Subgroups — #10948849

Order 768.

460 conjugacy classes of subgroups.

The subgroup lattice has 168 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	4	yes	maximal, complemented
3	Q₈×C₃⋊C₁₆	C₂	1	4	yes	maximal, complemented
4	#69883	C₂	1	4	yes	maximal
5	#69744	C₂	1	2	yes	maximal
6	#33500	—	3	1	no	maximal, Sylow
7	C₃×#533	C₄	1	1	yes	characteristic
8	C₃×#446	C₄	1	4	yes	—
9	C₃×C₈×Q₈	C₂²	1	4	yes	—
10	C₃×#424	C₄	1	2	yes	—
11	C₃×#423	C₄	1	1	yes	characteristic
12	C₃×#405	C₂²	1	2	yes	—
13	C₂×C₃×C₄×C₈	C₂²	1	1	yes	characteristic
14	#4549	C₂²	1	4	yes	—
15	C₄⋊(C₃⋊C₁₆)	C₂²	1	2	yes	—
16	C₄⋊(C₃⋊C₁₆)	C₂²	1	8	yes	—
17	C₄⋊(C₃⋊C₁₆)	C₂²	1	2	yes	—
18	#4461	C₂²	1	4	yes	—
19	C₄×C₃⋊C₁₆	C₂²	1	4	yes	—
20	C₄×C₃⋊C₁₆	C₂²	1	4	yes	—
21	#2960	D₃	1	1	yes	characteristic
22	#2183	—	3	4	no	—
23	Q₈×C₁₆	—	3	4	no	—
24	#2164	—	3	4	no	—
25	#2162	—	3	2	no	—
26	C₂×C₃×M₁₆	C₂×C₄	1	2	yes	—
27	C₂²×C₃×C₈	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₃×C₈	C₂³	1	1	yes	characteristic
29	C₃×C₄⋊Q₈	C₈	1	2	yes	—
30	C₃×#125	C₈	1	2	yes	—
31	C₃×C₂²⋊Q₈	C₈	1	4	yes	—
32	C₃×C₄×Q₈	C₂×C₄	1	4	yes	—
33	C₃×#117	C₂×C₄	1	2	yes	—
34	C₂×C₃×C₄²	C₂×C₄	1	1	yes	characteristic
35	C₃×C₄⋊C₈	C₂×C₄	1	4	yes	—
36	C₃×C₄⋊C₈	C₂×C₄	1	8	yes	—
37	C₃×#97	C₂×C₄	1	4	yes	—
38	C₃×C₄×C₈	C₂×C₄	1	2	yes	—
39	C₃×C₄×C₈	C₂³	1	2	yes	—
40	C₃×C₄×C₈	C₂³	1	4	yes	—
41	C₂×C₃⋊C₁₆	C₂³	1	4	yes	—
42	C₂×C₃⋊C₁₆	C₂³	1	4	yes	—
43	#533	Dic₃	1	1	yes	characteristic
44	#446	Dic₃	1	4	yes	—
45	C₈×Q₈	C₂×D₃	1	4	yes	—
46	#424	Dic₃	1	2	yes	—
47	#423	Dic₃	1	1	yes	characteristic
48	#405	C₂×D₃	1	2	yes	—
49	C₂×C₄×C₈	C₂×D₃	1	1	yes	characteristic
50	C₄⋊C₁₆	—	3	2	no	—
51	C₄⋊C₁₆	—	3	2	no	—
52	C₄⋊C₁₆	—	3	8	no	—
53	C₂²⋊C₁₆	—	3	4	no	—
54	#346	—	3	4	no	—
55	C₄×C₁₆	—	3	4	no	—
56	C₄×C₁₆	—	3	4	no	—
57	C₂×C₃×Q₈	C₂×C₈	1	4	yes	—
58	C₂²×C₃×C₄	C₂×C₈	1	2	yes	—
59	C₂²×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
60	C₃×M₁₆	—	4	2	no	—
61	C₂×C₃×C₈	—	2	2	no	—
62	C₂×C₃×C₈	—	4	1	no	—
63	C₂×C₃×C₈	C₂²×C₄	1	1	yes	characteristic
64	C₂×C₃×C₈	C₂²×C₄	1	2	yes	—
65	C₂×C₃×C₈	C₂²×C₄	1	4	yes	—
66	C₂×C₃×C₈	C₂⁴	1	1	yes	characteristic
67	C₃×C₄⋊C₄	C₂×C₈	1	4	yes	—
68	C₃×C₄⋊C₄	C₂×C₈	1	8	yes	—
69	C₃×C₄⋊C₄	C₂×C₈	1	4	yes	—
70	C₃×C₂²⋊C₄	C₂×C₈	1	4	yes	—
71	C₃×C₄²	C₂×C₈	1	1	yes	characteristic
72	C₃×C₄²	C₂×C₈	1	1	yes	characteristic
73	C₃×C₄²	C₂²×C₄	1	2	yes	—
74	C₃×C₄²	C₂²×C₄	1	4	yes	—
75	C₃⋊C₁₆	—	2	4	no	—
76	C₃⋊C₁₆	—	2	4	no	—
77	C₂×M₁₆	C₂×Dic₃	1	2	yes	—
78	C₂²×C₈	C₂×Dic₃	1	1	yes	characteristic
79	C₂²×C₈	C₂²×D₃	1	1	yes	characteristic
80	C₄⋊Q₈	C₃⋊C₈	1	2	yes	—
81	#125	C₃⋊C₈	1	2	yes	—
82	C₂²⋊Q₈	C₃⋊C₈	1	4	yes	—
83	C₄×Q₈	C₂×Dic₃	1	4	yes	—
84	#117	C₂×Dic₃	1	2	yes	—
85	C₂×C₄²	C₂×Dic₃	1	1	yes	characteristic
86	C₂×C₁₆	—	3	4	no	—
87	C₂×C₁₆	—	3	4	no	—
88	C₄⋊C₈	C₂×Dic₃	1	4	yes	—
89	C₄⋊C₈	C₂×Dic₃	1	8	yes	—
90	#97	C₂×Dic₃	1	4	yes	—
91	C₄×C₈	C₂×Dic₃	1	2	yes	—
92	C₄×C₈	C₂²×D₃	1	2	yes	—
93	C₄×C₈	C₂²×D₃	1	4	yes	—
94	C₂³×C₃	C₂²×C₈	1	1	yes	characteristic
95	C₃×Q₈	—	2	8	no	—
96	C₂×C₃×C₄	—	2	1	no	—
97	C₂×C₃×C₄	—	2	1	no	—
98	C₂×C₃×C₄	—	2	4	no	—
99	C₂×C₃×C₄	C₂²×C₈	1	1	yes	characteristic
100	C₂×C₃×C₄	C₂²×C₈	1	2	yes	—
101	C₂×C₃×C₄	C₂²×C₈	1	8	yes	—
102	C₂×C₃×C₄	C₂²×C₈	1	2	yes	—
103	C₂×C₃×C₄	C₂³×C₄	1	1	yes	characteristic
104	C₃×C₈	—	2	1	no	—
105	C₃×C₈	—	2	2	no	—
106	C₃×C₈	—	2	4	no	—
107	C₃×C₈	2₋¹⁺⁴	1	2	yes	—
108	C₂×Q₈	C₂×C₃⋊C₈	1	4	yes	—
109	C₂²×C₄	C₂×C₃⋊C₈	1	2	yes	—
110	C₂²×C₄	C₂²×Dic₃	1	1	yes	characteristic
111	M₁₆	—	4	2	no	—
112	C₂×C₈	—	2	2	no	—
113	C₂×C₈	—	4	1	no	—
114	C₂×C₈	C₂²×Dic₃	1	1	yes	characteristic
115	C₂×C₈	C₂²×Dic₃	1	2	yes	—
116	C₂×C₈	C₂²×Dic₃	1	4	yes	—
117	C₂×C₈	C₂³×D₃	1	1	yes	characteristic
118	C₄⋊C₄	C₂×C₃⋊C₈	1	4	yes	—
119	C₄⋊C₄	C₂×C₃⋊C₈	1	8	yes	—
120	C₄⋊C₄	C₂×C₃⋊C₈	1	4	yes	—
121	C₂²⋊C₄	C₂×C₃⋊C₈	1	4	yes	—
122	C₄²	C₂×C₃⋊C₈	1	1	yes	characteristic
123	C₄²	C₂×C₃⋊C₈	1	1	yes	characteristic
124	C₄²	C₂²×Dic₃	1	2	yes	—
125	C₄²	C₂²×Dic₃	1	4	yes	—
126	C₁₆	—	6	4	no	—
127	C₁₆	—	6	4	no	—
128	C₂²×C₃	—	2	2	no	—
129	C₂²×C₃	—	2	1	no	—
130	C₂²×C₃	C₂³×C₈	1	1	yes	characteristic
131	C₃×C₄	—	2	2	no	—
132	C₃×C₄	—	2	8	no	—
133	C₃×C₄	—	2	1	no	—
134	C₃×C₄	#504	1	4	yes	—
135	C₃×C₄	#519	1	1	yes	characteristic
136	C₃×C₄	#568	1	1	yes	characteristic
137	C₂³	C₂²×C₃⋊C₈	1	1	yes	characteristic
138	Q₈	—	2	8	no	—
139	C₂×C₄	—	2	4	no	—
140	C₂×C₄	—	2	1	no	—
141	C₂×C₄	—	2	1	no	—
142	C₂×C₄	C₂²×C₃⋊C₈	1	1	yes	characteristic
143	C₂×C₄	C₂²×C₃⋊C₈	1	2	yes	—
144	C₂×C₄	C₂²×C₃⋊C₈	1	8	yes	—
145	C₂×C₄	C₂²×C₃⋊C₈	1	2	yes	—
146	C₂×C₄	C₂³×Dic₃	1	1	yes	characteristic
147	C₈	—	2	1	no	—
148	C₈	—	2	2	no	—
149	C₈	—	2	4	no	—
150	C₈	#1007	1	2	yes	—
151	C₂×C₃	—	4	1	no	—
152	C₂×C₃	#2982	1	1	yes	characteristic
153	C₂×C₃	C₂×#504	1	2	yes	—
154	C₂²	—	2	1	no	—
155	C₂²	—	2	2	no	—
156	C₂²	C₂³×C₃⋊C₈	1	1	yes	characteristic
157	C₄	—	2	1	no	—
158	C₄	—	2	2	no	—
159	C₄	—	2	8	no	—
160	C₄	#5140	1	4	yes	—
161	C₄	#5812	1	1	yes	characteristic
162	C₄	#5819	1	1	yes	characteristic
163	C₃	#33500	1	1	yes	characteristic, complemented, Sylow
164	C₂	—	4	1	no	—
165	C₂	#82435	1	1	yes	characteristic
166	C₂	C₂×#5140	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	#33500
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
3. Q₈×C₃⋊C₁₆	C₂	semidirect
163. C₃	#33500	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
Q₈×C₃⋊C₁₆	2
#69883	2
#69744	2
#33500	3