Subgroups — C₅×C₈×C₄⋊Q₈ #11944431

Order 1280.

886 conjugacy classes of subgroups.

The subgroup lattice has 260 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×C₈×Q₈	C₂	1	2	yes	maximal
2	C₄×C₅×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic
3	C₅×#1984	C₂	1	2	yes	maximal
4	C₅×#1939	C₂	1	4	yes	maximal
5	C₅×#1842	C₂	1	1	yes	maximal, characteristic
6	C₅×C₈×C₄⋊C₄	C₂	1	4	yes	maximal
7	C₄²×C₅×C₈	C₂	1	1	yes	maximal, characteristic
8	C₂×C₅×C₄⋊Q₈	C₄	1	1	yes	characteristic
9	C₂×C₄×C₅×Q₈	C₂²	1	2	yes	—
10	C₅×C₈×Q₈	—	2	8	no	—
11	C₂×C₅×C₄⋊C₈	C₂²	1	4	yes	—
12	C₂×C₅×C₄⋊C₈	C₂²	1	2	yes	—
13	C₂×C₅×C₄⋊C₈	C₂²	1	4	yes	—
14	C₂×C₄×C₅×C₈	C₂²	1	2	yes	—
15	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
16	C₂×C₄×C₅×C₈	C₂²	1	4	yes	—
17	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
18	C₂×C₄×C₅×C₈	C₂²	1	2	yes	—
19	C₅×#391	C₄	1	2	yes	—
20	C₅×#389	C₄	1	4	yes	—
21	C₅×#384	C₄	1	1	yes	characteristic
22	C₄×C₅×C₄⋊C₄	C₂²	1	4	yes	—
23	C₄³×C₅	C₂²	1	1	yes	characteristic
24	C₅×#336	C₂²	1	8	yes	—
25	C₈×C₄⋊Q₈	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
26	C₂²×C₅×Q₈	C₂×C₄	1	2	yes	—
27	C₂²×C₅×C₈	C₂³	1	4	yes	—
28	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
29	C₂²×C₅×C₈	C₂³	1	2	yes	—
30	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
31	C₅×C₄⋊Q₈	C₈	1	8	yes	direct product factor
32	C₄×C₅×Q₈	—	2	4	no	—
33	C₄×C₅×Q₈	—	2	4	no	—
34	C₂×C₅×C₄⋊C₄	C₂×C₄	1	4	yes	—
35	C₂×C₅×C₄⋊C₄	C₂×C₄	1	2	yes	—
36	C₂×C₅×C₄⋊C₄	C₂×C₄	1	4	yes	—
37	C₂×C₅×C₄⋊C₄	C₂×C₄	1	4	yes	—
38	C₂×C₄²×C₅	C₂×C₄	1	2	yes	—
39	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
40	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
41	C₂×C₄²×C₅	C₂³	1	4	yes	—
42	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
43	C₂×C₄²×C₅	C₂³	1	2	yes	—
44	C₅×C₄⋊C₈	—	2	8	no	—
45	C₅×C₄⋊C₈	—	2	4	no	—
46	C₅×C₄⋊C₈	—	2	8	no	—
47	C₄×C₅×C₈	—	2	8	no	—
48	C₄×C₅×C₈	—	2	2	no	—
49	C₄×C₅×C₈	—	2	4	no	—
50	C₄×C₅×C₈	D₄	1	4	yes	—
51	C₄×C₅×C₈	Q₈	1	8	yes	complemented
52	C₅×#95	C₂×C₄	1	8	yes	—
53	C₂×C₈×Q₈	C₂×C₅	1	2	yes	—
54	C₄×C₄⋊Q₈	C₂×C₅	1	1	yes	characteristic
55	#1984	C₂×C₅	1	2	yes	—
56	#1939	C₂×C₅	1	4	yes	—
57	#1842	C₂×C₅	1	1	yes	characteristic
58	C₈×C₄⋊C₄	C₂×C₅	1	4	yes	—
59	C₄²×C₈	C₂×C₅	1	1	yes	characteristic
60	C₂×C₅×Q₈	—	2	4	no	—
61	C₂×C₅×Q₈	—	2	4	no	—
62	C₂×C₅×Q₈	C₂×C₈	1	8	yes	—
63	C₂²×C₄×C₅	C₂²×C₄	1	4	yes	—
64	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
65	C₂²×C₄×C₅	C₂²×C₄	1	2	yes	—
66	C₂²×C₄×C₅	C₂²×C₄	1	4	yes	—
67	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
68	C₂²×C₄×C₅	C₂²×C₄	1	2	yes	—
69	C₂²×C₄×C₅	C₂⁴	1	1	yes	characteristic
70	C₂×C₅×C₈	—	2	2	no	—
71	C₂×C₅×C₈	—	2	2	no	—
72	C₂×C₅×C₈	—	2	2	no	—
73	C₂×C₅×C₈	—	2	4	no	—
74	C₂×C₅×C₈	—	2	8	no	—
75	C₂×C₅×C₈	C₂×D₄	1	2	yes	—
76	C₂×C₅×C₈	C₂×D₄	1	2	yes	—
77	C₂×C₅×C₈	C₂×Q₈	1	4	yes	—
78	C₂×C₅×C₈	C₂×Q₈	1	4	yes	—
79	C₅×C₄⋊C₄	—	2	8	no	—
80	C₅×C₄⋊C₄	—	2	4	no	—
81	C₅×C₄⋊C₄	—	2	8	no	—
82	C₅×C₄⋊C₄	C₂×C₈	1	16	yes	—
83	C₄²×C₅	—	2	2	no	—
84	C₄²×C₅	—	2	4	no	—
85	C₄²×C₅	—	2	4	no	—
86	C₄²×C₅	—	2	4	no	—
87	C₄²×C₅	C₂×C₈	1	4	yes	—
88	C₄²×C₅	C₂×D₄	1	2	yes	—
89	C₄²×C₅	C₂×Q₈	1	4	yes	—
90	C₄²×C₅	#41	1	2	yes	—
91	C₄²×C₅	#41	1	4	yes	—
92	C₂×C₄⋊Q₈	C₄×C₅	1	1	yes	characteristic
93	C₂×C₄×Q₈	C₂²×C₅	1	2	yes	—
94	C₈×Q₈	—	2	8	no	—
95	C₂×C₄⋊C₈	C₂²×C₅	1	2	yes	—
96	C₂×C₄⋊C₈	C₂²×C₅	1	4	yes	—
97	C₂×C₄⋊C₈	C₂²×C₅	1	4	yes	—
98	C₂×C₄×C₈	C₂²×C₅	1	4	yes	—
99	C₂×C₄×C₈	C₂²×C₅	1	2	yes	—
100	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
101	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
102	C₂×C₄×C₈	C₂²×C₅	1	2	yes	—
103	#391	C₄×C₅	1	2	yes	—
104	#389	C₄×C₅	1	4	yes	—
105	#384	C₄×C₅	1	1	yes	characteristic
106	C₄×C₄⋊C₄	C₂²×C₅	1	4	yes	—
107	C₄³	C₂²×C₅	1	1	yes	characteristic
108	#336	C₂²×C₅	1	8	yes	—
109	C₂³×C₅	C₂³×C₄	1	1	yes	characteristic
110	C₅×Q₈	—	2	16	no	—
111	C₂×C₄×C₅	—	2	4	no	—
112	C₂×C₄×C₅	—	2	2	no	—
113	C₂×C₄×C₅	—	2	2	no	—
114	C₂×C₄×C₅	—	2	2	no	—
115	C₂×C₄×C₅	—	2	4	no	—
116	C₂×C₄×C₅	—	2	4	no	—
117	C₂×C₄×C₅	—	2	8	no	—
118	C₂×C₄×C₅	C₄×D₄	1	2	yes	—
119	C₂×C₄×C₅	C₄×D₄	1	2	yes	—
120	C₂×C₄×C₅	C₄×Q₈	1	4	yes	—
121	C₂×C₄×C₅	C₄×Q₈	1	4	yes	—
122	C₂×C₄×C₅	C₂²×C₈	1	8	yes	—
123	C₂×C₄×C₅	C₂²×C₈	1	2	yes	—
124	C₂×C₄×C₅	C₂²×C₈	1	4	yes	—
125	C₂×C₄×C₅	#131	1	2	yes	—
126	C₂×C₄×C₅	#131	1	4	yes	—
127	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
128	C₂×C₄×C₅	C₂²×Q₈	1	2	yes	—
129	C₂×C₄×C₅	C₂×#41	1	2	yes	—
130	C₂×C₄×C₅	C₂×#41	1	1	yes	characteristic
131	C₅×C₈	—	2	2	no	—
132	C₅×C₈	—	2	4	no	—
133	C₅×C₈	—	4	4	no	—
134	C₅×C₈	C₄⋊Q₈	1	4	yes	direct product factor
135	C₂²×Q₈	C₂×C₄×C₅	1	2	yes	—
136	C₂²×C₈	C₂³×C₅	1	4	yes	—
137	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
138	C₂²×C₈	C₂³×C₅	1	2	yes	—
139	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
140	C₄⋊Q₈	C₅×C₈	1	8	yes	direct product factor
141	C₄×Q₈	—	2	4	no	—
142	C₄×Q₈	—	2	4	no	—
143	C₂×C₄⋊C₄	C₂×C₄×C₅	1	2	yes	—
144	C₂×C₄⋊C₄	C₂×C₄×C₅	1	4	yes	—
145	C₂×C₄⋊C₄	C₂×C₄×C₅	1	4	yes	—
146	C₂×C₄⋊C₄	C₂×C₄×C₅	1	4	yes	—
147	C₂×C₄²	C₂×C₄×C₅	1	2	yes	—
148	C₂×C₄²	C₂×C₄×C₅	1	1	yes	characteristic
149	C₂×C₄²	C₂×C₄×C₅	1	1	yes	characteristic
150	C₂×C₄²	C₂³×C₅	1	4	yes	—
151	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
152	C₂×C₄²	C₂³×C₅	1	2	yes	—
153	C₄⋊C₈	—	2	8	no	—
154	C₄⋊C₈	—	2	4	no	—
155	C₄⋊C₈	—	2	8	no	—
156	C₄×C₈	—	2	8	no	—
157	C₄×C₈	—	2	2	no	—
158	C₄×C₈	—	2	4	no	—
159	C₄×C₈	C₅×D₄	1	4	yes	—
160	C₄×C₈	C₅×Q₈	1	8	yes	complemented
161	#95	C₂×C₄×C₅	1	8	yes	—
162	C₂²×C₅	C₂×C₄×D₄	1	1	yes	characteristic
163	C₂²×C₅	C₂×C₄×Q₈	1	2	yes	—
164	C₂²×C₅	C₂³×C₈	1	1	yes	characteristic
165	C₂²×C₅	C₂×#131	1	1	yes	characteristic
166	C₂²×C₅	C₂×#131	1	2	yes	—
167	C₄×C₅	—	2	8	no	—
168	C₄×C₅	—	2	2	no	—
169	C₄×C₅	—	2	4	no	—
170	C₄×C₅	—	4	4	no	—
171	C₄×C₅	C₈×D₄	1	4	yes	—
172	C₄×C₅	C₈×Q₈	1	8	yes	complemented
173	C₄×C₅	C₂×C₄⋊Q₈	1	1	yes	characteristic
174	C₄×C₅	#532	1	1	yes	characteristic
175	C₄×C₅	#533	1	2	yes	—
176	C₂×Q₈	—	2	4	no	—
177	C₂×Q₈	—	2	4	no	—
178	C₂×Q₈	C₂×C₅×C₈	1	8	yes	—
179	C₂²×C₄	C₂²×C₄×C₅	1	4	yes	—
180	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
181	C₂²×C₄	C₂²×C₄×C₅	1	2	yes	—
182	C₂²×C₄	C₂²×C₄×C₅	1	4	yes	—
183	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
184	C₂²×C₄	C₂²×C₄×C₅	1	2	yes	—
185	C₂²×C₄	C₂⁴×C₅	1	1	yes	characteristic
186	C₂×C₈	—	2	2	no	—
187	C₂×C₈	—	2	2	no	—
188	C₂×C₈	—	2	2	no	—
189	C₂×C₈	—	2	4	no	—
190	C₂×C₈	—	2	8	no	—
191	C₂×C₈	C₂×C₅×D₄	1	2	yes	—
192	C₂×C₈	C₂×C₅×D₄	1	2	yes	—
193	C₂×C₈	C₂×C₅×Q₈	1	4	yes	—
194	C₂×C₈	C₂×C₅×Q₈	1	4	yes	—
195	C₄⋊C₄	—	2	8	no	—
196	C₄⋊C₄	—	2	4	no	—
197	C₄⋊C₄	—	2	8	no	—
198	C₄⋊C₄	C₂×C₅×C₈	1	16	yes	—
199	C₄²	—	2	2	no	—
200	C₄²	—	2	4	no	—
201	C₄²	—	2	4	no	—
202	C₄²	—	2	4	no	—
203	C₄²	C₂×C₅×C₈	1	4	yes	—
204	C₄²	C₂×C₅×D₄	1	2	yes	—
205	C₄²	C₂×C₅×Q₈	1	4	yes	—
206	C₄²	C₅×#41	1	2	yes	—
207	C₄²	C₅×#41	1	4	yes	—
208	C₂×C₅	C₄×C₄⋊Q₈	1	1	yes	characteristic
209	C₂×C₅	C₂×C₈×D₄	1	1	yes	characteristic
210	C₂×C₅	#2931	1	1	yes	characteristic
211	C₂×C₅	C₂×C₈×Q₈	1	2	yes	—
212	C₂×C₅	#2960	1	2	yes	—
213	C₂³	C₂³×C₄×C₅	1	1	yes	characteristic
214	Q₈	—	2	16	no	—
215	C₂×C₄	—	2	4	no	—
216	C₂×C₄	—	2	2	no	—
217	C₂×C₄	—	2	2	no	—
218	C₂×C₄	—	2	2	no	—
219	C₂×C₄	—	2	4	no	—
220	C₂×C₄	—	2	4	no	—
221	C₂×C₄	—	2	8	no	—
222	C₂×C₄	C₄×C₅×D₄	1	2	yes	—
223	C₂×C₄	C₄×C₅×D₄	1	2	yes	—
224	C₂×C₄	C₄×C₅×Q₈	1	4	yes	—
225	C₂×C₄	C₄×C₅×Q₈	1	4	yes	—
226	C₂×C₄	C₂²×C₅×C₈	1	8	yes	—
227	C₂×C₄	C₂²×C₅×C₈	1	2	yes	—
228	C₂×C₄	C₂²×C₅×C₈	1	4	yes	—
229	C₂×C₄	C₅×#131	1	2	yes	—
230	C₂×C₄	C₅×#131	1	4	yes	—
231	C₂×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
232	C₂×C₄	C₂²×C₅×Q₈	1	2	yes	—
233	C₂×C₄	C₂×C₅×#41	1	2	yes	—
234	C₂×C₄	C₂×C₅×#41	1	1	yes	characteristic
235	C₈	—	2	2	no	—
236	C₈	—	2	4	no	—
237	C₈	—	4	4	no	—
238	C₈	C₅×C₄⋊Q₈	1	4	yes	direct product factor
239	C₅	C₈×C₄⋊Q₈	1	1	yes	characteristic, direct product factor, Sylow
240	C₂²	C₂×C₄×C₅×D₄	1	1	yes	characteristic
241	C₂²	C₂×C₄×C₅×Q₈	1	2	yes	—
242	C₂²	C₂³×C₅×C₈	1	1	yes	characteristic
243	C₂²	C₂×C₅×#131	1	1	yes	characteristic
244	C₂²	C₂×C₅×#131	1	2	yes	—
245	C₄	—	2	8	no	—
246	C₄	—	2	2	no	—
247	C₄	—	2	4	no	—
248	C₄	—	4	4	no	—
249	C₄	C₅×C₈×D₄	1	4	yes	—
250	C₄	C₅×C₈×Q₈	1	8	yes	complemented
251	C₄	C₂×C₅×C₄⋊Q₈	1	1	yes	characteristic
252	C₄	C₅×#532	1	1	yes	characteristic
253	C₄	C₅×#533	1	2	yes	—
254	C₂	C₄×C₅×C₄⋊Q₈	1	1	yes	characteristic
255	C₂	C₂×C₅×C₈×D₄	1	1	yes	characteristic
256	C₂	C₅×#2931	1	1	yes	characteristic
257	C₂	C₂×C₅×C₈×Q₈	1	2	yes	—
258	C₂	C₅×#2960	1	2	yes	—

	subgroup	G/H
center	C₂²×C₅×C₈	C₂³
commutator	C₂²	C₂³×C₅×C₈
Frattini, cosocle	C₂²×C₄	C₂⁴×C₅
socle, prime radical	C₂³×C₅	C₂³×C₄
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₅×C₈ ◁ G

Complements

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

N	H ≅ G/N	type
25. C₈×C₄⋊Q₈	C₅	direct (N × H)
31. C₅×C₄⋊Q₈	C₈	direct (N × H)
51. C₄×C₅×C₈	Q₈	semidirect
134. C₅×C₈	C₄⋊Q₈	direct (N × H)
140. C₄⋊Q₈	C₅×C₈	direct (N × H)
160. C₄×C₈	C₅×Q₈	semidirect
172. C₄×C₅	C₈×Q₈	semidirect
238. C₈	C₅×C₄⋊Q₈	direct (N × H)
239. C₅	C₈×C₄⋊Q₈	direct (N × H)
250. C₄	C₅×C₈×Q₈	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₅×C₈×Q₈	2
C₄×C₅×C₄⋊Q₈	2
C₅×#1984	2
C₅×#1939	2
C₅×#1842	2
C₅×C₈×C₄⋊C₄	2
C₄²×C₅×C₈	2
C₈×C₄⋊Q₈	5

Subgroups — C5×C8×C4⋊Q8 #11944431

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×C₈×C₄⋊Q₈ #11944431