Subgroups — C₂×C₃×#551 #87353

Order 384.

1328 conjugacy classes of subgroups.

The subgroup lattice has 276 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#551	C₂	1	16	yes	maximal, direct product factor
2	C₂×C₃×#126	C₂	1	2	yes	maximal, complemented
3	C₂×C₃×#124	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₃×#123	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₃×#123	C₂	1	2	yes	maximal, complemented
6	C₂×C₃×C₂²⋊Q₈	C₂	1	2	yes	maximal, complemented
7	C₂×C₃×#120	C₂	1	1	yes	maximal, characteristic
8	C₂×C₃×C₄×D₄	C₂	1	2	yes	maximal, complemented
9	C₂×C₃×#117	C₂	1	2	yes	maximal, complemented
10	C₂²×C₃×C₂²⋊C₄	C₂	1	2	yes	maximal, complemented
11	C₂×#551	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
12	C₂⁵×C₃	C₂²	1	1	yes	characteristic
13	C₂²×C₃×Q₈	C₂²	1	1	yes	characteristic, complemented
14	C₂²×C₃×D₄	C₂²	1	2	yes	—
15	C₂²×C₃×D₄	C₂²	1	1	yes	characteristic
16	C₂³×C₃×C₄	C₂²	1	2	yes	—
17	C₂³×C₃×C₄	C₂²	1	2	yes	—
18	C₂³×C₃×C₄	C₂²	1	1	yes	characteristic, complemented
19	C₃×#126	C₂²	1	16	yes	complemented
20	C₃×#124	C₂²	1	8	yes	complemented
21	C₃×#123	C₂²	1	8	yes	complemented
22	C₃×#123	C₂²	1	16	yes	complemented
23	C₃×C₂²⋊Q₈	C₂²	1	16	yes	complemented
24	C₃×#120	C₂²	1	8	yes	—
25	C₃×C₄×D₄	C₂²	1	16	yes	complemented
26	C₃×#117	C₂²	1	16	yes	complemented
27	C₂×C₃×C₄⋊C₄	C₂²	1	2	yes	complemented
28	C₂×C₃×C₄⋊C₄	C₂²	1	2	yes	complemented
29	C₂×C₃×C₄⋊C₄	C₂²	1	2	yes	complemented
30	C₂×C₃×C₄⋊C₄	C₂²	1	2	yes	complemented
31	C₂×C₃×C₂²⋊C₄	C₂²	1	2	yes	—
32	C₂×C₃×C₂²⋊C₄	C₂²	1	2	yes	—
33	C₂×C₃×C₂²⋊C₄	C₂²	1	2	yes	—
34	C₂×C₃×C₂²⋊C₄	C₂²	1	2	yes	—
35	C₂×C₃×C₂²⋊C₄	C₂²	1	2	yes	—
36	C₂×C₃×C₂²⋊C₄	C₂²	1	1	yes	characteristic
37	C₂×C₃×C₂²⋊C₄	C₂²	1	16	yes	complemented
38	C₂×C₃×C₂²⋊C₄	C₂²	1	2	yes	complemented
39	C₂×C₃×C₂²⋊C₄	C₂²	1	1	yes	characteristic, complemented
40	C₂×C₃×C₄²	C₂²	1	1	yes	characteristic, complemented
41	C₂×C₃×C₄²	C₂²	1	2	yes	complemented
42	#551	C₂×C₃	1	16	yes	direct product factor
43	C₂×#126	C₂×C₃	1	2	yes	complemented
44	C₂×#124	C₂×C₃	1	1	yes	characteristic, complemented
45	C₂×#123	C₂×C₃	1	1	yes	characteristic, complemented
46	C₂×#123	C₂×C₃	1	2	yes	complemented
47	C₂×C₂²⋊Q₈	C₂×C₃	1	2	yes	complemented
48	C₂×#120	C₂×C₃	1	1	yes	characteristic
49	C₂×C₄×D₄	C₂×C₃	1	2	yes	complemented
50	C₂×#117	C₂×C₃	1	2	yes	complemented
51	C₂²×C₂²⋊C₄	C₂×C₃	1	2	yes	complemented
52	C₂⁴×C₃	—	2	8	no	—
53	C₂⁴×C₃	—	4	2	no	—
54	C₂⁴×C₃	C₂³	1	1	yes	characteristic
55	C₂⁴×C₃	C₂³	1	1	yes	characteristic
56	C₂⁴×C₃	C₂³	1	4	yes	—
57	C₂⁴×C₃	C₂³	1	2	yes	—
58	C₂×C₃×Q₈	—	4	2	no	—
59	C₂×C₃×Q₈	C₂³	1	4	yes	complemented
60	C₂×C₃×D₄	—	2	8	no	—
61	C₂×C₃×D₄	—	4	2	no	—
62	C₂×C₃×D₄	C₂³	1	8	yes	—
63	C₂×C₃×D₄	C₂³	1	4	yes	—
64	C₂²×C₃×C₄	—	2	8	no	—
65	C₂²×C₃×C₄	—	4	4	no	—
66	C₂²×C₃×C₄	—	4	2	no	—
67	C₂²×C₃×C₄	C₂³	1	8	yes	—
68	C₂²×C₃×C₄	C₂³	1	2	yes	—
69	C₂²×C₃×C₄	C₂³	1	2	yes	—
70	C₂²×C₃×C₄	C₂³	1	2	yes	—
71	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
72	C₂²×C₃×C₄	C₂³	1	2	yes	—
73	C₂²×C₃×C₄	C₂³	1	2	yes	—
74	C₂²×C₃×C₄	C₂³	1	8	yes	—
75	C₂²×C₃×C₄	C₂³	1	4	yes	complemented
76	C₃×C₄⋊C₄	C₂³	1	8	yes	complemented
77	C₃×C₄⋊C₄	C₂³	1	8	yes	complemented
78	C₃×C₄⋊C₄	C₂³	1	8	yes	complemented
79	C₃×C₄⋊C₄	C₂³	1	8	yes	complemented
80	C₃×C₂²⋊C₄	C₂³	1	8	yes	—
81	C₃×C₂²⋊C₄	C₂³	1	8	yes	—
82	C₃×C₂²⋊C₄	C₂³	1	8	yes	—
83	C₃×C₂²⋊C₄	C₂³	1	8	yes	—
84	C₃×C₂²⋊C₄	C₂³	1	8	yes	—
85	C₃×C₂²⋊C₄	C₂³	1	4	yes	—
86	C₃×C₂²⋊C₄	C₂³	1	8	yes	complemented
87	C₃×C₂²⋊C₄	C₂³	1	4	yes	complemented
88	C₃×C₄²	C₂³	1	4	yes	complemented
89	C₃×C₄²	C₂³	1	8	yes	complemented
90	C₂⁵	C₂²×C₃	1	1	yes	characteristic
91	C₂²×Q₈	C₂²×C₃	1	1	yes	characteristic, complemented
92	C₂²×D₄	C₂²×C₃	1	2	yes	—
93	C₂²×D₄	C₂²×C₃	1	1	yes	characteristic
94	C₂³×C₄	C₂²×C₃	1	2	yes	—
95	C₂³×C₄	C₂²×C₃	1	2	yes	—
96	C₂³×C₄	C₂²×C₃	1	1	yes	characteristic, complemented
97	#126	C₂²×C₃	1	16	yes	complemented
98	#124	C₂²×C₃	1	8	yes	complemented
99	#123	C₂²×C₃	1	8	yes	complemented
100	#123	C₂²×C₃	1	16	yes	complemented
101	C₂²⋊Q₈	C₂²×C₃	1	16	yes	complemented
102	#120	C₂²×C₃	1	8	yes	—
103	C₄×D₄	C₂²×C₃	1	16	yes	complemented
104	#117	C₂²×C₃	1	16	yes	complemented
105	C₂×C₄⋊C₄	C₂²×C₃	1	2	yes	complemented
106	C₂×C₄⋊C₄	C₂²×C₃	1	2	yes	complemented
107	C₂×C₄⋊C₄	C₂²×C₃	1	2	yes	complemented
108	C₂×C₄⋊C₄	C₂²×C₃	1	2	yes	complemented
109	C₂×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
110	C₂×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
111	C₂×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
112	C₂×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
113	C₂×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
114	C₂×C₂²⋊C₄	C₂²×C₃	1	1	yes	characteristic
115	C₂×C₂²⋊C₄	C₂²×C₃	1	16	yes	complemented
116	C₂×C₂²⋊C₄	C₂²×C₃	1	2	yes	complemented
117	C₂×C₂²⋊C₄	C₂²×C₃	1	1	yes	characteristic, complemented
118	C₂×C₄²	C₂²×C₃	1	1	yes	characteristic, complemented
119	C₂×C₄²	C₂²×C₃	1	2	yes	complemented
120	C₂³×C₃	—	2	4	no	—
121	C₂³×C₃	—	2	4	no	—
122	C₂³×C₃	—	2	4	no	—
123	C₂³×C₃	—	2	16	no	—
124	C₂³×C₃	—	2	4	no	—
125	C₂³×C₃	—	2	2	no	—
126	C₂³×C₃	—	2	2	no	—
127	C₂³×C₃	—	4	4	no	—
128	C₂³×C₃	—	4	16	no	—
129	C₂³×C₃	#41	1	8	yes	—
130	C₂³×C₃	C₂⁴	1	2	yes	—
131	C₂³×C₃	C₂⁴	1	2	yes	—
132	C₂³×C₃	C₂⁴	1	4	yes	—
133	C₂³×C₃	C₂⁴	1	1	yes	characteristic
134	C₃×Q₈	—	4	4	no	—
135	C₃×D₄	—	2	16	no	—
136	C₃×D₄	—	4	4	no	—
137	C₂×C₃×C₄	—	2	16	no	—
138	C₂×C₃×C₄	—	2	4	no	—
139	C₂×C₃×C₄	—	2	4	no	—
140	C₂×C₃×C₄	—	2	4	no	—
141	C₂×C₃×C₄	—	2	4	no	—
142	C₂×C₃×C₄	—	2	4	no	—
143	C₂×C₃×C₄	—	2	2	no	—
144	C₂×C₃×C₄	—	4	8	no	—
145	C₂×C₃×C₄	—	4	4	no	—
146	C₂×C₃×C₄	C₂⁴	1	4	yes	—
147	C₂×C₃×C₄	C₂⁴	1	4	yes	—
148	C₂×C₃×C₄	C₂⁴	1	4	yes	—
149	C₂×C₃×C₄	C₂⁴	1	2	yes	—
150	C₂×C₃×C₄	C₂⁴	1	4	yes	—
151	C₂×C₃×C₄	C₂⁴	1	4	yes	—
152	C₂⁴	—	2	8	no	—
153	C₂⁴	—	4	2	no	—
154	C₂⁴	C₂³×C₃	1	1	yes	characteristic
155	C₂⁴	C₂³×C₃	1	1	yes	characteristic
156	C₂⁴	C₂³×C₃	1	4	yes	—
157	C₂⁴	C₂³×C₃	1	2	yes	—
158	C₂×Q₈	—	4	2	no	—
159	C₂×Q₈	C₂³×C₃	1	4	yes	complemented
160	C₂×D₄	—	2	8	no	—
161	C₂×D₄	—	4	2	no	—
162	C₂×D₄	C₂³×C₃	1	8	yes	—
163	C₂×D₄	C₂³×C₃	1	4	yes	—
164	C₂²×C₄	—	2	8	no	—
165	C₂²×C₄	—	4	4	no	—
166	C₂²×C₄	—	4	2	no	—
167	C₂²×C₄	C₂³×C₃	1	2	yes	—
168	C₂²×C₄	C₂³×C₃	1	2	yes	—
169	C₂²×C₄	C₂³×C₃	1	2	yes	—
170	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
171	C₂²×C₄	C₂³×C₃	1	2	yes	—
172	C₂²×C₄	C₂³×C₃	1	2	yes	—
173	C₂²×C₄	C₂³×C₃	1	8	yes	—
174	C₂²×C₄	C₂³×C₃	1	8	yes	—
175	C₂²×C₄	C₂³×C₃	1	4	yes	complemented
176	C₄⋊C₄	C₂³×C₃	1	8	yes	complemented
177	C₄⋊C₄	C₂³×C₃	1	8	yes	complemented
178	C₄⋊C₄	C₂³×C₃	1	8	yes	complemented
179	C₄⋊C₄	C₂³×C₃	1	8	yes	complemented
180	C₂²⋊C₄	C₂³×C₃	1	8	yes	—
181	C₂²⋊C₄	C₂³×C₃	1	8	yes	—
182	C₂²⋊C₄	C₂³×C₃	1	8	yes	—
183	C₂²⋊C₄	C₂³×C₃	1	8	yes	—
184	C₂²⋊C₄	C₂³×C₃	1	8	yes	—
185	C₂²⋊C₄	C₂³×C₃	1	4	yes	—
186	C₂²⋊C₄	C₂³×C₃	1	8	yes	complemented
187	C₂²⋊C₄	C₂³×C₃	1	4	yes	complemented
188	C₄²	C₂³×C₃	1	4	yes	complemented
189	C₄²	C₂³×C₃	1	8	yes	complemented
190	C₂²×C₃	—	2	16	no	—
191	C₂²×C₃	—	2	4	no	—
192	C₂²×C₃	—	2	4	no	—
193	C₂²×C₃	—	2	4	no	—
194	C₂²×C₃	—	2	4	no	—
195	C₂²×C₃	—	2	2	no	—
196	C₂²×C₃	—	2	2	no	—
197	C₂²×C₃	—	4	16	no	—
198	C₂²×C₃	—	4	4	no	—
199	C₂²×C₃	—	4	4	no	—
200	C₂²×C₃	C₂×#41	1	8	yes	—
201	C₂²×C₃	C₂×#41	1	4	yes	—
202	C₂²×C₃	H(5,𝔽₂)	1	2	yes	—
203	C₂²×C₃	C₂⁵	1	1	yes	characteristic
204	C₃×C₄	—	2	4	no	—
205	C₃×C₄	—	2	4	no	—
206	C₃×C₄	—	2	4	no	—
207	C₃×C₄	—	2	4	no	—
208	C₃×C₄	—	2	4	no	—
209	C₃×C₄	—	2	2	no	—
210	C₂³	—	2	4	no	—
211	C₂³	—	2	4	no	—
212	C₂³	—	2	4	no	—
213	C₂³	—	2	16	no	—
214	C₂³	—	2	4	no	—
215	C₂³	—	2	2	no	—
216	C₂³	—	2	2	no	—
217	C₂³	—	4	4	no	—
218	C₂³	—	4	16	no	—
219	C₂³	C₃×#41	1	8	yes	—
220	C₂³	C₂⁴×C₃	1	2	yes	—
221	C₂³	C₂⁴×C₃	1	2	yes	—
222	C₂³	C₂⁴×C₃	1	4	yes	—
223	C₂³	C₂⁴×C₃	1	1	yes	characteristic
224	Q₈	—	4	4	no	—
225	D₄	—	2	16	no	—
226	D₄	—	4	4	no	—
227	C₂×C₄	—	2	16	no	—
228	C₂×C₄	—	2	4	no	—
229	C₂×C₄	—	2	4	no	—
230	C₂×C₄	—	2	4	no	—
231	C₂×C₄	—	2	4	no	—
232	C₂×C₄	—	2	4	no	—
233	C₂×C₄	—	2	2	no	—
234	C₂×C₄	—	4	8	no	—
235	C₂×C₄	—	4	4	no	—
236	C₂×C₄	C₂⁴×C₃	1	4	yes	—
237	C₂×C₄	C₂⁴×C₃	1	4	yes	—
238	C₂×C₄	C₂⁴×C₃	1	4	yes	—
239	C₂×C₄	C₂⁴×C₃	1	2	yes	—
240	C₂×C₄	C₂⁴×C₃	1	4	yes	—
241	C₂×C₄	C₂⁴×C₃	1	4	yes	—
242	C₂×C₃	—	2	8	no	—
243	C₂×C₃	—	4	2	no	—
244	C₂×C₃	—	4	2	no	—
245	C₂×C₃	#551	1	4	yes	direct product factor
246	C₂×C₃	C₂²×#41	1	2	yes	—
247	C₂×C₃	C₂×H(5,𝔽₂)	1	1	yes	characteristic
248	C₂²	—	2	4	no	—
249	C₂²	—	2	4	no	—
250	C₂²	—	2	16	no	—
251	C₂²	—	2	4	no	—
252	C₂²	—	2	4	no	—
253	C₂²	—	2	2	no	—
254	C₂²	—	2	2	no	—
255	C₂²	—	4	16	no	—
256	C₂²	—	4	4	no	—
257	C₂²	—	4	4	no	—
258	C₂²	C₂×C₃×#41	1	8	yes	—
259	C₂²	C₂×C₃×#41	1	4	yes	—
260	C₂²	C₃×H(5,𝔽₂)	1	2	yes	—
261	C₂²	C₂⁵×C₃	1	1	yes	characteristic
262	C₄	—	2	4	no	—
263	C₄	—	2	4	no	—
264	C₄	—	2	4	no	—
265	C₄	—	2	4	no	—
266	C₄	—	2	4	no	—
267	C₄	—	2	2	no	—
268	C₃	C₂×#551	1	1	yes	characteristic, direct product factor, Sylow
269	C₂	—	2	8	no	—
270	C₂	—	4	2	no	—
271	C₂	—	4	2	no	—
272	C₂	C₃×#551	1	4	yes	direct product factor
273	C₂	C₂²×C₃×#41	1	2	yes	—
274	C₂	C₂×C₃×H(5,𝔽₂)	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₃	C₂⁴
commutator, Frattini, cosocle	C₂²	C₂⁵×C₃
prime radical	C₂×C₃×#120	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₃×#551	C₂	direct (N × H)
2. C₂×C₃×#126	C₂	semidirect
3. C₂×C₃×#124	C₂	semidirect
4. C₂×C₃×#123	C₂	semidirect
5. C₂×C₃×#123	C₂	semidirect
6. C₂×C₃×C₂²⋊Q₈	C₂	semidirect
8. C₂×C₃×C₄×D₄	C₂	semidirect
9. C₂×C₃×#117	C₂	semidirect
10. C₂²×C₃×C₂²⋊C₄	C₂	semidirect
11. C₂×#551	C₃	direct (N × H)
13. C₂²×C₃×Q₈	C₂²	semidirect
18. C₂³×C₃×C₄	C₂²	semidirect
19. C₃×#126	C₂²	semidirect
20. C₃×#124	C₂²	semidirect
21. C₃×#123	C₂²	semidirect
22. C₃×#123	C₂²	semidirect
23. C₃×C₂²⋊Q₈	C₂²	semidirect
25. C₃×C₄×D₄	C₂²	semidirect
26. C₃×#117	C₂²	semidirect
27. C₂×C₃×C₄⋊C₄	C₂²	semidirect
28. C₂×C₃×C₄⋊C₄	C₂²	semidirect
29. C₂×C₃×C₄⋊C₄	C₂²	semidirect
30. C₂×C₃×C₄⋊C₄	C₂²	semidirect
37. C₂×C₃×C₂²⋊C₄	C₂²	semidirect
38. C₂×C₃×C₂²⋊C₄	C₂²	semidirect
39. C₂×C₃×C₂²⋊C₄	C₂²	semidirect
40. C₂×C₃×C₄²	C₂²	semidirect
41. C₂×C₃×C₄²	C₂²	semidirect
42. #551	C₂×C₃	direct (N × H)
43. C₂×#126	C₂×C₃	semidirect
44. C₂×#124	C₂×C₃	semidirect
45. C₂×#123	C₂×C₃	semidirect
46. C₂×#123	C₂×C₃	semidirect
47. C₂×C₂²⋊Q₈	C₂×C₃	semidirect
49. C₂×C₄×D₄	C₂×C₃	semidirect
50. C₂×#117	C₂×C₃	semidirect
51. C₂²×C₂²⋊C₄	C₂×C₃	semidirect
59. C₂×C₃×Q₈	C₂³	semidirect
75. C₂²×C₃×C₄	C₂³	semidirect
76. C₃×C₄⋊C₄	C₂³	semidirect
77. C₃×C₄⋊C₄	C₂³	semidirect
78. C₃×C₄⋊C₄	C₂³	semidirect
79. C₃×C₄⋊C₄	C₂³	semidirect
86. C₃×C₂²⋊C₄	C₂³	semidirect
87. C₃×C₂²⋊C₄	C₂³	semidirect
88. C₃×C₄²	C₂³	semidirect
89. C₃×C₄²	C₂³	semidirect
91. C₂²×Q₈	C₂²×C₃	semidirect
96. C₂³×C₄	C₂²×C₃	semidirect
97. #126	C₂²×C₃	semidirect
98. #124	C₂²×C₃	semidirect
99. #123	C₂²×C₃	semidirect
100. #123	C₂²×C₃	semidirect
101. C₂²⋊Q₈	C₂²×C₃	semidirect
103. C₄×D₄	C₂²×C₃	semidirect
104. #117	C₂²×C₃	semidirect
105. C₂×C₄⋊C₄	C₂²×C₃	semidirect
106. C₂×C₄⋊C₄	C₂²×C₃	semidirect
107. C₂×C₄⋊C₄	C₂²×C₃	semidirect
108. C₂×C₄⋊C₄	C₂²×C₃	semidirect
115. C₂×C₂²⋊C₄	C₂²×C₃	semidirect
116. C₂×C₂²⋊C₄	C₂²×C₃	semidirect
117. C₂×C₂²⋊C₄	C₂²×C₃	semidirect
118. C₂×C₄²	C₂²×C₃	semidirect
119. C₂×C₄²	C₂²×C₃	semidirect
159. C₂×Q₈	C₂³×C₃	semidirect
175. C₂²×C₄	C₂³×C₃	semidirect
176. C₄⋊C₄	C₂³×C₃	semidirect
177. C₄⋊C₄	C₂³×C₃	semidirect
178. C₄⋊C₄	C₂³×C₃	semidirect
179. C₄⋊C₄	C₂³×C₃	semidirect
186. C₂²⋊C₄	C₂³×C₃	semidirect
187. C₂²⋊C₄	C₂³×C₃	semidirect
188. C₄²	C₂³×C₃	semidirect
189. C₄²	C₂³×C₃	semidirect
245. C₂×C₃	#551	direct (N × H)
268. C₃	C₂×#551	direct (N × H)
272. C₂	C₃×#551	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×#551	2
C₂×C₃×#126	2
C₂×C₃×#124	2
C₂×C₃×#123	2
C₂×C₃×C₂²⋊Q₈	2
C₂×C₃×#120	2
C₂×C₃×C₄×D₄	2
C₂×C₃×#117	2
C₂²×C₃×C₂²⋊C₄	2
C₂×#551	3

Subgroups — C2×C3×#551 #87353

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×#551 #87353