Subgroups — #11784028

Order 1152.

486 conjugacy classes of subgroups.

The subgroup lattice has 246 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×#63395	C₂	1	2	yes	maximal
3	C₄×C₄⋊Dic₉	C₂	1	1	yes	maximal, characteristic
4	C₂×#63362	C₂	1	1	yes	maximal, characteristic
5	#10588206	C₂	1	2	yes	maximal
6	#71631	—	3	1	no	maximal
7	C₂²×Q₈×C₉	C₂²	1	1	yes	characteristic
8	C₄×Q₈×C₉	—	2	2	no	—
9	C₄×Q₈×C₉	C₄	1	4	yes	complemented
10	C₂×C₉×C₄⋊C₄	—	2	1	no	—
11	C₂×C₉×C₄⋊C₄	C₂²	1	1	yes	characteristic
12	C₂×C₄²×C₉	—	2	1	no	—
13	C₂×C₄²×C₉	C₂²	1	1	yes	characteristic
14	C₂×C₄⋊Dic₉	C₂²	1	2	yes	—
15	C₂×C₄×Dic₉	—	2	1	no	—
16	C₂²×C₉⋊C₈	C₂²	1	2	yes	—
17	#63395	C₄	1	8	yes	complemented
18	C₉⋊#95	—	2	1	no	—
19	#63362	—	2	2	no	—
20	C₂×C₃×C₄×Q₈	D₃	1	1	yes	characteristic
21	C₂×#835	—	3	2	no	—
22	C₄×C₄⋊Dic₃	—	3	1	no	—
23	C₂×#803	—	3	1	no	—
24	#4529	—	3	2	no	—
25	C₂×Q₈×C₉	—	2	1	no	—
26	C₂×Q₈×C₉	—	2	2	no	—
27	C₂×Q₈×C₉	C₂×C₄	1	4	yes	—
28	C₂×Q₈×C₉	C₂×C₄	1	2	yes	—
29	C₂²×C₄×C₉	—	2	1	no	—
30	C₂²×C₄×C₉	—	2	1	no	—
31	C₂²×C₄×C₉	D₄	1	1	yes	characteristic
32	C₂²×C₄×C₉	D₄	1	1	yes	characteristic
33	C₂²×C₄×C₉	C₂³	1	1	yes	characteristic
34	C₂²×Dic₉	—	2	2	no	—
35	C₉×C₄⋊C₄	—	2	4	no	—
36	C₉×C₄⋊C₄	—	2	1	no	—
37	C₉×C₄⋊C₄	C₂×C₄	1	2	yes	—
38	C₄²×C₉	—	2	4	no	—
39	C₄²×C₉	C₂×C₄	1	2	yes	—
40	C₄²×C₉	D₄	1	2	yes	—
41	C₄⋊Dic₉	—	2	2	no	—
42	C₄⋊Dic₉	C₂×C₄	1	4	yes	—
43	C₄×Dic₉	—	4	2	no	—
44	C₂×C₉⋊C₈	—	2	2	no	—
45	C₂×C₉⋊C₈	—	2	2	no	—
46	C₂×C₉⋊C₈	C₂×C₄	1	4	yes	—
47	#1763	—	9	1	no	Sylow
48	C₂²×C₃×Q₈	C₂×D₃	1	1	yes	characteristic
49	C₃×C₄×Q₈	—	2	2	no	—
50	C₃×C₄×Q₈	Dic₃	1	4	yes	—
51	C₂×C₃×C₄⋊C₄	—	2	1	no	—
52	C₂×C₃×C₄⋊C₄	C₂×D₃	1	1	yes	characteristic
53	C₂×C₃×C₄²	—	2	1	no	—
54	C₂×C₃×C₄²	C₂×D₃	1	1	yes	characteristic
55	C₂×C₄⋊Dic₃	—	3	2	no	—
56	C₂×C₄×Dic₃	—	6	1	no	—
57	C₂²×C₃⋊C₈	—	3	2	no	—
58	#835	—	3	8	no	—
59	C₃⋊#95	—	6	1	no	—
60	#803	—	6	2	no	—
61	C₂³×C₉	C₂×D₄	1	1	yes	characteristic
62	Q₈×C₉	—	2	2	no	—
63	Q₈×C₉	—	2	4	no	—
64	Q₈×C₉	C₄²	1	4	yes	complemented
65	C₂×C₄×C₉	—	2	2	no	—
66	C₂×C₄×C₉	—	2	4	no	—
67	C₂×C₄×C₉	—	2	1	no	—
68	C₂×C₄×C₉	—	2	2	no	—
69	C₂×C₄×C₉	—	2	1	no	—
70	C₂×C₄×C₉	—	2	2	no	—
71	C₂×C₄×C₉	—	4	1	no	—
72	C₂×C₄×C₉	—	4	1	no	—
73	C₂×C₄×C₉	C₂²⋊C₄	1	2	yes	—
74	C₂×C₄×C₉	C₂²⋊C₄	1	2	yes	—
75	C₂×C₄×C₉	C₂²×C₄	1	2	yes	—
76	C₂×C₄×C₉	C₂²×C₄	1	1	yes	characteristic
77	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
78	C₂×C₄×C₉	#41	1	2	yes	—
79	C₂×Dic₉	—	2	4	no	—
80	C₂×Dic₉	—	4	2	no	—
81	C₂×Dic₉	—	4	2	no	—
82	C₉⋊C₈	—	2	4	no	—
83	C₂×C₄×Q₈	D₉	1	1	yes	characteristic
84	C₂×#103	—	9	2	no	—
85	C₂×#97	—	9	1	no	—
86	C₄×C₄⋊C₄	—	9	1	no	—
87	#340	—	9	2	no	—
88	C₂×C₃×Q₈	—	2	2	no	—
89	C₂×C₃×Q₈	—	2	1	no	—
90	C₂×C₃×Q₈	C₄×D₃	1	4	yes	—
91	C₂×C₃×Q₈	C₂×Dic₃	1	2	yes	—
92	C₂²×C₃×C₄	—	2	1	no	—
93	C₂²×C₃×C₄	—	2	1	no	—
94	C₂²×C₃×C₄	#67	1	1	yes	characteristic
95	C₂²×C₃×C₄	#67	1	1	yes	characteristic
96	C₂²×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
97	C₂²×Dic₃	—	6	2	no	—
98	C₃×C₄⋊C₄	—	2	4	no	—
99	C₃×C₄⋊C₄	—	2	1	no	—
100	C₃×C₄⋊C₄	C₂×Dic₃	1	2	yes	—
101	C₃×C₄²	—	2	4	no	—
102	C₃×C₄²	C₂×Dic₃	1	2	yes	—
103	C₃×C₄²	#67	1	2	yes	—
104	C₄⋊Dic₃	—	3	4	no	—
105	C₄⋊Dic₃	—	6	2	no	—
106	C₄×Dic₃	—	12	2	no	—
107	C₂×C₃⋊C₈	—	3	4	no	—
108	C₂×C₃⋊C₈	—	6	2	no	—
109	C₂×C₃⋊C₈	—	6	2	no	—
110	C₂²×C₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
111	C₂²×C₉	C₄×D₄	1	2	yes	—
112	C₂²×C₉	AGL(1,ℤ/8)	1	1	yes	characteristic
113	C₂²×C₉	#137	1	2	yes	—
114	C₂²×C₉	#137	1	1	yes	characteristic
115	C₄×C₉	—	2	4	no	—
116	C₄×C₉	—	2	2	no	—
117	C₄×C₉	—	2	2	no	—
118	C₄×C₉	—	4	2	no	—
119	C₄×C₉	C₂×C₄²	1	1	yes	characteristic
120	C₄×C₉	#117	1	1	yes	characteristic
121	C₄×C₉	C₄×D₄	1	2	yes	—
122	Dic₉	—	4	4	no	—
123	C₂²×Q₈	C₂×D₉	1	1	yes	characteristic
124	C₂²×C₈	—	9	2	no	—
125	C₄×Q₈	—	2	2	no	—
126	C₄×Q₈	Dic₉	1	4	yes	complemented
127	C₂×C₄⋊C₄	—	2	1	no	—
128	C₂×C₄⋊C₄	—	9	2	no	—
129	C₂×C₄⋊C₄	C₂×D₉	1	1	yes	characteristic
130	C₂×C₄²	—	2	1	no	—
131	C₂×C₄²	—	18	1	no	—
132	C₂×C₄²	C₂×D₉	1	1	yes	characteristic
133	#103	—	9	8	no	—
134	#97	—	18	2	no	—
135	#95	—	18	1	no	—
136	C₂³×C₃	C₂×#67	1	1	yes	characteristic
137	C₃×Q₈	—	2	2	no	—
138	C₃×Q₈	—	2	4	no	—
139	C₃×Q₈	C₄×Dic₃	1	4	yes	—
140	C₂×C₃×C₄	—	2	4	no	—
141	C₂×C₃×C₄	—	2	2	no	—
142	C₂×C₃×C₄	—	2	1	no	—
143	C₂×C₃×C₄	—	2	1	no	—
144	C₂×C₃×C₄	—	2	2	no	—
145	C₂×C₃×C₄	—	2	2	no	—
146	C₂×C₃×C₄	—	4	1	no	—
147	C₂×C₃×C₄	—	4	1	no	—
148	C₂×C₃×C₄	#217	1	2	yes	—
149	C₂×C₃×C₄	#217	1	2	yes	—
150	C₂×C₃×C₄	C₂×C₄×D₃	1	2	yes	—
151	C₂×C₃×C₄	#235	1	2	yes	—
152	C₂×C₃×C₄	C₂²×Dic₃	1	1	yes	characteristic
153	C₂×C₃×C₄	C₂×#67	1	1	yes	characteristic
154	C₂×Dic₃	—	6	4	no	—
155	C₂×Dic₃	—	12	2	no	—
156	C₂×Dic₃	—	12	2	no	—
157	C₃⋊C₈	—	6	4	no	—
158	C₂×C₉	C₄×C₂²⋊C₄	1	1	yes	characteristic
159	C₂×C₉	#417	1	1	yes	characteristic
160	C₂×C₉	#419	1	1	yes	characteristic
161	C₂×C₉	#440	1	2	yes	—
162	C₂×C₉	#441	1	2	yes	—
163	C₂×Q₈	—	2	2	no	—
164	C₂×Q₈	—	2	1	no	—
165	C₂×Q₈	C₄×D₉	1	4	yes	—
166	C₂×Q₈	C₂×Dic₉	1	2	yes	—
167	C₂²×C₄	—	2	1	no	—
168	C₂²×C₄	—	2	1	no	—
169	C₂²×C₄	—	18	2	no	—
170	C₂²×C₄	#611	1	1	yes	characteristic
171	C₂²×C₄	#611	1	1	yes	characteristic
172	C₂²×C₄	C₂²×D₉	1	1	yes	characteristic
173	C₂×C₈	—	9	4	no	—
174	C₂×C₈	—	18	2	no	—
175	C₂×C₈	—	18	2	no	—
176	C₄⋊C₄	—	2	4	no	—
177	C₄⋊C₄	—	2	1	no	—
178	C₄⋊C₄	—	9	4	no	—
179	C₄⋊C₄	—	18	2	no	—
180	C₄⋊C₄	C₂×Dic₉	1	2	yes	—
181	C₄²	—	2	4	no	—
182	C₄²	—	36	2	no	—
183	C₄²	C₂×Dic₉	1	2	yes	—
184	C₄²	#611	1	2	yes	—
185	C₂²×C₃	C₄×#67	1	2	yes	—
186	C₂²×C₃	#942	1	2	yes	—
187	C₂²×C₃	#949	1	1	yes	characteristic
188	C₂²×C₃	#951	1	1	yes	characteristic
189	C₂²×C₃	C₂×#217	1	1	yes	characteristic
190	C₃×C₄	—	2	4	no	—
191	C₃×C₄	—	2	2	no	—
192	C₃×C₄	—	2	2	no	—
193	C₃×C₄	—	4	2	no	—
194	C₃×C₄	C₂×C₄×Dic₃	1	1	yes	characteristic
195	C₃×C₄	#926	1	1	yes	characteristic
196	C₃×C₄	C₄×#67	1	2	yes	—
197	Dic₃	—	12	4	no	—
198	C₉	#1763	1	1	yes	characteristic, complemented, Sylow
199	C₂³	C₂×#611	1	1	yes	characteristic
200	Q₈	—	2	2	no	—
201	Q₈	—	2	4	no	—
202	Q₈	C₄×Dic₉	1	4	yes	complemented
203	C₂×C₄	—	2	4	no	—
204	C₂×C₄	—	2	2	no	—
205	C₂×C₄	—	2	1	no	—
206	C₂×C₄	—	2	1	no	—
207	C₂×C₄	—	2	2	no	—
208	C₂×C₄	—	2	2	no	—
209	C₂×C₄	—	4	1	no	—
210	C₂×C₄	—	4	1	no	—
211	C₂×C₄	—	18	4	no	—
212	C₂×C₄	—	36	2	no	—
213	C₂×C₄	—	36	2	no	—
214	C₂×C₄	#3676	1	2	yes	—
215	C₂×C₄	#3676	1	2	yes	—
216	C₂×C₄	C₂×C₄×D₉	1	2	yes	—
217	C₂×C₄	#3697	1	2	yes	—
218	C₂×C₄	C₂²×Dic₉	1	1	yes	characteristic
219	C₂×C₄	C₂×#611	1	1	yes	characteristic
220	C₈	—	18	4	no	—
221	C₂×C₃	#5026	1	2	yes	—
222	C₂×C₃	#5030	1	2	yes	—
223	C₂×C₃	C₄×#217	1	1	yes	characteristic
224	C₂×C₃	#5225	1	1	yes	characteristic
225	C₂×C₃	#5232	1	1	yes	characteristic
226	C₂²	C₄×#611	1	2	yes	—
227	C₂²	#63505	1	2	yes	—
228	C₂²	#63511	1	1	yes	characteristic
229	C₂²	#63512	1	1	yes	characteristic
230	C₂²	C₂×#3676	1	1	yes	characteristic
231	C₄	—	2	4	no	—
232	C₄	—	2	2	no	—
233	C₄	—	2	2	no	—
234	C₄	—	4	2	no	—
235	C₄	—	36	4	no	—
236	C₄	C₂×C₄×Dic₉	1	1	yes	characteristic
237	C₄	#63488	1	1	yes	characteristic
238	C₄	C₄×#611	1	2	yes	—
239	C₃	#71631	1	1	yes	characteristic
240	C₂	#10588706	1	2	yes	—
241	C₂	#10588709	1	2	yes	—
242	C₂	C₄×#3676	1	1	yes	characteristic
243	C₂	#10588906	1	1	yes	characteristic
244	C₂	#10588916	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×#611
commutator	C₄×C₉	C₂×C₄²
Frattini	C₂²×C₃×C₄	C₂²×D₃
Fitting	C₂×C₄×Q₈×C₉	C₂
socle, prime radical	C₂³×C₃	C₂×#67
cosocle	C₂²×C₄×C₉	C₂³
hypercenter	C₂×C₄×Q₈	D₉
hypocenter	C₉	#1763
LCS	G ▷ C₄×C₉ ▷ C₂×C₉ ▷ C₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×Q₈

Complements

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

N	H ≅ G/N	type
9. C₄×Q₈×C₉	C₄	semidirect
17. #63395	C₄	semidirect
64. Q₈×C₉	C₄²	semidirect
126. C₄×Q₈	Dic₉	semidirect
198. C₉	#1763	semidirect
202. Q₈	C₄×Dic₉	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×#63395	2
C₄×C₄⋊Dic₉	2
C₂×#63362	2
#10588206	2
#71631	3