Subgroups — #11791423

Order 1152.

486 conjugacy classes of subgroups.

The subgroup lattice has 216 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#63431	C₂	1	1	yes	maximal, characteristic
2	C₄×C₉×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic
3	C₂×#63368	C₂	1	4	yes	maximal
4	C₂×C₄⋊(C₉⋊C₈)	C₂	1	1	yes	maximal, characteristic
5	#69486	—	3	1	no	maximal
6	C₂²×Dic₁₈	C₂²	1	2	yes	—
7	C₂×C₉×C₄⋊C₄	C₂²	1	2	yes	—
8	C₂×C₄²×C₉	—	2	1	no	—
9	C₂×C₄²×C₉	C₂²	1	1	yes	characteristic
10	C₂×C₄⋊Dic₉	—	2	1	no	—
11	C₂×C₄⋊Dic₉	—	2	1	no	—
12	C₂²×C₉⋊C₈	C₂²	1	2	yes	—
13	#63431	—	2	2	no	—
14	#63431	C₄	1	4	yes	complemented
15	C₉×#95	—	2	1	no	—
16	#63368	—	2	8	no	—
17	C₄⋊(C₉⋊C₈)	—	2	2	no	—
18	C₂×#869	—	3	1	no	—
19	C₃×C₄×C₄⋊C₄	D₃	1	1	yes	characteristic
20	C₂×#810	—	3	4	no	—
21	C₂×C₄⋊(C₃⋊C₈)	—	3	1	no	—
22	C₂²×C₄×C₉	—	2	2	no	—
23	C₂²×C₄×C₉	D₄	1	1	yes	characteristic
24	C₂²×C₄×C₉	D₄	1	1	yes	characteristic
25	C₂²×C₄×C₉	C₂³	1	1	yes	characteristic
26	C₂²×Dic₉	—	2	2	no	—
27	C₂×Dic₁₈	—	2	4	no	—
28	C₂×Dic₁₈	—	2	4	no	—
29	C₂×Dic₁₈	—	2	2	no	—
30	C₂×Dic₁₈	C₂×C₄	1	4	yes	—
31	C₉×C₄⋊C₄	—	2	2	no	—
32	C₉×C₄⋊C₄	D₄	1	4	yes	—
33	C₄²×C₉	—	4	2	no	—
34	C₄²×C₉	C₂×C₄	1	2	yes	—
35	C₄²×C₉	D₄	1	2	yes	—
36	C₄⋊Dic₉	—	2	4	no	—
37	C₄⋊Dic₉	—	2	4	no	—
38	C₂×C₉⋊C₈	—	2	2	no	—
39	C₂×C₉⋊C₈	—	2	4	no	—
40	#1981	—	9	1	no	Sylow
41	C₂²×Dic₆	—	3	2	no	—
42	C₂×C₃×C₄⋊C₄	C₂×D₃	1	2	yes	—
43	C₂×C₃×C₄²	—	2	1	no	—
44	C₂×C₃×C₄²	C₂×D₃	1	1	yes	characteristic
45	C₂×C₄⋊Dic₃	—	6	1	no	—
46	C₂×C₄⋊Dic₃	—	6	1	no	—
47	C₂²×C₃⋊C₈	—	3	2	no	—
48	#869	—	3	4	no	—
49	#869	—	6	2	no	—
50	C₃×#95	—	2	1	no	—
51	#810	—	6	8	no	—
52	C₄⋊(C₃⋊C₈)	—	6	2	no	—
53	C₂³×C₉	C₂×D₄	1	1	yes	characteristic
54	C₂×C₄×C₉	—	2	2	no	—
55	C₂×C₄×C₉	—	2	4	no	—
56	C₂×C₄×C₉	—	4	2	no	—
57	C₂×C₄×C₉	—	4	2	no	—
58	C₂×C₄×C₉	C₂²⋊C₄	1	2	yes	—
59	C₂×C₄×C₉	C₂²⋊C₄	1	2	yes	—
60	C₂×C₄×C₉	SD₁₆	1	2	yes	—
61	C₂×C₄×C₉	Q₁₆	1	2	yes	—
62	C₂×C₄×C₉	C₂²×C₄	1	1	yes	characteristic
63	C₂×C₄×C₉	C₂×D₄	1	2	yes	—
64	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
65	C₂×C₄×C₉	#41	1	2	yes	—
66	C₂×Dic₉	—	2	4	no	—
67	C₂×Dic₉	—	4	4	no	—
68	Dic₁₈	—	2	8	no	—
69	Dic₁₈	—	4	2	no	—
70	Dic₁₈	—	4	2	no	—
71	C₉⋊C₈	—	4	2	no	—
72	C₂×C₄⋊Q₈	—	9	1	no	—
73	C₂×C₄⋊C₈	—	9	1	no	—
74	C₂×#103	—	9	4	no	—
75	C₄×C₄⋊C₄	D₉	1	1	yes	characteristic
76	C₂²×C₃×C₄	—	2	2	no	—
77	C₂²×C₃×C₄	#67	1	1	yes	characteristic
78	C₂²×C₃×C₄	#67	1	1	yes	characteristic
79	C₂²×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
80	C₂²×Dic₃	—	6	2	no	—
81	C₂×Dic₆	—	3	4	no	—
82	C₂×Dic₆	—	6	4	no	—
83	C₂×Dic₆	—	6	4	no	—
84	C₂×Dic₆	—	6	2	no	—
85	C₃×C₄⋊C₄	—	2	2	no	—
86	C₃×C₄⋊C₄	D₁₂	1	4	yes	—
87	C₃×C₄²	—	4	2	no	—
88	C₃×C₄²	C₄×D₃	1	2	yes	—
89	C₃×C₄²	D₁₂	1	2	yes	—
90	C₄⋊Dic₃	—	6	4	no	—
91	C₄⋊Dic₃	—	6	4	no	—
92	C₂×C₃⋊C₈	—	6	2	no	—
93	C₂×C₃⋊C₈	—	6	4	no	—
94	C₂²×C₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
95	C₂²×C₉	#121	1	2	yes	—
96	C₂²×C₉	C₂×SD₁₆	1	1	yes	characteristic
97	C₂²×C₉	C₂×Q₁₆	1	1	yes	characteristic
98	C₂²×C₉	#137	1	2	yes	—
99	C₄×C₉	—	2	2	no	—
100	C₄×C₉	—	4	4	no	—
101	C₄×C₉	#103	1	4	yes	complemented
102	C₄×C₉	C₄×D₄	1	2	yes	—
103	C₄×C₉	#124	1	1	yes	characteristic
104	C₄×C₉	#127	1	1	yes	characteristic
105	Dic₉	—	4	4	no	—
106	C₂²×Q₈	—	9	2	no	—
107	C₂²×C₈	—	9	2	no	—
108	C₄⋊Q₈	—	9	4	no	—
109	C₄⋊Q₈	—	18	2	no	—
110	C₂×C₄⋊C₄	—	18	1	no	—
111	C₂×C₄⋊C₄	—	18	1	no	—
112	C₂×C₄⋊C₄	C₂×D₉	1	2	yes	—
113	C₂×C₄²	—	2	1	no	—
114	C₂×C₄²	C₂×D₉	1	1	yes	characteristic
115	C₄⋊C₈	—	18	2	no	—
116	#103	—	18	8	no	—
117	#95	—	2	1	no	—
118	C₂³×C₃	C₂×#67	1	1	yes	characteristic
119	C₂×C₃×C₄	—	2	2	no	—
120	C₂×C₃×C₄	—	2	4	no	—
121	C₂×C₃×C₄	—	4	2	no	—
122	C₂×C₃×C₄	—	4	2	no	—
123	C₂×C₃×C₄	(C₂×D₃)⋊C₄	1	2	yes	—
124	C₂×C₃×C₄	(C₂×D₃)⋊C₄	1	2	yes	—
125	C₂×C₃×C₄	#214	1	2	yes	—
126	C₂×C₃×C₄	#216	1	2	yes	—
127	C₂×C₃×C₄	C₂×C₄×D₃	1	1	yes	characteristic
128	C₂×C₃×C₄	C₂×D₁₂	1	2	yes	—
129	C₂×C₃×C₄	C₂×D₁₂	1	1	yes	characteristic
130	C₂×C₃×C₄	#235	1	2	yes	—
131	C₂×Dic₃	—	6	4	no	—
132	C₂×Dic₃	—	12	4	no	—
133	Dic₆	—	6	8	no	—
134	Dic₆	—	12	2	no	—
135	Dic₆	—	12	2	no	—
136	C₃⋊C₈	—	12	2	no	—
137	C₂×C₉	#390	1	1	yes	characteristic
138	C₂×C₉	C₂×#103	1	1	yes	characteristic
139	C₂×C₉	#419	1	1	yes	characteristic
140	C₂×C₉	#461	1	2	yes	—
141	C₂×C₉	#462	1	2	yes	—
142	C₂×Q₈	—	9	4	no	—
143	C₂×Q₈	—	18	2	no	—
144	C₂×Q₈	—	18	4	no	—
145	C₂×Q₈	—	18	4	no	—
146	C₂²×C₄	—	2	2	no	—
147	C₂²×C₄	—	18	2	no	—
148	C₂²×C₄	#611	1	1	yes	characteristic
149	C₂²×C₄	#611	1	1	yes	characteristic
150	C₂²×C₄	C₂²×D₉	1	1	yes	characteristic
151	C₂×C₈	—	18	4	no	—
152	C₂×C₈	—	18	2	no	—
153	C₄⋊C₄	—	2	2	no	—
154	C₄⋊C₄	—	18	4	no	—
155	C₄⋊C₄	—	18	4	no	—
156	C₄⋊C₄	D₃₆	1	4	yes	—
157	C₄²	—	4	2	no	—
158	C₄²	C₄×D₉	1	2	yes	—
159	C₄²	D₃₆	1	2	yes	—
160	C₂²×C₃	C₂×(C₂×D₃)⋊C₄	1	1	yes	characteristic
161	C₂²×C₃	#930	1	2	yes	—
162	C₂²×C₃	C₂×#214	1	1	yes	characteristic
163	C₂²×C₃	C₂×#216	1	1	yes	characteristic
164	C₂²×C₃	#951	1	2	yes	—
165	C₃×C₄	—	2	2	no	—
166	C₃×C₄	—	4	4	no	—
167	C₃×C₄	#810	1	4	yes	—
168	C₃×C₄	C₄×D₁₂	1	2	yes	—
169	C₃×C₄	#874	1	1	yes	characteristic
170	C₃×C₄	#875	1	1	yes	characteristic
171	Dic₃	—	12	4	no	—
172	C₉	#1981	1	1	yes	characteristic, complemented, Sylow
173	C₂³	C₂×#611	1	1	yes	characteristic
174	Q₈	—	18	8	no	—
175	Q₈	—	36	2	no	—
176	Q₈	—	36	2	no	—
177	C₂×C₄	—	2	2	no	—
178	C₂×C₄	—	2	4	no	—
179	C₂×C₄	—	4	2	no	—
180	C₂×C₄	—	4	2	no	—
181	C₂×C₄	—	18	4	no	—
182	C₂×C₄	—	36	4	no	—
183	C₂×C₄	(C₂×D₉)⋊C₄	1	2	yes	—
184	C₂×C₄	(C₂×D₉)⋊C₄	1	2	yes	—
185	C₂×C₄	#3672	1	2	yes	—
186	C₂×C₄	#3674	1	2	yes	—
187	C₂×C₄	C₂×C₄×D₉	1	1	yes	characteristic
188	C₂×C₄	C₂×D₃₆	1	2	yes	—
189	C₂×C₄	C₂×D₃₆	1	1	yes	characteristic
190	C₂×C₄	#3697	1	2	yes	—
191	C₈	—	36	2	no	—
192	C₂×C₃	#4939	1	1	yes	characteristic
193	C₂×C₃	C₂×#810	1	1	yes	characteristic
194	C₂×C₃	#5004	1	1	yes	characteristic
195	C₂×C₃	#5019	1	2	yes	—
196	C₂×C₃	#5031	1	2	yes	—
197	C₂²	C₂×(C₂×D₉)⋊C₄	1	1	yes	characteristic
198	C₂²	#63492	1	2	yes	—
199	C₂²	C₂×#3672	1	1	yes	characteristic
200	C₂²	C₂×#3674	1	1	yes	characteristic
201	C₂²	#63511	1	2	yes	—
202	C₄	—	2	2	no	—
203	C₄	—	4	4	no	—
204	C₄	—	36	4	no	—
205	C₄	#63368	1	4	yes	complemented
206	C₄	C₄×D₃₆	1	2	yes	—
207	C₄	#63436	1	1	yes	characteristic
208	C₄	#63437	1	1	yes	characteristic
209	C₃	#69486	1	1	yes	characteristic
210	C₂	#10588619	1	1	yes	characteristic
211	C₂	C₂×#63368	1	1	yes	characteristic
212	C₂	#10588681	1	1	yes	characteristic
213	C₂	#10588695	1	2	yes	—
214	C₂	#10588707	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×#611
commutator	C₂×C₄×C₉	C₂²×C₄
Frattini	C₂²×C₃×C₄	C₂²×D₃
Fitting	C₄×C₉×C₄⋊C₄	C₂
socle, prime radical	C₂³×C₃	C₂×#67
cosocle	C₂²×C₄×C₉	C₂³
hypercenter	C₄×C₄⋊C₄	D₉
hypocenter	C₉	#1981
LCS	G ▷ C₂×C₄×C₉ ▷ C₂×C₉ ▷ C₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₄×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
14. #63431	C₄	semidirect
101. C₄×C₉	#103	semidirect
172. C₉	#1981	semidirect
205. C₄	#63368	semidirect

Maximal subgroups

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

subgroup	index
C₂×#63431	2
C₄×C₉×C₄⋊C₄	2
C₂×#63368	2
C₂×C₄⋊(C₉⋊C₈)	2
#69486	3