Subgroups — #10711534

Order 768.

366 conjugacy classes of subgroups.

The subgroup lattice has 170 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₄²×C₈	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄⋊(C₃⋊C₁₆)	C₂	1	2	yes	maximal
3	#67657	C₂	1	4	yes	maximal
4	#12879	—	3	1	no	maximal, Sylow
5	C₂×C₃×C₄×C₈	C₄	1	2	yes	—
6	C₂×C₃×C₄×C₈	C₄	1	1	yes	characteristic
7	C₂×C₃×C₄×C₈	C₂²	1	2	yes	—
8	C₂×C₃×C₄×C₈	C₂²	1	1	yes	characteristic
9	C₃×C₄³	C₄	1	1	yes	characteristic
10	C₂²×C₃⋊C₁₆	C₂²	1	4	yes	—
11	C₄⋊(C₃⋊C₁₆)	—	2	4	no	—
12	C₂×C₄⋊C₁₆	—	3	2	no	—
13	C₄²×C₈	D₃	1	1	yes	characteristic
14	#1374	—	3	4	no	—
15	C₂²×C₃×C₈	C₂×C₄	1	1	yes	characteristic
16	C₂²×C₃×C₈	C₂×C₄	1	2	yes	—
17	C₂²×C₃×C₈	C₂³	1	1	yes	characteristic
18	C₂×C₃×C₄²	C₈	1	2	yes	—
19	C₂×C₃×C₄²	C₈	1	1	yes	characteristic
20	C₂×C₃×C₄²	C₈	1	1	yes	characteristic
21	C₂×C₃×C₄²	C₂×C₄	1	2	yes	—
22	C₂×C₃×C₄²	C₂×C₄	1	1	yes	characteristic
23	C₃×C₄×C₈	—	2	4	no	—
24	C₃×C₄×C₈	—	2	4	no	—
25	C₃×C₄×C₈	—	2	2	no	—
26	C₃×C₄×C₈	D₄	1	2	yes	—
27	C₃×C₄×C₈	Q₈	1	2	yes	—
28	C₂×C₃⋊C₁₆	—	2	4	no	—
29	C₂×C₃⋊C₁₆	—	2	8	no	—
30	C₂²×C₁₆	—	3	4	no	—
31	C₂×C₄×C₈	Dic₃	1	2	yes	—
32	C₂×C₄×C₈	Dic₃	1	1	yes	characteristic
33	C₂×C₄×C₈	C₂×D₃	1	2	yes	—
34	C₂×C₄×C₈	C₂×D₃	1	1	yes	characteristic
35	C₄³	Dic₃	1	1	yes	characteristic
36	C₄⋊C₁₆	—	6	4	no	—
37	C₂²×C₃×C₄	C₂×C₈	1	1	yes	characteristic
38	C₂²×C₃×C₄	C₂×C₈	1	2	yes	—
39	C₂²×C₃×C₄	C₂×C₈	1	2	yes	—
40	C₂²×C₃×C₄	C₂×C₈	1	1	yes	characteristic
41	C₂²×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
42	C₂×C₃×C₈	—	2	2	no	—
43	C₂×C₃×C₈	—	2	2	no	—
44	C₂×C₃×C₈	—	2	2	no	—
45	C₂×C₃×C₈	C₄⋊C₄	1	2	yes	—
46	C₂×C₃×C₈	M₁₆	1	4	yes	—
47	C₂×C₃×C₈	C₂×D₄	1	1	yes	characteristic
48	C₂×C₃×C₈	C₂×Q₈	1	1	yes	characteristic
49	C₂×C₃×C₈	#41	1	4	yes	—
50	C₃×C₄²	—	2	4	no	—
51	C₃×C₄²	—	2	2	no	—
52	C₃×C₄²	—	2	2	no	—
53	C₃×C₄²	—	2	2	no	—
54	C₃×C₄²	C₁₆	1	4	yes	complemented
55	C₃×C₄²	C₄⋊C₄	1	2	yes	—
56	C₃×C₄²	M₁₆	1	2	yes	—
57	C₃⋊C₁₆	—	4	4	no	—
58	C₂²×C₈	C₂×Dic₃	1	2	yes	—
59	C₂²×C₈	C₂×Dic₃	1	1	yes	characteristic
60	C₂²×C₈	C₂²×D₃	1	1	yes	characteristic
61	C₂×C₄²	C₃⋊C₈	1	2	yes	—
62	C₂×C₄²	C₃⋊C₈	1	1	yes	characteristic
63	C₂×C₄²	C₃⋊C₈	1	1	yes	characteristic
64	C₂×C₄²	C₂×Dic₃	1	2	yes	—
65	C₂×C₄²	C₂×Dic₃	1	1	yes	characteristic
66	C₂×C₁₆	—	6	4	no	—
67	C₂×C₁₆	—	6	8	no	—
68	C₄×C₈	—	2	4	no	—
69	C₄×C₈	—	2	4	no	—
70	C₄×C₈	—	2	2	no	—
71	C₄×C₈	Dic₆	1	2	yes	—
72	C₄×C₈	D₁₂	1	2	yes	—
73	C₂³×C₃	C₂²×C₈	1	1	yes	characteristic
74	C₂×C₃×C₄	—	2	2	no	—
75	C₂×C₃×C₄	—	2	2	no	—
76	C₂×C₃×C₄	—	2	2	no	—
77	C₂×C₃×C₄	—	2	2	no	—
78	C₂×C₃×C₄	—	2	2	no	—
79	C₂×C₃×C₄	C₄⋊C₈	1	2	yes	—
80	C₂×C₃×C₄	C₄⋊C₈	1	2	yes	—
81	C₂×C₃×C₄	C₂×C₁₆	1	4	yes	—
82	C₂×C₃×C₄	C₂×C₁₆	1	2	yes	—
83	C₂×C₃×C₄	M₃₂	1	4	yes	—
84	C₂×C₃×C₄	M₃₂	1	2	yes	—
85	C₂×C₃×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
86	C₂×C₃×C₄	#117	1	2	yes	—
87	C₂×C₃×C₄	C₂×M₁₆	1	2	yes	—
88	C₂×C₃×C₄	C₂×M₁₆	1	1	yes	characteristic
89	C₃×C₈	—	2	2	no	—
90	C₃×C₈	—	2	4	no	—
91	C₃×C₈	#124	1	2	yes	—
92	C₃×C₈	#125	1	2	yes	—
93	C₂²×C₄	C₂×C₃⋊C₈	1	1	yes	characteristic
94	C₂²×C₄	C₂×C₃⋊C₈	1	2	yes	—
95	C₂²×C₄	C₂×C₃⋊C₈	1	2	yes	—
96	C₂²×C₄	C₂×C₃⋊C₈	1	1	yes	characteristic
97	C₂²×C₄	C₂²×Dic₃	1	1	yes	characteristic
98	C₂×C₈	—	2	2	no	—
99	C₂×C₈	—	2	2	no	—
100	C₂×C₈	—	2	2	no	—
101	C₂×C₈	#208	1	4	yes	—
102	C₂×C₈	C₄⋊Dic₃	1	2	yes	—
103	C₂×C₈	C₂×Dic₆	1	1	yes	characteristic
104	C₂×C₈	C₂×D₁₂	1	1	yes	characteristic
105	C₂×C₈	#235	1	4	yes	—
106	C₄²	—	2	4	no	—
107	C₄²	—	2	2	no	—
108	C₄²	—	2	2	no	—
109	C₄²	—	2	2	no	—
110	C₄²	C₃⋊C₁₆	1	4	yes	complemented
111	C₄²	#208	1	2	yes	—
112	C₄²	C₄⋊Dic₃	1	2	yes	—
113	C₁₆	—	12	4	no	—
114	C₂²×C₃	C₂×C₄⋊C₈	1	1	yes	characteristic
115	C₂²×C₃	#431	1	2	yes	—
116	C₂²×C₃	C₂²×C₁₆	1	1	yes	characteristic
117	C₂²×C₃	C₂×M₃₂	1	1	yes	characteristic
118	C₂²×C₃	C₂×M₃₂	1	2	yes	—
119	C₃×C₄	—	2	2	no	—
120	C₃×C₄	—	2	4	no	—
121	C₃×C₄	—	2	4	no	—
122	C₃×C₄	C₄⋊C₁₆	1	4	yes	—
123	C₃×C₄	#382	1	1	yes	characteristic
124	C₃×C₄	#423	1	1	yes	characteristic
125	C₃×C₄	#432	1	2	yes	—
126	C₂³	C₂²×C₃⋊C₈	1	1	yes	characteristic
127	C₂×C₄	—	2	2	no	—
128	C₂×C₄	—	2	2	no	—
129	C₂×C₄	—	2	2	no	—
130	C₂×C₄	—	2	2	no	—
131	C₂×C₄	—	2	2	no	—
132	C₂×C₄	C₄⋊(C₃⋊C₈)	1	2	yes	—
133	C₂×C₄	C₄⋊(C₃⋊C₈)	1	2	yes	—
134	C₂×C₄	C₂×C₃⋊C₁₆	1	4	yes	—
135	C₂×C₄	C₂×C₃⋊C₁₆	1	2	yes	—
136	C₂×C₄	#812	1	4	yes	—
137	C₂×C₄	#812	1	2	yes	—
138	C₂×C₄	C₂×#208	1	2	yes	—
139	C₂×C₄	C₂×#208	1	1	yes	characteristic
140	C₂×C₄	C₂×C₄⋊Dic₃	1	1	yes	characteristic
141	C₂×C₄	#926	1	2	yes	—
142	C₈	—	2	2	no	—
143	C₈	—	2	4	no	—
144	C₈	#870	1	2	yes	—
145	C₈	#875	1	2	yes	—
146	C₂×C₃	#1831	1	1	yes	characteristic
147	C₂×C₃	C₂×C₄⋊C₁₆	1	1	yes	characteristic
148	C₂×C₃	#2150	1	1	yes	characteristic
149	C₂×C₃	#2162	1	2	yes	—
150	C₂×C₃	#2163	1	2	yes	—
151	C₂²	C₂×C₄⋊(C₃⋊C₈)	1	1	yes	characteristic
152	C₂²	#4925	1	2	yes	—
153	C₂²	C₂²×C₃⋊C₁₆	1	1	yes	characteristic
154	C₂²	C₂×#812	1	1	yes	characteristic
155	C₂²	C₂×#812	1	2	yes	—
156	C₄	—	2	2	no	—
157	C₄	—	2	4	no	—
158	C₄	—	2	4	no	—
159	C₄	C₄⋊(C₃⋊C₁₆)	1	4	yes	—
160	C₄	#4924	1	1	yes	characteristic
161	C₄	#4926	1	2	yes	—
162	C₄	#4936	1	1	yes	characteristic
163	C₃	#12879	1	1	yes	characteristic, complemented, Sylow
164	C₂	#69392	1	1	yes	characteristic
165	C₂	C₂×C₄⋊(C₃⋊C₁₆)	1	1	yes	characteristic
166	C₂	#69743	1	1	yes	characteristic
167	C₂	#69744	1	2	yes	—
168	C₂	#69745	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₈	C₂²×D₃
commutator	C₂²×C₃	C₂²×C₁₆
Fitting	C₃×C₄²×C₈	C₂
socle, prime radical	C₂³×C₃	C₂²×C₈
cosocle	C₂²×C₃×C₈	C₂³
hypercenter	C₄²×C₈	D₃
hypocenter	C₃	#12879
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₈ ◁ C₄²×C₈

Complements

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

N	H ≅ G/N	type
54. C₃×C₄²	C₁₆	semidirect
110. C₄²	C₃⋊C₁₆	semidirect
163. C₃	#12879	semidirect

Maximal subgroups

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

subgroup	index
C₃×C₄²×C₈	2
C₂×C₄⋊(C₃⋊C₁₆)	2
#67657	2
#12879	3