Subgroups — C₂×C₄×C₉⋊H(3,𝔽₃) #14517777

Order 1944.

664 conjugacy classes of subgroups.

The subgroup lattice has 180 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₉⋊H(3,𝔽₃)	C₂	1	1	yes	maximal, characteristic
2	C₄×C₉⋊H(3,𝔽₃)	C₂	1	2	yes	maximal, direct product factor
3	C₂×C₃×C₄×3₋¹⁺²	C₃	1	3	yes	maximal, complemented
4	C₂×C₃×C₄×H(3,𝔽₃)	C₃	1	1	yes	maximal, characteristic
5	C₂×C₃²×C₄×C₉	C₃	1	1	yes	maximal, characteristic, complemented
6	C₂×C₄×C₃²⋊C₉	C₃	1	6	yes	maximal, complemented
7	C₂×C₄×C₃²⋊C₉	C₃	1	2	yes	maximal, complemented
8	C₂×C₉⋊H(3,𝔽₃)	C₄	1	2	yes	direct product factor
9	C₂×C₉⋊H(3,𝔽₃)	C₂²	1	1	yes	characteristic
10	C₂²×C₃×3₋¹⁺²	C₂×C₃	1	3	yes	—
11	C₂²×C₃×H(3,𝔽₃)	C₂×C₃	1	1	yes	characteristic
12	C₂²×C₃²×C₉	C₂×C₃	1	1	yes	characteristic
13	C₃×C₄×3₋¹⁺²	C₂×C₃	1	6	yes	complemented
14	C₃×C₄×H(3,𝔽₃)	C₂×C₃	1	2	yes	—
15	C₃²×C₄×C₉	C₂×C₃	1	2	yes	complemented
16	C₂²×C₃²⋊C₉	C₂×C₃	1	6	yes	—
17	C₂²×C₃²⋊C₉	C₂×C₃	1	2	yes	—
18	C₄×C₃²⋊C₉	C₂×C₃	1	12	yes	complemented
19	C₄×C₃²⋊C₉	C₂×C₃	1	4	yes	complemented
20	C₉⋊H(3,𝔽₃)	C₂×C₄	1	1	yes	characteristic, direct product factor, Sylow
21	C₂×C₃³×C₄	C₃²	1	3	yes	—
22	C₂×C₃³×C₄	C₃²	1	1	yes	characteristic
23	C₂×C₄×3₋¹⁺²	—	3	9	no	—
24	C₂×C₄×H(3,𝔽₃)	—	3	3	no	—
25	C₂×C₃×C₄×C₉	—	3	3	no	—
26	C₂×C₃×C₄×C₉	C₃²	1	2	yes	—
27	C₂×C₃×C₄×C₉	C₃²	1	6	yes	—
28	C₂×C₃×C₄×C₉	C₃²	1	1	yes	characteristic
29	C₂×C₃×3₋¹⁺²	C₃×C₄	1	6	yes	complemented
30	C₂×C₃×3₋¹⁺²	C₂²×C₃	1	3	yes	—
31	C₂×C₃×H(3,𝔽₃)	C₃×C₄	1	2	yes	—
32	C₂×C₃×H(3,𝔽₃)	C₂²×C₃	1	1	yes	characteristic
33	C₂×C₃²×C₉	C₃×C₄	1	2	yes	complemented
34	C₂×C₃²×C₉	C₂²×C₃	1	1	yes	characteristic
35	C₂×C₃²⋊C₉	C₃×C₄	1	12	yes	complemented
36	C₂×C₃²⋊C₉	C₃×C₄	1	4	yes	complemented
37	C₂×C₃²⋊C₉	C₂²×C₃	1	6	yes	—
38	C₂×C₃²⋊C₉	C₂²×C₃	1	2	yes	—
39	C₂²×C₃³	C₂×C₃²	1	3	yes	—
40	C₂²×C₃³	C₂×C₃²	1	1	yes	characteristic
41	C₃³×C₄	C₂×C₃²	1	6	yes	—
42	C₃³×C₄	C₂×C₃²	1	2	yes	—
43	C₂²×3₋¹⁺²	—	3	9	no	—
44	C₂²×H(3,𝔽₃)	—	3	3	no	—
45	C₂²×C₃×C₉	—	3	3	no	—
46	C₂²×C₃×C₉	C₂×C₃²	1	2	yes	—
47	C₂²×C₃×C₉	C₂×C₃²	1	6	yes	—
48	C₂²×C₃×C₉	C₂×C₃²	1	1	yes	characteristic
49	C₄×3₋¹⁺²	—	3	18	no	—
50	C₄×H(3,𝔽₃)	—	3	6	no	—
51	C₃×C₄×C₉	—	3	6	no	—
52	C₃×C₄×C₉	C₂×C₃²	1	4	yes	—
53	C₃×C₄×C₉	C₂×C₃²	1	12	yes	—
54	C₃×C₄×C₉	C₂×C₃²	1	2	yes	—
55	C₃×3₋¹⁺²	C₂×C₃×C₄	1	3	yes	complemented
56	C₃×H(3,𝔽₃)	C₂×C₃×C₄	1	1	yes	characteristic
57	C₃²×C₉	C₂×C₃×C₄	1	1	yes	characteristic, complemented
58	C₃²⋊C₉	C₂×C₃×C₄	1	6	yes	complemented
59	C₃²⋊C₉	C₂×C₃×C₄	1	2	yes	complemented
60	C₂×C₃²×C₄	—	3	1	no	—
61	C₂×C₃²×C₄	—	3	2	no	—
62	C₂×C₃²×C₄	—	3	3	no	—
63	C₂×C₃²×C₄	—	3	3	no	—
64	C₂×C₃²×C₄	—	3	6	no	—
65	C₂×C₃²×C₄	3₋¹⁺²	1	3	yes	complemented
66	C₂×C₃²×C₄	C₃³	1	1	yes	characteristic
67	C₂×C₄×C₉	—	3	2	no	—
68	C₂×C₄×C₉	—	3	6	no	—
69	C₂×C₄×C₉	H(3,𝔽₃)	1	3	yes	complemented
70	C₂×C₃³	C₃²×C₄	1	6	yes	—
71	C₂×C₃³	C₃²×C₄	1	2	yes	—
72	C₂×C₃³	C₂²×C₃²	1	3	yes	—
73	C₂×C₃³	C₂²×C₃²	1	1	yes	characteristic
74	C₂×3₋¹⁺²	—	3	18	no	—
75	C₂×3₋¹⁺²	—	3	9	no	—
76	C₂×H(3,𝔽₃)	—	3	6	no	—
77	C₂×H(3,𝔽₃)	—	3	3	no	—
78	C₂×C₃×C₉	—	3	6	no	—
79	C₂×C₃×C₉	—	3	3	no	—
80	C₂×C₃×C₉	C₃²×C₄	1	4	yes	—
81	C₂×C₃×C₉	C₃²×C₄	1	12	yes	—
82	C₂×C₃×C₉	C₃²×C₄	1	2	yes	—
83	C₂×C₃×C₉	C₂²×C₃²	1	2	yes	—
84	C₂×C₃×C₉	C₂²×C₃²	1	6	yes	—
85	C₂×C₃×C₉	C₂²×C₃²	1	1	yes	characteristic
86	C₂²×C₃²	—	3	1	no	—
87	C₂²×C₃²	—	3	2	no	—
88	C₂²×C₃²	—	3	3	no	—
89	C₂²×C₃²	—	3	3	no	—
90	C₂²×C₃²	—	3	6	no	—
91	C₂²×C₃²	C₂×3₋¹⁺²	1	3	yes	—
92	C₂²×C₃²	C₂×C₃³	1	1	yes	characteristic
93	C₃²×C₄	—	3	2	no	—
94	C₃²×C₄	—	3	4	no	—
95	C₃²×C₄	—	3	6	no	—
96	C₃²×C₄	—	3	6	no	—
97	C₃²×C₄	—	3	12	no	—
98	C₃²×C₄	C₂×3₋¹⁺²	1	6	yes	complemented
99	C₃²×C₄	C₂×C₃³	1	2	yes	—
100	C₂²×C₉	—	3	2	no	—
101	C₂²×C₉	—	3	6	no	—
102	C₂²×C₉	C₂×H(3,𝔽₃)	1	3	yes	—
103	C₄×C₉	—	3	4	no	—
104	C₄×C₉	—	3	12	no	—
105	C₄×C₉	C₂×H(3,𝔽₃)	1	6	yes	complemented
106	C₃³	C₂×C₃²×C₄	1	3	yes	—
107	C₃³	C₂×C₃²×C₄	1	1	yes	characteristic
108	3₋¹⁺²	—	3	9	no	—
109	H(3,𝔽₃)	—	3	3	no	—
110	C₃×C₉	—	3	3	no	—
111	C₃×C₉	C₂×C₃²×C₄	1	2	yes	—
112	C₃×C₉	C₂×C₃²×C₄	1	6	yes	—
113	C₃×C₉	C₂×C₃²×C₄	1	1	yes	characteristic
114	C₂×C₃×C₄	—	3	3	no	—
115	C₂×C₃×C₄	—	9	3	no	—
116	C₂×C₃×C₄	C₃×H(3,𝔽₃)	1	1	yes	characteristic
117	C₂×C₃×C₄	C₃×3₋¹⁺²	1	1	yes	characteristic
118	C₂×C₃×C₄	#737	1	2	yes	—
119	C₂×C₃²	—	3	2	no	—
120	C₂×C₃²	—	3	4	no	—
121	C₂×C₃²	—	3	1	no	—
122	C₂×C₃²	—	3	2	no	—
123	C₂×C₃²	—	3	6	no	—
124	C₂×C₃²	—	3	6	no	—
125	C₂×C₃²	—	3	12	no	—
126	C₂×C₃²	—	3	3	no	—
127	C₂×C₃²	—	3	3	no	—
128	C₂×C₃²	—	3	6	no	—
129	C₂×C₃²	C₄×3₋¹⁺²	1	6	yes	complemented
130	C₂×C₃²	C₂²×3₋¹⁺²	1	3	yes	—
131	C₂×C₃²	C₃³×C₄	1	2	yes	—
132	C₂×C₃²	C₂²×C₃³	1	1	yes	characteristic
133	C₂×C₉	—	3	4	no	—
134	C₂×C₉	—	3	12	no	—
135	C₂×C₉	—	3	2	no	—
136	C₂×C₉	—	3	6	no	—
137	C₂×C₉	C₄×H(3,𝔽₃)	1	6	yes	complemented
138	C₂×C₉	C₂²×H(3,𝔽₃)	1	3	yes	—
139	C₂²×C₃	—	3	3	no	—
140	C₂²×C₃	—	9	3	no	—
141	C₂²×C₃	C₂×C₃×H(3,𝔽₃)	1	1	yes	characteristic
142	C₂²×C₃	C₂×C₃×3₋¹⁺²	1	1	yes	characteristic
143	C₂²×C₃	C₂×#737	1	2	yes	—
144	C₃×C₄	—	3	6	no	—
145	C₃×C₄	—	9	6	no	—
146	C₃×C₄	C₂×C₃×H(3,𝔽₃)	1	2	yes	—
147	C₃×C₄	C₂×C₃×3₋¹⁺²	1	2	yes	—
148	C₃×C₄	C₂×#737	1	4	yes	—
149	C₃²	—	3	1	no	—
150	C₃²	—	3	2	no	—
151	C₃²	—	3	3	no	—
152	C₃²	—	3	3	no	—
153	C₃²	—	3	6	no	—
154	C₃²	C₂×C₄×3₋¹⁺²	1	3	yes	complemented
155	C₃²	C₂×C₃³×C₄	1	1	yes	characteristic
156	C₉	—	3	2	no	—
157	C₉	—	3	6	no	—
158	C₉	C₂×C₄×H(3,𝔽₃)	1	3	yes	complemented
159	C₂×C₄	C₉⋊H(3,𝔽₃)	1	1	yes	characteristic, direct product factor, Sylow
160	C₂×C₃	—	3	6	no	—
161	C₂×C₃	—	3	3	no	—
162	C₂×C₃	—	9	6	no	—
163	C₂×C₃	—	9	3	no	—
164	C₂×C₃	C₃×C₄×H(3,𝔽₃)	1	2	yes	—
165	C₂×C₃	C₃×C₄×3₋¹⁺²	1	2	yes	—
166	C₂×C₃	C₄×#737	1	4	yes	—
167	C₂×C₃	C₂²×C₃×H(3,𝔽₃)	1	1	yes	characteristic
168	C₂×C₃	C₂²×C₃×3₋¹⁺²	1	1	yes	characteristic
169	C₂×C₃	C₂²×#737	1	2	yes	—
170	C₂²	C₂×C₉⋊H(3,𝔽₃)	1	1	yes	characteristic
171	C₄	C₂×C₉⋊H(3,𝔽₃)	1	2	yes	direct product factor
172	C₃	—	3	3	no	—
173	C₃	—	9	3	no	—
174	C₃	C₂×C₃×C₄×H(3,𝔽₃)	1	1	yes	characteristic
175	C₃	C₂×C₃×C₄×3₋¹⁺²	1	1	yes	characteristic
176	C₃	C₂×C₄×#737	1	2	yes	—
177	C₂	C₄×C₉⋊H(3,𝔽₃)	1	2	yes	direct product factor
178	C₂	C₂²×C₉⋊H(3,𝔽₃)	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₃²×C₄	C₃³
commutator	C₃²	C₂×C₃³×C₄
Frattini, cosocle	C₂×C₃²	C₂²×C₃³
socle	C₂²×C₃²	C₂×C₃³
prime radical	C₂²×C₃×H(3,𝔽₃)	C₂×C₃
LCS	G ▷ C₃² ▷ C₁
UCS	C₁ ◁ C₂×C₃²×C₄ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₄×C₉⋊H(3,𝔽₃)	C₂	direct (N × H)
3. C₂×C₃×C₄×3₋¹⁺²	C₃	semidirect
5. C₂×C₃²×C₄×C₉	C₃	semidirect
6. C₂×C₄×C₃²⋊C₉	C₃	semidirect
7. C₂×C₄×C₃²⋊C₉	C₃	semidirect
8. C₂×C₉⋊H(3,𝔽₃)	C₄	direct (N × H)
13. C₃×C₄×3₋¹⁺²	C₂×C₃	semidirect
15. C₃²×C₄×C₉	C₂×C₃	semidirect
18. C₄×C₃²⋊C₉	C₂×C₃	semidirect
19. C₄×C₃²⋊C₉	C₂×C₃	semidirect
20. C₉⋊H(3,𝔽₃)	C₂×C₄	direct (N × H)
29. C₂×C₃×3₋¹⁺²	C₃×C₄	semidirect
33. C₂×C₃²×C₉	C₃×C₄	semidirect
35. C₂×C₃²⋊C₉	C₃×C₄	semidirect
36. C₂×C₃²⋊C₉	C₃×C₄	semidirect
55. C₃×3₋¹⁺²	C₂×C₃×C₄	semidirect
57. C₃²×C₉	C₂×C₃×C₄	semidirect
58. C₃²⋊C₉	C₂×C₃×C₄	semidirect
59. C₃²⋊C₉	C₂×C₃×C₄	semidirect
65. C₂×C₃²×C₄	3₋¹⁺²	semidirect
69. C₂×C₄×C₉	H(3,𝔽₃)	semidirect
98. C₃²×C₄	C₂×3₋¹⁺²	semidirect
105. C₄×C₉	C₂×H(3,𝔽₃)	semidirect
129. C₂×C₃²	C₄×3₋¹⁺²	semidirect
137. C₂×C₉	C₄×H(3,𝔽₃)	semidirect
154. C₃²	C₂×C₄×3₋¹⁺²	semidirect
158. C₉	C₂×C₄×H(3,𝔽₃)	semidirect
159. C₂×C₄	C₉⋊H(3,𝔽₃)	direct (N × H)
171. C₄	C₂×C₉⋊H(3,𝔽₃)	direct (N × H)
177. C₂	C₄×C₉⋊H(3,𝔽₃)	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×C₉⋊H(3,𝔽₃)	2
C₄×C₉⋊H(3,𝔽₃)	2
C₂×C₃×C₄×3₋¹⁺²	3
C₂×C₃×C₄×H(3,𝔽₃)	3
C₂×C₃²×C₄×C₉	3
C₂×C₄×C₃²⋊C₉	3

Subgroups — C2×C4×C9⋊H(3,𝔽3) #14517777

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₄×C₉⋊H(3,𝔽₃) #14517777