Subgroups — C₄×D₄×C₉² #72244142

Order 2592.

1081 conjugacy classes of subgroups.

The subgroup lattice has 156 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₄×C₉²	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₄×C₉²	C₂	1	2	yes	maximal, complemented
3	C₉²×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
4	C₉²×C₂²⋊C₄	C₂	1	2	yes	maximal, complemented
5	C₄²×C₉²	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃×C₄×D₄×C₉	C₃	1	4	yes	maximal
7	C₂³×C₉²	C₂²	1	2	yes	—
8	D₄×C₉²	C₄	1	4	yes	direct product factor
9	C₂×C₄×C₉²	—	2	4	no	—
10	C₂×C₄×C₉²	C₂²	1	1	yes	characteristic
11	C₂×C₄×C₉²	C₂²	1	2	yes	—
12	C₂×C₄×C₉²	C₂²	1	1	yes	characteristic
13	C₂×C₄×C₉²	C₂²	1	1	yes	characteristic
14	C₂×C₃×D₄×C₉	C₂×C₃	1	4	yes	—
15	C₂²×C₃×C₄×C₉	C₂×C₃	1	8	yes	—
16	C₃×C₉×C₄⋊C₄	C₂×C₃	1	4	yes	—
17	C₃×C₉×C₂²⋊C₄	C₂×C₃	1	8	yes	—
18	C₃×C₄²×C₉	C₂×C₃	1	4	yes	—
19	C₂²×C₉²	—	2	2	no	—
20	C₂²×C₉²	—	2	2	no	—
21	C₂²×C₉²	C₂×C₄	1	4	yes	complemented
22	C₂²×C₉²	C₂³	1	1	yes	characteristic
23	C₄×C₉²	—	2	2	no	—
24	C₄×C₉²	—	2	1	no	—
25	C₄×C₉²	C₂×C₄	1	2	yes	complemented
26	C₄×C₉²	D₄	1	2	yes	direct product factor
27	C₃²×C₄×D₄	C₃²	1	1	yes	characteristic
28	C₄×D₄×C₉	C₉	1	12	yes	direct product factor
29	C₂³×C₃×C₉	C₂²×C₃	1	8	yes	—
30	C₃×D₄×C₉	C₃×C₄	1	16	yes	—
31	C₂×C₃×C₄×C₉	—	2	16	no	—
32	C₂×C₃×C₄×C₉	C₂²×C₃	1	8	yes	—
33	C₂×C₃×C₄×C₉	C₂²×C₃	1	4	yes	—
34	C₂×C₃×C₄×C₉	C₂²×C₃	1	4	yes	—
35	C₂×C₃×C₄×C₉	C₂²×C₃	1	4	yes	—
36	C₂×C₉²	—	2	4	no	—
37	C₂×C₉²	C₂²×C₄	1	1	yes	characteristic
38	C₂×C₉²	C₂×D₄	1	1	yes	characteristic
39	C₂×C₉²	#41	1	1	yes	characteristic
40	C₂×C₃²×D₄	C₂×C₃²	1	1	yes	characteristic
41	C₂²×C₃²×C₄	C₂×C₃²	1	2	yes	—
42	C₃²×C₄⋊C₄	C₂×C₃²	1	1	yes	characteristic
43	C₃²×C₂²⋊C₄	C₂×C₃²	1	2	yes	—
44	C₃²×C₄²	C₂×C₃²	1	1	yes	characteristic
45	C₂×D₄×C₉	C₂×C₉	1	12	yes	—
46	C₂²×C₄×C₉	C₂×C₉	1	24	yes	complemented
47	C₉×C₄⋊C₄	C₂×C₉	1	12	yes	complemented
48	C₉×C₂²⋊C₄	C₂×C₉	1	24	yes	complemented
49	C₄²×C₉	C₂×C₉	1	12	yes	complemented
50	C₂²×C₃×C₉	—	2	8	no	—
51	C₂²×C₃×C₉	—	2	8	no	—
52	C₂²×C₃×C₉	C₂×C₃×C₄	1	16	yes	—
53	C₂²×C₃×C₉	C₂³×C₃	1	4	yes	—
54	C₃×C₄×C₉	—	2	8	no	—
55	C₃×C₄×C₉	—	2	4	no	—
56	C₃×C₄×C₉	C₂×C₃×C₄	1	8	yes	—
57	C₃×C₄×C₉	C₃×D₄	1	8	yes	—
58	C₃×C₄×D₄	C₃×C₉	1	4	yes	—
59	C₉²	C₄×D₄	1	1	yes	characteristic, direct product factor, Sylow
60	C₂³×C₃²	C₂²×C₃²	1	2	yes	—
61	C₃²×D₄	C₃²×C₄	1	4	yes	—
62	C₂×C₃²×C₄	—	2	4	no	—
63	C₂×C₃²×C₄	C₂²×C₃²	1	1	yes	characteristic
64	C₂×C₃²×C₄	C₂²×C₃²	1	2	yes	—
65	C₂×C₃²×C₄	C₂²×C₃²	1	1	yes	characteristic
66	C₂×C₃²×C₄	C₂²×C₃²	1	1	yes	characteristic
67	C₂³×C₉	C₂²×C₉	1	24	yes	—
68	D₄×C₉	C₄×C₉	1	48	yes	direct product factor
69	C₂×C₄×C₉	—	2	48	no	—
70	C₂×C₄×C₉	C₂²×C₉	1	12	yes	—
71	C₂×C₄×C₉	C₂²×C₉	1	24	yes	—
72	C₂×C₄×C₉	C₂²×C₉	1	12	yes	—
73	C₂×C₄×C₉	C₂²×C₉	1	12	yes	—
74	C₂×C₃×C₉	—	2	16	no	—
75	C₂×C₃×C₉	C₂²×C₃×C₄	1	4	yes	—
76	C₂×C₃×C₉	C₂×C₃×D₄	1	4	yes	—
77	C₂×C₃×C₉	C₃×#41	1	4	yes	—
78	C₂×C₃×D₄	C₂×C₃×C₉	1	4	yes	—
79	C₂²×C₃×C₄	C₂×C₃×C₉	1	8	yes	—
80	C₃×C₄⋊C₄	C₂×C₃×C₉	1	4	yes	—
81	C₃×C₂²⋊C₄	C₂×C₃×C₉	1	8	yes	—
82	C₃×C₄²	C₂×C₃×C₉	1	4	yes	—
83	C₂²×C₃²	—	2	2	no	—
84	C₂²×C₃²	—	2	2	no	—
85	C₂²×C₃²	C₂×C₃²×C₄	1	4	yes	—
86	C₂²×C₃²	C₂³×C₃²	1	1	yes	characteristic
87	C₃²×C₄	—	2	2	no	—
88	C₃²×C₄	—	2	1	no	—
89	C₃²×C₄	C₂×C₃²×C₄	1	2	yes	—
90	C₃²×C₄	C₃²×D₄	1	2	yes	—
91	C₂²×C₉	—	2	24	no	—
92	C₂²×C₉	—	2	24	no	—
93	C₂²×C₉	C₂×C₄×C₉	1	48	yes	complemented
94	C₂²×C₉	C₂³×C₉	1	12	yes	—
95	C₄×C₉	—	2	24	no	—
96	C₄×C₉	—	2	12	no	—
97	C₄×C₉	C₂×C₄×C₉	1	24	yes	complemented
98	C₄×C₉	D₄×C₉	1	24	yes	direct product factor
99	C₄×D₄	C₉²	1	1	yes	characteristic, direct product factor, Sylow
100	C₃×C₉	C₃×C₄×D₄	1	4	yes	—
101	C₂³×C₃	C₂²×C₃×C₉	1	8	yes	—
102	C₃×D₄	C₃×C₄×C₉	1	16	yes	—
103	C₂×C₃×C₄	—	2	16	no	—
104	C₂×C₃×C₄	C₂²×C₃×C₉	1	4	yes	—
105	C₂×C₃×C₄	C₂²×C₃×C₉	1	8	yes	—
106	C₂×C₃×C₄	C₂²×C₃×C₉	1	4	yes	—
107	C₂×C₃×C₄	C₂²×C₃×C₉	1	4	yes	—
108	C₂×C₃²	—	2	4	no	—
109	C₂×C₃²	C₂²×C₃²×C₄	1	1	yes	characteristic
110	C₂×C₃²	C₂×C₃²×D₄	1	1	yes	characteristic
111	C₂×C₃²	C₃²×#41	1	1	yes	characteristic
112	C₂×C₉	—	2	48	no	—
113	C₂×C₉	C₂²×C₄×C₉	1	12	yes	—
114	C₂×C₉	C₂×D₄×C₉	1	12	yes	—
115	C₂×C₉	C₉×#41	1	12	yes	—
116	C₂×D₄	C₂×C₉²	1	1	yes	characteristic
117	C₂²×C₄	C₂×C₉²	1	2	yes	complemented
118	C₄⋊C₄	C₂×C₉²	1	1	yes	characteristic, complemented
119	C₂²⋊C₄	C₂×C₉²	1	2	yes	complemented
120	C₄²	C₂×C₉²	1	1	yes	characteristic, complemented
121	C₂²×C₃	—	2	8	no	—
122	C₂²×C₃	—	2	8	no	—
123	C₂²×C₃	C₂×C₃×C₄×C₉	1	16	yes	—
124	C₂²×C₃	C₂³×C₃×C₉	1	4	yes	—
125	C₃×C₄	—	2	8	no	—
126	C₃×C₄	—	2	4	no	—
127	C₃×C₄	C₂×C₃×C₄×C₉	1	8	yes	—
128	C₃×C₄	C₃×D₄×C₉	1	8	yes	—
129	C₃²	C₃²×C₄×D₄	1	1	yes	characteristic
130	C₉	C₄×D₄×C₉	1	12	yes	direct product factor
131	C₂³	C₂²×C₉²	1	2	yes	—
132	D₄	C₄×C₉²	1	4	yes	direct product factor
133	C₂×C₄	—	2	4	no	—
134	C₂×C₄	C₂²×C₉²	1	1	yes	characteristic
135	C₂×C₄	C₂²×C₉²	1	2	yes	—
136	C₂×C₄	C₂²×C₉²	1	1	yes	characteristic
137	C₂×C₄	C₂²×C₉²	1	1	yes	characteristic
138	C₂×C₃	—	2	16	no	—
139	C₂×C₃	C₂²×C₃×C₄×C₉	1	4	yes	—
140	C₂×C₃	C₂×C₃×D₄×C₉	1	4	yes	—
141	C₂×C₃	C₃×C₉×#41	1	4	yes	—
142	C₂²	—	2	2	no	—
143	C₂²	—	2	2	no	—
144	C₂²	C₂×C₄×C₉²	1	4	yes	complemented
145	C₂²	C₂³×C₉²	1	1	yes	characteristic
146	C₄	—	2	2	no	—
147	C₄	—	2	1	no	—
148	C₄	C₂×C₄×C₉²	1	2	yes	complemented
149	C₄	D₄×C₉²	1	2	yes	direct product factor
150	C₃	C₃×C₄×D₄×C₉	1	4	yes	—
151	C₂	—	2	4	no	—
152	C₂	C₂²×C₄×C₉²	1	1	yes	characteristic
153	C₂	C₂×D₄×C₉²	1	1	yes	characteristic
154	C₂	C₉²×#41	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₉²	C₂²
commutator	C₂	C₂²×C₄×C₉²
Frattini, socle, cosocle	C₂²×C₃²	C₂³×C₃²
prime radical	C₂×C₃²×D₄	C₂×C₃²
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₉² ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂²×C₄×C₉²	C₂	semidirect
3. C₉²×C₄⋊C₄	C₂	semidirect
4. C₉²×C₂²⋊C₄	C₂	semidirect
5. C₄²×C₉²	C₂	semidirect
8. D₄×C₉²	C₄	direct (N × H)
21. C₂²×C₉²	C₂×C₄	semidirect
25. C₄×C₉²	C₂×C₄	semidirect
26. C₄×C₉²	D₄	direct (N × H)
28. C₄×D₄×C₉	C₉	direct (N × H)
46. C₂²×C₄×C₉	C₂×C₉	semidirect
47. C₉×C₄⋊C₄	C₂×C₉	semidirect
48. C₉×C₂²⋊C₄	C₂×C₉	semidirect
49. C₄²×C₉	C₂×C₉	semidirect
59. C₉²	C₄×D₄	direct (N × H)
68. D₄×C₉	C₄×C₉	direct (N × H)
93. C₂²×C₉	C₂×C₄×C₉	semidirect
97. C₄×C₉	C₂×C₄×C₉	semidirect
98. C₄×C₉	D₄×C₉	direct (N × H)
99. C₄×D₄	C₉²	direct (N × H)
117. C₂²×C₄	C₂×C₉²	semidirect
118. C₄⋊C₄	C₂×C₉²	semidirect
119. C₂²⋊C₄	C₂×C₉²	semidirect
120. C₄²	C₂×C₉²	semidirect
130. C₉	C₄×D₄×C₉	direct (N × H)
132. D₄	C₄×C₉²	direct (N × H)
144. C₂²	C₂×C₄×C₉²	semidirect
148. C₄	C₂×C₄×C₉²	semidirect
149. C₄	D₄×C₉²	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×D₄×C₉²	2
C₂²×C₄×C₉²	2
C₉²×C₄⋊C₄	2
C₉²×C₂²⋊C₄	2
C₄²×C₉²	2
C₃×C₄×D₄×C₉	3

Subgroups — C4×D4×C92 #72244142

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×D₄×C₉² #72244142