Subgroups — C₃×Q₈×C₃⋊C₁₆ #11797997

Order 1152.

319 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	C₃²×C₈×Q₈	C₂	1	1	yes	maximal, characteristic
2	C₃×C₄⋊(C₃⋊C₁₆)	C₂	1	3	yes	maximal
3	C₃×C₄×C₃⋊C₁₆	C₂	1	3	yes	maximal
4	C₃×Q₈×C₁₆	—	3	1	no	maximal
5	Q₈×C₃⋊C₁₆	C₃	1	1	yes	maximal, characteristic, direct product factor
6	C₃²×C₄×Q₈	C₄	1	1	yes	characteristic
7	C₃²×C₄⋊C₈	C₄	1	3	yes	—
8	C₃²×C₄×C₈	C₂²	1	3	yes	—
9	C₂×C₃×C₃⋊C₁₆	C₂²	1	3	yes	—
10	C₂×C₃×C₃⋊C₁₆	C₂²	1	1	yes	characteristic
11	C₃×C₈×Q₈	—	2	1	no	—
12	C₃×C₈×Q₈	D₃	1	1	yes	characteristic
13	C₃×C₈×Q₈	C₂×C₃	1	1	yes	characteristic
14	C₃×C₄⋊C₁₆	—	3	3	no	—
15	C₃×C₄×C₁₆	—	3	3	no	—
16	C₄⋊(C₃⋊C₁₆)	C₂×C₃	1	3	yes	—
17	C₄×C₃⋊C₁₆	C₂×C₃	1	3	yes	—
18	C₂×C₃²×Q₈	C₈	1	1	yes	characteristic
19	C₂×C₃²×C₈	C₂×C₄	1	3	yes	—
20	C₂×C₃²×C₈	C₂³	1	1	yes	characteristic
21	C₃²×C₄⋊C₄	C₈	1	3	yes	—
22	C₃²×C₄²	C₂×C₄	1	3	yes	—
23	C₃×C₃⋊C₁₆	—	2	3	no	—
24	C₃×C₃⋊C₁₆	Q₈	1	2	yes	direct product factor
25	Q₈×C₁₆	—	3	1	no	Sylow
26	C₃×C₄×Q₈	—	2	1	no	—
27	C₃×C₄×Q₈	Dic₃	1	1	yes	characteristic
28	C₃×C₄×Q₈	C₃×C₄	1	1	yes	characteristic
29	C₂×C₃×C₁₆	—	3	1	no	—
30	C₂×C₃×C₁₆	—	3	3	no	—
31	C₃×C₄⋊C₈	—	2	3	no	—
32	C₃×C₄⋊C₈	Dic₃	1	3	yes	—
33	C₃×C₄⋊C₈	C₃×C₄	1	3	yes	—
34	C₃×C₄×C₈	—	2	3	no	—
35	C₃×C₄×C₈	C₂×D₃	1	3	yes	—
36	C₃×C₄×C₈	C₂²×C₃	1	3	yes	—
37	C₂×C₃⋊C₁₆	C₂²×C₃	1	3	yes	—
38	C₂×C₃⋊C₁₆	C₂²×C₃	1	1	yes	characteristic
39	C₃²×Q₈	C₁₆	1	4	yes	complemented
40	C₂×C₃²×C₄	C₂×C₈	1	3	yes	—
41	C₂×C₃²×C₄	C₂×C₈	1	3	yes	—
42	C₂×C₃²×C₄	C₂²×C₄	1	1	yes	characteristic
43	C₃²×C₈	—	2	3	no	—
44	C₃²×C₈	C₂×Q₈	1	1	yes	characteristic
45	C₃²×C₈	#41	1	1	yes	characteristic
46	C₈×Q₈	C₃×D₃	1	1	yes	characteristic
47	C₄⋊C₁₆	—	3	3	no	—
48	C₄×C₁₆	—	3	3	no	—
49	C₂×C₃×Q₈	—	2	1	no	—
50	C₂×C₃×Q₈	C₃⋊C₈	1	1	yes	characteristic
51	C₂×C₃×Q₈	C₃×C₈	1	1	yes	characteristic
52	C₂×C₃×C₈	—	2	3	no	—
53	C₂×C₃×C₈	—	2	1	no	—
54	C₂×C₃×C₈	C₂×Dic₃	1	3	yes	—
55	C₂×C₃×C₈	C₂×C₃×C₄	1	3	yes	—
56	C₂×C₃×C₈	C₂²×D₃	1	1	yes	characteristic
57	C₂×C₃×C₈	C₂³×C₃	1	1	yes	characteristic
58	C₃×C₄⋊C₄	—	2	3	no	—
59	C₃×C₄⋊C₄	C₃⋊C₈	1	3	yes	—
60	C₃×C₄⋊C₄	C₃×C₈	1	3	yes	—
61	C₃×C₄²	—	2	3	no	—
62	C₃×C₄²	C₂×Dic₃	1	3	yes	—
63	C₃×C₄²	C₂×C₃×C₄	1	3	yes	—
64	C₃×C₁₆	—	3	2	no	—
65	C₃×C₁₆	—	6	3	no	—
66	C₃⋊C₁₆	—	2	3	no	—
67	C₃⋊C₁₆	C₃×Q₈	1	2	yes	direct product factor
68	C₂²×C₃²	C₂²×C₈	1	1	yes	characteristic
69	C₃²×C₄	—	2	3	no	—
70	C₃²×C₄	C₂×C₁₆	1	6	yes	—
71	C₃²×C₄	C₄×Q₈	1	1	yes	characteristic
72	C₃²×C₄	#131	1	1	yes	characteristic
73	C₄×Q₈	C₃×Dic₃	1	1	yes	characteristic
74	C₂×C₁₆	—	3	1	no	—
75	C₂×C₁₆	—	3	3	no	—
76	C₄⋊C₈	C₃×Dic₃	1	3	yes	—
77	C₄×C₈	C₂×C₃×D₃	1	3	yes	—
78	C₃×Q₈	—	2	4	no	—
79	C₃×Q₈	C₃⋊C₁₆	1	4	yes	direct product factor
80	C₃×Q₈	C₃×C₁₆	1	4	yes	complemented
81	C₂×C₃×C₄	—	2	3	no	—
82	C₂×C₃×C₄	—	2	3	no	—
83	C₂×C₃×C₄	—	2	1	no	—
84	C₂×C₃×C₄	C₂×C₃⋊C₈	1	3	yes	—
85	C₂×C₃×C₄	C₂×C₃⋊C₈	1	3	yes	—
86	C₂×C₃×C₄	C₂×C₃×C₈	1	3	yes	—
87	C₂×C₃×C₄	C₂×C₃×C₈	1	3	yes	—
88	C₂×C₃×C₄	C₂²×Dic₃	1	1	yes	characteristic
89	C₂×C₃×C₄	C₂²×C₃×C₄	1	1	yes	characteristic
90	C₃×C₈	—	2	3	no	—
91	C₃×C₈	—	2	3	no	—
92	C₃×C₈	—	2	1	no	—
93	C₃×C₈	—	2	1	no	—
94	C₃×C₈	—	4	3	no	—
95	C₃×C₈	D₃×Q₈	1	1	yes	characteristic
96	C₃×C₈	#239	1	1	yes	characteristic
97	C₃×C₈	C₂×C₃×Q₈	1	1	yes	characteristic
98	C₃×C₈	C₃×#41	1	1	yes	characteristic
99	C₂×C₃²	C₈×Q₈	1	1	yes	characteristic
100	C₂×C₃²	C₂²×C₁₆	1	1	yes	characteristic
101	C₂×C₃²	#504	1	1	yes	characteristic
102	C₂×Q₈	C₃×C₃⋊C₈	1	1	yes	characteristic
103	C₂×C₈	C₂×C₃×Dic₃	1	3	yes	—
104	C₂×C₈	C₂²×C₃×D₃	1	1	yes	characteristic
105	C₄⋊C₄	C₃×C₃⋊C₈	1	3	yes	—
106	C₄²	C₂×C₃×Dic₃	1	3	yes	—
107	C₁₆	—	3	2	no	—
108	C₁₆	—	6	3	no	—
109	C₂²×C₃	—	2	1	no	—
110	C₂²×C₃	C₂²×C₃⋊C₈	1	1	yes	characteristic
111	C₂²×C₃	C₂²×C₃×C₈	1	1	yes	characteristic
112	C₃×C₄	—	2	3	no	—
113	C₃×C₄	—	2	3	no	—
114	C₃×C₄	—	2	1	no	—
115	C₃×C₄	—	2	1	no	—
116	C₃×C₄	—	2	6	no	—
117	C₃×C₄	—	4	3	no	—
118	C₃×C₄	C₂×C₃⋊C₁₆	1	6	yes	—
119	C₃×C₄	C₂×C₃×C₁₆	1	6	yes	—
120	C₃×C₄	Q₈×Dic₃	1	1	yes	characteristic
121	C₃×C₄	#948	1	1	yes	characteristic
122	C₃×C₄	C₃×C₄×Q₈	1	1	yes	characteristic
123	C₃×C₄	C₃×#131	1	1	yes	characteristic
124	C₃²	Q₈×C₁₆	1	1	yes	characteristic, complemented, Sylow
125	Q₈	C₃×C₃⋊C₁₆	1	4	yes	direct product factor
126	C₂×C₄	C₂×C₃×C₃⋊C₈	1	3	yes	—
127	C₂×C₄	C₂×C₃×C₃⋊C₈	1	3	yes	—
128	C₂×C₄	C₂²×C₃×Dic₃	1	1	yes	characteristic
129	C₈	—	2	3	no	—
130	C₈	C₃×D₃×Q₈	1	1	yes	characteristic
131	C₈	C₃×#239	1	1	yes	characteristic
132	C₂×C₃	—	2	1	no	—
133	C₂×C₃	—	2	1	no	—
134	C₂×C₃	—	2	1	no	—
135	C₂×C₃	Q₈×C₃⋊C₈	1	1	yes	characteristic
136	C₂×C₃	C₂²×C₃⋊C₁₆	1	1	yes	characteristic
137	C₂×C₃	#5140	1	1	yes	characteristic
138	C₂×C₃	C₃×C₈×Q₈	1	1	yes	characteristic
139	C₂×C₃	C₂²×C₃×C₁₆	1	1	yes	characteristic
140	C₂×C₃	C₃×#504	1	1	yes	characteristic
141	C₂²	C₂²×C₃×C₃⋊C₈	1	1	yes	characteristic
142	C₄	—	2	3	no	—
143	C₄	C₂×C₃×C₃⋊C₁₆	1	6	yes	—
144	C₄	C₃×Q₈×Dic₃	1	1	yes	characteristic
145	C₄	C₃×#948	1	1	yes	characteristic
146	C₃	—	2	1	no	—
147	C₃	Q₈×C₃⋊C₁₆	1	1	yes	characteristic, direct product factor
148	C₃	C₃×Q₈×C₁₆	1	1	yes	characteristic, complemented
149	C₂	C₃×Q₈×C₃⋊C₈	1	1	yes	characteristic
150	C₂	C₂²×C₃×C₃⋊C₁₆	1	1	yes	characteristic
151	C₂	C₃×#5140	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₃×C₈	C₂²×D₃
commutator	C₂×C₃	C₂²×C₃×C₁₆
Frattini	C₂×C₈	C₂²×C₃×D₃
Fitting	C₃²×C₈×Q₈	C₂
socle, prime radical	C₂²×C₃²	C₂²×C₈
cosocle	C₂×C₃×C₈	C₂³×C₃
hypercenter	C₃×C₈×Q₈	D₃
hypocenter	C₃	C₃×Q₈×C₁₆
LCS	G ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₃×C₈ ◁ C₃×C₈×Q₈

Complements

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

N	H ≅ G/N	type
5. Q₈×C₃⋊C₁₆	C₃	direct (N × H)
24. C₃×C₃⋊C₁₆	Q₈	direct (N × H)
39. C₃²×Q₈	C₁₆	semidirect
67. C₃⋊C₁₆	C₃×Q₈	direct (N × H)
79. C₃×Q₈	C₃⋊C₁₆	direct (N × H)
80. C₃×Q₈	C₃×C₁₆	semidirect
124. C₃²	Q₈×C₁₆	semidirect
125. Q₈	C₃×C₃⋊C₁₆	direct (N × H)
147. C₃	Q₈×C₃⋊C₁₆	direct (N × H)
148. C₃	C₃×Q₈×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×Q₈	2
C₃×C₄⋊(C₃⋊C₁₆)	2
C₃×C₄×C₃⋊C₁₆	2
C₃×Q₈×C₁₆	3
Q₈×C₃⋊C₁₆	3

Subgroups — C3×Q8×C3⋊C16 #11797997

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×Q₈×C₃⋊C₁₆ #11797997