Subgroups — #11785953

Order 1152.

390 conjugacy classes of subgroups.

The subgroup lattice has 198 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	C₄²×Dic₉	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄⋊(C₉⋊C₈)	C₂	1	1	yes	maximal, characteristic
4	#10588228	C₂	1	4	yes	maximal
5	#70443	—	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	C₄⋊(C₉⋊C₈)	—	2	2	no	—
15	C₂×C₃×C₄⋊C₈	D₃	1	1	yes	characteristic
16	C₄²×Dic₃	—	3	1	no	—
17	C₂×C₄⋊(C₃⋊C₈)	—	3	1	no	—
18	#4550	—	3	4	no	—
19	C₂²×C₄×C₉	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₄×C₉	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₄×C₉	C₂³	1	1	yes	characteristic
22	C₂²×Dic₉	C₂×C₄	1	2	yes	—
23	C₂²×Dic₉	C₂×C₄	1	2	yes	—
24	C₂×C₈×C₉	—	2	2	no	—
25	C₂×C₈×C₉	—	2	4	no	—
26	C₄²×C₉	D₄	1	2	yes	—
27	C₄²×C₉	Q₈	1	2	yes	—
28	C₄×Dic₉	—	2	2	no	—
29	C₄×Dic₉	—	2	2	no	—
30	C₄×Dic₉	—	2	2	no	—
31	C₄×Dic₉	—	2	4	no	—
32	C₄×Dic₉	C₈	1	4	yes	complemented
33	C₂×C₉⋊C₈	—	2	2	no	—
34	C₂×C₉⋊C₈	—	2	4	no	—
35	#1831	—	9	1	no	Sylow
36	C₂²×C₃×C₈	C₂×D₃	1	2	yes	—
37	C₂×C₃×C₄²	C₂×D₃	1	1	yes	characteristic
38	C₂×C₄×Dic₃	—	3	1	no	—
39	C₂×C₄×Dic₃	—	3	2	no	—
40	C₂×C₄×Dic₃	—	3	1	no	—
41	C₂×C₄×Dic₃	—	3	2	no	—
42	C₂²×C₃⋊C₈	—	3	2	no	—
43	C₃×C₄⋊C₈	—	2	2	no	—
44	C₄⋊(C₃⋊C₈)	—	6	2	no	—
45	C₂³×C₉	C₂²×C₄	1	1	yes	characteristic
46	C₂×C₄×C₉	—	2	2	no	—
47	C₂×C₄×C₉	C₄⋊C₄	1	2	yes	—
48	C₂×C₄×C₉	C₄⋊C₄	1	2	yes	—
49	C₂×C₄×C₉	C₂×C₈	1	2	yes	—
50	C₂×C₄×C₉	M₁₆	1	2	yes	—
51	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
52	C₂×C₄×C₉	C₂×Q₈	1	1	yes	characteristic
53	C₂×C₄×C₉	#41	1	2	yes	—
54	C₂×C₄×C₉	#41	1	2	yes	—
55	C₂×Dic₉	—	2	2	no	—
56	C₂×Dic₉	—	2	2	no	—
57	C₂×Dic₉	—	2	2	no	—
58	C₂×Dic₉	—	2	2	no	—
59	C₂×Dic₉	C₂×C₈	1	4	yes	—
60	C₂×Dic₉	M₁₆	1	4	yes	—
61	C₈×C₉	—	4	2	no	—
62	C₉⋊C₈	—	4	2	no	—
63	C₂×C₄⋊C₈	—	9	1	no	—
64	C₂×C₄⋊C₈	D₉	1	1	yes	characteristic
65	C₄³	—	9	1	no	—
66	#336	—	9	4	no	—
67	C₂²×C₃×C₄	C₄×D₃	1	1	yes	characteristic
68	C₂²×C₃×C₄	C₄×D₃	1	1	yes	characteristic
69	C₂²×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
70	C₂²×Dic₃	—	3	2	no	—
71	C₂²×Dic₃	—	3	2	no	—
72	C₂×C₃×C₈	—	2	4	no	—
73	C₂×C₃×C₈	—	2	2	no	—
74	C₃×C₄²	Dic₆	1	2	yes	—
75	C₃×C₄²	#67	1	2	yes	—
76	C₄×Dic₃	—	3	4	no	—
77	C₄×Dic₃	—	6	4	no	—
78	C₄×Dic₃	—	6	2	no	—
79	C₄×Dic₃	—	6	2	no	—
80	C₄×Dic₃	—	6	2	no	—
81	C₂×C₃⋊C₈	—	6	4	no	—
82	C₂×C₃⋊C₈	—	6	2	no	—
83	C₂²×C₉	C₂×C₄⋊C₄	1	1	yes	characteristic
84	C₂²×C₉	#117	1	2	yes	—
85	C₂²×C₉	C₂²×C₈	1	1	yes	characteristic
86	C₂²×C₉	C₂×M₁₆	1	1	yes	characteristic
87	C₂²×C₉	C₂×M₁₆	1	2	yes	—
88	C₄×C₉	—	2	2	no	—
89	C₄×C₉	C₄⋊C₈	1	4	yes	—
90	C₄×C₉	#124	1	1	yes	characteristic
91	C₄×C₉	#124	1	1	yes	characteristic
92	C₄×C₉	#125	1	2	yes	—
93	Dic₉	—	2	4	no	—
94	Dic₉	—	2	4	no	—
95	C₂²×C₈	—	9	2	no	—
96	C₂²×C₈	C₂×D₉	1	2	yes	—
97	C₂×C₄²	—	9	1	no	—
98	C₂×C₄²	—	9	2	no	—
99	C₂×C₄²	—	9	1	no	—
100	C₂×C₄²	—	9	2	no	—
101	C₂×C₄²	C₂×D₉	1	1	yes	characteristic
102	C₄⋊C₈	—	2	2	no	—
103	C₄⋊C₈	—	18	2	no	—
104	C₂³×C₃	C₂×C₄×D₃	1	1	yes	characteristic
105	C₂×C₃×C₄	—	2	2	no	—
106	C₂×C₃×C₄	D₃×C₈	1	2	yes	—
107	C₂×C₃×C₄	ΓL(1,𝔽₂₅)	1	2	yes	—
108	C₂×C₃×C₄	Dic₃⋊C₄	1	2	yes	—
109	C₂×C₃×C₄	Dic₃⋊C₄	1	2	yes	—
110	C₂×C₃×C₄	C₂×Dic₆	1	1	yes	characteristic
111	C₂×C₃×C₄	#237	1	2	yes	—
112	C₂×C₃×C₄	#239	1	2	yes	—
113	C₂×C₃×C₄	C₂×#67	1	1	yes	characteristic
114	C₂×Dic₃	—	3	4	no	—
115	C₂×Dic₃	—	3	4	no	—
116	C₂×Dic₃	—	6	2	no	—
117	C₂×Dic₃	—	6	2	no	—
118	C₂×Dic₃	—	6	2	no	—
119	C₂×Dic₃	—	6	2	no	—
120	C₃×C₈	—	4	2	no	—
121	C₃⋊C₈	—	12	2	no	—
122	C₂×C₉	#382	1	1	yes	characteristic
123	C₂×C₉	C₂×C₄⋊C₈	1	1	yes	characteristic
124	C₂×C₉	#423	1	1	yes	characteristic
125	C₂×C₉	#431	1	2	yes	—
126	C₂×C₉	#432	1	2	yes	—
127	C₂²×C₄	—	9	2	no	—
128	C₂²×C₄	—	9	2	no	—
129	C₂²×C₄	C₄×D₉	1	1	yes	characteristic
130	C₂²×C₄	C₄×D₉	1	1	yes	characteristic
131	C₂²×C₄	C₂²×D₉	1	1	yes	characteristic
132	C₂×C₈	—	2	4	no	—
133	C₂×C₈	—	2	2	no	—
134	C₂×C₈	—	18	4	no	—
135	C₂×C₈	—	18	2	no	—
136	C₄²	—	9	4	no	—
137	C₄²	—	18	4	no	—
138	C₄²	—	18	2	no	—
139	C₄²	—	18	2	no	—
140	C₄²	—	18	2	no	—
141	C₄²	Dic₁₈	1	2	yes	—
142	C₄²	#611	1	2	yes	—
143	C₂²×C₃	#892	1	2	yes	—
144	C₂²×C₃	C₂×D₃×C₈	1	1	yes	characteristic
145	C₂²×C₃	C₂×ΓL(1,𝔽₂₅)	1	1	yes	characteristic
146	C₂²×C₃	D₃×M₁₆	1	2	yes	—
147	C₂²×C₃	C₂×Dic₃⋊C₄	1	1	yes	characteristic
148	C₃×C₄	—	2	2	no	—
149	C₃×C₄	Dic₃⋊C₈	1	4	yes	—
150	C₃×C₄	#890	1	2	yes	—
151	C₃×C₄	#936	1	1	yes	characteristic
152	C₃×C₄	#947	1	1	yes	characteristic
153	Dic₃	—	6	4	no	—
154	Dic₃	—	6	4	no	—
155	C₉	#1831	1	1	yes	characteristic, complemented, Sylow
156	C₂³	C₂×C₄×D₉	1	1	yes	characteristic
157	C₂×C₄	—	2	2	no	—
158	C₂×C₄	—	9	4	no	—
159	C₂×C₄	—	9	4	no	—
160	C₂×C₄	—	18	2	no	—
161	C₂×C₄	—	18	2	no	—
162	C₂×C₄	—	18	2	no	—
163	C₂×C₄	—	18	2	no	—
164	C₂×C₄	C₈×D₉	1	2	yes	—
165	C₂×C₄	#3663	1	2	yes	—
166	C₂×C₄	Dic₉⋊C₄	1	2	yes	—
167	C₂×C₄	Dic₉⋊C₄	1	2	yes	—
168	C₂×C₄	C₂×Dic₁₈	1	1	yes	characteristic
169	C₂×C₄	#3699	1	2	yes	—
170	C₂×C₄	#3701	1	2	yes	—
171	C₂×C₄	C₂×#611	1	1	yes	characteristic
172	C₈	—	4	2	no	—
173	C₈	—	36	2	no	—
174	C₂×C₃	#4833	1	2	yes	—
175	C₂×C₃	#4835	1	2	yes	—
176	C₂×C₃	#4977	1	1	yes	characteristic
177	C₂×C₃	C₂×Dic₃⋊C₈	1	1	yes	characteristic
178	C₂×C₃	#5118	1	1	yes	characteristic
179	C₂²	#63454	1	2	yes	—
180	C₂²	C₂×C₈×D₉	1	1	yes	characteristic
181	C₂²	C₂×#3663	1	1	yes	characteristic
182	C₂²	M₁₆×D₉	1	2	yes	—
183	C₂²	C₂×Dic₉⋊C₄	1	1	yes	characteristic
184	C₄	—	2	2	no	—
185	C₄	—	18	4	no	—
186	C₄	—	18	4	no	—
187	C₄	Dic₉⋊C₈	1	4	yes	—
188	C₄	#63452	1	2	yes	—
189	C₄	#63498	1	1	yes	characteristic
190	C₄	#63509	1	1	yes	characteristic
191	C₃	#70443	1	1	yes	characteristic
192	C₂	#10588517	1	2	yes	—
193	C₂	#10588519	1	2	yes	—
194	C₂	#10588657	1	1	yes	characteristic
195	C₂	C₂×Dic₉⋊C₈	1	1	yes	characteristic
196	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	#1831
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
32. C₄×Dic₉	C₈	semidirect
155. C₉	#1831	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
C₄²×Dic₉	2
C₂×C₄⋊(C₉⋊C₈)	2
#10588228	2
#70443	3