Subgroups — #10704178

Order 768.

298 conjugacy classes of subgroups.

The subgroup lattice has 136 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₄×M₃₂	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×C₃⋊C₁₆	C₂	1	1	yes	maximal, characteristic
3	#69724	C₂	1	1	yes	maximal, characteristic
4	#67499	C₂	1	4	yes	maximal, complemented
5	#11401	—	3	1	no	maximal, Sylow
6	C₂×C₃×M₃₂	C₄	1	2	yes	—
7	C₂×C₃×C₄×C₈	C₂²	1	1	yes	characteristic
8	C₂²×C₃⋊C₁₆	C₄	1	2	yes	—
9	#4925	C₄	1	2	yes	—
10	C₃×#346	C₄	1	2	yes	—
11	C₃×C₄×C₁₆	C₂²	1	2	yes	—
12	C₄×C₃⋊C₁₆	C₄	1	2	yes	—
13	C₄×C₃⋊C₁₆	C₂²	1	2	yes	—
14	#4455	C₄	1	2	yes	—
15	C₈×C₃⋊C₈	C₂²	1	2	yes	—
16	C₄×M₃₂	D₃	1	1	yes	characteristic
17	C₂×C₄×C₁₆	—	3	1	no	—
18	#1450	—	3	1	no	—
19	#1312	—	3	4	no	—
20	C₂²×C₃×C₈	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₃×C₈	C₂×C₄	1	1	yes	characteristic
22	C₂×C₃×C₄²	C₂×C₄	1	1	yes	characteristic
23	C₃×M₃₂	—	2	4	no	—
24	C₂×C₃×C₁₆	C₂×C₄	1	4	yes	—
25	C₃×C₄×C₈	C₂×C₄	1	1	yes	characteristic
26	C₃×C₄×C₈	C₂×C₄	1	1	yes	characteristic
27	C₃×C₄×C₈	C₂×C₄	1	1	yes	characteristic
28	C₃×C₄×C₈	C₂³	1	1	yes	characteristic
29	#830	C₈	1	4	yes	—
30	C₂×C₃⋊C₁₆	—	2	4	no	—
31	C₂×C₃⋊C₁₆	C₂×C₄	1	4	yes	—
32	C₄⋊(C₃⋊C₈)	C₈	1	4	yes	—
33	C₄×C₃⋊C₈	C₂×C₄	1	4	yes	—
34	C₂×M₃₂	Dic₃	1	2	yes	—
35	C₂²×C₁₆	—	3	2	no	—
36	#431	—	3	2	no	—
37	C₂×C₄×C₈	C₂×D₃	1	1	yes	characteristic
38	#346	Dic₃	1	2	yes	—
39	C₄×C₁₆	—	3	2	no	—
40	C₄×C₁₆	—	3	2	no	—
41	C₄×C₁₆	C₂×D₃	1	2	yes	—
42	#322	—	3	2	no	—
43	C₈²	—	3	2	no	—
44	C₂²×C₃×C₄	C₄²	1	1	yes	characteristic
45	C₂²×C₃×C₄	C₂×C₈	1	2	yes	—
46	C₂×C₃×C₈	C₄²	1	1	yes	characteristic
47	C₂×C₃×C₈	C₄²	1	1	yes	characteristic
48	C₂×C₃×C₈	M₁₆	1	4	yes	—
49	C₂×C₃×C₈	M₁₆	1	4	yes	—
50	C₂×C₃×C₈	C₂²×C₄	1	1	yes	characteristic
51	C₂×C₃×C₈	C₂²×C₄	1	1	yes	characteristic
52	C₃×C₄²	C₄²	1	1	yes	characteristic
53	C₃×C₄²	C₂×C₈	1	2	yes	—
54	C₃×C₄²	C₂²×C₄	1	1	yes	characteristic
55	C₂×C₃⋊C₈	C₂×C₈	1	8	yes	—
56	C₃×C₁₆	—	2	4	no	—
57	C₃⋊C₁₆	—	2	4	no	—
58	C₂²×C₈	C₄×D₃	1	1	yes	characteristic
59	C₂²×C₈	C₂×Dic₃	1	1	yes	characteristic
60	C₂×C₄²	C₄×D₃	1	1	yes	characteristic
61	M₃₂	—	2	4	no	—
62	C₂×C₁₆	—	3	4	no	—
63	C₂×C₁₆	—	6	4	no	—
64	C₂×C₁₆	C₂×Dic₃	1	4	yes	—
65	C₄⋊C₈	—	3	4	no	—
66	C₂²⋊C₈	—	3	4	no	—
67	C₄×C₈	—	3	4	no	—
68	C₄×C₈	C₄×D₃	1	1	yes	characteristic
69	C₄×C₈	C₄×D₃	1	1	yes	characteristic
70	C₄×C₈	C₂×Dic₃	1	1	yes	characteristic
71	C₄×C₈	C₂²×D₃	1	1	yes	characteristic
72	C₂³×C₃	C₄×C₈	1	1	yes	characteristic
73	C₂×C₃×C₄	—	2	4	no	—
74	C₂×C₃×C₄	C₄×C₈	1	2	yes	—
75	C₂×C₃×C₄	C₄×C₈	1	1	yes	characteristic
76	C₂×C₃×C₄	#97	1	2	yes	—
77	C₂×C₃×C₄	#97	1	2	yes	—
78	C₂×C₃×C₄	C₂×C₄²	1	1	yes	characteristic
79	C₂×C₃×C₄	C₂²×C₈	1	2	yes	—
80	C₃×C₈	#97	1	2	yes	—
81	C₃×C₈	#97	1	2	yes	—
82	C₃×C₈	C₂×M₁₆	1	2	yes	—
83	C₃×C₈	C₂×M₁₆	1	2	yes	—
84	C₃⋊C₈	—	2	8	no	—
85	C₂²×C₄	D₃×C₈	1	2	yes	—
86	C₂²×C₄	C₄×Dic₃	1	1	yes	characteristic
87	C₂×C₈	—	3	8	no	—
88	C₂×C₈	ΓL(1,𝔽₂₅)	1	4	yes	—
89	C₂×C₈	#208	1	4	yes	—
90	C₂×C₈	C₄×Dic₃	1	1	yes	characteristic
91	C₂×C₈	C₄×Dic₃	1	1	yes	characteristic
92	C₂×C₈	C₂×C₄×D₃	1	1	yes	characteristic
93	C₂×C₈	C₂²×Dic₃	1	1	yes	characteristic
94	C₄²	D₃×C₈	1	2	yes	—
95	C₄²	C₄×Dic₃	1	1	yes	characteristic
96	C₄²	C₂×C₄×D₃	1	1	yes	characteristic
97	C₁₆	—	2	4	no	—
98	C₁₆	—	6	4	no	—
99	C₂²×C₃	—	2	2	no	—
100	C₂²×C₃	#322	1	2	yes	—
101	C₂²×C₃	C₂×C₄×C₈	1	1	yes	characteristic
102	C₃×C₄	—	2	2	no	—
103	C₃×C₄	#322	1	2	yes	—
104	C₃×C₄	C₂×#97	1	1	yes	characteristic
105	C₃×C₄	C₂×#97	1	1	yes	characteristic
106	C₃×C₄	#504	1	4	yes	—
107	C₂³	C₈×Dic₃	1	1	yes	characteristic
108	C₂×C₄	—	2	4	no	—
109	C₂×C₄	#803	1	2	yes	—
110	C₂×C₄	C₈×Dic₃	1	2	yes	—
111	C₂×C₄	C₈×Dic₃	1	1	yes	characteristic
112	C₂×C₄	#815	1	2	yes	—
113	C₂×C₄	C₂×D₃×C₈	1	2	yes	—
114	C₂×C₄	C₂×C₄×Dic₃	1	1	yes	characteristic
115	C₈	—	6	8	no	—
116	C₈	#803	1	2	yes	—
117	C₈	#815	1	2	yes	—
118	C₈	C₂×ΓL(1,𝔽₂₅)	1	2	yes	—
119	C₈	C₂×#208	1	2	yes	—
120	C₂×C₃	—	2	2	no	—
121	C₂×C₃	C₂×#322	1	1	yes	characteristic
122	C₂×C₃	#2108	1	2	yes	—
123	C₂²	—	2	2	no	—
124	C₂²	#4454	1	2	yes	—
125	C₂²	C₂×C₈×Dic₃	1	1	yes	characteristic
126	C₄	—	2	2	no	—
127	C₄	#4454	1	2	yes	—
128	C₄	#4907	1	4	yes	—
129	C₄	C₂×#803	1	1	yes	characteristic
130	C₄	C₂×#815	1	1	yes	characteristic
131	C₃	#11401	1	1	yes	characteristic, complemented, Sylow
132	C₂	—	2	2	no	—
133	C₂	C₂×#4454	1	1	yes	characteristic
134	C₂	#71419	1	2	yes	—

	subgroup	G/H
center, Frattini	C₄×C₈	C₂²×D₃
commutator, socle	C₂²×C₃	C₂×C₄×C₈
Fitting	C₃×C₄×M₃₂	C₂
prime radical	C₂³×C₃	C₄×C₈
cosocle	C₃×C₄×C₈	C₂³
hypercenter	C₄×M₃₂	D₃
hypocenter	C₃	#11401
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₄×C₈ ◁ C₄×M₃₂

Complements

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

N	H ≅ G/N	type
4. #67499	C₂	semidirect
131. C₃	#11401	semidirect

Maximal subgroups

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

subgroup	index
C₃×C₄×M₃₂	2
C₂×C₄×C₃⋊C₁₆	2
#69724	2
#67499	2
#11401	3