Subgroups — C₃×#5735 #10595832

Order 576.

426 conjugacy classes of subgroups.

The subgroup lattice has 262 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃×#947	C₂	1	1	yes	maximal, characteristic
3	C₃×#946	C₂	1	2	yes	maximal, complemented
4	C₃×#944	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×#898	C₂	1	2	yes	maximal, complemented
6	C₃×#896	C₂	1	2	yes	maximal, complemented
7	C₃×#894	C₂	1	2	yes	maximal
8	C₃×#889	C₂	1	2	yes	maximal, complemented
9	C₃×#876	C₂	1	2	yes	maximal, complemented
10	C₃×#563	—	3	1	no	maximal
11	#5735	C₃	1	1	yes	maximal, characteristic, direct product factor
12	C₂×C₃²×Q₈	C₂²	1	1	yes	characteristic
13	C₂×C₃²×Q₈	C₂²	1	1	yes	characteristic
14	C₂×C₃×D₁₂	C₂²	1	1	yes	characteristic
15	C₂×C₃×C₄×D₃	C₂²	1	2	yes	—
16	C₂×C₃×Dic₆	C₂²	1	1	yes	characteristic
17	C₃²×C₄⋊C₄	C₂²	1	2	yes	—
18	C₃²×C₄⋊C₄	C₂²	1	2	yes	—
19	C₃²×C₄²	C₂²	1	1	yes	characteristic
20	C₃×(C₂×D₃)⋊C₄	C₂²	1	2	yes	—
21	C₃×(C₂×D₃)⋊C₄	C₂²	1	2	yes	—
22	C₃×(C₂×D₃)⋊C₄	C₂²	1	2	yes	—
23	C₃×(C₂×D₃)⋊C₄	C₂²	1	2	yes	—
24	C₃×(C₂×D₃)⋊C₄	C₂²	1	2	yes	—
25	C₃×C₄⋊Dic₃	C₂²	1	2	yes	—
26	C₃×Dic₃⋊C₄	C₂²	1	2	yes	—
27	C₃×Dic₃⋊C₄	C₂²	1	2	yes	—
28	C₃×Dic₃⋊C₄	C₂²	1	2	yes	—
29	C₃×Dic₃⋊C₄	C₂²	1	2	yes	—
30	C₃×Dic₃⋊C₄	C₂²	1	2	yes	—
31	C₃×C₄×Dic₃	C₂²	1	1	yes	characteristic
32	C₃×C₄×Dic₃	C₂²	1	1	yes	characteristic
33	C₃×C₄⋊Q₈	—	2	1	no	—
34	C₃×C₄⋊Q₈	—	3	1	no	—
35	C₃×C₄⋊Q₈	D₃	1	1	yes	characteristic, complemented
36	C₃×C₄⋊Q₈	C₂×C₃	1	1	yes	characteristic, complemented
37	C₃×#126	—	3	2	no	—
38	C₃×#126	—	3	2	no	—
39	C₃×#125	—	3	2	no	—
40	C₃×#124	—	3	1	no	—
41	C₃×#123	—	3	2	no	—
42	C₃×C₂²⋊Q₈	—	3	2	no	—
43	C₃×C₂²⋊Q₈	—	3	2	no	—
44	#947	C₂×C₃	1	1	yes	characteristic
45	#946	C₂×C₃	1	2	yes	complemented
46	#944	C₂×C₃	1	1	yes	characteristic, complemented
47	#898	C₂×C₃	1	2	yes	complemented
48	#896	C₂×C₃	1	2	yes	complemented
49	#894	C₂×C₃	1	2	yes	—
50	#889	C₂×C₃	1	2	yes	complemented
51	#876	C₂×C₃	1	2	yes	complemented
52	C₂²×C₃×D₃	C₂³	1	2	yes	—
53	C₃²×Q₈	—	4	1	no	—
54	C₃²×Q₈	—	4	1	no	—
55	C₂×C₃²×C₄	C₂³	1	1	yes	characteristic
56	C₂×C₃²×C₄	C₂³	1	1	yes	characteristic
57	C₂×C₃²×C₄	C₂³	1	2	yes	—
58	C₂×C₃²×C₄	C₂³	1	1	yes	characteristic
59	C₂×C₃²×C₄	C₂³	1	2	yes	—
60	C₂×C₃×Dic₃	C₂³	1	2	yes	—
61	C₂×C₃×Dic₃	C₂³	1	2	yes	—
62	C₂×C₃×Dic₃	C₂³	1	2	yes	—
63	C₃×D₁₂	—	4	1	no	—
64	C₃×C₄×D₃	—	4	2	no	—
65	C₃×Dic₆	—	4	1	no	—
66	#563	—	3	1	no	Sylow
67	C₂×C₃×Q₈	—	2	1	no	—
68	C₂×C₃×Q₈	—	2	1	no	—
69	C₂×C₃×Q₈	—	3	1	no	—
70	C₂×C₃×Q₈	C₂×D₃	1	1	yes	characteristic
71	C₂×C₃×Q₈	C₂×D₃	1	1	yes	characteristic
72	C₂×C₃×Q₈	C₂²×C₃	1	1	yes	characteristic
73	C₂×C₃×Q₈	C₂²×C₃	1	1	yes	characteristic
74	C₂×C₃×D₄	—	3	1	no	—
75	C₂²×C₃×C₄	—	3	2	no	—
76	C₂×D₁₂	C₂²×C₃	1	1	yes	characteristic
77	C₂×C₄×D₃	C₂²×C₃	1	2	yes	—
78	C₂×Dic₆	C₂²×C₃	1	1	yes	characteristic
79	C₃×C₄⋊C₄	—	2	2	no	—
80	C₃×C₄⋊C₄	—	2	2	no	—
81	C₃×C₄⋊C₄	—	3	2	no	—
82	C₃×C₄⋊C₄	—	3	2	no	—
83	C₃×C₄⋊C₄	—	3	2	no	—
84	C₃×C₄⋊C₄	—	3	2	no	—
85	C₃×C₄⋊C₄	—	3	2	no	—
86	C₃×C₄⋊C₄	—	3	2	no	—
87	C₃×C₄⋊C₄	C₂×D₃	1	2	yes	—
88	C₃×C₄⋊C₄	C₂×D₃	1	2	yes	—
89	C₃×C₄⋊C₄	C₂²×C₃	1	2	yes	—
90	C₃×C₄⋊C₄	C₂²×C₃	1	2	yes	—
91	C₃×C₂²⋊C₄	—	3	2	no	—
92	C₃×C₂²⋊C₄	—	3	2	no	—
93	C₃×C₂²⋊C₄	—	3	2	no	—
94	C₃×C₂²⋊C₄	—	3	2	no	—
95	C₃×C₂²⋊C₄	—	3	2	no	—
96	C₃×C₄²	—	2	1	no	—
97	C₃×C₄²	—	3	1	no	—
98	C₃×C₄²	—	3	1	no	—
99	C₃×C₄²	C₂×D₃	1	1	yes	characteristic
100	C₃×C₄²	C₂²×C₃	1	1	yes	characteristic
101	(C₂×D₃)⋊C₄	C₂²×C₃	1	2	yes	—
102	(C₂×D₃)⋊C₄	C₂²×C₃	1	2	yes	—
103	(C₂×D₃)⋊C₄	C₂²×C₃	1	2	yes	—
104	(C₂×D₃)⋊C₄	C₂²×C₃	1	2	yes	—
105	(C₂×D₃)⋊C₄	C₂²×C₃	1	2	yes	—
106	C₄⋊Dic₃	C₂²×C₃	1	2	yes	—
107	Dic₃⋊C₄	C₂²×C₃	1	2	yes	—
108	Dic₃⋊C₄	C₂²×C₃	1	2	yes	—
109	Dic₃⋊C₄	C₂²×C₃	1	2	yes	—
110	Dic₃⋊C₄	C₂²×C₃	1	2	yes	—
111	Dic₃⋊C₄	C₂²×C₃	1	2	yes	—
112	C₄×Dic₃	C₂²×C₃	1	1	yes	characteristic
113	C₄×Dic₃	C₂²×C₃	1	1	yes	characteristic
114	C₂²×C₃²	C₂⁴	1	1	yes	characteristic
115	C₂×C₃×D₃	—	2	2	no	—
116	C₂×C₃×D₃	—	2	4	no	—
117	C₃²×C₄	—	2	1	no	—
118	C₃²×C₄	—	2	2	no	—
119	C₃²×C₄	—	2	1	no	—
120	C₃²×C₄	—	2	2	no	—
121	C₃²×C₄	—	2	1	no	—
122	C₃×Dic₃	—	2	2	no	—
123	C₃×Dic₃	—	2	2	no	—
124	C₃×Dic₃	—	2	2	no	—
125	C₄⋊Q₈	—	3	1	no	—
126	C₄⋊Q₈	C₃×D₃	1	1	yes	characteristic, complemented
127	#126	—	3	2	no	—
128	#126	—	3	2	no	—
129	#125	—	3	2	no	—
130	#124	—	3	1	no	—
131	#123	—	3	2	no	—
132	C₂²⋊Q₈	—	3	2	no	—
133	C₂²⋊Q₈	—	3	2	no	—
134	C₂³×C₃	—	3	2	no	—
135	C₂²×D₃	C₂³×C₃	1	2	yes	—
136	C₃×Q₈	—	4	1	no	—
137	C₃×Q₈	—	4	1	no	—
138	C₃×Q₈	—	4	1	no	—
139	C₃×Q₈	—	4	1	no	—
140	C₃×Q₈	—	4	2	no	—
141	C₃×Q₈	—	4	2	no	—
142	C₃×Q₈	—	12	1	no	—
143	C₃×D₄	—	12	1	no	—
144	C₂×C₃×C₄	—	2	2	no	—
145	C₂×C₃×C₄	—	2	1	no	—
146	C₂×C₃×C₄	—	2	1	no	—
147	C₂×C₃×C₄	—	2	1	no	—
148	C₂×C₃×C₄	—	2	2	no	—
149	C₂×C₃×C₄	—	3	2	no	—
150	C₂×C₃×C₄	—	3	2	no	—
151	C₂×C₃×C₄	—	3	2	no	—
152	C₂×C₃×C₄	—	12	2	no	—
153	C₂×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
154	C₂×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
155	C₂×C₃×C₄	C₂²×D₃	1	2	yes	—
156	C₂×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
157	C₂×C₃×C₄	C₂²×D₃	1	2	yes	—
158	C₂×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
159	C₂×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
160	C₂×C₃×C₄	C₂³×C₃	1	2	yes	—
161	C₂×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
162	C₂×C₃×C₄	C₂³×C₃	1	2	yes	—
163	C₂×Dic₃	C₂³×C₃	1	2	yes	—
164	C₂×Dic₃	C₂³×C₃	1	2	yes	—
165	C₂×Dic₃	C₂³×C₃	1	2	yes	—
166	D₁₂	—	4	1	no	—
167	C₄×D₃	—	4	2	no	—
168	Dic₆	—	4	1	no	—
169	C₂×C₃²	H(5,𝔽₂)	1	1	yes	characteristic
170	C₂×C₃²	2₋¹⁺⁴	1	2	yes	—
171	C₃×D₃	—	4	2	no	—
172	C₂×Q₈	—	3	1	no	—
173	C₂×Q₈	C₂×C₃×D₃	1	1	yes	characteristic
174	C₂×Q₈	C₂×C₃×D₃	1	1	yes	characteristic
175	C₂×D₄	—	3	1	no	—
176	C₂²×C₄	—	3	2	no	—
177	C₄⋊C₄	—	3	2	no	—
178	C₄⋊C₄	—	3	2	no	—
179	C₄⋊C₄	—	3	2	no	—
180	C₄⋊C₄	—	3	2	no	—
181	C₄⋊C₄	—	3	2	no	—
182	C₄⋊C₄	—	3	2	no	—
183	C₄⋊C₄	C₂×C₃×D₃	1	2	yes	—
184	C₄⋊C₄	C₂×C₃×D₃	1	2	yes	—
185	C₂²⋊C₄	—	3	2	no	—
186	C₂²⋊C₄	—	3	2	no	—
187	C₂²⋊C₄	—	3	2	no	—
188	C₂²⋊C₄	—	3	2	no	—
189	C₂²⋊C₄	—	3	2	no	—
190	C₄²	—	3	1	no	—
191	C₄²	—	3	1	no	—
192	C₄²	C₂×C₃×D₃	1	1	yes	characteristic
193	C₂²×C₃	—	2	1	no	—
194	C₂²×C₃	—	6	2	no	—
195	C₂²×C₃	—	6	4	no	—
196	C₂²×C₃	C₂³×D₃	1	1	yes	characteristic
197	C₂²×C₃	C₂⁴×C₃	1	1	yes	characteristic
198	C₂×D₃	—	2	2	no	—
199	C₂×D₃	—	2	4	no	—
200	C₃×C₄	—	2	2	no	—
201	C₃×C₄	—	2	4	no	—
202	C₃×C₄	—	2	2	no	—
203	C₃×C₄	—	2	1	no	—
204	C₃×C₄	—	2	1	no	—
205	C₃×C₄	—	2	1	no	—
206	C₃×C₄	—	2	2	no	—
207	C₃×C₄	—	2	1	no	—
208	C₃×C₄	—	2	2	no	—
209	C₃×C₄	—	2	1	no	—
210	C₃×C₄	—	2	1	no	—
211	C₃×C₄	—	2	2	no	—
212	C₃×C₄	—	4	2	no	—
213	C₃×C₄	—	4	1	no	—
214	C₃×C₄	—	4	1	no	—
215	C₃×C₄	—	6	2	no	—
216	C₃×C₄	—	6	2	no	—
217	C₃×C₄	—	6	2	no	—
218	Dic₃	—	2	2	no	—
219	Dic₃	—	2	2	no	—
220	Dic₃	—	2	2	no	—
221	C₃²	#563	1	1	yes	characteristic, complemented, Sylow
222	C₂³	—	3	2	no	—
223	Q₈	—	4	1	no	—
224	Q₈	—	4	1	no	—
225	Q₈	—	12	1	no	—
226	D₄	—	12	1	no	—
227	C₂×C₄	—	3	2	no	—
228	C₂×C₄	—	3	2	no	—
229	C₂×C₄	—	3	2	no	—
230	C₂×C₄	—	12	2	no	—
231	C₂×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
232	C₂×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
233	C₂×C₄	C₂²×C₃×D₃	1	2	yes	—
234	C₂×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
235	C₂×C₄	C₂²×C₃×D₃	1	2	yes	—
236	C₂×C₃	—	2	1	no	—
237	C₂×C₃	—	2	2	no	—
238	C₂×C₃	—	12	2	no	—
239	C₂×C₃	#1004	1	1	yes	characteristic
240	C₂×C₃	#1007	1	2	yes	—
241	C₂×C₃	C₃×H(5,𝔽₂)	1	1	yes	characteristic
242	C₂×C₃	C₃×2₋¹⁺⁴	1	2	yes	—
243	D₃	—	4	2	no	—
244	C₂²	—	6	2	no	—
245	C₂²	—	6	4	no	—
246	C₂²	C₂³×C₃×D₃	1	1	yes	characteristic
247	C₄	—	2	2	no	—
248	C₄	—	2	1	no	—
249	C₄	—	2	1	no	—
250	C₄	—	2	1	no	—
251	C₄	—	2	2	no	—
252	C₄	—	6	2	no	—
253	C₄	—	6	2	no	—
254	C₄	—	6	2	no	—
255	C₃	—	2	1	no	—
256	C₃	#5735	1	1	yes	characteristic, direct product factor
257	C₃	C₃×#563	1	1	yes	characteristic, complemented
258	C₂	—	12	2	no	—
259	C₂	C₃×#1004	1	1	yes	characteristic
260	C₂	C₃×#1007	1	2	yes	—

	subgroup	G/H
center	C₂²×C₃	C₂³×D₃
commutator, cosocle	C₂²×C₃	C₂⁴×C₃
Frattini	C₂²	C₂³×C₃×D₃
Fitting	C₃²×C₄⋊Q₈	C₂
socle	C₂²×C₃²	C₂⁴
prime radical	C₂×C₃×D₁₂	C₂²
hypercenter	C₃×C₄⋊Q₈	D₃
hypocenter	C₃	C₃×#563
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₃ ◁ C₃×C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
1. C₃²×C₄⋊Q₈	C₂	semidirect
3. C₃×#946	C₂	semidirect
4. C₃×#944	C₂	semidirect
5. C₃×#898	C₂	semidirect
6. C₃×#896	C₂	semidirect
8. C₃×#889	C₂	semidirect
9. C₃×#876	C₂	semidirect
11. #5735	C₃	direct (N × H)
35. C₃×C₄⋊Q₈	D₃	semidirect
36. C₃×C₄⋊Q₈	C₂×C₃	semidirect
45. #946	C₂×C₃	semidirect
46. #944	C₂×C₃	semidirect
47. #898	C₂×C₃	semidirect
48. #896	C₂×C₃	semidirect
50. #889	C₂×C₃	semidirect
51. #876	C₂×C₃	semidirect
126. C₄⋊Q₈	C₃×D₃	semidirect
221. C₃²	#563	semidirect
256. C₃	#5735	direct (N × H)
257. C₃	C₃×#563	semidirect

Maximal subgroups

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

subgroup	index
C₃²×C₄⋊Q₈	2
C₃×#947	2
C₃×#946	2
C₃×#944	2
C₃×#898	2
C₃×#896	2
C₃×#894	2
C₃×#889	2
C₃×#876	2
C₃×#563	3
#5735	3

Subgroups — C3×#5735 #10595832

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#5735 #10595832