Subgroups — C₄×#88532 #13144226

Order 1728.

710 conjugacy classes of subgroups.

The subgroup lattice has 186 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₄²×C₉	C₂	1	1	yes	maximal, characteristic
2	C₂×#88532	C₂	1	2	yes	maximal
3	#11720516	C₂	1	2	yes	maximal, complemented
4	C₄×C₃⋊(C₉⋊C₈)	C₂	1	2	yes	maximal, complemented
5	C₄×#3748	—	3	1	no	maximal
6	C₄×#3667	—	3	3	no	maximal
7	C₂²×C₃×C₄×C₉	C₄	1	2	yes	—
8	C₂²×C₃×C₄×C₉	C₂²	1	1	yes	characteristic
9	C₃×C₄²×C₉	C₄	1	2	yes	—
10	C₃×C₄²×C₉	C₂²	1	1	yes	characteristic
11	C₃×C₄²×C₉	C₂²	1	1	yes	characteristic
12	#88532	C₄	1	8	yes	direct product factor
13	C₂×C₃⋊(C₉⋊C₈)	C₂²	1	4	yes	—
14	C₂×C₃²×C₄²	D₃	1	1	yes	characteristic
15	C₂×#3748	—	3	2	no	—
16	#63630	—	3	2	no	—
17	C₄×C₃⋊(C₃⋊C₈)	—	3	2	no	—
18	C₂×C₄²×C₉	D₃	1	3	yes	—
19	C₂×#3667	—	3	6	no	—
20	#63362	—	3	6	no	—
21	C₄×C₉⋊C₈	—	3	6	no	—
22	C₂³×C₃×C₉	C₂×C₄	1	1	yes	characteristic
23	C₂×C₃×C₄×C₉	—	2	4	no	—
24	C₂×C₃×C₄×C₉	C₂×C₄	1	2	yes	—
25	C₂×C₃×C₄×C₉	C₂×C₄	1	1	yes	characteristic
26	C₂×C₃×C₄×C₉	C₂×C₄	1	4	yes	—
27	C₂×C₃×C₄×C₉	C₂×C₄	1	2	yes	—
28	C₂×C₃×C₄×C₉	C₂³	1	1	yes	characteristic
29	C₃⋊(C₉⋊C₈)	C₂×C₄	1	8	yes	complemented
30	C₄×#208	—	9	3	no	—
31	C₄×#208	—	9	1	no	—
32	C₂²×C₃²×C₄	Dic₃	1	2	yes	—
33	C₂²×C₃²×C₄	C₂×D₃	1	1	yes	characteristic
34	C₃²×C₄²	Dic₃	1	2	yes	—
35	C₃²×C₄²	C₂×D₃	1	1	yes	characteristic
36	C₃²×C₄²	C₂×D₃	1	1	yes	characteristic
37	#3748	—	3	8	no	—
38	C₂×C₃⋊(C₃⋊C₈)	—	3	4	no	—
39	C₂²×C₄×C₉	Dic₃	1	6	yes	—
40	C₂²×C₄×C₉	C₂×D₃	1	3	yes	—
41	C₄²×C₉	Dic₃	1	6	yes	—
42	C₄²×C₉	C₂×D₃	1	3	yes	—
43	C₄²×C₉	C₂×D₃	1	3	yes	—
44	#3667	—	3	24	no	—
45	C₂×C₉⋊C₈	—	3	12	no	—
46	C₂²×C₃×C₉	—	2	2	no	—
47	C₂²×C₃×C₉	C₄²	1	2	yes	—
48	C₂²×C₃×C₉	C₂²×C₄	1	1	yes	characteristic
49	C₃×C₄×C₉	—	2	2	no	—
50	C₃×C₄×C₉	C₄²	1	2	yes	—
51	C₃×C₄×C₉	M₁₆	1	4	yes	complemented
52	C₃×C₄×C₉	C₂²×C₄	1	2	yes	—
53	C₂×C₃×C₄²	D₉	1	3	yes	—
54	C₂×C₃×C₄²	C₃⋊D₃	1	1	yes	characteristic
55	C₂×#208	—	9	6	no	—
56	C₂×#208	—	9	2	no	—
57	#803	—	9	6	no	—
58	#803	—	9	2	no	—
59	C₄×C₃⋊C₈	—	9	6	no	—
60	C₄×C₃⋊C₈	—	9	2	no	—
61	C₂³×C₃²	C₂×Dic₃	1	1	yes	characteristic
62	C₂×C₃²×C₄	—	2	4	no	—
63	C₂×C₃²×C₄	C₄×D₃	1	4	yes	—
64	C₂×C₃²×C₄	C₂×Dic₃	1	2	yes	—
65	C₂×C₃²×C₄	C₂×Dic₃	1	2	yes	—
66	C₂×C₃²×C₄	C₂×Dic₃	1	1	yes	characteristic
67	C₂×C₃²×C₄	C₂²×D₃	1	1	yes	characteristic
68	C₂³×C₉	C₂×Dic₃	1	3	yes	—
69	C₃⋊(C₃⋊C₈)	—	3	8	no	—
70	C₂×C₄×C₉	—	2	12	no	—
71	C₂×C₄×C₉	C₄×D₃	1	12	yes	—
72	C₂×C₄×C₉	C₂×Dic₃	1	6	yes	—
73	C₂×C₄×C₉	C₂×Dic₃	1	6	yes	—
74	C₂×C₄×C₉	C₂×Dic₃	1	3	yes	—
75	C₂×C₄×C₉	C₂²×D₃	1	3	yes	—
76	C₉⋊C₈	—	3	24	no	—
77	C₄×M₁₆	—	27	1	no	Sylow
78	C₂×C₃×C₉	—	2	2	no	—
79	C₂×C₃×C₉	C₂×C₄²	1	1	yes	characteristic
80	C₂×C₃×C₉	C₂×M₁₆	1	2	yes	—
81	C₂²×C₃×C₄	Dic₉	1	6	yes	—
82	C₂²×C₃×C₄	C₂×D₉	1	3	yes	—
83	C₂²×C₃×C₄	C₃⋊Dic₃	1	2	yes	—
84	C₂²×C₃×C₄	C₂×C₃⋊D₃	1	1	yes	characteristic
85	C₃×C₄²	Dic₉	1	6	yes	—
86	C₃×C₄²	C₂×D₉	1	3	yes	—
87	C₃×C₄²	C₂×D₉	1	3	yes	—
88	C₃×C₄²	C₃⋊Dic₃	1	2	yes	—
89	C₃×C₄²	C₂×C₃⋊D₃	1	1	yes	characteristic
90	C₃×C₄²	C₂×C₃⋊D₃	1	1	yes	characteristic
91	#208	—	9	24	no	—
92	#208	—	9	8	no	—
93	C₂×C₃⋊C₈	—	9	12	no	—
94	C₂×C₃⋊C₈	—	9	4	no	—
95	C₂²×C₃²	—	2	2	no	—
96	C₂²×C₃²	C₄×Dic₃	1	2	yes	—
97	C₂²×C₃²	C₂²×Dic₃	1	1	yes	characteristic
98	C₃²×C₄	—	2	2	no	—
99	C₃²×C₄	#208	1	4	yes	—
100	C₃²×C₄	C₄×Dic₃	1	2	yes	—
101	C₃²×C₄	C₂×C₄×D₃	1	2	yes	—
102	C₂²×C₉	—	2	6	no	—
103	C₂²×C₉	C₄×Dic₃	1	6	yes	—
104	C₂²×C₉	C₂²×Dic₃	1	3	yes	—
105	C₄×C₉	—	2	6	no	—
106	C₄×C₉	#208	1	12	yes	complemented
107	C₄×C₉	C₄×Dic₃	1	6	yes	—
108	C₄×C₉	C₂×C₄×D₃	1	6	yes	—
109	C₂×M₁₆	—	27	2	no	—
110	C₂×C₄²	C₃⋊D₉	1	1	yes	characteristic
111	#97	—	27	2	no	—
112	C₄×C₈	—	27	2	no	—
113	C₃×C₉	C₄×M₁₆	1	1	yes	characteristic, complemented, Sylow
114	C₂³×C₃	C₂×Dic₉	1	3	yes	—
115	C₂³×C₃	C₂×C₃⋊Dic₃	1	1	yes	characteristic
116	C₂×C₃×C₄	—	2	12	no	—
117	C₂×C₃×C₄	—	2	4	no	—
118	C₂×C₃×C₄	C₄×D₉	1	12	yes	—
119	C₂×C₃×C₄	C₂×Dic₉	1	6	yes	—
120	C₂×C₃×C₄	C₂×Dic₉	1	6	yes	—
121	C₂×C₃×C₄	C₂×Dic₉	1	3	yes	—
122	C₂×C₃×C₄	C₂²×D₉	1	3	yes	—
123	C₂×C₃×C₄	C₄×C₃⋊D₃	1	4	yes	—
124	C₂×C₃×C₄	C₂×C₃⋊Dic₃	1	2	yes	—
125	C₂×C₃×C₄	C₂×C₃⋊Dic₃	1	2	yes	—
126	C₂×C₃×C₄	C₂×C₃⋊Dic₃	1	1	yes	characteristic
127	C₂×C₃×C₄	C₂²×C₃⋊D₃	1	1	yes	characteristic
128	C₃⋊C₈	—	9	24	no	—
129	C₃⋊C₈	—	9	8	no	—
130	C₂×C₃²	—	2	2	no	—
131	C₂×C₃²	C₂×#208	1	2	yes	—
132	C₂×C₃²	C₂×C₄×Dic₃	1	1	yes	characteristic
133	C₂×C₉	—	2	6	no	—
134	C₂×C₉	C₂×#208	1	6	yes	—
135	C₂×C₉	C₂×C₄×Dic₃	1	3	yes	—
136	C₂²×C₄	C₃⋊Dic₉	1	2	yes	—
137	C₂²×C₄	C₂×C₃⋊D₉	1	1	yes	characteristic
138	M₁₆	—	27	8	no	—
139	C₂×C₈	—	27	4	no	—
140	C₄²	C₃⋊Dic₉	1	2	yes	—
141	C₄²	C₂×C₃⋊D₉	1	1	yes	characteristic
142	C₄²	C₂×C₃⋊D₉	1	1	yes	characteristic
143	C₂²×C₃	—	2	2	no	—
144	C₂²×C₃	—	2	6	no	—
145	C₂²×C₃	C₄×Dic₉	1	6	yes	—
146	C₂²×C₃	C₂²×Dic₉	1	3	yes	—
147	C₂²×C₃	C₄×C₃⋊Dic₃	1	2	yes	—
148	C₂²×C₃	C₂²×C₃⋊Dic₃	1	1	yes	characteristic
149	C₃×C₄	—	2	2	no	—
150	C₃×C₄	—	2	6	no	—
151	C₃×C₄	#3667	1	12	yes	complemented
152	C₃×C₄	C₄×Dic₉	1	6	yes	—
153	C₃×C₄	C₂×C₄×D₉	1	6	yes	—
154	C₃×C₄	#3748	1	4	yes	—
155	C₃×C₄	C₄×C₃⋊Dic₃	1	2	yes	—
156	C₃×C₄	C₂×C₄×C₃⋊D₃	1	2	yes	—
157	C₃²	C₄×#208	1	1	yes	characteristic
158	C₉	C₄×#208	1	3	yes	complemented
159	C₂³	C₂×C₃⋊Dic₉	1	1	yes	characteristic
160	C₂×C₄	—	2	4	no	—
161	C₂×C₄	C₄×C₃⋊D₉	1	4	yes	—
162	C₂×C₄	C₂×C₃⋊Dic₉	1	2	yes	—
163	C₂×C₄	C₂×C₃⋊Dic₉	1	2	yes	—
164	C₂×C₄	C₂×C₃⋊Dic₉	1	1	yes	characteristic
165	C₂×C₄	C₂²×C₃⋊D₉	1	1	yes	characteristic
166	C₈	—	27	8	no	—
167	C₂×C₃	—	2	2	no	—
168	C₂×C₃	—	2	6	no	—
169	C₂×C₃	C₂×#3667	1	6	yes	—
170	C₂×C₃	C₂×C₄×Dic₉	1	3	yes	—
171	C₂×C₃	C₂×#3748	1	2	yes	—
172	C₂×C₃	C₂×C₄×C₃⋊Dic₃	1	1	yes	characteristic
173	C₂²	—	2	2	no	—
174	C₂²	C₄×C₃⋊Dic₉	1	2	yes	—
175	C₂²	C₂²×C₃⋊Dic₉	1	1	yes	characteristic
176	C₄	—	2	2	no	—
177	C₄	#88532	1	4	yes	direct product factor
178	C₄	C₄×C₃⋊Dic₉	1	2	yes	—
179	C₄	C₂×C₄×C₃⋊D₉	1	2	yes	—
180	C₃	C₄×#3667	1	3	yes	complemented
181	C₃	C₄×#3748	1	1	yes	characteristic
182	C₂	—	2	2	no	—
183	C₂	C₂×#88532	1	2	yes	—
184	C₂	C₂×C₄×C₃⋊Dic₉	1	1	yes	characteristic

	subgroup	G/H
center	C₄²	C₂×C₃⋊D₉
commutator	C₂×C₃×C₉	C₂×C₄²
Frattini	C₂×C₃×C₄	C₂²×C₃⋊D₃
Fitting	C₂×C₃×C₄²×C₉	C₂
socle	C₂²×C₃²	C₂²×Dic₃
prime radical	C₂³×C₃²	C₂×Dic₃
cosocle	C₂×C₃×C₄×C₉	C₂³
hypercenter	C₂×C₄²	C₃⋊D₉
hypocenter	C₃×C₉	C₄×M₁₆
LCS	G ▷ C₂×C₃×C₉ ▷ C₃×C₉
UCS	C₁ ◁ C₄² ◁ C₂×C₄²

Complements

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

N	H ≅ G/N	type
3. #11720516	C₂	semidirect
4. C₄×C₃⋊(C₉⋊C₈)	C₂	semidirect
12. #88532	C₄	direct (N × H)
29. C₃⋊(C₉⋊C₈)	C₂×C₄	semidirect
51. C₃×C₄×C₉	M₁₆	semidirect
106. C₄×C₉	#208	semidirect
113. C₃×C₉	C₄×M₁₆	semidirect
151. C₃×C₄	#3667	semidirect
158. C₉	C₄×#208	semidirect
177. C₄	#88532	direct (N × H)
180. C₃	C₄×#3667	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃×C₄²×C₉	2
C₂×#88532	2
#11720516	2
C₄×C₃⋊(C₉⋊C₈)	2
C₄×#3748	3
C₄×#3667	3

Subgroups — C4×#88532 #13144226

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#88532 #13144226