Subgroups — D₄×C₃⋊(C₃⋊C₁₆) #11797435

Order 1152.

522 conjugacy classes of subgroups.

The subgroup lattice has 162 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²×C₈×D₄	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₃⋊(C₃⋊C₁₆)	C₂	1	2	yes	maximal, complemented
3	#10589415	C₂	1	2	yes	maximal, complemented
4	C₄⋊(C₃⋊(C₃⋊C₁₆))	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄×C₃⋊(C₃⋊C₁₆)	C₂	1	1	yes	maximal, characteristic, complemented
6	D₄×C₃⋊C₁₆	—	3	4	no	maximal
7	C₂²×C₃²×C₈	C₂²	1	2	yes	—
8	C₃²×C₄×D₄	C₄	1	1	yes	characteristic
9	C₃²×C₄⋊C₈	C₄	1	1	yes	characteristic
10	C₃²×C₂²⋊C₈	C₄	1	2	yes	—
11	C₃²×C₄×C₈	C₂²	1	1	yes	characteristic
12	C₂×C₃⋊(C₃⋊C₁₆)	—	2	4	no	—
13	C₂×C₃⋊(C₃⋊C₁₆)	C₂²	1	2	yes	—
14	C₂×C₃⋊(C₃⋊C₁₆)	C₂²	1	1	yes	characteristic
15	C₂×C₃⋊(C₃⋊C₁₆)	C₂²	1	1	yes	characteristic
16	C₃×C₈×D₄	D₃	1	4	yes	—
17	C₂²×C₃⋊C₁₆	—	3	8	no	—
18	#4549	—	3	8	no	—
19	C₄⋊(C₃⋊C₁₆)	—	3	4	no	—
20	C₄×C₃⋊C₁₆	—	3	4	no	—
21	C₂×C₃²×D₄	C₈	1	1	yes	characteristic
22	C₂²×C₃²×C₄	C₂×C₄	1	2	yes	—
23	C₂×C₃²×C₈	—	2	2	no	—
24	C₂×C₃²×C₈	—	2	2	no	—
25	C₂×C₃²×C₈	C₂×C₄	1	1	yes	characteristic
26	C₂×C₃²×C₈	C₂×C₄	1	2	yes	—
27	C₂×C₃²×C₈	C₂³	1	1	yes	characteristic
28	C₃²×C₄⋊C₄	C₈	1	1	yes	characteristic
29	C₃²×C₂²⋊C₄	C₈	1	2	yes	—
30	C₃²×C₄²	C₂×C₄	1	1	yes	characteristic
31	C₃⋊(C₃⋊C₁₆)	—	2	2	no	—
32	C₃⋊(C₃⋊C₁₆)	—	2	1	no	—
33	C₃⋊(C₃⋊C₁₆)	D₄	1	2	yes	direct product factor
34	D₄×C₁₆	—	9	1	no	Sylow
35	C₂²×C₃×C₈	C₂×D₃	1	8	yes	—
36	C₃×C₄×D₄	Dic₃	1	4	yes	—
37	C₃×C₄⋊C₈	Dic₃	1	4	yes	—
38	C₃×C₂²⋊C₈	Dic₃	1	8	yes	—
39	C₃×C₄×C₈	C₂×D₃	1	4	yes	—
40	C₂×C₃⋊C₁₆	—	3	4	no	—
41	C₂×C₃⋊C₁₆	—	3	4	no	—
42	C₂×C₃⋊C₁₆	—	3	8	no	—
43	C₂×C₃⋊C₁₆	—	6	16	no	—
44	C₂³×C₃²	C₂×C₈	1	2	yes	—
45	C₃²×D₄	C₁₆	1	4	yes	complemented
46	C₂×C₃²×C₄	—	2	2	no	—
47	C₂×C₃²×C₄	—	2	2	no	—
48	C₂×C₃²×C₄	C₂×C₈	1	1	yes	characteristic
49	C₂×C₃²×C₄	C₂×C₈	1	2	yes	—
50	C₂×C₃²×C₄	C₂×C₈	1	1	yes	characteristic
51	C₂×C₃²×C₄	C₂²×C₄	1	1	yes	characteristic
52	C₃²×C₈	—	2	2	no	—
53	C₃²×C₈	—	2	1	no	—
54	C₃²×C₈	C₂×D₄	1	1	yes	characteristic
55	C₃²×C₈	#41	1	1	yes	characteristic
56	C₂²×C₁₆	—	9	2	no	—
57	C₈×D₄	C₃⋊D₃	1	1	yes	characteristic
58	C₄⋊C₁₆	—	9	1	no	—
59	C₂²⋊C₁₆	—	9	2	no	—
60	C₄×C₁₆	—	9	1	no	—
61	C₂×C₃×D₄	C₃⋊C₈	1	4	yes	—
62	C₂²×C₃×C₄	C₂×Dic₃	1	8	yes	—
63	C₂×C₃×C₈	—	2	8	no	—
64	C₂×C₃×C₈	—	2	8	no	—
65	C₂×C₃×C₈	C₂×Dic₃	1	4	yes	—
66	C₂×C₃×C₈	C₂×Dic₃	1	8	yes	—
67	C₂×C₃×C₈	C₂²×D₃	1	4	yes	—
68	C₃×C₄⋊C₄	C₃⋊C₈	1	4	yes	—
69	C₃×C₂²⋊C₄	C₃⋊C₈	1	8	yes	—
70	C₃×C₄²	C₂×Dic₃	1	4	yes	—
71	C₃⋊C₁₆	—	3	8	no	—
72	C₃⋊C₁₆	—	6	8	no	—
73	C₃⋊C₁₆	—	6	4	no	—
74	C₂²×C₃²	—	2	2	no	—
75	C₂²×C₃²	—	2	2	no	—
76	C₂²×C₃²	C₂×C₁₆	1	4	yes	complemented
77	C₂²×C₃²	C₂²×C₈	1	1	yes	characteristic
78	C₃²×C₄	—	2	2	no	—
79	C₃²×C₄	—	2	1	no	—
80	C₃²×C₄	C₂×C₁₆	1	2	yes	complemented
81	C₃²×C₄	C₄×D₄	1	1	yes	characteristic
82	C₃²×C₄	#131	1	1	yes	characteristic
83	C₂²×C₈	C₂×C₃⋊D₃	1	2	yes	—
84	C₄×D₄	C₃⋊Dic₃	1	1	yes	characteristic
85	C₂×C₁₆	—	9	1	no	—
86	C₂×C₁₆	—	9	1	no	—
87	C₂×C₁₆	—	9	2	no	—
88	C₂×C₁₆	—	18	4	no	—
89	C₄⋊C₈	C₃⋊Dic₃	1	1	yes	characteristic
90	C₂²⋊C₈	C₃⋊Dic₃	1	2	yes	—
91	C₄×C₈	C₂×C₃⋊D₃	1	1	yes	characteristic
92	C₂³×C₃	C₂×C₃⋊C₈	1	8	yes	—
93	C₃×D₄	C₃⋊C₁₆	1	16	yes	complemented
94	C₂×C₃×C₄	—	2	8	no	—
95	C₂×C₃×C₄	—	2	8	no	—
96	C₂×C₃×C₄	C₂×C₃⋊C₈	1	4	yes	—
97	C₂×C₃×C₄	C₂×C₃⋊C₈	1	8	yes	—
98	C₂×C₃×C₄	C₂×C₃⋊C₈	1	4	yes	—
99	C₂×C₃×C₄	C₂²×Dic₃	1	4	yes	—
100	C₃×C₈	—	2	8	no	—
101	C₃×C₈	—	2	4	no	—
102	C₃×C₈	D₃×D₄	1	4	yes	—
103	C₃×C₈	#237	1	4	yes	—
104	C₂×C₃²	—	2	4	no	—
105	C₂×C₃²	C₈×D₄	1	1	yes	characteristic
106	C₂×C₃²	C₂²×C₁₆	1	1	yes	characteristic
107	C₂×C₃²	#504	1	1	yes	characteristic
108	C₂×D₄	C₃⋊(C₃⋊C₈)	1	1	yes	characteristic
109	C₂²×C₄	C₂×C₃⋊Dic₃	1	2	yes	—
110	C₂×C₈	—	2	2	no	—
111	C₂×C₈	—	2	2	no	—
112	C₂×C₈	C₂×C₃⋊Dic₃	1	1	yes	characteristic
113	C₂×C₈	C₂×C₃⋊Dic₃	1	2	yes	—
114	C₂×C₈	C₂²×C₃⋊D₃	1	1	yes	characteristic
115	C₄⋊C₄	C₃⋊(C₃⋊C₈)	1	1	yes	characteristic
116	C₂²⋊C₄	C₃⋊(C₃⋊C₈)	1	2	yes	—
117	C₄²	C₂×C₃⋊Dic₃	1	1	yes	characteristic
118	C₁₆	—	9	2	no	—
119	C₁₆	—	18	2	no	—
120	C₁₆	—	18	1	no	—
121	C₂²×C₃	—	2	8	no	—
122	C₂²×C₃	—	2	8	no	—
123	C₂²×C₃	C₂×C₃⋊C₁₆	1	16	yes	complemented
124	C₂²×C₃	C₂²×C₃⋊C₈	1	4	yes	—
125	C₃×C₄	—	2	4	no	—
126	C₃×C₄	—	2	8	no	—
127	C₃×C₄	C₂×C₃⋊C₁₆	1	8	yes	complemented
128	C₃×C₄	D₄×Dic₃	1	4	yes	—
129	C₃×C₄	#948	1	4	yes	—
130	C₃²	D₄×C₁₆	1	1	yes	characteristic, complemented, Sylow
131	C₂³	C₂×C₃⋊(C₃⋊C₈)	1	2	yes	—
132	D₄	C₃⋊(C₃⋊C₁₆)	1	4	yes	direct product factor
133	C₂×C₄	—	2	2	no	—
134	C₂×C₄	—	2	2	no	—
135	C₂×C₄	C₂×C₃⋊(C₃⋊C₈)	1	1	yes	characteristic
136	C₂×C₄	C₂×C₃⋊(C₃⋊C₈)	1	2	yes	—
137	C₂×C₄	C₂×C₃⋊(C₃⋊C₈)	1	1	yes	characteristic
138	C₂×C₄	C₂²×C₃⋊Dic₃	1	1	yes	characteristic
139	C₈	—	2	2	no	—
140	C₈	—	2	1	no	—
141	C₈	D₄×C₃⋊D₃	1	1	yes	characteristic
142	C₈	#3830	1	1	yes	characteristic
143	C₂×C₃	—	2	16	no	—
144	C₂×C₃	D₄×C₃⋊C₈	1	4	yes	—
145	C₂×C₃	C₂²×C₃⋊C₁₆	1	4	yes	—
146	C₂×C₃	#5140	1	4	yes	—
147	C₂²	—	2	2	no	—
148	C₂²	—	2	2	no	—
149	C₂²	C₂×C₃⋊(C₃⋊C₁₆)	1	4	yes	complemented
150	C₂²	C₂²×C₃⋊(C₃⋊C₈)	1	1	yes	characteristic
151	C₄	—	2	1	no	—
152	C₄	—	2	2	no	—
153	C₄	C₂×C₃⋊(C₃⋊C₁₆)	1	2	yes	complemented
154	C₄	D₄×C₃⋊Dic₃	1	1	yes	characteristic
155	C₄	#64157	1	1	yes	characteristic
156	C₃	D₄×C₃⋊C₁₆	1	4	yes	complemented
157	C₂	—	2	4	no	—
158	C₂	D₄×C₃⋊(C₃⋊C₈)	1	1	yes	characteristic
159	C₂	C₂²×C₃⋊(C₃⋊C₁₆)	1	1	yes	characteristic
160	C₂	#10592839	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₈	C₂²×C₃⋊D₃
commutator	C₂×C₃²	C₂²×C₁₆
Fitting	C₃²×C₈×D₄	C₂
socle	C₂²×C₃²	C₂²×C₈
prime radical	C₂×C₃²×D₄	C₈
cosocle	C₂×C₃²×C₈	C₂³
hypercenter	C₈×D₄	C₃⋊D₃
hypocenter	C₃²	D₄×C₁₆
LCS	G ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂×C₈ ◁ C₈×D₄

Complements

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

N	H ≅ G/N	type
2. C₂²×C₃⋊(C₃⋊C₁₆)	C₂	semidirect
3. #10589415	C₂	semidirect
4. C₄⋊(C₃⋊(C₃⋊C₁₆))	C₂	semidirect
5. C₄×C₃⋊(C₃⋊C₁₆)	C₂	semidirect
33. C₃⋊(C₃⋊C₁₆)	D₄	direct (N × H)
45. C₃²×D₄	C₁₆	semidirect
76. C₂²×C₃²	C₂×C₁₆	semidirect
80. C₃²×C₄	C₂×C₁₆	semidirect
93. C₃×D₄	C₃⋊C₁₆	semidirect
123. C₂²×C₃	C₂×C₃⋊C₁₆	semidirect
127. C₃×C₄	C₂×C₃⋊C₁₆	semidirect
130. C₃²	D₄×C₁₆	semidirect
132. D₄	C₃⋊(C₃⋊C₁₆)	direct (N × H)
149. C₂²	C₂×C₃⋊(C₃⋊C₁₆)	semidirect
153. C₄	C₂×C₃⋊(C₃⋊C₁₆)	semidirect
156. C₃	D₄×C₃⋊C₁₆	semidirect

Maximal subgroups

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

subgroup	index
C₃²×C₈×D₄	2
C₂²×C₃⋊(C₃⋊C₁₆)	2
#10589415	2
C₄⋊(C₃⋊(C₃⋊C₁₆))	2
C₄×C₃⋊(C₃⋊C₁₆)	2
D₄×C₃⋊C₁₆	3

Subgroups — D4×C3⋊(C3⋊C16) #11797435

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₄×C₃⋊(C₃⋊C₁₆) #11797435