Subgroups — #11785264

Order 1152.

438 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	C₂×C₄×Dic₁₈	C₂	1	1	yes	maximal, characteristic
2	C₉×#382	C₂	1	1	yes	maximal, characteristic
3	C₂×#63368	C₂	1	2	yes	maximal
4	C₂×C₄⋊(C₉⋊C₈)	C₂	1	1	yes	maximal, characteristic
5	#10588206	C₂	1	2	yes	maximal
6	#69559	—	3	1	no	maximal
7	C₂²×Dic₁₈	C₂²	1	1	yes	characteristic
8	C₂×C₉×C₄⋊C₄	C₂²	1	2	yes	—
9	C₂×C₄²×C₉	C₂²	1	1	yes	characteristic
10	C₂×C₄⋊Dic₉	C₂²	1	1	yes	characteristic
11	C₂×Dic₉⋊C₄	—	2	1	no	—
12	C₂×C₄×Dic₉	—	2	1	no	—
13	C₂²×C₉⋊C₈	C₂²	1	2	yes	—
14	C₄×Dic₁₈	—	2	2	no	—
15	C₄×Dic₁₈	C₄	1	4	yes	complemented
16	C₉×#95	—	2	1	no	—
17	C₉×#95	—	2	1	no	—
18	#63368	—	2	4	no	—
19	C₄⋊(C₉⋊C₈)	—	2	2	no	—
20	C₂×C₄×Dic₆	—	3	1	no	—
21	C₃×#382	D₃	1	1	yes	characteristic
22	C₂×#810	—	3	2	no	—
23	C₂×C₄⋊(C₃⋊C₈)	—	3	1	no	—
24	#4529	—	3	2	no	—
25	C₂²×C₄×C₉	—	2	2	no	—
26	C₂²×C₄×C₉	D₄	1	1	yes	characteristic
27	C₂²×C₄×C₉	D₄	1	1	yes	characteristic
28	C₂²×C₄×C₉	C₂³	1	1	yes	characteristic
29	C₂²×Dic₉	—	2	1	no	—
30	C₂²×Dic₉	—	2	1	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	C₉×C₄⋊C₄	—	2	2	no	—
38	C₉×C₄⋊C₄	—	2	2	no	—
39	C₄²×C₉	C₂×C₄	1	2	yes	—
40	C₄²×C₉	D₄	1	2	yes	—
41	C₄⋊Dic₉	—	2	1	no	—
42	C₄⋊Dic₉	C₂×C₄	1	2	yes	—
43	Dic₉⋊C₄	—	2	4	no	—
44	C₄×Dic₉	—	2	4	no	—
45	C₂×C₉⋊C₈	—	2	2	no	—
46	C₂×C₉⋊C₈	—	2	2	no	—
47	C₂×C₉⋊C₈	—	2	2	no	—
48	#1805	—	9	1	no	Sylow
49	C₂²×Dic₆	—	3	1	no	—
50	C₂×C₃×C₄⋊C₄	C₂×D₃	1	2	yes	—
51	C₂×C₃×C₄²	C₂×D₃	1	1	yes	characteristic
52	C₂×C₄⋊Dic₃	—	3	1	no	—
53	C₂×Dic₃⋊C₄	—	6	1	no	—
54	C₂×C₄×Dic₃	—	6	1	no	—
55	C₂²×C₃⋊C₈	—	3	2	no	—
56	C₄×Dic₆	—	3	4	no	—
57	C₄×Dic₆	—	6	2	no	—
58	C₃×#95	—	2	1	no	—
59	C₃×#95	—	2	1	no	—
60	#810	—	6	4	no	—
61	C₄⋊(C₃⋊C₈)	—	6	2	no	—
62	C₂³×C₉	C₂×D₄	1	1	yes	characteristic
63	C₂×C₄×C₉	—	2	2	no	—
64	C₂×C₄×C₉	—	2	2	no	—
65	C₂×C₄×C₉	—	4	2	no	—
66	C₂×C₄×C₉	—	4	2	no	—
67	C₂×C₄×C₉	—	4	2	no	—
68	C₂×C₄×C₉	C₂²⋊C₄	1	2	yes	—
69	C₂×C₄×C₉	C₂²⋊C₄	1	2	yes	—
70	C₂×C₄×C₉	C₂²×C₄	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	C₂×Q₈	1	1	yes	characteristic
74	C₂×C₄×C₉	#41	1	1	yes	characteristic
75	C₂×C₄×C₉	#41	1	1	yes	characteristic
76	C₂×Dic₉	—	2	2	no	—
77	C₂×Dic₉	—	2	2	no	—
78	C₂×Dic₉	—	2	2	no	—
79	C₂×Dic₉	—	2	2	no	—
80	C₂×Dic₉	—	4	2	no	—
81	Dic₁₈	—	2	2	no	—
82	Dic₁₈	—	2	2	no	—
83	Dic₁₈	—	2	2	no	—
84	Dic₁₈	C₄⋊C₄	1	4	yes	—
85	C₉⋊C₈	—	4	2	no	—
86	C₂×C₄×Q₈	—	9	1	no	—
87	C₂×C₄⋊C₈	—	9	1	no	—
88	C₂×#103	—	9	2	no	—
89	#382	D₉	1	1	yes	characteristic
90	#340	—	9	2	no	—
91	C₂²×C₃×C₄	—	2	2	no	—
92	C₂²×C₃×C₄	#67	1	1	yes	characteristic
93	C₂²×C₃×C₄	#67	1	1	yes	characteristic
94	C₂²×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
95	C₂²×Dic₃	—	6	1	no	—
96	C₂²×Dic₃	—	6	1	no	—
97	C₂×Dic₆	—	3	2	no	—
98	C₂×Dic₆	—	3	2	no	—
99	C₂×Dic₆	—	3	2	no	—
100	C₂×Dic₆	—	6	1	no	—
101	C₂×Dic₆	—	6	1	no	—
102	C₂×Dic₆	—	6	1	no	—
103	C₃×C₄⋊C₄	—	2	2	no	—
104	C₃×C₄⋊C₄	—	2	2	no	—
105	C₃×C₄²	C₄×D₃	1	2	yes	—
106	C₃×C₄²	D₁₂	1	2	yes	—
107	C₄⋊Dic₃	—	3	2	no	—
108	C₄⋊Dic₃	—	6	1	no	—
109	Dic₃⋊C₄	—	6	4	no	—
110	C₄×Dic₃	—	6	4	no	—
111	C₂×C₃⋊C₈	—	6	2	no	—
112	C₂×C₃⋊C₈	—	6	2	no	—
113	C₂×C₃⋊C₈	—	6	2	no	—
114	C₂²×C₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
115	C₂²×C₉	#121	1	1	yes	characteristic
116	C₂²×C₉	C₂²⋊Q₈	1	1	yes	characteristic
117	C₂²×C₉	#135	1	2	yes	—
118	C₂²×C₉	#137	1	2	yes	—
119	C₄×C₉	—	2	2	no	—
120	C₄×C₉	—	2	2	no	—
121	C₄×C₉	—	8	2	no	—
122	C₄×C₉	C₂×C₄⋊C₄	1	1	yes	characteristic
123	C₄×C₉	#117	1	1	yes	characteristic
124	C₄×C₉	#121	1	1	yes	characteristic
125	C₄×C₉	C₂²⋊Q₈	1	1	yes	characteristic
126	Dic₉	—	2	4	no	—
127	Dic₉	—	4	2	no	—
128	C₂²×Q₈	—	9	1	no	—
129	C₂²×C₈	—	9	2	no	—
130	C₄×Q₈	—	9	4	no	—
131	C₄×Q₈	—	18	2	no	—
132	C₂×C₄⋊C₄	—	9	1	no	—
133	C₂×C₄⋊C₄	—	18	1	no	—
134	C₂×C₄⋊C₄	C₂×D₉	1	2	yes	—
135	C₂×C₄²	—	18	1	no	—
136	C₂×C₄²	C₂×D₉	1	1	yes	characteristic
137	C₄⋊C₈	—	18	2	no	—
138	#103	—	18	4	no	—
139	#95	—	2	1	no	—
140	#95	—	2	1	no	—
141	C₂³×C₃	C₂×#67	1	1	yes	characteristic
142	C₂×C₃×C₄	—	2	2	no	—
143	C₂×C₃×C₄	—	2	2	no	—
144	C₂×C₃×C₄	—	4	2	no	—
145	C₂×C₃×C₄	—	4	2	no	—
146	C₂×C₃×C₄	—	4	2	no	—
147	C₂×C₃×C₄	(C₂×D₃)⋊C₄	1	2	yes	—
148	C₂×C₃×C₄	(C₂×D₃)⋊C₄	1	2	yes	—
149	C₂×C₃×C₄	C₂×C₄×D₃	1	1	yes	characteristic
150	C₂×C₃×C₄	C₂×D₁₂	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	D₃×Q₈	1	1	yes	characteristic
154	C₂×C₃×C₄	#239	1	1	yes	characteristic
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	—	12	2	no	—
160	Dic₆	—	3	4	no	—
161	Dic₆	—	6	2	no	—
162	Dic₆	—	6	2	no	—
163	Dic₆	—	6	2	no	—
164	C₃⋊C₈	—	12	2	no	—
165	C₂×C₉	#380	1	1	yes	characteristic
166	C₂×C₉	#416	1	1	yes	characteristic
167	C₂×C₉	#419	1	1	yes	characteristic
168	C₂×C₉	#464	1	2	yes	—
169	C₂×C₉	#479	1	2	yes	—
170	C₂×Q₈	—	9	2	no	—
171	C₂×Q₈	—	9	2	no	—
172	C₂×Q₈	—	9	2	no	—
173	C₂×Q₈	—	18	1	no	—
174	C₂×Q₈	—	18	1	no	—
175	C₂×Q₈	—	18	1	no	—
176	C₂²×C₄	—	2	2	no	—
177	C₂²×C₄	—	18	1	no	—
178	C₂²×C₄	—	18	1	no	—
179	C₂²×C₄	#611	1	1	yes	characteristic
180	C₂²×C₄	#611	1	1	yes	characteristic
181	C₂²×C₄	C₂²×D₉	1	1	yes	characteristic
182	C₂×C₈	—	18	2	no	—
183	C₂×C₈	—	18	2	no	—
184	C₂×C₈	—	18	2	no	—
185	C₄⋊C₄	—	2	2	no	—
186	C₄⋊C₄	—	2	2	no	—
187	C₄⋊C₄	—	9	2	no	—
188	C₄⋊C₄	—	18	4	no	—
189	C₄⋊C₄	—	18	1	no	—
190	C₄²	—	18	4	no	—
191	C₄²	C₄×D₉	1	2	yes	—
192	C₄²	D₃₆	1	2	yes	—
193	C₂²×C₃	C₂×(C₂×D₃)⋊C₄	1	1	yes	characteristic
194	C₂²×C₃	#938	1	1	yes	characteristic
195	C₂²×C₃	#946	1	1	yes	characteristic
196	C₂²×C₃	#950	1	2	yes	—
197	C₂²×C₃	#951	1	2	yes	—
198	C₃×C₄	—	2	2	no	—
199	C₃×C₄	—	2	2	no	—
200	C₃×C₄	—	8	2	no	—
201	C₃×C₄	D₃×C₄⋊C₄	1	1	yes	characteristic
202	C₃×C₄	#892	1	1	yes	characteristic
203	C₃×C₄	#895	1	1	yes	characteristic
204	C₃×C₄	#897	1	1	yes	characteristic
205	Dic₃	—	6	4	no	—
206	Dic₃	—	12	2	no	—
207	C₉	#1805	1	1	yes	characteristic, complemented, Sylow
208	C₂³	C₂×#611	1	1	yes	characteristic
209	Q₈	—	9	4	no	—
210	Q₈	—	18	2	no	—
211	Q₈	—	18	2	no	—
212	Q₈	—	18	2	no	—
213	C₂×C₄	—	2	2	no	—
214	C₂×C₄	—	2	2	no	—
215	C₂×C₄	—	4	2	no	—
216	C₂×C₄	—	4	2	no	—
217	C₂×C₄	—	4	2	no	—
218	C₂×C₄	—	18	2	no	—
219	C₂×C₄	—	18	2	no	—
220	C₂×C₄	—	18	2	no	—
221	C₂×C₄	—	18	2	no	—
222	C₂×C₄	—	36	2	no	—
223	C₂×C₄	(C₂×D₉)⋊C₄	1	2	yes	—
224	C₂×C₄	(C₂×D₉)⋊C₄	1	2	yes	—
225	C₂×C₄	C₂×C₄×D₉	1	1	yes	characteristic
226	C₂×C₄	C₂×D₃₆	1	1	yes	characteristic
227	C₂×C₄	D₄×D₉	1	1	yes	characteristic
228	C₂×C₄	#3699	1	1	yes	characteristic
229	C₂×C₄	Q₈×D₉	1	1	yes	characteristic
230	C₂×C₄	#3701	1	1	yes	characteristic
231	C₈	—	36	2	no	—
232	C₂×C₃	#4987	1	1	yes	characteristic
233	C₂×C₃	#5002	1	1	yes	characteristic
234	C₂×C₃	#5004	1	1	yes	characteristic
235	C₂×C₃	#5054	1	2	yes	—
236	C₂×C₃	#5063	1	2	yes	—
237	C₂²	C₂×(C₂×D₉)⋊C₄	1	1	yes	characteristic
238	C₂²	#63500	1	1	yes	characteristic
239	C₂²	#63508	1	1	yes	characteristic
240	C₂²	#63511	1	2	yes	—
241	C₂²	#63513	1	2	yes	—
242	C₄	—	2	2	no	—
243	C₄	—	2	2	no	—
244	C₄	—	8	2	no	—
245	C₄	—	18	4	no	—
246	C₄	—	36	2	no	—
247	C₄	C₄⋊C₄×D₉	1	1	yes	characteristic
248	C₄	#63454	1	1	yes	characteristic
249	C₄	#63457	1	1	yes	characteristic
250	C₄	#63459	1	1	yes	characteristic
251	C₃	#69559	1	1	yes	characteristic
252	C₂	#10588667	1	1	yes	characteristic
253	C₂	#10588681	1	1	yes	characteristic
254	C₂	#10588683	1	1	yes	characteristic
255	C₂	#10588736	1	2	yes	—
256	C₂	#10588745	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×#611
commutator	C₂×C₄×C₉	C₂²×C₄
Frattini	C₂²×C₃×C₄	C₂²×D₃
Fitting	C₉×#382	C₂
socle, prime radical	C₂³×C₃	C₂×#67
cosocle	C₂²×C₄×C₉	C₂³
hypercenter	#382	D₉
hypocenter	C₉	#1805
LCS	G ▷ C₂×C₄×C₉ ▷ C₂×C₉ ▷ C₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #382

Complements

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

N	H ≅ G/N	type
15. C₄×Dic₁₈	C₄	semidirect
207. C₉	#1805	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
C₉×#382	2
C₂×#63368	2
C₂×C₄⋊(C₉⋊C₈)	2
#10588206	2
#69559	3