Subgroups — #11790063

Order 1152.

372 conjugacy classes of subgroups.

The subgroup lattice has 225 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₉×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄⋊(C₉⋊C₈)	C₂	1	2	yes	maximal
3	C₂×C₄⋊(C₉⋊C₈)	C₂	1	1	yes	maximal, characteristic
4	C₂×#63362	C₂	1	1	yes	maximal, characteristic
5	#10588200	C₂	1	2	yes	maximal
6	#69485	—	3	1	no	maximal
7	C₂×C₉×C₄⋊C₄	C₄	1	2	yes	—
8	C₂×C₄²×C₉	C₂²	1	2	yes	—
9	C₂×C₄²×C₉	C₂²	1	1	yes	characteristic
10	C₂²×C₉⋊C₈	C₂²	1	2	yes	—
11	C₂²×C₉⋊C₈	C₂²	1	1	yes	characteristic
12	C₂²×C₉⋊C₈	C₂²	1	1	yes	characteristic
13	C₉×#95	C₄	1	1	yes	characteristic
14	C₉×#95	C₄	1	1	yes	characteristic
15	C₄⋊(C₉⋊C₈)	—	2	2	no	—
16	C₄⋊(C₉⋊C₈)	—	2	4	no	—
17	#63362	—	2	2	no	—
18	C₃×C₄×C₄⋊C₄	D₃	1	1	yes	characteristic
19	C₂×C₄⋊(C₃⋊C₈)	—	3	2	no	—
20	C₂×C₄⋊(C₃⋊C₈)	—	3	1	no	—
21	C₂×#803	—	3	1	no	—
22	#4524	—	3	2	no	—
23	C₂²×C₄×C₉	C₂×C₄	1	2	yes	—
24	C₂²×C₄×C₉	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₄×C₉	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₄×C₉	C₂×C₄	1	2	yes	—
27	C₂²×C₄×C₉	C₂³	1	1	yes	characteristic
28	C₉×C₄⋊C₄	—	2	4	no	—
29	C₄²×C₉	—	2	2	no	—
30	C₄²×C₉	—	2	2	no	—
31	C₄²×C₉	D₄	1	2	yes	—
32	C₄²×C₉	Q₈	1	2	yes	—
33	C₂×C₉⋊C₈	—	2	1	no	—
34	C₂×C₉⋊C₈	—	2	1	no	—
35	C₂×C₉⋊C₈	—	2	1	no	—
36	C₂×C₉⋊C₈	—	2	1	no	—
37	C₂×C₉⋊C₈	—	2	2	no	—
38	C₂×C₉⋊C₈	—	2	2	no	—
39	C₂×C₉⋊C₈	—	2	2	no	—
40	C₂×C₉⋊C₈	D₄	1	2	yes	—
41	C₂×C₉⋊C₈	Q₈	1	2	yes	—
42	#1940	—	9	1	no	Sylow
43	C₂×C₃×C₄⋊C₄	Dic₃	1	2	yes	—
44	C₂×C₃×C₄²	C₂×D₃	1	2	yes	—
45	C₂×C₃×C₄²	C₂×D₃	1	1	yes	characteristic
46	C₂²×C₃⋊C₈	—	3	1	no	—
47	C₂²×C₃⋊C₈	—	3	1	no	—
48	C₂²×C₃⋊C₈	—	3	2	no	—
49	C₃×#95	Dic₃	1	1	yes	characteristic
50	C₃×#95	Dic₃	1	1	yes	characteristic
51	C₄⋊(C₃⋊C₈)	—	6	4	no	—
52	C₄⋊(C₃⋊C₈)	—	6	2	no	—
53	#803	—	6	2	no	—
54	C₂³×C₉	C₂²×C₄	1	1	yes	characteristic
55	C₂×C₄×C₉	—	2	2	no	—
56	C₂×C₄×C₉	—	2	2	no	—
57	C₂×C₄×C₉	—	2	2	no	—
58	C₂×C₄×C₉	—	2	2	no	—
59	C₂×C₄×C₉	—	2	2	no	—
60	C₂×C₄×C₉	—	2	2	no	—
61	C₂×C₄×C₉	—	2	2	no	—
62	C₂×C₄×C₉	C₄⋊C₄	1	2	yes	—
63	C₂×C₄×C₉	C₄⋊C₄	1	2	yes	—
64	C₂×C₄×C₉	M₁₆	1	4	yes	—
65	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
66	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
67	C₂×C₄×C₉	C₂×Q₈	1	1	yes	characteristic
68	C₂×C₄×C₉	C₂×Q₈	1	1	yes	characteristic
69	C₂×C₄×C₉	#41	1	1	yes	characteristic
70	C₂×C₄×C₉	#41	1	1	yes	characteristic
71	C₉⋊C₈	—	2	2	no	—
72	C₉⋊C₈	—	2	2	no	—
73	C₉⋊C₈	—	4	2	no	—
74	C₂×C₄⋊C₈	—	9	2	no	—
75	C₂×C₄⋊C₈	—	9	1	no	—
76	C₂×#97	—	9	1	no	—
77	C₄×C₄⋊C₄	D₉	1	1	yes	characteristic
78	#336	—	9	2	no	—
79	C₂²×C₃×C₄	C₂×Dic₃	1	2	yes	—
80	C₂²×C₃×C₄	C₂×Dic₃	1	2	yes	—
81	C₂²×C₃×C₄	C₂×Dic₃	1	1	yes	characteristic
82	C₂²×C₃×C₄	C₂×Dic₃	1	1	yes	characteristic
83	C₂²×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
84	C₃×C₄⋊C₄	—	2	4	no	—
85	C₃×C₄²	—	2	2	no	—
86	C₃×C₄²	—	2	2	no	—
87	C₃×C₄²	Dic₆	1	2	yes	—
88	C₃×C₄²	D₁₂	1	2	yes	—
89	C₂×C₃⋊C₈	—	3	2	no	—
90	C₂×C₃⋊C₈	—	3	2	no	—
91	C₂×C₃⋊C₈	—	6	1	no	—
92	C₂×C₃⋊C₈	—	6	1	no	—
93	C₂×C₃⋊C₈	—	6	1	no	—
94	C₂×C₃⋊C₈	—	6	1	no	—
95	C₂×C₃⋊C₈	—	6	2	no	—
96	C₂×C₃⋊C₈	—	6	2	no	—
97	C₂×C₃⋊C₈	—	6	2	no	—
98	C₂²×C₉	C₂×C₄⋊C₄	1	1	yes	characteristic
99	C₂²×C₉	C₄×D₄	1	1	yes	characteristic
100	C₂²×C₉	C₄×Q₈	1	1	yes	characteristic
101	C₂²×C₉	C₂×M₁₆	1	2	yes	—
102	C₂²×C₉	#131	1	2	yes	—
103	C₄×C₉	—	2	2	no	—
104	C₄×C₉	—	2	2	no	—
105	C₄×C₉	—	2	4	no	—
106	C₄×C₉	—	4	2	no	—
107	C₄×C₉	#121	1	1	yes	characteristic
108	C₄×C₉	C₂²⋊Q₈	1	1	yes	characteristic
109	C₄×C₉	#125	1	1	yes	characteristic
110	C₄×C₉	C₄⋊Q₈	1	1	yes	characteristic
111	C₂²×C₈	—	9	1	no	—
112	C₂²×C₈	—	9	1	no	—
113	C₂²×C₈	—	9	2	no	—
114	C₂×C₄⋊C₄	Dic₉	1	2	yes	—
115	C₂×C₄²	C₂×D₉	1	2	yes	—
116	C₂×C₄²	C₂×D₉	1	1	yes	characteristic
117	C₄⋊C₈	—	18	4	no	—
118	C₄⋊C₈	—	18	2	no	—
119	#97	—	18	2	no	—
120	#95	Dic₉	1	1	yes	characteristic
121	#95	Dic₉	1	1	yes	characteristic
122	C₂³×C₃	C₂²×Dic₃	1	1	yes	characteristic
123	C₂×C₃×C₄	—	2	2	no	—
124	C₂×C₃×C₄	—	2	2	no	—
125	C₂×C₃×C₄	—	2	2	no	—
126	C₂×C₃×C₄	—	2	2	no	—
127	C₂×C₃×C₄	—	2	2	no	—
128	C₂×C₃×C₄	—	2	2	no	—
129	C₂×C₃×C₄	—	2	2	no	—
130	C₂×C₃×C₄	#208	1	4	yes	—
131	C₂×C₃×C₄	C₄⋊Dic₃	1	2	yes	—
132	C₂×C₃×C₄	C₄⋊Dic₃	1	2	yes	—
133	C₂×C₃×C₄	C₂×Dic₆	1	1	yes	characteristic
134	C₂×C₃×C₄	C₂×D₁₂	1	1	yes	characteristic
135	C₂×C₃×C₄	D₃×D₄	1	1	yes	characteristic
136	C₂×C₃×C₄	#237	1	1	yes	characteristic
137	C₂×C₃×C₄	D₃×Q₈	1	1	yes	characteristic
138	C₂×C₃×C₄	#239	1	1	yes	characteristic
139	C₃⋊C₈	—	6	2	no	—
140	C₃⋊C₈	—	6	2	no	—
141	C₃⋊C₈	—	12	2	no	—
142	C₂×C₉	#389	1	1	yes	characteristic
143	C₂×C₉	#423	1	1	yes	characteristic
144	C₂×C₉	#424	1	1	yes	characteristic
145	C₂×C₉	#435	1	2	yes	—
146	C₂×C₉	#446	1	2	yes	—
147	C₂²×C₄	C₂×Dic₉	1	2	yes	—
148	C₂²×C₄	C₂×Dic₉	1	2	yes	—
149	C₂²×C₄	C₂×Dic₉	1	1	yes	characteristic
150	C₂²×C₄	C₂×Dic₉	1	1	yes	characteristic
151	C₂²×C₄	C₂²×D₉	1	1	yes	characteristic
152	C₂×C₈	—	9	2	no	—
153	C₂×C₈	—	9	2	no	—
154	C₂×C₈	—	18	1	no	—
155	C₂×C₈	—	18	1	no	—
156	C₂×C₈	—	18	1	no	—
157	C₂×C₈	—	18	1	no	—
158	C₂×C₈	—	18	2	no	—
159	C₂×C₈	—	18	2	no	—
160	C₂×C₈	—	18	2	no	—
161	C₄⋊C₄	—	2	4	no	—
162	C₄²	—	2	2	no	—
163	C₄²	—	2	2	no	—
164	C₄²	Dic₁₈	1	2	yes	—
165	C₄²	D₃₆	1	2	yes	—
166	C₂²×C₃	C₂×#208	1	2	yes	—
167	C₂²×C₃	C₂×C₄⋊Dic₃	1	1	yes	characteristic
168	C₂²×C₃	D₄×Dic₃	1	1	yes	characteristic
169	C₂²×C₃	Q₈×Dic₃	1	1	yes	characteristic
170	C₂²×C₃	#948	1	2	yes	—
171	C₃×C₄	—	2	2	no	—
172	C₃×C₄	—	2	2	no	—
173	C₃×C₄	—	2	4	no	—
174	C₃×C₄	—	4	2	no	—
175	C₃×C₄	#888	1	1	yes	characteristic
176	C₃×C₄	#890	1	1	yes	characteristic
177	C₃×C₄	#895	1	1	yes	characteristic
178	C₃×C₄	#897	1	1	yes	characteristic
179	C₉	#1940	1	1	yes	characteristic, complemented, Sylow
180	C₂³	C₂²×Dic₉	1	1	yes	characteristic
181	C₂×C₄	—	2	2	no	—
182	C₂×C₄	—	2	2	no	—
183	C₂×C₄	—	2	2	no	—
184	C₂×C₄	—	2	2	no	—
185	C₂×C₄	—	2	2	no	—
186	C₂×C₄	—	2	2	no	—
187	C₂×C₄	—	2	2	no	—
188	C₂×C₄	#3667	1	4	yes	—
189	C₂×C₄	C₄⋊Dic₉	1	2	yes	—
190	C₂×C₄	C₄⋊Dic₉	1	2	yes	—
191	C₂×C₄	C₂×Dic₁₈	1	1	yes	characteristic
192	C₂×C₄	C₂×D₃₆	1	1	yes	characteristic
193	C₂×C₄	D₄×D₉	1	1	yes	characteristic
194	C₂×C₄	#3699	1	1	yes	characteristic
195	C₂×C₄	Q₈×D₉	1	1	yes	characteristic
196	C₂×C₄	#3701	1	1	yes	characteristic
197	C₈	—	18	2	no	—
198	C₈	—	18	2	no	—
199	C₈	—	36	2	no	—
200	C₂×C₃	#4924	1	1	yes	characteristic
201	C₂×C₃	#4984	1	1	yes	characteristic
202	C₂×C₃	#4999	1	1	yes	characteristic
203	C₂×C₃	#5011	1	2	yes	—
204	C₂×C₃	#5024	1	2	yes	—
205	C₂²	C₂×#3667	1	2	yes	—
206	C₂²	C₂×C₄⋊Dic₉	1	1	yes	characteristic
207	C₂²	D₄×Dic₉	1	1	yes	characteristic
208	C₂²	Q₈×Dic₉	1	1	yes	characteristic
209	C₂²	#63510	1	2	yes	—
210	C₄	—	2	2	no	—
211	C₄	—	2	4	no	—
212	C₄	—	2	2	no	—
213	C₄	—	4	2	no	—
214	C₄	#63450	1	1	yes	characteristic
215	C₄	#63452	1	1	yes	characteristic
216	C₄	#63457	1	1	yes	characteristic
217	C₄	#63459	1	1	yes	characteristic
218	C₃	#69485	1	1	yes	characteristic
219	C₂	#10588604	1	1	yes	characteristic
220	C₂	#10588664	1	1	yes	characteristic
221	C₂	#10588677	1	1	yes	characteristic
222	C₂	#10588688	1	2	yes	—
223	C₂	#10588701	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	C₂²×D₉
commutator	C₂³×C₉	C₂²×C₄
Frattini	C₂²×C₃×C₄	C₂²×D₃
Fitting	C₄×C₉×C₄⋊C₄	C₂
socle, prime radical	C₂³×C₃	C₂²×Dic₃
cosocle	C₂²×C₄×C₉	C₂³
hypercenter	C₄×C₄⋊C₄	D₉
hypocenter	C₉	#1940
LCS	G ▷ C₂³×C₉ ▷ C₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₄×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
179. C₉	#1940	semidirect

Maximal subgroups

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

subgroup	index
C₄×C₉×C₄⋊C₄	2
C₂×C₄⋊(C₉⋊C₈)	2
C₂×#63362	2
#10588200	2
#69485	3