Subgroups — #140841

Order 512.

406 conjugacy classes of subgroups.

The subgroup lattice has 274 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21705	C₂	1	1	yes	maximal, characteristic
2	C₄×#364	C₂	1	1	yes	maximal, characteristic, complemented
3	#11604	C₂	1	2	yes	maximal, complemented
4	#11602	C₂	1	2	yes	maximal, complemented
5	#8266	C₂	1	1	yes	maximal, characteristic, complemented
6	#3147	C₄	1	1	yes	characteristic
7	C₂×#443	—	2	1	no	—
8	C₂×#443	C₂²	1	2	yes	—
9	#2931	—	2	1	no	—
10	C₂×#364	C₂²	1	1	yes	characteristic
11	C₂×#364	C₂²	1	1	yes	characteristic
12	C₄×M₃₂	—	2	1	no	—
13	#1974	—	2	1	no	—
14	#1974	C₄	1	2	yes	—
15	#1962	—	2	1	no	—
16	#1848	C₄	1	1	yes	characteristic
17	C₄²×C₈	C₂²	1	1	yes	characteristic
18	#1564	—	2	1	no	—
19	#1450	C₂²	1	2	yes	—
20	#1376	—	2	1	no	—
21	#1336	—	2	4	no	—
22	#1335	—	2	4	no	—
23	#1276	—	2	1	no	—
24	C₂×#135	—	2	1	no	—
25	C₂×#135	C₂×C₄	1	2	yes	—
26	C₂×#131	—	2	2	no	—
27	#532	—	2	1	no	—
28	C₂×M₃₂	—	2	1	no	—
29	C₂×M₃₂	—	2	1	no	—
30	#495	—	2	1	no	—
31	#495	C₈	1	2	yes	complemented
32	#494	C₈	1	1	yes	characteristic, complemented
33	#493	C₈	1	1	yes	characteristic, complemented
34	#492	—	2	1	no	—
35	#443	—	2	2	no	—
36	#443	—	2	2	no	—
37	#443	—	2	2	no	—
38	#443	—	2	2	no	—
39	#443	—	4	1	no	—
40	#443	—	4	1	no	—
41	#443	—	4	1	no	—
42	#443	—	4	1	no	—
43	#436	—	4	2	no	—
44	C₈×D₄	—	4	2	no	—
45	#431	—	2	2	no	—
46	C₂×#108	—	2	1	no	—
47	C₂×#108	C₂×C₄	1	2	yes	—
48	#423	—	2	1	no	—
49	C₂×#104	—	2	2	no	—
50	C₂×#104	—	2	2	no	—
51	#408	—	2	2	no	—
52	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
53	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
54	C₂×C₄×C₈	D₄	1	1	yes	characteristic
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	C₄³	D₄	1	1	yes	characteristic
59	#364	—	2	1	no	—
60	#364	—	2	1	no	—
61	#364	D₄	1	2	yes	complemented
62	#364	D₄	1	2	yes	complemented
63	#346	—	4	1	no	—
64	C₄×C₁₆	—	4	1	no	—
65	#335	—	2	2	no	—
66	#334	—	2	2	no	—
67	#322	—	2	2	no	—
68	C₈²	—	2	2	no	—
69	C₂×#41	—	2	2	no	—
70	#135	—	2	2	no	—
71	#135	—	2	2	no	—
72	#135	—	2	2	no	—
73	#135	—	2	2	no	—
74	#135	—	4	1	no	—
75	#135	—	4	1	no	—
76	#135	—	4	1	no	—
77	#135	—	4	1	no	—
78	C₂×Q₁₆	—	2	1	no	—
79	C₂×Q₁₆	C₂×C₈	1	2	yes	—
80	C₂×SD₁₆	—	2	2	no	—
81	C₂×SD₁₆	—	2	2	no	—
82	C₂×D₈	—	2	1	no	—
83	C₂×D₈	C₂×C₈	1	2	yes	—
84	#131	—	4	2	no	—
85	#131	—	4	2	no	—
86	#131	—	4	2	no	—
87	#131	—	4	2	no	—
88	C₂×M₁₆	—	2	2	no	—
89	C₂×M₁₆	—	4	2	no	—
90	C₂²×C₈	—	4	2	no	—
91	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
92	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
93	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
94	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
95	C₄⋊Q₈	—	2	1	no	—
96	#127	—	2	1	no	—
97	#124	—	2	2	no	—
98	#121	—	4	2	no	—
99	C₄×D₄	—	4	2	no	—
100	C₂×C₄²	—	4	1	no	—
101	C₂×C₄²	C₂²⋊C₄	1	1	yes	characteristic
102	C₂×C₄²	C₂²⋊C₄	1	1	yes	characteristic
103	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
104	M₃₂	—	2	2	no	—
105	M₃₂	—	2	2	no	—
106	M₃₂	—	4	1	no	—
107	M₃₂	—	4	1	no	—
108	C₂×C₁₆	—	4	1	no	—
109	C₂×C₁₆	—	4	1	no	—
110	#108	—	2	2	no	—
111	#108	—	2	2	no	—
112	#108	—	4	1	no	—
113	#108	—	4	1	no	—
114	C₄⋊C₈	—	2	4	no	—
115	C₄⋊C₈	—	4	2	no	—
116	#104	—	4	2	no	—
117	#104	—	4	2	no	—
118	#104	—	4	2	no	—
119	#104	—	4	2	no	—
120	C₂²⋊C₈	—	4	4	no	—
121	C₂²⋊C₈	—	4	2	no	—
122	C₄×C₈	—	2	1	no	—
123	C₄×C₈	—	2	1	no	—
124	C₄×C₈	—	2	1	no	—
125	C₄×C₈	—	2	1	no	—
126	C₄×C₈	—	2	1	no	—
127	C₄×C₈	—	2	1	no	—
128	C₄×C₈	—	4	2	no	—
129	C₄×C₈	C₂×C₈	1	1	yes	characteristic
130	C₄×C₈	C₂×C₈	1	1	yes	characteristic
131	C₄×C₈	M₁₆	1	1	yes	characteristic
132	C₄×C₈	M₁₆	1	1	yes	characteristic
133	C₄×C₈	D₈	1	1	yes	characteristic
134	C₄×C₈	D₈	1	1	yes	characteristic
135	C₄×C₈	SD₁₆	1	1	yes	characteristic
136	C₄×C₈	SD₁₆	1	1	yes	characteristic
137	C₄×C₈	C₂×D₄	1	1	yes	characteristic
138	C₄×C₈	C₂×D₄	1	1	yes	characteristic
139	C₄×C₈	#41	1	1	yes	characteristic
140	C₄×C₈	#41	1	1	yes	characteristic
141	#41	—	4	2	no	—
142	#41	—	4	2	no	—
143	#41	—	4	2	no	—
144	#41	—	4	2	no	—
145	C₂×Q₈	—	2	2	no	—
146	C₂×D₄	—	2	2	no	—
147	C₂×D₄	—	4	2	no	—
148	C₂×D₄	—	4	2	no	—
149	C₂²×C₄	—	2	1	no	—
150	C₂²×C₄	—	2	1	no	—
151	C₂²×C₄	—	2	1	no	—
152	C₂²×C₄	—	4	2	no	—
153	C₂²×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
154	Q₁₆	—	2	2	no	—
155	Q₁₆	—	2	2	no	—
156	Q₁₆	—	4	1	no	—
157	Q₁₆	—	4	1	no	—
158	SD₁₆	—	4	2	no	—
159	SD₁₆	—	4	2	no	—
160	SD₁₆	—	4	2	no	—
161	SD₁₆	—	4	2	no	—
162	D₈	—	2	2	no	—
163	D₈	—	2	2	no	—
164	D₈	—	4	1	no	—
165	D₈	—	4	1	no	—
166	M₁₆	—	4	2	no	—
167	M₁₆	—	4	2	no	—
168	M₁₆	—	8	2	no	—
169	M₁₆	—	8	2	no	—
170	C₂×C₈	—	2	1	no	—
171	C₂×C₈	—	2	1	no	—
172	C₂×C₈	—	2	1	no	—
173	C₂×C₈	—	4	4	no	—
174	C₂×C₈	—	4	2	no	—
175	C₂×C₈	—	8	2	no	—
176	C₂×C₈	—	8	2	no	—
177	C₂×C₈	C₂²⋊C₈	1	1	yes	characteristic
178	C₂×C₈	C₂²⋊C₈	1	1	yes	characteristic
179	C₂×C₈	#102	1	1	yes	characteristic
180	C₂×C₈	#102	1	1	yes	characteristic
181	C₂×C₈	#104	1	1	yes	characteristic
182	C₂×C₈	#104	1	1	yes	characteristic
183	C₂×C₈	C₄×D₄	1	1	yes	characteristic
184	C₂×C₈	C₄×D₄	1	1	yes	characteristic
185	C₂×C₈	#121	1	1	yes	characteristic
186	C₂×C₈	#121	1	1	yes	characteristic
187	C₂×C₈	C₂²×C₈	1	1	yes	characteristic
188	C₂×C₈	C₂×M₁₆	1	1	yes	characteristic
189	C₂×C₈	#131	1	1	yes	characteristic
190	C₂×C₈	#131	1	1	yes	characteristic
191	C₂×C₈	C₂×D₈	1	1	yes	characteristic
192	C₂×C₈	C₂×SD₁₆	1	1	yes	characteristic
193	C₂×C₈	AGL(1,ℤ/8)	1	1	yes	characteristic
194	C₂×C₈	AGL(1,ℤ/8)	1	1	yes	characteristic
195	C₄⋊C₄	—	4	2	no	—
196	C₂²⋊C₄	—	4	4	no	—
197	C₄²	—	2	1	no	—
198	C₄²	—	2	1	no	—
199	C₄²	—	2	1	no	—
200	C₄²	—	4	4	no	—
201	C₄²	C₂²⋊C₈	1	1	yes	characteristic
202	C₄²	C₂²⋊C₈	1	1	yes	characteristic
203	C₄²	#102	1	1	yes	characteristic
204	C₄²	#102	1	1	yes	characteristic
205	C₄²	#124	1	1	yes	characteristic
206	C₄²	#127	1	1	yes	characteristic
207	C₁₆	—	4	2	no	—
208	C₁₆	—	4	2	no	—
209	C₂³	—	4	2	no	—
210	C₂³	#421	1	1	yes	characteristic
211	Q₈	—	4	2	no	—
212	Q₈	—	4	2	no	—
213	D₄	—	4	2	no	—
214	D₄	—	4	2	no	—
215	D₄	—	8	4	no	—
216	D₄	—	8	2	no	—
217	D₄	—	8	2	no	—
218	C₂×C₄	—	2	1	no	—
219	C₂×C₄	—	2	1	no	—
220	C₂×C₄	—	2	1	no	—
221	C₂×C₄	—	2	1	no	—
222	C₂×C₄	—	2	1	no	—
223	C₂×C₄	—	2	1	no	—
224	C₂×C₄	—	4	2	no	—
225	C₂×C₄	—	4	2	no	—
226	C₂×C₄	—	4	2	no	—
227	C₂×C₄	—	4	2	no	—
228	C₂×C₄	—	8	2	no	—
229	C₂×C₄	—	8	2	no	—
230	C₂×C₄	#390	1	1	yes	characteristic
231	C₂×C₄	C₂×C₂²⋊C₈	1	1	yes	characteristic
232	C₂×C₄	#408	1	1	yes	characteristic
233	C₂×C₄	C₂×#102	1	1	yes	characteristic
234	C₂×C₄	#418	1	1	yes	characteristic
235	C₂×C₄	C₂×#104	1	1	yes	characteristic
236	C₈	—	4	1	no	—
237	C₈	—	8	4	no	—
238	C₈	—	8	2	no	—
239	C₈	D₄⋊C₈	1	1	yes	characteristic
240	C₈	D₄⋊C₈	1	1	yes	characteristic
241	C₈	D₄⋊C₈	1	1	yes	characteristic
242	C₈	D₄⋊C₈	1	1	yes	characteristic
243	C₈	C₈×D₄	1	1	yes	characteristic
244	C₈	C₈×D₄	1	1	yes	characteristic
245	C₈	#436	1	1	yes	characteristic
246	C₈	#436	1	1	yes	characteristic
247	C₈	#459	1	1	yes	characteristic
248	C₈	#459	1	1	yes	characteristic
249	C₈	#460	1	1	yes	characteristic
250	C₈	#460	1	1	yes	characteristic
251	C₂²	—	4	1	no	—
252	C₂²	—	8	4	no	—
253	C₂²	—	8	2	no	—
254	C₂²	#1478	1	1	yes	characteristic
255	C₂²	#1480	1	1	yes	characteristic
256	C₂²	#1980	1	1	yes	characteristic
257	C₄	—	2	2	no	—
258	C₄	—	2	2	no	—
259	C₄	—	2	2	no	—
260	C₄	—	4	1	no	—
261	C₄	—	4	1	no	—
262	C₄	—	4	1	no	—
263	C₄	—	8	2	no	—
264	C₄	C₂×D₄⋊C₈	1	1	yes	characteristic
265	C₄	#1484	1	1	yes	characteristic
266	C₄	#1954	1	1	yes	characteristic
267	C₄	#1982	1	1	yes	characteristic
268	C₂	—	2	1	no	—
269	C₂	—	2	1	no	—
270	C₂	—	2	1	no	—
271	C₂	—	16	2	no	—
272	C₂	#9674	1	1	yes	characteristic

	subgroup	G/H
center	C₈	#459
commutator	C₂×C₈	C₂²×C₈
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂	#9674
prime radical	#3147	C₄
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈ ◁ C₂²×C₈ ◁ C₄²×C₈ ◁ 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
2. C₄×#364	C₂	semidirect
3. #11604	C₂	semidirect
4. #11602	C₂	semidirect
5. #8266	C₂	semidirect
31. #495	C₈	semidirect
32. #494	C₈	semidirect
33. #493	C₈	semidirect
61. #364	D₄	semidirect
62. #364	D₄	semidirect

Maximal subgroups

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

subgroup	index
#21705	2
C₄×#364	2
#11604	2
#11602	2
#8266	2