Subgroups — C₃×#1803 #72518

Order 384.

308 conjugacy classes of subgroups.

The subgroup lattice has 190 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₄×Q₈	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃×#103	C₂	1	2	yes	maximal
4	C₃×#384	C₂	1	1	yes	maximal, characteristic
5	C₃×#340	C₂	1	2	yes	maximal
6	#1803	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
7	C₂²×C₃×Q₈	C₂²	1	1	yes	characteristic
8	C₂²×C₃×C₈	C₂²	1	2	yes	—
9	C₃×C₄×Q₈	—	2	2	no	—
10	C₃×C₄×Q₈	C₄	1	4	yes	complemented
11	C₂×C₃×C₄⋊C₄	—	2	1	no	—
12	C₂×C₃×C₄⋊C₄	—	2	1	no	—
13	C₂×C₃×C₄⋊C₄	—	2	1	no	—
14	C₂×C₃×C₄⋊C₄	C₂²	1	2	yes	—
15	C₂×C₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
16	C₂×C₃×C₄²	—	2	1	no	—
17	C₂×C₃×C₄²	C₂²	1	1	yes	characteristic
18	C₃×C₄⋊C₈	—	2	2	no	—
19	C₃×#103	—	2	4	no	—
20	C₂×C₄×Q₈	C₂×C₃	1	1	yes	characteristic
21	C₂×C₄⋊C₈	C₂×C₃	1	1	yes	characteristic
22	C₂×#103	C₂×C₃	1	2	yes	—
23	#384	C₂×C₃	1	1	yes	characteristic
24	#340	C₂×C₃	1	2	yes	—
25	C₂×C₃×Q₈	—	2	1	no	—
26	C₂×C₃×Q₈	—	2	1	no	—
27	C₂×C₃×Q₈	—	2	1	no	—
28	C₂×C₃×Q₈	C₂×C₄	1	2	yes	—
29	C₂×C₃×Q₈	D₄	1	2	yes	—
30	C₂×C₃×Q₈	Q₈	1	2	yes	—
31	C₂²×C₃×C₄	—	2	1	no	—
32	C₂²×C₃×C₄	—	2	1	no	—
33	C₂²×C₃×C₄	—	2	2	no	—
34	C₂²×C₃×C₄	D₄	1	1	yes	characteristic
35	C₂²×C₃×C₄	D₄	1	1	yes	characteristic
36	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
37	C₂×C₃×C₈	—	2	2	no	—
38	C₂×C₃×C₈	—	2	2	no	—
39	C₂×C₃×C₈	—	2	2	no	—
40	C₃×C₄⋊C₄	—	2	4	no	—
41	C₃×C₄⋊C₄	—	2	1	no	—
42	C₃×C₄⋊C₄	—	2	2	no	—
43	C₃×C₄⋊C₄	—	2	2	no	—
44	C₃×C₄⋊C₄	—	4	2	no	—
45	C₃×C₄⋊C₄	—	4	2	no	—
46	C₃×C₄⋊C₄	C₂×C₄	1	2	yes	—
47	C₃×C₄²	—	2	4	no	—
48	C₃×C₄²	C₂×C₄	1	2	yes	—
49	C₃×C₄²	D₄	1	2	yes	—
50	C₂²×Q₈	C₂²×C₃	1	1	yes	characteristic
51	C₂²×C₈	C₂²×C₃	1	2	yes	—
52	C₄×Q₈	—	2	2	no	—
53	C₄×Q₈	C₃×C₄	1	4	yes	complemented
54	C₂×C₄⋊C₄	—	2	1	no	—
55	C₂×C₄⋊C₄	—	2	1	no	—
56	C₂×C₄⋊C₄	—	2	1	no	—
57	C₂×C₄⋊C₄	C₂²×C₃	1	2	yes	—
58	C₂×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
59	C₂×C₄²	—	2	1	no	—
60	C₂×C₄²	C₂²×C₃	1	1	yes	characteristic
61	C₄⋊C₈	—	2	2	no	—
62	#103	—	2	4	no	—
63	C₂³×C₃	C₂×D₄	1	1	yes	characteristic
64	C₃×Q₈	—	2	2	no	—
65	C₃×Q₈	—	2	2	no	—
66	C₃×Q₈	—	2	2	no	—
67	C₃×Q₈	C₄⋊C₄	1	4	yes	complemented
68	C₂×C₃×C₄	—	2	2	no	—
69	C₂×C₃×C₄	—	2	2	no	—
70	C₂×C₃×C₄	—	2	2	no	—
71	C₂×C₃×C₄	—	2	1	no	—
72	C₂×C₃×C₄	—	2	1	no	—
73	C₂×C₃×C₄	—	2	2	no	—
74	C₂×C₃×C₄	—	4	1	no	—
75	C₂×C₃×C₄	—	4	1	no	—
76	C₂×C₃×C₄	—	4	2	no	—
77	C₂×C₃×C₄	—	4	2	no	—
78	C₂×C₃×C₄	—	4	2	no	—
79	C₂×C₃×C₄	C₂²⋊C₄	1	2	yes	—
80	C₂×C₃×C₄	C₂²⋊C₄	1	2	yes	—
81	C₂×C₃×C₄	SD₁₆	1	2	yes	—
82	C₂×C₃×C₄	Q₁₆	1	2	yes	—
83	C₂×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
84	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
85	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
86	C₂×C₃×C₄	C₂×Q₈	1	1	yes	characteristic
87	C₂×C₃×C₄	#41	1	1	yes	characteristic
88	C₂×C₃×C₄	#41	1	1	yes	characteristic
89	C₃×C₈	—	4	2	no	—
90	C₂×Q₈	—	2	1	no	—
91	C₂×Q₈	—	2	1	no	—
92	C₂×Q₈	—	2	1	no	—
93	C₂×Q₈	C₂×C₃×C₄	1	2	yes	—
94	C₂×Q₈	C₃×D₄	1	2	yes	—
95	C₂×Q₈	C₃×Q₈	1	2	yes	—
96	C₂²×C₄	—	2	1	no	—
97	C₂²×C₄	—	2	1	no	—
98	C₂²×C₄	—	2	2	no	—
99	C₂²×C₄	C₃×D₄	1	1	yes	characteristic
100	C₂²×C₄	C₃×D₄	1	1	yes	characteristic
101	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
102	C₂×C₈	—	2	2	no	—
103	C₂×C₈	—	2	2	no	—
104	C₂×C₈	—	2	2	no	—
105	C₄⋊C₄	—	2	4	no	—
106	C₄⋊C₄	—	2	1	no	—
107	C₄⋊C₄	—	2	2	no	—
108	C₄⋊C₄	—	2	2	no	—
109	C₄⋊C₄	—	4	2	no	—
110	C₄⋊C₄	—	4	2	no	—
111	C₄⋊C₄	C₂×C₃×C₄	1	2	yes	—
112	C₄²	—	2	4	no	—
113	C₄²	C₂×C₃×C₄	1	2	yes	—
114	C₄²	C₃×D₄	1	2	yes	—
115	C₂²×C₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
116	C₂²×C₃	#121	1	1	yes	characteristic
117	C₂²×C₃	C₂²⋊Q₈	1	1	yes	characteristic
118	C₂²×C₃	C₂×SD₁₆	1	1	yes	characteristic
119	C₂²×C₃	C₂×Q₁₆	1	1	yes	characteristic
120	C₂²×C₃	AGL(1,ℤ/8)	1	1	yes	characteristic
121	C₂²×C₃	#137	1	1	yes	characteristic
122	C₃×C₄	—	2	4	no	—
123	C₃×C₄	—	2	2	no	—
124	C₃×C₄	—	4	2	no	—
125	C₃×C₄	—	8	2	no	—
126	C₃×C₄	#103	1	4	yes	complemented
127	C₃×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
128	C₃×C₄	#117	1	1	yes	characteristic
129	C₃×C₄	#121	1	1	yes	characteristic
130	C₃×C₄	C₂²⋊Q₈	1	1	yes	characteristic
131	C₂³	C₂×C₃×D₄	1	1	yes	characteristic
132	Q₈	—	2	2	no	—
133	Q₈	—	2	2	no	—
134	Q₈	—	2	2	no	—
135	Q₈	C₃×C₄⋊C₄	1	4	yes	complemented
136	C₂×C₄	—	2	2	no	—
137	C₂×C₄	—	2	2	no	—
138	C₂×C₄	—	2	2	no	—
139	C₂×C₄	—	2	1	no	—
140	C₂×C₄	—	2	1	no	—
141	C₂×C₄	—	2	2	no	—
142	C₂×C₄	—	4	1	no	—
143	C₂×C₄	—	4	1	no	—
144	C₂×C₄	—	4	2	no	—
145	C₂×C₄	—	4	2	no	—
146	C₂×C₄	—	4	2	no	—
147	C₂×C₄	C₃×C₂²⋊C₄	1	2	yes	—
148	C₂×C₄	C₃×C₂²⋊C₄	1	2	yes	—
149	C₂×C₄	C₃×SD₁₆	1	2	yes	—
150	C₂×C₄	C₃×Q₁₆	1	2	yes	—
151	C₂×C₄	C₂²×C₃×C₄	1	1	yes	characteristic
152	C₂×C₄	C₂×C₃×D₄	1	1	yes	characteristic
153	C₂×C₄	C₂×C₃×D₄	1	1	yes	characteristic
154	C₂×C₄	C₂×C₃×Q₈	1	1	yes	characteristic
155	C₂×C₄	C₃×#41	1	1	yes	characteristic
156	C₂×C₄	C₃×#41	1	1	yes	characteristic
157	C₈	—	4	2	no	—
158	C₂×C₃	#380	1	1	yes	characteristic
159	C₂×C₃	C₂×#103	1	1	yes	characteristic
160	C₂×C₃	#417	1	1	yes	characteristic
161	C₂×C₃	#460	1	1	yes	characteristic
162	C₂×C₃	#462	1	1	yes	characteristic
163	C₂×C₃	#475	1	1	yes	characteristic
164	C₂×C₃	#477	1	1	yes	characteristic
165	C₂²	C₂×C₃×C₂²⋊C₄	1	1	yes	characteristic
166	C₂²	C₃×#121	1	1	yes	characteristic
167	C₂²	C₃×C₂²⋊Q₈	1	1	yes	characteristic
168	C₂²	C₂×C₃×SD₁₆	1	1	yes	characteristic
169	C₂²	C₂×C₃×Q₁₆	1	1	yes	characteristic
170	C₂²	C₃×AGL(1,ℤ/8)	1	1	yes	characteristic
171	C₂²	C₃×#137	1	1	yes	characteristic
172	C₄	—	2	4	no	—
173	C₄	—	2	2	no	—
174	C₄	—	4	2	no	—
175	C₄	—	8	2	no	—
176	C₄	C₃×#103	1	4	yes	complemented
177	C₄	C₂×C₃×C₄⋊C₄	1	1	yes	characteristic
178	C₄	C₃×#117	1	1	yes	characteristic
179	C₄	C₃×#121	1	1	yes	characteristic
180	C₄	C₃×C₂²⋊Q₈	1	1	yes	characteristic
181	C₃	#1803	1	1	yes	characteristic, direct product factor, Sylow
182	C₂	C₃×#380	1	1	yes	characteristic
183	C₂	C₂×C₃×#103	1	1	yes	characteristic
184	C₂	C₃×#417	1	1	yes	characteristic
185	C₂	C₃×#460	1	1	yes	characteristic
186	C₂	C₃×#462	1	1	yes	characteristic
187	C₂	C₃×#475	1	1	yes	characteristic
188	C₂	C₃×#477	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂³×C₃	C₂×D₄
commutator	C₂×C₄	C₂²×C₃×C₄
Frattini, cosocle	C₂²×C₄	C₂³×C₃
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₃ ◁ C₂×C₃×C₄² ◁ G

Complements

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

N	H ≅ G/N	type
6. #1803	C₃	direct (N × H)
10. C₃×C₄×Q₈	C₄	semidirect
53. C₄×Q₈	C₃×C₄	semidirect
67. C₃×Q₈	C₄⋊C₄	semidirect
126. C₃×C₄	#103	semidirect
135. Q₈	C₃×C₄⋊C₄	semidirect
176. C₄	C₃×#103	semidirect
181. C₃	#1803	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×C₃×C₄×Q₈	2
C₂×C₃×C₄⋊C₈	2
C₂×C₃×#103	2
C₃×#384	2
C₃×#340	2
#1803	3

Subgroups — C3×#1803 #72518

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#1803 #72518