Subgroups — C₂²×D₃×#41 #87563

Order 384.

6520 conjugacy classes of subgroups.

The subgroup lattice has 198 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	C₂×D₃×#41	C₂	1	48	yes	maximal, direct product factor
3	C₂²×#239	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂²×D₃×Q₈	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂²×#237	C₂	1	3	yes	maximal, complemented
6	C₂²×D₃×D₄	C₂	1	3	yes	maximal, complemented
7	C₂²×#235	C₂	1	3	yes	maximal, complemented
8	C₂³×C₄×D₃	C₂	1	3	yes	maximal, complemented
9	C₂³×#41	—	3	1	no	maximal, Sylow
10	C₂⁴×D₃	C₂²	1	3	yes	—
11	C₂×C₃×#41	C₂²	1	24	yes	complemented
12	C₂²×C₃×Q₈	C₂²	1	1	yes	characteristic, complemented
13	C₂²×C₃×D₄	C₂²	1	3	yes	complemented
14	C₂³×C₃×C₄	C₂²	1	3	yes	complemented
15	C₂²×#67	C₂²	1	6	yes	complemented
16	C₂³×Dic₃	C₂²	1	3	yes	complemented
17	D₃×#41	C₂²	1	256	yes	direct product factor
18	C₂×#239	C₂²	1	24	yes	complemented
19	C₂×D₃×Q₈	C₂²	1	24	yes	complemented
20	C₂×#237	C₂²	1	72	yes	complemented
21	C₂×D₃×D₄	C₂²	1	72	yes	complemented
22	C₂×#235	C₂²	1	72	yes	complemented
23	C₂²×D₁₂	C₂²	1	3	yes	complemented
24	C₂²×C₄×D₃	C₂²	1	1	yes	characteristic
25	C₂²×C₄×D₃	C₂²	1	3	yes	—
26	C₂²×C₄×D₃	C₂²	1	72	yes	complemented
27	C₂²×C₄×D₃	C₂²	1	3	yes	complemented
28	C₂²×C₄×D₃	C₂²	1	3	yes	complemented
29	C₂²×Dic₆	C₂²	1	3	yes	complemented
30	C₂²×#41	—	3	48	no	—
31	C₂²×#41	—	3	1	no	—
32	C₂²×#41	—	3	3	no	—
33	C₂²×#41	—	3	3	no	—
34	C₂²×#41	D₃	1	1	yes	characteristic, direct product factor
35	C₂³×Q₈	—	3	1	no	—
36	C₂³×D₄	—	3	3	no	—
37	C₂⁴×C₄	—	3	3	no	—
38	C₂⁴×C₃	C₂³	1	3	yes	—
39	C₂³×D₃	—	2	24	no	—
40	C₂³×D₃	C₂³	1	3	yes	—
41	C₂³×D₃	C₂³	1	36	yes	—
42	C₂³×D₃	C₂³	1	1	yes	characteristic
43	C₃×#41	C₂³	1	64	yes	complemented
44	C₂×C₃×Q₈	C₂³	1	12	yes	complemented
45	C₂×C₃×D₄	C₂³	1	36	yes	complemented
46	C₂²×C₃×C₄	C₂³	1	3	yes	—
47	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
48	C₂²×C₃×C₄	C₂³	1	36	yes	complemented
49	C₂×#67	C₂³	1	72	yes	complemented
50	C₂²×Dic₃	C₂³	1	3	yes	—
51	C₂²×Dic₃	C₂³	1	1	yes	characteristic
52	C₂²×Dic₃	C₂³	1	36	yes	complemented
53	#239	C₂³	1	64	yes	complemented
54	D₃×Q₈	C₂³	1	64	yes	complemented
55	#237	C₂³	1	192	yes	complemented
56	D₃×D₄	C₂³	1	192	yes	complemented
57	#235	C₂³	1	192	yes	complemented
58	C₂×D₁₂	C₂³	1	36	yes	complemented
59	C₂×C₄×D₃	C₂³	1	12	yes	—
60	C₂×C₄×D₃	C₂³	1	36	yes	—
61	C₂×C₄×D₃	C₂³	1	192	yes	complemented
62	C₂×C₄×D₃	C₂³	1	36	yes	complemented
63	C₂×C₄×D₃	C₂³	1	36	yes	complemented
64	C₂×Dic₆	C₂³	1	36	yes	complemented
65	C₂⁵	—	3	3	no	—
66	C₂×#41	—	3	256	no	—
67	C₂×#41	—	3	24	no	—
68	C₂×#41	—	3	72	no	—
69	C₂×#41	—	3	72	no	—
70	C₂×#41	C₂×D₃	1	24	yes	direct product factor
71	C₂²×Q₈	—	3	24	no	—
72	C₂²×Q₈	—	3	3	no	—
73	C₂²×Q₈	C₂×D₃	1	1	yes	characteristic, complemented
74	C₂²×D₄	—	3	6	no	—
75	C₂²×D₄	—	3	72	no	—
76	C₂²×D₄	—	3	3	no	—
77	C₂²×D₄	C₂×D₃	1	3	yes	complemented
78	C₂³×C₄	—	3	3	no	—
79	C₂³×C₄	—	3	3	no	—
80	C₂³×C₄	—	3	3	no	—
81	C₂³×C₄	—	3	1	no	—
82	C₂³×C₄	—	3	3	no	—
83	C₂³×C₄	—	3	72	no	—
84	C₂³×C₄	C₂×D₃	1	3	yes	complemented
85	C₂³×C₃	—	2	12	no	—
86	C₂³×C₃	C₂⁴	1	18	yes	—
87	C₂³×C₃	C₂⁴	1	1	yes	characteristic
88	C₂²×D₃	—	2	12	no	—
89	C₂²×D₃	—	2	144	no	—
90	C₂²×D₃	#41	1	8	yes	direct product factor
91	C₂²×D₃	C₂⁴	1	18	yes	—
92	C₂²×D₃	C₂⁴	1	48	yes	—
93	C₂²×D₃	C₂⁴	1	6	yes	—
94	C₃×Q₈	C₂⁴	1	16	yes	complemented
95	C₃×D₄	C₂⁴	1	48	yes	complemented
96	C₂×C₃×C₄	C₂⁴	1	18	yes	—
97	C₂×C₃×C₄	C₂⁴	1	6	yes	—
98	C₂×C₃×C₄	C₂⁴	1	48	yes	complemented
99	#67	C₂⁴	1	96	yes	complemented
100	C₂×Dic₃	C₂⁴	1	18	yes	—
101	C₂×Dic₃	C₂⁴	1	6	yes	—
102	C₂×Dic₃	C₂⁴	1	48	yes	complemented
103	D₁₂	C₂⁴	1	48	yes	complemented
104	C₄×D₃	C₂⁴	1	16	yes	—
105	C₄×D₃	C₂⁴	1	48	yes	—
106	C₄×D₃	C₂⁴	1	48	yes	complemented
107	C₄×D₃	C₂⁴	1	48	yes	complemented
108	Dic₆	C₂⁴	1	48	yes	complemented
109	C₂⁴	—	3	1	no	—
110	C₂⁴	—	3	36	no	—
111	C₂⁴	—	3	3	no	—
112	C₂⁴	—	6	24	no	—
113	C₂⁴	C₂²×D₃	1	3	yes	—
114	#41	—	3	64	no	—
115	#41	—	3	192	no	—
116	#41	—	3	192	no	—
117	#41	C₂²×D₃	1	64	yes	direct product factor
118	C₂×Q₈	—	3	64	no	—
119	C₂×Q₈	—	3	36	no	—
120	C₂×Q₈	C₂²×D₃	1	12	yes	complemented
121	C₂×D₄	—	3	72	no	—
122	C₂×D₄	—	3	192	no	—
123	C₂×D₄	—	3	36	no	—
124	C₂×D₄	C₂²×D₃	1	36	yes	complemented
125	C₂²×C₄	—	3	1	no	—
126	C₂²×C₄	—	3	3	no	—
127	C₂²×C₄	—	3	36	no	—
128	C₂²×C₄	—	3	12	no	—
129	C₂²×C₄	—	3	36	no	—
130	C₂²×C₄	—	3	36	no	—
131	C₂²×C₄	—	3	36	no	—
132	C₂²×C₄	—	3	192	no	—
133	C₂²×C₄	C₂²×D₃	1	3	yes	—
134	C₂²×C₄	C₂²×D₃	1	1	yes	characteristic
135	C₂²×C₄	C₂²×D₃	1	36	yes	complemented
136	C₂²×C₃	—	2	36	no	—
137	C₂²×C₃	C₂×#41	1	4	yes	complemented
138	C₂²×C₃	C₂⁵	1	12	yes	—
139	C₂²×C₃	C₂⁵	1	3	yes	—
140	C₂×D₃	—	2	36	no	—
141	C₂×D₃	—	2	96	no	—
142	C₂×D₃	C₂×#41	1	24	yes	direct product factor
143	C₂×D₃	C₂⁵	1	12	yes	—
144	C₂×D₃	C₂⁵	1	4	yes	—
145	C₃×C₄	C₂⁵	1	12	yes	—
146	C₃×C₄	C₂⁵	1	4	yes	—
147	Dic₃	C₂⁵	1	12	yes	—
148	Dic₃	C₂⁵	1	4	yes	—
149	C₂³	—	2	12	no	—
150	C₂³	—	3	8	no	—
151	C₂³	—	3	6	no	—
152	C₂³	—	3	48	no	—
153	C₂³	—	3	18	no	—
154	C₂³	—	6	144	no	—
155	C₂³	—	6	12	no	—
156	C₂³	C₂³×D₃	1	18	yes	—
157	C₂³	C₂³×D₃	1	1	yes	characteristic
158	Q₈	—	3	48	no	—
159	Q₈	C₂³×D₃	1	16	yes	complemented
160	D₄	—	3	96	no	—
161	D₄	—	3	48	no	—
162	D₄	C₂³×D₃	1	48	yes	complemented
163	C₂×C₄	—	3	6	no	—
164	C₂×C₄	—	3	18	no	—
165	C₂×C₄	—	3	48	no	—
166	C₂×C₄	—	3	16	no	—
167	C₂×C₄	—	3	48	no	—
168	C₂×C₄	—	3	48	no	—
169	C₂×C₄	—	3	48	no	—
170	C₂×C₄	C₂³×D₃	1	6	yes	—
171	C₂×C₄	C₂³×D₃	1	18	yes	—
172	C₂×C₄	C₂³×D₃	1	48	yes	complemented
173	C₂×C₃	—	2	12	no	—
174	C₂×C₃	C₂²×#41	1	6	yes	complemented
175	C₂×C₃	C₂⁶	1	1	yes	characteristic
176	D₃	—	2	12	no	—
177	D₃	C₂²×#41	1	8	yes	direct product factor
178	C₂²	—	2	36	no	—
179	C₂²	—	3	24	no	—
180	C₂²	—	3	4	no	—
181	C₂²	—	3	12	no	—
182	C₂²	—	6	96	no	—
183	C₂²	—	6	36	no	—
184	C₂²	D₃×#41	1	4	yes	direct product factor
185	C₂²	C₂⁴×D₃	1	12	yes	—
186	C₂²	C₂⁴×D₃	1	3	yes	—
187	C₄	—	3	4	no	—
188	C₄	—	3	12	no	—
189	C₄	C₂⁴×D₃	1	4	yes	—
190	C₄	C₂⁴×D₃	1	12	yes	—
191	C₃	C₂³×#41	1	1	yes	characteristic, complemented, Sylow
192	C₂	—	2	12	no	—
193	C₂	—	3	8	no	—
194	C₂	—	6	12	no	—
195	C₂	C₂×D₃×#41	1	6	yes	direct product factor
196	C₂	C₂⁵×D₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂²×D₃
commutator, cosocle	C₂×C₃	C₂⁶
Frattini	C₂	C₂⁵×D₃
Fitting	C₂²×C₃×#41	C₂
socle	C₂³×C₃	C₂⁴
hypercenter	C₂²×#41	D₃
hypocenter	C₃	C₂³×#41
LCS	G ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×#41

Complements

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

N	H ≅ G/N	type
1. C₂²×C₃×#41	C₂	semidirect
2. C₂×D₃×#41	C₂	direct (N × H)
3. C₂²×#239	C₂	semidirect
4. C₂²×D₃×Q₈	C₂	semidirect
5. C₂²×#237	C₂	semidirect
6. C₂²×D₃×D₄	C₂	semidirect
7. C₂²×#235	C₂	semidirect
8. C₂³×C₄×D₃	C₂	semidirect
11. C₂×C₃×#41	C₂²	semidirect
12. C₂²×C₃×Q₈	C₂²	semidirect
13. C₂²×C₃×D₄	C₂²	semidirect
14. C₂³×C₃×C₄	C₂²	semidirect
15. C₂²×#67	C₂²	semidirect
16. C₂³×Dic₃	C₂²	semidirect
17. D₃×#41	C₂²	direct (N × H)
18. C₂×#239	C₂²	semidirect
19. C₂×D₃×Q₈	C₂²	semidirect
20. C₂×#237	C₂²	semidirect
21. C₂×D₃×D₄	C₂²	semidirect
22. C₂×#235	C₂²	semidirect
23. C₂²×D₁₂	C₂²	semidirect
26. C₂²×C₄×D₃	C₂²	semidirect
27. C₂²×C₄×D₃	C₂²	semidirect
28. C₂²×C₄×D₃	C₂²	semidirect
29. C₂²×Dic₆	C₂²	semidirect
34. C₂²×#41	D₃	direct (N × H)
43. C₃×#41	C₂³	semidirect
44. C₂×C₃×Q₈	C₂³	semidirect
45. C₂×C₃×D₄	C₂³	semidirect
48. C₂²×C₃×C₄	C₂³	semidirect
49. C₂×#67	C₂³	semidirect
52. C₂²×Dic₃	C₂³	semidirect
53. #239	C₂³	semidirect
54. D₃×Q₈	C₂³	semidirect
55. #237	C₂³	semidirect
56. D₃×D₄	C₂³	semidirect
57. #235	C₂³	semidirect
58. C₂×D₁₂	C₂³	semidirect
61. C₂×C₄×D₃	C₂³	semidirect
62. C₂×C₄×D₃	C₂³	semidirect
63. C₂×C₄×D₃	C₂³	semidirect
64. C₂×Dic₆	C₂³	semidirect
70. C₂×#41	C₂×D₃	direct (N × H)
73. C₂²×Q₈	C₂×D₃	semidirect
77. C₂²×D₄	C₂×D₃	semidirect
84. C₂³×C₄	C₂×D₃	semidirect
90. C₂²×D₃	#41	direct (N × H)
94. C₃×Q₈	C₂⁴	semidirect
95. C₃×D₄	C₂⁴	semidirect
98. C₂×C₃×C₄	C₂⁴	semidirect
99. #67	C₂⁴	semidirect
102. C₂×Dic₃	C₂⁴	semidirect
103. D₁₂	C₂⁴	semidirect
106. C₄×D₃	C₂⁴	semidirect
107. C₄×D₃	C₂⁴	semidirect
108. Dic₆	C₂⁴	semidirect
117. #41	C₂²×D₃	direct (N × H)
120. C₂×Q₈	C₂²×D₃	semidirect
124. C₂×D₄	C₂²×D₃	semidirect
135. C₂²×C₄	C₂²×D₃	semidirect
137. C₂²×C₃	C₂×#41	semidirect
142. C₂×D₃	C₂×#41	direct (N × H)
159. Q₈	C₂³×D₃	semidirect
162. D₄	C₂³×D₃	semidirect
172. C₂×C₄	C₂³×D₃	semidirect
174. C₂×C₃	C₂²×#41	semidirect
177. D₃	C₂²×#41	direct (N × H)
184. C₂²	D₃×#41	direct (N × H)
191. C₃	C₂³×#41	semidirect
195. C₂	C₂×D₃×#41	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
C₂×D₃×#41	2
C₂²×#239	2
C₂²×D₃×Q₈	2
C₂²×#237	2
C₂²×D₃×D₄	2
C₂²×#235	2
C₂³×C₄×D₃	2
C₂³×#41	3

Subgroups — C22×D3×#41 #87563

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×D₃×#41 #87563