Subgroups — C₃²×#1388 #11768437

Order 1152.

372 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	C₂×C₃²×M₃₂	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃²×#108	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃²×#108	C₂	1	1	yes	maximal, characteristic
4	C₃×#1388	C₃	1	4	yes	maximal, direct product factor
5	C₂×C₃²×M₁₆	—	2	1	no	—
6	C₂×C₃²×M₁₆	—	2	1	no	—
7	C₂²×C₃²×C₈	C₂²	1	1	yes	characteristic
8	C₃²×M₃₂	—	2	1	no	—
9	C₃²×M₃₂	C₄	1	2	yes	—
10	C₂×C₃²×C₁₆	—	2	1	no	—
11	C₃²×#108	—	2	1	no	—
12	C₃²×#108	—	2	1	no	—
13	C₃²×#108	C₄	1	2	yes	—
14	C₃²×#108	C₄	1	2	yes	—
15	C₂×C₃×M₃₂	C₂×C₃	1	4	yes	—
16	C₂×C₃×#108	C₂×C₃	1	4	yes	—
17	C₂×C₃×#108	C₂×C₃	1	4	yes	—
18	C₂²×C₃²×C₄	D₄	1	1	yes	characteristic
19	C₃²×M₁₆	—	2	2	no	—
20	C₃²×M₁₆	—	2	2	no	—
21	C₃²×M₁₆	—	4	1	no	—
22	C₃²×M₁₆	—	4	1	no	—
23	C₂×C₃²×C₈	—	4	1	no	—
24	C₂×C₃²×C₈	—	4	1	no	—
25	C₂×C₃²×C₈	C₂×C₄	1	1	yes	characteristic
26	C₂×C₃²×C₈	C₂×C₄	1	1	yes	characteristic
27	C₂×C₃²×C₈	C₂×C₄	1	1	yes	characteristic
28	C₂×C₃²×C₈	D₄	1	1	yes	characteristic
29	C₂×C₃²×C₈	D₄	1	1	yes	characteristic
30	C₂×C₃²×C₈	Q₈	1	1	yes	characteristic
31	C₃²×C₁₆	—	2	2	no	—
32	#1388	C₃²	1	1	yes	characteristic, direct product factor, Sylow
33	C₂×C₃×M₁₆	—	2	4	no	—
34	C₂×C₃×M₁₆	—	2	4	no	—
35	C₂²×C₃×C₈	C₂²×C₃	1	4	yes	—
36	C₃×M₃₂	—	2	4	no	—
37	C₃×M₃₂	C₃×C₄	1	8	yes	—
38	C₂×C₃×C₁₆	—	2	4	no	—
39	C₃×#108	—	2	4	no	—
40	C₃×#108	—	2	4	no	—
41	C₃×#108	C₃×C₄	1	8	yes	—
42	C₃×#108	C₃×C₄	1	8	yes	—
43	C₂³×C₃²	Q₁₆	1	1	yes	characteristic
44	C₂×C₃²×C₄	C₂²⋊C₄	1	1	yes	characteristic
45	C₂×C₃²×C₄	C₂²⋊C₄	1	1	yes	characteristic
46	C₂×C₃²×C₄	C₄⋊C₄	1	1	yes	characteristic
47	C₂×C₃²×C₄	D₈	1	1	yes	characteristic
48	C₂×C₃²×C₄	SD₁₆	1	1	yes	characteristic
49	C₂×C₃²×C₄	SD₁₆	1	1	yes	characteristic
50	C₃²×C₈	—	4	2	no	—
51	C₃²×C₈	—	4	2	no	—
52	C₃²×C₈	C₄²	1	1	yes	characteristic
53	C₃²×C₈	C₂²⋊C₄	1	1	yes	characteristic
54	C₃²×C₈	C₄⋊C₄	1	1	yes	characteristic
55	C₃²×C₈	C₄⋊C₄	1	1	yes	characteristic
56	C₂×M₃₂	C₂×C₃²	1	1	yes	characteristic
57	C₂×#108	C₂×C₃²	1	1	yes	characteristic
58	C₂×#108	C₂×C₃²	1	1	yes	characteristic
59	C₂²×C₃×C₄	C₃×D₄	1	4	yes	—
60	C₃×M₁₆	—	2	8	no	—
61	C₃×M₁₆	—	2	8	no	—
62	C₃×M₁₆	—	4	4	no	—
63	C₃×M₁₆	—	4	4	no	—
64	C₂×C₃×C₈	—	4	4	no	—
65	C₂×C₃×C₈	—	4	4	no	—
66	C₂×C₃×C₈	C₂×C₃×C₄	1	4	yes	—
67	C₂×C₃×C₈	C₂×C₃×C₄	1	4	yes	—
68	C₂×C₃×C₈	C₂×C₃×C₄	1	4	yes	—
69	C₂×C₃×C₈	C₃×D₄	1	4	yes	—
70	C₂×C₃×C₈	C₃×D₄	1	4	yes	—
71	C₂×C₃×C₈	C₃×Q₈	1	4	yes	—
72	C₃×C₁₆	—	2	8	no	—
73	C₂²×C₃²	—	4	1	no	—
74	C₂²×C₃²	#103	1	1	yes	characteristic
75	C₂²×C₃²	#103	1	1	yes	characteristic
76	C₂²×C₃²	#107	1	1	yes	characteristic
77	C₃²×C₄	#95	1	1	yes	characteristic
78	C₃²×C₄	#102	1	1	yes	characteristic
79	C₃²×C₄	#102	1	1	yes	characteristic
80	C₃²×C₄	#106	1	1	yes	characteristic
81	C₂×M₁₆	—	2	1	no	—
82	C₂×M₁₆	—	2	1	no	—
83	C₂²×C₈	C₂²×C₃²	1	1	yes	characteristic
84	M₃₂	—	2	1	no	—
85	M₃₂	C₃²×C₄	1	2	yes	—
86	C₂×C₁₆	—	2	1	no	—
87	#108	—	2	1	no	—
88	#108	—	2	1	no	—
89	#108	C₃²×C₄	1	2	yes	—
90	#108	C₃²×C₄	1	2	yes	—
91	C₂³×C₃	C₃×Q₁₆	1	4	yes	—
92	C₂×C₃×C₄	C₃×C₂²⋊C₄	1	4	yes	—
93	C₂×C₃×C₄	C₃×C₂²⋊C₄	1	4	yes	—
94	C₂×C₃×C₄	C₃×C₄⋊C₄	1	4	yes	—
95	C₂×C₃×C₄	C₃×D₈	1	4	yes	—
96	C₂×C₃×C₄	C₃×SD₁₆	1	4	yes	—
97	C₂×C₃×C₄	C₃×SD₁₆	1	4	yes	—
98	C₃×C₈	—	4	8	no	—
99	C₃×C₈	—	4	8	no	—
100	C₃×C₈	C₃×C₄²	1	4	yes	—
101	C₃×C₈	C₃×C₂²⋊C₄	1	4	yes	—
102	C₃×C₈	C₃×C₄⋊C₄	1	4	yes	—
103	C₃×C₈	C₃×C₄⋊C₄	1	4	yes	—
104	C₂×C₃²	—	2	1	no	—
105	C₂×C₃²	—	2	1	no	—
106	C₂×C₃²	—	2	1	no	—
107	C₂×C₃²	#340	1	1	yes	characteristic
108	C₂²×C₄	C₃²×D₄	1	1	yes	characteristic
109	M₁₆	—	2	2	no	—
110	M₁₆	—	2	2	no	—
111	M₁₆	—	4	1	no	—
112	M₁₆	—	4	1	no	—
113	C₂×C₈	—	4	1	no	—
114	C₂×C₈	—	4	1	no	—
115	C₂×C₈	C₂×C₃²×C₄	1	1	yes	characteristic
116	C₂×C₈	C₂×C₃²×C₄	1	1	yes	characteristic
117	C₂×C₈	C₂×C₃²×C₄	1	1	yes	characteristic
118	C₂×C₈	C₃²×D₄	1	1	yes	characteristic
119	C₂×C₈	C₃²×D₄	1	1	yes	characteristic
120	C₂×C₈	C₃²×Q₈	1	1	yes	characteristic
121	C₁₆	—	2	2	no	—
122	C₂²×C₃	—	4	4	no	—
123	C₂²×C₃	C₃×#103	1	4	yes	—
124	C₂²×C₃	C₃×#103	1	4	yes	—
125	C₂²×C₃	C₃×#107	1	4	yes	—
126	C₃×C₄	C₃×#95	1	4	yes	—
127	C₃×C₄	C₃×#102	1	4	yes	—
128	C₃×C₄	C₃×#102	1	4	yes	—
129	C₃×C₄	C₃×#106	1	4	yes	—
130	C₃²	#1388	1	1	yes	characteristic, direct product factor, Sylow
131	C₂³	C₃²×Q₁₆	1	1	yes	characteristic
132	C₂×C₄	C₃²×C₂²⋊C₄	1	1	yes	characteristic
133	C₂×C₄	C₃²×C₂²⋊C₄	1	1	yes	characteristic
134	C₂×C₄	C₃²×C₄⋊C₄	1	1	yes	characteristic
135	C₂×C₄	C₃²×D₈	1	1	yes	characteristic
136	C₂×C₄	C₃²×SD₁₆	1	1	yes	characteristic
137	C₂×C₄	C₃²×SD₁₆	1	1	yes	characteristic
138	C₈	—	4	2	no	—
139	C₈	—	4	2	no	—
140	C₈	C₃²×C₄²	1	1	yes	characteristic
141	C₈	C₃²×C₂²⋊C₄	1	1	yes	characteristic
142	C₈	C₃²×C₄⋊C₄	1	1	yes	characteristic
143	C₈	C₃²×C₄⋊C₄	1	1	yes	characteristic
144	C₂×C₃	—	2	4	no	—
145	C₂×C₃	—	2	4	no	—
146	C₂×C₃	—	2	4	no	—
147	C₂×C₃	C₃×#340	1	4	yes	—
148	C₂²	—	4	1	no	—
149	C₂²	C₃²×#103	1	1	yes	characteristic
150	C₂²	C₃²×#103	1	1	yes	characteristic
151	C₂²	C₃²×#107	1	1	yes	characteristic
152	C₄	C₃²×#95	1	1	yes	characteristic
153	C₄	C₃²×#102	1	1	yes	characteristic
154	C₄	C₃²×#102	1	1	yes	characteristic
155	C₄	C₃²×#106	1	1	yes	characteristic
156	C₃	C₃×#1388	1	4	yes	direct product factor
157	C₂	—	2	1	no	—
158	C₂	—	2	1	no	—
159	C₂	—	2	1	no	—
160	C₂	C₃²×#340	1	1	yes	characteristic

	subgroup	G/H
center	C₃²×C₄	#102
commutator	C₈	C₃²×C₄²
Frattini, cosocle	C₂²×C₈	C₂²×C₃²
socle	C₂×C₃²	#340
prime radical	C₂³×C₃²	Q₁₆
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₃²×C₄ ◁ C₂²×C₃²×C₄ ◁ C₂²×C₃²×C₈ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₃×#1388	C₃	direct (N × H)
32. #1388	C₃²	direct (N × H)
130. C₃²	#1388	direct (N × H)
156. C₃	C₃×#1388	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×M₃₂	2
C₂×C₃²×#108	2
C₃×#1388	3

Subgroups — C32×#1388 #11768437

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃²×#1388 #11768437