Subgroups — #13146096

Order 1728.

1652 conjugacy classes of subgroups.

The subgroup lattice has 168 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃³×C₈	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×C₃⋊(C₃⋊Dic₃)	C₂	1	1	yes	maximal, characteristic
3	C₂²×C₃⋊(C₃⋊(C₃⋊C₈))	C₂	1	1	yes	maximal, characteristic
4	#10589416	—	3	13	no	maximal
5	C₂²×C₃³×C₄	C₂²	1	1	yes	characteristic
6	C₂²×C₃⋊(C₃⋊Dic₃)	C₄	1	2	yes	—
7	C₂×C₃³×C₈	—	2	1	no	—
8	C₂×C₃³×C₈	—	2	1	no	—
9	C₂×C₃³×C₈	C₄	1	2	yes	complemented
10	C₄×C₃⋊(C₃⋊Dic₃)	—	2	1	no	—
11	C₄×C₃⋊(C₃⋊Dic₃)	—	2	1	no	—
12	C₂×C₃⋊(C₃⋊(C₃⋊C₈))	—	2	1	no	—
13	C₂×C₃⋊(C₃⋊(C₃⋊C₈))	—	2	1	no	—
14	C₂×C₃⋊(C₃⋊(C₃⋊C₈))	C₄	1	2	yes	complemented
15	C₂²×C₃²×C₈	D₃	1	13	yes	—
16	C₂×C₄×C₃⋊Dic₃	—	3	13	no	—
17	C₂²×C₃⋊(C₃⋊C₈)	—	3	13	no	—
18	C₂³×C₃³	C₂×C₄	1	1	yes	characteristic
19	C₂×C₃³×C₄	C₂×C₄	1	1	yes	characteristic
20	C₂×C₃³×C₄	C₂×C₄	1	1	yes	characteristic
21	C₂×C₃³×C₄	D₄	1	1	yes	characteristic
22	C₂×C₃³×C₄	D₄	1	1	yes	characteristic
23	C₂×C₃³×C₄	D₄	1	1	yes	characteristic
24	C₂×C₃³×C₄	Q₈	1	1	yes	characteristic
25	C₂×C₃⋊(C₃⋊Dic₃)	—	2	2	no	—
26	C₂×C₃⋊(C₃⋊Dic₃)	—	2	2	no	—
27	C₂×C₃⋊(C₃⋊Dic₃)	C₈	1	4	yes	complemented
28	C₃³×C₈	—	2	2	no	—
29	C₃⋊(C₃⋊(C₃⋊C₈))	—	2	2	no	—
30	#4550	—	9	13	no	—
31	C₂²×C₃²×C₄	C₂×D₃	1	13	yes	—
32	C₂²×C₃⋊Dic₃	—	3	26	no	—
33	C₂×C₃²×C₈	—	2	13	no	—
34	C₂×C₃²×C₈	—	2	13	no	—
35	C₂×C₃²×C₈	Dic₃	1	26	yes	complemented
36	C₄×C₃⋊Dic₃	—	6	13	no	—
37	C₄×C₃⋊Dic₃	—	6	13	no	—
38	C₂×C₃⋊(C₃⋊C₈)	—	3	26	no	—
39	C₂×C₃⋊(C₃⋊C₈)	—	6	13	no	—
40	C₂×C₃⋊(C₃⋊C₈)	—	6	13	no	—
41	C₂²×C₃³	C₄²	1	1	yes	characteristic
42	C₂²×C₃³	C₂²⋊C₄	1	1	yes	characteristic
43	C₂²×C₃³	C₄⋊C₄	1	1	yes	characteristic
44	C₂²×C₃³	C₂×C₈	1	2	yes	—
45	C₂²×C₃³	M₁₆	1	2	yes	—
46	C₃³×C₄	C₂²⋊C₄	1	1	yes	characteristic
47	C₃³×C₄	C₂²⋊C₄	1	1	yes	characteristic
48	C₃³×C₄	C₄⋊C₄	1	1	yes	characteristic
49	C₃³×C₄	C₄⋊C₄	1	1	yes	characteristic
50	C₃⋊(C₃⋊Dic₃)	—	2	4	no	—
51	C₂²×C₃×C₈	C₃⋊D₃	1	13	yes	—
52	C₂×C₄×Dic₃	—	9	13	no	—
53	C₂²×C₃⋊C₈	—	9	13	no	—
54	C₂³×C₃²	C₄×D₃	1	13	yes	—
55	C₂×C₃²×C₄	Dic₆	1	13	yes	—
56	C₂×C₃²×C₄	C₄×D₃	1	13	yes	—
57	C₂×C₃²×C₄	D₁₂	1	13	yes	—
58	C₂×C₃²×C₄	C₂×Dic₃	1	13	yes	—
59	C₂×C₃²×C₄	#67	1	13	yes	—
60	C₂×C₃²×C₄	#67	1	13	yes	—
61	C₂×C₃⋊Dic₃	—	3	52	no	—
62	C₂×C₃⋊Dic₃	—	6	26	no	—
63	C₂×C₃⋊Dic₃	—	6	26	no	—
64	C₃²×C₈	—	2	26	no	—
65	C₃⋊(C₃⋊C₈)	—	6	26	no	—
66	#336	—	27	1	no	Sylow
67	C₂×C₃³	#95	1	1	yes	characteristic
68	C₂×C₃³	C₄×C₈	1	1	yes	characteristic
69	C₂×C₃³	#97	1	1	yes	characteristic
70	C₂×C₃³	C₂²⋊C₈	1	2	yes	—
71	C₂×C₃³	C₄⋊C₈	1	2	yes	—
72	C₂²×C₃×C₄	C₂×C₃⋊D₃	1	13	yes	—
73	C₂²×Dic₃	—	9	26	no	—
74	C₂×C₃×C₈	—	2	13	no	—
75	C₂×C₃×C₈	—	2	13	no	—
76	C₂×C₃×C₈	C₃⋊Dic₃	1	26	yes	complemented
77	C₄×Dic₃	—	18	13	no	—
78	C₄×Dic₃	—	18	13	no	—
79	C₂×C₃⋊C₈	—	9	26	no	—
80	C₂×C₃⋊C₈	—	18	13	no	—
81	C₂×C₃⋊C₈	—	18	13	no	—
82	C₂²×C₃²	D₃×C₈	1	26	yes	—
83	C₂²×C₃²	ΓL(1,𝔽₂₅)	1	26	yes	—
84	C₂²×C₃²	C₄×Dic₃	1	13	yes	—
85	C₂²×C₃²	Dic₃⋊C₄	1	13	yes	—
86	C₂²×C₃²	(C₂×D₃)⋊C₄	1	13	yes	—
87	C₃²×C₄	Dic₃⋊C₄	1	13	yes	—
88	C₃²×C₄	C₄⋊Dic₃	1	13	yes	—
89	C₃²×C₄	(C₂×D₃)⋊C₄	1	13	yes	—
90	C₃²×C₄	#217	1	13	yes	—
91	C₃⋊Dic₃	—	6	52	no	—
92	C₂²×C₈	—	27	1	no	—
93	C₂²×C₈	C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
94	C₂×C₄²	—	27	1	no	—
95	C₃³	#336	1	1	yes	characteristic, complemented, Sylow
96	C₂³×C₃	C₄×C₃⋊D₃	1	13	yes	—
97	C₂×C₃×C₄	#634	1	13	yes	—
98	C₂×C₃×C₄	C₄×C₃⋊D₃	1	13	yes	—
99	C₂×C₃×C₄	C₄⋊(C₃⋊D₃)	1	13	yes	—
100	C₂×C₃×C₄	C₂×C₃⋊Dic₃	1	13	yes	—
101	C₂×C₃×C₄	#638	1	13	yes	—
102	C₂×C₃×C₄	#638	1	13	yes	—
103	C₂×Dic₃	—	9	52	no	—
104	C₂×Dic₃	—	18	26	no	—
105	C₂×Dic₃	—	18	26	no	—
106	C₃×C₈	—	2	26	no	—
107	C₃⋊C₈	—	18	26	no	—
108	C₂×C₃²	C₈×Dic₃	1	13	yes	—
109	C₂×C₃²	Dic₃⋊C₈	1	26	yes	—
110	C₂×C₃²	#815	1	13	yes	—
111	C₂×C₃²	(C₂×D₃)⋊C₈	1	26	yes	—
112	C₂×C₃²	C₃⋊#95	1	13	yes	—
113	C₂²×C₄	—	27	2	no	—
114	C₂²×C₄	C₂×C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
115	C₂×C₈	—	2	1	no	—
116	C₂×C₈	—	2	1	no	—
117	C₂×C₈	—	27	2	no	—
118	C₂×C₈	—	54	1	no	—
119	C₂×C₈	—	54	1	no	—
120	C₂×C₈	C₃⋊(C₃⋊Dic₃)	1	2	yes	complemented
121	C₄²	—	54	1	no	—
122	C₄²	—	54	1	no	—
123	C₂²×C₃	C₈×C₃⋊D₃	1	26	yes	—
124	C₂²×C₃	#3743	1	26	yes	—
125	C₂²×C₃	C₄×C₃⋊Dic₃	1	13	yes	—
126	C₂²×C₃	#3750	1	13	yes	—
127	C₂²×C₃	#3752	1	13	yes	—
128	C₃×C₄	#3750	1	13	yes	—
129	C₃×C₄	C₄⋊(C₃⋊Dic₃)	1	13	yes	—
130	C₃×C₄	#3752	1	13	yes	—
131	C₃×C₄	#3757	1	13	yes	—
132	Dic₃	—	18	52	no	—
133	C₃²	#4550	1	13	yes	complemented
134	C₂³	C₄×C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
135	C₂×C₄	—	27	4	no	—
136	C₂×C₄	—	54	2	no	—
137	C₂×C₄	—	54	2	no	—
138	C₂×C₄	#6308	1	1	yes	characteristic
139	C₂×C₄	C₄×C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
140	C₂×C₄	C₄⋊(C₃⋊(C₃⋊D₃))	1	1	yes	characteristic
141	C₂×C₄	C₂×C₃⋊(C₃⋊Dic₃)	1	1	yes	characteristic
142	C₂×C₄	#6312	1	1	yes	characteristic
143	C₂×C₄	#6312	1	1	yes	characteristic
144	C₈	—	2	2	no	—
145	C₈	—	54	2	no	—
146	C₂×C₃	C₈×C₃⋊Dic₃	1	13	yes	—
147	C₂×C₃	#63641	1	26	yes	—
148	C₂×C₃	#63642	1	13	yes	—
149	C₂×C₃	#63647	1	26	yes	—
150	C₂×C₃	#63658	1	13	yes	—
151	C₂²	C₈×C₃⋊(C₃⋊D₃)	1	2	yes	—
152	C₂²	#88850	1	2	yes	—
153	C₂²	C₄×C₃⋊(C₃⋊Dic₃)	1	1	yes	characteristic
154	C₂²	#88857	1	1	yes	characteristic
155	C₂²	#88859	1	1	yes	characteristic
156	C₄	—	54	4	no	—
157	C₄	#88857	1	1	yes	characteristic
158	C₄	C₄⋊(C₃⋊(C₃⋊Dic₃))	1	1	yes	characteristic
159	C₄	#88859	1	1	yes	characteristic
160	C₄	#88864	1	1	yes	characteristic
161	C₃	#10589416	1	13	yes	complemented
162	C₂	C₈×C₃⋊(C₃⋊Dic₃)	1	1	yes	characteristic
163	C₂	#11722184	1	2	yes	—
164	C₂	#11722185	1	1	yes	characteristic
165	C₂	#11722190	1	2	yes	—
166	C₂	#11722201	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂×C₃⋊(C₃⋊D₃)
commutator	C₂×C₃³	C₄×C₈
Fitting	C₂²×C₃³×C₈	C₂
socle, prime radical	C₂³×C₃³	C₂×C₄
cosocle	C₂²×C₃³×C₄	C₂²
hypercenter	C₂²×C₈	C₃⋊(C₃⋊D₃)
hypocenter	C₃³	#336
LCS	G ▷ C₂×C₃³ ▷ C₃³
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₈

Complements

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

N	H ≅ G/N	type
9. C₂×C₃³×C₈	C₄	semidirect
14. C₂×C₃⋊(C₃⋊(C₃⋊C₈))	C₄	semidirect
27. C₂×C₃⋊(C₃⋊Dic₃)	C₈	semidirect
35. C₂×C₃²×C₈	Dic₃	semidirect
76. C₂×C₃×C₈	C₃⋊Dic₃	semidirect
95. C₃³	#336	semidirect
120. C₂×C₈	C₃⋊(C₃⋊Dic₃)	semidirect
133. C₃²	#4550	semidirect
161. C₃	#10589416	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₃³×C₈	2
C₂×C₄×C₃⋊(C₃⋊Dic₃)	2
C₂²×C₃⋊(C₃⋊(C₃⋊C₈))	2
#10589416	3