Subgroups — C₄×C₄⋊(3₋¹⁺²⋊C₄) #13142440

Order 1728.

636 conjugacy classes of subgroups.

The subgroup lattice has 255 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	C₂×C₄⋊(3₋¹⁺²⋊C₄)	C₂	1	2	yes	maximal
3	C₂×C₄×3₋¹⁺²⋊C₄	C₂	1	2	yes	maximal
4	3₋¹⁺²⋊#95	C₂	1	2	yes	maximal
5	C₃×C₄×C₄⋊Dic₃	—	3	1	no	maximal
6	C₄×C₄⋊Dic₉	C₃	1	1	yes	maximal, characteristic, complemented
7	C₂²×C₄×3₋¹⁺²	C₂²	1	1	yes	characteristic
8	C₂²×C₄×3₋¹⁺²	C₂²	1	1	yes	characteristic
9	C₂²×C₄×3₋¹⁺²	C₂²	1	1	yes	characteristic
10	C₂²×3₋¹⁺²⋊C₄	C₂²	1	4	yes	—
11	C₄²×3₋¹⁺²	C₄	1	4	yes	complemented
12	C₄⋊(3₋¹⁺²⋊C₄)	C₄	1	8	yes	direct product factor
13	C₄×3₋¹⁺²⋊C₄	—	2	4	no	—
14	C₂×C₃²×C₄²	D₃	1	1	yes	characteristic
15	C₂×C₃×C₄⋊Dic₃	—	3	2	no	—
16	C₂×C₃×C₄×Dic₃	—	3	2	no	—
17	C₃×C₃⋊#95	—	3	2	no	—
18	C₂×C₄²×C₉	—	2	1	no	—
19	C₂×C₄²×C₉	C₂×C₃	1	1	yes	characteristic
20	C₂×C₄⋊Dic₉	C₂×C₃	1	2	yes	—
21	C₂×C₄×Dic₉	C₂×C₃	1	2	yes	—
22	C₉⋊#95	C₂×C₃	1	2	yes	—
23	C₂³×3₋¹⁺²	C₂³	1	1	yes	characteristic
24	C₂×C₄×3₋¹⁺²	—	2	2	no	—
25	C₂×C₄×3₋¹⁺²	C₂×C₄	1	2	yes	—
26	C₂×C₄×3₋¹⁺²	C₂×C₄	1	4	yes	—
27	C₂×C₄×3₋¹⁺²	C₂×C₄	1	2	yes	—
28	C₂×C₄×3₋¹⁺²	C₂×C₄	1	2	yes	—
29	C₂×C₄×3₋¹⁺²	D₄	1	2	yes	—
30	C₂×C₄×3₋¹⁺²	Q₈	1	2	yes	—
31	C₂×3₋¹⁺²⋊C₄	—	2	4	no	—
32	C₂×3₋¹⁺²⋊C₄	—	2	4	no	—
33	C₂×3₋¹⁺²⋊C₄	C₂×C₄	1	8	yes	—
34	C₃×C₄×C₄⋊C₄	—	9	1	no	—
35	C₄×C₄⋊Dic₃	—	3	1	no	—
36	C₂²×C₃²×C₄	C₂×D₃	1	1	yes	characteristic
37	C₂²×C₃²×C₄	C₂×D₃	1	1	yes	characteristic
38	C₂²×C₃²×C₄	C₂×D₃	1	1	yes	characteristic
39	C₂²×C₃×Dic₃	—	3	4	no	—
40	C₃²×C₄²	Dic₃	1	4	yes	—
41	C₃×C₄⋊Dic₃	—	3	8	no	—
42	C₃×C₄×Dic₃	—	6	4	no	—
43	C₂²×C₄×C₉	—	2	1	no	—
44	C₂²×C₄×C₉	—	2	1	no	—
45	C₂²×C₄×C₉	—	2	1	no	—
46	C₂²×C₄×C₉	C₂²×C₃	1	1	yes	characteristic
47	C₂²×C₄×C₉	C₂²×C₃	1	1	yes	characteristic
48	C₂²×C₄×C₉	C₂²×C₃	1	1	yes	characteristic
49	C₂²×Dic₉	C₂²×C₃	1	4	yes	—
50	C₄²×C₉	—	2	4	no	—
51	C₄²×C₉	C₃×C₄	1	4	yes	complemented
52	C₄⋊Dic₉	C₃×C₄	1	8	yes	complemented
53	C₄×Dic₉	—	2	4	no	—
54	C₂²×3₋¹⁺²	C₂²×C₄	1	1	yes	characteristic
55	C₂²×3₋¹⁺²	C₂²×C₄	1	2	yes	—
56	C₂²×3₋¹⁺²	C₂×D₄	1	1	yes	characteristic
57	C₂²×3₋¹⁺²	C₂×Q₈	1	1	yes	characteristic
58	C₂²×3₋¹⁺²	#41	1	2	yes	—
59	C₄×3₋¹⁺²	—	2	2	no	—
60	C₄×3₋¹⁺²	C₄²	1	4	yes	complemented
61	C₄×3₋¹⁺²	C₄⋊C₄	1	4	yes	complemented
62	3₋¹⁺²⋊C₄	—	2	8	no	—
63	C₂×C₃×C₄⋊C₄	—	9	2	no	—
64	C₂×C₃×C₄²	—	3	1	no	—
65	C₂×C₃×C₄²	—	9	2	no	—
66	C₂×C₃×C₄²	C₃×D₃	1	1	yes	characteristic
67	C₂×C₄⋊Dic₃	—	3	2	no	—
68	C₂×C₄×Dic₃	—	3	2	no	—
69	C₃×#95	—	9	2	no	—
70	C₃⋊#95	—	3	2	no	—
71	C₂³×C₃²	C₂²×D₃	1	1	yes	characteristic
72	C₂×C₃²×C₄	—	2	2	no	—
73	C₂×C₃²×C₄	Dic₆	1	2	yes	—
74	C₂×C₃²×C₄	C₄×D₃	1	4	yes	—
75	C₂×C₃²×C₄	D₁₂	1	2	yes	—
76	C₂×C₃²×C₄	C₂×Dic₃	1	2	yes	—
77	C₂×C₃²×C₄	C₂×Dic₃	1	2	yes	—
78	C₂×C₃²×C₄	C₂×Dic₃	1	2	yes	—
79	C₂×C₃×Dic₃	—	3	8	no	—
80	C₂×C₃×Dic₃	—	6	4	no	—
81	C₂×C₃×Dic₃	—	6	4	no	—
82	C₂³×C₉	—	2	1	no	—
83	C₂³×C₉	C₂³×C₃	1	1	yes	characteristic
84	C₂×C₄×C₉	—	2	4	no	—
85	C₂×C₄×C₉	—	2	2	no	—
86	C₂×C₄×C₉	—	2	2	no	—
87	C₂×C₄×C₉	—	2	4	no	—
88	C₂×C₄×C₉	—	2	2	no	—
89	C₂×C₄×C₉	—	2	2	no	—
90	C₂×C₄×C₉	—	2	2	no	—
91	C₂×C₄×C₉	—	2	2	no	—
92	C₂×C₄×C₉	C₂×C₃×C₄	1	4	yes	—
93	C₂×C₄×C₉	C₂×C₃×C₄	1	2	yes	—
94	C₂×C₄×C₉	C₂×C₃×C₄	1	2	yes	—
95	C₂×C₄×C₉	C₂×C₃×C₄	1	2	yes	—
96	C₂×C₄×C₉	C₃×D₄	1	2	yes	—
97	C₂×C₄×C₉	C₃×Q₈	1	2	yes	—
98	C₂×Dic₉	—	2	4	no	—
99	C₂×Dic₉	—	2	4	no	—
100	C₂×Dic₉	C₂×C₃×C₄	1	8	yes	—
101	C₄×C₄⋊C₄	—	9	1	no	Sylow
102	C₂×3₋¹⁺²	C₂×C₄²	1	1	yes	characteristic
103	C₂×3₋¹⁺²	C₂×C₄⋊C₄	1	1	yes	characteristic
104	C₂×3₋¹⁺²	#117	1	1	yes	characteristic
105	C₂×3₋¹⁺²	C₄×D₄	1	2	yes	—
106	C₂×3₋¹⁺²	C₄×Q₈	1	2	yes	—
107	C₂²×C₃×C₄	—	3	1	no	—
108	C₂²×C₃×C₄	—	3	1	no	—
109	C₂²×C₃×C₄	—	3	1	no	—
110	C₂²×C₃×C₄	—	9	4	no	—
111	C₂²×C₃×C₄	C₂×C₃×D₃	1	1	yes	characteristic
112	C₂²×C₃×C₄	C₂×C₃×D₃	1	1	yes	characteristic
113	C₂²×C₃×C₄	C₂×C₃×D₃	1	1	yes	characteristic
114	C₂²×Dic₃	—	3	4	no	—
115	C₃×C₄⋊C₄	—	9	8	no	—
116	C₃×C₄²	—	3	4	no	—
117	C₃×C₄²	—	18	4	no	—
118	C₃×C₄²	C₃×Dic₃	1	4	yes	—
119	C₄⋊Dic₃	—	3	8	no	—
120	C₄×Dic₃	—	6	4	no	—
121	C₂²×C₃²	C₂×Dic₆	1	1	yes	characteristic
122	C₂²×C₃²	C₂×C₄×D₃	1	2	yes	—
123	C₂²×C₃²	C₂×D₁₂	1	1	yes	characteristic
124	C₂²×C₃²	#235	1	2	yes	—
125	C₂²×C₃²	C₂²×Dic₃	1	1	yes	characteristic
126	C₃²×C₄	—	2	2	no	—
127	C₃²×C₄	C₄×Dic₃	1	4	yes	—
128	C₃²×C₄	C₄⋊Dic₃	1	4	yes	—
129	C₃×Dic₃	—	6	8	no	—
130	C₂²×C₉	—	2	1	no	—
131	C₂²×C₉	—	2	2	no	—
132	C₂²×C₉	—	2	1	no	—
133	C₂²×C₉	—	2	2	no	—
134	C₂²×C₉	—	2	1	no	—
135	C₂²×C₉	C₂²×C₃×C₄	1	2	yes	—
136	C₂²×C₉	C₂²×C₃×C₄	1	1	yes	characteristic
137	C₂²×C₉	C₂×C₃×D₄	1	1	yes	characteristic
138	C₂²×C₉	C₂×C₃×Q₈	1	1	yes	characteristic
139	C₂²×C₉	C₃×#41	1	2	yes	—
140	C₄×C₉	—	2	4	no	—
141	C₄×C₉	—	2	2	no	—
142	C₄×C₉	—	2	4	no	—
143	C₄×C₉	—	2	4	no	—
144	C₄×C₉	C₃×C₄²	1	4	yes	complemented
145	C₄×C₉	C₃×C₄⋊C₄	1	4	yes	complemented
146	Dic₉	—	2	8	no	—
147	C₂×C₄⋊C₄	—	9	2	no	—
148	C₂×C₄²	—	9	2	no	—
149	C₂×C₄²	AGL(1,ℤ/9)	1	1	yes	characteristic
150	#95	—	9	2	no	—
151	3₋¹⁺²	C₄×C₄⋊C₄	1	1	yes	characteristic, complemented, Sylow
152	C₂³×C₃	—	3	1	no	—
153	C₂³×C₃	C₂²×C₃×D₃	1	1	yes	characteristic
154	C₂×C₃×C₄	—	2	2	no	—
155	C₂×C₃×C₄	—	3	2	no	—
156	C₂×C₃×C₄	—	3	4	no	—
157	C₂×C₃×C₄	—	3	2	no	—
158	C₂×C₃×C₄	—	3	2	no	—
159	C₂×C₃×C₄	—	3	2	no	—
160	C₂×C₃×C₄	—	3	2	no	—
161	C₂×C₃×C₄	—	6	2	no	—
162	C₂×C₃×C₄	—	9	8	no	—
163	C₂×C₃×C₄	—	18	4	no	—
164	C₂×C₃×C₄	—	18	4	no	—
165	C₂×C₃×C₄	C₃×Dic₆	1	2	yes	—
166	C₂×C₃×C₄	C₃×C₄×D₃	1	4	yes	—
167	C₂×C₃×C₄	C₃×D₁₂	1	2	yes	—
168	C₂×C₃×C₄	C₂×C₃×Dic₃	1	2	yes	—
169	C₂×C₃×C₄	C₂×C₃×Dic₃	1	2	yes	—
170	C₂×C₃×C₄	C₂×C₃×Dic₃	1	2	yes	—
171	C₂×Dic₃	—	3	8	no	—
172	C₂×Dic₃	—	6	4	no	—
173	C₂×Dic₃	—	6	4	no	—
174	C₂×C₃²	C₄×Dic₆	1	2	yes	—
175	C₂×C₃²	C₄×D₁₂	1	2	yes	—
176	C₂×C₃²	C₂×C₄×Dic₃	1	1	yes	characteristic
177	C₂×C₃²	C₂×C₄⋊Dic₃	1	1	yes	characteristic
178	C₂×C₃²	#926	1	1	yes	characteristic
179	C₂×C₉	—	2	1	no	—
180	C₂×C₉	—	2	1	no	—
181	C₂×C₉	—	2	1	no	—
182	C₂×C₉	—	2	2	no	—
183	C₂×C₉	—	2	2	no	—
184	C₂×C₉	C₂×C₃×C₄²	1	1	yes	characteristic
185	C₂×C₉	C₂×C₃×C₄⋊C₄	1	1	yes	characteristic
186	C₂×C₉	C₃×#117	1	1	yes	characteristic
187	C₂×C₉	C₃×C₄×D₄	1	2	yes	—
188	C₂×C₉	C₃×C₄×Q₈	1	2	yes	—
189	C₂²×C₄	—	9	4	no	—
190	C₂²×C₄	C₂×AGL(1,ℤ/9)	1	1	yes	characteristic
191	C₂²×C₄	C₂×AGL(1,ℤ/9)	1	1	yes	characteristic
192	C₂²×C₄	C₂×AGL(1,ℤ/9)	1	1	yes	characteristic
193	C₄⋊C₄	—	9	8	no	—
194	C₄²	—	18	4	no	—
195	C₄²	3₋¹⁺²⋊C₄	1	4	yes	complemented
196	C₂²×C₃	—	3	1	no	—
197	C₂²×C₃	—	3	2	no	—
198	C₂²×C₃	—	3	1	no	—
199	C₂²×C₃	—	3	2	no	—
200	C₂²×C₃	—	3	1	no	—
201	C₂²×C₃	C₂×C₃×Dic₆	1	1	yes	characteristic
202	C₂²×C₃	C₂×C₃×C₄×D₃	1	2	yes	—
203	C₂²×C₃	C₂×C₃×D₁₂	1	1	yes	characteristic
204	C₂²×C₃	C₃×#235	1	2	yes	—
205	C₂²×C₃	C₂²×C₃×Dic₃	1	1	yes	characteristic
206	C₃×C₄	—	2	2	no	—
207	C₃×C₄	—	3	4	no	—
208	C₃×C₄	—	3	4	no	—
209	C₃×C₄	—	6	2	no	—
210	C₃×C₄	—	18	8	no	—
211	C₃×C₄	C₃×C₄×Dic₃	1	4	yes	—
212	C₃×C₄	C₃×C₄⋊Dic₃	1	4	yes	—
213	Dic₃	—	6	8	no	—
214	C₃²	C₄×C₄⋊Dic₃	1	1	yes	characteristic
215	C₉	—	2	1	no	—
216	C₉	C₃×C₄×C₄⋊C₄	1	1	yes	characteristic, complemented
217	C₂³	C₂²×AGL(1,ℤ/9)	1	1	yes	characteristic
218	C₂×C₄	—	2	2	no	—
219	C₂×C₄	—	9	8	no	—
220	C₂×C₄	—	18	4	no	—
221	C₂×C₄	—	18	4	no	—
222	C₂×C₄	3₋¹⁺²⋊Q₈	1	2	yes	—
223	C₂×C₄	C₄×AGL(1,ℤ/9)	1	4	yes	—
224	C₂×C₄	C₄⋊AGL(1,ℤ/9)	1	2	yes	—
225	C₂×C₄	C₂×3₋¹⁺²⋊C₄	1	2	yes	—
226	C₂×C₄	C₂×3₋¹⁺²⋊C₄	1	2	yes	—
227	C₂×C₄	C₂×3₋¹⁺²⋊C₄	1	2	yes	—
228	C₂×C₃	—	3	1	no	—
229	C₂×C₃	—	3	1	no	—
230	C₂×C₃	—	3	1	no	—
231	C₂×C₃	—	3	2	no	—
232	C₂×C₃	—	3	2	no	—
233	C₂×C₃	C₃×C₄×Dic₆	1	2	yes	—
234	C₂×C₃	C₃×C₄×D₁₂	1	2	yes	—
235	C₂×C₃	C₂×C₃×C₄×Dic₃	1	1	yes	characteristic
236	C₂×C₃	C₂×C₃×C₄⋊Dic₃	1	1	yes	characteristic
237	C₂×C₃	C₃×#926	1	1	yes	characteristic
238	C₂²	C₂×3₋¹⁺²⋊Q₈	1	1	yes	characteristic
239	C₂²	C₂×C₄×AGL(1,ℤ/9)	1	2	yes	—
240	C₂²	C₂×C₄⋊AGL(1,ℤ/9)	1	1	yes	characteristic
241	C₂²	#3697⋊C₃	1	2	yes	—
242	C₂²	C₂²×3₋¹⁺²⋊C₄	1	1	yes	characteristic
243	C₄	—	2	2	no	—
244	C₄	—	18	8	no	—
245	C₄	C₄×3₋¹⁺²⋊C₄	1	4	yes	complemented
246	C₄	C₄⋊(3₋¹⁺²⋊C₄)	1	4	yes	direct product factor
247	C₃	—	3	1	no	—
248	C₃	C₃×C₄×C₄⋊Dic₃	1	1	yes	characteristic
249	C₂	C₄×3₋¹⁺²⋊Q₈	1	2	yes	—
250	C₂	C₄×C₄⋊AGL(1,ℤ/9)	1	2	yes	—
251	C₂	C₂×C₄×3₋¹⁺²⋊C₄	1	1	yes	characteristic
252	C₂	C₂×C₄⋊(3₋¹⁺²⋊C₄)	1	1	yes	characteristic
253	C₂	#63488⋊C₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×AGL(1,ℤ/9)
commutator	C₂×C₉	C₂×C₃×C₄²
Frattini, socle	C₂³×C₃	C₂²×C₃×D₃
Fitting	C₂×C₄²×3₋¹⁺²	C₂
prime radical	C₂³×C₃²	C₂²×D₃
cosocle	C₂³×C₉	C₂³×C₃
hypercenter	C₂×C₄²	AGL(1,ℤ/9)
hypocenter	C₉	C₃×C₄×C₄⋊C₄
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
6. C₄×C₄⋊Dic₉	C₃	semidirect
11. C₄²×3₋¹⁺²	C₄	semidirect
12. C₄⋊(3₋¹⁺²⋊C₄)	C₄	direct (N × H)
51. C₄²×C₉	C₃×C₄	semidirect
52. C₄⋊Dic₉	C₃×C₄	semidirect
60. C₄×3₋¹⁺²	C₄²	semidirect
61. C₄×3₋¹⁺²	C₄⋊C₄	semidirect
144. C₄×C₉	C₃×C₄²	semidirect
145. C₄×C₉	C₃×C₄⋊C₄	semidirect
151. 3₋¹⁺²	C₄×C₄⋊C₄	semidirect
195. C₄²	3₋¹⁺²⋊C₄	semidirect
216. C₉	C₃×C₄×C₄⋊C₄	semidirect
245. C₄	C₄×3₋¹⁺²⋊C₄	semidirect
246. C₄	C₄⋊(3₋¹⁺²⋊C₄)	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
C₂×C₄⋊(3₋¹⁺²⋊C₄)	2
C₂×C₄×3₋¹⁺²⋊C₄	2
3₋¹⁺²⋊#95	2
C₃×C₄×C₄⋊Dic₃	3
C₄×C₄⋊Dic₉	3

Subgroups — C4×C4⋊(3−1+2⋊C4) #13142440

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×C₄⋊(3₋¹⁺²⋊C₄) #13142440