Subgroups — #11783272

Order 1152.

672 conjugacy classes of subgroups.

The subgroup lattice has 282 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₉×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic
2	C₂×#63488	C₂	1	1	yes	maximal, characteristic
3	C₂²×#3667	C₂	1	1	yes	maximal, characteristic
4	#10588206	C₂	1	4	yes	maximal, complemented
5	#71558	—	3	1	no	maximal
6	C₂³×C₄×C₉	—	2	1	no	—
7	C₂³×C₄×C₉	C₂²	1	1	yes	characteristic
8	C₂×C₉×C₄⋊C₄	—	2	1	no	—
9	C₂×C₉×C₄⋊C₄	—	2	2	no	—
10	C₂×C₉×C₄⋊C₄	C₄	1	4	yes	complemented
11	C₂×C₉×C₄⋊C₄	C₂²	1	2	yes	—
12	C₂×#3676	—	2	1	no	—
13	#63488	—	2	2	no	—
14	#63488	C₄	1	4	yes	complemented
15	C₂×C₄⋊Dic₉	C₂²	1	2	yes	—
16	C₂×C₄×Dic₉	—	2	1	no	—
17	C₂×#3667	—	2	2	no	—
18	C₂×#3667	—	2	2	no	—
19	C₂×#3667	C₄	1	4	yes	complemented
20	C₂²×C₉⋊C₈	C₂²	1	2	yes	—
21	C₂²×C₃×C₄⋊C₄	D₃	1	1	yes	characteristic
22	C₂×#926	—	3	1	no	—
23	C₂²×#208	—	3	1	no	—
24	#4529	—	3	4	no	—
25	C₂⁴×C₉	D₄	1	1	yes	characteristic
26	C₂²×C₄×C₉	—	2	4	no	—
27	C₂²×C₄×C₉	—	2	2	no	—
28	C₂²×C₄×C₉	—	4	4	no	—
29	C₂²×C₄×C₉	C₂×C₄	1	2	yes	—
30	C₂²×C₄×C₉	C₂×C₄	1	2	yes	—
31	C₂²×C₄×C₉	C₂×C₄	1	2	yes	—
32	C₂²×C₄×C₉	D₄	1	1	yes	characteristic
33	C₂²×C₄×C₉	D₄	1	2	yes	—
34	C₂²×C₄×C₉	D₄	1	2	yes	—
35	C₂²×C₄×C₉	Q₈	1	2	yes	—
36	C₂²×C₄×C₉	C₂³	1	1	yes	characteristic
37	C₂²×Dic₉	—	2	2	no	—
38	C₉×C₄⋊C₄	—	2	4	no	—
39	C₉×C₄⋊C₄	—	2	2	no	—
40	C₉×C₄⋊C₄	C₂×C₄	1	4	yes	—
41	#3676	—	2	4	no	—
42	C₄⋊Dic₉	—	2	2	no	—
43	C₄⋊Dic₉	C₂×C₄	1	4	yes	complemented
44	C₄×Dic₉	—	2	4	no	—
45	#3667	—	2	8	no	—
46	C₂×C₉⋊C₈	—	2	2	no	—
47	C₂×C₉⋊C₈	—	2	2	no	—
48	C₂×C₉⋊C₈	C₂×C₄	1	4	yes	complemented
49	#1736	—	9	1	no	Sylow
50	C₂³×C₃×C₄	—	2	1	no	—
51	C₂³×C₃×C₄	C₂×D₃	1	1	yes	characteristic
52	C₂×C₃×C₄⋊C₄	—	2	1	no	—
53	C₂×C₃×C₄⋊C₄	—	2	2	no	—
54	C₂×C₃×C₄⋊C₄	Dic₃	1	4	yes	—
55	C₂×C₃×C₄⋊C₄	C₂×D₃	1	2	yes	—
56	C₂×#217	—	6	1	no	—
57	#926	—	3	4	no	—
58	#926	—	6	2	no	—
59	C₂×C₄⋊Dic₃	—	3	2	no	—
60	C₂×C₄×Dic₃	—	6	1	no	—
61	C₂×#208	—	3	4	no	—
62	C₂×#208	—	6	2	no	—
63	C₂×#208	—	6	2	no	—
64	C₂²×C₃⋊C₈	—	3	2	no	—
65	C₂³×C₉	—	2	2	no	—
66	C₂³×C₉	—	2	2	no	—
67	C₂³×C₉	C₂²⋊C₄	1	2	yes	—
68	C₂³×C₉	C₂²⋊C₄	1	2	yes	—
69	C₂³×C₉	C₄⋊C₄	1	2	yes	—
70	C₂³×C₉	C₂×D₄	1	1	yes	characteristic
71	C₂×C₄×C₉	—	2	4	no	—
72	C₂×C₄×C₉	—	4	8	no	—
73	C₂×C₄×C₉	—	4	4	no	—
74	C₂×C₄×C₉	C₄²	1	4	yes	—
75	C₂×C₄×C₉	C₂²⋊C₄	1	2	yes	—
76	C₂×C₄×C₉	C₂²⋊C₄	1	4	yes	—
77	C₂×C₄×C₉	C₂²⋊C₄	1	2	yes	—
78	C₂×C₄×C₉	C₄⋊C₄	1	2	yes	—
79	C₂×C₄×C₉	C₄⋊C₄	1	4	yes	—
80	C₂×C₄×C₉	C₄⋊C₄	1	4	yes	—
81	C₂×C₄×C₉	C₂²×C₄	1	1	yes	characteristic
82	C₂×C₄×C₉	C₂²×C₄	1	1	yes	characteristic
83	C₂×C₄×C₉	C₂²×C₄	1	1	yes	characteristic
84	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
85	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
86	C₂×C₄×C₉	C₂×Q₈	1	1	yes	characteristic
87	C₂×Dic₉	—	2	4	no	—
88	C₂×Dic₉	—	4	2	no	—
89	C₂×Dic₉	—	4	2	no	—
90	C₉⋊C₈	—	2	4	no	—
91	C₂²×M₁₆	—	9	1	no	—
92	C₂×#117	—	9	1	no	—
93	C₂²×C₄⋊C₄	D₉	1	1	yes	characteristic
94	#340	—	9	4	no	—
95	C₂⁴×C₃	#67	1	1	yes	characteristic
96	C₂²×C₃×C₄	—	2	4	no	—
97	C₂²×C₃×C₄	—	2	2	no	—
98	C₂²×C₃×C₄	—	4	4	no	—
99	C₂²×C₃×C₄	Dic₆	1	2	yes	—
100	C₂²×C₃×C₄	C₄×D₃	1	2	yes	—
101	C₂²×C₃×C₄	C₄×D₃	1	2	yes	—
102	C₂²×C₃×C₄	D₁₂	1	2	yes	—
103	C₂²×C₃×C₄	C₂×Dic₃	1	2	yes	—
104	C₂²×C₃×C₄	#67	1	1	yes	characteristic
105	C₂²×C₃×C₄	#67	1	2	yes	—
106	C₂²×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
107	C₂²×Dic₃	—	6	2	no	—
108	C₃×C₄⋊C₄	—	2	4	no	—
109	C₃×C₄⋊C₄	—	2	2	no	—
110	C₃×C₄⋊C₄	C₂×Dic₃	1	4	yes	—
111	#217	—	6	4	no	—
112	C₄⋊Dic₃	—	3	4	no	—
113	C₄⋊Dic₃	—	6	2	no	—
114	C₄×Dic₃	—	6	4	no	—
115	#208	—	6	8	no	—
116	C₂×C₃⋊C₈	—	3	4	no	—
117	C₂×C₃⋊C₈	—	6	2	no	—
118	C₂×C₃⋊C₈	—	6	2	no	—
119	C₂²×C₉	—	2	4	no	—
120	C₂²×C₉	—	2	2	no	—
121	C₂²×C₉	—	2	2	no	—
122	C₂²×C₉	—	2	4	no	—
123	C₂²×C₉	#95	1	4	yes	—
124	C₂²×C₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
125	C₂²×C₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
126	C₂²×C₉	C₂×C₄⋊C₄	1	1	yes	characteristic
127	C₂²×C₉	AGL(1,ℤ/8)	1	2	yes	—
128	C₂²×C₉	#137	1	2	yes	—
129	C₄×C₉	—	4	4	no	—
130	C₄×C₉	#95	1	4	yes	—
131	C₄×C₉	C₂×C₄²	1	1	yes	characteristic
132	C₄×C₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
133	C₄×C₉	C₂×C₄⋊C₄	1	1	yes	characteristic
134	C₄×C₉	C₂×C₄⋊C₄	1	1	yes	characteristic
135	Dic₉	—	4	4	no	—
136	C₂³×C₄	—	2	1	no	—
137	C₂³×C₄	C₂×D₉	1	1	yes	characteristic
138	C₂×M₁₆	—	9	4	no	—
139	C₂×M₁₆	—	18	2	no	—
140	C₂×M₁₆	—	18	2	no	—
141	C₂²×C₈	—	9	2	no	—
142	#117	—	9	4	no	—
143	#117	—	18	2	no	—
144	C₂×C₄⋊C₄	—	2	1	no	—
145	C₂×C₄⋊C₄	—	2	2	no	—
146	C₂×C₄⋊C₄	—	9	2	no	—
147	C₂×C₄⋊C₄	Dic₉	1	4	yes	complemented
148	C₂×C₄⋊C₄	C₂×D₉	1	2	yes	—
149	C₂×C₂²⋊C₄	—	18	1	no	—
150	C₂×C₄²	—	18	1	no	—
151	C₂³×C₃	—	2	2	no	—
152	C₂³×C₃	—	2	2	no	—
153	C₂³×C₃	Dic₃⋊C₄	1	2	yes	—
154	C₂³×C₃	(C₂×D₃)⋊C₄	1	2	yes	—
155	C₂³×C₃	#217	1	2	yes	—
156	C₂³×C₃	C₂×#67	1	1	yes	characteristic
157	C₂×C₃×C₄	—	2	4	no	—
158	C₂×C₃×C₄	—	4	8	no	—
159	C₂×C₃×C₄	—	4	4	no	—
160	C₂×C₃×C₄	C₄×Dic₃	1	4	yes	—
161	C₂×C₃×C₄	Dic₃⋊C₄	1	2	yes	—
162	C₂×C₃×C₄	Dic₃⋊C₄	1	4	yes	—
163	C₂×C₃×C₄	C₄⋊Dic₃	1	4	yes	—
164	C₂×C₃×C₄	(C₂×D₃)⋊C₄	1	2	yes	—
165	C₂×C₃×C₄	(C₂×D₃)⋊C₄	1	4	yes	—
166	C₂×C₃×C₄	#217	1	2	yes	—
167	C₂×C₃×C₄	C₂×Dic₆	1	1	yes	characteristic
168	C₂×C₃×C₄	C₂×C₄×D₃	1	1	yes	characteristic
169	C₂×C₃×C₄	C₂×C₄×D₃	1	1	yes	characteristic
170	C₂×C₃×C₄	C₂×D₁₂	1	1	yes	characteristic
171	C₂×C₃×C₄	C₂²×Dic₃	1	1	yes	characteristic
172	C₂×C₃×C₄	C₂×#67	1	1	yes	characteristic
173	C₂×Dic₃	—	6	4	no	—
174	C₂×Dic₃	—	12	2	no	—
175	C₂×Dic₃	—	12	2	no	—
176	C₃⋊C₈	—	6	4	no	—
177	C₂×C₉	—	2	4	no	—
178	C₂×C₉	C₂×#95	1	1	yes	characteristic
179	C₂×C₉	#417	1	2	yes	—
180	C₂×C₉	#418	1	1	yes	characteristic
181	C₂×C₉	#419	1	1	yes	characteristic
182	C₂×C₉	#428	1	2	yes	—
183	C₂⁴	#611	1	1	yes	characteristic
184	C₂²×C₄	—	2	4	no	—
185	C₂²×C₄	—	2	2	no	—
186	C₂²×C₄	—	4	4	no	—
187	C₂²×C₄	—	18	2	no	—
188	C₂²×C₄	Dic₁₈	1	2	yes	—
189	C₂²×C₄	C₄×D₉	1	2	yes	—
190	C₂²×C₄	C₄×D₉	1	2	yes	—
191	C₂²×C₄	D₃₆	1	2	yes	—
192	C₂²×C₄	C₂×Dic₉	1	2	yes	—
193	C₂²×C₄	#611	1	1	yes	characteristic
194	C₂²×C₄	#611	1	2	yes	—
195	C₂²×C₄	C₂²×D₉	1	1	yes	characteristic
196	M₁₆	—	18	8	no	—
197	C₂×C₈	—	9	4	no	—
198	C₂×C₈	—	18	2	no	—
199	C₂×C₈	—	18	2	no	—
200	C₄⋊C₄	—	2	4	no	—
201	C₄⋊C₄	—	2	2	no	—
202	C₄⋊C₄	—	9	4	no	—
203	C₄⋊C₄	—	18	2	no	—
204	C₄⋊C₄	C₂×Dic₉	1	4	yes	—
205	C₂²⋊C₄	—	18	4	no	—
206	C₄²	—	18	4	no	—
207	C₂²×C₃	—	2	2	no	—
208	C₂²×C₃	—	2	4	no	—
209	C₂²×C₃	—	2	4	no	—
210	C₂²×C₃	—	2	2	no	—
211	C₂²×C₃	C₃⋊#95	1	4	yes	—
212	C₂²×C₃	C₂×Dic₃⋊C₄	1	1	yes	characteristic
213	C₂²×C₃	C₂×(C₂×D₃)⋊C₄	1	1	yes	characteristic
214	C₂²×C₃	#932	1	2	yes	—
215	C₂²×C₃	#942	1	2	yes	—
216	C₂²×C₃	C₂×#217	1	1	yes	characteristic
217	C₃×C₄	—	4	4	no	—
218	C₃×C₄	C₃⋊#95	1	4	yes	—
219	C₃×C₄	C₂×C₄×Dic₃	1	1	yes	characteristic
220	C₃×C₄	C₂×Dic₃⋊C₄	1	1	yes	characteristic
221	C₃×C₄	C₂×C₄⋊Dic₃	1	1	yes	characteristic
222	C₃×C₄	C₂×(C₂×D₃)⋊C₄	1	1	yes	characteristic
223	Dic₃	—	12	4	no	—
224	C₉	#1736	1	1	yes	characteristic, complemented, Sylow
225	C₂³	—	2	2	no	—
226	C₂³	—	2	2	no	—
227	C₂³	Dic₉⋊C₄	1	2	yes	—
228	C₂³	(C₂×D₉)⋊C₄	1	2	yes	—
229	C₂³	#3676	1	2	yes	—
230	C₂³	C₂×#611	1	1	yes	characteristic
231	C₂×C₄	—	2	4	no	—
232	C₂×C₄	—	4	8	no	—
233	C₂×C₄	—	4	4	no	—
234	C₂×C₄	—	18	4	no	—
235	C₂×C₄	—	36	2	no	—
236	C₂×C₄	—	36	2	no	—
237	C₂×C₄	C₄×Dic₉	1	4	yes	—
238	C₂×C₄	Dic₉⋊C₄	1	2	yes	—
239	C₂×C₄	Dic₉⋊C₄	1	4	yes	—
240	C₂×C₄	C₄⋊Dic₉	1	4	yes	—
241	C₂×C₄	(C₂×D₉)⋊C₄	1	4	yes	—
242	C₂×C₄	(C₂×D₉)⋊C₄	1	2	yes	—
243	C₂×C₄	#3676	1	2	yes	—
244	C₂×C₄	C₂×Dic₁₈	1	1	yes	characteristic
245	C₂×C₄	C₂×C₄×D₉	1	1	yes	characteristic
246	C₂×C₄	C₂×C₄×D₉	1	1	yes	characteristic
247	C₂×C₄	C₂×D₃₆	1	1	yes	characteristic
248	C₂×C₄	C₂²×Dic₉	1	1	yes	characteristic
249	C₂×C₄	C₂×#611	1	1	yes	characteristic
250	C₈	—	18	4	no	—
251	C₂×C₃	—	2	4	no	—
252	C₂×C₃	#4964	1	2	yes	—
253	C₂×C₃	#4966	1	2	yes	—
254	C₂×C₃	C₂×C₃⋊#95	1	1	yes	characteristic
255	C₂×C₃	#5215	1	1	yes	characteristic
256	C₂×C₃	#5225	1	1	yes	characteristic
257	C₂²	—	2	2	no	—
258	C₂²	—	2	4	no	—
259	C₂²	—	2	4	no	—
260	C₂²	—	2	2	no	—
261	C₂²	C₉⋊#95	1	4	yes	—
262	C₂²	C₂×Dic₉⋊C₄	1	1	yes	characteristic
263	C₂²	C₂×(C₂×D₉)⋊C₄	1	1	yes	characteristic
264	C₂²	#63495	1	2	yes	—
265	C₂²	#63505	1	2	yes	—
266	C₂²	C₂×#3676	1	1	yes	characteristic
267	C₄	—	4	4	no	—
268	C₄	—	36	4	no	—
269	C₄	C₉⋊#95	1	4	yes	—
270	C₄	C₂×C₄×Dic₉	1	1	yes	characteristic
271	C₄	C₂×Dic₉⋊C₄	1	1	yes	characteristic
272	C₄	C₂×C₄⋊Dic₉	1	1	yes	characteristic
273	C₄	C₂×(C₂×D₉)⋊C₄	1	1	yes	characteristic
274	C₃	#71558	1	1	yes	characteristic
275	C₂	—	2	4	no	—
276	C₂	#10588644	1	2	yes	—
277	C₂	#10588649	1	2	yes	—
278	C₂	C₂×C₉⋊#95	1	1	yes	characteristic
279	C₂	#10588896	1	1	yes	characteristic
280	C₂	#10588906	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×#611
commutator	C₄×C₉	C₂×C₄²
Frattini	C₂²×C₃×C₄	C₂²×D₃
Fitting	C₂²×C₉×C₄⋊C₄	C₂
socle	C₂³×C₃	C₂×#67
prime radical	C₂⁴×C₃	#67
cosocle	C₂²×C₄×C₉	C₂³
hypercenter	C₂²×C₄⋊C₄	D₉
hypocenter	C₉	#1736
LCS	G ▷ C₄×C₉ ▷ C₂×C₉ ▷ C₉
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂²×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
4. #10588206	C₂	semidirect
10. C₂×C₉×C₄⋊C₄	C₄	semidirect
14. #63488	C₄	semidirect
19. C₂×#3667	C₄	semidirect
43. C₄⋊Dic₉	C₂×C₄	semidirect
48. C₂×C₉⋊C₈	C₂×C₄	semidirect
147. C₂×C₄⋊C₄	Dic₉	semidirect
224. C₉	#1736	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₉×C₄⋊C₄	2
C₂×#63488	2
C₂²×#3667	2
#10588206	2
#71558	3