Subgroups — C₂×D₃×M₃₂ #82005

Order 384.

400 conjugacy classes of subgroups.

The subgroup lattice has 176 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×D₃×C₈	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×M₃₂	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#812	C₂	1	1	yes	maximal, characteristic, complemented
4	D₃×M₃₂	C₂	1	8	yes	maximal, direct product factor
5	C₂×#798	C₂	1	2	yes	maximal, complemented
6	C₂×D₃×C₁₆	C₂	1	2	yes	maximal, complemented
7	C₂²×M₃₂	—	3	1	no	maximal, Sylow
8	C₂²×C₄×D₃	C₄	1	1	yes	characteristic
9	C₂²×C₃×C₈	C₂²	1	1	yes	characteristic
10	C₂²×C₃⋊C₈	C₄	1	1	yes	characteristic
11	C₂×D₃×C₈	C₄	1	4	yes	—
12	C₂×D₃×C₈	C₄	1	1	yes	characteristic
13	C₂×D₃×C₈	C₄	1	1	yes	characteristic
14	C₂×D₃×C₈	C₂²	1	1	yes	characteristic
15	C₂×D₃×C₈	C₂²	1	1	yes	characteristic
16	C₂×D₃×C₈	C₂²	1	4	yes	—
17	C₃×M₃₂	C₂²	1	4	yes	complemented
18	C₂×C₃×C₁₆	C₂²	1	2	yes	complemented
19	#812	C₂²	1	4	yes	complemented
20	C₂×C₃⋊C₁₆	C₂²	1	2	yes	complemented
21	#798	C₂²	1	8	yes	complemented
22	D₃×C₁₆	C₂²	1	8	yes	complemented
23	C₂³×C₈	—	3	1	no	—
24	C₂×M₃₂	—	3	1	no	—
25	C₂×M₃₂	—	3	8	no	—
26	C₂×M₃₂	—	3	2	no	—
27	C₂×M₃₂	D₃	1	1	yes	characteristic, direct product factor
28	C₂²×C₁₆	—	3	2	no	—
29	C₂³×D₃	C₈	1	1	yes	characteristic
30	C₂²×C₃×C₄	C₂×C₄	1	1	yes	characteristic
31	C₂²×Dic₃	C₈	1	1	yes	characteristic
32	C₂×C₄×D₃	C₈	1	1	yes	characteristic
33	C₂×C₄×D₃	C₈	1	4	yes	—
34	C₂×C₄×D₃	C₈	1	1	yes	characteristic
35	C₂×C₄×D₃	C₂×C₄	1	1	yes	characteristic
36	C₂×C₄×D₃	C₂×C₄	1	4	yes	—
37	C₂×C₄×D₃	C₂×C₄	1	1	yes	characteristic
38	C₂×C₃×C₈	C₂×C₄	1	1	yes	characteristic
39	C₂×C₃×C₈	C₂×C₄	1	2	yes	—
40	C₂×C₃×C₈	C₂³	1	2	yes	—
41	C₂×C₃×C₈	C₂³	1	1	yes	characteristic
42	C₂×C₃⋊C₈	C₂×C₄	1	1	yes	characteristic
43	C₂×C₃⋊C₈	C₂×C₄	1	1	yes	characteristic
44	C₂×C₃⋊C₈	C₂×C₄	1	4	yes	—
45	D₃×C₈	C₂×C₄	1	2	yes	—
46	D₃×C₈	C₂×C₄	1	2	yes	—
47	D₃×C₈	C₂×C₄	1	4	yes	—
48	D₃×C₈	C₂×C₄	1	4	yes	—
49	D₃×C₈	C₂³	1	2	yes	—
50	D₃×C₈	C₂³	1	2	yes	—
51	C₃×C₁₆	C₂³	1	4	yes	complemented
52	C₃⋊C₁₆	C₂³	1	4	yes	complemented
53	C₂³×C₄	—	3	1	no	—
54	C₂²×C₈	—	3	1	no	—
55	C₂²×C₈	—	3	4	no	—
56	C₂²×C₈	—	3	1	no	—
57	C₂²×C₈	—	3	1	no	—
58	C₂²×C₈	—	3	4	no	—
59	C₂²×C₈	—	3	1	no	—
60	C₂²×C₈	—	3	1	no	—
61	C₂²×C₈	C₂×D₃	1	1	yes	characteristic
62	M₃₂	—	3	4	no	—
63	M₃₂	—	3	8	no	—
64	M₃₂	C₂×D₃	1	4	yes	direct product factor
65	C₂×C₁₆	—	3	2	no	—
66	C₂×C₁₆	—	3	8	no	—
67	C₂×C₁₆	C₂×D₃	1	2	yes	complemented
68	C₂³×C₃	C₂×C₈	1	1	yes	characteristic
69	C₂²×D₃	—	2	4	no	—
70	C₂²×D₃	C₂×C₈	1	1	yes	characteristic
71	C₂²×D₃	C₂×C₈	1	4	yes	—
72	C₂²×D₃	C₂×C₈	1	1	yes	characteristic
73	C₂×C₃×C₄	C₂×C₈	1	2	yes	—
74	C₂×C₃×C₄	C₂×C₈	1	1	yes	characteristic
75	C₂×C₃×C₄	C₂²×C₄	1	2	yes	—
76	C₂×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
77	C₂×Dic₃	C₂×C₈	1	4	yes	—
78	C₂×Dic₃	C₂×C₈	1	1	yes	characteristic
79	C₂×Dic₃	C₂×C₈	1	1	yes	characteristic
80	C₄×D₃	C₂×C₈	1	2	yes	—
81	C₄×D₃	C₂×C₈	1	4	yes	—
82	C₄×D₃	C₂×C₈	1	2	yes	—
83	C₄×D₃	C₂×C₈	1	4	yes	—
84	C₄×D₃	C₂²×C₄	1	2	yes	—
85	C₄×D₃	C₂²×C₄	1	2	yes	—
86	C₃×C₈	C₂²×C₄	1	2	yes	—
87	C₃×C₈	C₂²×C₄	1	1	yes	characteristic
88	C₃×C₈	C₂⁴	1	1	yes	characteristic
89	C₃⋊C₈	C₂²×C₄	1	2	yes	—
90	C₃⋊C₈	C₂²×C₄	1	2	yes	—
91	C₂⁴	—	3	1	no	—
92	C₂²×C₄	—	3	1	no	—
93	C₂²×C₄	—	3	1	no	—
94	C₂²×C₄	—	3	4	no	—
95	C₂²×C₄	—	3	1	no	—
96	C₂²×C₄	—	3	4	no	—
97	C₂²×C₄	—	3	1	no	—
98	C₂²×C₄	—	3	1	no	—
99	C₂²×C₄	C₄×D₃	1	1	yes	characteristic
100	C₂×C₈	—	3	4	no	—
101	C₂×C₈	—	3	1	no	—
102	C₂×C₈	—	3	1	no	—
103	C₂×C₈	—	3	4	no	—
104	C₂×C₈	—	3	4	no	—
105	C₂×C₈	—	3	2	no	—
106	C₂×C₈	—	3	2	no	—
107	C₂×C₈	—	3	2	no	—
108	C₂×C₈	—	3	2	no	—
109	C₂×C₈	C₄×D₃	1	1	yes	characteristic
110	C₂×C₈	C₄×D₃	1	2	yes	—
111	C₂×C₈	C₂²×D₃	1	2	yes	—
112	C₂×C₈	C₂²×D₃	1	1	yes	characteristic
113	C₁₆	—	3	4	no	—
114	C₁₆	C₂²×D₃	1	4	yes	complemented
115	C₂²×C₃	—	2	2	no	—
116	C₂²×C₃	C₂²×C₈	1	2	yes	—
117	C₂²×C₃	C₂²×C₈	1	1	yes	characteristic
118	C₂×D₃	—	2	2	no	—
119	C₂×D₃	—	2	8	no	—
120	C₂×D₃	M₃₂	1	4	yes	direct product factor
121	C₂×D₃	C₂²×C₈	1	2	yes	—
122	C₂×D₃	C₂²×C₈	1	2	yes	—
123	C₃×C₄	C₂²×C₈	1	2	yes	—
124	C₃×C₄	C₂²×C₈	1	1	yes	characteristic
125	C₃×C₄	C₂³×C₄	1	1	yes	characteristic
126	Dic₃	C₂²×C₈	1	2	yes	—
127	Dic₃	C₂²×C₈	1	2	yes	—
128	C₂³	—	3	1	no	—
129	C₂³	—	3	4	no	—
130	C₂³	—	3	1	no	—
131	C₂³	—	6	4	no	—
132	C₂³	D₃×C₈	1	1	yes	characteristic
133	C₂×C₄	—	3	4	no	—
134	C₂×C₄	—	3	1	no	—
135	C₂×C₄	—	3	1	no	—
136	C₂×C₄	—	3	4	no	—
137	C₂×C₄	—	3	2	no	—
138	C₂×C₄	—	3	4	no	—
139	C₂×C₄	—	3	2	no	—
140	C₂×C₄	—	3	2	no	—
141	C₂×C₄	—	3	2	no	—
142	C₂×C₄	D₃×C₈	1	2	yes	—
143	C₂×C₄	D₃×C₈	1	1	yes	characteristic
144	C₂×C₄	C₂×C₄×D₃	1	2	yes	—
145	C₂×C₄	C₂×C₄×D₃	1	1	yes	characteristic
146	C₈	—	3	2	no	—
147	C₈	—	3	2	no	—
148	C₈	C₂×C₄×D₃	1	2	yes	—
149	C₈	C₂×C₄×D₃	1	1	yes	characteristic
150	C₈	C₂³×D₃	1	1	yes	characteristic
151	C₂×C₃	—	2	2	no	—
152	C₂×C₃	C₂×M₃₂	1	2	yes	complemented
153	C₂×C₃	C₂³×C₈	1	1	yes	characteristic
154	D₃	—	2	2	no	—
155	D₃	C₂×M₃₂	1	4	yes	direct product factor
156	C₂²	—	2	2	no	—
157	C₂²	—	3	4	no	—
158	C₂²	—	3	2	no	—
159	C₂²	—	3	2	no	—
160	C₂²	—	6	8	no	—
161	C₂²	—	6	2	no	—
162	C₂²	C₂×D₃×C₈	1	2	yes	—
163	C₂²	C₂×D₃×C₈	1	1	yes	characteristic
164	C₄	—	3	2	no	—
165	C₄	—	3	2	no	—
166	C₄	C₂×D₃×C₈	1	2	yes	—
167	C₄	C₂×D₃×C₈	1	1	yes	characteristic
168	C₄	C₂²×C₄×D₃	1	1	yes	characteristic
169	C₃	C₂²×M₃₂	1	1	yes	characteristic, complemented, Sylow
170	C₂	—	2	2	no	—
171	C₂	—	3	4	no	—
172	C₂	—	6	2	no	—
173	C₂	D₃×M₃₂	1	2	yes	direct product factor
174	C₂	C₂²×D₃×C₈	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	C₂²×D₃
commutator	C₂×C₃	C₂³×C₈
Frattini	C₈	C₂³×D₃
Fitting	C₂×C₃×M₃₂	C₂
socle	C₂²×C₃	C₂²×C₈
prime radical	C₂³×D₃	C₈
cosocle	C₃×C₈	C₂⁴
hypercenter	C₂×M₃₂	D₃
hypocenter	C₃	C₂²×M₃₂
LCS	G ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₈ ◁ C₂×M₃₂

Complements

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

N	H ≅ G/N	type
2. C₂×C₃×M₃₂	C₂	semidirect
3. C₂×#812	C₂	semidirect
4. D₃×M₃₂	C₂	direct (N × H)
5. C₂×#798	C₂	semidirect
6. C₂×D₃×C₁₆	C₂	semidirect
17. C₃×M₃₂	C₂²	semidirect
18. C₂×C₃×C₁₆	C₂²	semidirect
19. #812	C₂²	semidirect
20. C₂×C₃⋊C₁₆	C₂²	semidirect
21. #798	C₂²	semidirect
22. D₃×C₁₆	C₂²	semidirect
27. C₂×M₃₂	D₃	direct (N × H)
51. C₃×C₁₆	C₂³	semidirect
52. C₃⋊C₁₆	C₂³	semidirect
64. M₃₂	C₂×D₃	direct (N × H)
67. C₂×C₁₆	C₂×D₃	semidirect
114. C₁₆	C₂²×D₃	semidirect
120. C₂×D₃	M₃₂	direct (N × H)
152. C₂×C₃	C₂×M₃₂	semidirect
155. D₃	C₂×M₃₂	direct (N × H)
169. C₃	C₂²×M₃₂	semidirect
173. C₂	D₃×M₃₂	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×D₃×C₈	2
C₂×C₃×M₃₂	2
C₂×#812	2
D₃×M₃₂	2
C₂×#798	2
C₂×D₃×C₁₆	2
C₂²×M₃₂	3

Subgroups — C2×D3×M32 #82005

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×D₃×M₃₂ #82005