Subgroups — C₈×#6821 #14418895

Order 1920.

912 conjugacy classes of subgroups.

The subgroup lattice has 304 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#6821	C₂	1	1	yes	maximal, characteristic
2	C₃×C₅×C₈×Q₈	C₂	1	1	yes	maximal, characteristic, complemented
3	Q₈×C₃⋊#168	C₂	1	1	yes	maximal, characteristic, complemented
4	#11750935	C₂	1	3	yes	maximal, complemented
5	#11750934	C₂	1	3	yes	maximal, complemented
6	C₈×D₆₀	C₂	1	3	yes	maximal, complemented
7	C₄×C₈×D₁₅	C₂	1	3	yes	maximal, complemented
8	C₈×#713	—	3	1	no	maximal, Hall
9	C₂×#6821	C₄	1	1	yes	characteristic
10	C₃×C₄×C₅×Q₈	C₂²	1	1	yes	characteristic
11	Q₈×Dic₁₅	C₄	1	1	yes	characteristic
12	C₂×C₈×D₁₅	C₂²	1	3	yes	—
13	C₂×C₈×D₁₅	C₂²	1	3	yes	—
14	#91782	C₄	1	3	yes	—
15	#91781	C₄	1	3	yes	—
16	C₄×D₆₀	C₂²	1	3	yes	—
17	C₄²×D₁₅	C₂²	1	3	yes	—
18	C₃×C₅×C₄⋊C₈	C₂²	1	3	yes	—
19	C₃×C₄×C₅×C₈	C₂²	1	3	yes	—
20	#91104	C₂²	1	6	yes	—
21	C₈×Dic₁₅	C₂²	1	3	yes	—
22	C₈×Dic₁₅	C₂²	1	1	yes	characteristic
23	C₄⋊(C₃⋊#168)	C₂²	1	3	yes	—
24	C₄×C₃⋊#168	C₂²	1	3	yes	—
25	C₈×#239	—	5	1	no	maximal, Hall
26	C₄×#713	—	3	1	no	—
27	C₅×C₈×Q₈	D₃	1	1	yes	characteristic, complemented
28	Q₈×#168	—	3	1	no	—
29	#65123	—	3	3	no	—
30	#65122	—	3	3	no	—
31	C₈×D₂₀	—	3	3	no	—
32	C₄×C₈×D₅	—	3	3	no	—
33	C₂×C₃×C₅×Q₈	C₂×C₄	1	1	yes	characteristic
34	#6821	C₈	1	8	yes	direct product factor
35	C₂×D₆₀	C₂×C₄	1	3	yes	—
36	C₂×C₄×D₁₅	C₂×C₄	1	3	yes	—
37	C₂×C₄×D₁₅	C₂×C₄	1	3	yes	—
38	C₂×C₄×D₁₅	C₂³	1	3	yes	—
39	C₂×C₃×C₅×C₈	C₂³	1	3	yes	—
40	C₂×C₃×C₅×C₈	C₂³	1	1	yes	characteristic
41	C₃×C₅×C₄⋊C₄	C₂×C₄	1	3	yes	—
42	C₃×C₄²×C₅	C₂³	1	3	yes	—
43	#6714	C₂×C₄	1	6	yes	—
44	C₄⋊Dic₁₅	C₂×C₄	1	3	yes	—
45	C₄×Dic₁₅	C₂×C₄	1	3	yes	—
46	C₄×Dic₁₅	C₂×C₄	1	3	yes	—
47	C₄×Dic₁₅	C₂³	1	1	yes	characteristic
48	C₂×C₃⋊#168	C₂³	1	3	yes	—
49	C₂×C₃⋊#168	C₂³	1	1	yes	characteristic
50	C₈×D₁₅	—	2	6	no	—
51	C₈×D₁₅	—	2	6	no	—
52	C₄×#239	—	5	1	no	—
53	C₃×C₈×Q₈	D₅	1	1	yes	characteristic, complemented
54	Q₈×C₃⋊C₈	—	5	1	no	—
55	#4834	—	5	3	no	—
56	#4833	—	5	3	no	—
57	C₈×D₁₂	—	5	3	no	—
58	C₄×D₃×C₈	—	5	3	no	—
59	C₂×#713	—	3	1	no	—
60	C₄×C₅×Q₈	C₂×D₃	1	1	yes	characteristic
61	Q₈×Dic₅	—	3	1	no	—
62	C₂×C₈×D₅	—	3	3	no	—
63	C₂×C₈×D₅	—	3	3	no	—
64	#4039	—	3	3	no	—
65	#4038	—	3	3	no	—
66	C₄×D₂₀	—	3	3	no	—
67	C₄²×D₅	—	3	3	no	—
68	C₅×C₄⋊C₈	C₂×D₃	1	3	yes	—
69	C₄×C₅×C₈	C₂×D₃	1	3	yes	—
70	#3952	—	3	6	no	—
71	C₈×Dic₅	—	3	3	no	—
72	C₈×Dic₅	—	3	1	no	—
73	C₄⋊#168	—	3	3	no	—
74	C₄×#168	—	3	3	no	—
75	C₈×#41	—	15	1	no	Sylow
76	C₂²×D₁₅	C₂²×C₄	1	3	yes	—
77	C₃×C₅×Q₈	C₂×C₈	1	4	yes	complemented
78	C₂×C₃×C₄×C₅	C₂²×C₄	1	3	yes	—
79	C₂×C₃×C₄×C₅	C₂²×C₄	1	3	yes	—
80	C₂×C₃×C₄×C₅	C₂⁴	1	1	yes	characteristic
81	C₂×Dic₁₅	C₂²×C₄	1	3	yes	—
82	C₂×Dic₁₅	C₂²×C₄	1	1	yes	characteristic
83	C₂×Dic₁₅	C₂²×C₄	1	1	yes	characteristic
84	D₆₀	C₂×C₈	1	12	yes	complemented
85	C₄×D₁₅	—	2	3	no	—
86	C₄×D₁₅	—	2	3	no	—
87	C₄×D₁₅	—	2	6	no	—
88	C₄×D₁₅	C₂×C₈	1	12	yes	complemented
89	C₃×C₅×C₈	—	2	3	no	—
90	C₃×C₅×C₈	#41	1	2	yes	complemented
91	C₃⋊#168	—	2	3	no	—
92	C₃⋊#168	#41	1	2	yes	complemented
93	C₂×#239	—	5	1	no	—
94	C₃×C₄×Q₈	C₂×D₅	1	1	yes	characteristic
95	Q₈×Dic₃	—	5	1	no	—
96	C₂×D₃×C₈	—	5	3	no	—
97	C₂×D₃×C₈	—	5	3	no	—
98	#893	—	5	3	no	—
99	#892	—	5	3	no	—
100	C₄×D₁₂	—	5	3	no	—
101	C₄²×D₃	—	5	3	no	—
102	C₃×C₄⋊C₈	C₂×D₅	1	3	yes	—
103	C₃×C₄×C₈	C₂×D₅	1	3	yes	—
104	(C₂×D₃)⋊C₈	—	5	6	no	—
105	C₈×Dic₃	—	5	1	no	—
106	C₈×Dic₃	—	5	3	no	—
107	C₄⋊(C₃⋊C₈)	—	5	3	no	—
108	C₄×C₃⋊C₈	—	5	3	no	—
109	C₂×C₅×Q₈	C₄×D₃	1	1	yes	characteristic
110	#713	—	3	8	no	—
111	C₂×D₂₀	—	3	3	no	—
112	C₂×C₄×D₅	—	3	3	no	—
113	C₂×C₄×D₅	—	3	3	no	—
114	C₂×C₄×D₅	—	3	3	no	—
115	C₂×C₅×C₈	C₂²×D₃	1	3	yes	—
116	C₂×C₅×C₈	C₂²×D₃	1	1	yes	characteristic
117	C₅×C₄⋊C₄	C₄×D₃	1	3	yes	—
118	C₄²×C₅	C₂²×D₃	1	3	yes	—
119	#685	—	3	6	no	—
120	C₄⋊Dic₅	—	3	3	no	—
121	C₄×Dic₅	—	3	3	no	—
122	C₄×Dic₅	—	3	3	no	—
123	C₄×Dic₅	—	3	1	no	—
124	C₂×#168	—	3	1	no	—
125	C₂×#168	—	3	3	no	—
126	C₈×D₅	—	6	6	no	—
127	C₈×D₅	—	6	6	no	—
128	C₄×#41	—	15	1	no	—
129	C₈×Q₈	—	15	1	no	—
130	C₈×Q₈	D₁₅	1	1	yes	characteristic, complemented
131	C₈×D₄	—	15	3	no	—
132	C₈×D₄	—	15	3	no	—
133	#431	—	15	3	no	—
134	C₂×C₄×C₈	—	15	3	no	—
135	C₂²×C₃×C₅	C₂³×C₄	1	1	yes	characteristic
136	C₂×D₁₅	—	2	3	no	—
137	C₂×D₁₅	—	2	3	no	—
138	C₂×D₁₅	C₂²×C₈	1	6	yes	—
139	C₃×C₄×C₅	—	2	3	no	—
140	C₃×C₄×C₅	C₂²×C₈	1	6	yes	—
141	C₃×C₄×C₅	C₂×#41	1	1	yes	characteristic
142	C₃×C₄×C₅	C₂×#41	1	1	yes	characteristic
143	Dic₁₅	—	2	3	no	—
144	Dic₁₅	C₂²×C₈	1	2	yes	—
145	Dic₁₅	#131	1	2	yes	—
146	C₂×C₃×Q₈	C₄×D₅	1	1	yes	characteristic
147	#239	—	5	8	no	—
148	C₂×D₁₂	—	5	3	no	—
149	C₂×C₄×D₃	—	5	3	no	—
150	C₂×C₄×D₃	—	5	3	no	—
151	C₂×C₄×D₃	—	5	3	no	—
152	C₂×C₃×C₈	C₂²×D₅	1	3	yes	—
153	C₂×C₃×C₈	C₂²×D₅	1	1	yes	characteristic
154	C₃×C₄⋊C₄	C₄×D₅	1	3	yes	—
155	C₃×C₄²	C₂²×D₅	1	3	yes	—
156	(C₂×D₃)⋊C₄	—	5	6	no	—
157	C₄⋊Dic₃	—	5	3	no	—
158	C₄×Dic₃	—	5	3	no	—
159	C₄×Dic₃	—	5	3	no	—
160	C₄×Dic₃	—	5	1	no	—
161	C₂×C₃⋊C₈	—	5	1	no	—
162	C₂×C₃⋊C₈	—	5	3	no	—
163	D₃×C₈	—	10	6	no	—
164	D₃×C₈	—	10	6	no	—
165	C₂²×D₅	—	3	3	no	—
166	C₅×Q₈	D₃×C₈	1	4	yes	complemented
167	C₂×C₄×C₅	C₂×C₄×D₃	1	3	yes	—
168	C₂×C₄×C₅	C₂×C₄×D₃	1	3	yes	—
169	C₂×C₄×C₅	C₂³×D₃	1	1	yes	characteristic
170	C₂×Dic₅	—	3	1	no	—
171	C₂×Dic₅	—	3	1	no	—
172	C₂×Dic₅	—	3	3	no	—
173	D₂₀	—	3	12	no	—
174	C₄×D₅	—	3	12	no	—
175	C₄×D₅	—	6	6	no	—
176	C₄×D₅	—	6	3	no	—
177	C₄×D₅	—	6	3	no	—
178	C₅×C₈	—	2	3	no	—
179	C₅×C₈	#239	1	2	yes	complemented
180	#168	—	3	2	no	—
181	#168	—	6	3	no	—
182	C₂×#41	—	15	1	no	—
183	C₂²×C₈	—	15	3	no	—
184	C₂²×C₈	—	15	3	no	—
185	C₄×Q₈	—	15	1	no	—
186	C₄×Q₈	C₂×D₁₅	1	1	yes	characteristic
187	C₄×D₄	—	15	3	no	—
188	C₄×D₄	—	15	3	no	—
189	#117	—	15	3	no	—
190	C₂×C₄²	—	15	3	no	—
191	C₄⋊C₈	—	15	3	no	—
192	C₄⋊C₈	C₂×D₁₅	1	3	yes	—
193	C₂²⋊C₈	—	15	6	no	—
194	C₄×C₈	—	15	3	no	—
195	C₄×C₈	—	15	1	no	—
196	C₄×C₈	—	15	3	no	—
197	C₄×C₈	C₂×D₁₅	1	3	yes	—
198	C₂×C₃×C₅	C₄×#41	1	1	yes	characteristic
199	C₂×C₃×C₅	C₂³×C₈	1	1	yes	characteristic
200	C₂×C₃×C₅	C₂×#131	1	1	yes	characteristic
201	D₁₅	—	2	6	no	—
202	C₂²×D₃	—	5	3	no	—
203	C₃×Q₈	C₈×D₅	1	4	yes	complemented
204	C₂×C₃×C₄	C₂×C₄×D₅	1	3	yes	—
205	C₂×C₃×C₄	C₂×C₄×D₅	1	3	yes	—
206	C₂×C₃×C₄	C₂³×D₅	1	1	yes	characteristic
207	C₂×Dic₃	—	5	1	no	—
208	C₂×Dic₃	—	5	1	no	—
209	C₂×Dic₃	—	5	3	no	—
210	D₁₂	—	5	12	no	—
211	C₄×D₃	—	5	12	no	—
212	C₄×D₃	—	10	6	no	—
213	C₄×D₃	—	10	3	no	—
214	C₄×D₃	—	10	3	no	—
215	C₃×C₈	—	2	3	no	—
216	C₃×C₈	#713	1	2	yes	complemented
217	C₃⋊C₈	—	5	2	no	—
218	C₃⋊C₈	—	10	3	no	—
219	C₂²×C₅	C₂²×C₄×D₃	1	1	yes	characteristic
220	C₂×D₅	—	3	6	no	—
221	C₂×D₅	—	6	3	no	—
222	C₂×D₅	—	6	3	no	—
223	C₄×C₅	—	2	3	no	—
224	C₄×C₅	C₂×D₃×C₈	1	6	yes	—
225	C₄×C₅	C₂×#239	1	1	yes	characteristic
226	C₄×C₅	D₃×#41	1	1	yes	characteristic
227	Dic₅	—	3	2	no	—
228	Dic₅	—	3	2	no	—
229	Dic₅	—	6	3	no	—
230	#41	—	15	8	no	—
231	C₂×Q₈	C₄×D₁₅	1	1	yes	characteristic
232	C₂×D₄	—	15	3	no	—
233	C₂²×C₄	—	15	3	no	—
234	C₂²×C₄	—	15	3	no	—
235	C₂²×C₄	—	15	3	no	—
236	C₂×C₈	—	15	1	no	—
237	C₂×C₈	—	15	3	no	—
238	C₂×C₈	—	30	6	no	—
239	C₂×C₈	—	30	6	no	—
240	C₂×C₈	C₂²×D₁₅	1	3	yes	—
241	C₂×C₈	C₂²×D₁₅	1	1	yes	characteristic
242	C₄⋊C₄	—	15	3	no	—
243	C₄⋊C₄	C₄×D₁₅	1	3	yes	—
244	C₂²⋊C₄	—	15	6	no	—
245	C₄²	—	15	3	no	—
246	C₄²	—	15	3	no	—
247	C₄²	—	15	1	no	—
248	C₄²	C₂²×D₁₅	1	3	yes	—
249	C₃×C₅	C₈×#41	1	1	yes	characteristic, complemented, Hall
250	C₂²×C₃	C₂²×C₄×D₅	1	1	yes	characteristic
251	C₂×D₃	—	5	6	no	—
252	C₂×D₃	—	10	3	no	—
253	C₂×D₃	—	10	3	no	—
254	C₃×C₄	—	2	3	no	—
255	C₃×C₄	C₂×C₈×D₅	1	6	yes	—
256	C₃×C₄	C₂×#713	1	1	yes	characteristic
257	C₃×C₄	D₅×#41	1	1	yes	characteristic
258	Dic₃	—	5	2	no	—
259	Dic₃	—	5	2	no	—
260	Dic₃	—	10	3	no	—
261	C₂×C₅	C₄×#239	1	1	yes	characteristic
262	C₂×C₅	C₂²×D₃×C₈	1	1	yes	characteristic
263	C₂×C₅	D₃×#131	1	1	yes	characteristic
264	D₅	—	6	6	no	—
265	C₂³	—	15	3	no	—
266	Q₈	C₈×D₁₅	1	4	yes	complemented
267	D₄	—	15	12	no	—
268	C₂×C₄	—	15	1	no	—
269	C₂×C₄	—	15	1	no	—
270	C₂×C₄	—	15	12	no	—
271	C₂×C₄	—	15	3	no	—
272	C₂×C₄	—	30	6	no	—
273	C₂×C₄	—	30	3	no	—
274	C₂×C₄	—	30	3	no	—
275	C₂×C₄	C₂×C₄×D₁₅	1	3	yes	—
276	C₂×C₄	C₂×C₄×D₁₅	1	3	yes	—
277	C₂×C₄	C₂³×D₁₅	1	1	yes	characteristic
278	C₈	—	2	3	no	—
279	C₈	—	15	2	no	—
280	C₈	—	30	3	no	—
281	C₈	#6821	1	2	yes	direct product factor
282	C₂×C₃	C₄×#713	1	1	yes	characteristic
283	C₂×C₃	C₂²×C₈×D₅	1	1	yes	characteristic
284	C₂×C₃	D₅×#131	1	1	yes	characteristic
285	D₃	—	10	6	no	—
286	C₅	C₈×#239	1	1	yes	characteristic, complemented, Sylow
287	C₂²	—	15	6	no	—
288	C₂²	—	30	3	no	—
289	C₂²	—	30	3	no	—
290	C₂²	C₂²×C₄×D₁₅	1	1	yes	characteristic
291	C₄	—	2	3	no	—
292	C₄	—	15	2	no	—
293	C₄	—	15	2	no	—
294	C₄	—	30	3	no	—
295	C₄	C₂×C₈×D₁₅	1	6	yes	—
296	C₄	C₂×#6821	1	1	yes	characteristic
297	C₄	#41×D₁₅	1	1	yes	characteristic
298	C₃	C₈×#713	1	1	yes	characteristic, complemented, Sylow
299	C₂	—	30	6	no	—
300	C₂	C₄×#6821	1	1	yes	characteristic
301	C₂	C₂²×C₈×D₁₅	1	1	yes	characteristic
302	C₂	D₁₅×#131	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	C₂²×D₁₅
commutator	C₂×C₃×C₅	C₂³×C₈
Frattini	C₂×C₄	C₂³×D₁₅
Fitting	C₃×C₅×C₈×Q₈	C₂
socle	C₂²×C₃×C₅	C₂³×C₄
prime radical	C₂×#6821	C₄
cosocle	C₂×C₃×C₄×C₅	C₂⁴
hypercenter	C₈×Q₈	D₁₅
hypocenter	C₃×C₅	C₈×#41
LCS	G ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂×C₈ ◁ C₈×Q₈

Complements

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

N	H ≅ G/N	type
2. C₃×C₅×C₈×Q₈	C₂	semidirect
3. Q₈×C₃⋊#168	C₂	semidirect
4. #11750935	C₂	semidirect
5. #11750934	C₂	semidirect
6. C₈×D₆₀	C₂	semidirect
7. C₄×C₈×D₁₅	C₂	semidirect
27. C₅×C₈×Q₈	D₃	semidirect
34. #6821	C₈	direct (N × H)
53. C₃×C₈×Q₈	D₅	semidirect
77. C₃×C₅×Q₈	C₂×C₈	semidirect
84. D₆₀	C₂×C₈	semidirect
88. C₄×D₁₅	C₂×C₈	semidirect
90. C₃×C₅×C₈	#41	semidirect
92. C₃⋊#168	#41	semidirect
130. C₈×Q₈	D₁₅	semidirect
166. C₅×Q₈	D₃×C₈	semidirect
179. C₅×C₈	#239	semidirect
203. C₃×Q₈	C₈×D₅	semidirect
216. C₃×C₈	#713	semidirect
249. C₃×C₅	C₈×#41	semidirect
266. Q₈	C₈×D₁₅	semidirect
281. C₈	#6821	direct (N × H)
286. C₅	C₈×#239	semidirect
298. C₃	C₈×#713	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₄×#6821	2
C₃×C₅×C₈×Q₈	2
Q₈×C₃⋊#168	2
#11750935	2
#11750934	2
C₈×D₆₀	2
C₄×C₈×D₁₅	2
C₈×#713	3
C₈×#239	5

Subgroups — C8×#6821 #14418895

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₈×#6821 #14418895