Subgroups — C₂×C₃×#547 #87348

Order 384.

1632 conjugacy classes of subgroups.

The subgroup lattice has 240 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃×#41	C₂	1	2	yes	maximal, complemented
2	C₃×#547	C₂	1	16	yes	maximal, direct product factor
3	C₂×C₃×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₃×#123	C₂	1	4	yes	maximal, complemented
5	C₂×C₃×C₂²⋊Q₈	C₂	1	2	yes	maximal, complemented
6	C₂×C₃×#121	C₂	1	2	yes	maximal, complemented
7	C₂×C₃×C₄×D₄	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₃×C₄×D₄	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×C₃×C₄⋊C₄	C₂	1	2	yes	maximal, complemented
10	C₂×#547	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
11	C₂×C₃×#41	—	2	16	no	—
12	C₂×C₃×#41	C₂²	1	16	yes	complemented
13	C₂²×C₃×Q₈	C₂²	1	2	yes	—
14	C₂²×C₃×D₄	C₂²	1	4	yes	—
15	C₂²×C₃×D₄	C₂²	1	1	yes	characteristic
16	C₂²×C₃×D₄	C₂²	1	1	yes	characteristic
17	C₂³×C₃×C₄	C₂²	1	4	yes	—
18	C₂³×C₃×C₄	C₂²	1	2	yes	—
19	C₂³×C₃×C₄	C₂²	1	2	yes	—
20	C₃×C₄⋊Q₈	C₂²	1	8	yes	complemented
21	C₃×#123	C₂²	1	32	yes	complemented
22	C₃×C₂²⋊Q₈	C₂²	1	16	yes	complemented
23	C₃×#121	C₂²	1	16	yes	complemented
24	C₃×C₄×D₄	C₂²	1	8	yes	complemented
25	C₃×C₄×D₄	C₂²	1	8	yes	complemented
26	C₂×C₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
27	C₂×C₃×C₄⋊C₄	C₂²	1	2	yes	—
28	C₂×C₃×C₄⋊C₄	C₂²	1	4	yes	—
29	C₂×C₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
30	C₂×C₃×C₄⋊C₄	C₂²	1	2	yes	—
31	C₂×C₃×C₄⋊C₄	C₂²	1	16	yes	complemented
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	4	yes	—
35	C₂×C₃×C₄²	C₂²	1	1	yes	characteristic
36	C₂²×#41	C₂×C₃	1	2	yes	complemented
37	#547	C₂×C₃	1	16	yes	direct product factor
38	C₂×C₄⋊Q₈	C₂×C₃	1	1	yes	characteristic, complemented
39	C₂×#123	C₂×C₃	1	4	yes	complemented
40	C₂×C₂²⋊Q₈	C₂×C₃	1	2	yes	complemented
41	C₂×#121	C₂×C₃	1	2	yes	complemented
42	C₂×C₄×D₄	C₂×C₃	1	1	yes	characteristic, complemented
43	C₂×C₄×D₄	C₂×C₃	1	1	yes	characteristic, complemented
44	C₂²×C₄⋊C₄	C₂×C₃	1	2	yes	complemented
45	C₂⁴×C₃	C₂³	1	2	yes	—
46	C₂⁴×C₃	C₂³	1	2	yes	—
47	C₃×#41	—	2	64	no	—
48	C₂×C₃×Q₈	—	2	8	no	—
49	C₂×C₃×Q₈	C₂³	1	8	yes	—
50	C₂×C₃×D₄	—	2	16	no	—
51	C₂×C₃×D₄	—	2	4	no	—
52	C₂×C₃×D₄	D₄	1	8	yes	complemented
53	C₂×C₃×D₄	C₂³	1	4	yes	—
54	C₂×C₃×D₄	C₂³	1	16	yes	—
55	C₂×C₃×D₄	C₂³	1	4	yes	—
56	C₂²×C₃×C₄	—	2	16	no	—
57	C₂²×C₃×C₄	—	2	8	no	—
58	C₂²×C₃×C₄	—	2	8	no	—
59	C₂²×C₃×C₄	C₂³	1	8	yes	—
60	C₂²×C₃×C₄	C₂³	1	16	yes	—
61	C₂²×C₃×C₄	C₂³	1	8	yes	—
62	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
63	C₂²×C₃×C₄	C₂³	1	2	yes	—
64	C₂²×C₃×C₄	C₂³	1	2	yes	—
65	C₂²×C₃×C₄	C₂³	1	4	yes	—
66	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
67	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
68	C₃×C₄⋊C₄	C₂³	1	4	yes	—
69	C₃×C₄⋊C₄	C₂³	1	8	yes	—
70	C₃×C₄⋊C₄	C₂³	1	16	yes	—
71	C₃×C₄⋊C₄	C₂³	1	4	yes	—
72	C₃×C₄⋊C₄	C₂³	1	8	yes	—
73	C₃×C₂²⋊C₄	C₂³	1	8	yes	—
74	C₃×C₂²⋊C₄	C₂³	1	8	yes	—
75	C₃×C₂²⋊C₄	C₂³	1	16	yes	—
76	C₃×C₄²	C₂³	1	4	yes	—
77	C₂×#41	—	2	16	no	—
78	C₂×#41	C₂²×C₃	1	16	yes	complemented
79	C₂²×Q₈	C₂²×C₃	1	2	yes	—
80	C₂²×D₄	C₂²×C₃	1	1	yes	characteristic
81	C₂²×D₄	C₂²×C₃	1	4	yes	—
82	C₂²×D₄	C₂²×C₃	1	1	yes	characteristic
83	C₂³×C₄	C₂²×C₃	1	2	yes	—
84	C₂³×C₄	C₂²×C₃	1	4	yes	—
85	C₂³×C₄	C₂²×C₃	1	2	yes	—
86	C₄⋊Q₈	C₂²×C₃	1	8	yes	complemented
87	#123	C₂²×C₃	1	32	yes	complemented
88	C₂²⋊Q₈	C₂²×C₃	1	16	yes	complemented
89	#121	C₂²×C₃	1	16	yes	complemented
90	C₄×D₄	C₂²×C₃	1	8	yes	complemented
91	C₄×D₄	C₂²×C₃	1	8	yes	complemented
92	C₂×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
93	C₂×C₄⋊C₄	C₂²×C₃	1	2	yes	—
94	C₂×C₄⋊C₄	C₂²×C₃	1	4	yes	—
95	C₂×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
96	C₂×C₄⋊C₄	C₂²×C₃	1	2	yes	—
97	C₂×C₄⋊C₄	C₂²×C₃	1	16	yes	complemented
98	C₂×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
99	C₂×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
100	C₂×C₂²⋊C₄	C₂²×C₃	1	4	yes	—
101	C₂×C₄²	C₂²×C₃	1	1	yes	characteristic
102	C₂³×C₃	—	2	4	no	—
103	C₂³×C₃	—	2	4	no	—
104	C₂³×C₃	—	2	4	no	—
105	C₂³×C₃	—	2	4	no	—
106	C₂³×C₃	—	2	4	no	—
107	C₂³×C₃	C₂×D₄	1	8	yes	—
108	C₂³×C₃	C₂⁴	1	4	yes	—
109	C₂³×C₃	C₂⁴	1	4	yes	—
110	C₂³×C₃	C₂⁴	1	1	yes	characteristic
111	C₃×Q₈	—	2	16	no	—
112	C₃×D₄	—	2	32	no	—
113	C₃×D₄	—	2	8	no	—
114	C₃×D₄	C₂×D₄	1	16	yes	complemented
115	C₂×C₃×C₄	—	2	4	no	—
116	C₂×C₃×C₄	—	2	8	no	—
117	C₂×C₃×C₄	—	2	32	no	—
118	C₂×C₃×C₄	—	2	4	no	—
119	C₂×C₃×C₄	—	2	2	no	—
120	C₂×C₃×C₄	—	2	16	no	—
121	C₂×C₃×C₄	—	2	16	no	—
122	C₂×C₃×C₄	C₂×D₄	1	4	yes	—
123	C₂×C₃×C₄	#41	1	4	yes	complemented
124	C₂×C₃×C₄	C₂⁴	1	2	yes	—
125	C₂×C₃×C₄	C₂⁴	1	4	yes	—
126	C₂×C₃×C₄	C₂⁴	1	4	yes	—
127	C₂×C₃×C₄	C₂⁴	1	8	yes	—
128	C₂×C₃×C₄	C₂⁴	1	2	yes	—
129	C₂×C₃×C₄	C₂⁴	1	2	yes	—
130	C₂⁴	C₂³×C₃	1	2	yes	—
131	C₂⁴	C₂³×C₃	1	2	yes	—
132	#41	—	2	64	no	—
133	C₂×Q₈	—	2	8	no	—
134	C₂×Q₈	C₂³×C₃	1	8	yes	—
135	C₂×D₄	—	2	16	no	—
136	C₂×D₄	—	2	4	no	—
137	C₂×D₄	C₃×D₄	1	8	yes	complemented
138	C₂×D₄	C₂³×C₃	1	4	yes	—
139	C₂×D₄	C₂³×C₃	1	16	yes	—
140	C₂×D₄	C₂³×C₃	1	4	yes	—
141	C₂²×C₄	—	2	16	no	—
142	C₂²×C₄	—	2	8	no	—
143	C₂²×C₄	—	2	8	no	—
144	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
145	C₂²×C₄	C₂³×C₃	1	2	yes	—
146	C₂²×C₄	C₂³×C₃	1	2	yes	—
147	C₂²×C₄	C₂³×C₃	1	4	yes	—
148	C₂²×C₄	C₂³×C₃	1	8	yes	—
149	C₂²×C₄	C₂³×C₃	1	16	yes	—
150	C₂²×C₄	C₂³×C₃	1	8	yes	—
151	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
152	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
153	C₄⋊C₄	C₂³×C₃	1	4	yes	—
154	C₄⋊C₄	C₂³×C₃	1	8	yes	—
155	C₄⋊C₄	C₂³×C₃	1	16	yes	—
156	C₄⋊C₄	C₂³×C₃	1	4	yes	—
157	C₄⋊C₄	C₂³×C₃	1	8	yes	—
158	C₂²⋊C₄	C₂³×C₃	1	8	yes	—
159	C₂²⋊C₄	C₂³×C₃	1	8	yes	—
160	C₂²⋊C₄	C₂³×C₃	1	16	yes	—
161	C₄²	C₂³×C₃	1	4	yes	—
162	C₂²×C₃	—	2	16	no	—
163	C₂²×C₃	—	2	4	no	—
164	C₂²×C₃	—	2	4	no	—
165	C₂²×C₃	—	2	4	no	—
166	C₂²×C₃	—	2	4	no	—
167	C₂²×C₃	—	2	4	no	—
168	C₂²×C₃	—	4	8	no	—
169	C₂²×C₃	C₂²×D₄	1	8	yes	—
170	C₂²×C₃	C₂²×D₄	1	2	yes	—
171	C₂²×C₃	C₂×#41	1	2	yes	—
172	C₂²×C₃	2₋¹⁺⁴	1	2	yes	—
173	C₂²×C₃	C₂⁵	1	1	yes	characteristic
174	C₃×C₄	—	2	4	no	—
175	C₃×C₄	—	2	8	no	—
176	C₃×C₄	—	2	4	no	—
177	C₃×C₄	—	2	2	no	—
178	C₃×C₄	C₂²×D₄	1	4	yes	—
179	C₃×C₄	C₂×#41	1	4	yes	complemented
180	C₂³	—	2	4	no	—
181	C₂³	—	2	4	no	—
182	C₂³	—	2	4	no	—
183	C₂³	—	2	4	no	—
184	C₂³	—	2	4	no	—
185	C₂³	C₂×C₃×D₄	1	8	yes	—
186	C₂³	C₂⁴×C₃	1	4	yes	—
187	C₂³	C₂⁴×C₃	1	4	yes	—
188	C₂³	C₂⁴×C₃	1	1	yes	characteristic
189	Q₈	—	2	16	no	—
190	D₄	—	2	32	no	—
191	D₄	—	2	8	no	—
192	D₄	C₂×C₃×D₄	1	16	yes	complemented
193	C₂×C₄	—	2	4	no	—
194	C₂×C₄	—	2	8	no	—
195	C₂×C₄	—	2	32	no	—
196	C₂×C₄	—	2	4	no	—
197	C₂×C₄	—	2	2	no	—
198	C₂×C₄	—	2	16	no	—
199	C₂×C₄	—	2	16	no	—
200	C₂×C₄	C₂×C₃×D₄	1	4	yes	—
201	C₂×C₄	C₃×#41	1	4	yes	complemented
202	C₂×C₄	C₂⁴×C₃	1	2	yes	—
203	C₂×C₄	C₂⁴×C₃	1	4	yes	—
204	C₂×C₄	C₂⁴×C₃	1	4	yes	—
205	C₂×C₄	C₂⁴×C₃	1	8	yes	—
206	C₂×C₄	C₂⁴×C₃	1	2	yes	—
207	C₂×C₄	C₂⁴×C₃	1	2	yes	—
208	C₂×C₃	—	2	8	no	—
209	C₂×C₃	—	4	4	no	—
210	C₂×C₃	#547	1	4	yes	direct product factor
211	C₂×C₃	C₂³×D₄	1	1	yes	characteristic
212	C₂×C₃	C₂²×#41	1	1	yes	characteristic
213	C₂×C₃	C₂×2₋¹⁺⁴	1	1	yes	characteristic
214	C₂²	—	2	4	no	—
215	C₂²	—	2	4	no	—
216	C₂²	—	2	16	no	—
217	C₂²	—	2	4	no	—
218	C₂²	—	2	4	no	—
219	C₂²	—	2	4	no	—
220	C₂²	—	4	8	no	—
221	C₂²	C₂²×C₃×D₄	1	8	yes	—
222	C₂²	C₂²×C₃×D₄	1	2	yes	—
223	C₂²	C₂×C₃×#41	1	2	yes	—
224	C₂²	C₃×2₋¹⁺⁴	1	2	yes	—
225	C₂²	C₂⁵×C₃	1	1	yes	characteristic
226	C₄	—	2	4	no	—
227	C₄	—	2	8	no	—
228	C₄	—	2	4	no	—
229	C₄	—	2	2	no	—
230	C₄	C₂²×C₃×D₄	1	4	yes	—
231	C₄	C₂×C₃×#41	1	4	yes	complemented
232	C₃	C₂×#547	1	1	yes	characteristic, direct product factor, Sylow
233	C₂	—	2	8	no	—
234	C₂	—	4	4	no	—
235	C₂	C₃×#547	1	4	yes	direct product factor
236	C₂	C₂³×C₃×D₄	1	1	yes	characteristic
237	C₂	C₂²×C₃×#41	1	1	yes	characteristic
238	C₂	C₂×C₃×2₋¹⁺⁴	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₃	C₂⁴
commutator, Frattini, cosocle	C₂²	C₂⁵×C₃
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. C₂²×C₃×#41	C₂	semidirect
2. C₃×#547	C₂	direct (N × H)
3. C₂×C₃×C₄⋊Q₈	C₂	semidirect
4. C₂×C₃×#123	C₂	semidirect
5. C₂×C₃×C₂²⋊Q₈	C₂	semidirect
6. C₂×C₃×#121	C₂	semidirect
7. C₂×C₃×C₄×D₄	C₂	semidirect
8. C₂×C₃×C₄×D₄	C₂	semidirect
9. C₂²×C₃×C₄⋊C₄	C₂	semidirect
10. C₂×#547	C₃	direct (N × H)
12. C₂×C₃×#41	C₂²	semidirect
20. C₃×C₄⋊Q₈	C₂²	semidirect
21. C₃×#123	C₂²	semidirect
22. C₃×C₂²⋊Q₈	C₂²	semidirect
23. C₃×#121	C₂²	semidirect
24. C₃×C₄×D₄	C₂²	semidirect
25. C₃×C₄×D₄	C₂²	semidirect
31. C₂×C₃×C₄⋊C₄	C₂²	semidirect
36. C₂²×#41	C₂×C₃	semidirect
37. #547	C₂×C₃	direct (N × H)
38. C₂×C₄⋊Q₈	C₂×C₃	semidirect
39. C₂×#123	C₂×C₃	semidirect
40. C₂×C₂²⋊Q₈	C₂×C₃	semidirect
41. C₂×#121	C₂×C₃	semidirect
42. C₂×C₄×D₄	C₂×C₃	semidirect
43. C₂×C₄×D₄	C₂×C₃	semidirect
44. C₂²×C₄⋊C₄	C₂×C₃	semidirect
52. C₂×C₃×D₄	D₄	semidirect
78. C₂×#41	C₂²×C₃	semidirect
86. C₄⋊Q₈	C₂²×C₃	semidirect
87. #123	C₂²×C₃	semidirect
88. C₂²⋊Q₈	C₂²×C₃	semidirect
89. #121	C₂²×C₃	semidirect
90. C₄×D₄	C₂²×C₃	semidirect
91. C₄×D₄	C₂²×C₃	semidirect
97. C₂×C₄⋊C₄	C₂²×C₃	semidirect
114. C₃×D₄	C₂×D₄	semidirect
123. C₂×C₃×C₄	#41	semidirect
137. C₂×D₄	C₃×D₄	semidirect
179. C₃×C₄	C₂×#41	semidirect
192. D₄	C₂×C₃×D₄	semidirect
201. C₂×C₄	C₃×#41	semidirect
210. C₂×C₃	#547	direct (N × H)
231. C₄	C₂×C₃×#41	semidirect
232. C₃	C₂×#547	direct (N × H)
235. C₂	C₃×#547	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₃×#41	2
C₃×#547	2
C₂×C₃×C₄⋊Q₈	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×#547	3

Subgroups — C2×C3×#547 #87348

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×#547 #87348