Subgroups — C₂×#2168 #33430

Order 256.

377 conjugacy classes of subgroups.

The subgroup lattice has 168 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, complemented
2	C₂³×C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₈×D₄	C₂	1	1	yes	maximal, characteristic
4	#2168	C₂	1	8	yes	maximal, direct product factor
5	C₂×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₂²⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₂²⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#346	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂³×C₈	C₂²	1	1	yes	characteristic
10	C₂³×C₈	C₂²	1	1	yes	characteristic
11	C₂×C₄×D₄	C₄	1	1	yes	characteristic
12	C₂×M₃₂	—	2	4	no	—
13	C₂×M₃₂	C₂²	1	4	yes	complemented
14	C₂²×C₁₆	—	2	4	no	—
15	C₂²×C₁₆	C₂²	1	1	yes	characteristic
16	C₂²×C₁₆	C₂²	1	1	yes	characteristic
17	C₂²×C₁₆	C₂²	1	1	yes	characteristic
18	C₂²×C₁₆	C₂²	1	1	yes	characteristic
19	C₂²×C₁₆	C₂²	1	4	yes	complemented
20	C₈×D₄	C₄	1	4	yes	—
21	C₈×D₄	C₂²	1	4	yes	—
22	C₂×C₄⋊C₈	C₄	1	1	yes	characteristic
23	C₂×C₂²⋊C₈	C₄	1	1	yes	characteristic
24	C₂×C₂²⋊C₈	C₄	1	1	yes	characteristic
25	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
26	C₄⋊C₁₆	C₂²	1	4	yes	complemented
27	C₂²⋊C₁₆	C₂²	1	4	yes	complemented
28	C₂²⋊C₁₆	C₂²	1	4	yes	complemented
29	#346	C₂²	1	4	yes	complemented
30	C₂²×D₄	C₈	1	1	yes	characteristic
31	C₂³×C₄	C₂×C₄	1	1	yes	characteristic
32	C₂³×C₄	C₂×C₄	1	1	yes	characteristic
33	C₂²×C₈	—	2	2	no	—
34	C₂²×C₈	—	2	2	no	—
35	C₂²×C₈	—	2	2	no	—
36	C₂²×C₈	—	2	2	no	—
37	C₂²×C₈	C₂×C₄	1	1	yes	characteristic
38	C₂²×C₈	C₂×C₄	1	1	yes	characteristic
39	C₂²×C₈	C₂×C₄	1	2	yes	—
40	C₂²×C₈	C₂×C₄	1	2	yes	—
41	C₂²×C₈	C₂×C₄	1	1	yes	characteristic
42	C₂²×C₈	C₂³	1	2	yes	—
43	C₂²×C₈	C₂³	1	2	yes	—
44	C₂²×C₈	C₂³	1	1	yes	characteristic
45	C₄×D₄	C₈	1	4	yes	—
46	C₄×D₄	C₂×C₄	1	4	yes	—
47	C₂×C₄⋊C₄	C₈	1	1	yes	characteristic
48	C₂×C₂²⋊C₄	C₈	1	1	yes	characteristic
49	C₂×C₂²⋊C₄	C₈	1	1	yes	characteristic
50	C₂×C₄²	C₂×C₄	1	1	yes	characteristic
51	M₃₂	—	2	8	no	—
52	C₂×C₁₆	—	2	2	no	—
53	C₂×C₁₆	—	2	2	no	—
54	C₂×C₁₆	—	2	2	no	—
55	C₂×C₁₆	—	2	8	no	—
56	C₂×C₁₆	D₄	1	4	yes	complemented
57	C₂×C₁₆	C₂³	1	2	yes	—
58	C₂×C₁₆	C₂³	1	2	yes	—
59	C₂×C₁₆	C₂³	1	2	yes	—
60	C₂×C₁₆	C₂³	1	2	yes	—
61	C₄⋊C₈	C₂×C₄	1	4	yes	—
62	C₂²⋊C₈	C₂×C₄	1	4	yes	—
63	C₂²⋊C₈	C₂×C₄	1	4	yes	—
64	C₄×C₈	C₂×C₄	1	2	yes	—
65	C₄×C₈	C₂³	1	2	yes	—
66	C₂⁴	C₂×C₈	1	1	yes	characteristic
67	C₂⁴	C₂×C₈	1	1	yes	characteristic
68	C₂×D₄	—	2	4	no	—
69	C₂×D₄	C₂×C₈	1	4	yes	—
70	C₂²×C₄	—	2	2	no	—
71	C₂²×C₄	—	2	2	no	—
72	C₂²×C₄	—	2	2	no	—
73	C₂²×C₄	—	2	2	no	—
74	C₂²×C₄	C₂×C₈	1	1	yes	characteristic
75	C₂²×C₄	C₂×C₈	1	1	yes	characteristic
76	C₂²×C₄	C₂×C₈	1	1	yes	characteristic
77	C₂²×C₄	C₂×C₈	1	1	yes	characteristic
78	C₂²×C₄	C₂×C₈	1	2	yes	—
79	C₂²×C₄	C₂×C₈	1	2	yes	—
80	C₂²×C₄	C₂²×C₄	1	2	yes	—
81	C₂²×C₄	C₂²×C₄	1	2	yes	—
82	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
83	C₂×C₈	—	2	2	no	—
84	C₂×C₈	—	2	2	no	—
85	C₂×C₈	—	2	2	no	—
86	C₂×C₈	—	2	4	no	—
87	C₂×C₈	—	2	4	no	—
88	C₂×C₈	—	2	2	no	—
89	C₂×C₈	—	2	2	no	—
90	C₂×C₈	—	2	2	no	—
91	C₂×C₈	—	2	2	no	—
92	C₂×C₈	C₂²×C₄	1	2	yes	—
93	C₂×C₈	C₂²×C₄	1	2	yes	—
94	C₂×C₈	C₂²×C₄	1	2	yes	—
95	C₂×C₈	C₂²×C₄	1	1	yes	characteristic
96	C₂×C₈	C₂×D₄	1	2	yes	—
97	C₂×C₈	#41	1	2	yes	—
98	C₂×C₈	C₂⁴	1	1	yes	characteristic
99	C₄⋊C₄	C₂×C₈	1	4	yes	—
100	C₂²⋊C₄	C₂×C₈	1	4	yes	—
101	C₂²⋊C₄	C₂×C₈	1	4	yes	—
102	C₄²	C₂×C₈	1	2	yes	—
103	C₄²	C₂²×C₄	1	2	yes	—
104	C₁₆	—	2	2	no	—
105	C₁₆	—	2	2	no	—
106	C₁₆	—	2	2	no	—
107	C₁₆	C₂×D₄	1	4	yes	complemented
108	C₂³	—	2	2	no	—
109	C₂³	—	2	2	no	—
110	C₂³	—	2	2	no	—
111	C₂³	—	2	2	no	—
112	C₂³	—	2	2	no	—
113	C₂³	M₃₂	1	4	yes	complemented
114	C₂³	C₂²×C₈	1	2	yes	—
115	C₂³	C₂²×C₈	1	2	yes	—
116	C₂³	C₂²×C₈	1	1	yes	characteristic
117	D₄	—	2	8	no	—
118	C₂×C₄	—	2	2	no	—
119	C₂×C₄	—	2	2	no	—
120	C₂×C₄	—	2	2	no	—
121	C₂×C₄	—	2	2	no	—
122	C₂×C₄	—	2	2	no	—
123	C₂×C₄	—	2	2	no	—
124	C₂×C₄	—	2	2	no	—
125	C₂×C₄	—	2	2	no	—
126	C₂×C₄	—	2	4	no	—
127	C₂×C₄	—	2	4	no	—
128	C₂×C₄	C₄×D₄	1	2	yes	—
129	C₂×C₄	C₂²×C₈	1	2	yes	—
130	C₂×C₄	C₂²×C₈	1	2	yes	—
131	C₂×C₄	C₂²×C₈	1	2	yes	—
132	C₂×C₄	C₂²×C₈	1	2	yes	—
133	C₂×C₄	C₂²×C₈	1	1	yes	characteristic
134	C₂×C₄	#131	1	2	yes	—
135	C₂×C₄	C₂³×C₄	1	1	yes	characteristic
136	C₈	—	2	2	no	—
137	C₈	—	2	2	no	—
138	C₈	—	2	2	no	—
139	C₈	C₄×D₄	1	2	yes	—
140	C₈	C₂²×D₄	1	1	yes	characteristic
141	C₈	C₂×#41	1	1	yes	characteristic
142	C₂²	—	2	2	no	—
143	C₂²	—	2	2	no	—
144	C₂²	—	2	8	no	—
145	C₂²	—	2	2	no	—
146	C₂²	—	2	2	no	—
147	C₂²	—	2	2	no	—
148	C₂²	—	4	4	no	—
149	C₂²	C₈×D₄	1	2	yes	—
150	C₂²	C₂×M₃₂	1	2	yes	—
151	C₂²	C₂×M₃₂	1	4	yes	complemented
152	C₂²	#504	1	2	yes	—
153	C₂²	C₂³×C₈	1	1	yes	characteristic
154	C₄	—	2	2	no	—
155	C₄	—	2	2	no	—
156	C₄	—	2	2	no	—
157	C₄	—	2	2	no	—
158	C₄	C₈×D₄	1	2	yes	—
159	C₄	C₂×C₄×D₄	1	1	yes	characteristic
160	C₄	C₂×#131	1	1	yes	characteristic
161	C₂	—	2	4	no	—
162	C₂	—	4	2	no	—
163	C₂	#2168	1	4	yes	direct product factor
164	C₂	C₂×C₈×D₄	1	1	yes	characteristic
165	C₂	C₂²×M₃₂	1	1	yes	characteristic
166	C₂	C₂×#504	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₈	C₂³
commutator	C₂²	C₂³×C₈
Frattini, cosocle	C₂×C₈	C₂⁴
socle	C₂³	C₂²×C₈
prime radical	C₂²×D₄	C₈
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₈ ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
1. C₂²×M₃₂	C₂	semidirect
2. C₂³×C₁₆	C₂	semidirect
4. #2168	C₂	direct (N × H)
5. C₂×C₄⋊C₁₆	C₂	semidirect
6. C₂×C₂²⋊C₁₆	C₂	semidirect
7. C₂×C₂²⋊C₁₆	C₂	semidirect
8. C₂×#346	C₂	semidirect
13. C₂×M₃₂	C₂²	semidirect
19. C₂²×C₁₆	C₂²	semidirect
26. C₄⋊C₁₆	C₂²	semidirect
27. C₂²⋊C₁₆	C₂²	semidirect
28. C₂²⋊C₁₆	C₂²	semidirect
29. #346	C₂²	semidirect
56. C₂×C₁₆	D₄	semidirect
107. C₁₆	C₂×D₄	semidirect
113. C₂³	M₃₂	semidirect
151. C₂²	C₂×M₃₂	semidirect
163. C₂	#2168	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
C₂³×C₁₆	2
C₂×C₈×D₄	2
#2168	2
C₂×C₄⋊C₁₆	2
C₂×C₂²⋊C₁₆	2
C₂×#346	2

Subgroups — C2×#2168 #33430

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#2168 #33430