Subgroups — #10948832

Order 768.

486 conjugacy classes of subgroups.

The subgroup lattice has 178 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	8	yes	maximal, complemented
3	#69745	C₂	1	4	yes	maximal
4	#69744	C₂	1	2	yes	maximal
5	#33497	—	3	1	no	maximal, Sylow
6	C₂×C₃×C₄×Q₈	C₄	1	1	yes	characteristic
7	C₃×C₈×Q₈	C₄	1	4	yes	—
8	C₃×C₈×Q₈	C₂²	1	4	yes	—
9	C₂×C₃×C₄⋊C₈	C₄	1	2	yes	—
10	C₂×C₃×C₄⋊C₈	C₄	1	1	yes	characteristic
11	C₂×C₃×C₄×C₈	C₂²	1	1	yes	characteristic
12	C₂×C₃×C₄×C₈	C₂²	1	2	yes	—
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	8	yes	—
17	C₄×C₃⋊C₁₆	C₂²	1	4	yes	—
18	C₂×C₈×Q₈	D₃	1	1	yes	characteristic
19	#2183	—	3	8	no	—
20	#2163	—	3	4	no	—
21	#2162	—	3	2	no	—
22	C₂²×C₃×Q₈	C₈	1	1	yes	characteristic
23	C₂²×C₃×C₈	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₃×C₈	C₂×C₄	1	2	yes	—
25	C₂²×C₃×C₈	C₂³	1	1	yes	characteristic
26	C₃×C₄×Q₈	C₈	1	4	yes	—
27	C₃×C₄×Q₈	C₂×C₄	1	4	yes	—
28	C₂×C₃×C₄⋊C₄	C₈	1	2	yes	—
29	C₂×C₃×C₄⋊C₄	C₈	1	1	yes	characteristic
30	C₂×C₃×C₄²	C₂×C₄	1	2	yes	—
31	C₂×C₃×C₄²	C₂×C₄	1	1	yes	characteristic
32	C₃×C₄⋊C₈	C₂×C₄	1	8	yes	—
33	C₃×C₄⋊C₈	C₂×C₄	1	4	yes	—
34	C₃×C₄×C₈	C₂×C₄	1	2	yes	—
35	C₃×C₄×C₈	C₂×C₄	1	4	yes	—
36	C₃×C₄×C₈	C₂³	1	2	yes	—
37	C₃×C₄×C₈	C₂³	1	4	yes	—
38	C₂×C₃⋊C₁₆	C₂³	1	8	yes	—
39	C₂×C₄×Q₈	Dic₃	1	1	yes	characteristic
40	C₈×Q₈	Dic₃	1	4	yes	—
41	C₈×Q₈	C₂×D₃	1	4	yes	—
42	C₂×C₄⋊C₈	Dic₃	1	2	yes	—
43	C₂×C₄⋊C₈	Dic₃	1	1	yes	characteristic
44	C₂×C₄×C₈	C₂×D₃	1	1	yes	characteristic
45	C₂×C₄×C₈	C₂×D₃	1	2	yes	—
46	C₄⋊C₁₆	—	3	4	no	—
47	C₄⋊C₁₆	—	3	8	no	—
48	C₂²⋊C₁₆	—	3	4	no	—
49	#346	—	3	8	no	—
50	C₄×C₁₆	—	3	4	no	—
51	C₂×C₃×Q₈	—	2	4	no	—
52	C₂×C₃×Q₈	C₂×C₈	1	4	yes	—
53	C₂²×C₃×C₄	C₂×C₈	1	2	yes	—
54	C₂²×C₃×C₄	C₂×C₈	1	2	yes	—
55	C₂²×C₃×C₄	C₂×C₈	1	1	yes	characteristic
56	C₂²×C₃×C₄	C₂×C₈	1	1	yes	characteristic
57	C₂²×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
58	C₂×C₃×C₈	—	2	2	no	—
59	C₂×C₃×C₈	—	4	1	no	—
60	C₂×C₃×C₈	—	4	2	no	—
61	C₂×C₃×C₈	C₂²×C₄	1	1	yes	characteristic
62	C₂×C₃×C₈	C₂²×C₄	1	2	yes	—
63	C₂×C₃×C₈	C₂²×C₄	1	4	yes	—
64	C₂×C₃×C₈	C₂⁴	1	1	yes	characteristic
65	C₃×C₄⋊C₄	C₂×C₈	1	8	yes	—
66	C₃×C₄⋊C₄	C₂×C₈	1	4	yes	—
67	C₃×C₄²	C₂×C₈	1	4	yes	—
68	C₃×C₄²	C₂×C₈	1	2	yes	—
69	C₃×C₄²	C₂²×C₄	1	4	yes	—
70	C₃×C₄²	C₂²×C₄	1	2	yes	—
71	C₃⋊C₁₆	—	2	8	no	—
72	C₂²×Q₈	C₃⋊C₈	1	1	yes	characteristic
73	C₂²×C₈	C₂×Dic₃	1	1	yes	characteristic
74	C₂²×C₈	C₂×Dic₃	1	2	yes	—
75	C₂²×C₈	C₂²×D₃	1	1	yes	characteristic
76	C₄×Q₈	C₃⋊C₈	1	4	yes	—
77	C₄×Q₈	C₂×Dic₃	1	4	yes	—
78	C₂×C₄⋊C₄	C₃⋊C₈	1	2	yes	—
79	C₂×C₄⋊C₄	C₃⋊C₈	1	1	yes	characteristic
80	C₂×C₄²	C₂×Dic₃	1	2	yes	—
81	C₂×C₄²	C₂×Dic₃	1	1	yes	characteristic
82	C₂×C₁₆	—	3	8	no	—
83	C₄⋊C₈	C₂×Dic₃	1	8	yes	—
84	C₄⋊C₈	C₂×Dic₃	1	4	yes	—
85	C₄×C₈	C₂×Dic₃	1	2	yes	—
86	C₄×C₈	C₂×Dic₃	1	4	yes	—
87	C₄×C₈	C₂²×D₃	1	2	yes	—
88	C₄×C₈	C₂²×D₃	1	4	yes	—
89	C₂³×C₃	C₂²×C₈	1	1	yes	characteristic
90	C₃×Q₈	—	2	8	no	—
91	C₂×C₃×C₄	—	2	4	no	—
92	C₂×C₃×C₄	—	2	1	no	—
93	C₂×C₃×C₄	—	2	1	no	—
94	C₂×C₃×C₄	—	4	2	no	—
95	C₂×C₃×C₄	—	4	1	no	—
96	C₂×C₃×C₄	M₃₂	1	4	yes	—
97	C₂×C₃×C₄	C₂²×C₈	1	1	yes	characteristic
98	C₂×C₃×C₄	C₂²×C₈	1	4	yes	—
99	C₂×C₃×C₄	C₂²×C₈	1	2	yes	—
100	C₂×C₃×C₄	C₂²×C₈	1	4	yes	—
101	C₂×C₃×C₄	C₂²×C₈	1	2	yes	—
102	C₂×C₃×C₄	C₂³×C₄	1	1	yes	characteristic
103	C₃×C₈	—	2	1	no	—
104	C₃×C₈	—	2	2	no	—
105	C₃×C₈	—	2	4	no	—
106	C₃×C₈	2₋¹⁺⁴	1	2	yes	—
107	C₂×Q₈	—	2	4	no	—
108	C₂×Q₈	C₂×C₃⋊C₈	1	4	yes	—
109	C₂²×C₄	C₂×C₃⋊C₈	1	2	yes	—
110	C₂²×C₄	C₂×C₃⋊C₈	1	2	yes	—
111	C₂²×C₄	C₂×C₃⋊C₈	1	1	yes	characteristic
112	C₂²×C₄	C₂×C₃⋊C₈	1	1	yes	characteristic
113	C₂²×C₄	C₂²×Dic₃	1	1	yes	characteristic
114	C₂×C₈	—	2	2	no	—
115	C₂×C₈	—	4	1	no	—
116	C₂×C₈	—	4	2	no	—
117	C₂×C₈	C₂²×Dic₃	1	1	yes	characteristic
118	C₂×C₈	C₂²×Dic₃	1	2	yes	—
119	C₂×C₈	C₂²×Dic₃	1	4	yes	—
120	C₂×C₈	C₂³×D₃	1	1	yes	characteristic
121	C₄⋊C₄	C₂×C₃⋊C₈	1	8	yes	—
122	C₄⋊C₄	C₂×C₃⋊C₈	1	4	yes	—
123	C₄²	C₂×C₃⋊C₈	1	4	yes	—
124	C₄²	C₂×C₃⋊C₈	1	2	yes	—
125	C₄²	C₂²×Dic₃	1	4	yes	—
126	C₄²	C₂²×Dic₃	1	2	yes	—
127	C₁₆	—	6	8	no	—
128	C₂²×C₃	—	2	1	no	—
129	C₂²×C₃	—	2	1	no	—
130	C₂²×C₃	C₂×M₃₂	1	2	yes	—
131	C₂²×C₃	C₂³×C₈	1	1	yes	characteristic
132	C₃×C₄	—	2	4	no	—
133	C₃×C₄	—	2	4	no	—
134	C₃×C₄	—	2	2	no	—
135	C₃×C₄	—	2	1	no	—
136	C₃×C₄	C₂×M₃₂	1	4	yes	—
137	C₃×C₄	#519	1	1	yes	characteristic
138	C₃×C₄	#568	1	1	yes	characteristic
139	C₂³	C₂²×C₃⋊C₈	1	1	yes	characteristic
140	Q₈	—	2	8	no	—
141	C₂×C₄	—	2	4	no	—
142	C₂×C₄	—	2	1	no	—
143	C₂×C₄	—	2	1	no	—
144	C₂×C₄	—	4	2	no	—
145	C₂×C₄	—	4	1	no	—
146	C₂×C₄	#812	1	4	yes	—
147	C₂×C₄	C₂²×C₃⋊C₈	1	1	yes	characteristic
148	C₂×C₄	C₂²×C₃⋊C₈	1	4	yes	—
149	C₂×C₄	C₂²×C₃⋊C₈	1	2	yes	—
150	C₂×C₄	C₂²×C₃⋊C₈	1	4	yes	—
151	C₂×C₄	C₂²×C₃⋊C₈	1	2	yes	—
152	C₂×C₄	C₂³×Dic₃	1	1	yes	characteristic
153	C₈	—	2	1	no	—
154	C₈	—	2	2	no	—
155	C₈	—	2	4	no	—
156	C₈	#1007	1	2	yes	—
157	C₂×C₃	—	2	2	no	—
158	C₂×C₃	#2982	1	1	yes	characteristic
159	C₂×C₃	C₂²×M₃₂	1	1	yes	characteristic
160	C₂×C₃	#3407	1	1	yes	characteristic
161	C₂²	—	2	1	no	—
162	C₂²	—	2	1	no	—
163	C₂²	C₂×#812	1	2	yes	—
164	C₂²	C₂³×C₃⋊C₈	1	1	yes	characteristic
165	C₄	—	2	1	no	—
166	C₄	—	2	4	no	—
167	C₄	—	2	4	no	—
168	C₄	—	2	2	no	—
169	C₄	C₂×#812	1	4	yes	—
170	C₄	#5812	1	1	yes	characteristic
171	C₄	#5819	1	1	yes	characteristic
172	C₃	#33497	1	1	yes	characteristic, complemented, Sylow
173	C₂	—	2	2	no	—
174	C₂	#82435	1	1	yes	characteristic
175	C₂	C₂²×#812	1	1	yes	characteristic
176	C₂	#83780	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	#33497
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
172. C₃	#33497	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
#69745	2
#69744	2
#33497	3