Subgroups — C₂×#12015 #483352

Order 512.

574 conjugacy classes of subgroups.

The subgroup lattice has 239 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#2162	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic
3	C₂×C₈×M₁₆	C₂	1	1	yes	maximal, characteristic
4	#12015	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	C₂³×C₁₆	—	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	#2162	—	2	2	no	—
12	#2162	C₂²	1	4	yes	—
13	C₂×C₄⋊C₁₆	—	2	2	no	—
14	C₂×C₄⋊C₁₆	—	2	1	no	—
15	C₂×C₄⋊C₁₆	C₂²	1	2	yes	—
16	C₂×C₄⋊C₁₆	C₂²	1	4	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×C₄×C₁₆	—	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	1	yes	characteristic
28	C₂²×M₁₆	C₈	1	1	yes	characteristic
29	C₂³×C₈	D₄	1	1	yes	characteristic
30	C₂³×C₈	Q₈	1	1	yes	characteristic
31	C₂²×C₄²	C₂×C₄	1	1	yes	characteristic
32	C₂²×C₁₆	—	2	4	no	—
33	C₂²×C₁₆	—	2	1	no	—
34	C₂²×C₁₆	—	2	1	no	—
35	C₂²×C₁₆	—	2	2	no	—
36	C₂²×C₁₆	—	4	4	no	—
37	#431	—	2	2	no	—
38	#431	C₂×C₄	1	4	yes	—
39	C₂×C₄⋊C₈	—	2	1	no	—
40	C₂×C₂²⋊C₈	—	2	1	no	—
41	C₄×M₁₆	C₈	1	4	yes	—
42	C₄×M₁₆	C₂×C₄	1	4	yes	—
43	C₂×#97	C₈	1	2	yes	—
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	C₄×C₁₆	—	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	4	no	—
70	C₂×M₁₆	C₁₆	1	8	yes	complemented
71	C₂×M₁₆	C₂×C₈	1	4	yes	—
72	C₂×M₁₆	C₂×C₈	1	4	yes	—
73	C₂²×C₈	—	2	2	no	—
74	C₂²×C₈	—	2	2	no	—
75	C₂²×C₈	—	2	2	no	—
76	C₂²×C₈	—	4	2	no	—
77	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
78	C₂²×C₈	C₄⋊C₄	1	2	yes	—
79	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
80	C₂²×C₈	C₄⋊C₄	1	2	yes	—
81	C₂²×C₈	C₂×C₈	1	2	yes	—
82	C₂²×C₈	C₂×C₈	1	2	yes	—
83	C₂²×C₈	C₂×D₄	1	2	yes	—
84	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
85	C₂²×C₈	C₂×Q₈	1	1	yes	characteristic
86	C₂²×C₈	C₂×Q₈	1	2	yes	—
87	C₂×C₄²	C₄⋊C₄	1	1	yes	characteristic
88	C₂×C₄²	C₄⋊C₄	1	2	yes	—
89	C₂×C₄²	C₂×C₈	1	1	yes	characteristic
90	C₂×C₄²	C₂×C₈	1	2	yes	—
91	C₂×C₄²	M₁₆	1	1	yes	characteristic
92	C₂×C₄²	M₁₆	1	2	yes	—
93	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
94	C₂×C₄²	C₂²×C₄	1	2	yes	—
95	C₂×C₁₆	—	2	2	no	—
96	C₂×C₁₆	—	2	2	no	—
97	C₂×C₁₆	—	2	4	no	—
98	C₂×C₁₆	—	4	8	no	—
99	C₂×C₁₆	—	4	2	no	—
100	C₂×C₁₆	—	4	4	no	—
101	C₂×C₁₆	—	4	2	no	—
102	C₄⋊C₈	—	2	4	no	—
103	C₂²⋊C₈	—	2	4	no	—
104	#97	C₂×C₈	1	8	yes	—
105	C₄×C₈	—	2	2	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	C₂×C₈	1	4	yes	—
110	C₄×C₈	C₂²×C₄	1	4	yes	—
111	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
112	C₄×C₈	C₂²×C₄	1	2	yes	—
113	C₄×C₈	C₂²×C₄	1	4	yes	—
114	C₄×C₈	C₂×D₄	1	2	yes	—
115	C₄×C₈	C₂×D₄	1	1	yes	characteristic
116	C₄×C₈	C₂×Q₈	1	2	yes	—
117	C₄×C₈	C₂×Q₈	1	1	yes	characteristic
118	C₄×C₈	C₂⁴	1	1	yes	characteristic
119	C₂⁴	C₄⋊C₈	1	1	yes	characteristic
120	C₂²×C₄	—	2	2	no	—
121	C₂²×C₄	—	2	2	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	2	yes	—
127	C₂²×C₄	C₄⋊C₈	1	2	yes	—
128	C₂²×C₄	C₄⋊C₈	1	1	yes	characteristic
129	C₂²×C₄	C₂×C₁₆	1	4	yes	—
130	C₂²×C₄	M₃₂	1	4	yes	—
131	C₂²×C₄	C₂×C₄⋊C₄	1	2	yes	—
132	C₂²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
133	C₂²×C₄	C₂²×C₈	1	1	yes	characteristic
134	C₂²×C₄	C₂²×C₈	1	2	yes	—
135	C₂²×C₄	C₂×M₁₆	1	1	yes	characteristic
136	C₂²×C₄	C₂×M₁₆	1	2	yes	—
137	M₁₆	—	2	8	no	—
138	M₁₆	C₂×C₁₆	1	16	yes	complemented
139	C₂×C₈	—	2	2	no	—
140	C₂×C₈	—	2	2	no	—
141	C₂×C₈	—	2	2	no	—
142	C₂×C₈	—	2	4	no	—
143	C₂×C₈	—	2	2	no	—
144	C₂×C₈	—	2	2	no	—
145	C₂×C₈	—	2	4	no	—
146	C₂×C₈	—	2	4	no	—
147	C₂×C₈	—	4	2	no	—
148	C₂×C₈	—	4	2	no	—
149	C₂×C₈	—	4	4	no	—
150	C₂×C₈	C₂×C₁₆	1	8	yes	complemented
151	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
152	C₂×C₈	C₂×C₄⋊C₄	1	2	yes	—
153	C₂×C₈	C₂×C₄⋊C₄	1	2	yes	—
154	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
155	C₂×C₈	C₂²×C₈	1	4	yes	—
156	C₂×C₈	C₂²×C₈	1	4	yes	—
157	C₂×C₈	AGL(1,ℤ/8)	1	2	yes	—
158	C₂×C₈	#137	1	2	yes	—
159	C₂×C₈	C₂²×D₄	1	1	yes	characteristic
160	C₂×C₈	C₂²×Q₈	1	1	yes	characteristic
161	C₄²	C₄⋊C₈	1	2	yes	—
162	C₄²	C₄⋊C₈	1	2	yes	—
163	C₄²	C₄⋊C₈	1	2	yes	—
164	C₄²	C₂×C₄⋊C₄	1	2	yes	—
165	C₄²	C₂×C₄⋊C₄	1	1	yes	characteristic
166	C₄²	C₂²×C₈	1	1	yes	characteristic
167	C₄²	C₂²×C₈	1	2	yes	—
168	C₄²	C₂×M₁₆	1	1	yes	characteristic
169	C₄²	C₂×M₁₆	1	2	yes	—
170	C₄²	C₂³×C₄	1	1	yes	characteristic
171	C₁₆	—	4	2	no	—
172	C₁₆	—	4	4	no	—
173	C₁₆	—	4	2	no	—
174	C₂³	—	2	2	no	—
175	C₂³	—	2	2	no	—
176	C₂³	C₄⋊C₁₆	1	4	yes	—
177	C₂³	C₂×C₄⋊C₈	1	2	yes	—
178	C₂³	C₂×C₄⋊C₈	1	1	yes	characteristic
179	C₂×C₄	—	2	2	no	—
180	C₂×C₄	—	2	2	no	—
181	C₂×C₄	—	2	2	no	—
182	C₂×C₄	—	2	4	no	—
183	C₂×C₄	—	2	2	no	—
184	C₂×C₄	—	2	2	no	—
185	C₂×C₄	—	4	2	no	—
186	C₂×C₄	—	4	2	no	—
187	C₂×C₄	C₄⋊C₁₆	1	8	yes	—
188	C₂×C₄	C₄⋊C₁₆	1	4	yes	—
189	C₂×C₄	C₂×C₄⋊C₈	1	1	yes	characteristic
190	C₂×C₄	C₂×C₄⋊C₈	1	2	yes	—
191	C₂×C₄	C₂×C₄⋊C₈	1	2	yes	—
192	C₂×C₄	C₂×C₄⋊C₈	1	1	yes	characteristic
193	C₂×C₄	C₂×C₄⋊C₈	1	1	yes	characteristic
194	C₂×C₄	C₂×C₄⋊C₈	1	2	yes	—
195	C₂×C₄	#428	1	2	yes	—
196	C₂×C₄	#430	1	2	yes	—
197	C₂×C₄	C₂²×C₁₆	1	2	yes	—
198	C₂×C₄	C₂²×C₁₆	1	4	yes	—
199	C₂×C₄	C₂×M₃₂	1	2	yes	—
200	C₂×C₄	C₂×M₃₂	1	4	yes	—
201	C₂×C₄	C₂²×C₄⋊C₄	1	1	yes	characteristic
202	C₂×C₄	C₂³×C₈	1	1	yes	characteristic
203	C₂×C₄	C₂²×M₁₆	1	1	yes	characteristic
204	C₈	—	2	4	no	—
205	C₈	—	2	2	no	—
206	C₈	—	2	2	no	—
207	C₈	—	2	1	no	—
208	C₈	—	2	1	no	—
209	C₈	—	4	4	no	—
210	C₈	#428	1	2	yes	—
211	C₈	C₂²×C₁₆	1	8	yes	complemented
212	C₈	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
213	C₈	C₂×#137	1	1	yes	characteristic
214	C₂²	—	2	2	no	—
215	C₂²	—	2	2	no	—
216	C₂²	—	2	8	no	—
217	C₂²	#1565	1	2	yes	—
218	C₂²	C₂×C₄⋊C₁₆	1	4	yes	—
219	C₂²	C₂×C₄⋊C₁₆	1	2	yes	—
220	C₂²	#2153	1	2	yes	—
221	C₂²	C₂²×C₄⋊C₈	1	1	yes	characteristic
222	C₄	—	2	1	no	—
223	C₄	—	2	1	no	—
224	C₄	—	2	2	no	—
225	C₄	—	2	2	no	—
226	C₄	#1565	1	2	yes	—
227	C₄	C₂×C₄⋊C₁₆	1	2	yes	—
228	C₄	C₂×C₄⋊C₁₆	1	4	yes	—
229	C₄	C₂×#428	1	1	yes	characteristic
230	C₄	C₂×#430	1	1	yes	characteristic
231	C₄	C₂³×C₁₆	1	1	yes	characteristic
232	C₄	C₂²×M₃₂	1	1	yes	characteristic
233	C₂	—	2	4	no	—
234	C₂	#12015	1	4	yes	direct product factor
235	C₂	C₂×#1565	1	1	yes	characteristic
236	C₂	C₂²×C₄⋊C₁₆	1	1	yes	characteristic
237	C₂	C₂×#2153	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₈	C₂×D₄
commutator	C₄	C₂³×C₁₆
Frattini, cosocle	C₄×C₈	C₂⁴
socle	C₂³	C₂×C₄⋊C₈
prime radical	C₂⁴	C₄⋊C₈
LCS	G ▷ 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. #12015	C₂	direct (N × H)
5. C₂×#1370	C₂	semidirect
6. C₂×#1367	C₂	semidirect
25. #1370	C₂²	semidirect
26. #1367	C₂²	semidirect
70. C₂×M₁₆	C₁₆	semidirect
138. M₁₆	C₂×C₁₆	semidirect
150. C₂×C₈	C₂×C₁₆	semidirect
211. C₈	C₂²×C₁₆	semidirect
234. C₂	#12015	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×#2162	2
C₂²×C₄⋊C₁₆	2
C₂×C₈×M₁₆	2
#12015	2
C₂×#1370	2
C₂×#1367	2

Subgroups — C2×#12015 #483352

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#12015 #483352