Subgroups — D₄⋊(C₃⋊(C₉⋊C₈)) #13135896

Order 1728.

410 conjugacy classes of subgroups.

The subgroup lattice has 222 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	C₄⋊(C₃⋊(C₉⋊C₈))	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄×C₃⋊(C₉⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
4	D₄⋊(C₃⋊(C₃⋊C₈))	—	3	1	no	maximal
5	D₄⋊(C₉⋊C₈)	—	3	3	no	maximal
6	C₂×C₃×D₄×C₉	C₄	1	1	yes	characteristic
7	C₂²×C₃×C₄×C₉	—	2	1	no	—
8	C₃×C₉×C₄⋊C₄	C₄	1	1	yes	characteristic
9	C₃×C₉×C₂²⋊C₄	—	2	1	no	—
10	C₃×C₄²×C₉	C₂²	1	1	yes	characteristic
11	C₂×C₃⋊(C₉⋊C₈)	—	2	1	no	—
12	C₂×C₃⋊(C₉⋊C₈)	—	2	1	no	—
13	C₃²×C₄×D₄	D₃	1	1	yes	characteristic
14	C₄⋊(C₃⋊(C₃⋊C₈))	—	3	1	no	—
15	C₄×C₃⋊(C₃⋊C₈)	—	3	1	no	—
16	C₄×D₄×C₉	D₃	1	3	yes	—
17	C₄⋊(C₉⋊C₈)	—	3	3	no	—
18	C₄×C₉⋊C₈	—	3	3	no	—
19	C₂³×C₃×C₉	—	2	1	no	—
20	C₃×D₄×C₉	—	2	1	no	—
21	C₃×D₄×C₉	C₈	1	2	yes	complemented
22	C₂×C₃×C₄×C₉	—	2	1	no	—
23	C₂×C₃×C₄×C₉	—	4	1	no	—
24	C₂×C₃×C₄×C₉	—	4	1	no	—
25	C₂×C₃×C₄×C₉	C₂×C₄	1	1	yes	characteristic
26	C₂×C₃×C₄×C₉	D₄	1	1	yes	characteristic
27	C₂×C₃×C₄×C₉	D₄	1	1	yes	characteristic
28	C₃⋊(C₉⋊C₈)	—	2	2	no	—
29	C₃⋊(C₉⋊C₈)	—	4	1	no	—
30	D₄⋊(C₃⋊C₈)	—	9	3	no	—
31	D₄⋊(C₃⋊C₈)	—	9	1	no	—
32	C₂×C₃²×D₄	Dic₃	1	1	yes	characteristic
33	C₂²×C₃²×C₄	—	2	1	no	—
34	C₃²×C₄⋊C₄	Dic₃	1	1	yes	characteristic
35	C₃²×C₂²⋊C₄	—	2	1	no	—
36	C₃²×C₄²	C₂×D₃	1	1	yes	characteristic
37	C₂×C₃⋊(C₃⋊C₈)	—	6	1	no	—
38	C₂×C₃⋊(C₃⋊C₈)	—	6	1	no	—
39	C₂×D₄×C₉	Dic₃	1	3	yes	—
40	C₂²×C₄×C₉	—	2	3	no	—
41	C₉×C₄⋊C₄	Dic₃	1	3	yes	—
42	C₉×C₂²⋊C₄	—	2	3	no	—
43	C₄²×C₉	C₂×D₃	1	3	yes	—
44	C₂×C₉⋊C₈	—	6	3	no	—
45	C₂×C₉⋊C₈	—	6	3	no	—
46	C₂²×C₃×C₉	—	2	2	no	—
47	C₂²×C₃×C₉	—	4	1	no	—
48	C₂²×C₃×C₉	—	4	1	no	—
49	C₂²×C₃×C₉	C₂²⋊C₄	1	1	yes	characteristic
50	C₃×C₄×C₉	—	2	1	no	—
51	C₃×C₄×C₉	—	4	1	no	—
52	C₃×C₄×C₉	C₂×C₈	1	1	yes	characteristic
53	C₃×C₄×C₉	M₁₆	1	1	yes	characteristic
54	C₃×C₄×C₉	D₈	1	1	yes	characteristic
55	C₃×C₄×C₉	SD₁₆	1	1	yes	characteristic
56	C₃×C₄×D₄	D₉	1	3	yes	—
57	C₃×C₄×D₄	C₃⋊D₃	1	1	yes	characteristic
58	C₄⋊(C₃⋊C₈)	—	9	3	no	—
59	C₄⋊(C₃⋊C₈)	—	9	1	no	—
60	C₄×C₃⋊C₈	—	9	3	no	—
61	C₄×C₃⋊C₈	—	9	1	no	—
62	C₂³×C₃²	—	2	1	no	—
63	C₃²×D₄	—	2	1	no	—
64	C₃²×D₄	C₃⋊C₈	1	2	yes	—
65	C₂×C₃²×C₄	—	2	1	no	—
66	C₂×C₃²×C₄	—	4	1	no	—
67	C₂×C₃²×C₄	—	4	1	no	—
68	C₂×C₃²×C₄	C₂×Dic₃	1	1	yes	characteristic
69	C₂×C₃²×C₄	#67	1	1	yes	characteristic
70	C₂×C₃²×C₄	#67	1	1	yes	characteristic
71	C₂³×C₉	—	2	3	no	—
72	C₃⋊(C₃⋊C₈)	—	6	2	no	—
73	C₃⋊(C₃⋊C₈)	—	12	1	no	—
74	D₄×C₉	—	2	3	no	—
75	D₄×C₉	C₃⋊C₈	1	6	yes	complemented
76	C₂×C₄×C₉	—	2	3	no	—
77	C₂×C₄×C₉	—	4	3	no	—
78	C₂×C₄×C₉	—	4	3	no	—
79	C₂×C₄×C₉	C₂×Dic₃	1	3	yes	—
80	C₂×C₄×C₉	#67	1	3	yes	—
81	C₂×C₄×C₉	#67	1	3	yes	—
82	C₉⋊C₈	—	6	6	no	—
83	C₉⋊C₈	—	12	3	no	—
84	D₄⋊C₈	—	27	1	no	Sylow
85	C₂×C₃×C₉	—	4	2	no	—
86	C₂×C₃×C₉	C₂²⋊C₈	1	1	yes	characteristic
87	C₂×C₃×C₉	#102	1	1	yes	characteristic
88	C₂×C₃×C₉	#104	1	1	yes	characteristic
89	C₂×C₃×D₄	Dic₉	1	3	yes	—
90	C₂×C₃×D₄	C₃⋊Dic₃	1	1	yes	characteristic
91	C₂²×C₃×C₄	—	2	3	no	—
92	C₂²×C₃×C₄	—	2	1	no	—
93	C₃×C₄⋊C₄	Dic₉	1	3	yes	—
94	C₃×C₄⋊C₄	C₃⋊Dic₃	1	1	yes	characteristic
95	C₃×C₂²⋊C₄	—	2	3	no	—
96	C₃×C₂²⋊C₄	—	2	1	no	—
97	C₃×C₄²	C₂×D₉	1	3	yes	—
98	C₃×C₄²	C₂×C₃⋊D₃	1	1	yes	characteristic
99	C₂×C₃⋊C₈	—	18	3	no	—
100	C₂×C₃⋊C₈	—	18	1	no	—
101	C₂×C₃⋊C₈	—	18	3	no	—
102	C₂×C₃⋊C₈	—	18	1	no	—
103	C₂²×C₃²	—	2	2	no	—
104	C₂²×C₃²	—	4	1	no	—
105	C₂²×C₃²	—	4	1	no	—
106	C₂²×C₃²	#217	1	1	yes	characteristic
107	C₃²×C₄	—	2	1	no	—
108	C₃²×C₄	—	4	1	no	—
109	C₃²×C₄	C₂×C₃⋊C₈	1	1	yes	characteristic
110	C₃²×C₄	#208	1	1	yes	characteristic
111	C₃²×C₄	#213	1	1	yes	characteristic
112	C₃²×C₄	#214	1	1	yes	characteristic
113	C₂²×C₉	—	2	6	no	—
114	C₂²×C₉	—	4	3	no	—
115	C₂²×C₉	—	4	3	no	—
116	C₂²×C₉	#217	1	3	yes	—
117	C₄×C₉	—	2	3	no	—
118	C₄×C₉	—	4	3	no	—
119	C₄×C₉	C₂×C₃⋊C₈	1	3	yes	—
120	C₄×C₉	#208	1	3	yes	—
121	C₄×C₉	#213	1	3	yes	—
122	C₄×C₉	#214	1	3	yes	—
123	C₄×D₄	C₃⋊D₉	1	1	yes	characteristic
124	C₄⋊C₈	—	27	1	no	—
125	C₄×C₈	—	27	1	no	—
126	C₃×C₉	D₄⋊C₈	1	1	yes	characteristic, complemented, Sylow
127	C₂³×C₃	—	2	3	no	—
128	C₂³×C₃	—	2	1	no	—
129	C₃×D₄	—	2	3	no	—
130	C₃×D₄	—	2	1	no	—
131	C₃×D₄	C₉⋊C₈	1	6	yes	complemented
132	C₃×D₄	C₃⋊(C₃⋊C₈)	1	2	yes	—
133	C₂×C₃×C₄	—	2	3	no	—
134	C₂×C₃×C₄	—	2	1	no	—
135	C₂×C₃×C₄	—	4	3	no	—
136	C₂×C₃×C₄	—	4	3	no	—
137	C₂×C₃×C₄	—	4	1	no	—
138	C₂×C₃×C₄	—	4	1	no	—
139	C₂×C₃×C₄	C₂×Dic₉	1	3	yes	—
140	C₂×C₃×C₄	#611	1	3	yes	—
141	C₂×C₃×C₄	#611	1	3	yes	—
142	C₂×C₃×C₄	C₂×C₃⋊Dic₃	1	1	yes	characteristic
143	C₂×C₃×C₄	#638	1	1	yes	characteristic
144	C₂×C₃×C₄	#638	1	1	yes	characteristic
145	C₃⋊C₈	—	18	2	no	—
146	C₃⋊C₈	—	18	6	no	—
147	C₃⋊C₈	—	36	3	no	—
148	C₃⋊C₈	—	36	1	no	—
149	C₂×C₃²	—	4	2	no	—
150	C₂×C₃²	#830	1	1	yes	characteristic
151	C₂×C₃²	#832	1	1	yes	characteristic
152	C₂×C₃²	#837	1	1	yes	characteristic
153	C₂×C₉	—	4	6	no	—
154	C₂×C₉	#830	1	3	yes	—
155	C₂×C₉	#832	1	3	yes	—
156	C₂×C₉	#837	1	3	yes	—
157	C₂×D₄	C₃⋊Dic₉	1	1	yes	characteristic
158	C₂²×C₄	—	2	1	no	—
159	C₂×C₈	—	54	1	no	—
160	C₂×C₈	—	54	1	no	—
161	C₄⋊C₄	C₃⋊Dic₉	1	1	yes	characteristic
162	C₂²⋊C₄	—	2	1	no	—
163	C₄²	C₂×C₃⋊D₉	1	1	yes	characteristic
164	C₂²×C₃	—	2	6	no	—
165	C₂²×C₃	—	2	2	no	—
166	C₂²×C₃	—	4	3	no	—
167	C₂²×C₃	—	4	3	no	—
168	C₂²×C₃	—	4	1	no	—
169	C₂²×C₃	—	4	1	no	—
170	C₂²×C₃	#3676	1	3	yes	—
171	C₂²×C₃	#3757	1	1	yes	characteristic
172	C₃×C₄	—	2	3	no	—
173	C₃×C₄	—	2	1	no	—
174	C₃×C₄	—	4	3	no	—
175	C₃×C₄	—	4	1	no	—
176	C₃×C₄	C₂×C₉⋊C₈	1	3	yes	—
177	C₃×C₄	#3667	1	3	yes	—
178	C₃×C₄	#3672	1	3	yes	—
179	C₃×C₄	#3673	1	3	yes	—
180	C₃×C₄	C₂×C₃⋊(C₃⋊C₈)	1	1	yes	characteristic
181	C₃×C₄	#3748	1	1	yes	characteristic
182	C₃×C₄	#3753	1	1	yes	characteristic
183	C₃×C₄	#3754	1	1	yes	characteristic
184	C₃²	D₄⋊(C₃⋊C₈)	1	1	yes	characteristic
185	C₉	D₄⋊(C₃⋊C₈)	1	3	yes	complemented
186	C₂³	—	2	1	no	—
187	D₄	—	2	1	no	—
188	D₄	C₃⋊(C₉⋊C₈)	1	2	yes	complemented
189	C₂×C₄	—	2	1	no	—
190	C₂×C₄	—	4	1	no	—
191	C₂×C₄	—	4	1	no	—
192	C₂×C₄	C₂×C₃⋊Dic₉	1	1	yes	characteristic
193	C₂×C₄	#6233	1	1	yes	characteristic
194	C₂×C₄	#6233	1	1	yes	characteristic
195	C₈	—	54	2	no	—
196	C₈	—	108	1	no	—
197	C₂×C₃	—	4	6	no	—
198	C₂×C₃	—	4	2	no	—
199	C₂×C₃	#63389	1	3	yes	—
200	C₂×C₃	#63392	1	3	yes	—
201	C₂×C₃	#63396	1	3	yes	—
202	C₂×C₃	#63657	1	1	yes	characteristic
203	C₂×C₃	#63659	1	1	yes	characteristic
204	C₂×C₃	#63664	1	1	yes	characteristic
205	C₂²	—	2	2	no	—
206	C₂²	—	4	1	no	—
207	C₂²	—	4	1	no	—
208	C₂²	#88551	1	1	yes	characteristic
209	C₄	—	2	1	no	—
210	C₄	—	4	1	no	—
211	C₄	C₂×C₃⋊(C₉⋊C₈)	1	1	yes	characteristic
212	C₄	#88532	1	1	yes	characteristic
213	C₄	#88543	1	1	yes	characteristic
214	C₄	#88544	1	1	yes	characteristic
215	C₃	D₄⋊(C₉⋊C₈)	1	3	yes	complemented
216	C₃	D₄⋊(C₃⋊(C₃⋊C₈))	1	1	yes	characteristic
217	C₂	—	4	2	no	—
218	C₂	#11720571	1	1	yes	characteristic
219	C₂	#11720576	1	1	yes	characteristic
220	C₂	#11720585	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#6233
commutator	C₃×C₄×C₉	C₂×C₈
Frattini	C₃×C₄²	C₂×C₃⋊D₃
Fitting	C₃×C₄×D₄×C₉	C₂
socle	C₂²×C₃²	#217
prime radical	C₂×C₃²×D₄	Dic₃
cosocle	C₃×C₄²×C₉	C₂²
hypercenter	C₄×D₄	C₃⋊D₉
hypocenter	C₃×C₉	D₄⋊C₈
LCS	G ▷ C₃×C₄×C₉ ▷ C₂×C₃×C₉ ▷ C₃×C₉
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×D₄

Complements

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

N	H ≅ G/N	type
2. C₄⋊(C₃⋊(C₉⋊C₈))	C₂	semidirect
3. C₄×C₃⋊(C₉⋊C₈)	C₂	semidirect
21. C₃×D₄×C₉	C₈	semidirect
75. D₄×C₉	C₃⋊C₈	semidirect
126. C₃×C₉	D₄⋊C₈	semidirect
131. C₃×D₄	C₉⋊C₈	semidirect
185. C₉	D₄⋊(C₃⋊C₈)	semidirect
188. D₄	C₃⋊(C₉⋊C₈)	semidirect
215. 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
C₄⋊(C₃⋊(C₉⋊C₈))	2
C₄×C₃⋊(C₉⋊C₈)	2
D₄⋊(C₃⋊(C₃⋊C₈))	3
D₄⋊(C₉⋊C₈)	3

Subgroups — D4⋊(C3⋊(C9⋊C8)) #13135896

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₄⋊(C₃⋊(C₉⋊C₈)) #13135896