Subgroups — #11776181

Order 1152.

318 conjugacy classes of subgroups.

The subgroup lattice has 228 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₉×#533	C₂	1	1	yes	maximal, characteristic
2	#10588605	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄⋊(C₉⋊C₈)	C₂	1	1	yes	maximal, characteristic
4	#10588224	C₂	1	2	yes	maximal, complemented
5	#10588223	C₂	1	2	yes	maximal, complemented
6	#71828	—	3	1	no	maximal
7	C₉×C₄⋊Q₈	C₂²	1	1	yes	characteristic
8	C₉×C₄⋊Q₈	C₂²	1	1	yes	characteristic
9	C₉×#125	—	2	1	no	—
10	C₉×C₂²⋊Q₈	—	2	1	no	—
11	C₉×C₂²⋊Q₈	C₄	1	2	yes	—
12	C₄×Q₈×C₉	—	2	1	no	—
13	C₄×Q₈×C₉	C₄	1	2	yes	—
14	C₉×#117	—	2	1	no	—
15	C₂×C₄²×C₉	C₂²	1	1	yes	characteristic
16	C₂²×C₉⋊C₈	—	2	1	no	—
17	#63389	—	2	1	no	—
18	C₄⋊(C₉⋊C₈)	—	2	1	no	—
19	C₄⋊(C₉⋊C₈)	C₂²	1	2	yes	—
20	C₄⋊(C₉⋊C₈)	C₂²	1	2	yes	—
21	C₄×C₉⋊C₈	—	2	1	no	—
22	C₃×#533	D₃	1	1	yes	characteristic
23	#4925	—	3	1	no	—
24	C₂×C₄⋊(C₃⋊C₈)	—	3	1	no	—
25	#4547	—	3	2	no	—
26	#4546	—	3	2	no	—
27	C₂×Q₈×C₉	—	2	1	no	—
28	C₂×Q₈×C₉	C₂×C₄	1	2	yes	—
29	C₂²×C₄×C₉	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₄×C₉	D₄	1	1	yes	characteristic
31	C₂²×C₄×C₉	D₄	1	1	yes	characteristic
32	C₉×C₄⋊C₄	—	2	2	no	—
33	C₉×C₄⋊C₄	—	2	2	no	—
34	C₉×C₄⋊C₄	—	2	1	no	—
35	C₉×C₄⋊C₄	—	2	1	no	—
36	C₉×C₄⋊C₄	—	2	1	no	—
37	C₉×C₄⋊C₄	C₂×C₄	1	2	yes	—
38	C₉×C₂²⋊C₄	—	2	2	no	—
39	C₄²×C₉	—	2	2	no	—
40	C₄²×C₉	C₂×C₄	1	1	yes	characteristic
41	C₄²×C₉	D₄	1	1	yes	characteristic
42	C₄²×C₉	D₄	1	1	yes	characteristic
43	C₄²×C₉	C₂³	1	1	yes	characteristic
44	C₂×C₉⋊C₈	—	2	2	no	—
45	C₂×C₉⋊C₈	—	2	2	no	—
46	C₂×C₉⋊C₈	—	4	2	no	—
47	#1542	—	9	1	no	Sylow
48	C₃×C₄⋊Q₈	C₂×D₃	1	1	yes	characteristic
49	C₃×C₄⋊Q₈	C₂×D₃	1	1	yes	characteristic
50	C₃×#125	—	2	1	no	—
51	C₃×C₂²⋊Q₈	—	2	1	no	—
52	C₃×C₂²⋊Q₈	Dic₃	1	2	yes	—
53	C₃×C₄×Q₈	—	2	1	no	—
54	C₃×C₄×Q₈	Dic₃	1	2	yes	—
55	C₃×#117	—	2	1	no	—
56	C₂×C₃×C₄²	C₂×D₃	1	1	yes	characteristic
57	C₂²×C₃⋊C₈	—	6	1	no	—
58	#830	—	6	1	no	—
59	C₄⋊(C₃⋊C₈)	—	3	2	no	—
60	C₄⋊(C₃⋊C₈)	—	3	2	no	—
61	C₄⋊(C₃⋊C₈)	—	6	1	no	—
62	C₄×C₃⋊C₈	—	6	1	no	—
63	C₂³×C₉	C₂²⋊C₄	1	1	yes	characteristic
64	Q₈×C₉	—	4	2	no	—
65	Q₈×C₉	—	4	2	no	—
66	C₂×C₄×C₉	—	2	2	no	—
67	C₂×C₄×C₉	—	2	2	no	—
68	C₂×C₄×C₉	—	2	2	no	—
69	C₂×C₄×C₉	—	4	1	no	—
70	C₂×C₄×C₉	C₂²⋊C₄	1	1	yes	characteristic
71	C₂×C₄×C₉	C₂²⋊C₄	1	1	yes	characteristic
72	C₂×C₄×C₉	C₂²⋊C₄	1	1	yes	characteristic
73	C₂×C₄×C₉	SD₁₆	1	2	yes	—
74	C₂×C₄×C₉	Q₁₆	1	2	yes	—
75	C₂×C₄×C₉	C₂²×C₄	1	1	yes	characteristic
76	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
77	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
78	C₉⋊C₈	—	4	2	no	—
79	C₉⋊C₈	—	4	2	no	—
80	#533	D₉	1	1	yes	characteristic
81	#431	—	9	1	no	—
82	C₂×C₄⋊C₈	—	9	1	no	—
83	#333	—	9	2	no	—
84	#332	—	9	2	no	—
85	C₂×C₃×Q₈	—	2	1	no	—
86	C₂×C₃×Q₈	C₂×Dic₃	1	2	yes	—
87	C₂²×C₃×C₄	C₂×Dic₃	1	1	yes	characteristic
88	C₂²×C₃×C₄	#67	1	1	yes	characteristic
89	C₂²×C₃×C₄	#67	1	1	yes	characteristic
90	C₃×C₄⋊C₄	—	2	2	no	—
91	C₃×C₄⋊C₄	—	2	2	no	—
92	C₃×C₄⋊C₄	—	2	1	no	—
93	C₃×C₄⋊C₄	—	2	1	no	—
94	C₃×C₄⋊C₄	—	2	1	no	—
95	C₃×C₄⋊C₄	C₂×Dic₃	1	2	yes	—
96	C₃×C₂²⋊C₄	—	2	2	no	—
97	C₃×C₄²	—	2	2	no	—
98	C₃×C₄²	C₂×Dic₃	1	1	yes	characteristic
99	C₃×C₄²	#67	1	1	yes	characteristic
100	C₃×C₄²	#67	1	1	yes	characteristic
101	C₃×C₄²	C₂²×D₃	1	1	yes	characteristic
102	C₂×C₃⋊C₈	—	6	2	no	—
103	C₂×C₃⋊C₈	—	6	2	no	—
104	C₂×C₃⋊C₈	—	12	2	no	—
105	C₂²×C₉	—	2	1	no	—
106	C₂²×C₉	—	2	1	no	—
107	C₂²×C₉	#103	1	2	yes	—
108	C₂²×C₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
109	C₄×C₉	—	2	1	no	—
110	C₄×C₉	—	2	1	no	—
111	C₄×C₉	—	2	1	no	—
112	C₄×C₉	—	4	2	no	—
113	C₄×C₉	—	4	2	no	—
114	C₄×C₉	#103	1	2	yes	—
115	C₄×C₉	C₂×SD₁₆	1	1	yes	characteristic
116	C₄×C₉	C₂×Q₁₆	1	1	yes	characteristic
117	C₄×C₉	#135	1	2	yes	—
118	C₂²×C₈	—	18	1	no	—
119	C₄⋊Q₈	C₂×D₉	1	1	yes	characteristic
120	C₄⋊Q₈	C₂×D₉	1	1	yes	characteristic
121	#125	—	2	1	no	—
122	C₂²⋊Q₈	—	2	1	no	—
123	C₂²⋊Q₈	Dic₉	1	2	yes	—
124	C₄×Q₈	—	2	1	no	—
125	C₄×Q₈	Dic₉	1	2	yes	—
126	#117	—	2	1	no	—
127	C₂×C₄²	C₂×D₉	1	1	yes	characteristic
128	C₄⋊C₈	—	9	2	no	—
129	C₄⋊C₈	—	9	2	no	—
130	C₄⋊C₈	—	18	1	no	—
131	C₂²⋊C₈	—	18	1	no	—
132	C₄×C₈	—	18	1	no	—
133	C₂³×C₃	#217	1	1	yes	characteristic
134	C₃×Q₈	—	4	2	no	—
135	C₃×Q₈	—	4	2	no	—
136	C₂×C₃×C₄	—	2	2	no	—
137	C₂×C₃×C₄	—	2	2	no	—
138	C₂×C₃×C₄	—	2	2	no	—
139	C₂×C₃×C₄	—	4	1	no	—
140	C₂×C₃×C₄	#215	1	2	yes	—
141	C₂×C₃×C₄	#216	1	2	yes	—
142	C₂×C₃×C₄	#217	1	1	yes	characteristic
143	C₂×C₃×C₄	#217	1	1	yes	characteristic
144	C₂×C₃×C₄	#217	1	1	yes	characteristic
145	C₂×C₃×C₄	C₂²×Dic₃	1	1	yes	characteristic
146	C₂×C₃×C₄	C₂×#67	1	1	yes	characteristic
147	C₂×C₃×C₄	C₂×#67	1	1	yes	characteristic
148	C₃⋊C₈	—	12	2	no	—
149	C₃⋊C₈	—	12	2	no	—
150	C₂×C₉	—	2	2	no	—
151	C₂×C₉	#413	1	1	yes	characteristic
152	C₂×C₉	C₂×#103	1	1	yes	characteristic
153	C₂×C₉	#416	1	1	yes	characteristic
154	C₂×Q₈	—	2	1	no	—
155	C₂×Q₈	C₂×Dic₉	1	2	yes	—
156	C₂²×C₄	C₂×Dic₉	1	1	yes	characteristic
157	C₂²×C₄	#611	1	1	yes	characteristic
158	C₂²×C₄	#611	1	1	yes	characteristic
159	C₂×C₈	—	18	2	no	—
160	C₂×C₈	—	18	2	no	—
161	C₂×C₈	—	36	2	no	—
162	C₄⋊C₄	—	2	2	no	—
163	C₄⋊C₄	—	2	2	no	—
164	C₄⋊C₄	—	2	1	no	—
165	C₄⋊C₄	—	2	1	no	—
166	C₄⋊C₄	—	2	1	no	—
167	C₄⋊C₄	C₂×Dic₉	1	2	yes	—
168	C₂²⋊C₄	—	2	2	no	—
169	C₄²	—	2	2	no	—
170	C₄²	C₂×Dic₉	1	1	yes	characteristic
171	C₄²	#611	1	1	yes	characteristic
172	C₄²	#611	1	1	yes	characteristic
173	C₄²	C₂²×D₉	1	1	yes	characteristic
174	C₂²×C₃	—	2	1	no	—
175	C₂²×C₃	—	2	1	no	—
176	C₂²×C₃	#835	1	2	yes	—
177	C₂²×C₃	C₂×#217	1	1	yes	characteristic
178	C₃×C₄	—	2	1	no	—
179	C₃×C₄	—	2	1	no	—
180	C₃×C₄	—	2	1	no	—
181	C₃×C₄	—	4	2	no	—
182	C₃×C₄	—	4	2	no	—
183	C₃×C₄	#835	1	2	yes	—
184	C₃×C₄	C₂×#215	1	1	yes	characteristic
185	C₃×C₄	C₂×#216	1	1	yes	characteristic
186	C₃×C₄	#950	1	2	yes	—
187	C₉	#1542	1	1	yes	characteristic, complemented, Sylow
188	C₂³	#3676	1	1	yes	characteristic
189	Q₈	—	4	2	no	—
190	Q₈	—	4	2	no	—
191	C₂×C₄	—	2	2	no	—
192	C₂×C₄	—	2	2	no	—
193	C₂×C₄	—	2	2	no	—
194	C₂×C₄	—	4	1	no	—
195	C₂×C₄	#3674	1	2	yes	—
196	C₂×C₄	#3675	1	2	yes	—
197	C₂×C₄	#3676	1	1	yes	characteristic
198	C₂×C₄	#3676	1	1	yes	characteristic
199	C₂×C₄	#3676	1	1	yes	characteristic
200	C₂×C₄	C₂²×Dic₉	1	1	yes	characteristic
201	C₂×C₄	C₂×#611	1	1	yes	characteristic
202	C₂×C₄	C₂×#611	1	1	yes	characteristic
203	C₈	—	36	2	no	—
204	C₈	—	36	2	no	—
205	C₂×C₃	—	2	2	no	—
206	C₂×C₃	C₂×#835	1	1	yes	characteristic
207	C₂×C₃	#5233	1	1	yes	characteristic
208	C₂×C₃	#5237	1	1	yes	characteristic
209	C₂²	—	2	1	no	—
210	C₂²	—	2	1	no	—
211	C₂²	#63395	1	2	yes	—
212	C₂²	C₂×#3676	1	1	yes	characteristic
213	C₄	—	2	1	no	—
214	C₄	—	2	1	no	—
215	C₄	—	2	1	no	—
216	C₄	—	4	2	no	—
217	C₄	—	4	2	no	—
218	C₄	#63395	1	2	yes	—
219	C₄	C₂×#3674	1	1	yes	characteristic
220	C₄	C₂×#3675	1	1	yes	characteristic
221	C₄	#63513	1	2	yes	—
222	C₃	#71828	1	1	yes	characteristic
223	C₂	—	2	2	no	—
224	C₂	C₂×#63395	1	1	yes	characteristic
225	C₂	#10588915	1	1	yes	characteristic
226	C₂	#10588917	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#3676
commutator	C₂×C₄×C₉	C₂²×C₄
Frattini	C₃×C₄²	C₂²×D₃
Fitting	C₉×#533	C₂
socle	C₂²×C₃	C₂×#217
prime radical	C₂³×C₃	#217
cosocle	C₄²×C₉	C₂³
hypercenter	#533	D₉
hypocenter	C₉	#1542
LCS	G ▷ C₂×C₄×C₉ ▷ C₂²×C₉ ▷ C₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #533

Complements

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

N	H ≅ G/N	type
4. #10588224	C₂	semidirect
5. #10588223	C₂	semidirect
187. C₉	#1542	semidirect

Maximal subgroups

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

subgroup	index
C₉×#533	2
#10588605	2
C₂×C₄⋊(C₉⋊C₈)	2
#10588224	2
#10588223	2
#71828	3