Subgroups — C₂×D₄⋊#11758185 #29589712

Order 15616.

470 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×C₄⋊#11758185	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₄×#11758185	C₂	1	1	yes	maximal, characteristic, complemented
3	D₄⋊#11758185	C₂	1	4	yes	maximal, direct product factor
4	C₂×C₈×D₄×C₆₁	C₂	1	1	yes	maximal, characteristic
5	C₂²×#11758185	—	2	1	no	—
6	C₂²×#11758185	—	2	1	no	—
7	C₄⋊#11758185	—	2	1	no	—
8	C₄⋊#11758185	C₂²	1	2	yes	complemented
9	C₄×#11758185	—	2	1	no	—
10	C₄×#11758185	C₂²	1	2	yes	complemented
11	C₂³×C₈×C₆₁	—	2	1	no	—
12	C₂×C₄×D₄×C₆₁	C₄	1	1	yes	characteristic
13	C₈×D₄×C₆₁	—	2	1	no	—
14	C₈×D₄×C₆₁	—	2	1	no	—
15	C₈×D₄×C₆₁	C₄	1	2	yes	—
16	C₈×D₄×C₆₁	C₂²	1	2	yes	—
17	C₂×C₄⋊C₈×C₆₁	C₄	1	1	yes	characteristic
18	C₂×C₂²⋊C₈×C₆₁	—	2	1	no	—
19	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
20	C₂×#11758185	—	2	4	no	—
21	C₂×#11758185	—	2	2	no	—
22	C₂×#11758185	—	2	2	no	—
23	C₂×#11758185	—	4	2	no	—
24	C₂²×D₄×C₆₁	C₈	1	1	yes	characteristic
25	C₂³×C₄×C₆₁	—	2	1	no	—
26	C₂²×C₈×C₆₁	—	2	1	no	—
27	C₂²×C₈×C₆₁	—	2	2	no	—
28	C₂²×C₈×C₆₁	—	2	2	no	—
29	C₂²×C₈×C₆₁	—	4	2	no	—
30	C₂²×C₈×C₆₁	—	4	2	no	—
31	C₂²×C₈×C₆₁	D₄	1	1	yes	characteristic
32	C₂²×C₈×C₆₁	D₄	1	1	yes	characteristic
33	C₄×D₄×C₆₁	—	2	1	no	—
34	C₄×D₄×C₆₁	—	2	1	no	—
35	C₄×D₄×C₆₁	C₈	1	2	yes	—
36	C₄×D₄×C₆₁	C₂×C₄	1	2	yes	—
37	C₂×C₄⋊C₄×C₆₁	C₈	1	1	yes	characteristic
38	C₂×C₂²⋊C₄×C₆₁	—	2	1	no	—
39	C₂×C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
40	C₄⋊C₈×C₆₁	—	2	1	no	—
41	C₄⋊C₈×C₆₁	C₂×C₄	1	2	yes	—
42	C₂²⋊C₈×C₆₁	—	2	2	no	—
43	C₂²⋊C₈×C₆₁	—	2	2	no	—
44	C₄×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
45	C₄×C₈×C₆₁	D₄	1	1	yes	characteristic
46	C₄×C₈×C₆₁	D₄	1	1	yes	characteristic
47	C₄×C₈×C₆₁	C₂³	1	1	yes	characteristic
48	C₂⁴×C₆₁	—	2	1	no	—
49	C₂×D₄×C₆₁	—	2	1	no	—
50	C₂×D₄×C₆₁	—	2	2	no	—
51	C₂×D₄×C₆₁	C₁₆	1	4	yes	complemented
52	C₂×D₄×C₆₁	C₂×C₈	1	2	yes	—
53	C₂²×C₄×C₆₁	—	2	1	no	—
54	C₂²×C₄×C₆₁	—	2	2	no	—
55	C₂²×C₄×C₆₁	—	2	2	no	—
56	C₂²×C₄×C₆₁	—	4	2	no	—
57	C₂²×C₄×C₆₁	—	4	2	no	—
58	C₂²×C₄×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
59	C₂²×C₄×C₆₁	C₂×C₈	1	1	yes	characteristic
60	C₂²×C₄×C₆₁	M₁₆	1	1	yes	characteristic
61	C₂×C₈×C₆₁	—	2	2	no	—
62	C₂×C₈×C₆₁	—	2	2	no	—
63	C₂×C₈×C₆₁	—	4	2	no	—
64	C₂×C₈×C₆₁	—	4	4	no	—
65	C₂×C₈×C₆₁	—	4	2	no	—
66	C₂×C₈×C₆₁	—	4	2	no	—
67	C₂×C₈×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
68	C₂×C₈×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
69	C₂×C₈×C₆₁	D₈	1	2	yes	—
70	C₂×C₈×C₆₁	SD₁₆	1	2	yes	—
71	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
72	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
73	C₄⋊C₄×C₆₁	—	2	1	no	—
74	C₄⋊C₄×C₆₁	C₂×C₈	1	2	yes	—
75	C₂²⋊C₄×C₆₁	—	2	2	no	—
76	C₂²⋊C₄×C₆₁	—	2	2	no	—
77	C₄²×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
78	C₄²×C₆₁	C₂×C₈	1	1	yes	characteristic
79	C₄²×C₆₁	M₁₆	1	1	yes	characteristic
80	C₄²×C₆₁	C₂²×C₄	1	1	yes	characteristic
81	#11758185	—	2	4	no	—
82	#11758185	—	4	2	no	—
83	C₂³×C₆₁	—	2	4	no	—
84	C₂³×C₆₁	—	2	2	no	—
85	C₂³×C₆₁	—	4	2	no	—
86	C₂³×C₆₁	—	4	2	no	—
87	C₂³×C₆₁	C₂²⋊C₈	1	1	yes	characteristic
88	D₄×C₆₁	—	2	4	no	—
89	D₄×C₆₁	—	2	2	no	—
90	D₄×C₆₁	C₂×C₁₆	1	4	yes	complemented
91	C₂×C₄×C₆₁	—	2	2	no	—
92	C₂×C₄×C₆₁	—	2	2	no	—
93	C₂×C₄×C₆₁	—	4	2	no	—
94	C₂×C₄×C₆₁	—	4	4	no	—
95	C₂×C₄×C₆₁	—	4	2	no	—
96	C₂×C₄×C₆₁	—	4	2	no	—
97	C₂×C₄×C₆₁	C₂²⋊C₈	1	1	yes	characteristic
98	C₂×C₄×C₆₁	C₂²⋊C₈	1	1	yes	characteristic
99	C₂×C₄×C₆₁	C₂²⋊C₈	1	1	yes	characteristic
100	C₂×C₄×C₆₁	#102	1	2	yes	—
101	C₂×C₄×C₆₁	#104	1	2	yes	—
102	C₂×C₄×C₆₁	C₂×C₁₆	1	2	yes	—
103	C₂×C₄×C₆₁	M₃₂	1	2	yes	—
104	C₂×C₄×C₆₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
105	C₂×C₄×C₆₁	C₂²×C₈	1	1	yes	characteristic
106	C₂×C₄×C₆₁	C₂×M₁₆	1	1	yes	characteristic
107	C₈×C₆₁	—	2	1	no	—
108	C₈×C₆₁	—	2	1	no	—
109	C₈×C₆₁	—	4	2	no	—
110	C₈×C₆₁	#102	1	2	yes	—
111	C₈×C₆₁	C₂×D₈	1	1	yes	characteristic
112	C₈×C₆₁	C₂×SD₁₆	1	1	yes	characteristic
113	C₂×D₄⋊C₁₆	—	61	1	no	maximal, Sylow
114	C₂²×C₆₁	—	2	4	no	—
115	C₂²×C₆₁	—	4	8	no	—
116	C₂²×C₆₁	—	4	2	no	—
117	C₂²×C₆₁	—	4	2	no	—
118	C₂²×C₆₁	D₄⋊C₈	1	2	yes	—
119	C₂²×C₆₁	C₂²⋊C₁₆	1	2	yes	—
120	C₂²×C₆₁	#350	1	2	yes	—
121	C₂²×C₆₁	C₂×C₂²⋊C₈	1	1	yes	characteristic
122	C₄×C₆₁	—	2	1	no	—
123	C₄×C₆₁	—	2	1	no	—
124	C₄×C₆₁	—	4	2	no	—
125	C₄×C₆₁	D₄⋊C₈	1	2	yes	—
126	C₄×C₆₁	C₂²⋊C₁₆	1	2	yes	—
127	C₄×C₆₁	C₂×#102	1	1	yes	characteristic
128	C₄×C₆₁	C₂×#104	1	1	yes	characteristic
129	C₄×C₆₁	C₂²×C₁₆	1	1	yes	characteristic
130	C₄×C₆₁	C₂×M₃₂	1	1	yes	characteristic
131	C₂×C₈×D₄	D₆₁	1	1	yes	characteristic
132	C₂×C₄⋊C₁₆	—	61	1	no	—
133	C₂×C₄×C₁₆	—	61	1	no	—
134	D₄⋊C₁₆	—	61	4	no	—
135	C₂×C₆₁	—	4	4	no	—
136	C₂×C₆₁	D₄⋊C₁₆	1	4	yes	complemented
137	C₂×C₆₁	C₂×D₄⋊C₈	1	1	yes	characteristic
138	C₂×C₆₁	C₂×C₂²⋊C₁₆	1	1	yes	characteristic
139	C₂×C₆₁	C₂×#350	1	1	yes	characteristic
140	C₂³×C₈	—	2	1	no	—
141	C₂×C₄×D₄	Dic₆₁	1	1	yes	characteristic
142	C₂²×C₁₆	—	122	1	no	—
143	C₂²×C₁₆	—	122	1	no	—
144	C₈×D₄	—	2	1	no	—
145	C₈×D₄	—	2	1	no	—
146	C₈×D₄	Dic₆₁	1	2	yes	—
147	C₈×D₄	C₂×D₆₁	1	2	yes	—
148	C₂×C₄⋊C₈	Dic₆₁	1	1	yes	characteristic
149	C₂×C₂²⋊C₈	—	2	1	no	—
150	C₂×C₄×C₈	C₂×D₆₁	1	1	yes	characteristic
151	C₄⋊C₁₆	—	61	2	no	—
152	C₄⋊C₁₆	—	122	1	no	—
153	C₄×C₁₆	—	61	2	no	—
154	C₄×C₁₆	—	122	1	no	—
155	C₆₁	C₂×D₄⋊C₁₆	1	1	yes	characteristic, complemented, Sylow
156	C₂²×D₄	#92418	1	1	yes	characteristic
157	C₂³×C₄	—	2	1	no	—
158	C₂²×C₈	—	2	1	no	—
159	C₂²×C₈	—	2	2	no	—
160	C₂²×C₈	—	2	2	no	—
161	C₂²×C₈	—	4	2	no	—
162	C₂²×C₈	—	4	2	no	—
163	C₂²×C₈	C₂²⋊D₆₁	1	1	yes	characteristic
164	C₂²×C₈	C₂²⋊D₆₁	1	1	yes	characteristic
165	C₄×D₄	—	2	1	no	—
166	C₄×D₄	—	2	1	no	—
167	C₄×D₄	#92418	1	2	yes	—
168	C₄×D₄	C₂×Dic₆₁	1	2	yes	—
169	C₂×C₄⋊C₄	#92418	1	1	yes	characteristic
170	C₂×C₂²⋊C₄	—	2	1	no	—
171	C₂×C₄²	C₂×Dic₆₁	1	1	yes	characteristic
172	C₂×C₁₆	—	122	4	no	—
173	C₂×C₁₆	—	122	2	no	—
174	C₂×C₁₆	—	122	2	no	—
175	C₂×C₁₆	—	244	2	no	—
176	C₄⋊C₈	—	2	1	no	—
177	C₄⋊C₈	C₂×Dic₆₁	1	2	yes	—
178	C₂²⋊C₈	—	2	2	no	—
179	C₂²⋊C₈	—	2	2	no	—
180	C₄×C₈	C₂×Dic₆₁	1	1	yes	characteristic
181	C₄×C₈	C₂²⋊D₆₁	1	1	yes	characteristic
182	C₄×C₈	C₂²⋊D₆₁	1	1	yes	characteristic
183	C₄×C₈	C₂²×D₆₁	1	1	yes	characteristic
184	C₂⁴	—	2	1	no	—
185	C₂×D₄	—	2	1	no	—
186	C₂×D₄	—	2	2	no	—
187	C₂×D₄	#11758185	1	4	yes	complemented
188	C₂×D₄	C₂×#92418	1	2	yes	—
189	C₂²×C₄	—	2	1	no	—
190	C₂²×C₄	—	2	2	no	—
191	C₂²×C₄	—	2	2	no	—
192	C₂²×C₄	—	4	2	no	—
193	C₂²×C₄	—	4	2	no	—
194	C₂²×C₄	C₂×#92418	1	1	yes	characteristic
195	C₂²×C₄	#11758194	1	1	yes	characteristic
196	C₂²×C₄	C₂²⋊Dic₆₁	1	1	yes	characteristic
197	C₂×C₈	—	2	2	no	—
198	C₂×C₈	—	2	2	no	—
199	C₂×C₈	—	4	2	no	—
200	C₂×C₈	—	4	4	no	—
201	C₂×C₈	—	4	2	no	—
202	C₂×C₈	—	4	2	no	—
203	C₂×C₈	#11758199	1	2	yes	—
204	C₂×C₈	#11758200	1	2	yes	—
205	C₂×C₈	C₂²⋊Dic₆₁	1	1	yes	characteristic
206	C₂×C₈	C₂²⋊Dic₆₁	1	1	yes	characteristic
207	C₂×C₈	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
208	C₂×C₈	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
209	C₄⋊C₄	—	2	1	no	—
210	C₄⋊C₄	C₂×#92418	1	2	yes	—
211	C₂²⋊C₄	—	2	2	no	—
212	C₂²⋊C₄	—	2	2	no	—
213	C₄²	C₂×#92418	1	1	yes	characteristic
214	C₄²	#11758194	1	1	yes	characteristic
215	C₄²	C₂²⋊Dic₆₁	1	1	yes	characteristic
216	C₄²	C₂²×Dic₆₁	1	1	yes	characteristic
217	C₁₆	—	122	4	no	—
218	C₁₆	—	244	2	no	—
219	C₂³	—	2	2	no	—
220	C₂³	—	2	4	no	—
221	C₂³	—	4	2	no	—
222	C₂³	—	4	2	no	—
223	C₂³	C₂²⋊#92418	1	1	yes	characteristic
224	D₄	—	2	4	no	—
225	D₄	—	2	2	no	—
226	D₄	C₂×#11758185	1	4	yes	complemented
227	C₂×C₄	—	2	2	no	—
228	C₂×C₄	—	2	2	no	—
229	C₂×C₄	—	4	2	no	—
230	C₂×C₄	—	4	4	no	—
231	C₂×C₄	—	4	2	no	—
232	C₂×C₄	—	4	2	no	—
233	C₂×C₄	C₂²⋊#92418	1	1	yes	characteristic
234	C₂×C₄	C₂²⋊#92418	1	1	yes	characteristic
235	C₂×C₄	C₂²⋊#92418	1	1	yes	characteristic
236	C₂×C₄	#14519698	1	2	yes	—
237	C₂×C₄	#14519704	1	2	yes	—
238	C₂×C₄	C₂×#11758185	1	2	yes	—
239	C₂×C₄	#14519717	1	2	yes	—
240	C₂×C₄	C₂×C₂²⋊Dic₆₁	1	1	yes	characteristic
241	C₂×C₄	C₂²×#92418	1	1	yes	characteristic
242	C₂×C₄	C₂×#11758194	1	1	yes	characteristic
243	C₈	—	2	1	no	—
244	C₈	—	2	1	no	—
245	C₈	—	4	2	no	—
246	C₈	#14519698	1	2	yes	—
247	C₈	C₂×#11758199	1	1	yes	characteristic
248	C₈	C₂×#11758200	1	1	yes	characteristic
249	C₂²	—	2	4	no	—
250	C₂²	—	4	2	no	—
251	C₂²	—	4	2	no	—
252	C₂²	—	4	8	no	—
253	C₂²	D₄⋊#92418	1	2	yes	—
254	C₂²	C₂²⋊#11758185	1	2	yes	—
255	C₂²	#16839546	1	2	yes	—
256	C₂²	C₂×C₂²⋊#92418	1	1	yes	characteristic
257	C₄	—	2	1	no	—
258	C₄	—	2	1	no	—
259	C₄	—	4	2	no	—
260	C₄	D₄⋊#92418	1	2	yes	—
261	C₄	C₂²⋊#11758185	1	2	yes	—
262	C₄	C₂×#14519698	1	1	yes	characteristic
263	C₄	C₂×#14519704	1	1	yes	characteristic
264	C₄	C₂²×#11758185	1	1	yes	characteristic
265	C₄	C₂×#14519717	1	1	yes	characteristic
266	C₂	—	4	4	no	—
267	C₂	D₄⋊#11758185	1	4	yes	direct product factor
268	C₂	C₂×D₄⋊#92418	1	1	yes	characteristic
269	C₂	C₂×C₂²⋊#11758185	1	1	yes	characteristic
270	C₂	C₂×#16839546	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₈	C₂²⋊D₆₁
commutator	C₄×C₆₁	C₂²×C₁₆
Frattini	C₄×C₈	C₂²×D₆₁
Fitting	C₂×C₈×D₄×C₆₁	C₂
socle	C₂³×C₆₁	C₂²⋊C₈
prime radical	C₂²×D₄×C₆₁	C₈
cosocle	C₄×C₈×C₆₁	C₂³
hypercenter	C₂×C₈×D₄	D₆₁
hypocenter	C₆₁	C₂×D₄⋊C₁₆
LCS	G ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂²×C₈ ◁ C₂×C₄×C₈ ◁ C₂×C₈×D₄

Complements

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

N	H ≅ G/N	type
1. C₂×C₄⋊#11758185	C₂	semidirect
2. C₂×C₄×#11758185	C₂	semidirect
3. D₄⋊#11758185	C₂	direct (N × H)
8. C₄⋊#11758185	C₂²	semidirect
10. C₄×#11758185	C₂²	semidirect
51. C₂×D₄×C₆₁	C₁₆	semidirect
90. D₄×C₆₁	C₂×C₁₆	semidirect
136. C₂×C₆₁	D₄⋊C₁₆	semidirect
155. C₆₁	C₂×D₄⋊C₁₆	semidirect
187. C₂×D₄	#11758185	semidirect
226. D₄	C₂×#11758185	semidirect
267. C₂	D₄⋊#11758185	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⋊#11758185	2
C₂×C₄×#11758185	2
D₄⋊#11758185	2
C₂×C₈×D₄×C₆₁	2
C₂×D₄⋊C₁₆	61

Subgroups — C2×D4⋊#11758185 #29589712

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×D₄⋊#11758185 #29589712