Subgroups — C₉×#2058 #11766779

Order 1152.

423 conjugacy classes of subgroups.

The subgroup lattice has 243 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₉×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic
2	C₂×C₉×#106	C₂	1	2	yes	maximal
3	C₉×#389	C₂	1	1	yes	maximal, characteristic
4	C₉×#340	C₂	1	2	yes	maximal
5	C₉×#336	C₂	1	1	yes	maximal, characteristic
6	C₃×#2058	C₃	1	1	yes	maximal, characteristic
7	C₂²×Q₈×C₉	—	2	1	no	—
8	C₂²×C₈×C₉	C₂²	1	2	yes	—
9	C₉×C₄⋊Q₈	—	2	4	no	—
10	C₂×C₉×C₄⋊C₄	—	2	1	no	—
11	C₂×C₉×C₄⋊C₄	—	2	1	no	—
12	C₂×C₉×C₄⋊C₄	C₂²	1	1	yes	characteristic
13	C₂×C₉×C₄⋊C₄	C₂²	1	2	yes	—
14	C₂×C₉×C₄⋊C₄	C₂²	1	1	yes	characteristic
15	C₂×C₄²×C₉	C₂²	1	1	yes	characteristic
16	C₉×#106	—	2	4	no	—
17	C₉×#95	—	2	1	no	—
18	C₂×C₃×C₄⋊Q₈	C₂×C₃	1	1	yes	characteristic
19	C₂×C₃×#106	C₂×C₃	1	2	yes	—
20	C₃×#389	C₂×C₃	1	1	yes	characteristic
21	C₃×#340	C₂×C₃	1	2	yes	—
22	C₃×#336	C₂×C₃	1	1	yes	characteristic
23	C₂×Q₈×C₉	—	2	4	no	—
24	C₂×Q₈×C₉	—	4	2	no	—
25	C₂×Q₈×C₉	—	4	2	no	—
26	C₂²×C₄×C₉	—	2	1	no	—
27	C₂²×C₄×C₉	—	2	2	no	—
28	C₂²×C₄×C₉	—	2	1	no	—
29	C₂²×C₄×C₉	D₄	1	1	yes	characteristic
30	C₂²×C₄×C₉	D₄	1	1	yes	characteristic
31	C₂²×C₄×C₉	C₂³	1	1	yes	characteristic
32	C₂×C₈×C₉	—	2	2	no	—
33	C₂×C₈×C₉	—	2	2	no	—
34	C₂×C₈×C₉	Q₈	1	4	yes	complemented
35	C₉×C₄⋊C₄	—	2	4	no	—
36	C₉×C₄⋊C₄	—	2	2	no	—
37	C₉×C₄⋊C₄	—	2	1	no	—
38	C₉×C₄⋊C₄	—	2	2	no	—
39	C₉×C₄⋊C₄	—	4	2	no	—
40	C₉×C₄⋊C₄	D₄	1	4	yes	—
41	C₉×C₄⋊C₄	Q₈	1	2	yes	complemented
42	C₄²×C₉	—	2	2	no	—
43	#2058	C₉	1	1	yes	characteristic, direct product factor, Sylow
44	C₂²×C₃×Q₈	—	2	1	no	—
45	C₂²×C₃×C₈	C₂²×C₃	1	2	yes	—
46	C₃×C₄⋊Q₈	—	2	4	no	—
47	C₂×C₃×C₄⋊C₄	—	2	1	no	—
48	C₂×C₃×C₄⋊C₄	—	2	1	no	—
49	C₂×C₃×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
50	C₂×C₃×C₄⋊C₄	C₂²×C₃	1	2	yes	—
51	C₂×C₃×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
52	C₂×C₃×C₄²	C₂²×C₃	1	1	yes	characteristic
53	C₃×#106	—	2	4	no	—
54	C₃×#95	—	2	1	no	—
55	C₂³×C₉	C₂×D₄	1	1	yes	characteristic
56	Q₈×C₉	—	4	8	no	—
57	C₂×C₄×C₉	—	2	1	no	—
58	C₂×C₄×C₉	—	2	1	no	—
59	C₂×C₄×C₉	—	2	1	no	—
60	C₂×C₄×C₉	—	2	1	no	—
61	C₂×C₄×C₉	—	2	4	no	—
62	C₂×C₄×C₉	—	2	2	no	—
63	C₂×C₄×C₉	—	4	1	no	—
64	C₂×C₄×C₉	—	4	2	no	—
65	C₂×C₄×C₉	—	4	1	no	—
66	C₂×C₄×C₉	—	4	2	no	—
67	C₂×C₄×C₉	—	4	1	no	—
68	C₂×C₄×C₉	—	4	1	no	—
69	C₂×C₄×C₉	—	4	1	no	—
70	C₂×C₄×C₉	SD₁₆	1	2	yes	—
71	C₂×C₄×C₉	SD₁₆	1	2	yes	—
72	C₂×C₄×C₉	C₂×D₄	1	2	yes	—
73	C₂×C₄×C₉	C₂×Q₈	1	2	yes	—
74	C₂×C₄×C₉	C₂×Q₈	1	1	yes	characteristic
75	C₂×C₄×C₉	#41	1	1	yes	characteristic
76	C₈×C₉	—	2	4	no	—
77	C₂×C₄⋊Q₈	C₂×C₉	1	1	yes	characteristic
78	C₂×#106	C₂×C₉	1	2	yes	—
79	#389	C₂×C₉	1	1	yes	characteristic
80	#340	C₂×C₉	1	2	yes	—
81	#336	C₂×C₉	1	1	yes	characteristic
82	C₂×C₃×Q₈	—	2	4	no	—
83	C₂×C₃×Q₈	—	4	2	no	—
84	C₂×C₃×Q₈	—	4	2	no	—
85	C₂²×C₃×C₄	—	2	1	no	—
86	C₂²×C₃×C₄	—	2	2	no	—
87	C₂²×C₃×C₄	—	2	1	no	—
88	C₂²×C₃×C₄	C₃×D₄	1	1	yes	characteristic
89	C₂²×C₃×C₄	C₃×D₄	1	1	yes	characteristic
90	C₂²×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
91	C₂×C₃×C₈	—	2	2	no	—
92	C₂×C₃×C₈	—	2	2	no	—
93	C₂×C₃×C₈	C₃×Q₈	1	4	yes	—
94	C₃×C₄⋊C₄	—	2	4	no	—
95	C₃×C₄⋊C₄	—	2	2	no	—
96	C₃×C₄⋊C₄	—	2	1	no	—
97	C₃×C₄⋊C₄	—	2	2	no	—
98	C₃×C₄⋊C₄	—	4	2	no	—
99	C₃×C₄⋊C₄	C₃×D₄	1	4	yes	—
100	C₃×C₄⋊C₄	C₃×Q₈	1	2	yes	—
101	C₃×C₄²	—	2	2	no	—
102	C₂²×C₉	#120	1	1	yes	characteristic
103	C₂²×C₉	C₂²⋊Q₈	1	1	yes	characteristic
104	C₂²×C₉	C₄⋊Q₈	1	1	yes	characteristic
105	C₂²×C₉	C₂×SD₁₆	1	1	yes	characteristic
106	C₂²×C₉	C₂×SD₁₆	1	1	yes	characteristic
107	C₂²×C₉	AGL(1,ℤ/8)	1	1	yes	characteristic
108	C₂²×C₉	#137	1	1	yes	characteristic
109	C₄×C₉	—	2	2	no	—
110	C₄×C₉	—	2	2	no	—
111	C₄×C₉	—	4	4	no	—
112	C₄×C₉	—	4	2	no	—
113	C₄×C₉	—	8	1	no	—
114	C₄×C₉	C₂²⋊Q₈	1	2	yes	—
115	C₄×C₉	C₄⋊Q₈	1	2	yes	—
116	C₂²×Q₈	—	2	1	no	—
117	C₂²×C₈	C₂²×C₉	1	2	yes	—
118	C₄⋊Q₈	—	2	4	no	—
119	C₂×C₄⋊C₄	—	2	1	no	—
120	C₂×C₄⋊C₄	—	2	1	no	—
121	C₂×C₄⋊C₄	C₂²×C₉	1	1	yes	characteristic
122	C₂×C₄⋊C₄	C₂²×C₉	1	2	yes	—
123	C₂×C₄⋊C₄	C₂²×C₉	1	1	yes	characteristic
124	C₂×C₄²	C₂²×C₉	1	1	yes	characteristic
125	#106	—	2	4	no	—
126	#95	—	2	1	no	—
127	C₂³×C₃	C₂×C₃×D₄	1	1	yes	characteristic
128	C₃×Q₈	—	4	8	no	—
129	C₂×C₃×C₄	—	2	1	no	—
130	C₂×C₃×C₄	—	2	1	no	—
131	C₂×C₃×C₄	—	2	1	no	—
132	C₂×C₃×C₄	—	2	1	no	—
133	C₂×C₃×C₄	—	2	4	no	—
134	C₂×C₃×C₄	—	2	2	no	—
135	C₂×C₃×C₄	—	4	1	no	—
136	C₂×C₃×C₄	—	4	2	no	—
137	C₂×C₃×C₄	—	4	1	no	—
138	C₂×C₃×C₄	—	4	2	no	—
139	C₂×C₃×C₄	—	4	1	no	—
140	C₂×C₃×C₄	—	4	1	no	—
141	C₂×C₃×C₄	—	4	1	no	—
142	C₂×C₃×C₄	C₃×SD₁₆	1	2	yes	—
143	C₂×C₃×C₄	C₃×SD₁₆	1	2	yes	—
144	C₂×C₃×C₄	C₂×C₃×D₄	1	2	yes	—
145	C₂×C₃×C₄	C₂×C₃×Q₈	1	2	yes	—
146	C₂×C₃×C₄	C₂×C₃×Q₈	1	1	yes	characteristic
147	C₂×C₃×C₄	C₃×#41	1	1	yes	characteristic
148	C₃×C₈	—	2	4	no	—
149	C₂×C₉	#395	1	1	yes	characteristic
150	C₂×C₉	#448	1	1	yes	characteristic
151	C₂×C₉	#450	1	1	yes	characteristic
152	C₂×C₉	#475	1	1	yes	characteristic
153	C₂×C₉	#476	1	1	yes	characteristic
154	C₂×C₉	#498	1	1	yes	characteristic
155	C₂×C₉	#501	1	1	yes	characteristic
156	C₂×Q₈	—	2	4	no	—
157	C₂×Q₈	—	4	2	no	—
158	C₂×Q₈	—	4	2	no	—
159	C₂²×C₄	—	2	1	no	—
160	C₂²×C₄	—	2	2	no	—
161	C₂²×C₄	—	2	1	no	—
162	C₂²×C₄	D₄×C₉	1	1	yes	characteristic
163	C₂²×C₄	D₄×C₉	1	1	yes	characteristic
164	C₂²×C₄	C₂³×C₉	1	1	yes	characteristic
165	C₂×C₈	—	2	2	no	—
166	C₂×C₈	—	2	2	no	—
167	C₂×C₈	Q₈×C₉	1	4	yes	complemented
168	C₄⋊C₄	—	2	4	no	—
169	C₄⋊C₄	—	2	2	no	—
170	C₄⋊C₄	—	2	1	no	—
171	C₄⋊C₄	—	2	2	no	—
172	C₄⋊C₄	—	4	2	no	—
173	C₄⋊C₄	D₄×C₉	1	4	yes	—
174	C₄⋊C₄	Q₈×C₉	1	2	yes	complemented
175	C₄²	—	2	2	no	—
176	C₂²×C₃	C₃×#120	1	1	yes	characteristic
177	C₂²×C₃	C₃×C₂²⋊Q₈	1	1	yes	characteristic
178	C₂²×C₃	C₃×C₄⋊Q₈	1	1	yes	characteristic
179	C₂²×C₃	C₂×C₃×SD₁₆	1	1	yes	characteristic
180	C₂²×C₃	C₂×C₃×SD₁₆	1	1	yes	characteristic
181	C₂²×C₃	C₃×AGL(1,ℤ/8)	1	1	yes	characteristic
182	C₂²×C₃	C₃×#137	1	1	yes	characteristic
183	C₃×C₄	—	2	2	no	—
184	C₃×C₄	—	2	2	no	—
185	C₃×C₄	—	4	4	no	—
186	C₃×C₄	—	4	2	no	—
187	C₃×C₄	—	8	1	no	—
188	C₃×C₄	C₃×C₂²⋊Q₈	1	2	yes	—
189	C₃×C₄	C₃×C₄⋊Q₈	1	2	yes	—
190	C₉	#2058	1	1	yes	characteristic, direct product factor, Sylow
191	C₂³	C₂×D₄×C₉	1	1	yes	characteristic
192	Q₈	—	4	8	no	—
193	C₂×C₄	—	2	1	no	—
194	C₂×C₄	—	2	1	no	—
195	C₂×C₄	—	2	1	no	—
196	C₂×C₄	—	2	1	no	—
197	C₂×C₄	—	2	4	no	—
198	C₂×C₄	—	2	2	no	—
199	C₂×C₄	—	4	1	no	—
200	C₂×C₄	—	4	2	no	—
201	C₂×C₄	—	4	2	no	—
202	C₂×C₄	—	4	1	no	—
203	C₂×C₄	—	4	1	no	—
204	C₂×C₄	—	4	1	no	—
205	C₂×C₄	—	4	1	no	—
206	C₂×C₄	C₉×SD₁₆	1	2	yes	—
207	C₂×C₄	C₉×SD₁₆	1	2	yes	—
208	C₂×C₄	C₂×D₄×C₉	1	2	yes	—
209	C₂×C₄	C₂×Q₈×C₉	1	2	yes	—
210	C₂×C₄	C₂×Q₈×C₉	1	1	yes	characteristic
211	C₂×C₄	C₉×#41	1	1	yes	characteristic
212	C₈	—	2	4	no	—
213	C₂×C₃	C₃×#395	1	1	yes	characteristic
214	C₂×C₃	C₃×#448	1	1	yes	characteristic
215	C₂×C₃	C₃×#450	1	1	yes	characteristic
216	C₂×C₃	C₃×#475	1	1	yes	characteristic
217	C₂×C₃	C₃×#476	1	1	yes	characteristic
218	C₂×C₃	C₃×#498	1	1	yes	characteristic
219	C₂×C₃	C₃×#501	1	1	yes	characteristic
220	C₂²	C₉×#120	1	1	yes	characteristic
221	C₂²	C₉×C₂²⋊Q₈	1	1	yes	characteristic
222	C₂²	C₉×C₄⋊Q₈	1	1	yes	characteristic
223	C₂²	C₂×C₉×SD₁₆	1	1	yes	characteristic
224	C₂²	C₂×C₉×SD₁₆	1	1	yes	characteristic
225	C₂²	C₉×AGL(1,ℤ/8)	1	1	yes	characteristic
226	C₂²	C₉×#137	1	1	yes	characteristic
227	C₄	—	2	2	no	—
228	C₄	—	2	2	no	—
229	C₄	—	4	4	no	—
230	C₄	—	4	2	no	—
231	C₄	—	8	1	no	—
232	C₄	C₉×C₂²⋊Q₈	1	2	yes	—
233	C₄	C₉×C₄⋊Q₈	1	2	yes	—
234	C₃	C₃×#2058	1	1	yes	characteristic
235	C₂	C₉×#395	1	1	yes	characteristic
236	C₂	C₉×#448	1	1	yes	characteristic
237	C₂	C₉×#450	1	1	yes	characteristic
238	C₂	C₉×#475	1	1	yes	characteristic
239	C₂	C₉×#476	1	1	yes	characteristic
240	C₂	C₉×#498	1	1	yes	characteristic
241	C₂	C₉×#501	1	1	yes	characteristic

	subgroup	G/H
center	C₂³×C₉	C₂×D₄
commutator	C₂²×C₄	C₂³×C₉
Frattini, cosocle	C₂²×C₃×C₄	C₂³×C₃
socle, prime radical	C₂³×C₃	C₂×C₃×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₉ ◁ C₂×C₄²×C₉ ◁ G

Complements

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

N	H ≅ G/N	type
34. C₂×C₈×C₉	Q₈	semidirect
41. C₉×C₄⋊C₄	Q₈	semidirect
43. #2058	C₉	direct (N × H)
167. C₂×C₈	Q₈×C₉	semidirect
174. C₄⋊C₄	Q₈×C₉	semidirect
190. C₉	#2058	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₉×C₄⋊Q₈	2
C₂×C₉×#106	2
C₉×#389	2
C₉×#340	2
C₉×#336	2
C₃×#2058	3

Subgroups — C9×#2058 #11766779

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₉×#2058 #11766779