Subgroups — C₂×#12907 #494156

Order 512.

826 conjugacy classes of subgroups.

The subgroup lattice has 178 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	2	yes	maximal
2	C₂²×#366	C₂	1	2	yes	maximal
3	C₂²×C₄×C₁₆	C₂	1	1	yes	maximal, characteristic
4	C₂×#1846	C₂	1	2	yes	maximal
5	#12907	C₂	1	8	yes	maximal, direct product factor
6	C₂³×C₁₆	C₂²	1	1	yes	characteristic
7	C₂³×C₁₆	C₂²	1	1	yes	characteristic
8	C₂²×#107	C₂²	1	4	yes	—
9	C₂²×#106	—	2	2	no	—
10	C₂²×C₄×C₈	C₂²	1	1	yes	characteristic
11	C₂×#382	—	2	2	no	—
12	C₂×#367	—	2	8	no	—
13	C₂×#367	C₂²	1	8	yes	—
14	C₂×#366	—	2	8	no	—
15	C₂×#366	C₂²	1	8	yes	—
16	C₂×C₄×C₁₆	C₄	1	8	yes	complemented
17	C₂×C₄×C₁₆	C₂²	1	4	yes	—
18	#1846	—	2	4	no	—
19	#1846	C₂²	1	8	yes	—
20	C₂³×C₈	D₄	1	1	yes	characteristic
21	C₂³×C₈	C₂³	1	1	yes	characteristic
22	C₂²×C₄⋊C₄	—	2	4	no	—
23	C₂²×C₄²	D₄	1	1	yes	characteristic
24	C₂²×C₁₆	C₂×C₄	1	4	yes	—
25	C₂²×C₁₆	C₂×C₄	1	4	yes	—
26	C₂²×C₁₆	D₄	1	4	yes	—
27	C₂²×C₁₆	D₄	1	4	yes	—
28	C₂²×C₁₆	Q₈	1	4	yes	—
29	C₂²×C₁₆	Q₈	1	4	yes	—
30	C₂²×C₁₆	C₂³	1	2	yes	—
31	C₂²×C₁₆	C₂³	1	2	yes	—
32	C₂×#107	—	2	8	no	—
33	C₂×#107	—	2	8	no	—
34	C₂×#107	—	2	4	no	—
35	C₂×#107	C₂³	1	8	yes	—
36	C₂×#106	—	2	8	no	—
37	C₂×#106	—	4	8	no	—
38	C₂×C₄×C₈	C₂×C₄	1	4	yes	—
39	C₂×C₄×C₈	D₄	1	2	yes	—
40	C₂×C₄×C₈	Q₈	1	4	yes	—
41	C₂×C₄×C₈	C₂³	1	2	yes	—
42	#382	—	2	8	no	—
43	#382	—	4	4	no	—
44	C₂×#95	—	4	4	no	—
45	#367	—	2	16	no	—
46	#366	—	2	16	no	—
47	C₄×C₁₆	C₂×C₄	1	16	yes	complemented
48	C₂³×C₄	—	4	4	no	—
49	C₂³×C₄	D₈	1	2	yes	—
50	C₂³×C₄	C₂×D₄	1	1	yes	characteristic
51	C₂²×C₈	C₄⋊C₄	1	4	yes	—
52	C₂²×C₈	D₈	1	4	yes	—
53	C₂²×C₈	Q₁₆	1	4	yes	—
54	C₂²×C₈	C₂²×C₄	1	2	yes	—
55	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
56	C₂²×C₈	C₂×D₄	1	2	yes	—
57	C₂²×C₈	C₂×D₄	1	2	yes	—
58	C₂²×C₈	C₂×D₄	1	2	yes	—
59	C₂²×C₈	C₂×Q₈	1	2	yes	—
60	C₂²×C₈	C₂×Q₈	1	2	yes	—
61	C₂²×C₈	C₂×Q₈	1	2	yes	—
62	C₂²×C₈	C₂⁴	1	1	yes	characteristic
63	C₂×C₄⋊C₄	—	2	8	no	—
64	C₂×C₄⋊C₄	—	4	8	no	—
65	C₂×C₄⋊C₄	—	4	8	no	—
66	C₂×C₄⋊C₄	—	4	4	no	—
67	C₂×C₄²	C₄⋊C₄	1	4	yes	—
68	C₂×C₄²	D₈	1	2	yes	—
69	C₂×C₄²	Q₁₆	1	4	yes	—
70	C₂×C₄²	C₂×D₄	1	2	yes	—
71	C₂×C₁₆	C₄⋊C₄	1	16	yes	—
72	C₂×C₁₆	C₄⋊C₄	1	16	yes	—
73	C₂×C₁₆	C₂²×C₄	1	4	yes	—
74	C₂×C₁₆	C₂²×C₄	1	4	yes	—
75	C₂×C₁₆	C₂×D₄	1	4	yes	—
76	C₂×C₁₆	C₂×D₄	1	4	yes	—
77	C₂×C₁₆	C₂×Q₈	1	4	yes	—
78	C₂×C₁₆	C₂×Q₈	1	4	yes	—
79	#107	—	2	8	no	—
80	#107	—	2	8	no	—
81	#107	—	4	8	no	—
82	#106	—	4	16	no	—
83	C₄×C₈	C₄⋊C₄	1	8	yes	—
84	C₄×C₈	C₂²×C₄	1	4	yes	—
85	C₄×C₈	C₂×Q₈	1	4	yes	—
86	#95	—	4	16	no	—
87	C₂⁴	C₂×D₈	1	1	yes	characteristic
88	C₂²×C₄	—	2	8	no	—
89	C₂²×C₄	—	4	8	no	—
90	C₂²×C₄	—	8	8	no	—
91	C₂²×C₄	—	8	16	no	—
92	C₂²×C₄	#107	1	8	yes	—
93	C₂²×C₄	C₂×C₄⋊C₄	1	2	yes	—
94	C₂²×C₄	#127	1	2	yes	—
95	C₂²×C₄	C₄⋊Q₈	1	2	yes	—
96	C₂²×C₄	C₂×D₈	1	4	yes	—
97	C₂²×C₄	C₂×D₈	1	2	yes	—
98	C₂²×C₄	C₂×D₈	1	1	yes	characteristic
99	C₂²×C₄	C₂×Q₁₆	1	2	yes	—
100	C₂²×C₄	C₂×Q₁₆	1	2	yes	—
101	C₂²×C₄	C₂²×D₄	1	1	yes	characteristic
102	C₂×C₈	#107	1	16	yes	—
103	C₂×C₈	C₂×C₄⋊C₄	1	4	yes	—
104	C₂×C₈	C₂×C₄⋊C₄	1	4	yes	—
105	C₂×C₈	C₂×C₄⋊C₄	1	2	yes	—
106	C₂×C₈	C₂×C₄⋊C₄	1	4	yes	—
107	C₂×C₈	#127	1	2	yes	—
108	C₂×C₈	C₄⋊Q₈	1	4	yes	—
109	C₂×C₈	C₄⋊Q₈	1	4	yes	—
110	C₂×C₈	C₄⋊Q₈	1	2	yes	—
111	C₂×C₈	C₂×D₈	1	4	yes	—
112	C₂×C₈	C₂×Q₁₆	1	4	yes	—
113	C₂×C₈	C₂³×C₄	1	1	yes	characteristic
114	C₂×C₈	C₂²×D₄	1	1	yes	characteristic
115	C₂×C₈	C₂²×D₄	1	1	yes	characteristic
116	C₂×C₈	C₂²×Q₈	1	1	yes	characteristic
117	C₂×C₈	C₂²×Q₈	1	1	yes	characteristic
118	C₂×C₈	C₂²×Q₈	1	1	yes	characteristic
119	C₄⋊C₄	—	4	8	no	—
120	C₄⋊C₄	—	4	8	no	—
121	C₄⋊C₄	—	8	8	no	—
122	C₄²	#107	1	8	yes	—
123	C₄²	C₂×C₄⋊C₄	1	4	yes	—
124	C₄²	C₂×Q₁₆	1	4	yes	—
125	C₁₆	C₂×C₄⋊C₄	1	8	yes	—
126	C₁₆	C₂×C₄⋊C₄	1	8	yes	—
127	C₂³	C₂×#107	1	2	yes	—
128	C₂³	#493	1	2	yes	—
129	C₂³	#500	1	2	yes	—
130	C₂³	#508	1	8	yes	—
131	C₂³	C₂²×D₈	1	1	yes	characteristic
132	C₂×C₄	—	2	8	no	—
133	C₂×C₄	—	2	16	no	—
134	C₂×C₄	—	8	8	no	—
135	C₂×C₄	—	8	16	no	—
136	C₂×C₄	—	16	16	no	—
137	C₂×C₄	#384	1	4	yes	—
138	C₂×C₄	C₂×#107	1	8	yes	—
139	C₂×C₄	C₂×#107	1	2	yes	—
140	C₂×C₄	C₂×#107	1	4	yes	—
141	C₂×C₄	#493	1	2	yes	—
142	C₂×C₄	#494	1	4	yes	—
143	C₂×C₄	#500	1	2	yes	—
144	C₂×C₄	#500	1	4	yes	—
145	C₂×C₄	C₂²×C₄⋊C₄	1	1	yes	characteristic
146	C₂×C₄	C₂×#127	1	1	yes	characteristic
147	C₂×C₄	C₂×C₄⋊Q₈	1	1	yes	characteristic
148	C₂×C₄	C₂²×D₈	1	1	yes	characteristic
149	C₂×C₄	C₂²×Q₁₆	1	1	yes	characteristic
150	C₂×C₄	C₂²×Q₁₆	1	1	yes	characteristic
151	C₈	#384	1	4	yes	—
152	C₈	C₂×#107	1	8	yes	—
153	C₈	C₂²×C₄⋊C₄	1	1	yes	characteristic
154	C₈	C₂²×C₄⋊C₄	1	1	yes	characteristic
155	C₈	C₂×C₄⋊Q₈	1	1	yes	characteristic
156	C₈	C₂×C₄⋊Q₈	1	1	yes	characteristic
157	C₂²	#1845	1	4	yes	—
158	C₂²	#2158	1	8	yes	—
159	C₂²	#2249	1	8	yes	—
160	C₂²	#2253	1	8	yes	—
161	C₂²	C₂²×#107	1	1	yes	characteristic
162	C₂²	C₂×#493	1	1	yes	characteristic
163	C₂²	C₂×#500	1	1	yes	characteristic
164	C₂²	C₂×#508	1	4	yes	—
165	C₄	—	2	8	no	—
166	C₄	—	16	8	no	—
167	C₄	#1845	1	4	yes	—
168	C₄	C₂×#384	1	1	yes	characteristic
169	C₄	C₂²×#107	1	1	yes	characteristic
170	C₄	C₂×#494	1	1	yes	characteristic
171	C₄	C₂×#500	1	1	yes	characteristic
172	C₂	#12907	1	8	yes	direct product factor
173	C₂	C₂×#1845	1	1	yes	characteristic
174	C₂	C₂×#2158	1	2	yes	—
175	C₂	C₂×#2249	1	2	yes	—
176	C₂	C₂×#2253	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
5. #12907	C₂	direct (N × H)
16. C₂×C₄×C₁₆	C₄	semidirect
47. C₄×C₁₆	C₂×C₄	semidirect
172. C₂	#12907	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×#366	2
C₂²×C₄×C₁₆	2
C₂×#1846	2
#12907	2

Subgroups — C2×#12907 #494156

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#12907 #494156