Subgroups — C₂×#12083 #483818

Order 512.

470 conjugacy classes of subgroups.

The subgroup lattice has 185 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×Q₃₂	C₂	1	1	yes	maximal, characteristic
2	C₂×#1591	C₂	1	2	yes	maximal
3	#12083	C₂	1	8	yes	maximal, direct product factor
4	C₂×#1371	C₂	1	1	yes	maximal, characteristic
5	C₂×#1332	C₂	1	2	yes	maximal
6	C₂×C₈×C₁₆	C₂	1	1	yes	maximal, characteristic
7	C₂²×Q₃₂	C₄	1	1	yes	characteristic
8	C₂×#446	—	2	2	no	—
9	C₂×C₄×Q₁₆	C₂²	1	2	yes	—
10	C₄×Q₃₂	C₄	1	4	yes	—
11	C₄×Q₃₂	C₂²	1	4	yes	—
12	C₂×#366	C₄	1	1	yes	characteristic
13	C₂×#358	C₄	1	2	yes	—
14	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
15	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
16	#1591	—	2	4	no	—
17	#1591	C₂²	1	8	yes	—
18	C₂×#335	C₂²	1	2	yes	—
19	C₂×Q₈⋊C₈	—	2	2	no	—
20	C₂×C₈²	C₂²	1	1	yes	characteristic
21	#1371	C₂²	1	4	yes	—
22	#1332	C₂²	1	8	yes	—
23	C₈×C₁₆	C₂²	1	4	yes	—
24	C₂²×Q₁₆	C₂×C₄	1	2	yes	—
25	C₂×C₄×Q₈	—	2	2	no	—
26	C₂×Q₃₂	—	2	4	no	—
27	C₂×Q₃₂	C₂×C₄	1	4	yes	—
28	C₂²×C₁₆	—	2	1	no	—
29	C₂²×C₁₆	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₁₆	C₂×C₄	1	1	yes	characteristic
31	#446	—	2	8	no	—
32	#446	—	4	4	no	—
33	C₄×Q₁₆	—	2	2	no	—
34	C₄×Q₁₆	—	2	2	no	—
35	C₄×Q₁₆	C₂×C₄	1	4	yes	—
36	C₄×Q₁₆	C₂³	1	4	yes	—
37	C₂×#107	C₂×C₄	1	2	yes	—
38	C₂×C₄⋊C₈	—	2	2	no	—
39	C₂×C₄⋊C₈	—	4	2	no	—
40	C₂×#103	—	2	2	no	—
41	C₂×#97	—	4	2	no	—
42	C₂×C₄×C₈	D₄	1	1	yes	characteristic
43	C₂×C₄×C₈	D₄	1	1	yes	characteristic
44	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
45	#366	C₂×C₄	1	4	yes	—
46	#358	C₂×C₄	1	8	yes	—
47	C₄×C₁₆	—	2	1	no	—
48	C₄×C₁₆	C₂×C₄	1	2	yes	—
49	C₄×C₁₆	C₂³	1	2	yes	—
50	C₄×C₁₆	C₂³	1	2	yes	—
51	#335	—	2	2	no	—
52	#335	C₂³	1	4	yes	—
53	Q₈⋊C₈	—	2	8	no	—
54	C₈²	D₄	1	2	yes	—
55	C₈²	C₂³	1	2	yes	—
56	C₂²×Q₈	—	2	2	no	—
57	C₂×Q₁₆	—	2	2	no	—
58	C₂×Q₁₆	—	2	4	no	—
59	C₂×Q₁₆	—	2	4	no	—
60	C₂×Q₁₆	C₂²×C₄	1	4	yes	—
61	C₂²×C₈	—	2	1	no	—
62	C₂²×C₈	—	4	2	no	—
63	C₂²×C₈	D₈	1	1	yes	characteristic
64	C₂²×C₈	D₈	1	1	yes	characteristic
65	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
66	C₂²×C₈	#41	1	1	yes	characteristic
67	C₄×Q₈	—	2	4	no	—
68	C₄×Q₈	—	2	4	no	—
69	C₄×Q₈	—	4	2	no	—
70	C₄×Q₈	—	4	2	no	—
71	C₂×C₄⋊C₄	—	2	2	no	—
72	C₂×C₄⋊C₄	—	4	2	no	—
73	C₂×C₄²	—	4	2	no	—
74	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
75	Q₃₂	—	2	8	no	—
76	C₂×C₁₆	—	2	4	no	—
77	C₂×C₁₆	—	2	2	no	—
78	C₂×C₁₆	—	2	2	no	—
79	C₂×C₁₆	M₁₆	1	4	yes	—
80	C₂×C₁₆	C₂²×C₄	1	2	yes	—
81	C₂×C₁₆	C₂²×C₄	1	2	yes	—
82	#107	—	2	2	no	—
83	#107	C₂²×C₄	1	4	yes	—
84	C₄⋊C₈	—	2	4	no	—
85	C₄⋊C₈	—	4	8	no	—
86	C₄⋊C₈	—	4	2	no	—
87	#103	—	2	4	no	—
88	#103	—	2	4	no	—
89	#97	—	4	8	no	—
90	C₄×C₈	—	2	1	no	—
91	C₄×C₈	D₈	1	2	yes	—
92	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
93	C₄×C₈	C₂×D₄	1	2	yes	—
94	C₄×C₈	C₂×D₄	1	1	yes	characteristic
95	C₄×C₈	C₂×D₄	1	2	yes	—
96	C₄×C₈	#41	1	1	yes	characteristic
97	C₄×C₈	C₂⁴	1	1	yes	characteristic
98	C₂×Q₈	—	2	4	no	—
99	C₂×Q₈	—	4	2	no	—
100	C₂×Q₈	—	4	4	no	—
101	C₂×Q₈	—	4	4	no	—
102	C₂²×C₄	—	4	2	no	—
103	C₂²×C₄	—	4	2	no	—
104	C₂²×C₄	C₄×D₄	1	1	yes	characteristic
105	C₂²×C₄	C₂×D₈	1	1	yes	characteristic
106	C₂²×C₄	#135	1	1	yes	characteristic
107	Q₁₆	—	2	4	no	—
108	Q₁₆	—	2	4	no	—
109	Q₁₆	—	4	4	no	—
110	C₂×C₈	—	2	4	no	—
111	C₂×C₈	—	2	2	no	—
112	C₂×C₈	—	2	2	no	—
113	C₂×C₈	—	2	2	no	—
114	C₂×C₈	—	4	4	no	—
115	C₂×C₈	—	8	4	no	—
116	C₂×C₈	Q₃₂	1	4	yes	complemented
117	C₂×C₈	C₄×D₄	1	1	yes	characteristic
118	C₂×C₈	C₄×D₄	1	1	yes	characteristic
119	C₂×C₈	C₂×M₁₆	1	2	yes	—
120	C₂×C₈	#131	1	2	yes	—
121	C₂×C₈	C₂×D₈	1	2	yes	—
122	C₂×C₈	C₂×D₈	1	2	yes	—
123	C₂×C₈	C₂³×C₄	1	1	yes	characteristic
124	C₂×C₈	C₂×#41	1	1	yes	characteristic
125	C₄⋊C₄	—	2	4	no	—
126	C₄⋊C₄	—	4	4	no	—
127	C₄⋊C₄	—	4	2	no	—
128	C₄⋊C₄	—	4	4	no	—
129	C₄²	—	4	4	no	—
130	C₄²	—	4	4	no	—
131	C₄²	C₄×D₄	1	1	yes	characteristic
132	C₄²	C₂×D₈	1	1	yes	characteristic
133	C₄²	#135	1	1	yes	characteristic
134	C₄²	C₂²×D₄	1	1	yes	characteristic
135	C₁₆	—	2	4	no	—
136	C₁₆	—	2	2	no	—
137	C₁₆	C₂×M₁₆	1	4	yes	—
138	C₂³	C₄×D₈	1	1	yes	characteristic
139	Q₈	—	4	4	no	—
140	Q₈	—	4	4	no	—
141	Q₈	—	8	4	no	—
142	C₂×C₄	—	2	2	no	—
143	C₂×C₄	—	4	4	no	—
144	C₂×C₄	—	4	4	no	—
145	C₂×C₄	—	8	4	no	—
146	C₂×C₄	—	8	4	no	—
147	C₂×C₄	#436	1	2	yes	—
148	C₂×C₄	C₄×D₈	1	1	yes	characteristic
149	C₂×C₄	C₄×D₈	1	1	yes	characteristic
150	C₂×C₄	C₄×D₈	1	1	yes	characteristic
151	C₂×C₄	#443	1	2	yes	—
152	C₂×C₄	C₂×Q₃₂	1	2	yes	—
153	C₂×C₄	#508	1	2	yes	—
154	C₂×C₄	C₂×C₄×D₄	1	1	yes	characteristic
155	C₂×C₄	C₂²×D₈	1	1	yes	characteristic
156	C₂×C₄	C₂×#135	1	1	yes	characteristic
157	C₈	—	2	4	no	—
158	C₈	—	2	2	no	—
159	C₈	—	2	1	no	—
160	C₈	—	2	1	no	—
161	C₈	—	8	4	no	—
162	C₈	#436	1	2	yes	—
163	C₈	C₂×Q₃₂	1	4	yes	complemented
164	C₈	C₂²×M₁₆	1	1	yes	characteristic
165	C₈	C₂×#131	1	1	yes	characteristic
166	C₂²	#1589	1	2	yes	—
167	C₂²	C₄×Q₃₂	1	2	yes	—
168	C₂²	#2178	1	2	yes	—
169	C₂²	C₂×C₄×D₈	1	1	yes	characteristic
170	C₄	—	2	1	no	—
171	C₄	—	2	1	no	—
172	C₄	—	8	4	no	—
173	C₄	—	8	4	no	—
174	C₄	#1589	1	2	yes	—
175	C₄	C₄×Q₃₂	1	2	yes	—
176	C₄	C₂×#436	1	1	yes	characteristic
177	C₄	C₂×#443	1	1	yes	characteristic
178	C₄	C₂²×Q₃₂	1	1	yes	characteristic
179	C₄	C₂×#508	1	1	yes	characteristic
180	C₂	#12083	1	4	yes	direct product factor
181	C₂	C₂×#1589	1	1	yes	characteristic
182	C₂	C₂×C₄×Q₃₂	1	1	yes	characteristic
183	C₂	C₂×#2178	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×D₈
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₄×C₈	C₂⁴
socle, prime radical	C₂³	C₄×D₈
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ C₂×C₈² ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
3. #12083	C₂	direct (N × H)
116. C₂×C₈	Q₃₂	semidirect
163. C₈	C₂×Q₃₂	semidirect
180. C₂	#12083	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×C₄×Q₃₂	2
C₂×#1591	2
#12083	2
C₂×#1371	2
C₂×#1332	2
C₂×C₈×C₁₆	2

Subgroups — C2×#12083 #483818

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#12083 #483818