Subgroups — Q₈×#504 #7622647

Order 512.

1348 conjugacy classes of subgroups.

The subgroup lattice has 184 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Q₈×#131	C₂	1	1	yes	maximal, characteristic
2	Q₈×M₃₂	C₂	1	3	yes	maximal, complemented
3	#33476	C₂	1	9	yes	maximal, complemented
4	#33474	C₂	1	9	yes	maximal, complemented
5	C₂×Q₈×C₁₆	C₂	1	3	yes	maximal, complemented
6	#33398	C₂	1	3	yes	maximal
7	C₄×#504	C₂	1	3	yes	maximal
8	Q₈×#41	C₄	1	1	yes	characteristic
9	C₂×#504	C₂²	1	3	yes	—
10	C₂×#504	C₂²	1	1	yes	characteristic
11	Q₈×M₁₆	C₄	1	3	yes	—
12	#2962	C₄	1	9	yes	—
13	#2960	C₂²	1	9	yes	—
14	C₂×C₈×Q₈	C₂²	1	3	yes	—
15	#2906	C₄	1	3	yes	—
16	C₄×#131	C₂²	1	3	yes	—
17	#2183	C₂²	1	18	yes	—
18	#2183	C₂²	1	18	yes	—
19	Q₈×C₁₆	C₂²	1	1	yes	characteristic
20	Q₈×C₁₆	C₂²	1	9	yes	—
21	Q₈×C₁₆	C₂²	1	9	yes	—
22	Q₈×C₁₆	C₂²	1	3	yes	—
23	Q₈×C₁₆	C₂²	1	6	yes	—
24	#2151	C₂²	1	18	yes	—
25	#2150	C₂²	1	9	yes	—
26	C₂×C₄⋊C₁₆	C₂²	1	9	yes	—
27	#2108	C₂²	1	9	yes	—
28	#2108	C₂²	1	9	yes	—
29	C₄×M₃₂	C₂²	1	9	yes	—
30	C₂×C₄×C₁₆	C₂²	1	9	yes	—
31	C₂×#131	C₂×C₄	1	3	yes	—
32	C₂×#131	C₂³	1	1	yes	characteristic
33	Q₈²	C₈	1	1	yes	characteristic
34	#557	C₈	1	3	yes	—
35	#554	C₈	1	9	yes	—
36	D₄×Q₈	C₈	1	3	yes	—
37	#533	C₂×C₄	1	9	yes	—
38	C₄×#41	C₂×C₄	1	3	yes	—
39	C₂×C₄×Q₈	C₂×C₄	1	3	yes	—
40	#504	—	2	12	no	—
41	#504	Q₈	1	8	yes	direct product factor
42	C₂×M₃₂	C₂³	1	9	yes	—
43	C₂×M₃₂	C₂³	1	3	yes	—
44	C₂²×C₁₆	C₂³	1	9	yes	—
45	C₂²×C₁₆	C₂³	1	3	yes	—
46	#446	C₂×C₄	1	18	yes	—
47	#446	C₂×C₄	1	18	yes	—
48	C₈×Q₈	C₂×C₄	1	9	yes	—
49	C₈×Q₈	C₂×C₄	1	3	yes	—
50	C₈×Q₈	C₂³	1	1	yes	characteristic
51	C₈×Q₈	C₂³	1	9	yes	—
52	C₈×Q₈	C₂³	1	6	yes	—
53	#424	C₂×C₄	1	18	yes	—
54	#423	C₂×C₄	1	9	yes	—
55	C₂×C₄⋊C₈	C₂×C₄	1	9	yes	—
56	#405	C₂×C₄	1	9	yes	—
57	#405	C₂³	1	9	yes	—
58	C₄×M₁₆	C₂×C₄	1	9	yes	—
59	C₂×C₄×C₈	C₂³	1	9	yes	—
60	C₄⋊C₁₆	C₂³	1	3	yes	—
61	C₄⋊C₁₆	C₂³	1	9	yes	—
62	C₄⋊C₁₆	C₂³	1	18	yes	—
63	C₄⋊C₁₆	C₂³	1	18	yes	—
64	#346	C₂³	1	18	yes	—
65	C₄×C₁₆	C₂³	1	9	yes	—
66	C₄×C₁₆	C₂³	1	9	yes	—
67	C₄×C₁₆	C₂³	1	3	yes	—
68	C₄×C₁₆	C₂³	1	9	yes	—
69	C₂×#41	C₂²×C₄	1	1	yes	characteristic
70	C₂²×Q₈	C₂×C₈	1	3	yes	—
71	#131	—	2	12	no	—
72	#131	C₂×Q₈	1	4	yes	—
73	#131	#41	1	4	yes	—
74	C₂×M₁₆	C₂²×C₄	1	9	yes	—
75	C₂×M₁₆	C₂²×C₄	1	3	yes	—
76	C₂²×C₈	C₂²×C₄	1	9	yes	—
77	C₂²×C₈	C₂⁴	1	3	yes	—
78	C₄⋊Q₈	C₂×C₈	1	9	yes	—
79	C₄⋊Q₈	C₂×C₈	1	9	yes	—
80	#125	C₂×C₈	1	18	yes	—
81	C₂²⋊Q₈	C₂×C₈	1	18	yes	—
82	C₂²⋊Q₈	C₂×C₈	1	18	yes	—
83	C₄×Q₈	C₂×C₈	1	3	yes	—
84	C₄×Q₈	C₂×C₈	1	9	yes	—
85	C₄×Q₈	C₂×C₈	1	3	yes	—
86	C₄×Q₈	C₂×C₈	1	3	yes	—
87	C₄×Q₈	C₂²×C₄	1	1	yes	characteristic
88	C₄×Q₈	C₂²×C₄	1	9	yes	—
89	C₄×Q₈	C₂²×C₄	1	6	yes	—
90	C₄×D₄	C₂×C₈	1	9	yes	—
91	C₄×D₄	C₂×C₈	1	9	yes	—
92	#117	C₂²×C₄	1	9	yes	—
93	C₂×C₄⋊C₄	C₂×C₈	1	9	yes	—
94	C₂×C₄²	C₂²×C₄	1	9	yes	—
95	M₃₂	—	2	18	no	—
96	M₃₂	C₂×Q₈	1	12	yes	complemented
97	C₂×C₁₆	—	2	18	no	—
98	C₂×C₁₆	C₂×Q₈	1	12	yes	complemented
99	C₂×C₁₆	C₂⁴	1	3	yes	—
100	C₂×C₁₆	C₂⁴	1	9	yes	—
101	C₂×C₁₆	C₂⁴	1	3	yes	—
102	C₂×C₁₆	C₂⁴	1	1	yes	characteristic
103	C₄⋊C₈	C₂²×C₄	1	3	yes	—
104	C₄⋊C₈	C₂²×C₄	1	9	yes	—
105	C₄⋊C₈	C₂²×C₄	1	18	yes	—
106	C₄⋊C₈	C₂²×C₄	1	18	yes	—
107	#97	C₂²×C₄	1	18	yes	—
108	C₄×C₈	C₂²×C₄	1	9	yes	—
109	C₄×C₈	C₂²×C₄	1	9	yes	—
110	C₄×C₈	C₂⁴	1	3	yes	—
111	C₄×C₈	C₂⁴	1	9	yes	—
112	#41	C₄×Q₈	1	4	yes	—
113	#41	#131	1	4	yes	—
114	C₂×Q₈	—	2	12	no	—
115	C₂×Q₈	C₂²×C₈	1	9	yes	—
116	C₂×Q₈	C₂²×C₈	1	6	yes	—
117	C₂×Q₈	C₂²×C₈	1	1	yes	characteristic
118	C₂×Q₈	C₂²×C₈	1	1	yes	characteristic
119	C₂×D₄	C₂²×C₈	1	3	yes	—
120	C₂²×C₄	C₂²×C₈	1	9	yes	—
121	C₂²×C₄	C₂²×C₈	1	9	yes	—
122	C₂²×C₄	C₂³×C₄	1	3	yes	—
123	M₁₆	—	2	18	no	—
124	M₁₆	C₄×Q₈	1	12	yes	—
125	C₂×C₈	—	2	18	no	—
126	C₂×C₈	C₂³×C₄	1	3	yes	—
127	C₂×C₈	C₂³×C₄	1	9	yes	—
128	C₂×C₈	C₂³×C₄	1	3	yes	—
129	C₂×C₈	C₂²×Q₈	1	6	yes	—
130	C₂×C₈	C₂×#41	1	6	yes	—
131	C₂×C₈	C₂⁵	1	1	yes	characteristic
132	C₄⋊C₄	C₂²×C₈	1	3	yes	—
133	C₄⋊C₄	C₂²×C₈	1	18	yes	—
134	C₄⋊C₄	C₂²×C₈	1	9	yes	—
135	C₄⋊C₄	C₂²×C₈	1	9	yes	—
136	C₄⋊C₄	C₂²×C₈	1	9	yes	—
137	C₄⋊C₄	C₂²×C₈	1	18	yes	—
138	C₂²⋊C₄	C₂²×C₈	1	18	yes	—
139	C₄²	C₂²×C₈	1	9	yes	—
140	C₄²	C₂²×C₈	1	9	yes	—
141	C₄²	C₂³×C₄	1	3	yes	—
142	C₄²	C₂³×C₄	1	9	yes	—
143	C₁₆	—	2	3	no	—
144	C₁₆	—	2	9	no	—
145	C₁₆	C₂²×Q₈	1	2	yes	—
146	C₁₆	C₂²×Q₈	1	6	yes	—
147	C₂³	C₂³×C₈	1	3	yes	—
148	Q₈	—	2	18	no	—
149	Q₈	—	4	6	no	—
150	Q₈	C₈×Q₈	1	4	yes	—
151	Q₈	#504	1	4	yes	direct product factor
152	D₄	C₈×Q₈	1	12	yes	—
153	C₂×C₄	—	2	18	no	—
154	C₂×C₄	—	4	9	no	—
155	C₂×C₄	C₂×C₄×Q₈	1	6	yes	—
156	C₂×C₄	C₂³×C₈	1	3	yes	—
157	C₂×C₄	C₂³×C₈	1	9	yes	—
158	C₂×C₄	C₂³×C₈	1	9	yes	—
159	C₂×C₄	C₂³×C₈	1	3	yes	—
160	C₂×C₄	C₂³×C₈	1	3	yes	—
161	C₂×C₄	C₂×#131	1	6	yes	—
162	C₂×C₄	C₂⁴×C₄	1	1	yes	characteristic
163	C₈	—	2	3	no	—
164	C₈	—	2	9	no	—
165	C₈	C₂×C₄×Q₈	1	6	yes	—
166	C₈	C₂³×Q₈	1	1	yes	characteristic
167	C₈	C₂²×#41	1	1	yes	characteristic
168	C₂²	—	2	3	no	—
169	C₂²	—	2	3	no	—
170	C₂²	C₂×C₈×Q₈	1	6	yes	—
171	C₂²	C₂⁴×C₈	1	1	yes	characteristic
172	C₄	—	2	9	no	—
173	C₄	—	2	3	no	—
174	C₄	—	2	9	no	—
175	C₄	C₂×C₈×Q₈	1	6	yes	—
176	C₄	C₂×#504	1	6	yes	—
177	C₄	C₂²×C₄×Q₈	1	1	yes	characteristic
178	C₄	C₂²×#131	1	1	yes	characteristic
179	C₂	—	2	6	no	—
180	C₂	C₂²×C₈×Q₈	1	1	yes	characteristic
181	C₂	C₂²×#504	1	1	yes	characteristic
182	C₂	#62624	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₁₆	C₂⁴
commutator, socle	C₂²	C₂⁴×C₈
Frattini, cosocle	C₂×C₈	C₂⁵
prime radical	C₂×#41	C₂²×C₄
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₁₆ ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
2. Q₈×M₃₂	C₂	semidirect
3. #33476	C₂	semidirect
4. #33474	C₂	semidirect
5. C₂×Q₈×C₁₆	C₂	semidirect
41. #504	Q₈	direct (N × H)
96. M₃₂	C₂×Q₈	semidirect
98. C₂×C₁₆	C₂×Q₈	semidirect
151. Q₈	#504	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×#131	2
Q₈×M₃₂	2
#33476	2
#33474	2
C₂×Q₈×C₁₆	2
#33398	2
C₄×#504	2

Subgroups — Q8×#504 #7622647

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Q₈×#504 #7622647