Subgroups — C₃×#5728 #10595825

Order 576.

478 conjugacy classes of subgroups.

The subgroup lattice has 251 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×#946	C₂	1	2	yes	maximal
3	C₃×Q₈×Dic₃	C₂	1	2	yes	maximal, complemented
4	C₃×#897	C₂	1	2	yes	maximal
5	C₃×#892	C₂	1	2	yes	maximal
6	C₃×#890	C₂	1	2	yes	maximal, complemented
7	C₃×#887	C₂	1	2	yes	maximal, complemented
8	C₃×C₄²×D₃	C₂	1	1	yes	maximal, characteristic
9	C₃×#869	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃×#533	—	3	1	no	maximal
11	#5728	C₃	1	1	yes	maximal, characteristic, direct product factor
12	C₂×C₃²×Q₈	C₂²	1	2	yes	—
13	C₂×C₃×C₄×D₃	C₂²	1	1	yes	characteristic
14	C₂×C₃×C₄×D₃	C₂²	1	1	yes	characteristic
15	C₂×C₃×C₄×D₃	C₂²	1	1	yes	characteristic
16	C₂×C₃×Dic₆	C₂²	1	2	yes	—
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	4	yes	—
21	C₃×C₄⋊Dic₃	C₂²	1	2	yes	—
22	C₃×C₄⋊Dic₃	C₂²	1	4	yes	—
23	C₃×C₄⋊Dic₃	C₂²	1	2	yes	—
24	C₃×Dic₃⋊C₄	C₂²	1	4	yes	—
25	C₃×C₄×Dic₃	C₂²	1	1	yes	characteristic
26	C₃×C₄×Dic₃	C₂²	1	4	yes	—
27	C₃×C₄×Dic₃	C₂²	1	1	yes	characteristic
28	C₃×C₄×Dic₃	C₂²	1	1	yes	characteristic
29	C₃×C₄⋊Q₈	—	2	1	no	—
30	C₃×C₄⋊Q₈	—	3	1	no	—
31	C₃×C₄⋊Q₈	D₃	1	1	yes	characteristic, complemented
32	C₃×C₄⋊Q₈	C₂×C₃	1	1	yes	characteristic, complemented
33	C₃×#125	—	3	2	no	—
34	C₃×C₂²⋊Q₈	—	3	2	no	—
35	C₃×C₂²⋊Q₈	—	3	2	no	—
36	C₃×C₄×Q₈	—	3	2	no	—
37	C₃×C₄×Q₈	—	3	2	no	—
38	C₃×#117	—	3	2	no	—
39	C₂×C₃×C₄²	—	3	1	no	—
40	#946	C₂×C₃	1	2	yes	—
41	Q₈×Dic₃	C₂×C₃	1	2	yes	complemented
42	#897	C₂×C₃	1	2	yes	—
43	#892	C₂×C₃	1	2	yes	—
44	#890	C₂×C₃	1	2	yes	complemented
45	#887	C₂×C₃	1	2	yes	complemented
46	C₄²×D₃	C₂×C₃	1	1	yes	characteristic
47	#869	C₂×C₃	1	1	yes	characteristic, complemented
48	C₂²×C₃×D₃	C₂³	1	1	yes	characteristic
49	C₃²×Q₈	—	2	4	no	—
50	C₂×C₃²×C₄	C₂³	1	4	yes	—
51	C₂×C₃²×C₄	C₂³	1	1	yes	characteristic
52	C₂×C₃²×C₄	C₂³	1	1	yes	characteristic
53	C₂×C₃²×C₄	C₂³	1	1	yes	characteristic
54	C₂×C₃×Dic₃	C₂³	1	1	yes	characteristic
55	C₂×C₃×Dic₃	C₂³	1	1	yes	characteristic
56	C₂×C₃×Dic₃	C₂³	1	4	yes	—
57	C₂×C₃×Dic₃	C₂³	1	1	yes	characteristic
58	C₃×C₄×D₃	—	2	2	no	—
59	C₃×C₄×D₃	—	2	2	no	—
60	C₃×C₄×D₃	Q₈	1	4	yes	complemented
61	C₃×Dic₆	—	2	4	no	—
62	#533	—	3	1	no	Sylow
63	C₂×C₃×Q₈	—	2	2	no	—
64	C₂×C₃×Q₈	—	3	2	no	—
65	C₂×C₃×Q₈	C₂×D₃	1	2	yes	—
66	C₂×C₃×Q₈	C₂²×C₃	1	2	yes	—
67	C₂²×C₃×C₄	—	3	1	no	—
68	C₂²×C₃×C₄	—	3	1	no	—
69	C₂²×C₃×C₄	—	3	1	no	—
70	C₂×C₄×D₃	C₂²×C₃	1	1	yes	characteristic
71	C₂×C₄×D₃	C₂²×C₃	1	1	yes	characteristic
72	C₂×C₄×D₃	C₂²×C₃	1	1	yes	characteristic
73	C₂×Dic₆	C₂²×C₃	1	2	yes	—
74	C₃×C₄⋊C₄	—	2	2	no	—
75	C₃×C₄⋊C₄	—	2	2	no	—
76	C₃×C₄⋊C₄	—	3	4	no	—
77	C₃×C₄⋊C₄	—	3	2	no	—
78	C₃×C₄⋊C₄	—	3	4	no	—
79	C₃×C₄⋊C₄	—	3	2	no	—
80	C₃×C₄⋊C₄	C₂×D₃	1	2	yes	—
81	C₃×C₄⋊C₄	C₂×D₃	1	2	yes	—
82	C₃×C₄⋊C₄	C₂²×C₃	1	2	yes	—
83	C₃×C₄⋊C₄	C₂²×C₃	1	2	yes	—
84	C₃×C₂²⋊C₄	—	3	4	no	—
85	C₃×C₄²	—	2	1	no	—
86	C₃×C₄²	—	3	1	no	—
87	C₃×C₄²	—	3	1	no	—
88	C₃×C₄²	—	3	1	no	—
89	C₃×C₄²	—	3	4	no	—
90	C₃×C₄²	C₂×D₃	1	1	yes	characteristic
91	C₃×C₄²	C₂²×C₃	1	1	yes	characteristic
92	(C₂×D₃)⋊C₄	C₂²×C₃	1	4	yes	—
93	C₄⋊Dic₃	C₂²×C₃	1	2	yes	—
94	C₄⋊Dic₃	C₂²×C₃	1	2	yes	—
95	C₄⋊Dic₃	C₂²×C₃	1	4	yes	—
96	Dic₃⋊C₄	C₂²×C₃	1	4	yes	—
97	C₄×Dic₃	C₂²×C₃	1	1	yes	characteristic
98	C₄×Dic₃	C₂²×C₃	1	1	yes	characteristic
99	C₄×Dic₃	C₂²×C₃	1	1	yes	characteristic
100	C₄×Dic₃	C₂²×C₃	1	4	yes	—
101	C₂²×C₃²	C₂⁴	1	1	yes	characteristic
102	C₂×C₃×D₃	—	2	1	no	—
103	C₂×C₃×D₃	—	2	1	no	—
104	C₂×C₃×D₃	C₂×Q₈	1	2	yes	—
105	C₃²×C₄	—	2	4	no	—
106	C₃²×C₄	C₂×Q₈	1	2	yes	—
107	C₃²×C₄	#41	1	2	yes	—
108	C₃²×C₄	#41	1	2	yes	—
109	C₃×Dic₃	—	2	4	no	—
110	C₃×Dic₃	—	2	1	no	—
111	C₃×Dic₃	—	2	1	no	—
112	C₃×Dic₃	C₂×Q₈	1	2	yes	—
113	C₄⋊Q₈	—	3	1	no	—
114	C₄⋊Q₈	C₃×D₃	1	1	yes	characteristic, complemented
115	#125	—	3	2	no	—
116	C₂²⋊Q₈	—	3	2	no	—
117	C₂²⋊Q₈	—	3	2	no	—
118	C₄×Q₈	—	3	2	no	—
119	C₄×Q₈	—	3	2	no	—
120	#117	—	3	2	no	—
121	C₂×C₄²	—	3	1	no	—
122	C₂³×C₃	—	3	1	no	—
123	C₂²×D₃	C₂³×C₃	1	1	yes	characteristic
124	C₃×Q₈	—	2	4	no	—
125	C₃×Q₈	—	2	4	no	—
126	C₃×Q₈	—	4	4	no	—
127	C₃×Q₈	—	6	4	no	—
128	C₂×C₃×C₄	—	2	1	no	—
129	C₂×C₃×C₄	—	2	1	no	—
130	C₂×C₃×C₄	—	2	1	no	—
131	C₂×C₃×C₄	—	2	4	no	—
132	C₂×C₃×C₄	—	3	1	no	—
133	C₂×C₃×C₄	—	3	4	no	—
134	C₂×C₃×C₄	—	3	1	no	—
135	C₂×C₃×C₄	—	3	1	no	—
136	C₂×C₃×C₄	—	3	4	no	—
137	C₂×C₃×C₄	—	6	2	no	—
138	C₂×C₃×C₄	—	6	2	no	—
139	C₂×C₃×C₄	C₂²×D₃	1	4	yes	—
140	C₂×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
141	C₂×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
142	C₂×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
143	C₂×C₃×C₄	C₂³×C₃	1	4	yes	—
144	C₂×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
145	C₂×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
146	C₂×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
147	C₂×Dic₃	C₂³×C₃	1	1	yes	characteristic
148	C₂×Dic₃	C₂³×C₃	1	1	yes	characteristic
149	C₂×Dic₃	C₂³×C₃	1	4	yes	—
150	C₂×Dic₃	C₂³×C₃	1	1	yes	characteristic
151	C₄×D₃	—	2	2	no	—
152	C₄×D₃	—	2	2	no	—
153	C₄×D₃	C₃×Q₈	1	4	yes	complemented
154	Dic₆	—	2	4	no	—
155	C₂×C₃²	C₂²×Q₈	1	1	yes	characteristic
156	C₂×C₃²	C₂×#41	1	1	yes	characteristic
157	C₂×C₃²	C₂×#41	1	1	yes	characteristic
158	C₃×D₃	—	2	2	no	—
159	C₂×Q₈	—	3	2	no	—
160	C₂×Q₈	C₂×C₃×D₃	1	2	yes	—
161	C₂²×C₄	—	3	1	no	—
162	C₂²×C₄	—	3	1	no	—
163	C₂²×C₄	—	3	1	no	—
164	C₄⋊C₄	—	3	2	no	—
165	C₄⋊C₄	—	3	4	no	—
166	C₄⋊C₄	—	3	4	no	—
167	C₄⋊C₄	—	3	2	no	—
168	C₄⋊C₄	C₂×C₃×D₃	1	2	yes	—
169	C₄⋊C₄	C₂×C₃×D₃	1	2	yes	—
170	C₂²⋊C₄	—	3	4	no	—
171	C₄²	—	3	1	no	—
172	C₄²	—	3	1	no	—
173	C₄²	—	3	1	no	—
174	C₄²	—	3	4	no	—
175	C₄²	C₂×C₃×D₃	1	1	yes	characteristic
176	C₂²×C₃	—	2	1	no	—
177	C₂²×C₃	—	3	2	no	—
178	C₂²×C₃	—	6	1	no	—
179	C₂²×C₃	—	6	1	no	—
180	C₂²×C₃	C₂³×D₃	1	1	yes	characteristic
181	C₂²×C₃	C₂⁴×C₃	1	1	yes	characteristic
182	C₂×D₃	—	2	1	no	—
183	C₂×D₃	—	2	1	no	—
184	C₂×D₃	C₂×C₃×Q₈	1	2	yes	—
185	C₃×C₄	—	2	4	no	—
186	C₃×C₄	—	2	4	no	—
187	C₃×C₄	—	2	2	no	—
188	C₃×C₄	—	2	2	no	—
189	C₃×C₄	—	2	2	no	—
190	C₃×C₄	—	3	2	no	—
191	C₃×C₄	—	4	4	no	—
192	C₃×C₄	—	6	1	no	—
193	C₃×C₄	—	6	1	no	—
194	C₃×C₄	—	6	4	no	—
195	C₃×C₄	#237	1	2	yes	—
196	C₃×C₄	D₃×Q₈	1	2	yes	—
197	C₃×C₄	#239	1	2	yes	—
198	C₃×C₄	C₂×C₃×Q₈	1	2	yes	—
199	C₃×C₄	C₃×#41	1	2	yes	—
200	C₃×C₄	C₃×#41	1	2	yes	—
201	Dic₃	—	2	4	no	—
202	Dic₃	—	2	1	no	—
203	Dic₃	—	2	1	no	—
204	Dic₃	C₂×C₃×Q₈	1	2	yes	—
205	C₃²	#533	1	1	yes	characteristic, complemented, Sylow
206	C₂³	—	3	1	no	—
207	Q₈	—	2	4	no	—
208	Q₈	—	6	4	no	—
209	C₂×C₄	—	3	1	no	—
210	C₂×C₄	—	3	4	no	—
211	C₂×C₄	—	3	1	no	—
212	C₂×C₄	—	3	1	no	—
213	C₂×C₄	—	3	4	no	—
214	C₂×C₄	—	6	2	no	—
215	C₂×C₄	—	6	2	no	—
216	C₂×C₄	C₂²×C₃×D₃	1	4	yes	—
217	C₂×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
218	C₂×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
219	C₂×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
220	C₂×C₃	—	2	1	no	—
221	C₂×C₃	—	2	1	no	—
222	C₂×C₃	—	2	1	no	—
223	C₂×C₃	—	6	2	no	—
224	C₂×C₃	C₂×#237	1	1	yes	characteristic
225	C₂×C₃	C₂×D₃×Q₈	1	1	yes	characteristic
226	C₂×C₃	C₂×#239	1	1	yes	characteristic
227	C₂×C₃	C₂²×C₃×Q₈	1	1	yes	characteristic
228	C₂×C₃	C₂×C₃×#41	1	1	yes	characteristic
229	C₂×C₃	C₂×C₃×#41	1	1	yes	characteristic
230	D₃	—	2	2	no	—
231	C₂²	—	3	2	no	—
232	C₂²	—	6	1	no	—
233	C₂²	—	6	1	no	—
234	C₂²	C₂³×C₃×D₃	1	1	yes	characteristic
235	C₄	—	2	4	no	—
236	C₄	—	3	2	no	—
237	C₄	—	6	1	no	—
238	C₄	—	6	1	no	—
239	C₄	—	6	4	no	—
240	C₄	C₃×#237	1	2	yes	—
241	C₄	C₃×D₃×Q₈	1	2	yes	—
242	C₄	C₃×#239	1	2	yes	—
243	C₃	—	2	1	no	—
244	C₃	#5728	1	1	yes	characteristic, direct product factor
245	C₃	C₃×#533	1	1	yes	characteristic, complemented
246	C₂	—	6	2	no	—
247	C₂	C₂×C₃×#237	1	1	yes	characteristic
248	C₂	C₂×C₃×D₃×Q₈	1	1	yes	characteristic
249	C₂	C₂×C₃×#239	1	1	yes	characteristic

	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	C₂²×C₃×D₃	C₂³
hypercenter	C₃×C₄⋊Q₈	D₃
hypocenter	C₃	C₃×#533
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×Q₈×Dic₃	C₂	semidirect
6. C₃×#890	C₂	semidirect
7. C₃×#887	C₂	semidirect
9. C₃×#869	C₂	semidirect
11. #5728	C₃	direct (N × H)
31. C₃×C₄⋊Q₈	D₃	semidirect
32. C₃×C₄⋊Q₈	C₂×C₃	semidirect
41. Q₈×Dic₃	C₂×C₃	semidirect
44. #890	C₂×C₃	semidirect
45. #887	C₂×C₃	semidirect
47. #869	C₂×C₃	semidirect
60. C₃×C₄×D₃	Q₈	semidirect
114. C₄⋊Q₈	C₃×D₃	semidirect
153. C₄×D₃	C₃×Q₈	semidirect
205. C₃²	#533	semidirect
244. C₃	#5728	direct (N × H)
245. C₃	C₃×#533	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×#946	2
C₃×Q₈×Dic₃	2
C₃×#897	2
C₃×#892	2
C₃×#890	2
C₃×#887	2
C₃×C₄²×D₃	2
C₃×#869	2
C₃×#533	3
#5728	3

Subgroups — C3×#5728 #10595825

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#5728 #10595825