Subgroups — C₄×#88525 #13143756

Order 1728.

670 conjugacy classes of subgroups.

The subgroup lattice has 228 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#88525	C₂	1	1	yes	maximal, characteristic
2	C₄×C₄⋊(C₃⋊D₉)	C₂	1	2	yes	maximal, complemented
3	C₃×C₄×C₈×C₉	C₂	1	1	yes	maximal, characteristic, complemented
4	#11720543	C₂	1	2	yes	maximal, complemented
5	#11720541	C₂	1	1	yes	maximal, characteristic, complemented
6	C₄×#3745	—	3	1	no	maximal
7	C₄×D₇₂	—	3	3	no	maximal
8	C₂×C₄⋊(C₃⋊D₉)	C₂²	1	2	yes	—
9	C₂×C₄×C₃⋊D₉	—	2	2	no	—
10	C₂×C₃×C₈×C₉	C₂²	1	1	yes	characteristic
11	C₂×C₃×C₈×C₉	C₂²	1	1	yes	characteristic
12	C₃×C₄²×C₉	C₂²	1	1	yes	characteristic
13	#88536	—	2	2	no	—
14	C₄⋊(C₃⋊Dic₉)	C₂²	1	2	yes	—
15	#88525	C₄	1	4	yes	direct product factor
16	C₂×#3745	—	3	1	no	—
17	C₄×C₄⋊(C₃⋊D₃)	—	3	2	no	—
18	C₃²×C₄×C₈	D₃	1	1	yes	characteristic
19	#63648	—	3	2	no	—
20	#63645	—	3	1	no	—
21	C₂×D₇₂	—	3	3	no	—
22	C₄×D₃₆	—	3	6	no	—
23	C₄×C₈×C₉	D₃	1	3	yes	complemented
24	#63380	—	3	6	no	—
25	#63378	—	3	3	no	—
26	C₂²×C₃⋊D₉	—	2	2	no	—
27	C₂×C₃×C₄×C₉	D₄	1	1	yes	characteristic
28	C₂×C₃×C₄×C₉	D₄	1	1	yes	characteristic
29	C₂×C₃×C₄×C₉	C₂³	1	1	yes	characteristic
30	C₂×C₃⋊Dic₉	—	2	2	no	—
31	C₄⋊(C₃⋊D₉)	—	2	2	no	—
32	C₄⋊(C₃⋊D₉)	C₂×C₄	1	4	yes	complemented
33	C₄×C₃⋊D₉	—	4	4	no	—
34	C₃×C₈×C₉	—	2	1	no	—
35	C₃×C₈×C₉	C₂×C₄	1	2	yes	complemented
36	C₄×D₂₄	—	9	3	no	—
37	C₄×D₂₄	—	9	1	no	—
38	C₂×C₄⋊(C₃⋊D₃)	—	3	2	no	—
39	C₂×C₄×C₃⋊D₃	—	6	2	no	—
40	C₂×C₃²×C₈	C₂×D₃	1	1	yes	characteristic
41	C₂×C₃²×C₈	C₂×D₃	1	1	yes	characteristic
42	C₃²×C₄²	C₂×D₃	1	1	yes	characteristic
43	#3752	—	6	2	no	—
44	C₄⋊(C₃⋊Dic₃)	—	3	2	no	—
45	#3745	—	3	4	no	—
46	C₂×D₃₆	—	3	6	no	—
47	C₂×C₄×D₉	—	6	6	no	—
48	C₂×C₈×C₉	C₂×D₃	1	3	yes	—
49	C₂×C₈×C₉	C₂×D₃	1	3	yes	—
50	C₄²×C₉	C₂×D₃	1	3	yes	—
51	(C₂×D₉)⋊C₄	—	6	6	no	—
52	C₄⋊Dic₉	—	3	6	no	—
53	D₇₂	—	3	12	no	—
54	C₂²×C₃×C₉	C₂×D₄	1	1	yes	characteristic
55	C₂×C₃⋊D₉	—	2	4	no	—
56	C₂×C₃⋊D₉	—	4	2	no	—
57	C₂×C₃⋊D₉	—	4	2	no	—
58	C₃×C₄×C₉	—	2	1	no	—
59	C₃×C₄×C₉	D₈	1	2	yes	complemented
60	C₃×C₄×C₉	C₂²×C₄	1	1	yes	characteristic
61	C₃×C₄×C₉	#41	1	1	yes	characteristic
62	C₃⋊Dic₉	—	4	2	no	—
63	C₂×D₂₄	—	9	3	no	—
64	C₂×D₂₄	—	9	1	no	—
65	C₄×D₁₂	—	9	6	no	—
66	C₄×D₁₂	—	9	2	no	—
67	C₃×C₄×C₈	D₉	1	3	yes	complemented
68	C₃×C₄×C₈	C₃⋊D₃	1	1	yes	characteristic
69	#821	—	9	6	no	—
70	#821	—	9	2	no	—
71	#818	—	9	3	no	—
72	#818	—	9	1	no	—
73	C₂²×C₃⋊D₃	—	6	2	no	—
74	C₂×C₃²×C₄	D₁₂	1	1	yes	characteristic
75	C₂×C₃²×C₄	D₁₂	1	1	yes	characteristic
76	C₂×C₃²×C₄	C₂²×D₃	1	1	yes	characteristic
77	C₂×C₃⋊Dic₃	—	6	2	no	—
78	C₄⋊(C₃⋊D₃)	—	3	4	no	—
79	C₄⋊(C₃⋊D₃)	—	6	2	no	—
80	C₄×C₃⋊D₃	—	12	4	no	—
81	C₂²×D₉	—	6	6	no	—
82	C₃²×C₈	—	2	1	no	—
83	C₃²×C₈	C₄×D₃	1	2	yes	—
84	C₂×C₄×C₉	D₁₂	1	3	yes	—
85	C₂×C₄×C₉	D₁₂	1	3	yes	—
86	C₂×C₄×C₉	C₂²×D₃	1	3	yes	—
87	C₂×Dic₉	—	6	6	no	—
88	D₃₆	—	3	12	no	—
89	D₃₆	—	6	6	no	—
90	C₄×D₉	—	12	12	no	—
91	C₈×C₉	—	2	3	no	—
92	C₈×C₉	C₄×D₃	1	6	yes	complemented
93	C₄×D₈	—	27	1	no	Sylow
94	C₂×C₃×C₉	C₄×D₄	1	1	yes	characteristic
95	C₂×C₃×C₉	C₂×D₈	1	1	yes	characteristic
96	C₂×C₃×C₉	#135	1	1	yes	characteristic
97	C₃⋊D₉	—	4	4	no	—
98	C₂×D₁₂	—	9	6	no	—
99	C₂×D₁₂	—	9	2	no	—
100	C₂×C₄×D₃	—	18	6	no	—
101	C₂×C₄×D₃	—	18	2	no	—
102	C₂×C₃×C₈	C₂×D₉	1	3	yes	—
103	C₂×C₃×C₈	C₂×D₉	1	3	yes	—
104	C₂×C₃×C₈	C₂×C₃⋊D₃	1	1	yes	characteristic
105	C₂×C₃×C₈	C₂×C₃⋊D₃	1	1	yes	characteristic
106	C₃×C₄²	C₂×D₉	1	3	yes	—
107	C₃×C₄²	C₂×C₃⋊D₃	1	1	yes	characteristic
108	(C₂×D₃)⋊C₄	—	18	6	no	—
109	(C₂×D₃)⋊C₄	—	18	2	no	—
110	C₄⋊Dic₃	—	9	6	no	—
111	C₄⋊Dic₃	—	9	2	no	—
112	D₂₄	—	9	12	no	—
113	D₂₄	—	9	4	no	—
114	C₂²×C₃²	C₂×D₁₂	1	1	yes	characteristic
115	C₂×C₃⋊D₃	—	6	4	no	—
116	C₂×C₃⋊D₃	—	12	2	no	—
117	C₂×C₃⋊D₃	—	12	2	no	—
118	C₃²×C₄	—	2	1	no	—
119	C₃²×C₄	D₂₄	1	2	yes	—
120	C₃²×C₄	C₂×C₄×D₃	1	1	yes	characteristic
121	C₃²×C₄	#235	1	1	yes	characteristic
122	C₃⋊Dic₃	—	12	2	no	—
123	C₂²×C₉	C₂×D₁₂	1	3	yes	—
124	C₂×D₉	—	6	12	no	—
125	C₂×D₉	—	12	6	no	—
126	C₂×D₉	—	12	6	no	—
127	C₄×C₉	—	2	3	no	—
128	C₄×C₉	D₂₄	1	6	yes	complemented
129	C₄×C₉	C₂×C₄×D₃	1	3	yes	—
130	C₄×C₉	#235	1	3	yes	—
131	Dic₉	—	12	6	no	—
132	C₂×D₈	—	27	1	no	—
133	C₄×D₄	—	27	2	no	—
134	#107	—	27	1	no	—
135	#102	—	27	2	no	—
136	C₄×C₈	C₃⋊D₉	1	1	yes	characteristic, complemented
137	C₃×C₉	C₄×D₈	1	1	yes	characteristic, complemented, Sylow
138	C₂²×D₃	—	18	6	no	—
139	C₂²×D₃	—	18	2	no	—
140	C₂×C₃×C₄	D₃₆	1	3	yes	—
141	C₂×C₃×C₄	D₃₆	1	3	yes	—
142	C₂×C₃×C₄	C₂²×D₉	1	3	yes	—
143	C₂×C₃×C₄	C₄⋊(C₃⋊D₃)	1	1	yes	characteristic
144	C₂×C₃×C₄	C₄⋊(C₃⋊D₃)	1	1	yes	characteristic
145	C₂×C₃×C₄	C₂²×C₃⋊D₃	1	1	yes	characteristic
146	C₂×Dic₃	—	18	6	no	—
147	C₂×Dic₃	—	18	2	no	—
148	D₁₂	—	9	12	no	—
149	D₁₂	—	9	4	no	—
150	D₁₂	—	18	6	no	—
151	D₁₂	—	18	2	no	—
152	C₄×D₃	—	36	12	no	—
153	C₄×D₃	—	36	4	no	—
154	C₃×C₈	—	2	3	no	—
155	C₃×C₈	—	2	1	no	—
156	C₃×C₈	C₄×D₉	1	6	yes	complemented
157	C₃×C₈	C₄×C₃⋊D₃	1	2	yes	—
158	C₂×C₃²	C₄×D₁₂	1	1	yes	characteristic
159	C₂×C₃²	C₂×D₂₄	1	1	yes	characteristic
160	C₂×C₃²	#904	1	1	yes	characteristic
161	C₃⋊D₃	—	12	4	no	—
162	C₂×C₉	C₄×D₁₂	1	3	yes	—
163	C₂×C₉	C₂×D₂₄	1	3	yes	—
164	C₂×C₉	#904	1	3	yes	—
165	D₉	—	12	12	no	—
166	C₂×D₄	—	27	2	no	—
167	C₂²×C₄	—	54	2	no	—
168	D₈	—	27	4	no	—
169	C₂×C₈	C₂×C₃⋊D₉	1	1	yes	characteristic
170	C₂×C₈	C₂×C₃⋊D₉	1	1	yes	characteristic
171	C₄⋊C₄	—	27	2	no	—
172	C₂²⋊C₄	—	54	2	no	—
173	C₄²	C₂×C₃⋊D₉	1	1	yes	characteristic
174	C₂²×C₃	C₂×D₃₆	1	3	yes	—
175	C₂²×C₃	C₂×C₄⋊(C₃⋊D₃)	1	1	yes	characteristic
176	C₂×D₃	—	18	4	no	—
177	C₂×D₃	—	18	12	no	—
178	C₂×D₃	—	36	2	no	—
179	C₂×D₃	—	36	2	no	—
180	C₂×D₃	—	36	6	no	—
181	C₂×D₃	—	36	6	no	—
182	C₃×C₄	—	2	1	no	—
183	C₃×C₄	—	2	3	no	—
184	C₃×C₄	D₇₂	1	6	yes	complemented
185	C₃×C₄	C₂×C₄×D₉	1	3	yes	—
186	C₃×C₄	#3697	1	3	yes	—
187	C₃×C₄	#3745	1	2	yes	—
188	C₃×C₄	C₂×C₄×C₃⋊D₃	1	1	yes	characteristic
189	C₃×C₄	#3828	1	1	yes	characteristic
190	Dic₃	—	36	2	no	—
191	Dic₃	—	36	6	no	—
192	C₃²	C₄×D₂₄	1	1	yes	characteristic
193	C₉	C₄×D₂₄	1	3	yes	complemented
194	C₂³	—	54	2	no	—
195	D₄	—	27	4	no	—
196	D₄	—	54	2	no	—
197	C₂×C₄	—	54	2	no	—
198	C₂×C₄	—	108	4	no	—
199	C₂×C₄	C₄⋊(C₃⋊D₉)	1	1	yes	characteristic
200	C₂×C₄	C₄⋊(C₃⋊D₉)	1	1	yes	characteristic
201	C₂×C₄	C₂²×C₃⋊D₉	1	1	yes	characteristic
202	C₈	—	2	1	no	—
203	C₈	C₄×C₃⋊D₉	1	2	yes	complemented
204	C₂×C₃	C₄×D₃₆	1	3	yes	—
205	C₂×C₃	C₂×D₇₂	1	3	yes	—
206	C₂×C₃	#63467	1	3	yes	—
207	C₂×C₃	C₄×C₄⋊(C₃⋊D₃)	1	1	yes	characteristic
208	C₂×C₃	C₂×#3745	1	1	yes	characteristic
209	C₂×C₃	#64113	1	1	yes	characteristic
210	D₃	—	36	4	no	—
211	D₃	—	36	12	no	—
212	C₂²	—	54	4	no	—
213	C₂²	—	108	2	no	—
214	C₂²	—	108	2	no	—
215	C₂²	C₂×C₄⋊(C₃⋊D₉)	1	1	yes	characteristic
216	C₄	—	2	1	no	—
217	C₄	—	108	2	no	—
218	C₄	#88525	1	2	yes	direct product factor
219	C₄	C₂×C₄×C₃⋊D₉	1	1	yes	characteristic
220	C₄	#88736	1	1	yes	characteristic
221	C₃	C₄×D₇₂	1	3	yes	complemented
222	C₃	C₄×#3745	1	1	yes	characteristic
223	C₂	—	108	4	no	—
224	C₂	C₄×C₄⋊(C₃⋊D₉)	1	1	yes	characteristic
225	C₂	C₂×#88525	1	1	yes	characteristic
226	C₂	#11721773	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₄⋊(C₃⋊D₉)
commutator	C₃×C₄×C₉	C₂²×C₄
Frattini	C₂×C₃×C₄	C₂²×C₃⋊D₃
Fitting	C₃×C₄×C₈×C₉	C₂
socle	C₂²×C₃²	C₂×D₁₂
prime radical	C₂×#88525	C₂
cosocle	C₂×C₃×C₄×C₉	C₂³
hypercenter	C₄×C₈	C₃⋊D₉
hypocenter	C₃×C₉	C₄×D₈
LCS	G ▷ C₃×C₄×C₉ ▷ C₂×C₃×C₉ ▷ C₃×C₉
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×C₈

Complements

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

N	H ≅ G/N	type
2. C₄×C₄⋊(C₃⋊D₉)	C₂	semidirect
3. C₃×C₄×C₈×C₉	C₂	semidirect
4. #11720543	C₂	semidirect
5. #11720541	C₂	semidirect
15. #88525	C₄	direct (N × H)
23. C₄×C₈×C₉	D₃	semidirect
32. C₄⋊(C₃⋊D₉)	C₂×C₄	semidirect
35. C₃×C₈×C₉	C₂×C₄	semidirect
59. C₃×C₄×C₉	D₈	semidirect
67. C₃×C₄×C₈	D₉	semidirect
92. C₈×C₉	C₄×D₃	semidirect
128. C₄×C₉	D₂₄	semidirect
136. C₄×C₈	C₃⋊D₉	semidirect
137. C₃×C₉	C₄×D₈	semidirect
156. C₃×C₈	C₄×D₉	semidirect
184. C₃×C₄	D₇₂	semidirect
193. C₉	C₄×D₂₄	semidirect
203. C₈	C₄×C₃⋊D₉	semidirect
218. C₄	#88525	direct (N × H)
221. C₃	C₄×D₇₂	semidirect

Maximal subgroups

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

subgroup	index
C₂×#88525	2
C₄×C₄⋊(C₃⋊D₉)	2
C₃×C₄×C₈×C₉	2
#11720543	2
#11720541	2
C₄×#3745	3
C₄×D₇₂	3

Subgroups — C4×#88525 #13143756

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#88525 #13143756