Subgroups — C₂×C₃×#539 #87335

Order 384.

1144 conjugacy classes of subgroups.

The subgroup lattice has 268 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#539	C₂	1	16	yes	maximal, direct product factor
2	C₂×C₃×#126	C₂	1	2	yes	maximal, complemented
3	C₂×C₃×#125	C₂	1	2	yes	maximal, complemented
4	C₂×C₃×#123	C₂	1	2	yes	maximal, 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×C₂²⋊Q₈	C₂	1	2	yes	maximal, complemented
8	C₂×C₃×#121	C₂	1	1	yes	maximal, characteristic
9	C₂×C₃×C₄×D₄	C₂	1	2	yes	maximal, complemented
10	C₂²×C₃×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×#539	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
12	C₂²×C₃×Q₈	C₂²	1	1	yes	characteristic
13	C₂²×C₃×D₄	C₂²	1	2	yes	—
14	C₂²×C₃×D₄	C₂²	1	1	yes	characteristic
15	C₂³×C₃×C₄	C₂²	1	1	yes	characteristic
16	C₂³×C₃×C₄	C₂²	1	1	yes	characteristic
17	C₂³×C₃×C₄	C₂²	1	1	yes	characteristic
18	C₂³×C₃×C₄	C₂²	1	2	yes	—
19	C₃×#126	C₂²	1	16	yes	complemented
20	C₃×#125	C₂²	1	16	yes	complemented
21	C₃×#123	C₂²	1	16	yes	complemented
22	C₃×#123	C₂²	1	16	yes	complemented
23	C₃×C₂²⋊Q₈	C₂²	1	8	yes	complemented
24	C₃×C₂²⋊Q₈	C₂²	1	16	yes	complemented
25	C₃×#121	C₂²	1	8	yes	—
26	C₃×C₄×D₄	C₂²	1	16	yes	complemented
27	C₂×C₃×C₄⋊C₄	C₂²	1	2	yes	—
28	C₂×C₃×C₄⋊C₄	C₂²	1	2	yes	—
29	C₂×C₃×C₄⋊C₄	C₂²	1	2	yes	—
30	C₂×C₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
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	8	yes	complemented
36	C₂×C₃×C₂²⋊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×C₄²	C₂²	1	2	yes	—
42	#539	C₂×C₃	1	16	yes	direct product factor
43	C₂×#126	C₂×C₃	1	2	yes	complemented
44	C₂×#125	C₂×C₃	1	2	yes	complemented
45	C₂×#123	C₂×C₃	1	2	yes	complemented
46	C₂×#123	C₂×C₃	1	2	yes	complemented
47	C₂×C₂²⋊Q₈	C₂×C₃	1	1	yes	characteristic, complemented
48	C₂×C₂²⋊Q₈	C₂×C₃	1	2	yes	complemented
49	C₂×#121	C₂×C₃	1	1	yes	characteristic
50	C₂×C₄×D₄	C₂×C₃	1	2	yes	complemented
51	C₂²×C₄⋊C₄	C₂×C₃	1	1	yes	characteristic, complemented
52	C₂⁴×C₃	C₂³	1	2	yes	—
53	C₂⁴×C₃	C₂³	1	1	yes	characteristic
54	C₂×C₃×Q₈	—	4	2	no	—
55	C₂×C₃×Q₈	C₂³	1	4	yes	—
56	C₂×C₃×D₄	—	2	8	no	—
57	C₂×C₃×D₄	—	4	2	no	—
58	C₂×C₃×D₄	C₂³	1	8	yes	—
59	C₂×C₃×D₄	C₂³	1	4	yes	—
60	C₂²×C₃×C₄	—	2	4	no	—
61	C₂²×C₃×C₄	—	4	2	no	—
62	C₂²×C₃×C₄	—	4	4	no	—
63	C₂²×C₃×C₄	—	4	2	no	—
64	C₂²×C₃×C₄	C₂³	1	4	yes	—
65	C₂²×C₃×C₄	C₂³	1	2	yes	—
66	C₂²×C₃×C₄	C₂³	1	2	yes	—
67	C₂²×C₃×C₄	C₂³	1	2	yes	—
68	C₂²×C₃×C₄	C₂³	1	2	yes	—
69	C₂²×C₃×C₄	C₂³	1	2	yes	—
70	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
71	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
72	C₂²×C₃×C₄	C₂³	1	4	yes	—
73	C₂²×C₃×C₄	C₂³	1	4	yes	—
74	C₂²×C₃×C₄	C₂³	1	8	yes	—
75	C₃×C₄⋊C₄	C₂³	1	8	yes	—
76	C₃×C₄⋊C₄	C₂³	1	8	yes	—
77	C₃×C₄⋊C₄	C₂³	1	8	yes	—
78	C₃×C₄⋊C₄	C₂³	1	4	yes	—
79	C₃×C₄⋊C₄	C₂³	1	8	yes	—
80	C₃×C₄⋊C₄	C₂³	1	8	yes	—
81	C₃×C₄⋊C₄	C₂³	1	4	yes	—
82	C₃×C₄⋊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	8	yes	—
86	C₃×C₂²⋊C₄	C₂³	1	8	yes	—
87	C₃×C₂²⋊C₄	C₂³	1	8	yes	—
88	C₃×C₄²	C₂³	1	8	yes	—
89	C₂²×Q₈	C₂²×C₃	1	1	yes	characteristic
90	C₂²×D₄	C₂²×C₃	1	2	yes	—
91	C₂²×D₄	C₂²×C₃	1	1	yes	characteristic
92	C₂³×C₄	C₂²×C₃	1	1	yes	characteristic
93	C₂³×C₄	C₂²×C₃	1	1	yes	characteristic
94	C₂³×C₄	C₂²×C₃	1	1	yes	characteristic
95	C₂³×C₄	C₂²×C₃	1	2	yes	—
96	#126	C₂²×C₃	1	16	yes	complemented
97	#125	C₂²×C₃	1	16	yes	complemented
98	#123	C₂²×C₃	1	16	yes	complemented
99	#123	C₂²×C₃	1	16	yes	complemented
100	C₂²⋊Q₈	C₂²×C₃	1	8	yes	complemented
101	C₂²⋊Q₈	C₂²×C₃	1	16	yes	complemented
102	#121	C₂²×C₃	1	8	yes	—
103	C₄×D₄	C₂²×C₃	1	16	yes	complemented
104	C₂×C₄⋊C₄	C₂²×C₃	1	2	yes	—
105	C₂×C₄⋊C₄	C₂²×C₃	1	2	yes	—
106	C₂×C₄⋊C₄	C₂²×C₃	1	2	yes	—
107	C₂×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
108	C₂×C₄⋊C₄	C₂²×C₃	1	2	yes	—
109	C₂×C₄⋊C₄	C₂²×C₃	1	2	yes	—
110	C₂×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
111	C₂×C₄⋊C₄	C₂²×C₃	1	2	yes	—
112	C₂×C₄⋊C₄	C₂²×C₃	1	8	yes	complemented
113	C₂×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
114	C₂×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
115	C₂×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
116	C₂×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
117	C₂×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
118	C₂×C₄²	C₂²×C₃	1	2	yes	—
119	C₂³×C₃	—	2	2	no	—
120	C₂³×C₃	—	2	2	no	—
121	C₂³×C₃	—	2	4	no	—
122	C₂³×C₃	—	2	4	no	—
123	C₂³×C₃	—	2	4	no	—
124	C₂³×C₃	#41	1	4	yes	—
125	C₂³×C₃	C₂⁴	1	4	yes	—
126	C₂³×C₃	C₂⁴	1	2	yes	—
127	C₂³×C₃	C₂⁴	1	1	yes	characteristic
128	C₃×Q₈	—	4	4	no	—
129	C₃×D₄	—	2	16	no	—
130	C₃×D₄	—	4	4	no	—
131	C₂×C₃×C₄	—	2	8	no	—
132	C₂×C₃×C₄	—	2	4	no	—
133	C₂×C₃×C₄	—	2	4	no	—
134	C₂×C₃×C₄	—	2	4	no	—
135	C₂×C₃×C₄	—	2	2	no	—
136	C₂×C₃×C₄	—	2	4	no	—
137	C₂×C₃×C₄	—	2	2	no	—
138	C₂×C₃×C₄	—	2	4	no	—
139	C₂×C₃×C₄	—	4	4	no	—
140	C₂×C₃×C₄	—	4	8	no	—
141	C₂×C₃×C₄	—	4	4	no	—
142	C₂×C₃×C₄	C₂⁴	1	4	yes	—
143	C₂×C₃×C₄	C₂⁴	1	4	yes	—
144	C₂×C₃×C₄	C₂⁴	1	4	yes	—
145	C₂×C₃×C₄	C₂⁴	1	4	yes	—
146	C₂×C₃×C₄	C₂⁴	1	4	yes	—
147	C₂×C₃×C₄	C₂⁴	1	2	yes	—
148	C₂×C₃×C₄	C₂⁴	1	2	yes	—
149	C₂⁴	C₂³×C₃	1	2	yes	—
150	C₂⁴	C₂³×C₃	1	1	yes	characteristic
151	C₂×Q₈	—	4	2	no	—
152	C₂×Q₈	C₂³×C₃	1	4	yes	—
153	C₂×D₄	—	2	8	no	—
154	C₂×D₄	—	4	2	no	—
155	C₂×D₄	C₂³×C₃	1	8	yes	—
156	C₂×D₄	C₂³×C₃	1	4	yes	—
157	C₂²×C₄	—	2	4	no	—
158	C₂²×C₄	—	4	2	no	—
159	C₂²×C₄	—	4	4	no	—
160	C₂²×C₄	—	4	2	no	—
161	C₂²×C₄	C₂³×C₃	1	2	yes	—
162	C₂²×C₄	C₂³×C₃	1	2	yes	—
163	C₂²×C₄	C₂³×C₃	1	2	yes	—
164	C₂²×C₄	C₂³×C₃	1	2	yes	—
165	C₂²×C₄	C₂³×C₃	1	2	yes	—
166	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
167	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
168	C₂²×C₄	C₂³×C₃	1	4	yes	—
169	C₂²×C₄	C₂³×C₃	1	4	yes	—
170	C₂²×C₄	C₂³×C₃	1	4	yes	—
171	C₂²×C₄	C₂³×C₃	1	8	yes	—
172	C₄⋊C₄	C₂³×C₃	1	8	yes	—
173	C₄⋊C₄	C₂³×C₃	1	8	yes	—
174	C₄⋊C₄	C₂³×C₃	1	8	yes	—
175	C₄⋊C₄	C₂³×C₃	1	4	yes	—
176	C₄⋊C₄	C₂³×C₃	1	8	yes	—
177	C₄⋊C₄	C₂³×C₃	1	8	yes	—
178	C₄⋊C₄	C₂³×C₃	1	4	yes	—
179	C₄⋊C₄	C₂³×C₃	1	8	yes	—
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	C₄²	C₂³×C₃	1	8	yes	—
186	C₂²×C₃	—	2	8	no	—
187	C₂²×C₃	—	2	2	no	—
188	C₂²×C₃	—	2	2	no	—
189	C₂²×C₃	—	2	4	no	—
190	C₂²×C₃	—	2	4	no	—
191	C₂²×C₃	—	2	4	no	—
192	C₂²×C₃	—	4	8	no	—
193	C₂²×C₃	C₂×#41	1	4	yes	—
194	C₂²×C₃	C₂×#41	1	2	yes	—
195	C₂²×C₃	H(5,𝔽₂)	1	2	yes	—
196	C₂²×C₃	2₋¹⁺⁴	1	2	yes	—
197	C₂²×C₃	C₂⁵	1	1	yes	characteristic
198	C₃×C₄	—	2	4	no	—
199	C₃×C₄	—	2	4	no	—
200	C₃×C₄	—	2	4	no	—
201	C₃×C₄	—	2	2	no	—
202	C₃×C₄	—	2	4	no	—
203	C₃×C₄	—	2	2	no	—
204	C₃×C₄	—	2	4	no	—
205	C₂³	—	2	2	no	—
206	C₂³	—	2	2	no	—
207	C₂³	—	2	4	no	—
208	C₂³	—	2	4	no	—
209	C₂³	—	2	4	no	—
210	C₂³	C₃×#41	1	4	yes	—
211	C₂³	C₂⁴×C₃	1	4	yes	—
212	C₂³	C₂⁴×C₃	1	2	yes	—
213	C₂³	C₂⁴×C₃	1	1	yes	characteristic
214	Q₈	—	4	4	no	—
215	D₄	—	2	16	no	—
216	D₄	—	4	4	no	—
217	C₂×C₄	—	2	8	no	—
218	C₂×C₄	—	2	4	no	—
219	C₂×C₄	—	2	4	no	—
220	C₂×C₄	—	2	4	no	—
221	C₂×C₄	—	2	2	no	—
222	C₂×C₄	—	2	4	no	—
223	C₂×C₄	—	2	2	no	—
224	C₂×C₄	—	2	4	no	—
225	C₂×C₄	—	4	4	no	—
226	C₂×C₄	—	4	8	no	—
227	C₂×C₄	—	4	4	no	—
228	C₂×C₄	C₂⁴×C₃	1	4	yes	—
229	C₂×C₄	C₂⁴×C₃	1	4	yes	—
230	C₂×C₄	C₂⁴×C₃	1	4	yes	—
231	C₂×C₄	C₂⁴×C₃	1	4	yes	—
232	C₂×C₄	C₂⁴×C₃	1	4	yes	—
233	C₂×C₄	C₂⁴×C₃	1	2	yes	—
234	C₂×C₄	C₂⁴×C₃	1	2	yes	—
235	C₂×C₃	—	2	4	no	—
236	C₂×C₃	—	4	4	no	—
237	C₂×C₃	#539	1	4	yes	direct product factor
238	C₂×C₃	C₂²×#41	1	1	yes	characteristic
239	C₂×C₃	C₂×H(5,𝔽₂)	1	1	yes	characteristic
240	C₂×C₃	C₂×2₋¹⁺⁴	1	1	yes	characteristic
241	C₂²	—	2	2	no	—
242	C₂²	—	2	2	no	—
243	C₂²	—	2	8	no	—
244	C₂²	—	2	4	no	—
245	C₂²	—	2	4	no	—
246	C₂²	—	2	4	no	—
247	C₂²	—	4	8	no	—
248	C₂²	C₂×C₃×#41	1	4	yes	—
249	C₂²	C₂×C₃×#41	1	2	yes	—
250	C₂²	C₃×H(5,𝔽₂)	1	2	yes	—
251	C₂²	C₃×2₋¹⁺⁴	1	2	yes	—
252	C₂²	C₂⁵×C₃	1	1	yes	characteristic
253	C₄	—	2	4	no	—
254	C₄	—	2	4	no	—
255	C₄	—	2	4	no	—
256	C₄	—	2	2	no	—
257	C₄	—	2	4	no	—
258	C₄	—	2	2	no	—
259	C₄	—	2	4	no	—
260	C₃	C₂×#539	1	1	yes	characteristic, direct product factor, Sylow
261	C₂	—	2	4	no	—
262	C₂	—	4	4	no	—
263	C₂	C₃×#539	1	4	yes	direct product factor
264	C₂	C₂²×C₃×#41	1	1	yes	characteristic
265	C₂	C₂×C₃×H(5,𝔽₂)	1	1	yes	characteristic
266	C₂	C₂×C₃×2₋¹⁺⁴	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₃	C₂⁴
commutator, Frattini, cosocle	C₂²	C₂⁵×C₃
prime radical	C₂×C₃×#121	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×#539	C₂	direct (N × H)
2. C₂×C₃×#126	C₂	semidirect
3. C₂×C₃×#125	C₂	semidirect
4. C₂×C₃×#123	C₂	semidirect
5. C₂×C₃×#123	C₂	semidirect
6. C₂×C₃×C₂²⋊Q₈	C₂	semidirect
7. C₂×C₃×C₂²⋊Q₈	C₂	semidirect
9. C₂×C₃×C₄×D₄	C₂	semidirect
10. C₂²×C₃×C₄⋊C₄	C₂	semidirect
11. C₂×#539	C₃	direct (N × H)
19. C₃×#126	C₂²	semidirect
20. C₃×#125	C₂²	semidirect
21. C₃×#123	C₂²	semidirect
22. C₃×#123	C₂²	semidirect
23. C₃×C₂²⋊Q₈	C₂²	semidirect
24. C₃×C₂²⋊Q₈	C₂²	semidirect
26. C₃×C₄×D₄	C₂²	semidirect
35. C₂×C₃×C₄⋊C₄	C₂²	semidirect
42. #539	C₂×C₃	direct (N × H)
43. C₂×#126	C₂×C₃	semidirect
44. C₂×#125	C₂×C₃	semidirect
45. C₂×#123	C₂×C₃	semidirect
46. C₂×#123	C₂×C₃	semidirect
47. C₂×C₂²⋊Q₈	C₂×C₃	semidirect
48. C₂×C₂²⋊Q₈	C₂×C₃	semidirect
50. C₂×C₄×D₄	C₂×C₃	semidirect
51. C₂²×C₄⋊C₄	C₂×C₃	semidirect
96. #126	C₂²×C₃	semidirect
97. #125	C₂²×C₃	semidirect
98. #123	C₂²×C₃	semidirect
99. #123	C₂²×C₃	semidirect
100. C₂²⋊Q₈	C₂²×C₃	semidirect
101. C₂²⋊Q₈	C₂²×C₃	semidirect
103. C₄×D₄	C₂²×C₃	semidirect
112. C₂×C₄⋊C₄	C₂²×C₃	semidirect
237. C₂×C₃	#539	direct (N × H)
260. C₃	C₂×#539	direct (N × H)
263. C₂	C₃×#539	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×#539	2
C₂×C₃×#126	2
C₂×C₃×#125	2
C₂×C₃×#123	2
C₂×C₃×C₂²⋊Q₈	2
C₂×C₃×#121	2
C₂×C₃×C₄×D₄	2
C₂²×C₃×C₄⋊C₄	2
C₂×#539	3

Subgroups — C2×C3×#539 #87335

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×#539 #87335