Subgroups — #11800875

Order 1152.

1044 conjugacy classes of subgroups.

The subgroup lattice has 180 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₉×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic
2	C₂×#63488	C₂	1	2	yes	maximal
3	C₄×#3676	C₂	1	4	yes	maximal
4	C₄⋊C₄×Dic₉	C₂	1	8	yes	maximal, complemented
5	#82144	—	3	1	no	maximal
6	C₂³×C₄×C₉	C₂²	1	2	yes	—
7	C₂³×C₄×C₉	C₂²	1	1	yes	characteristic
8	C₂×C₉×C₄⋊C₄	C₄	1	8	yes	complemented
9	C₂×C₉×C₄⋊C₄	C₂²	1	4	yes	—
10	C₂×#3676	C₂²	1	4	yes	—
11	#63488	C₄	1	16	yes	complemented
12	C₂×C₄⋊Dic₉	C₂²	1	4	yes	—
13	C₂×C₄×Dic₉	C₂²	1	8	yes	—
14	C₂×C₄×Dic₉	C₂²	1	4	yes	—
15	C₉⋊#95	C₂²	1	8	yes	—
16	C₂²×C₃×C₄⋊C₄	D₃	1	1	yes	characteristic
17	C₂×#926	—	3	2	no	—
18	C₄×#217	—	3	4	no	—
19	Dic₃×C₄⋊C₄	—	3	8	no	—
20	C₂⁴×C₉	C₂³	1	1	yes	characteristic
21	C₂²×C₄×C₉	—	4	4	no	—
22	C₂²×C₄×C₉	C₂×C₄	1	8	yes	—
23	C₂²×C₄×C₉	C₂×C₄	1	8	yes	—
24	C₂²×C₄×C₉	C₂×C₄	1	4	yes	—
25	C₂²×C₄×C₉	C₂³	1	4	yes	—
26	C₂²×C₄×C₉	C₂³	1	2	yes	—
27	C₂²×Dic₉	C₂³	1	8	yes	—
28	C₉×C₄⋊C₄	C₂×C₄	1	16	yes	—
29	#3676	C₂×C₄	1	16	yes	—
30	C₄⋊Dic₉	C₂×C₄	1	16	yes	complemented
31	C₄×Dic₉	—	4	8	no	—
32	C₄×Dic₉	C₂×C₄	1	16	yes	complemented
33	#2274	—	9	1	no	Sylow
34	C₂³×C₃×C₄	C₂×D₃	1	2	yes	—
35	C₂³×C₃×C₄	C₂×D₃	1	1	yes	characteristic
36	C₂×C₃×C₄⋊C₄	Dic₃	1	8	yes	—
37	C₂×C₃×C₄⋊C₄	C₂×D₃	1	4	yes	—
38	C₂×#217	—	3	4	no	—
39	#926	—	3	16	no	—
40	C₂×C₄⋊Dic₃	—	3	4	no	—
41	C₂×C₄×Dic₃	—	3	4	no	—
42	C₂×C₄×Dic₃	—	3	8	no	—
43	C₃⋊#95	—	3	8	no	—
44	C₂³×C₉	—	2	2	no	—
45	C₂³×C₉	—	2	2	no	—
46	C₂³×C₉	C₂²×C₄	1	4	yes	—
47	C₂³×C₉	C₂²×C₄	1	2	yes	—
48	C₂³×C₉	C₂⁴	1	1	yes	characteristic
49	C₂×C₄×C₉	—	2	8	no	—
50	C₂×C₄×C₉	—	4	8	no	—
51	C₂×C₄×C₉	C₄²	1	16	yes	complemented
52	C₂×C₄×C₉	C₂²×C₄	1	4	yes	—
53	C₂×C₄×C₉	C₂²×C₄	1	4	yes	—
54	C₂×C₄×C₉	C₂²×C₄	1	8	yes	—
55	C₂×C₄×C₉	C₂²×C₄	1	4	yes	—
56	C₂×Dic₉	—	2	8	no	—
57	C₂×Dic₉	—	2	8	no	—
58	C₂×Dic₉	C₂²×C₄	1	16	yes	—
59	C₂×#117	—	9	2	no	—
60	C₂²×C₄⋊C₄	D₉	1	1	yes	characteristic
61	C₄×C₄⋊C₄	—	9	8	no	—
62	C₄×C₂²⋊C₄	—	9	4	no	—
63	C₂⁴×C₃	C₂²×D₃	1	1	yes	characteristic
64	C₂²×C₃×C₄	—	4	4	no	—
65	C₂²×C₃×C₄	C₄×D₃	1	8	yes	—
66	C₂²×C₃×C₄	C₂×Dic₃	1	8	yes	—
67	C₂²×C₃×C₄	C₂×Dic₃	1	4	yes	—
68	C₂²×C₃×C₄	C₂²×D₃	1	4	yes	—
69	C₂²×C₃×C₄	C₂²×D₃	1	2	yes	—
70	C₂²×Dic₃	—	3	8	no	—
71	C₃×C₄⋊C₄	C₂×Dic₃	1	16	yes	—
72	#217	—	3	16	no	—
73	C₄⋊Dic₃	—	3	16	no	—
74	C₄×Dic₃	—	3	16	no	—
75	C₄×Dic₃	—	12	8	no	—
76	C₂²×C₉	—	2	2	no	—
77	C₂²×C₉	—	2	2	no	—
78	C₂²×C₉	—	2	8	no	—
79	C₂²×C₉	C₂×C₄²	1	4	yes	—
80	C₂²×C₉	C₂³×C₄	1	2	yes	—
81	C₂²×C₉	C₂³×C₄	1	1	yes	characteristic
82	C₂²×C₉	H(5,𝔽₂)	1	2	yes	—
83	C₂²×C₉	2₋¹⁺⁴	1	2	yes	—
84	C₄×C₉	—	2	8	no	—
85	C₄×C₉	C₂×C₄²	1	8	yes	—
86	Dic₉	—	2	16	no	—
87	C₂³×C₄	C₂×D₉	1	2	yes	—
88	C₂³×C₄	C₂×D₉	1	1	yes	characteristic
89	#117	—	9	16	no	—
90	C₂×C₄⋊C₄	—	9	4	no	—
91	C₂×C₄⋊C₄	Dic₉	1	8	yes	complemented
92	C₂×C₄⋊C₄	C₂×D₉	1	4	yes	—
93	C₂×C₂²⋊C₄	—	9	4	no	—
94	C₂×C₄²	—	9	4	no	—
95	C₂×C₄²	—	9	8	no	—
96	#95	—	9	8	no	—
97	C₂³×C₃	—	2	2	no	—
98	C₂³×C₃	—	2	2	no	—
99	C₂³×C₃	C₂×C₄×D₃	1	4	yes	—
100	C₂³×C₃	C₂²×Dic₃	1	2	yes	—
101	C₂³×C₃	C₂³×D₃	1	1	yes	characteristic
102	C₂×C₃×C₄	—	2	8	no	—
103	C₂×C₃×C₄	—	4	8	no	—
104	C₂×C₃×C₄	C₄×Dic₃	1	16	yes	—
105	C₂×C₃×C₄	C₂×C₄×D₃	1	4	yes	—
106	C₂×C₃×C₄	C₂×C₄×D₃	1	4	yes	—
107	C₂×C₃×C₄	C₂²×Dic₃	1	8	yes	—
108	C₂×C₃×C₄	C₂²×Dic₃	1	4	yes	—
109	C₂×Dic₃	—	3	16	no	—
110	C₂×Dic₃	—	6	8	no	—
111	C₂×Dic₃	—	6	8	no	—
112	C₂×C₉	—	2	4	no	—
113	C₂×C₉	C₂²×C₄²	1	1	yes	characteristic
114	C₂×C₉	#518	1	1	yes	characteristic
115	C₂×C₉	#519	1	1	yes	characteristic
116	C₂×C₉	#520	1	4	yes	—
117	C₂⁴	C₂²×D₉	1	1	yes	characteristic
118	C₂²×C₄	—	4	4	no	—
119	C₂²×C₄	—	9	8	no	—
120	C₂²×C₄	C₄×D₉	1	8	yes	—
121	C₂²×C₄	C₂×Dic₉	1	8	yes	—
122	C₂²×C₄	C₂×Dic₉	1	4	yes	—
123	C₂²×C₄	C₂²×D₉	1	4	yes	—
124	C₂²×C₄	C₂²×D₉	1	2	yes	—
125	C₄⋊C₄	—	9	16	no	—
126	C₄⋊C₄	C₂×Dic₉	1	16	yes	—
127	C₂²⋊C₄	—	9	16	no	—
128	C₄²	—	9	16	no	—
129	C₄²	—	36	8	no	—
130	C₂²×C₃	—	2	8	no	—
131	C₂²×C₃	—	2	2	no	—
132	C₂²×C₃	—	2	2	no	—
133	C₂²×C₃	C₂×C₄×Dic₃	1	4	yes	—
134	C₂²×C₃	C₂²×C₄×D₃	1	2	yes	—
135	C₂²×C₃	#1004	1	2	yes	—
136	C₂²×C₃	#1007	1	2	yes	—
137	C₂²×C₃	C₂³×Dic₃	1	1	yes	characteristic
138	C₃×C₄	—	2	8	no	—
139	C₃×C₄	C₂×C₄×Dic₃	1	8	yes	—
140	Dic₃	—	6	16	no	—
141	C₉	#2274	1	1	yes	characteristic, complemented, Sylow
142	C₂³	—	2	2	no	—
143	C₂³	—	2	2	no	—
144	C₂³	C₂×C₄×D₉	1	4	yes	—
145	C₂³	C₂²×Dic₉	1	2	yes	—
146	C₂³	C₂³×D₉	1	1	yes	characteristic
147	C₂×C₄	—	2	8	no	—
148	C₂×C₄	—	4	8	no	—
149	C₂×C₄	—	9	16	no	—
150	C₂×C₄	—	18	8	no	—
151	C₂×C₄	—	18	8	no	—
152	C₂×C₄	C₄×Dic₉	1	16	yes	complemented
153	C₂×C₄	C₂×C₄×D₉	1	4	yes	—
154	C₂×C₄	C₂×C₄×D₉	1	4	yes	—
155	C₂×C₄	C₂²×Dic₉	1	8	yes	—
156	C₂×C₄	C₂²×Dic₉	1	4	yes	—
157	C₂×C₃	—	2	4	no	—
158	C₂×C₃	#5504	1	4	yes	—
159	C₂×C₃	C₂²×C₄×Dic₃	1	1	yes	characteristic
160	C₂×C₃	#5798	1	1	yes	characteristic
161	C₂×C₃	#5812	1	1	yes	characteristic
162	C₂²	—	2	8	no	—
163	C₂²	—	2	2	no	—
164	C₂²	—	2	2	no	—
165	C₂²	C₂×C₄×Dic₉	1	4	yes	—
166	C₂²	C₂²×C₄×D₉	1	2	yes	—
167	C₂²	#63710	1	2	yes	—
168	C₂²	#63713	1	2	yes	—
169	C₂²	C₂³×Dic₉	1	1	yes	characteristic
170	C₄	—	2	8	no	—
171	C₄	—	18	16	no	—
172	C₄	C₂×C₄×Dic₉	1	8	yes	—
173	C₃	#82144	1	1	yes	characteristic
174	C₂	—	2	4	no	—
175	C₂	#10589606	1	4	yes	—
176	C₂	C₂²×C₄×Dic₉	1	1	yes	characteristic
177	C₂	#10589900	1	1	yes	characteristic
178	C₂	#10589914	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂³×D₉
commutator	C₂×C₉	C₂²×C₄²
Frattini, socle	C₂³×C₃	C₂³×D₃
Fitting	C₂²×C₉×C₄⋊C₄	C₂
prime radical	C₂⁴×C₃	C₂²×D₃
cosocle	C₂³×C₉	C₂⁴
hypercenter	C₂²×C₄⋊C₄	D₉
hypocenter	C₉	#2274
LCS	G ▷ C₂×C₉ ▷ C₉
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
4. C₄⋊C₄×Dic₉	C₂	semidirect
8. C₂×C₉×C₄⋊C₄	C₄	semidirect
11. #63488	C₄	semidirect
30. C₄⋊Dic₉	C₂×C₄	semidirect
32. C₄×Dic₉	C₂×C₄	semidirect
51. C₂×C₄×C₉	C₄²	semidirect
91. C₂×C₄⋊C₄	Dic₉	semidirect
141. C₉	#2274	semidirect
152. C₂×C₄	C₄×Dic₉	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₉×C₄⋊C₄	2
C₂×#63488	2
C₄×#3676	2
C₄⋊C₄×Dic₉	2
#82144	3