Subgroups — C₂×C₈×Dic₂₄₄ #29649385

Order 15616.

734 conjugacy classes of subgroups.

The subgroup lattice has 258 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₈×Dic₁₂₂	C₂	1	2	yes	maximal
2	C₂×C₄×Dic₂₄₄	C₂	1	1	yes	maximal, characteristic
3	C₈×Dic₂₄₄	C₂	1	8	yes	maximal, direct product factor
4	C₂×#16839515	C₂	1	1	yes	maximal, characteristic
5	C₂×#16839494	C₂	1	2	yes	maximal
6	C₂×C₈²×C₆₁	C₂	1	1	yes	maximal, characteristic
7	C₂²×Dic₂₄₄	C₄	1	1	yes	characteristic
8	C₂×C₄×Dic₁₂₂	C₂²	1	2	yes	—
9	C₈×Dic₁₂₂	—	2	4	no	—
10	C₈×Dic₁₂₂	C₂²	1	8	yes	—
11	C₄×Dic₂₄₄	C₄	1	4	yes	—
12	C₄×Dic₂₄₄	C₂²	1	4	yes	—
13	C₂×#14519711	C₄	1	1	yes	characteristic
14	C₂×C₄⋊#92418	C₂²	1	2	yes	—
15	C₂×#14519706	—	2	2	no	—
16	C₂×#14519702	C₄	1	2	yes	—
17	C₂×C₈×Dic₆₁	—	2	2	no	—
18	#16839515	C₂²	1	4	yes	—
19	#16839494	C₂²	1	8	yes	—
20	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
21	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
22	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
23	C₈²×C₆₁	C₂²	1	4	yes	—
24	C₂²×Dic₁₂₂	C₂×C₄	1	2	yes	—
25	C₂×Dic₂₄₄	C₈	1	8	yes	direct product factor
26	C₂×Dic₂₄₄	C₂×C₄	1	4	yes	—
27	C₂²×#92418	—	2	2	no	—
28	C₄×Dic₁₂₂	—	2	2	no	—
29	C₄×Dic₁₂₂	—	2	2	no	—
30	C₄×Dic₁₂₂	C₂×C₄	1	4	yes	—
31	C₄×Dic₁₂₂	C₂³	1	4	yes	—
32	C₂×C₄⋊Dic₆₁	C₂×C₄	1	2	yes	—
33	C₂×#11758196	—	2	2	no	—
34	C₂×C₄×Dic₆₁	—	2	2	no	—
35	#14519711	C₂×C₄	1	4	yes	—
36	C₄⋊#92418	—	2	2	no	—
37	C₄⋊#92418	C₂³	1	4	yes	—
38	#14519706	—	2	8	no	—
39	#14519702	C₂×C₄	1	8	yes	—
40	C₈×Dic₆₁	—	2	8	no	—
41	C₂²×C₈×C₆₁	—	2	1	no	—
42	C₂²×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
43	C₂²×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
44	C₂²×C₈×C₆₁	D₄	1	1	yes	characteristic
45	C₂²×C₈×C₆₁	D₄	1	1	yes	characteristic
46	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
47	C₄×C₈×C₆₁	—	2	1	no	—
48	C₄×C₈×C₆₁	C₂×C₄	1	2	yes	—
49	C₄×C₈×C₆₁	D₄	1	2	yes	—
50	C₄×C₈×C₆₁	C₂³	1	2	yes	—
51	C₄×C₈×C₆₁	C₂³	1	2	yes	—
52	C₄×C₈×C₆₁	C₂³	1	2	yes	—
53	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
54	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
55	C₂²×C₄×C₆₁	#41	1	1	yes	characteristic
56	C₂²×Dic₆₁	—	2	2	no	—
57	C₂²×Dic₆₁	—	2	2	no	—
58	C₂×Dic₁₂₂	—	2	2	no	—
59	C₂×Dic₁₂₂	—	2	4	no	—
60	C₂×Dic₁₂₂	C₂×C₈	1	8	yes	—
61	C₂×Dic₁₂₂	C₂²×C₄	1	4	yes	—
62	C₂×C₈×C₆₁	—	2	4	no	—
63	C₂×C₈×C₆₁	—	2	2	no	—
64	C₂×C₈×C₆₁	—	2	2	no	—
65	C₂×C₈×C₆₁	—	2	2	no	—
66	C₂×C₈×C₆₁	C₂×C₈	1	4	yes	—
67	C₂×C₈×C₆₁	Q₁₆	1	4	yes	complemented
68	C₂×C₈×C₆₁	C₂²×C₄	1	2	yes	—
69	C₂×C₈×C₆₁	C₂²×C₄	1	2	yes	—
70	C₂×C₈×C₆₁	C₂×D₄	1	2	yes	—
71	C₂×C₈×C₆₁	C₂×D₄	1	2	yes	—
72	C₄²×C₆₁	C₂²×C₄	1	1	yes	characteristic
73	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
74	C₄²×C₆₁	#41	1	1	yes	characteristic
75	C₄²×C₆₁	C₂⁴	1	1	yes	characteristic
76	C₄⋊Dic₆₁	—	2	2	no	—
77	C₄⋊Dic₆₁	C₂²×C₄	1	4	yes	—
78	#11758196	—	2	4	no	—
79	#11758196	—	2	4	no	—
80	C₄×Dic₆₁	—	2	4	no	—
81	C₄×Dic₆₁	—	2	4	no	—
82	C₂×#92418	—	2	4	no	—
83	C₂×#92418	—	4	4	no	—
84	Dic₂₄₄	C₂×C₈	1	16	yes	direct product factor
85	C₂³×C₆₁	C₄×D₄	1	1	yes	characteristic
86	C₂×C₄×C₆₁	—	2	2	no	—
87	C₂×C₄×C₆₁	C₄×D₄	1	1	yes	characteristic
88	C₂×C₄×C₆₁	C₄×D₄	1	1	yes	characteristic
89	C₂×C₄×C₆₁	C₄×D₄	1	1	yes	characteristic
90	C₂×C₄×C₆₁	C₂²×C₈	1	2	yes	—
91	C₂×C₄×C₆₁	#131	1	2	yes	—
92	C₂×C₄×C₆₁	C₂×Q₁₆	1	2	yes	—
93	C₂×C₄×C₆₁	#135	1	2	yes	—
94	C₂×C₄×C₆₁	C₂³×C₄	1	1	yes	characteristic
95	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
96	C₂×C₄×C₆₁	C₂×#41	1	1	yes	characteristic
97	C₂×Dic₆₁	—	2	4	no	—
98	C₂×Dic₆₁	—	2	8	no	—
99	C₂×Dic₆₁	—	2	4	no	—
100	C₂×Dic₆₁	—	4	4	no	—
101	Dic₁₂₂	—	2	4	no	—
102	Dic₁₂₂	—	2	8	no	—
103	Dic₁₂₂	C₂²×C₈	1	8	yes	—
104	C₈×C₆₁	—	2	4	no	—
105	C₈×C₆₁	—	2	2	no	—
106	C₈×C₆₁	—	2	2	no	—
107	C₈×C₆₁	C₂²×C₈	1	4	yes	—
108	C₈×C₆₁	C₂×Q₁₆	1	4	yes	complemented
109	#92418	—	4	4	no	—
110	C₂×C₈×Q₁₆	—	61	1	no	maximal, Sylow
111	C₂²×C₆₁	C₈×D₄	1	2	yes	—
112	C₂²×C₆₁	C₄×Q₁₆	1	2	yes	—
113	C₂²×C₆₁	#443	1	2	yes	—
114	C₂²×C₆₁	C₂×C₄×D₄	1	1	yes	characteristic
115	C₄×C₆₁	—	2	1	no	—
116	C₄×C₆₁	—	2	1	no	—
117	C₄×C₆₁	C₈×D₄	1	2	yes	—
118	C₄×C₆₁	C₄×Q₁₆	1	2	yes	—
119	C₄×C₆₁	C₂³×C₈	1	1	yes	characteristic
120	C₄×C₆₁	C₂×#131	1	1	yes	characteristic
121	C₄×C₆₁	C₂²×Q₁₆	1	1	yes	characteristic
122	C₄×C₆₁	C₂×#135	1	1	yes	characteristic
123	Dic₆₁	—	2	8	no	—
124	Dic₆₁	—	4	4	no	—
125	C₂×C₈×Q₈	—	61	2	no	—
126	C₂×C₄×Q₁₆	—	61	1	no	—
127	C₈×Q₁₆	—	61	8	no	—
128	C₂×#335	—	61	1	no	—
129	C₂×Q₈⋊C₈	—	61	2	no	—
130	C₂×C₈²	D₆₁	1	1	yes	characteristic
131	C₂×C₆₁	C₈×Q₁₆	1	4	yes	complemented
132	C₂×C₆₁	C₂×C₈×D₄	1	1	yes	characteristic
133	C₂×C₆₁	C₂×C₄×Q₁₆	1	1	yes	characteristic
134	C₂×C₆₁	C₂×#443	1	1	yes	characteristic
135	C₂²×Q₁₆	—	61	1	no	—
136	C₂×C₄×Q₈	—	61	2	no	—
137	C₈×Q₈	—	61	8	no	—
138	C₈×Q₈	—	122	4	no	—
139	C₄×Q₁₆	—	61	4	no	—
140	C₄×Q₁₆	—	61	4	no	—
141	C₂×#107	—	61	1	no	—
142	C₂×C₄⋊C₈	—	61	2	no	—
143	C₂×C₄⋊C₈	—	122	2	no	—
144	C₂×#103	—	61	2	no	—
145	C₂×C₄×C₈	—	122	2	no	—
146	C₂×C₄×C₈	C₂×D₆₁	1	1	yes	characteristic
147	C₂×C₄×C₈	C₂×D₆₁	1	1	yes	characteristic
148	C₂×C₄×C₈	C₂×D₆₁	1	1	yes	characteristic
149	#335	—	61	4	no	—
150	Q₈⋊C₈	—	61	8	no	—
151	C₈²	C₂×D₆₁	1	4	yes	—
152	C₆₁	C₂×C₈×Q₁₆	1	1	yes	characteristic, complemented, Sylow
153	C₂²×Q₈	—	61	2	no	—
154	C₂×Q₁₆	—	61	8	no	—
155	C₂×Q₁₆	—	61	4	no	—
156	C₂²×C₈	—	2	1	no	—
157	C₂²×C₈	—	122	2	no	—
158	C₂²×C₈	C₄×D₆₁	1	1	yes	characteristic
159	C₂²×C₈	C₄×D₆₁	1	1	yes	characteristic
160	C₂²×C₈	D₂₄₄	1	1	yes	characteristic
161	C₂²×C₈	D₂₄₄	1	1	yes	characteristic
162	C₄×Q₈	—	61	4	no	—
163	C₄×Q₈	—	61	4	no	—
164	C₄×Q₈	—	122	2	no	—
165	C₄×Q₈	—	122	2	no	—
166	C₂×C₄⋊C₄	—	61	2	no	—
167	C₂×C₄⋊C₄	—	122	2	no	—
168	C₂×C₄²	—	122	2	no	—
169	C₂×C₄²	C₂²×D₆₁	1	1	yes	characteristic
170	#107	—	61	4	no	—
171	C₄⋊C₈	—	61	4	no	—
172	C₄⋊C₈	—	122	8	no	—
173	C₄⋊C₈	—	122	2	no	—
174	#103	—	61	8	no	—
175	C₄×C₈	—	2	1	no	—
176	C₄×C₈	—	122	8	no	—
177	C₄×C₈	C₄×D₆₁	1	2	yes	—
178	C₄×C₈	D₂₄₄	1	2	yes	—
179	C₄×C₈	C₂²×D₆₁	1	2	yes	—
180	C₄×C₈	C₂²×D₆₁	1	2	yes	—
181	C₄×C₈	C₂²×D₆₁	1	2	yes	—
182	C₂×Q₈	—	61	4	no	—
183	C₂×Q₈	—	61	8	no	—
184	C₂×Q₈	—	122	2	no	—
185	C₂×Q₈	—	122	4	no	—
186	C₂²×C₄	—	122	2	no	—
187	C₂²×C₄	—	122	2	no	—
188	C₂²×C₄	C₂×C₄×D₆₁	1	1	yes	characteristic
189	C₂²×C₄	C₂×D₂₄₄	1	1	yes	characteristic
190	C₂²×C₄	#11758222	1	1	yes	characteristic
191	Q₁₆	—	61	16	no	—
192	C₂×C₈	—	2	4	no	—
193	C₂×C₈	—	2	2	no	—
194	C₂×C₈	—	2	2	no	—
195	C₂×C₈	—	2	2	no	—
196	C₂×C₈	—	122	4	no	—
197	C₂×C₈	—	244	4	no	—
198	C₂×C₈	C₈×D₆₁	1	4	yes	—
199	C₂×C₈	Dic₂₄₄	1	4	yes	direct product factor
200	C₂×C₈	C₂×C₄×D₆₁	1	2	yes	—
201	C₂×C₈	C₂×C₄×D₆₁	1	2	yes	—
202	C₂×C₈	C₂×D₂₄₄	1	2	yes	—
203	C₂×C₈	C₂×D₂₄₄	1	2	yes	—
204	C₄⋊C₄	—	61	4	no	—
205	C₄⋊C₄	—	122	4	no	—
206	C₄⋊C₄	—	122	2	no	—
207	C₄⋊C₄	—	122	4	no	—
208	C₄²	—	122	4	no	—
209	C₄²	—	122	4	no	—
210	C₄²	C₂×C₄×D₆₁	1	1	yes	characteristic
211	C₄²	C₂×D₂₄₄	1	1	yes	characteristic
212	C₄²	#11758222	1	1	yes	characteristic
213	C₄²	C₂³×D₆₁	1	1	yes	characteristic
214	C₂³	C₄×D₂₄₄	1	1	yes	characteristic
215	Q₈	—	61	8	no	—
216	Q₈	—	122	4	no	—
217	Q₈	—	122	8	no	—
218	C₂×C₄	—	2	2	no	—
219	C₂×C₄	—	122	4	no	—
220	C₂×C₄	—	122	8	no	—
221	C₂×C₄	—	122	4	no	—
222	C₂×C₄	—	244	4	no	—
223	C₂×C₄	C₄×D₂₄₄	1	1	yes	characteristic
224	C₂×C₄	C₄×D₂₄₄	1	1	yes	characteristic
225	C₂×C₄	C₄×D₂₄₄	1	1	yes	characteristic
226	C₂×C₄	C₂×C₈×D₆₁	1	2	yes	—
227	C₂×C₄	#14519783	1	2	yes	—
228	C₂×C₄	C₂×Dic₂₄₄	1	2	yes	—
229	C₂×C₄	#14519799	1	2	yes	—
230	C₂×C₄	C₂²×C₄×D₆₁	1	1	yes	characteristic
231	C₂×C₄	C₂²×D₂₄₄	1	1	yes	characteristic
232	C₂×C₄	C₂×#11758222	1	1	yes	characteristic
233	C₈	—	2	4	no	—
234	C₈	—	2	2	no	—
235	C₈	—	2	2	no	—
236	C₈	—	244	4	no	—
237	C₈	C₂×C₈×D₆₁	1	4	yes	—
238	C₈	C₂×Dic₂₄₄	1	4	yes	direct product factor
239	C₂²	C₈×D₂₄₄	1	2	yes	—
240	C₂²	C₄×Dic₂₄₄	1	2	yes	—
241	C₂²	#16839880	1	2	yes	—
242	C₂²	C₂×C₄×D₂₄₄	1	1	yes	characteristic
243	C₄	—	2	1	no	—
244	C₄	—	2	1	no	—
245	C₄	—	122	8	no	—
246	C₄	—	244	4	no	—
247	C₄	C₈×D₂₄₄	1	2	yes	—
248	C₄	C₄×Dic₂₄₄	1	2	yes	—
249	C₄	C₂²×C₈×D₆₁	1	1	yes	characteristic
250	C₄	C₂×#14519783	1	1	yes	characteristic
251	C₄	C₂²×Dic₂₄₄	1	1	yes	characteristic
252	C₄	C₂×#14519799	1	1	yes	characteristic
253	C₂	C₈×Dic₂₄₄	1	4	yes	direct product factor
254	C₂	C₂×C₈×D₂₄₄	1	1	yes	characteristic
255	C₂	C₂×C₄×Dic₂₄₄	1	1	yes	characteristic
256	C₂	C₂×#16839880	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₈	D₂₄₄
commutator	C₄×C₆₁	C₂³×C₈
Frattini	C₄²	C₂³×D₆₁
Fitting	C₂×C₈²×C₆₁	C₂
socle, prime radical	C₂³×C₆₁	C₄×D₄
cosocle	C₄²×C₆₁	C₂⁴
hypercenter	C₂×C₈²	D₆₁
hypocenter	C₆₁	C₂×C₈×Q₁₆
LCS	G ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂²×C₈ ◁ C₂×C₄×C₈ ◁ C₂×C₈²

Complements

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

N	H ≅ G/N	type
3. C₈×Dic₂₄₄	C₂	direct (N × H)
25. C₂×Dic₂₄₄	C₈	direct (N × H)
67. C₂×C₈×C₆₁	Q₁₆	semidirect
84. Dic₂₄₄	C₂×C₈	direct (N × H)
108. C₈×C₆₁	C₂×Q₁₆	semidirect
131. C₂×C₆₁	C₈×Q₁₆	semidirect
152. C₆₁	C₂×C₈×Q₁₆	semidirect
199. C₂×C₈	Dic₂₄₄	direct (N × H)
238. C₈	C₂×Dic₂₄₄	direct (N × H)
253. C₂	C₈×Dic₂₄₄	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×C₈×Dic₁₂₂	2
C₂×C₄×Dic₂₄₄	2
C₈×Dic₂₄₄	2
C₂×#16839515	2
C₂×#16839494	2
C₂×C₈²×C₆₁	2
C₂×C₈×Q₁₆	61

Subgroups — C2×C8×Dic244 #29649385

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₈×Dic₂₄₄ #29649385