Subgroups — C₂×Dic₃×#41 #87258

Order 384.

1728 conjugacy classes of subgroups.

The subgroup lattice has 144 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
2	Dic₃×#41	C₂	1	16	yes	maximal, direct product factor
3	C₂×Q₈×Dic₃	C₂	1	2	yes	maximal, complemented
4	C₂×D₄×Dic₃	C₂	1	6	yes	maximal, complemented
5	C₂×#926	C₂	1	3	yes	maximal, complemented
6	C₂²×C₄×Dic₃	C₂	1	3	yes	maximal, complemented
7	C₂×C₄×#41	—	3	1	no	maximal, Sylow
8	C₂×C₃×#41	C₄	1	16	yes	complemented
9	C₂×C₃×#41	C₂²	1	8	yes	—
10	C₂²×C₃×Q₈	C₂²	1	1	yes	characteristic
11	C₂²×C₃×D₄	C₂²	1	3	yes	—
12	C₂³×C₃×C₄	C₂²	1	3	yes	—
13	C₂³×Dic₃	C₂²	1	6	yes	—
14	C₂×#217	C₂²	1	6	yes	—
15	Q₈×Dic₃	C₂²	1	16	yes	complemented
16	D₄×Dic₃	C₂²	1	48	yes	complemented
17	#926	C₂²	1	24	yes	complemented
18	C₂×C₄⋊Dic₃	C₂²	1	6	yes	—
19	C₂×C₄×Dic₃	C₂²	1	1	yes	characteristic
20	C₂×C₄×Dic₃	C₂²	1	3	yes	—
21	C₂×C₄×Dic₃	C₂²	1	6	yes	—
22	C₂×C₄×Dic₃	C₂²	1	24	yes	complemented
23	C₂²×#41	D₃	1	1	yes	characteristic
24	C₄×#41	—	3	16	no	—
25	C₂×C₄×Q₈	—	3	2	no	—
26	C₂×C₄×D₄	—	3	6	no	—
27	C₂×#117	—	3	3	no	—
28	C₂²×C₄²	—	3	3	no	—
29	C₂⁴×C₃	C₂³	1	3	yes	—
30	C₃×#41	C₂×C₄	1	64	yes	complemented
31	C₂×C₃×Q₈	C₂×C₄	1	8	yes	complemented
32	C₂×C₃×Q₈	C₂³	1	4	yes	—
33	C₂×C₃×D₄	C₂×C₄	1	24	yes	complemented
34	C₂×C₃×D₄	C₂³	1	12	yes	—
35	C₂²×C₃×C₄	C₂×C₄	1	24	yes	complemented
36	C₂²×C₃×C₄	C₂³	1	3	yes	—
37	C₂²×C₃×C₄	C₂³	1	12	yes	—
38	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
39	C₂²×Dic₃	—	2	24	no	—
40	C₂²×Dic₃	C₂³	1	6	yes	—
41	C₂²×Dic₃	C₂³	1	24	yes	—
42	C₂²×Dic₃	C₂³	1	2	yes	—
43	#217	C₂³	1	24	yes	—
44	C₄⋊Dic₃	C₂³	1	24	yes	—
45	C₄×Dic₃	C₂³	1	4	yes	—
46	C₄×Dic₃	C₂³	1	12	yes	—
47	C₄×Dic₃	C₂³	1	24	yes	—
48	C₂×#41	Dic₃	1	16	yes	direct product factor
49	C₂×#41	C₂×D₃	1	8	yes	—
50	C₂²×Q₈	C₂×D₃	1	1	yes	characteristic
51	C₂²×D₄	C₂×D₃	1	3	yes	—
52	C₂³×C₄	—	3	6	no	—
53	C₂³×C₄	C₂×D₃	1	3	yes	—
54	C₄×Q₈	—	3	16	no	—
55	C₄×D₄	—	3	48	no	—
56	#117	—	3	24	no	—
57	C₂×C₄⋊C₄	—	3	6	no	—
58	C₂×C₂²⋊C₄	—	3	6	no	—
59	C₂×C₄²	—	3	6	no	—
60	C₂×C₄²	—	3	24	no	—
61	C₂×C₄²	—	3	3	no	—
62	C₂×C₄²	—	3	1	no	—
63	C₂³×C₃	—	2	12	no	—
64	C₂³×C₃	C₂²×C₄	1	12	yes	—
65	C₂³×C₃	C₂⁴	1	6	yes	—
66	C₂³×C₃	C₂⁴	1	1	yes	characteristic
67	C₃×Q₈	C₂²×C₄	1	16	yes	complemented
68	C₃×D₄	C₂²×C₄	1	48	yes	complemented
69	C₂×C₃×C₄	C₂²×C₄	1	4	yes	—
70	C₂×C₃×C₄	C₂²×C₄	1	12	yes	—
71	C₂×C₃×C₄	C₂²×C₄	1	48	yes	complemented
72	C₂×C₃×C₄	C₂⁴	1	2	yes	—
73	C₂×C₃×C₄	C₂⁴	1	6	yes	—
74	C₂×Dic₃	—	2	12	no	—
75	C₂×Dic₃	—	2	48	no	—
76	C₂×Dic₃	#41	1	8	yes	direct product factor
77	C₂×Dic₃	C₂⁴	1	12	yes	—
78	C₂×Dic₃	C₂⁴	1	4	yes	—
79	C₂⁴	C₂²×D₃	1	3	yes	—
80	#41	C₂×Dic₃	1	64	yes	direct product factor
81	C₂×Q₈	C₂×Dic₃	1	8	yes	complemented
82	C₂×Q₈	C₂²×D₃	1	4	yes	—
83	C₂×D₄	C₂×Dic₃	1	24	yes	complemented
84	C₂×D₄	C₂²×D₃	1	12	yes	—
85	C₂²×C₄	—	3	2	no	—
86	C₂²×C₄	—	3	6	no	—
87	C₂²×C₄	—	3	24	no	—
88	C₂²×C₄	—	6	24	no	—
89	C₂²×C₄	C₂×Dic₃	1	24	yes	complemented
90	C₂²×C₄	C₂²×D₃	1	3	yes	—
91	C₂²×C₄	C₂²×D₃	1	12	yes	—
92	C₂²×C₄	C₂²×D₃	1	1	yes	characteristic
93	C₄⋊C₄	—	3	24	no	—
94	C₂²⋊C₄	—	3	24	no	—
95	C₄²	—	3	4	no	—
96	C₄²	—	3	12	no	—
97	C₄²	—	3	24	no	—
98	C₂²×C₃	—	2	24	no	—
99	C₂²×C₃	—	2	12	no	—
100	C₂²×C₃	C₂³×C₄	1	12	yes	—
101	C₂²×C₃	C₂³×C₄	1	2	yes	—
102	C₂²×C₃	C₂×#41	1	4	yes	—
103	C₂²×C₃	C₂⁵	1	1	yes	characteristic
104	C₃×C₄	C₂³×C₄	1	4	yes	—
105	C₃×C₄	C₂³×C₄	1	12	yes	—
106	Dic₃	—	2	12	no	—
107	Dic₃	C₂×#41	1	8	yes	direct product factor
108	C₂³	—	2	12	no	—
109	C₂³	C₂²×Dic₃	1	12	yes	—
110	C₂³	C₂³×D₃	1	6	yes	—
111	C₂³	C₂³×D₃	1	1	yes	characteristic
112	Q₈	C₂²×Dic₃	1	16	yes	complemented
113	D₄	C₂²×Dic₃	1	48	yes	complemented
114	C₂×C₄	—	3	8	no	—
115	C₂×C₄	—	3	4	no	—
116	C₂×C₄	—	3	12	no	—
117	C₂×C₄	—	6	48	no	—
118	C₂×C₄	—	6	12	no	—
119	C₂×C₄	C₂²×Dic₃	1	4	yes	—
120	C₂×C₄	C₂²×Dic₃	1	12	yes	—
121	C₂×C₄	C₂²×Dic₃	1	48	yes	complemented
122	C₂×C₄	C₂³×D₃	1	2	yes	—
123	C₂×C₄	C₂³×D₃	1	6	yes	—
124	C₂×C₃	—	2	12	no	—
125	C₂×C₃	C₄×#41	1	4	yes	complemented
126	C₂×C₃	C₂⁴×C₄	1	1	yes	characteristic
127	C₂×C₃	C₂²×#41	1	2	yes	—
128	C₂²	—	2	12	no	—
129	C₂²	—	2	24	no	—
130	C₂²	D₃×#41	1	4	yes	—
131	C₂²	C₂³×Dic₃	1	12	yes	—
132	C₂²	C₂³×Dic₃	1	2	yes	—
133	C₂²	C₂⁴×D₃	1	1	yes	characteristic
134	C₄	—	3	8	no	—
135	C₄	—	6	12	no	—
136	C₄	C₂³×Dic₃	1	4	yes	—
137	C₄	C₂³×Dic₃	1	12	yes	—
138	C₃	C₂×C₄×#41	1	1	yes	characteristic, complemented, Sylow
139	C₂	—	2	12	no	—
140	C₂	Dic₃×#41	1	4	yes	direct product factor
141	C₂	C₂×D₃×#41	1	2	yes	—
142	C₂	C₂⁴×Dic₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂²×D₃
commutator	C₂×C₃	C₂⁴×C₄
Frattini	C₂²	C₂⁴×D₃
Fitting, prime radical	C₂²×C₃×#41	C₂
socle	C₂³×C₃	C₂⁴
cosocle	C₂²×C₃	C₂⁵
hypercenter	C₂²×#41	D₃
hypocenter	C₃	C₂×C₄×#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
2. Dic₃×#41	C₂	direct (N × H)
3. C₂×Q₈×Dic₃	C₂	semidirect
4. C₂×D₄×Dic₃	C₂	semidirect
5. C₂×#926	C₂	semidirect
6. C₂²×C₄×Dic₃	C₂	semidirect
8. C₂×C₃×#41	C₄	semidirect
15. Q₈×Dic₃	C₂²	semidirect
16. D₄×Dic₃	C₂²	semidirect
17. #926	C₂²	semidirect
22. C₂×C₄×Dic₃	C₂²	semidirect
30. C₃×#41	C₂×C₄	semidirect
31. C₂×C₃×Q₈	C₂×C₄	semidirect
33. C₂×C₃×D₄	C₂×C₄	semidirect
35. C₂²×C₃×C₄	C₂×C₄	semidirect
48. C₂×#41	Dic₃	direct (N × H)
67. C₃×Q₈	C₂²×C₄	semidirect
68. C₃×D₄	C₂²×C₄	semidirect
71. C₂×C₃×C₄	C₂²×C₄	semidirect
76. C₂×Dic₃	#41	direct (N × H)
80. #41	C₂×Dic₃	direct (N × H)
81. C₂×Q₈	C₂×Dic₃	semidirect
83. C₂×D₄	C₂×Dic₃	semidirect
89. C₂²×C₄	C₂×Dic₃	semidirect
107. Dic₃	C₂×#41	direct (N × H)
112. Q₈	C₂²×Dic₃	semidirect
113. D₄	C₂²×Dic₃	semidirect
121. C₂×C₄	C₂²×Dic₃	semidirect
125. C₂×C₃	C₄×#41	semidirect
138. C₃	C₂×C₄×#41	semidirect
140. C₂	Dic₃×#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
Dic₃×#41	2
C₂×Q₈×Dic₃	2
C₂×D₄×Dic₃	2
C₂×#926	2
C₂²×C₄×Dic₃	2
C₂×C₄×#41	3

Subgroups — C2×Dic3×#41 #87258

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×Dic₃×#41 #87258