Subgroups — #11790558

Order 1152.

594 conjugacy classes of subgroups.

The subgroup lattice has 252 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×D₄×C₉	C₂	1	1	yes	maximal, characteristic
2	C₂×#63389	C₂	1	4	yes	maximal, complemented
3	C₂×C₄⋊(C₉⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₄×C₉⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
5	#71614	—	3	1	no	maximal
6	C₂²×D₄×C₉	C₄	1	1	yes	characteristic
7	C₂³×C₄×C₉	C₂²	1	2	yes	—
8	C₄×D₄×C₉	—	2	2	no	—
9	C₄×D₄×C₉	—	2	2	no	—
10	C₂×C₉×C₄⋊C₄	C₄	1	1	yes	characteristic
11	C₂×C₉×C₂²⋊C₄	C₄	1	2	yes	—
12	C₂×C₄²×C₉	C₂²	1	1	yes	characteristic
13	C₂²×C₉⋊C₈	C₂²	1	2	yes	—
14	C₂²×C₉⋊C₈	C₂²	1	2	yes	—
15	#63389	—	2	8	no	—
16	C₄⋊(C₉⋊C₈)	—	2	2	no	—
17	C₄×C₉⋊C₈	—	2	2	no	—
18	C₂×C₃×C₄×D₄	D₃	1	1	yes	characteristic
19	C₂×#830	—	3	4	no	—
20	C₂×C₄⋊(C₃⋊C₈)	—	3	1	no	—
21	C₂×C₄×C₃⋊C₈	—	3	1	no	—
22	C₂⁴×C₉	C₂×C₄	1	2	yes	—
23	C₂×D₄×C₉	—	2	2	no	—
24	C₂×D₄×C₉	—	2	2	no	—
25	C₂×D₄×C₉	C₈	1	4	yes	complemented
26	C₂²×C₄×C₉	—	2	2	no	—
27	C₂²×C₄×C₉	—	2	2	no	—
28	C₂²×C₄×C₉	—	2	4	no	—
29	C₂²×C₄×C₉	—	2	4	no	—
30	C₂²×C₄×C₉	C₂×C₄	1	2	yes	—
31	C₂²×C₄×C₉	C₂×C₄	1	1	yes	characteristic
32	C₂²×C₄×C₉	C₂×C₄	1	1	yes	characteristic
33	C₂²×C₄×C₉	C₂³	1	1	yes	characteristic
34	C₉×C₄⋊C₄	—	2	2	no	—
35	C₉×C₂²⋊C₄	—	2	4	no	—
36	C₄²×C₉	D₄	1	2	yes	—
37	C₄²×C₉	D₄	1	2	yes	—
38	C₂×C₉⋊C₈	—	2	2	no	—
39	C₂×C₉⋊C₈	—	2	2	no	—
40	C₂×C₉⋊C₈	—	2	2	no	—
41	C₂×C₉⋊C₈	—	2	4	no	—
42	C₂×C₉⋊C₈	D₄	1	4	yes	complemented
43	#1954	—	9	1	no	Sylow
44	C₂²×C₃×D₄	Dic₃	1	1	yes	characteristic
45	C₂³×C₃×C₄	C₂×D₃	1	2	yes	—
46	C₃×C₄×D₄	—	2	2	no	—
47	C₃×C₄×D₄	—	2	2	no	—
48	C₂×C₃×C₄⋊C₄	Dic₃	1	1	yes	characteristic
49	C₂×C₃×C₂²⋊C₄	Dic₃	1	2	yes	—
50	C₂×C₃×C₄²	C₂×D₃	1	1	yes	characteristic
51	C₂²×C₃⋊C₈	—	3	2	no	—
52	C₂²×C₃⋊C₈	—	3	2	no	—
53	#830	—	6	8	no	—
54	C₄⋊(C₃⋊C₈)	—	6	2	no	—
55	C₄×C₃⋊C₈	—	6	2	no	—
56	C₂³×C₉	—	2	2	no	—
57	C₂³×C₉	—	2	4	no	—
58	C₂³×C₉	—	2	2	no	—
59	C₂³×C₉	—	2	4	no	—
60	C₂³×C₉	C₂×C₈	1	4	yes	—
61	C₂³×C₉	C₂²×C₄	1	1	yes	characteristic
62	D₄×C₉	—	2	8	no	—
63	C₂×C₄×C₉	—	2	2	no	—
64	C₂×C₄×C₉	—	2	2	no	—
65	C₂×C₄×C₉	—	2	4	no	—
66	C₂×C₄×C₉	—	4	4	no	—
67	C₂×C₄×C₉	—	4	2	no	—
68	C₂×C₄×C₉	—	4	2	no	—
69	C₂×C₄×C₉	C₂²⋊C₄	1	2	yes	—
70	C₂×C₄×C₉	C₂²⋊C₄	1	2	yes	—
71	C₂×C₄×C₉	C₂×C₈	1	2	yes	—
72	C₂×C₄×C₉	M₁₆	1	2	yes	—
73	C₂×C₄×C₉	C₂×D₄	1	2	yes	—
74	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
75	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
76	C₂×C₄×C₉	#41	1	2	yes	—
77	C₉⋊C₈	—	2	4	no	—
78	C₉⋊C₈	—	4	2	no	—
79	C₂×C₄×D₄	D₉	1	1	yes	characteristic
80	C₂×C₄⋊C₈	—	9	1	no	—
81	C₂×C₂²⋊C₈	—	9	4	no	—
82	C₂×C₄×C₈	—	9	1	no	—
83	C₂⁴×C₃	C₂×Dic₃	1	2	yes	—
84	C₂×C₃×D₄	—	2	2	no	—
85	C₂×C₃×D₄	—	2	2	no	—
86	C₂×C₃×D₄	C₃⋊C₈	1	4	yes	—
87	C₂²×C₃×C₄	—	2	4	no	—
88	C₂²×C₃×C₄	—	2	4	no	—
89	C₂²×C₃×C₄	—	2	2	no	—
90	C₂²×C₃×C₄	—	2	2	no	—
91	C₂²×C₃×C₄	C₂×Dic₃	1	2	yes	—
92	C₂²×C₃×C₄	C₂×Dic₃	1	1	yes	characteristic
93	C₂²×C₃×C₄	C₂×Dic₃	1	1	yes	characteristic
94	C₂²×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
95	C₃×C₄⋊C₄	—	2	2	no	—
96	C₃×C₂²⋊C₄	—	2	4	no	—
97	C₃×C₄²	#67	1	2	yes	—
98	C₃×C₄²	#67	1	2	yes	—
99	C₂×C₃⋊C₈	—	3	4	no	—
100	C₂×C₃⋊C₈	—	6	2	no	—
101	C₂×C₃⋊C₈	—	6	2	no	—
102	C₂×C₃⋊C₈	—	6	4	no	—
103	C₂×C₃⋊C₈	—	6	2	no	—
104	C₂²×C₉	—	2	4	no	—
105	C₂²×C₉	—	2	8	no	—
106	C₂²×C₉	—	4	2	no	—
107	C₂²×C₉	—	4	2	no	—
108	C₂²×C₉	—	4	4	no	—
109	C₂²×C₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
110	C₂²×C₉	C₄×D₄	1	2	yes	—
111	C₂²×C₉	C₂²×C₈	1	1	yes	characteristic
112	C₂²×C₉	C₂×M₁₆	1	1	yes	characteristic
113	C₂²×C₉	#131	1	2	yes	—
114	C₄×C₉	—	2	2	no	—
115	C₄×C₉	—	4	2	no	—
116	C₄×C₉	C₂²⋊C₈	1	4	yes	complemented
117	C₄×C₉	#121	1	2	yes	—
118	C₄×C₉	#124	1	1	yes	characteristic
119	C₄×C₉	#127	1	1	yes	characteristic
120	C₂²×D₄	Dic₉	1	1	yes	characteristic
121	C₂³×C₄	C₂×D₉	1	2	yes	—
122	C₂²×C₈	—	9	2	no	—
123	C₂²×C₈	—	9	2	no	—
124	C₄×D₄	—	2	2	no	—
125	C₄×D₄	—	2	2	no	—
126	C₂×C₄⋊C₄	Dic₉	1	1	yes	characteristic
127	C₂×C₂²⋊C₄	Dic₉	1	2	yes	—
128	C₂×C₄²	C₂×D₉	1	1	yes	characteristic
129	C₄⋊C₈	—	18	2	no	—
130	C₂²⋊C₈	—	18	8	no	—
131	C₄×C₈	—	18	2	no	—
132	C₂³×C₃	—	2	2	no	—
133	C₂³×C₃	—	2	4	no	—
134	C₂³×C₃	—	2	4	no	—
135	C₂³×C₃	—	2	2	no	—
136	C₂³×C₃	C₂×C₃⋊C₈	1	4	yes	—
137	C₂³×C₃	C₂²×Dic₃	1	1	yes	characteristic
138	C₃×D₄	—	2	8	no	—
139	C₂×C₃×C₄	—	2	2	no	—
140	C₂×C₃×C₄	—	2	4	no	—
141	C₂×C₃×C₄	—	2	2	no	—
142	C₂×C₃×C₄	—	4	2	no	—
143	C₂×C₃×C₄	—	4	4	no	—
144	C₂×C₃×C₄	—	4	2	no	—
145	C₂×C₃×C₄	C₂×C₃⋊C₈	1	2	yes	—
146	C₂×C₃×C₄	#208	1	2	yes	—
147	C₂×C₃×C₄	#217	1	2	yes	—
148	C₂×C₃×C₄	#217	1	2	yes	—
149	C₂×C₃×C₄	D₃×D₄	1	2	yes	—
150	C₂×C₃×C₄	#237	1	2	yes	—
151	C₂×C₃×C₄	C₂×#67	1	1	yes	characteristic
152	C₂×C₃×C₄	C₂×#67	1	1	yes	characteristic
153	C₃⋊C₈	—	6	4	no	—
154	C₃⋊C₈	—	12	2	no	—
155	C₂×C₉	—	4	4	no	—
156	C₂×C₉	#390	1	1	yes	characteristic
157	C₂×C₉	C₂×C₂²⋊C₈	1	1	yes	characteristic
158	C₂×C₉	#408	1	1	yes	characteristic
159	C₂×C₉	C₈×D₄	1	2	yes	—
160	C₂×C₉	#436	1	2	yes	—
161	C₂⁴	C₂×Dic₉	1	2	yes	—
162	C₂×D₄	—	2	2	no	—
163	C₂×D₄	—	2	2	no	—
164	C₂×D₄	C₉⋊C₈	1	4	yes	complemented
165	C₂²×C₄	—	2	4	no	—
166	C₂²×C₄	—	2	4	no	—
167	C₂²×C₄	—	2	2	no	—
168	C₂²×C₄	—	2	2	no	—
169	C₂²×C₄	C₂×Dic₉	1	2	yes	—
170	C₂²×C₄	C₂×Dic₉	1	1	yes	characteristic
171	C₂²×C₄	C₂×Dic₉	1	1	yes	characteristic
172	C₂²×C₄	C₂²×D₉	1	1	yes	characteristic
173	C₂×C₈	—	9	4	no	—
174	C₂×C₈	—	18	2	no	—
175	C₂×C₈	—	18	2	no	—
176	C₂×C₈	—	18	4	no	—
177	C₂×C₈	—	18	2	no	—
178	C₄⋊C₄	—	2	2	no	—
179	C₂²⋊C₄	—	2	4	no	—
180	C₄²	#611	1	2	yes	—
181	C₄²	#611	1	2	yes	—
182	C₂²×C₃	—	2	4	no	—
183	C₂²×C₃	—	2	8	no	—
184	C₂²×C₃	—	4	2	no	—
185	C₂²×C₃	—	4	2	no	—
186	C₂²×C₃	—	4	4	no	—
187	C₂²×C₃	C₂²×C₃⋊C₈	1	1	yes	characteristic
188	C₂²×C₃	C₂×#208	1	1	yes	characteristic
189	C₂²×C₃	D₄×Dic₃	1	2	yes	—
190	C₂²×C₃	#948	1	2	yes	—
191	C₂²×C₃	C₂×#217	1	1	yes	characteristic
192	C₃×C₄	—	2	2	no	—
193	C₃×C₄	—	4	2	no	—
194	C₃×C₄	#830	1	4	yes	—
195	C₃×C₄	#936	1	1	yes	characteristic
196	C₃×C₄	#938	1	2	yes	—
197	C₃×C₄	#940	1	1	yes	characteristic
198	C₉	#1954	1	1	yes	characteristic, complemented, Sylow
199	C₂³	—	2	4	no	—
200	C₂³	—	2	4	no	—
201	C₂³	—	2	2	no	—
202	C₂³	—	2	2	no	—
203	C₂³	C₂×C₉⋊C₈	1	4	yes	—
204	C₂³	C₂²×Dic₉	1	1	yes	characteristic
205	D₄	—	2	8	no	—
206	C₂×C₄	—	2	2	no	—
207	C₂×C₄	—	2	4	no	—
208	C₂×C₄	—	2	2	no	—
209	C₂×C₄	—	4	2	no	—
210	C₂×C₄	—	4	4	no	—
211	C₂×C₄	—	4	2	no	—
212	C₂×C₄	C₂×C₉⋊C₈	1	2	yes	—
213	C₂×C₄	#3667	1	2	yes	—
214	C₂×C₄	#3676	1	2	yes	—
215	C₂×C₄	#3676	1	2	yes	—
216	C₂×C₄	D₄×D₉	1	2	yes	—
217	C₂×C₄	#3699	1	2	yes	—
218	C₂×C₄	C₂×#611	1	1	yes	characteristic
219	C₂×C₄	C₂×#611	1	1	yes	characteristic
220	C₈	—	18	4	no	—
221	C₈	—	36	2	no	—
222	C₂×C₃	—	4	4	no	—
223	C₂×C₃	D₄×C₃⋊C₈	1	2	yes	—
224	C₂×C₃	#5012	1	2	yes	—
225	C₂×C₃	C₂×#830	1	1	yes	characteristic
226	C₂×C₃	#5221	1	1	yes	characteristic
227	C₂×C₃	#5234	1	1	yes	characteristic
228	C₂²	—	2	8	no	—
229	C₂²	—	2	4	no	—
230	C₂²	—	4	4	no	—
231	C₂²	—	4	2	no	—
232	C₂²	—	4	2	no	—
233	C₂²	C₂²×C₉⋊C₈	1	1	yes	characteristic
234	C₂²	C₂×#3667	1	1	yes	characteristic
235	C₂²	D₄×Dic₉	1	2	yes	—
236	C₂²	#63510	1	2	yes	—
237	C₂²	C₂×#3676	1	1	yes	characteristic
238	C₄	—	2	2	no	—
239	C₄	—	4	2	no	—
240	C₄	#63389	1	4	yes	complemented
241	C₄	#63498	1	1	yes	characteristic
242	C₄	#63500	1	2	yes	—
243	C₄	#63502	1	1	yes	characteristic
244	C₃	#71614	1	1	yes	characteristic
245	C₂	—	4	4	no	—
246	C₂	D₄×C₉⋊C₈	1	2	yes	—
247	C₂	#10588689	1	2	yes	—
248	C₂	C₂×#63389	1	1	yes	characteristic
249	C₂	#10588901	1	1	yes	characteristic
250	C₂	#10588912	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×C₄×D₄×C₉	C₂
socle	C₂³×C₃	C₂²×Dic₃
prime radical	C₂²×C₃×D₄	Dic₃
cosocle	C₂²×C₄×C₉	C₂³
hypercenter	C₂×C₄×D₄	D₉
hypocenter	C₉	#1954
LCS	G ▷ C₂²×C₉ ▷ C₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄×D₄

Complements

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

N	H ≅ G/N	type
2. C₂×#63389	C₂	semidirect
3. C₂×C₄⋊(C₉⋊C₈)	C₂	semidirect
4. C₂×C₄×C₉⋊C₈	C₂	semidirect
25. C₂×D₄×C₉	C₈	semidirect
42. C₂×C₉⋊C₈	D₄	semidirect
116. C₄×C₉	C₂²⋊C₈	semidirect
164. C₂×D₄	C₉⋊C₈	semidirect
198. C₉	#1954	semidirect
240. C₄	#63389	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×D₄×C₉	2
C₂×#63389	2
C₂×C₄⋊(C₉⋊C₈)	2
C₂×C₄×C₉⋊C₈	2
#71614	3