Subgroups — C₂×#12017 #483353

Order 512.

536 conjugacy classes of subgroups.

The subgroup lattice has 233 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×#2150	C₂	1	1	yes	maximal, characteristic
3	C₂×C₈×M₁₆	C₂	1	1	yes	maximal, characteristic
4	#12017	C₂	1	8	yes	maximal, direct product factor
5	C₂×#1370	C₂	1	2	yes	maximal, complemented
6	C₂×#1367	C₂	1	2	yes	maximal, complemented
7	C₂²×M₃₂	—	2	1	no	—
8	C₂×#431	C₄	1	1	yes	characteristic
9	C₂×C₄×M₁₆	C₄	1	1	yes	characteristic
10	C₂²×C₄×C₈	C₂²	1	1	yes	characteristic
11	#2163	—	2	2	no	—
12	#2163	C₂²	1	4	yes	—
13	#2150	—	2	2	no	—
14	#2150	C₂²	1	4	yes	—
15	C₂×C₄⋊C₁₆	—	2	1	no	—
16	C₂×C₄⋊C₁₆	C₂²	1	2	yes	—
17	C₂×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
18	C₂×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
19	C₂×C₂²⋊C₁₆	—	2	1	no	—
20	C₂×#346	—	2	1	no	—
21	C₈×M₁₆	C₄	1	4	yes	—
22	C₈×M₁₆	C₂²	1	4	yes	—
23	C₂×#322	C₄	1	2	yes	—
24	C₂×C₈²	C₂²	1	2	yes	—
25	#1370	C₂²	1	8	yes	complemented
26	#1367	C₂²	1	8	yes	complemented
27	C₂²×M₁₆	C₈	1	2	yes	—
28	C₂³×C₈	D₄	1	1	yes	characteristic
29	C₂³×C₈	Q₈	1	1	yes	characteristic
30	C₂²×C₄²	C₂×C₄	1	1	yes	characteristic
31	C₂×M₃₂	—	2	4	no	—
32	C₂×M₃₂	—	4	4	no	—
33	C₂²×C₁₆	—	2	1	no	—
34	C₂²×C₁₆	—	2	1	no	—
35	C₂²×C₁₆	—	2	2	no	—
36	#431	—	2	2	no	—
37	#431	C₂×C₄	1	4	yes	—
38	C₂×C₄⋊C₈	—	2	1	no	—
39	C₂×C₂²⋊C₈	—	2	1	no	—
40	C₄×M₁₆	C₈	1	4	yes	—
41	C₄×M₁₆	C₂×C₄	1	4	yes	—
42	C₂×#97	C₈	1	1	yes	characteristic
43	C₂×#97	C₈	1	1	yes	characteristic
44	C₂×C₄×C₈	C₂×C₄	1	2	yes	—
45	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
46	C₂×C₄×C₈	C₂×C₄	1	2	yes	—
47	C₂×C₄×C₈	C₂×C₄	1	2	yes	—
48	C₂×C₄×C₈	D₄	1	2	yes	—
49	C₂×C₄×C₈	D₄	1	1	yes	characteristic
50	C₂×C₄×C₈	Q₈	1	1	yes	characteristic
51	C₂×C₄×C₈	Q₈	1	2	yes	—
52	C₂×C₄×C₈	C₂³	1	2	yes	—
53	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
54	C₄⋊C₁₆	—	2	4	no	—
55	C₄⋊C₁₆	—	2	2	no	—
56	C₄⋊C₁₆	—	2	1	no	—
57	C₄⋊C₁₆	—	2	1	no	—
58	C₄⋊C₁₆	C₂³	1	2	yes	—
59	C₄⋊C₁₆	C₂³	1	4	yes	—
60	C₄⋊C₁₆	C₂³	1	2	yes	—
61	C₂²⋊C₁₆	—	2	4	no	—
62	#346	—	2	4	no	—
63	#322	C₂×C₄	1	8	yes	—
64	C₈²	C₂×C₄	1	4	yes	—
65	C₈²	C₂³	1	4	yes	—
66	C₂³×C₄	C₄⋊C₄	1	1	yes	characteristic
67	C₂³×C₄	C₂×C₈	1	1	yes	characteristic
68	C₂³×C₄	M₁₆	1	1	yes	characteristic
69	C₂×M₁₆	—	2	8	no	—
70	C₂×M₁₆	C₂×C₈	1	8	yes	—
71	C₂²×C₈	—	2	2	no	—
72	C₂²×C₈	—	2	2	no	—
73	C₂²×C₈	—	2	2	no	—
74	C₂²×C₈	—	4	2	no	—
75	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
76	C₂²×C₈	C₄⋊C₄	1	2	yes	—
77	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
78	C₂²×C₈	C₄⋊C₄	1	2	yes	—
79	C₂²×C₈	C₂×C₈	1	2	yes	—
80	C₂²×C₈	C₂×C₈	1	2	yes	—
81	C₂²×C₈	C₂×D₄	1	2	yes	—
82	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
83	C₂²×C₈	C₂×Q₈	1	1	yes	characteristic
84	C₂²×C₈	C₂×Q₈	1	2	yes	—
85	C₂×C₄²	C₄⋊C₄	1	1	yes	characteristic
86	C₂×C₄²	C₄⋊C₄	1	2	yes	—
87	C₂×C₄²	C₂×C₈	1	1	yes	characteristic
88	C₂×C₄²	C₂×C₈	1	2	yes	—
89	C₂×C₄²	M₁₆	1	1	yes	characteristic
90	C₂×C₄²	M₁₆	1	2	yes	—
91	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
92	C₂×C₄²	C₂²×C₄	1	2	yes	—
93	M₃₂	—	4	8	no	—
94	C₂×C₁₆	—	2	2	no	—
95	C₂×C₁₆	—	2	2	no	—
96	C₂×C₁₆	—	2	4	no	—
97	C₂×C₁₆	—	4	2	no	—
98	C₂×C₁₆	—	4	4	no	—
99	C₂×C₁₆	—	4	2	no	—
100	C₄⋊C₈	—	2	4	no	—
101	C₂²⋊C₈	—	2	4	no	—
102	#97	C₂×C₈	1	4	yes	—
103	#97	C₂×C₈	1	4	yes	—
104	C₄×C₈	—	2	2	no	—
105	C₄×C₈	C₄⋊C₄	1	2	yes	—
106	C₄×C₈	C₄⋊C₄	1	2	yes	—
107	C₄×C₈	C₄⋊C₄	1	2	yes	—
108	C₄×C₈	C₂×C₈	1	4	yes	—
109	C₄×C₈	C₂²×C₄	1	4	yes	—
110	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
111	C₄×C₈	C₂²×C₄	1	2	yes	—
112	C₄×C₈	C₂²×C₄	1	4	yes	—
113	C₄×C₈	C₂×D₄	1	2	yes	—
114	C₄×C₈	C₂×D₄	1	1	yes	characteristic
115	C₄×C₈	C₂×Q₈	1	2	yes	—
116	C₄×C₈	C₂×Q₈	1	1	yes	characteristic
117	C₄×C₈	C₂⁴	1	1	yes	characteristic
118	C₂⁴	C₄⋊C₈	1	1	yes	characteristic
119	C₂²×C₄	—	2	2	no	—
120	C₂²×C₄	—	2	2	no	—
121	C₂²×C₄	—	2	4	no	—
122	C₂²×C₄	—	4	2	no	—
123	C₂²×C₄	C₄⋊C₈	1	1	yes	characteristic
124	C₂²×C₄	C₄⋊C₈	1	1	yes	characteristic
125	C₂²×C₄	C₄⋊C₈	1	2	yes	—
126	C₂²×C₄	C₄⋊C₈	1	1	yes	characteristic
127	C₂²×C₄	C₄⋊C₈	1	2	yes	—
128	C₂²×C₄	C₄⋊C₈	1	2	yes	—
129	C₂²×C₄	C₂×C₄⋊C₄	1	2	yes	—
130	C₂²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
131	C₂²×C₄	C₂²×C₈	1	2	yes	—
132	C₂²×C₄	C₂²×C₈	1	1	yes	characteristic
133	C₂²×C₄	C₂×M₁₆	1	1	yes	characteristic
134	C₂²×C₄	C₂×M₁₆	1	2	yes	—
135	M₁₆	—	2	16	no	—
136	C₂×C₈	—	2	2	no	—
137	C₂×C₈	—	2	2	no	—
138	C₂×C₈	—	2	2	no	—
139	C₂×C₈	—	2	4	no	—
140	C₂×C₈	—	2	2	no	—
141	C₂×C₈	—	2	2	no	—
142	C₂×C₈	—	2	4	no	—
143	C₂×C₈	—	2	4	no	—
144	C₂×C₈	—	4	2	no	—
145	C₂×C₈	—	4	2	no	—
146	C₂×C₈	—	4	4	no	—
147	C₂×C₈	M₃₂	1	8	yes	complemented
148	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
149	C₂×C₈	C₂×C₄⋊C₄	1	2	yes	—
150	C₂×C₈	C₂×C₄⋊C₄	1	2	yes	—
151	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
152	C₂×C₈	C₂²×C₈	1	4	yes	—
153	C₂×C₈	C₂²×C₈	1	4	yes	—
154	C₂×C₈	AGL(1,ℤ/8)	1	2	yes	—
155	C₂×C₈	#137	1	2	yes	—
156	C₂×C₈	C₂²×D₄	1	1	yes	characteristic
157	C₂×C₈	C₂²×Q₈	1	1	yes	characteristic
158	C₄²	C₄⋊C₈	1	2	yes	—
159	C₄²	C₄⋊C₈	1	2	yes	—
160	C₄²	C₄⋊C₈	1	2	yes	—
161	C₄²	C₂×C₄⋊C₄	1	2	yes	—
162	C₄²	C₂×C₄⋊C₄	1	1	yes	characteristic
163	C₄²	C₂²×C₈	1	1	yes	characteristic
164	C₄²	C₂²×C₈	1	2	yes	—
165	C₄²	C₂×M₁₆	1	1	yes	characteristic
166	C₄²	C₂×M₁₆	1	2	yes	—
167	C₄²	C₂³×C₄	1	1	yes	characteristic
168	C₁₆	—	4	2	no	—
169	C₁₆	—	4	4	no	—
170	C₁₆	—	4	2	no	—
171	C₂³	—	2	2	no	—
172	C₂³	—	2	2	no	—
173	C₂³	—	2	2	no	—
174	C₂³	C₂×C₄⋊C₈	1	2	yes	—
175	C₂³	C₂×C₄⋊C₈	1	1	yes	characteristic
176	C₂×C₄	—	2	2	no	—
177	C₂×C₄	—	2	2	no	—
178	C₂×C₄	—	2	2	no	—
179	C₂×C₄	—	2	2	no	—
180	C₂×C₄	—	2	4	no	—
181	C₂×C₄	—	2	4	no	—
182	C₂×C₄	—	2	2	no	—
183	C₂×C₄	—	2	2	no	—
184	C₂×C₄	—	2	4	no	—
185	C₂×C₄	—	4	2	no	—
186	C₂×C₄	—	4	2	no	—
187	C₂×C₄	C₂×C₄⋊C₈	1	1	yes	characteristic
188	C₂×C₄	C₂×C₄⋊C₈	1	2	yes	—
189	C₂×C₄	C₂×C₄⋊C₈	1	2	yes	—
190	C₂×C₄	C₂×C₄⋊C₈	1	2	yes	—
191	C₂×C₄	C₂×C₄⋊C₈	1	1	yes	characteristic
192	C₂×C₄	C₂×C₄⋊C₈	1	1	yes	characteristic
193	C₂×C₄	#428	1	2	yes	—
194	C₂×C₄	#430	1	2	yes	—
195	C₂×C₄	C₂×M₃₂	1	4	yes	—
196	C₂×C₄	C₂²×C₄⋊C₄	1	1	yes	characteristic
197	C₂×C₄	C₂³×C₈	1	1	yes	characteristic
198	C₂×C₄	C₂²×M₁₆	1	1	yes	characteristic
199	C₈	—	2	4	no	—
200	C₈	—	2	2	no	—
201	C₈	—	2	2	no	—
202	C₈	—	2	1	no	—
203	C₈	—	2	1	no	—
204	C₈	—	4	4	no	—
205	C₈	#428	1	2	yes	—
206	C₈	C₂×M₃₂	1	8	yes	complemented
207	C₈	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
208	C₈	C₂×#137	1	1	yes	characteristic
209	C₂²	—	2	2	no	—
210	C₂²	—	2	2	no	—
211	C₂²	—	2	2	no	—
212	C₂²	—	4	4	no	—
213	C₂²	#1565	1	2	yes	—
214	C₂²	#2150	1	2	yes	—
215	C₂²	#2153	1	2	yes	—
216	C₂²	C₂²×C₄⋊C₈	1	1	yes	characteristic
217	C₄	—	2	2	no	—
218	C₄	—	2	1	no	—
219	C₄	—	2	1	no	—
220	C₄	—	2	2	no	—
221	C₄	—	2	2	no	—
222	C₄	#1565	1	2	yes	—
223	C₄	#2150	1	2	yes	—
224	C₄	C₂×#428	1	1	yes	characteristic
225	C₄	C₂×#430	1	1	yes	characteristic
226	C₄	C₂²×M₃₂	1	2	yes	—
227	C₂	—	4	2	no	—
228	C₂	#12017	1	4	yes	direct product factor
229	C₂	C₂×#1565	1	1	yes	characteristic
230	C₂	C₂×#2150	1	1	yes	characteristic
231	C₂	C₂×#2153	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₈	C₂×D₄
commutator	C₂×C₄	C₂³×C₈
Frattini, cosocle	C₄×C₈	C₂⁴
socle	C₂³	C₂×C₄⋊C₈
prime radical	C₂⁴	C₄⋊C₈
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₈ ◁ 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. #12017	C₂	direct (N × H)
5. C₂×#1370	C₂	semidirect
6. C₂×#1367	C₂	semidirect
25. #1370	C₂²	semidirect
26. #1367	C₂²	semidirect
147. C₂×C₈	M₃₂	semidirect
206. C₈	C₂×M₃₂	semidirect
228. C₂	#12017	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×#2150	2
C₂×C₈×M₁₆	2
#12017	2
C₂×#1370	2
C₂×#1367	2

Subgroups — C2×#12017 #483353

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#12017 #483353