Subgroups — #11786174

Order 1152.

450 conjugacy classes of subgroups.

The subgroup lattice has 201 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₉×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic
2	C₂×Dic₉⋊C₈	C₂	1	4	yes	maximal
3	C₄²×Dic₉	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄⋊(C₉⋊C₈)	C₂	1	1	yes	maximal, characteristic
5	#70445	—	3	1	no	maximal
6	C₂²×C₈×C₉	C₂²	1	2	yes	—
7	C₂×C₄²×C₉	C₂²	1	1	yes	characteristic
8	C₂×C₄×Dic₉	C₄	1	1	yes	characteristic
9	C₂×C₄×Dic₉	C₄	1	2	yes	—
10	C₂×C₄×Dic₉	C₄	1	1	yes	characteristic
11	C₂×C₄×Dic₉	C₂²	1	2	yes	—
12	C₂²×C₉⋊C₈	C₂²	1	2	yes	—
13	C₉×C₄⋊C₈	—	2	2	no	—
14	Dic₉⋊C₈	—	2	8	no	—
15	C₄⋊(C₉⋊C₈)	—	2	2	no	—
16	C₂×C₃×C₄⋊C₈	D₃	1	1	yes	characteristic
17	C₂×Dic₃⋊C₈	—	3	4	no	—
18	C₄²×Dic₃	—	3	1	no	—
19	C₂×C₄⋊(C₃⋊C₈)	—	3	1	no	—
20	C₂²×C₄×C₉	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₄×C₉	C₂×C₄	1	1	yes	characteristic
22	C₂²×C₄×C₉	C₂³	1	1	yes	characteristic
23	C₂²×Dic₉	C₂×C₄	1	2	yes	—
24	C₂²×Dic₉	C₂×C₄	1	2	yes	—
25	C₂×C₈×C₉	—	2	4	no	—
26	C₂×C₈×C₉	—	2	2	no	—
27	C₄²×C₉	D₄	1	2	yes	—
28	C₄²×C₉	Q₈	1	2	yes	—
29	C₄×Dic₉	—	2	2	no	—
30	C₄×Dic₉	—	2	4	no	—
31	C₄×Dic₉	C₈	1	4	yes	complemented
32	C₄×Dic₉	D₄	1	4	yes	—
33	C₄×Dic₉	Q₈	1	4	yes	—
34	C₂×C₉⋊C₈	—	2	2	no	—
35	C₂×C₉⋊C₈	—	2	4	no	—
36	#1842	—	9	1	no	Sylow
37	C₂²×C₃×C₈	C₂×D₃	1	2	yes	—
38	C₂×C₃×C₄²	C₂×D₃	1	1	yes	characteristic
39	C₂×C₄×Dic₃	—	3	1	no	—
40	C₂×C₄×Dic₃	—	3	2	no	—
41	C₂×C₄×Dic₃	—	3	2	no	—
42	C₂×C₄×Dic₃	—	3	1	no	—
43	C₂²×C₃⋊C₈	—	3	2	no	—
44	C₃×C₄⋊C₈	—	2	2	no	—
45	Dic₃⋊C₈	—	6	8	no	—
46	C₄⋊(C₃⋊C₈)	—	6	2	no	—
47	C₂³×C₉	C₂²×C₄	1	1	yes	characteristic
48	C₂×C₄×C₉	—	2	2	no	—
49	C₂×C₄×C₉	C₄⋊C₄	1	2	yes	—
50	C₂×C₄×C₉	C₄⋊C₄	1	2	yes	—
51	C₂×C₄×C₉	C₂×C₈	1	2	yes	—
52	C₂×C₄×C₉	M₁₆	1	2	yes	—
53	C₂×C₄×C₉	C₂×D₄	1	2	yes	—
54	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
55	C₂×C₄×C₉	C₂×Q₈	1	1	yes	characteristic
56	C₂×C₄×C₉	C₂×Q₈	1	2	yes	—
57	C₂×Dic₉	—	2	2	no	—
58	C₂×Dic₉	—	2	2	no	—
59	C₂×Dic₉	C₄⋊C₄	1	4	yes	—
60	C₂×Dic₉	C₄⋊C₄	1	4	yes	—
61	C₂×Dic₉	C₂×C₈	1	4	yes	—
62	C₂×Dic₉	M₁₆	1	4	yes	—
63	C₈×C₉	—	4	2	no	—
64	C₉⋊C₈	—	4	2	no	—
65	C₂×C₄⋊C₈	—	9	4	no	—
66	C₂×C₄⋊C₈	—	9	1	no	—
67	C₂×C₄⋊C₈	D₉	1	1	yes	characteristic
68	C₄³	—	9	1	no	—
69	C₂²×C₃×C₄	C₄×D₃	1	1	yes	characteristic
70	C₂²×C₃×C₄	C₄×D₃	1	1	yes	characteristic
71	C₂²×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
72	C₂²×Dic₃	—	3	2	no	—
73	C₂²×Dic₃	—	3	2	no	—
74	C₂×C₃×C₈	—	2	4	no	—
75	C₂×C₃×C₈	—	2	2	no	—
76	C₃×C₄²	Dic₆	1	2	yes	—
77	C₃×C₄²	#67	1	2	yes	—
78	C₄×Dic₃	—	3	4	no	—
79	C₄×Dic₃	—	3	4	no	—
80	C₄×Dic₃	—	3	4	no	—
81	C₄×Dic₃	—	6	4	no	—
82	C₄×Dic₃	—	6	2	no	—
83	C₂×C₃⋊C₈	—	6	4	no	—
84	C₂×C₃⋊C₈	—	6	2	no	—
85	C₂²×C₉	C₂×C₄⋊C₄	1	2	yes	—
86	C₂²×C₉	C₂×C₄⋊C₄	1	1	yes	characteristic
87	C₂²×C₉	C₂²×C₈	1	1	yes	characteristic
88	C₂²×C₉	C₂×M₁₆	1	1	yes	characteristic
89	C₂²×C₉	C₂×M₁₆	1	2	yes	—
90	C₄×C₉	—	2	2	no	—
91	C₄×C₉	C₄⋊C₈	1	4	yes	complemented
92	C₄×C₉	#127	1	1	yes	characteristic
93	C₄×C₉	C₄⋊Q₈	1	2	yes	—
94	C₄×C₉	C₄⋊Q₈	1	1	yes	characteristic
95	Dic₉	—	2	4	no	—
96	Dic₉	C₄⋊C₈	1	8	yes	complemented
97	C₂²×C₈	—	9	2	no	—
98	C₂²×C₈	C₂×D₉	1	2	yes	—
99	C₂×C₄²	—	9	1	no	—
100	C₂×C₄²	—	9	2	no	—
101	C₂×C₄²	—	9	2	no	—
102	C₂×C₄²	—	9	1	no	—
103	C₂×C₄²	C₂×D₉	1	1	yes	characteristic
104	C₄⋊C₈	—	2	2	no	—
105	C₄⋊C₈	—	18	8	no	—
106	C₄⋊C₈	—	18	2	no	—
107	C₂³×C₃	C₂×C₄×D₃	1	1	yes	characteristic
108	C₂×C₃×C₄	—	2	2	no	—
109	C₂×C₃×C₄	D₃×C₈	1	2	yes	—
110	C₂×C₃×C₄	ΓL(1,𝔽₂₅)	1	2	yes	—
111	C₂×C₃×C₄	Dic₃⋊C₄	1	2	yes	—
112	C₂×C₃×C₄	Dic₃⋊C₄	1	2	yes	—
113	C₂×C₃×C₄	C₂×Dic₆	1	1	yes	characteristic
114	C₂×C₃×C₄	D₃×D₄	1	2	yes	—
115	C₂×C₃×C₄	D₃×Q₈	1	2	yes	—
116	C₂×C₃×C₄	C₂×#67	1	1	yes	characteristic
117	C₂×Dic₃	—	3	4	no	—
118	C₂×Dic₃	—	3	4	no	—
119	C₂×Dic₃	—	3	4	no	—
120	C₂×Dic₃	—	3	4	no	—
121	C₂×Dic₃	—	6	2	no	—
122	C₂×Dic₃	—	6	2	no	—
123	C₃×C₈	—	4	2	no	—
124	C₃⋊C₈	—	12	2	no	—
125	C₂×C₉	#384	1	1	yes	characteristic
126	C₂×C₉	C₂×C₄⋊C₈	1	2	yes	—
127	C₂×C₉	C₂×C₄⋊C₈	1	1	yes	characteristic
128	C₂×C₉	#423	1	2	yes	—
129	C₂×C₉	#423	1	1	yes	characteristic
130	C₂²×C₄	—	9	2	no	—
131	C₂²×C₄	—	9	2	no	—
132	C₂²×C₄	C₄×D₉	1	1	yes	characteristic
133	C₂²×C₄	C₄×D₉	1	1	yes	characteristic
134	C₂²×C₄	C₂²×D₉	1	1	yes	characteristic
135	C₂×C₈	—	2	4	no	—
136	C₂×C₈	—	2	2	no	—
137	C₂×C₈	—	18	4	no	—
138	C₂×C₈	—	18	2	no	—
139	C₄²	—	9	4	no	—
140	C₄²	—	9	4	no	—
141	C₄²	—	9	4	no	—
142	C₄²	—	18	4	no	—
143	C₄²	—	18	2	no	—
144	C₄²	Dic₁₈	1	2	yes	—
145	C₄²	#611	1	2	yes	—
146	C₂²×C₃	D₃×C₄⋊C₄	1	2	yes	—
147	C₂²×C₃	C₂×D₃×C₈	1	1	yes	characteristic
148	C₂²×C₃	C₂×ΓL(1,𝔽₂₅)	1	1	yes	characteristic
149	C₂²×C₃	D₃×M₁₆	1	2	yes	—
150	C₂²×C₃	C₂×Dic₃⋊C₄	1	1	yes	characteristic
151	C₃×C₄	—	2	2	no	—
152	C₃×C₄	Dic₃⋊C₈	1	4	yes	—
153	C₃×C₄	#888	1	2	yes	—
154	C₃×C₄	#940	1	1	yes	characteristic
155	C₃×C₄	#944	1	1	yes	characteristic
156	Dic₃	—	3	8	no	—
157	Dic₃	—	6	4	no	—
158	C₉	#1842	1	1	yes	characteristic, complemented, Sylow
159	C₂³	C₂×C₄×D₉	1	1	yes	characteristic
160	C₂×C₄	—	2	2	no	—
161	C₂×C₄	—	9	4	no	—
162	C₂×C₄	—	9	4	no	—
163	C₂×C₄	—	9	4	no	—
164	C₂×C₄	—	9	4	no	—
165	C₂×C₄	—	18	2	no	—
166	C₂×C₄	—	18	2	no	—
167	C₂×C₄	C₈×D₉	1	2	yes	—
168	C₂×C₄	#3663	1	2	yes	—
169	C₂×C₄	Dic₉⋊C₄	1	2	yes	—
170	C₂×C₄	Dic₉⋊C₄	1	2	yes	—
171	C₂×C₄	C₂×Dic₁₈	1	1	yes	characteristic
172	C₂×C₄	D₄×D₉	1	2	yes	—
173	C₂×C₄	Q₈×D₉	1	2	yes	—
174	C₂×C₄	C₂×#611	1	1	yes	characteristic
175	C₈	—	4	2	no	—
176	C₈	—	36	2	no	—
177	C₂×C₃	D₃×C₄⋊C₈	1	2	yes	—
178	C₂×C₃	#4837	1	2	yes	—
179	C₂×C₃	#4975	1	1	yes	characteristic
180	C₂×C₃	C₂×Dic₃⋊C₈	1	1	yes	characteristic
181	C₂×C₃	#5118	1	1	yes	characteristic
182	C₂²	C₄⋊C₄×D₉	1	2	yes	—
183	C₂²	C₂×C₈×D₉	1	1	yes	characteristic
184	C₂²	C₂×#3663	1	1	yes	characteristic
185	C₂²	M₁₆×D₉	1	2	yes	—
186	C₂²	C₂×Dic₉⋊C₄	1	1	yes	characteristic
187	C₄	—	2	2	no	—
188	C₄	—	9	8	no	—
189	C₄	—	18	4	no	—
190	C₄	Dic₉⋊C₈	1	4	yes	complemented
191	C₄	#63450	1	2	yes	—
192	C₄	#63502	1	1	yes	characteristic
193	C₄	#63506	1	1	yes	characteristic
194	C₃	#70445	1	1	yes	characteristic
195	C₂	D₉×C₄⋊C₈	1	2	yes	—
196	C₂	#10588521	1	2	yes	—
197	C₂	#10588655	1	1	yes	characteristic
198	C₂	C₂×Dic₉⋊C₈	1	1	yes	characteristic
199	C₂	#10588802	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂²×D₉
commutator	C₂²×C₉	C₂²×C₈
Frattini	C₂²×C₃×C₄	C₂²×D₃
Fitting	C₂×C₉×C₄⋊C₈	C₂
socle, prime radical	C₂³×C₃	C₂×C₄×D₃
cosocle	C₂²×C₄×C₉	C₂³
hypercenter	C₂×C₄⋊C₈	D₉
hypocenter	C₉	#1842
LCS	G ▷ C₂²×C₉ ▷ C₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄⋊C₈

Complements

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

N	H ≅ G/N	type
31. C₄×Dic₉	C₈	semidirect
91. C₄×C₉	C₄⋊C₈	semidirect
96. Dic₉	C₄⋊C₈	semidirect
158. C₉	#1842	semidirect
190. C₄	Dic₉⋊C₈	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₉×C₄⋊C₈	2
C₂×Dic₉⋊C₈	2
C₄²×Dic₉	2
C₂×C₄⋊(C₉⋊C₈)	2
#70445	3