Subgroups — C₄×#88521 #13143746

Order 1728.

550 conjugacy classes of subgroups.

The subgroup lattice has 204 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#88521	C₂	1	1	yes	maximal, characteristic
2	C₄×#6226	C₂	1	2	yes	maximal
3	C₃×C₄×C₈×C₉	C₂	1	1	yes	maximal, characteristic
4	#11720541	C₂	1	1	yes	maximal, characteristic
5	#11720539	C₂	1	2	yes	maximal
6	C₄×#3746	—	3	1	no	maximal
7	C₄×Dic₃₆	—	3	3	no	maximal
8	C₂×#6226	C₂²	1	2	yes	—
9	C₂×C₃×C₈×C₉	C₂²	1	1	yes	characteristic
10	C₂×C₃×C₈×C₉	C₂²	1	1	yes	characteristic
11	C₃×C₄²×C₉	C₂²	1	1	yes	characteristic
12	C₄⋊(C₃⋊Dic₉)	C₂²	1	2	yes	—
13	#88534	—	2	2	no	—
14	C₄×C₃⋊Dic₉	—	2	2	no	—
15	#88521	C₄	1	4	yes	direct product factor
16	C₂×#3746	—	3	1	no	—
17	C₄×#634	—	3	2	no	—
18	C₃²×C₄×C₈	D₃	1	1	yes	characteristic
19	#63645	—	3	1	no	—
20	#63643	—	3	2	no	—
21	C₂×Dic₃₆	—	3	3	no	—
22	C₄×Dic₁₈	—	3	6	no	—
23	C₄×C₈×C₉	D₃	1	3	yes	—
24	#63378	—	3	3	no	—
25	#63376	—	3	6	no	—
26	C₂×C₃×C₄×C₉	D₄	1	1	yes	characteristic
27	C₂×C₃×C₄×C₉	D₄	1	1	yes	characteristic
28	C₂×C₃×C₄×C₉	C₂³	1	1	yes	characteristic
29	C₂×C₃⋊Dic₉	—	2	2	no	—
30	C₂×C₃⋊Dic₉	—	2	2	no	—
31	#6226	—	2	2	no	—
32	#6226	C₂×C₄	1	4	yes	—
33	C₃×C₈×C₉	—	2	1	no	—
34	C₃×C₈×C₉	C₂×C₄	1	2	yes	—
35	C₄×Dic₁₂	—	9	3	no	—
36	C₄×Dic₁₂	—	9	1	no	—
37	C₂×#634	—	3	2	no	—
38	C₂×C₃²×C₈	C₂×D₃	1	1	yes	characteristic
39	C₂×C₃²×C₈	C₂×D₃	1	1	yes	characteristic
40	C₃²×C₄²	C₂×D₃	1	1	yes	characteristic
41	C₄⋊(C₃⋊Dic₃)	—	3	2	no	—
42	#3750	—	6	2	no	—
43	C₄×C₃⋊Dic₃	—	6	2	no	—
44	#3746	—	3	4	no	—
45	C₂×Dic₁₈	—	3	6	no	—
46	C₂×C₈×C₉	C₂×D₃	1	3	yes	—
47	C₂×C₈×C₉	C₂×D₃	1	3	yes	—
48	C₄²×C₉	C₂×D₃	1	3	yes	—
49	C₄⋊Dic₉	—	3	6	no	—
50	Dic₉⋊C₄	—	6	6	no	—
51	C₄×Dic₉	—	6	6	no	—
52	Dic₃₆	—	3	12	no	—
53	C₂²×C₃×C₉	C₂×D₄	1	1	yes	characteristic
54	C₃×C₄×C₉	—	2	1	no	—
55	C₃×C₄×C₉	Q₁₆	1	2	yes	complemented
56	C₃×C₄×C₉	C₂²×C₄	1	1	yes	characteristic
57	C₃×C₄×C₉	#41	1	1	yes	characteristic
58	C₃⋊Dic₉	—	2	4	no	—
59	C₃⋊Dic₉	—	4	2	no	—
60	C₂×Dic₁₂	—	9	3	no	—
61	C₂×Dic₁₂	—	9	1	no	—
62	C₄×Dic₆	—	9	6	no	—
63	C₄×Dic₆	—	9	2	no	—
64	C₃×C₄×C₈	D₉	1	3	yes	—
65	C₃×C₄×C₈	C₃⋊D₃	1	1	yes	characteristic
66	#818	—	9	3	no	—
67	#818	—	9	1	no	—
68	#816	—	9	6	no	—
69	#816	—	9	2	no	—
70	C₂×C₃²×C₄	D₁₂	1	1	yes	characteristic
71	C₂×C₃²×C₄	D₁₂	1	1	yes	characteristic
72	C₂×C₃²×C₄	C₂²×D₃	1	1	yes	characteristic
73	C₂×C₃⋊Dic₃	—	6	2	no	—
74	C₂×C₃⋊Dic₃	—	6	2	no	—
75	#634	—	3	4	no	—
76	#634	—	6	2	no	—
77	C₃²×C₈	—	2	1	no	—
78	C₃²×C₈	C₄×D₃	1	2	yes	—
79	C₂×C₄×C₉	D₁₂	1	3	yes	—
80	C₂×C₄×C₉	D₁₂	1	3	yes	—
81	C₂×C₄×C₉	C₂²×D₃	1	3	yes	—
82	C₂×Dic₉	—	6	6	no	—
83	C₂×Dic₉	—	6	6	no	—
84	Dic₁₈	—	3	12	no	—
85	Dic₁₈	—	6	6	no	—
86	C₈×C₉	—	2	3	no	—
87	C₈×C₉	C₄×D₃	1	6	yes	—
88	C₄×Q₁₆	—	27	1	no	Sylow
89	C₂×C₃×C₉	C₄×D₄	1	1	yes	characteristic
90	C₂×C₃×C₉	C₂×Q₁₆	1	1	yes	characteristic
91	C₂×C₃×C₉	#135	1	1	yes	characteristic
92	C₂×Dic₆	—	9	6	no	—
93	C₂×Dic₆	—	9	2	no	—
94	C₂×C₃×C₈	C₂×D₉	1	3	yes	—
95	C₂×C₃×C₈	C₂×D₉	1	3	yes	—
96	C₂×C₃×C₈	C₂×C₃⋊D₃	1	1	yes	characteristic
97	C₂×C₃×C₈	C₂×C₃⋊D₃	1	1	yes	characteristic
98	C₃×C₄²	C₂×D₉	1	3	yes	—
99	C₃×C₄²	C₂×C₃⋊D₃	1	1	yes	characteristic
100	C₄⋊Dic₃	—	9	6	no	—
101	C₄⋊Dic₃	—	9	2	no	—
102	Dic₃⋊C₄	—	18	6	no	—
103	Dic₃⋊C₄	—	18	2	no	—
104	C₄×Dic₃	—	18	6	no	—
105	C₄×Dic₃	—	18	2	no	—
106	Dic₁₂	—	9	12	no	—
107	Dic₁₂	—	9	4	no	—
108	C₂²×C₃²	C₂×D₁₂	1	1	yes	characteristic
109	C₃²×C₄	—	2	1	no	—
110	C₃²×C₄	Dic₁₂	1	2	yes	—
111	C₃²×C₄	C₂×C₄×D₃	1	1	yes	characteristic
112	C₃²×C₄	#235	1	1	yes	characteristic
113	C₃⋊Dic₃	—	6	4	no	—
114	C₃⋊Dic₃	—	12	2	no	—
115	C₂²×C₉	C₂×D₁₂	1	3	yes	—
116	C₄×C₉	—	2	3	no	—
117	C₄×C₉	Dic₁₂	1	6	yes	complemented
118	C₄×C₉	C₂×C₄×D₃	1	3	yes	—
119	C₄×C₉	#235	1	3	yes	—
120	Dic₉	—	6	12	no	—
121	Dic₉	—	12	6	no	—
122	C₂×Q₁₆	—	27	1	no	—
123	C₄×Q₈	—	27	2	no	—
124	#107	—	27	1	no	—
125	#103	—	27	2	no	—
126	C₄×C₈	C₃⋊D₉	1	1	yes	characteristic
127	C₃×C₉	C₄×Q₁₆	1	1	yes	characteristic, complemented, Sylow
128	C₂×C₃×C₄	D₃₆	1	3	yes	—
129	C₂×C₃×C₄	D₃₆	1	3	yes	—
130	C₂×C₃×C₄	C₂²×D₉	1	3	yes	—
131	C₂×C₃×C₄	C₄⋊(C₃⋊D₃)	1	1	yes	characteristic
132	C₂×C₃×C₄	C₄⋊(C₃⋊D₃)	1	1	yes	characteristic
133	C₂×C₃×C₄	C₂²×C₃⋊D₃	1	1	yes	characteristic
134	C₂×Dic₃	—	18	6	no	—
135	C₂×Dic₃	—	18	2	no	—
136	C₂×Dic₃	—	18	6	no	—
137	C₂×Dic₃	—	18	2	no	—
138	Dic₆	—	9	12	no	—
139	Dic₆	—	9	4	no	—
140	Dic₆	—	18	6	no	—
141	Dic₆	—	18	2	no	—
142	C₃×C₈	—	2	3	no	—
143	C₃×C₈	—	2	1	no	—
144	C₃×C₈	C₄×D₉	1	6	yes	—
145	C₃×C₈	C₄×C₃⋊D₃	1	2	yes	—
146	C₂×C₃²	C₄×D₁₂	1	1	yes	characteristic
147	C₂×C₃²	#904	1	1	yes	characteristic
148	C₂×C₃²	C₂×Dic₁₂	1	1	yes	characteristic
149	C₂×C₉	C₄×D₁₂	1	3	yes	—
150	C₂×C₉	#904	1	3	yes	—
151	C₂×C₉	C₂×Dic₁₂	1	3	yes	—
152	C₂×Q₈	—	27	2	no	—
153	Q₁₆	—	27	4	no	—
154	C₂×C₈	C₂×C₃⋊D₉	1	1	yes	characteristic
155	C₂×C₈	C₂×C₃⋊D₉	1	1	yes	characteristic
156	C₄⋊C₄	—	27	2	no	—
157	C₄⋊C₄	—	54	2	no	—
158	C₄²	—	54	2	no	—
159	C₄²	C₂×C₃⋊D₉	1	1	yes	characteristic
160	C₂²×C₃	C₂×D₃₆	1	3	yes	—
161	C₂²×C₃	C₂×C₄⋊(C₃⋊D₃)	1	1	yes	characteristic
162	C₃×C₄	—	2	1	no	—
163	C₃×C₄	—	2	3	no	—
164	C₃×C₄	Dic₃₆	1	6	yes	complemented
165	C₃×C₄	C₂×C₄×D₉	1	3	yes	—
166	C₃×C₄	#3697	1	3	yes	—
167	C₃×C₄	#3746	1	2	yes	—
168	C₃×C₄	C₂×C₄×C₃⋊D₃	1	1	yes	characteristic
169	C₃×C₄	#3828	1	1	yes	characteristic
170	Dic₃	—	18	4	no	—
171	Dic₃	—	18	12	no	—
172	Dic₃	—	36	2	no	—
173	Dic₃	—	36	6	no	—
174	C₃²	C₄×Dic₁₂	1	1	yes	characteristic
175	C₉	C₄×Dic₁₂	1	3	yes	complemented
176	Q₈	—	27	4	no	—
177	Q₈	—	54	2	no	—
178	C₂×C₄	—	54	2	no	—
179	C₂×C₄	—	54	2	no	—
180	C₂×C₄	C₄⋊(C₃⋊D₉)	1	1	yes	characteristic
181	C₂×C₄	C₄⋊(C₃⋊D₉)	1	1	yes	characteristic
182	C₂×C₄	C₂²×C₃⋊D₉	1	1	yes	characteristic
183	C₈	—	2	1	no	—
184	C₈	C₄×C₃⋊D₉	1	2	yes	—
185	C₂×C₃	C₄×D₃₆	1	3	yes	—
186	C₂×C₃	C₂×Dic₃₆	1	3	yes	—
187	C₂×C₃	#63467	1	3	yes	—
188	C₂×C₃	C₄×C₄⋊(C₃⋊D₃)	1	1	yes	characteristic
189	C₂×C₃	#64113	1	1	yes	characteristic
190	C₂×C₃	C₂×#3746	1	1	yes	characteristic
191	C₂²	C₂×C₄⋊(C₃⋊D₉)	1	1	yes	characteristic
192	C₄	—	2	1	no	—
193	C₄	—	54	4	no	—
194	C₄	—	108	2	no	—
195	C₄	#88521	1	2	yes	direct product factor
196	C₄	C₂×C₄×C₃⋊D₉	1	1	yes	characteristic
197	C₄	#88736	1	1	yes	characteristic
198	C₃	C₄×Dic₃₆	1	3	yes	complemented
199	C₃	C₄×#3746	1	1	yes	characteristic
200	C₂	C₄×C₄⋊(C₃⋊D₉)	1	1	yes	characteristic
201	C₂	C₂×#88521	1	1	yes	characteristic
202	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, prime radical	C₂²×C₃²	C₂×D₁₂
cosocle	C₂×C₃×C₄×C₉	C₂³
hypercenter	C₄×C₈	C₃⋊D₉
hypocenter	C₃×C₉	C₄×Q₁₆
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
15. #88521	C₄	direct (N × H)
55. C₃×C₄×C₉	Q₁₆	semidirect
117. C₄×C₉	Dic₁₂	semidirect
127. C₃×C₉	C₄×Q₁₆	semidirect
164. C₃×C₄	Dic₃₆	semidirect
175. C₉	C₄×Dic₁₂	semidirect
195. C₄	#88521	direct (N × H)
198. C₃	C₄×Dic₃₆	semidirect

Maximal subgroups

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

subgroup	index
C₂×#88521	2
C₄×#6226	2
C₃×C₄×C₈×C₉	2
#11720541	2
#11720539	2
C₄×#3746	3
C₄×Dic₃₆	3

Subgroups — C4×#88521 #13143746

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#88521 #13143746