Subgroups — C₂×Q₈×M₁₆ #60190

Order 256.

1077 conjugacy classes of subgroups.

The subgroup lattice has 144 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄×Q₈	C₂	1	1	yes	maximal, characteristic
2	Q₈×M₁₆	C₂	1	16	yes	maximal, direct product factor
3	C₂×#446	C₂	1	6	yes	maximal, complemented
4	C₂×C₈×Q₈	C₂	1	2	yes	maximal, complemented
5	C₂×#423	C₂	1	3	yes	maximal
6	C₂×C₄×M₁₆	C₂	1	3	yes	maximal
7	C₂³×Q₈	C₄	1	1	yes	characteristic
8	C₂²×M₁₆	C₂²	1	3	yes	—
9	C₂²×M₁₆	C₂²	1	1	yes	characteristic
10	C₂×C₄×Q₈	C₄	1	3	yes	—
11	C₂×C₄×Q₈	C₄	1	8	yes	—
12	C₂×C₄×Q₈	C₄	1	1	yes	characteristic
13	C₂×C₄×Q₈	C₂²	1	1	yes	characteristic
14	C₂×C₄×Q₈	C₂²	1	3	yes	—
15	C₂×C₄×Q₈	C₂²	1	8	yes	—
16	C₂²×C₄⋊C₄	C₄	1	3	yes	—
17	C₂²×C₄²	C₂²	1	3	yes	—
18	#446	C₂²	1	48	yes	complemented
19	C₈×Q₈	C₂²	1	16	yes	complemented
20	#423	C₂²	1	24	yes	—
21	C₂×C₄⋊C₈	C₂²	1	6	yes	—
22	C₂×C₄⋊C₈	C₂²	1	6	yes	—
23	C₄×M₁₆	C₂²	1	24	yes	—
24	C₂×#97	C₂²	1	6	yes	—
25	C₂×C₄×C₈	C₂²	1	6	yes	—
26	C₂²×Q₈	—	2	8	no	—
27	C₂²×Q₈	C₂×C₄	1	8	yes	—
28	C₂²×Q₈	C₂×C₄	1	3	yes	—
29	C₂²×Q₈	C₂×C₄	1	1	yes	characteristic
30	C₂³×C₄	C₂×C₄	1	3	yes	—
31	C₂³×C₄	C₂×C₄	1	3	yes	—
32	C₂³×C₄	C₂³	1	1	yes	characteristic
33	C₂×M₁₆	—	2	12	no	—
34	C₂×M₁₆	Q₈	1	8	yes	direct product factor
35	C₂×M₁₆	C₂³	1	12	yes	—
36	C₂×M₁₆	C₂³	1	4	yes	—
37	C₂²×C₈	C₂³	1	6	yes	—
38	C₂²×C₈	C₂³	1	2	yes	—
39	C₄×Q₈	C₂×C₄	1	12	yes	—
40	C₄×Q₈	C₂×C₄	1	24	yes	—
41	C₄×Q₈	C₂×C₄	1	4	yes	—
42	C₄×Q₈	C₂×C₄	1	8	yes	—
43	C₄×Q₈	C₂³	1	4	yes	—
44	C₄×Q₈	C₂³	1	12	yes	—
45	C₂×C₄⋊C₄	C₂×C₄	1	6	yes	—
46	C₂×C₄⋊C₄	C₂×C₄	1	3	yes	—
47	C₂×C₄⋊C₄	C₂×C₄	1	3	yes	—
48	C₂×C₄⋊C₄	C₂×C₄	1	24	yes	—
49	C₂×C₄²	C₂×C₄	1	3	yes	—
50	C₂×C₄²	C₂×C₄	1	3	yes	—
51	C₂×C₄²	C₂×C₄	1	12	yes	—
52	C₂×C₄²	C₂³	1	12	yes	—
53	C₂×C₄²	C₂³	1	3	yes	—
54	C₂×C₄²	C₂³	1	3	yes	—
55	C₄⋊C₈	C₂³	1	24	yes	—
56	C₄⋊C₈	C₂³	1	24	yes	—
57	#97	C₂³	1	24	yes	—
58	C₄×C₈	C₂³	1	24	yes	—
59	C₂⁴	C₂²×C₄	1	1	yes	characteristic
60	C₂×Q₈	—	2	32	no	—
61	C₂×Q₈	—	2	12	no	—
62	C₂×Q₈	M₁₆	1	8	yes	direct product factor
63	C₂×Q₈	C₂²×C₄	1	12	yes	—
64	C₂×Q₈	C₂²×C₄	1	4	yes	—
65	C₂²×C₄	—	2	12	no	—
66	C₂²×C₄	—	4	6	no	—
67	C₂²×C₄	C₂²×C₄	1	3	yes	—
68	C₂²×C₄	C₂²×C₄	1	3	yes	—
69	C₂²×C₄	C₂²×C₄	1	3	yes	—
70	C₂²×C₄	C₂²×C₄	1	12	yes	—
71	C₂²×C₄	C₂²×C₄	1	12	yes	—
72	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
73	C₂²×C₄	C₂²×C₄	1	2	yes	—
74	C₂²×C₄	C₂²×C₄	1	3	yes	—
75	C₂²×C₄	C₂×Q₈	1	4	yes	—
76	C₂²×C₄	#41	1	4	yes	—
77	C₂²×C₄	C₂⁴	1	2	yes	—
78	C₂²×C₄	C₂⁴	1	1	yes	characteristic
79	M₁₆	—	2	24	no	—
80	M₁₆	C₂×Q₈	1	16	yes	direct product factor
81	C₂×C₈	—	2	12	no	—
82	C₂×C₈	C₂×Q₈	1	8	yes	complemented
83	C₂×C₈	C₂⁴	1	12	yes	—
84	C₂×C₈	C₂⁴	1	4	yes	—
85	C₄⋊C₄	C₂²×C₄	1	24	yes	—
86	C₄⋊C₄	C₂²×C₄	1	12	yes	—
87	C₄⋊C₄	C₂²×C₄	1	12	yes	—
88	C₄²	C₂²×C₄	1	12	yes	—
89	C₄²	C₂²×C₄	1	6	yes	—
90	C₄²	C₂²×C₄	1	12	yes	—
91	C₄²	C₂²×C₄	1	6	yes	—
92	C₄²	C₂⁴	1	6	yes	—
93	C₄²	C₂⁴	1	6	yes	—
94	C₂³	—	2	2	no	—
95	C₂³	—	2	2	no	—
96	C₂³	C₄×Q₈	1	4	yes	—
97	C₂³	C₂³×C₄	1	2	yes	—
98	C₂³	C₂³×C₄	1	1	yes	characteristic
99	Q₈	—	2	24	no	—
100	Q₈	C₂×M₁₆	1	16	yes	direct product factor
101	C₂×C₄	—	2	6	no	—
102	C₂×C₄	—	2	6	no	—
103	C₂×C₄	—	2	6	no	—
104	C₂×C₄	—	2	24	no	—
105	C₂×C₄	—	4	12	no	—
106	C₂×C₄	C₄×Q₈	1	8	yes	—
107	C₂×C₄	C₄×Q₈	1	4	yes	—
108	C₂×C₄	C₂×M₁₆	1	12	yes	—
109	C₂×C₄	C₂³×C₄	1	6	yes	—
110	C₂×C₄	C₂³×C₄	1	6	yes	—
111	C₂×C₄	C₂³×C₄	1	6	yes	—
112	C₂×C₄	C₂³×C₄	1	2	yes	—
113	C₂×C₄	C₂³×C₄	1	1	yes	characteristic
114	C₂×C₄	C₂³×C₄	1	6	yes	—
115	C₂×C₄	C₂²×Q₈	1	2	yes	—
116	C₂×C₄	C₂²×Q₈	1	4	yes	—
117	C₂×C₄	C₂×#41	1	2	yes	—
118	C₂×C₄	C₂×#41	1	4	yes	—
119	C₂×C₄	C₂⁵	1	1	yes	characteristic
120	C₈	—	2	12	no	—
121	C₈	C₂²×Q₈	1	8	yes	complemented
122	C₂²	—	2	8	no	—
123	C₂²	—	2	2	no	—
124	C₂²	—	2	2	no	—
125	C₂²	C₂×C₄×Q₈	1	4	yes	—
126	C₂²	C₂×C₄×Q₈	1	2	yes	—
127	C₂²	C₂²×M₁₆	1	2	yes	—
128	C₂²	#568	1	2	yes	—
129	C₂²	C₂⁴×C₄	1	1	yes	characteristic
130	C₄	—	2	6	no	—
131	C₄	—	2	6	no	—
132	C₄	—	2	6	no	—
133	C₄	C₂×C₄×Q₈	1	4	yes	—
134	C₄	C₂×C₄×Q₈	1	2	yes	—
135	C₄	C₂²×M₁₆	1	12	yes	—
136	C₄	C₂³×Q₈	1	1	yes	characteristic
137	C₄	C₂²×#41	1	1	yes	characteristic
138	C₂	—	2	4	no	—
139	C₂	Q₈×M₁₆	1	4	yes	direct product factor
140	C₂	C₂²×C₄×Q₈	1	1	yes	characteristic
141	C₂	C₂³×M₁₆	1	1	yes	characteristic
142	C₂	C₂×#568	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂⁴
commutator	C₂²	C₂⁴×C₄
Frattini, cosocle	C₂×C₄	C₂⁵
socle	C₂³	C₂³×C₄
prime radical	C₂⁴	C₂²×C₄
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
2. Q₈×M₁₆	C₂	direct (N × H)
3. C₂×#446	C₂	semidirect
4. C₂×C₈×Q₈	C₂	semidirect
18. #446	C₂²	semidirect
19. C₈×Q₈	C₂²	semidirect
34. C₂×M₁₆	Q₈	direct (N × H)
62. C₂×Q₈	M₁₆	direct (N × H)
80. M₁₆	C₂×Q₈	direct (N × H)
82. C₂×C₈	C₂×Q₈	semidirect
100. Q₈	C₂×M₁₆	direct (N × H)
121. C₈	C₂²×Q₈	semidirect
139. C₂	Q₈×M₁₆	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₄×Q₈	2
Q₈×M₁₆	2
C₂×#446	2
C₂×C₈×Q₈	2
C₂×#423	2
C₂×C₄×M₁₆	2

Subgroups — C2×Q8×M16 #60190

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×Q₈×M₁₆ #60190