Subgroups — C₃²×#5812 #13179186

Order 1728.

2220 conjugacy classes of subgroups.

The subgroup lattice has 176 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃³×Q₈	C₂	1	1	yes	maximal, characteristic
2	C₃²×Q₈×Dic₃	C₂	1	8	yes	maximal, complemented
3	C₃²×#926	C₂	1	6	yes	maximal
4	C₃²×#519	—	3	1	no	maximal
5	C₃×#5812	C₃	1	4	yes	maximal, direct product factor
6	C₂×C₃³×Q₈	C₄	1	8	yes	complemented
7	C₂×C₃³×Q₈	C₂²	1	4	yes	—
8	C₂²×C₃³×C₄	C₂²	1	3	yes	—
9	C₃²×#217	C₂²	1	4	yes	—
10	C₃²×C₄⋊Dic₃	C₂²	1	12	yes	—
11	C₃²×C₄×Dic₃	C₂²	1	12	yes	—
12	C₂²×C₃²×Q₈	—	2	4	no	—
13	C₂²×C₃²×Q₈	D₃	1	1	yes	characteristic
14	C₂²×C₃²×Q₈	C₂×C₃	1	4	yes	—
15	C₃²×C₄×Q₈	—	3	8	no	—
16	C₃²×#117	—	3	6	no	—
17	C₃×Q₈×Dic₃	C₂×C₃	1	32	yes	complemented
18	C₃×#926	C₂×C₃	1	24	yes	—
19	C₂³×C₃³	C₂³	1	1	yes	characteristic
20	C₃³×Q₈	C₂×C₄	1	16	yes	—
21	C₂×C₃³×C₄	C₂×C₄	1	12	yes	—
22	C₂×C₃³×C₄	C₂³	1	6	yes	—
23	C₂×C₃²×Dic₃	C₂³	1	8	yes	—
24	C₃×#519	—	3	4	no	—
25	#5812	C₃²	1	1	yes	characteristic, direct product factor
26	C₂×C₃²×Q₈	—	2	32	no	—
27	C₂×C₃²×Q₈	—	2	16	no	—
28	C₂×C₃²×Q₈	Dic₃	1	8	yes	complemented
29	C₂×C₃²×Q₈	C₃×C₄	1	32	yes	complemented
30	C₂×C₃²×Q₈	C₂×D₃	1	4	yes	—
31	C₂×C₃²×Q₈	C₂²×C₃	1	16	yes	—
32	C₂²×C₃²×C₄	—	2	12	no	—
33	C₂²×C₃²×C₄	C₂×D₃	1	3	yes	—
34	C₂²×C₃²×C₄	C₂²×C₃	1	12	yes	—
35	C₃²×C₄⋊C₄	—	3	12	no	—
36	C₃²×C₂²⋊C₄	—	3	4	no	—
37	C₃²×C₄²	—	3	12	no	—
38	C₃×#217	C₂²×C₃	1	16	yes	—
39	C₃×C₄⋊Dic₃	C₂²×C₃	1	48	yes	—
40	C₃×C₄×Dic₃	C₂²×C₃	1	48	yes	—
41	C₂²×C₃³	—	2	2	no	—
42	C₂²×C₃³	C₂²×C₄	1	2	yes	—
43	C₂²×C₃³	C₂⁴	1	1	yes	characteristic
44	C₃³×C₄	C₂²×C₄	1	12	yes	—
45	C₃²×Dic₃	—	2	8	no	—
46	C₂²×C₃×Q₈	—	2	4	no	—
47	C₂²×C₃×Q₈	C₃×D₃	1	4	yes	—
48	C₂²×C₃×Q₈	C₂×C₃²	1	1	yes	characteristic
49	C₃×C₄×Q₈	—	3	32	no	—
50	C₃×#117	—	3	24	no	—
51	Q₈×Dic₃	C₂×C₃²	1	8	yes	complemented
52	#926	C₂×C₃²	1	6	yes	—
53	C₂³×C₃²	—	2	4	no	—
54	C₂³×C₃²	C₂²×D₃	1	1	yes	characteristic
55	C₂³×C₃²	C₂³×C₃	1	4	yes	—
56	C₃²×Q₈	—	2	64	no	—
57	C₃²×Q₈	C₂×Dic₃	1	16	yes	—
58	C₃²×Q₈	C₂×C₃×C₄	1	64	yes	—
59	C₂×C₃²×C₄	—	2	48	no	—
60	C₂×C₃²×C₄	—	2	24	no	—
61	C₂×C₃²×C₄	—	3	8	no	—
62	C₂×C₃²×C₄	C₂×Dic₃	1	12	yes	—
63	C₂×C₃²×C₄	C₂×C₃×C₄	1	48	yes	—
64	C₂×C₃²×C₄	C₂²×D₃	1	6	yes	—
65	C₂×C₃²×C₄	C₂³×C₃	1	24	yes	—
66	C₂×C₃×Dic₃	C₂³×C₃	1	32	yes	—
67	#519	—	3	1	no	Sylow
68	C₂×C₃³	—	2	2	no	—
69	C₂×C₃³	C₂³×C₄	1	1	yes	characteristic
70	C₂×C₃³	2₋¹⁺⁴	1	2	yes	—
71	C₂×C₃×Q₈	—	2	32	no	—
72	C₂×C₃×Q₈	—	2	16	no	—
73	C₂×C₃×Q₈	C₃×Dic₃	1	32	yes	complemented
74	C₂×C₃×Q₈	C₃²×C₄	1	8	yes	complemented
75	C₂×C₃×Q₈	C₂×C₃×D₃	1	16	yes	—
76	C₂×C₃×Q₈	C₂²×C₃²	1	4	yes	—
77	C₂²×C₃×C₄	—	2	12	no	—
78	C₂²×C₃×C₄	C₂×C₃×D₃	1	12	yes	—
79	C₂²×C₃×C₄	C₂²×C₃²	1	3	yes	—
80	C₃×C₄⋊C₄	—	3	48	no	—
81	C₃×C₂²⋊C₄	—	3	16	no	—
82	C₃×C₄²	—	3	48	no	—
83	#217	C₂²×C₃²	1	4	yes	—
84	C₄⋊Dic₃	C₂²×C₃²	1	12	yes	—
85	C₄×Dic₃	C₂²×C₃²	1	12	yes	—
86	C₂²×C₃²	—	2	16	no	—
87	C₂²×C₃²	—	2	2	no	—
88	C₂²×C₃²	—	2	8	no	—
89	C₂²×C₃²	—	2	4	no	—
90	C₂²×C₃²	—	2	8	no	—
91	C₂²×C₃²	C₂²×Dic₃	1	2	yes	—
92	C₂²×C₃²	C₂²×C₃×C₄	1	8	yes	—
93	C₂²×C₃²	C₂³×D₃	1	1	yes	characteristic
94	C₂²×C₃²	C₂⁴×C₃	1	4	yes	—
95	C₃²×C₄	—	2	48	no	—
96	C₃²×C₄	—	6	8	no	—
97	C₃²×C₄	C₂²×Dic₃	1	12	yes	—
98	C₃²×C₄	C₂²×C₃×C₄	1	48	yes	—
99	C₃×Dic₃	—	2	32	no	—
100	C₂²×Q₈	C₃²×D₃	1	1	yes	characteristic
101	C₄×Q₈	—	3	8	no	—
102	#117	—	3	6	no	—
103	C₃³	#519	1	1	yes	characteristic, complemented, Sylow
104	C₂³×C₃	—	2	4	no	—
105	C₂³×C₃	C₂²×C₃×D₃	1	4	yes	—
106	C₂³×C₃	C₂³×C₃²	1	1	yes	characteristic
107	C₃×Q₈	—	2	64	no	—
108	C₃×Q₈	C₂×C₃×Dic₃	1	64	yes	—
109	C₃×Q₈	C₂×C₃²×C₄	1	16	yes	—
110	C₂×C₃×C₄	—	2	48	no	—
111	C₂×C₃×C₄	—	2	24	no	—
112	C₂×C₃×C₄	—	3	32	no	—
113	C₂×C₃×C₄	C₂×C₃×Dic₃	1	48	yes	—
114	C₂×C₃×C₄	C₂×C₃²×C₄	1	12	yes	—
115	C₂×C₃×C₄	C₂²×C₃×D₃	1	24	yes	—
116	C₂×C₃×C₄	C₂³×C₃²	1	6	yes	—
117	C₂×Dic₃	C₂³×C₃²	1	8	yes	—
118	C₂×C₃²	—	2	16	no	—
119	C₂×C₃²	—	2	2	no	—
120	C₂×C₃²	—	2	8	no	—
121	C₂×C₃²	—	2	8	no	—
122	C₂×C₃²	—	2	4	no	—
123	C₂×C₃²	#1007	1	2	yes	—
124	C₂×C₃²	C₂³×Dic₃	1	1	yes	characteristic
125	C₂×C₃²	C₂³×C₃×C₄	1	4	yes	—
126	C₂×C₃²	C₃×2₋¹⁺⁴	1	8	yes	—
127	C₂×Q₈	C₃²×Dic₃	1	8	yes	complemented
128	C₂×Q₈	C₂×C₃²×D₃	1	4	yes	—
129	C₂²×C₄	C₂×C₃²×D₃	1	3	yes	—
130	C₄⋊C₄	—	3	12	no	—
131	C₂²⋊C₄	—	3	4	no	—
132	C₄²	—	3	12	no	—
133	C₂²×C₃	—	2	16	no	—
134	C₂²×C₃	—	2	8	no	—
135	C₂²×C₃	—	2	2	no	—
136	C₂²×C₃	—	2	4	no	—
137	C₂²×C₃	—	2	8	no	—
138	C₂²×C₃	C₂²×C₃×Dic₃	1	8	yes	—
139	C₂²×C₃	C₂²×C₃²×C₄	1	2	yes	—
140	C₂²×C₃	C₂³×C₃×D₃	1	4	yes	—
141	C₂²×C₃	C₂⁴×C₃²	1	1	yes	characteristic
142	C₃×C₄	—	2	48	no	—
143	C₃×C₄	—	6	32	no	—
144	C₃×C₄	C₂²×C₃×Dic₃	1	48	yes	—
145	C₃×C₄	C₂²×C₃²×C₄	1	12	yes	—
146	Dic₃	—	2	8	no	—
147	C₃²	—	2	4	no	—
148	C₃²	#5812	1	1	yes	characteristic, direct product factor
149	C₃²	C₃×#519	1	4	yes	complemented
150	C₂³	C₂²×C₃²×D₃	1	1	yes	characteristic
151	Q₈	C₂×C₃²×Dic₃	1	16	yes	—
152	C₂×C₄	—	3	8	no	—
153	C₂×C₄	C₂×C₃²×Dic₃	1	12	yes	—
154	C₂×C₄	C₂²×C₃²×D₃	1	6	yes	—
155	C₂×C₃	—	2	16	no	—
156	C₂×C₃	—	2	8	no	—
157	C₂×C₃	—	2	2	no	—
158	C₂×C₃	—	2	8	no	—
159	C₂×C₃	—	2	4	no	—
160	C₂×C₃	C₃×#1007	1	8	yes	—
161	C₂×C₃	C₂³×C₃×Dic₃	1	4	yes	—
162	C₂×C₃	C₂³×C₃²×C₄	1	1	yes	characteristic
163	C₂×C₃	C₃²×2₋¹⁺⁴	1	2	yes	—
164	C₂²	—	2	2	no	—
165	C₂²	C₂²×C₃²×Dic₃	1	2	yes	—
166	C₂²	C₂³×C₃²×D₃	1	1	yes	characteristic
167	C₄	—	6	8	no	—
168	C₄	C₂²×C₃²×Dic₃	1	12	yes	—
169	C₃	—	2	4	no	—
170	C₃	C₃×#5812	1	4	yes	direct product factor
171	C₃	C₃²×#519	1	1	yes	characteristic, complemented
172	C₂	—	2	2	no	—
173	C₂	C₃²×#1007	1	2	yes	—
174	C₂	C₂³×C₃²×Dic₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₃²	C₂³×D₃
commutator	C₂×C₃	C₂³×C₃²×C₄
Frattini	C₂²	C₂³×C₃²×D₃
Fitting	C₂²×C₃³×Q₈	C₂
socle	C₂²×C₃³	C₂⁴
prime radical	C₂³×C₃³	C₂³
cosocle	C₂²×C₃	C₂⁴×C₃²
hypercenter	C₂²×C₃²×Q₈	D₃
hypocenter	C₃	C₃²×#519
LCS	G ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₃² ◁ C₂²×C₃²×Q₈

Complements

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

N	H ≅ G/N	type
2. C₃²×Q₈×Dic₃	C₂	semidirect
5. C₃×#5812	C₃	direct (N × H)
6. C₂×C₃³×Q₈	C₄	semidirect
17. C₃×Q₈×Dic₃	C₂×C₃	semidirect
25. #5812	C₃²	direct (N × H)
28. C₂×C₃²×Q₈	Dic₃	semidirect
29. C₂×C₃²×Q₈	C₃×C₄	semidirect
51. Q₈×Dic₃	C₂×C₃²	semidirect
73. C₂×C₃×Q₈	C₃×Dic₃	semidirect
74. C₂×C₃×Q₈	C₃²×C₄	semidirect
103. C₃³	#519	semidirect
127. C₂×Q₈	C₃²×Dic₃	semidirect
148. C₃²	#5812	direct (N × H)
149. C₃²	C₃×#519	semidirect
170. C₃	C₃×#5812	direct (N × H)
171. C₃	C₃²×#519	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₃³×Q₈	2
C₃²×Q₈×Dic₃	2
C₃²×#926	2
C₃²×#519	3
C₃×#5812	3

Subgroups — C32×#5812 #13179186

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃²×#5812 #13179186