Subgroups — #10795572

Order 768.

1350 conjugacy classes of subgroups.

The subgroup lattice has 138 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₄³	C₂	1	1	yes	maximal, characteristic
2	C₂×#4924	C₂	1	6	yes	maximal
3	#69394	C₂	1	8	yes	maximal, complemented
4	#20791	—	3	1	no	maximal, Sylow
5	C₂²×C₃×C₄²	C₄	1	4	yes	—
6	C₂²×C₃×C₄²	C₂²	1	3	yes	—
7	C₂²×#208	C₂²	1	4	yes	—
8	C₃×C₄³	C₄	1	3	yes	—
9	C₃×C₄³	C₄	1	1	yes	characteristic
10	C₃×C₄³	C₂²	1	4	yes	—
11	#4924	—	2	24	no	—
12	C₂×C₄⋊(C₃⋊C₈)	C₂²	1	24	yes	—
13	C₂×#423	—	3	6	no	—
14	C₂×C₄³	D₃	1	1	yes	characteristic
15	#1842	—	3	8	no	—
16	C₂³×C₃×C₄	C₂×C₄	1	6	yes	—
17	C₂³×C₃×C₄	C₂³	1	1	yes	characteristic
18	C₂×C₃×C₄²	—	2	16	no	—
19	C₂×C₃×C₄²	C₂×C₄	1	12	yes	—
20	C₂×C₃×C₄²	C₂×C₄	1	3	yes	—
21	C₂×C₃×C₄²	C₂×C₄	1	3	yes	—
22	C₂×C₃×C₄²	C₂×C₄	1	4	yes	—
23	C₂×C₃×C₄²	D₄	1	12	yes	—
24	C₂×C₃×C₄²	Q₈	1	12	yes	—
25	C₂×C₃×C₄²	C₂³	1	6	yes	—
26	C₂×#208	—	2	24	no	—
27	C₂²×C₃⋊C₈	C₂³	1	8	yes	—
28	C₄⋊(C₃⋊C₈)	—	2	48	no	—
29	C₂²×M₁₆	—	3	4	no	—
30	C₂²×C₄²	Dic₃	1	4	yes	—
31	C₂²×C₄²	C₂×D₃	1	3	yes	—
32	#423	—	6	24	no	—
33	C₂×C₄⋊C₈	—	3	24	no	—
34	C₄³	Dic₃	1	3	yes	—
35	C₄³	Dic₃	1	1	yes	characteristic
36	C₄³	C₂×D₃	1	4	yes	—
37	C₂⁴×C₃	C₂²×C₄	1	1	yes	characteristic
38	C₂²×C₃×C₄	—	2	24	no	—
39	C₂²×C₃×C₄	C₄⋊C₄	1	24	yes	—
40	C₂²×C₃×C₄	C₂²×C₄	1	6	yes	—
41	C₂²×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
42	C₂²×C₃×C₄	C₂²×C₄	1	6	yes	—
43	C₂²×C₃×C₄	C₂×D₄	1	6	yes	—
44	C₂²×C₃×C₄	C₂×Q₈	1	6	yes	—
45	C₂²×C₃×C₄	C₂⁴	1	1	yes	characteristic
46	C₃×C₄²	—	2	24	no	—
47	C₃×C₄²	C₄⋊C₄	1	12	yes	—
48	C₃×C₄²	C₄⋊C₄	1	12	yes	—
49	C₃×C₄²	M₁₆	1	16	yes	complemented
50	C₃×C₄²	C₂×D₄	1	12	yes	—
51	C₃×C₄²	C₂×Q₈	1	12	yes	—
52	#208	—	4	16	no	—
53	C₂×C₃⋊C₈	—	2	24	no	—
54	C₂³×C₄	C₂×Dic₃	1	6	yes	—
55	C₂³×C₄	C₂²×D₃	1	1	yes	characteristic
56	C₂×M₁₆	—	6	24	no	—
57	C₂²×C₈	—	3	8	no	—
58	C₂×C₄²	—	2	16	no	—
59	C₂×C₄²	Dic₆	1	12	yes	—
60	C₂×C₄²	D₁₂	1	12	yes	—
61	C₂×C₄²	C₂×Dic₃	1	12	yes	—
62	C₂×C₄²	C₂×Dic₃	1	4	yes	—
63	C₂×C₄²	C₂×Dic₃	1	3	yes	—
64	C₂×C₄²	C₂×Dic₃	1	3	yes	—
65	C₂×C₄²	C₂²×D₃	1	6	yes	—
66	C₄⋊C₈	—	6	48	no	—
67	C₂³×C₃	—	2	4	no	—
68	C₂³×C₃	C₂×C₄⋊C₄	1	6	yes	—
69	C₂³×C₃	C₂³×C₄	1	1	yes	characteristic
70	C₂×C₃×C₄	—	2	12	no	—
71	C₂×C₃×C₄	—	2	48	no	—
72	C₂×C₃×C₄	C₂×C₄⋊C₄	1	12	yes	—
73	C₂×C₃×C₄	C₂×C₄⋊C₄	1	6	yes	—
74	C₂×C₃×C₄	C₂×C₄⋊C₄	1	12	yes	—
75	C₂×C₃×C₄	#127	1	4	yes	—
76	C₂×C₃×C₄	C₄⋊Q₈	1	12	yes	—
77	C₂×C₃×C₄	C₂×M₁₆	1	24	yes	—
78	C₂×C₃×C₄	C₂²×D₄	1	3	yes	—
79	C₂×C₃×C₄	C₂²×Q₈	1	3	yes	—
80	C₃⋊C₈	—	4	8	no	—
81	C₂⁴	C₂²×Dic₃	1	1	yes	characteristic
82	C₂²×C₄	—	2	24	no	—
83	C₂²×C₄	C₄⋊Dic₃	1	24	yes	—
84	C₂²×C₄	C₂×Dic₆	1	6	yes	—
85	C₂²×C₄	C₂×D₁₂	1	6	yes	—
86	C₂²×C₄	C₂²×Dic₃	1	6	yes	—
87	C₂²×C₄	C₂²×Dic₃	1	6	yes	—
88	C₂²×C₄	C₂²×Dic₃	1	1	yes	characteristic
89	C₂²×C₄	C₂³×D₃	1	1	yes	characteristic
90	M₁₆	—	12	16	no	—
91	C₂×C₈	—	6	24	no	—
92	C₄²	—	2	24	no	—
93	C₄²	#208	1	16	yes	complemented
94	C₄²	C₄⋊Dic₃	1	12	yes	—
95	C₄²	C₄⋊Dic₃	1	12	yes	—
96	C₄²	C₂×Dic₆	1	12	yes	—
97	C₄²	C₂×D₁₂	1	12	yes	—
98	C₂²×C₃	—	2	12	no	—
99	C₂²×C₃	#384	1	4	yes	—
100	C₂²×C₃	C₂²×C₄⋊C₄	1	3	yes	—
101	C₂²×C₃	C₂²×M₁₆	1	4	yes	—
102	C₃×C₄	—	2	12	no	—
103	C₃×C₄	#384	1	4	yes	—
104	C₃×C₄	#423	1	24	yes	complemented
105	C₃×C₄	C₂×#127	1	1	yes	characteristic
106	C₃×C₄	C₂×C₄⋊Q₈	1	3	yes	—
107	C₂³	—	2	4	no	—
108	C₂³	C₂×C₄⋊Dic₃	1	6	yes	—
109	C₂³	C₂³×Dic₃	1	1	yes	characteristic
110	C₂×C₄	—	2	12	no	—
111	C₂×C₄	—	2	48	no	—
112	C₂×C₄	#869	1	12	yes	—
113	C₂×C₄	#874	1	4	yes	—
114	C₂×C₄	C₂×#208	1	24	yes	—
115	C₂×C₄	C₂×C₄⋊Dic₃	1	12	yes	—
116	C₂×C₄	C₂×C₄⋊Dic₃	1	12	yes	—
117	C₂×C₄	C₂×C₄⋊Dic₃	1	6	yes	—
118	C₂×C₄	C₂²×Dic₆	1	3	yes	—
119	C₂×C₄	C₂²×D₁₂	1	3	yes	—
120	C₈	—	12	8	no	—
121	C₂×C₃	—	2	4	no	—
122	C₂×C₃	C₂×#384	1	1	yes	characteristic
123	C₂×C₃	C₂×#423	1	6	yes	—
124	C₂²	—	2	12	no	—
125	C₂²	#4935	1	4	yes	—
126	C₂²	C₂²×#208	1	4	yes	—
127	C₂²	C₂²×C₄⋊Dic₃	1	3	yes	—
128	C₄	—	2	12	no	—
129	C₄	#4924	1	24	yes	complemented
130	C₄	#4935	1	4	yes	—
131	C₄	C₂×#869	1	3	yes	—
132	C₄	C₂×#874	1	1	yes	characteristic
133	C₃	#20791	1	1	yes	characteristic, complemented, Sylow
134	C₂	—	2	4	no	—
135	C₂	C₂×#4924	1	6	yes	—
136	C₂	C₂×#4935	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₃
commutator, socle	C₂³×C₃	C₂³×C₄
Fitting	C₂×C₃×C₄³	C₂
prime radical	C₂⁴×C₃	C₂²×C₄
cosocle	C₂²×C₃×C₄	C₂⁴
hypercenter	C₂×C₄³	D₃
hypocenter	C₃	#20791
LCS	G ▷ C₂³×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄³

Complements

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

N	H ≅ G/N	type
3. #69394	C₂	semidirect
49. C₃×C₄²	M₁₆	semidirect
93. C₄²	#208	semidirect
104. C₃×C₄	#423	semidirect
129. C₄	#4924	semidirect
133. C₃	#20791	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃×C₄³	2
C₂×#4924	2
#69394	2
#20791	3