Subgroups — C₂²×#63372 #11858611

Order 1152.

720 conjugacy classes of subgroups.

The subgroup lattice has 165 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₈×C₉	C₂	1	1	yes	maximal, characteristic
2	C₂²×#3667	C₂	1	2	yes	maximal
3	C₂×#63372	C₂	1	12	yes	maximal, direct product factor
4	C₂²×#819	—	3	1	no	maximal
5	C₂³×C₄×C₉	C₂²	1	1	yes	characteristic
6	C₂²×C₈×C₉	C₄	1	6	yes	—
7	C₂²×C₈×C₉	C₄	1	2	yes	—
8	C₂²×C₈×C₉	C₂²	1	6	yes	—
9	C₂×#3667	—	2	6	no	—
10	C₂×#3667	C₂²	1	12	yes	—
11	C₂²×C₉⋊C₈	—	2	2	no	—
12	#63372	C₂²	1	16	yes	direct product factor
13	C₂³×C₃×C₈	D₃	1	1	yes	characteristic
14	C₂²×#208	—	3	2	no	—
15	C₂×#819	—	3	12	no	—
16	C₂⁴×C₉	Q₈	1	1	yes	characteristic
17	C₂²×C₄×C₉	C₂×C₄	1	3	yes	—
18	C₂²×C₄×C₉	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₄×C₉	D₄	1	1	yes	characteristic
20	C₂²×C₄×C₉	D₄	1	3	yes	—
21	C₂²×C₄×C₉	Q₈	1	3	yes	—
22	C₂²×C₄×C₉	C₂³	1	3	yes	—
23	C₂×C₈×C₉	C₂×C₄	1	12	yes	—
24	C₂×C₈×C₉	C₂×C₄	1	12	yes	—
25	C₂×C₈×C₉	C₂³	1	4	yes	—
26	#3667	—	2	12	no	—
27	#3667	C₂³	1	8	yes	—
28	C₂×C₉⋊C₈	—	2	12	no	—
29	C₂²×#108	—	9	1	no	Sylow
30	C₂³×C₃×C₄	C₂×D₃	1	1	yes	characteristic
31	C₂²×C₃×C₈	Dic₃	1	6	yes	—
32	C₂²×C₃×C₈	Dic₃	1	2	yes	—
33	C₂²×C₃×C₈	C₂×D₃	1	6	yes	—
34	C₂×#208	—	3	12	no	—
35	C₂×#208	—	6	6	no	—
36	C₂²×C₃⋊C₈	—	6	2	no	—
37	#819	—	3	16	no	—
38	C₂³×C₉	—	2	4	no	—
39	C₂³×C₉	C₄⋊C₄	1	3	yes	—
40	C₂³×C₉	C₄⋊C₄	1	1	yes	characteristic
41	C₂³×C₉	C₂×Q₈	1	3	yes	—
42	C₂×C₄×C₉	C₄⋊C₄	1	6	yes	—
43	C₂×C₄×C₉	C₄⋊C₄	1	3	yes	—
44	C₂×C₄×C₉	C₄⋊C₄	1	3	yes	—
45	C₂×C₄×C₉	C₂²×C₄	1	3	yes	—
46	C₂×C₄×C₉	C₂²×C₄	1	3	yes	—
47	C₂×C₄×C₉	C₂×D₄	1	3	yes	—
48	C₂×C₄×C₉	C₂×D₄	1	3	yes	—
49	C₂×C₄×C₉	C₂×Q₈	1	3	yes	—
50	C₂×C₄×C₉	C₂⁴	1	1	yes	characteristic
51	C₈×C₉	C₂²×C₄	1	8	yes	—
52	C₉⋊C₈	—	2	8	no	—
53	C₂²×M₁₆	—	9	2	no	—
54	C₂³×C₈	D₉	1	1	yes	characteristic
55	C₂×#108	—	9	12	no	—
56	C₂⁴×C₃	Dic₆	1	1	yes	characteristic
57	C₂²×C₃×C₄	Dic₆	1	3	yes	—
58	C₂²×C₃×C₄	D₁₂	1	3	yes	—
59	C₂²×C₃×C₄	D₁₂	1	1	yes	characteristic
60	C₂²×C₃×C₄	C₂×Dic₃	1	3	yes	—
61	C₂²×C₃×C₄	C₂×Dic₃	1	1	yes	characteristic
62	C₂²×C₃×C₄	C₂²×D₃	1	3	yes	—
63	C₂×C₃×C₈	C₂×Dic₃	1	12	yes	—
64	C₂×C₃×C₈	C₂×Dic₃	1	12	yes	—
65	C₂×C₃×C₈	C₂²×D₃	1	4	yes	—
66	#208	—	3	8	no	—
67	#208	—	6	12	no	—
68	C₂×C₃⋊C₈	—	6	12	no	—
69	C₂²×C₉	—	2	6	no	—
70	C₂²×C₉	—	2	6	no	—
71	C₂²×C₉	#108	1	4	yes	complemented
72	C₂²×C₉	C₂×C₄⋊C₄	1	3	yes	—
73	C₂²×C₉	C₂×C₄⋊C₄	1	3	yes	—
74	C₂²×C₉	C₂²×Q₈	1	1	yes	characteristic
75	C₄×C₉	C₂×C₄⋊C₄	1	3	yes	—
76	C₄×C₉	C₂×C₄⋊C₄	1	3	yes	—
77	C₄×C₉	C₂³×C₄	1	1	yes	characteristic
78	C₄×C₉	C₂²×D₄	1	1	yes	characteristic
79	C₂³×C₄	C₂×D₉	1	1	yes	characteristic
80	C₂×M₁₆	—	9	12	no	—
81	C₂×M₁₆	—	18	6	no	—
82	C₂²×C₈	—	18	2	no	—
83	C₂²×C₈	Dic₉	1	2	yes	—
84	C₂²×C₈	Dic₉	1	6	yes	—
85	C₂²×C₈	C₂×D₉	1	6	yes	—
86	#108	—	9	16	no	—
87	C₂³×C₃	—	2	4	no	—
88	C₂³×C₃	C₄⋊Dic₃	1	3	yes	—
89	C₂³×C₃	C₄⋊Dic₃	1	1	yes	characteristic
90	C₂³×C₃	C₂×Dic₆	1	3	yes	—
91	C₂×C₃×C₄	C₄⋊Dic₃	1	6	yes	—
92	C₂×C₃×C₄	C₄⋊Dic₃	1	3	yes	—
93	C₂×C₃×C₄	C₄⋊Dic₃	1	3	yes	—
94	C₂×C₃×C₄	C₂×Dic₆	1	3	yes	—
95	C₂×C₃×C₄	C₂×D₁₂	1	3	yes	—
96	C₂×C₃×C₄	C₂×D₁₂	1	3	yes	—
97	C₂×C₃×C₄	C₂²×Dic₃	1	3	yes	—
98	C₂×C₃×C₄	C₂²×Dic₃	1	3	yes	—
99	C₂×C₃×C₄	C₂³×D₃	1	1	yes	characteristic
100	C₃×C₈	C₂²×Dic₃	1	8	yes	—
101	C₃⋊C₈	—	6	8	no	—
102	C₂×C₉	—	2	3	no	—
103	C₂×C₉	—	2	1	no	—
104	C₂×C₉	C₂×#108	1	6	yes	complemented
105	C₂×C₉	C₂²×C₄⋊C₄	1	1	yes	characteristic
106	C₂⁴	Dic₁₈	1	1	yes	characteristic
107	C₂²×C₄	Dic₁₈	1	3	yes	—
108	C₂²×C₄	D₃₆	1	3	yes	—
109	C₂²×C₄	D₃₆	1	1	yes	characteristic
110	C₂²×C₄	C₂×Dic₉	1	1	yes	characteristic
111	C₂²×C₄	C₂×Dic₉	1	3	yes	—
112	C₂²×C₄	C₂²×D₉	1	3	yes	—
113	M₁₆	—	9	8	no	—
114	M₁₆	—	18	12	no	—
115	C₂×C₈	—	18	12	no	—
116	C₂×C₈	C₂×Dic₉	1	12	yes	—
117	C₂×C₈	C₂×Dic₉	1	12	yes	—
118	C₂×C₈	C₂²×D₉	1	4	yes	—
119	C₂²×C₃	—	2	6	no	—
120	C₂²×C₃	—	2	6	no	—
121	C₂²×C₃	#819	1	4	yes	—
122	C₂²×C₃	C₂×C₄⋊Dic₃	1	3	yes	—
123	C₂²×C₃	C₂×C₄⋊Dic₃	1	3	yes	—
124	C₂²×C₃	C₂²×Dic₆	1	1	yes	characteristic
125	C₃×C₄	C₂×C₄⋊Dic₃	1	3	yes	—
126	C₃×C₄	C₂×C₄⋊Dic₃	1	3	yes	—
127	C₃×C₄	C₂²×D₁₂	1	1	yes	characteristic
128	C₃×C₄	C₂³×Dic₃	1	1	yes	characteristic
129	C₉	C₂²×#108	1	1	yes	characteristic, complemented, Sylow
130	C₂³	—	2	4	no	—
131	C₂³	C₄⋊Dic₉	1	3	yes	—
132	C₂³	C₄⋊Dic₉	1	1	yes	characteristic
133	C₂³	C₂×Dic₁₈	1	3	yes	—
134	C₂×C₄	C₄⋊Dic₉	1	6	yes	—
135	C₂×C₄	C₄⋊Dic₉	1	3	yes	—
136	C₂×C₄	C₄⋊Dic₉	1	3	yes	—
137	C₂×C₄	C₂×Dic₁₈	1	3	yes	—
138	C₂×C₄	C₂×D₃₆	1	3	yes	—
139	C₂×C₄	C₂×D₃₆	1	3	yes	—
140	C₂×C₄	C₂²×Dic₉	1	3	yes	—
141	C₂×C₄	C₂²×Dic₉	1	3	yes	—
142	C₂×C₄	C₂³×D₉	1	1	yes	characteristic
143	C₈	—	18	8	no	—
144	C₈	C₂²×Dic₉	1	8	yes	—
145	C₂×C₃	—	2	3	no	—
146	C₂×C₃	—	2	1	no	—
147	C₂×C₃	C₂×#819	1	6	yes	—
148	C₂×C₃	C₂²×C₄⋊Dic₃	1	1	yes	characteristic
149	C₂²	—	2	6	no	—
150	C₂²	—	2	6	no	—
151	C₂²	#63372	1	4	yes	direct product factor
152	C₂²	C₂×C₄⋊Dic₉	1	3	yes	—
153	C₂²	C₂×C₄⋊Dic₉	1	3	yes	—
154	C₂²	C₂²×Dic₁₈	1	1	yes	characteristic
155	C₄	C₂×C₄⋊Dic₉	1	3	yes	—
156	C₄	C₂×C₄⋊Dic₉	1	3	yes	—
157	C₄	C₂²×D₃₆	1	1	yes	characteristic
158	C₄	C₂³×Dic₉	1	1	yes	characteristic
159	C₃	C₂²×#819	1	1	yes	characteristic
160	C₂	—	2	1	no	—
161	C₂	—	2	3	no	—
162	C₂	C₂×#63372	1	6	yes	direct product factor
163	C₂	C₂²×C₄⋊Dic₉	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	D₃₆
commutator	C₄×C₉	C₂³×C₄
Frattini	C₂×C₃×C₄	C₂³×D₃
Fitting	C₂³×C₈×C₉	C₂
socle	C₂³×C₃	C₄⋊Dic₃
prime radical	C₂⁴×C₃	Dic₆
cosocle	C₂×C₄×C₉	C₂⁴
hypercenter	C₂³×C₈	D₉
hypocenter	C₉	C₂²×#108
LCS	G ▷ C₄×C₉ ▷ C₂×C₉ ▷ C₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₂³×C₄ ◁ C₂³×C₈

Complements

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

N	H ≅ G/N	type
3. C₂×#63372	C₂	direct (N × H)
12. #63372	C₂²	direct (N × H)
71. C₂²×C₉	#108	semidirect
104. C₂×C₉	C₂×#108	semidirect
129. C₉	C₂²×#108	semidirect
151. C₂²	#63372	direct (N × H)
162. C₂	C₂×#63372	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×C₈×C₉	2
C₂²×#3667	2
C₂×#63372	2
C₂²×#819	3

Subgroups — C22×#63372 #11858611

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#63372 #11858611