Subgroups — C₃²×#1579 #11768628

Order 1152.

480 conjugacy classes of subgroups.

The subgroup lattice has 150 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²×C₈×Q₈	C₂	1	2	yes	maximal
2	C₃²×C₄×Q₁₆	C₂	1	1	yes	maximal, characteristic
3	C₃²×#334	C₂	1	1	yes	maximal, characteristic
4	C₃²×Q₈⋊C₈	C₂	1	2	yes	maximal
5	C₃²×#322	C₂	1	1	yes	maximal, characteristic
6	C₃×#1579	C₃	1	4	yes	maximal, direct product factor
7	C₂×C₃²×Q₁₆	C₄	1	1	yes	characteristic
8	C₃²×C₄×Q₈	C₂²	1	2	yes	—
9	C₃²×#107	C₄	1	1	yes	characteristic
10	C₃²×C₄⋊C₈	—	2	2	no	—
11	C₃²×C₄⋊C₈	C₂²	1	2	yes	—
12	C₃²×#103	C₄	1	2	yes	—
13	C₃²×C₄×C₈	—	2	2	no	—
14	C₃²×C₄×C₈	C₂²	1	1	yes	characteristic
15	C₃²×C₄×C₈	C₂²	1	1	yes	characteristic
16	C₃²×C₄×C₈	C₂²	1	1	yes	characteristic
17	C₃×C₈×Q₈	C₂×C₃	1	8	yes	—
18	C₃×C₄×Q₁₆	C₂×C₃	1	4	yes	—
19	C₃×#334	C₂×C₃	1	4	yes	—
20	C₃×Q₈⋊C₈	C₂×C₃	1	8	yes	—
21	C₃×#322	C₂×C₃	1	4	yes	—
22	C₂×C₃²×Q₈	C₂×C₄	1	2	yes	—
23	C₃²×Q₁₆	C₈	1	4	yes	complemented
24	C₂×C₃²×C₈	—	2	2	no	—
25	C₂×C₃²×C₈	—	2	1	no	—
26	C₂×C₃²×C₈	C₂×C₄	1	1	yes	characteristic
27	C₂×C₃²×C₈	C₂×C₄	1	1	yes	characteristic
28	C₂×C₃²×C₈	D₄	1	2	yes	—
29	C₃²×C₄⋊C₄	—	2	2	no	—
30	C₃²×C₄⋊C₄	C₂×C₄	1	2	yes	—
31	C₃²×C₄²	—	2	2	no	—
32	C₃²×C₄²	C₂³	1	1	yes	characteristic
33	#1579	C₃²	1	1	yes	characteristic, direct product factor, Sylow
34	C₂×C₃×Q₁₆	C₃×C₄	1	4	yes	—
35	C₃×C₄×Q₈	C₂²×C₃	1	8	yes	—
36	C₃×#107	C₃×C₄	1	4	yes	—
37	C₃×C₄⋊C₈	—	2	8	no	—
38	C₃×C₄⋊C₈	C₂²×C₃	1	8	yes	—
39	C₃×#103	C₃×C₄	1	8	yes	—
40	C₃×C₄×C₈	—	2	8	no	—
41	C₃×C₄×C₈	C₂²×C₃	1	4	yes	—
42	C₃×C₄×C₈	C₂²×C₃	1	4	yes	—
43	C₃×C₄×C₈	C₂²×C₃	1	4	yes	—
44	C₃²×Q₈	—	2	2	no	—
45	C₃²×Q₈	C₂×C₈	1	4	yes	—
46	C₂×C₃²×C₄	—	2	2	no	—
47	C₂×C₃²×C₄	—	2	2	no	—
48	C₂×C₃²×C₄	C₂²×C₄	1	1	yes	characteristic
49	C₂×C₃²×C₄	C₂×D₄	1	1	yes	characteristic
50	C₂×C₃²×C₄	#41	1	1	yes	characteristic
51	C₃²×C₈	—	2	1	no	—
52	C₃²×C₈	—	2	2	no	—
53	C₃²×C₈	—	4	1	no	—
54	C₃²×C₈	—	4	2	no	—
55	C₃²×C₈	C₂×C₈	1	2	yes	—
56	C₈×Q₈	C₂×C₃²	1	2	yes	—
57	C₄×Q₁₆	C₂×C₃²	1	1	yes	characteristic
58	#334	C₂×C₃²	1	1	yes	characteristic
59	Q₈⋊C₈	C₂×C₃²	1	2	yes	—
60	#322	C₂×C₃²	1	1	yes	characteristic
61	C₂×C₃×Q₈	C₂×C₃×C₄	1	8	yes	—
62	C₃×Q₁₆	C₃×C₈	1	16	yes	complemented
63	C₂×C₃×C₈	—	2	8	no	—
64	C₂×C₃×C₈	—	2	4	no	—
65	C₂×C₃×C₈	C₂×C₃×C₄	1	4	yes	—
66	C₂×C₃×C₈	C₂×C₃×C₄	1	4	yes	—
67	C₂×C₃×C₈	C₃×D₄	1	8	yes	—
68	C₃×C₄⋊C₄	—	2	8	no	—
69	C₃×C₄⋊C₄	C₂×C₃×C₄	1	8	yes	—
70	C₃×C₄²	—	2	8	no	—
71	C₃×C₄²	C₂³×C₃	1	4	yes	—
72	C₂²×C₃²	C₄×D₄	1	1	yes	characteristic
73	C₃²×C₄	—	2	4	no	—
74	C₃²×C₄	—	2	1	no	—
75	C₃²×C₄	—	4	2	no	—
76	C₃²×C₄	C₂²×C₈	1	1	yes	characteristic
77	C₃²×C₄	#131	1	1	yes	characteristic
78	C₃²×C₄	#137	1	2	yes	—
79	C₂×Q₁₆	C₃²×C₄	1	1	yes	characteristic
80	C₄×Q₈	C₂²×C₃²	1	2	yes	—
81	#107	C₃²×C₄	1	1	yes	characteristic
82	C₄⋊C₈	—	2	2	no	—
83	C₄⋊C₈	C₂²×C₃²	1	2	yes	—
84	#103	C₃²×C₄	1	2	yes	—
85	C₄×C₈	—	2	2	no	—
86	C₄×C₈	C₂²×C₃²	1	1	yes	characteristic
87	C₄×C₈	C₂²×C₃²	1	1	yes	characteristic
88	C₄×C₈	C₂²×C₃²	1	1	yes	characteristic
89	C₃×Q₈	—	2	8	no	—
90	C₃×Q₈	C₂×C₃×C₈	1	16	yes	—
91	C₂×C₃×C₄	—	2	8	no	—
92	C₂×C₃×C₄	—	2	8	no	—
93	C₂×C₃×C₄	C₂²×C₃×C₄	1	4	yes	—
94	C₂×C₃×C₄	C₂×C₃×D₄	1	4	yes	—
95	C₂×C₃×C₄	C₃×#41	1	4	yes	—
96	C₃×C₈	—	2	4	no	—
97	C₃×C₈	—	2	8	no	—
98	C₃×C₈	—	4	4	no	—
99	C₃×C₈	—	4	8	no	—
100	C₃×C₈	C₂×C₃×C₈	1	8	yes	—
101	C₂×C₃²	C₈×D₄	1	1	yes	characteristic
102	C₂×C₃²	#441	1	1	yes	characteristic
103	C₂×C₃²	#444	1	1	yes	characteristic
104	C₂×Q₈	C₂×C₃²×C₄	1	2	yes	—
105	Q₁₆	C₃²×C₈	1	4	yes	complemented
106	C₂×C₈	—	2	2	no	—
107	C₂×C₈	—	2	1	no	—
108	C₂×C₈	C₂×C₃²×C₄	1	1	yes	characteristic
109	C₂×C₈	C₂×C₃²×C₄	1	1	yes	characteristic
110	C₂×C₈	C₃²×D₄	1	2	yes	—
111	C₄⋊C₄	—	2	2	no	—
112	C₄⋊C₄	C₂×C₃²×C₄	1	2	yes	—
113	C₄²	—	2	2	no	—
114	C₄²	C₂³×C₃²	1	1	yes	characteristic
115	C₂²×C₃	C₃×C₄×D₄	1	4	yes	—
116	C₃×C₄	—	2	16	no	—
117	C₃×C₄	—	2	4	no	—
118	C₃×C₄	—	4	8	no	—
119	C₃×C₄	C₂²×C₃×C₈	1	4	yes	—
120	C₃×C₄	C₃×#131	1	4	yes	—
121	C₃×C₄	C₃×#137	1	8	yes	—
122	C₃²	#1579	1	1	yes	characteristic, direct product factor, Sylow
123	Q₈	—	2	2	no	—
124	Q₈	C₂×C₃²×C₈	1	4	yes	—
125	C₂×C₄	—	2	2	no	—
126	C₂×C₄	—	2	2	no	—
127	C₂×C₄	C₂²×C₃²×C₄	1	1	yes	characteristic
128	C₂×C₄	C₂×C₃²×D₄	1	1	yes	characteristic
129	C₂×C₄	C₃²×#41	1	1	yes	characteristic
130	C₈	—	2	1	no	—
131	C₈	—	2	2	no	—
132	C₈	—	4	1	no	—
133	C₈	—	4	2	no	—
134	C₈	C₂×C₃²×C₈	1	2	yes	—
135	C₂×C₃	C₃×C₈×D₄	1	4	yes	—
136	C₂×C₃	C₃×#441	1	4	yes	—
137	C₂×C₃	C₃×#444	1	4	yes	—
138	C₂²	C₃²×C₄×D₄	1	1	yes	characteristic
139	C₄	—	2	4	no	—
140	C₄	—	2	1	no	—
141	C₄	—	4	2	no	—
142	C₄	C₂²×C₃²×C₈	1	1	yes	characteristic
143	C₄	C₃²×#131	1	1	yes	characteristic
144	C₄	C₃²×#137	1	2	yes	—
145	C₃	C₃×#1579	1	4	yes	direct product factor
146	C₂	C₃²×C₈×D₄	1	1	yes	characteristic
147	C₂	C₃²×#441	1	1	yes	characteristic
148	C₂	C₃²×#444	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₃²×C₄	C₂×D₄
commutator	C₄	C₂²×C₃²×C₈
Frattini, cosocle	C₄²	C₂³×C₃²
socle, prime radical	C₂²×C₃²	C₄×D₄
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₃²×C₄ ◁ C₃²×C₄×C₈ ◁ G

Complements

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

N	H ≅ G/N	type
6. C₃×#1579	C₃	direct (N × H)
23. C₃²×Q₁₆	C₈	semidirect
33. #1579	C₃²	direct (N × H)
62. C₃×Q₁₆	C₃×C₈	semidirect
105. Q₁₆	C₃²×C₈	semidirect
122. C₃²	#1579	direct (N × H)
145. C₃	C₃×#1579	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃²×C₈×Q₈	2
C₃²×C₄×Q₁₆	2
C₃²×#334	2
C₃²×Q₈⋊C₈	2
C₃²×#322	2
C₃×#1579	3

Subgroups — C32×#1579 #11768628

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃²×#1579 #11768628