Subgroups — C₃²×#1277 #11768326

Order 1152.

540 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	2	yes	maximal
2	C₂×C₃²×C₄×C₈	C₂	1	1	yes	maximal, characteristic
3	C₃×#1277	C₃	1	4	yes	maximal, direct product factor
4	C₂²×C₃²×C₈	—	2	2	no	—
5	C₂²×C₃²×C₈	C₄	1	2	yes	—
6	C₂×C₃²×C₄²	C₂²	1	1	yes	characteristic
7	C₃²×C₄⋊C₈	—	2	2	no	—
8	C₃²×C₄⋊C₈	C₄	1	4	yes	—
9	C₃²×C₄×C₈	—	2	2	no	—
10	C₂×C₃×C₄⋊C₈	C₂×C₃	1	8	yes	—
11	C₂×C₃×C₄×C₈	C₂×C₃	1	4	yes	—
12	C₂²×C₃²×C₄	C₂×C₄	1	1	yes	characteristic
13	C₂²×C₃²×C₄	D₄	1	1	yes	characteristic
14	C₂²×C₃²×C₄	Q₈	1	1	yes	characteristic
15	C₂×C₃²×C₈	—	2	2	no	—
16	C₂×C₃²×C₈	—	2	2	no	—
17	C₂×C₃²×C₈	—	2	4	no	—
18	C₂×C₃²×C₈	—	4	2	no	—
19	C₂×C₃²×C₈	—	4	2	no	—
20	C₂×C₃²×C₈	C₈	1	4	yes	complemented
21	C₃²×C₄²	C₂×C₄	1	2	yes	—
22	C₃²×C₄²	D₄	1	2	yes	—
23	#1277	C₃²	1	1	yes	characteristic, direct product factor, Sylow
24	C₂²×C₃×C₈	—	2	8	no	—
25	C₂²×C₃×C₈	C₃×C₄	1	8	yes	—
26	C₂×C₃×C₄²	C₂²×C₃	1	4	yes	—
27	C₃×C₄⋊C₈	—	2	8	no	—
28	C₃×C₄⋊C₈	C₃×C₄	1	16	yes	—
29	C₃×C₄×C₈	—	2	8	no	—
30	C₂³×C₃²	C₄⋊C₄	1	1	yes	characteristic
31	C₂×C₃²×C₄	—	2	2	no	—
32	C₂×C₃²×C₄	C₄²	1	1	yes	characteristic
33	C₂×C₃²×C₄	C₂²⋊C₄	1	1	yes	characteristic
34	C₂×C₃²×C₄	C₂²⋊C₄	1	2	yes	—
35	C₂×C₃²×C₄	C₄⋊C₄	1	2	yes	—
36	C₂×C₃²×C₄	C₂×C₈	1	2	yes	—
37	C₂×C₃²×C₄	M₁₆	1	2	yes	—
38	C₂×C₃²×C₄	D₈	1	1	yes	characteristic
39	C₂×C₃²×C₄	SD₁₆	1	2	yes	—
40	C₂×C₃²×C₄	Q₁₆	1	1	yes	characteristic
41	C₃²×C₈	—	2	4	no	—
42	C₃²×C₈	—	4	4	no	—
43	C₂×C₄⋊C₈	C₂×C₃²	1	2	yes	—
44	C₂×C₄×C₈	C₂×C₃²	1	1	yes	characteristic
45	C₂²×C₃×C₄	C₂×C₃×C₄	1	4	yes	—
46	C₂²×C₃×C₄	C₃×D₄	1	4	yes	—
47	C₂²×C₃×C₄	C₃×Q₈	1	4	yes	—
48	C₂×C₃×C₈	—	2	8	no	—
49	C₂×C₃×C₈	—	2	8	no	—
50	C₂×C₃×C₈	—	2	16	no	—
51	C₂×C₃×C₈	—	4	8	no	—
52	C₂×C₃×C₈	—	4	8	no	—
53	C₂×C₃×C₈	C₃×C₈	1	16	yes	complemented
54	C₃×C₄²	C₂×C₃×C₄	1	8	yes	—
55	C₃×C₄²	C₃×D₄	1	8	yes	—
56	C₂²×C₃²	#95	1	1	yes	characteristic
57	C₂²×C₃²	C₄⋊C₈	1	2	yes	—
58	C₂²×C₃²	#106	1	1	yes	characteristic
59	C₂²×C₃²	#107	1	1	yes	characteristic
60	C₂²×C₃²	#108	1	2	yes	—
61	C₃²×C₄	—	2	2	no	—
62	C₃²×C₄	C₄×C₈	1	1	yes	characteristic
63	C₃²×C₄	#97	1	1	yes	characteristic
64	C₃²×C₄	C₂²⋊C₈	1	2	yes	—
65	C₃²×C₄	#102	1	2	yes	—
66	C₃²×C₄	#103	1	2	yes	—
67	C₂²×C₈	—	2	2	no	—
68	C₂²×C₈	C₃²×C₄	1	2	yes	—
69	C₂×C₄²	C₂²×C₃²	1	1	yes	characteristic
70	C₄⋊C₈	—	2	2	no	—
71	C₄⋊C₈	C₃²×C₄	1	4	yes	—
72	C₄×C₈	—	2	2	no	—
73	C₂³×C₃	C₃×C₄⋊C₄	1	4	yes	—
74	C₂×C₃×C₄	—	2	8	no	—
75	C₂×C₃×C₄	C₃×C₄²	1	4	yes	—
76	C₂×C₃×C₄	C₃×C₂²⋊C₄	1	4	yes	—
77	C₂×C₃×C₄	C₃×C₂²⋊C₄	1	8	yes	—
78	C₂×C₃×C₄	C₃×C₄⋊C₄	1	8	yes	—
79	C₂×C₃×C₄	C₂×C₃×C₈	1	8	yes	—
80	C₂×C₃×C₄	C₃×M₁₆	1	8	yes	—
81	C₂×C₃×C₄	C₃×D₈	1	4	yes	—
82	C₂×C₃×C₄	C₃×SD₁₆	1	8	yes	—
83	C₂×C₃×C₄	C₃×Q₁₆	1	4	yes	—
84	C₃×C₈	—	2	16	no	—
85	C₃×C₈	—	4	16	no	—
86	C₂×C₃²	#334	1	2	yes	—
87	C₂×C₃²	#335	1	2	yes	—
88	C₂×C₃²	#336	1	1	yes	characteristic
89	C₂×C₃²	#340	1	1	yes	characteristic
90	C₂×C₃²	#341	1	1	yes	characteristic
91	C₂²×C₄	C₂×C₃²×C₄	1	1	yes	characteristic
92	C₂²×C₄	C₃²×D₄	1	1	yes	characteristic
93	C₂²×C₄	C₃²×Q₈	1	1	yes	characteristic
94	C₂×C₈	—	2	2	no	—
95	C₂×C₈	—	2	2	no	—
96	C₂×C₈	—	2	4	no	—
97	C₂×C₈	—	4	2	no	—
98	C₂×C₈	—	4	2	no	—
99	C₂×C₈	C₃²×C₈	1	4	yes	complemented
100	C₄²	C₂×C₃²×C₄	1	2	yes	—
101	C₄²	C₃²×D₄	1	2	yes	—
102	C₂²×C₃	C₃×#95	1	4	yes	—
103	C₂²×C₃	C₃×C₄⋊C₈	1	8	yes	—
104	C₂²×C₃	C₃×#106	1	4	yes	—
105	C₂²×C₃	C₃×#107	1	4	yes	—
106	C₂²×C₃	C₃×#108	1	8	yes	—
107	C₃×C₄	—	2	8	no	—
108	C₃×C₄	C₃×C₄×C₈	1	4	yes	—
109	C₃×C₄	C₃×#97	1	4	yes	—
110	C₃×C₄	C₃×C₂²⋊C₈	1	8	yes	—
111	C₃×C₄	C₃×#102	1	8	yes	—
112	C₃×C₄	C₃×#103	1	8	yes	—
113	C₃²	#1277	1	1	yes	characteristic, direct product factor, Sylow
114	C₂³	C₃²×C₄⋊C₄	1	1	yes	characteristic
115	C₂×C₄	—	2	2	no	—
116	C₂×C₄	C₃²×C₄²	1	1	yes	characteristic
117	C₂×C₄	C₃²×C₂²⋊C₄	1	1	yes	characteristic
118	C₂×C₄	C₃²×C₂²⋊C₄	1	2	yes	—
119	C₂×C₄	C₃²×C₄⋊C₄	1	2	yes	—
120	C₂×C₄	C₂×C₃²×C₈	1	2	yes	—
121	C₂×C₄	C₃²×M₁₆	1	2	yes	—
122	C₂×C₄	C₃²×D₈	1	1	yes	characteristic
123	C₂×C₄	C₃²×SD₁₆	1	2	yes	—
124	C₂×C₄	C₃²×Q₁₆	1	1	yes	characteristic
125	C₈	—	2	4	no	—
126	C₈	—	4	4	no	—
127	C₂×C₃	C₃×#334	1	8	yes	—
128	C₂×C₃	C₃×#335	1	8	yes	—
129	C₂×C₃	C₃×#336	1	4	yes	—
130	C₂×C₃	C₃×#340	1	4	yes	—
131	C₂×C₃	C₃×#341	1	4	yes	—
132	C₂²	C₃²×#95	1	1	yes	characteristic
133	C₂²	C₃²×C₄⋊C₈	1	2	yes	—
134	C₂²	C₃²×#106	1	1	yes	characteristic
135	C₂²	C₃²×#107	1	1	yes	characteristic
136	C₂²	C₃²×#108	1	2	yes	—
137	C₄	—	2	2	no	—
138	C₄	C₃²×C₄×C₈	1	1	yes	characteristic
139	C₄	C₃²×#97	1	1	yes	characteristic
140	C₄	C₃²×C₂²⋊C₈	1	2	yes	—
141	C₄	C₃²×#102	1	2	yes	—
142	C₄	C₃²×#103	1	2	yes	—
143	C₃	C₃×#1277	1	4	yes	direct product factor
144	C₂	C₃²×#334	1	2	yes	—
145	C₂	C₃²×#335	1	2	yes	—
146	C₂	C₃²×#336	1	1	yes	characteristic
147	C₂	C₃²×#340	1	1	yes	characteristic
148	C₂	C₃²×#341	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₃²×C₄	D₄
commutator	C₄	C₃²×C₄×C₈
Frattini, cosocle	C₂×C₄²	C₂²×C₃²
socle, prime radical	C₂³×C₃²	C₄⋊C₄
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃²×C₄ ◁ C₂×C₃²×C₄² ◁ G

Complements

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

N	H ≅ G/N	type
3. C₃×#1277	C₃	direct (N × H)
20. C₂×C₃²×C₈	C₈	semidirect
23. #1277	C₃²	direct (N × H)
53. C₂×C₃×C₈	C₃×C₈	semidirect
99. C₂×C₈	C₃²×C₈	semidirect
113. C₃²	#1277	direct (N × H)
143. C₃	C₃×#1277	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
C₂×C₃²×C₄×C₈	2
C₃×#1277	3

Subgroups — C32×#1277 #11768326

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃²×#1277 #11768326