Subgroups — #11785207

Order 1152.

462 conjugacy classes of subgroups.

The subgroup lattice has 285 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×Dic₁₈	C₂	1	1	yes	maximal, characteristic
2	C₂×C₉×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic
3	C₂×#63376	C₂	1	2	yes	maximal
4	#10588615	C₂	1	1	yes	maximal, characteristic
5	#10588229	C₂	1	2	yes	maximal
6	#70485	—	3	1	no	maximal
7	C₂²×Dic₁₈	C₂²	1	1	yes	characteristic
8	C₂²×C₈×C₉	C₂²	1	2	yes	—
9	C₂×C₄²×C₉	C₂²	1	1	yes	characteristic
10	C₂×C₄⋊Dic₉	—	2	1	no	—
11	C₂×C₄⋊Dic₉	—	2	1	no	—
12	C₂×C₄⋊Dic₉	C₂²	1	2	yes	—
13	C₂×C₄⋊Dic₉	C₂²	1	1	yes	characteristic
14	C₂×Dic₉⋊C₄	—	2	1	no	—
15	C₂×C₄×Dic₉	—	2	1	no	—
16	C₄×Dic₁₈	—	2	2	no	—
17	C₄×Dic₁₈	C₄	1	4	yes	complemented
18	C₉×C₄⋊C₈	—	2	2	no	—
19	#63376	—	2	4	no	—
20	C₂×C₄×Dic₆	—	3	1	no	—
21	C₂×C₃×C₄⋊C₈	D₃	1	1	yes	characteristic
22	C₂×#816	—	3	2	no	—
23	#4935	—	3	1	no	—
24	#4551	—	3	2	no	—
25	C₂²×C₄×C₉	D₄	1	1	yes	characteristic
26	C₂²×C₄×C₉	D₄	1	1	yes	characteristic
27	C₂²×C₄×C₉	C₂³	1	1	yes	characteristic
28	C₂²×Dic₉	—	2	1	no	—
29	C₂²×Dic₉	—	2	1	no	—
30	C₂²×Dic₉	—	2	2	no	—
31	C₂×Dic₁₈	—	2	1	no	—
32	C₂×Dic₁₈	—	2	1	no	—
33	C₂×Dic₁₈	—	2	1	no	—
34	C₂×Dic₁₈	C₂×C₄	1	2	yes	—
35	C₂×Dic₁₈	D₄	1	2	yes	—
36	C₂×Dic₁₈	Q₈	1	2	yes	—
37	C₂×C₈×C₉	—	2	2	no	—
38	C₂×C₈×C₉	—	2	2	no	—
39	C₂×C₈×C₉	—	2	2	no	—
40	C₄²×C₉	C₂×C₄	1	2	yes	—
41	C₄²×C₉	D₄	1	2	yes	—
42	C₄⋊Dic₉	—	2	1	no	—
43	C₄⋊Dic₉	—	2	2	no	—
44	C₄⋊Dic₉	—	2	2	no	—
45	C₄⋊Dic₉	—	4	2	no	—
46	C₄⋊Dic₉	—	4	2	no	—
47	C₄⋊Dic₉	C₂×C₄	1	2	yes	—
48	Dic₉⋊C₄	—	2	4	no	—
49	C₄×Dic₉	—	2	4	no	—
50	#1803	—	9	1	no	Sylow
51	C₂²×Dic₆	—	3	1	no	—
52	C₂²×C₃×C₈	C₂×D₃	1	2	yes	—
53	C₂×C₃×C₄²	C₂×D₃	1	1	yes	characteristic
54	C₂×C₄⋊Dic₃	—	3	1	no	—
55	C₂×C₄⋊Dic₃	—	3	2	no	—
56	C₂×C₄⋊Dic₃	—	6	1	no	—
57	C₂×C₄⋊Dic₃	—	6	1	no	—
58	C₂×Dic₃⋊C₄	—	6	1	no	—
59	C₂×C₄×Dic₃	—	6	1	no	—
60	C₄×Dic₆	—	3	4	no	—
61	C₄×Dic₆	—	6	2	no	—
62	C₃×C₄⋊C₈	—	2	2	no	—
63	#816	—	6	4	no	—
64	C₂³×C₉	C₂×D₄	1	1	yes	characteristic
65	C₂×C₄×C₉	—	2	1	no	—
66	C₂×C₄×C₉	—	2	1	no	—
67	C₂×C₄×C₉	C₂²⋊C₄	1	2	yes	—
68	C₂×C₄×C₉	C₂²⋊C₄	1	2	yes	—
69	C₂×C₄×C₉	SD₁₆	1	2	yes	—
70	C₂×C₄×C₉	Q₁₆	1	2	yes	—
71	C₂×C₄×C₉	C₂²×C₄	1	1	yes	characteristic
72	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
73	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
74	C₂×C₄×C₉	C₂×Q₈	1	1	yes	characteristic
75	C₂×C₄×C₉	#41	1	1	yes	characteristic
76	C₂×C₄×C₉	#41	1	1	yes	characteristic
77	C₂×Dic₉	—	2	2	no	—
78	C₂×Dic₉	—	2	2	no	—
79	C₂×Dic₉	—	2	2	no	—
80	C₂×Dic₉	—	2	2	no	—
81	C₂×Dic₉	—	4	1	no	—
82	C₂×Dic₉	—	4	1	no	—
83	C₂×Dic₉	—	4	2	no	—
84	C₂×Dic₉	—	4	2	no	—
85	C₂×Dic₉	—	4	2	no	—
86	Dic₁₈	—	2	2	no	—
87	Dic₁₈	—	2	2	no	—
88	Dic₁₈	—	2	2	no	—
89	Dic₁₈	C₄⋊C₄	1	4	yes	complemented
90	C₈×C₉	—	4	2	no	—
91	C₂×C₄×Q₈	—	9	1	no	—
92	C₂×C₄⋊C₈	D₉	1	1	yes	characteristic
93	C₂×#103	—	9	2	no	—
94	#384	—	9	1	no	—
95	#340	—	9	2	no	—
96	C₂²×C₃×C₄	D₁₂	1	1	yes	characteristic
97	C₂²×C₃×C₄	D₁₂	1	1	yes	characteristic
98	C₂²×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
99	C₂²×Dic₃	—	6	1	no	—
100	C₂²×Dic₃	—	6	1	no	—
101	C₂²×Dic₃	—	6	2	no	—
102	C₂×Dic₆	—	3	2	no	—
103	C₂×Dic₆	—	3	2	no	—
104	C₂×Dic₆	—	3	2	no	—
105	C₂×Dic₆	—	6	1	no	—
106	C₂×Dic₆	—	6	1	no	—
107	C₂×Dic₆	—	6	1	no	—
108	C₂×C₃×C₈	—	2	2	no	—
109	C₂×C₃×C₈	—	2	2	no	—
110	C₂×C₃×C₈	—	2	2	no	—
111	C₃×C₄²	C₄×D₃	1	2	yes	—
112	C₃×C₄²	#67	1	2	yes	—
113	C₄⋊Dic₃	—	3	2	no	—
114	C₄⋊Dic₃	—	6	1	no	—
115	C₄⋊Dic₃	—	6	2	no	—
116	C₄⋊Dic₃	—	6	2	no	—
117	C₄⋊Dic₃	—	12	2	no	—
118	C₄⋊Dic₃	—	12	2	no	—
119	Dic₃⋊C₄	—	6	4	no	—
120	C₄×Dic₃	—	6	4	no	—
121	C₂²×C₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
122	C₂²×C₉	#121	1	1	yes	characteristic
123	C₂²×C₉	C₂²⋊Q₈	1	1	yes	characteristic
124	C₂²×C₉	C₂×SD₁₆	1	1	yes	characteristic
125	C₂²×C₉	C₂×Q₁₆	1	1	yes	characteristic
126	C₂²×C₉	AGL(1,ℤ/8)	1	1	yes	characteristic
127	C₂²×C₉	#137	1	1	yes	characteristic
128	C₄×C₉	—	2	2	no	—
129	C₄×C₉	#103	1	4	yes	complemented
130	C₄×C₉	C₂×C₄⋊C₄	1	1	yes	characteristic
131	C₄×C₉	#117	1	1	yes	characteristic
132	C₄×C₉	#121	1	1	yes	characteristic
133	C₄×C₉	C₂²⋊Q₈	1	1	yes	characteristic
134	Dic₉	—	2	4	no	—
135	Dic₉	—	4	2	no	—
136	Dic₉	—	8	2	no	—
137	C₂²×Q₈	—	9	1	no	—
138	C₂²×C₈	C₂×D₉	1	2	yes	—
139	C₄×Q₈	—	9	4	no	—
140	C₄×Q₈	—	18	2	no	—
141	C₂×C₄⋊C₄	—	9	1	no	—
142	C₂×C₄⋊C₄	—	9	2	no	—
143	C₂×C₄⋊C₄	—	18	1	no	—
144	C₂×C₄⋊C₄	—	18	1	no	—
145	C₂×C₄⋊C₄	—	18	1	no	—
146	C₂×C₄²	—	18	1	no	—
147	C₂×C₄²	C₂×D₉	1	1	yes	characteristic
148	C₄⋊C₈	—	2	2	no	—
149	#103	—	18	4	no	—
150	C₂³×C₃	C₂×D₁₂	1	1	yes	characteristic
151	C₂×C₃×C₄	—	2	1	no	—
152	C₂×C₃×C₄	—	2	1	no	—
153	C₂×C₃×C₄	#204	1	2	yes	—
154	C₂×C₃×C₄	Dic₁₂	1	2	yes	—
155	C₂×C₃×C₄	(C₂×D₃)⋊C₄	1	2	yes	—
156	C₂×C₃×C₄	(C₂×D₃)⋊C₄	1	2	yes	—
157	C₂×C₃×C₄	C₂×C₄×D₃	1	1	yes	characteristic
158	C₂×C₃×C₄	D₃×D₄	1	1	yes	characteristic
159	C₂×C₃×C₄	#237	1	1	yes	characteristic
160	C₂×C₃×C₄	D₃×Q₈	1	1	yes	characteristic
161	C₂×C₃×C₄	#239	1	1	yes	characteristic
162	C₂×C₃×C₄	C₂×#67	1	1	yes	characteristic
163	C₂×Dic₃	—	6	2	no	—
164	C₂×Dic₃	—	6	2	no	—
165	C₂×Dic₃	—	6	2	no	—
166	C₂×Dic₃	—	6	2	no	—
167	C₂×Dic₃	—	12	1	no	—
168	C₂×Dic₃	—	12	1	no	—
169	C₂×Dic₃	—	12	2	no	—
170	C₂×Dic₃	—	12	2	no	—
171	C₂×Dic₃	—	12	2	no	—
172	Dic₆	—	3	4	no	—
173	Dic₆	—	6	2	no	—
174	Dic₆	—	6	2	no	—
175	Dic₆	—	6	2	no	—
176	C₃×C₈	—	4	2	no	—
177	C₂×C₉	#380	1	1	yes	characteristic
178	C₂×C₉	C₂×#103	1	1	yes	characteristic
179	C₂×C₉	#417	1	1	yes	characteristic
180	C₂×C₉	#460	1	1	yes	characteristic
181	C₂×C₉	#462	1	1	yes	characteristic
182	C₂×C₉	#475	1	1	yes	characteristic
183	C₂×C₉	#477	1	1	yes	characteristic
184	C₂×Q₈	—	9	2	no	—
185	C₂×Q₈	—	9	2	no	—
186	C₂×Q₈	—	9	2	no	—
187	C₂×Q₈	—	18	1	no	—
188	C₂×Q₈	—	18	1	no	—
189	C₂×Q₈	—	18	1	no	—
190	C₂²×C₄	—	18	1	no	—
191	C₂²×C₄	—	18	1	no	—
192	C₂²×C₄	—	18	2	no	—
193	C₂²×C₄	D₃₆	1	1	yes	characteristic
194	C₂²×C₄	D₃₆	1	1	yes	characteristic
195	C₂²×C₄	C₂²×D₉	1	1	yes	characteristic
196	C₂×C₈	—	2	2	no	—
197	C₂×C₈	—	2	2	no	—
198	C₂×C₈	—	2	2	no	—
199	C₄⋊C₄	—	9	2	no	—
200	C₄⋊C₄	—	18	4	no	—
201	C₄⋊C₄	—	18	1	no	—
202	C₄⋊C₄	—	18	2	no	—
203	C₄⋊C₄	—	18	2	no	—
204	C₄⋊C₄	—	36	2	no	—
205	C₄⋊C₄	—	36	2	no	—
206	C₄²	—	18	4	no	—
207	C₄²	C₄×D₉	1	2	yes	—
208	C₄²	#611	1	2	yes	—
209	C₂²×C₃	#895	1	1	yes	characteristic
210	C₂²×C₃	#897	1	1	yes	characteristic
211	C₂²×C₃	C₂×#204	1	1	yes	characteristic
212	C₂²×C₃	C₂×Dic₁₂	1	1	yes	characteristic
213	C₂²×C₃	#908	1	1	yes	characteristic
214	C₂²×C₃	#909	1	1	yes	characteristic
215	C₂²×C₃	C₂×(C₂×D₃)⋊C₄	1	1	yes	characteristic
216	C₃×C₄	—	2	2	no	—
217	C₃×C₄	#816	1	4	yes	—
218	C₃×C₄	D₃×C₄⋊C₄	1	1	yes	characteristic
219	C₃×C₄	#892	1	1	yes	characteristic
220	C₃×C₄	#938	1	1	yes	characteristic
221	C₃×C₄	#946	1	1	yes	characteristic
222	Dic₃	—	6	4	no	—
223	Dic₃	—	12	2	no	—
224	Dic₃	—	24	2	no	—
225	C₉	#1803	1	1	yes	characteristic, complemented, Sylow
226	C₂³	C₂×D₃₆	1	1	yes	characteristic
227	Q₈	—	9	4	no	—
228	Q₈	—	18	2	no	—
229	Q₈	—	18	2	no	—
230	Q₈	—	18	2	no	—
231	C₂×C₄	—	2	1	no	—
232	C₂×C₄	—	2	1	no	—
233	C₂×C₄	—	18	2	no	—
234	C₂×C₄	—	18	2	no	—
235	C₂×C₄	—	18	2	no	—
236	C₂×C₄	—	18	2	no	—
237	C₂×C₄	—	36	1	no	—
238	C₂×C₄	—	36	1	no	—
239	C₂×C₄	—	36	2	no	—
240	C₂×C₄	—	36	2	no	—
241	C₂×C₄	—	36	2	no	—
242	C₂×C₄	Dic₃₆	1	2	yes	—
243	C₂×C₄	#3664	1	2	yes	—
244	C₂×C₄	(C₂×D₉)⋊C₄	1	2	yes	—
245	C₂×C₄	(C₂×D₉)⋊C₄	1	2	yes	—
246	C₂×C₄	C₂×C₄×D₉	1	1	yes	characteristic
247	C₂×C₄	D₄×D₉	1	1	yes	characteristic
248	C₂×C₄	#3699	1	1	yes	characteristic
249	C₂×C₄	Q₈×D₉	1	1	yes	characteristic
250	C₂×C₄	#3701	1	1	yes	characteristic
251	C₂×C₄	C₂×#611	1	1	yes	characteristic
252	C₈	—	4	2	no	—
253	C₂×C₃	#4848	1	1	yes	characteristic
254	C₂×C₃	#4849	1	1	yes	characteristic
255	C₂×C₃	#4850	1	1	yes	characteristic
256	C₂×C₃	#4851	1	1	yes	characteristic
257	C₂×C₃	#4987	1	1	yes	characteristic
258	C₂×C₃	C₂×#816	1	1	yes	characteristic
259	C₂×C₃	#5133	1	1	yes	characteristic
260	C₂²	#63457	1	1	yes	characteristic
261	C₂²	#63459	1	1	yes	characteristic
262	C₂²	C₂×Dic₃₆	1	1	yes	characteristic
263	C₂²	C₂×#3664	1	1	yes	characteristic
264	C₂²	#63470	1	1	yes	characteristic
265	C₂²	#63471	1	1	yes	characteristic
266	C₂²	C₂×(C₂×D₉)⋊C₄	1	1	yes	characteristic
267	C₄	—	2	2	no	—
268	C₄	—	18	4	no	—
269	C₄	—	36	2	no	—
270	C₄	—	72	2	no	—
271	C₄	#63376	1	4	yes	complemented
272	C₄	C₄⋊C₄×D₉	1	1	yes	characteristic
273	C₄	#63454	1	1	yes	characteristic
274	C₄	#63500	1	1	yes	characteristic
275	C₄	#63508	1	1	yes	characteristic
276	C₃	#70485	1	1	yes	characteristic
277	C₂	#10588512	1	1	yes	characteristic
278	C₂	#10588513	1	1	yes	characteristic
279	C₂	#10588514	1	1	yes	characteristic
280	C₂	#10588515	1	1	yes	characteristic
281	C₂	#10588667	1	1	yes	characteristic
282	C₂	C₂×#63376	1	1	yes	characteristic
283	C₂	#10588815	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×D₃₆
commutator	C₂×C₄×C₉	C₂²×C₄
Frattini	C₂²×C₃×C₄	C₂²×D₃
Fitting	C₂×C₉×C₄⋊C₈	C₂
socle, prime radical	C₂³×C₃	C₂×D₁₂
cosocle	C₂²×C₄×C₉	C₂³
hypercenter	C₂×C₄⋊C₈	D₉
hypocenter	C₉	#1803
LCS	G ▷ C₂×C₄×C₉ ▷ C₂×C₉ ▷ C₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊C₈

Complements

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

N	H ≅ G/N	type
17. C₄×Dic₁₈	C₄	semidirect
89. Dic₁₈	C₄⋊C₄	semidirect
129. C₄×C₉	#103	semidirect
225. C₉	#1803	semidirect
271. C₄	#63376	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×Dic₁₈	2
C₂×C₉×C₄⋊C₈	2
C₂×#63376	2
#10588615	2
#10588229	2
#70485	3