Subgroups — C₂×C₄⋊(C₃⋊(C₃⋊C₁₆)) #11796972

Order 1152.

552 conjugacy classes of subgroups.

The subgroup lattice has 150 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃²×C₄×C₈	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₃⋊(C₃⋊C₁₆)	C₂	1	2	yes	maximal
3	C₄⋊(C₃⋊(C₃⋊C₁₆))	C₂	1	4	yes	maximal, direct product factor
4	C₂×C₄⋊(C₃⋊C₁₆)	—	3	4	no	maximal
5	C₂²×C₃²×C₈	C₄	1	1	yes	characteristic
6	C₂²×C₃²×C₈	C₂²	1	1	yes	characteristic
7	C₂×C₃²×C₄²	C₄	1	1	yes	characteristic
8	C₃²×C₄×C₈	C₄	1	2	yes	—
9	C₃²×C₄×C₈	C₂²	1	2	yes	—
10	C₂×C₃⋊(C₃⋊C₁₆)	—	2	4	no	—
11	C₂×C₃⋊(C₃⋊C₁₆)	C₂²	1	4	yes	—
12	C₂×C₃×C₄×C₈	D₃	1	4	yes	—
13	C₂²×C₃⋊C₁₆	—	3	8	no	—
14	C₄⋊(C₃⋊C₁₆)	—	3	16	no	—
15	C₂²×C₃²×C₄	C₈	1	1	yes	characteristic
16	C₂²×C₃²×C₄	C₈	1	1	yes	characteristic
17	C₂²×C₃²×C₄	C₂×C₄	1	1	yes	characteristic
18	C₂×C₃²×C₈	—	2	2	no	—
19	C₂×C₃²×C₈	C₂×C₄	1	2	yes	—
20	C₂×C₃²×C₈	C₂×C₄	1	1	yes	characteristic
21	C₂×C₃²×C₈	D₄	1	2	yes	—
22	C₂×C₃²×C₈	Q₈	1	2	yes	—
23	C₂×C₃²×C₈	C₂³	1	1	yes	characteristic
24	C₃²×C₄²	C₈	1	2	yes	—
25	C₃²×C₄²	C₂×C₄	1	2	yes	—
26	C₃⋊(C₃⋊C₁₆)	—	2	4	no	—
27	C₂×C₄⋊C₁₆	—	9	1	no	Sylow
28	C₂²×C₃×C₈	Dic₃	1	4	yes	—
29	C₂²×C₃×C₈	C₂×D₃	1	4	yes	—
30	C₂×C₃×C₄²	Dic₃	1	4	yes	—
31	C₃×C₄×C₈	Dic₃	1	8	yes	—
32	C₃×C₄×C₈	C₂×D₃	1	8	yes	—
33	C₂×C₃⋊C₁₆	—	3	16	no	—
34	C₂×C₃⋊C₁₆	—	6	16	no	—
35	C₂³×C₃²	C₂×C₈	1	1	yes	characteristic
36	C₂×C₃²×C₄	—	2	2	no	—
37	C₂×C₃²×C₄	C₁₆	1	4	yes	complemented
38	C₂×C₃²×C₄	C₄⋊C₄	1	2	yes	—
39	C₂×C₃²×C₄	C₂×C₈	1	2	yes	—
40	C₂×C₃²×C₄	C₂×C₈	1	1	yes	characteristic
41	C₂×C₃²×C₄	C₂×C₈	1	2	yes	—
42	C₂×C₃²×C₄	M₁₆	1	2	yes	—
43	C₂×C₃²×C₄	C₂²×C₄	1	1	yes	characteristic
44	C₃²×C₈	—	2	2	no	—
45	C₃²×C₈	C₄⋊C₄	1	2	yes	—
46	C₃²×C₈	C₂×D₄	1	1	yes	characteristic
47	C₃²×C₈	C₂×Q₈	1	1	yes	characteristic
48	C₂²×C₁₆	—	9	2	no	—
49	C₂×C₄×C₈	C₃⋊D₃	1	1	yes	characteristic
50	C₄⋊C₁₆	—	9	4	no	—
51	C₂²×C₃×C₄	C₃⋊C₈	1	4	yes	—
52	C₂²×C₃×C₄	C₃⋊C₈	1	4	yes	—
53	C₂²×C₃×C₄	C₂×Dic₃	1	4	yes	—
54	C₂×C₃×C₈	—	2	8	no	—
55	C₂×C₃×C₈	Dic₆	1	8	yes	—
56	C₂×C₃×C₈	D₁₂	1	8	yes	—
57	C₂×C₃×C₈	C₂×Dic₃	1	8	yes	—
58	C₂×C₃×C₈	C₂×Dic₃	1	4	yes	—
59	C₂×C₃×C₈	C₂²×D₃	1	4	yes	—
60	C₃×C₄²	C₃⋊C₈	1	8	yes	—
61	C₃×C₄²	C₂×Dic₃	1	8	yes	—
62	C₃⋊C₁₆	—	6	16	no	—
63	C₂²×C₃²	C₄⋊C₈	1	2	yes	—
64	C₂²×C₃²	C₂×C₁₆	1	2	yes	—
65	C₂²×C₃²	M₃₂	1	2	yes	—
66	C₂²×C₃²	C₂²×C₈	1	1	yes	characteristic
67	C₃²×C₄	—	2	2	no	—
68	C₃²×C₄	C₄⋊C₈	1	2	yes	—
69	C₃²×C₄	C₂×C₁₆	1	4	yes	complemented
70	C₃²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
71	C₃²×C₄	C₂×M₁₆	1	1	yes	characteristic
72	C₂²×C₈	C₃⋊Dic₃	1	1	yes	characteristic
73	C₂²×C₈	C₂×C₃⋊D₃	1	1	yes	characteristic
74	C₂×C₄²	C₃⋊Dic₃	1	1	yes	characteristic
75	C₂×C₁₆	—	9	4	no	—
76	C₂×C₁₆	—	18	4	no	—
77	C₄×C₈	C₃⋊Dic₃	1	2	yes	—
78	C₄×C₈	C₂×C₃⋊D₃	1	2	yes	—
79	C₂³×C₃	C₂×C₃⋊C₈	1	4	yes	—
80	C₂×C₃×C₄	—	2	8	no	—
81	C₂×C₃×C₄	C₃⋊C₁₆	1	16	yes	complemented
82	C₂×C₃×C₄	C₂×C₃⋊C₈	1	8	yes	—
83	C₂×C₃×C₄	C₂×C₃⋊C₈	1	4	yes	—
84	C₂×C₃×C₄	C₂×C₃⋊C₈	1	8	yes	—
85	C₂×C₃×C₄	#208	1	8	yes	—
86	C₂×C₃×C₄	C₄⋊Dic₃	1	8	yes	—
87	C₂×C₃×C₄	C₂²×Dic₃	1	4	yes	—
88	C₃×C₈	—	2	8	no	—
89	C₃×C₈	C₄⋊Dic₃	1	8	yes	—
90	C₃×C₈	C₂×Dic₆	1	4	yes	—
91	C₃×C₈	C₂×D₁₂	1	4	yes	—
92	C₂×C₃²	C₄⋊C₁₆	1	4	yes	complemented
93	C₂×C₃²	C₂×C₄⋊C₈	1	1	yes	characteristic
94	C₂×C₃²	C₂²×C₁₆	1	1	yes	characteristic
95	C₂×C₃²	C₂×M₃₂	1	1	yes	characteristic
96	C₂²×C₄	C₃⋊(C₃⋊C₈)	1	1	yes	characteristic
97	C₂²×C₄	C₃⋊(C₃⋊C₈)	1	1	yes	characteristic
98	C₂²×C₄	C₂×C₃⋊Dic₃	1	1	yes	characteristic
99	C₂×C₈	—	2	2	no	—
100	C₂×C₈	#634	1	2	yes	—
101	C₂×C₈	C₄⋊(C₃⋊D₃)	1	2	yes	—
102	C₂×C₈	C₂×C₃⋊Dic₃	1	2	yes	—
103	C₂×C₈	C₂×C₃⋊Dic₃	1	1	yes	characteristic
104	C₂×C₈	C₂²×C₃⋊D₃	1	1	yes	characteristic
105	C₄²	C₃⋊(C₃⋊C₈)	1	2	yes	—
106	C₄²	C₂×C₃⋊Dic₃	1	2	yes	—
107	C₁₆	—	18	4	no	—
108	C₂²×C₃	C₄⋊(C₃⋊C₈)	1	8	yes	—
109	C₂²×C₃	C₂×C₃⋊C₁₆	1	8	yes	—
110	C₂²×C₃	#812	1	8	yes	—
111	C₂²×C₃	C₂²×C₃⋊C₈	1	4	yes	—
112	C₃×C₄	—	2	8	no	—
113	C₃×C₄	C₄⋊(C₃⋊C₈)	1	8	yes	—
114	C₃×C₄	C₂×C₃⋊C₁₆	1	16	yes	complemented
115	C₃×C₄	C₂×#208	1	4	yes	—
116	C₃×C₄	C₂×C₄⋊Dic₃	1	4	yes	—
117	C₃²	C₂×C₄⋊C₁₆	1	1	yes	characteristic, complemented, Sylow
118	C₂³	C₂×C₃⋊(C₃⋊C₈)	1	1	yes	characteristic
119	C₂×C₄	—	2	2	no	—
120	C₂×C₄	C₃⋊(C₃⋊C₁₆)	1	4	yes	complemented
121	C₂×C₄	C₂×C₃⋊(C₃⋊C₈)	1	2	yes	—
122	C₂×C₄	C₂×C₃⋊(C₃⋊C₈)	1	1	yes	characteristic
123	C₂×C₄	C₂×C₃⋊(C₃⋊C₈)	1	2	yes	—
124	C₂×C₄	#3748	1	2	yes	—
125	C₂×C₄	C₄⋊(C₃⋊Dic₃)	1	2	yes	—
126	C₂×C₄	C₂²×C₃⋊Dic₃	1	1	yes	characteristic
127	C₈	—	2	2	no	—
128	C₈	C₄⋊(C₃⋊Dic₃)	1	2	yes	—
129	C₈	C₂×#634	1	1	yes	characteristic
130	C₈	C₂×C₄⋊(C₃⋊D₃)	1	1	yes	characteristic
131	C₂×C₃	C₄⋊(C₃⋊C₁₆)	1	16	yes	complemented
132	C₂×C₃	C₂×C₄⋊(C₃⋊C₈)	1	4	yes	—
133	C₂×C₃	C₂²×C₃⋊C₁₆	1	4	yes	—
134	C₂×C₃	C₂×#812	1	4	yes	—
135	C₂²	C₄⋊(C₃⋊(C₃⋊C₈))	1	2	yes	—
136	C₂²	C₂×C₃⋊(C₃⋊C₁₆)	1	2	yes	—
137	C₂²	#63639	1	2	yes	—
138	C₂²	C₂²×C₃⋊(C₃⋊C₈)	1	1	yes	characteristic
139	C₄	—	2	2	no	—
140	C₄	C₄⋊(C₃⋊(C₃⋊C₈))	1	2	yes	—
141	C₄	C₂×C₃⋊(C₃⋊C₁₆)	1	4	yes	complemented
142	C₄	C₂×#3748	1	1	yes	characteristic
143	C₄	C₂×C₄⋊(C₃⋊Dic₃)	1	1	yes	characteristic
144	C₃	C₂×C₄⋊(C₃⋊C₁₆)	1	4	yes	complemented
145	C₂	C₄⋊(C₃⋊(C₃⋊C₁₆))	1	4	yes	direct product factor
146	C₂	C₂×C₄⋊(C₃⋊(C₃⋊C₈))	1	1	yes	characteristic
147	C₂	C₂²×C₃⋊(C₃⋊C₁₆)	1	1	yes	characteristic
148	C₂	C₂×#63639	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₈	C₂×C₃⋊D₃
commutator	C₂×C₃²	C₂²×C₁₆
Frattini	C₂×C₈	C₂²×C₃⋊D₃
Fitting	C₂×C₃²×C₄×C₈	C₂
socle, prime radical	C₂³×C₃²	C₂×C₈
cosocle	C₂×C₃²×C₈	C₂³
hypercenter	C₂×C₄×C₈	C₃⋊D₃
hypocenter	C₃²	C₂×C₄⋊C₁₆
LCS	G ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂²×C₈ ◁ C₂×C₄×C₈

Complements

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

N	H ≅ G/N	type
3. C₄⋊(C₃⋊(C₃⋊C₁₆))	C₂	direct (N × H)
37. C₂×C₃²×C₄	C₁₆	semidirect
69. C₃²×C₄	C₂×C₁₆	semidirect
81. C₂×C₃×C₄	C₃⋊C₁₆	semidirect
92. C₂×C₃²	C₄⋊C₁₆	semidirect
114. C₃×C₄	C₂×C₃⋊C₁₆	semidirect
117. C₃²	C₂×C₄⋊C₁₆	semidirect
120. C₂×C₄	C₃⋊(C₃⋊C₁₆)	semidirect
131. C₂×C₃	C₄⋊(C₃⋊C₁₆)	semidirect
141. C₄	C₂×C₃⋊(C₃⋊C₁₆)	semidirect
144. C₃	C₂×C₄⋊(C₃⋊C₁₆)	semidirect
145. C₂	C₄⋊(C₃⋊(C₃⋊C₁₆))	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×C₃²×C₄×C₈	2
C₂²×C₃⋊(C₃⋊C₁₆)	2
C₄⋊(C₃⋊(C₃⋊C₁₆))	2
C₂×C₄⋊(C₃⋊C₁₆)	3

Subgroups — C2×C4⋊(C3⋊(C3⋊C16)) #11796972

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₄⋊(C₃⋊(C₃⋊C₁₆)) #11796972