Subgroups — #13135898

Order 1728.

350 conjugacy classes of subgroups.

The subgroup lattice has 192 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₄×Q₈×C₉	C₂	1	1	yes	maximal, characteristic
2	C₄⋊(C₃⋊(C₉⋊C₈))	C₂	1	1	yes	maximal, characteristic
3	C₄×C₃⋊(C₉⋊C₈)	C₂	1	1	yes	maximal, characteristic
4	#10589401	—	3	1	no	maximal
5	#10588212	—	3	3	no	maximal
6	C₂×C₃×Q₈×C₉	C₄	1	1	yes	characteristic
7	C₃×C₉×C₄⋊C₄	—	2	1	no	—
8	C₃×C₉×C₄⋊C₄	C₄	1	1	yes	characteristic
9	C₃×C₄²×C₉	—	2	1	no	—
10	C₃×C₄²×C₉	C₂²	1	1	yes	characteristic
11	C₂×C₃⋊(C₉⋊C₈)	—	2	1	no	—
12	C₂×C₃⋊(C₉⋊C₈)	—	2	1	no	—
13	C₃²×C₄×Q₈	D₃	1	1	yes	characteristic
14	C₄⋊(C₃⋊(C₃⋊C₈))	—	3	1	no	—
15	C₄×C₃⋊(C₃⋊C₈)	—	3	1	no	—
16	C₄×Q₈×C₉	D₃	1	3	yes	—
17	C₄⋊(C₉⋊C₈)	—	3	3	no	—
18	C₄×C₉⋊C₈	—	3	3	no	—
19	C₃×Q₈×C₉	—	2	1	no	—
20	C₃×Q₈×C₉	C₈	1	2	yes	complemented
21	C₂×C₃×C₄×C₉	—	2	1	no	—
22	C₂×C₃×C₄×C₉	—	2	1	no	—
23	C₂×C₃×C₄×C₉	C₂×C₄	1	1	yes	characteristic
24	C₂×C₃×C₄×C₉	D₄	1	1	yes	characteristic
25	C₂×C₃×C₄×C₉	D₄	1	1	yes	characteristic
26	C₃⋊(C₉⋊C₈)	—	2	2	no	—
27	C₃⋊(C₉⋊C₈)	—	4	1	no	—
28	#4535	—	9	3	no	—
29	#4535	—	9	1	no	—
30	C₂×C₃²×Q₈	Dic₃	1	1	yes	characteristic
31	C₃²×C₄⋊C₄	—	2	1	no	—
32	C₃²×C₄⋊C₄	Dic₃	1	1	yes	characteristic
33	C₃²×C₄²	—	2	1	no	—
34	C₃²×C₄²	C₂×D₃	1	1	yes	characteristic
35	C₂×C₃⋊(C₃⋊C₈)	—	6	1	no	—
36	C₂×C₃⋊(C₃⋊C₈)	—	6	1	no	—
37	C₂×Q₈×C₉	Dic₃	1	3	yes	—
38	C₉×C₄⋊C₄	—	2	3	no	—
39	C₉×C₄⋊C₄	Dic₃	1	3	yes	—
40	C₄²×C₉	—	2	3	no	—
41	C₄²×C₉	C₂×D₃	1	3	yes	—
42	C₂×C₉⋊C₈	—	6	3	no	—
43	C₂×C₉⋊C₈	—	6	3	no	—
44	C₂²×C₃×C₉	C₂²⋊C₄	1	1	yes	characteristic
45	C₃×C₄×C₉	—	2	2	no	—
46	C₃×C₄×C₉	—	2	1	no	—
47	C₃×C₄×C₉	—	4	1	no	—
48	C₃×C₄×C₉	C₂×C₈	1	1	yes	characteristic
49	C₃×C₄×C₉	M₁₆	1	1	yes	characteristic
50	C₃×C₄×C₉	SD₁₆	1	1	yes	characteristic
51	C₃×C₄×C₉	Q₁₆	1	1	yes	characteristic
52	C₃×C₄×Q₈	D₉	1	3	yes	—
53	C₃×C₄×Q₈	C₃⋊D₃	1	1	yes	characteristic
54	C₄⋊(C₃⋊C₈)	—	9	3	no	—
55	C₄⋊(C₃⋊C₈)	—	9	1	no	—
56	C₄×C₃⋊C₈	—	9	3	no	—
57	C₄×C₃⋊C₈	—	9	1	no	—
58	C₃²×Q₈	—	2	1	no	—
59	C₃²×Q₈	C₃⋊C₈	1	2	yes	—
60	C₂×C₃²×C₄	—	2	1	no	—
61	C₂×C₃²×C₄	—	2	1	no	—
62	C₂×C₃²×C₄	C₂×Dic₃	1	1	yes	characteristic
63	C₂×C₃²×C₄	#67	1	1	yes	characteristic
64	C₂×C₃²×C₄	#67	1	1	yes	characteristic
65	C₃⋊(C₃⋊C₈)	—	6	2	no	—
66	C₃⋊(C₃⋊C₈)	—	12	1	no	—
67	Q₈×C₉	—	2	3	no	—
68	Q₈×C₉	C₃⋊C₈	1	6	yes	complemented
69	C₂×C₄×C₉	—	2	3	no	—
70	C₂×C₄×C₉	—	2	3	no	—
71	C₂×C₄×C₉	C₂×Dic₃	1	3	yes	—
72	C₂×C₄×C₉	#67	1	3	yes	—
73	C₂×C₄×C₉	#67	1	3	yes	—
74	C₉⋊C₈	—	6	6	no	—
75	C₉⋊C₈	—	12	3	no	—
76	Q₈⋊C₈	—	27	1	no	Sylow
77	C₂×C₃×C₉	C₂²⋊C₈	1	1	yes	characteristic
78	C₂×C₃×C₉	#103	1	1	yes	characteristic
79	C₂×C₃×C₉	#104	1	1	yes	characteristic
80	C₂×C₃×Q₈	Dic₉	1	3	yes	—
81	C₂×C₃×Q₈	C₃⋊Dic₃	1	1	yes	characteristic
82	C₃×C₄⋊C₄	—	2	3	no	—
83	C₃×C₄⋊C₄	—	2	1	no	—
84	C₃×C₄⋊C₄	Dic₉	1	3	yes	—
85	C₃×C₄⋊C₄	C₃⋊Dic₃	1	1	yes	characteristic
86	C₃×C₄²	—	2	3	no	—
87	C₃×C₄²	—	2	1	no	—
88	C₃×C₄²	C₂×D₉	1	3	yes	—
89	C₃×C₄²	C₂×C₃⋊D₃	1	1	yes	characteristic
90	C₂×C₃⋊C₈	—	18	3	no	—
91	C₂×C₃⋊C₈	—	18	1	no	—
92	C₂×C₃⋊C₈	—	18	3	no	—
93	C₂×C₃⋊C₈	—	18	1	no	—
94	C₂²×C₃²	#217	1	1	yes	characteristic
95	C₃²×C₄	—	2	2	no	—
96	C₃²×C₄	—	2	1	no	—
97	C₃²×C₄	—	4	1	no	—
98	C₃²×C₄	C₂×C₃⋊C₈	1	1	yes	characteristic
99	C₃²×C₄	#208	1	1	yes	characteristic
100	C₃²×C₄	#215	1	1	yes	characteristic
101	C₃²×C₄	#216	1	1	yes	characteristic
102	C₂²×C₉	#217	1	3	yes	—
103	C₄×C₉	—	2	6	no	—
104	C₄×C₉	—	2	3	no	—
105	C₄×C₉	—	4	3	no	—
106	C₄×C₉	C₂×C₃⋊C₈	1	3	yes	—
107	C₄×C₉	#208	1	3	yes	—
108	C₄×C₉	#215	1	3	yes	—
109	C₄×C₉	#216	1	3	yes	—
110	C₄×Q₈	C₃⋊D₉	1	1	yes	characteristic
111	C₄⋊C₈	—	27	1	no	—
112	C₄×C₈	—	27	1	no	—
113	C₃×C₉	Q₈⋊C₈	1	1	yes	characteristic, complemented, Sylow
114	C₃×Q₈	—	2	3	no	—
115	C₃×Q₈	—	2	1	no	—
116	C₃×Q₈	C₉⋊C₈	1	6	yes	complemented
117	C₃×Q₈	C₃⋊(C₃⋊C₈)	1	2	yes	—
118	C₂×C₃×C₄	—	2	3	no	—
119	C₂×C₃×C₄	—	2	1	no	—
120	C₂×C₃×C₄	—	2	3	no	—
121	C₂×C₃×C₄	—	2	1	no	—
122	C₂×C₃×C₄	C₂×Dic₉	1	3	yes	—
123	C₂×C₃×C₄	#611	1	3	yes	—
124	C₂×C₃×C₄	#611	1	3	yes	—
125	C₂×C₃×C₄	C₂×C₃⋊Dic₃	1	1	yes	characteristic
126	C₂×C₃×C₄	#638	1	1	yes	characteristic
127	C₂×C₃×C₄	#638	1	1	yes	characteristic
128	C₃⋊C₈	—	18	2	no	—
129	C₃⋊C₈	—	18	6	no	—
130	C₃⋊C₈	—	36	3	no	—
131	C₃⋊C₈	—	36	1	no	—
132	C₂×C₃²	#830	1	1	yes	characteristic
133	C₂×C₃²	#835	1	1	yes	characteristic
134	C₂×C₃²	#837	1	1	yes	characteristic
135	C₂×C₉	#830	1	3	yes	—
136	C₂×C₉	#835	1	3	yes	—
137	C₂×C₉	#837	1	3	yes	—
138	C₂×Q₈	C₃⋊Dic₉	1	1	yes	characteristic
139	C₂×C₈	—	54	1	no	—
140	C₂×C₈	—	54	1	no	—
141	C₄⋊C₄	—	2	1	no	—
142	C₄⋊C₄	C₃⋊Dic₉	1	1	yes	characteristic
143	C₄²	—	2	1	no	—
144	C₄²	C₂×C₃⋊D₉	1	1	yes	characteristic
145	C₂²×C₃	#3676	1	3	yes	—
146	C₂²×C₃	#3757	1	1	yes	characteristic
147	C₃×C₄	—	2	6	no	—
148	C₃×C₄	—	2	3	no	—
149	C₃×C₄	—	2	2	no	—
150	C₃×C₄	—	2	1	no	—
151	C₃×C₄	—	4	3	no	—
152	C₃×C₄	—	4	1	no	—
153	C₃×C₄	C₂×C₉⋊C₈	1	3	yes	—
154	C₃×C₄	#3667	1	3	yes	—
155	C₃×C₄	#3674	1	3	yes	—
156	C₃×C₄	#3675	1	3	yes	—
157	C₃×C₄	C₂×C₃⋊(C₃⋊C₈)	1	1	yes	characteristic
158	C₃×C₄	#3748	1	1	yes	characteristic
159	C₃×C₄	#3755	1	1	yes	characteristic
160	C₃×C₄	#3756	1	1	yes	characteristic
161	C₃²	#4535	1	1	yes	characteristic
162	C₉	#4535	1	3	yes	complemented
163	Q₈	—	2	1	no	—
164	Q₈	C₃⋊(C₉⋊C₈)	1	2	yes	complemented
165	C₂×C₄	—	2	1	no	—
166	C₂×C₄	—	2	1	no	—
167	C₂×C₄	C₂×C₃⋊Dic₉	1	1	yes	characteristic
168	C₂×C₄	#6233	1	1	yes	characteristic
169	C₂×C₄	#6233	1	1	yes	characteristic
170	C₈	—	54	2	no	—
171	C₈	—	108	1	no	—
172	C₂×C₃	#63389	1	3	yes	—
173	C₂×C₃	#63395	1	3	yes	—
174	C₂×C₃	#63396	1	3	yes	—
175	C₂×C₃	#63657	1	1	yes	characteristic
176	C₂×C₃	#63662	1	1	yes	characteristic
177	C₂×C₃	#63664	1	1	yes	characteristic
178	C₂²	#88551	1	1	yes	characteristic
179	C₄	—	2	2	no	—
180	C₄	—	2	1	no	—
181	C₄	—	4	1	no	—
182	C₄	C₂×C₃⋊(C₉⋊C₈)	1	1	yes	characteristic
183	C₄	#88532	1	1	yes	characteristic
184	C₄	#88547	1	1	yes	characteristic
185	C₄	#88548	1	1	yes	characteristic
186	C₃	#10588212	1	3	yes	complemented
187	C₃	#10589401	1	1	yes	characteristic
188	C₂	#11720571	1	1	yes	characteristic
189	C₂	#11720582	1	1	yes	characteristic
190	C₂	#11720585	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#6233
commutator	C₃×C₄×C₉	C₂×C₈
Frattini	C₃×C₄²	C₂×C₃⋊D₃
Fitting	C₃×C₄×Q₈×C₉	C₂
socle, prime radical	C₂²×C₃²	#217
cosocle	C₃×C₄²×C₉	C₂²
hypercenter	C₄×Q₈	C₃⋊D₉
hypocenter	C₃×C₉	Q₈⋊C₈
LCS	G ▷ C₃×C₄×C₉ ▷ C₂×C₃×C₉ ▷ C₃×C₉
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×Q₈

Complements

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

N	H ≅ G/N	type
20. C₃×Q₈×C₉	C₈	semidirect
68. Q₈×C₉	C₃⋊C₈	semidirect
113. C₃×C₉	Q₈⋊C₈	semidirect
116. C₃×Q₈	C₉⋊C₈	semidirect
162. C₉	#4535	semidirect
164. Q₈	C₃⋊(C₉⋊C₈)	semidirect
186. C₃	#10588212	semidirect

Maximal subgroups

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

subgroup	index
C₃×C₄×Q₈×C₉	2
C₄⋊(C₃⋊(C₉⋊C₈))	2
C₄×C₃⋊(C₉⋊C₈)	2
#10589401	3
#10588212	3