Subgroups — C₃²×#538 #10596339

Order 576.

750 conjugacy classes of subgroups.

The subgroup lattice has 225 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	2	yes	maximal, complemented
3	C₃²×#123	C₂	1	2	yes	maximal, complemented
4	C₃²×C₂²⋊Q₈	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃²×#121	C₂	1	2	yes	maximal, complemented
6	C₃²×#121	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃²×#120	C₂	1	2	yes	maximal, complemented
8	C₃²×C₄×D₄	C₂	1	2	yes	maximal, complemented
9	C₂×C₃²×C₂²⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃×#538	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	2	yes	—
15	C₂×C₃²×D₄	C₂²	1	2	yes	—
16	C₂×C₃²×D₄	C₂²	1	1	yes	characteristic, complemented
17	C₂²×C₃²×C₄	C₂²	1	1	yes	characteristic
18	C₂²×C₃²×C₄	C₂²	1	1	yes	characteristic
19	C₂²×C₃²×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×C₄⋊C₄	C₂²	1	1	yes	characteristic, complemented
23	C₃²×C₄⋊C₄	C₂²	1	1	yes	characteristic, complemented
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	2	yes	—
28	C₃²×C₂²⋊C₄	C₂²	1	2	yes	—
29	C₃²×C₂²⋊C₄	C₂²	1	2	yes	complemented
30	C₃²×C₂²⋊C₄	C₂²	1	2	yes	complemented
31	C₃²×C₄²	C₂²	1	2	yes	complemented
32	C₃×#126	C₂×C₃	1	8	yes	complemented
33	C₃×#124	C₂×C₃	1	8	yes	complemented
34	C₃×#123	C₂×C₃	1	8	yes	complemented
35	C₃×C₂²⋊Q₈	C₂×C₃	1	4	yes	complemented
36	C₃×#121	C₂×C₃	1	8	yes	complemented
37	C₃×#121	C₂×C₃	1	4	yes	complemented
38	C₃×#120	C₂×C₃	1	8	yes	complemented
39	C₃×C₄×D₄	C₂×C₃	1	8	yes	complemented
40	C₂×C₃×C₂²⋊C₄	C₂×C₃	1	4	yes	complemented
41	C₂³×C₃²	—	2	2	no	—
42	C₂³×C₃²	—	4	1	no	—
43	C₂³×C₃²	—	4	1	no	—
44	C₂³×C₃²	C₂³	1	2	yes	—
45	C₂³×C₃²	C₂³	1	2	yes	—
46	C₂³×C₃²	C₂³	1	1	yes	characteristic
47	C₃²×Q₈	—	4	1	no	—
48	C₃²×D₄	—	2	4	no	—
49	C₃²×D₄	—	4	2	no	—
50	C₃²×D₄	—	4	2	no	—
51	C₃²×D₄	—	4	1	no	—
52	C₂×C₃²×C₄	—	4	1	no	—
53	C₂×C₃²×C₄	—	4	1	no	—
54	C₂×C₃²×C₄	—	4	2	no	—
55	C₂×C₃²×C₄	C₂³	1	2	yes	—
56	C₂×C₃²×C₄	C₂³	1	2	yes	—
57	C₂×C₃²×C₄	C₂³	1	2	yes	—
58	C₂×C₃²×C₄	C₂³	1	1	yes	characteristic
59	C₂×C₃²×C₄	C₂³	1	1	yes	characteristic
60	C₂×C₃²×C₄	C₂³	1	2	yes	—
61	#538	C₃²	1	1	yes	characteristic, direct product factor, Sylow
62	C₂⁴×C₃	C₂²×C₃	1	4	yes	—
63	C₂×C₃×Q₈	C₂²×C₃	1	4	yes	complemented
64	C₂×C₃×D₄	C₂²×C₃	1	8	yes	—
65	C₂×C₃×D₄	C₂²×C₃	1	8	yes	—
66	C₂×C₃×D₄	C₂²×C₃	1	8	yes	—
67	C₂×C₃×D₄	C₂²×C₃	1	4	yes	complemented
68	C₂²×C₃×C₄	C₂²×C₃	1	4	yes	—
69	C₂²×C₃×C₄	C₂²×C₃	1	4	yes	—
70	C₂²×C₃×C₄	C₂²×C₃	1	8	yes	complemented
71	C₃×C₄⋊C₄	C₂²×C₃	1	8	yes	complemented
72	C₃×C₄⋊C₄	C₂²×C₃	1	8	yes	complemented
73	C₃×C₄⋊C₄	C₂²×C₃	1	4	yes	complemented
74	C₃×C₄⋊C₄	C₂²×C₃	1	4	yes	complemented
75	C₃×C₂²⋊C₄	C₂²×C₃	1	8	yes	—
76	C₃×C₂²⋊C₄	C₂²×C₃	1	8	yes	—
77	C₃×C₂²⋊C₄	C₂²×C₃	1	8	yes	—
78	C₃×C₂²⋊C₄	C₂²×C₃	1	8	yes	—
79	C₃×C₂²⋊C₄	C₂²×C₃	1	8	yes	—
80	C₃×C₂²⋊C₄	C₂²×C₃	1	8	yes	complemented
81	C₃×C₂²⋊C₄	C₂²×C₃	1	8	yes	complemented
82	C₃×C₄²	C₂²×C₃	1	8	yes	complemented
83	C₂²×C₃²	—	2	1	no	—
84	C₂²×C₃²	—	2	1	no	—
85	C₂²×C₃²	—	2	2	no	—
86	C₂²×C₃²	—	2	2	no	—
87	C₂²×C₃²	—	2	2	no	—
88	C₂²×C₃²	—	2	2	no	—
89	C₂²×C₃²	—	2	2	no	—
90	C₂²×C₃²	—	2	2	no	—
91	C₂²×C₃²	—	4	4	no	—
92	C₂²×C₃²	#41	1	2	yes	—
93	C₂²×C₃²	C₂⁴	1	1	yes	characteristic
94	C₃²×C₄	—	2	2	no	—
95	C₃²×C₄	—	2	2	no	—
96	C₃²×C₄	—	2	1	no	—
97	C₃²×C₄	—	2	2	no	—
98	C₃²×C₄	—	2	1	no	—
99	C₃²×C₄	—	2	2	no	—
100	#126	C₂×C₃²	1	2	yes	complemented
101	#124	C₂×C₃²	1	2	yes	complemented
102	#123	C₂×C₃²	1	2	yes	complemented
103	C₂²⋊Q₈	C₂×C₃²	1	1	yes	characteristic, complemented
104	#121	C₂×C₃²	1	2	yes	complemented
105	#121	C₂×C₃²	1	1	yes	characteristic, complemented
106	#120	C₂×C₃²	1	2	yes	complemented
107	C₄×D₄	C₂×C₃²	1	2	yes	complemented
108	C₂×C₂²⋊C₄	C₂×C₃²	1	1	yes	characteristic, complemented
109	C₂³×C₃	—	2	8	no	—
110	C₂³×C₃	—	4	4	no	—
111	C₂³×C₃	—	4	4	no	—
112	C₂³×C₃	C₂³×C₃	1	8	yes	—
113	C₂³×C₃	C₂³×C₃	1	8	yes	—
114	C₂³×C₃	C₂³×C₃	1	4	yes	—
115	C₃×Q₈	—	4	4	no	—
116	C₃×D₄	—	2	16	no	—
117	C₃×D₄	—	4	8	no	—
118	C₃×D₄	—	4	8	no	—
119	C₃×D₄	—	4	4	no	—
120	C₂×C₃×C₄	—	4	4	no	—
121	C₂×C₃×C₄	—	4	4	no	—
122	C₂×C₃×C₄	—	4	8	no	—
123	C₂×C₃×C₄	C₂³×C₃	1	8	yes	—
124	C₂×C₃×C₄	C₂³×C₃	1	8	yes	—
125	C₂×C₃×C₄	C₂³×C₃	1	8	yes	—
126	C₂×C₃×C₄	C₂³×C₃	1	4	yes	—
127	C₂×C₃×C₄	C₂³×C₃	1	4	yes	—
128	C₂×C₃×C₄	C₂³×C₃	1	8	yes	—
129	C₂×C₃²	—	2	2	no	—
130	C₂×C₃²	—	4	2	no	—
131	C₂×C₃²	—	4	2	no	—
132	C₂×C₃²	C₂×#41	1	1	yes	characteristic
133	C₂×C₃²	H(5,𝔽₂)	1	1	yes	characteristic
134	C₂×C₃²	H(5,𝔽₂)	1	1	yes	characteristic
135	C₂⁴	C₂²×C₃²	1	1	yes	characteristic
136	C₂×Q₈	C₂²×C₃²	1	1	yes	characteristic, complemented
137	C₂×D₄	C₂²×C₃²	1	2	yes	—
138	C₂×D₄	C₂²×C₃²	1	2	yes	—
139	C₂×D₄	C₂²×C₃²	1	2	yes	—
140	C₂×D₄	C₂²×C₃²	1	1	yes	characteristic, complemented
141	C₂²×C₄	C₂²×C₃²	1	1	yes	characteristic
142	C₂²×C₄	C₂²×C₃²	1	1	yes	characteristic
143	C₂²×C₄	C₂²×C₃²	1	2	yes	complemented
144	C₄⋊C₄	C₂²×C₃²	1	2	yes	complemented
145	C₄⋊C₄	C₂²×C₃²	1	2	yes	complemented
146	C₄⋊C₄	C₂²×C₃²	1	1	yes	characteristic, complemented
147	C₄⋊C₄	C₂²×C₃²	1	1	yes	characteristic, complemented
148	C₂²⋊C₄	C₂²×C₃²	1	2	yes	—
149	C₂²⋊C₄	C₂²×C₃²	1	2	yes	—
150	C₂²⋊C₄	C₂²×C₃²	1	2	yes	—
151	C₂²⋊C₄	C₂²×C₃²	1	2	yes	—
152	C₂²⋊C₄	C₂²×C₃²	1	2	yes	—
153	C₂²⋊C₄	C₂²×C₃²	1	2	yes	complemented
154	C₂²⋊C₄	C₂²×C₃²	1	2	yes	complemented
155	C₄²	C₂²×C₃²	1	2	yes	complemented
156	C₂²×C₃	—	2	4	no	—
157	C₂²×C₃	—	2	4	no	—
158	C₂²×C₃	—	2	8	no	—
159	C₂²×C₃	—	2	8	no	—
160	C₂²×C₃	—	2	8	no	—
161	C₂²×C₃	—	2	8	no	—
162	C₂²×C₃	—	2	8	no	—
163	C₂²×C₃	—	2	8	no	—
164	C₂²×C₃	—	4	16	no	—
165	C₂²×C₃	C₃×#41	1	8	yes	—
166	C₂²×C₃	C₂⁴×C₃	1	4	yes	—
167	C₃×C₄	—	2	8	no	—
168	C₃×C₄	—	2	8	no	—
169	C₃×C₄	—	2	4	no	—
170	C₃×C₄	—	2	8	no	—
171	C₃×C₄	—	2	4	no	—
172	C₃×C₄	—	2	8	no	—
173	C₃²	#538	1	1	yes	characteristic, direct product factor, Sylow
174	C₂³	—	2	2	no	—
175	C₂³	—	4	1	no	—
176	C₂³	—	4	1	no	—
177	C₂³	C₂³×C₃²	1	2	yes	—
178	C₂³	C₂³×C₃²	1	2	yes	—
179	C₂³	C₂³×C₃²	1	1	yes	characteristic
180	Q₈	—	4	1	no	—
181	D₄	—	2	4	no	—
182	D₄	—	4	2	no	—
183	D₄	—	4	2	no	—
184	D₄	—	4	1	no	—
185	C₂×C₄	—	4	1	no	—
186	C₂×C₄	—	4	1	no	—
187	C₂×C₄	—	4	2	no	—
188	C₂×C₄	C₂³×C₃²	1	2	yes	—
189	C₂×C₄	C₂³×C₃²	1	2	yes	—
190	C₂×C₄	C₂³×C₃²	1	2	yes	—
191	C₂×C₄	C₂³×C₃²	1	1	yes	characteristic
192	C₂×C₄	C₂³×C₃²	1	1	yes	characteristic
193	C₂×C₄	C₂³×C₃²	1	2	yes	—
194	C₂×C₃	—	2	8	no	—
195	C₂×C₃	—	4	8	no	—
196	C₂×C₃	—	4	8	no	—
197	C₂×C₃	C₂×C₃×#41	1	4	yes	—
198	C₂×C₃	C₃×H(5,𝔽₂)	1	4	yes	—
199	C₂×C₃	C₃×H(5,𝔽₂)	1	4	yes	—
200	C₂²	—	2	1	no	—
201	C₂²	—	2	1	no	—
202	C₂²	—	2	2	no	—
203	C₂²	—	2	2	no	—
204	C₂²	—	2	2	no	—
205	C₂²	—	2	2	no	—
206	C₂²	—	2	2	no	—
207	C₂²	—	2	2	no	—
208	C₂²	—	4	4	no	—
209	C₂²	C₃²×#41	1	2	yes	—
210	C₂²	C₂⁴×C₃²	1	1	yes	characteristic
211	C₄	—	2	2	no	—
212	C₄	—	2	2	no	—
213	C₄	—	2	1	no	—
214	C₄	—	2	2	no	—
215	C₄	—	2	1	no	—
216	C₄	—	2	2	no	—
217	C₃	C₃×#538	1	4	yes	direct product factor
218	C₂	—	2	2	no	—
219	C₂	—	4	2	no	—
220	C₂	—	4	2	no	—
221	C₂	C₂×C₃²×#41	1	1	yes	characteristic
222	C₂	C₃²×H(5,𝔽₂)	1	1	yes	characteristic
223	C₂	C₃²×H(5,𝔽₂)	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃²	C₂⁴
commutator, Frattini, cosocle	C₂²	C₂⁴×C₃²
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×C₂²⋊Q₈	C₂	semidirect
5. C₃²×#121	C₂	semidirect
6. C₃²×#121	C₂	semidirect
7. C₃²×#120	C₂	semidirect
8. C₃²×C₄×D₄	C₂	semidirect
9. C₂×C₃²×C₂²⋊C₄	C₂	semidirect
10. C₃×#538	C₃	direct (N × H)
12. C₂×C₃²×Q₈	C₂²	semidirect
16. C₂×C₃²×D₄	C₂²	semidirect
19. C₂²×C₃²×C₄	C₂²	semidirect
20. C₃²×C₄⋊C₄	C₂²	semidirect
21. C₃²×C₄⋊C₄	C₂²	semidirect
22. C₃²×C₄⋊C₄	C₂²	semidirect
23. C₃²×C₄⋊C₄	C₂²	semidirect
29. C₃²×C₂²⋊C₄	C₂²	semidirect
30. C₃²×C₂²⋊C₄	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×C₂²⋊Q₈	C₂×C₃	semidirect
36. C₃×#121	C₂×C₃	semidirect
37. C₃×#121	C₂×C₃	semidirect
38. C₃×#120	C₂×C₃	semidirect
39. C₃×C₄×D₄	C₂×C₃	semidirect
40. C₂×C₃×C₂²⋊C₄	C₂×C₃	semidirect
61. #538	C₃²	direct (N × H)
63. C₂×C₃×Q₈	C₂²×C₃	semidirect
67. C₂×C₃×D₄	C₂²×C₃	semidirect
70. C₂²×C₃×C₄	C₂²×C₃	semidirect
71. C₃×C₄⋊C₄	C₂²×C₃	semidirect
72. C₃×C₄⋊C₄	C₂²×C₃	semidirect
73. C₃×C₄⋊C₄	C₂²×C₃	semidirect
74. C₃×C₄⋊C₄	C₂²×C₃	semidirect
80. C₃×C₂²⋊C₄	C₂²×C₃	semidirect
81. C₃×C₂²⋊C₄	C₂²×C₃	semidirect
82. C₃×C₄²	C₂²×C₃	semidirect
100. #126	C₂×C₃²	semidirect
101. #124	C₂×C₃²	semidirect
102. #123	C₂×C₃²	semidirect
103. C₂²⋊Q₈	C₂×C₃²	semidirect
104. #121	C₂×C₃²	semidirect
105. #121	C₂×C₃²	semidirect
106. #120	C₂×C₃²	semidirect
107. C₄×D₄	C₂×C₃²	semidirect
108. C₂×C₂²⋊C₄	C₂×C₃²	semidirect
136. C₂×Q₈	C₂²×C₃²	semidirect
140. C₂×D₄	C₂²×C₃²	semidirect
143. C₂²×C₄	C₂²×C₃²	semidirect
144. C₄⋊C₄	C₂²×C₃²	semidirect
145. C₄⋊C₄	C₂²×C₃²	semidirect
146. C₄⋊C₄	C₂²×C₃²	semidirect
147. C₄⋊C₄	C₂²×C₃²	semidirect
153. C₂²⋊C₄	C₂²×C₃²	semidirect
154. C₂²⋊C₄	C₂²×C₃²	semidirect
155. C₄²	C₂²×C₃²	semidirect
173. C₃²	#538	direct (N × H)
217. C₃	C₃×#538	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×#121	2
C₃²×#120	2
C₃²×C₄×D₄	2
C₂×C₃²×C₂²⋊C₄	2
C₃×#538	3

Subgroups — C32×#538 #10596339

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃²×#538 #10596339