Subgroups — C₂×#12905 #494154

Order 512.

914 conjugacy classes of subgroups.

The subgroup lattice has 162 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#366	C₂	1	4	yes	maximal
2	C₂²×C₄×C₁₆	C₂	1	1	yes	maximal, characteristic
3	C₂×#1845	C₂	1	2	yes	maximal
4	#12905	C₂	1	8	yes	maximal, direct product factor
5	C₂³×C₁₆	C₂²	1	2	yes	—
6	C₂²×#107	—	2	2	no	—
7	C₂²×#107	C₂²	1	4	yes	—
8	C₂²×C₄×C₈	C₂²	1	1	yes	characteristic
9	C₂×#384	—	2	2	no	—
10	C₂×#366	—	2	16	no	—
11	C₂×#366	C₂²	1	16	yes	—
12	C₂×C₄×C₁₆	C₄	1	8	yes	complemented
13	C₂×C₄×C₁₆	C₂²	1	4	yes	—
14	#1845	—	2	4	no	—
15	#1845	C₂²	1	8	yes	—
16	C₂³×C₈	D₄	1	1	yes	characteristic
17	C₂³×C₈	C₂³	1	1	yes	characteristic
18	C₂²×C₄⋊C₄	—	2	4	no	—
19	C₂²×C₄⋊C₄	—	4	4	no	—
20	C₂²×C₄²	D₄	1	1	yes	characteristic
21	C₂²×C₁₆	C₂×C₄	1	8	yes	—
22	C₂²×C₁₆	D₄	1	8	yes	—
23	C₂²×C₁₆	Q₈	1	8	yes	—
24	C₂²×C₁₆	C₂³	1	4	yes	—
25	C₂×#107	—	2	16	no	—
26	C₂×#107	—	2	4	no	—
27	C₂×#107	—	2	8	no	—
28	C₂×#107	—	4	8	no	—
29	C₂×#107	C₂³	1	8	yes	—
30	C₂×C₄×C₈	C₂×C₄	1	4	yes	—
31	C₂×C₄×C₈	D₄	1	2	yes	—
32	C₂×C₄×C₈	Q₈	1	4	yes	—
33	C₂×C₄×C₈	C₂³	1	2	yes	—
34	#384	—	2	8	no	—
35	#384	—	4	4	no	—
36	#366	—	2	32	no	—
37	C₄×C₁₆	C₂×C₄	1	16	yes	complemented
38	C₂³×C₄	—	4	4	no	—
39	C₂³×C₄	D₈	1	2	yes	—
40	C₂³×C₄	C₂×D₄	1	1	yes	characteristic
41	C₂²×C₈	C₄⋊C₄	1	4	yes	—
42	C₂²×C₈	D₈	1	4	yes	—
43	C₂²×C₈	Q₁₆	1	4	yes	—
44	C₂²×C₈	C₂²×C₄	1	2	yes	—
45	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
46	C₂²×C₈	C₂×D₄	1	4	yes	—
47	C₂²×C₈	C₂×D₄	1	2	yes	—
48	C₂²×C₈	C₂×Q₈	1	4	yes	—
49	C₂²×C₈	C₂×Q₈	1	2	yes	—
50	C₂²×C₈	C₂⁴	1	1	yes	characteristic
51	C₂×C₄⋊C₄	—	2	8	no	—
52	C₂×C₄⋊C₄	—	4	8	no	—
53	C₂×C₄⋊C₄	—	4	8	no	—
54	C₂×C₄⋊C₄	—	4	4	no	—
55	C₂×C₄⋊C₄	—	4	16	no	—
56	C₂×C₄⋊C₄	—	8	16	no	—
57	C₂×C₄²	C₄⋊C₄	1	4	yes	—
58	C₂×C₄²	D₈	1	2	yes	—
59	C₂×C₄²	Q₁₆	1	4	yes	—
60	C₂×C₄²	C₂×D₄	1	2	yes	—
61	C₂×C₁₆	C₄⋊C₄	1	32	yes	complemented
62	C₂×C₁₆	C₂²×C₄	1	8	yes	—
63	C₂×C₁₆	C₂×D₄	1	8	yes	—
64	C₂×C₁₆	C₂×Q₈	1	8	yes	—
65	#107	—	2	16	no	—
66	#107	—	4	8	no	—
67	#107	—	4	16	no	—
68	C₄×C₈	C₄⋊C₄	1	8	yes	—
69	C₄×C₈	C₂²×C₄	1	4	yes	—
70	C₄×C₈	C₂×Q₈	1	4	yes	—
71	C₂⁴	C₂×D₈	1	1	yes	characteristic
72	C₂²×C₄	—	4	8	no	—
73	C₂²×C₄	—	8	8	no	—
74	C₂²×C₄	—	8	16	no	—
75	C₂²×C₄	#107	1	8	yes	—
76	C₂²×C₄	D₁₆	1	8	yes	—
77	C₂²×C₄	Q₃₂	1	8	yes	—
78	C₂²×C₄	C₂×C₄⋊C₄	1	2	yes	—
79	C₂²×C₄	#127	1	2	yes	—
80	C₂²×C₄	C₄⋊Q₈	1	2	yes	—
81	C₂²×C₄	C₂×D₈	1	2	yes	—
82	C₂²×C₄	C₂×D₈	1	1	yes	characteristic
83	C₂²×C₄	C₂×D₈	1	4	yes	—
84	C₂²×C₄	C₂×Q₁₆	1	2	yes	—
85	C₂²×C₄	C₂×Q₁₆	1	2	yes	—
86	C₂²×C₄	C₂²×D₄	1	1	yes	characteristic
87	C₂×C₈	#107	1	16	yes	—
88	C₂×C₈	C₂×C₄⋊C₄	1	8	yes	—
89	C₂×C₈	C₂×C₄⋊C₄	1	2	yes	—
90	C₂×C₈	C₂×C₄⋊C₄	1	4	yes	—
91	C₂×C₈	#127	1	2	yes	—
92	C₂×C₈	C₄⋊Q₈	1	8	yes	—
93	C₂×C₈	C₄⋊Q₈	1	2	yes	—
94	C₂×C₈	C₂×D₈	1	4	yes	—
95	C₂×C₈	C₂×Q₁₆	1	4	yes	—
96	C₂×C₈	C₂³×C₄	1	1	yes	characteristic
97	C₂×C₈	C₂²×D₄	1	2	yes	—
98	C₂×C₈	C₂²×Q₈	1	2	yes	—
99	C₂×C₈	C₂²×Q₈	1	1	yes	characteristic
100	C₄⋊C₄	—	4	8	no	—
101	C₄⋊C₄	—	4	8	no	—
102	C₄⋊C₄	—	8	8	no	—
103	C₄⋊C₄	—	8	32	no	—
104	C₄²	#107	1	8	yes	—
105	C₄²	C₂×C₄⋊C₄	1	4	yes	—
106	C₄²	C₂×Q₁₆	1	4	yes	—
107	C₁₆	C₂×C₄⋊C₄	1	16	yes	complemented
108	C₂³	C₂×#107	1	2	yes	—
109	C₂³	#493	1	2	yes	—
110	C₂³	#500	1	2	yes	—
111	C₂³	C₂×D₁₆	1	4	yes	—
112	C₂³	C₂×Q₃₂	1	4	yes	—
113	C₂³	C₂²×D₈	1	1	yes	characteristic
114	C₂×C₄	—	8	8	no	—
115	C₂×C₄	—	8	16	no	—
116	C₂×C₄	—	16	16	no	—
117	C₂×C₄	#366	1	32	yes	complemented
118	C₂×C₄	#384	1	4	yes	—
119	C₂×C₄	C₂×#107	1	2	yes	—
120	C₂×C₄	C₂×#107	1	4	yes	—
121	C₂×C₄	C₂×#107	1	8	yes	—
122	C₂×C₄	#493	1	2	yes	—
123	C₂×C₄	#494	1	4	yes	—
124	C₂×C₄	#500	1	2	yes	—
125	C₂×C₄	#500	1	4	yes	—
126	C₂×C₄	C₂×D₁₆	1	8	yes	—
127	C₂×C₄	C₂×Q₃₂	1	8	yes	—
128	C₂×C₄	C₂²×C₄⋊C₄	1	1	yes	characteristic
129	C₂×C₄	C₂×#127	1	1	yes	characteristic
130	C₂×C₄	C₂×C₄⋊Q₈	1	1	yes	characteristic
131	C₂×C₄	C₂²×D₈	1	1	yes	characteristic
132	C₂×C₄	C₂²×Q₁₆	1	1	yes	characteristic
133	C₂×C₄	C₂²×Q₁₆	1	1	yes	characteristic
134	C₈	#384	1	4	yes	—
135	C₈	C₂×#107	1	8	yes	—
136	C₈	C₂²×C₄⋊C₄	1	2	yes	—
137	C₈	C₂×C₄⋊Q₈	1	2	yes	—
138	C₂²	#1845	1	4	yes	—
139	C₂²	C₂×#366	1	8	yes	—
140	C₂²	#2246	1	4	yes	—
141	C₂²	#2247	1	4	yes	—
142	C₂²	#2252	1	8	yes	—
143	C₂²	C₂²×#107	1	1	yes	characteristic
144	C₂²	C₂×#493	1	1	yes	characteristic
145	C₂²	C₂×#500	1	1	yes	characteristic
146	C₂²	C₂²×D₁₆	1	2	yes	—
147	C₂²	C₂²×Q₃₂	1	2	yes	—
148	C₄	—	16	8	no	—
149	C₄	#1845	1	4	yes	—
150	C₄	C₂×#366	1	16	yes	complemented
151	C₄	C₂×#384	1	1	yes	characteristic
152	C₄	C₂²×#107	1	1	yes	characteristic
153	C₄	C₂×#494	1	1	yes	characteristic
154	C₄	C₂×#500	1	1	yes	characteristic
155	C₂	#12905	1	8	yes	direct product factor
156	C₂	C₂×#1845	1	1	yes	characteristic
157	C₂	C₂²×#366	1	2	yes	—
158	C₂	C₂×#2246	1	1	yes	characteristic
159	C₂	C₂×#2247	1	1	yes	characteristic
160	C₂	C₂×#2252	1	2	yes	—

	subgroup	G/H
center, socle, prime radical	C₂⁴	C₂×D₈
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂²×C₄² ◁ C₂²×C₄×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. #12905	C₂	direct (N × H)
12. C₂×C₄×C₁₆	C₄	semidirect
37. C₄×C₁₆	C₂×C₄	semidirect
61. C₂×C₁₆	C₄⋊C₄	semidirect
107. C₁₆	C₂×C₄⋊C₄	semidirect
117. C₂×C₄	#366	semidirect
150. C₄	C₂×#366	semidirect
155. C₂	#12905	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×#366	2
C₂²×C₄×C₁₆	2
C₂×#1845	2
#12905	2

Subgroups — C2×#12905 #494154

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#12905 #494154