Subgroups — C₃²×#551 #10596352

Order 576.

744 conjugacy classes of subgroups.

The subgroup lattice has 207 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²×#126	C₂	1	2	yes	maximal, complemented
2	C₃²×#124	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃²×#123	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃²×#123	C₂	1	2	yes	maximal, complemented
5	C₃²×C₂²⋊Q₈	C₂	1	2	yes	maximal, complemented
6	C₃²×#120	C₂	1	1	yes	maximal, characteristic
7	C₃²×C₄×D₄	C₂	1	2	yes	maximal, complemented
8	C₃²×#117	C₂	1	2	yes	maximal, complemented
9	C₂×C₃²×C₂²⋊C₄	C₂	1	2	yes	maximal, complemented
10	C₃×#551	C₃	1	4	yes	maximal, direct product factor
11	C₂⁴×C₃²	C₂²	1	1	yes	characteristic
12	C₂×C₃²×Q₈	C₂²	1	1	yes	characteristic, complemented
13	C₂×C₃²×D₄	C₂²	1	2	yes	—
14	C₂×C₃²×D₄	C₂²	1	1	yes	characteristic
15	C₂²×C₃²×C₄	C₂²	1	2	yes	—
16	C₂²×C₃²×C₄	C₂²	1	2	yes	—
17	C₂²×C₃²×C₄	C₂²	1	1	yes	characteristic, complemented
18	C₃²×C₄⋊C₄	C₂²	1	2	yes	complemented
19	C₃²×C₄⋊C₄	C₂²	1	2	yes	complemented
20	C₃²×C₄⋊C₄	C₂²	1	2	yes	complemented
21	C₃²×C₄⋊C₄	C₂²	1	2	yes	complemented
22	C₃²×C₂²⋊C₄	C₂²	1	2	yes	—
23	C₃²×C₂²⋊C₄	C₂²	1	2	yes	—
24	C₃²×C₂²⋊C₄	C₂²	1	2	yes	—
25	C₃²×C₂²⋊C₄	C₂²	1	2	yes	—
26	C₃²×C₂²⋊C₄	C₂²	1	2	yes	—
27	C₃²×C₂²⋊C₄	C₂²	1	1	yes	characteristic
28	C₃²×C₂²⋊C₄	C₂²	1	2	yes	complemented
29	C₃²×C₂²⋊C₄	C₂²	1	1	yes	characteristic, complemented
30	C₃²×C₄²	C₂²	1	1	yes	characteristic, complemented
31	C₃²×C₄²	C₂²	1	2	yes	complemented
32	C₃×#126	C₂×C₃	1	8	yes	complemented
33	C₃×#124	C₂×C₃	1	4	yes	complemented
34	C₃×#123	C₂×C₃	1	4	yes	complemented
35	C₃×#123	C₂×C₃	1	8	yes	complemented
36	C₃×C₂²⋊Q₈	C₂×C₃	1	8	yes	complemented
37	C₃×#120	C₂×C₃	1	4	yes	—
38	C₃×C₄×D₄	C₂×C₃	1	8	yes	complemented
39	C₃×#117	C₂×C₃	1	8	yes	complemented
40	C₂×C₃×C₂²⋊C₄	C₂×C₃	1	8	yes	complemented
41	C₂³×C₃²	—	2	4	no	—
42	C₂³×C₃²	—	4	1	no	—
43	C₂³×C₃²	C₂³	1	1	yes	characteristic
44	C₂³×C₃²	C₂³	1	1	yes	characteristic
45	C₂³×C₃²	C₂³	1	2	yes	—
46	C₃²×Q₈	—	4	1	no	—
47	C₃²×D₄	—	2	4	no	—
48	C₃²×D₄	—	4	1	no	—
49	C₂×C₃²×C₄	—	2	4	no	—
50	C₂×C₃²×C₄	—	4	2	no	—
51	C₂×C₃²×C₄	—	4	1	no	—
52	C₂×C₃²×C₄	C₂³	1	2	yes	—
53	C₂×C₃²×C₄	C₂³	1	2	yes	—
54	C₂×C₃²×C₄	C₂³	1	2	yes	—
55	C₂×C₃²×C₄	C₂³	1	1	yes	characteristic
56	C₂×C₃²×C₄	C₂³	1	2	yes	—
57	C₂×C₃²×C₄	C₂³	1	2	yes	—
58	#551	C₃²	1	1	yes	characteristic, direct product factor, Sylow
59	C₂⁴×C₃	C₂²×C₃	1	4	yes	—
60	C₂×C₃×Q₈	C₂²×C₃	1	4	yes	complemented
61	C₂×C₃×D₄	C₂²×C₃	1	8	yes	—
62	C₂×C₃×D₄	C₂²×C₃	1	4	yes	—
63	C₂²×C₃×C₄	C₂²×C₃	1	8	yes	—
64	C₂²×C₃×C₄	C₂²×C₃	1	8	yes	—
65	C₂²×C₃×C₄	C₂²×C₃	1	4	yes	complemented
66	C₃×C₄⋊C₄	C₂²×C₃	1	8	yes	complemented
67	C₃×C₄⋊C₄	C₂²×C₃	1	8	yes	complemented
68	C₃×C₄⋊C₄	C₂²×C₃	1	8	yes	complemented
69	C₃×C₄⋊C₄	C₂²×C₃	1	8	yes	complemented
70	C₃×C₂²⋊C₄	C₂²×C₃	1	8	yes	—
71	C₃×C₂²⋊C₄	C₂²×C₃	1	8	yes	—
72	C₃×C₂²⋊C₄	C₂²×C₃	1	8	yes	—
73	C₃×C₂²⋊C₄	C₂²×C₃	1	8	yes	—
74	C₃×C₂²⋊C₄	C₂²×C₃	1	8	yes	—
75	C₃×C₂²⋊C₄	C₂²×C₃	1	4	yes	—
76	C₃×C₂²⋊C₄	C₂²×C₃	1	8	yes	complemented
77	C₃×C₂²⋊C₄	C₂²×C₃	1	4	yes	complemented
78	C₃×C₄²	C₂²×C₃	1	4	yes	complemented
79	C₃×C₄²	C₂²×C₃	1	8	yes	complemented
80	C₂²×C₃²	—	2	2	no	—
81	C₂²×C₃²	—	2	2	no	—
82	C₂²×C₃²	—	2	2	no	—
83	C₂²×C₃²	—	2	2	no	—
84	C₂²×C₃²	—	2	1	no	—
85	C₂²×C₃²	—	2	1	no	—
86	C₂²×C₃²	—	4	4	no	—
87	C₂²×C₃²	#41	1	4	yes	—
88	C₂²×C₃²	C₂⁴	1	1	yes	characteristic
89	C₃²×C₄	—	2	2	no	—
90	C₃²×C₄	—	2	2	no	—
91	C₃²×C₄	—	2	2	no	—
92	C₃²×C₄	—	2	2	no	—
93	C₃²×C₄	—	2	2	no	—
94	C₃²×C₄	—	2	1	no	—
95	#126	C₂×C₃²	1	2	yes	complemented
96	#124	C₂×C₃²	1	1	yes	characteristic, complemented
97	#123	C₂×C₃²	1	1	yes	characteristic, complemented
98	#123	C₂×C₃²	1	2	yes	complemented
99	C₂²⋊Q₈	C₂×C₃²	1	2	yes	complemented
100	#120	C₂×C₃²	1	1	yes	characteristic
101	C₄×D₄	C₂×C₃²	1	2	yes	complemented
102	#117	C₂×C₃²	1	2	yes	complemented
103	C₂×C₂²⋊C₄	C₂×C₃²	1	2	yes	complemented
104	C₂³×C₃	—	2	16	no	—
105	C₂³×C₃	—	4	4	no	—
106	C₂³×C₃	C₂³×C₃	1	4	yes	—
107	C₂³×C₃	C₂³×C₃	1	4	yes	—
108	C₂³×C₃	C₂³×C₃	1	8	yes	—
109	C₃×Q₈	—	4	4	no	—
110	C₃×D₄	—	2	16	no	—
111	C₃×D₄	—	4	4	no	—
112	C₂×C₃×C₄	—	2	16	no	—
113	C₂×C₃×C₄	—	4	8	no	—
114	C₂×C₃×C₄	—	4	4	no	—
115	C₂×C₃×C₄	C₂³×C₃	1	8	yes	—
116	C₂×C₃×C₄	C₂³×C₃	1	8	yes	—
117	C₂×C₃×C₄	C₂³×C₃	1	8	yes	—
118	C₂×C₃×C₄	C₂³×C₃	1	4	yes	—
119	C₂×C₃×C₄	C₂³×C₃	1	8	yes	—
120	C₂×C₃×C₄	C₂³×C₃	1	8	yes	—
121	C₂×C₃²	—	2	4	no	—
122	C₂×C₃²	—	4	1	no	—
123	C₂×C₃²	—	4	1	no	—
124	C₂×C₃²	C₂×#41	1	2	yes	—
125	C₂×C₃²	H(5,𝔽₂)	1	1	yes	characteristic
126	C₂⁴	C₂²×C₃²	1	1	yes	characteristic
127	C₂×Q₈	C₂²×C₃²	1	1	yes	characteristic, complemented
128	C₂×D₄	C₂²×C₃²	1	2	yes	—
129	C₂×D₄	C₂²×C₃²	1	1	yes	characteristic
130	C₂²×C₄	C₂²×C₃²	1	2	yes	—
131	C₂²×C₄	C₂²×C₃²	1	2	yes	—
132	C₂²×C₄	C₂²×C₃²	1	1	yes	characteristic, complemented
133	C₄⋊C₄	C₂²×C₃²	1	2	yes	complemented
134	C₄⋊C₄	C₂²×C₃²	1	2	yes	complemented
135	C₄⋊C₄	C₂²×C₃²	1	2	yes	complemented
136	C₄⋊C₄	C₂²×C₃²	1	2	yes	complemented
137	C₂²⋊C₄	C₂²×C₃²	1	2	yes	—
138	C₂²⋊C₄	C₂²×C₃²	1	2	yes	—
139	C₂²⋊C₄	C₂²×C₃²	1	2	yes	—
140	C₂²⋊C₄	C₂²×C₃²	1	2	yes	—
141	C₂²⋊C₄	C₂²×C₃²	1	2	yes	—
142	C₂²⋊C₄	C₂²×C₃²	1	1	yes	characteristic
143	C₂²⋊C₄	C₂²×C₃²	1	2	yes	complemented
144	C₂²⋊C₄	C₂²×C₃²	1	1	yes	characteristic, complemented
145	C₄²	C₂²×C₃²	1	1	yes	characteristic, complemented
146	C₄²	C₂²×C₃²	1	2	yes	complemented
147	C₂²×C₃	—	2	8	no	—
148	C₂²×C₃	—	2	8	no	—
149	C₂²×C₃	—	2	8	no	—
150	C₂²×C₃	—	2	8	no	—
151	C₂²×C₃	—	2	4	no	—
152	C₂²×C₃	—	2	4	no	—
153	C₂²×C₃	—	4	16	no	—
154	C₂²×C₃	C₃×#41	1	16	yes	—
155	C₂²×C₃	C₂⁴×C₃	1	4	yes	—
156	C₃×C₄	—	2	8	no	—
157	C₃×C₄	—	2	8	no	—
158	C₃×C₄	—	2	8	no	—
159	C₃×C₄	—	2	8	no	—
160	C₃×C₄	—	2	8	no	—
161	C₃×C₄	—	2	4	no	—
162	C₃²	#551	1	1	yes	characteristic, direct product factor, Sylow
163	C₂³	—	2	4	no	—
164	C₂³	—	4	1	no	—
165	C₂³	C₂³×C₃²	1	1	yes	characteristic
166	C₂³	C₂³×C₃²	1	1	yes	characteristic
167	C₂³	C₂³×C₃²	1	2	yes	—
168	Q₈	—	4	1	no	—
169	D₄	—	2	4	no	—
170	D₄	—	4	1	no	—
171	C₂×C₄	—	2	4	no	—
172	C₂×C₄	—	4	2	no	—
173	C₂×C₄	—	4	1	no	—
174	C₂×C₄	C₂³×C₃²	1	2	yes	—
175	C₂×C₄	C₂³×C₃²	1	2	yes	—
176	C₂×C₄	C₂³×C₃²	1	2	yes	—
177	C₂×C₄	C₂³×C₃²	1	1	yes	characteristic
178	C₂×C₄	C₂³×C₃²	1	2	yes	—
179	C₂×C₄	C₂³×C₃²	1	2	yes	—
180	C₂×C₃	—	2	16	no	—
181	C₂×C₃	—	4	4	no	—
182	C₂×C₃	—	4	4	no	—
183	C₂×C₃	C₂×C₃×#41	1	8	yes	—
184	C₂×C₃	C₃×H(5,𝔽₂)	1	4	yes	—
185	C₂²	—	2	2	no	—
186	C₂²	—	2	2	no	—
187	C₂²	—	2	2	no	—
188	C₂²	—	2	2	no	—
189	C₂²	—	2	1	no	—
190	C₂²	—	2	1	no	—
191	C₂²	—	4	4	no	—
192	C₂²	C₃²×#41	1	4	yes	—
193	C₂²	C₂⁴×C₃²	1	1	yes	characteristic
194	C₄	—	2	2	no	—
195	C₄	—	2	2	no	—
196	C₄	—	2	2	no	—
197	C₄	—	2	2	no	—
198	C₄	—	2	2	no	—
199	C₄	—	2	1	no	—
200	C₃	C₃×#551	1	4	yes	direct product factor
201	C₂	—	2	4	no	—
202	C₂	—	4	1	no	—
203	C₂	—	4	1	no	—
204	C₂	C₂×C₃²×#41	1	2	yes	—
205	C₂	C₃²×H(5,𝔽₂)	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃²	C₂⁴
commutator, Frattini, cosocle	C₂²	C₂⁴×C₃²
prime radical	C₃²×#120	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
1. C₃²×#126	C₂	semidirect
2. C₃²×#124	C₂	semidirect
3. C₃²×#123	C₂	semidirect
4. C₃²×#123	C₂	semidirect
5. C₃²×C₂²⋊Q₈	C₂	semidirect
7. C₃²×C₄×D₄	C₂	semidirect
8. C₃²×#117	C₂	semidirect
9. C₂×C₃²×C₂²⋊C₄	C₂	semidirect
10. C₃×#551	C₃	direct (N × H)
12. C₂×C₃²×Q₈	C₂²	semidirect
17. C₂²×C₃²×C₄	C₂²	semidirect
18. C₃²×C₄⋊C₄	C₂²	semidirect
19. C₃²×C₄⋊C₄	C₂²	semidirect
20. C₃²×C₄⋊C₄	C₂²	semidirect
21. C₃²×C₄⋊C₄	C₂²	semidirect
28. C₃²×C₂²⋊C₄	C₂²	semidirect
29. C₃²×C₂²⋊C₄	C₂²	semidirect
30. C₃²×C₄²	C₂²	semidirect
31. C₃²×C₄²	C₂²	semidirect
32. C₃×#126	C₂×C₃	semidirect
33. C₃×#124	C₂×C₃	semidirect
34. C₃×#123	C₂×C₃	semidirect
35. C₃×#123	C₂×C₃	semidirect
36. C₃×C₂²⋊Q₈	C₂×C₃	semidirect
38. C₃×C₄×D₄	C₂×C₃	semidirect
39. C₃×#117	C₂×C₃	semidirect
40. C₂×C₃×C₂²⋊C₄	C₂×C₃	semidirect
58. #551	C₃²	direct (N × H)
60. C₂×C₃×Q₈	C₂²×C₃	semidirect
65. C₂²×C₃×C₄	C₂²×C₃	semidirect
66. C₃×C₄⋊C₄	C₂²×C₃	semidirect
67. C₃×C₄⋊C₄	C₂²×C₃	semidirect
68. C₃×C₄⋊C₄	C₂²×C₃	semidirect
69. C₃×C₄⋊C₄	C₂²×C₃	semidirect
76. C₃×C₂²⋊C₄	C₂²×C₃	semidirect
77. C₃×C₂²⋊C₄	C₂²×C₃	semidirect
78. C₃×C₄²	C₂²×C₃	semidirect
79. C₃×C₄²	C₂²×C₃	semidirect
95. #126	C₂×C₃²	semidirect
96. #124	C₂×C₃²	semidirect
97. #123	C₂×C₃²	semidirect
98. #123	C₂×C₃²	semidirect
99. C₂²⋊Q₈	C₂×C₃²	semidirect
101. C₄×D₄	C₂×C₃²	semidirect
102. #117	C₂×C₃²	semidirect
103. C₂×C₂²⋊C₄	C₂×C₃²	semidirect
127. C₂×Q₈	C₂²×C₃²	semidirect
132. C₂²×C₄	C₂²×C₃²	semidirect
133. C₄⋊C₄	C₂²×C₃²	semidirect
134. C₄⋊C₄	C₂²×C₃²	semidirect
135. C₄⋊C₄	C₂²×C₃²	semidirect
136. C₄⋊C₄	C₂²×C₃²	semidirect
143. C₂²⋊C₄	C₂²×C₃²	semidirect
144. C₂²⋊C₄	C₂²×C₃²	semidirect
145. C₄²	C₂²×C₃²	semidirect
146. C₄²	C₂²×C₃²	semidirect
162. C₃²	#551	direct (N × H)
200. C₃	C₃×#551	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×#126	2
C₃²×#124	2
C₃²×#123	2
C₃²×C₂²⋊Q₈	2
C₃²×#120	2
C₃²×C₄×D₄	2
C₃²×#117	2
C₂×C₃²×C₂²⋊C₄	2
C₃×#551	3

Subgroups — C32×#551 #10596352

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃²×#551 #10596352