Subgroups — C₂×C₃×#548 #87349

Order 384.

1620 conjugacy classes of subgroups.

The subgroup lattice has 212 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	1	yes	maximal, characteristic, complemented
2	C₂³×C₃×Q₈	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×#548	C₂	1	16	yes	maximal, direct product factor
4	C₂×C₃×#124	C₂	1	3	yes	maximal, complemented
5	C₂×C₃×C₂²⋊Q₈	C₂	1	3	yes	maximal, complemented
6	C₂×C₃×#121	C₂	1	3	yes	maximal, complemented
7	C₂×C₃×C₄×Q₈	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₃×C₄×D₄	C₂	1	3	yes	maximal, complemented
9	C₂×#548	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
10	C₂×C₃×#41	—	2	8	no	—
11	C₂×C₃×#41	C₂²	1	8	yes	complemented
12	C₂²×C₃×Q₈	—	2	8	no	—
13	C₂²×C₃×Q₈	C₂²	1	3	yes	—
14	C₂²×C₃×Q₈	C₂²	1	1	yes	characteristic
15	C₂²×C₃×Q₈	C₂²	1	8	yes	complemented
16	C₂²×C₃×D₄	C₂²	1	3	yes	—
17	C₂²×C₃×D₄	C₂²	1	3	yes	—
18	C₂³×C₃×C₄	C₂²	1	3	yes	—
19	C₂³×C₃×C₄	C₂²	1	3	yes	—
20	C₃×#124	C₂²	1	24	yes	complemented
21	C₃×C₂²⋊Q₈	C₂²	1	24	yes	complemented
22	C₃×#121	C₂²	1	24	yes	complemented
23	C₃×C₄×Q₈	C₂²	1	8	yes	complemented
24	C₃×C₄×D₄	C₂²	1	24	yes	complemented
25	C₂×C₃×C₄⋊C₄	C₂²	1	3	yes	—
26	C₂×C₃×C₄⋊C₄	C₂²	1	3	yes	—
27	C₂×C₃×C₂²⋊C₄	C₂²	1	1	yes	characteristic
28	C₂×C₃×C₂²⋊C₄	C₂²	1	3	yes	—
29	C₂×C₃×C₂²⋊C₄	C₂²	1	6	yes	—
30	C₂×C₃×C₄²	C₂²	1	3	yes	—
31	C₂²×#41	C₂×C₃	1	1	yes	characteristic, complemented
32	C₂³×Q₈	C₂×C₃	1	1	yes	characteristic, complemented
33	#548	C₂×C₃	1	16	yes	direct product factor
34	C₂×#124	C₂×C₃	1	3	yes	complemented
35	C₂×C₂²⋊Q₈	C₂×C₃	1	3	yes	complemented
36	C₂×#121	C₂×C₃	1	3	yes	complemented
37	C₂×C₄×Q₈	C₂×C₃	1	1	yes	characteristic, complemented
38	C₂×C₄×D₄	C₂×C₃	1	3	yes	complemented
39	C₂⁴×C₃	C₂³	1	3	yes	—
40	C₂⁴×C₃	C₂³	1	1	yes	characteristic
41	C₃×#41	—	2	32	no	—
42	C₂×C₃×Q₈	—	2	32	no	—
43	C₂×C₃×Q₈	—	2	12	no	—
44	C₂×C₃×Q₈	D₄	1	8	yes	complemented
45	C₂×C₃×Q₈	C₂³	1	12	yes	—
46	C₂×C₃×Q₈	C₂³	1	4	yes	—
47	C₂×C₃×D₄	—	2	12	no	—
48	C₂×C₃×D₄	—	2	12	no	—
49	C₂×C₃×D₄	C₂³	1	12	yes	—
50	C₂×C₃×D₄	C₂³	1	12	yes	—
51	C₂²×C₃×C₄	—	2	12	no	—
52	C₂²×C₃×C₄	—	2	12	no	—
53	C₂²×C₃×C₄	C₂³	1	12	yes	—
54	C₂²×C₃×C₄	C₂³	1	12	yes	—
55	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
56	C₂²×C₃×C₄	C₂³	1	3	yes	—
57	C₂²×C₃×C₄	C₂³	1	3	yes	—
58	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
59	C₂²×C₃×C₄	C₂³	1	3	yes	—
60	C₃×C₄⋊C₄	C₂³	1	12	yes	—
61	C₃×C₄⋊C₄	C₂³	1	12	yes	—
62	C₃×C₂²⋊C₄	C₂³	1	4	yes	—
63	C₃×C₂²⋊C₄	C₂³	1	12	yes	—
64	C₃×C₂²⋊C₄	C₂³	1	24	yes	—
65	C₃×C₄²	C₂³	1	12	yes	—
66	C₂×#41	—	2	8	no	—
67	C₂×#41	C₂²×C₃	1	8	yes	complemented
68	C₂²×Q₈	—	2	8	no	—
69	C₂²×Q₈	C₂²×C₃	1	3	yes	—
70	C₂²×Q₈	C₂²×C₃	1	1	yes	characteristic
71	C₂²×Q₈	C₂²×C₃	1	8	yes	complemented
72	C₂²×D₄	C₂²×C₃	1	3	yes	—
73	C₂²×D₄	C₂²×C₃	1	3	yes	—
74	C₂³×C₄	C₂²×C₃	1	3	yes	—
75	C₂³×C₄	C₂²×C₃	1	3	yes	—
76	#124	C₂²×C₃	1	24	yes	complemented
77	C₂²⋊Q₈	C₂²×C₃	1	24	yes	complemented
78	#121	C₂²×C₃	1	24	yes	complemented
79	C₄×Q₈	C₂²×C₃	1	8	yes	complemented
80	C₄×D₄	C₂²×C₃	1	24	yes	complemented
81	C₂×C₄⋊C₄	C₂²×C₃	1	3	yes	—
82	C₂×C₄⋊C₄	C₂²×C₃	1	3	yes	—
83	C₂×C₂²⋊C₄	C₂²×C₃	1	1	yes	characteristic
84	C₂×C₂²⋊C₄	C₂²×C₃	1	3	yes	—
85	C₂×C₂²⋊C₄	C₂²×C₃	1	6	yes	—
86	C₂×C₄²	C₂²×C₃	1	3	yes	—
87	C₂³×C₃	—	2	2	no	—
88	C₂³×C₃	—	2	2	no	—
89	C₂³×C₃	—	2	6	no	—
90	C₂³×C₃	—	2	6	no	—
91	C₂³×C₃	—	2	6	no	—
92	C₂³×C₃	#41	1	4	yes	complemented
93	C₂³×C₃	C₂⁴	1	6	yes	—
94	C₂³×C₃	C₂⁴	1	2	yes	—
95	C₂³×C₃	C₂⁴	1	1	yes	characteristic
96	C₃×Q₈	—	2	24	no	—
97	C₃×Q₈	C₂×D₄	1	16	yes	complemented
98	C₃×D₄	—	2	24	no	—
99	C₃×D₄	—	2	24	no	—
100	C₂×C₃×C₄	—	2	2	no	—
101	C₂×C₃×C₄	—	2	6	no	—
102	C₂×C₃×C₄	—	2	6	no	—
103	C₂×C₃×C₄	—	2	2	no	—
104	C₂×C₃×C₄	—	2	24	no	—
105	C₂×C₃×C₄	—	2	24	no	—
106	C₂×C₃×C₄	C₂×D₄	1	12	yes	—
107	C₂×C₃×C₄	C₂⁴	1	2	yes	—
108	C₂×C₃×C₄	C₂⁴	1	6	yes	—
109	C₂×C₃×C₄	C₂⁴	1	6	yes	—
110	C₂×C₃×C₄	C₂⁴	1	2	yes	—
111	C₂×C₃×C₄	C₂⁴	1	6	yes	—
112	C₂⁴	C₂³×C₃	1	3	yes	—
113	C₂⁴	C₂³×C₃	1	1	yes	characteristic
114	#41	—	2	32	no	—
115	C₂×Q₈	—	2	32	no	—
116	C₂×Q₈	—	2	12	no	—
117	C₂×Q₈	C₃×D₄	1	8	yes	complemented
118	C₂×Q₈	C₂³×C₃	1	12	yes	—
119	C₂×Q₈	C₂³×C₃	1	4	yes	—
120	C₂×D₄	—	2	12	no	—
121	C₂×D₄	—	2	12	no	—
122	C₂×D₄	C₂³×C₃	1	12	yes	—
123	C₂×D₄	C₂³×C₃	1	12	yes	—
124	C₂²×C₄	—	2	12	no	—
125	C₂²×C₄	—	2	12	no	—
126	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
127	C₂²×C₄	C₂³×C₃	1	3	yes	—
128	C₂²×C₄	C₂³×C₃	1	3	yes	—
129	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
130	C₂²×C₄	C₂³×C₃	1	12	yes	—
131	C₂²×C₄	C₂³×C₃	1	12	yes	—
132	C₂²×C₄	C₂³×C₃	1	3	yes	—
133	C₄⋊C₄	C₂³×C₃	1	12	yes	—
134	C₄⋊C₄	C₂³×C₃	1	12	yes	—
135	C₂²⋊C₄	C₂³×C₃	1	4	yes	—
136	C₂²⋊C₄	C₂³×C₃	1	12	yes	—
137	C₂²⋊C₄	C₂³×C₃	1	24	yes	—
138	C₄²	C₂³×C₃	1	12	yes	—
139	C₂²×C₃	—	2	8	no	—
140	C₂²×C₃	—	2	2	no	—
141	C₂²×C₃	—	2	2	no	—
142	C₂²×C₃	—	2	6	no	—
143	C₂²×C₃	—	2	6	no	—
144	C₂²×C₃	—	2	6	no	—
145	C₂²×C₃	—	4	12	no	—
146	C₂²×C₃	C₂²×D₄	1	2	yes	—
147	C₂²×C₃	C₂×#41	1	2	yes	—
148	C₂²×C₃	C₂×#41	1	4	yes	complemented
149	C₂²×C₃	2₋¹⁺⁴	1	2	yes	—
150	C₂²×C₃	C₂⁵	1	1	yes	characteristic
151	C₃×C₄	—	2	2	no	—
152	C₃×C₄	—	2	6	no	—
153	C₃×C₄	—	2	6	no	—
154	C₃×C₄	—	2	2	no	—
155	C₃×C₄	C₂²×D₄	1	12	yes	—
156	C₂³	—	2	2	no	—
157	C₂³	—	2	2	no	—
158	C₂³	—	2	6	no	—
159	C₂³	—	2	6	no	—
160	C₂³	—	2	6	no	—
161	C₂³	C₃×#41	1	4	yes	complemented
162	C₂³	C₂⁴×C₃	1	6	yes	—
163	C₂³	C₂⁴×C₃	1	2	yes	—
164	C₂³	C₂⁴×C₃	1	1	yes	characteristic
165	Q₈	—	2	24	no	—
166	Q₈	C₂×C₃×D₄	1	16	yes	complemented
167	D₄	—	2	24	no	—
168	D₄	—	2	24	no	—
169	C₂×C₄	—	2	2	no	—
170	C₂×C₄	—	2	6	no	—
171	C₂×C₄	—	2	6	no	—
172	C₂×C₄	—	2	2	no	—
173	C₂×C₄	—	2	24	no	—
174	C₂×C₄	—	2	24	no	—
175	C₂×C₄	C₂×C₃×D₄	1	12	yes	—
176	C₂×C₄	C₂⁴×C₃	1	2	yes	—
177	C₂×C₄	C₂⁴×C₃	1	6	yes	—
178	C₂×C₄	C₂⁴×C₃	1	6	yes	—
179	C₂×C₄	C₂⁴×C₃	1	2	yes	—
180	C₂×C₄	C₂⁴×C₃	1	6	yes	—
181	C₂×C₃	—	2	4	no	—
182	C₂×C₃	—	4	6	no	—
183	C₂×C₃	#548	1	4	yes	direct product factor
184	C₂×C₃	C₂³×D₄	1	1	yes	characteristic
185	C₂×C₃	C₂²×#41	1	1	yes	characteristic
186	C₂×C₃	C₂×2₋¹⁺⁴	1	1	yes	characteristic
187	C₂²	—	2	2	no	—
188	C₂²	—	2	2	no	—
189	C₂²	—	2	8	no	—
190	C₂²	—	2	6	no	—
191	C₂²	—	2	6	no	—
192	C₂²	—	2	6	no	—
193	C₂²	—	4	12	no	—
194	C₂²	C₂²×C₃×D₄	1	2	yes	—
195	C₂²	C₂×C₃×#41	1	2	yes	—
196	C₂²	C₂×C₃×#41	1	4	yes	complemented
197	C₂²	C₃×2₋¹⁺⁴	1	2	yes	—
198	C₂²	C₂⁵×C₃	1	1	yes	characteristic
199	C₄	—	2	2	no	—
200	C₄	—	2	6	no	—
201	C₄	—	2	6	no	—
202	C₄	—	2	2	no	—
203	C₄	C₂²×C₃×D₄	1	12	yes	—
204	C₃	C₂×#548	1	1	yes	characteristic, direct product factor, Sylow
205	C₂	—	2	4	no	—
206	C₂	—	4	6	no	—
207	C₂	C₃×#548	1	4	yes	direct product factor
208	C₂	C₂³×C₃×D₄	1	1	yes	characteristic
209	C₂	C₂²×C₃×#41	1	1	yes	characteristic
210	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. C₂³×C₃×Q₈	C₂	semidirect
3. C₃×#548	C₂	direct (N × H)
4. C₂×C₃×#124	C₂	semidirect
5. C₂×C₃×C₂²⋊Q₈	C₂	semidirect
6. C₂×C₃×#121	C₂	semidirect
7. C₂×C₃×C₄×Q₈	C₂	semidirect
8. C₂×C₃×C₄×D₄	C₂	semidirect
9. C₂×#548	C₃	direct (N × H)
11. C₂×C₃×#41	C₂²	semidirect
15. C₂²×C₃×Q₈	C₂²	semidirect
20. C₃×#124	C₂²	semidirect
21. C₃×C₂²⋊Q₈	C₂²	semidirect
22. C₃×#121	C₂²	semidirect
23. C₃×C₄×Q₈	C₂²	semidirect
24. C₃×C₄×D₄	C₂²	semidirect
31. C₂²×#41	C₂×C₃	semidirect
32. C₂³×Q₈	C₂×C₃	semidirect
33. #548	C₂×C₃	direct (N × H)
34. C₂×#124	C₂×C₃	semidirect
35. C₂×C₂²⋊Q₈	C₂×C₃	semidirect
36. C₂×#121	C₂×C₃	semidirect
37. C₂×C₄×Q₈	C₂×C₃	semidirect
38. C₂×C₄×D₄	C₂×C₃	semidirect
44. C₂×C₃×Q₈	D₄	semidirect
67. C₂×#41	C₂²×C₃	semidirect
71. C₂²×Q₈	C₂²×C₃	semidirect
76. #124	C₂²×C₃	semidirect
77. C₂²⋊Q₈	C₂²×C₃	semidirect
78. #121	C₂²×C₃	semidirect
79. C₄×Q₈	C₂²×C₃	semidirect
80. C₄×D₄	C₂²×C₃	semidirect
92. C₂³×C₃	#41	semidirect
97. C₃×Q₈	C₂×D₄	semidirect
117. C₂×Q₈	C₃×D₄	semidirect
148. C₂²×C₃	C₂×#41	semidirect
161. C₂³	C₃×#41	semidirect
166. Q₈	C₂×C₃×D₄	semidirect
183. C₂×C₃	#548	direct (N × H)
196. C₂²	C₂×C₃×#41	semidirect
204. C₃	C₂×#548	direct (N × H)
207. C₂	C₃×#548	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
C₂³×C₃×Q₈	2
C₃×#548	2
C₂×C₃×#124	2
C₂×C₃×C₂²⋊Q₈	2
C₂×C₃×#121	2
C₂×C₃×C₄×Q₈	2
C₂×C₃×C₄×D₄	2
C₂×#548	3

Subgroups — C2×C3×#548 #87349

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×#548 #87349