Subgroups — #11791654

Order 1152.

390 conjugacy classes of subgroups.

The subgroup lattice has 225 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₉×#125	C₂	1	1	yes	maximal, characteristic
2	C₄×C₄⋊Dic₉	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄⋊(C₉⋊C₈)	C₂	1	1	yes	maximal, characteristic
4	#10588206	C₂	1	2	yes	maximal
5	#10588206	C₂	1	2	yes	maximal
6	#71738	—	3	1	no	maximal
7	C₉×#125	—	2	2	no	—
8	C₉×#125	C₄	1	4	yes	complemented
9	C₂×C₉×C₄⋊C₄	—	2	1	no	—
10	C₂×C₉×C₄⋊C₄	—	2	1	no	—
11	C₂×C₉×C₄⋊C₄	C₂²	1	1	yes	characteristic
12	C₂×C₉×C₄⋊C₄	C₂²	1	1	yes	characteristic
13	C₂×C₄²×C₉	C₂²	1	1	yes	characteristic
14	C₂×C₄⋊Dic₉	C₂²	1	2	yes	—
15	C₂×C₄×Dic₉	—	2	1	no	—
16	C₂²×C₉⋊C₈	C₂²	1	2	yes	—
17	C₉⋊#95	—	2	1	no	—
18	C₄⋊(C₉⋊C₈)	—	2	2	no	—
19	C₂×C₃×#125	D₃	1	1	yes	characteristic
20	C₄×C₄⋊Dic₃	—	3	1	no	—
21	C₂×C₄⋊(C₃⋊C₈)	—	3	1	no	—
22	#4529	—	3	2	no	—
23	#4529	—	3	2	no	—
24	C₂²×C₄×C₉	—	2	1	no	—
25	C₂²×C₄×C₉	—	2	1	no	—
26	C₂²×C₄×C₉	D₄	1	1	yes	characteristic
27	C₂²×C₄×C₉	D₄	1	1	yes	characteristic
28	C₂²×C₄×C₉	C₂³	1	1	yes	characteristic
29	C₂²×Dic₉	—	2	2	no	—
30	C₉×C₄⋊C₄	—	2	4	no	—
31	C₉×C₄⋊C₄	—	2	4	no	—
32	C₉×C₄⋊C₄	—	2	1	no	—
33	C₉×C₄⋊C₄	—	2	1	no	—
34	C₉×C₄⋊C₄	C₂×C₄	1	2	yes	—
35	C₉×C₄⋊C₄	C₂×C₄	1	2	yes	—
36	C₄²×C₉	C₂×C₄	1	2	yes	—
37	C₄²×C₉	D₄	1	2	yes	—
38	C₄⋊Dic₉	—	2	2	no	—
39	C₄⋊Dic₉	Q₈	1	4	yes	—
40	C₄×Dic₉	—	4	2	no	—
41	C₂×C₉⋊C₈	—	2	2	no	—
42	C₂×C₉⋊C₈	—	2	4	no	—
43	#1989	—	9	1	no	Sylow
44	C₃×#125	—	2	2	no	—
45	C₃×#125	Dic₃	1	4	yes	—
46	C₂×C₃×C₄⋊C₄	—	2	1	no	—
47	C₂×C₃×C₄⋊C₄	—	2	1	no	—
48	C₂×C₃×C₄⋊C₄	C₂×D₃	1	1	yes	characteristic
49	C₂×C₃×C₄⋊C₄	C₂×D₃	1	1	yes	characteristic
50	C₂×C₃×C₄²	C₂×D₃	1	1	yes	characteristic
51	C₂×C₄⋊Dic₃	—	3	2	no	—
52	C₂×C₄×Dic₃	—	6	1	no	—
53	C₂²×C₃⋊C₈	—	3	2	no	—
54	C₃⋊#95	—	6	1	no	—
55	C₄⋊(C₃⋊C₈)	—	6	2	no	—
56	C₂³×C₉	C₂×D₄	1	1	yes	characteristic
57	C₂×C₄×C₉	—	2	2	no	—
58	C₂×C₄×C₉	—	2	1	no	—
59	C₂×C₄×C₉	—	2	1	no	—
60	C₂×C₄×C₉	—	2	2	no	—
61	C₂×C₄×C₉	—	2	2	no	—
62	C₂×C₄×C₉	—	4	2	no	—
63	C₂×C₄×C₉	—	4	1	no	—
64	C₂×C₄×C₉	—	4	1	no	—
65	C₂×C₄×C₉	C₂²⋊C₄	1	2	yes	—
66	C₂×C₄×C₉	C₂²⋊C₄	1	2	yes	—
67	C₂×C₄×C₉	C₂²×C₄	1	1	yes	characteristic
68	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
69	C₂×C₄×C₉	C₂×Q₈	1	2	yes	—
70	C₂×C₄×C₉	#41	1	2	yes	—
71	C₂×Dic₉	—	2	4	no	—
72	C₂×Dic₉	—	4	2	no	—
73	C₂×Dic₉	—	4	2	no	—
74	C₉⋊C₈	—	4	2	no	—
75	C₂×#125	D₉	1	1	yes	characteristic
76	C₂×C₄⋊C₈	—	9	1	no	—
77	C₄×C₄⋊C₄	—	9	1	no	—
78	#340	—	9	2	no	—
79	#340	—	9	2	no	—
80	C₂²×C₃×C₄	—	2	1	no	—
81	C₂²×C₃×C₄	—	2	1	no	—
82	C₂²×C₃×C₄	#67	1	1	yes	characteristic
83	C₂²×C₃×C₄	#67	1	1	yes	characteristic
84	C₂²×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
85	C₂²×Dic₃	—	6	2	no	—
86	C₃×C₄⋊C₄	—	2	4	no	—
87	C₃×C₄⋊C₄	—	2	4	no	—
88	C₃×C₄⋊C₄	—	2	1	no	—
89	C₃×C₄⋊C₄	—	2	1	no	—
90	C₃×C₄⋊C₄	C₂×Dic₃	1	2	yes	—
91	C₃×C₄⋊C₄	C₂×Dic₃	1	2	yes	—
92	C₃×C₄²	C₂×Dic₃	1	2	yes	—
93	C₃×C₄²	#67	1	2	yes	—
94	C₄⋊Dic₃	—	3	4	no	—
95	C₄⋊Dic₃	—	6	2	no	—
96	C₄×Dic₃	—	12	2	no	—
97	C₂×C₃⋊C₈	—	6	4	no	—
98	C₂×C₃⋊C₈	—	6	2	no	—
99	C₂²×C₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
100	C₂²×C₉	C₂²⋊Q₈	1	2	yes	—
101	C₂²×C₉	#135	1	2	yes	—
102	C₂²×C₉	AGL(1,ℤ/8)	1	1	yes	characteristic
103	C₂²×C₉	#137	1	1	yes	characteristic
104	C₄×C₉	—	2	2	no	—
105	C₄×C₉	—	2	2	no	—
106	C₄×C₉	—	4	2	no	—
107	C₄×C₉	—	4	2	no	—
108	C₄×C₉	C₄×Q₈	1	2	yes	—
109	C₄×C₉	#124	1	1	yes	characteristic
110	C₄×C₉	C₄⋊Q₈	1	1	yes	characteristic
111	Dic₉	—	4	4	no	—
112	C₂²×C₈	—	9	2	no	—
113	#125	—	2	2	no	—
114	#125	Dic₉	1	4	yes	complemented
115	C₂×C₄⋊C₄	—	2	1	no	—
116	C₂×C₄⋊C₄	—	2	1	no	—
117	C₂×C₄⋊C₄	—	9	2	no	—
118	C₂×C₄⋊C₄	C₂×D₉	1	1	yes	characteristic
119	C₂×C₄⋊C₄	C₂×D₉	1	1	yes	characteristic
120	C₂×C₄²	—	18	1	no	—
121	C₂×C₄²	C₂×D₉	1	1	yes	characteristic
122	C₄⋊C₈	—	18	2	no	—
123	#95	—	18	1	no	—
124	C₂³×C₃	C₂×#67	1	1	yes	characteristic
125	C₂×C₃×C₄	—	2	1	no	—
126	C₂×C₃×C₄	—	2	2	no	—
127	C₂×C₃×C₄	—	2	1	no	—
128	C₂×C₃×C₄	—	2	2	no	—
129	C₂×C₃×C₄	—	2	2	no	—
130	C₂×C₃×C₄	—	4	1	no	—
131	C₂×C₃×C₄	—	4	2	no	—
132	C₂×C₃×C₄	—	4	1	no	—
133	C₂×C₃×C₄	#217	1	2	yes	—
134	C₂×C₃×C₄	#217	1	2	yes	—
135	C₂×C₃×C₄	D₃×Q₈	1	2	yes	—
136	C₂×C₃×C₄	#239	1	2	yes	—
137	C₂×C₃×C₄	C₂²×Dic₃	1	1	yes	characteristic
138	C₂×C₃×C₄	C₂×#67	1	1	yes	characteristic
139	C₂×Dic₃	—	6	4	no	—
140	C₂×Dic₃	—	12	2	no	—
141	C₂×Dic₃	—	12	2	no	—
142	C₃⋊C₈	—	12	2	no	—
143	C₂×C₉	#391	1	1	yes	characteristic
144	C₂×C₉	#416	1	1	yes	characteristic
145	C₂×C₉	#417	1	1	yes	characteristic
146	C₂×C₉	#478	1	2	yes	—
147	C₂×C₉	#479	1	2	yes	—
148	C₂²×C₄	—	2	1	no	—
149	C₂²×C₄	—	2	1	no	—
150	C₂²×C₄	—	18	2	no	—
151	C₂²×C₄	#611	1	1	yes	characteristic
152	C₂²×C₄	#611	1	1	yes	characteristic
153	C₂²×C₄	C₂²×D₉	1	1	yes	characteristic
154	C₂×C₈	—	18	4	no	—
155	C₂×C₈	—	18	2	no	—
156	C₄⋊C₄	—	2	4	no	—
157	C₄⋊C₄	—	2	4	no	—
158	C₄⋊C₄	—	2	1	no	—
159	C₄⋊C₄	—	2	1	no	—
160	C₄⋊C₄	—	9	4	no	—
161	C₄⋊C₄	—	18	2	no	—
162	C₄⋊C₄	C₂×Dic₉	1	2	yes	—
163	C₄⋊C₄	C₂×Dic₉	1	2	yes	—
164	C₄²	—	36	2	no	—
165	C₄²	C₂×Dic₉	1	2	yes	—
166	C₄²	#611	1	2	yes	—
167	C₂²×C₃	#946	1	2	yes	—
168	C₂²×C₃	#949	1	1	yes	characteristic
169	C₂²×C₃	#950	1	2	yes	—
170	C₂²×C₃	#951	1	1	yes	characteristic
171	C₂²×C₃	C₂×#217	1	1	yes	characteristic
172	C₃×C₄	—	2	2	no	—
173	C₃×C₄	—	2	2	no	—
174	C₃×C₄	—	4	2	no	—
175	C₃×C₄	—	4	2	no	—
176	C₃×C₄	#944	1	1	yes	characteristic
177	C₃×C₄	Q₈×Dic₃	1	2	yes	—
178	C₃×C₄	#947	1	1	yes	characteristic
179	Dic₃	—	12	4	no	—
180	C₉	#1989	1	1	yes	characteristic, complemented, Sylow
181	C₂³	C₂×#611	1	1	yes	characteristic
182	C₂×C₄	—	2	1	no	—
183	C₂×C₄	—	2	1	no	—
184	C₂×C₄	—	2	2	no	—
185	C₂×C₄	—	2	2	no	—
186	C₂×C₄	—	2	2	no	—
187	C₂×C₄	—	4	1	no	—
188	C₂×C₄	—	4	1	no	—
189	C₂×C₄	—	4	2	no	—
190	C₂×C₄	—	18	4	no	—
191	C₂×C₄	—	36	2	no	—
192	C₂×C₄	—	36	2	no	—
193	C₂×C₄	#3676	1	2	yes	—
194	C₂×C₄	#3676	1	2	yes	—
195	C₂×C₄	Q₈×D₉	1	2	yes	—
196	C₂×C₄	#3701	1	2	yes	—
197	C₂×C₄	C₂²×Dic₉	1	1	yes	characteristic
198	C₂×C₄	C₂×#611	1	1	yes	characteristic
199	C₈	—	36	2	no	—
200	C₂×C₃	#5063	1	2	yes	—
201	C₂×C₃	#5066	1	2	yes	—
202	C₂×C₃	#5229	1	1	yes	characteristic
203	C₂×C₃	#5232	1	1	yes	characteristic
204	C₂×C₃	#5233	1	1	yes	characteristic
205	C₂²	#63508	1	2	yes	—
206	C₂²	#63511	1	1	yes	characteristic
207	C₂²	#63512	1	1	yes	characteristic
208	C₂²	#63513	1	2	yes	—
209	C₂²	C₂×#3676	1	1	yes	characteristic
210	C₄	—	2	2	no	—
211	C₄	—	2	2	no	—
212	C₄	—	4	2	no	—
213	C₄	—	4	2	no	—
214	C₄	—	36	4	no	—
215	C₄	#63506	1	1	yes	characteristic
216	C₄	Q₈×Dic₉	1	2	yes	—
217	C₄	#63509	1	1	yes	characteristic
218	C₃	#71738	1	1	yes	characteristic
219	C₂	#10588745	1	2	yes	—
220	C₂	#10588747	1	2	yes	—
221	C₂	#10588909	1	1	yes	characteristic
222	C₂	#10588916	1	1	yes	characteristic
223	C₂	#10588917	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×#611
commutator	C₂×C₄×C₉	C₂²×C₄
Frattini	C₂²×C₃×C₄	C₂²×D₃
Fitting	C₂×C₉×#125	C₂
socle, prime radical	C₂³×C₃	C₂×#67
cosocle	C₂²×C₄×C₉	C₂³
hypercenter	C₂×#125	D₉
hypocenter	C₉	#1989
LCS	G ▷ C₂×C₄×C₉ ▷ C₂×C₉ ▷ C₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#125

Complements

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

N	H ≅ G/N	type
8. C₉×#125	C₄	semidirect
114. #125	Dic₉	semidirect
180. C₉	#1989	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₉×#125	2
C₄×C₄⋊Dic₉	2
C₂×C₄⋊(C₉⋊C₈)	2
#10588206	2
#71738	3