Subgroups — C₁₆×C₄⋊Q₈ #496133

Order 512.

592 conjugacy classes of subgroups.

The subgroup lattice has 180 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Q₈×C₁₆	C₂	1	2	yes	maximal
2	C₈×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic
3	#13263	C₂	1	2	yes	maximal
4	#13159	C₂	1	4	yes	maximal
5	#12898	C₂	1	1	yes	maximal, characteristic
6	C₁₆×C₄⋊C₄	C₂	1	4	yes	maximal
7	C₄²×C₁₆	C₂	1	1	yes	maximal, characteristic
8	C₂×C₈×Q₈	C₂²	1	2	yes	—
9	C₄×C₄⋊Q₈	C₄	1	1	yes	characteristic
10	Q₈×C₁₆	—	2	8	no	—
11	C₂×C₄⋊C₁₆	C₂²	1	4	yes	—
12	C₂×C₄⋊C₁₆	C₂²	1	2	yes	—
13	C₂×C₄⋊C₁₆	C₂²	1	4	yes	—
14	C₂×C₄×C₁₆	C₂²	1	2	yes	—
15	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
16	C₂×C₄×C₁₆	C₂²	1	4	yes	—
17	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
18	C₂×C₄×C₁₆	C₂²	1	2	yes	—
19	#1984	C₄	1	2	yes	—
20	#1939	C₄	1	4	yes	—
21	#1842	C₄	1	1	yes	characteristic
22	C₈×C₄⋊C₄	C₂²	1	4	yes	—
23	C₄²×C₈	C₂²	1	1	yes	characteristic
24	#1374	C₂²	1	8	yes	—
25	C₂×C₄⋊Q₈	C₈	1	1	yes	characteristic
26	C₂×C₄×Q₈	C₂×C₄	1	2	yes	—
27	C₂²×C₁₆	C₂³	1	4	yes	—
28	C₂²×C₁₆	C₂³	1	1	yes	characteristic
29	C₂²×C₁₆	C₂³	1	2	yes	—
30	C₂²×C₁₆	C₂³	1	1	yes	characteristic
31	C₈×Q₈	—	2	4	no	—
32	C₈×Q₈	—	2	4	no	—
33	C₂×C₄⋊C₈	C₂×C₄	1	2	yes	—
34	C₂×C₄⋊C₈	C₂×C₄	1	4	yes	—
35	C₂×C₄⋊C₈	C₂×C₄	1	4	yes	—
36	C₂×C₄×C₈	C₂×C₄	1	2	yes	—
37	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
38	C₂×C₄×C₈	C₂³	1	4	yes	—
39	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
40	C₂×C₄×C₈	C₂³	1	2	yes	—
41	#391	C₈	1	2	yes	—
42	#389	C₈	1	4	yes	—
43	#384	C₈	1	1	yes	characteristic
44	C₄×C₄⋊C₄	C₂×C₄	1	4	yes	—
45	C₄³	C₂×C₄	1	1	yes	characteristic
46	C₄⋊C₁₆	—	2	8	no	—
47	C₄⋊C₁₆	—	2	4	no	—
48	C₄⋊C₁₆	—	2	8	no	—
49	C₄×C₁₆	—	2	8	no	—
50	C₄×C₁₆	—	2	2	no	—
51	C₄×C₁₆	—	2	4	no	—
52	C₄×C₁₆	D₄	1	4	yes	—
53	C₄×C₁₆	Q₈	1	8	yes	complemented
54	#336	C₂×C₄	1	8	yes	—
55	C₂²×Q₈	C₂×C₈	1	2	yes	—
56	C₂²×C₈	C₂²×C₄	1	4	yes	—
57	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
58	C₂²×C₈	C₂²×C₄	1	2	yes	—
59	C₂²×C₈	C₂⁴	1	1	yes	characteristic
60	C₄⋊Q₈	C₁₆	1	8	yes	direct product factor
61	C₄×Q₈	—	2	4	no	—
62	C₄×Q₈	—	2	4	no	—
63	C₂×C₄⋊C₄	C₂×C₈	1	2	yes	—
64	C₂×C₄⋊C₄	C₂×C₈	1	4	yes	—
65	C₂×C₄⋊C₄	C₂×C₈	1	4	yes	—
66	C₂×C₄⋊C₄	C₂×C₈	1	4	yes	—
67	C₂×C₄²	C₂×C₈	1	2	yes	—
68	C₂×C₄²	C₂×C₈	1	1	yes	characteristic
69	C₂×C₄²	C₂×C₈	1	1	yes	characteristic
70	C₂×C₄²	C₂²×C₄	1	4	yes	—
71	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
72	C₂×C₄²	C₂²×C₄	1	2	yes	—
73	C₂×C₁₆	—	2	2	no	—
74	C₂×C₁₆	—	2	2	no	—
75	C₂×C₁₆	—	2	2	no	—
76	C₂×C₁₆	—	2	4	no	—
77	C₂×C₁₆	—	2	8	no	—
78	C₂×C₁₆	C₂×D₄	1	2	yes	—
79	C₂×C₁₆	C₂×D₄	1	2	yes	—
80	C₂×C₁₆	C₂×Q₈	1	4	yes	—
81	C₂×C₁₆	C₂×Q₈	1	4	yes	—
82	C₄⋊C₈	—	2	8	no	—
83	C₄⋊C₈	—	2	4	no	—
84	C₄⋊C₈	—	2	8	no	—
85	C₄×C₈	—	2	4	no	—
86	C₄×C₈	—	2	4	no	—
87	C₄×C₈	—	2	2	no	—
88	C₄×C₈	—	2	4	no	—
89	C₄×C₈	C₂×D₄	1	2	yes	—
90	C₄×C₈	C₂×Q₈	1	4	yes	—
91	C₄×C₈	#41	1	2	yes	—
92	C₄×C₈	#41	1	4	yes	—
93	#95	C₂×C₈	1	8	yes	—
94	C₂×Q₈	—	2	4	no	—
95	C₂×Q₈	—	2	4	no	—
96	C₂×Q₈	C₂×C₁₆	1	8	yes	—
97	C₂²×C₄	C₂²×C₈	1	4	yes	—
98	C₂²×C₄	C₂²×C₈	1	1	yes	characteristic
99	C₂²×C₄	C₂²×C₈	1	2	yes	—
100	C₂²×C₄	C₂²×C₈	1	4	yes	—
101	C₂²×C₄	C₂²×C₈	1	1	yes	characteristic
102	C₂²×C₄	C₂²×C₈	1	2	yes	—
103	C₂²×C₄	C₂³×C₄	1	1	yes	characteristic
104	C₂×C₈	—	2	2	no	—
105	C₂×C₈	—	2	2	no	—
106	C₂×C₈	—	2	2	no	—
107	C₂×C₈	—	2	4	no	—
108	C₂×C₈	—	2	8	no	—
109	C₂×C₈	C₄×D₄	1	2	yes	—
110	C₂×C₈	C₄×Q₈	1	4	yes	—
111	C₂×C₈	C₂²×D₄	1	1	yes	characteristic
112	C₂×C₈	C₂²×Q₈	1	2	yes	—
113	C₂×C₈	C₂×#41	1	2	yes	—
114	C₂×C₈	C₂×#41	1	1	yes	characteristic
115	C₄⋊C₄	—	2	8	no	—
116	C₄⋊C₄	—	2	4	no	—
117	C₄⋊C₄	—	2	8	no	—
118	C₄⋊C₄	C₂×C₁₆	1	16	yes	—
119	C₄²	—	2	4	no	—
120	C₄²	—	2	4	no	—
121	C₄²	—	2	2	no	—
122	C₄²	—	2	4	no	—
123	C₄²	C₂×C₁₆	1	4	yes	—
124	C₄²	C₄×D₄	1	2	yes	—
125	C₄²	C₄×Q₈	1	4	yes	—
126	C₄²	#131	1	2	yes	—
127	C₄²	#131	1	4	yes	—
128	C₁₆	—	2	2	no	—
129	C₁₆	—	2	4	no	—
130	C₁₆	—	4	4	no	—
131	C₁₆	C₄⋊Q₈	1	4	yes	direct product factor
132	C₂³	C₂³×C₈	1	1	yes	characteristic
133	Q₈	—	2	16	no	—
134	C₂×C₄	—	2	2	no	—
135	C₂×C₄	—	2	2	no	—
136	C₂×C₄	—	2	2	no	—
137	C₂×C₄	—	2	4	no	—
138	C₂×C₄	—	2	4	no	—
139	C₂×C₄	—	2	8	no	—
140	C₂×C₄	—	2	4	no	—
141	C₂×C₄	C₈×D₄	1	2	yes	—
142	C₂×C₄	C₈×D₄	1	2	yes	—
143	C₂×C₄	C₈×Q₈	1	4	yes	—
144	C₂×C₄	C₈×Q₈	1	4	yes	—
145	C₂×C₄	C₂²×C₁₆	1	8	yes	—
146	C₂×C₄	C₂²×C₁₆	1	2	yes	—
147	C₂×C₄	C₂²×C₁₆	1	4	yes	—
148	C₂×C₄	#504	1	2	yes	—
149	C₂×C₄	#504	1	4	yes	—
150	C₂×C₄	C₂×C₄×D₄	1	1	yes	characteristic
151	C₂×C₄	C₂×C₄×Q₈	1	2	yes	—
152	C₂×C₄	C₂×#131	1	2	yes	—
153	C₂×C₄	C₂×#131	1	1	yes	characteristic
154	C₈	—	2	2	no	—
155	C₈	—	2	4	no	—
156	C₈	—	4	4	no	—
157	C₈	C₂×C₄⋊Q₈	1	1	yes	characteristic
158	C₈	#532	1	1	yes	characteristic
159	C₈	#533	1	2	yes	—
160	C₂²	C₂×C₈×D₄	1	1	yes	characteristic
161	C₂²	C₂×C₈×Q₈	1	2	yes	—
162	C₂²	C₂³×C₁₆	1	1	yes	characteristic
163	C₂²	C₂×#504	1	1	yes	characteristic
164	C₂²	C₂×#504	1	2	yes	—
165	C₄	—	2	8	no	—
166	C₄	—	2	2	no	—
167	C₄	—	2	4	no	—
168	C₄	—	4	4	no	—
169	C₄	D₄×C₁₆	1	4	yes	—
170	C₄	Q₈×C₁₆	1	8	yes	complemented
171	C₄	C₄×C₄⋊Q₈	1	1	yes	characteristic
172	C₄	#2931	1	1	yes	characteristic
173	C₄	#2960	1	2	yes	—
174	C₂	C₈×C₄⋊Q₈	1	1	yes	characteristic
175	C₂	C₂×D₄×C₁₆	1	1	yes	characteristic
176	C₂	#33434	1	1	yes	characteristic
177	C₂	C₂×Q₈×C₁₆	1	2	yes	—
178	C₂	#33474	1	2	yes	—

	subgroup	G/H
center	C₂²×C₁₆	C₂³
commutator	C₂²	C₂³×C₁₆
Frattini, cosocle	C₂²×C₈	C₂⁴
socle, prime radical	C₂³	C₂³×C₈
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₁₆ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
53. C₄×C₁₆	Q₈	semidirect
60. C₄⋊Q₈	C₁₆	direct (N × H)
131. C₁₆	C₄⋊Q₈	direct (N × H)
170. C₄	Q₈×C₁₆	semidirect

Maximal subgroups

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

subgroup	index
C₂×Q₈×C₁₆	2
C₈×C₄⋊Q₈	2
#13263	2
#13159	2
#12898	2
C₁₆×C₄⋊C₄	2
C₄²×C₁₆	2

Subgroups — C16×C4⋊Q8 #496133

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₆×C₄⋊Q₈ #496133