Subgroups — #10948836

Order 768.

494 conjugacy classes of subgroups.

The subgroup lattice has 182 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₈×Q₈	C₂	1	1	yes	maximal, characteristic
2	#70949	C₂	1	6	yes	maximal, complemented
3	Q₈×C₃⋊C₁₆	C₂	1	2	yes	maximal, complemented
4	#69745	C₂	1	3	yes	maximal
5	#69744	C₂	1	3	yes	maximal
6	#33498	—	3	1	no	maximal, Sylow
7	C₂×C₃×C₄×Q₈	C₄	1	1	yes	characteristic
8	C₃×C₈×Q₈	C₄	1	3	yes	—
9	C₃×C₈×Q₈	C₄	1	1	yes	characteristic
10	C₃×C₈×Q₈	C₂²	1	1	yes	characteristic
11	C₃×C₈×Q₈	C₂²	1	3	yes	—
12	C₂×C₃×C₄⋊C₈	C₄	1	3	yes	—
13	C₂×C₃×C₄×C₈	C₂²	1	3	yes	—
14	#4549	C₂²	1	1	yes	characteristic
15	#4549	C₂²	1	3	yes	—
16	C₄⋊(C₃⋊C₁₆)	C₂²	1	6	yes	—
17	C₄⋊(C₃⋊C₁₆)	C₂²	1	6	yes	—
18	#4461	C₂²	1	6	yes	—
19	C₄×C₃⋊C₁₆	C₂²	1	6	yes	—
20	C₂×C₈×Q₈	D₃	1	1	yes	characteristic
21	#2183	—	3	6	no	—
22	Q₈×C₁₆	—	3	2	no	—
23	#2163	—	3	3	no	—
24	#2162	—	3	3	no	—
25	C₂²×C₃×Q₈	C₈	1	1	yes	characteristic
26	C₂²×C₃×C₈	C₂×C₄	1	3	yes	—
27	C₂²×C₃×C₈	C₂³	1	1	yes	characteristic
28	C₃×C₄×Q₈	C₈	1	3	yes	—
29	C₃×C₄×Q₈	C₈	1	1	yes	characteristic
30	C₃×C₄×Q₈	C₂×C₄	1	3	yes	—
31	C₃×C₄×Q₈	C₂×C₄	1	1	yes	characteristic
32	C₂×C₃×C₄⋊C₄	C₈	1	3	yes	—
33	C₂×C₃×C₄²	C₂×C₄	1	3	yes	—
34	C₃×C₄⋊C₈	C₂×C₄	1	3	yes	—
35	C₃×C₄⋊C₈	C₂×C₄	1	6	yes	—
36	C₃×C₄⋊C₈	C₂×C₄	1	3	yes	—
37	C₃×C₄×C₈	C₂×C₄	1	3	yes	—
38	C₃×C₄×C₈	C₂×C₄	1	3	yes	—
39	C₃×C₄×C₈	C₂³	1	3	yes	—
40	C₃×C₄×C₈	C₂³	1	3	yes	—
41	C₂×C₃⋊C₁₆	C₂³	1	2	yes	—
42	C₂×C₃⋊C₁₆	C₂³	1	6	yes	—
43	C₂×C₄×Q₈	Dic₃	1	1	yes	characteristic
44	C₈×Q₈	Dic₃	1	3	yes	—
45	C₈×Q₈	Dic₃	1	1	yes	characteristic
46	C₈×Q₈	C₂×D₃	1	1	yes	characteristic
47	C₈×Q₈	C₂×D₃	1	3	yes	—
48	C₂×C₄⋊C₈	Dic₃	1	3	yes	—
49	C₂×C₄×C₈	C₂×D₃	1	3	yes	—
50	C₄⋊C₁₆	—	3	6	no	—
51	C₄⋊C₁₆	—	3	6	no	—
52	C₂²⋊C₁₆	—	3	1	no	—
53	C₂²⋊C₁₆	—	3	3	no	—
54	#346	—	3	6	no	—
55	C₄×C₁₆	—	3	6	no	—
56	C₂×C₃×Q₈	—	2	4	no	—
57	C₂×C₃×Q₈	C₂×C₈	1	3	yes	—
58	C₂×C₃×Q₈	C₂×C₈	1	1	yes	characteristic
59	C₂²×C₃×C₄	C₂×C₈	1	3	yes	—
60	C₂²×C₃×C₄	C₂×C₈	1	3	yes	—
61	C₂²×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
62	C₂×C₃×C₈	—	2	2	no	—
63	C₂×C₃×C₈	—	4	3	no	—
64	C₂×C₃×C₈	C₂²×C₄	1	1	yes	characteristic
65	C₂×C₃×C₈	C₂²×C₄	1	3	yes	—
66	C₂×C₃×C₈	C₂²×C₄	1	3	yes	—
67	C₂×C₃×C₈	C₂⁴	1	1	yes	characteristic
68	C₃×C₄⋊C₄	C₂×C₈	1	6	yes	—
69	C₃×C₄⋊C₄	C₂×C₈	1	3	yes	—
70	C₃×C₄⋊C₄	C₂×C₈	1	3	yes	—
71	C₃×C₄²	C₂×C₈	1	3	yes	—
72	C₃×C₄²	C₂×C₈	1	3	yes	—
73	C₃×C₄²	C₂²×C₄	1	3	yes	—
74	C₃×C₄²	C₂²×C₄	1	3	yes	—
75	C₃⋊C₁₆	—	2	2	no	—
76	C₃⋊C₁₆	—	2	6	no	—
77	C₂²×Q₈	C₃⋊C₈	1	1	yes	characteristic
78	C₂²×C₈	C₂×Dic₃	1	3	yes	—
79	C₂²×C₈	C₂²×D₃	1	1	yes	characteristic
80	C₄×Q₈	C₃⋊C₈	1	3	yes	—
81	C₄×Q₈	C₃⋊C₈	1	1	yes	characteristic
82	C₄×Q₈	C₂×Dic₃	1	3	yes	—
83	C₄×Q₈	C₂×Dic₃	1	1	yes	characteristic
84	C₂×C₄⋊C₄	C₃⋊C₈	1	3	yes	—
85	C₂×C₄²	C₂×Dic₃	1	3	yes	—
86	C₂×C₁₆	—	3	2	no	—
87	C₂×C₁₆	—	3	6	no	—
88	C₄⋊C₈	C₂×Dic₃	1	3	yes	—
89	C₄⋊C₈	C₂×Dic₃	1	6	yes	—
90	C₄⋊C₈	C₂×Dic₃	1	3	yes	—
91	C₄×C₈	C₂×Dic₃	1	3	yes	—
92	C₄×C₈	C₂×Dic₃	1	3	yes	—
93	C₄×C₈	C₂²×D₃	1	3	yes	—
94	C₄×C₈	C₂²×D₃	1	3	yes	—
95	C₂³×C₃	C₂²×C₈	1	1	yes	characteristic
96	C₃×Q₈	—	2	6	no	—
97	C₃×Q₈	M₃₂	1	4	yes	—
98	C₂×C₃×C₄	—	2	6	no	—
99	C₂×C₃×C₄	—	2	1	no	—
100	C₂×C₃×C₄	—	2	1	no	—
101	C₂×C₃×C₄	—	2	6	no	—
102	C₂×C₃×C₄	C₂²×C₈	1	1	yes	characteristic
103	C₂×C₃×C₄	C₂²×C₈	1	3	yes	—
104	C₂×C₃×C₄	C₂²×C₈	1	3	yes	—
105	C₂×C₃×C₄	C₂²×C₈	1	3	yes	—
106	C₂×C₃×C₄	C₂²×C₈	1	3	yes	—
107	C₂×C₃×C₄	C₂³×C₄	1	1	yes	characteristic
108	C₃×C₈	—	2	1	no	—
109	C₃×C₈	—	2	3	no	—
110	C₃×C₈	—	2	3	no	—
111	C₃×C₈	2₋¹⁺⁴	1	2	yes	—
112	C₂×Q₈	—	2	4	no	—
113	C₂×Q₈	C₂×C₃⋊C₈	1	3	yes	—
114	C₂×Q₈	C₂×C₃⋊C₈	1	1	yes	characteristic
115	C₂²×C₄	C₂×C₃⋊C₈	1	3	yes	—
116	C₂²×C₄	C₂×C₃⋊C₈	1	3	yes	—
117	C₂²×C₄	C₂²×Dic₃	1	1	yes	characteristic
118	C₂×C₈	—	2	2	no	—
119	C₂×C₈	—	4	3	no	—
120	C₂×C₈	C₂²×Dic₃	1	1	yes	characteristic
121	C₂×C₈	C₂²×Dic₃	1	3	yes	—
122	C₂×C₈	C₂²×Dic₃	1	3	yes	—
123	C₂×C₈	C₂³×D₃	1	1	yes	characteristic
124	C₄⋊C₄	C₂×C₃⋊C₈	1	6	yes	—
125	C₄⋊C₄	C₂×C₃⋊C₈	1	3	yes	—
126	C₄⋊C₄	C₂×C₃⋊C₈	1	3	yes	—
127	C₄²	C₂×C₃⋊C₈	1	3	yes	—
128	C₄²	C₂×C₃⋊C₈	1	3	yes	—
129	C₄²	C₂²×Dic₃	1	3	yes	—
130	C₄²	C₂²×Dic₃	1	3	yes	—
131	C₁₆	—	6	2	no	—
132	C₁₆	—	6	6	no	—
133	C₂²×C₃	—	2	1	no	—
134	C₂²×C₃	—	2	1	no	—
135	C₂²×C₃	#504	1	2	yes	—
136	C₂²×C₃	C₂³×C₈	1	1	yes	characteristic
137	C₃×C₄	—	2	3	no	—
138	C₃×C₄	—	2	3	no	—
139	C₃×C₄	—	2	3	no	—
140	C₃×C₄	—	2	1	no	—
141	C₃×C₄	C₂×M₃₂	1	6	yes	—
142	C₃×C₄	#519	1	1	yes	characteristic
143	C₃×C₄	#568	1	1	yes	characteristic
144	C₂³	C₂²×C₃⋊C₈	1	1	yes	characteristic
145	Q₈	—	2	6	no	—
146	Q₈	#812	1	4	yes	—
147	C₂×C₄	—	2	6	no	—
148	C₂×C₄	—	2	1	no	—
149	C₂×C₄	—	2	1	no	—
150	C₂×C₄	—	2	6	no	—
151	C₂×C₄	C₂²×C₃⋊C₈	1	1	yes	characteristic
152	C₂×C₄	C₂²×C₃⋊C₈	1	3	yes	—
153	C₂×C₄	C₂²×C₃⋊C₈	1	3	yes	—
154	C₂×C₄	C₂²×C₃⋊C₈	1	3	yes	—
155	C₂×C₄	C₂²×C₃⋊C₈	1	3	yes	—
156	C₂×C₄	C₂³×Dic₃	1	1	yes	characteristic
157	C₈	—	2	1	no	—
158	C₈	—	2	3	no	—
159	C₈	—	2	3	no	—
160	C₈	#1007	1	2	yes	—
161	C₂×C₃	—	2	2	no	—
162	C₂×C₃	#2982	1	1	yes	characteristic
163	C₂×C₃	C₂²×M₃₂	1	1	yes	characteristic
164	C₂×C₃	C₂×#504	1	1	yes	characteristic
165	C₂²	—	2	1	no	—
166	C₂²	—	2	1	no	—
167	C₂²	#5140	1	2	yes	—
168	C₂²	C₂³×C₃⋊C₈	1	1	yes	characteristic
169	C₄	—	2	1	no	—
170	C₄	—	2	3	no	—
171	C₄	—	2	3	no	—
172	C₄	—	2	3	no	—
173	C₄	C₂×#812	1	6	yes	—
174	C₄	#5812	1	1	yes	characteristic
175	C₄	#5819	1	1	yes	characteristic
176	C₃	#33498	1	1	yes	characteristic, complemented, Sylow
177	C₂	—	2	2	no	—
178	C₂	#82435	1	1	yes	characteristic
179	C₂	C₂²×#812	1	1	yes	characteristic
180	C₂	C₂×#5140	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₈	C₂³×D₃
commutator, socle	C₂²×C₃	C₂³×C₈
Fitting	C₂×C₃×C₈×Q₈	C₂
prime radical	C₂³×C₃	C₂²×C₈
cosocle	C₂×C₃×C₈	C₂⁴
hypercenter	C₂×C₈×Q₈	D₃
hypocenter	C₃	#33498
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₈×Q₈

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
176. C₃	#33498	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃×C₈×Q₈	2
#70949	2
Q₈×C₃⋊C₁₆	2
#69745	2
#69744	2
#33498	3