Subgroups — #11797472

Order 1152.

246 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	C₈×D₄×C₉	C₂	1	1	yes	maximal, characteristic
2	C₂×#63371	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×C₉⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
4	#10588226	C₂	1	1	yes	maximal, characteristic, complemented
5	#10588226	C₂	1	1	yes	maximal, characteristic, complemented
6	C₄⋊(C₉⋊C₁₆)	C₂	1	1	yes	maximal, characteristic, complemented
7	#10588138	C₂	1	1	yes	maximal, characteristic, complemented
8	#70857	—	3	1	no	maximal
9	C₂²×C₈×C₉	C₂²	1	1	yes	characteristic
10	C₂²×C₈×C₉	C₂²	1	1	yes	characteristic
11	C₄×D₄×C₉	C₄	1	1	yes	characteristic
12	C₉×C₄⋊C₈	C₄	1	1	yes	characteristic
13	C₉×C₂²⋊C₈	C₄	1	1	yes	characteristic
14	C₉×C₂²⋊C₈	C₄	1	1	yes	characteristic
15	C₄×C₈×C₉	C₂²	1	1	yes	characteristic
16	#63371	—	2	2	no	—
17	C₂×C₉⋊C₁₆	—	2	2	no	—
18	C₂×C₉⋊C₁₆	C₂²	1	1	yes	characteristic
19	C₂×C₉⋊C₁₆	C₂²	1	1	yes	characteristic
20	C₂×C₉⋊C₁₆	C₂²	1	1	yes	characteristic
21	C₂×C₉⋊C₁₆	C₂²	1	1	yes	characteristic
22	C₃×C₈×D₄	D₃	1	1	yes	characteristic
23	C₂×#812	—	3	1	no	—
24	C₂²×C₃⋊C₁₆	—	3	1	no	—
25	#4549	—	3	1	no	—
26	#4549	—	3	1	no	—
27	C₄⋊(C₃⋊C₁₆)	—	3	1	no	—
28	#4461	—	3	1	no	—
29	C₂×D₄×C₉	C₈	1	1	yes	characteristic
30	C₂²×C₄×C₉	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₄×C₉	C₂×C₄	1	1	yes	characteristic
32	C₂×C₈×C₉	—	2	1	no	—
33	C₂×C₈×C₉	—	2	1	no	—
34	C₂×C₈×C₉	—	2	1	no	—
35	C₂×C₈×C₉	—	2	1	no	—
36	C₂×C₈×C₉	C₂×C₄	1	1	yes	characteristic
37	C₂×C₈×C₉	C₂×C₄	1	1	yes	characteristic
38	C₂×C₈×C₉	C₂×C₄	1	1	yes	characteristic
39	C₂×C₈×C₉	C₂³	1	1	yes	characteristic
40	C₉×C₄⋊C₄	C₈	1	1	yes	characteristic
41	C₉×C₂²⋊C₄	C₈	1	1	yes	characteristic
42	C₉×C₂²⋊C₄	C₈	1	1	yes	characteristic
43	C₄²×C₉	C₂×C₄	1	1	yes	characteristic
44	C₉⋊C₁₆	—	2	1	no	—
45	C₉⋊C₁₆	—	2	1	no	—
46	C₉⋊C₁₆	—	2	1	no	—
47	C₉⋊C₁₆	D₄	1	2	yes	complemented
48	#2168	—	9	1	no	Sylow
49	C₂²×C₃×C₈	C₂×D₃	1	1	yes	characteristic
50	C₂²×C₃×C₈	C₂×D₃	1	1	yes	characteristic
51	C₃×C₄×D₄	Dic₃	1	1	yes	characteristic
52	C₃×C₄⋊C₈	Dic₃	1	1	yes	characteristic
53	C₃×C₂²⋊C₈	Dic₃	1	1	yes	characteristic
54	C₃×C₂²⋊C₈	Dic₃	1	1	yes	characteristic
55	C₃×C₄×C₈	C₂×D₃	1	1	yes	characteristic
56	#812	—	6	2	no	—
57	C₂×C₃⋊C₁₆	—	3	1	no	—
58	C₂×C₃⋊C₁₆	—	3	1	no	—
59	C₂×C₃⋊C₁₆	—	3	1	no	—
60	C₂×C₃⋊C₁₆	—	3	1	no	—
61	C₂×C₃⋊C₁₆	—	6	2	no	—
62	C₂³×C₉	C₂×C₈	1	1	yes	characteristic
63	C₂³×C₉	C₂×C₈	1	1	yes	characteristic
64	D₄×C₉	—	2	2	no	—
65	C₂×C₄×C₉	—	2	1	no	—
66	C₂×C₄×C₉	—	2	1	no	—
67	C₂×C₄×C₉	—	2	1	no	—
68	C₂×C₄×C₉	—	2	1	no	—
69	C₂×C₄×C₉	C₂×C₈	1	1	yes	characteristic
70	C₂×C₄×C₉	C₂×C₈	1	1	yes	characteristic
71	C₂×C₄×C₉	C₂×C₈	1	1	yes	characteristic
72	C₂×C₄×C₉	C₂×C₈	1	1	yes	characteristic
73	C₂×C₄×C₉	C₂²×C₄	1	1	yes	characteristic
74	C₈×C₉	—	2	1	no	—
75	C₈×C₉	—	2	1	no	—
76	C₈×C₉	—	2	1	no	—
77	C₈×C₉	C₂×D₄	1	1	yes	characteristic
78	C₈×C₉	#41	1	1	yes	characteristic
79	C₂×M₃₂	—	9	1	no	—
80	C₂²×C₁₆	—	9	1	no	—
81	C₈×D₄	D₉	1	1	yes	characteristic
82	C₄⋊C₁₆	—	9	1	no	—
83	C₂²⋊C₁₆	—	9	1	no	—
84	C₂²⋊C₁₆	—	9	1	no	—
85	#346	—	9	1	no	—
86	C₂×C₃×D₄	C₃⋊C₈	1	1	yes	characteristic
87	C₂²×C₃×C₄	C₂×Dic₃	1	1	yes	characteristic
88	C₂²×C₃×C₄	C₂×Dic₃	1	1	yes	characteristic
89	C₂×C₃×C₈	—	2	1	no	—
90	C₂×C₃×C₈	—	2	1	no	—
91	C₂×C₃×C₈	—	2	1	no	—
92	C₂×C₃×C₈	—	2	1	no	—
93	C₂×C₃×C₈	C₂×Dic₃	1	1	yes	characteristic
94	C₂×C₃×C₈	C₂×Dic₃	1	1	yes	characteristic
95	C₂×C₃×C₈	C₂×Dic₃	1	1	yes	characteristic
96	C₂×C₃×C₈	C₂²×D₃	1	1	yes	characteristic
97	C₃×C₄⋊C₄	C₃⋊C₈	1	1	yes	characteristic
98	C₃×C₂²⋊C₄	C₃⋊C₈	1	1	yes	characteristic
99	C₃×C₂²⋊C₄	C₃⋊C₈	1	1	yes	characteristic
100	C₃×C₄²	C₂×Dic₃	1	1	yes	characteristic
101	C₃⋊C₁₆	—	3	2	no	—
102	C₃⋊C₁₆	—	6	1	no	—
103	C₃⋊C₁₆	—	6	1	no	—
104	C₃⋊C₁₆	—	6	1	no	—
105	C₂²×C₉	—	2	1	no	—
106	C₂²×C₉	—	2	1	no	—
107	C₂²×C₉	—	2	1	no	—
108	C₂²×C₉	—	2	1	no	—
109	C₂²×C₉	—	2	1	no	—
110	C₂²×C₉	M₃₂	1	2	yes	complemented
111	C₂²×C₉	C₂²×C₈	1	1	yes	characteristic
112	C₄×C₉	—	2	1	no	—
113	C₄×C₉	—	2	1	no	—
114	C₄×C₉	—	2	1	no	—
115	C₄×C₉	—	2	1	no	—
116	C₄×C₉	C₄×D₄	1	1	yes	characteristic
117	C₄×C₉	#131	1	1	yes	characteristic
118	C₂²×C₈	C₂×D₉	1	1	yes	characteristic
119	C₂²×C₈	C₂×D₉	1	1	yes	characteristic
120	C₄×D₄	Dic₉	1	1	yes	characteristic
121	M₃₂	—	18	2	no	—
122	C₂×C₁₆	—	9	1	no	—
123	C₂×C₁₆	—	9	1	no	—
124	C₂×C₁₆	—	9	1	no	—
125	C₂×C₁₆	—	9	1	no	—
126	C₂×C₁₆	—	18	2	no	—
127	C₄⋊C₈	Dic₉	1	1	yes	characteristic
128	C₂²⋊C₈	Dic₉	1	1	yes	characteristic
129	C₂²⋊C₈	Dic₉	1	1	yes	characteristic
130	C₄×C₈	C₂×D₉	1	1	yes	characteristic
131	C₂³×C₃	C₂×C₃⋊C₈	1	1	yes	characteristic
132	C₂³×C₃	C₂×C₃⋊C₈	1	1	yes	characteristic
133	C₃×D₄	—	2	2	no	—
134	C₂×C₃×C₄	—	2	1	no	—
135	C₂×C₃×C₄	—	2	1	no	—
136	C₂×C₃×C₄	—	2	1	no	—
137	C₂×C₃×C₄	—	2	1	no	—
138	C₂×C₃×C₄	C₂×C₃⋊C₈	1	1	yes	characteristic
139	C₂×C₃×C₄	C₂×C₃⋊C₈	1	1	yes	characteristic
140	C₂×C₃×C₄	C₂×C₃⋊C₈	1	1	yes	characteristic
141	C₂×C₃×C₄	C₂×C₃⋊C₈	1	1	yes	characteristic
142	C₂×C₃×C₄	C₂²×Dic₃	1	1	yes	characteristic
143	C₃×C₈	—	2	1	no	—
144	C₃×C₈	—	2	1	no	—
145	C₃×C₈	—	2	1	no	—
146	C₃×C₈	D₃×D₄	1	1	yes	characteristic
147	C₃×C₈	#237	1	1	yes	characteristic
148	C₂×C₉	—	2	2	no	—
149	C₂×C₉	—	4	1	no	—
150	C₂×C₉	C₈×D₄	1	1	yes	characteristic
151	C₂×C₉	C₂×M₃₂	1	1	yes	characteristic
152	C₂×C₉	#504	1	1	yes	characteristic
153	C₂×D₄	C₉⋊C₈	1	1	yes	characteristic
154	C₂²×C₄	C₂×Dic₉	1	1	yes	characteristic
155	C₂²×C₄	C₂×Dic₉	1	1	yes	characteristic
156	C₂×C₈	—	2	1	no	—
157	C₂×C₈	—	2	1	no	—
158	C₂×C₈	—	2	1	no	—
159	C₂×C₈	—	2	1	no	—
160	C₂×C₈	C₂×Dic₉	1	1	yes	characteristic
161	C₂×C₈	C₂×Dic₉	1	1	yes	characteristic
162	C₂×C₈	C₂×Dic₉	1	1	yes	characteristic
163	C₂×C₈	C₂²×D₉	1	1	yes	characteristic
164	C₄⋊C₄	C₉⋊C₈	1	1	yes	characteristic
165	C₂²⋊C₄	C₉⋊C₈	1	1	yes	characteristic
166	C₂²⋊C₄	C₉⋊C₈	1	1	yes	characteristic
167	C₄²	C₂×Dic₉	1	1	yes	characteristic
168	C₁₆	—	9	2	no	—
169	C₁₆	—	18	1	no	—
170	C₁₆	—	18	1	no	—
171	C₁₆	—	18	1	no	—
172	C₂²×C₃	—	2	1	no	—
173	C₂²×C₃	—	2	1	no	—
174	C₂²×C₃	—	2	1	no	—
175	C₂²×C₃	—	2	1	no	—
176	C₂²×C₃	—	2	1	no	—
177	C₂²×C₃	#812	1	2	yes	—
178	C₂²×C₃	C₂²×C₃⋊C₈	1	1	yes	characteristic
179	C₃×C₄	—	2	1	no	—
180	C₃×C₄	—	2	1	no	—
181	C₃×C₄	—	2	1	no	—
182	C₃×C₄	—	2	1	no	—
183	C₃×C₄	D₄×Dic₃	1	1	yes	characteristic
184	C₃×C₄	#948	1	1	yes	characteristic
185	C₉	#2168	1	1	yes	characteristic, complemented, Sylow
186	C₂³	C₂×C₉⋊C₈	1	1	yes	characteristic
187	C₂³	C₂×C₉⋊C₈	1	1	yes	characteristic
188	D₄	—	2	2	no	—
189	C₂×C₄	—	2	1	no	—
190	C₂×C₄	—	2	1	no	—
191	C₂×C₄	—	2	1	no	—
192	C₂×C₄	—	2	1	no	—
193	C₂×C₄	C₂×C₉⋊C₈	1	1	yes	characteristic
194	C₂×C₄	C₂×C₉⋊C₈	1	1	yes	characteristic
195	C₂×C₄	C₂×C₉⋊C₈	1	1	yes	characteristic
196	C₂×C₄	C₂×C₉⋊C₈	1	1	yes	characteristic
197	C₂×C₄	C₂²×Dic₉	1	1	yes	characteristic
198	C₈	—	2	1	no	—
199	C₈	—	2	1	no	—
200	C₈	—	2	1	no	—
201	C₈	D₄×D₉	1	1	yes	characteristic
202	C₈	#3699	1	1	yes	characteristic
203	C₂×C₃	—	2	2	no	—
204	C₂×C₃	—	4	1	no	—
205	C₂×C₃	D₄×C₃⋊C₈	1	1	yes	characteristic
206	C₂×C₃	C₂×#812	1	1	yes	characteristic
207	C₂×C₃	#5140	1	1	yes	characteristic
208	C₂²	—	2	1	no	—
209	C₂²	—	2	1	no	—
210	C₂²	—	2	1	no	—
211	C₂²	—	2	1	no	—
212	C₂²	—	2	1	no	—
213	C₂²	#63371	1	2	yes	complemented
214	C₂²	C₂²×C₉⋊C₈	1	1	yes	characteristic
215	C₄	—	2	1	no	—
216	C₄	—	2	1	no	—
217	C₄	—	2	1	no	—
218	C₄	—	2	1	no	—
219	C₄	D₄×Dic₉	1	1	yes	characteristic
220	C₄	#63510	1	1	yes	characteristic
221	C₃	#70857	1	1	yes	characteristic
222	C₂	—	2	2	no	—
223	C₂	—	4	1	no	—
224	C₂	D₄×C₉⋊C₈	1	1	yes	characteristic
225	C₂	C₂×#63371	1	1	yes	characteristic
226	C₂	#10588820	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	C₂²×D₉
commutator	C₂²×C₉	C₂²×C₈
Frattini	C₂×C₃×C₈	C₂²×D₃
Fitting	C₈×D₄×C₉	C₂
socle	C₂²×C₃	C₂²×C₃⋊C₈
prime radical	C₂×C₃×D₄	C₃⋊C₈
cosocle	C₂×C₈×C₉	C₂³
hypercenter	C₈×D₄	D₉
hypocenter	C₉	#2168
LCS	G ▷ C₂²×C₉ ▷ C₉
UCS	C₁ ◁ C₂×C₈ ◁ C₈×D₄

Complements

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

N	H ≅ G/N	type
2. C₂×#63371	C₂	semidirect
3. C₂²×C₉⋊C₁₆	C₂	semidirect
4. #10588226	C₂	semidirect
5. #10588226	C₂	semidirect
6. C₄⋊(C₉⋊C₁₆)	C₂	semidirect
7. #10588138	C₂	semidirect
47. C₉⋊C₁₆	D₄	semidirect
110. C₂²×C₉	M₃₂	semidirect
185. C₉	#2168	semidirect
213. C₂²	#63371	semidirect

Maximal subgroups

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

subgroup	index
C₈×D₄×C₉	2
C₂×#63371	2
C₂²×C₉⋊C₁₆	2
#10588226	2
C₄⋊(C₉⋊C₁₆)	2
#10588138	2
#70857	3