Subgroups — C₃×Q₈×C₃⋊(C₃⋊C₈) #13163061

Order 1728.

892 conjugacy classes of subgroups.

The subgroup lattice has 178 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃³×C₄×Q₈	C₂	1	1	yes	maximal, characteristic
2	C₃×C₄⋊(C₃⋊(C₃⋊C₈))	C₂	1	3	yes	maximal
3	C₃×C₄×C₃⋊(C₃⋊C₈)	C₂	1	3	yes	maximal
4	Q₈×C₃⋊(C₃⋊C₈)	C₃	1	1	yes	maximal, characteristic, direct product factor
5	C₃×Q₈×C₃⋊C₈	—	3	4	no	maximal
6	C₂×C₃³×Q₈	C₄	1	1	yes	characteristic
7	C₃³×C₄⋊C₄	C₄	1	3	yes	—
8	C₃³×C₄²	C₂²	1	3	yes	—
9	C₂×C₃×C₃⋊(C₃⋊C₈)	C₂²	1	3	yes	—
10	C₂×C₃×C₃⋊(C₃⋊C₈)	C₂²	1	1	yes	characteristic
11	C₃²×C₄×Q₈	—	2	4	no	—
12	C₃²×C₄×Q₈	D₃	1	4	yes	—
13	C₃²×C₄×Q₈	C₂×C₃	1	1	yes	characteristic
14	C₄⋊(C₃⋊(C₃⋊C₈))	C₂×C₃	1	3	yes	—
15	C₄×C₃⋊(C₃⋊C₈)	C₂×C₃	1	3	yes	—
16	C₃×C₄⋊(C₃⋊C₈)	—	3	12	no	—
17	C₃×C₄×C₃⋊C₈	—	3	12	no	—
18	C₃³×Q₈	C₈	1	4	yes	complemented
19	C₂×C₃³×C₄	C₂×C₄	1	3	yes	—
20	C₂×C₃³×C₄	C₂×C₄	1	3	yes	—
21	C₂×C₃³×C₄	C₂³	1	1	yes	characteristic
22	C₃×C₃⋊(C₃⋊C₈)	—	2	3	no	—
23	C₃×C₃⋊(C₃⋊C₈)	Q₈	1	2	yes	direct product factor
24	C₃×C₈×Q₈	—	9	1	no	—
25	Q₈×C₃⋊C₈	—	3	4	no	—
26	C₂×C₃²×Q₈	—	2	4	no	—
27	C₂×C₃²×Q₈	Dic₃	1	4	yes	—
28	C₂×C₃²×Q₈	C₃×C₄	1	1	yes	characteristic
29	C₃²×C₄⋊C₄	—	2	12	no	—
30	C₃²×C₄⋊C₄	Dic₃	1	12	yes	—
31	C₃²×C₄⋊C₄	C₃×C₄	1	3	yes	—
32	C₃²×C₄²	—	2	12	no	—
33	C₃²×C₄²	C₂×D₃	1	12	yes	—
34	C₃²×C₄²	C₂²×C₃	1	3	yes	—
35	C₂×C₃⋊(C₃⋊C₈)	C₂²×C₃	1	3	yes	—
36	C₂×C₃⋊(C₃⋊C₈)	C₂²×C₃	1	1	yes	characteristic
37	C₂×C₃×C₃⋊C₈	—	3	4	no	—
38	C₂×C₃×C₃⋊C₈	—	3	12	no	—
39	C₂²×C₃³	C₂²×C₄	1	1	yes	characteristic
40	C₃³×C₄	—	2	3	no	—
41	C₃³×C₄	C₂×C₈	1	6	yes	—
42	C₃³×C₄	C₂×Q₈	1	1	yes	characteristic
43	C₃³×C₄	#41	1	1	yes	characteristic
44	C₃×C₄×Q₈	—	2	4	no	—
45	C₃×C₄×Q₈	C₃×D₃	1	4	yes	—
46	C₃×C₄×Q₈	C₃⋊D₃	1	1	yes	characteristic
47	C₃×C₄⋊C₈	—	9	3	no	—
48	C₃×C₄×C₈	—	9	3	no	—
49	C₄⋊(C₃⋊C₈)	—	3	12	no	—
50	C₄×C₃⋊C₈	—	3	12	no	—
51	C₃²×Q₈	—	2	16	no	—
52	C₃²×Q₈	C₃⋊C₈	1	16	yes	complemented
53	C₃²×Q₈	C₃×C₈	1	4	yes	complemented
54	C₂×C₃²×C₄	—	2	12	no	—
55	C₂×C₃²×C₄	—	2	12	no	—
56	C₂×C₃²×C₄	—	2	4	no	—
57	C₂×C₃²×C₄	C₂×Dic₃	1	12	yes	—
58	C₂×C₃²×C₄	C₂×Dic₃	1	12	yes	—
59	C₂×C₃²×C₄	C₂×C₃×C₄	1	3	yes	—
60	C₂×C₃²×C₄	C₂×C₃×C₄	1	3	yes	—
61	C₂×C₃²×C₄	C₂²×D₃	1	4	yes	—
62	C₂×C₃²×C₄	C₂³×C₃	1	1	yes	characteristic
63	C₃⋊(C₃⋊C₈)	—	2	3	no	—
64	C₃⋊(C₃⋊C₈)	C₃×Q₈	1	2	yes	direct product factor
65	C₃×C₃⋊C₈	—	3	8	no	—
66	C₃×C₃⋊C₈	—	6	12	no	—
67	C₈×Q₈	—	9	1	no	Sylow
68	C₂×C₃³	C₄×Q₈	1	1	yes	characteristic
69	C₂×C₃³	C₂²×C₈	1	1	yes	characteristic
70	C₂×C₃³	#131	1	1	yes	characteristic
71	C₂×C₃×Q₈	—	2	4	no	—
72	C₂×C₃×Q₈	C₃×Dic₃	1	4	yes	—
73	C₂×C₃×Q₈	C₃⋊Dic₃	1	1	yes	characteristic
74	C₂×C₃×C₈	—	9	1	no	—
75	C₂×C₃×C₈	—	9	3	no	—
76	C₃×C₄⋊C₄	—	2	12	no	—
77	C₃×C₄⋊C₄	C₃×Dic₃	1	12	yes	—
78	C₃×C₄⋊C₄	C₃⋊Dic₃	1	3	yes	—
79	C₃×C₄²	—	2	12	no	—
80	C₃×C₄²	C₂×C₃×D₃	1	12	yes	—
81	C₃×C₄²	C₂×C₃⋊D₃	1	3	yes	—
82	C₂×C₃⋊C₈	—	3	4	no	—
83	C₂×C₃⋊C₈	—	3	12	no	—
84	C₂²×C₃²	—	2	4	no	—
85	C₂²×C₃²	C₂²×Dic₃	1	4	yes	—
86	C₂²×C₃²	C₂²×C₃×C₄	1	1	yes	characteristic
87	C₃²×C₄	—	2	3	no	—
88	C₃²×C₄	—	2	24	no	—
89	C₃²×C₄	—	2	4	no	—
90	C₃²×C₄	—	2	4	no	—
91	C₃²×C₄	—	2	12	no	—
92	C₃²×C₄	—	4	12	no	—
93	C₃²×C₄	C₂×C₃⋊C₈	1	24	yes	—
94	C₃²×C₄	C₂×C₃×C₈	1	6	yes	—
95	C₃²×C₄	D₃×Q₈	1	4	yes	—
96	C₃²×C₄	#239	1	4	yes	—
97	C₃²×C₄	C₂×C₃×Q₈	1	1	yes	characteristic
98	C₃²×C₄	C₃×#41	1	1	yes	characteristic
99	C₄×Q₈	C₃×C₃⋊D₃	1	1	yes	characteristic
100	C₄⋊C₈	—	9	3	no	—
101	C₄×C₈	—	9	3	no	—
102	C₃³	C₈×Q₈	1	1	yes	characteristic, complemented, Sylow
103	C₃×Q₈	—	2	16	no	—
104	C₃×Q₈	C₃×C₃⋊C₈	1	16	yes	complemented
105	C₃×Q₈	C₃⋊(C₃⋊C₈)	1	4	yes	direct product factor
106	C₂×C₃×C₄	—	2	12	no	—
107	C₂×C₃×C₄	—	2	12	no	—
108	C₂×C₃×C₄	—	2	4	no	—
109	C₂×C₃×C₄	C₂×C₃×Dic₃	1	12	yes	—
110	C₂×C₃×C₄	C₂×C₃×Dic₃	1	12	yes	—
111	C₂×C₃×C₄	C₂×C₃⋊Dic₃	1	3	yes	—
112	C₂×C₃×C₄	C₂×C₃⋊Dic₃	1	3	yes	—
113	C₂×C₃×C₄	C₂²×C₃×D₃	1	4	yes	—
114	C₂×C₃×C₄	C₂²×C₃⋊D₃	1	1	yes	characteristic
115	C₃×C₈	—	9	2	no	—
116	C₃×C₈	—	18	3	no	—
117	C₃⋊C₈	—	3	8	no	—
118	C₃⋊C₈	—	6	12	no	—
119	C₂×C₃²	—	2	4	no	—
120	C₂×C₃²	—	2	4	no	—
121	C₂×C₃²	—	2	4	no	—
122	C₂×C₃²	C₂²×C₃⋊C₈	1	4	yes	—
123	C₂×C₃²	Q₈×Dic₃	1	4	yes	—
124	C₂×C₃²	#948	1	4	yes	—
125	C₂×C₃²	C₃×C₄×Q₈	1	1	yes	characteristic
126	C₂×C₃²	C₂²×C₃×C₈	1	1	yes	characteristic
127	C₂×C₃²	C₃×#131	1	1	yes	characteristic
128	C₂×Q₈	C₃×C₃⋊Dic₃	1	1	yes	characteristic
129	C₂×C₈	—	9	1	no	—
130	C₂×C₈	—	9	3	no	—
131	C₄⋊C₄	C₃×C₃⋊Dic₃	1	3	yes	—
132	C₄²	C₂×C₃×C₃⋊D₃	1	3	yes	—
133	C₂²×C₃	—	2	4	no	—
134	C₂²×C₃	C₂²×C₃×Dic₃	1	4	yes	—
135	C₂²×C₃	C₂²×C₃⋊Dic₃	1	1	yes	characteristic
136	C₃×C₄	—	2	12	no	—
137	C₃×C₄	—	2	3	no	—
138	C₃×C₄	—	2	24	no	—
139	C₃×C₄	—	2	4	no	—
140	C₃×C₄	—	2	4	no	—
141	C₃×C₄	—	4	12	no	—
142	C₃×C₄	C₂×C₃×C₃⋊C₈	1	24	yes	—
143	C₃×C₄	C₂×C₃⋊(C₃⋊C₈)	1	6	yes	—
144	C₃×C₄	C₃×D₃×Q₈	1	4	yes	—
145	C₃×C₄	C₃×#239	1	4	yes	—
146	C₃×C₄	Q₈×C₃⋊D₃	1	1	yes	characteristic
147	C₃×C₄	#3832	1	1	yes	characteristic
148	C₃²	—	2	4	no	—
149	C₃²	Q₈×C₃⋊C₈	1	4	yes	complemented
150	C₃²	C₃×C₈×Q₈	1	1	yes	characteristic, complemented
151	Q₈	C₃×C₃⋊(C₃⋊C₈)	1	4	yes	direct product factor
152	C₂×C₄	C₂×C₃×C₃⋊Dic₃	1	3	yes	—
153	C₂×C₄	C₂×C₃×C₃⋊Dic₃	1	3	yes	—
154	C₂×C₄	C₂²×C₃×C₃⋊D₃	1	1	yes	characteristic
155	C₈	—	9	2	no	—
156	C₈	—	18	3	no	—
157	C₂×C₃	—	2	4	no	—
158	C₂×C₃	—	2	4	no	—
159	C₂×C₃	—	2	4	no	—
160	C₂×C₃	C₂²×C₃×C₃⋊C₈	1	4	yes	—
161	C₂×C₃	C₃×Q₈×Dic₃	1	4	yes	—
162	C₂×C₃	C₃×#948	1	4	yes	—
163	C₂×C₃	C₂²×C₃⋊(C₃⋊C₈)	1	1	yes	characteristic
164	C₂×C₃	Q₈×C₃⋊Dic₃	1	1	yes	characteristic
165	C₂×C₃	#64157	1	1	yes	characteristic
166	C₂²	C₂²×C₃×C₃⋊Dic₃	1	1	yes	characteristic
167	C₄	—	2	3	no	—
168	C₄	C₂×C₃×C₃⋊(C₃⋊C₈)	1	6	yes	—
169	C₄	C₃×Q₈×C₃⋊D₃	1	1	yes	characteristic
170	C₄	C₃×#3832	1	1	yes	characteristic
171	C₃	—	2	4	no	—
172	C₃	C₃×Q₈×C₃⋊C₈	1	4	yes	complemented
173	C₃	Q₈×C₃⋊(C₃⋊C₈)	1	1	yes	characteristic, direct product factor
174	C₂	C₂²×C₃×C₃⋊(C₃⋊C₈)	1	1	yes	characteristic
175	C₂	C₃×Q₈×C₃⋊Dic₃	1	1	yes	characteristic
176	C₂	C₃×#64157	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₃×C₄	C₂²×C₃⋊D₃
commutator	C₂×C₃²	C₂²×C₃×C₈
Frattini	C₂×C₄	C₂²×C₃×C₃⋊D₃
Fitting	C₃³×C₄×Q₈	C₂
socle, prime radical	C₂²×C₃³	C₂²×C₄
cosocle	C₂×C₃²×C₄	C₂³×C₃
hypercenter	C₃×C₄×Q₈	C₃⋊D₃
hypocenter	C₃²	C₃×C₈×Q₈
LCS	G ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂×C₃×C₄ ◁ C₃×C₄×Q₈

Complements

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

N	H ≅ G/N	type
4. Q₈×C₃⋊(C₃⋊C₈)	C₃	direct (N × H)
18. C₃³×Q₈	C₈	semidirect
23. C₃×C₃⋊(C₃⋊C₈)	Q₈	direct (N × H)
52. C₃²×Q₈	C₃⋊C₈	semidirect
53. C₃²×Q₈	C₃×C₈	semidirect
64. C₃⋊(C₃⋊C₈)	C₃×Q₈	direct (N × H)
102. C₃³	C₈×Q₈	semidirect
104. C₃×Q₈	C₃×C₃⋊C₈	semidirect
105. C₃×Q₈	C₃⋊(C₃⋊C₈)	direct (N × H)
149. C₃²	Q₈×C₃⋊C₈	semidirect
150. C₃²	C₃×C₈×Q₈	semidirect
151. Q₈	C₃×C₃⋊(C₃⋊C₈)	direct (N × H)
172. C₃	C₃×Q₈×C₃⋊C₈	semidirect
173. C₃	Q₈×C₃⋊(C₃⋊C₈)	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃³×C₄×Q₈	2
C₃×C₄⋊(C₃⋊(C₃⋊C₈))	2
C₃×C₄×C₃⋊(C₃⋊C₈)	2
Q₈×C₃⋊(C₃⋊C₈)	3
C₃×Q₈×C₃⋊C₈	3

Subgroups — C3×Q8×C3⋊(C3⋊C8) #13163061

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×Q₈×C₃⋊(C₃⋊C₈) #13163061