Subgroups — M₁₆×D₈ #18034

Order 256.

397 conjugacy classes of subgroups.

The subgroup lattice has 189 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×D₈	C₂	1	1	yes	maximal, characteristic
2	D₄×M₁₆	C₂	1	2	yes	maximal, complemented
3	#1589	C₂	1	2	yes	maximal, complemented
4	#1581	C₂	1	4	yes	maximal, complemented
5	C₈×D₈	C₂	1	2	yes	maximal, complemented
6	#1567	C₂	1	1	yes	maximal, characteristic, complemented
7	#1486	C₂	1	2	yes	maximal, complemented
8	C₈×M₁₆	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×D₈	C₄	1	1	yes	characteristic
10	C₂²×M₁₆	—	2	2	no	—
11	C₂×C₄×D₄	C₂²	1	2	yes	—
12	C₄×D₈	C₄	1	2	yes	—
13	C₄×D₈	C₄	1	1	yes	characteristic
14	C₄×D₈	C₄	1	1	yes	characteristic
15	C₄×D₈	C₂²	1	1	yes	characteristic
16	C₄×D₈	C₂²	1	2	yes	—
17	C₄×D₈	C₂²	1	1	yes	characteristic
18	#436	C₂²	1	4	yes	complemented
19	#435	—	2	4	no	—
20	C₈×D₄	C₂²	1	4	yes	complemented
21	#431	C₂²	1	1	yes	characteristic
22	C₂×#107	C₄	1	1	yes	characteristic
23	#423	C₂²	1	2	yes	—
24	C₂×#102	C₄	1	2	yes	—
25	#407	—	2	2	no	—
26	C₄×M₁₆	C₂²	1	1	yes	characteristic
27	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
28	#335	C₂²	1	2	yes	complemented
29	#335	C₂²	1	2	yes	complemented
30	D₄⋊C₈	C₂²	1	4	yes	complemented
31	D₄⋊C₈	C₂²	1	4	yes	complemented
32	#322	C₂²	1	2	yes	complemented
33	C₈²	C₂²	1	2	yes	complemented
34	C₂²×D₄	C₂×C₄	1	2	yes	—
35	C₂³×C₄	—	2	2	no	—
36	C₂×D₈	—	2	4	no	—
37	C₂×D₈	C₂×C₄	1	2	yes	—
38	C₂×D₈	C₂×C₄	1	1	yes	characteristic
39	C₂×D₈	C₂×C₄	1	1	yes	characteristic
40	C₂×M₁₆	—	2	2	no	—
41	C₂×M₁₆	—	2	4	no	—
42	C₂×M₁₆	—	4	8	no	—
43	C₂×M₁₆	D₄	1	1	yes	characteristic
44	C₂×M₁₆	D₄	1	1	yes	characteristic
45	C₂²×C₈	—	2	4	no	—
46	C₂²×C₈	C₂×C₄	1	1	yes	characteristic
47	C₂²×C₈	C₂×C₄	1	1	yes	characteristic
48	C₄×D₄	—	2	2	no	—
49	C₄×D₄	—	2	2	no	—
50	C₄×D₄	C₂×C₄	1	2	yes	—
51	C₄×D₄	C₂×C₄	1	2	yes	—
52	C₄×D₄	C₂³	1	2	yes	—
53	C₄×D₄	C₂³	1	2	yes	—
54	C₂×C₄⋊C₄	C₂×C₄	1	2	yes	—
55	C₂×C₂²⋊C₄	—	2	2	no	—
56	C₂×C₄²	C₂³	1	1	yes	characteristic
57	#107	C₂×C₄	1	2	yes	—
58	#107	C₂×C₄	1	1	yes	characteristic
59	#107	C₂×C₄	1	1	yes	characteristic
60	C₄⋊C₈	—	2	2	no	—
61	C₄⋊C₈	—	2	1	no	—
62	C₄⋊C₈	C₂³	1	4	yes	—
63	#102	C₂×C₄	1	2	yes	—
64	#102	C₂×C₄	1	2	yes	—
65	#102	C₂×C₄	1	2	yes	—
66	#102	C₂×C₄	1	2	yes	—
67	C₂²⋊C₈	—	2	1	no	—
68	C₂²⋊C₈	—	2	4	no	—
69	C₂²⋊C₈	—	2	4	no	—
70	#97	D₄	1	2	yes	—
71	C₄×C₈	C₂×C₄	1	1	yes	characteristic
72	C₄×C₈	C₂×C₄	1	1	yes	characteristic
73	C₄×C₈	C₂³	1	2	yes	—
74	C₄×C₈	C₂³	1	1	yes	characteristic
75	C₄×C₈	C₂³	1	1	yes	characteristic
76	C₄×C₈	C₂³	1	2	yes	—
77	C₂⁴	—	2	2	no	—
78	C₂×D₄	—	2	4	no	—
79	C₂×D₄	—	2	2	no	—
80	C₂×D₄	—	4	2	no	—
81	C₂×D₄	C₂²×C₄	1	2	yes	—
82	C₂×D₄	C₂²×C₄	1	2	yes	—
83	C₂²×C₄	—	2	2	no	—
84	C₂²×C₄	—	2	2	no	—
85	C₂²×C₄	—	2	2	no	—
86	C₂²×C₄	—	2	2	no	—
87	C₂²×C₄	—	2	2	no	—
88	C₂²×C₄	—	4	4	no	—
89	C₂²×C₄	—	4	4	no	—
90	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
91	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
92	C₂²×C₄	#41	1	1	yes	characteristic
93	D₈	—	2	4	no	—
94	D₈	—	2	2	no	—
95	D₈	M₁₆	1	4	yes	direct product factor
96	M₁₆	—	2	2	no	—
97	M₁₆	—	4	4	no	—
98	M₁₆	—	4	8	no	—
99	M₁₆	D₈	1	4	yes	direct product factor
100	C₂×C₈	—	2	2	no	—
101	C₂×C₈	—	2	4	no	—
102	C₂×C₈	—	2	2	no	—
103	C₂×C₈	—	4	1	no	—
104	C₂×C₈	—	4	8	no	—
105	C₂×C₈	C₂²×C₄	1	1	yes	characteristic
106	C₂×C₈	C₂²×C₄	1	1	yes	characteristic
107	C₂×C₈	C₂²×C₄	1	1	yes	characteristic
108	C₂×C₈	C₂²×C₄	1	1	yes	characteristic
109	C₂×C₈	C₂×D₄	1	2	yes	—
110	C₂×C₈	C₂×D₄	1	2	yes	—
111	C₄⋊C₄	—	2	2	no	—
112	C₄⋊C₄	C₂²×C₄	1	2	yes	—
113	C₄⋊C₄	C₂²×C₄	1	2	yes	—
114	C₂²⋊C₄	—	2	2	no	—
115	C₂²⋊C₄	—	2	2	no	—
116	C₂²⋊C₄	—	2	2	no	—
117	C₂²⋊C₄	—	2	2	no	—
118	C₄²	C₂²×C₄	1	1	yes	characteristic
119	C₄²	C₂×D₄	1	1	yes	characteristic
120	C₄²	#41	1	1	yes	characteristic
121	C₄²	C₂⁴	1	1	yes	characteristic
122	C₂³	—	2	2	no	—
123	C₂³	—	2	2	no	—
124	C₂³	—	4	4	no	—
125	C₂³	—	4	4	no	—
126	C₂³	—	8	2	no	—
127	C₂³	C₄×D₄	1	1	yes	characteristic
128	D₄	—	2	2	no	—
129	D₄	—	2	2	no	—
130	D₄	—	2	2	no	—
131	D₄	—	4	4	no	—
132	D₄	C₂×M₁₆	1	4	yes	complemented
133	C₂×C₄	—	2	1	no	—
134	C₂×C₄	—	2	1	no	—
135	C₂×C₄	—	2	2	no	—
136	C₂×C₄	—	2	2	no	—
137	C₂×C₄	—	4	1	no	—
138	C₂×C₄	—	4	4	no	—
139	C₂×C₄	—	4	2	no	—
140	C₂×C₄	—	4	2	no	—
141	C₂×C₄	—	4	2	no	—
142	C₂×C₄	—	4	2	no	—
143	C₂×C₄	—	4	4	no	—
144	C₂×C₄	—	4	4	no	—
145	C₂×C₄	C₄×D₄	1	1	yes	characteristic
146	C₂×C₄	C₄×D₄	1	1	yes	characteristic
147	C₂×C₄	C₄×D₄	1	1	yes	characteristic
148	C₂×C₄	C₂×D₈	1	2	yes	—
149	C₂×C₄	#135	1	2	yes	—
150	C₂×C₄	C₂³×C₄	1	1	yes	characteristic
151	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
152	C₂×C₄	C₂×#41	1	1	yes	characteristic
153	C₈	—	2	1	no	—
154	C₈	—	2	1	no	—
155	C₈	—	2	1	no	—
156	C₈	—	2	2	no	—
157	C₈	—	4	2	no	—
158	C₈	—	4	4	no	—
159	C₈	C₂×M₁₆	1	2	yes	complemented
160	C₈	C₂×D₈	1	4	yes	complemented
161	C₂²	—	2	1	no	—
162	C₂²	—	2	1	no	—
163	C₂²	—	2	4	no	—
164	C₂²	—	4	2	no	—
165	C₂²	—	4	2	no	—
166	C₂²	—	4	2	no	—
167	C₂²	—	4	2	no	—
168	C₂²	—	4	2	no	—
169	C₂²	—	8	8	no	—
170	C₂²	C₄×D₈	1	2	yes	—
171	C₂²	C₂×C₄×D₄	1	1	yes	characteristic
172	C₄	—	2	1	no	—
173	C₄	—	2	1	no	—
174	C₄	—	2	1	no	—
175	C₄	—	4	2	no	—
176	C₄	—	4	2	no	—
177	C₄	C₄×D₈	1	2	yes	—
178	C₄	C₂²×M₁₆	1	1	yes	characteristic
179	C₄	#568	1	1	yes	characteristic
180	C₄	C₂²×D₈	1	1	yes	characteristic
181	C₄	C₂×#135	1	1	yes	characteristic
182	C₂	—	2	2	no	—
183	C₂	—	4	4	no	—
184	C₂	—	8	2	no	—
185	C₂	D₄×M₁₆	1	1	yes	characteristic
186	C₂	C₂×C₄×D₈	1	1	yes	characteristic
187	C₂	#2957	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄
Frattini, cosocle	C₄²	C₂⁴
socle	C₂²	C₂×C₄×D₄
prime radical	C₂²×D₈	C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₄×M₁₆ ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

Complements

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

N	H ≅ G/N	type
2. D₄×M₁₆	C₂	semidirect
3. #1589	C₂	semidirect
4. #1581	C₂	semidirect
5. C₈×D₈	C₂	semidirect
6. #1567	C₂	semidirect
7. #1486	C₂	semidirect
8. C₈×M₁₆	C₂	semidirect
18. #436	C₂²	semidirect
20. C₈×D₄	C₂²	semidirect
28. #335	C₂²	semidirect
29. #335	C₂²	semidirect
30. D₄⋊C₈	C₂²	semidirect
31. D₄⋊C₈	C₂²	semidirect
32. #322	C₂²	semidirect
33. C₈²	C₂²	semidirect
95. D₈	M₁₆	direct (N × H)
99. M₁₆	D₈	direct (N × H)
132. D₄	C₂×M₁₆	semidirect
159. C₈	C₂×M₁₆	semidirect
160. C₈	C₂×D₈	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×D₈	2
D₄×M₁₆	2
#1589	2
#1581	2
C₈×D₈	2
#1567	2
#1486	2
C₈×M₁₆	2

Subgroups — M16×D8 #18034

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — M₁₆×D₈ #18034