Subgroups — C₂×#1383 #12562

Order 256.

307 conjugacy classes of subgroups.

The subgroup lattice has 152 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×M₃₂	C₂	1	1	yes	maximal, characteristic
2	C₂²×#108	C₂	1	1	yes	maximal, characteristic
3	C₂²×#106	C₂	1	1	yes	maximal, characteristic
4	#1383	C₂	1	4	yes	maximal, direct product factor
5	C₂²×M₁₆	—	2	1	no	—
6	C₂³×C₈	C₂²	1	1	yes	characteristic
7	C₂²×C₄⋊C₄	—	2	1	no	—
8	C₂×M₃₂	—	2	2	no	—
9	C₂×M₃₂	—	2	2	no	—
10	C₂×M₃₂	C₄	1	2	yes	complemented
11	C₂×M₃₂	C₂²	1	2	yes	—
12	C₂²×C₁₆	C₄	1	2	yes	complemented
13	C₂×#108	—	2	1	no	—
14	C₂×#108	—	2	2	no	—
15	C₂×#108	C₄	1	4	yes	complemented
16	C₂×#108	C₂²	1	2	yes	—
17	C₂×#106	—	2	1	no	—
18	C₂×#106	—	2	2	no	—
19	C₂×#106	C₄	1	4	yes	—
20	C₂×#106	C₂²	1	2	yes	—
21	C₂³×C₄	—	4	1	no	—
22	C₂³×C₄	D₄	1	1	yes	characteristic
23	C₂×M₁₆	—	2	4	no	—
24	C₂×M₁₆	—	2	2	no	—
25	C₂×M₁₆	—	4	2	no	—
26	C₂×M₁₆	—	4	1	no	—
27	C₂²×C₈	—	4	1	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	2	yes	—
31	C₂²×C₈	C₂×C₄	1	2	yes	—
32	C₂²×C₈	D₄	1	1	yes	characteristic
33	C₂²×C₈	D₄	1	2	yes	—
34	C₂²×C₈	D₄	1	2	yes	—
35	C₂²×C₈	Q₈	1	1	yes	characteristic
36	C₂²×C₈	Q₈	1	1	yes	characteristic
37	C₂²×C₈	C₂³	1	1	yes	characteristic
38	C₂×C₄⋊C₄	—	2	4	no	—
39	C₂×C₄⋊C₄	—	2	2	no	—
40	C₂×C₄⋊C₄	—	4	2	no	—
41	C₂×C₄⋊C₄	—	4	1	no	—
42	M₃₂	—	2	4	no	—
43	M₃₂	—	2	2	no	—
44	M₃₂	—	2	2	no	—
45	C₂×C₁₆	—	2	4	no	—
46	C₂×C₁₆	C₂×C₄	1	4	yes	complemented
47	#108	—	2	4	no	—
48	#108	—	2	2	no	—
49	#108	C₂×C₄	1	4	yes	complemented
50	#106	—	2	4	no	—
51	#106	—	2	2	no	—
52	#106	C₂×C₄	1	4	yes	—
53	C₂⁴	SD₁₆	1	1	yes	characteristic
54	C₂²×C₄	—	4	4	no	—
55	C₂²×C₄	—	4	2	no	—
56	C₂²×C₄	—	8	4	no	—
57	C₂²×C₄	C₂²⋊C₄	1	2	yes	—
58	C₂²×C₄	C₂²⋊C₄	1	2	yes	—
59	C₂²×C₄	C₄⋊C₄	1	1	yes	characteristic
60	C₂²×C₄	C₄⋊C₄	1	1	yes	characteristic
61	C₂²×C₄	D₈	1	2	yes	—
62	C₂²×C₄	SD₁₆	1	1	yes	characteristic
63	C₂²×C₄	SD₁₆	1	1	yes	characteristic
64	C₂²×C₄	SD₁₆	1	1	yes	characteristic
65	C₂²×C₄	Q₁₆	1	2	yes	—
66	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
67	M₁₆	—	2	4	no	—
68	M₁₆	—	4	4	no	—
69	M₁₆	—	4	2	no	—
70	C₂×C₈	—	4	4	no	—
71	C₂×C₈	—	4	2	no	—
72	C₂×C₈	C₄²	1	2	yes	—
73	C₂×C₈	C₄²	1	2	yes	—
74	C₂×C₈	C₂²⋊C₄	1	2	yes	—
75	C₂×C₈	C₂²⋊C₄	1	2	yes	—
76	C₂×C₈	C₂²⋊C₄	1	2	yes	—
77	C₂×C₈	C₂²⋊C₄	1	2	yes	—
78	C₂×C₈	C₄⋊C₄	1	1	yes	characteristic
79	C₂×C₈	C₄⋊C₄	1	1	yes	characteristic
80	C₂×C₈	C₄⋊C₄	1	2	yes	—
81	C₂×C₈	C₄⋊C₄	1	2	yes	—
82	C₂×C₈	C₄⋊C₄	1	2	yes	—
83	C₂×C₈	C₄⋊C₄	1	2	yes	—
84	C₂×C₈	C₂²×C₄	1	1	yes	characteristic
85	C₂×C₈	C₂²×C₄	1	1	yes	characteristic
86	C₂×C₈	C₂²×C₄	1	1	yes	characteristic
87	C₂×C₈	C₂×D₄	1	1	yes	characteristic
88	C₂×C₈	C₂×D₄	1	1	yes	characteristic
89	C₂×C₈	C₂×Q₈	1	1	yes	characteristic
90	C₄⋊C₄	—	2	4	no	—
91	C₄⋊C₄	—	4	4	no	—
92	C₄⋊C₄	—	4	2	no	—
93	C₁₆	—	2	4	no	—
94	C₂³	—	2	4	no	—
95	C₂³	#102	1	2	yes	—
96	C₂³	#103	1	2	yes	—
97	C₂³	#106	1	1	yes	characteristic
98	C₂³	#106	1	1	yes	characteristic
99	C₂³	C₂×SD₁₆	1	1	yes	characteristic
100	C₂×C₄	—	4	4	no	—
101	C₂×C₄	—	8	8	no	—
102	C₂×C₄	—	8	4	no	—
103	C₂×C₄	#95	1	2	yes	—
104	C₂×C₄	#95	1	2	yes	—
105	C₂×C₄	#102	1	2	yes	—
106	C₂×C₄	#102	1	2	yes	—
107	C₂×C₄	#102	1	2	yes	—
108	C₂×C₄	#103	1	2	yes	—
109	C₂×C₄	#103	1	2	yes	—
110	C₂×C₄	#103	1	2	yes	—
111	C₂×C₄	#106	1	1	yes	characteristic
112	C₂×C₄	#106	1	1	yes	characteristic
113	C₂×C₄	#107	1	2	yes	—
114	C₂×C₄	#107	1	2	yes	—
115	C₂×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
116	C₂×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
117	C₂×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
118	C₂×C₄	C₂×D₈	1	1	yes	characteristic
119	C₂×C₄	C₂×SD₁₆	1	1	yes	characteristic
120	C₂×C₄	C₂×Q₁₆	1	1	yes	characteristic
121	C₈	—	4	4	no	—
122	C₈	#95	1	2	yes	—
123	C₈	#95	1	2	yes	—
124	C₈	C₂×C₄²	1	1	yes	characteristic
125	C₈	C₂×C₂²⋊C₄	1	1	yes	characteristic
126	C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
127	C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
128	C₂²	—	2	4	no	—
129	C₂²	—	2	4	no	—
130	C₂²	—	2	2	no	—
131	C₂²	—	2	2	no	—
132	C₂²	#340	1	2	yes	—
133	C₂²	#340	1	2	yes	—
134	C₂²	#365	1	4	yes	—
135	C₂²	C₂×#102	1	1	yes	characteristic
136	C₂²	C₂×#103	1	1	yes	characteristic
137	C₂²	C₂×#106	1	1	yes	characteristic
138	C₄	—	8	4	no	—
139	C₄	#340	1	2	yes	—
140	C₄	#340	1	2	yes	—
141	C₄	C₂×#95	1	1	yes	characteristic
142	C₄	C₂×#102	1	1	yes	characteristic
143	C₄	C₂×#103	1	1	yes	characteristic
144	C₄	C₂×#107	1	1	yes	characteristic
145	C₂	—	2	1	no	—
146	C₂	—	2	2	no	—
147	C₂	—	2	1	no	—
148	C₂	#1383	1	4	yes	direct product factor
149	C₂	C₂×#340	1	1	yes	characteristic
150	C₂	C₂×#365	1	2	yes	—

	subgroup	G/H
center, socle	C₂³	#106
commutator	C₈	C₂×C₄²
Frattini, cosocle	C₂²×C₈	C₂³
prime radical	C₂⁴	SD₁₆
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
4. #1383	C₂	direct (N × H)
10. C₂×M₃₂	C₄	semidirect
12. C₂²×C₁₆	C₄	semidirect
15. C₂×#108	C₄	semidirect
46. C₂×C₁₆	C₂×C₄	semidirect
49. #108	C₂×C₄	semidirect
148. C₂	#1383	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×M₃₂	2
C₂²×#108	2
C₂²×#106	2
#1383	2

Subgroups — C2×#1383 #12562

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#1383 #12562