Subgroups — C₂×#12906 #494155

Order 512.

914 conjugacy classes of subgroups.

The subgroup lattice has 156 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#367	C₂	1	4	yes	maximal
2	C₂²×C₄×C₁₆	C₂	1	1	yes	maximal, characteristic
3	C₂×#1845	C₂	1	2	yes	maximal
4	#12906	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×#367	—	2	16	no	—
11	C₂×#367	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	#367	—	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	SD₃₂	1	16	yes	—
77	C₂²×C₄	C₂×C₄⋊C₄	1	2	yes	—
78	C₂²×C₄	#127	1	2	yes	—
79	C₂²×C₄	C₄⋊Q₈	1	2	yes	—
80	C₂²×C₄	C₂×D₈	1	2	yes	—
81	C₂²×C₄	C₂×D₈	1	1	yes	characteristic
82	C₂²×C₄	C₂×D₈	1	4	yes	—
83	C₂²×C₄	C₂×Q₁₆	1	2	yes	—
84	C₂²×C₄	C₂×Q₁₆	1	2	yes	—
85	C₂²×C₄	C₂²×D₄	1	1	yes	characteristic
86	C₂×C₈	#107	1	16	yes	—
87	C₂×C₈	C₂×C₄⋊C₄	1	8	yes	—
88	C₂×C₈	C₂×C₄⋊C₄	1	2	yes	—
89	C₂×C₈	C₂×C₄⋊C₄	1	4	yes	—
90	C₂×C₈	#127	1	2	yes	—
91	C₂×C₈	C₄⋊Q₈	1	8	yes	—
92	C₂×C₈	C₄⋊Q₈	1	2	yes	—
93	C₂×C₈	C₂×D₈	1	4	yes	—
94	C₂×C₈	C₂×Q₁₆	1	4	yes	—
95	C₂×C₈	C₂³×C₄	1	1	yes	characteristic
96	C₂×C₈	C₂²×D₄	1	2	yes	—
97	C₂×C₈	C₂²×Q₈	1	2	yes	—
98	C₂×C₈	C₂²×Q₈	1	1	yes	characteristic
99	C₄⋊C₄	—	4	8	no	—
100	C₄⋊C₄	—	4	8	no	—
101	C₄⋊C₄	—	8	8	no	—
102	C₄⋊C₄	—	8	32	no	—
103	C₄²	#107	1	8	yes	—
104	C₄²	C₂×C₄⋊C₄	1	4	yes	—
105	C₄²	C₂×Q₁₆	1	4	yes	—
106	C₁₆	C₂×C₄⋊C₄	1	16	yes	complemented
107	C₂³	C₂×#107	1	2	yes	—
108	C₂³	#493	1	2	yes	—
109	C₂³	#500	1	2	yes	—
110	C₂³	C₂×SD₃₂	1	8	yes	—
111	C₂³	C₂²×D₈	1	1	yes	characteristic
112	C₂×C₄	—	8	8	no	—
113	C₂×C₄	—	8	16	no	—
114	C₂×C₄	—	16	16	no	—
115	C₂×C₄	#367	1	32	yes	complemented
116	C₂×C₄	#384	1	4	yes	—
117	C₂×C₄	C₂×#107	1	2	yes	—
118	C₂×C₄	C₂×#107	1	4	yes	—
119	C₂×C₄	C₂×#107	1	8	yes	—
120	C₂×C₄	#493	1	2	yes	—
121	C₂×C₄	#494	1	4	yes	—
122	C₂×C₄	#500	1	2	yes	—
123	C₂×C₄	#500	1	4	yes	—
124	C₂×C₄	C₂×SD₃₂	1	16	yes	—
125	C₂×C₄	C₂²×C₄⋊C₄	1	1	yes	characteristic
126	C₂×C₄	C₂×#127	1	1	yes	characteristic
127	C₂×C₄	C₂×C₄⋊Q₈	1	1	yes	characteristic
128	C₂×C₄	C₂²×D₈	1	1	yes	characteristic
129	C₂×C₄	C₂²×Q₁₆	1	1	yes	characteristic
130	C₂×C₄	C₂²×Q₁₆	1	1	yes	characteristic
131	C₈	#384	1	4	yes	—
132	C₈	C₂×#107	1	8	yes	—
133	C₈	C₂²×C₄⋊C₄	1	2	yes	—
134	C₈	C₂×C₄⋊Q₈	1	2	yes	—
135	C₂²	#1845	1	4	yes	—
136	C₂²	C₂×#367	1	8	yes	—
137	C₂²	#2248	1	8	yes	—
138	C₂²	#2254	1	8	yes	—
139	C₂²	C₂²×#107	1	1	yes	characteristic
140	C₂²	C₂×#493	1	1	yes	characteristic
141	C₂²	C₂×#500	1	1	yes	characteristic
142	C₂²	C₂²×SD₃₂	1	4	yes	—
143	C₄	—	16	8	no	—
144	C₄	#1845	1	4	yes	—
145	C₄	C₂×#367	1	16	yes	complemented
146	C₄	C₂×#384	1	1	yes	characteristic
147	C₄	C₂²×#107	1	1	yes	characteristic
148	C₄	C₂×#494	1	1	yes	characteristic
149	C₄	C₂×#500	1	1	yes	characteristic
150	C₂	#12906	1	8	yes	direct product factor
151	C₂	C₂×#1845	1	1	yes	characteristic
152	C₂	C₂²×#367	1	2	yes	—
153	C₂	C₂×#2248	1	2	yes	—
154	C₂	C₂×#2254	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. #12906	C₂	direct (N × H)
12. C₂×C₄×C₁₆	C₄	semidirect
37. C₄×C₁₆	C₂×C₄	semidirect
61. C₂×C₁₆	C₄⋊C₄	semidirect
106. C₁₆	C₂×C₄⋊C₄	semidirect
115. C₂×C₄	#367	semidirect
145. C₄	C₂×#367	semidirect
150. C₂	#12906	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×#367	2
C₂²×C₄×C₁₆	2
C₂×#1845	2
#12906	2

Subgroups — C2×#12906 #494155

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#12906 #494155