Subgroups — C₂×#21461415 #30612293

Order 15616.

894 conjugacy classes of subgroups.

The subgroup lattice has 272 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₄×#92420	C₂	1	2	yes	maximal, complemented
2	#21461415	C₂	1	8	yes	maximal, direct product factor
3	C₂×#16840623	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×D₄⋊#92420	C₂	1	2	yes	maximal, complemented
5	C₂×#16840603	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×D₈×Dic₆₁	C₂	1	1	yes	maximal, characteristic
7	C₂³×#92420	—	2	2	no	—
8	D₄×#92420	—	2	4	no	—
9	D₄×#92420	C₂²	1	8	yes	complemented
10	C₂×C₄⋊#92420	C₂²	1	2	yes	—
11	C₂×C₂²⋊#92420	—	2	2	no	—
12	C₂×C₄×#92420	C₂²	1	1	yes	characteristic
13	C₂×C₄×#92420	C₂²	1	1	yes	characteristic
14	#16840623	C₂²	1	4	yes	complemented
15	D₄⋊#92420	C₂²	1	8	yes	complemented
16	#16840603	C₂²	1	4	yes	complemented
17	C₂×D₄×Dic₆₁	C₂²	1	2	yes	—
18	D₈×Dic₆₁	C₄	1	4	yes	—
19	D₈×Dic₆₁	C₂²	1	4	yes	—
20	C₂×#14519711	C₄	1	1	yes	characteristic
21	C₂×#14519698	C₄	1	2	yes	—
22	C₂×C₈×Dic₆₁	C₂²	1	1	yes	characteristic
23	C₂²×D₈×C₆₁	C₄	1	1	yes	characteristic
24	C₂²×#92420	—	2	2	no	—
25	C₂²×#92420	—	2	1	no	—
26	C₂²×#92420	—	2	8	no	—
27	C₂²×#92420	—	4	8	no	—
28	C₂²×#92420	D₄	1	2	yes	—
29	C₄⋊#92420	—	2	2	no	—
30	C₄⋊#92420	C₂³	1	4	yes	—
31	C₂²⋊#92420	—	2	8	no	—
32	C₄×#92420	—	2	1	no	—
33	C₄×#92420	D₄	1	2	yes	—
34	C₄×#92420	C₂³	1	2	yes	—
35	C₄×#92420	C₂³	1	2	yes	—
36	C₂³×Dic₆₁	—	2	2	no	—
37	C₂²×#92418	C₂×C₄	1	1	yes	characteristic
38	D₄×Dic₆₁	—	2	2	no	—
39	D₄×Dic₆₁	—	2	2	no	—
40	D₄×Dic₆₁	C₂×C₄	1	4	yes	—
41	D₄×Dic₆₁	C₂³	1	4	yes	—
42	C₂×C₄⋊Dic₆₁	C₂×C₄	1	2	yes	—
43	C₂×C₂²⋊Dic₆₁	—	2	2	no	—
44	C₂×C₄×Dic₆₁	C₂³	1	1	yes	characteristic
45	#14519711	C₂×C₄	1	4	yes	—
46	#14519698	C₂×C₄	1	8	yes	—
47	C₈×Dic₆₁	C₂×C₄	1	2	yes	—
48	C₈×Dic₆₁	C₂³	1	2	yes	—
49	C₂²×D₄×C₆₁	C₂×C₄	1	2	yes	—
50	C₂×D₈×C₆₁	C₈	1	8	yes	complemented
51	C₂×D₈×C₆₁	C₂×C₄	1	4	yes	—
52	C₂²×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
53	C₂⁴×C₆₁	—	2	2	no	—
54	C₂×D₄×C₆₁	—	2	2	no	—
55	C₂×D₄×C₆₁	—	2	4	no	—
56	C₂×D₄×C₆₁	C₂×C₈	1	8	yes	complemented
57	C₂×D₄×C₆₁	C₂²×C₄	1	4	yes	—
58	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
59	C₂²×Dic₆₁	—	2	2	no	—
60	C₂²×Dic₆₁	—	2	4	no	—
61	C₂²×Dic₆₁	—	2	4	no	—
62	C₂²×Dic₆₁	—	4	4	no	—
63	C₂²×Dic₆₁	—	4	4	no	—
64	C₂²×Dic₆₁	C₂×D₄	1	1	yes	characteristic
65	C₂²×Dic₆₁	#41	1	1	yes	characteristic
66	C₂×#92420	—	2	4	no	—
67	C₂×#92420	—	2	4	no	—
68	C₂×#92420	—	2	2	no	—
69	C₂×#92420	—	4	16	no	—
70	C₂×#92420	—	4	2	no	—
71	C₂×#92420	—	4	4	no	—
72	C₂×#92420	C₂×D₄	1	4	yes	—
73	D₈×C₆₁	C₂×C₈	1	16	yes	complemented
74	C₂×C₈×C₆₁	C₂×C₈	1	4	yes	complemented
75	C₂×C₈×C₆₁	C₂²×C₄	1	2	yes	—
76	C₂²⋊Dic₆₁	—	2	4	no	—
77	C₂²⋊Dic₆₁	—	2	4	no	—
78	C₄⋊Dic₆₁	—	2	2	no	—
79	C₄⋊Dic₆₁	C₂²×C₄	1	4	yes	—
80	C₄×Dic₆₁	C₂²×C₄	1	1	yes	characteristic
81	C₄×Dic₆₁	C₂×D₄	1	1	yes	characteristic
82	C₄×Dic₆₁	#41	1	1	yes	characteristic
83	C₄×Dic₆₁	C₂⁴	1	1	yes	characteristic
84	C₂×#92418	—	2	2	no	—
85	C₂×#92418	C₂²×C₄	1	2	yes	—
86	C₂³×C₆₁	—	2	4	no	—
87	C₂³×C₆₁	—	2	8	no	—
88	C₂³×C₆₁	—	4	4	no	—
89	C₂³×C₆₁	—	4	4	no	—
90	C₂³×C₆₁	C₄×D₄	1	1	yes	characteristic
91	D₄×C₆₁	—	2	4	no	—
92	D₄×C₆₁	—	2	8	no	—
93	D₄×C₆₁	C₂²×C₈	1	8	yes	complemented
94	C₂×C₄×C₆₁	C₄×D₄	1	1	yes	characteristic
95	C₂×C₄×C₆₁	C₂²×C₈	1	2	yes	—
96	C₂×C₄×C₆₁	#131	1	2	yes	—
97	C₂×C₄×C₆₁	C₂³×C₄	1	1	yes	characteristic
98	C₂×Dic₆₁	—	2	2	no	—
99	C₂×Dic₆₁	—	2	4	no	—
100	C₂×Dic₆₁	—	4	4	no	—
101	C₂×Dic₆₁	—	4	4	no	—
102	C₂×Dic₆₁	—	4	4	no	—
103	C₂×Dic₆₁	—	4	8	no	—
104	C₂×Dic₆₁	C₄×D₄	1	1	yes	characteristic
105	C₂×Dic₆₁	C₄×D₄	1	1	yes	characteristic
106	C₂×Dic₆₁	AGL(1,ℤ/8)	1	4	yes	—
107	C₂×Dic₆₁	C₂²×D₄	1	1	yes	characteristic
108	C₂×Dic₆₁	C₂×#41	1	1	yes	characteristic
109	#92420	—	2	4	no	—
110	#92420	—	4	2	no	—
111	#92420	—	4	4	no	—
112	C₈×C₆₁	C₂²×C₈	1	4	yes	complemented
113	#92418	—	2	2	no	—
114	C₂×#1580	—	61	1	no	maximal, Sylow
115	C₂²×C₆₁	—	2	8	no	—
116	C₂²×C₆₁	—	4	16	no	—
117	C₂²×C₆₁	—	4	4	no	—
118	C₂²×C₆₁	—	4	4	no	—
119	C₂²×C₆₁	C₈×D₄	1	2	yes	—
120	C₂²×C₆₁	#442	1	2	yes	—
121	C₂²×C₆₁	#444	1	2	yes	—
122	C₂²×C₆₁	C₂×C₄×D₄	1	1	yes	characteristic
123	C₄×C₆₁	C₈×D₄	1	2	yes	—
124	C₄×C₆₁	C₂³×C₈	1	1	yes	characteristic
125	C₄×C₆₁	C₂×#131	1	1	yes	characteristic
126	Dic₆₁	—	2	1	no	—
127	Dic₆₁	—	2	1	no	—
128	Dic₆₁	—	4	4	no	—
129	Dic₆₁	#442	1	2	yes	—
130	Dic₆₁	C₂×AGL(1,ℤ/8)	1	2	yes	—
131	C₂×C₄×D₈	—	61	1	no	—
132	C₂×C₈×D₄	—	61	2	no	—
133	#1580	—	61	8	no	—
134	C₂×#334	—	61	1	no	—
135	C₂×D₄⋊C₈	—	61	2	no	—
136	C₂×#322	—	61	1	no	—
137	C₂×C₆₁	—	4	8	no	—
138	C₂×C₆₁	#1580	1	4	yes	complemented
139	C₂×C₆₁	C₂×C₈×D₄	1	1	yes	characteristic
140	C₂×C₆₁	C₂×#442	1	1	yes	characteristic
141	C₂×C₆₁	C₂×#444	1	1	yes	characteristic
142	C₂²×D₈	#6923	1	1	yes	characteristic
143	C₂³×C₈	—	122	2	no	—
144	C₂×C₄×D₄	—	61	2	no	—
145	C₄×D₈	—	61	4	no	—
146	C₄×D₈	—	61	4	no	—
147	C₈×D₄	—	61	8	no	—
148	C₈×D₄	—	122	4	no	—
149	C₂×#107	—	61	1	no	—
150	C₂×C₄⋊C₈	—	61	2	no	—
151	C₂×#102	—	61	2	no	—
152	C₂×C₂²⋊C₈	—	122	2	no	—
153	C₂×C₄×C₈	—	61	1	no	—
154	C₂×C₄×C₈	—	61	1	no	—
155	C₂×C₄×C₈	—	61	1	no	—
156	#334	—	61	4	no	—
157	D₄⋊C₈	—	61	8	no	—
158	#322	—	61	4	no	—
159	C₆₁	C₂×#1580	1	1	yes	characteristic, complemented, Sylow
160	C₂²×D₄	C₂×#6923	1	2	yes	—
161	C₂³×C₄	—	122	2	no	—
162	C₂×D₈	#92420	1	8	yes	complemented
163	C₂×D₈	C₂×#6923	1	4	yes	—
164	C₂²×C₈	—	61	2	no	—
165	C₂²×C₈	—	61	1	no	—
166	C₂²×C₈	—	122	8	no	—
167	C₂²×C₈	—	122	2	no	—
168	C₂²×C₈	—	122	1	no	—
169	C₂²×C₈	—	244	8	no	—
170	C₂²×C₈	C₂×#6923	1	1	yes	characteristic
171	C₄×D₄	—	61	4	no	—
172	C₄×D₄	—	61	4	no	—
173	C₄×D₄	—	122	2	no	—
174	C₄×D₄	—	122	2	no	—
175	C₂×C₄⋊C₄	—	61	2	no	—
176	C₂×C₂²⋊C₄	—	122	2	no	—
177	C₂×C₄²	—	61	1	no	—
178	#107	—	61	4	no	—
179	C₄⋊C₈	—	61	4	no	—
180	C₄⋊C₈	—	122	2	no	—
181	#102	—	61	8	no	—
182	C₂²⋊C₈	—	122	8	no	—
183	C₄×C₈	—	61	2	no	—
184	C₄×C₈	—	61	2	no	—
185	C₄×C₈	—	61	2	no	—
186	C₄×C₈	—	61	2	no	—
187	C₄×C₈	—	61	2	no	—
188	C₄×C₈	—	122	1	no	—
189	C₂⁴	—	2	2	no	—
190	C₂×D₄	—	2	2	no	—
191	C₂×D₄	—	2	4	no	—
192	C₂×D₄	C₂×#92420	1	8	yes	complemented
193	C₂×D₄	C₂²×#6923	1	4	yes	—
194	C₂²×C₄	—	61	1	no	—
195	C₂²×C₄	—	61	1	no	—
196	C₂²×C₄	—	122	4	no	—
197	C₂²×C₄	—	122	4	no	—
198	C₂²×C₄	—	122	2	no	—
199	C₂²×C₄	—	244	4	no	—
200	C₂²×C₄	—	244	4	no	—
201	C₂²×C₄	C₂²×#6923	1	1	yes	characteristic
202	D₈	C₂×#92420	1	16	yes	complemented
203	C₂×C₈	—	61	4	no	—
204	C₂×C₈	—	61	2	no	—
205	C₂×C₈	—	122	4	no	—
206	C₂×C₈	—	122	2	no	—
207	C₂×C₈	—	122	2	no	—
208	C₂×C₈	—	122	4	no	—
209	C₂×C₈	—	244	16	no	—
210	C₂×C₈	—	244	4	no	—
211	C₂×C₈	—	244	2	no	—
212	C₂×C₈	C₂×#92420	1	4	yes	complemented
213	C₂×C₈	C₂²×#6923	1	2	yes	—
214	C₄⋊C₄	—	61	4	no	—
215	C₄⋊C₄	—	122	2	no	—
216	C₂²⋊C₄	—	122	4	no	—
217	C₂²⋊C₄	—	122	4	no	—
218	C₄²	—	61	1	no	—
219	C₄²	—	61	1	no	—
220	C₄²	—	61	1	no	—
221	C₄²	—	61	1	no	—
222	C₂³	—	2	8	no	—
223	C₂³	—	2	4	no	—
224	C₂³	—	4	4	no	—
225	C₂³	—	4	4	no	—
226	C₂³	D₄×#6923	1	1	yes	characteristic
227	D₄	—	2	8	no	—
228	D₄	—	2	4	no	—
229	D₄	C₂²×#92420	1	8	yes	complemented
230	C₂×C₄	—	61	4	no	—
231	C₂×C₄	—	61	1	no	—
232	C₂×C₄	—	61	1	no	—
233	C₂×C₄	—	61	1	no	—
234	C₂×C₄	—	61	1	no	—
235	C₂×C₄	—	122	2	no	—
236	C₂×C₄	—	122	4	no	—
237	C₂×C₄	—	244	4	no	—
238	C₂×C₄	—	244	4	no	—
239	C₂×C₄	—	244	8	no	—
240	C₂×C₄	—	244	4	no	—
241	C₂×C₄	D₄×#6923	1	1	yes	characteristic
242	C₂×C₄	C₂²×#92420	1	2	yes	—
243	C₂×C₄	#14519864	1	2	yes	—
244	C₂×C₄	C₂³×#6923	1	1	yes	characteristic
245	C₈	—	122	4	no	—
246	C₈	—	122	2	no	—
247	C₈	—	244	4	no	—
248	C₈	—	244	2	no	—
249	C₈	C₂²×#92420	1	4	yes	complemented
250	C₂²	—	2	8	no	—
251	C₂²	—	4	4	no	—
252	C₂²	—	4	4	no	—
253	C₂²	—	4	16	no	—
254	C₂²	D₄×#92420	1	2	yes	—
255	C₂²	#16840803	1	2	yes	—
256	C₂²	#16840809	1	2	yes	—
257	C₂²	C₂×D₄×#6923	1	1	yes	characteristic
258	C₄	—	61	2	no	—
259	C₄	—	61	2	no	—
260	C₄	—	122	1	no	—
261	C₄	—	122	1	no	—
262	C₄	—	244	4	no	—
263	C₄	D₄×#92420	1	2	yes	—
264	C₄	C₂³×#92420	1	1	yes	characteristic
265	C₄	C₂×#14519864	1	1	yes	characteristic
266	C₂	—	4	8	no	—
267	C₂	#21461415	1	4	yes	direct product factor
268	C₂	C₂×D₄×#92420	1	1	yes	characteristic
269	C₂	C₂×#16840803	1	1	yes	characteristic
270	C₂	C₂×#16840809	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	D₄×#6923
commutator	C₄×C₆₁	C₂³×C₈
Frattini	C₂×C₄	C₂³×#6923
Fitting, prime radical	C₂²×D₈×C₆₁	C₄
socle	C₂³×C₆₁	C₄×D₄
cosocle	C₄×Dic₆₁	C₂⁴
hypercenter	C₂²×D₈	#6923
hypocenter	C₆₁	C₂×#1580
LCS	G ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×D₈

Complements

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

N	H ≅ G/N	type
1. C₂×D₄×#92420	C₂	semidirect
2. #21461415	C₂	direct (N × H)
3. C₂×#16840623	C₂	semidirect
4. C₂×D₄⋊#92420	C₂	semidirect
5. C₂×#16840603	C₂	semidirect
9. D₄×#92420	C₂²	semidirect
14. #16840623	C₂²	semidirect
15. D₄⋊#92420	C₂²	semidirect
16. #16840603	C₂²	semidirect
50. C₂×D₈×C₆₁	C₈	semidirect
56. C₂×D₄×C₆₁	C₂×C₈	semidirect
73. D₈×C₆₁	C₂×C₈	semidirect
74. C₂×C₈×C₆₁	C₂×C₈	semidirect
93. D₄×C₆₁	C₂²×C₈	semidirect
112. C₈×C₆₁	C₂²×C₈	semidirect
138. C₂×C₆₁	#1580	semidirect
159. C₆₁	C₂×#1580	semidirect
162. C₂×D₈	#92420	semidirect
192. C₂×D₄	C₂×#92420	semidirect
202. D₈	C₂×#92420	semidirect
212. C₂×C₈	C₂×#92420	semidirect
229. D₄	C₂²×#92420	semidirect
249. C₈	C₂²×#92420	semidirect
267. C₂	#21461415	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×D₄×#92420	2
#21461415	2
C₂×#16840623	2
C₂×D₄⋊#92420	2
C₂×#16840603	2
C₂×D₈×Dic₆₁	2
C₂×#1580	61

Subgroups — C2×#21461415 #30612293

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#21461415 #30612293