Subgroups — #11784241

Order 1152.

444 conjugacy classes of subgroups.

The subgroup lattice has 255 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Dic₉⋊C₈	C₂	1	1	yes	maximal, characteristic
2	C₂×C₈×Dic₉	C₂	1	1	yes	maximal, characteristic
3	C₂×C₈×Dic₉	C₂	1	1	yes	maximal, characteristic
4	C₄×Dic₉⋊C₄	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄×C₉⋊C₈	C₂	1	1	yes	maximal, characteristic
6	C₉×#336	C₂	1	1	yes	maximal, characteristic
7	#10588228	C₂	1	1	yes	maximal, characteristic
8	#68651	—	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×Dic₉⋊C₄	C₄	1	2	yes	—
13	C₂×C₄×Dic₉	C₂²	1	1	yes	characteristic
14	C₂×C₄×Dic₉	C₂²	1	1	yes	characteristic
15	C₂²×C₉⋊C₈	C₂²	1	1	yes	characteristic
16	C₂²×C₉⋊C₈	C₂²	1	1	yes	characteristic
17	C₉⋊#95	C₄	1	2	yes	—
18	Dic₉⋊C₈	C₄	1	4	yes	complemented
19	C₈×Dic₉	—	2	2	no	—
20	C₈×Dic₉	C₄	1	4	yes	complemented
21	C₄×C₉⋊C₈	—	2	2	no	—
22	C₂×Dic₃⋊C₈	—	3	1	no	—
23	C₂×C₈×Dic₃	—	3	1	no	—
24	C₂×C₈×Dic₃	—	3	1	no	—
25	C₄×Dic₃⋊C₄	—	3	1	no	—
26	C₂×C₄×C₃⋊C₈	—	3	1	no	—
27	C₃×#336	D₃	1	1	yes	characteristic
28	#4550	—	3	1	no	—
29	C₂²×C₄×C₉	C₂×C₄	1	2	yes	—
30	C₂²×C₄×C₉	C₂³	1	1	yes	characteristic
31	C₂²×Dic₉	C₂×C₄	1	1	yes	characteristic
32	C₂²×Dic₉	C₂×C₄	1	2	yes	—
33	C₂²×Dic₉	C₂×C₄	1	1	yes	characteristic
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	—	2	1	no	—
41	C₄²×C₉	—	2	1	no	—
42	Dic₉⋊C₄	C₈	1	8	yes	complemented
43	C₄×Dic₉	—	2	1	no	—
44	C₄×Dic₉	—	2	1	no	—
45	C₄×Dic₉	C₂×C₄	1	2	yes	—
46	C₄×Dic₉	C₂×C₄	1	2	yes	—
47	C₂×C₉⋊C₈	—	2	1	no	—
48	C₂×C₉⋊C₈	—	2	1	no	—
49	C₂×C₉⋊C₈	C₂×C₄	1	2	yes	—
50	C₂×C₉⋊C₈	C₂×C₄	1	2	yes	—
51	C₂×C₉⋊C₈	D₄	1	2	yes	—
52	C₂×C₉⋊C₈	Q₈	1	2	yes	—
53	C₈×C₄⋊C₄	—	9	1	no	Sylow
54	C₂²×C₃×C₈	C₂×D₃	1	1	yes	characteristic
55	C₂²×C₃×C₈	C₂×D₃	1	1	yes	characteristic
56	C₂×C₃×C₄²	C₂×D₃	1	1	yes	characteristic
57	C₂×Dic₃⋊C₄	—	3	2	no	—
58	C₂×C₄×Dic₃	—	3	1	no	—
59	C₂×C₄×Dic₃	—	3	1	no	—
60	C₂²×C₃⋊C₈	—	3	1	no	—
61	C₂²×C₃⋊C₈	—	3	1	no	—
62	C₃⋊#95	—	3	2	no	—
63	Dic₃⋊C₈	—	3	4	no	—
64	C₈×Dic₃	—	3	4	no	—
65	C₈×Dic₃	—	6	2	no	—
66	C₄×C₃⋊C₈	—	6	2	no	—
67	C₂³×C₉	C₂²×C₄	1	1	yes	characteristic
68	C₂×C₄×C₉	—	2	2	no	—
69	C₂×C₄×C₉	—	2	2	no	—
70	C₂×C₄×C₉	C₂×C₈	1	4	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	C₄²	1	2	yes	—
81	C₂×Dic₉	C₄²	1	2	yes	—
82	C₂×Dic₉	C₂×C₈	1	4	yes	—
83	C₂×Dic₉	C₂×C₈	1	4	yes	—
84	C₈×C₉	—	2	2	no	—
85	C₈×C₉	—	2	2	no	—
86	C₉⋊C₈	—	2	2	no	—
87	C₉⋊C₈	C₄⋊C₄	1	4	yes	complemented
88	C₂×C₄⋊C₈	—	9	1	no	—
89	C₂×C₄×C₈	—	9	1	no	—
90	C₂×C₄×C₈	—	9	1	no	—
91	C₂×C₄×C₈	—	9	1	no	—
92	C₄×C₄⋊C₄	—	9	1	no	—
93	#336	—	9	1	no	—
94	#336	D₉	1	1	yes	characteristic
95	C₂²×C₃×C₄	C₄×D₃	1	2	yes	—
96	C₂²×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
97	C₂²×Dic₃	—	3	1	no	—
98	C₂²×Dic₃	—	3	1	no	—
99	C₂²×Dic₃	—	3	2	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	—	2	1	no	—
104	C₂×C₃×C₈	C₄×D₃	1	2	yes	—
105	C₂×C₃×C₈	C₄×D₃	1	2	yes	—
106	C₃×C₄²	—	2	1	no	—
107	C₃×C₄²	—	2	1	no	—
108	Dic₃⋊C₄	—	3	8	no	—
109	C₄×Dic₃	—	3	2	no	—
110	C₄×Dic₃	—	3	2	no	—
111	C₄×Dic₃	—	6	1	no	—
112	C₄×Dic₃	—	6	1	no	—
113	C₂×C₃⋊C₈	—	3	2	no	—
114	C₂×C₃⋊C₈	—	3	2	no	—
115	C₂×C₃⋊C₈	—	3	2	no	—
116	C₂×C₃⋊C₈	—	3	2	no	—
117	C₂×C₃⋊C₈	—	6	1	no	—
118	C₂×C₃⋊C₈	—	6	1	no	—
119	C₂²×C₉	C₂×C₄²	1	1	yes	characteristic
120	C₂²×C₉	C₄×D₄	1	1	yes	characteristic
121	C₂²×C₉	C₄×Q₈	1	1	yes	characteristic
122	C₂²×C₉	C₂²×C₈	1	2	yes	—
123	C₂²×C₉	#131	1	2	yes	—
124	C₄×C₉	—	2	4	no	—
125	C₄×C₉	C₂×C₄⋊C₄	1	1	yes	characteristic
126	C₄×C₉	#117	1	1	yes	characteristic
127	C₄×C₉	C₄×D₄	1	1	yes	characteristic
128	C₄×C₉	C₄×Q₈	1	1	yes	characteristic
129	Dic₉	—	2	2	no	—
130	Dic₉	—	2	4	no	—
131	Dic₉	C₄×C₈	1	4	yes	complemented
132	C₂²×C₈	—	9	1	no	—
133	C₂²×C₈	—	9	1	no	—
134	C₂²×C₈	C₂×D₉	1	1	yes	characteristic
135	C₂²×C₈	C₂×D₉	1	1	yes	characteristic
136	C₂×C₄⋊C₄	—	9	2	no	—
137	C₂×C₄²	—	9	1	no	—
138	C₂×C₄²	—	9	1	no	—
139	C₂×C₄²	C₂×D₉	1	1	yes	characteristic
140	C₄⋊C₈	—	9	4	no	—
141	C₄×C₈	—	9	4	no	—
142	C₄×C₈	—	18	2	no	—
143	C₄×C₈	—	18	2	no	—
144	#95	—	9	2	no	—
145	C₂³×C₃	C₂×C₄×D₃	1	1	yes	characteristic
146	C₂×C₃×C₄	—	2	2	no	—
147	C₂×C₃×C₄	—	2	2	no	—
148	C₂×C₃×C₄	D₃×C₈	1	4	yes	—
149	C₂×C₃×C₄	C₂×C₄×D₃	1	1	yes	characteristic
150	C₂×C₃×C₄	C₂×C₄×D₃	1	1	yes	characteristic
151	C₂×C₃×C₄	D₃×D₄	1	1	yes	characteristic
152	C₂×C₃×C₄	#237	1	1	yes	characteristic
153	C₂×C₃×C₄	#237	1	1	yes	characteristic
154	C₂×C₃×C₄	D₃×Q₈	1	1	yes	characteristic
155	C₂×Dic₃	—	3	2	no	—
156	C₂×Dic₃	—	3	4	no	—
157	C₂×Dic₃	—	3	2	no	—
158	C₂×Dic₃	—	3	4	no	—
159	C₂×Dic₃	—	6	2	no	—
160	C₂×Dic₃	—	6	2	no	—
161	C₂×Dic₃	—	6	2	no	—
162	C₃×C₈	—	2	2	no	—
163	C₃×C₈	—	2	2	no	—
164	C₃⋊C₈	—	3	4	no	—
165	C₃⋊C₈	—	6	2	no	—
166	C₂×C₉	C₄×C₄⋊C₄	1	1	yes	characteristic
167	C₂×C₉	C₂×C₄×C₈	1	1	yes	characteristic
168	C₂×C₉	#405	1	1	yes	characteristic
169	C₂×C₉	C₈×D₄	1	2	yes	—
170	C₂×C₉	C₈×Q₈	1	2	yes	—
171	C₂²×C₄	—	9	1	no	—
172	C₂²×C₄	—	9	1	no	—
173	C₂²×C₄	—	9	2	no	—
174	C₂²×C₄	C₄×D₉	1	2	yes	—
175	C₂²×C₄	C₂²×D₉	1	1	yes	characteristic
176	C₂×C₈	—	2	1	no	—
177	C₂×C₈	—	2	1	no	—
178	C₂×C₈	—	2	1	no	—
179	C₂×C₈	—	2	1	no	—
180	C₂×C₈	—	9	2	no	—
181	C₂×C₈	—	9	2	no	—
182	C₂×C₈	—	9	2	no	—
183	C₂×C₈	—	9	2	no	—
184	C₂×C₈	—	18	1	no	—
185	C₂×C₈	—	18	1	no	—
186	C₂×C₈	C₄×D₉	1	2	yes	—
187	C₂×C₈	C₄×D₉	1	2	yes	—
188	C₄⋊C₄	—	9	8	no	—
189	C₄²	—	2	1	no	—
190	C₄²	—	2	1	no	—
191	C₄²	—	9	2	no	—
192	C₄²	—	9	2	no	—
193	C₄²	—	18	1	no	—
194	C₄²	—	18	1	no	—
195	C₂²×C₃	C₄²×D₃	1	1	yes	characteristic
196	C₂²×C₃	#67⋊C₄	1	1	yes	characteristic
197	C₂²×C₃	#887	1	1	yes	characteristic
198	C₂²×C₃	C₂×D₃×C₈	1	2	yes	—
199	C₂²×C₃	#907	1	2	yes	—
200	C₃×C₄	—	2	4	no	—
201	C₃×C₄	#877	1	1	yes	characteristic
202	C₃×C₄	#67⋊C₄	1	1	yes	characteristic
203	C₃×C₄	#887	1	1	yes	characteristic
204	C₃×C₄	D₃×C₄⋊C₄	1	1	yes	characteristic
205	Dic₃	—	3	4	no	—
206	Dic₃	—	6	2	no	—
207	Dic₃	—	6	4	no	—
208	C₉	C₈×C₄⋊C₄	1	1	yes	characteristic, complemented, Sylow
209	C₂³	C₂×C₄×D₉	1	1	yes	characteristic
210	C₂×C₄	—	2	2	no	—
211	C₂×C₄	—	2	2	no	—
212	C₂×C₄	—	9	2	no	—
213	C₂×C₄	—	9	4	no	—
214	C₂×C₄	—	9	2	no	—
215	C₂×C₄	—	9	4	no	—
216	C₂×C₄	—	18	2	no	—
217	C₂×C₄	—	18	2	no	—
218	C₂×C₄	—	18	2	no	—
219	C₂×C₄	C₈×D₉	1	4	yes	—
220	C₂×C₄	C₂×C₄×D₉	1	1	yes	characteristic
221	C₂×C₄	C₂×C₄×D₉	1	1	yes	characteristic
222	C₂×C₄	D₄×D₉	1	1	yes	characteristic
223	C₂×C₄	#3699	1	1	yes	characteristic
224	C₂×C₄	#3699	1	1	yes	characteristic
225	C₂×C₄	Q₈×D₉	1	1	yes	characteristic
226	C₈	—	2	2	no	—
227	C₈	—	2	2	no	—
228	C₈	—	9	4	no	—
229	C₈	—	18	2	no	—
230	C₂×C₃	#4649	1	1	yes	characteristic
231	C₂×C₃	C₄×D₃×C₈	1	1	yes	characteristic
232	C₂×C₃	#4708	1	1	yes	characteristic
233	C₂×C₃	#67⋊C₈	1	2	yes	—
234	C₂×C₃	#4830	1	2	yes	—
235	C₂²	C₄²×D₉	1	1	yes	characteristic
236	C₂²	#611⋊C₄	1	1	yes	characteristic
237	C₂²	#63449	1	1	yes	characteristic
238	C₂²	C₂×C₈×D₉	1	2	yes	—
239	C₂²	#63469	1	2	yes	—
240	C₄	—	2	4	no	—
241	C₄	—	9	4	no	—
242	C₄	—	18	2	no	—
243	C₄	—	18	4	no	—
244	C₄	#63439	1	1	yes	characteristic
245	C₄	#611⋊C₄	1	1	yes	characteristic
246	C₄	#63449	1	1	yes	characteristic
247	C₄	C₄⋊C₄×D₉	1	1	yes	characteristic
248	C₃	#68651	1	1	yes	characteristic
249	C₂	#10588329	1	1	yes	characteristic
250	C₂	C₄×C₈×D₉	1	1	yes	characteristic
251	C₂	#10588389	1	1	yes	characteristic
252	C₂	#611⋊C₈	1	2	yes	—
253	C₂	#10588508	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	C₂²×D₉
commutator	C₂×C₉	C₂×C₄×C₈
Frattini	C₂²×C₃×C₄	C₂²×D₃
Fitting	C₉×#336	C₂
socle, prime radical	C₂³×C₃	C₂×C₄×D₃
cosocle	C₂²×C₄×C₉	C₂³
hypercenter	#336	D₉
hypocenter	C₉	C₈×C₄⋊C₄
LCS	G ▷ C₂×C₉ ▷ C₉
UCS	C₁ ◁ C₂²×C₄ ◁ #336

Complements

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

N	H ≅ G/N	type
18. Dic₉⋊C₈	C₄	semidirect
20. C₈×Dic₉	C₄	semidirect
42. Dic₉⋊C₄	C₈	semidirect
87. C₉⋊C₈	C₄⋊C₄	semidirect
131. Dic₉	C₄×C₈	semidirect
208. C₉	C₈×C₄⋊C₄	semidirect

Maximal subgroups

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

subgroup	index
C₂×Dic₉⋊C₈	2
C₂×C₈×Dic₉	2
C₄×Dic₉⋊C₄	2
C₂×C₄×C₉⋊C₈	2
C₉×#336	2
#10588228	2
#68651	3