Subgroups — C₂×#63378⋊C₃ #13143140

Order 1728.

428 conjugacy classes of subgroups.

The subgroup lattice has 208 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₈×3₋¹⁺²	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄⋊(3₋¹⁺²⋊C₄)	C₂	1	2	yes	maximal
3	#63378⋊C₃	C₂	1	4	yes	maximal, direct product factor
4	C₂×C₃×#818	—	3	1	no	maximal
5	C₂×#63378	C₃	1	1	yes	maximal, characteristic, complemented
6	C₂²×C₄×3₋¹⁺²	C₂²	1	1	yes	characteristic
7	C₂²×3₋¹⁺²⋊C₄	—	2	2	no	—
8	C₂×C₈×3₋¹⁺²	C₄	1	4	yes	complemented
9	C₂×C₈×3₋¹⁺²	C₂²	1	2	yes	—
10	C₄⋊(3₋¹⁺²⋊C₄)	—	2	2	no	—
11	C₄⋊(3₋¹⁺²⋊C₄)	C₂²	1	4	yes	—
12	C₂²×C₃²×C₈	D₃	1	1	yes	characteristic
13	C₂×C₃×C₄⋊Dic₃	—	3	2	no	—
14	C₃×#818	—	3	4	no	—
15	C₂²×C₈×C₉	—	2	1	no	—
16	C₂²×C₈×C₉	C₂×C₃	1	1	yes	characteristic
17	C₂×C₄⋊Dic₉	C₂×C₃	1	2	yes	—
18	#63378	C₂×C₃	1	4	yes	complemented
19	C₂³×3₋¹⁺²	D₄	1	1	yes	characteristic
20	C₂×C₄×3₋¹⁺²	C₂×C₄	1	2	yes	—
21	C₂×C₄×3₋¹⁺²	D₄	1	1	yes	characteristic
22	C₂×C₄×3₋¹⁺²	Q₈	1	2	yes	—
23	C₂×C₄×3₋¹⁺²	C₂³	1	1	yes	characteristic
24	C₂×3₋¹⁺²⋊C₄	—	2	4	no	—
25	C₂×3₋¹⁺²⋊C₄	—	4	4	no	—
26	C₈×3₋¹⁺²	C₂×C₄	1	4	yes	complemented
27	C₂×C₃×#107	—	9	1	no	—
28	C₂×#818	—	3	1	no	—
29	C₂²×C₃²×C₄	C₂×D₃	1	1	yes	characteristic
30	C₂²×C₃×Dic₃	—	6	2	no	—
31	C₂×C₃²×C₈	Dic₃	1	4	yes	—
32	C₂×C₃²×C₈	C₂×D₃	1	2	yes	—
33	C₃×C₄⋊Dic₃	—	3	4	no	—
34	C₃×C₄⋊Dic₃	—	6	2	no	—
35	C₂²×C₄×C₉	—	2	1	no	—
36	C₂²×C₄×C₉	C₂²×C₃	1	1	yes	characteristic
37	C₂²×Dic₉	—	2	2	no	—
38	C₂×C₈×C₉	—	2	4	no	—
39	C₂×C₈×C₉	—	2	2	no	—
40	C₂×C₈×C₉	C₃×C₄	1	4	yes	complemented
41	C₂×C₈×C₉	C₂²×C₃	1	2	yes	—
42	C₄⋊Dic₉	—	2	2	no	—
43	C₄⋊Dic₉	C₂²×C₃	1	4	yes	—
44	C₂²×3₋¹⁺²	C₄⋊C₄	1	2	yes	—
45	C₂²×3₋¹⁺²	D₈	1	2	yes	—
46	C₂²×3₋¹⁺²	Q₁₆	1	2	yes	—
47	C₂²×3₋¹⁺²	C₂×D₄	1	1	yes	characteristic
48	C₄×3₋¹⁺²	C₄⋊C₄	1	2	yes	—
49	C₄×3₋¹⁺²	C₂²×C₄	1	1	yes	characteristic
50	C₄×3₋¹⁺²	C₂×Q₈	1	1	yes	characteristic
51	3₋¹⁺²⋊C₄	—	4	4	no	—
52	C₂²×C₃×C₈	—	3	1	no	—
53	C₂²×C₃×C₈	C₃×D₃	1	1	yes	characteristic
54	C₂×C₃×C₄⋊C₄	—	9	2	no	—
55	C₂×C₄⋊Dic₃	—	3	2	no	—
56	C₃×#107	—	9	4	no	—
57	#818	—	3	4	no	—
58	C₂³×C₃²	D₁₂	1	1	yes	characteristic
59	C₂×C₃²×C₄	Dic₆	1	2	yes	—
60	C₂×C₃²×C₄	D₁₂	1	1	yes	characteristic
61	C₂×C₃²×C₄	C₂×Dic₃	1	2	yes	—
62	C₂×C₃²×C₄	C₂²×D₃	1	1	yes	characteristic
63	C₂×C₃×Dic₃	—	6	4	no	—
64	C₂×C₃×Dic₃	—	12	4	no	—
65	C₂³×C₉	—	2	1	no	—
66	C₂³×C₉	C₃×D₄	1	1	yes	characteristic
67	C₃²×C₈	C₂×Dic₃	1	4	yes	—
68	C₂×C₄×C₉	—	2	2	no	—
69	C₂×C₄×C₉	—	2	1	no	—
70	C₂×C₄×C₉	—	2	2	no	—
71	C₂×C₄×C₉	—	2	1	no	—
72	C₂×C₄×C₉	C₂×C₃×C₄	1	2	yes	—
73	C₂×C₄×C₉	C₃×D₄	1	1	yes	characteristic
74	C₂×C₄×C₉	C₃×Q₈	1	2	yes	—
75	C₂×C₄×C₉	C₂³×C₃	1	1	yes	characteristic
76	C₂×Dic₉	—	2	4	no	—
77	C₂×Dic₉	—	4	4	no	—
78	C₈×C₉	—	2	4	no	—
79	C₈×C₉	C₂×C₃×C₄	1	4	yes	complemented
80	C₂×#107	—	9	1	no	Sylow
81	C₂×3₋¹⁺²	#107	1	4	yes	complemented
82	C₂×3₋¹⁺²	C₂×C₄⋊C₄	1	1	yes	characteristic
83	C₂×3₋¹⁺²	C₂×D₈	1	1	yes	characteristic
84	C₂×3₋¹⁺²	C₂×Q₁₆	1	1	yes	characteristic
85	C₂²×C₃×C₄	—	3	1	no	—
86	C₂²×C₃×C₄	—	18	2	no	—
87	C₂²×C₃×C₄	C₂×C₃×D₃	1	1	yes	characteristic
88	C₂²×Dic₃	—	6	2	no	—
89	C₂×C₃×C₈	—	3	4	no	—
90	C₂×C₃×C₈	—	3	2	no	—
91	C₂×C₃×C₈	C₃×Dic₃	1	4	yes	—
92	C₂×C₃×C₈	C₂×C₃×D₃	1	2	yes	—
93	C₃×C₄⋊C₄	—	9	4	no	—
94	C₃×C₄⋊C₄	—	18	2	no	—
95	C₄⋊Dic₃	—	3	4	no	—
96	C₄⋊Dic₃	—	6	2	no	—
97	C₂²×C₃²	D₂₄	1	2	yes	—
98	C₂²×C₃²	Dic₁₂	1	2	yes	—
99	C₂²×C₃²	C₄⋊Dic₃	1	2	yes	—
100	C₂²×C₃²	C₂×D₁₂	1	1	yes	characteristic
101	C₃²×C₄	C₄⋊Dic₃	1	2	yes	—
102	C₃²×C₄	C₂×Dic₆	1	1	yes	characteristic
103	C₃²×C₄	C₂²×Dic₃	1	1	yes	characteristic
104	C₃×Dic₃	—	12	4	no	—
105	C₂²×C₉	—	2	2	no	—
106	C₂²×C₉	—	2	2	no	—
107	C₂²×C₉	—	2	2	no	—
108	C₂²×C₉	—	2	1	no	—
109	C₂²×C₉	C₃×C₄⋊C₄	1	2	yes	—
110	C₂²×C₉	C₃×D₈	1	2	yes	—
111	C₂²×C₉	C₃×Q₁₆	1	2	yes	—
112	C₂²×C₉	C₂×C₃×D₄	1	1	yes	characteristic
113	C₄×C₉	—	2	2	no	—
114	C₄×C₉	—	2	1	no	—
115	C₄×C₉	—	2	1	no	—
116	C₄×C₉	C₃×C₄⋊C₄	1	2	yes	—
117	C₄×C₉	C₂²×C₃×C₄	1	1	yes	characteristic
118	C₄×C₉	C₂×C₃×Q₈	1	1	yes	characteristic
119	Dic₉	—	4	4	no	—
120	C₂²×C₈	AGL(1,ℤ/9)	1	1	yes	characteristic
121	C₂×C₄⋊C₄	—	9	2	no	—
122	#107	—	9	4	no	—
123	3₋¹⁺²	C₂×#107	1	1	yes	characteristic, complemented, Sylow
124	C₂³×C₃	—	3	1	no	—
125	C₂³×C₃	C₃×D₁₂	1	1	yes	characteristic
126	C₂×C₃×C₄	—	3	2	no	—
127	C₂×C₃×C₄	—	3	1	no	—
128	C₂×C₃×C₄	—	3	2	no	—
129	C₂×C₃×C₄	—	3	1	no	—
130	C₂×C₃×C₄	—	18	4	no	—
131	C₂×C₃×C₄	—	36	4	no	—
132	C₂×C₃×C₄	C₃×Dic₆	1	2	yes	—
133	C₂×C₃×C₄	C₃×D₁₂	1	1	yes	characteristic
134	C₂×C₃×C₄	C₂×C₃×Dic₃	1	2	yes	—
135	C₂×C₃×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
136	C₂×Dic₃	—	6	4	no	—
137	C₂×Dic₃	—	12	4	no	—
138	C₃×C₈	—	3	4	no	—
139	C₃×C₈	C₂×C₃×Dic₃	1	4	yes	—
140	C₂×C₃²	#818	1	4	yes	—
141	C₂×C₃²	C₂×D₂₄	1	1	yes	characteristic
142	C₂×C₃²	C₂×Dic₁₂	1	1	yes	characteristic
143	C₂×C₃²	C₂×C₄⋊Dic₃	1	1	yes	characteristic
144	C₂×C₉	—	2	1	no	—
145	C₂×C₉	—	2	1	no	—
146	C₂×C₉	—	2	1	no	—
147	C₂×C₉	—	2	4	no	—
148	C₂×C₉	C₃×#107	1	4	yes	complemented
149	C₂×C₉	C₂×C₃×C₄⋊C₄	1	1	yes	characteristic
150	C₂×C₉	C₂×C₃×D₈	1	1	yes	characteristic
151	C₂×C₉	C₂×C₃×Q₁₆	1	1	yes	characteristic
152	C₂²×C₄	—	18	2	no	—
153	C₂²×C₄	C₂×AGL(1,ℤ/9)	1	1	yes	characteristic
154	C₂×C₈	3₋¹⁺²⋊C₄	1	4	yes	complemented
155	C₂×C₈	C₂×AGL(1,ℤ/9)	1	2	yes	—
156	C₄⋊C₄	—	9	4	no	—
157	C₄⋊C₄	—	18	2	no	—
158	C₂²×C₃	—	3	1	no	—
159	C₂²×C₃	—	3	2	no	—
160	C₂²×C₃	—	3	2	no	—
161	C₂²×C₃	—	3	2	no	—
162	C₂²×C₃	C₃×D₂₄	1	2	yes	—
163	C₂²×C₃	C₃×Dic₁₂	1	2	yes	—
164	C₂²×C₃	C₃×C₄⋊Dic₃	1	2	yes	—
165	C₂²×C₃	C₂×C₃×D₁₂	1	1	yes	characteristic
166	C₃×C₄	—	3	1	no	—
167	C₃×C₄	—	3	1	no	—
168	C₃×C₄	—	3	2	no	—
169	C₃×C₄	—	36	4	no	—
170	C₃×C₄	C₃×C₄⋊Dic₃	1	2	yes	—
171	C₃×C₄	C₂×C₃×Dic₆	1	1	yes	characteristic
172	C₃×C₄	C₂²×C₃×Dic₃	1	1	yes	characteristic
173	Dic₃	—	12	4	no	—
174	C₃²	C₂×#818	1	1	yes	characteristic
175	C₉	—	2	1	no	—
176	C₉	C₂×C₃×#107	1	1	yes	characteristic, complemented
177	C₂³	C₄⋊AGL(1,ℤ/9)	1	1	yes	characteristic
178	C₂×C₄	—	18	4	no	—
179	C₂×C₄	—	36	4	no	—
180	C₂×C₄	3₋¹⁺²⋊Q₈	1	2	yes	—
181	C₂×C₄	C₄⋊AGL(1,ℤ/9)	1	1	yes	characteristic
182	C₂×C₄	C₂×3₋¹⁺²⋊C₄	1	2	yes	—
183	C₂×C₄	C₂²×AGL(1,ℤ/9)	1	1	yes	characteristic
184	C₈	C₂×3₋¹⁺²⋊C₄	1	4	yes	complemented
185	C₂×C₃	—	3	1	no	—
186	C₂×C₃	—	3	1	no	—
187	C₂×C₃	—	3	1	no	—
188	C₂×C₃	—	3	4	no	—
189	C₂×C₃	C₃×#818	1	4	yes	—
190	C₂×C₃	C₂×C₃×D₂₄	1	1	yes	characteristic
191	C₂×C₃	C₂×C₃×Dic₁₂	1	1	yes	characteristic
192	C₂×C₃	C₂×C₃×C₄⋊Dic₃	1	1	yes	characteristic
193	C₂²	Dic₃₆⋊C₃	1	2	yes	—
194	C₂²	D₇₂⋊C₃	1	2	yes	—
195	C₂²	C₄⋊(3₋¹⁺²⋊C₄)	1	2	yes	—
196	C₂²	C₂×C₄⋊AGL(1,ℤ/9)	1	1	yes	characteristic
197	C₄	—	36	4	no	—
198	C₄	C₄⋊(3₋¹⁺²⋊C₄)	1	2	yes	—
199	C₄	C₂×3₋¹⁺²⋊Q₈	1	1	yes	characteristic
200	C₄	C₂²×3₋¹⁺²⋊C₄	1	1	yes	characteristic
201	C₃	—	3	1	no	—
202	C₃	C₂×C₃×#818	1	1	yes	characteristic
203	C₂	#63378⋊C₃	1	4	yes	direct product factor
204	C₂	C₂×Dic₃₆⋊C₃	1	1	yes	characteristic
205	C₂	C₂×D₇₂⋊C₃	1	1	yes	characteristic
206	C₂	C₂×C₄⋊(3₋¹⁺²⋊C₄)	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₄⋊AGL(1,ℤ/9)
commutator	C₄×C₉	C₂²×C₃×C₄
Frattini	C₂×C₃×C₄	C₂²×C₃×D₃
Fitting	C₂²×C₈×3₋¹⁺²	C₂
socle	C₂³×C₃	C₃×D₁₂
prime radical	C₂³×C₃²	D₁₂
cosocle	C₂×C₄×C₉	C₂³×C₃
hypercenter	C₂²×C₈	AGL(1,ℤ/9)
hypocenter	C₉	C₂×C₃×#107
LCS	G ▷ C₄×C₉ ▷ C₂×C₉ ▷ C₉
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×C₈

Complements

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

N	H ≅ G/N	type
3. #63378⋊C₃	C₂	direct (N × H)
5. C₂×#63378	C₃	semidirect
8. C₂×C₈×3₋¹⁺²	C₄	semidirect
18. #63378	C₂×C₃	semidirect
26. C₈×3₋¹⁺²	C₂×C₄	semidirect
40. C₂×C₈×C₉	C₃×C₄	semidirect
79. C₈×C₉	C₂×C₃×C₄	semidirect
81. C₂×3₋¹⁺²	#107	semidirect
123. 3₋¹⁺²	C₂×#107	semidirect
148. C₂×C₉	C₃×#107	semidirect
154. C₂×C₈	3₋¹⁺²⋊C₄	semidirect
176. C₉	C₂×C₃×#107	semidirect
184. C₈	C₂×3₋¹⁺²⋊C₄	semidirect
203. C₂	#63378⋊C₃	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×C₈×3₋¹⁺²	2
C₂×C₄⋊(3₋¹⁺²⋊C₄)	2
#63378⋊C₃	2
C₂×C₃×#818	3
C₂×#63378	3

Subgroups — C2×#63378⋊C3 #13143140

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#63378⋊C₃ #13143140