Subgroups — C₂×#12904 #494151

Order 512.

766 conjugacy classes of subgroups.

The subgroup lattice has 234 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#365	C₂	1	4	yes	maximal
2	C₂×C₄×M₃₂	C₂	1	1	yes	maximal, characteristic
3	C₂×#1848	C₂	1	1	yes	maximal, characteristic
4	C₂×#1846	C₂	1	1	yes	maximal, characteristic
5	#12904	C₂	1	8	yes	maximal, direct product factor
6	C₂²×M₃₂	C₂²	1	2	yes	—
7	C₂²×#108	—	2	1	no	—
8	C₂²×#108	C₂²	1	2	yes	—
9	C₂²×#107	—	2	1	no	—
10	C₂²×#106	C₂²	1	2	yes	—
11	C₂×#423	—	2	1	no	—
12	C₂²×C₄×C₈	C₂²	1	1	yes	characteristic
13	C₂×#382	—	2	1	no	—
14	C₂×#365	—	2	16	no	—
15	C₂×#365	C₂²	1	16	yes	—
16	C₄×M₃₂	C₄	1	4	yes	complemented
17	C₄×M₃₂	C₂²	1	4	yes	—
18	C₂×#346	C₄	1	2	yes	complemented
19	C₂×C₄×C₁₆	C₄	1	2	yes	complemented
20	#1848	—	2	2	no	—
21	#1848	C₂²	1	4	yes	—
22	#1846	—	2	2	no	—
23	#1846	C₂²	1	4	yes	—
24	C₂²×M₁₆	—	2	2	no	—
25	C₂³×C₈	D₄	1	1	yes	characteristic
26	C₂³×C₈	C₂³	1	1	yes	characteristic
27	C₂²×C₄⋊C₄	—	2	2	no	—
28	C₂²×C₄²	D₄	1	1	yes	characteristic
29	C₂×M₃₂	C₂×C₄	1	4	yes	—
30	C₂×M₃₂	D₄	1	8	yes	—
31	C₂×M₃₂	Q₈	1	8	yes	—
32	C₂×M₃₂	C₂³	1	4	yes	—
33	C₂²×C₁₆	C₂×C₄	1	4	yes	—
34	C₂×#108	—	2	8	no	—
35	C₂×#108	—	2	2	no	—
36	C₂×#108	—	2	4	no	—
37	C₂×#108	—	4	4	no	—
38	C₂×#108	C₂³	1	4	yes	—
39	C₂×#107	—	2	4	no	—
40	C₂×#107	—	4	4	no	—
41	C₂×#106	—	2	8	no	—
42	C₂×#106	—	2	2	no	—
43	C₂×#106	C₂³	1	4	yes	—
44	#423	—	2	4	no	—
45	#423	—	4	2	no	—
46	C₂×C₄⋊C₈	—	4	2	no	—
47	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
48	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
49	C₂×C₄×C₈	C₂×C₄	1	2	yes	—
50	C₂×C₄×C₈	D₄	1	2	yes	—
51	C₂×C₄×C₈	Q₈	1	2	yes	—
52	C₂×C₄×C₈	Q₈	1	1	yes	characteristic
53	C₂×C₄×C₈	Q₈	1	1	yes	characteristic
54	C₂×C₄×C₈	C₂³	1	2	yes	—
55	#382	—	2	4	no	—
56	#382	—	4	2	no	—
57	C₂×#95	—	4	2	no	—
58	#365	—	2	32	no	—
59	#346	C₂×C₄	1	8	yes	complemented
60	C₄×C₁₆	C₂×C₄	1	8	yes	complemented
61	C₂³×C₄	—	4	2	no	—
62	C₂³×C₄	SD₁₆	1	2	yes	—
63	C₂³×C₄	C₂×D₄	1	1	yes	characteristic
64	C₂×M₁₆	—	2	4	no	—
65	C₂×M₁₆	—	4	2	no	—
66	C₂×M₁₆	—	4	4	no	—
67	C₂×M₁₆	—	4	4	no	—
68	C₂²×C₈	—	4	2	no	—
69	C₂²×C₈	C₄⋊C₄	1	2	yes	—
70	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
71	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
72	C₂²×C₈	SD₁₆	1	4	yes	—
73	C₂²×C₈	SD₁₆	1	4	yes	—
74	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
75	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
76	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
77	C₂²×C₈	C₂×D₄	1	4	yes	—
78	C₂²×C₈	C₂×D₄	1	2	yes	—
79	C₂²×C₈	C₂×Q₈	1	4	yes	—
80	C₂²×C₈	C₂×Q₈	1	1	yes	characteristic
81	C₂²×C₈	C₂×Q₈	1	1	yes	characteristic
82	C₂²×C₈	C₂⁴	1	1	yes	characteristic
83	C₂×C₄⋊C₄	—	2	4	no	—
84	C₂×C₄⋊C₄	—	4	4	no	—
85	C₂×C₄⋊C₄	—	4	4	no	—
86	C₂×C₄⋊C₄	—	4	2	no	—
87	C₂×C₄²	C₄⋊C₄	1	1	yes	characteristic
88	C₂×C₄²	C₄⋊C₄	1	1	yes	characteristic
89	C₂×C₄²	C₄⋊C₄	1	2	yes	—
90	C₂×C₄²	SD₁₆	1	2	yes	—
91	C₂×C₄²	SD₁₆	1	2	yes	—
92	C₂×C₄²	SD₁₆	1	2	yes	—
93	C₂×C₄²	C₂×D₄	1	2	yes	—
94	M₃₂	C₄⋊C₄	1	16	yes	—
95	M₃₂	C₂×D₄	1	8	yes	—
96	M₃₂	C₂×Q₈	1	8	yes	—
97	C₂×C₁₆	C₄⋊C₄	1	16	yes	—
98	C₂×C₁₆	C₂²×C₄	1	8	yes	—
99	#108	—	2	8	no	—
100	#108	—	4	4	no	—
101	#108	—	4	8	no	—
102	#107	—	4	8	no	—
103	#106	—	2	8	no	—
104	#106	—	4	4	no	—
105	C₄⋊C₈	—	4	8	no	—
106	C₄×C₈	C₄⋊C₄	1	2	yes	—
107	C₄×C₈	C₄⋊C₄	1	2	yes	—
108	C₄×C₈	C₄⋊C₄	1	2	yes	—
109	C₄×C₈	C₄⋊C₄	1	2	yes	—
110	C₄×C₈	C₂²×C₄	1	2	yes	—
111	C₄×C₈	C₂²×C₄	1	2	yes	—
112	C₄×C₈	C₂×Q₈	1	2	yes	—
113	C₄×C₈	C₂×Q₈	1	2	yes	—
114	#95	—	4	8	no	—
115	C₂⁴	C₂×SD₁₆	1	1	yes	characteristic
116	C₂²×C₄	—	4	4	no	—
117	C₂²×C₄	—	4	4	no	—
118	C₂²×C₄	—	8	8	no	—
119	C₂²×C₄	—	8	4	no	—
120	C₂²×C₄	#106	1	2	yes	—
121	C₂²×C₄	#106	1	2	yes	—
122	C₂²×C₄	#106	1	4	yes	—
123	C₂²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
124	C₂²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
125	C₂²×C₄	#127	1	2	yes	—
126	C₂²×C₄	C₄⋊Q₈	1	2	yes	—
127	C₂²×C₄	C₂×SD₁₆	1	4	yes	—
128	C₂²×C₄	C₂×SD₁₆	1	2	yes	—
129	C₂²×C₄	C₂×SD₁₆	1	2	yes	—
130	C₂²×C₄	C₂×SD₁₆	1	1	yes	characteristic
131	C₂²×C₄	C₂×SD₁₆	1	1	yes	characteristic
132	C₂²×C₄	C₂×SD₁₆	1	1	yes	characteristic
133	C₂²×C₄	C₂²×D₄	1	1	yes	characteristic
134	M₁₆	—	4	4	no	—
135	M₁₆	—	4	4	no	—
136	M₁₆	—	8	4	no	—
137	C₂×C₈	—	4	4	no	—
138	C₂×C₈	—	8	4	no	—
139	C₂×C₈	#106	1	8	yes	—
140	C₂×C₈	#106	1	4	yes	—
141	C₂×C₈	#106	1	4	yes	—
142	C₂×C₈	C₂×C₄⋊C₄	1	4	yes	—
143	C₂×C₈	C₂×C₄⋊C₄	1	4	yes	—
144	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
145	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
146	C₂×C₈	C₂×C₄⋊C₄	1	2	yes	—
147	C₂×C₈	C₂×C₄⋊C₄	1	2	yes	—
148	C₂×C₈	#127	1	2	yes	—
149	C₂×C₈	C₄⋊Q₈	1	4	yes	—
150	C₂×C₈	C₄⋊Q₈	1	4	yes	—
151	C₂×C₈	C₄⋊Q₈	1	2	yes	—
152	C₂×C₈	C₂×SD₁₆	1	4	yes	—
153	C₂×C₈	C₂×SD₁₆	1	4	yes	—
154	C₂×C₈	C₂³×C₄	1	1	yes	characteristic
155	C₂×C₈	C₂²×D₄	1	2	yes	—
156	C₂×C₈	C₂²×Q₈	1	2	yes	—
157	C₂×C₈	C₂²×Q₈	1	1	yes	characteristic
158	C₄⋊C₄	—	4	4	no	—
159	C₄⋊C₄	—	4	4	no	—
160	C₄⋊C₄	—	8	4	no	—
161	C₄²	#106	1	4	yes	—
162	C₄²	#106	1	2	yes	—
163	C₄²	#106	1	2	yes	—
164	C₄²	C₂×C₄⋊C₄	1	2	yes	—
165	C₄²	C₂×C₄⋊C₄	1	2	yes	—
166	C₄²	C₂×SD₁₆	1	4	yes	—
167	C₁₆	C₂×C₄⋊C₄	1	16	yes	—
168	C₂³	—	2	4	no	—
169	C₂³	C₂×#106	1	1	yes	characteristic
170	C₂³	C₂×#106	1	1	yes	characteristic
171	C₂³	#492	1	2	yes	—
172	C₂³	#498	1	2	yes	—
173	C₂³	C₂²×SD₁₆	1	1	yes	characteristic
174	C₂×C₄	—	2	4	no	—
175	C₂×C₄	—	2	4	no	—
176	C₂×C₄	—	4	4	no	—
177	C₂×C₄	—	4	4	no	—
178	C₂×C₄	—	8	8	no	—
179	C₂×C₄	—	8	4	no	—
180	C₂×C₄	—	16	8	no	—
181	C₂×C₄	#384	1	2	yes	—
182	C₂×C₄	#384	1	2	yes	—
183	C₂×C₄	C₂×#106	1	4	yes	—
184	C₂×C₄	C₂×#106	1	4	yes	—
185	C₂×C₄	C₂×#106	1	1	yes	characteristic
186	C₂×C₄	C₂×#106	1	1	yes	characteristic
187	C₂×C₄	C₂×#106	1	2	yes	—
188	C₂×C₄	C₂×#106	1	2	yes	—
189	C₂×C₄	#492	1	2	yes	—
190	C₂×C₄	#492	1	2	yes	—
191	C₂×C₄	#492	1	2	yes	—
192	C₂×C₄	#498	1	2	yes	—
193	C₂×C₄	#498	1	2	yes	—
194	C₂×C₄	#498	1	2	yes	—
195	C₂×C₄	C₂²×C₄⋊C₄	1	1	yes	characteristic
196	C₂×C₄	C₂×#127	1	1	yes	characteristic
197	C₂×C₄	C₂×C₄⋊Q₈	1	1	yes	characteristic
198	C₂×C₄	C₂²×SD₁₆	1	1	yes	characteristic
199	C₂×C₄	C₂²×SD₁₆	1	1	yes	characteristic
200	C₂×C₄	C₂²×SD₁₆	1	1	yes	characteristic
201	C₈	—	8	4	no	—
202	C₈	#384	1	2	yes	—
203	C₈	#384	1	2	yes	—
204	C₈	C₂×#106	1	4	yes	—
205	C₈	C₂×#106	1	4	yes	—
206	C₈	C₂²×C₄⋊C₄	1	2	yes	—
207	C₈	C₂×C₄⋊Q₈	1	2	yes	—
208	C₂²	—	2	4	no	—
209	C₂²	—	2	4	no	—
210	C₂²	—	2	2	no	—
211	C₂²	—	2	2	no	—
212	C₂²	#1844	1	2	yes	—
213	C₂²	#1844	1	2	yes	—
214	C₂²	#2155	1	4	yes	—
215	C₂²	C₂²×#106	1	1	yes	characteristic
216	C₂²	C₂×#492	1	1	yes	characteristic
217	C₂²	C₂×#498	1	1	yes	characteristic
218	C₄	—	2	4	no	—
219	C₄	—	2	4	no	—
220	C₄	—	16	4	no	—
221	C₄	#1844	1	2	yes	—
222	C₄	#1844	1	2	yes	—
223	C₄	C₂×#384	1	1	yes	characteristic
224	C₄	C₂²×#106	1	1	yes	characteristic
225	C₄	C₂×#492	1	1	yes	characteristic
226	C₄	C₂×#498	1	1	yes	characteristic
227	C₂	—	2	1	no	—
228	C₂	—	2	2	no	—
229	C₂	—	2	1	no	—
230	C₂	#12904	1	4	yes	direct product factor
231	C₂	C₂×#1844	1	1	yes	characteristic
232	C₂	C₂×#2155	1	2	yes	—

	subgroup	G/H
center, socle	C₂³	C₂×#106
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂⁴	C₂×SD₁₆
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ 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. #12904	C₂	direct (N × H)
16. C₄×M₃₂	C₄	semidirect
18. C₂×#346	C₄	semidirect
19. C₂×C₄×C₁₆	C₄	semidirect
59. #346	C₂×C₄	semidirect
60. C₄×C₁₆	C₂×C₄	semidirect
230. C₂	#12904	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×#365	2
C₂×C₄×M₃₂	2
C₂×#1848	2
C₂×#1846	2
#12904	2

Subgroups — C2×#12904 #494151

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#12904 #494151