Subgroups — #11784253

Order 1152.

444 conjugacy classes of subgroups.

The subgroup lattice has 207 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×C₈×Dic₉	C₂	1	2	yes	maximal
3	C₄×C₄⋊Dic₉	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄×C₉⋊C₈	C₂	1	1	yes	maximal, characteristic
5	#10588228	C₂	1	2	yes	maximal
6	#70465	—	3	1	no	maximal
7	C₂²×C₈×C₉	C₂²	1	2	yes	—
8	C₂×C₄²×C₉	C₂²	1	1	yes	characteristic
9	C₂×C₄⋊Dic₉	C₄	1	2	yes	—
10	C₂×C₄×Dic₉	C₂²	1	2	yes	—
11	C₂²×C₉⋊C₈	C₂²	1	1	yes	characteristic
12	C₂²×C₉⋊C₈	C₂²	1	1	yes	characteristic
13	C₉×C₄⋊C₈	C₄	1	4	yes	complemented
14	C₉⋊#95	C₄	1	2	yes	—
15	C₈×Dic₉	—	2	4	no	—
16	C₄×C₉⋊C₈	C₄	1	4	yes	complemented
17	C₂×C₃×C₄⋊C₈	D₃	1	1	yes	characteristic
18	C₂×C₈×Dic₃	—	3	2	no	—
19	C₄×C₄⋊Dic₃	—	3	1	no	—
20	C₂×C₄×C₃⋊C₈	—	3	1	no	—
21	#4550	—	3	2	no	—
22	C₂²×C₄×C₉	C₂×C₄	1	1	yes	characteristic
23	C₂²×C₄×C₉	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₄×C₉	C₂³	1	1	yes	characteristic
25	C₂²×Dic₉	C₂×C₄	1	4	yes	—
26	C₂×C₈×C₉	—	2	2	no	—
27	C₂×C₈×C₉	—	2	2	no	—
28	C₂×C₈×C₉	C₂×C₄	1	4	yes	—
29	C₄²×C₉	C₂×C₄	1	2	yes	—
30	C₄²×C₉	C₂×C₄	1	2	yes	—
31	C₄⋊Dic₉	C₈	1	8	yes	complemented
32	C₄×Dic₉	—	2	2	no	—
33	C₄×Dic₉	—	2	2	no	—
34	C₂×C₉⋊C₈	—	2	1	no	—
35	C₂×C₉⋊C₈	—	2	1	no	—
36	C₂×C₉⋊C₈	C₂×C₄	1	2	yes	—
37	C₂×C₉⋊C₈	C₂×C₄	1	2	yes	—
38	C₂×C₉⋊C₈	D₄	1	2	yes	—
39	C₂×C₉⋊C₈	Q₈	1	2	yes	—
40	C₈×C₄⋊C₄	—	9	1	no	Sylow
41	C₂²×C₃×C₈	C₂×D₃	1	2	yes	—
42	C₂×C₃×C₄²	C₂×D₃	1	1	yes	characteristic
43	C₂×C₄⋊Dic₃	—	3	2	no	—
44	C₂×C₄×Dic₃	—	3	2	no	—
45	C₂²×C₃⋊C₈	—	3	1	no	—
46	C₂²×C₃⋊C₈	—	3	1	no	—
47	C₃×C₄⋊C₈	Dic₃	1	4	yes	—
48	C₃⋊#95	—	3	2	no	—
49	C₈×Dic₃	—	6	4	no	—
50	C₄×C₃⋊C₈	—	3	4	no	—
51	C₂³×C₉	C₂²×C₄	1	1	yes	characteristic
52	C₂×C₄×C₉	—	2	2	no	—
53	C₂×C₄×C₉	C₄²	1	2	yes	—
54	C₂×C₄×C₉	C₄²	1	2	yes	—
55	C₂×C₄×C₉	C₂×C₈	1	4	yes	—
56	C₂×C₄×C₉	C₂²×C₄	1	1	yes	characteristic
57	C₂×C₄×C₉	C₂²×C₄	1	1	yes	characteristic
58	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
59	C₂×C₄×C₉	C₂×Q₈	1	1	yes	characteristic
60	C₂×C₄×C₉	#41	1	1	yes	characteristic
61	C₂×C₄×C₉	#41	1	1	yes	characteristic
62	C₂×Dic₉	—	2	4	no	—
63	C₂×Dic₉	—	2	4	no	—
64	C₂×Dic₉	C₂×C₈	1	8	yes	—
65	C₈×C₉	—	2	4	no	—
66	C₉⋊C₈	—	2	2	no	—
67	C₉⋊C₈	C₄⋊C₄	1	4	yes	complemented
68	C₂×C₄⋊C₈	D₉	1	1	yes	characteristic
69	C₂×C₄×C₈	—	9	1	no	—
70	C₂×C₄×C₈	—	9	2	no	—
71	C₄×C₄⋊C₄	—	9	1	no	—
72	#336	—	9	2	no	—
73	C₂²×C₃×C₄	C₄×D₃	1	1	yes	characteristic
74	C₂²×C₃×C₄	C₄×D₃	1	1	yes	characteristic
75	C₂²×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
76	C₂²×Dic₃	—	3	4	no	—
77	C₂×C₃×C₈	—	2	2	no	—
78	C₂×C₃×C₈	—	2	2	no	—
79	C₂×C₃×C₈	C₂×Dic₃	1	4	yes	—
80	C₃×C₄²	C₄×D₃	1	2	yes	—
81	C₃×C₄²	C₂×Dic₃	1	2	yes	—
82	C₄⋊Dic₃	—	3	8	no	—
83	C₄×Dic₃	—	6	2	no	—
84	C₄×Dic₃	—	6	2	no	—
85	C₂×C₃⋊C₈	—	3	2	no	—
86	C₂×C₃⋊C₈	—	3	2	no	—
87	C₂×C₃⋊C₈	—	3	2	no	—
88	C₂×C₃⋊C₈	—	3	2	no	—
89	C₂×C₃⋊C₈	—	6	1	no	—
90	C₂×C₃⋊C₈	—	6	1	no	—
91	C₂²×C₉	C₂×C₄²	1	1	yes	characteristic
92	C₂²×C₉	C₄×D₄	1	1	yes	characteristic
93	C₂²×C₉	C₄×Q₈	1	1	yes	characteristic
94	C₂²×C₉	C₂²×C₈	1	2	yes	—
95	C₂²×C₉	#131	1	2	yes	—
96	C₄×C₉	—	2	2	no	—
97	C₄×C₉	C₄×C₈	1	4	yes	complemented
98	C₄×C₉	C₂×C₄⋊C₄	1	1	yes	characteristic
99	C₄×C₉	#117	1	1	yes	characteristic
100	C₄×C₉	C₄×D₄	1	1	yes	characteristic
101	C₄×C₉	C₄×Q₈	1	1	yes	characteristic
102	Dic₉	—	2	8	no	—
103	C₂²×C₈	—	9	1	no	—
104	C₂²×C₈	—	9	1	no	—
105	C₂²×C₈	C₂×D₉	1	2	yes	—
106	C₂×C₄⋊C₄	—	9	2	no	—
107	C₂×C₄²	—	9	2	no	—
108	C₂×C₄²	C₂×D₉	1	1	yes	characteristic
109	C₄⋊C₈	Dic₉	1	4	yes	complemented
110	C₄×C₈	—	9	4	no	—
111	C₄×C₈	—	18	4	no	—
112	#95	—	9	2	no	—
113	C₂³×C₃	C₂×C₄×D₃	1	1	yes	characteristic
114	C₂×C₃×C₄	—	2	2	no	—
115	C₂×C₃×C₄	D₃×C₈	1	4	yes	—
116	C₂×C₃×C₄	C₄×Dic₃	1	2	yes	—
117	C₂×C₃×C₄	C₄×Dic₃	1	2	yes	—
118	C₂×C₃×C₄	C₂×C₄×D₃	1	1	yes	characteristic
119	C₂×C₃×C₄	D₃×D₄	1	1	yes	characteristic
120	C₂×C₃×C₄	#237	1	1	yes	characteristic
121	C₂×C₃×C₄	D₃×Q₈	1	1	yes	characteristic
122	C₂×C₃×C₄	#239	1	1	yes	characteristic
123	C₂×C₃×C₄	C₂²×Dic₃	1	1	yes	characteristic
124	C₂×Dic₃	—	3	8	no	—
125	C₂×Dic₃	—	6	4	no	—
126	C₂×Dic₃	—	6	4	no	—
127	C₃×C₈	—	2	4	no	—
128	C₃⋊C₈	—	3	4	no	—
129	C₃⋊C₈	—	6	2	no	—
130	C₂×C₉	C₄×C₄⋊C₄	1	1	yes	characteristic
131	C₂×C₉	C₂×C₄×C₈	1	1	yes	characteristic
132	C₂×C₉	#405	1	1	yes	characteristic
133	C₂×C₉	C₈×D₄	1	2	yes	—
134	C₂×C₉	C₈×Q₈	1	2	yes	—
135	C₂²×C₄	—	9	4	no	—
136	C₂²×C₄	C₄×D₉	1	1	yes	characteristic
137	C₂²×C₄	C₄×D₉	1	1	yes	characteristic
138	C₂²×C₄	C₂²×D₉	1	1	yes	characteristic
139	C₂×C₈	—	2	2	no	—
140	C₂×C₈	—	2	2	no	—
141	C₂×C₈	—	9	2	no	—
142	C₂×C₈	—	9	2	no	—
143	C₂×C₈	—	9	2	no	—
144	C₂×C₈	—	9	2	no	—
145	C₂×C₈	—	18	1	no	—
146	C₂×C₈	—	18	1	no	—
147	C₂×C₈	C₂×Dic₉	1	4	yes	—
148	C₄⋊C₄	—	9	8	no	—
149	C₄²	—	18	2	no	—
150	C₄²	—	18	2	no	—
151	C₄²	C₄×D₉	1	2	yes	—
152	C₄²	C₂×Dic₉	1	2	yes	—
153	C₂²×C₃	#887	1	1	yes	characteristic
154	C₂²×C₃	#893	1	1	yes	characteristic
155	C₂²×C₃	C₂×D₃×C₈	1	2	yes	—
156	C₂²×C₃	#907	1	2	yes	—
157	C₂²×C₃	C₂×C₄×Dic₃	1	1	yes	characteristic
158	C₃×C₄	—	2	2	no	—
159	C₃×C₄	C₈×Dic₃	1	4	yes	—
160	C₃×C₄	D₃×C₄⋊C₄	1	1	yes	characteristic
161	C₃×C₄	#892	1	1	yes	characteristic
162	C₃×C₄	D₄×Dic₃	1	1	yes	characteristic
163	C₃×C₄	Q₈×Dic₃	1	1	yes	characteristic
164	Dic₃	—	6	8	no	—
165	C₉	C₈×C₄⋊C₄	1	1	yes	characteristic, complemented, Sylow
166	C₂³	C₂×C₄×D₉	1	1	yes	characteristic
167	C₂×C₄	—	2	2	no	—
168	C₂×C₄	—	9	8	no	—
169	C₂×C₄	—	18	4	no	—
170	C₂×C₄	—	18	4	no	—
171	C₂×C₄	C₈×D₉	1	4	yes	—
172	C₂×C₄	C₄×Dic₉	1	2	yes	—
173	C₂×C₄	C₄×Dic₉	1	2	yes	—
174	C₂×C₄	C₂×C₄×D₉	1	1	yes	characteristic
175	C₂×C₄	D₄×D₉	1	1	yes	characteristic
176	C₂×C₄	#3699	1	1	yes	characteristic
177	C₂×C₄	Q₈×D₉	1	1	yes	characteristic
178	C₂×C₄	#3701	1	1	yes	characteristic
179	C₂×C₄	C₂²×Dic₉	1	1	yes	characteristic
180	C₈	—	2	4	no	—
181	C₈	—	9	4	no	—
182	C₈	—	18	2	no	—
183	C₂×C₃	#4830	1	2	yes	—
184	C₂×C₃	#4834	1	2	yes	—
185	C₂×C₃	Dic₃×C₄⋊C₄	1	1	yes	characteristic
186	C₂×C₃	C₂×C₈×Dic₃	1	1	yes	characteristic
187	C₂×C₃	#5122	1	1	yes	characteristic
188	C₂²	#63449	1	1	yes	characteristic
189	C₂²	#63455	1	1	yes	characteristic
190	C₂²	C₂×C₈×D₉	1	2	yes	—
191	C₂²	#63469	1	2	yes	—
192	C₂²	C₂×C₄×Dic₉	1	1	yes	characteristic
193	C₄	—	2	2	no	—
194	C₄	—	18	8	no	—
195	C₄	C₈×Dic₉	1	4	yes	complemented
196	C₄	C₄⋊C₄×D₉	1	1	yes	characteristic
197	C₄	#63454	1	1	yes	characteristic
198	C₄	D₄×Dic₉	1	1	yes	characteristic
199	C₄	Q₈×Dic₉	1	1	yes	characteristic
200	C₃	#70465	1	1	yes	characteristic
201	C₂	#10588508	1	2	yes	—
202	C₂	#10588518	1	2	yes	—
203	C₂	C₄⋊C₄×Dic₉	1	1	yes	characteristic
204	C₂	C₂×C₈×Dic₉	1	1	yes	characteristic
205	C₂	#10588797	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂²×D₉
commutator	C₂×C₉	C₂×C₄×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	C₈×C₄⋊C₄
LCS	G ▷ C₂×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
13. C₉×C₄⋊C₈	C₄	semidirect
16. C₄×C₉⋊C₈	C₄	semidirect
31. C₄⋊Dic₉	C₈	semidirect
67. C₉⋊C₈	C₄⋊C₄	semidirect
97. C₄×C₉	C₄×C₈	semidirect
109. C₄⋊C₈	Dic₉	semidirect
165. C₉	C₈×C₄⋊C₄	semidirect
195. C₄	C₈×Dic₉	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×C₈×Dic₉	2
C₄×C₄⋊Dic₉	2
C₂×C₄×C₉⋊C₈	2
#10588228	2
#70465	3