Subgroups — C₂×(3₋¹⁺²⋊C₄)⋊C₈ #13143135

Order 1728.

452 conjugacy classes of subgroups.

The subgroup lattice has 256 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₈×3₋¹⁺²	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×3₋¹⁺²⋊C₄	C₂	1	1	yes	maximal, characteristic
3	C₂²×3₋¹⁺²⋊C₈	C₂	1	1	yes	maximal, characteristic
4	(3₋¹⁺²⋊C₄)⋊C₈	C₂	1	4	yes	maximal, direct product factor
5	C₂×C₃×Dic₃⋊C₈	—	3	1	no	maximal
6	C₂×Dic₉⋊C₈	C₃	1	1	yes	maximal, characteristic, complemented
7	C₂²×C₄×3₋¹⁺²	C₂²	1	1	yes	characteristic
8	C₂²×3₋¹⁺²⋊C₄	C₄	1	1	yes	characteristic
9	C₂²×3₋¹⁺²⋊C₄	C₄	1	1	yes	characteristic
10	C₂×C₈×3₋¹⁺²	—	2	2	no	—
11	C₂×C₈×3₋¹⁺²	C₂²	1	2	yes	—
12	C₄×3₋¹⁺²⋊C₄	C₄	1	2	yes	—
13	C₄×3₋¹⁺²⋊C₄	C₂²	1	2	yes	—
14	C₂×3₋¹⁺²⋊C₈	—	2	2	no	—
15	C₂×3₋¹⁺²⋊C₈	C₂²	1	2	yes	—
16	C₂²×C₃²×C₈	D₃	1	1	yes	characteristic
17	C₂×C₃×C₄×Dic₃	—	3	1	no	—
18	C₂²×C₃×C₃⋊C₈	—	3	1	no	—
19	C₃×Dic₃⋊C₈	—	3	4	no	—
20	C₂²×C₈×C₉	—	2	1	no	—
21	C₂²×C₈×C₉	C₂×C₃	1	1	yes	characteristic
22	C₂×C₄×Dic₉	C₂×C₃	1	1	yes	characteristic
23	C₂²×C₉⋊C₈	C₂×C₃	1	1	yes	characteristic
24	Dic₉⋊C₈	C₂×C₃	1	4	yes	complemented
25	C₂³×3₋¹⁺²	C₂×C₄	1	1	yes	characteristic
26	C₂×C₄×3₋¹⁺²	C₂×C₄	1	1	yes	characteristic
27	C₂×C₄×3₋¹⁺²	D₄	1	2	yes	—
28	C₂×C₄×3₋¹⁺²	Q₈	1	2	yes	—
29	C₂×C₄×3₋¹⁺²	C₂³	1	1	yes	characteristic
30	C₂×3₋¹⁺²⋊C₄	—	2	2	no	—
31	C₂×3₋¹⁺²⋊C₄	C₈	1	4	yes	complemented
32	C₂×3₋¹⁺²⋊C₄	C₂×C₄	1	2	yes	—
33	C₂×3₋¹⁺²⋊C₄	C₂×C₄	1	2	yes	—
34	C₈×3₋¹⁺²	—	2	2	no	—
35	3₋¹⁺²⋊C₈	—	2	2	no	—
36	C₂×C₃×C₄⋊C₈	—	9	1	no	—
37	C₂×Dic₃⋊C₈	—	3	1	no	—
38	C₂²×C₃²×C₄	C₂×D₃	1	1	yes	characteristic
39	C₂²×C₃×Dic₃	—	3	1	no	—
40	C₂²×C₃×Dic₃	—	3	1	no	—
41	C₂×C₃²×C₈	—	2	2	no	—
42	C₂×C₃²×C₈	C₂×D₃	1	2	yes	—
43	C₃×C₄×Dic₃	—	3	2	no	—
44	C₃×C₄×Dic₃	—	3	2	no	—
45	C₂×C₃×C₃⋊C₈	—	3	2	no	—
46	C₂×C₃×C₃⋊C₈	—	6	2	no	—
47	C₂²×C₄×C₉	—	2	1	no	—
48	C₂²×C₄×C₉	C₂²×C₃	1	1	yes	characteristic
49	C₂²×Dic₉	C₃×C₄	1	1	yes	characteristic
50	C₂²×Dic₉	C₃×C₄	1	1	yes	characteristic
51	C₂×C₈×C₉	—	2	4	no	—
52	C₂×C₈×C₉	—	2	2	no	—
53	C₂×C₈×C₉	—	2	2	no	—
54	C₂×C₈×C₉	C₂²×C₃	1	2	yes	—
55	C₄×Dic₉	C₃×C₄	1	2	yes	—
56	C₄×Dic₉	C₂²×C₃	1	2	yes	—
57	C₂×C₉⋊C₈	—	2	2	no	—
58	C₂×C₉⋊C₈	C₂²×C₃	1	2	yes	—
59	C₂²×3₋¹⁺²	C₄⋊C₄	1	2	yes	—
60	C₂²×3₋¹⁺²	C₂×C₈	1	2	yes	—
61	C₂²×3₋¹⁺²	M₁₆	1	2	yes	—
62	C₂²×3₋¹⁺²	C₂²×C₄	1	1	yes	characteristic
63	C₄×3₋¹⁺²	C₄⋊C₄	1	2	yes	—
64	C₄×3₋¹⁺²	C₂×D₄	1	1	yes	characteristic
65	C₄×3₋¹⁺²	C₂×Q₈	1	1	yes	characteristic
66	3₋¹⁺²⋊C₄	—	2	2	no	—
67	3₋¹⁺²⋊C₄	C₂×C₈	1	4	yes	complemented
68	C₂²×C₃×C₈	—	3	1	no	—
69	C₂²×C₃×C₈	—	9	1	no	—
70	C₂²×C₃×C₈	C₃×D₃	1	1	yes	characteristic
71	C₂×C₃×C₄²	—	9	1	no	—
72	C₂×C₄×Dic₃	—	3	1	no	—
73	C₂²×C₃⋊C₈	—	3	1	no	—
74	C₃×C₄⋊C₈	—	9	4	no	—
75	Dic₃⋊C₈	—	3	4	no	—
76	C₂³×C₃²	C₄×D₃	1	1	yes	characteristic
77	C₂×C₃²×C₄	Dic₆	1	2	yes	—
78	C₂×C₃²×C₄	C₄×D₃	1	1	yes	characteristic
79	C₂×C₃²×C₄	#67	1	2	yes	—
80	C₂×C₃²×C₄	C₂²×D₃	1	1	yes	characteristic
81	C₂×C₃×Dic₃	—	3	4	no	—
82	C₂×C₃×Dic₃	—	3	2	no	—
83	C₂×C₃×Dic₃	—	3	2	no	—
84	C₂×C₃×Dic₃	—	6	2	no	—
85	C₂³×C₉	—	2	1	no	—
86	C₂³×C₉	C₂×C₃×C₄	1	1	yes	characteristic
87	C₃²×C₈	—	2	2	no	—
88	C₃×C₃⋊C₈	—	6	2	no	—
89	C₂×C₄×C₉	—	2	2	no	—
90	C₂×C₄×C₉	—	2	1	no	—
91	C₂×C₄×C₉	—	2	2	no	—
92	C₂×C₄×C₉	—	2	1	no	—
93	C₂×C₄×C₉	C₂×C₃×C₄	1	1	yes	characteristic
94	C₂×C₄×C₉	C₃×D₄	1	2	yes	—
95	C₂×C₄×C₉	C₃×Q₈	1	2	yes	—
96	C₂×C₄×C₉	C₂³×C₃	1	1	yes	characteristic
97	C₂×Dic₉	—	2	2	no	—
98	C₂×Dic₉	C₃×C₈	1	4	yes	complemented
99	C₂×Dic₉	C₂×C₃×C₄	1	2	yes	—
100	C₂×Dic₉	C₂×C₃×C₄	1	2	yes	—
101	C₈×C₉	—	2	4	no	—
102	C₈×C₉	—	2	2	no	—
103	C₉⋊C₈	—	2	2	no	—
104	C₂×C₄⋊C₈	—	9	1	no	Sylow
105	C₂×3₋¹⁺²	C₄⋊C₈	1	4	yes	complemented
106	C₂×3₋¹⁺²	C₂×C₄⋊C₄	1	1	yes	characteristic
107	C₂×3₋¹⁺²	C₂²×C₈	1	1	yes	characteristic
108	C₂×3₋¹⁺²	C₂×M₁₆	1	1	yes	characteristic
109	C₂²×C₃×C₄	—	3	1	no	—
110	C₂²×C₃×C₄	—	9	1	no	—
111	C₂²×C₃×C₄	—	9	1	no	—
112	C₂²×C₃×C₄	C₂×C₃×D₃	1	1	yes	characteristic
113	C₂²×Dic₃	—	3	1	no	—
114	C₂²×Dic₃	—	3	1	no	—
115	C₂×C₃×C₈	—	2	2	no	—
116	C₂×C₃×C₈	—	3	2	no	—
117	C₂×C₃×C₈	—	6	2	no	—
118	C₂×C₃×C₈	—	9	2	no	—
119	C₂×C₃×C₈	—	18	2	no	—
120	C₂×C₃×C₈	C₂×C₃×D₃	1	2	yes	—
121	C₃×C₄²	—	9	2	no	—
122	C₃×C₄²	—	9	2	no	—
123	C₄×Dic₃	—	3	2	no	—
124	C₄×Dic₃	—	3	2	no	—
125	C₂×C₃⋊C₈	—	3	2	no	—
126	C₂×C₃⋊C₈	—	6	2	no	—
127	C₂²×C₃²	D₃×C₈	1	2	yes	—
128	C₂²×C₃²	ΓL(1,𝔽₂₅)	1	2	yes	—
129	C₂²×C₃²	Dic₃⋊C₄	1	2	yes	—
130	C₂²×C₃²	C₂×C₄×D₃	1	1	yes	characteristic
131	C₃²×C₄	Dic₃⋊C₄	1	2	yes	—
132	C₃²×C₄	C₂×Dic₆	1	1	yes	characteristic
133	C₃²×C₄	C₂×#67	1	1	yes	characteristic
134	C₃×Dic₃	—	3	4	no	—
135	C₃×Dic₃	—	6	2	no	—
136	C₂²×C₉	—	2	2	no	—
137	C₂²×C₉	—	2	2	no	—
138	C₂²×C₉	—	2	2	no	—
139	C₂²×C₉	—	2	1	no	—
140	C₂²×C₉	C₃×C₄⋊C₄	1	2	yes	—
141	C₂²×C₉	C₂×C₃×C₈	1	2	yes	—
142	C₂²×C₉	C₃×M₁₆	1	2	yes	—
143	C₂²×C₉	C₂²×C₃×C₄	1	1	yes	characteristic
144	C₄×C₉	—	2	2	no	—
145	C₄×C₉	—	2	1	no	—
146	C₄×C₉	—	2	1	no	—
147	C₄×C₉	C₃×C₄⋊C₄	1	2	yes	—
148	C₄×C₉	C₂×C₃×D₄	1	1	yes	characteristic
149	C₄×C₉	C₂×C₃×Q₈	1	1	yes	characteristic
150	Dic₉	—	2	2	no	—
151	Dic₉	C₂×C₃×C₈	1	4	yes	complemented
152	C₂²×C₈	—	9	1	no	—
153	C₂²×C₈	AGL(1,ℤ/9)	1	1	yes	characteristic
154	C₂×C₄²	—	9	1	no	—
155	C₄⋊C₈	—	9	4	no	—
156	3₋¹⁺²	C₂×C₄⋊C₈	1	1	yes	characteristic, complemented, Sylow
157	C₂³×C₃	—	3	1	no	—
158	C₂³×C₃	C₃×C₄×D₃	1	1	yes	characteristic
159	C₂×C₃×C₄	—	3	2	no	—
160	C₂×C₃×C₄	—	3	1	no	—
161	C₂×C₃×C₄	—	3	2	no	—
162	C₂×C₃×C₄	—	3	1	no	—
163	C₂×C₃×C₄	—	9	2	no	—
164	C₂×C₃×C₄	—	9	4	no	—
165	C₂×C₃×C₄	—	9	2	no	—
166	C₂×C₃×C₄	—	18	2	no	—
167	C₂×C₃×C₄	C₃×Dic₆	1	2	yes	—
168	C₂×C₃×C₄	C₃×C₄×D₃	1	1	yes	characteristic
169	C₂×C₃×C₄	C₃×#67	1	2	yes	—
170	C₂×C₃×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
171	C₂×Dic₃	—	3	4	no	—
172	C₂×Dic₃	—	3	2	no	—
173	C₂×Dic₃	—	3	2	no	—
174	C₂×Dic₃	—	6	2	no	—
175	C₃×C₈	—	2	2	no	—
176	C₃×C₈	—	6	2	no	—
177	C₃×C₈	—	18	2	no	—
178	C₃⋊C₈	—	6	2	no	—
179	C₂×C₃²	Dic₃⋊C₈	1	4	yes	—
180	C₂×C₃²	C₂×D₃×C₈	1	1	yes	characteristic
181	C₂×C₃²	C₂×ΓL(1,𝔽₂₅)	1	1	yes	characteristic
182	C₂×C₃²	C₂×Dic₃⋊C₄	1	1	yes	characteristic
183	C₂×C₉	—	2	1	no	—
184	C₂×C₉	—	2	1	no	—
185	C₂×C₉	—	2	1	no	—
186	C₂×C₉	—	2	4	no	—
187	C₂×C₉	C₃×C₄⋊C₈	1	4	yes	complemented
188	C₂×C₉	C₂×C₃×C₄⋊C₄	1	1	yes	characteristic
189	C₂×C₉	C₂²×C₃×C₈	1	1	yes	characteristic
190	C₂×C₉	C₂×C₃×M₁₆	1	1	yes	characteristic
191	C₂²×C₄	—	9	1	no	—
192	C₂²×C₄	—	9	1	no	—
193	C₂²×C₄	C₂×AGL(1,ℤ/9)	1	1	yes	characteristic
194	C₂×C₈	—	2	2	no	—
195	C₂×C₈	—	9	2	no	—
196	C₂×C₈	—	18	2	no	—
197	C₂×C₈	C₂×AGL(1,ℤ/9)	1	2	yes	—
198	C₄²	—	9	2	no	—
199	C₄²	—	9	2	no	—
200	C₂²×C₃	—	3	1	no	—
201	C₂²×C₃	—	3	2	no	—
202	C₂²×C₃	—	3	2	no	—
203	C₂²×C₃	—	3	2	no	—
204	C₂²×C₃	C₃×D₃×C₈	1	2	yes	—
205	C₂²×C₃	C₃×ΓL(1,𝔽₂₅)	1	2	yes	—
206	C₂²×C₃	C₃×Dic₃⋊C₄	1	2	yes	—
207	C₂²×C₃	C₂×C₃×C₄×D₃	1	1	yes	characteristic
208	C₃×C₄	—	3	1	no	—
209	C₃×C₄	—	3	1	no	—
210	C₃×C₄	—	3	2	no	—
211	C₃×C₄	—	9	4	no	—
212	C₃×C₄	—	18	2	no	—
213	C₃×C₄	C₃×Dic₃⋊C₄	1	2	yes	—
214	C₃×C₄	C₂×C₃×Dic₆	1	1	yes	characteristic
215	C₃×C₄	C₂×C₃×#67	1	1	yes	characteristic
216	Dic₃	—	3	4	no	—
217	Dic₃	—	6	2	no	—
218	C₃²	C₂×Dic₃⋊C₈	1	1	yes	characteristic
219	C₉	—	2	1	no	—
220	C₉	C₂×C₃×C₄⋊C₈	1	1	yes	characteristic, complemented
221	C₂³	C₄×AGL(1,ℤ/9)	1	1	yes	characteristic
222	C₂×C₄	—	9	2	no	—
223	C₂×C₄	—	9	4	no	—
224	C₂×C₄	—	9	2	no	—
225	C₂×C₄	—	18	2	no	—
226	C₂×C₄	3₋¹⁺²⋊Q₈	1	2	yes	—
227	C₂×C₄	C₄×AGL(1,ℤ/9)	1	1	yes	characteristic
228	C₂×C₄	#611⋊C₃	1	2	yes	—
229	C₂×C₄	C₂²×AGL(1,ℤ/9)	1	1	yes	characteristic
230	C₈	—	2	2	no	—
231	C₈	—	18	2	no	—
232	C₂×C₃	—	3	1	no	—
233	C₂×C₃	—	3	1	no	—
234	C₂×C₃	—	3	1	no	—
235	C₂×C₃	—	3	4	no	—
236	C₂×C₃	C₃×Dic₃⋊C₈	1	4	yes	—
237	C₂×C₃	C₂×C₃×D₃×C₈	1	1	yes	characteristic
238	C₂×C₃	C₂×C₃×ΓL(1,𝔽₂₅)	1	1	yes	characteristic
239	C₂×C₃	C₂×C₃×Dic₃⋊C₄	1	1	yes	characteristic
240	C₂²	C₈×AGL(1,ℤ/9)	1	2	yes	—
241	C₂²	#3663⋊C₃	1	2	yes	—
242	C₂²	(3₋¹⁺²⋊C₄)⋊C₄	1	2	yes	—
243	C₂²	C₂×C₄×AGL(1,ℤ/9)	1	1	yes	characteristic
244	C₄	—	9	4	no	—
245	C₄	—	18	2	no	—
246	C₄	(3₋¹⁺²⋊C₄)⋊C₄	1	2	yes	—
247	C₄	C₂×3₋¹⁺²⋊Q₈	1	1	yes	characteristic
248	C₄	C₂×#611⋊C₃	1	1	yes	characteristic
249	C₃	—	3	1	no	—
250	C₃	C₂×C₃×Dic₃⋊C₈	1	1	yes	characteristic
251	C₂	(3₋¹⁺²⋊C₄)⋊C₈	1	4	yes	direct product factor
252	C₂	C₂×C₈×AGL(1,ℤ/9)	1	1	yes	characteristic
253	C₂	C₂×#3663⋊C₃	1	1	yes	characteristic
254	C₂	C₂×(3₋¹⁺²⋊C₄)⋊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	C₂²×C₈×3₋¹⁺²	C₂
socle	C₂³×C₃	C₃×C₄×D₃
prime radical	C₂³×C₃²	C₄×D₃
cosocle	C₂×C₄×C₉	C₂³×C₃
hypercenter	C₂²×C₈	AGL(1,ℤ/9)
hypocenter	C₉	C₂×C₃×C₄⋊C₈
LCS	G ▷ C₂×C₉ ▷ C₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₈

Complements

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

N	H ≅ G/N	type
4. (3₋¹⁺²⋊C₄)⋊C₈	C₂	direct (N × H)
6. C₂×Dic₉⋊C₈	C₃	semidirect
24. Dic₉⋊C₈	C₂×C₃	semidirect
31. C₂×3₋¹⁺²⋊C₄	C₈	semidirect
67. 3₋¹⁺²⋊C₄	C₂×C₈	semidirect
98. C₂×Dic₉	C₃×C₈	semidirect
105. C₂×3₋¹⁺²	C₄⋊C₈	semidirect
151. Dic₉	C₂×C₃×C₈	semidirect
156. 3₋¹⁺²	C₂×C₄⋊C₈	semidirect
187. C₂×C₉	C₃×C₄⋊C₈	semidirect
220. C₉	C₂×C₃×C₄⋊C₈	semidirect
251. C₂	(3₋¹⁺²⋊C₄)⋊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
C₂²×C₈×3₋¹⁺²	2
C₂×C₄×3₋¹⁺²⋊C₄	2
C₂²×3₋¹⁺²⋊C₈	2
(3₋¹⁺²⋊C₄)⋊C₈	2
C₂×C₃×Dic₃⋊C₈	3
C₂×Dic₉⋊C₈	3

Subgroups — C2×(3−1+2⋊C4)⋊C8 #13143135

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×(3₋¹⁺²⋊C₄)⋊C₈ #13143135