Subgroups — C₃×#5625 #10595722

Order 576.

421 conjugacy classes of subgroups.

The subgroup lattice has 298 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²×C₂²⋊Q₈	C₂	1	1	yes	maximal, characteristic
2	C₃×Q₈×Dic₃	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×#944	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×#926	C₂	1	1	yes	maximal, characteristic
5	C₃×#924	C₂	1	1	yes	maximal, characteristic
6	C₃×#890	C₂	1	2	yes	maximal, complemented
7	C₃×#890	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃×#889	C₂	1	2	yes	maximal, complemented
9	C₃×#887	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃×#879	C₂	1	2	yes	maximal
11	C₃×#879	C₂	1	2	yes	maximal
12	C₃×#541	—	3	1	no	maximal
13	#5625	C₃	1	1	yes	maximal, characteristic, direct product factor
14	C₂×C₃²×Q₈	C₂²	1	1	yes	characteristic
15	C₂²×C₃²×C₄	C₂²	1	1	yes	characteristic
16	C₂×C₃×Dic₆	C₂²	1	1	yes	characteristic
17	C₃²×C₄⋊C₄	C₂²	1	1	yes	characteristic
18	C₃²×C₄⋊C₄	C₂²	1	2	yes	—
19	C₃²×C₂²⋊C₄	C₂²	1	2	yes	—
20	C₃×#217	C₂²	1	2	yes	—
21	C₃×#217	C₂²	1	1	yes	characteristic
22	C₃×#217	C₂²	1	1	yes	characteristic
23	C₃×C₄⋊Dic₃	C₂²	1	2	yes	—
24	C₃×C₄⋊Dic₃	C₂²	1	2	yes	—
25	C₃×C₄⋊Dic₃	C₂²	1	1	yes	characteristic
26	C₃×C₄⋊Dic₃	C₂²	1	1	yes	characteristic
27	C₃×C₄⋊Dic₃	C₂²	1	1	yes	characteristic
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×Dic₃⋊C₄	C₂²	1	2	yes	—
32	C₃×Dic₃⋊C₄	C₂²	1	2	yes	—
33	C₃×C₄×Dic₃	C₂²	1	2	yes	—
34	C₃×C₄×Dic₃	C₂²	1	2	yes	—
35	C₃×C₄×Dic₃	C₂²	1	1	yes	characteristic
36	C₃×C₄×Dic₃	C₂²	1	1	yes	characteristic
37	C₃×C₄⋊Q₈	—	3	1	no	—
38	C₃×#126	—	3	2	no	—
39	C₃×#126	—	3	2	no	—
40	C₃×#125	—	3	2	no	—
41	C₃×#125	—	3	1	no	—
42	C₃×#125	—	3	2	no	—
43	C₃×C₂²⋊Q₈	—	2	1	no	—
44	C₃×C₂²⋊Q₈	—	3	1	no	—
45	C₃×C₂²⋊Q₈	D₃	1	1	yes	characteristic
46	C₃×C₂²⋊Q₈	C₂×C₃	1	1	yes	characteristic
47	C₃×C₄×Q₈	—	3	1	no	—
48	C₃×C₄×Q₈	—	3	1	no	—
49	C₃×#117	—	3	1	no	—
50	Q₈×Dic₃	C₂×C₃	1	1	yes	characteristic, complemented
51	#944	C₂×C₃	1	1	yes	characteristic, complemented
52	#926	C₂×C₃	1	1	yes	characteristic
53	#924	C₂×C₃	1	1	yes	characteristic
54	#890	C₂×C₃	1	2	yes	complemented
55	#890	C₂×C₃	1	1	yes	characteristic, complemented
56	#889	C₂×C₃	1	2	yes	complemented
57	#887	C₂×C₃	1	1	yes	characteristic, complemented
58	#879	C₂×C₃	1	2	yes	—
59	#879	C₂×C₃	1	2	yes	—
60	C₂³×C₃²	C₂³	1	1	yes	characteristic
61	C₃²×Q₈	—	2	2	no	—
62	C₂×C₃²×C₄	—	2	2	no	—
63	C₂×C₃²×C₄	C₂³	1	2	yes	—
64	C₂×C₃²×C₄	C₂³	1	2	yes	—
65	C₂×C₃²×C₄	C₂³	1	1	yes	characteristic
66	C₂×C₃²×C₄	C₂³	1	1	yes	characteristic
67	C₂×C₃×Dic₃	C₂³	1	1	yes	characteristic
68	C₂×C₃×Dic₃	C₂³	1	1	yes	characteristic
69	C₂×C₃×Dic₃	C₂³	1	1	yes	characteristic
70	C₂×C₃×Dic₃	C₂³	1	2	yes	—
71	C₂×C₃×Dic₃	C₂³	1	2	yes	—
72	C₂×C₃×Dic₃	C₂³	1	1	yes	characteristic
73	C₃×Dic₆	—	2	2	no	—
74	#541	—	3	1	no	Sylow
75	C₂×C₃×Q₈	—	2	1	no	—
76	C₂×C₃×Q₈	—	3	1	no	—
77	C₂×C₃×Q₈	C₂×D₃	1	1	yes	characteristic
78	C₂×C₃×Q₈	C₂²×C₃	1	1	yes	characteristic
79	C₂²×C₃×C₄	—	2	1	no	—
80	C₂²×C₃×C₄	C₂×D₃	1	1	yes	characteristic
81	C₂²×C₃×C₄	C₂²×C₃	1	1	yes	characteristic
82	C₂×Dic₆	C₂²×C₃	1	1	yes	characteristic
83	C₃×C₄⋊C₄	—	2	1	no	—
84	C₃×C₄⋊C₄	—	2	2	no	—
85	C₃×C₄⋊C₄	—	3	2	no	—
86	C₃×C₄⋊C₄	—	3	2	no	—
87	C₃×C₄⋊C₄	—	3	2	no	—
88	C₃×C₄⋊C₄	—	3	2	no	—
89	C₃×C₄⋊C₄	—	3	2	no	—
90	C₃×C₄⋊C₄	—	3	1	no	—
91	C₃×C₄⋊C₄	—	3	1	no	—
92	C₃×C₄⋊C₄	—	3	1	no	—
93	C₃×C₄⋊C₄	—	3	2	no	—
94	C₃×C₄⋊C₄	—	3	2	no	—
95	C₃×C₄⋊C₄	C₂×D₃	1	1	yes	characteristic
96	C₃×C₄⋊C₄	C₂×D₃	1	2	yes	—
97	C₃×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
98	C₃×C₄⋊C₄	C₂²×C₃	1	2	yes	—
99	C₃×C₂²⋊C₄	—	2	2	no	—
100	C₃×C₂²⋊C₄	—	3	1	no	—
101	C₃×C₂²⋊C₄	—	3	1	no	—
102	C₃×C₂²⋊C₄	—	3	2	no	—
103	C₃×C₂²⋊C₄	C₂×D₃	1	2	yes	—
104	C₃×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
105	C₃×C₄²	—	3	1	no	—
106	C₃×C₄²	—	3	1	no	—
107	C₃×C₄²	—	3	2	no	—
108	C₃×C₄²	—	3	2	no	—
109	#217	C₂²×C₃	1	2	yes	—
110	#217	C₂²×C₃	1	1	yes	characteristic
111	#217	C₂²×C₃	1	1	yes	characteristic
112	C₄⋊Dic₃	C₂²×C₃	1	2	yes	—
113	C₄⋊Dic₃	C₂²×C₃	1	2	yes	—
114	C₄⋊Dic₃	C₂²×C₃	1	1	yes	characteristic
115	C₄⋊Dic₃	C₂²×C₃	1	1	yes	characteristic
116	C₄⋊Dic₃	C₂²×C₃	1	1	yes	characteristic
117	Dic₃⋊C₄	C₂²×C₃	1	2	yes	—
118	Dic₃⋊C₄	C₂²×C₃	1	2	yes	—
119	Dic₃⋊C₄	C₂²×C₃	1	2	yes	—
120	Dic₃⋊C₄	C₂²×C₃	1	2	yes	—
121	Dic₃⋊C₄	C₂²×C₃	1	2	yes	—
122	C₄×Dic₃	C₂²×C₃	1	2	yes	—
123	C₄×Dic₃	C₂²×C₃	1	2	yes	—
124	C₄×Dic₃	C₂²×C₃	1	1	yes	characteristic
125	C₄×Dic₃	C₂²×C₃	1	1	yes	characteristic
126	C₂²×C₃²	—	2	1	no	—
127	C₂²×C₃²	—	2	1	no	—
128	C₂²×C₃²	—	2	1	no	—
129	C₂²×C₃²	C₂⁴	1	1	yes	characteristic
130	C₃²×C₄	—	2	1	no	—
131	C₃²×C₄	—	2	2	no	—
132	C₃²×C₄	—	2	2	no	—
133	C₃²×C₄	#41	1	2	yes	—
134	C₃×Dic₃	—	2	2	no	—
135	C₃×Dic₃	—	2	1	no	—
136	C₃×Dic₃	—	2	2	no	—
137	C₃×Dic₃	—	2	1	no	—
138	C₃×Dic₃	—	2	1	no	—
139	C₃×Dic₃	—	2	1	no	—
140	C₄⋊Q₈	—	3	1	no	—
141	#126	—	3	2	no	—
142	#126	—	3	2	no	—
143	#125	—	3	2	no	—
144	#125	—	3	1	no	—
145	#125	—	3	2	no	—
146	C₂²⋊Q₈	—	3	1	no	—
147	C₂²⋊Q₈	C₃×D₃	1	1	yes	characteristic
148	C₄×Q₈	—	3	1	no	—
149	C₄×Q₈	—	3	1	no	—
150	#117	—	3	1	no	—
151	C₂³×C₃	—	2	1	no	—
152	C₂³×C₃	C₂²×D₃	1	1	yes	characteristic
153	C₂³×C₃	C₂³×C₃	1	1	yes	characteristic
154	C₃×Q₈	—	2	2	no	—
155	C₃×Q₈	—	2	2	no	—
156	C₃×Q₈	—	4	2	no	—
157	C₃×Q₈	—	6	2	no	—
158	C₂×C₃×C₄	—	2	2	no	—
159	C₂×C₃×C₄	—	2	2	no	—
160	C₂×C₃×C₄	—	2	1	no	—
161	C₂×C₃×C₄	—	2	2	no	—
162	C₂×C₃×C₄	—	2	2	no	—
163	C₂×C₃×C₄	—	2	1	no	—
164	C₂×C₃×C₄	—	3	1	no	—
165	C₂×C₃×C₄	—	3	1	no	—
166	C₂×C₃×C₄	—	3	1	no	—
167	C₂×C₃×C₄	—	3	2	no	—
168	C₂×C₃×C₄	—	3	1	no	—
169	C₂×C₃×C₄	—	3	2	no	—
170	C₂×C₃×C₄	—	4	2	no	—
171	C₂×C₃×C₄	C₂²×D₃	1	2	yes	—
172	C₂×C₃×C₄	C₂²×D₃	1	2	yes	—
173	C₂×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
174	C₂×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
175	C₂×C₃×C₄	C₂³×C₃	1	2	yes	—
176	C₂×C₃×C₄	C₂³×C₃	1	2	yes	—
177	C₂×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
178	C₂×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
179	C₂×Dic₃	C₂³×C₃	1	1	yes	characteristic
180	C₂×Dic₃	C₂³×C₃	1	1	yes	characteristic
181	C₂×Dic₃	C₂³×C₃	1	1	yes	characteristic
182	C₂×Dic₃	C₂³×C₃	1	2	yes	—
183	C₂×Dic₃	C₂³×C₃	1	2	yes	—
184	C₂×Dic₃	C₂³×C₃	1	1	yes	characteristic
185	Dic₆	—	2	2	no	—
186	C₂×C₃²	—	4	1	no	—
187	C₂×C₃²	C₂×#41	1	1	yes	characteristic
188	C₂×C₃²	2₋¹⁺⁴	1	1	yes	characteristic
189	C₂×C₃²	2₋¹⁺⁴	1	1	yes	characteristic
190	C₂×Q₈	—	3	1	no	—
191	C₂×Q₈	C₂×C₃×D₃	1	1	yes	characteristic
192	C₂²×C₄	C₂×C₃×D₃	1	1	yes	characteristic
193	C₄⋊C₄	—	3	1	no	—
194	C₄⋊C₄	—	3	1	no	—
195	C₄⋊C₄	—	3	1	no	—
196	C₄⋊C₄	—	3	2	no	—
197	C₄⋊C₄	—	3	2	no	—
198	C₄⋊C₄	—	3	2	no	—
199	C₄⋊C₄	—	3	2	no	—
200	C₄⋊C₄	—	3	2	no	—
201	C₄⋊C₄	—	3	2	no	—
202	C₄⋊C₄	—	3	2	no	—
203	C₄⋊C₄	C₂×C₃×D₃	1	1	yes	characteristic
204	C₄⋊C₄	C₂×C₃×D₃	1	2	yes	—
205	C₂²⋊C₄	—	3	1	no	—
206	C₂²⋊C₄	—	3	1	no	—
207	C₂²⋊C₄	—	3	2	no	—
208	C₂²⋊C₄	C₂×C₃×D₃	1	2	yes	—
209	C₄²	—	3	1	no	—
210	C₄²	—	3	1	no	—
211	C₄²	—	3	2	no	—
212	C₄²	—	3	2	no	—
213	C₂²×C₃	—	2	2	no	—
214	C₂²×C₃	—	2	1	no	—
215	C₂²×C₃	—	2	1	no	—
216	C₂²×C₃	—	2	1	no	—
217	C₂²×C₃	—	2	1	no	—
218	C₂²×C₃	—	2	1	no	—
219	C₂²×C₃	—	2	1	no	—
220	C₂²×C₃	—	2	1	no	—
221	C₂²×C₃	—	4	1	no	—
222	C₂²×C₃	—	4	1	no	—
223	C₂²×C₃	C₂³×D₃	1	1	yes	characteristic
224	C₂²×C₃	C₂⁴×C₃	1	1	yes	characteristic
225	C₃×C₄	—	2	1	no	—
226	C₃×C₄	—	2	2	no	—
227	C₃×C₄	—	2	2	no	—
228	C₃×C₄	—	2	1	no	—
229	C₃×C₄	—	2	2	no	—
230	C₃×C₄	—	2	2	no	—
231	C₃×C₄	—	2	2	no	—
232	C₃×C₄	—	4	1	no	—
233	C₃×C₄	—	4	2	no	—
234	C₃×C₄	—	4	2	no	—
235	C₃×C₄	—	6	2	no	—
236	C₃×C₄	—	6	1	no	—
237	C₃×C₄	—	6	1	no	—
238	C₃×C₄	—	6	2	no	—
239	C₃×C₄	—	6	1	no	—
240	C₃×C₄	—	6	1	no	—
241	C₃×C₄	#237	1	2	yes	—
242	C₃×C₄	C₃×#41	1	2	yes	—
243	Dic₃	—	2	2	no	—
244	Dic₃	—	2	1	no	—
245	Dic₃	—	2	2	no	—
246	Dic₃	—	2	1	no	—
247	Dic₃	—	2	1	no	—
248	Dic₃	—	2	1	no	—
249	C₃²	#541	1	1	yes	characteristic, complemented, Sylow
250	C₂³	C₂²×C₃×D₃	1	1	yes	characteristic
251	Q₈	—	2	2	no	—
252	Q₈	—	6	2	no	—
253	C₂×C₄	—	2	2	no	—
254	C₂×C₄	—	3	1	no	—
255	C₂×C₄	—	3	1	no	—
256	C₂×C₄	—	3	1	no	—
257	C₂×C₄	—	3	2	no	—
258	C₂×C₄	—	3	1	no	—
259	C₂×C₄	—	3	2	no	—
260	C₂×C₄	C₂²×C₃×D₃	1	2	yes	—
261	C₂×C₄	C₂²×C₃×D₃	1	2	yes	—
262	C₂×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
263	C₂×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
264	C₂×C₃	—	2	1	no	—
265	C₂×C₃	—	2	1	no	—
266	C₂×C₃	—	2	1	no	—
267	C₂×C₃	—	4	2	no	—
268	C₂×C₃	—	4	1	no	—
269	C₂×C₃	—	4	1	no	—
270	C₂×C₃	C₂×#237	1	1	yes	characteristic
271	C₂×C₃	#1007	1	1	yes	characteristic
272	C₂×C₃	#1010	1	1	yes	characteristic
273	C₂×C₃	C₂×C₃×#41	1	1	yes	characteristic
274	C₂×C₃	C₃×2₋¹⁺⁴	1	1	yes	characteristic
275	C₂×C₃	C₃×2₋¹⁺⁴	1	1	yes	characteristic
276	C₂²	—	2	1	no	—
277	C₂²	—	2	1	no	—
278	C₂²	—	2	1	no	—
279	C₂²	C₂³×C₃×D₃	1	1	yes	characteristic
280	C₄	—	2	1	no	—
281	C₄	—	2	2	no	—
282	C₄	—	2	2	no	—
283	C₄	—	6	2	no	—
284	C₄	—	6	1	no	—
285	C₄	—	6	1	no	—
286	C₄	—	6	2	no	—
287	C₄	—	6	1	no	—
288	C₄	—	6	1	no	—
289	C₄	C₃×#237	1	2	yes	—
290	C₃	—	2	1	no	—
291	C₃	#5625	1	1	yes	characteristic, direct product factor
292	C₃	C₃×#541	1	1	yes	characteristic, complemented
293	C₂	—	4	1	no	—
294	C₂	C₂×C₃×#237	1	1	yes	characteristic
295	C₂	C₃×#1007	1	1	yes	characteristic
296	C₂	C₃×#1010	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×C₂²⋊Q₈	C₂
socle	C₂²×C₃²	C₂⁴
prime radical	C₂³×C₃²	C₂³
hypercenter	C₃×C₂²⋊Q₈	D₃
hypocenter	C₃	C₃×#541
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₃ ◁ C₃×C₂²⋊Q₈

Complements

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

N	H ≅ G/N	type
2. C₃×Q₈×Dic₃	C₂	semidirect
3. C₃×#944	C₂	semidirect
6. C₃×#890	C₂	semidirect
7. C₃×#890	C₂	semidirect
8. C₃×#889	C₂	semidirect
9. C₃×#887	C₂	semidirect
13. #5625	C₃	direct (N × H)
50. Q₈×Dic₃	C₂×C₃	semidirect
51. #944	C₂×C₃	semidirect
54. #890	C₂×C₃	semidirect
55. #890	C₂×C₃	semidirect
56. #889	C₂×C₃	semidirect
57. #887	C₂×C₃	semidirect
249. C₃²	#541	semidirect
291. C₃	#5625	direct (N × H)
292. C₃	C₃×#541	semidirect

Maximal subgroups

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

subgroup	index
C₃²×C₂²⋊Q₈	2
C₃×Q₈×Dic₃	2
C₃×#944	2
C₃×#926	2
C₃×#924	2
C₃×#890	2
C₃×#889	2
C₃×#887	2
C₃×#879	2
C₃×#541	3
#5625	3

Subgroups — C3×#5625 #10595722

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#5625 #10595722