Subgroups — C₂×C₄⋊(3₋¹⁺²⋊C₈) #13142430

Order 1728.

468 conjugacy classes of subgroups.

The subgroup lattice has 239 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄²×3₋¹⁺²	C₂	1	1	yes	maximal, characteristic
2	C₂²×3₋¹⁺²⋊C₈	C₂	1	2	yes	maximal
3	C₄⋊(3₋¹⁺²⋊C₈)	C₂	1	4	yes	maximal, direct product factor
4	C₂×C₃×C₄⋊(C₃⋊C₈)	—	3	1	no	maximal
5	C₂×C₄⋊(C₉⋊C₈)	C₃	1	1	yes	maximal, characteristic, complemented
6	C₂²×C₄×3₋¹⁺²	C₄	1	1	yes	characteristic
7	C₂²×C₄×3₋¹⁺²	C₄	1	1	yes	characteristic
8	C₂²×C₄×3₋¹⁺²	C₂²	1	1	yes	characteristic
9	C₄²×3₋¹⁺²	C₄	1	2	yes	—
10	C₄²×3₋¹⁺²	C₂²	1	2	yes	—
11	C₂×3₋¹⁺²⋊C₈	—	2	4	no	—
12	C₂×3₋¹⁺²⋊C₈	C₂²	1	4	yes	—
13	C₂×C₃²×C₄²	D₃	1	1	yes	characteristic
14	C₂²×C₃×C₃⋊C₈	—	3	2	no	—
15	C₃×C₄⋊(C₃⋊C₈)	—	3	4	no	—
16	C₂×C₄²×C₉	—	2	1	no	—
17	C₂×C₄²×C₉	C₂×C₃	1	1	yes	characteristic
18	C₂²×C₉⋊C₈	C₂×C₃	1	2	yes	—
19	C₄⋊(C₉⋊C₈)	C₂×C₃	1	4	yes	complemented
20	C₂³×3₋¹⁺²	C₂×C₄	1	1	yes	characteristic
21	C₂×C₄×3₋¹⁺²	—	2	2	no	—
22	C₂×C₄×3₋¹⁺²	C₈	1	4	yes	complemented
23	C₂×C₄×3₋¹⁺²	C₂×C₄	1	2	yes	—
24	C₂×C₄×3₋¹⁺²	C₂×C₄	1	1	yes	characteristic
25	C₂×C₄×3₋¹⁺²	C₂×C₄	1	2	yes	—
26	C₂×C₄×3₋¹⁺²	D₄	1	2	yes	—
27	C₂×C₄×3₋¹⁺²	Q₈	1	2	yes	—
28	C₂×C₄×3₋¹⁺²	C₂³	1	1	yes	characteristic
29	3₋¹⁺²⋊C₈	—	2	4	no	—
30	C₂×C₃×C₄⋊C₈	—	9	1	no	—
31	C₂×C₄⋊(C₃⋊C₈)	—	3	1	no	—
32	C₂²×C₃²×C₄	Dic₃	1	1	yes	characteristic
33	C₂²×C₃²×C₄	Dic₃	1	1	yes	characteristic
34	C₂²×C₃²×C₄	C₂×D₃	1	1	yes	characteristic
35	C₃²×C₄²	Dic₃	1	2	yes	—
36	C₃²×C₄²	C₂×D₃	1	2	yes	—
37	C₂×C₃×C₃⋊C₈	—	3	4	no	—
38	C₂×C₃×C₃⋊C₈	—	6	4	no	—
39	C₂²×C₄×C₉	—	2	1	no	—
40	C₂²×C₄×C₉	—	2	1	no	—
41	C₂²×C₄×C₉	—	2	1	no	—
42	C₂²×C₄×C₉	C₃×C₄	1	1	yes	characteristic
43	C₂²×C₄×C₉	C₃×C₄	1	1	yes	characteristic
44	C₂²×C₄×C₉	C₂²×C₃	1	1	yes	characteristic
45	C₄²×C₉	—	2	2	no	—
46	C₄²×C₉	—	2	2	no	—
47	C₄²×C₉	C₃×C₄	1	2	yes	—
48	C₄²×C₉	C₂²×C₃	1	2	yes	—
49	C₂×C₉⋊C₈	—	2	4	no	—
50	C₂×C₉⋊C₈	C₂²×C₃	1	4	yes	—
51	C₂²×3₋¹⁺²	C₄⋊C₄	1	2	yes	—
52	C₂²×3₋¹⁺²	C₂×C₈	1	2	yes	—
53	C₂²×3₋¹⁺²	M₁₆	1	2	yes	—
54	C₂²×3₋¹⁺²	C₂²×C₄	1	1	yes	characteristic
55	C₄×3₋¹⁺²	—	2	2	no	—
56	C₄×3₋¹⁺²	C₄⋊C₄	1	2	yes	—
57	C₄×3₋¹⁺²	C₂×C₈	1	4	yes	complemented
58	C₄×3₋¹⁺²	C₂×D₄	1	1	yes	characteristic
59	C₄×3₋¹⁺²	C₂×Q₈	1	1	yes	characteristic
60	C₂²×C₃×C₈	—	9	2	no	—
61	C₂×C₃×C₄²	—	3	1	no	—
62	C₂×C₃×C₄²	C₃×D₃	1	1	yes	characteristic
63	C₂²×C₃⋊C₈	—	3	2	no	—
64	C₃×C₄⋊C₈	—	9	4	no	—
65	C₄⋊(C₃⋊C₈)	—	3	4	no	—
66	C₂³×C₃²	C₂×Dic₃	1	1	yes	characteristic
67	C₂×C₃²×C₄	—	2	2	no	—
68	C₂×C₃²×C₄	C₃⋊C₈	1	4	yes	—
69	C₂×C₃²×C₄	Dic₆	1	2	yes	—
70	C₂×C₃²×C₄	D₁₂	1	2	yes	—
71	C₂×C₃²×C₄	C₂×Dic₃	1	2	yes	—
72	C₂×C₃²×C₄	C₂×Dic₃	1	2	yes	—
73	C₂×C₃²×C₄	C₂×Dic₃	1	1	yes	characteristic
74	C₂×C₃²×C₄	C₂²×D₃	1	1	yes	characteristic
75	C₂³×C₉	—	2	1	no	—
76	C₂³×C₉	C₂×C₃×C₄	1	1	yes	characteristic
77	C₃×C₃⋊C₈	—	6	4	no	—
78	C₂×C₄×C₉	—	2	4	no	—
79	C₂×C₄×C₉	—	2	2	no	—
80	C₂×C₄×C₉	—	2	4	no	—
81	C₂×C₄×C₉	—	2	2	no	—
82	C₂×C₄×C₉	—	2	2	no	—
83	C₂×C₄×C₉	—	2	2	no	—
84	C₂×C₄×C₉	—	2	2	no	—
85	C₂×C₄×C₉	—	2	1	no	—
86	C₂×C₄×C₉	—	2	1	no	—
87	C₂×C₄×C₉	C₃×C₈	1	4	yes	complemented
88	C₂×C₄×C₉	C₂×C₃×C₄	1	2	yes	—
89	C₂×C₄×C₉	C₂×C₃×C₄	1	1	yes	characteristic
90	C₂×C₄×C₉	C₂×C₃×C₄	1	2	yes	—
91	C₂×C₄×C₉	C₃×D₄	1	2	yes	—
92	C₂×C₄×C₉	C₃×Q₈	1	2	yes	—
93	C₂×C₄×C₉	C₂³×C₃	1	1	yes	characteristic
94	C₉⋊C₈	—	2	4	no	—
95	C₂×C₄⋊C₈	—	9	1	no	Sylow
96	C₂×3₋¹⁺²	C₄⋊C₈	1	4	yes	complemented
97	C₂×3₋¹⁺²	C₂×C₄⋊C₄	1	1	yes	characteristic
98	C₂×3₋¹⁺²	C₂²×C₈	1	1	yes	characteristic
99	C₂×3₋¹⁺²	C₂×M₁₆	1	1	yes	characteristic
100	C₂²×C₃×C₄	—	3	1	no	—
101	C₂²×C₃×C₄	—	3	1	no	—
102	C₂²×C₃×C₄	—	3	1	no	—
103	C₂²×C₃×C₄	C₃×Dic₃	1	1	yes	characteristic
104	C₂²×C₃×C₄	C₃×Dic₃	1	1	yes	characteristic
105	C₂²×C₃×C₄	C₂×C₃×D₃	1	1	yes	characteristic
106	C₂×C₃×C₈	—	9	4	no	—
107	C₂×C₃×C₈	—	18	4	no	—
108	C₃×C₄²	—	3	2	no	—
109	C₃×C₄²	—	3	2	no	—
110	C₃×C₄²	C₃×Dic₃	1	2	yes	—
111	C₃×C₄²	C₂×C₃×D₃	1	2	yes	—
112	C₂×C₃⋊C₈	—	3	4	no	—
113	C₂×C₃⋊C₈	—	6	4	no	—
114	C₂²×C₃²	C₂×C₃⋊C₈	1	2	yes	—
115	C₂²×C₃²	#208	1	2	yes	—
116	C₂²×C₃²	C₄⋊Dic₃	1	2	yes	—
117	C₂²×C₃²	C₂²×Dic₃	1	1	yes	characteristic
118	C₃²×C₄	—	2	2	no	—
119	C₃²×C₄	C₂×C₃⋊C₈	1	4	yes	—
120	C₃²×C₄	C₄⋊Dic₃	1	2	yes	—
121	C₃²×C₄	C₂×Dic₆	1	1	yes	characteristic
122	C₃²×C₄	C₂×D₁₂	1	1	yes	characteristic
123	C₂²×C₉	—	2	2	no	—
124	C₂²×C₉	—	2	2	no	—
125	C₂²×C₉	—	2	2	no	—
126	C₂²×C₉	—	2	1	no	—
127	C₂²×C₉	C₃×C₄⋊C₄	1	2	yes	—
128	C₂²×C₉	C₂×C₃×C₈	1	2	yes	—
129	C₂²×C₉	C₃×M₁₆	1	2	yes	—
130	C₂²×C₉	C₂²×C₃×C₄	1	1	yes	characteristic
131	C₄×C₉	—	2	4	no	—
132	C₄×C₉	—	2	2	no	—
133	C₄×C₉	—	2	4	no	—
134	C₄×C₉	—	2	2	no	—
135	C₄×C₉	—	2	1	no	—
136	C₄×C₉	—	2	1	no	—
137	C₄×C₉	C₃×C₄⋊C₄	1	2	yes	—
138	C₄×C₉	C₂×C₃×C₈	1	4	yes	complemented
139	C₄×C₉	C₂×C₃×D₄	1	1	yes	characteristic
140	C₄×C₉	C₂×C₃×Q₈	1	1	yes	characteristic
141	C₂²×C₈	—	9	2	no	—
142	C₂×C₄²	AGL(1,ℤ/9)	1	1	yes	characteristic
143	C₄⋊C₈	—	9	4	no	—
144	3₋¹⁺²	C₂×C₄⋊C₈	1	1	yes	characteristic, complemented, Sylow
145	C₂³×C₃	—	3	1	no	—
146	C₂³×C₃	C₂×C₃×Dic₃	1	1	yes	characteristic
147	C₂×C₃×C₄	—	2	2	no	—
148	C₂×C₃×C₄	—	3	4	no	—
149	C₂×C₃×C₄	—	3	2	no	—
150	C₂×C₃×C₄	—	3	2	no	—
151	C₂×C₃×C₄	—	3	2	no	—
152	C₂×C₃×C₄	—	3	1	no	—
153	C₂×C₃×C₄	—	3	2	no	—
154	C₂×C₃×C₄	—	3	1	no	—
155	C₂×C₃×C₄	—	6	2	no	—
156	C₂×C₃×C₄	C₃×C₃⋊C₈	1	4	yes	—
157	C₂×C₃×C₄	C₃×Dic₆	1	2	yes	—
158	C₂×C₃×C₄	C₃×D₁₂	1	2	yes	—
159	C₂×C₃×C₄	C₂×C₃×Dic₃	1	2	yes	—
160	C₂×C₃×C₄	C₂×C₃×Dic₃	1	2	yes	—
161	C₂×C₃×C₄	C₂×C₃×Dic₃	1	1	yes	characteristic
162	C₂×C₃×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
163	C₃×C₈	—	18	4	no	—
164	C₃⋊C₈	—	6	4	no	—
165	C₂×C₃²	C₄⋊(C₃⋊C₈)	1	4	yes	—
166	C₂×C₃²	C₂²×C₃⋊C₈	1	1	yes	characteristic
167	C₂×C₃²	C₂×#208	1	1	yes	characteristic
168	C₂×C₃²	C₂×C₄⋊Dic₃	1	1	yes	characteristic
169	C₂×C₉	—	2	1	no	—
170	C₂×C₉	—	2	1	no	—
171	C₂×C₉	—	2	1	no	—
172	C₂×C₉	—	2	4	no	—
173	C₂×C₉	C₃×C₄⋊C₈	1	4	yes	complemented
174	C₂×C₉	C₂×C₃×C₄⋊C₄	1	1	yes	characteristic
175	C₂×C₉	C₂²×C₃×C₈	1	1	yes	characteristic
176	C₂×C₉	C₂×C₃×M₁₆	1	1	yes	characteristic
177	C₂²×C₄	3₋¹⁺²⋊C₄	1	1	yes	characteristic
178	C₂²×C₄	3₋¹⁺²⋊C₄	1	1	yes	characteristic
179	C₂²×C₄	C₂×AGL(1,ℤ/9)	1	1	yes	characteristic
180	C₂×C₈	—	9	4	no	—
181	C₂×C₈	—	18	4	no	—
182	C₄²	3₋¹⁺²⋊C₄	1	2	yes	—
183	C₄²	C₂×AGL(1,ℤ/9)	1	2	yes	—
184	C₂²×C₃	—	3	1	no	—
185	C₂²×C₃	—	3	2	no	—
186	C₂²×C₃	—	3	2	no	—
187	C₂²×C₃	—	3	2	no	—
188	C₂²×C₃	C₂×C₃×C₃⋊C₈	1	2	yes	—
189	C₂²×C₃	C₃×#208	1	2	yes	—
190	C₂²×C₃	C₃×C₄⋊Dic₃	1	2	yes	—
191	C₂²×C₃	C₂²×C₃×Dic₃	1	1	yes	characteristic
192	C₃×C₄	—	2	2	no	—
193	C₃×C₄	—	3	1	no	—
194	C₃×C₄	—	3	1	no	—
195	C₃×C₄	—	3	4	no	—
196	C₃×C₄	—	3	2	no	—
197	C₃×C₄	—	6	2	no	—
198	C₃×C₄	C₂×C₃×C₃⋊C₈	1	4	yes	—
199	C₃×C₄	C₃×C₄⋊Dic₃	1	2	yes	—
200	C₃×C₄	C₂×C₃×Dic₆	1	1	yes	characteristic
201	C₃×C₄	C₂×C₃×D₁₂	1	1	yes	characteristic
202	C₃²	C₂×C₄⋊(C₃⋊C₈)	1	1	yes	characteristic
203	C₉	—	2	1	no	—
204	C₉	C₂×C₃×C₄⋊C₈	1	1	yes	characteristic, complemented
205	C₂³	C₂×3₋¹⁺²⋊C₄	1	1	yes	characteristic
206	C₂×C₄	—	2	2	no	—
207	C₂×C₄	3₋¹⁺²⋊C₈	1	4	yes	complemented
208	C₂×C₄	3₋¹⁺²⋊Q₈	1	2	yes	—
209	C₂×C₄	C₄⋊AGL(1,ℤ/9)	1	2	yes	—
210	C₂×C₄	C₂×3₋¹⁺²⋊C₄	1	2	yes	—
211	C₂×C₄	C₂×3₋¹⁺²⋊C₄	1	2	yes	—
212	C₂×C₄	C₂×3₋¹⁺²⋊C₄	1	1	yes	characteristic
213	C₂×C₄	C₂²×AGL(1,ℤ/9)	1	1	yes	characteristic
214	C₈	—	18	4	no	—
215	C₂×C₃	—	3	1	no	—
216	C₂×C₃	—	3	1	no	—
217	C₂×C₃	—	3	1	no	—
218	C₂×C₃	—	3	4	no	—
219	C₂×C₃	C₃×C₄⋊(C₃⋊C₈)	1	4	yes	—
220	C₂×C₃	C₂²×C₃×C₃⋊C₈	1	1	yes	characteristic
221	C₂×C₃	C₂×C₃×#208	1	1	yes	characteristic
222	C₂×C₃	C₂×C₃×C₄⋊Dic₃	1	1	yes	characteristic
223	C₂²	C₂×3₋¹⁺²⋊C₈	1	2	yes	—
224	C₂²	#3667⋊C₃	1	2	yes	—
225	C₂²	C₄⋊(3₋¹⁺²⋊C₄)	1	2	yes	—
226	C₂²	C₂²×3₋¹⁺²⋊C₄	1	1	yes	characteristic
227	C₄	—	2	2	no	—
228	C₄	C₂×3₋¹⁺²⋊C₈	1	4	yes	complemented
229	C₄	C₄⋊(3₋¹⁺²⋊C₄)	1	2	yes	—
230	C₄	C₂×3₋¹⁺²⋊Q₈	1	1	yes	characteristic
231	C₄	C₂×C₄⋊AGL(1,ℤ/9)	1	1	yes	characteristic
232	C₃	—	3	1	no	—
233	C₃	C₂×C₃×C₄⋊(C₃⋊C₈)	1	1	yes	characteristic
234	C₂	C₄⋊(3₋¹⁺²⋊C₈)	1	4	yes	direct product factor
235	C₂	C₂²×3₋¹⁺²⋊C₈	1	1	yes	characteristic
236	C₂	C₂×#3667⋊C₃	1	1	yes	characteristic
237	C₂	C₂×C₄⋊(3₋¹⁺²⋊C₄)	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×AGL(1,ℤ/9)
commutator	C₂×C₉	C₂²×C₃×C₈
Frattini	C₂×C₃×C₄	C₂²×C₃×D₃
Fitting	C₂×C₄²×3₋¹⁺²	C₂
socle	C₂³×C₃	C₂×C₃×Dic₃
prime radical	C₂³×C₃²	C₂×Dic₃
cosocle	C₂×C₄×C₉	C₂³×C₃
hypercenter	C₂×C₄²	AGL(1,ℤ/9)
hypocenter	C₉	C₂×C₃×C₄⋊C₈
LCS	G ▷ C₂×C₉ ▷ C₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄²

Complements

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

N	H ≅ G/N	type
3. C₄⋊(3₋¹⁺²⋊C₈)	C₂	direct (N × H)
5. C₂×C₄⋊(C₉⋊C₈)	C₃	semidirect
19. C₄⋊(C₉⋊C₈)	C₂×C₃	semidirect
22. C₂×C₄×3₋¹⁺²	C₈	semidirect
57. C₄×3₋¹⁺²	C₂×C₈	semidirect
87. C₂×C₄×C₉	C₃×C₈	semidirect
96. C₂×3₋¹⁺²	C₄⋊C₈	semidirect
138. C₄×C₉	C₂×C₃×C₈	semidirect
144. 3₋¹⁺²	C₂×C₄⋊C₈	semidirect
173. C₂×C₉	C₃×C₄⋊C₈	semidirect
204. C₉	C₂×C₃×C₄⋊C₈	semidirect
207. C₂×C₄	3₋¹⁺²⋊C₈	semidirect
228. C₄	C₂×3₋¹⁺²⋊C₈	semidirect
234. C₂	C₄⋊(3₋¹⁺²⋊C₈)	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₄²×3₋¹⁺²	2
C₂²×3₋¹⁺²⋊C₈	2
C₄⋊(3₋¹⁺²⋊C₈)	2
C₂×C₃×C₄⋊(C₃⋊C₈)	3
C₂×C₄⋊(C₉⋊C₈)	3

Subgroups — C2×C4⋊(3−1+2⋊C8) #13142430

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₄⋊(3₋¹⁺²⋊C₈) #13142430