Subgroups — C₂²×#63510 #11918455

Order 1152.

2256 conjugacy classes of subgroups.

The subgroup lattice has 153 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₉×#41	C₂	1	1	yes	maximal, characteristic
2	C₂×#63510	C₂	1	24	yes	maximal, direct product factor
3	C₂²×#3667	C₂	1	3	yes	maximal, complemented
4	C₂³×C₉⋊C₈	C₂	1	3	yes	maximal, complemented
5	C₂²×#948	—	3	1	no	maximal
6	C₂×C₉×#41	C₄	1	12	yes	—
7	C₂×C₉×#41	C₂²	1	12	yes	—
8	C₂²×Q₈×C₉	C₄	1	1	yes	characteristic
9	C₂²×D₄×C₉	C₄	1	3	yes	—
10	C₂³×C₄×C₉	C₂²	1	3	yes	—
11	#63510	C₂²	1	64	yes	direct product factor
12	C₂×#3667	C₂²	1	36	yes	complemented
13	C₂²×C₉⋊C₈	C₂²	1	1	yes	characteristic
14	C₂²×C₉⋊C₈	C₂²	1	3	yes	—
15	C₂²×C₉⋊C₈	C₂²	1	36	yes	complemented
16	C₂²×C₃×#41	D₃	1	1	yes	characteristic
17	C₂×#948	—	3	24	no	—
18	C₂²×#208	—	3	3	no	—
19	C₂³×C₃⋊C₈	—	3	3	no	—
20	C₂⁴×C₉	C₂×C₄	1	3	yes	—
21	C₉×#41	C₂×C₄	1	48	yes	—
22	C₉×#41	C₂³	1	16	yes	—
23	C₂×Q₈×C₉	C₂×C₄	1	12	yes	—
24	C₂×D₄×C₉	C₂×C₄	1	36	yes	—
25	C₂²×C₄×C₉	C₂×C₄	1	3	yes	—
26	C₂²×C₄×C₉	C₂×C₄	1	18	yes	—
27	C₂²×C₄×C₉	C₂³	1	1	yes	characteristic
28	C₂²×C₄×C₉	C₂³	1	18	yes	—
29	#3667	C₂³	1	48	yes	complemented
30	C₂×C₉⋊C₈	C₂³	1	6	yes	—
31	C₂×C₉⋊C₈	C₂³	1	18	yes	—
32	C₂×C₉⋊C₈	C₂³	1	48	yes	complemented
33	C₂²×#131	—	9	1	no	Sylow
34	C₂×C₃×#41	Dic₃	1	12	yes	—
35	C₂×C₃×#41	C₂×D₃	1	12	yes	—
36	C₂²×C₃×Q₈	Dic₃	1	1	yes	characteristic
37	C₂²×C₃×D₄	Dic₃	1	3	yes	—
38	C₂³×C₃×C₄	C₂×D₃	1	3	yes	—
39	#948	—	3	64	no	—
40	C₂×#208	—	3	36	no	—
41	C₂²×C₃⋊C₈	—	3	1	no	—
42	C₂²×C₃⋊C₈	—	3	3	no	—
43	C₂²×C₃⋊C₈	—	3	36	no	—
44	C₂³×C₉	—	2	12	no	—
45	C₂³×C₉	C₂²×C₄	1	18	yes	—
46	C₂³×C₉	C₂²×C₄	1	1	yes	characteristic
47	Q₈×C₉	C₂²×C₄	1	16	yes	—
48	D₄×C₉	C₂²×C₄	1	48	yes	—
49	C₂×C₄×C₉	C₂²×C₄	1	3	yes	—
50	C₂×C₄×C₉	C₂²×C₄	1	18	yes	—
51	C₂×C₄×C₉	C₂²×C₄	1	36	yes	—
52	C₂×C₄×C₉	C₂⁴	1	3	yes	—
53	C₂×C₄×C₉	C₂⁴	1	12	yes	—
54	C₉⋊C₈	C₂⁴	1	4	yes	—
55	C₉⋊C₈	C₂⁴	1	12	yes	—
56	C₂²×#41	D₉	1	1	yes	characteristic
57	C₂×#131	—	9	24	no	—
58	C₂²×M₁₆	—	9	3	no	—
59	C₂³×C₈	—	9	3	no	—
60	C₂⁴×C₃	C₂×Dic₃	1	3	yes	—
61	C₃×#41	C₂×Dic₃	1	48	yes	—
62	C₃×#41	C₂²×D₃	1	16	yes	—
63	C₂×C₃×Q₈	C₂×Dic₃	1	12	yes	—
64	C₂×C₃×D₄	C₂×Dic₃	1	36	yes	—
65	C₂²×C₃×C₄	C₂×Dic₃	1	3	yes	—
66	C₂²×C₃×C₄	C₂×Dic₃	1	18	yes	—
67	C₂²×C₃×C₄	C₂²×D₃	1	18	yes	—
68	C₂²×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
69	#208	—	3	48	no	—
70	C₂×C₃⋊C₈	—	3	6	no	—
71	C₂×C₃⋊C₈	—	3	18	no	—
72	C₂×C₃⋊C₈	—	3	48	no	—
73	C₂²×C₉	—	2	36	no	—
74	C₂²×C₉	#131	1	4	yes	complemented
75	C₂²×C₉	C₂³×C₄	1	12	yes	—
76	C₂²×C₉	C₂³×C₄	1	3	yes	—
77	C₄×C₉	C₂³×C₄	1	3	yes	—
78	C₄×C₉	C₂³×C₄	1	12	yes	—
79	C₄×C₉	C₂⁵	1	1	yes	characteristic
80	C₂×#41	Dic₉	1	12	yes	—
81	C₂×#41	C₂×D₉	1	12	yes	—
82	C₂²×Q₈	Dic₉	1	1	yes	characteristic
83	C₂²×D₄	Dic₉	1	3	yes	—
84	C₂³×C₄	C₂×D₉	1	3	yes	—
85	#131	—	9	64	no	—
86	C₂×M₁₆	—	9	36	no	—
87	C₂²×C₈	—	9	36	no	—
88	C₂²×C₈	—	9	1	no	—
89	C₂²×C₈	—	9	3	no	—
90	C₂³×C₃	—	2	12	no	—
91	C₂³×C₃	C₂²×Dic₃	1	18	yes	—
92	C₂³×C₃	C₂²×Dic₃	1	1	yes	characteristic
93	C₃×Q₈	C₂²×Dic₃	1	16	yes	—
94	C₃×D₄	C₂²×Dic₃	1	48	yes	—
95	C₂×C₃×C₄	C₂²×Dic₃	1	3	yes	—
96	C₂×C₃×C₄	C₂²×Dic₃	1	18	yes	—
97	C₂×C₃×C₄	C₂²×Dic₃	1	36	yes	—
98	C₂×C₃×C₄	C₂³×D₃	1	3	yes	—
99	C₂×C₃×C₄	C₂³×D₃	1	12	yes	—
100	C₃⋊C₈	—	3	4	no	—
101	C₃⋊C₈	—	3	12	no	—
102	C₂×C₉	—	2	12	no	—
103	C₂×C₉	C₂×#131	1	6	yes	complemented
104	C₂×C₉	C₂⁴×C₄	1	1	yes	characteristic
105	C₂⁴	C₂×Dic₉	1	3	yes	—
106	#41	C₂×Dic₉	1	48	yes	—
107	#41	C₂²×D₉	1	16	yes	—
108	C₂×Q₈	C₂×Dic₉	1	12	yes	—
109	C₂×D₄	C₂×Dic₉	1	36	yes	—
110	C₂²×C₄	C₂×Dic₉	1	3	yes	—
111	C₂²×C₄	C₂×Dic₉	1	18	yes	—
112	C₂²×C₄	C₂²×D₉	1	18	yes	—
113	C₂²×C₄	C₂²×D₉	1	1	yes	characteristic
114	M₁₆	—	9	48	no	—
115	C₂×C₈	—	9	6	no	—
116	C₂×C₈	—	9	18	no	—
117	C₂×C₈	—	9	48	no	—
118	C₂²×C₃	—	2	36	no	—
119	C₂²×C₃	#948	1	4	yes	—
120	C₂²×C₃	C₂³×Dic₃	1	12	yes	—
121	C₂²×C₃	C₂³×Dic₃	1	3	yes	—
122	C₃×C₄	C₂³×Dic₃	1	3	yes	—
123	C₃×C₄	C₂³×Dic₃	1	12	yes	—
124	C₃×C₄	C₂⁴×D₃	1	1	yes	characteristic
125	C₉	C₂²×#131	1	1	yes	characteristic, complemented, Sylow
126	C₂³	—	2	12	no	—
127	C₂³	C₂²×Dic₉	1	18	yes	—
128	C₂³	C₂²×Dic₉	1	1	yes	characteristic
129	Q₈	C₂²×Dic₉	1	16	yes	—
130	D₄	C₂²×Dic₉	1	48	yes	—
131	C₂×C₄	C₂²×Dic₉	1	3	yes	—
132	C₂×C₄	C₂²×Dic₉	1	18	yes	—
133	C₂×C₄	C₂²×Dic₉	1	36	yes	—
134	C₂×C₄	C₂³×D₉	1	3	yes	—
135	C₂×C₄	C₂³×D₉	1	12	yes	—
136	C₈	—	9	4	no	—
137	C₈	—	9	12	no	—
138	C₂×C₃	—	2	12	no	—
139	C₂×C₃	C₂×#948	1	6	yes	—
140	C₂×C₃	C₂⁴×Dic₃	1	1	yes	characteristic
141	C₂²	—	2	36	no	—
142	C₂²	#63510	1	4	yes	direct product factor
143	C₂²	C₂³×Dic₉	1	12	yes	—
144	C₂²	C₂³×Dic₉	1	3	yes	—
145	C₄	C₂³×Dic₉	1	3	yes	—
146	C₄	C₂³×Dic₉	1	12	yes	—
147	C₄	C₂⁴×D₉	1	1	yes	characteristic
148	C₃	C₂²×#948	1	1	yes	characteristic
149	C₂	—	2	12	no	—
150	C₂	C₂×#63510	1	6	yes	direct product factor
151	C₂	C₂⁴×Dic₉	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂²×D₉
commutator	C₂×C₉	C₂⁴×C₄
Frattini	C₃×C₄	C₂⁴×D₃
Fitting	C₂²×C₉×#41	C₂
socle	C₂³×C₃	C₂²×Dic₃
prime radical	C₂²×C₃×#41	D₃
cosocle	C₄×C₉	C₂⁵
hypercenter	C₂²×#41	D₉
hypocenter	C₉	C₂²×#131
LCS	G ▷ C₂×C₉ ▷ C₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×#41

Complements

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

N	H ≅ G/N	type
2. C₂×#63510	C₂	direct (N × H)
3. C₂²×#3667	C₂	semidirect
4. C₂³×C₉⋊C₈	C₂	semidirect
11. #63510	C₂²	direct (N × H)
12. C₂×#3667	C₂²	semidirect
15. C₂²×C₉⋊C₈	C₂²	semidirect
29. #3667	C₂³	semidirect
32. C₂×C₉⋊C₈	C₂³	semidirect
74. C₂²×C₉	#131	semidirect
103. C₂×C₉	C₂×#131	semidirect
125. C₉	C₂²×#131	semidirect
142. C₂²	#63510	direct (N × H)
150. C₂	C₂×#63510	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₉×#41	2
C₂×#63510	2
C₂²×#3667	2
C₂³×C₉⋊C₈	2
C₂²×#948	3

Subgroups — C22×#63510 #11918455

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#63510 #11918455