Subgroups — C₂×C₃×#436 #84792

Order 384.

552 conjugacy classes of subgroups.

The subgroup lattice has 172 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃×M₁₆	C₂	1	2	yes	maximal, complemented
2	C₂×C₃×C₄×D₄	C₂	1	1	yes	maximal, characteristic
3	C₃×#436	C₂	1	8	yes	maximal, direct product factor
4	C₂×C₃×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₃×C₂²⋊C₈	C₂	1	2	yes	maximal, complemented
6	C₂×C₃×C₄×C₈	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#436	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
8	C₂²×C₃×D₄	C₄	1	1	yes	characteristic
9	C₂³×C₃×C₄	C₂²	1	2	yes	—
10	C₂×C₃×M₁₆	—	2	8	no	—
11	C₂×C₃×M₁₆	C₂²	1	8	yes	complemented
12	C₂²×C₃×C₈	C₂²	1	2	yes	—
13	C₂²×C₃×C₈	C₂²	1	1	yes	characteristic
14	C₂²×C₃×C₈	C₂²	1	1	yes	characteristic
15	C₃×C₄×D₄	C₄	1	4	yes	—
16	C₃×C₄×D₄	C₂²	1	4	yes	—
17	C₂×C₃×C₄⋊C₄	C₄	1	1	yes	characteristic
18	C₂×C₃×C₂²⋊C₄	C₄	1	2	yes	—
19	C₂×C₃×C₄²	C₂²	1	1	yes	characteristic
20	C₃×C₄⋊C₈	C₂²	1	4	yes	complemented
21	C₃×C₂²⋊C₈	C₂²	1	8	yes	complemented
22	C₃×C₄×C₈	C₂²	1	4	yes	complemented
23	C₂²×M₁₆	C₂×C₃	1	2	yes	complemented
24	C₂×C₄×D₄	C₂×C₃	1	1	yes	characteristic
25	#436	C₂×C₃	1	8	yes	direct product factor
26	C₂×C₄⋊C₈	C₂×C₃	1	1	yes	characteristic, complemented
27	C₂×C₂²⋊C₈	C₂×C₃	1	2	yes	complemented
28	C₂×C₄×C₈	C₂×C₃	1	1	yes	characteristic, complemented
29	C₂⁴×C₃	C₂×C₄	1	2	yes	—
30	C₂×C₃×D₄	—	2	4	no	—
31	C₂×C₃×D₄	C₂×C₄	1	4	yes	—
32	C₂²×C₃×C₄	—	2	4	no	—
33	C₂²×C₃×C₄	—	2	4	no	—
34	C₂²×C₃×C₄	C₂×C₄	1	1	yes	characteristic
35	C₂²×C₃×C₄	C₂×C₄	1	2	yes	—
36	C₂²×C₃×C₄	C₂×C₄	1	4	yes	—
37	C₂²×C₃×C₄	C₂×C₄	1	1	yes	characteristic
38	C₂²×C₃×C₄	C₂³	1	4	yes	—
39	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
40	C₃×M₁₆	—	2	16	no	—
41	C₂×C₃×C₈	—	2	4	no	—
42	C₂×C₃×C₈	—	2	2	no	—
43	C₂×C₃×C₈	D₄	1	4	yes	complemented
44	C₂×C₃×C₈	C₂³	1	2	yes	—
45	C₂×C₃×C₈	C₂³	1	4	yes	—
46	C₂×C₃×C₈	C₂³	1	2	yes	—
47	C₃×C₄⋊C₄	C₂×C₄	1	4	yes	—
48	C₃×C₂²⋊C₄	C₂×C₄	1	8	yes	—
49	C₃×C₄²	C₂×C₄	1	2	yes	—
50	C₃×C₄²	C₂³	1	2	yes	—
51	C₂²×D₄	C₃×C₄	1	1	yes	characteristic
52	C₂³×C₄	C₂²×C₃	1	2	yes	—
53	C₂×M₁₆	—	2	8	no	—
54	C₂×M₁₆	C₂²×C₃	1	8	yes	complemented
55	C₂²×C₈	C₂²×C₃	1	2	yes	—
56	C₂²×C₈	C₂²×C₃	1	1	yes	characteristic
57	C₂²×C₈	C₂²×C₃	1	1	yes	characteristic
58	C₄×D₄	C₃×C₄	1	4	yes	—
59	C₄×D₄	C₂²×C₃	1	4	yes	—
60	C₂×C₄⋊C₄	C₃×C₄	1	1	yes	characteristic
61	C₂×C₂²⋊C₄	C₃×C₄	1	2	yes	—
62	C₂×C₄²	C₂²×C₃	1	1	yes	characteristic
63	C₄⋊C₈	C₂²×C₃	1	4	yes	complemented
64	C₂²⋊C₈	C₂²×C₃	1	8	yes	complemented
65	C₄×C₈	C₂²×C₃	1	4	yes	complemented
66	C₂³×C₃	—	2	4	no	—
67	C₂³×C₃	—	2	4	no	—
68	C₂³×C₃	—	2	4	no	—
69	C₂³×C₃	C₂²×C₄	1	4	yes	—
70	C₂³×C₃	C₂²×C₄	1	1	yes	characteristic
71	C₃×D₄	—	2	8	no	—
72	C₂×C₃×C₄	—	2	4	no	—
73	C₂×C₃×C₄	—	2	2	no	—
74	C₂×C₃×C₄	—	2	8	no	—
75	C₂×C₃×C₄	—	2	4	no	—
76	C₂×C₃×C₄	—	2	4	no	—
77	C₂×C₃×C₄	M₁₆	1	4	yes	complemented
78	C₂×C₃×C₄	C₂²×C₄	1	2	yes	—
79	C₂×C₃×C₄	C₂²×C₄	1	4	yes	—
80	C₂×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
81	C₂×C₃×C₄	C₂²×C₄	1	2	yes	—
82	C₂×C₃×C₄	C₂×D₄	1	2	yes	—
83	C₂×C₃×C₄	#41	1	2	yes	—
84	C₂×C₃×C₄	C₂⁴	1	1	yes	characteristic
85	C₃×C₈	—	2	4	no	—
86	C₃×C₈	—	2	2	no	—
87	C₃×C₈	C₂×D₄	1	4	yes	complemented
88	C₂⁴	C₂×C₃×C₄	1	2	yes	—
89	C₂×D₄	—	2	4	no	—
90	C₂×D₄	C₂×C₃×C₄	1	4	yes	—
91	C₂²×C₄	—	2	4	no	—
92	C₂²×C₄	—	2	4	no	—
93	C₂²×C₄	C₂×C₃×C₄	1	1	yes	characteristic
94	C₂²×C₄	C₂×C₃×C₄	1	2	yes	—
95	C₂²×C₄	C₂×C₃×C₄	1	4	yes	—
96	C₂²×C₄	C₂×C₃×C₄	1	1	yes	characteristic
97	C₂²×C₄	C₂³×C₃	1	4	yes	—
98	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
99	M₁₆	—	2	16	no	—
100	C₂×C₈	—	2	4	no	—
101	C₂×C₈	—	2	2	no	—
102	C₂×C₈	C₃×D₄	1	4	yes	complemented
103	C₂×C₈	C₂³×C₃	1	2	yes	—
104	C₂×C₈	C₂³×C₃	1	4	yes	—
105	C₂×C₈	C₂³×C₃	1	2	yes	—
106	C₄⋊C₄	C₂×C₃×C₄	1	4	yes	—
107	C₂²⋊C₄	C₂×C₃×C₄	1	8	yes	—
108	C₄²	C₂×C₃×C₄	1	2	yes	—
109	C₄²	C₂³×C₃	1	2	yes	—
110	C₂²×C₃	—	2	4	no	—
111	C₂²×C₃	—	2	4	no	—
112	C₂²×C₃	—	2	4	no	—
113	C₂²×C₃	—	4	8	no	—
114	C₂²×C₃	C₄×D₄	1	2	yes	—
115	C₂²×C₃	C₂×M₁₆	1	2	yes	—
116	C₂²×C₃	#131	1	2	yes	—
117	C₂²×C₃	C₂³×C₄	1	1	yes	characteristic
118	C₃×C₄	—	2	4	no	—
119	C₃×C₄	—	2	2	no	—
120	C₃×C₄	C₄×D₄	1	2	yes	—
121	C₃×C₄	C₂×M₁₆	1	4	yes	complemented
122	C₃×C₄	C₂²×D₄	1	1	yes	characteristic
123	C₃×C₄	C₂×#41	1	1	yes	characteristic
124	C₂³	—	2	4	no	—
125	C₂³	—	2	4	no	—
126	C₂³	—	2	4	no	—
127	C₂³	C₂²×C₃×C₄	1	4	yes	—
128	C₂³	C₂²×C₃×C₄	1	1	yes	characteristic
129	D₄	—	2	8	no	—
130	C₂×C₄	—	2	4	no	—
131	C₂×C₄	—	2	2	no	—
132	C₂×C₄	—	2	8	no	—
133	C₂×C₄	—	2	4	no	—
134	C₂×C₄	—	2	4	no	—
135	C₂×C₄	C₃×M₁₆	1	4	yes	complemented
136	C₂×C₄	C₂²×C₃×C₄	1	2	yes	—
137	C₂×C₄	C₂²×C₃×C₄	1	4	yes	—
138	C₂×C₄	C₂²×C₃×C₄	1	1	yes	characteristic
139	C₂×C₄	C₂²×C₃×C₄	1	2	yes	—
140	C₂×C₄	C₂×C₃×D₄	1	2	yes	—
141	C₂×C₄	C₃×#41	1	2	yes	—
142	C₂×C₄	C₂⁴×C₃	1	1	yes	characteristic
143	C₈	—	2	4	no	—
144	C₈	—	2	2	no	—
145	C₈	C₂×C₃×D₄	1	4	yes	complemented
146	C₂×C₃	—	4	4	no	—
147	C₂×C₃	#436	1	4	yes	direct product factor
148	C₂×C₃	C₂×C₄×D₄	1	1	yes	characteristic
149	C₂×C₃	C₂²×M₁₆	1	1	yes	characteristic
150	C₂×C₃	C₂×#131	1	1	yes	characteristic
151	C₂²	—	2	4	no	—
152	C₂²	—	2	4	no	—
153	C₂²	—	2	4	no	—
154	C₂²	—	4	8	no	—
155	C₂²	C₃×C₄×D₄	1	2	yes	—
156	C₂²	C₂×C₃×M₁₆	1	2	yes	—
157	C₂²	C₃×#131	1	2	yes	—
158	C₂²	C₂³×C₃×C₄	1	1	yes	characteristic
159	C₄	—	2	2	no	—
160	C₄	—	2	4	no	—
161	C₄	C₃×C₄×D₄	1	2	yes	—
162	C₄	C₂×C₃×M₁₆	1	4	yes	complemented
163	C₄	C₂²×C₃×D₄	1	1	yes	characteristic
164	C₄	C₂×C₃×#41	1	1	yes	characteristic
165	C₃	C₂×#436	1	1	yes	characteristic, direct product factor, Sylow
166	C₂	—	4	4	no	—
167	C₂	C₃×#436	1	4	yes	direct product factor
168	C₂	C₂×C₃×C₄×D₄	1	1	yes	characteristic
169	C₂	C₂²×C₃×M₁₆	1	1	yes	characteristic
170	C₂	C₂×C₃×#131	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₃×C₄	C₂³
commutator	C₂²	C₂³×C₃×C₄
Frattini, cosocle	C₂×C₄	C₂⁴×C₃
socle	C₂³×C₃	C₂²×C₄
prime radical	C₂²×C₃×D₄	C₄
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₃×C₄ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂²×C₃×M₁₆	C₂	semidirect
3. C₃×#436	C₂	direct (N × H)
4. C₂×C₃×C₄⋊C₈	C₂	semidirect
5. C₂×C₃×C₂²⋊C₈	C₂	semidirect
6. C₂×C₃×C₄×C₈	C₂	semidirect
7. C₂×#436	C₃	direct (N × H)
11. C₂×C₃×M₁₆	C₂²	semidirect
20. C₃×C₄⋊C₈	C₂²	semidirect
21. C₃×C₂²⋊C₈	C₂²	semidirect
22. C₃×C₄×C₈	C₂²	semidirect
23. C₂²×M₁₆	C₂×C₃	semidirect
25. #436	C₂×C₃	direct (N × H)
26. C₂×C₄⋊C₈	C₂×C₃	semidirect
27. C₂×C₂²⋊C₈	C₂×C₃	semidirect
28. C₂×C₄×C₈	C₂×C₃	semidirect
43. C₂×C₃×C₈	D₄	semidirect
54. C₂×M₁₆	C₂²×C₃	semidirect
63. C₄⋊C₈	C₂²×C₃	semidirect
64. C₂²⋊C₈	C₂²×C₃	semidirect
65. C₄×C₈	C₂²×C₃	semidirect
77. C₂×C₃×C₄	M₁₆	semidirect
87. C₃×C₈	C₂×D₄	semidirect
102. C₂×C₈	C₃×D₄	semidirect
121. C₃×C₄	C₂×M₁₆	semidirect
135. C₂×C₄	C₃×M₁₆	semidirect
145. C₈	C₂×C₃×D₄	semidirect
147. C₂×C₃	#436	direct (N × H)
162. C₄	C₂×C₃×M₁₆	semidirect
165. C₃	C₂×#436	direct (N × H)
167. C₂	C₃×#436	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×M₁₆	2
C₂×C₃×C₄×D₄	2
C₃×#436	2
C₂×C₃×C₄⋊C₈	2
C₂×C₃×C₂²⋊C₈	2
C₂×C₃×C₄×C₈	2
C₂×#436	3

Subgroups — C2×C3×#436 #84792

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×#436 #84792