Subgroups — C₄×#88551 #13144799

Order 1728.

1290 conjugacy classes of subgroups.

The subgroup lattice has 258 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₃×C₄×C₉	C₂	1	1	yes	maximal, characteristic
2	C₂×#88551	C₂	1	2	yes	maximal
3	C₂×C₄×C₃⋊Dic₉	C₂	1	2	yes	maximal, complemented
4	#11720572	C₂	1	2	yes	maximal, complemented
5	C₄×#3757	—	3	1	no	maximal
6	C₄×#3676	—	3	3	no	maximal
7	C₂⁴×C₃×C₉	C₂²	1	1	yes	characteristic
8	C₂²×C₃×C₄×C₉	—	2	4	no	—
9	C₂²×C₃×C₄×C₉	C₄	1	4	yes	complemented
10	C₂²×C₃×C₄×C₉	C₂²	1	1	yes	characteristic
11	C₂²×C₃×C₄×C₉	C₂²	1	1	yes	characteristic
12	C₂²×C₃⋊Dic₉	C₂²	1	4	yes	—
13	#88551	C₄	1	8	yes	direct product factor
14	C₄×C₃⋊Dic₉	—	2	4	no	—
15	C₂³×C₃²×C₄	D₃	1	1	yes	characteristic
16	C₂×#3757	—	3	2	no	—
17	C₂×C₄×C₃⋊Dic₃	—	3	2	no	—
18	C₂³×C₄×C₉	D₃	1	3	yes	—
19	#63658	—	3	2	no	—
20	C₂×#3676	—	3	6	no	—
21	C₂×C₄×Dic₉	—	3	6	no	—
22	C₉⋊#95	—	3	6	no	—
23	C₂³×C₃×C₉	—	2	2	no	—
24	C₂³×C₃×C₉	—	2	2	no	—
25	C₂³×C₃×C₉	C₂×C₄	1	4	yes	—
26	C₂³×C₃×C₉	C₂×C₄	1	2	yes	—
27	C₂³×C₃×C₉	C₂³	1	1	yes	characteristic
28	C₂×C₃×C₄×C₉	—	2	2	no	—
29	C₂×C₃×C₄×C₉	—	2	8	no	—
30	C₂×C₃×C₄×C₉	C₂×C₄	1	2	yes	—
31	C₂×C₃×C₄×C₉	C₂×C₄	1	2	yes	—
32	C₂×C₃×C₄×C₉	D₄	1	4	yes	—
33	C₂×C₃⋊Dic₉	—	2	4	no	—
34	C₂×C₃⋊Dic₉	—	2	4	no	—
35	C₂×C₃⋊Dic₉	C₂×C₄	1	8	yes	complemented
36	C₄×#217	—	9	3	no	—
37	C₄×#217	—	9	1	no	—
38	C₂⁴×C₃²	C₂×D₃	1	1	yes	characteristic
39	C₂²×C₃²×C₄	—	2	4	no	—
40	C₂²×C₃²×C₄	Dic₃	1	4	yes	—
41	C₂²×C₃²×C₄	C₂×D₃	1	1	yes	characteristic
42	C₂²×C₃²×C₄	C₂×D₃	1	1	yes	characteristic
43	C₂²×C₃⋊Dic₃	—	3	4	no	—
44	C₂⁴×C₉	C₂×D₃	1	3	yes	—
45	#3757	—	3	8	no	—
46	C₄×C₃⋊Dic₃	—	6	4	no	—
47	C₂²×C₄×C₉	—	2	12	no	—
48	C₂²×C₄×C₉	Dic₃	1	12	yes	complemented
49	C₂²×C₄×C₉	C₂×D₃	1	3	yes	—
50	C₂²×C₄×C₉	C₂×D₃	1	3	yes	—
51	C₂²×Dic₉	—	3	12	no	—
52	#3676	—	3	24	no	—
53	C₄×Dic₉	—	6	12	no	—
54	C₂²×C₃×C₉	—	2	2	no	—
55	C₂²×C₃×C₉	—	2	2	no	—
56	C₂²×C₃×C₉	—	2	8	no	—
57	C₂²×C₃×C₉	C₄²	1	4	yes	complemented
58	C₂²×C₃×C₉	C₂²×C₄	1	1	yes	characteristic
59	C₂²×C₃×C₉	C₂²×C₄	1	2	yes	—
60	C₂²×C₃×C₉	C₂×D₄	1	2	yes	—
61	C₂²×C₃×C₉	#41	1	2	yes	—
62	C₃×C₄×C₉	—	2	2	no	—
63	C₃×C₄×C₉	C₂²⋊C₄	1	4	yes	complemented
64	C₃⋊Dic₉	—	2	8	no	—
65	C₂³×C₃×C₄	D₉	1	3	yes	—
66	C₂³×C₃×C₄	C₃⋊D₃	1	1	yes	characteristic
67	C₂×#217	—	9	6	no	—
68	C₂×#217	—	9	2	no	—
69	C₂×C₄×Dic₃	—	9	6	no	—
70	C₂×C₄×Dic₃	—	9	2	no	—
71	C₃⋊#95	—	9	6	no	—
72	C₃⋊#95	—	9	2	no	—
73	C₂³×C₃²	—	2	2	no	—
74	C₂³×C₃²	—	2	2	no	—
75	C₂³×C₃²	C₄×D₃	1	4	yes	—
76	C₂³×C₃²	C₂×Dic₃	1	2	yes	—
77	C₂³×C₃²	C₂²×D₃	1	1	yes	characteristic
78	C₂×C₃²×C₄	—	2	2	no	—
79	C₂×C₃²×C₄	—	2	8	no	—
80	C₂×C₃²×C₄	C₂×Dic₃	1	2	yes	—
81	C₂×C₃²×C₄	C₂×Dic₃	1	2	yes	—
82	C₂×C₃²×C₄	#67	1	4	yes	—
83	C₂×C₃⋊Dic₃	—	3	8	no	—
84	C₂×C₃⋊Dic₃	—	6	4	no	—
85	C₂×C₃⋊Dic₃	—	6	4	no	—
86	C₂³×C₉	—	2	6	no	—
87	C₂³×C₉	—	2	6	no	—
88	C₂³×C₉	C₄×D₃	1	12	yes	—
89	C₂³×C₉	C₂×Dic₃	1	6	yes	—
90	C₂³×C₉	C₂²×D₃	1	3	yes	—
91	C₂×C₄×C₉	—	2	6	no	—
92	C₂×C₄×C₉	—	2	24	no	—
93	C₂×C₄×C₉	C₂×Dic₃	1	6	yes	—
94	C₂×C₄×C₉	C₂×Dic₃	1	6	yes	—
95	C₂×C₄×C₉	#67	1	12	yes	—
96	C₂×Dic₉	—	3	24	no	—
97	C₂×Dic₉	—	6	12	no	—
98	C₂×Dic₉	—	6	12	no	—
99	C₄×C₂²⋊C₄	—	27	1	no	Sylow
100	C₂×C₃×C₉	—	2	4	no	—
101	C₂×C₃×C₉	C₂×C₄²	1	1	yes	characteristic
102	C₂×C₃×C₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
103	C₂×C₃×C₉	#117	1	1	yes	characteristic
104	C₂×C₃×C₉	C₄×D₄	1	4	yes	—
105	C₂⁴×C₃	C₂×D₉	1	3	yes	—
106	C₂⁴×C₃	C₂×C₃⋊D₃	1	1	yes	characteristic
107	C₂²×C₃×C₄	—	2	12	no	—
108	C₂²×C₃×C₄	—	2	4	no	—
109	C₂²×C₃×C₄	Dic₉	1	12	yes	complemented
110	C₂²×C₃×C₄	C₂×D₉	1	3	yes	—
111	C₂²×C₃×C₄	C₂×D₉	1	3	yes	—
112	C₂²×C₃×C₄	C₃⋊Dic₃	1	4	yes	—
113	C₂²×C₃×C₄	C₂×C₃⋊D₃	1	1	yes	characteristic
114	C₂²×C₃×C₄	C₂×C₃⋊D₃	1	1	yes	characteristic
115	C₂²×Dic₃	—	9	12	no	—
116	C₂²×Dic₃	—	9	4	no	—
117	#217	—	9	24	no	—
118	#217	—	9	8	no	—
119	C₄×Dic₃	—	18	12	no	—
120	C₄×Dic₃	—	18	4	no	—
121	C₂²×C₃²	—	2	2	no	—
122	C₂²×C₃²	—	2	8	no	—
123	C₂²×C₃²	—	2	2	no	—
124	C₂²×C₃²	C₄×Dic₃	1	4	yes	—
125	C₂²×C₃²	C₂×C₄×D₃	1	2	yes	—
126	C₂²×C₃²	#235	1	2	yes	—
127	C₂²×C₃²	C₂²×Dic₃	1	1	yes	characteristic
128	C₂²×C₃²	C₂×#67	1	2	yes	—
129	C₃²×C₄	—	2	2	no	—
130	C₃²×C₄	#217	1	4	yes	—
131	C₃⋊Dic₃	—	6	8	no	—
132	C₂²×C₉	—	2	6	no	—
133	C₂²×C₉	—	2	24	no	—
134	C₂²×C₉	—	2	6	no	—
135	C₂²×C₉	C₄×Dic₃	1	12	yes	complemented
136	C₂²×C₉	C₂×C₄×D₃	1	6	yes	—
137	C₂²×C₉	#235	1	6	yes	—
138	C₂²×C₉	C₂²×Dic₃	1	3	yes	—
139	C₂²×C₉	C₂×#67	1	6	yes	—
140	C₄×C₉	—	2	6	no	—
141	C₄×C₉	#217	1	12	yes	complemented
142	Dic₉	—	6	24	no	—
143	C₂³×C₄	C₃⋊D₉	1	1	yes	characteristic
144	C₂×C₂²⋊C₄	—	27	2	no	—
145	C₂×C₄²	—	27	2	no	—
146	#95	—	27	2	no	—
147	C₃×C₉	C₄×C₂²⋊C₄	1	1	yes	characteristic, complemented, Sylow
148	C₂³×C₃	—	2	6	no	—
149	C₂³×C₃	—	2	6	no	—
150	C₂³×C₃	—	2	2	no	—
151	C₂³×C₃	—	2	2	no	—
152	C₂³×C₃	C₄×D₉	1	12	yes	—
153	C₂³×C₃	C₂×Dic₉	1	6	yes	—
154	C₂³×C₃	C₂²×D₉	1	3	yes	—
155	C₂³×C₃	C₄×C₃⋊D₃	1	4	yes	—
156	C₂³×C₃	C₂×C₃⋊Dic₃	1	2	yes	—
157	C₂³×C₃	C₂²×C₃⋊D₃	1	1	yes	characteristic
158	C₂×C₃×C₄	—	2	6	no	—
159	C₂×C₃×C₄	—	2	2	no	—
160	C₂×C₃×C₄	—	2	24	no	—
161	C₂×C₃×C₄	—	2	8	no	—
162	C₂×C₃×C₄	C₂×Dic₉	1	6	yes	—
163	C₂×C₃×C₄	C₂×Dic₉	1	6	yes	—
164	C₂×C₃×C₄	#611	1	12	yes	—
165	C₂×C₃×C₄	C₂×C₃⋊Dic₃	1	2	yes	—
166	C₂×C₃×C₄	C₂×C₃⋊Dic₃	1	2	yes	—
167	C₂×C₃×C₄	#638	1	4	yes	—
168	C₂×Dic₃	—	9	24	no	—
169	C₂×Dic₃	—	9	8	no	—
170	C₂×Dic₃	—	18	12	no	—
171	C₂×Dic₃	—	18	12	no	—
172	C₂×Dic₃	—	18	4	no	—
173	C₂×Dic₃	—	18	4	no	—
174	C₂×C₃²	—	2	4	no	—
175	C₂×C₃²	C₂×C₄×Dic₃	1	1	yes	characteristic
176	C₂×C₃²	#926	1	1	yes	characteristic
177	C₂×C₃²	C₄×#67	1	4	yes	—
178	C₂×C₃²	C₂×#217	1	1	yes	characteristic
179	C₂×C₉	—	2	12	no	—
180	C₂×C₉	C₂×C₄×Dic₃	1	3	yes	—
181	C₂×C₉	#926	1	3	yes	—
182	C₂×C₉	C₄×#67	1	12	yes	—
183	C₂×C₉	C₂×#217	1	3	yes	—
184	C₂⁴	C₂×C₃⋊D₉	1	1	yes	characteristic
185	C₂²×C₄	—	2	4	no	—
186	C₂²×C₄	—	27	4	no	—
187	C₂²×C₄	C₃⋊Dic₉	1	4	yes	complemented
188	C₂²×C₄	C₂×C₃⋊D₉	1	1	yes	characteristic
189	C₂²×C₄	C₂×C₃⋊D₉	1	1	yes	characteristic
190	C₂²⋊C₄	—	27	8	no	—
191	C₄²	—	54	4	no	—
192	C₂²×C₃	—	2	6	no	—
193	C₂²×C₃	—	2	24	no	—
194	C₂²×C₃	—	2	6	no	—
195	C₂²×C₃	—	2	2	no	—
196	C₂²×C₃	—	2	8	no	—
197	C₂²×C₃	—	2	2	no	—
198	C₂²×C₃	C₄×Dic₉	1	12	yes	complemented
199	C₂²×C₃	C₂×C₄×D₉	1	6	yes	—
200	C₂²×C₃	#3697	1	6	yes	—
201	C₂²×C₃	C₂²×Dic₉	1	3	yes	—
202	C₂²×C₃	C₂×#611	1	6	yes	—
203	C₂²×C₃	C₄×C₃⋊Dic₃	1	4	yes	—
204	C₂²×C₃	C₂×C₄×C₃⋊D₃	1	2	yes	—
205	C₂²×C₃	#3828	1	2	yes	—
206	C₂²×C₃	C₂²×C₃⋊Dic₃	1	1	yes	characteristic
207	C₂²×C₃	C₂×#638	1	2	yes	—
208	C₃×C₄	—	2	6	no	—
209	C₃×C₄	—	2	2	no	—
210	C₃×C₄	#3676	1	12	yes	complemented
211	C₃×C₄	#3757	1	4	yes	—
212	Dic₃	—	18	24	no	—
213	Dic₃	—	18	8	no	—
214	C₃²	C₄×#217	1	1	yes	characteristic
215	C₉	C₄×#217	1	3	yes	complemented
216	C₂³	—	2	2	no	—
217	C₂³	—	2	2	no	—
218	C₂³	C₄×C₃⋊D₉	1	4	yes	—
219	C₂³	C₂×C₃⋊Dic₉	1	2	yes	—
220	C₂³	C₂²×C₃⋊D₉	1	1	yes	characteristic
221	C₂×C₄	—	2	2	no	—
222	C₂×C₄	—	2	8	no	—
223	C₂×C₄	—	27	8	no	—
224	C₂×C₄	—	54	4	no	—
225	C₂×C₄	—	54	4	no	—
226	C₂×C₄	C₂×C₃⋊Dic₉	1	2	yes	—
227	C₂×C₄	C₂×C₃⋊Dic₉	1	2	yes	—
228	C₂×C₄	#6233	1	4	yes	—
229	C₂×C₃	—	2	4	no	—
230	C₂×C₃	—	2	12	no	—
231	C₂×C₃	C₂×C₄×Dic₉	1	3	yes	—
232	C₂×C₃	#63488	1	3	yes	—
233	C₂×C₃	C₄×#611	1	12	yes	—
234	C₂×C₃	C₂×#3676	1	3	yes	—
235	C₂×C₃	C₂×C₄×C₃⋊Dic₃	1	1	yes	characteristic
236	C₂×C₃	#64135	1	1	yes	characteristic
237	C₂×C₃	C₄×#638	1	4	yes	—
238	C₂×C₃	C₂×#3757	1	1	yes	characteristic
239	C₂²	—	2	2	no	—
240	C₂²	—	2	8	no	—
241	C₂²	—	2	2	no	—
242	C₂²	C₄×C₃⋊Dic₉	1	4	yes	complemented
243	C₂²	C₂×C₄×C₃⋊D₉	1	2	yes	—
244	C₂²	#88736	1	2	yes	—
245	C₂²	C₂²×C₃⋊Dic₉	1	1	yes	characteristic
246	C₂²	C₂×#6233	1	2	yes	—
247	C₄	—	2	2	no	—
248	C₄	—	54	8	no	—
249	C₄	#88551	1	4	yes	direct product factor
250	C₃	C₄×#3676	1	3	yes	complemented
251	C₃	C₄×#3757	1	1	yes	characteristic
252	C₂	—	2	4	no	—
253	C₂	C₂×C₄×C₃⋊Dic₉	1	1	yes	characteristic
254	C₂	#11721815	1	1	yes	characteristic
255	C₂	C₄×#6233	1	4	yes	—
256	C₂	C₂×#88551	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×C₃⋊D₉
commutator	C₂×C₃×C₉	C₂×C₄²
Frattini	C₂³×C₃	C₂²×C₃⋊D₃
Fitting	C₂³×C₃×C₄×C₉	C₂
socle	C₂³×C₃²	C₂²×D₃
prime radical	C₂⁴×C₃²	C₂×D₃
cosocle	C₂³×C₃×C₉	C₂³
hypercenter	C₂³×C₄	C₃⋊D₉
hypocenter	C₃×C₉	C₄×C₂²⋊C₄
LCS	G ▷ C₂×C₃×C₉ ▷ C₃×C₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₂³×C₄

Complements

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

N	H ≅ G/N	type
3. C₂×C₄×C₃⋊Dic₉	C₂	semidirect
4. #11720572	C₂	semidirect
9. C₂²×C₃×C₄×C₉	C₄	semidirect
13. #88551	C₄	direct (N × H)
35. C₂×C₃⋊Dic₉	C₂×C₄	semidirect
48. C₂²×C₄×C₉	Dic₃	semidirect
57. C₂²×C₃×C₉	C₄²	semidirect
63. C₃×C₄×C₉	C₂²⋊C₄	semidirect
109. C₂²×C₃×C₄	Dic₉	semidirect
135. C₂²×C₉	C₄×Dic₃	semidirect
141. C₄×C₉	#217	semidirect
147. C₃×C₉	C₄×C₂²⋊C₄	semidirect
187. C₂²×C₄	C₃⋊Dic₉	semidirect
198. C₂²×C₃	C₄×Dic₉	semidirect
210. C₃×C₄	#3676	semidirect
215. C₉	C₄×#217	semidirect
242. C₂²	C₄×C₃⋊Dic₉	semidirect
249. C₄	#88551	direct (N × H)
250. C₃	C₄×#3676	semidirect

Maximal subgroups

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

subgroup	index
C₂³×C₃×C₄×C₉	2
C₂×#88551	2
C₂×C₄×C₃⋊Dic₉	2
#11720572	2
C₄×#3757	3
C₄×#3676	3

Subgroups — C4×#88551 #13144799

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#88551 #13144799