Subgroups — C₃²×#2171 #11769220

Order 1152.

432 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	C₃²×#504	C₂	1	2	yes	maximal, complemented
2	C₃²×#443	C₂	1	1	yes	maximal, characteristic
3	C₃²×#364	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃²×#350	C₂	1	2	yes	maximal, complemented
5	C₃²×#346	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃×#2171	C₃	1	4	yes	maximal, direct product factor
7	C₃²×#135	C₄	1	1	yes	characteristic
8	C₃²×#131	C₂²	1	2	yes	—
9	C₃²×M₃₂	—	2	2	no	—
10	C₃²×M₃₂	C₂²	1	2	yes	—
11	C₂×C₃²×C₁₆	—	2	2	no	—
12	C₂×C₃²×C₁₆	C₂²	1	1	yes	characteristic
13	C₂×C₃²×C₁₆	C₂²	1	1	yes	characteristic
14	C₃²×#108	C₄	1	1	yes	characteristic
15	C₃²×#104	C₄	1	2	yes	—
16	C₃²×C₄×C₈	C₂²	1	1	yes	characteristic
17	C₃×#504	C₂×C₃	1	8	yes	complemented
18	C₃×#443	C₂×C₃	1	4	yes	—
19	C₃×#364	C₂×C₃	1	4	yes	complemented
20	C₃×#350	C₂×C₃	1	8	yes	complemented
21	C₃×#346	C₂×C₃	1	4	yes	complemented
22	C₃²×#41	C₂×C₄	1	2	yes	—
23	C₃²×Q₁₆	C₈	1	1	yes	characteristic
24	C₃²×SD₁₆	C₈	1	2	yes	—
25	C₃²×D₈	C₈	1	1	yes	characteristic
26	C₃²×M₁₆	—	2	2	no	—
27	C₃²×M₁₆	C₂×C₄	1	2	yes	—
28	C₂×C₃²×C₈	—	2	2	no	—
29	C₂×C₃²×C₈	C₂×C₄	1	1	yes	characteristic
30	C₂×C₃²×C₈	C₂³	1	1	yes	characteristic
31	C₃²×C₄²	C₂×C₄	1	1	yes	characteristic
32	C₃²×C₁₆	—	2	2	no	—
33	C₃²×C₁₆	—	2	1	no	—
34	C₃²×C₁₆	D₄	1	2	yes	—
35	#2171	C₃²	1	1	yes	characteristic, direct product factor, Sylow
36	C₃×#135	C₃×C₄	1	4	yes	—
37	C₃×#131	C₂²×C₃	1	8	yes	—
38	C₃×M₃₂	—	2	8	no	—
39	C₃×M₃₂	C₂²×C₃	1	8	yes	—
40	C₂×C₃×C₁₆	—	2	8	no	—
41	C₂×C₃×C₁₆	C₂²×C₃	1	4	yes	—
42	C₂×C₃×C₁₆	C₂²×C₃	1	4	yes	—
43	C₃×#108	C₃×C₄	1	4	yes	—
44	C₃×#104	C₃×C₄	1	8	yes	—
45	C₃×C₄×C₈	C₂²×C₃	1	4	yes	—
46	C₃²×Q₈	C₂×C₈	1	2	yes	—
47	C₃²×D₄	—	2	2	no	—
48	C₃²×D₄	C₂×C₈	1	2	yes	—
49	C₂×C₃²×C₄	—	2	2	no	—
50	C₂×C₃²×C₄	—	2	1	no	—
51	C₂×C₃²×C₄	C₂²×C₄	1	1	yes	characteristic
52	C₃²×C₈	—	2	2	no	—
53	C₃²×C₈	C₂×C₈	1	1	yes	characteristic
54	C₃²×C₈	C₂×C₈	1	1	yes	characteristic
55	C₃²×C₈	C₂×D₄	1	1	yes	characteristic
56	C₃²×C₈	#41	1	1	yes	characteristic
57	#504	C₂×C₃²	1	2	yes	complemented
58	#443	C₂×C₃²	1	1	yes	characteristic
59	#364	C₂×C₃²	1	1	yes	characteristic, complemented
60	#350	C₂×C₃²	1	2	yes	complemented
61	#346	C₂×C₃²	1	1	yes	characteristic, complemented
62	C₃×#41	C₂×C₃×C₄	1	8	yes	—
63	C₃×Q₁₆	C₃×C₈	1	4	yes	—
64	C₃×SD₁₆	C₃×C₈	1	8	yes	—
65	C₃×D₈	C₃×C₈	1	4	yes	—
66	C₃×M₁₆	—	2	8	no	—
67	C₃×M₁₆	C₂×C₃×C₄	1	8	yes	—
68	C₂×C₃×C₈	—	2	8	no	—
69	C₂×C₃×C₈	C₂×C₃×C₄	1	4	yes	—
70	C₂×C₃×C₈	C₂³×C₃	1	4	yes	—
71	C₃×C₄²	C₂×C₃×C₄	1	4	yes	—
72	C₃×C₁₆	—	2	8	no	—
73	C₃×C₁₆	—	2	4	no	—
74	C₃×C₁₆	C₃×D₄	1	8	yes	—
75	C₂²×C₃²	—	2	2	no	—
76	C₂²×C₃²	#131	1	1	yes	characteristic
77	C₃²×C₄	—	2	2	no	—
78	C₃²×C₄	—	4	1	no	—
79	C₃²×C₄	C₄×D₄	1	1	yes	characteristic
80	C₃²×C₄	C₂²×C₈	1	1	yes	characteristic
81	#135	C₃²×C₄	1	1	yes	characteristic
82	#131	C₂²×C₃²	1	2	yes	—
83	M₃₂	—	2	2	no	—
84	M₃₂	C₂²×C₃²	1	2	yes	—
85	C₂×C₁₆	—	2	2	no	—
86	C₂×C₁₆	C₂²×C₃²	1	1	yes	characteristic
87	C₂×C₁₆	C₂²×C₃²	1	1	yes	characteristic
88	#108	C₃²×C₄	1	1	yes	characteristic
89	#104	C₃²×C₄	1	2	yes	—
90	C₄×C₈	C₂²×C₃²	1	1	yes	characteristic
91	C₃×Q₈	C₂×C₃×C₈	1	8	yes	—
92	C₃×D₄	—	2	8	no	—
93	C₃×D₄	C₂×C₃×C₈	1	8	yes	—
94	C₂×C₃×C₄	—	2	8	no	—
95	C₂×C₃×C₄	—	2	4	no	—
96	C₂×C₃×C₄	C₂²×C₃×C₄	1	4	yes	—
97	C₃×C₈	—	2	8	no	—
98	C₃×C₈	C₂×C₃×C₈	1	4	yes	—
99	C₃×C₈	C₂×C₃×C₈	1	4	yes	—
100	C₃×C₈	C₂×C₃×D₄	1	4	yes	—
101	C₃×C₈	C₃×#41	1	4	yes	—
102	C₂×C₃²	—	2	1	no	—
103	C₂×C₃²	—	4	2	no	—
104	C₂×C₃²	C₈×D₄	1	1	yes	characteristic
105	#41	C₂×C₃²×C₄	1	2	yes	—
106	Q₁₆	C₃²×C₈	1	1	yes	characteristic
107	SD₁₆	C₃²×C₈	1	2	yes	—
108	D₈	C₃²×C₈	1	1	yes	characteristic
109	M₁₆	—	2	2	no	—
110	M₁₆	C₂×C₃²×C₄	1	2	yes	—
111	C₂×C₈	—	2	2	no	—
112	C₂×C₈	C₂×C₃²×C₄	1	1	yes	characteristic
113	C₂×C₈	C₂³×C₃²	1	1	yes	characteristic
114	C₄²	C₂×C₃²×C₄	1	1	yes	characteristic
115	C₁₆	—	2	2	no	—
116	C₁₆	—	2	1	no	—
117	C₁₆	C₃²×D₄	1	2	yes	—
118	C₂²×C₃	—	2	8	no	—
119	C₂²×C₃	C₃×#131	1	4	yes	—
120	C₃×C₄	—	2	8	no	—
121	C₃×C₄	—	4	4	no	—
122	C₃×C₄	C₃×C₄×D₄	1	4	yes	—
123	C₃×C₄	C₂²×C₃×C₈	1	4	yes	—
124	C₃²	#2171	1	1	yes	characteristic, direct product factor, Sylow
125	Q₈	C₂×C₃²×C₈	1	2	yes	—
126	D₄	—	2	2	no	—
127	D₄	C₂×C₃²×C₈	1	2	yes	—
128	C₂×C₄	—	2	2	no	—
129	C₂×C₄	—	2	1	no	—
130	C₂×C₄	C₂²×C₃²×C₄	1	1	yes	characteristic
131	C₈	—	2	2	no	—
132	C₈	C₂×C₃²×C₈	1	1	yes	characteristic
133	C₈	C₂×C₃²×C₈	1	1	yes	characteristic
134	C₈	C₂×C₃²×D₄	1	1	yes	characteristic
135	C₈	C₃²×#41	1	1	yes	characteristic
136	C₂×C₃	—	2	4	no	—
137	C₂×C₃	—	4	8	no	—
138	C₂×C₃	C₃×C₈×D₄	1	4	yes	—
139	C₂²	—	2	2	no	—
140	C₂²	C₃²×#131	1	1	yes	characteristic
141	C₄	—	2	2	no	—
142	C₄	—	4	1	no	—
143	C₄	C₃²×C₄×D₄	1	1	yes	characteristic
144	C₄	C₂²×C₃²×C₈	1	1	yes	characteristic
145	C₃	C₃×#2171	1	4	yes	direct product factor
146	C₂	—	2	1	no	—
147	C₂	—	4	2	no	—
148	C₂	C₃²×C₈×D₄	1	1	yes	characteristic

	subgroup	G/H
center	C₃²×C₈	C₂×D₄
commutator	C₄	C₂²×C₃²×C₈
Frattini, cosocle	C₂×C₈	C₂³×C₃²
socle	C₂×C₃²	C₈×D₄
prime radical	C₃²×#135	C₄
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₃²×C₈ ◁ C₂×C₃²×C₁₆ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₃²×#504	C₂	semidirect
3. C₃²×#364	C₂	semidirect
4. C₃²×#350	C₂	semidirect
5. C₃²×#346	C₂	semidirect
6. C₃×#2171	C₃	direct (N × H)
17. C₃×#504	C₂×C₃	semidirect
19. C₃×#364	C₂×C₃	semidirect
20. C₃×#350	C₂×C₃	semidirect
21. C₃×#346	C₂×C₃	semidirect
35. #2171	C₃²	direct (N × H)
57. #504	C₂×C₃²	semidirect
59. #364	C₂×C₃²	semidirect
60. #350	C₂×C₃²	semidirect
61. #346	C₂×C₃²	semidirect
124. C₃²	#2171	direct (N × H)
145. C₃	C₃×#2171	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃²×#504	2
C₃²×#443	2
C₃²×#364	2
C₃²×#350	2
C₃²×#346	2
C₃×#2171	3

Subgroups — C32×#2171 #11769220

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃²×#2171 #11769220