Subgroups — #11791031

Order 1152.

576 conjugacy classes of subgroups.

The subgroup lattice has 258 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₉×D₈	C₂	1	1	yes	maximal, characteristic
2	#10588901	C₂	1	2	yes	maximal, complemented
3	C₂×#63392	C₂	1	2	yes	maximal, complemented
4	C₂×#63378	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₈×Dic₉	C₂	1	1	yes	maximal, characteristic, complemented
6	#72038	—	3	1	no	maximal
7	C₂²×D₄×C₉	C₂²	1	2	yes	—
8	C₂×C₉×D₈	—	2	2	no	—
9	C₂×C₉×D₈	—	2	2	no	—
10	C₂×C₉×D₈	C₄	1	4	yes	complemented
11	C₂²×C₈×C₉	C₂²	1	1	yes	characteristic
12	C₂×#3676	—	2	4	no	—
13	C₂×C₄⋊Dic₉	C₂²	1	2	yes	—
14	C₂×C₄×Dic₉	C₂²	1	1	yes	characteristic
15	C₂²×C₉⋊C₈	C₂²	1	1	yes	characteristic
16	#63392	—	2	4	no	—
17	#63378	—	2	2	no	—
18	C₈×Dic₉	—	2	2	no	—
19	C₂²×C₃×D₈	D₃	1	1	yes	characteristic
20	#5221	—	3	2	no	—
21	C₂×#832	—	3	2	no	—
22	C₂×#818	—	3	1	no	—
23	C₂×C₈×Dic₃	—	3	1	no	—
24	C₂⁴×C₉	—	2	2	no	—
25	C₂×D₄×C₉	—	2	2	no	—
26	C₂×D₄×C₉	—	2	2	no	—
27	C₂×D₄×C₉	—	2	2	no	—
28	C₂×D₄×C₉	—	2	2	no	—
29	C₂×D₄×C₉	—	2	2	no	—
30	C₂×D₄×C₉	C₂×C₄	1	4	yes	—
31	C₂²×C₄×C₉	C₂³	1	1	yes	characteristic
32	C₂²×Dic₉	—	2	2	no	—
33	C₂²×Dic₉	D₄	1	2	yes	—
34	C₉×D₈	—	2	8	no	—
35	C₂×C₈×C₉	C₂×C₄	1	2	yes	—
36	C₂×C₈×C₉	D₄	1	2	yes	—
37	C₂×C₈×C₉	D₄	1	2	yes	—
38	#3676	—	4	8	no	—
39	C₄⋊Dic₉	—	2	2	no	—
40	C₄⋊Dic₉	—	2	2	no	—
41	C₄×Dic₉	—	2	1	no	—
42	C₄×Dic₉	—	2	1	no	—
43	C₂×C₉⋊C₈	—	2	1	no	—
44	C₂×C₉⋊C₈	—	2	1	no	—
45	C₂×C₉⋊C₈	D₄	1	2	yes	—
46	#1970	—	9	1	no	Sylow
47	C₂²×C₃×D₄	C₂×D₃	1	2	yes	—
48	C₂×C₃×D₈	—	2	2	no	—
49	C₂×C₃×D₈	—	2	2	no	—
50	C₂×C₃×D₈	Dic₃	1	4	yes	—
51	C₂²×C₃×C₈	C₂×D₃	1	1	yes	characteristic
52	C₂×#217	—	6	4	no	—
53	C₂×C₄⋊Dic₃	—	3	2	no	—
54	C₂×C₄×Dic₃	—	3	1	no	—
55	C₂²×C₃⋊C₈	—	3	1	no	—
56	#832	—	6	4	no	—
57	#818	—	6	2	no	—
58	C₈×Dic₃	—	6	2	no	—
59	C₂³×C₉	—	2	4	no	—
60	C₂³×C₉	—	4	2	no	—
61	C₂³×C₉	—	4	2	no	—
62	C₂³×C₉	—	4	4	no	—
63	C₂³×C₉	—	4	4	no	—
64	C₂³×C₉	C₂×D₄	1	1	yes	characteristic
65	D₄×C₉	—	2	4	no	—
66	D₄×C₉	—	2	4	no	—
67	D₄×C₉	—	4	2	no	—
68	D₄×C₉	—	4	2	no	—
69	C₂×C₄×C₉	C₂²×C₄	1	1	yes	characteristic
70	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
71	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
72	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
73	C₂×C₄×C₉	#41	1	1	yes	characteristic
74	C₂×C₄×C₉	#41	1	1	yes	characteristic
75	C₂×Dic₉	—	2	2	no	—
76	C₂×Dic₉	—	2	2	no	—
77	C₂×Dic₉	—	4	2	no	—
78	C₂×Dic₉	—	4	4	no	—
79	C₂×Dic₉	D₈	1	4	yes	complemented
80	C₈×C₉	C₂²⋊C₄	1	4	yes	complemented
81	C₉⋊C₈	—	2	2	no	—
82	C₂²×D₈	D₉	1	1	yes	characteristic
83	C₂×#107	—	9	1	no	—
84	C₂×#102	—	9	2	no	—
85	C₂×C₄×C₈	—	9	1	no	—
86	#390	—	9	2	no	—
87	C₂⁴×C₃	—	2	2	no	—
88	C₂×C₃×D₄	—	2	2	no	—
89	C₂×C₃×D₄	—	2	2	no	—
90	C₂×C₃×D₄	—	2	2	no	—
91	C₂×C₃×D₄	—	2	2	no	—
92	C₂×C₃×D₄	—	2	2	no	—
93	C₂×C₃×D₄	C₂×Dic₃	1	4	yes	—
94	C₂²×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
95	C₂²×Dic₃	—	3	2	no	—
96	C₂²×Dic₃	—	6	2	no	—
97	C₃×D₈	—	2	8	no	—
98	C₂×C₃×C₈	C₂×Dic₃	1	2	yes	—
99	C₂×C₃×C₈	#67	1	2	yes	—
100	C₂×C₃×C₈	#67	1	2	yes	—
101	#217	—	12	8	no	—
102	C₄⋊Dic₃	—	6	2	no	—
103	C₄⋊Dic₃	—	6	2	no	—
104	C₄×Dic₃	—	6	1	no	—
105	C₄×Dic₃	—	6	1	no	—
106	C₂×C₃⋊C₈	—	3	2	no	—
107	C₂×C₃⋊C₈	—	6	1	no	—
108	C₂×C₃⋊C₈	—	6	1	no	—
109	C₂²×C₉	—	4	4	no	—
110	C₂²×C₉	—	4	8	no	—
111	C₂²×C₉	—	8	4	no	—
112	C₂²×C₉	—	8	2	no	—
113	C₂²×C₉	—	8	2	no	—
114	C₂²×C₉	C₄×D₄	1	1	yes	characteristic
115	C₂²×C₉	#121	1	1	yes	characteristic
116	C₂²×C₉	#127	1	1	yes	characteristic
117	C₂²×C₉	C₂×D₈	1	2	yes	—
118	C₂²×C₉	#135	1	2	yes	—
119	C₄×C₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
120	C₄×C₉	C₄×D₄	1	1	yes	characteristic
121	C₄×C₉	#121	1	1	yes	characteristic
122	C₄×C₉	#124	1	1	yes	characteristic
123	Dic₉	—	2	4	no	—
124	Dic₉	—	8	2	no	—
125	C₂²×D₄	C₂×D₉	1	2	yes	—
126	C₂×D₈	—	2	2	no	—
127	C₂×D₈	—	2	2	no	—
128	C₂×D₈	Dic₉	1	4	yes	complemented
129	C₂²×C₈	—	9	1	no	—
130	C₂²×C₈	C₂×D₉	1	1	yes	characteristic
131	C₂×C₄⋊C₄	—	9	2	no	—
132	C₂×C₂²⋊C₄	—	18	4	no	—
133	C₂×C₄²	—	9	1	no	—
134	#107	—	18	2	no	—
135	#102	—	18	4	no	—
136	C₄×C₈	—	18	2	no	—
137	C₂³×C₃	—	2	4	no	—
138	C₂³×C₃	—	4	4	no	—
139	C₂³×C₃	—	4	4	no	—
140	C₂³×C₃	—	4	2	no	—
141	C₂³×C₃	—	4	2	no	—
142	C₂³×C₃	D₃×D₄	1	1	yes	characteristic
143	C₃×D₄	—	2	4	no	—
144	C₃×D₄	—	2	4	no	—
145	C₃×D₄	—	4	2	no	—
146	C₃×D₄	—	4	2	no	—
147	C₂×C₃×C₄	D₃×D₄	1	1	yes	characteristic
148	C₂×C₃×C₄	#237	1	1	yes	characteristic
149	C₂×C₃×C₄	#237	1	1	yes	characteristic
150	C₂×C₃×C₄	C₂²×Dic₃	1	1	yes	characteristic
151	C₂×C₃×C₄	C₂×#67	1	1	yes	characteristic
152	C₂×C₃×C₄	C₂×#67	1	1	yes	characteristic
153	C₂×Dic₃	—	3	4	no	—
154	C₂×Dic₃	—	6	2	no	—
155	C₂×Dic₃	—	6	2	no	—
156	C₂×Dic₃	—	12	4	no	—
157	C₂×Dic₃	—	12	2	no	—
158	C₃×C₈	#217	1	4	yes	—
159	C₃⋊C₈	—	6	2	no	—
160	C₂×C₉	—	8	4	no	—
161	C₂×C₉	#390	1	1	yes	characteristic
162	C₂×C₉	C₄×D₈	1	2	yes	—
163	C₂×C₉	#466	1	2	yes	—
164	C₂×C₉	#493	1	1	yes	characteristic
165	C₂×C₉	#495	1	1	yes	characteristic
166	C₂⁴	—	2	2	no	—
167	C₂×D₄	—	2	2	no	—
168	C₂×D₄	—	2	2	no	—
169	C₂×D₄	—	2	2	no	—
170	C₂×D₄	—	2	2	no	—
171	C₂×D₄	—	2	2	no	—
172	C₂×D₄	C₂×Dic₉	1	4	yes	—
173	C₂²×C₄	—	9	2	no	—
174	C₂²×C₄	—	18	2	no	—
175	C₂²×C₄	C₂²×D₉	1	1	yes	characteristic
176	D₈	—	2	8	no	—
177	C₂×C₈	—	9	2	no	—
178	C₂×C₈	—	18	1	no	—
179	C₂×C₈	—	18	1	no	—
180	C₂×C₈	C₂×Dic₉	1	2	yes	—
181	C₂×C₈	#611	1	2	yes	—
182	C₂×C₈	#611	1	2	yes	—
183	C₄⋊C₄	—	18	2	no	—
184	C₄⋊C₄	—	18	2	no	—
185	C₂²⋊C₄	—	36	8	no	—
186	C₄²	—	18	1	no	—
187	C₄²	—	18	1	no	—
188	C₂²×C₃	—	4	8	no	—
189	C₂²×C₃	—	4	4	no	—
190	C₂²×C₃	—	8	2	no	—
191	C₂²×C₃	—	8	2	no	—
192	C₂²×C₃	—	8	4	no	—
193	C₂²×C₃	D₃×D₈	1	2	yes	—
194	C₂²×C₃	#912	1	2	yes	—
195	C₂²×C₃	D₄×Dic₃	1	1	yes	characteristic
196	C₂²×C₃	#938	1	1	yes	characteristic
197	C₂²×C₃	#940	1	1	yes	characteristic
198	C₃×C₄	D₄×Dic₃	1	1	yes	characteristic
199	C₃×C₄	#936	1	1	yes	characteristic
200	C₃×C₄	#938	1	1	yes	characteristic
201	C₃×C₄	C₂×#217	1	1	yes	characteristic
202	Dic₃	—	6	4	no	—
203	Dic₃	—	24	2	no	—
204	C₉	#1970	1	1	yes	characteristic, complemented, Sylow
205	C₂³	—	2	4	no	—
206	C₂³	—	4	2	no	—
207	C₂³	—	4	2	no	—
208	C₂³	—	4	4	no	—
209	C₂³	—	4	4	no	—
210	C₂³	D₄×D₉	1	1	yes	characteristic
211	D₄	—	2	4	no	—
212	D₄	—	2	4	no	—
213	D₄	—	4	2	no	—
214	D₄	—	4	2	no	—
215	C₂×C₄	—	9	4	no	—
216	C₂×C₄	—	18	2	no	—
217	C₂×C₄	—	18	2	no	—
218	C₂×C₄	—	36	2	no	—
219	C₂×C₄	—	36	4	no	—
220	C₂×C₄	D₄×D₉	1	1	yes	characteristic
221	C₂×C₄	#3699	1	1	yes	characteristic
222	C₂×C₄	#3699	1	1	yes	characteristic
223	C₂×C₄	C₂²×Dic₉	1	1	yes	characteristic
224	C₂×C₄	C₂×#611	1	1	yes	characteristic
225	C₂×C₄	C₂×#611	1	1	yes	characteristic
226	C₈	—	18	2	no	—
227	C₈	#3676	1	4	yes	complemented
228	C₂×C₃	—	8	4	no	—
229	C₂×C₃	Dic₃×D₈	1	2	yes	—
230	C₂×C₃	#5151	1	1	yes	characteristic
231	C₂×C₃	#5155	1	1	yes	characteristic
232	C₂×C₃	#5157	1	2	yes	—
233	C₂×C₃	#5221	1	1	yes	characteristic
234	C₂²	—	4	8	no	—
235	C₂²	—	4	4	no	—
236	C₂²	—	8	2	no	—
237	C₂²	—	8	2	no	—
238	C₂²	—	8	4	no	—
239	C₂²	D₈×D₉	1	2	yes	—
240	C₂²	#63474	1	2	yes	—
241	C₂²	D₄×Dic₉	1	1	yes	characteristic
242	C₂²	#63500	1	1	yes	characteristic
243	C₂²	#63502	1	1	yes	characteristic
244	C₄	—	18	4	no	—
245	C₄	—	72	2	no	—
246	C₄	D₄×Dic₉	1	1	yes	characteristic
247	C₄	#63498	1	1	yes	characteristic
248	C₄	#63500	1	1	yes	characteristic
249	C₄	C₂×#3676	1	1	yes	characteristic
250	C₃	#72038	1	1	yes	characteristic
251	C₂	—	8	4	no	—
252	C₂	D₈×Dic₉	1	2	yes	—
253	C₂	#10588831	1	1	yes	characteristic
254	C₂	#10588835	1	1	yes	characteristic
255	C₂	#10588837	1	2	yes	—
256	C₂	#10588901	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	D₄×D₉
commutator	C₂×C₄×C₉	C₂²×C₄
Frattini	C₂²×C₃×C₄	C₂²×D₃
Fitting	C₂²×C₉×D₈	C₂
socle	C₂³×C₃	D₃×D₄
prime radical	C₂²×C₃×D₈	D₃
cosocle	C₂²×C₄×C₉	C₂³
hypercenter	C₂²×D₈	D₉
hypocenter	C₉	#1970
LCS	G ▷ C₂×C₄×C₉ ▷ C₂×C₉ ▷ C₉
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×D₈

Complements

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

N	H ≅ G/N	type
2. #10588901	C₂	semidirect
3. C₂×#63392	C₂	semidirect
4. C₂×#63378	C₂	semidirect
5. C₂×C₈×Dic₉	C₂	semidirect
10. C₂×C₉×D₈	C₄	semidirect
79. C₂×Dic₉	D₈	semidirect
80. C₈×C₉	C₂²⋊C₄	semidirect
128. C₂×D₈	Dic₉	semidirect
204. C₉	#1970	semidirect
227. C₈	#3676	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₉×D₈	2
#10588901	2
C₂×#63392	2
C₂×#63378	2
C₂×C₈×Dic₉	2
#72038	3