Subgroups — Q₈×#14581728 #35423297

Order 21248.

658 conjugacy classes of subgroups.

The subgroup lattice has 208 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Q₈×#13050574	C₂	1	2	yes	maximal
2	#24908239	C₂	1	6	yes	maximal
3	#24907635	C₂	1	3	yes	maximal
4	C₄×#14581728	C₂	1	3	yes	maximal
5	C₂×C₈×Q₈×C₈₃	C₂	1	1	yes	maximal, characteristic
6	#19127832	—	2	6	no	—
7	Q₈×C₈₃⋊C₈	—	2	2	no	—
8	C₂×#14581728	C₂²	1	3	yes	—
9	C₂×#14581728	C₂²	1	6	yes	—
10	C₂×#14581728	C₂²	1	1	yes	characteristic
11	#19127725	C₂²	1	6	yes	—
12	C₄×#13050574	C₂²	1	6	yes	—
13	#19127466	C₂²	1	6	yes	—
14	C₂×C₄×Q₈×C₈₃	C₂²	1	1	yes	characteristic
15	C₈×Q₈×C₈₃	C₄	1	6	yes	—
16	C₈×Q₈×C₈₃	C₄	1	2	yes	—
17	C₂×C₄⋊C₈×C₈₃	C₂²	1	3	yes	—
18	C₂×C₄×C₈×C₈₃	C₂²	1	3	yes	—
19	C₂×#13050574	C₂³	1	6	yes	—
20	C₂×#13050574	C₂³	1	2	yes	—
21	#14581728	—	2	6	no	—
22	#14581728	—	4	6	no	—
23	#14581728	Q₈	1	4	yes	direct product factor
24	C₄⋊(C₈₃⋊C₈)	—	2	6	no	—
25	C₄⋊(C₈₃⋊C₈)	—	2	6	no	—
26	#14581703	—	2	6	no	—
27	C₄×C₈₃⋊C₈	—	2	6	no	—
28	C₂²×Q₈×C₈₃	Q₈	1	1	yes	characteristic
29	C₂²×C₈×C₈₃	C₂³	1	3	yes	—
30	C₂²×C₈×C₈₃	C₂³	1	1	yes	characteristic
31	C₄×Q₈×C₈₃	C₂×C₄	1	3	yes	—
32	C₄×Q₈×C₈₃	C₂×C₄	1	1	yes	characteristic
33	C₄×Q₈×C₈₃	D₄	1	1	yes	characteristic
34	C₄×Q₈×C₈₃	D₄	1	3	yes	—
35	C₂×C₄⋊C₄×C₈₃	Q₈	1	3	yes	—
36	C₂×C₄²×C₈₃	C₂³	1	3	yes	—
37	C₄⋊C₈×C₈₃	C₂×C₄	1	6	yes	—
38	C₄⋊C₈×C₈₃	C₂×C₄	1	6	yes	—
39	C₄×C₈×C₈₃	C₂×C₄	1	6	yes	—
40	C₄×C₈×C₈₃	C₂×C₄	1	6	yes	—
41	#13050574	—	2	2	no	—
42	#13050574	—	2	6	no	—
43	#13050574	—	2	6	no	—
44	#13050574	C₂×Q₈	1	4	yes	—
45	C₂×C₈₃⋊C₈	—	2	2	no	—
46	C₂×C₈₃⋊C₈	—	2	6	no	—
47	C₂×Q₈×C₈₃	—	2	4	no	—
48	C₂×Q₈×C₈₃	C₄⋊C₄	1	3	yes	—
49	C₂×Q₈×C₈₃	C₄⋊C₄	1	1	yes	characteristic
50	C₂²×C₄×C₈₃	C₂×Q₈	1	3	yes	—
51	C₂²×C₄×C₈₃	C₂×Q₈	1	3	yes	—
52	C₂²×C₄×C₈₃	C₂⁴	1	1	yes	characteristic
53	C₂×C₈×C₈₃	—	2	3	no	—
54	C₂×C₈×C₈₃	—	2	3	no	—
55	C₂×C₈×C₈₃	C₂²×C₄	1	6	yes	—
56	C₂×C₈×C₈₃	C₂²×C₄	1	2	yes	—
57	C₂×C₈×C₈₃	C₂×Q₈	1	2	yes	—
58	C₂×C₈×C₈₃	#41	1	2	yes	—
59	C₄⋊C₄×C₈₃	C₄⋊C₄	1	6	yes	—
60	C₄⋊C₄×C₈₃	C₄⋊C₄	1	3	yes	—
61	C₄⋊C₄×C₈₃	C₄⋊C₄	1	3	yes	—
62	C₄²×C₈₃	C₂²×C₄	1	3	yes	—
63	C₄²×C₈₃	C₂²×C₄	1	3	yes	—
64	C₄²×C₈₃	C₂×D₄	1	3	yes	—
65	C₄²×C₈₃	C₂×D₄	1	3	yes	—
66	C₂³×C₈₃	C₂²×Q₈	1	1	yes	characteristic
67	Q₈×C₈₃	—	2	6	no	—
68	Q₈×C₈₃	#108	1	4	yes	complemented
69	C₂×C₄×C₈₃	—	2	6	no	—
70	C₂×C₄×C₈₃	—	4	3	no	—
71	C₂×C₄×C₈₃	C₂×C₄⋊C₄	1	3	yes	—
72	C₂×C₄×C₈₃	C₂×C₄⋊C₄	1	3	yes	—
73	C₂×C₄×C₈₃	C₂×C₄⋊C₄	1	3	yes	—
74	C₂×C₄×C₈₃	C₂×C₄⋊C₄	1	3	yes	—
75	C₂×C₄×C₈₃	C₂³×C₄	1	1	yes	characteristic
76	C₂×C₄×C₈₃	C₂²×D₄	1	1	yes	characteristic
77	C₂×C₄×C₈₃	C₂²×Q₈	1	1	yes	characteristic
78	C₂×C₄×C₈₃	C₂×#41	1	1	yes	characteristic
79	C₂×C₄×C₈₃	H(5,𝔽₂)	1	1	yes	characteristic
80	C₂×C₄×C₈₃	2₋¹⁺⁴	1	1	yes	characteristic
81	C₈×C₈₃	—	2	6	no	—
82	C₈×C₈₃	C₄×Q₈	1	4	yes	—
83	C₈₃⋊C₈	—	2	4	no	—
84	C₈₃⋊C₈	—	4	6	no	—
85	C₂²×C₈₃	—	2	1	no	—
86	C₂²×C₈₃	—	2	1	no	—
87	C₂²×C₈₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
88	C₂²×C₈₃	#557	1	1	yes	characteristic
89	C₂²×C₈₃	Q₈²	1	1	yes	characteristic
90	C₄×C₈₃	—	2	3	no	—
91	C₄×C₈₃	—	2	3	no	—
92	C₄×C₈₃	—	2	3	no	—
93	C₄×C₈₃	C₂×#108	1	6	yes	—
94	C₄×C₈₃	C₂×C₄×Q₈	1	1	yes	characteristic
95	C₄×C₈₃	#520	1	1	yes	characteristic
96	C₄×C₈₃	D₄×Q₈	1	1	yes	characteristic
97	C₄×C₈₃	#550	1	1	yes	characteristic
98	Q₈×#108	—	83	1	no	maximal, Sylow
99	C₂×C₈₃	—	2	1	no	—
100	C₂×C₈₃	—	2	1	no	—
101	C₂×C₈₃	Q₈×C₄⋊C₄	1	1	yes	characteristic
102	C₂×C₈₃	C₂²×#108	1	1	yes	characteristic
103	C₂×C₈₃	#2916	1	1	yes	characteristic
104	Q₈×M₁₆	—	83	2	no	—
105	C₂×C₈×Q₈	D₈₃	1	1	yes	characteristic
106	#1950	—	83	6	no	—
107	#1848	—	83	3	no	—
108	C₄×#108	—	83	3	no	—
109	C₈₃	Q₈×#108	1	1	yes	characteristic, complemented, Sylow
110	C₂×C₄×Q₈	C₂×D₈₃	1	1	yes	characteristic
111	#446	—	166	6	no	—
112	C₈×Q₈	—	166	2	no	—
113	C₈×Q₈	Dic₈₃	1	6	yes	—
114	C₈×Q₈	Dic₈₃	1	2	yes	—
115	C₂×#108	—	83	6	no	—
116	C₂×#108	—	83	1	no	—
117	C₂×#108	—	83	3	no	—
118	#423	—	83	6	no	—
119	C₂×C₄⋊C₈	C₂×D₈₃	1	3	yes	—
120	C₄×M₁₆	—	83	6	no	—
121	C₂×C₄×C₈	C₂×D₈₃	1	3	yes	—
122	#341	—	83	6	no	—
123	C₂²×Q₈	Dic₁₆₆	1	1	yes	characteristic
124	C₂×M₁₆	—	83	2	no	—
125	C₂×M₁₆	—	83	6	no	—
126	C₂²×C₈	C₂²×D₈₃	1	3	yes	—
127	C₂²×C₈	C₂²×D₈₃	1	1	yes	characteristic
128	C₄×Q₈	D₃₃₂	1	1	yes	characteristic
129	C₄×Q₈	D₃₃₂	1	3	yes	—
130	C₄×Q₈	C₂×Dic₈₃	1	3	yes	—
131	C₄×Q₈	C₂×Dic₈₃	1	1	yes	characteristic
132	C₂×C₄⋊C₄	Dic₁₆₆	1	3	yes	—
133	C₂×C₄²	C₂²×D₈₃	1	3	yes	—
134	#108	—	332	6	no	—
135	#108	—	83	4	no	—
136	#108	—	166	6	no	—
137	C₄⋊C₈	—	166	6	no	—
138	C₄⋊C₈	—	166	6	no	—
139	C₄⋊C₈	C₂×Dic₈₃	1	6	yes	—
140	C₄⋊C₈	C₂×Dic₈₃	1	6	yes	—
141	#97	—	166	6	no	—
142	C₄×C₈	—	166	6	no	—
143	C₄×C₈	C₂×Dic₈₃	1	6	yes	—
144	C₄×C₈	C₂×Dic₈₃	1	6	yes	—
145	C₂×Q₈	—	2	4	no	—
146	C₂×Q₈	C₄⋊Dic₈₃	1	3	yes	—
147	C₂×Q₈	C₄⋊Dic₈₃	1	1	yes	characteristic
148	C₂²×C₄	C₂×Dic₁₆₆	1	3	yes	—
149	C₂²×C₄	C₂×Dic₁₆₆	1	3	yes	—
150	C₂²×C₄	C₂³×D₈₃	1	1	yes	characteristic
151	M₁₆	—	83	4	no	—
152	M₁₆	—	166	6	no	—
153	M₁₆	—	166	6	no	—
154	M₁₆	—	166	2	no	—
155	C₂×C₈	—	2	3	no	—
156	C₂×C₈	—	2	3	no	—
157	C₂×C₈	—	166	6	no	—
158	C₂×C₈	—	166	2	no	—
159	C₂×C₈	C₂²×Dic₈₃	1	6	yes	—
160	C₂×C₈	C₂²×Dic₈₃	1	2	yes	—
161	C₂×C₈	Q₈×D₈₃	1	2	yes	—
162	C₂×C₈	#13050591	1	2	yes	—
163	C₄⋊C₄	C₄⋊Dic₈₃	1	6	yes	—
164	C₄⋊C₄	C₄⋊Dic₈₃	1	3	yes	—
165	C₄⋊C₄	C₄⋊Dic₈₃	1	3	yes	—
166	C₄²	C₂²×Dic₈₃	1	3	yes	—
167	C₄²	C₂²×Dic₈₃	1	3	yes	—
168	C₄²	C₂×D₃₃₂	1	3	yes	—
169	C₄²	C₂×D₃₃₂	1	3	yes	—
170	C₂³	C₂²×Dic₁₆₆	1	1	yes	characteristic
171	Q₈	—	2	6	no	—
172	Q₈	#14581728	1	4	yes	direct product factor
173	C₂×C₄	—	2	6	no	—
174	C₂×C₄	—	4	3	no	—
175	C₂×C₄	C₂×C₄⋊Dic₈₃	1	3	yes	—
176	C₂×C₄	C₂×C₄⋊Dic₈₃	1	3	yes	—
177	C₂×C₄	C₂×C₄⋊Dic₈₃	1	3	yes	—
178	C₂×C₄	C₂×C₄⋊Dic₈₃	1	3	yes	—
179	C₂×C₄	C₂³×Dic₈₃	1	1	yes	characteristic
180	C₂×C₄	C₂²×D₃₃₂	1	1	yes	characteristic
181	C₂×C₄	C₂×Q₈×D₈₃	1	1	yes	characteristic
182	C₂×C₄	C₂×#13050591	1	1	yes	characteristic
183	C₂×C₄	#14581838	1	1	yes	characteristic
184	C₂×C₄	#14581839	1	1	yes	characteristic
185	C₈	—	2	6	no	—
186	C₈	—	332	6	no	—
187	C₈	—	166	4	no	—
188	C₈	Q₈×Dic₈₃	1	4	yes	—
189	C₂²	—	2	1	no	—
190	C₂²	—	2	1	no	—
191	C₂²	C₂²×C₄⋊Dic₈₃	1	1	yes	characteristic
192	C₂²	#19128430	1	1	yes	characteristic
193	C₂²	Q₈×Dic₁₆₆	1	1	yes	characteristic
194	C₄	—	2	3	no	—
195	C₄	—	2	3	no	—
196	C₄	—	2	3	no	—
197	C₄	C₂×#14581728	1	6	yes	—
198	C₄	C₂×Q₈×Dic₈₃	1	1	yes	characteristic
199	C₄	#19128184	1	1	yes	characteristic
200	C₄	Q₈×D₃₃₂	1	1	yes	characteristic
201	C₄	#19128373	1	1	yes	characteristic
202	C₂	—	2	1	no	—
203	C₂	—	2	1	no	—
204	C₂	Q₈×C₄⋊Dic₈₃	1	1	yes	characteristic
205	C₂	C₂²×#14581728	1	1	yes	characteristic
206	C₂	#24916074	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×D₃₃₂
commutator	C₂×C₄×C₈₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₈₃
Fitting	C₂×C₈×Q₈×C₈₃	C₂
socle	C₂²×C₈₃	C₂²×C₄⋊C₄
prime radical	C₂³×C₈₃	C₂²×Q₈
cosocle	C₂²×C₄×C₈₃	C₂⁴
hypercenter	C₂×C₈×Q₈	D₈₃
hypocenter	C₈₃	Q₈×#108
LCS	G ▷ C₂×C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×Q₈ ◁ C₂×C₈×Q₈

Complements

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

N	H ≅ G/N	type
23. #14581728	Q₈	direct (N × H)
68. Q₈×C₈₃	#108	semidirect
109. C₈₃	Q₈×#108	semidirect
172. Q₈	#14581728	direct (N × H)

Maximal subgroups

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

subgroup	index
Q₈×#13050574	2
#24908239	2
#24907635	2
C₄×#14581728	2
C₂×C₈×Q₈×C₈₃	2
Q₈×#108	83

Subgroups — Q8×#14581728 #35423297

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Q₈×#14581728 #35423297