Subgroups — C₃²×#536 #10596337

Order 576.

810 conjugacy classes of subgroups.

The subgroup lattice has 159 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃²×#41	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₃²×Q₈	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃²×C₄⋊Q₈	C₂	1	2	yes	maximal, complemented
4	C₃²×#124	C₂	1	2	yes	maximal, complemented
5	C₃²×#123	C₂	1	4	yes	maximal, complemented
6	C₃²×C₂²⋊Q₈	C₂	1	4	yes	maximal, complemented
7	C₃²×#117	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃×#536	C₃	1	4	yes	maximal, direct product factor
9	C₃²×#41	—	2	4	no	—
10	C₂×C₃²×Q₈	—	2	4	no	—
11	C₂×C₃²×Q₈	C₂²	1	2	yes	—
12	C₂×C₃²×Q₈	C₂²	1	2	yes	—
13	C₂×C₃²×Q₈	C₂²	1	1	yes	characteristic
14	C₂×C₃²×D₄	C₂²	1	2	yes	—
15	C₂×C₃²×D₄	C₂²	1	1	yes	characteristic
16	C₂²×C₃²×C₄	C₂²	1	2	yes	—
17	C₂²×C₃²×C₄	C₂²	1	2	yes	—
18	C₂²×C₃²×C₄	C₂²	1	1	yes	characteristic
19	C₃²×C₄⋊C₄	C₂²	1	2	yes	—
20	C₃²×C₄⋊C₄	C₂²	1	8	yes	—
21	C₃²×C₂²⋊C₄	C₂²	1	2	yes	—
22	C₃²×C₂²⋊C₄	C₂²	1	8	yes	—
23	C₃²×C₄²	C₂²	1	2	yes	—
24	C₂×C₃×#41	C₂×C₃	1	4	yes	—
25	C₂²×C₃×Q₈	C₂×C₃	1	4	yes	complemented
26	C₃×C₄⋊Q₈	C₂×C₃	1	8	yes	complemented
27	C₃×#124	C₂×C₃	1	8	yes	complemented
28	C₃×#123	C₂×C₃	1	16	yes	complemented
29	C₃×C₂²⋊Q₈	C₂×C₃	1	16	yes	complemented
30	C₃×#117	C₂×C₃	1	4	yes	complemented
31	C₂³×C₃²	C₂³	1	2	yes	—
32	C₂³×C₃²	C₂³	1	1	yes	characteristic
33	C₃²×Q₈	—	2	4	no	—
34	C₃²×Q₈	—	2	4	no	—
35	C₃²×Q₈	—	2	2	no	—
36	C₃²×D₄	—	2	4	no	—
37	C₃²×D₄	—	2	2	no	—
38	C₂×C₃²×C₄	—	2	4	no	—
39	C₂×C₃²×C₄	—	2	4	no	—
40	C₂×C₃²×C₄	D₄	1	4	yes	—
41	C₂×C₃²×C₄	C₂³	1	4	yes	—
42	C₂×C₃²×C₄	C₂³	1	4	yes	—
43	C₂×C₃²×C₄	C₂³	1	2	yes	—
44	C₂×C₃²×C₄	C₂³	1	1	yes	characteristic
45	C₂×C₃²×C₄	C₂³	1	1	yes	characteristic
46	#536	C₃²	1	1	yes	characteristic, direct product factor, Sylow
47	C₃×#41	—	2	16	no	—
48	C₂×C₃×Q₈	—	2	16	no	—
49	C₂×C₃×Q₈	C₂²×C₃	1	8	yes	—
50	C₂×C₃×Q₈	C₂²×C₃	1	8	yes	—
51	C₂×C₃×Q₈	C₂²×C₃	1	4	yes	—
52	C₂×C₃×D₄	C₂²×C₃	1	8	yes	—
53	C₂×C₃×D₄	C₂²×C₃	1	4	yes	—
54	C₂²×C₃×C₄	C₂²×C₃	1	8	yes	—
55	C₂²×C₃×C₄	C₂²×C₃	1	8	yes	—
56	C₂²×C₃×C₄	C₂²×C₃	1	4	yes	—
57	C₃×C₄⋊C₄	C₂²×C₃	1	8	yes	—
58	C₃×C₄⋊C₄	C₂²×C₃	1	32	yes	—
59	C₃×C₂²⋊C₄	C₂²×C₃	1	8	yes	—
60	C₃×C₂²⋊C₄	C₂²×C₃	1	32	yes	—
61	C₃×C₄²	C₂²×C₃	1	8	yes	—
62	C₂²×C₃²	—	2	2	no	—
63	C₂²×C₃²	—	2	2	no	—
64	C₂²×C₃²	—	2	4	no	—
65	C₂²×C₃²	C₂×D₄	1	2	yes	—
66	C₂²×C₃²	C₂⁴	1	1	yes	characteristic
67	C₃²×C₄	—	2	2	no	—
68	C₃²×C₄	—	2	4	no	—
69	C₃²×C₄	—	2	4	no	—
70	C₃²×C₄	C₂×D₄	1	2	yes	—
71	C₃²×C₄	C₂×D₄	1	2	yes	—
72	C₂×#41	C₂×C₃²	1	1	yes	characteristic
73	C₂²×Q₈	C₂×C₃²	1	1	yes	characteristic, complemented
74	C₄⋊Q₈	C₂×C₃²	1	2	yes	complemented
75	#124	C₂×C₃²	1	2	yes	complemented
76	#123	C₂×C₃²	1	4	yes	complemented
77	C₂²⋊Q₈	C₂×C₃²	1	4	yes	complemented
78	#117	C₂×C₃²	1	1	yes	characteristic, complemented
79	C₂³×C₃	C₂³×C₃	1	8	yes	—
80	C₂³×C₃	C₂³×C₃	1	4	yes	—
81	C₃×Q₈	—	2	16	no	—
82	C₃×Q₈	—	2	16	no	—
83	C₃×Q₈	—	2	8	no	—
84	C₃×D₄	—	2	16	no	—
85	C₃×D₄	—	2	8	no	—
86	C₂×C₃×C₄	—	2	16	no	—
87	C₂×C₃×C₄	—	2	16	no	—
88	C₂×C₃×C₄	C₃×D₄	1	16	yes	—
89	C₂×C₃×C₄	C₂³×C₃	1	16	yes	—
90	C₂×C₃×C₄	C₂³×C₃	1	16	yes	—
91	C₂×C₃×C₄	C₂³×C₃	1	8	yes	—
92	C₂×C₃×C₄	C₂³×C₃	1	4	yes	—
93	C₂×C₃×C₄	C₂³×C₃	1	4	yes	—
94	C₂×C₃²	—	2	2	no	—
95	C₂×C₃²	—	4	2	no	—
96	C₂×C₃²	C₂²×D₄	1	1	yes	characteristic
97	C₂×C₃²	2₋¹⁺⁴	1	2	yes	—
98	#41	—	2	4	no	—
99	C₂×Q₈	—	2	4	no	—
100	C₂×Q₈	C₂²×C₃²	1	2	yes	—
101	C₂×Q₈	C₂²×C₃²	1	2	yes	—
102	C₂×Q₈	C₂²×C₃²	1	1	yes	characteristic
103	C₂×D₄	C₂²×C₃²	1	2	yes	—
104	C₂×D₄	C₂²×C₃²	1	1	yes	characteristic
105	C₂²×C₄	C₂²×C₃²	1	2	yes	—
106	C₂²×C₄	C₂²×C₃²	1	2	yes	—
107	C₂²×C₄	C₂²×C₃²	1	1	yes	characteristic
108	C₄⋊C₄	C₂²×C₃²	1	2	yes	—
109	C₄⋊C₄	C₂²×C₃²	1	8	yes	—
110	C₂²⋊C₄	C₂²×C₃²	1	2	yes	—
111	C₂²⋊C₄	C₂²×C₃²	1	8	yes	—
112	C₄²	C₂²×C₃²	1	2	yes	—
113	C₂²×C₃	—	2	8	no	—
114	C₂²×C₃	—	2	8	no	—
115	C₂²×C₃	—	2	16	no	—
116	C₂²×C₃	C₂×C₃×D₄	1	8	yes	—
117	C₂²×C₃	C₂⁴×C₃	1	4	yes	—
118	C₃×C₄	—	2	8	no	—
119	C₃×C₄	—	2	16	no	—
120	C₃×C₄	—	2	16	no	—
121	C₃×C₄	C₂×C₃×D₄	1	8	yes	—
122	C₃×C₄	C₂×C₃×D₄	1	8	yes	—
123	C₃²	#536	1	1	yes	characteristic, direct product factor, Sylow
124	C₂³	C₂³×C₃²	1	2	yes	—
125	C₂³	C₂³×C₃²	1	1	yes	characteristic
126	Q₈	—	2	4	no	—
127	Q₈	—	2	4	no	—
128	Q₈	—	2	2	no	—
129	D₄	—	2	4	no	—
130	D₄	—	2	2	no	—
131	C₂×C₄	—	2	4	no	—
132	C₂×C₄	—	2	4	no	—
133	C₂×C₄	C₃²×D₄	1	4	yes	—
134	C₂×C₄	C₂³×C₃²	1	4	yes	—
135	C₂×C₄	C₂³×C₃²	1	4	yes	—
136	C₂×C₄	C₂³×C₃²	1	2	yes	—
137	C₂×C₄	C₂³×C₃²	1	1	yes	characteristic
138	C₂×C₄	C₂³×C₃²	1	1	yes	characteristic
139	C₂×C₃	—	2	8	no	—
140	C₂×C₃	—	4	8	no	—
141	C₂×C₃	C₂²×C₃×D₄	1	4	yes	—
142	C₂×C₃	C₃×2₋¹⁺⁴	1	8	yes	—
143	C₂²	—	2	2	no	—
144	C₂²	—	2	2	no	—
145	C₂²	—	2	4	no	—
146	C₂²	C₂×C₃²×D₄	1	2	yes	—
147	C₂²	C₂⁴×C₃²	1	1	yes	characteristic
148	C₄	—	2	2	no	—
149	C₄	—	2	4	no	—
150	C₄	—	2	4	no	—
151	C₄	C₂×C₃²×D₄	1	2	yes	—
152	C₄	C₂×C₃²×D₄	1	2	yes	—
153	C₃	C₃×#536	1	4	yes	direct product factor
154	C₂	—	2	2	no	—
155	C₂	—	4	2	no	—
156	C₂	C₂²×C₃²×D₄	1	1	yes	characteristic
157	C₂	C₃²×2₋¹⁺⁴	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₃²	C₂⁴
commutator, Frattini, cosocle	C₂²	C₂⁴×C₃²
prime radical	C₂×C₃²×#41	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₃² ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂²×C₃²×Q₈	C₂	semidirect
3. C₃²×C₄⋊Q₈	C₂	semidirect
4. C₃²×#124	C₂	semidirect
5. C₃²×#123	C₂	semidirect
6. C₃²×C₂²⋊Q₈	C₂	semidirect
7. C₃²×#117	C₂	semidirect
8. C₃×#536	C₃	direct (N × H)
25. C₂²×C₃×Q₈	C₂×C₃	semidirect
26. C₃×C₄⋊Q₈	C₂×C₃	semidirect
27. C₃×#124	C₂×C₃	semidirect
28. C₃×#123	C₂×C₃	semidirect
29. C₃×C₂²⋊Q₈	C₂×C₃	semidirect
30. C₃×#117	C₂×C₃	semidirect
46. #536	C₃²	direct (N × H)
73. C₂²×Q₈	C₂×C₃²	semidirect
74. C₄⋊Q₈	C₂×C₃²	semidirect
75. #124	C₂×C₃²	semidirect
76. #123	C₂×C₃²	semidirect
77. C₂²⋊Q₈	C₂×C₃²	semidirect
78. #117	C₂×C₃²	semidirect
123. C₃²	#536	direct (N × H)
153. C₃	C₃×#536	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₃²×#41	2
C₂²×C₃²×Q₈	2
C₃²×C₄⋊Q₈	2
C₃²×#124	2
C₃²×#123	2
C₃²×C₂²⋊Q₈	2
C₃²×#117	2
C₃×#536	3

Subgroups — C32×#536 #10596337

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃²×#536 #10596337