Subgroups — #11805593

Order 1152.

828 conjugacy classes of subgroups.

The subgroup lattice has 183 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×Q₈×C₉	C₂	1	1	yes	maximal, characteristic
2	C₂×Q₈×Dic₉	C₂	1	2	yes	maximal
3	#10588664	C₂	1	6	yes	maximal
4	#10588616	C₂	1	3	yes	maximal
5	C₄×C₄⋊Dic₉	C₂	1	3	yes	maximal
6	#82460	—	3	1	no	maximal
7	C₂²×Q₈×C₉	C₂²	1	1	yes	characteristic
8	C₄×Q₈×C₉	C₄	1	8	yes	complemented
9	C₂×C₉×C₄⋊C₄	C₂²	1	3	yes	—
10	C₂×C₄²×C₉	C₂²	1	3	yes	—
11	Q₈×Dic₉	—	2	8	no	—
12	C₂×C₄⋊Dic₉	C₂²	1	6	yes	—
13	C₂×C₄⋊Dic₉	C₂²	1	6	yes	—
14	C₂×C₄×Dic₉	C₂²	1	6	yes	—
15	C₉⋊#95	C₂²	1	1	yes	characteristic
16	C₉⋊#95	C₂²	1	3	yes	—
17	C₉⋊#95	C₂²	1	6	yes	—
18	C₂×C₃×C₄×Q₈	D₃	1	1	yes	characteristic
19	C₂×Q₈×Dic₃	—	3	2	no	—
20	#4984	—	3	6	no	—
21	#4936	—	3	3	no	—
22	C₄×C₄⋊Dic₃	—	3	3	no	—
23	C₂×Q₈×C₉	C₂×C₄	1	4	yes	—
24	C₂×Q₈×C₉	D₄	1	4	yes	—
25	C₂×Q₈×C₉	Q₈	1	4	yes	—
26	C₂²×C₄×C₉	C₂³	1	1	yes	characteristic
27	C₂²×C₄×C₉	C₂³	1	3	yes	—
28	C₂²×C₄×C₉	C₂³	1	3	yes	—
29	C₂²×Dic₉	C₂³	1	6	yes	—
30	C₂²×Dic₉	C₂³	1	2	yes	—
31	C₉×C₄⋊C₄	C₂×C₄	1	12	yes	—
32	C₄²×C₉	C₂×C₄	1	12	yes	—
33	C₄⋊Dic₉	—	2	12	no	—
34	C₄⋊Dic₉	—	2	12	no	—
35	C₄×Dic₉	—	2	12	no	—
36	#2351	—	9	1	no	Sylow
37	C₂²×C₃×Q₈	C₂×D₃	1	1	yes	characteristic
38	C₃×C₄×Q₈	Dic₃	1	8	yes	—
39	C₂×C₃×C₄⋊C₄	C₂×D₃	1	3	yes	—
40	C₂×C₃×C₄²	C₂×D₃	1	3	yes	—
41	Q₈×Dic₃	—	6	8	no	—
42	C₂×C₄⋊Dic₃	—	3	6	no	—
43	C₂×C₄⋊Dic₃	—	3	6	no	—
44	C₂×C₄×Dic₃	—	3	6	no	—
45	C₃⋊#95	—	3	1	no	—
46	C₃⋊#95	—	3	3	no	—
47	C₃⋊#95	—	3	6	no	—
48	C₂³×C₉	C₂⁴	1	1	yes	characteristic
49	Q₈×C₉	C₄⋊C₄	1	16	yes	complemented
50	C₂×C₄×C₉	—	2	6	no	—
51	C₂×C₄×C₉	—	2	2	no	—
52	C₂×C₄×C₉	C₂²×C₄	1	2	yes	—
53	C₂×C₄×C₉	C₂²×C₄	1	6	yes	—
54	C₂×C₄×C₉	C₂²×C₄	1	6	yes	—
55	C₂×C₄×C₉	C₂×D₄	1	6	yes	—
56	C₂×C₄×C₉	C₂×Q₈	1	6	yes	—
57	C₂×Dic₉	—	2	2	no	—
58	C₂×Dic₉	—	2	2	no	—
59	C₂×Dic₉	—	2	6	no	—
60	C₂×Dic₉	—	2	6	no	—
61	C₂×Dic₉	—	2	6	no	—
62	C₂×Dic₉	#41	1	4	yes	—
63	C₂×C₄×Q₈	—	9	2	no	—
64	C₂×C₄×Q₈	D₉	1	1	yes	characteristic
65	#389	—	9	6	no	—
66	#382	—	9	3	no	—
67	C₄×C₄⋊C₄	—	9	3	no	—
68	C₂×C₃×Q₈	Dic₆	1	4	yes	—
69	C₂×C₃×Q₈	D₁₂	1	4	yes	—
70	C₂×C₃×Q₈	C₂×Dic₃	1	4	yes	—
71	C₂²×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
72	C₂²×C₃×C₄	C₂²×D₃	1	3	yes	—
73	C₂²×C₃×C₄	C₂²×D₃	1	3	yes	—
74	C₂²×Dic₃	—	3	2	no	—
75	C₂²×Dic₃	—	3	6	no	—
76	C₃×C₄⋊C₄	C₂×Dic₃	1	12	yes	—
77	C₃×C₄²	C₂×Dic₃	1	12	yes	—
78	C₄⋊Dic₃	—	6	12	no	—
79	C₄⋊Dic₃	—	6	12	no	—
80	C₄×Dic₃	—	6	12	no	—
81	C₂²×C₉	C₂³×C₄	1	1	yes	characteristic
82	C₂²×C₉	C₂²×D₄	1	1	yes	characteristic
83	C₂²×C₉	C₂²×Q₈	1	1	yes	characteristic
84	C₂²×C₉	C₂×#41	1	2	yes	—
85	C₂²×C₉	2₋¹⁺⁴	1	2	yes	—
86	C₄×C₉	—	2	6	no	—
87	C₄×C₉	—	2	2	no	—
88	C₄×C₉	C₂×C₄⋊C₄	1	12	yes	—
89	Dic₉	—	2	4	no	—
90	Dic₉	—	4	6	no	—
91	C₂²×Q₈	C₂×D₉	1	1	yes	characteristic
92	C₄×Q₈	—	18	8	no	—
93	C₄×Q₈	Dic₉	1	8	yes	complemented
94	C₂×C₄⋊C₄	—	9	6	no	—
95	C₂×C₄⋊C₄	—	9	6	no	—
96	C₂×C₄⋊C₄	C₂×D₉	1	3	yes	—
97	C₂×C₄²	—	9	6	no	—
98	C₂×C₄²	C₂×D₉	1	3	yes	—
99	#95	—	9	1	no	—
100	#95	—	9	3	no	—
101	#95	—	9	6	no	—
102	C₂³×C₃	C₂³×D₃	1	1	yes	characteristic
103	C₃×Q₈	C₄⋊Dic₃	1	16	yes	—
104	C₂×C₃×C₄	—	2	2	no	—
105	C₂×C₃×C₄	—	2	6	no	—
106	C₂×C₃×C₄	C₂×Dic₆	1	6	yes	—
107	C₂×C₃×C₄	C₂×D₁₂	1	6	yes	—
108	C₂×C₃×C₄	C₂²×Dic₃	1	2	yes	—
109	C₂×C₃×C₄	C₂²×Dic₃	1	6	yes	—
110	C₂×C₃×C₄	C₂²×Dic₃	1	6	yes	—
111	C₂×Dic₃	—	3	4	no	—
112	C₂×Dic₃	—	6	2	no	—
113	C₂×Dic₃	—	6	2	no	—
114	C₂×Dic₃	—	6	6	no	—
115	C₂×Dic₃	—	6	6	no	—
116	C₂×Dic₃	—	6	6	no	—
117	C₂×C₉	C₂²×C₄⋊C₄	1	1	yes	characteristic
118	C₂×C₉	C₄×#41	1	1	yes	characteristic
119	C₂×C₉	#519	1	1	yes	characteristic
120	C₂×C₉	#548	1	2	yes	—
121	C₂×C₉	#557	1	2	yes	—
122	C₂×Q₈	Dic₁₈	1	4	yes	—
123	C₂×Q₈	D₃₆	1	4	yes	—
124	C₂×Q₈	C₂×Dic₉	1	4	yes	—
125	C₂²×C₄	—	9	2	no	—
126	C₂²×C₄	—	9	6	no	—
127	C₂²×C₄	C₂²×D₉	1	1	yes	characteristic
128	C₂²×C₄	C₂²×D₉	1	3	yes	—
129	C₂²×C₄	C₂²×D₉	1	3	yes	—
130	C₄⋊C₄	—	18	12	no	—
131	C₄⋊C₄	—	18	12	no	—
132	C₄⋊C₄	C₂×Dic₉	1	12	yes	—
133	C₄²	—	18	12	no	—
134	C₄²	C₂×Dic₉	1	12	yes	—
135	C₂²×C₃	C₂²×Dic₆	1	1	yes	characteristic
136	C₂²×C₃	C₂²×D₁₂	1	1	yes	characteristic
137	C₂²×C₃	#1007	1	2	yes	—
138	C₂²×C₃	D₃×#41	1	2	yes	—
139	C₂²×C₃	C₂³×Dic₃	1	1	yes	characteristic
140	C₃×C₄	—	2	2	no	—
141	C₃×C₄	—	2	6	no	—
142	C₃×C₄	C₂×C₄⋊Dic₃	1	12	yes	—
143	Dic₃	—	6	4	no	—
144	Dic₃	—	12	6	no	—
145	C₉	#2351	1	1	yes	characteristic, complemented, Sylow
146	C₂³	C₂³×D₉	1	1	yes	characteristic
147	Q₈	C₄⋊Dic₉	1	16	yes	complemented
148	C₂×C₄	—	2	2	no	—
149	C₂×C₄	—	2	6	no	—
150	C₂×C₄	—	9	4	no	—
151	C₂×C₄	—	18	2	no	—
152	C₂×C₄	—	18	2	no	—
153	C₂×C₄	—	18	6	no	—
154	C₂×C₄	—	18	6	no	—
155	C₂×C₄	—	18	6	no	—
156	C₂×C₄	C₂×Dic₁₈	1	6	yes	—
157	C₂×C₄	C₂×D₃₆	1	6	yes	—
158	C₂×C₄	C₂²×Dic₉	1	2	yes	—
159	C₂×C₄	C₂²×Dic₉	1	6	yes	—
160	C₂×C₄	C₂²×Dic₉	1	6	yes	—
161	C₂×C₃	#5569	1	2	yes	—
162	C₂×C₃	#5576	1	2	yes	—
163	C₂×C₃	C₂²×C₄⋊Dic₃	1	1	yes	characteristic
164	C₂×C₃	#5812	1	1	yes	characteristic
165	C₂×C₃	Dic₃×#41	1	1	yes	characteristic
166	C₂²	C₂²×Dic₁₈	1	1	yes	characteristic
167	C₂²	C₂²×D₃₆	1	1	yes	characteristic
168	C₂²	#63713	1	2	yes	—
169	C₂²	#41×D₉	1	2	yes	—
170	C₂²	C₂³×Dic₉	1	1	yes	characteristic
171	C₄	—	2	2	no	—
172	C₄	—	2	6	no	—
173	C₄	—	18	4	no	—
174	C₄	—	36	6	no	—
175	C₄	C₂×C₄⋊Dic₉	1	12	yes	—
176	C₃	#82460	1	1	yes	characteristic
177	C₂	#10589671	1	2	yes	—
178	C₂	#10589678	1	2	yes	—
179	C₂	C₂²×C₄⋊Dic₉	1	1	yes	characteristic
180	C₂	#10589914	1	1	yes	characteristic
181	C₂	#41×Dic₉	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂³×D₉
commutator	C₂²×C₉	C₂³×C₄
Frattini, socle, prime radical	C₂³×C₃	C₂³×D₃
Fitting	C₂×C₄×Q₈×C₉	C₂
cosocle	C₂³×C₉	C₂⁴
hypercenter	C₂×C₄×Q₈	D₉
hypocenter	C₉	#2351
LCS	G ▷ C₂²×C₉ ▷ C₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×Q₈

Complements

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

N	H ≅ G/N	type
8. C₄×Q₈×C₉	C₄	semidirect
49. Q₈×C₉	C₄⋊C₄	semidirect
93. C₄×Q₈	Dic₉	semidirect
145. C₉	#2351	semidirect
147. Q₈	C₄⋊Dic₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×Q₈×C₉	2
C₂×Q₈×Dic₉	2
#10588664	2
#10588616	2
C₄×C₄⋊Dic₉	2
#82460	3