Subgroups — C₂×C₃×#529 #87323

Order 384.

1232 conjugacy classes of subgroups.

The subgroup lattice has 300 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#529	C₂	1	16	yes	maximal, direct product factor
2	C₂×C₃×#126	C₂	1	2	yes	maximal, complemented
3	C₂×C₃×#125	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₃×#124	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₃×#123	C₂	1	2	yes	maximal, complemented
6	C₂×C₃×C₂²⋊Q₈	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₃×#121	C₂	1	1	yes	maximal, characteristic
8	C₂×C₃×C₄×Q₈	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×C₃×C₄×D₄	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×C₃×C₄×D₄	C₂	1	2	yes	maximal, complemented
11	C₂×C₃×#117	C₂	1	2	yes	maximal, complemented
12	C₂²×C₃×C₄²	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂×#529	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
14	C₂²×C₃×Q₈	C₂²	1	1	yes	characteristic
15	C₂²×C₃×D₄	C₂²	1	2	yes	—
16	C₂²×C₃×D₄	C₂²	1	1	yes	characteristic
17	C₂³×C₃×C₄	C₂²	1	1	yes	characteristic
18	C₂³×C₃×C₄	C₂²	1	1	yes	characteristic
19	C₂³×C₃×C₄	C₂²	1	1	yes	characteristic
20	C₂³×C₃×C₄	C₂²	1	2	yes	—
21	C₃×#126	C₂²	1	16	yes	complemented
22	C₃×#125	C₂²	1	8	yes	complemented
23	C₃×#124	C₂²	1	8	yes	complemented
24	C₃×#123	C₂²	1	16	yes	complemented
25	C₃×C₂²⋊Q₈	C₂²	1	8	yes	complemented
26	C₃×#121	C₂²	1	8	yes	—
27	C₃×C₄×Q₈	C₂²	1	8	yes	complemented
28	C₃×C₄×D₄	C₂²	1	8	yes	complemented
29	C₃×C₄×D₄	C₂²	1	16	yes	complemented
30	C₃×#117	C₂²	1	16	yes	complemented
31	C₂×C₃×C₄⋊C₄	C₂²	1	2	yes	—
32	C₂×C₃×C₄⋊C₄	C₂²	1	2	yes	—
33	C₂×C₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
34	C₂×C₃×C₄⋊C₄	C₂²	1	2	yes	—
35	C₂×C₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
36	C₂×C₃×C₄⋊C₄	C₂²	1	2	yes	—
37	C₂×C₃×C₂²⋊C₄	C₂²	1	2	yes	—
38	C₂×C₃×C₂²⋊C₄	C₂²	1	2	yes	—
39	C₂×C₃×C₂²⋊C₄	C₂²	1	2	yes	—
40	C₂×C₃×C₂²⋊C₄	C₂²	1	2	yes	—
41	C₂×C₃×C₂²⋊C₄	C₂²	1	2	yes	—
42	C₂×C₃×C₄²	C₂²	1	1	yes	characteristic
43	C₂×C₃×C₄²	C₂²	1	1	yes	characteristic
44	C₂×C₃×C₄²	C₂²	1	1	yes	characteristic
45	C₂×C₃×C₄²	C₂²	1	1	yes	characteristic
46	C₂×C₃×C₄²	C₂²	1	2	yes	—
47	C₂×C₃×C₄²	C₂²	1	8	yes	complemented
48	#529	C₂×C₃	1	16	yes	direct product factor
49	C₂×#126	C₂×C₃	1	2	yes	complemented
50	C₂×#125	C₂×C₃	1	1	yes	characteristic, complemented
51	C₂×#124	C₂×C₃	1	1	yes	characteristic, complemented
52	C₂×#123	C₂×C₃	1	2	yes	complemented
53	C₂×C₂²⋊Q₈	C₂×C₃	1	1	yes	characteristic, complemented
54	C₂×#121	C₂×C₃	1	1	yes	characteristic
55	C₂×C₄×Q₈	C₂×C₃	1	1	yes	characteristic, complemented
56	C₂×C₄×D₄	C₂×C₃	1	2	yes	complemented
57	C₂×C₄×D₄	C₂×C₃	1	1	yes	characteristic, complemented
58	C₂×#117	C₂×C₃	1	2	yes	complemented
59	C₂²×C₄²	C₂×C₃	1	1	yes	characteristic, complemented
60	C₂⁴×C₃	C₂³	1	2	yes	—
61	C₂⁴×C₃	C₂³	1	1	yes	characteristic
62	C₂×C₃×Q₈	—	2	4	no	—
63	C₂×C₃×Q₈	C₂³	1	4	yes	—
64	C₂×C₃×D₄	—	2	8	no	—
65	C₂×C₃×D₄	—	2	4	no	—
66	C₂×C₃×D₄	C₂³	1	8	yes	—
67	C₂×C₃×D₄	C₂³	1	4	yes	—
68	C₂²×C₃×C₄	—	2	4	no	—
69	C₂²×C₃×C₄	—	2	4	no	—
70	C₂²×C₃×C₄	—	2	8	no	—
71	C₂²×C₃×C₄	—	2	4	no	—
72	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
73	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
74	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
75	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
76	C₂²×C₃×C₄	C₂³	1	2	yes	—
77	C₂²×C₃×C₄	C₂³	1	2	yes	—
78	C₂²×C₃×C₄	C₂³	1	4	yes	—
79	C₂²×C₃×C₄	C₂³	1	4	yes	—
80	C₂²×C₃×C₄	C₂³	1	4	yes	—
81	C₂²×C₃×C₄	C₂³	1	2	yes	—
82	C₂²×C₃×C₄	C₂³	1	8	yes	—
83	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
84	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
85	C₃×C₄⋊C₄	C₂³	1	8	yes	—
86	C₃×C₄⋊C₄	C₂³	1	8	yes	—
87	C₃×C₄⋊C₄	C₂³	1	4	yes	—
88	C₃×C₄⋊C₄	C₂³	1	8	yes	—
89	C₃×C₄⋊C₄	C₂³	1	4	yes	—
90	C₃×C₄⋊C₄	C₂³	1	8	yes	—
91	C₃×C₂²⋊C₄	C₂³	1	8	yes	—
92	C₃×C₂²⋊C₄	C₂³	1	8	yes	—
93	C₃×C₂²⋊C₄	C₂³	1	8	yes	—
94	C₃×C₂²⋊C₄	C₂³	1	8	yes	—
95	C₃×C₂²⋊C₄	C₂³	1	8	yes	—
96	C₃×C₄²	C₂³	1	4	yes	—
97	C₃×C₄²	C₂³	1	4	yes	—
98	C₃×C₄²	C₂³	1	4	yes	—
99	C₃×C₄²	C₂³	1	4	yes	—
100	C₃×C₄²	C₂³	1	8	yes	—
101	C₂²×Q₈	C₂²×C₃	1	1	yes	characteristic
102	C₂²×D₄	C₂²×C₃	1	2	yes	—
103	C₂²×D₄	C₂²×C₃	1	1	yes	characteristic
104	C₂³×C₄	C₂²×C₃	1	2	yes	—
105	C₂³×C₄	C₂²×C₃	1	1	yes	characteristic
106	C₂³×C₄	C₂²×C₃	1	1	yes	characteristic
107	C₂³×C₄	C₂²×C₃	1	1	yes	characteristic
108	#126	C₂²×C₃	1	16	yes	complemented
109	#125	C₂²×C₃	1	8	yes	complemented
110	#124	C₂²×C₃	1	8	yes	complemented
111	#123	C₂²×C₃	1	16	yes	complemented
112	C₂²⋊Q₈	C₂²×C₃	1	8	yes	complemented
113	#121	C₂²×C₃	1	8	yes	—
114	C₄×Q₈	C₂²×C₃	1	8	yes	complemented
115	C₄×D₄	C₂²×C₃	1	8	yes	complemented
116	C₄×D₄	C₂²×C₃	1	16	yes	complemented
117	#117	C₂²×C₃	1	16	yes	complemented
118	C₂×C₄⋊C₄	C₂²×C₃	1	2	yes	—
119	C₂×C₄⋊C₄	C₂²×C₃	1	2	yes	—
120	C₂×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
121	C₂×C₄⋊C₄	C₂²×C₃	1	2	yes	—
122	C₂×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
123	C₂×C₄⋊C₄	C₂²×C₃	1	2	yes	—
124	C₂×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
125	C₂×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
126	C₂×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
127	C₂×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
128	C₂×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
129	C₂×C₄²	C₂²×C₃	1	1	yes	characteristic
130	C₂×C₄²	C₂²×C₃	1	2	yes	—
131	C₂×C₄²	C₂²×C₃	1	1	yes	characteristic
132	C₂×C₄²	C₂²×C₃	1	1	yes	characteristic
133	C₂×C₄²	C₂²×C₃	1	1	yes	characteristic
134	C₂×C₄²	C₂²×C₃	1	8	yes	complemented
135	C₂³×C₃	—	2	2	no	—
136	C₂³×C₃	—	2	2	no	—
137	C₂³×C₃	—	2	4	no	—
138	C₂³×C₃	—	2	4	no	—
139	C₂³×C₃	—	2	4	no	—
140	C₂³×C₃	#41	1	4	yes	—
141	C₂³×C₃	C₂⁴	1	4	yes	—
142	C₂³×C₃	C₂⁴	1	2	yes	—
143	C₂³×C₃	C₂⁴	1	1	yes	characteristic
144	C₃×Q₈	—	2	8	no	—
145	C₃×D₄	—	2	16	no	—
146	C₃×D₄	—	2	8	no	—
147	C₂×C₃×C₄	—	2	8	no	—
148	C₂×C₃×C₄	—	2	2	no	—
149	C₂×C₃×C₄	—	2	2	no	—
150	C₂×C₃×C₄	—	2	2	no	—
151	C₂×C₃×C₄	—	2	2	no	—
152	C₂×C₃×C₄	—	2	4	no	—
153	C₂×C₃×C₄	—	2	4	no	—
154	C₂×C₃×C₄	—	2	4	no	—
155	C₂×C₃×C₄	—	2	8	no	—
156	C₂×C₃×C₄	—	2	16	no	—
157	C₂×C₃×C₄	—	2	8	no	—
158	C₂×C₃×C₄	#41	1	4	yes	—
159	C₂×C₃×C₄	#41	1	4	yes	—
160	C₂×C₃×C₄	C₂⁴	1	2	yes	—
161	C₂×C₃×C₄	C₂⁴	1	2	yes	—
162	C₂×C₃×C₄	C₂⁴	1	2	yes	—
163	C₂×C₃×C₄	C₂⁴	1	2	yes	—
164	C₂×C₃×C₄	C₂⁴	1	4	yes	—
165	C₂×C₃×C₄	C₂⁴	1	4	yes	—
166	C₂×C₃×C₄	C₂⁴	1	4	yes	—
167	C₂×C₃×C₄	C₂⁴	1	2	yes	—
168	C₂×C₃×C₄	C₂⁴	1	2	yes	—
169	C₂⁴	C₂³×C₃	1	2	yes	—
170	C₂⁴	C₂³×C₃	1	1	yes	characteristic
171	C₂×Q₈	—	2	4	no	—
172	C₂×Q₈	C₂³×C₃	1	4	yes	—
173	C₂×D₄	—	2	8	no	—
174	C₂×D₄	—	2	4	no	—
175	C₂×D₄	C₂³×C₃	1	8	yes	—
176	C₂×D₄	C₂³×C₃	1	4	yes	—
177	C₂²×C₄	—	2	4	no	—
178	C₂²×C₄	—	2	4	no	—
179	C₂²×C₄	—	2	8	no	—
180	C₂²×C₄	—	2	4	no	—
181	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
182	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
183	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
184	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
185	C₂²×C₄	C₂³×C₃	1	2	yes	—
186	C₂²×C₄	C₂³×C₃	1	2	yes	—
187	C₂²×C₄	C₂³×C₃	1	4	yes	—
188	C₂²×C₄	C₂³×C₃	1	4	yes	—
189	C₂²×C₄	C₂³×C₃	1	4	yes	—
190	C₂²×C₄	C₂³×C₃	1	2	yes	—
191	C₂²×C₄	C₂³×C₃	1	8	yes	—
192	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
193	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
194	C₄⋊C₄	C₂³×C₃	1	8	yes	—
195	C₄⋊C₄	C₂³×C₃	1	8	yes	—
196	C₄⋊C₄	C₂³×C₃	1	4	yes	—
197	C₄⋊C₄	C₂³×C₃	1	8	yes	—
198	C₄⋊C₄	C₂³×C₃	1	4	yes	—
199	C₄⋊C₄	C₂³×C₃	1	8	yes	—
200	C₂²⋊C₄	C₂³×C₃	1	8	yes	—
201	C₂²⋊C₄	C₂³×C₃	1	8	yes	—
202	C₂²⋊C₄	C₂³×C₃	1	8	yes	—
203	C₂²⋊C₄	C₂³×C₃	1	8	yes	—
204	C₂²⋊C₄	C₂³×C₃	1	8	yes	—
205	C₄²	C₂³×C₃	1	4	yes	—
206	C₄²	C₂³×C₃	1	4	yes	—
207	C₄²	C₂³×C₃	1	4	yes	—
208	C₄²	C₂³×C₃	1	4	yes	—
209	C₄²	C₂³×C₃	1	8	yes	—
210	C₂²×C₃	—	2	8	no	—
211	C₂²×C₃	—	2	2	no	—
212	C₂²×C₃	—	2	2	no	—
213	C₂²×C₃	—	2	4	no	—
214	C₂²×C₃	—	2	4	no	—
215	C₂²×C₃	—	2	4	no	—
216	C₂²×C₃	—	4	8	no	—
217	C₂²×C₃	C₂×#41	1	4	yes	—
218	C₂²×C₃	C₂×#41	1	2	yes	—
219	C₂²×C₃	C₂×#41	1	2	yes	—
220	C₂²×C₃	C₂×#41	1	2	yes	—
221	C₂²×C₃	C₂⁵	1	1	yes	characteristic
222	C₃×C₄	—	2	2	no	—
223	C₃×C₄	—	2	2	no	—
224	C₃×C₄	—	2	2	no	—
225	C₃×C₄	—	2	2	no	—
226	C₃×C₄	—	2	4	no	—
227	C₃×C₄	—	2	4	no	—
228	C₃×C₄	—	2	4	no	—
229	C₃×C₄	C₂×#41	1	4	yes	—
230	C₃×C₄	C₂×#41	1	4	yes	—
231	C₂³	—	2	2	no	—
232	C₂³	—	2	2	no	—
233	C₂³	—	2	4	no	—
234	C₂³	—	2	4	no	—
235	C₂³	—	2	4	no	—
236	C₂³	C₃×#41	1	4	yes	—
237	C₂³	C₂⁴×C₃	1	4	yes	—
238	C₂³	C₂⁴×C₃	1	2	yes	—
239	C₂³	C₂⁴×C₃	1	1	yes	characteristic
240	Q₈	—	2	8	no	—
241	D₄	—	2	16	no	—
242	D₄	—	2	8	no	—
243	C₂×C₄	—	2	8	no	—
244	C₂×C₄	—	2	2	no	—
245	C₂×C₄	—	2	2	no	—
246	C₂×C₄	—	2	2	no	—
247	C₂×C₄	—	2	2	no	—
248	C₂×C₄	—	2	4	no	—
249	C₂×C₄	—	2	4	no	—
250	C₂×C₄	—	2	4	no	—
251	C₂×C₄	—	2	8	no	—
252	C₂×C₄	—	2	16	no	—
253	C₂×C₄	—	2	8	no	—
254	C₂×C₄	C₃×#41	1	4	yes	—
255	C₂×C₄	C₃×#41	1	4	yes	—
256	C₂×C₄	C₂⁴×C₃	1	2	yes	—
257	C₂×C₄	C₂⁴×C₃	1	2	yes	—
258	C₂×C₄	C₂⁴×C₃	1	2	yes	—
259	C₂×C₄	C₂⁴×C₃	1	2	yes	—
260	C₂×C₄	C₂⁴×C₃	1	4	yes	—
261	C₂×C₄	C₂⁴×C₃	1	4	yes	—
262	C₂×C₄	C₂⁴×C₃	1	4	yes	—
263	C₂×C₄	C₂⁴×C₃	1	2	yes	—
264	C₂×C₄	C₂⁴×C₃	1	2	yes	—
265	C₂×C₃	—	2	4	no	—
266	C₂×C₃	—	4	4	no	—
267	C₂×C₃	#529	1	4	yes	direct product factor
268	C₂×C₃	C₂²×#41	1	1	yes	characteristic
269	C₂×C₃	C₂²×#41	1	1	yes	characteristic
270	C₂×C₃	C₂²×#41	1	1	yes	characteristic
271	C₂²	—	2	2	no	—
272	C₂²	—	2	2	no	—
273	C₂²	—	2	8	no	—
274	C₂²	—	2	4	no	—
275	C₂²	—	2	4	no	—
276	C₂²	—	2	4	no	—
277	C₂²	—	4	8	no	—
278	C₂²	C₂×C₃×#41	1	4	yes	—
279	C₂²	C₂×C₃×#41	1	2	yes	—
280	C₂²	C₂×C₃×#41	1	2	yes	—
281	C₂²	C₂×C₃×#41	1	2	yes	—
282	C₂²	C₂⁵×C₃	1	1	yes	characteristic
283	C₄	—	2	2	no	—
284	C₄	—	2	2	no	—
285	C₄	—	2	2	no	—
286	C₄	—	2	2	no	—
287	C₄	—	2	4	no	—
288	C₄	—	2	4	no	—
289	C₄	—	2	4	no	—
290	C₄	C₂×C₃×#41	1	4	yes	—
291	C₄	C₂×C₃×#41	1	4	yes	—
292	C₃	C₂×#529	1	1	yes	characteristic, direct product factor, Sylow
293	C₂	—	2	4	no	—
294	C₂	—	4	4	no	—
295	C₂	C₃×#529	1	4	yes	direct product factor
296	C₂	C₂²×C₃×#41	1	1	yes	characteristic
297	C₂	C₂²×C₃×#41	1	1	yes	characteristic
298	C₂	C₂²×C₃×#41	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₃×C₄	C₂³
commutator, Frattini, cosocle	C₂²	C₂⁵×C₃
socle	C₂³×C₃	C₂⁴
prime radical	C₂×C₃×#121	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₃×C₄ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₃×#529	C₂	direct (N × H)
2. C₂×C₃×#126	C₂	semidirect
3. C₂×C₃×#125	C₂	semidirect
4. C₂×C₃×#124	C₂	semidirect
5. C₂×C₃×#123	C₂	semidirect
6. C₂×C₃×C₂²⋊Q₈	C₂	semidirect
8. C₂×C₃×C₄×Q₈	C₂	semidirect
9. C₂×C₃×C₄×D₄	C₂	semidirect
10. C₂×C₃×C₄×D₄	C₂	semidirect
11. C₂×C₃×#117	C₂	semidirect
12. C₂²×C₃×C₄²	C₂	semidirect
13. C₂×#529	C₃	direct (N × H)
21. C₃×#126	C₂²	semidirect
22. C₃×#125	C₂²	semidirect
23. C₃×#124	C₂²	semidirect
24. C₃×#123	C₂²	semidirect
25. C₃×C₂²⋊Q₈	C₂²	semidirect
27. C₃×C₄×Q₈	C₂²	semidirect
28. C₃×C₄×D₄	C₂²	semidirect
29. C₃×C₄×D₄	C₂²	semidirect
30. C₃×#117	C₂²	semidirect
47. C₂×C₃×C₄²	C₂²	semidirect
48. #529	C₂×C₃	direct (N × H)
49. C₂×#126	C₂×C₃	semidirect
50. C₂×#125	C₂×C₃	semidirect
51. C₂×#124	C₂×C₃	semidirect
52. C₂×#123	C₂×C₃	semidirect
53. C₂×C₂²⋊Q₈	C₂×C₃	semidirect
55. C₂×C₄×Q₈	C₂×C₃	semidirect
56. C₂×C₄×D₄	C₂×C₃	semidirect
57. C₂×C₄×D₄	C₂×C₃	semidirect
58. C₂×#117	C₂×C₃	semidirect
59. C₂²×C₄²	C₂×C₃	semidirect
108. #126	C₂²×C₃	semidirect
109. #125	C₂²×C₃	semidirect
110. #124	C₂²×C₃	semidirect
111. #123	C₂²×C₃	semidirect
112. C₂²⋊Q₈	C₂²×C₃	semidirect
114. C₄×Q₈	C₂²×C₃	semidirect
115. C₄×D₄	C₂²×C₃	semidirect
116. C₄×D₄	C₂²×C₃	semidirect
117. #117	C₂²×C₃	semidirect
134. C₂×C₄²	C₂²×C₃	semidirect
267. C₂×C₃	#529	direct (N × H)
292. C₃	C₂×#529	direct (N × H)
295. C₂	C₃×#529	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃×#529	2
C₂×C₃×#126	2
C₂×C₃×#125	2
C₂×C₃×#124	2
C₂×C₃×#123	2
C₂×C₃×C₂²⋊Q₈	2
C₂×C₃×#121	2
C₂×C₃×C₄×Q₈	2
C₂×C₃×C₄×D₄	2
C₂×C₃×#117	2
C₂²×C₃×C₄²	2
C₂×#529	3

Subgroups — C2×C3×#529 #87323

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×#529 #87323