Subgroups — C₂×#1568 #17792

Order 256.

383 conjugacy classes of subgroups.

The subgroup lattice has 148 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#432	C₂	1	2	yes	maximal
2	C₂²×C₄×C₈	C₂	1	1	yes	maximal, characteristic
3	#1568	C₂	1	8	yes	maximal, direct product factor
4	C₂×#335	C₂	1	2	yes	maximal, complemented
5	C₂×#334	C₂	1	2	yes	maximal, complemented
6	C₂³×C₈	C₄	1	1	yes	characteristic
7	C₂³×C₈	C₄	1	1	yes	characteristic
8	C₂²×C₄²	C₂²	1	1	yes	characteristic
9	#432	—	2	4	no	—
10	#432	C₂²	1	8	yes	—
11	C₂×C₄⋊C₈	C₂²	1	4	yes	—
12	C₂×C₂²⋊C₈	—	2	2	no	—
13	C₂×#97	—	2	2	no	—
14	C₂×C₄×C₈	C₄	1	1	yes	characteristic
15	C₂×C₄×C₈	C₄	1	4	yes	—
16	C₂×C₄×C₈	C₄	1	1	yes	characteristic
17	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
18	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
19	C₂×C₄×C₈	C₂²	1	4	yes	—
20	#335	C₂²	1	8	yes	complemented
21	#334	C₂²	1	8	yes	complemented
22	C₂³×C₄	C₂×C₄	1	1	yes	characteristic
23	C₂³×C₄	D₄	1	1	yes	characteristic
24	C₂³×C₄	Q₈	1	1	yes	characteristic
25	C₂²×C₈	—	2	4	no	—
26	C₂²×C₈	—	2	4	no	—
27	C₂²×C₈	—	2	4	no	—
28	C₂²×C₈	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₈	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₈	C₂×C₄	1	4	yes	—
31	C₂²×C₈	C₂×C₄	1	4	yes	—
32	C₂²×C₈	C₂×C₄	1	1	yes	characteristic
33	C₂²×C₈	C₂×C₄	1	1	yes	characteristic
34	C₂×C₄²	C₂×C₄	1	2	yes	—
35	C₂×C₄²	C₂×C₄	1	1	yes	characteristic
36	C₂×C₄²	D₄	1	2	yes	—
37	C₂×C₄²	D₄	1	1	yes	characteristic
38	C₂×C₄²	Q₈	1	1	yes	characteristic
39	C₂×C₄²	Q₈	1	2	yes	—
40	C₂×C₄²	C₂³	1	2	yes	—
41	C₂×C₄²	C₂³	1	1	yes	characteristic
42	C₄⋊C₈	—	2	4	no	—
43	C₄⋊C₈	C₂³	1	8	yes	—
44	C₂²⋊C₈	—	2	8	no	—
45	#97	—	2	8	no	—
46	C₄×C₈	C₂×C₄	1	4	yes	—
47	C₄×C₈	C₂×C₄	1	2	yes	—
48	C₄×C₈	C₂×C₄	1	2	yes	—
49	C₄×C₈	C₂×C₄	1	4	yes	—
50	C₄×C₈	C₂³	1	2	yes	—
51	C₄×C₈	C₂³	1	2	yes	—
52	C₂⁴	C₄⋊C₄	1	1	yes	characteristic
53	C₂²×C₄	—	2	4	no	—
54	C₂²×C₄	—	2	2	no	—
55	C₂²×C₄	—	2	2	no	—
56	C₂²×C₄	—	2	4	no	—
57	C₂²×C₄	C₄⋊C₄	1	2	yes	—
58	C₂²×C₄	C₄⋊C₄	1	1	yes	characteristic
59	C₂²×C₄	C₄⋊C₄	1	1	yes	characteristic
60	C₂²×C₄	C₄⋊C₄	1	2	yes	—
61	C₂²×C₄	C₄⋊C₄	1	1	yes	characteristic
62	C₂²×C₄	C₄⋊C₄	1	2	yes	—
63	C₂²×C₄	C₂²×C₄	1	2	yes	—
64	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
65	C₂²×C₄	C₂×D₄	1	2	yes	—
66	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
67	C₂²×C₄	C₂×Q₈	1	2	yes	—
68	C₂²×C₄	C₂×Q₈	1	1	yes	characteristic
69	C₂×C₈	—	2	2	no	—
70	C₂×C₈	—	2	2	no	—
71	C₂×C₈	—	2	8	no	—
72	C₂×C₈	—	2	8	no	—
73	C₂×C₈	—	2	8	no	—
74	C₂×C₈	—	4	8	no	—
75	C₂×C₈	M₁₆	1	4	yes	—
76	C₂×C₈	M₁₆	1	4	yes	—
77	C₂×C₈	C₂²×C₄	1	2	yes	—
78	C₂×C₈	C₂²×C₄	1	2	yes	—
79	C₂×C₈	C₂²×C₄	1	2	yes	—
80	C₂×C₈	C₂²×C₄	1	2	yes	—
81	C₄²	C₄⋊C₄	1	2	yes	—
82	C₄²	C₄⋊C₄	1	2	yes	—
83	C₄²	C₄⋊C₄	1	2	yes	—
84	C₄²	C₂²×C₄	1	2	yes	—
85	C₄²	C₂²×C₄	1	1	yes	characteristic
86	C₄²	C₂×D₄	1	2	yes	—
87	C₄²	C₂×D₄	1	1	yes	characteristic
88	C₄²	C₂×Q₈	1	2	yes	—
89	C₄²	C₂×Q₈	1	1	yes	characteristic
90	C₄²	C₂⁴	1	1	yes	characteristic
91	C₂³	—	2	2	no	—
92	C₂³	—	2	2	no	—
93	C₂³	#108	1	4	yes	—
94	C₂³	C₂×C₄⋊C₄	1	2	yes	—
95	C₂³	C₂×C₄⋊C₄	1	1	yes	characteristic
96	C₂×C₄	—	2	2	no	—
97	C₂×C₄	—	2	2	no	—
98	C₂×C₄	—	2	4	no	—
99	C₂×C₄	—	2	2	no	—
100	C₂×C₄	—	2	4	no	—
101	C₂×C₄	—	2	2	no	—
102	C₂×C₄	—	2	2	no	—
103	C₂×C₄	—	2	2	no	—
104	C₂×C₄	—	2	4	no	—
105	C₂×C₄	—	2	4	no	—
106	C₂×C₄	—	2	4	no	—
107	C₂×C₄	C₂×C₄⋊C₄	1	2	yes	—
108	C₂×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
109	C₂×C₄	C₂×C₄⋊C₄	1	2	yes	—
110	C₂×C₄	C₂×C₄⋊C₄	1	2	yes	—
111	C₂×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
112	C₂×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
113	C₂×C₄	C₂×M₁₆	1	2	yes	—
114	C₂×C₄	C₂×M₁₆	1	2	yes	—
115	C₂×C₄	#135	1	4	yes	—
116	C₂×C₄	C₂³×C₄	1	1	yes	characteristic
117	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
118	C₂×C₄	C₂²×Q₈	1	1	yes	characteristic
119	C₈	—	2	2	no	—
120	C₈	—	2	2	no	—
121	C₈	—	4	8	no	—
122	C₈	C₂×M₁₆	1	4	yes	—
123	C₈	C₂×M₁₆	1	4	yes	—
124	C₂²	—	2	8	no	—
125	C₂²	—	2	2	no	—
126	C₂²	—	2	2	no	—
127	C₂²	#423	1	2	yes	—
128	C₂²	#427	1	2	yes	—
129	C₂²	C₂×#108	1	4	yes	—
130	C₂²	C₂×#108	1	2	yes	—
131	C₂²	C₂²×C₄⋊C₄	1	1	yes	characteristic
132	C₄	—	2	2	no	—
133	C₄	—	2	2	no	—
134	C₄	—	2	1	no	—
135	C₄	—	2	2	no	—
136	C₄	—	2	1	no	—
137	C₄	#423	1	2	yes	—
138	C₄	#427	1	2	yes	—
139	C₄	C₂²×M₁₆	1	1	yes	characteristic
140	C₄	C₂²×M₁₆	1	1	yes	characteristic
141	C₄	C₂×#135	1	2	yes	—
142	C₂	—	2	4	no	—
143	C₂	#1568	1	4	yes	direct product factor
144	C₂	C₂×#423	1	1	yes	characteristic
145	C₂	C₂×#427	1	1	yes	characteristic
146	C₂	C₂²×#108	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
3. #1568	C₂	direct (N × H)
4. C₂×#335	C₂	semidirect
5. C₂×#334	C₂	semidirect
20. #335	C₂²	semidirect
21. #334	C₂²	semidirect
143. C₂	#1568	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#432	2
C₂²×C₄×C₈	2
#1568	2
C₂×#335	2
C₂×#334	2

Subgroups — C2×#1568 #17792

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#1568 #17792