Subgroups — C₃×C₉×#332 #13135997

Order 1728.

320 conjugacy classes of subgroups.

The subgroup lattice has 174 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₉×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic
2	C₃×C₉×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic
3	C₃×C₉×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic
4	C₃²×#332	C₃	1	1	yes	maximal, characteristic
5	C₉×#332	C₃	1	3	yes	maximal, direct product factor
6	C₂×C₃×Q₈×C₉	—	2	1	no	—
7	C₂×C₃×C₈×C₉	—	2	1	no	—
8	C₂×C₃×C₈×C₉	—	2	1	no	—
9	C₃×C₉×C₄⋊C₄	—	2	1	no	—
10	C₃×C₉×C₄⋊C₄	C₄	1	2	yes	—
11	C₃×C₄²×C₉	C₂²	1	1	yes	characteristic
12	C₃²×C₄⋊Q₈	C₂×C₃	1	1	yes	characteristic
13	C₃²×C₄⋊C₈	C₂×C₃	1	1	yes	characteristic
14	C₃²×C₄⋊C₈	C₂×C₃	1	1	yes	characteristic
15	C₉×C₄⋊Q₈	C₂×C₃	1	3	yes	—
16	C₉×C₄⋊C₈	C₂×C₃	1	3	yes	—
17	C₉×C₄⋊C₈	C₂×C₃	1	3	yes	—
18	C₃×Q₈×C₉	—	4	1	no	—
19	C₃×Q₈×C₉	—	4	1	no	—
20	C₂×C₃×C₄×C₉	—	2	2	no	—
21	C₂×C₃×C₄×C₉	C₂×C₄	1	1	yes	characteristic
22	C₂×C₃×C₄×C₉	D₄	1	1	yes	characteristic
23	C₂×C₃×C₄×C₉	D₄	1	1	yes	characteristic
24	C₃×C₈×C₉	—	4	1	no	—
25	C₃×C₈×C₉	—	4	1	no	—
26	C₃×#332	C₉	1	3	yes	direct product factor
27	C₃×#332	C₃²	1	1	yes	characteristic
28	C₂×C₃²×Q₈	—	2	1	no	—
29	C₂×C₃²×C₈	—	2	1	no	—
30	C₂×C₃²×C₈	—	2	1	no	—
31	C₃²×C₄⋊C₄	—	2	1	no	—
32	C₃²×C₄⋊C₄	C₃×C₄	1	2	yes	—
33	C₃²×C₄²	C₂²×C₃	1	1	yes	characteristic
34	C₂×Q₈×C₉	—	2	3	no	—
35	C₂×C₈×C₉	—	2	3	no	—
36	C₂×C₈×C₉	—	2	3	no	—
37	C₉×C₄⋊C₄	—	2	3	no	—
38	C₉×C₄⋊C₄	C₃×C₄	1	6	yes	—
39	C₄²×C₉	C₂²×C₃	1	3	yes	—
40	C₂²×C₃×C₉	C₂²⋊C₄	1	1	yes	characteristic
41	C₃×C₄×C₉	—	2	1	no	—
42	C₃×C₄×C₉	—	4	2	no	—
43	C₃×C₄×C₉	D₈	1	1	yes	characteristic
44	C₃×C₄×C₉	SD₁₆	1	1	yes	characteristic
45	C₃×C₄×C₉	SD₁₆	1	1	yes	characteristic
46	C₃×C₄×C₉	Q₁₆	1	1	yes	characteristic
47	C₃×C₄⋊Q₈	C₂×C₉	1	3	yes	—
48	C₃×C₄⋊Q₈	C₂×C₃²	1	1	yes	characteristic
49	C₃×C₄⋊C₈	C₂×C₉	1	3	yes	—
50	C₃×C₄⋊C₈	C₂×C₉	1	3	yes	—
51	C₃×C₄⋊C₈	C₂×C₃²	1	1	yes	characteristic
52	C₃×C₄⋊C₈	C₂×C₃²	1	1	yes	characteristic
53	C₃²×Q₈	—	4	1	no	—
54	C₃²×Q₈	—	4	1	no	—
55	C₂×C₃²×C₄	—	2	2	no	—
56	C₂×C₃²×C₄	C₂×C₃×C₄	1	1	yes	characteristic
57	C₂×C₃²×C₄	C₃×D₄	1	1	yes	characteristic
58	C₂×C₃²×C₄	C₃×D₄	1	1	yes	characteristic
59	C₃²×C₈	—	4	1	no	—
60	C₃²×C₈	—	4	1	no	—
61	Q₈×C₉	—	4	3	no	—
62	Q₈×C₉	—	4	3	no	—
63	C₂×C₄×C₉	—	2	6	no	—
64	C₂×C₄×C₉	C₂×C₃×C₄	1	3	yes	—
65	C₂×C₄×C₉	C₃×D₄	1	3	yes	—
66	C₂×C₄×C₉	C₃×D₄	1	3	yes	—
67	C₈×C₉	—	4	3	no	—
68	C₈×C₉	—	4	3	no	—
69	#332	C₃×C₉	1	1	yes	characteristic, direct product factor, Sylow
70	C₂×C₃×C₉	#101	1	1	yes	characteristic
71	C₂×C₃×C₉	#102	1	1	yes	characteristic
72	C₂×C₃×C₉	#103	1	1	yes	characteristic
73	C₂×C₃×Q₈	—	2	3	no	—
74	C₂×C₃×Q₈	—	2	1	no	—
75	C₂×C₃×C₈	—	2	3	no	—
76	C₂×C₃×C₈	—	2	3	no	—
77	C₂×C₃×C₈	—	2	1	no	—
78	C₂×C₃×C₈	—	2	1	no	—
79	C₃×C₄⋊C₄	—	2	3	no	—
80	C₃×C₄⋊C₄	—	2	1	no	—
81	C₃×C₄⋊C₄	C₄×C₉	1	6	yes	—
82	C₃×C₄⋊C₄	C₃²×C₄	1	2	yes	—
83	C₃×C₄²	C₂²×C₉	1	3	yes	—
84	C₃×C₄²	C₂²×C₃²	1	1	yes	characteristic
85	C₂²×C₃²	C₃×C₂²⋊C₄	1	1	yes	characteristic
86	C₃²×C₄	—	2	1	no	—
87	C₃²×C₄	—	4	2	no	—
88	C₃²×C₄	C₃×D₈	1	1	yes	characteristic
89	C₃²×C₄	C₃×SD₁₆	1	1	yes	characteristic
90	C₃²×C₄	C₃×SD₁₆	1	1	yes	characteristic
91	C₃²×C₄	C₃×Q₁₆	1	1	yes	characteristic
92	C₂²×C₉	C₃×C₂²⋊C₄	1	3	yes	—
93	C₄×C₉	—	2	3	no	—
94	C₄×C₉	—	4	6	no	—
95	C₄×C₉	C₃×D₈	1	3	yes	—
96	C₄×C₉	C₃×SD₁₆	1	3	yes	—
97	C₄×C₉	C₃×SD₁₆	1	3	yes	—
98	C₄×C₉	C₃×Q₁₆	1	3	yes	—
99	C₄⋊Q₈	C₂×C₃×C₉	1	1	yes	characteristic
100	C₄⋊C₈	C₂×C₃×C₉	1	1	yes	characteristic
101	C₄⋊C₈	C₂×C₃×C₉	1	1	yes	characteristic
102	C₃×C₉	#332	1	1	yes	characteristic, direct product factor, Sylow
103	C₃×Q₈	—	4	3	no	—
104	C₃×Q₈	—	4	3	no	—
105	C₃×Q₈	—	4	1	no	—
106	C₃×Q₈	—	4	1	no	—
107	C₂×C₃×C₄	—	2	6	no	—
108	C₂×C₃×C₄	—	2	2	no	—
109	C₂×C₃×C₄	C₂×C₄×C₉	1	3	yes	—
110	C₂×C₃×C₄	D₄×C₉	1	3	yes	—
111	C₂×C₃×C₄	D₄×C₉	1	3	yes	—
112	C₂×C₃×C₄	C₂×C₃²×C₄	1	1	yes	characteristic
113	C₂×C₃×C₄	C₃²×D₄	1	1	yes	characteristic
114	C₂×C₃×C₄	C₃²×D₄	1	1	yes	characteristic
115	C₃×C₈	—	4	3	no	—
116	C₃×C₈	—	4	3	no	—
117	C₃×C₈	—	4	1	no	—
118	C₃×C₈	—	4	1	no	—
119	C₂×C₃²	C₃×#101	1	1	yes	characteristic
120	C₂×C₃²	C₃×#102	1	1	yes	characteristic
121	C₂×C₃²	C₃×#103	1	1	yes	characteristic
122	C₂×C₉	C₃×#101	1	3	yes	—
123	C₂×C₉	C₃×#102	1	3	yes	—
124	C₂×C₉	C₃×#103	1	3	yes	—
125	C₂×Q₈	—	2	1	no	—
126	C₂×C₈	—	2	1	no	—
127	C₂×C₈	—	2	1	no	—
128	C₄⋊C₄	—	2	1	no	—
129	C₄⋊C₄	C₃×C₄×C₉	1	2	yes	—
130	C₄²	C₂²×C₃×C₉	1	1	yes	characteristic
131	C₂²×C₃	C₉×C₂²⋊C₄	1	3	yes	—
132	C₂²×C₃	C₃²×C₂²⋊C₄	1	1	yes	characteristic
133	C₃×C₄	—	2	3	no	—
134	C₃×C₄	—	2	1	no	—
135	C₃×C₄	—	4	6	no	—
136	C₃×C₄	—	4	2	no	—
137	C₃×C₄	C₉×D₈	1	3	yes	—
138	C₃×C₄	C₉×SD₁₆	1	3	yes	—
139	C₃×C₄	C₉×SD₁₆	1	3	yes	—
140	C₃×C₄	C₉×Q₁₆	1	3	yes	—
141	C₃×C₄	C₃²×D₈	1	1	yes	characteristic
142	C₃×C₄	C₃²×SD₁₆	1	1	yes	characteristic
143	C₃×C₄	C₃²×SD₁₆	1	1	yes	characteristic
144	C₃×C₄	C₃²×Q₁₆	1	1	yes	characteristic
145	C₃²	C₃×#332	1	1	yes	characteristic
146	C₉	C₃×#332	1	3	yes	direct product factor
147	Q₈	—	4	1	no	—
148	Q₈	—	4	1	no	—
149	C₂×C₄	—	2	2	no	—
150	C₂×C₄	C₂×C₃×C₄×C₉	1	1	yes	characteristic
151	C₂×C₄	C₃×D₄×C₉	1	1	yes	characteristic
152	C₂×C₄	C₃×D₄×C₉	1	1	yes	characteristic
153	C₈	—	4	1	no	—
154	C₈	—	4	1	no	—
155	C₂×C₃	C₉×#101	1	3	yes	—
156	C₂×C₃	C₉×#102	1	3	yes	—
157	C₂×C₃	C₉×#103	1	3	yes	—
158	C₂×C₃	C₃²×#101	1	1	yes	characteristic
159	C₂×C₃	C₃²×#102	1	1	yes	characteristic
160	C₂×C₃	C₃²×#103	1	1	yes	characteristic
161	C₂²	C₃×C₉×C₂²⋊C₄	1	1	yes	characteristic
162	C₄	—	2	1	no	—
163	C₄	—	4	2	no	—
164	C₄	C₃×C₉×D₈	1	1	yes	characteristic
165	C₄	C₃×C₉×SD₁₆	1	1	yes	characteristic
166	C₄	C₃×C₉×SD₁₆	1	1	yes	characteristic
167	C₄	C₃×C₉×Q₁₆	1	1	yes	characteristic
168	C₃	C₉×#332	1	3	yes	direct product factor
169	C₃	C₃²×#332	1	1	yes	characteristic
170	C₂	C₃×C₉×#101	1	1	yes	characteristic
171	C₂	C₃×C₉×#102	1	1	yes	characteristic
172	C₂	C₃×C₉×#103	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₃×C₉	C₂²⋊C₄
commutator	C₂×C₄	C₂×C₃×C₄×C₉
Frattini, cosocle	C₃×C₄²	C₂²×C₃²
socle, prime radical	C₂²×C₃²	C₃×C₂²⋊C₄
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₃×C₉ ◁ C₃×C₄²×C₉ ◁ G

Complements

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

N	H ≅ G/N	type
5. C₉×#332	C₃	direct (N × H)
26. C₃×#332	C₉	direct (N × H)
69. #332	C₃×C₉	direct (N × H)
102. C₃×C₉	#332	direct (N × H)
146. C₉	C₃×#332	direct (N × H)
168. C₃	C₉×#332	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃×C₉×C₄⋊Q₈	2
C₃×C₉×C₄⋊C₈	2
C₃²×#332	3
C₉×#332	3

Subgroups — C3×C9×#332 #13135997

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₉×#332 #13135997