Subgroups — C₂×#12030 #483418

Order 512.

504 conjugacy classes of subgroups.

The subgroup lattice has 208 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#2163	C₂	1	1	yes	maximal, characteristic
2	C₂×#1566	C₂	1	1	yes	maximal, characteristic
3	C₂²×#334	C₂	1	1	yes	maximal, characteristic
4	#12030	C₂	1	8	yes	maximal, direct product factor
5	C₂×#1368	C₂	1	4	yes	maximal, complemented
6	C₂×#423	—	2	1	no	—
7	C₂²×C₄⋊C₈	—	2	1	no	—
8	C₂²×C₄×C₈	C₂²	1	1	yes	characteristic
9	#2163	C₄	1	4	yes	—
10	#2163	C₂²	1	4	yes	—
11	C₂×C₄⋊C₁₆	C₄	1	2	yes	—
12	C₂×C₂²⋊C₁₆	C₄	1	2	yes	—
13	C₂×#346	C₂²	1	2	yes	—
14	#1566	—	2	2	no	—
15	#1566	C₂²	1	4	yes	—
16	C₂×#334	—	2	1	no	—
17	C₂×#334	—	2	2	no	—
18	C₂×#334	—	2	1	no	—
19	C₂×#334	C₂²	1	2	yes	—
20	C₂×#334	C₂²	1	4	yes	—
21	C₂×#334	C₂²	1	2	yes	—
22	#1368	C₂²	1	16	yes	complemented
23	C₂²×M₁₆	—	4	1	no	—
24	C₂³×C₈	—	4	1	no	—
25	C₂³×C₈	C₂×C₄	1	1	yes	characteristic
26	C₂³×C₈	Q₈	1	1	yes	characteristic
27	C₂²×C₄²	D₄	1	1	yes	characteristic
28	C₂²×C₁₆	C₂×C₄	1	4	yes	—
29	#423	—	2	4	no	—
30	#423	—	4	2	no	—
31	C₂×C₄⋊C₈	—	2	2	no	—
32	C₂×C₄⋊C₈	—	2	4	no	—
33	C₂×C₄⋊C₈	—	2	2	no	—
34	C₂×C₄⋊C₈	—	4	2	no	—
35	C₂×C₄⋊C₈	—	4	1	no	—
36	C₂×C₄⋊C₈	—	4	1	no	—
37	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
38	C₂×C₄×C₈	C₂×C₄	1	2	yes	—
39	C₂×C₄×C₈	D₄	1	1	yes	characteristic
40	C₂×C₄×C₈	D₄	1	2	yes	—
41	C₂×C₄×C₈	Q₈	1	1	yes	characteristic
42	C₂×C₄×C₈	Q₈	1	2	yes	—
43	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
44	C₂×C₄×C₈	C₂³	1	2	yes	—
45	C₄⋊C₁₆	C₂×C₄	1	8	yes	—
46	C₂²⋊C₁₆	C₂×C₄	1	8	yes	—
47	#346	C₂×C₄	1	4	yes	—
48	#346	C₂³	1	4	yes	—
49	#334	—	2	2	no	—
50	#334	—	2	4	no	—
51	#334	—	2	4	no	—
52	#334	—	2	2	no	—
53	#334	C₂³	1	4	yes	—
54	#334	C₂³	1	4	yes	—
55	C₂³×C₄	C₄⋊C₄	1	1	yes	characteristic
56	C₂³×C₄	SD₁₆	1	1	yes	characteristic
57	C₂³×C₄	SD₁₆	1	1	yes	characteristic
58	C₂×M₁₆	—	4	4	no	—
59	C₂×M₁₆	—	8	4	no	—
60	C₂²×C₈	—	2	4	no	—
61	C₂²×C₈	—	4	2	no	—
62	C₂²×C₈	—	4	4	no	—
63	C₂²×C₈	—	4	2	no	—
64	C₂²×C₈	—	4	2	no	—
65	C₂²×C₈	—	8	4	no	—
66	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
67	C₂²×C₈	C₄⋊C₄	1	2	yes	—
68	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
69	C₂²×C₈	C₄⋊C₄	1	2	yes	—
70	C₂²×C₈	C₂²×C₄	1	2	yes	—
71	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
72	C₂²×C₈	C₂×Q₈	1	1	yes	characteristic
73	C₂²×C₈	C₂×Q₈	1	2	yes	—
74	C₂×C₄²	C₄⋊C₄	1	2	yes	—
75	C₂×C₄²	C₄⋊C₄	1	1	yes	characteristic
76	C₂×C₄²	SD₁₆	1	2	yes	—
77	C₂×C₄²	SD₁₆	1	1	yes	characteristic
78	C₂×C₄²	SD₁₆	1	1	yes	characteristic
79	C₂×C₄²	SD₁₆	1	2	yes	—
80	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
81	C₂×C₄²	C₂×D₄	1	2	yes	—
82	C₂×C₁₆	—	2	8	no	—
83	C₂×C₁₆	C₂²×C₄	1	8	yes	—
84	C₄⋊C₈	—	2	4	no	—
85	C₄⋊C₈	—	2	4	no	—
86	C₄⋊C₈	—	4	4	no	—
87	C₄⋊C₈	—	4	2	no	—
88	C₄⋊C₈	—	4	2	no	—
89	C₄⋊C₈	—	4	4	no	—
90	C₄×C₈	C₄⋊C₄	1	2	yes	—
91	C₄×C₈	C₄⋊C₄	1	2	yes	—
92	C₄×C₈	C₄⋊C₄	1	2	yes	—
93	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
94	C₄×C₈	C₂²×C₄	1	2	yes	—
95	C₄×C₈	C₂×D₄	1	2	yes	—
96	C₄×C₈	C₂×D₄	1	1	yes	characteristic
97	C₄×C₈	C₂×Q₈	1	2	yes	—
98	C₄×C₈	C₂×Q₈	1	1	yes	characteristic
99	C₄×C₈	C₂⁴	1	1	yes	characteristic
100	C₂⁴	#106	1	1	yes	characteristic
101	C₂²×C₄	—	2	4	no	—
102	C₂²×C₄	—	2	4	no	—
103	C₂²×C₄	—	2	2	no	—
104	C₂²×C₄	—	2	2	no	—
105	C₂²×C₄	#106	1	2	yes	—
106	C₂²×C₄	#106	1	2	yes	—
107	C₂²×C₄	#106	1	2	yes	—
108	C₂²×C₄	#106	1	1	yes	characteristic
109	C₂²×C₄	#106	1	1	yes	characteristic
110	C₂²×C₄	#106	1	1	yes	characteristic
111	C₂²×C₄	C₂×C₄⋊C₄	1	2	yes	—
112	C₂²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
113	C₂²×C₄	C₂×SD₁₆	1	2	yes	—
114	C₂²×C₄	C₂×SD₁₆	1	2	yes	—
115	C₂²×C₄	C₂×SD₁₆	1	1	yes	characteristic
116	C₂²×C₄	C₂×SD₁₆	1	1	yes	characteristic
117	M₁₆	—	8	8	no	—
118	C₂×C₈	—	2	2	no	—
119	C₂×C₈	—	2	2	no	—
120	C₂×C₈	—	2	2	no	—
121	C₂×C₈	—	2	4	no	—
122	C₂×C₈	—	2	4	no	—
123	C₂×C₈	—	4	2	no	—
124	C₂×C₈	—	4	2	no	—
125	C₂×C₈	—	4	4	no	—
126	C₂×C₈	—	4	4	no	—
127	C₂×C₈	—	8	4	no	—
128	C₂×C₈	—	8	8	no	—
129	C₂×C₈	—	8	4	no	—
130	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
131	C₂×C₈	C₂×C₄⋊C₄	1	2	yes	—
132	C₂×C₈	C₂×C₄⋊C₄	1	2	yes	—
133	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
134	C₂×C₈	#131	1	4	yes	—
135	C₂×C₈	C₂³×C₄	1	1	yes	characteristic
136	C₂×C₈	C₂²×Q₈	1	1	yes	characteristic
137	C₄²	#106	1	2	yes	—
138	C₄²	#106	1	2	yes	—
139	C₄²	#106	1	2	yes	—
140	C₄²	C₂×C₄⋊C₄	1	1	yes	characteristic
141	C₄²	C₂×C₄⋊C₄	1	2	yes	—
142	C₄²	C₂×SD₁₆	1	2	yes	—
143	C₄²	C₂×SD₁₆	1	1	yes	characteristic
144	C₄²	C₂×SD₁₆	1	1	yes	characteristic
145	C₄²	C₂×SD₁₆	1	2	yes	—
146	C₄²	C₂²×D₄	1	1	yes	characteristic
147	C₁₆	—	2	8	no	—
148	C₂³	—	2	2	no	—
149	C₂³	—	2	2	no	—
150	C₂³	#365	1	4	yes	—
151	C₂³	C₂×#106	1	2	yes	—
152	C₂³	C₂×#106	1	1	yes	characteristic
153	C₂×C₄	—	2	2	no	—
154	C₂×C₄	—	2	2	no	—
155	C₂×C₄	—	2	2	no	—
156	C₂×C₄	—	2	4	no	—
157	C₂×C₄	—	2	4	no	—
158	C₂×C₄	—	2	2	no	—
159	C₂×C₄	—	2	4	no	—
160	C₂×C₄	—	2	2	no	—
161	C₂×C₄	—	2	2	no	—
162	C₂×C₄	—	2	4	no	—
163	C₂×C₄	—	2	4	no	—
164	C₂×C₄	#365	1	4	yes	—
165	C₂×C₄	#424	1	2	yes	—
166	C₂×C₄	C₂×#106	1	1	yes	characteristic
167	C₂×C₄	C₂×#106	1	1	yes	characteristic
168	C₂×C₄	C₂×#106	1	2	yes	—
169	C₂×C₄	C₂×#106	1	2	yes	—
170	C₂×C₄	C₂×#106	1	1	yes	characteristic
171	C₂×C₄	C₂×#106	1	2	yes	—
172	C₂×C₄	#430	1	2	yes	—
173	C₂×C₄	C₂²×C₄⋊C₄	1	1	yes	characteristic
174	C₂×C₄	C₂²×SD₁₆	1	1	yes	characteristic
175	C₂×C₄	C₂²×SD₁₆	1	1	yes	characteristic
176	C₈	—	2	2	no	—
177	C₈	—	2	2	no	—
178	C₈	—	2	1	no	—
179	C₈	—	2	1	no	—
180	C₈	—	8	4	no	—
181	C₈	—	8	4	no	—
182	C₈	#424	1	2	yes	—
183	C₈	C₂×#131	1	2	yes	—
184	C₂²	—	2	2	no	—
185	C₂²	—	2	2	no	—
186	C₂²	—	2	8	no	—
187	C₂²	#1570	1	2	yes	—
188	C₂²	C₂×#365	1	4	yes	—
189	C₂²	C₂×#365	1	2	yes	—
190	C₂²	C₂×#365	1	2	yes	—
191	C₂²	C₂²×#106	1	1	yes	characteristic
192	C₄	—	2	1	no	—
193	C₄	—	2	1	no	—
194	C₄	—	2	1	no	—
195	C₄	—	2	2	no	—
196	C₄	—	2	1	no	—
197	C₄	—	2	2	no	—
198	C₄	#1570	1	2	yes	—
199	C₄	C₂×#365	1	4	yes	—
200	C₄	C₂×#424	1	1	yes	characteristic
201	C₄	C₂×#430	1	1	yes	characteristic
202	C₂	—	2	4	no	—
203	C₂	#12030	1	4	yes	direct product factor
204	C₂	C₂×#1570	1	1	yes	characteristic
205	C₂	C₂²×#365	1	1	yes	characteristic
206	C₂	C₂²×#365	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³	C₂×#106
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₄×C₈	C₂⁴
prime radical	C₂⁴	#106
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ C₂²×C₄×C₈ ◁ 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
4. #12030	C₂	direct (N × H)
5. C₂×#1368	C₂	semidirect
22. #1368	C₂²	semidirect
203. C₂	#12030	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×#2163	2
C₂×#1566	2
C₂²×#334	2
#12030	2
C₂×#1368	2

Subgroups — C2×#12030 #483418

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#12030 #483418