C₉	C₂×Dic₃	1	6	yes	—
86	C₃⋊(C₃⋊C₈)	—	3	2	no	—
87	C₃⋊(C₃⋊C₈)	—	6	2	no	—
88	C₃⋊(C₃⋊C₈)	—	6	1	no	—
89	D₄×C₉	C₃⋊C₈	1	12	yes	complemented
90	C₂×C₄×C₉	—	2	6	no	—
91	C₂×C₄×C₉	—	2	6	no	—
92	C₂×C₄×C₉	C₂×Dic₃	1	3	yes	—
93	C₂×C₄×C₉	C₂×Dic₃	1	6	yes	—
94	C₂×C₄×C₉	C₂×Dic₃	1	3	yes	—
95	C₂×C₄×C₉	C₂²×D₃	1	3	yes	—
96	C₉⋊C₈	—	3	6	no	—
97	C₉⋊C₈	—	6	6	no	—
98	C₉⋊C₈	—	6	3	no	—
99	C₈×D₄	—	27	1	no	Sylow
100	C₂×C₃×C₉	—	2	4	no	—
101	C₂×C₃×C₉	C₄×D₄	1	1	yes	characteristic
102	C₂×C₃×C₉	C₂²×C₈	1	1	yes	characteristic
103	C₂×C₃×C₉	#131	1	1	yes	characteristic
104	C₂×C₃×D₄	Dic₉	1	3	yes	—
105	C₂×C₃×D₄	C₃⋊Dic₃	1	1	yes	characteristic
106	C₂²×C₃×C₄	C₂×D₉	1	6	yes	—
107	C₂²×C₃×C₄	C₂×C₃⋊D₃	1	2	yes	—
108	C₃×C₄⋊C₄	Dic₉	1	3	yes	—
109	C₃×C₄⋊C₄	C₃⋊Dic₃	1	1	yes	characteristic
110	C₃×C₂²⋊C₄	Dic₉	1	6	yes	—
111	C₃×C₂²⋊C₄	C₃⋊Dic₃	1	2	yes	—
112	C₃×C₄²	C₂×D₉	1	3	yes	—
113	C₃×C₄²	C₂×C₃⋊D₃	1	1	yes	characteristic
114	C₂×C₃⋊C₈	—	9	3	no	—
115	C₂×C₃⋊C₈	—	9	1	no	—
116	C₂×C₃⋊C₈	—	9	3	no	—
117	C₂×C₃⋊C₈	—	9	6	no	—
118	C₂×C₃⋊C₈	—	9	1	no	—
119	C₂×C₃⋊C₈	—	9	2	no	—
120	C₂×C₃⋊C₈	—	18	4	no	—
121	C₂×C₃⋊C₈	—	18	12	no	—
122	C₂²×C₃²	—	2	2	no	—
123	C₂²×C₃²	—	2	2	no	—
124	C₂²×C₃²	C₂×C₃⋊C₈	1	4	yes	—
125	C₂²×C₃²	C₂²×Dic₃	1	1	yes	characteristic
126	C₃²×C₄	—	2	2	no	—
127	C₃²×C₄	—	2	1	no	—
128	C₃²×C₄	C₂×C₃⋊C₈	1	2	yes	—
129	C₃²×C₄	D₃×D₄	1	1	yes	characteristic
130	C₃²×C₄	#237	1	1	yes	characteristic
131	C₂²×C₉	—	2	6	no	—
132	C₂²×C₉	—	2	6	no	—
133	C₂²×C₉	C₂×C₃⋊C₈	1	12	yes	complemented
134	C₂²×C₉	C₂²×Dic₃	1	3	yes	—
135	C₄×C₉	—	2	6	no	—
136	C₄×C₉	—	2	3	no	—
137	C₄×C₉	C₂×C₃⋊C₈	1	6	yes	complemented
138	C₄×C₉	D₃×D₄	1	3	yes	—
139	C₄×C₉	#237	1	3	yes	—
140	C₂²×C₈	—	27	2	no	—
141	C₄×D₄	C₃⋊D₉	1	1	yes	characteristic
142	C₄⋊C₈	—	27	1	no	—
143	C₂²⋊C₈	—	27	2	no	—
144	C₄×C₈	—	27	1	no	—
145	C₃×C₉	C₈×D₄	1	1	yes	characteristic, complemented, Sylow
146	C₂³×C₃	C₂×Dic₉	1	6	yes	—
147	C₂³×C₃	C₂×C₃⋊Dic₃	1	2	yes	—
148	C₃×D₄	C₉⋊C₈	1	12	yes	complemented
149	C₃×D₄	C₃⋊(C₃⋊C₈)	1	4	yes	—
150	C₂×C₃×C₄	—	2	6	no	—
151	C₂×C₃×C₄	—	2	6	no	—
152	C₂×C₃×C₄	—	2	2	no	—
153	C₂×C₃×C₄	—	2	2	no	—
154	C₂×C₃×C₄	C₂×Dic₉	1	3	yes	—
155	C₂×C₃×C₄	C₂×Dic₉	1	6	yes	—
156	C₂×C₃×C₄	C₂×Dic₉	1	3	yes	—
157	C₂×C₃×C₄	C₂²×D₉	1	3	yes	—
158	C₂×C₃×C₄	C₂×C₃⋊Dic₃	1	1	yes	characteristic
159	C₂×C₃×C₄	C₂×C₃⋊Dic₃	1	2	yes	—
160	C₂×C₃×C₄	C₂×C₃⋊Dic₃	1	1	yes	characteristic
161	C₂×C₃×C₄	C₂²×C₃⋊D₃	1	1	yes	characteristic
162	C₃⋊C₈	—	9	6	no	—
163	C₃⋊C₈	—	9	2	no	—
164	C₃⋊C₈	—	18	6	no	—
165	C₃⋊C₈	—	18	3	no	—
166	C₃⋊C₈	—	18	2	no	—
167	C₃⋊C₈	—	18	1	no	—
168	C₂×C₃²	—	2	4	no	—
169	C₂×C₃²	C₂²×C₃⋊C₈	1	1	yes	characteristic
170	C₂×C₃²	D₄×Dic₃	1	1	yes	characteristic
171	C₂×C₃²	#948	1	1	yes	characteristic
172	C₂×C₉	—	2	12	no	—
173	C₂×C₉	C₂²×C₃⋊C₈	1	3	yes	—
174	C₂×C₉	D₄×Dic₃	1	3	yes	—
175	C₂×C₉	#948	1	3	yes	—
176	C₂×D₄	C₃⋊Dic₉	1	1	yes	characteristic
177	C₂²×C₄	C₂×C₃⋊D₉	1	2	yes	—
178	C₂×C₈	—	27	1	no	—
179	C₂×C₈	—	27	1	no	—
180	C₂×C₈	—	27	2	no	—
181	C₂×C₈	—	54	4	no	—
182	C₄⋊C₄	C₃⋊Dic₉	1	1	yes	characteristic
183	C₂²⋊C₄	C₃⋊Dic₉	1	2	yes	—
184	C₄²	C₂×C₃⋊D₉	1	1	yes	characteristic
185	C₂²×C₃	—	2	6	no	—
186	C₂²×C₃	—	2	6	no	—
187	C₂²×C₃	—	2	2	no	—
188	C₂²×C₃	—	2	2	no	—
189	C₂²×C₃	C₂×C₉⋊C₈	1	12	yes	complemented
190	C₂²×C₃	C₂²×Dic₉	1	3	yes	—
191	C₂²×C₃	C₂×C₃⋊(C₃⋊C₈)	1	4	yes	—
192	C₂²×C₃	C₂²×C₃⋊Dic₃	1	1	yes	characteristic
193	C₃×C₄	—	2	6	no	—
194	C₃×C₄	—	2	3	no	—
195	C₃×C₄	—	2	2	no	—
196	C₃×C₄	—	2	1	no	—
197	C₃×C₄	C₂×C₉⋊C₈	1	6	yes	complemented
198	C₃×C₄	D₄×D₉	1	3	yes	—
199	C₃×C₄	#3699	1	3	yes	—
200	C₃×C₄	C₂×C₃⋊(C₃⋊C₈)	1	2	yes	—
201	C₃×C₄	D₄×C₃⋊D₃	1	1	yes	characteristic
202	C₃×C₄	#3830	1	1	yes	characteristic
203	C₃²	D₄×C₃⋊C₈	1	1	yes	characteristic
204	C₉	D₄×C₃⋊C₈	1	3	yes	complemented
205	C₂³	C₂×C₃⋊Dic₉	1	2	yes	—
206	D₄	C₃⋊(C₉⋊C₈)	1	4	yes	direct product factor
207	C₂×C₄	—	2	2	no	—
208	C₂×C₄	—	2	2	no	—
209	C₂×C₄	C₂×C₃⋊Dic₉	1	1	yes	characteristic
210	C₂×C₄	C₂×C₃⋊Dic₉	1	2	yes	—
211	C₂×C₄	C₂×C₃⋊Dic₉	1	1	yes	characteristic
212	C₂×C₄	C₂²×C₃⋊D₉	1	1	yes	characteristic
213	C₈	—	27	2	no	—
214	C₈	—	54	2	no	—
215	C₈	—	54	1	no	—
216	C₂×C₃	—	2	12	no	—
217	C₂×C₃	—	2	4	no	—
218	C₂×C₃	C₂²×C₉⋊C₈	1	3	yes	—
219	C₂×C₃	D₄×Dic₉	1	3	yes	—
220	C₂×C₃	#63510	1	3	yes	—
221	C₂×C₃	C₂²×C₃⋊(C₃⋊C₈)	1	1	yes	characteristic
222	C₂×C₃	D₄×C₃⋊Dic₃	1	1	yes	characteristic
223	C₂×C₃	#64157	1	1	yes	characteristic
224	C₂²	—	2	2	no	—
225	C₂²	—	2	2	no	—
226	C₂²	C₂×C₃⋊(C₉⋊C₈)	1	4	yes	complemented
227	C₂²	C₂²×C₃⋊Dic₉	1	1	yes	characteristic
228	C₄	—	2	2	no	—
229	C₄	—	2	1	no	—
230	C₄	C₂×C₃⋊(C₉⋊C₈)	1	2	yes	complemented
231	C₄	D₄×C₃⋊D₉	1	1	yes	characteristic
232	C₄	#88742	1	1	yes	characteristic
233	C₃	D₄×C₉⋊C₈	1	3	yes	complemented
234	C₃	D₄×C₃⋊(C₃⋊C₈)	1	1	yes	characteristic
235	C₂	—	2	4	no	—
236	C₂	C₂²×C₃⋊(C₉⋊C₈)	1	1	yes	characteristic
237	C₂	D₄×C₃⋊Dic₉	1	1	yes	characteristic
238	C₂	#11721865	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×C₃⋊D₉
commutator	C₂×C₃×C₉	C₂²×C₈
Frattini	C₂×C₃×C₄	C₂²×C₃⋊D₃
Fitting	C₃×C₄×D₄×C₉	C₂
socle	C₂²×C₃²	C₂²×Dic₃
prime radical	C₂×C₃²×D₄	Dic₃
cosocle	C₂×C₃×C₄×C₉	C₂³
hypercenter	C₄×D₄	C₃⋊D₉
hypocenter	C₃×C₉	C₈×D₄
LCS	G ▷ C₂×C₃×C₉ ▷ C₃×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×C₃⋊(C₉⋊C₈)	C₂	semidirect
3. #11720571	C₂	semidirect
4. C₄⋊(C₃⋊(C₉⋊C₈))	C₂	semidirect
5. C₄×C₃⋊(C₉⋊C₈)	C₂	semidirect
28. C₃×D₄×C₉	C₈	semidirect
37. C₃⋊(C₉⋊C₈)	D₄	direct (N × H)
60. C₂²×C₃×C₉	C₂×C₈	semidirect
64. C₃×C₄×C₉	C₂×C₈	semidirect
89. D₄×C₉	C₃⋊C₈	semidirect
133. C₂²×C₉	C₂×C₃⋊C₈	semidirect
137. C₄×C₉	C₂×C₃⋊C₈	semidirect
145. C₃×C₉	C₈×D₄	semidirect
148. C₃×D₄	C₉⋊C₈	semidirect
189. C₂²×C₃	C₂×C₉⋊C₈	semidirect
197. C₃×C₄	C₂×C₉⋊C₈	semidirect
204. C₉	D₄×C₃⋊C₈	semidirect
206. D₄	C₃⋊(C₉⋊C₈)	direct (N × H)
226. C₂²	C₂×C₃⋊(C₉⋊C₈)	semidirect
230. C₄	C₂×C₃⋊(C₉⋊C₈)	semidirect
233. C₃	D₄×C₉⋊C₈	semidirect

Maximal subgroups

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

subgroup	index
C₃×C₄×D₄×C₉	2
C₂²×C₃⋊(C₉⋊C₈)	2
#11720571	2
C₄⋊(C₃⋊(C₉⋊C₈))	2
C₄×C₃⋊(C₉⋊C₈)	2
D₄×C₃⋊(C₃⋊C₈)	3
D₄×C₉⋊C₈	3

Subgroups — D4×C3⋊(C9⋊C8) #13144398

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₄×C₃⋊(C₉⋊C₈) #13144398