Subgroups — #10948853

Order 768.

444 conjugacy classes of subgroups.

The subgroup lattice has 160 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#2960	C₂	1	1	yes	maximal, characteristic
2	#70949	C₂	1	8	yes	maximal, complemented
3	#69883	C₂	1	4	yes	maximal
4	#69745	C₂	1	2	yes	maximal
5	#33501	—	3	1	no	maximal, Sylow
6	C₃×#533	C₄	1	1	yes	characteristic
7	C₃×#446	C₄	1	4	yes	—
8	C₃×C₈×Q₈	C₂²	1	4	yes	—
9	C₃×#424	C₄	1	2	yes	—
10	C₃×#423	C₄	1	1	yes	characteristic
11	C₃×#405	C₂²	1	2	yes	—
12	C₂×C₃×C₄×C₈	C₂²	1	1	yes	characteristic
13	#4549	C₂²	1	4	yes	—
14	C₄⋊(C₃⋊C₁₆)	C₂²	1	4	yes	—
15	C₄⋊(C₃⋊C₁₆)	C₂²	1	8	yes	—
16	#4461	C₂²	1	4	yes	—
17	#4461	C₂²	1	4	yes	—
18	C₄×C₃⋊C₁₆	C₂²	1	4	yes	—
19	#2960	D₃	1	1	yes	characteristic
20	#2183	—	3	8	no	—
21	#2164	—	3	4	no	—
22	#2163	—	3	2	no	—
23	C₂×C₃×M₁₆	C₂×C₄	1	2	yes	—
24	C₂²×C₃×C₈	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₃×C₈	C₂³	1	1	yes	characteristic
26	C₃×C₄⋊Q₈	C₈	1	2	yes	—
27	C₃×#125	C₈	1	2	yes	—
28	C₃×C₂²⋊Q₈	C₈	1	4	yes	—
29	C₃×C₄×Q₈	C₂×C₄	1	4	yes	—
30	C₃×#117	C₂×C₄	1	2	yes	—
31	C₂×C₃×C₄²	C₂×C₄	1	1	yes	characteristic
32	C₃×C₄⋊C₈	C₂×C₄	1	4	yes	—
33	C₃×C₄⋊C₈	C₂×C₄	1	8	yes	—
34	C₃×#97	C₂×C₄	1	4	yes	—
35	C₃×C₄×C₈	C₂×C₄	1	2	yes	—
36	C₃×C₄×C₈	C₂³	1	4	yes	—
37	C₃×C₄×C₈	C₂³	1	2	yes	—
38	C₂×C₃⋊C₁₆	C₂³	1	8	yes	—
39	#533	Dic₃	1	1	yes	characteristic
40	#446	Dic₃	1	4	yes	—
41	C₈×Q₈	C₂×D₃	1	4	yes	—
42	#424	Dic₃	1	2	yes	—
43	#423	Dic₃	1	1	yes	characteristic
44	#405	C₂×D₃	1	2	yes	—
45	C₂×C₄×C₈	C₂×D₃	1	1	yes	characteristic
46	C₄⋊C₁₆	—	3	4	no	—
47	C₄⋊C₁₆	—	3	8	no	—
48	C₂²⋊C₁₆	—	3	4	no	—
49	#346	—	3	4	no	—
50	#346	—	3	4	no	—
51	C₄×C₁₆	—	3	4	no	—
52	C₂×C₃×Q₈	C₂×C₈	1	4	yes	—
53	C₂²×C₃×C₄	C₂×C₈	1	2	yes	—
54	C₂²×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
55	C₃×M₁₆	—	4	2	no	—
56	C₂×C₃×C₈	—	2	2	no	—
57	C₂×C₃×C₈	—	4	1	no	—
58	C₂×C₃×C₈	C₂²×C₄	1	1	yes	characteristic
59	C₂×C₃×C₈	C₂²×C₄	1	2	yes	—
60	C₂×C₃×C₈	C₂²×C₄	1	4	yes	—
61	C₂×C₃×C₈	C₂⁴	1	1	yes	characteristic
62	C₃×C₄⋊C₄	C₂×C₈	1	4	yes	—
63	C₃×C₄⋊C₄	C₂×C₈	1	8	yes	—
64	C₃×C₄⋊C₄	C₂×C₈	1	4	yes	—
65	C₃×C₂²⋊C₄	C₂×C₈	1	4	yes	—
66	C₃×C₄²	C₂×C₈	1	1	yes	characteristic
67	C₃×C₄²	C₂×C₈	1	1	yes	characteristic
68	C₃×C₄²	C₂²×C₄	1	2	yes	—
69	C₃×C₄²	C₂²×C₄	1	4	yes	—
70	C₃⋊C₁₆	—	2	8	no	—
71	C₂×M₁₆	C₂×Dic₃	1	2	yes	—
72	C₂²×C₈	C₂×Dic₃	1	1	yes	characteristic
73	C₂²×C₈	C₂²×D₃	1	1	yes	characteristic
74	C₄⋊Q₈	C₃⋊C₈	1	2	yes	—
75	#125	C₃⋊C₈	1	2	yes	—
76	C₂²⋊Q₈	C₃⋊C₈	1	4	yes	—
77	C₄×Q₈	C₂×Dic₃	1	4	yes	—
78	#117	C₂×Dic₃	1	2	yes	—
79	C₂×C₄²	C₂×Dic₃	1	1	yes	characteristic
80	C₂×C₁₆	—	3	8	no	—
81	C₄⋊C₈	C₂×Dic₃	1	4	yes	—
82	C₄⋊C₈	C₂×Dic₃	1	8	yes	—
83	#97	C₂×Dic₃	1	4	yes	—
84	C₄×C₈	C₂×Dic₃	1	2	yes	—
85	C₄×C₈	C₂²×D₃	1	4	yes	—
86	C₄×C₈	C₂²×D₃	1	2	yes	—
87	C₂³×C₃	C₂²×C₈	1	1	yes	characteristic
88	C₃×Q₈	—	4	4	no	—
89	C₂×C₃×C₄	—	2	1	no	—
90	C₂×C₃×C₄	—	2	1	no	—
91	C₂×C₃×C₄	—	4	2	no	—
92	C₂×C₃×C₄	C₂²×C₈	1	1	yes	characteristic
93	C₂×C₃×C₄	C₂²×C₈	1	2	yes	—
94	C₂×C₃×C₄	C₂²×C₈	1	2	yes	—
95	C₂×C₃×C₄	C₂²×C₈	1	8	yes	—
96	C₂×C₃×C₄	C₂³×C₄	1	1	yes	characteristic
97	C₃×C₈	—	2	1	no	—
98	C₃×C₈	—	2	2	no	—
99	C₃×C₈	—	2	4	no	—
100	C₃×C₈	2₋¹⁺⁴	1	2	yes	—
101	C₂×Q₈	C₂×C₃⋊C₈	1	4	yes	—
102	C₂²×C₄	C₂×C₃⋊C₈	1	2	yes	—
103	C₂²×C₄	C₂²×Dic₃	1	1	yes	characteristic
104	M₁₆	—	4	2	no	—
105	C₂×C₈	—	2	2	no	—
106	C₂×C₈	—	4	1	no	—
107	C₂×C₈	C₂²×Dic₃	1	1	yes	characteristic
108	C₂×C₈	C₂²×Dic₃	1	2	yes	—
109	C₂×C₈	C₂²×Dic₃	1	4	yes	—
110	C₂×C₈	C₂³×D₃	1	1	yes	characteristic
111	C₄⋊C₄	C₂×C₃⋊C₈	1	4	yes	—
112	C₄⋊C₄	C₂×C₃⋊C₈	1	8	yes	—
113	C₄⋊C₄	C₂×C₃⋊C₈	1	4	yes	—
114	C₂²⋊C₄	C₂×C₃⋊C₈	1	4	yes	—
115	C₄²	C₂×C₃⋊C₈	1	1	yes	characteristic
116	C₄²	C₂×C₃⋊C₈	1	1	yes	characteristic
117	C₄²	C₂²×Dic₃	1	2	yes	—
118	C₄²	C₂²×Dic₃	1	4	yes	—
119	C₁₆	—	6	8	no	—
120	C₂²×C₃	—	2	2	no	—
121	C₂²×C₃	—	2	1	no	—
122	C₂²×C₃	C₂³×C₈	1	1	yes	characteristic
123	C₃×C₄	—	2	2	no	—
124	C₃×C₄	—	2	2	no	—
125	C₃×C₄	—	2	8	no	—
126	C₃×C₄	—	2	1	no	—
127	C₃×C₄	#519	1	1	yes	characteristic
128	C₃×C₄	#568	1	1	yes	characteristic
129	C₂³	C₂²×C₃⋊C₈	1	1	yes	characteristic
130	Q₈	—	4	4	no	—
131	C₂×C₄	—	2	1	no	—
132	C₂×C₄	—	2	1	no	—
133	C₂×C₄	—	4	2	no	—
134	C₂×C₄	C₂²×C₃⋊C₈	1	1	yes	characteristic
135	C₂×C₄	C₂²×C₃⋊C₈	1	2	yes	—
136	C₂×C₄	C₂²×C₃⋊C₈	1	2	yes	—
137	C₂×C₄	C₂²×C₃⋊C₈	1	8	yes	—
138	C₂×C₄	C₂³×Dic₃	1	1	yes	characteristic
139	C₈	—	2	1	no	—
140	C₈	—	2	2	no	—
141	C₈	—	2	4	no	—
142	C₈	#1007	1	2	yes	—
143	C₂×C₃	—	4	1	no	—
144	C₂×C₃	#2982	1	1	yes	characteristic
145	C₂×C₃	#3407	1	2	yes	—
146	C₂²	—	2	1	no	—
147	C₂²	—	2	2	no	—
148	C₂²	C₂³×C₃⋊C₈	1	1	yes	characteristic
149	C₄	—	2	1	no	—
150	C₄	—	2	2	no	—
151	C₄	—	2	2	no	—
152	C₄	—	2	8	no	—
153	C₄	#5812	1	1	yes	characteristic
154	C₄	#5819	1	1	yes	characteristic
155	C₃	#33501	1	1	yes	characteristic, complemented, Sylow
156	C₂	—	4	1	no	—
157	C₂	#82435	1	1	yes	characteristic
158	C₂	#83780	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₈	C₂³×D₃
commutator, socle	C₂²×C₃	C₂³×C₈
Fitting	C₃×#2960	C₂
prime radical	C₂³×C₃	C₂²×C₈
cosocle	C₂×C₃×C₈	C₂⁴
hypercenter	#2960	D₃
hypocenter	C₃	#33501
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₈ ◁ #2960

Complements

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

N	H ≅ G/N	type
2. #70949	C₂	semidirect
155. C₃	#33501	semidirect

Maximal subgroups

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

subgroup	index
C₃×#2960	2
#70949	2
#69883	2
#69745	2
#33501	3