Subgroups — C₈×#360 #128670

Order 512.

363 conjugacy classes of subgroups.

The subgroup lattice has 271 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₈×#135	C₂	1	1	yes	maximal, characteristic
2	C₄×#360	C₂	1	1	yes	maximal, characteristic
3	#11598	C₂	1	1	yes	maximal, characteristic
4	C₈×M₃₂	C₂	1	1	yes	maximal, characteristic, complemented
5	C₈×#106	C₂	1	1	yes	maximal, characteristic, complemented
6	#7175	C₂	1	1	yes	maximal, characteristic, complemented
7	#7158	C₂	1	1	yes	maximal, characteristic, complemented
8	C₈×#41	—	2	1	no	—
9	C₄×#135	C₂²	1	1	yes	characteristic
10	C₂×#360	C₄	1	2	yes	—
11	C₄×M₃₂	C₂²	1	1	yes	characteristic
12	C₄×M₃₂	C₂²	1	1	yes	characteristic
13	C₄×#106	C₂²	1	1	yes	characteristic
14	C₈×C₄⋊C₄	—	2	1	no	—
15	C₈×Q₁₆	C₄	1	1	yes	characteristic, complemented
16	C₈×Q₁₆	C₄	1	1	yes	characteristic, complemented
17	C₈×SD₁₆	—	2	1	no	—
18	C₈×SD₁₆	—	2	1	no	—
19	C₈×D₈	C₄	1	1	yes	characteristic, complemented
20	C₈×D₈	C₄	1	1	yes	characteristic, complemented
21	#1564	C₂²	1	1	yes	characteristic
22	C₂×#334	C₂²	1	1	yes	characteristic
23	#1476	—	2	1	no	—
24	C₂×C₈²	C₂²	1	1	yes	characteristic
25	#1386	C₄	1	2	yes	—
26	#1334	C₄	1	2	yes	complemented
27	#1333	C₄	1	2	yes	complemented
28	#1311	—	2	1	no	—
29	C₈×C₁₆	—	2	1	no	—
30	#1279	—	2	1	no	—
31	C₂×#135	C₂×C₄	1	1	yes	characteristic
32	C₄×#41	—	2	1	no	—
33	C₂×M₃₂	—	2	1	no	—
34	C₂×M₃₂	C₂×C₄	1	2	yes	—
35	C₈×Q₈	—	2	1	no	—
36	C₈×Q₈	—	2	1	no	—
37	C₄×Q₁₆	C₂×C₄	1	1	yes	characteristic
38	C₄×Q₁₆	C₂×C₄	1	1	yes	characteristic
39	C₄×SD₁₆	—	2	1	no	—
40	C₄×SD₁₆	—	2	1	no	—
41	C₄×D₈	C₂×C₄	1	1	yes	characteristic
42	C₄×D₈	C₂×C₄	1	1	yes	characteristic
43	C₈×D₄	—	2	1	no	—
44	C₈×D₄	—	2	1	no	—
45	C₈×D₄	—	4	1	no	—
46	C₈×D₄	—	4	1	no	—
47	#431	—	2	1	no	—
48	#431	—	4	1	no	—
49	#427	C₂×C₄	1	1	yes	characteristic
50	C₂×#106	C₂×C₄	1	2	yes	—
51	C₂×C₄⋊C₈	—	2	1	no	—
52	#416	—	2	1	no	—
53	C₂×C₄×C₈	—	4	1	no	—
54	C₂×C₄×C₈	—	4	1	no	—
55	C₂×C₄×C₈	D₄	1	1	yes	characteristic
56	C₂×C₄×C₈	D₄	1	1	yes	characteristic
57	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
58	C₄×C₄⋊C₄	—	2	1	no	—
59	#360	C₈	1	8	yes	direct product factor
60	#346	—	2	1	no	—
61	#346	C₂×C₄	1	2	yes	—
62	C₄×C₁₆	—	2	1	no	—
63	C₄×C₁₆	—	2	1	no	—
64	#340	—	2	1	no	—
65	#336	—	4	1	no	—
66	#335	C₂×C₄	1	1	yes	characteristic
67	#335	C₂×C₄	1	1	yes	characteristic
68	#334	—	2	1	no	—
69	#334	—	2	1	no	—
70	#334	C₂×C₄	1	2	yes	—
71	Q₈⋊C₈	—	2	1	no	—
72	Q₈⋊C₈	—	2	1	no	—
73	D₄⋊C₈	—	2	1	no	—
74	D₄⋊C₈	—	2	1	no	—
75	C₈²	C₂×C₄	1	1	yes	characteristic
76	C₈²	C₂×C₄	1	1	yes	characteristic
77	C₈²	D₄	1	1	yes	characteristic
78	C₈²	D₄	1	1	yes	characteristic
79	C₂×#41	—	2	1	no	—
80	#135	—	2	2	no	—
81	#135	C₂×C₈	1	4	yes	—
82	C₂×Q₁₆	C₄²	1	1	yes	characteristic
83	C₂×SD₁₆	—	2	1	no	—
84	C₂×D₈	C₄²	1	1	yes	characteristic
85	C₂²×C₈	—	2	1	no	—
86	C₂²×C₈	—	4	1	no	—
87	C₂²×C₈	—	4	1	no	—
88	C₂²×C₈	—	4	1	no	—
89	C₂²×C₈	D₈	1	1	yes	characteristic
90	C₂²×C₈	D₈	1	1	yes	characteristic
91	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
92	C₂²×C₈	#41	1	1	yes	characteristic
93	C₄×Q₈	—	2	1	no	—
94	C₄×Q₈	—	2	1	no	—
95	C₄×D₄	—	2	1	no	—
96	C₄×D₄	—	2	1	no	—
97	C₄×D₄	—	4	1	no	—
98	C₄×D₄	—	4	1	no	—
99	#117	—	2	1	no	—
100	#117	—	4	1	no	—
101	C₂×C₄⋊C₄	—	2	2	no	—
102	C₂×C₄²	—	4	1	no	—
103	C₂×C₄²	—	4	1	no	—
104	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
105	M₃₂	—	2	4	no	—
106	M₃₂	—	2	2	no	—
107	M₃₂	C₂×C₈	1	4	yes	complemented
108	C₂×C₁₆	—	2	2	no	—
109	C₂×C₁₆	—	2	2	no	—
110	#107	C₄²	1	1	yes	characteristic
111	#107	C₄²	1	1	yes	characteristic
112	#106	—	2	1	no	—
113	#106	—	2	2	no	—
114	#106	C₂×C₈	1	4	yes	complemented
115	C₄⋊C₈	—	2	1	no	—
116	C₄⋊C₈	—	2	1	no	—
117	C₄⋊C₈	—	2	2	no	—
118	C₄⋊C₈	—	4	1	no	—
119	C₄⋊C₈	—	4	1	no	—
120	C₄⋊C₈	—	4	1	no	—
121	#103	—	2	1	no	—
122	#103	—	2	1	no	—
123	#102	—	2	1	no	—
124	#102	—	2	1	no	—
125	C₂²⋊C₈	—	4	1	no	—
126	C₂²⋊C₈	—	4	1	no	—
127	C₂²⋊C₈	—	4	1	no	—
128	C₄×C₈	—	2	1	no	—
129	C₄×C₈	—	4	1	no	—
130	C₄×C₈	—	4	1	no	—
131	C₄×C₈	—	4	1	no	—
132	C₄×C₈	—	8	2	no	—
133	C₄×C₈	C₂²⋊C₄	1	1	yes	characteristic
134	C₄×C₈	C₂²⋊C₄	1	1	yes	characteristic
135	C₄×C₈	C₂²⋊C₄	1	1	yes	characteristic
136	C₄×C₈	C₂²⋊C₄	1	1	yes	characteristic
137	C₄×C₈	SD₁₆	1	1	yes	characteristic
138	C₄×C₈	SD₁₆	1	1	yes	characteristic
139	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
140	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
141	C₄×C₈	C₂×D₄	1	1	yes	characteristic
142	C₄×C₈	#41	1	1	yes	characteristic
143	#95	—	4	1	no	—
144	#41	—	2	4	no	—
145	#41	—	4	2	no	—
146	C₂×Q₈	—	2	1	no	—
147	C₂×D₄	—	2	1	no	—
148	C₂×D₄	—	4	1	no	—
149	C₂²×C₄	—	4	1	no	—
150	C₂²×C₄	—	4	2	no	—
151	C₂²×C₄	—	4	1	no	—
152	C₂²×C₄	—	4	1	no	—
153	C₂²×C₄	C₄×D₄	1	1	yes	characteristic
154	C₂²×C₄	C₂×D₈	1	1	yes	characteristic
155	C₂²×C₄	#135	1	1	yes	characteristic
156	Q₁₆	—	2	1	no	—
157	Q₁₆	C₄×C₈	1	2	yes	complemented
158	SD₁₆	—	2	4	no	—
159	D₈	—	2	1	no	—
160	D₈	C₄×C₈	1	2	yes	complemented
161	C₂×C₈	—	2	1	no	—
162	C₂×C₈	—	2	2	no	—
163	C₂×C₈	—	2	2	no	—
164	C₂×C₈	—	2	1	no	—
165	C₂×C₈	—	4	1	no	—
166	C₂×C₈	—	4	2	no	—
167	C₂×C₈	—	4	1	no	—
168	C₂×C₈	—	4	1	no	—
169	C₂×C₈	—	4	2	no	—
170	C₂×C₈	—	8	2	no	—
171	C₂×C₈	—	8	1	no	—
172	C₂×C₈	—	8	1	no	—
173	C₂×C₈	—	8	2	no	—
174	C₂×C₈	#102	1	1	yes	characteristic
175	C₂×C₈	#102	1	1	yes	characteristic
176	C₂×C₈	#102	1	1	yes	characteristic
177	C₂×C₈	#102	1	1	yes	characteristic
178	C₂×C₈	C₂×C₄²	1	1	yes	characteristic
179	C₂×C₈	#117	1	1	yes	characteristic
180	C₂×C₈	C₄×D₄	1	1	yes	characteristic
181	C₂×C₈	C₄×D₄	1	1	yes	characteristic
182	C₂×C₈	C₂²×C₈	1	2	yes	—
183	C₂×C₈	#131	1	2	yes	—
184	C₄⋊C₄	—	2	1	no	—
185	C₄⋊C₄	—	2	1	no	—
186	C₄⋊C₄	—	2	4	no	—
187	C₄⋊C₄	—	4	1	no	—
188	C₄⋊C₄	—	4	1	no	—
189	C₄⋊C₄	—	4	2	no	—
190	C₂²⋊C₄	—	4	1	no	—
191	C₂²⋊C₄	—	4	1	no	—
192	C₂²⋊C₄	—	4	1	no	—
193	C₄²	—	4	1	no	—
194	C₄²	—	4	1	no	—
195	C₄²	—	4	1	no	—
196	C₄²	—	8	1	no	—
197	C₄²	—	8	1	no	—
198	C₄²	C₂×C₂²⋊C₄	1	1	yes	characteristic
199	C₄²	C₄×D₄	1	1	yes	characteristic
200	C₄²	C₂×SD₁₆	1	1	yes	characteristic
201	C₄²	#135	1	1	yes	characteristic
202	C₁₆	—	2	4	no	—
203	C₁₆	—	2	4	no	—
204	C₂³	—	4	1	no	—
205	C₂³	C₄×D₈	1	1	yes	characteristic
206	Q₈	—	2	2	no	—
207	Q₈	—	4	1	no	—
208	D₄	—	2	2	no	—
209	D₄	—	4	4	no	—
210	D₄	—	4	1	no	—
211	C₂×C₄	—	2	1	no	—
212	C₂×C₄	—	2	1	no	—
213	C₂×C₄	—	4	1	no	—
214	C₂×C₄	—	4	1	no	—
215	C₂×C₄	—	4	4	no	—
216	C₂×C₄	—	4	4	no	—
217	C₂×C₄	—	4	1	no	—
218	C₂×C₄	—	4	1	no	—
219	C₂×C₄	—	8	2	no	—
220	C₂×C₄	—	8	2	no	—
221	C₂×C₄	—	8	2	no	—
222	C₂×C₄	—	8	1	no	—
223	C₂×C₄	—	8	1	no	—
224	C₂×C₄	C₄×C₂²⋊C₄	1	1	yes	characteristic
225	C₂×C₄	C₂×#102	1	1	yes	characteristic
226	C₂×C₄	#416	1	1	yes	characteristic
227	C₂×C₄	C₈×D₄	1	2	yes	—
228	C₂×C₄	C₄×D₈	1	1	yes	characteristic
229	C₂×C₄	C₄×SD₁₆	1	1	yes	characteristic
230	C₂×C₄	C₄×SD₁₆	1	1	yes	characteristic
231	C₂×C₄	#443	1	2	yes	—
232	C₈	—	2	2	no	—
233	C₈	—	2	1	no	—
234	C₈	—	2	1	no	—
235	C₈	—	2	1	no	—
236	C₈	—	2	1	no	—
237	C₈	—	2	1	no	—
238	C₈	—	4	1	no	—
239	C₈	—	8	2	no	—
240	C₈	—	8	1	no	—
241	C₈	—	8	1	no	—
242	C₈	#360	1	2	yes	direct product factor
243	C₈	C₂×C₄×C₈	1	1	yes	characteristic
244	C₈	#405	1	1	yes	characteristic
245	C₈	C₈×D₄	1	2	yes	—
246	C₂²	—	2	1	no	—
247	C₂²	—	2	1	no	—
248	C₂²	—	4	2	no	—
249	C₂²	—	8	1	no	—
250	C₂²	—	8	1	no	—
251	C₂²	C₈×D₈	1	2	yes	—
252	C₂²	C₄×#102	1	1	yes	characteristic
253	C₄	—	2	1	no	—
254	C₄	—	2	1	no	—
255	C₄	—	2	1	no	—
256	C₄	—	4	2	no	—
257	C₄	—	8	4	no	—
258	C₄	—	8	1	no	—
259	C₄	—	8	1	no	—
260	C₄	C₈×SD₁₆	1	2	yes	—
261	C₄	C₈×C₂²⋊C₄	1	1	yes	characteristic
262	C₄	#1764	1	1	yes	characteristic
263	C₄	C₂×#360	1	1	yes	characteristic
264	C₄	#2145	1	1	yes	characteristic
265	C₂	—	2	2	no	—
266	C₂	—	8	2	no	—
267	C₂	C₈×#102	1	1	yes	characteristic
268	C₂	C₄×#360	1	1	yes	characteristic
269	C₂	#12646	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	#102
commutator	C₈	C₂×C₄×C₈
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂²	C₄×#102
prime radical	C₂×#135	C₂×C₄
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ C₂×C₈² ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
4. C₈×M₃₂	C₂	semidirect
5. C₈×#106	C₂	semidirect
6. #7175	C₂	semidirect
7. #7158	C₂	semidirect
15. C₈×Q₁₆	C₄	semidirect
16. C₈×Q₁₆	C₄	semidirect
19. C₈×D₈	C₄	semidirect
20. C₈×D₈	C₄	semidirect
26. #1334	C₄	semidirect
27. #1333	C₄	semidirect
59. #360	C₈	direct (N × H)
107. M₃₂	C₂×C₈	semidirect
114. #106	C₂×C₈	semidirect
157. Q₁₆	C₄×C₈	semidirect
160. D₈	C₄×C₈	semidirect
242. C₈	#360	direct (N × H)

Maximal subgroups

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

subgroup	index
C₈×#135	2
C₄×#360	2
#11598	2
C₈×M₃₂	2
C₈×#106	2
#7175	2
#7158	2

Subgroups — C8×#360 #128670

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₈×#360 #128670