Subgroups — #11858338

Order 1152.

726 conjugacy classes of subgroups.

The subgroup lattice has 207 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₉×#41	C₂	1	1	yes	maximal, characteristic
2	C₂×#63510	C₂	1	2	yes	maximal
3	#10588677	C₂	1	6	yes	maximal, complemented
4	#10588604	C₂	1	3	yes	maximal, complemented
5	C₂×C₄⋊(C₉⋊C₈)	C₂	1	3	yes	maximal, complemented
6	#82512	—	3	1	no	maximal
7	C₂×C₉×#41	C₂²	1	1	yes	characteristic
8	C₄×Q₈×C₉	C₄	1	1	yes	characteristic
9	C₄×Q₈×C₉	C₄	1	1	yes	characteristic
10	C₄×D₄×C₉	C₄	1	3	yes	—
11	C₄×D₄×C₉	C₄	1	3	yes	—
12	C₉×#117	C₂²	1	3	yes	—
13	C₂×C₄²×C₉	C₂²	1	3	yes	—
14	#63510	—	2	8	no	—
15	C₂×#3667	C₂²	1	6	yes	—
16	C₂²×C₉⋊C₈	C₂²	1	6	yes	—
17	C₄⋊(C₉⋊C₈)	C₂²	1	6	yes	—
18	C₄⋊(C₉⋊C₈)	C₂²	1	6	yes	—
19	C₄⋊(C₉⋊C₈)	C₂²	1	1	yes	characteristic
20	C₄⋊(C₉⋊C₈)	C₂²	1	3	yes	—
21	C₃×C₄×#41	D₃	1	1	yes	characteristic
22	C₂×#948	—	3	2	no	—
23	#4999	—	3	6	no	—
24	#4924	—	3	3	no	—
25	C₂×C₄⋊(C₃⋊C₈)	—	3	3	no	—
26	C₉×#41	D₄	1	4	yes	—
27	C₉×#41	Q₈	1	4	yes	—
28	C₂×Q₈×C₉	C₂×C₄	1	1	yes	characteristic
29	C₂×D₄×C₉	C₂×C₄	1	3	yes	—
30	C₂²×C₄×C₉	C₂×C₄	1	3	yes	—
31	C₂²×C₄×C₉	C₂×C₄	1	3	yes	—
32	C₂²×C₄×C₉	C₂³	1	3	yes	—
33	C₉×C₄⋊C₄	C₂×C₄	1	3	yes	—
34	C₉×C₄⋊C₄	C₂×C₄	1	3	yes	—
35	C₉×C₂²⋊C₄	C₂×C₄	1	6	yes	—
36	C₄²×C₉	C₂×C₄	1	3	yes	—
37	C₄²×C₉	C₂×C₄	1	3	yes	—
38	C₄²×C₉	C₂³	1	1	yes	characteristic
39	C₄²×C₉	C₂³	1	3	yes	—
40	#3667	—	2	12	no	—
41	C₂×C₉⋊C₈	—	2	12	no	—
42	C₂×C₉⋊C₈	C₂³	1	2	yes	—
43	C₂×C₉⋊C₈	C₂³	1	6	yes	—
44	#2906	—	9	1	no	Sylow
45	C₂×C₃×#41	C₂×D₃	1	1	yes	characteristic
46	C₃×C₄×Q₈	Dic₃	1	1	yes	characteristic
47	C₃×C₄×Q₈	Dic₃	1	1	yes	characteristic
48	C₃×C₄×D₄	Dic₃	1	3	yes	—
49	C₃×C₄×D₄	Dic₃	1	3	yes	—
50	C₃×#117	C₂×D₃	1	3	yes	—
51	C₂×C₃×C₄²	C₂×D₃	1	3	yes	—
52	#948	—	6	8	no	—
53	C₂×#208	—	3	6	no	—
54	C₂²×C₃⋊C₈	—	3	6	no	—
55	C₄⋊(C₃⋊C₈)	—	3	1	no	—
56	C₄⋊(C₃⋊C₈)	—	3	3	no	—
57	C₄⋊(C₃⋊C₈)	—	3	6	no	—
58	C₄⋊(C₃⋊C₈)	—	3	6	no	—
59	C₂³×C₉	C₂²×C₄	1	3	yes	—
60	Q₈×C₉	C₄⋊C₄	1	4	yes	—
61	D₄×C₉	C₄⋊C₄	1	12	yes	—
62	C₂×C₄×C₉	—	2	6	no	—
63	C₂×C₄×C₉	—	4	3	no	—
64	C₂×C₄×C₉	C₂²×C₄	1	1	yes	characteristic
65	C₂×C₄×C₉	C₂²×C₄	1	3	yes	—
66	C₂×C₄×C₉	C₂²×C₄	1	3	yes	—
67	C₂×C₄×C₉	C₂²×C₄	1	3	yes	—
68	C₂×C₄×C₉	C₂²×C₄	1	1	yes	characteristic
69	C₂×C₄×C₉	C₂×D₄	1	6	yes	—
70	C₂×C₄×C₉	C₂×Q₈	1	6	yes	—
71	C₂×C₄×C₉	C₂⁴	1	1	yes	characteristic
72	C₉⋊C₈	—	2	2	no	—
73	C₉⋊C₈	—	2	6	no	—
74	C₂×#131	—	9	2	no	—
75	C₄×#41	D₉	1	1	yes	characteristic
76	#424	—	9	6	no	—
77	#423	—	9	3	no	—
78	C₂×C₄⋊C₈	—	9	3	no	—
79	C₃×#41	Dic₆	1	4	yes	—
80	C₃×#41	D₁₂	1	4	yes	—
81	C₂×C₃×Q₈	C₂×Dic₃	1	1	yes	characteristic
82	C₂×C₃×D₄	C₂×Dic₃	1	3	yes	—
83	C₂²×C₃×C₄	C₂×Dic₃	1	3	yes	—
84	C₂²×C₃×C₄	C₂×Dic₃	1	3	yes	—
85	C₂²×C₃×C₄	C₂²×D₃	1	3	yes	—
86	C₃×C₄⋊C₄	C₂×Dic₃	1	3	yes	—
87	C₃×C₄⋊C₄	C₂×Dic₃	1	3	yes	—
88	C₃×C₂²⋊C₄	C₂×Dic₃	1	6	yes	—
89	C₃×C₄²	C₂×Dic₃	1	3	yes	—
90	C₃×C₄²	C₂×Dic₃	1	3	yes	—
91	C₃×C₄²	C₂²×D₃	1	1	yes	characteristic
92	C₃×C₄²	C₂²×D₃	1	3	yes	—
93	#208	—	6	12	no	—
94	C₂×C₃⋊C₈	—	3	2	no	—
95	C₂×C₃⋊C₈	—	3	6	no	—
96	C₂×C₃⋊C₈	—	6	12	no	—
97	C₂²×C₉	—	2	3	no	—
98	C₂²×C₉	—	2	3	no	—
99	C₂²×C₉	C₂×C₄⋊C₄	1	6	yes	—
100	C₂²×C₉	C₂³×C₄	1	1	yes	characteristic
101	C₄×C₉	—	2	3	no	—
102	C₄×C₉	—	2	1	no	—
103	C₄×C₉	—	2	3	no	—
104	C₄×C₉	C₂×C₄⋊C₄	1	6	yes	—
105	C₄×C₉	#131	1	2	yes	complemented
106	C₄×C₉	C₂²×D₄	1	1	yes	characteristic
107	C₄×C₉	C₂²×Q₈	1	1	yes	characteristic
108	C₂×#41	C₂×D₉	1	1	yes	characteristic
109	#131	—	18	8	no	—
110	C₂×M₁₆	—	9	6	no	—
111	C₂²×C₈	—	9	6	no	—
112	C₄×Q₈	Dic₉	1	1	yes	characteristic
113	C₄×Q₈	Dic₉	1	1	yes	characteristic
114	C₄×D₄	Dic₉	1	3	yes	—
115	C₄×D₄	Dic₉	1	3	yes	—
116	#117	C₂×D₉	1	3	yes	—
117	C₂×C₄²	C₂×D₉	1	3	yes	—
118	C₄⋊C₈	—	9	1	no	—
119	C₄⋊C₈	—	9	3	no	—
120	C₄⋊C₈	—	9	6	no	—
121	C₄⋊C₈	—	9	6	no	—
122	C₂³×C₃	C₂²×Dic₃	1	3	yes	—
123	C₃×Q₈	C₄⋊Dic₃	1	4	yes	—
124	C₃×D₄	C₄⋊Dic₃	1	12	yes	—
125	C₂×C₃×C₄	—	2	6	no	—
126	C₂×C₃×C₄	—	4	3	no	—
127	C₂×C₃×C₄	C₂×Dic₆	1	6	yes	—
128	C₂×C₃×C₄	C₂×D₁₂	1	6	yes	—
129	C₂×C₃×C₄	C₂²×Dic₃	1	1	yes	characteristic
130	C₂×C₃×C₄	C₂²×Dic₃	1	3	yes	—
131	C₂×C₃×C₄	C₂²×Dic₃	1	3	yes	—
132	C₂×C₃×C₄	C₂²×Dic₃	1	3	yes	—
133	C₂×C₃×C₄	C₂²×Dic₃	1	1	yes	characteristic
134	C₂×C₃×C₄	C₂³×D₃	1	1	yes	characteristic
135	C₃⋊C₈	—	6	2	no	—
136	C₃⋊C₈	—	6	6	no	—
137	C₂×C₉	—	2	6	no	—
138	C₂×C₉	C₂²×C₄⋊C₄	1	1	yes	characteristic
139	C₂×C₉	C₂×#131	1	1	yes	characteristic
140	C₂×C₉	#568	1	1	yes	characteristic
141	#41	Dic₁₈	1	4	yes	—
142	#41	D₃₆	1	4	yes	—
143	C₂×Q₈	C₂×Dic₉	1	1	yes	characteristic
144	C₂×D₄	C₂×Dic₉	1	3	yes	—
145	C₂²×C₄	C₂×Dic₉	1	3	yes	—
146	C₂²×C₄	C₂×Dic₉	1	3	yes	—
147	C₂²×C₄	C₂²×D₉	1	3	yes	—
148	M₁₆	—	18	12	no	—
149	C₂×C₈	—	9	2	no	—
150	C₂×C₈	—	9	6	no	—
151	C₂×C₈	—	18	12	no	—
152	C₄⋊C₄	C₂×Dic₉	1	3	yes	—
153	C₄⋊C₄	C₂×Dic₉	1	3	yes	—
154	C₂²⋊C₄	C₂×Dic₉	1	6	yes	—
155	C₄²	C₂×Dic₉	1	3	yes	—
156	C₄²	C₂×Dic₉	1	3	yes	—
157	C₄²	C₂²×D₉	1	1	yes	characteristic
158	C₄²	C₂²×D₉	1	3	yes	—
159	C₂²×C₃	—	2	3	no	—
160	C₂²×C₃	—	2	3	no	—
161	C₂²×C₃	C₂×C₄⋊Dic₃	1	6	yes	—
162	C₂²×C₃	C₂³×Dic₃	1	1	yes	characteristic
163	C₃×C₄	—	2	3	no	—
164	C₃×C₄	—	2	1	no	—
165	C₃×C₄	—	2	3	no	—
166	C₃×C₄	C₂×C₄⋊Dic₃	1	6	yes	—
167	C₃×C₄	#948	1	2	yes	—
168	C₃×C₄	C₂²×Dic₆	1	1	yes	characteristic
169	C₃×C₄	C₂²×D₁₂	1	1	yes	characteristic
170	C₉	#2906	1	1	yes	characteristic, complemented, Sylow
171	C₂³	C₂²×Dic₉	1	3	yes	—
172	Q₈	C₄⋊Dic₉	1	4	yes	—
173	D₄	C₄⋊Dic₉	1	12	yes	—
174	C₂×C₄	—	2	6	no	—
175	C₂×C₄	—	4	3	no	—
176	C₂×C₄	C₂×Dic₁₈	1	6	yes	—
177	C₂×C₄	C₂×D₃₆	1	6	yes	—
178	C₂×C₄	C₂²×Dic₉	1	1	yes	characteristic
179	C₂×C₄	C₂²×Dic₉	1	3	yes	—
180	C₂×C₄	C₂²×Dic₉	1	3	yes	—
181	C₂×C₄	C₂²×Dic₉	1	1	yes	characteristic
182	C₂×C₄	C₂²×Dic₉	1	3	yes	—
183	C₂×C₄	C₂³×D₉	1	1	yes	characteristic
184	C₈	—	18	2	no	—
185	C₈	—	18	6	no	—
186	C₂×C₃	—	2	6	no	—
187	C₂×C₃	C₂²×C₄⋊Dic₃	1	1	yes	characteristic
188	C₂×C₃	C₂×#948	1	1	yes	characteristic
189	C₂×C₃	#5819	1	1	yes	characteristic
190	C₂²	—	2	3	no	—
191	C₂²	—	2	3	no	—
192	C₂²	C₂×C₄⋊Dic₉	1	6	yes	—
193	C₂²	C₂³×Dic₉	1	1	yes	characteristic
194	C₄	—	2	3	no	—
195	C₄	—	2	1	no	—
196	C₄	—	2	3	no	—
197	C₄	C₂×C₄⋊Dic₉	1	6	yes	—
198	C₄	#63510	1	2	yes	complemented
199	C₄	C₂²×Dic₁₈	1	1	yes	characteristic
200	C₄	C₂²×D₃₆	1	1	yes	characteristic
201	C₃	#82512	1	1	yes	characteristic
202	C₂	—	2	6	no	—
203	C₂	C₂²×C₄⋊Dic₉	1	1	yes	characteristic
204	C₂	C₂×#63510	1	1	yes	characteristic
205	C₂	#10589921	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂³×D₉
commutator	C₂²×C₉	C₂³×C₄
Frattini	C₂×C₃×C₄	C₂³×D₃
Fitting	C₄×C₉×#41	C₂
socle	C₂²×C₃	C₂³×Dic₃
prime radical	C₂×C₃×#41	C₂×D₃
cosocle	C₂×C₄×C₉	C₂⁴
hypercenter	C₄×#41	D₉
hypocenter	C₉	#2906
LCS	G ▷ C₂²×C₉ ▷ C₉
UCS	C₁ ◁ C₂×C₄ ◁ C₄×#41

Complements

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

N	H ≅ G/N	type
3. #10588677	C₂	semidirect
4. #10588604	C₂	semidirect
5. C₂×C₄⋊(C₉⋊C₈)	C₂	semidirect
105. C₄×C₉	#131	semidirect
170. C₉	#2906	semidirect
198. C₄	#63510	semidirect

Maximal subgroups

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

subgroup	index
C₄×C₉×#41	2
C₂×#63510	2
#10588677	2
#10588604	2
C₂×C₄⋊(C₉⋊C₈)	2
#82512	3