Subgroups — #11860864

Order 1152.

726 conjugacy classes of subgroups.

The subgroup lattice has 288 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#63467	C₂	1	1	yes	maximal, characteristic
2	#10589625	C₂	1	2	yes	maximal, complemented
3	C₉×#405	C₂	1	1	yes	maximal, characteristic, complemented
4	#10588815	C₂	1	2	yes	maximal, complemented
5	C₂×#63378	C₂	1	1	yes	maximal, characteristic, complemented
6	#10588391	C₂	1	4	yes	maximal, complemented
7	C₄×D₇₂	C₂	1	2	yes	maximal, complemented
8	C₄×Dic₃₆	C₂	1	2	yes	maximal, complemented
9	#78562	—	3	1	no	maximal
10	C₂×#3697	C₂²	1	2	yes	—
11	C₂×C₉×M₁₆	C₂²	1	1	yes	characteristic
12	C₂²×C₈×C₉	C₂²	1	1	yes	characteristic
13	C₉×#117	C₂²	1	1	yes	characteristic
14	C₂×C₄⋊Dic₉	C₂²	1	2	yes	—
15	#63467	C₄	1	8	yes	complemented
16	C₂×D₇₂	C₂²	1	1	yes	characteristic
17	C₂×#3664	C₂²	1	2	yes	—
18	C₂×Dic₃₆	C₂²	1	1	yes	characteristic
19	#63454	—	2	2	no	—
20	C₄⋊C₄×D₉	—	2	2	no	—
21	#611⋊C₄	—	2	4	no	—
22	C₄×D₃₆	C₂²	1	4	yes	—
23	C₄×Dic₁₈	C₂²	1	4	yes	—
24	C₉×#97	C₂²	1	2	yes	—
25	C₄×C₈×C₉	C₂²	1	2	yes	—
26	#63380	C₂²	1	4	yes	—
27	#63378	C₂²	1	2	yes	—
28	#63378	C₂²	1	2	yes	—
29	#63376	C₂²	1	4	yes	—
30	C₂×#904	—	3	1	no	—
31	#5523	—	3	2	no	—
32	C₃×#405	D₃	1	1	yes	characteristic
33	#5133	—	3	2	no	—
34	C₂×#818	—	3	1	no	—
35	#4710	—	3	4	no	—
36	C₄×Dic₁₂	—	3	2	no	—
37	C₄×D₂₄	—	3	2	no	—
38	C₂²×C₄×C₉	C₂³	1	1	yes	characteristic
39	C₂×#611	—	2	2	no	—
40	C₂²×Dic₉	—	2	2	no	—
41	#3697	—	2	4	no	—
42	#3697	C₂×C₄	1	8	yes	complemented
43	C₂×D₃₆	C₂³	1	2	yes	—
44	C₂×C₄×D₉	—	2	2	no	—
45	C₂×C₄×D₉	—	2	4	no	—
46	C₂×Dic₁₈	C₂³	1	2	yes	—
47	C₉×M₁₆	—	2	2	no	—
48	C₂×C₈×C₉	C₂×C₄	1	4	yes	complemented
49	C₂×C₈×C₉	C₂³	1	2	yes	—
50	C₂×C₈×C₉	C₂³	1	1	yes	characteristic
51	C₂×C₈×C₉	C₂³	1	1	yes	characteristic
52	C₉×C₄⋊C₄	D₄	1	1	yes	characteristic
53	C₉×C₄⋊C₄	D₄	1	1	yes	characteristic
54	C₉×C₂²⋊C₄	D₄	1	2	yes	—
55	C₄²×C₉	C₂³	1	2	yes	—
56	(C₂×D₉)⋊C₄	—	2	4	no	—
57	C₄⋊Dic₉	—	2	2	no	—
58	C₄⋊Dic₉	C₂³	1	4	yes	—
59	Dic₉⋊C₄	—	2	4	no	—
60	C₄×Dic₉	—	2	4	no	—
61	D₇₂	C₂×C₄	1	4	yes	complemented
62	#3664	C₂×C₄	1	8	yes	complemented
63	Dic₃₆	C₂×C₄	1	4	yes	complemented
64	#2951	—	9	1	no	Sylow
65	C₂×#235	—	3	2	no	—
66	C₂×C₃×M₁₆	C₂×D₃	1	1	yes	characteristic
67	C₂²×C₃×C₈	C₂×D₃	1	1	yes	characteristic
68	C₃×#117	C₂×D₃	1	1	yes	characteristic
69	C₂×C₄⋊Dic₃	—	3	2	no	—
70	C₂×Dic₁₂	—	3	1	no	—
71	#904	—	3	8	no	—
72	C₂×D₂₄	—	3	1	no	—
73	C₂×#204	—	3	2	no	—
74	#892	—	6	2	no	—
75	D₃×C₄⋊C₄	—	6	2	no	—
76	#67⋊C₄	—	6	4	no	—
77	C₄×D₁₂	—	3	4	no	—
78	C₄×Dic₆	—	3	4	no	—
79	C₃×#97	C₂×D₃	1	2	yes	—
80	C₃×C₄×C₈	C₂×D₃	1	2	yes	—
81	#821	—	3	4	no	—
82	#818	—	3	2	no	—
83	#818	—	3	2	no	—
84	#816	—	3	4	no	—
85	C₂³×C₉	C₂×D₄	1	1	yes	characteristic
86	C₂²×D₉	—	2	2	no	—
87	C₂×C₄×C₉	C₂²×C₄	1	2	yes	—
88	C₂×C₄×C₉	C₂×D₄	1	2	yes	—
89	C₂×C₄×C₉	C₂×D₄	1	2	yes	—
90	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
91	C₂×C₄×C₉	#41	1	2	yes	—
92	C₂×C₄×C₉	C₂⁴	1	1	yes	characteristic
93	#611	—	2	8	no	—
94	C₂×Dic₉	—	2	2	no	—
95	C₂×Dic₉	—	2	4	no	—
96	C₂×Dic₉	—	8	2	no	—
97	D₃₆	—	2	2	no	—
98	D₃₆	C₂²×C₄	1	4	yes	—
99	C₄×D₉	—	2	8	no	—
100	C₄×D₉	—	8	4	no	—
101	Dic₁₈	—	2	2	no	—
102	Dic₁₈	C₂²×C₄	1	4	yes	—
103	C₈×C₉	—	2	2	no	—
104	C₈×C₉	C₂²×C₄	1	2	yes	—
105	C₈×C₉	C₂²×C₄	1	2	yes	—
106	C₂×#135	—	9	1	no	—
107	#520	—	9	2	no	—
108	#440	—	9	4	no	—
109	C₄×Q₁₆	—	9	2	no	—
110	C₄×D₈	—	9	2	no	—
111	C₂×#107	—	9	1	no	—
112	#417	—	9	2	no	—
113	#405	D₉	1	1	yes	characteristic, complemented
114	C₂²×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
115	C₂×#67	—	6	2	no	—
116	C₂²×Dic₃	—	6	2	no	—
117	#235	—	3	8	no	—
118	#235	—	6	4	no	—
119	C₂×D₁₂	—	3	2	no	—
120	C₂×C₄×D₃	—	6	2	no	—
121	C₂×C₄×D₃	—	6	4	no	—
122	C₂×Dic₆	—	3	2	no	—
123	C₃×M₁₆	—	2	2	no	—
124	C₂×C₃×C₈	C₄×D₃	1	4	yes	—
125	C₂×C₃×C₈	C₂²×D₃	1	2	yes	—
126	C₂×C₃×C₈	C₂²×D₃	1	1	yes	characteristic
127	C₂×C₃×C₈	C₂²×D₃	1	1	yes	characteristic
128	C₃×C₄⋊C₄	D₁₂	1	1	yes	characteristic
129	C₃×C₄⋊C₄	D₁₂	1	1	yes	characteristic
130	C₃×C₂²⋊C₄	D₁₂	1	2	yes	—
131	C₃×C₄²	C₂²×D₃	1	2	yes	—
132	(C₂×D₃)⋊C₄	—	6	4	no	—
133	C₄⋊Dic₃	—	3	4	no	—
134	C₄⋊Dic₃	—	6	2	no	—
135	Dic₃⋊C₄	—	6	4	no	—
136	C₄×Dic₃	—	6	4	no	—
137	Dic₁₂	—	3	4	no	—
138	D₂₄	—	3	4	no	—
139	#204	—	3	8	no	—
140	C₂²×C₉	—	2	1	no	—
141	C₂²×C₉	—	2	1	no	—
142	C₂²×C₉	C₄×D₄	1	2	yes	—
143	C₂²×C₉	C₂²×D₄	1	1	yes	characteristic
144	C₂×D₉	—	2	4	no	—
145	C₂×D₉	—	4	2	no	—
146	C₂×D₉	—	4	2	no	—
147	C₄×C₉	—	2	2	no	—
148	C₄×C₉	—	2	2	no	—
149	C₄×C₉	C₄×D₄	1	2	yes	—
150	C₄×C₉	C₂³×C₄	1	1	yes	characteristic
151	C₄×C₉	C₂×#41	1	1	yes	characteristic
152	Dic₉	—	2	4	no	—
153	Dic₉	—	4	4	no	—
154	C₂×#41	—	9	2	no	—
155	#135	—	9	8	no	—
156	C₂×Q₁₆	—	9	1	no	—
157	C₂×SD₁₆	—	9	2	no	—
158	C₂×D₈	—	9	1	no	—
159	C₂×M₁₆	C₂×D₉	1	1	yes	characteristic
160	C₂²×C₈	C₂×D₉	1	1	yes	characteristic
161	C₄×Q₈	—	9	4	no	—
162	C₄×D₄	—	9	4	no	—
163	C₄×D₄	—	18	4	no	—
164	#117	—	18	2	no	—
165	#117	C₂×D₉	1	1	yes	characteristic
166	C₂×C₄⋊C₄	—	9	2	no	—
167	C₂×C₄⋊C₄	—	18	2	no	—
168	#107	—	9	2	no	—
169	#107	—	9	2	no	—
170	#103	—	9	4	no	—
171	#102	—	9	4	no	—
172	#97	C₂×D₉	1	2	yes	—
173	C₄×C₈	C₂×D₉	1	2	yes	—
174	C₂³×C₃	C₂×D₁₂	1	1	yes	characteristic
175	C₂²×D₃	—	6	2	no	—
176	C₂×C₃×C₄	C₂×C₄×D₃	1	2	yes	—
177	C₂×C₃×C₄	C₂×D₁₂	1	2	yes	—
178	C₂×C₃×C₄	C₂×D₁₂	1	2	yes	—
179	C₂×C₃×C₄	C₂×D₁₂	1	1	yes	characteristic
180	C₂×C₃×C₄	#235	1	2	yes	—
181	C₂×C₃×C₄	C₂³×D₃	1	1	yes	characteristic
182	#67	—	6	8	no	—
183	C₂×Dic₃	—	6	2	no	—
184	C₂×Dic₃	—	6	4	no	—
185	C₂×Dic₃	—	24	2	no	—
186	D₁₂	—	3	4	no	—
187	D₁₂	—	6	2	no	—
188	C₄×D₃	—	6	8	no	—
189	C₄×D₃	—	24	4	no	—
190	Dic₆	—	3	4	no	—
191	Dic₆	—	6	2	no	—
192	C₃×C₈	—	2	2	no	—
193	C₃×C₈	C₂×C₄×D₃	1	2	yes	—
194	C₃×C₈	C₂×C₄×D₃	1	2	yes	—
195	C₂×C₉	—	2	2	no	—
196	C₂×C₉	C₂×C₄×D₄	1	1	yes	characteristic
197	C₂×C₉	#576	1	1	yes	characteristic
198	C₂×C₉	#578	1	1	yes	characteristic
199	D₉	—	4	4	no	—
200	#41	—	9	8	no	—
201	#41	—	18	4	no	—
202	C₂×Q₈	—	9	2	no	—
203	C₂×D₄	—	9	2	no	—
204	C₂×D₄	—	18	2	no	—
205	C₂²×C₄	—	18	2	no	—
206	C₂²×C₄	—	18	2	no	—
207	C₂²×C₄	—	18	4	no	—
208	C₂²×C₄	C₂²×D₉	1	1	yes	characteristic
209	Q₁₆	—	9	4	no	—
210	SD₁₆	—	9	8	no	—
211	D₈	—	9	4	no	—
212	M₁₆	—	2	2	no	—
213	C₂×C₈	C₄×D₉	1	4	yes	complemented
214	C₂×C₈	C₂²×D₉	1	2	yes	—
215	C₂×C₈	C₂²×D₉	1	1	yes	characteristic
216	C₂×C₈	C₂²×D₉	1	1	yes	characteristic
217	C₄⋊C₄	—	9	4	no	—
218	C₄⋊C₄	—	18	2	no	—
219	C₄⋊C₄	—	18	4	no	—
220	C₄⋊C₄	D₃₆	1	1	yes	characteristic
221	C₄⋊C₄	D₃₆	1	1	yes	characteristic
222	C₂²⋊C₄	—	18	4	no	—
223	C₂²⋊C₄	D₃₆	1	2	yes	—
224	C₄²	—	18	4	no	—
225	C₄²	C₂²×D₉	1	2	yes	—
226	C₂²×C₃	—	2	1	no	—
227	C₂²×C₃	—	2	1	no	—
228	C₂²×C₃	C₄×D₁₂	1	2	yes	—
229	C₂²×C₃	C₂²×D₁₂	1	1	yes	characteristic
230	C₂×D₃	—	6	4	no	—
231	C₂×D₃	—	12	2	no	—
232	C₂×D₃	—	12	2	no	—
233	C₃×C₄	—	2	2	no	—
234	C₃×C₄	—	2	2	no	—
235	C₃×C₄	C₄×D₁₂	1	2	yes	—
236	C₃×C₄	C₂²×C₄×D₃	1	1	yes	characteristic
237	C₃×C₄	C₂×#235	1	1	yes	characteristic
238	Dic₃	—	6	4	no	—
239	Dic₃	—	12	4	no	—
240	C₉	#2951	1	1	yes	characteristic, complemented, Sylow
241	C₂³	—	18	2	no	—
242	C₂³	C₂×D₃₆	1	1	yes	characteristic
243	Q₈	—	9	4	no	—
244	Q₈	—	18	2	no	—
245	D₄	—	9	4	no	—
246	D₄	—	18	8	no	—
247	D₄	—	18	2	no	—
248	C₂×C₄	—	18	2	no	—
249	C₂×C₄	—	18	8	no	—
250	C₂×C₄	—	18	4	no	—
251	C₂×C₄	—	72	4	no	—
252	C₂×C₄	—	72	2	no	—
253	C₂×C₄	C₂×C₄×D₉	1	2	yes	—
254	C₂×C₄	C₂×D₃₆	1	2	yes	—
255	C₂×C₄	C₂×D₃₆	1	2	yes	—
256	C₂×C₄	C₂×D₃₆	1	1	yes	characteristic
257	C₂×C₄	#3697	1	2	yes	—
258	C₂×C₄	C₂³×D₉	1	1	yes	characteristic
259	C₈	—	2	2	no	—
260	C₈	C₂×C₄×D₉	1	2	yes	—
261	C₈	C₂×C₄×D₉	1	2	yes	—
262	C₂×C₃	—	2	2	no	—
263	C₂×C₃	C₂×C₄×D₁₂	1	1	yes	characteristic
264	C₂×C₃	#5752	1	1	yes	characteristic
265	C₂×C₃	#5753	1	1	yes	characteristic
266	D₃	—	12	4	no	—
267	C₂²	—	2	1	no	—
268	C₂²	—	2	1	no	—
269	C₂²	—	18	4	no	—
270	C₂²	—	36	2	no	—
271	C₂²	—	36	2	no	—
272	C₂²	C₄×D₃₆	1	2	yes	—
273	C₂²	C₂²×D₃₆	1	1	yes	characteristic
274	C₄	—	2	2	no	—
275	C₄	—	2	2	no	—
276	C₄	—	18	4	no	—
277	C₄	—	36	4	no	—
278	C₄	C₄×D₃₆	1	2	yes	—
279	C₄	C₂²×C₄×D₉	1	1	yes	characteristic
280	C₄	C₂×#3697	1	1	yes	characteristic
281	C₃	#78562	1	1	yes	characteristic
282	C₂	—	2	2	no	—
283	C₂	—	36	4	no	—
284	C₂	C₂×C₄×D₃₆	1	1	yes	characteristic
285	C₂	#10589854	1	1	yes	characteristic
286	C₂	#10589855	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×D₃₆
commutator	C₄×C₉	C₂³×C₄
Frattini	C₂×C₃×C₄	C₂³×D₃
Fitting	C₉×#405	C₂
socle	C₂²×C₃	C₂²×D₁₂
prime radical	C₂×#63467	C₂
cosocle	C₂×C₄×C₉	C₂⁴
hypercenter	#405	D₉
hypocenter	C₉	#2951
LCS	G ▷ C₄×C₉ ▷ C₂×C₉ ▷ C₉
UCS	C₁ ◁ C₂² ◁ #117 ◁ #405

Complements

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

N	H ≅ G/N	type
2. #10589625	C₂	semidirect
3. C₉×#405	C₂	semidirect
4. #10588815	C₂	semidirect
5. C₂×#63378	C₂	semidirect
6. #10588391	C₂	semidirect
7. C₄×D₇₂	C₂	semidirect
8. C₄×Dic₃₆	C₂	semidirect
15. #63467	C₄	semidirect
42. #3697	C₂×C₄	semidirect
48. C₂×C₈×C₉	C₂×C₄	semidirect
61. D₇₂	C₂×C₄	semidirect
62. #3664	C₂×C₄	semidirect
63. Dic₃₆	C₂×C₄	semidirect
113. #405	D₉	semidirect
213. C₂×C₈	C₄×D₉	semidirect
240. C₉	#2951	semidirect

Maximal subgroups

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

subgroup	index
C₂×#63467	2
#10589625	2
C₉×#405	2
#10588815	2
C₂×#63378	2
#10588391	2
C₄×D₇₂	2
C₄×Dic₃₆	2
#78562	3