Subgroups — #11805584

Order 1152.

828 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×Q₈×C₉	C₂	1	1	yes	maximal, characteristic
2	C₂×Q₈×Dic₉	C₂	1	1	yes	maximal, characteristic
3	C₂×Q₈×Dic₉	C₂	1	1	yes	maximal, characteristic
4	#10588657	C₂	1	3	yes	maximal
5	#10588652	C₂	1	3	yes	maximal
6	C₄×Dic₉⋊C₄	C₂	1	3	yes	maximal
7	#10588608	C₂	1	3	yes	maximal
8	#82461	—	3	1	no	maximal
9	C₂²×Q₈×C₉	C₂²	1	1	yes	characteristic
10	C₄×Q₈×C₉	—	2	4	no	—
11	C₂×C₉×C₄⋊C₄	C₂²	1	3	yes	—
12	C₂×C₄²×C₉	C₂²	1	3	yes	—
13	Q₈×Dic₉	—	2	4	no	—
14	Q₈×Dic₉	C₄	1	8	yes	complemented
15	C₂×C₄⋊Dic₉	C₂²	1	3	yes	—
16	C₂×C₄⋊Dic₉	C₂²	1	3	yes	—
17	C₂×Dic₉⋊C₄	C₂²	1	3	yes	—
18	C₂×Dic₉⋊C₄	C₂²	1	3	yes	—
19	C₂×C₄×Dic₉	C₂²	1	3	yes	—
20	C₂×C₄×Dic₉	C₂²	1	3	yes	—
21	C₉⋊#95	C₂²	1	6	yes	—
22	C₉⋊#95	C₂²	1	1	yes	characteristic
23	C₉⋊#95	C₂²	1	3	yes	—
24	C₂×C₃×C₄×Q₈	D₃	1	1	yes	characteristic
25	C₂×Q₈×Dic₃	—	3	1	no	—
26	C₂×Q₈×Dic₃	—	3	1	no	—
27	#4977	—	3	3	no	—
28	#4972	—	3	3	no	—
29	C₄×Dic₃⋊C₄	—	3	3	no	—
30	#4928	—	3	3	no	—
31	C₂×Q₈×C₉	C₂×C₄	1	4	yes	—
32	C₂×Q₈×C₉	D₄	1	4	yes	—
33	C₂×Q₈×C₉	Q₈	1	4	yes	—
34	C₂²×C₄×C₉	C₂³	1	3	yes	—
35	C₂²×C₄×C₉	C₂³	1	1	yes	characteristic
36	C₂²×C₄×C₉	C₂³	1	3	yes	—
37	C₂²×Dic₉	C₂³	1	3	yes	—
38	C₂²×Dic₉	C₂³	1	1	yes	characteristic
39	C₂²×Dic₉	C₂³	1	3	yes	—
40	C₂²×Dic₉	C₂³	1	1	yes	characteristic
41	C₉×C₄⋊C₄	—	2	6	no	—
42	C₄²×C₉	—	2	6	no	—
43	C₄⋊Dic₉	—	2	6	no	—
44	C₄⋊Dic₉	C₂×C₄	1	12	yes	—
45	Dic₉⋊C₄	—	2	6	no	—
46	Dic₉⋊C₄	—	2	6	no	—
47	C₄×Dic₉	—	2	6	no	—
48	C₄×Dic₉	C₂×C₄	1	12	yes	—
49	#2351	—	9	1	no	Sylow
50	C₂²×C₃×Q₈	C₂×D₃	1	1	yes	characteristic
51	C₃×C₄×Q₈	—	2	4	no	—
52	C₂×C₃×C₄⋊C₄	C₂×D₃	1	3	yes	—
53	C₂×C₃×C₄²	C₂×D₃	1	3	yes	—
54	Q₈×Dic₃	—	3	8	no	—
55	Q₈×Dic₃	—	6	4	no	—
56	C₂×C₄⋊Dic₃	—	3	3	no	—
57	C₂×C₄⋊Dic₃	—	3	3	no	—
58	C₂×Dic₃⋊C₄	—	3	3	no	—
59	C₂×Dic₃⋊C₄	—	3	3	no	—
60	C₂×C₄×Dic₃	—	3	3	no	—
61	C₂×C₄×Dic₃	—	3	3	no	—
62	C₃⋊#95	—	3	6	no	—
63	C₃⋊#95	—	3	1	no	—
64	C₃⋊#95	—	3	3	no	—
65	C₂³×C₉	C₂⁴	1	1	yes	characteristic
66	Q₈×C₉	C₄⋊C₄	1	16	yes	complemented
67	C₂×C₄×C₉	—	2	1	no	—
68	C₂×C₄×C₉	—	2	1	no	—
69	C₂×C₄×C₉	—	2	3	no	—
70	C₂×C₄×C₉	—	2	3	no	—
71	C₂×C₄×C₉	—	2	3	no	—
72	C₂×C₄×C₉	C₂²×C₄	1	6	yes	—
73	C₂×C₄×C₉	C₂×D₄	1	6	yes	—
74	C₂×C₄×C₉	C₂×Q₈	1	6	yes	—
75	C₂×C₄×C₉	#41	1	2	yes	—
76	C₂×Dic₉	—	2	1	no	—
77	C₂×Dic₉	—	2	3	no	—
78	C₂×Dic₉	—	2	3	no	—
79	C₂×Dic₉	—	2	1	no	—
80	C₂×Dic₉	—	2	1	no	—
81	C₂×Dic₉	—	2	1	no	—
82	C₂×Dic₉	—	2	3	no	—
83	C₂×Dic₉	—	2	3	no	—
84	C₂×Dic₉	—	2	3	no	—
85	C₂×Dic₉	C₂²×C₄	1	2	yes	—
86	C₂×Dic₉	C₂²×C₄	1	6	yes	—
87	C₂×Dic₉	#41	1	2	yes	—
88	C₂×C₄×Q₈	—	9	1	no	—
89	C₂×C₄×Q₈	—	9	1	no	—
90	C₂×C₄×Q₈	D₉	1	1	yes	characteristic
91	#389	—	9	3	no	—
92	#389	—	9	3	no	—
93	#382	—	9	3	no	—
94	C₄×C₄⋊C₄	—	9	3	no	—
95	C₂×C₃×Q₈	Dic₆	1	4	yes	—
96	C₂×C₃×Q₈	C₄×D₃	1	4	yes	—
97	C₂×C₃×Q₈	#67	1	4	yes	—
98	C₂²×C₃×C₄	C₂²×D₃	1	3	yes	—
99	C₂²×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
100	C₂²×C₃×C₄	C₂²×D₃	1	3	yes	—
101	C₂²×Dic₃	—	3	1	no	—
102	C₂²×Dic₃	—	3	3	no	—
103	C₂²×Dic₃	—	3	1	no	—
104	C₂²×Dic₃	—	3	3	no	—
105	C₃×C₄⋊C₄	—	2	6	no	—
106	C₃×C₄²	—	2	6	no	—
107	C₄⋊Dic₃	—	3	12	no	—
108	C₄⋊Dic₃	—	6	6	no	—
109	Dic₃⋊C₄	—	6	6	no	—
110	Dic₃⋊C₄	—	6	6	no	—
111	C₄×Dic₃	—	3	12	no	—
112	C₄×Dic₃	—	6	6	no	—
113	C₂²×C₉	C₂³×C₄	1	1	yes	characteristic
114	C₂²×C₉	C₂²×D₄	1	1	yes	characteristic
115	C₂²×C₉	C₂²×Q₈	1	1	yes	characteristic
116	C₂²×C₉	C₂×#41	1	1	yes	characteristic
117	C₂²×C₉	C₂×#41	1	1	yes	characteristic
118	C₂²×C₉	2₋¹⁺⁴	1	1	yes	characteristic
119	C₂²×C₉	2₋¹⁺⁴	1	1	yes	characteristic
120	C₄×C₉	—	2	2	no	—
121	C₄×C₉	—	4	3	no	—
122	C₄×C₉	C₂×C₄⋊C₄	1	12	yes	—
123	Dic₉	—	2	6	no	—
124	Dic₉	—	2	2	no	—
125	Dic₉	—	2	2	no	—
126	Dic₉	—	4	3	no	—
127	C₂²×Q₈	C₂×D₉	1	1	yes	characteristic
128	C₄×Q₈	—	2	4	no	—
129	C₄×Q₈	—	9	8	no	—
130	C₄×Q₈	—	18	4	no	—
131	C₂×C₄⋊C₄	—	9	3	no	—
132	C₂×C₄⋊C₄	—	9	3	no	—
133	C₂×C₄⋊C₄	—	9	3	no	—
134	C₂×C₄⋊C₄	—	9	3	no	—
135	C₂×C₄⋊C₄	C₂×D₉	1	3	yes	—
136	C₂×C₄²	—	9	3	no	—
137	C₂×C₄²	—	9	3	no	—
138	C₂×C₄²	C₂×D₉	1	3	yes	—
139	#95	—	9	1	no	—
140	#95	—	9	3	no	—
141	#95	—	9	6	no	—
142	C₂³×C₃	C₂³×D₃	1	1	yes	characteristic
143	C₃×Q₈	Dic₃⋊C₄	1	16	yes	—
144	C₂×C₃×C₄	—	2	1	no	—
145	C₂×C₃×C₄	—	2	1	no	—
146	C₂×C₃×C₄	—	2	3	no	—
147	C₂×C₃×C₄	—	2	3	no	—
148	C₂×C₃×C₄	—	2	3	no	—
149	C₂×C₃×C₄	C₂×Dic₆	1	6	yes	—
150	C₂×C₃×C₄	C₂×C₄×D₃	1	6	yes	—
151	C₂×C₃×C₄	#239	1	2	yes	—
152	C₂×C₃×C₄	C₂×#67	1	6	yes	—
153	C₂×Dic₃	—	3	2	no	—
154	C₂×Dic₃	—	3	6	no	—
155	C₂×Dic₃	—	3	2	no	—
156	C₂×Dic₃	—	6	3	no	—
157	C₂×Dic₃	—	6	1	no	—
158	C₂×Dic₃	—	6	3	no	—
159	C₂×Dic₃	—	6	1	no	—
160	C₂×Dic₃	—	6	1	no	—
161	C₂×Dic₃	—	6	1	no	—
162	C₂×Dic₃	—	6	3	no	—
163	C₂×Dic₃	—	6	3	no	—
164	C₂×Dic₃	—	6	3	no	—
165	C₂×C₉	C₂²×C₄⋊C₄	1	1	yes	characteristic
166	C₂×C₉	C₄×#41	1	1	yes	characteristic
167	C₂×C₉	#519	1	1	yes	characteristic
168	C₂×C₉	#548	1	1	yes	characteristic
169	C₂×C₉	#548	1	1	yes	characteristic
170	C₂×C₉	#557	1	1	yes	characteristic
171	C₂×C₉	#557	1	1	yes	characteristic
172	C₂×Q₈	Dic₁₈	1	4	yes	—
173	C₂×Q₈	C₄×D₉	1	4	yes	—
174	C₂×Q₈	#611	1	4	yes	—
175	C₂²×C₄	—	9	1	no	—
176	C₂²×C₄	—	9	3	no	—
177	C₂²×C₄	—	9	1	no	—
178	C₂²×C₄	—	9	3	no	—
179	C₂²×C₄	C₂²×D₉	1	3	yes	—
180	C₂²×C₄	C₂²×D₉	1	1	yes	characteristic
181	C₂²×C₄	C₂²×D₉	1	3	yes	—
182	C₄⋊C₄	—	2	6	no	—
183	C₄⋊C₄	—	9	12	no	—
184	C₄⋊C₄	—	18	6	no	—
185	C₄⋊C₄	—	18	6	no	—
186	C₄⋊C₄	—	18	6	no	—
187	C₄²	—	2	6	no	—
188	C₄²	—	9	12	no	—
189	C₄²	—	18	6	no	—
190	C₂²×C₃	C₂²×Dic₆	1	1	yes	characteristic
191	C₂²×C₃	C₂²×C₄×D₃	1	1	yes	characteristic
192	C₂²×C₃	C₂×#239	1	1	yes	characteristic
193	C₂²×C₃	#1007	1	1	yes	characteristic
194	C₂²×C₃	D₃×#41	1	1	yes	characteristic
195	C₂²×C₃	#1010	1	1	yes	characteristic
196	C₂²×C₃	C₂²×#67	1	1	yes	characteristic
197	C₃×C₄	—	2	2	no	—
198	C₃×C₄	—	4	3	no	—
199	C₃×C₄	C₂×Dic₃⋊C₄	1	12	yes	—
200	Dic₃	—	6	2	no	—
201	Dic₃	—	6	6	no	—
202	Dic₃	—	6	2	no	—
203	Dic₃	—	12	3	no	—
204	C₉	#2351	1	1	yes	characteristic, complemented, Sylow
205	C₂³	C₂³×D₉	1	1	yes	characteristic
206	Q₈	Dic₉⋊C₄	1	16	yes	complemented
207	C₂×C₄	—	2	1	no	—
208	C₂×C₄	—	2	1	no	—
209	C₂×C₄	—	2	3	no	—
210	C₂×C₄	—	2	3	no	—
211	C₂×C₄	—	2	3	no	—
212	C₂×C₄	—	9	2	no	—
213	C₂×C₄	—	9	6	no	—
214	C₂×C₄	—	9	2	no	—
215	C₂×C₄	—	18	3	no	—
216	C₂×C₄	—	18	1	no	—
217	C₂×C₄	—	18	3	no	—
218	C₂×C₄	—	18	1	no	—
219	C₂×C₄	—	18	1	no	—
220	C₂×C₄	—	18	1	no	—
221	C₂×C₄	—	18	3	no	—
222	C₂×C₄	—	18	3	no	—
223	C₂×C₄	—	18	3	no	—
224	C₂×C₄	C₂×Dic₁₈	1	6	yes	—
225	C₂×C₄	C₂×C₄×D₉	1	6	yes	—
226	C₂×C₄	#3701	1	2	yes	—
227	C₂×C₄	C₂×#611	1	6	yes	—
228	C₂×C₃	#5569	1	1	yes	characteristic
229	C₂×C₃	#5570	1	1	yes	characteristic
230	C₂×C₃	#5572	1	1	yes	characteristic
231	C₂×C₃	C₄×#239	1	1	yes	characteristic
232	C₂×C₃	C₂²×Dic₃⋊C₄	1	1	yes	characteristic
233	C₂×C₃	#5817	1	1	yes	characteristic
234	C₂×C₃	#5831	1	1	yes	characteristic
235	C₂²	C₂²×Dic₁₈	1	1	yes	characteristic
236	C₂²	C₂²×C₄×D₉	1	1	yes	characteristic
237	C₂²	C₂×#3701	1	1	yes	characteristic
238	C₂²	#63713	1	1	yes	characteristic
239	C₂²	#41×D₉	1	1	yes	characteristic
240	C₂²	#63716	1	1	yes	characteristic
241	C₂²	C₂²×#611	1	1	yes	characteristic
242	C₄	—	2	2	no	—
243	C₄	—	4	3	no	—
244	C₄	—	18	2	no	—
245	C₄	—	18	6	no	—
246	C₄	—	18	2	no	—
247	C₄	—	36	3	no	—
248	C₄	C₂×Dic₉⋊C₄	1	12	yes	—
249	C₃	#82461	1	1	yes	characteristic
250	C₂	#10589671	1	1	yes	characteristic
251	C₂	#10589672	1	1	yes	characteristic
252	C₂	#10589674	1	1	yes	characteristic
253	C₂	C₄×#3701	1	1	yes	characteristic
254	C₂	C₂²×Dic₉⋊C₄	1	1	yes	characteristic
255	C₂	#10589919	1	1	yes	characteristic
256	C₂	#10589933	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂³×D₉
commutator	C₂²×C₉	C₂³×C₄
Frattini, socle, prime radical	C₂³×C₃	C₂³×D₃
Fitting	C₂×C₄×Q₈×C₉	C₂
cosocle	C₂³×C₉	C₂⁴
hypercenter	C₂×C₄×Q₈	D₉
hypocenter	C₉	#2351
LCS	G ▷ C₂²×C₉ ▷ C₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×Q₈

Complements

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

N	H ≅ G/N	type
14. Q₈×Dic₉	C₄	semidirect
66. Q₈×C₉	C₄⋊C₄	semidirect
204. C₉	#2351	semidirect
206. Q₈	Dic₉⋊C₄	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×Q₈×C₉	2
C₂×Q₈×Dic₉	2
#10588657	2
#10588652	2
C₄×Dic₉⋊C₄	2
#10588608	2
#82461	3