Subgroups — C₂×#12909 #494162

Order 512.

758 conjugacy classes of subgroups.

The subgroup lattice has 200 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#368	C₂	1	4	yes	maximal
2	C₂²×C₄×C₁₆	C₂	1	1	yes	maximal, characteristic
3	C₂×#1848	C₂	1	2	yes	maximal
4	#12909	C₂	1	8	yes	maximal, direct product factor
5	C₂³×C₁₆	C₂²	1	2	yes	—
6	C₂²×#108	—	2	2	no	—
7	C₂²×#108	C₂²	1	4	yes	—
8	C₂×#423	—	2	2	no	—
9	C₂²×C₄×C₈	C₂²	1	1	yes	characteristic
10	C₂×#368	—	2	16	no	—
11	C₂×#368	C₂²	1	16	yes	—
12	C₂×C₄×C₁₆	C₄	1	2	yes	—
13	C₂×C₄×C₁₆	C₄	1	4	yes	—
14	C₂×C₄×C₁₆	C₄	1	2	yes	—
15	C₂×C₄×C₁₆	C₂²	1	4	yes	—
16	#1848	—	2	4	no	—
17	#1848	C₂²	1	8	yes	—
18	C₂²×M₁₆	—	2	4	no	—
19	C₂³×C₈	D₄	1	1	yes	characteristic
20	C₂³×C₈	C₂³	1	1	yes	characteristic
21	C₂²×C₄²	D₄	1	1	yes	characteristic
22	C₂²×C₁₆	C₂×C₄	1	4	yes	—
23	C₂²×C₁₆	C₂×C₄	1	4	yes	—
24	C₂²×C₁₆	D₄	1	8	yes	—
25	C₂²×C₁₆	Q₈	1	8	yes	—
26	C₂²×C₁₆	C₂³	1	4	yes	—
27	C₂×#108	—	2	16	no	—
28	C₂×#108	—	2	4	no	—
29	C₂×#108	—	2	8	no	—
30	C₂×#108	—	4	8	no	—
31	C₂×#108	C₂³	1	8	yes	—
32	#423	—	2	8	no	—
33	#423	—	4	4	no	—
34	C₂×C₄⋊C₈	—	4	4	no	—
35	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
36	C₂×C₄×C₈	C₂×C₄	1	2	yes	—
37	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
38	C₂×C₄×C₈	D₄	1	2	yes	—
39	C₂×C₄×C₈	Q₈	1	1	yes	characteristic
40	C₂×C₄×C₈	Q₈	1	2	yes	—
41	C₂×C₄×C₈	Q₈	1	1	yes	characteristic
42	C₂×C₄×C₈	C₂³	1	2	yes	—
43	#368	—	2	32	no	—
44	C₄×C₁₆	C₂×C₄	1	8	yes	—
45	C₄×C₁₆	C₂×C₄	1	8	yes	—
46	C₂³×C₄	Q₁₆	1	2	yes	—
47	C₂³×C₄	C₂×D₄	1	1	yes	characteristic
48	C₂×M₁₆	—	2	8	no	—
49	C₂×M₁₆	—	4	4	no	—
50	C₂×M₁₆	—	4	8	no	—
51	C₂×M₁₆	—	4	8	no	—
52	C₂²×C₈	—	4	4	no	—
53	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
54	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
55	C₂²×C₈	C₄⋊C₄	1	2	yes	—
56	C₂²×C₈	D₈	1	4	yes	—
57	C₂²×C₈	Q₁₆	1	4	yes	—
58	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
59	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
60	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
61	C₂²×C₈	C₂×D₄	1	4	yes	—
62	C₂²×C₈	C₂×D₄	1	2	yes	—
63	C₂²×C₈	C₂×Q₈	1	4	yes	—
64	C₂²×C₈	C₂×Q₈	1	1	yes	characteristic
65	C₂²×C₈	C₂×Q₈	1	1	yes	characteristic
66	C₂²×C₈	C₂⁴	1	1	yes	characteristic
67	C₂×C₄²	C₄⋊C₄	1	1	yes	characteristic
68	C₂×C₄²	C₄⋊C₄	1	2	yes	—
69	C₂×C₄²	C₄⋊C₄	1	1	yes	characteristic
70	C₂×C₄²	D₈	1	2	yes	—
71	C₂×C₄²	D₈	1	2	yes	—
72	C₂×C₄²	Q₁₆	1	2	yes	—
73	C₂×C₄²	C₂×D₄	1	2	yes	—
74	C₂×C₁₆	C₄⋊C₄	1	16	yes	—
75	C₂×C₁₆	C₄⋊C₄	1	16	yes	—
76	C₂×C₁₆	C₂²×C₄	1	8	yes	—
77	C₂×C₁₆	C₂×D₄	1	8	yes	—
78	C₂×C₁₆	C₂×Q₈	1	8	yes	—
79	#108	—	2	16	no	—
80	#108	—	4	8	no	—
81	#108	—	4	16	no	—
82	C₄⋊C₈	—	4	16	no	—
83	C₄×C₈	C₄⋊C₄	1	2	yes	—
84	C₄×C₈	C₄⋊C₄	1	2	yes	—
85	C₄×C₈	C₄⋊C₄	1	2	yes	—
86	C₄×C₈	C₄⋊C₄	1	2	yes	—
87	C₄×C₈	C₂²×C₄	1	2	yes	—
88	C₄×C₈	C₂²×C₄	1	2	yes	—
89	C₄×C₈	C₂×Q₈	1	2	yes	—
90	C₄×C₈	C₂×Q₈	1	2	yes	—
91	C₂⁴	C₂×Q₁₆	1	1	yes	characteristic
92	C₂²×C₄	—	2	8	no	—
93	C₂²×C₄	#107	1	2	yes	—
94	C₂²×C₄	#107	1	4	yes	—
95	C₂²×C₄	#107	1	2	yes	—
96	C₂²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
97	C₂²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
98	C₂²×C₄	#127	1	2	yes	—
99	C₂²×C₄	C₄⋊Q₈	1	2	yes	—
100	C₂²×C₄	C₂×D₈	1	2	yes	—
101	C₂²×C₄	C₂×D₈	1	1	yes	characteristic
102	C₂²×C₄	C₂×D₈	1	1	yes	characteristic
103	C₂²×C₄	C₂×Q₁₆	1	4	yes	—
104	C₂²×C₄	C₂×Q₁₆	1	2	yes	—
105	C₂²×C₄	C₂×Q₁₆	1	1	yes	characteristic
106	C₂²×C₄	C₂²×D₄	1	1	yes	characteristic
107	M₁₆	—	4	8	no	—
108	M₁₆	—	4	8	no	—
109	M₁₆	—	8	8	no	—
110	C₂×C₈	—	4	8	no	—
111	C₂×C₈	—	8	8	no	—
112	C₂×C₈	#107	1	8	yes	—
113	C₂×C₈	#107	1	4	yes	—
114	C₂×C₈	#107	1	4	yes	—
115	C₂×C₈	C₂×C₄⋊C₄	1	4	yes	—
116	C₂×C₈	C₂×C₄⋊C₄	1	4	yes	—
117	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
118	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
119	C₂×C₈	C₂×C₄⋊C₄	1	2	yes	—
120	C₂×C₈	C₂×C₄⋊C₄	1	2	yes	—
121	C₂×C₈	#127	1	2	yes	—
122	C₂×C₈	C₄⋊Q₈	1	4	yes	—
123	C₂×C₈	C₄⋊Q₈	1	4	yes	—
124	C₂×C₈	C₄⋊Q₈	1	2	yes	—
125	C₂×C₈	C₂×D₈	1	4	yes	—
126	C₂×C₈	C₂×Q₁₆	1	4	yes	—
127	C₂×C₈	C₂³×C₄	1	1	yes	characteristic
128	C₂×C₈	C₂²×D₄	1	2	yes	—
129	C₂×C₈	C₂²×Q₈	1	2	yes	—
130	C₂×C₈	C₂²×Q₈	1	1	yes	characteristic
131	C₄²	#107	1	2	yes	—
132	C₄²	#107	1	4	yes	—
133	C₄²	#107	1	2	yes	—
134	C₄²	C₂×C₄⋊C₄	1	2	yes	—
135	C₄²	C₂×C₄⋊C₄	1	2	yes	—
136	C₄²	C₂×D₈	1	4	yes	—
137	C₁₆	C₂×C₄⋊C₄	1	16	yes	—
138	C₂³	—	2	4	no	—
139	C₂³	C₂×#107	1	1	yes	characteristic
140	C₂³	C₂×#107	1	1	yes	characteristic
141	C₂³	#494	1	2	yes	—
142	C₂³	#500	1	2	yes	—
143	C₂³	C₂²×Q₁₆	1	1	yes	characteristic
144	C₂×C₄	—	2	4	no	—
145	C₂×C₄	—	2	8	no	—
146	C₂×C₄	—	2	8	no	—
147	C₂×C₄	#368	1	8	yes	complemented
148	C₂×C₄	#384	1	2	yes	—
149	C₂×C₄	#384	1	2	yes	—
150	C₂×C₄	C₂×#107	1	4	yes	—
151	C₂×C₄	C₂×#107	1	1	yes	characteristic
152	C₂×C₄	C₂×#107	1	1	yes	characteristic
153	C₂×C₄	C₂×#107	1	2	yes	—
154	C₂×C₄	C₂×#107	1	2	yes	—
155	C₂×C₄	C₂×#107	1	4	yes	—
156	C₂×C₄	#493	1	2	yes	—
157	C₂×C₄	#493	1	2	yes	—
158	C₂×C₄	#494	1	2	yes	—
159	C₂×C₄	#500	1	2	yes	—
160	C₂×C₄	#500	1	2	yes	—
161	C₂×C₄	#500	1	2	yes	—
162	C₂×C₄	C₂²×C₄⋊C₄	1	1	yes	characteristic
163	C₂×C₄	C₂×#127	1	1	yes	characteristic
164	C₂×C₄	C₂×C₄⋊Q₈	1	1	yes	characteristic
165	C₂×C₄	C₂²×D₈	1	1	yes	characteristic
166	C₂×C₄	C₂²×D₈	1	1	yes	characteristic
167	C₂×C₄	C₂²×Q₁₆	1	1	yes	characteristic
168	C₈	—	8	8	no	—
169	C₈	#384	1	2	yes	—
170	C₈	#384	1	2	yes	—
171	C₈	C₂×#107	1	4	yes	—
172	C₈	C₂×#107	1	4	yes	—
173	C₈	C₂²×C₄⋊C₄	1	2	yes	—
174	C₈	C₂×C₄⋊Q₈	1	2	yes	—
175	C₂²	—	2	4	no	—
176	C₂²	—	2	4	no	—
177	C₂²	—	2	2	no	—
178	C₂²	—	2	2	no	—
179	C₂²	#1845	1	2	yes	—
180	C₂²	#1845	1	2	yes	—
181	C₂²	C₂×#368	1	4	yes	—
182	C₂²	C₂²×#107	1	1	yes	characteristic
183	C₂²	C₂×#494	1	1	yes	characteristic
184	C₂²	C₂×#500	1	1	yes	characteristic
185	C₄	—	2	4	no	—
186	C₄	#1845	1	2	yes	—
187	C₄	#1845	1	2	yes	—
188	C₄	C₂×#368	1	8	yes	complemented
189	C₄	C₂×#384	1	1	yes	characteristic
190	C₄	C₂²×#107	1	1	yes	characteristic
191	C₄	C₂×#493	1	1	yes	characteristic
192	C₄	C₂×#500	1	1	yes	characteristic
193	C₂	—	2	2	no	—
194	C₂	—	2	1	no	—
195	C₂	—	2	1	no	—
196	C₂	#12909	1	4	yes	direct product factor
197	C₂	C₂×#1845	1	1	yes	characteristic
198	C₂	C₂²×#368	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	C₂×D₈
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂³	C₂×#107
prime radical	C₂⁴	C₂×Q₁₆
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ 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. #12909	C₂	direct (N × H)
147. C₂×C₄	#368	semidirect
188. C₄	C₂×#368	semidirect
196. C₂	#12909	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×#368	2
C₂²×C₄×C₁₆	2
C₂×#1848	2
#12909	2

Subgroups — C2×#12909 #494162

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#12909 #494162