Subgroups — #11784218

Order 1152.

390 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×C₉×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic
2	C₂×#63375	C₂	1	1	yes	maximal, characteristic
3	C₂×C₈×Dic₉	C₂	1	1	yes	maximal, characteristic
4	#10588612	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄⋊(C₉⋊C₈)	C₂	1	1	yes	maximal, characteristic
6	#10588228	C₂	1	1	yes	maximal, characteristic
7	#10588228	C₂	1	1	yes	maximal, characteristic
8	#70442	—	3	1	no	maximal
9	C₂²×C₈×C₉	C₂²	1	1	yes	characteristic
10	C₂²×C₈×C₉	C₂²	1	1	yes	characteristic
11	C₂×C₄²×C₉	C₂²	1	1	yes	characteristic
12	C₂×C₄×Dic₉	C₂²	1	1	yes	characteristic
13	C₂×C₄×Dic₉	C₂²	1	1	yes	characteristic
14	C₂²×C₉⋊C₈	C₂²	1	1	yes	characteristic
15	C₂²×C₉⋊C₈	C₂²	1	1	yes	characteristic
16	C₉×C₄⋊C₈	C₄	1	4	yes	complemented
17	C₉⋊#95	C₄	1	2	yes	—
18	C₉⋊#95	C₄	1	2	yes	—
19	#63375	—	2	2	no	—
20	C₈×Dic₉	—	2	2	no	—
21	C₄⋊(C₉⋊C₈)	C₄	1	4	yes	complemented
22	C₂×C₃×C₄⋊C₈	D₃	1	1	yes	characteristic
23	C₂×#815	—	3	1	no	—
24	C₂×C₈×Dic₃	—	3	1	no	—
25	#4932	—	3	1	no	—
26	C₂×C₄⋊(C₃⋊C₈)	—	3	1	no	—
27	#4550	—	3	1	no	—
28	#4550	—	3	1	no	—
29	C₂²×C₄×C₉	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₄×C₉	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₄×C₉	C₂³	1	1	yes	characteristic
32	C₂²×Dic₉	C₂×C₄	1	2	yes	—
33	C₂²×Dic₉	C₂×C₄	1	2	yes	—
34	C₂×C₈×C₉	—	2	1	no	—
35	C₂×C₈×C₉	—	2	1	no	—
36	C₂×C₈×C₉	—	2	1	no	—
37	C₂×C₈×C₉	—	2	1	no	—
38	C₂×C₈×C₉	C₂×C₄	1	2	yes	—
39	C₂×C₈×C₉	C₂×C₄	1	2	yes	—
40	C₄²×C₉	C₂×C₄	1	2	yes	—
41	C₄²×C₉	C₂×C₄	1	2	yes	—
42	C₄×Dic₉	—	2	1	no	—
43	C₄×Dic₉	—	2	1	no	—
44	C₄×Dic₉	D₄	1	2	yes	—
45	C₄×Dic₉	Q₈	1	2	yes	—
46	C₂×C₉⋊C₈	—	2	1	no	—
47	C₂×C₉⋊C₈	—	2	1	no	—
48	C₂×C₉⋊C₈	—	2	1	no	—
49	C₂×C₉⋊C₈	—	2	1	no	—
50	C₂×C₉⋊C₈	C₂×C₄	1	2	yes	—
51	C₂×C₉⋊C₈	C₂×C₄	1	2	yes	—
52	#1768	—	9	1	no	Sylow
53	C₂²×C₃×C₈	C₂×D₃	1	1	yes	characteristic
54	C₂²×C₃×C₈	C₂×D₃	1	1	yes	characteristic
55	C₂×C₃×C₄²	C₂×D₃	1	1	yes	characteristic
56	C₂×C₄×Dic₃	—	3	1	no	—
57	C₂×C₄×Dic₃	—	3	1	no	—
58	C₂²×C₃⋊C₈	—	3	1	no	—
59	C₂²×C₃⋊C₈	—	3	1	no	—
60	C₃×C₄⋊C₈	Dic₃	1	4	yes	—
61	C₃⋊#95	—	3	2	no	—
62	C₃⋊#95	—	3	2	no	—
63	#815	—	6	2	no	—
64	C₈×Dic₃	—	6	2	no	—
65	C₄⋊(C₃⋊C₈)	—	3	4	no	—
66	C₂³×C₉	C₂²×C₄	1	1	yes	characteristic
67	C₂×C₄×C₉	—	2	2	no	—
68	C₂×C₄×C₉	—	2	2	no	—
69	C₂×C₄×C₉	C₄²	1	2	yes	—
70	C₂×C₄×C₉	C₄²	1	2	yes	—
71	C₂×C₄×C₉	C₂²×C₄	1	1	yes	characteristic
72	C₂×C₄×C₉	C₂²×C₄	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	—	2	2	no	—
82	C₂×Dic₉	C₄⋊C₄	1	4	yes	—
83	C₈×C₉	—	2	2	no	—
84	C₈×C₉	—	2	2	no	—
85	C₉⋊C₈	—	2	2	no	—
86	C₉⋊C₈	—	2	2	no	—
87	C₂×C₄⋊C₈	—	9	1	no	—
88	C₂×C₄⋊C₈	D₉	1	1	yes	characteristic
89	C₂×#97	—	9	1	no	—
90	C₂×C₄×C₈	—	9	1	no	—
91	#376	—	9	1	no	—
92	#336	—	9	1	no	—
93	#336	—	9	1	no	—
94	C₂²×C₃×C₄	C₄×D₃	1	1	yes	characteristic
95	C₂²×C₃×C₄	C₄×D₃	1	1	yes	characteristic
96	C₂²×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
97	C₂²×Dic₃	—	3	2	no	—
98	C₂²×Dic₃	—	3	2	no	—
99	C₂×C₃×C₈	—	2	1	no	—
100	C₂×C₃×C₈	—	2	1	no	—
101	C₂×C₃×C₈	—	2	1	no	—
102	C₂×C₃×C₈	—	2	1	no	—
103	C₂×C₃×C₈	C₂×Dic₃	1	2	yes	—
104	C₂×C₃×C₈	C₂×Dic₃	1	2	yes	—
105	C₃×C₄²	C₄×D₃	1	2	yes	—
106	C₃×C₄²	C₂×Dic₃	1	2	yes	—
107	C₄×Dic₃	—	3	2	no	—
108	C₄×Dic₃	—	3	2	no	—
109	C₄×Dic₃	—	6	1	no	—
110	C₄×Dic₃	—	6	1	no	—
111	C₂×C₃⋊C₈	—	3	2	no	—
112	C₂×C₃⋊C₈	—	3	2	no	—
113	C₂×C₃⋊C₈	—	6	1	no	—
114	C₂×C₃⋊C₈	—	6	1	no	—
115	C₂×C₃⋊C₈	—	6	1	no	—
116	C₂×C₃⋊C₈	—	6	1	no	—
117	C₂²×C₉	C₂×C₄²	1	1	yes	characteristic
118	C₂²×C₉	C₂×C₄⋊C₄	1	1	yes	characteristic
119	C₂²×C₉	#117	1	1	yes	characteristic
120	C₂²×C₉	#131	1	2	yes	—
121	C₂²×C₉	#131	1	2	yes	—
122	C₄×C₉	—	2	2	no	—
123	C₄×C₉	—	2	2	no	—
124	C₄×C₉	C₄×D₄	1	1	yes	characteristic
125	C₄×C₉	C₄×D₄	1	1	yes	characteristic
126	C₄×C₉	C₄×Q₈	1	1	yes	characteristic
127	C₄×C₉	C₄×Q₈	1	1	yes	characteristic
128	Dic₉	—	2	4	no	—
129	Dic₉	—	4	2	no	—
130	C₂²×C₈	—	9	1	no	—
131	C₂²×C₈	—	9	1	no	—
132	C₂²×C₈	C₂×D₉	1	1	yes	characteristic
133	C₂²×C₈	C₂×D₉	1	1	yes	characteristic
134	C₂×C₄²	—	9	1	no	—
135	C₂×C₄²	—	9	1	no	—
136	C₂×C₄²	C₂×D₉	1	1	yes	characteristic
137	C₄⋊C₈	—	9	4	no	—
138	C₄⋊C₈	Dic₉	1	4	yes	complemented
139	#97	—	18	2	no	—
140	C₄×C₈	—	18	2	no	—
141	#95	—	9	2	no	—
142	#95	—	9	2	no	—
143	C₂³×C₃	C₂×C₄×D₃	1	1	yes	characteristic
144	C₂×C₃×C₄	—	2	2	no	—
145	C₂×C₃×C₄	—	2	2	no	—
146	C₂×C₃×C₄	C₄×Dic₃	1	2	yes	—
147	C₂×C₃×C₄	C₄×Dic₃	1	2	yes	—
148	C₂×C₃×C₄	C₂×C₄×D₃	1	1	yes	characteristic
149	C₂×C₃×C₄	D₃×D₄	1	1	yes	characteristic
150	C₂×C₃×C₄	#237	1	1	yes	characteristic
151	C₂×C₃×C₄	D₃×Q₈	1	1	yes	characteristic
152	C₂×C₃×C₄	#239	1	1	yes	characteristic
153	C₂×C₃×C₄	C₂²×Dic₃	1	1	yes	characteristic
154	C₂×Dic₃	—	3	4	no	—
155	C₂×Dic₃	—	6	2	no	—
156	C₂×Dic₃	—	6	2	no	—
157	C₂×Dic₃	—	6	2	no	—
158	C₂×Dic₃	—	6	2	no	—
159	C₂×Dic₃	—	6	2	no	—
160	C₃×C₈	—	2	2	no	—
161	C₃×C₈	—	2	2	no	—
162	C₃⋊C₈	—	6	2	no	—
163	C₃⋊C₈	—	6	2	no	—
164	C₂×C₉	C₄×C₄⋊C₄	1	1	yes	characteristic
165	C₂×C₉	#405	1	1	yes	characteristic
166	C₂×C₉	#405	1	1	yes	characteristic
167	C₂×C₉	#424	1	2	yes	—
168	C₂×C₉	#433	1	2	yes	—
169	C₂²×C₄	—	9	2	no	—
170	C₂²×C₄	—	9	2	no	—
171	C₂²×C₄	C₄×D₉	1	1	yes	characteristic
172	C₂²×C₄	C₄×D₉	1	1	yes	characteristic
173	C₂²×C₄	C₂²×D₉	1	1	yes	characteristic
174	C₂×C₈	—	2	1	no	—
175	C₂×C₈	—	2	1	no	—
176	C₂×C₈	—	2	1	no	—
177	C₂×C₈	—	2	1	no	—
178	C₂×C₈	—	9	2	no	—
179	C₂×C₈	—	9	2	no	—
180	C₂×C₈	—	18	1	no	—
181	C₂×C₈	—	18	1	no	—
182	C₂×C₈	—	18	1	no	—
183	C₂×C₈	—	18	1	no	—
184	C₂×C₈	C₂×Dic₉	1	2	yes	—
185	C₂×C₈	C₂×Dic₉	1	2	yes	—
186	C₄²	—	9	2	no	—
187	C₄²	—	9	2	no	—
188	C₄²	—	18	1	no	—
189	C₄²	—	18	1	no	—
190	C₄²	C₄×D₉	1	2	yes	—
191	C₄²	C₂×Dic₉	1	2	yes	—
192	C₂²×C₃	D₃×C₄⋊C₄	1	1	yes	characteristic
193	C₂²×C₃	#892	1	1	yes	characteristic
194	C₂²×C₃	#901	1	2	yes	—
195	C₂²×C₃	#907	1	2	yes	—
196	C₂²×C₃	C₂×C₄×Dic₃	1	1	yes	characteristic
197	C₃×C₄	—	2	2	no	—
198	C₃×C₄	—	2	2	no	—
199	C₃×C₄	#887	1	1	yes	characteristic
200	C₃×C₄	#893	1	1	yes	characteristic
201	C₃×C₄	D₄×Dic₃	1	1	yes	characteristic
202	C₃×C₄	Q₈×Dic₃	1	1	yes	characteristic
203	Dic₃	—	6	4	no	—
204	Dic₃	—	12	2	no	—
205	C₉	#1768	1	1	yes	characteristic, complemented, Sylow
206	C₂³	C₂×C₄×D₉	1	1	yes	characteristic
207	C₂×C₄	—	2	2	no	—
208	C₂×C₄	—	2	2	no	—
209	C₂×C₄	—	9	4	no	—
210	C₂×C₄	—	18	2	no	—
211	C₂×C₄	—	18	2	no	—
212	C₂×C₄	—	18	2	no	—
213	C₂×C₄	—	18	2	no	—
214	C₂×C₄	—	18	2	no	—
215	C₂×C₄	C₄×Dic₉	1	2	yes	—
216	C₂×C₄	C₄×Dic₉	1	2	yes	—
217	C₂×C₄	C₂×C₄×D₉	1	1	yes	characteristic
218	C₂×C₄	D₄×D₉	1	1	yes	characteristic
219	C₂×C₄	#3699	1	1	yes	characteristic
220	C₂×C₄	Q₈×D₉	1	1	yes	characteristic
221	C₂×C₄	#3701	1	1	yes	characteristic
222	C₂×C₄	C₂²×Dic₉	1	1	yes	characteristic
223	C₈	—	2	2	no	—
224	C₈	—	2	2	no	—
225	C₈	—	18	2	no	—
226	C₈	—	18	2	no	—
227	C₂×C₃	#4839	1	2	yes	—
228	C₂×C₃	#4840	1	2	yes	—
229	C₂×C₃	Dic₃×C₄⋊C₄	1	1	yes	characteristic
230	C₂×C₃	#5101	1	1	yes	characteristic
231	C₂×C₃	#5122	1	1	yes	characteristic
232	C₂²	C₄⋊C₄×D₉	1	1	yes	characteristic
233	C₂²	#63454	1	1	yes	characteristic
234	C₂²	#63464	1	2	yes	—
235	C₂²	#63469	1	2	yes	—
236	C₂²	C₂×C₄×Dic₉	1	1	yes	characteristic
237	C₄	—	2	2	no	—
238	C₄	—	2	2	no	—
239	C₄	—	18	4	no	—
240	C₄	—	36	2	no	—
241	C₄	#63449	1	1	yes	characteristic
242	C₄	#63455	1	1	yes	characteristic
243	C₄	D₄×Dic₉	1	1	yes	characteristic
244	C₄	Q₈×Dic₉	1	1	yes	characteristic
245	C₃	#70442	1	1	yes	characteristic
246	C₂	#10588523	1	2	yes	—
247	C₂	#10588524	1	2	yes	—
248	C₂	C₄⋊C₄×Dic₉	1	1	yes	characteristic
249	C₂	#10588782	1	1	yes	characteristic
250	C₂	#10588797	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂²×D₉
commutator	C₂²×C₉	C₂×C₄²
Frattini	C₂²×C₃×C₄	C₂²×D₃
Fitting	C₂×C₉×C₄⋊C₈	C₂
socle, prime radical	C₂³×C₃	C₂×C₄×D₃
cosocle	C₂²×C₄×C₉	C₂³
hypercenter	C₂×C₄⋊C₈	D₉
hypocenter	C₉	#1768
LCS	G ▷ C₂²×C₉ ▷ C₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄⋊C₈

Complements

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

N	H ≅ G/N	type
16. C₉×C₄⋊C₈	C₄	semidirect
21. C₄⋊(C₉⋊C₈)	C₄	semidirect
138. C₄⋊C₈	Dic₉	semidirect
205. C₉	#1768	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₉×C₄⋊C₈	2
C₂×#63375	2
C₂×C₈×Dic₉	2
#10588612	2
C₂×C₄⋊(C₉⋊C₈)	2
#10588228	2
#70442	3