Subgroups — C₃³×#533 #13179988

Order 1728.

3108 conjugacy classes of subgroups.

The subgroup lattice has 156 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	2	yes	maximal, complemented
2	C₃³×#125	C₂	1	2	yes	maximal, complemented
3	C₃³×C₂²⋊Q₈	C₂	1	4	yes	maximal
4	C₃³×C₄×Q₈	C₂	1	4	yes	maximal, complemented
5	C₃³×#117	C₂	1	2	yes	maximal
6	C₂×C₃³×C₄²	C₂	1	1	yes	maximal, characteristic
7	C₃²×#533	C₃	1	13	yes	maximal, direct product factor
8	C₂×C₃³×Q₈	C₂²	1	4	yes	—
9	C₂²×C₃³×C₄	C₂²	1	2	yes	—
10	C₂²×C₃³×C₄	C₂²	1	1	yes	characteristic
11	C₃³×C₄⋊C₄	C₂²	1	4	yes	—
12	C₃³×C₄⋊C₄	C₂²	1	8	yes	—
13	C₃³×C₄⋊C₄	C₂²	1	4	yes	—
14	C₃³×C₂²⋊C₄	C₂²	1	4	yes	—
15	C₃³×C₄²	C₂²	1	1	yes	characteristic
16	C₃³×C₄²	C₂²	1	1	yes	characteristic
17	C₃³×C₄²	C₂²	1	2	yes	—
18	C₃³×C₄²	C₂²	1	4	yes	—
19	C₃²×C₄⋊Q₈	C₂×C₃	1	26	yes	complemented
20	C₃²×#125	C₂×C₃	1	26	yes	complemented
21	C₃²×C₂²⋊Q₈	C₂×C₃	1	52	yes	—
22	C₃²×C₄×Q₈	C₂×C₃	1	52	yes	complemented
23	C₃²×#117	C₂×C₃	1	26	yes	—
24	C₂×C₃²×C₄²	C₂×C₃	1	13	yes	—
25	C₂³×C₃³	C₂³	1	1	yes	characteristic
26	C₃³×Q₈	—	2	8	no	—
27	C₂×C₃³×C₄	—	2	4	no	—
28	C₂×C₃³×C₄	Q₈	1	4	yes	complemented
29	C₂×C₃³×C₄	C₂³	1	2	yes	—
30	C₂×C₃³×C₄	C₂³	1	8	yes	—
31	C₂×C₃³×C₄	C₂³	1	1	yes	characteristic
32	C₂×C₃³×C₄	C₂³	1	2	yes	—
33	C₂×C₃³×C₄	C₂³	1	1	yes	characteristic
34	C₃×#533	C₃²	1	13	yes	direct product factor
35	C₂×C₃²×Q₈	C₂²×C₃	1	52	yes	—
36	C₂²×C₃²×C₄	C₂²×C₃	1	26	yes	—
37	C₂²×C₃²×C₄	C₂²×C₃	1	13	yes	—
38	C₃²×C₄⋊C₄	C₂²×C₃	1	52	yes	—
39	C₃²×C₄⋊C₄	C₂²×C₃	1	104	yes	—
40	C₃²×C₄⋊C₄	C₂²×C₃	1	52	yes	—
41	C₃²×C₂²⋊C₄	C₂²×C₃	1	52	yes	—
42	C₃²×C₄²	C₂²×C₃	1	26	yes	—
43	C₃²×C₄²	C₂²×C₃	1	13	yes	—
44	C₃²×C₄²	C₂²×C₃	1	13	yes	—
45	C₃²×C₄²	C₂²×C₃	1	52	yes	—
46	C₂²×C₃³	—	2	2	no	—
47	C₂²×C₃³	C₂×Q₈	1	2	yes	—
48	C₂²×C₃³	C₂⁴	1	1	yes	characteristic
49	C₃³×C₄	—	2	2	no	—
50	C₃³×C₄	—	2	8	no	—
51	C₃³×C₄	C₂×Q₈	1	2	yes	—
52	C₃³×C₄	C₂×Q₈	1	2	yes	—
53	C₃³×C₄	#41	1	4	yes	—
54	C₃×C₄⋊Q₈	C₂×C₃²	1	26	yes	complemented
55	C₃×#125	C₂×C₃²	1	26	yes	complemented
56	C₃×C₂²⋊Q₈	C₂×C₃²	1	52	yes	—
57	C₃×C₄×Q₈	C₂×C₃²	1	52	yes	complemented
58	C₃×#117	C₂×C₃²	1	26	yes	—
59	C₂×C₃×C₄²	C₂×C₃²	1	13	yes	—
60	C₂³×C₃²	C₂³×C₃	1	13	yes	—
61	C₃²×Q₈	—	2	104	no	—
62	C₂×C₃²×C₄	—	2	52	no	—
63	C₂×C₃²×C₄	C₃×Q₈	1	52	yes	complemented
64	C₂×C₃²×C₄	C₂³×C₃	1	26	yes	—
65	C₂×C₃²×C₄	C₂³×C₃	1	104	yes	—
66	C₂×C₃²×C₄	C₂³×C₃	1	13	yes	—
67	C₂×C₃²×C₄	C₂³×C₃	1	13	yes	—
68	C₂×C₃²×C₄	C₂³×C₃	1	26	yes	—
69	#533	C₃³	1	1	yes	characteristic, direct product factor, Sylow
70	C₂×C₃³	—	2	2	no	—
71	C₂×C₃³	C₂²×Q₈	1	1	yes	characteristic
72	C₂×C₃³	C₂×#41	1	2	yes	—
73	C₂×C₃×Q₈	C₂²×C₃²	1	52	yes	—
74	C₂²×C₃×C₄	C₂²×C₃²	1	26	yes	—
75	C₂²×C₃×C₄	C₂²×C₃²	1	13	yes	—
76	C₃×C₄⋊C₄	C₂²×C₃²	1	52	yes	—
77	C₃×C₄⋊C₄	C₂²×C₃²	1	104	yes	—
78	C₃×C₄⋊C₄	C₂²×C₃²	1	52	yes	—
79	C₃×C₂²⋊C₄	C₂²×C₃²	1	52	yes	—
80	C₃×C₄²	C₂²×C₃²	1	26	yes	—
81	C₃×C₄²	C₂²×C₃²	1	13	yes	—
82	C₃×C₄²	C₂²×C₃²	1	13	yes	—
83	C₃×C₄²	C₂²×C₃²	1	52	yes	—
84	C₂²×C₃²	—	2	26	no	—
85	C₂²×C₃²	C₂×C₃×Q₈	1	26	yes	—
86	C₂²×C₃²	C₂⁴×C₃	1	13	yes	—
87	C₃²×C₄	—	2	26	no	—
88	C₃²×C₄	—	2	104	no	—
89	C₃²×C₄	C₂×C₃×Q₈	1	26	yes	—
90	C₃²×C₄	C₂×C₃×Q₈	1	26	yes	—
91	C₃²×C₄	C₃×#41	1	52	yes	—
92	C₄⋊Q₈	C₂×C₃³	1	2	yes	complemented
93	#125	C₂×C₃³	1	2	yes	complemented
94	C₂²⋊Q₈	C₂×C₃³	1	4	yes	—
95	C₄×Q₈	C₂×C₃³	1	4	yes	complemented
96	#117	C₂×C₃³	1	2	yes	—
97	C₂×C₄²	C₂×C₃³	1	1	yes	characteristic
98	C₃³	#533	1	1	yes	characteristic, direct product factor, Sylow
99	C₂³×C₃	C₂³×C₃²	1	13	yes	—
100	C₃×Q₈	—	2	104	no	—
101	C₂×C₃×C₄	—	2	52	no	—
102	C₂×C₃×C₄	C₃²×Q₈	1	52	yes	complemented
103	C₂×C₃×C₄	C₂³×C₃²	1	26	yes	—
104	C₂×C₃×C₄	C₂³×C₃²	1	104	yes	—
105	C₂×C₃×C₄	C₂³×C₃²	1	13	yes	—
106	C₂×C₃×C₄	C₂³×C₃²	1	13	yes	—
107	C₂×C₃×C₄	C₂³×C₃²	1	26	yes	—
108	C₂×C₃²	—	2	26	no	—
109	C₂×C₃²	C₂²×C₃×Q₈	1	13	yes	—
110	C₂×C₃²	C₂×C₃×#41	1	26	yes	—
111	C₂×Q₈	C₂²×C₃³	1	4	yes	—
112	C₂²×C₄	C₂²×C₃³	1	2	yes	—
113	C₂²×C₄	C₂²×C₃³	1	1	yes	characteristic
114	C₄⋊C₄	C₂²×C₃³	1	4	yes	—
115	C₄⋊C₄	C₂²×C₃³	1	8	yes	—
116	C₄⋊C₄	C₂²×C₃³	1	4	yes	—
117	C₂²⋊C₄	C₂²×C₃³	1	4	yes	—
118	C₄²	C₂²×C₃³	1	2	yes	—
119	C₄²	C₂²×C₃³	1	1	yes	characteristic
120	C₄²	C₂²×C₃³	1	1	yes	characteristic
121	C₄²	C₂²×C₃³	1	4	yes	—
122	C₂²×C₃	—	2	26	no	—
123	C₂²×C₃	C₂×C₃²×Q₈	1	26	yes	—
124	C₂²×C₃	C₂⁴×C₃²	1	13	yes	—
125	C₃×C₄	—	2	26	no	—
126	C₃×C₄	—	2	104	no	—
127	C₃×C₄	C₂×C₃²×Q₈	1	26	yes	—
128	C₃×C₄	C₂×C₃²×Q₈	1	26	yes	—
129	C₃×C₄	C₃²×#41	1	52	yes	—
130	C₃²	C₃×#533	1	13	yes	direct product factor
131	C₂³	C₂³×C₃³	1	1	yes	characteristic
132	Q₈	—	2	8	no	—
133	C₂×C₄	—	2	4	no	—
134	C₂×C₄	C₃³×Q₈	1	4	yes	complemented
135	C₂×C₄	C₂³×C₃³	1	2	yes	—
136	C₂×C₄	C₂³×C₃³	1	8	yes	—
137	C₂×C₄	C₂³×C₃³	1	1	yes	characteristic
138	C₂×C₄	C₂³×C₃³	1	1	yes	characteristic
139	C₂×C₄	C₂³×C₃³	1	2	yes	—
140	C₂×C₃	—	2	26	no	—
141	C₂×C₃	C₂²×C₃²×Q₈	1	13	yes	—
142	C₂×C₃	C₂×C₃²×#41	1	26	yes	—
143	C₂²	—	2	2	no	—
144	C₂²	C₂×C₃³×Q₈	1	2	yes	—
145	C₂²	C₂⁴×C₃³	1	1	yes	characteristic
146	C₄	—	2	2	no	—
147	C₄	—	2	8	no	—
148	C₄	C₂×C₃³×Q₈	1	2	yes	—
149	C₄	C₂×C₃³×Q₈	1	2	yes	—
150	C₄	C₃³×#41	1	4	yes	—
151	C₃	C₃²×#533	1	13	yes	direct product factor
152	C₂	—	2	2	no	—
153	C₂	C₂²×C₃³×Q₈	1	1	yes	characteristic
154	C₂	C₂×C₃³×#41	1	2	yes	—

	subgroup	G/H
center	C₂×C₃³×C₄	C₂³
commutator, Frattini, cosocle	C₂²	C₂⁴×C₃³
socle	C₂²×C₃³	C₂⁴
prime radical	C₂³×C₃³	C₂³
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₃³×C₄ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₃³×C₄⋊Q₈	C₂	semidirect
2. C₃³×#125	C₂	semidirect
4. C₃³×C₄×Q₈	C₂	semidirect
7. C₃²×#533	C₃	direct (N × H)
19. C₃²×C₄⋊Q₈	C₂×C₃	semidirect
20. C₃²×#125	C₂×C₃	semidirect
22. C₃²×C₄×Q₈	C₂×C₃	semidirect
28. C₂×C₃³×C₄	Q₈	semidirect
34. C₃×#533	C₃²	direct (N × H)
54. C₃×C₄⋊Q₈	C₂×C₃²	semidirect
55. C₃×#125	C₂×C₃²	semidirect
57. C₃×C₄×Q₈	C₂×C₃²	semidirect
63. C₂×C₃²×C₄	C₃×Q₈	semidirect
69. #533	C₃³	direct (N × H)
92. C₄⋊Q₈	C₂×C₃³	semidirect
93. #125	C₂×C₃³	semidirect
95. C₄×Q₈	C₂×C₃³	semidirect
98. C₃³	#533	direct (N × H)
102. C₂×C₃×C₄	C₃²×Q₈	semidirect
130. C₃²	C₃×#533	direct (N × H)
134. C₂×C₄	C₃³×Q₈	semidirect
151. C₃	C₃²×#533	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
C₃³×#125	2
C₃³×C₂²⋊Q₈	2
C₃³×C₄×Q₈	2
C₃³×#117	2
C₂×C₃³×C₄²	2
C₃²×#533	3

Subgroups — C33×#533 #13179988

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃³×#533 #13179988