Subgroups — #496725

Order 512.

555 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#33418	C₂	1	1	yes	maximal, characteristic
2	#20818	C₂	1	1	yes	maximal, characteristic
3	C₂×#1848	C₂	1	1	yes	maximal, characteristic
4	#13185	C₂	1	4	yes	maximal, complemented
5	#12928	C₂	1	4	yes	maximal
6	#12904	C₂	1	2	yes	maximal, complemented
7	#12887	C₂	1	2	yes	maximal, complemented
8	C₂²×#108	—	2	1	no	—
9	C₂²×#108	C₂²	1	2	yes	—
10	C₂×#423	—	2	1	no	—
11	C₂²×C₄×C₈	C₂²	1	1	yes	characteristic
12	#2283	—	2	1	no	—
13	#2163	C₄	1	2	yes	complemented
14	#2163	C₄	1	2	yes	complemented
15	#2162	C₄	1	2	yes	complemented
16	#2162	C₄	1	2	yes	complemented
17	C₂×#365	C₂²	1	8	yes	—
18	#2150	C₂²	1	2	yes	—
19	#2112	C₂²	1	2	yes	—
20	C₄×M₃₂	C₂²	1	2	yes	—
21	#1848	—	2	1	no	—
22	#1848	—	2	1	no	—
23	#1848	C₂²	1	2	yes	—
24	#1848	C₂²	1	2	yes	—
25	#1846	C₂²	1	1	yes	characteristic
26	#1846	C₂²	1	1	yes	characteristic
27	#1835	—	2	1	no	—
28	#1808	—	2	1	no	—
29	#1807	C₂²	1	2	yes	—
30	C₄×#107	—	2	1	no	—
31	C₄×#106	C₂²	1	2	yes	—
32	#1383	C₂²	1	8	yes	—
33	C₂²×M₁₆	—	2	2	no	—
34	C₂³×C₈	D₄	1	1	yes	characteristic
35	C₂³×C₈	C₂³	1	1	yes	characteristic
36	C₂²×C₄²	D₄	1	1	yes	characteristic
37	C₂×M₃₂	C₂³	1	4	yes	—
38	C₂×#108	—	2	2	no	—
39	C₂×#108	—	2	2	no	—
40	C₂×#108	—	2	2	no	—
41	C₂×#108	—	4	4	no	—
42	C₂×#108	—	4	4	no	—
43	C₂×#108	C₂³	1	4	yes	—
44	C₂×#107	—	2	1	no	—
45	C₂×#107	—	2	1	no	—
46	C₂×#106	C₂³	1	2	yes	—
47	C₂×#106	C₂³	1	2	yes	—
48	#423	—	2	2	no	—
49	#423	—	2	2	no	—
50	#423	—	4	1	no	—
51	#423	—	4	1	no	—
52	C₂×C₄⋊C₈	—	4	1	no	—
53	C₂×C₄⋊C₈	—	4	1	no	—
54	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
55	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
56	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
57	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
58	C₂×C₄×C₈	D₄	1	1	yes	characteristic
59	C₂×C₄×C₈	D₄	1	1	yes	characteristic
60	C₂×C₄×C₈	Q₈	1	1	yes	characteristic
61	C₂×C₄×C₈	Q₈	1	1	yes	characteristic
62	C₂×C₄×C₈	Q₈	1	1	yes	characteristic
63	C₂×C₄×C₈	Q₈	1	1	yes	characteristic
64	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
65	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
66	#383	—	4	1	no	—
67	#382	—	2	1	no	—
68	#382	—	2	1	no	—
69	#381	—	4	1	no	—
70	#380	—	2	2	no	—
71	C₄×C₄⋊C₄	—	2	2	no	—
72	C₄×C₂²⋊C₄	—	4	1	no	—
73	#376	—	4	1	no	—
74	#365	—	4	8	no	—
75	C₄⋊C₁₆	C₂×C₄	1	4	yes	—
76	C₄⋊C₁₆	C₂×C₄	1	4	yes	—
77	C₂²⋊C₁₆	C₂×C₄	1	4	yes	—
78	C₂²⋊C₁₆	C₂×C₄	1	4	yes	—
79	#346	C₂×C₄	1	4	yes	—
80	C₄×C₁₆	C₂×C₄	1	4	yes	—
81	#340	—	2	2	no	—
82	#340	—	2	2	no	—
83	C₂³×C₄	SD₁₆	1	1	yes	characteristic
84	C₂³×C₄	SD₁₆	1	1	yes	characteristic
85	C₂³×C₄	C₂×D₄	1	1	yes	characteristic
86	C₂×M₁₆	—	2	4	no	—
87	C₂×M₁₆	—	4	2	no	—
88	C₂×M₁₆	—	4	4	no	—
89	C₂×M₁₆	—	4	4	no	—
90	C₂²×C₈	—	2	2	no	—
91	C₂²×C₈	—	2	2	no	—
92	C₂²×C₈	—	2	2	no	—
93	C₂²×C₈	—	2	2	no	—
94	C₂²×C₈	—	4	2	no	—
95	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
96	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
97	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
98	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
99	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
100	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
101	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
102	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
103	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
104	C₂²×C₈	C₂×Q₈	1	1	yes	characteristic
105	C₂²×C₈	C₂×Q₈	1	1	yes	characteristic
106	C₂²×C₈	C₂⁴	1	1	yes	characteristic
107	C₂×C₄⋊C₄	—	2	2	no	—
108	C₂×C₄⋊C₄	—	2	2	no	—
109	C₂×C₂²⋊C₄	—	4	2	no	—
110	C₂×C₄²	—	4	2	no	—
111	C₂×C₄²	C₄⋊C₄	1	1	yes	characteristic
112	C₂×C₄²	C₄⋊C₄	1	1	yes	characteristic
113	C₂×C₄²	C₄⋊C₄	1	1	yes	characteristic
114	C₂×C₄²	C₄⋊C₄	1	1	yes	characteristic
115	C₂×C₄²	SD₁₆	1	1	yes	characteristic
116	C₂×C₄²	SD₁₆	1	1	yes	characteristic
117	C₂×C₄²	SD₁₆	1	1	yes	characteristic
118	C₂×C₄²	SD₁₆	1	1	yes	characteristic
119	C₂×C₄²	SD₁₆	1	1	yes	characteristic
120	C₂×C₄²	SD₁₆	1	1	yes	characteristic
121	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
122	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
123	M₃₂	—	2	4	no	—
124	M₃₂	—	2	4	no	—
125	C₂×C₁₆	C₂²×C₄	1	8	yes	—
126	#108	—	2	8	no	—
127	#108	—	4	4	no	—
128	#108	—	4	4	no	—
129	#108	—	4	4	no	—
130	#107	—	4	2	no	—
131	#107	—	4	2	no	—
132	#106	—	2	4	no	—
133	#106	—	2	4	no	—
134	C₄⋊C₈	—	4	2	no	—
135	C₄⋊C₈	—	4	2	no	—
136	C₄⋊C₈	—	4	2	no	—
137	C₄⋊C₈	—	4	2	no	—
138	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
139	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
140	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
141	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
142	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
143	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
144	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
145	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
146	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
147	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
148	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
149	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
150	C₄×C₈	C₂×Q₈	1	1	yes	characteristic
151	C₄×C₈	C₂×Q₈	1	1	yes	characteristic
152	C₄×C₈	C₂×Q₈	1	1	yes	characteristic
153	C₄×C₈	C₂×Q₈	1	1	yes	characteristic
154	#95	—	4	2	no	—
155	#95	—	4	1	no	—
156	#95	—	4	1	no	—
157	#95	—	4	1	no	—
158	#95	—	4	1	no	—
159	#95	—	4	2	no	—
160	C₂⁴	C₂×SD₁₆	1	1	yes	characteristic
161	C₂²×C₄	—	2	1	no	—
162	C₂²×C₄	—	2	1	no	—
163	C₂²×C₄	—	2	1	no	—
164	C₂²×C₄	—	2	1	no	—
165	C₂²×C₄	—	4	2	no	—
166	C₂²×C₄	—	4	2	no	—
167	C₂²×C₄	—	4	2	no	—
168	C₂²×C₄	—	4	2	no	—
169	C₂²×C₄	#106	1	1	yes	characteristic
170	C₂²×C₄	#106	1	1	yes	characteristic
171	C₂²×C₄	#106	1	1	yes	characteristic
172	C₂²×C₄	#106	1	1	yes	characteristic
173	C₂²×C₄	#106	1	1	yes	characteristic
174	C₂²×C₄	#106	1	1	yes	characteristic
175	C₂²×C₄	#106	1	1	yes	characteristic
176	C₂²×C₄	#106	1	1	yes	characteristic
177	C₂²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
178	C₂²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
179	C₂²×C₄	C₂×SD₁₆	1	1	yes	characteristic
180	C₂²×C₄	C₂×SD₁₆	1	1	yes	characteristic
181	C₂²×C₄	C₂×SD₁₆	1	1	yes	characteristic
182	C₂²×C₄	C₂×SD₁₆	1	1	yes	characteristic
183	C₂²×C₄	C₂×SD₁₆	1	1	yes	characteristic
184	C₂²×C₄	C₂×SD₁₆	1	1	yes	characteristic
185	C₂²×C₄	C₂×SD₁₆	1	1	yes	characteristic
186	C₂²×C₄	C₂²×D₄	1	1	yes	characteristic
187	M₁₆	—	4	4	no	—
188	M₁₆	—	4	4	no	—
189	M₁₆	—	8	4	no	—
190	C₂×C₈	—	2	1	no	—
191	C₂×C₈	—	2	1	no	—
192	C₂×C₈	—	2	1	no	—
193	C₂×C₈	—	2	1	no	—
194	C₂×C₈	—	2	2	no	—
195	C₂×C₈	—	2	2	no	—
196	C₂×C₈	—	2	2	no	—
197	C₂×C₈	—	2	2	no	—
198	C₂×C₈	—	2	2	no	—
199	C₂×C₈	—	2	2	no	—
200	C₂×C₈	—	2	2	no	—
201	C₂×C₈	—	2	2	no	—
202	C₂×C₈	—	4	4	no	—
203	C₂×C₈	—	8	4	no	—
204	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
205	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
206	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
207	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
208	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
209	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
210	C₂×C₈	AGL(1,ℤ/8)	1	2	yes	—
211	C₂×C₈	#137	1	2	yes	—
212	C₂×C₈	C₂³×C₄	1	1	yes	characteristic
213	C₂×C₈	C₂²×Q₈	1	1	yes	characteristic
214	C₂×C₈	H(5,𝔽₂)	1	2	yes	—
215	C₂×C₈	2₋¹⁺⁴	1	2	yes	—
216	C₄⋊C₄	—	4	2	no	—
217	C₄⋊C₄	—	4	2	no	—
218	C₄⋊C₄	—	4	2	no	—
219	C₄⋊C₄	—	4	2	no	—
220	C₂²⋊C₄	—	8	4	no	—
221	C₄²	—	8	4	no	—
222	C₄²	#106	1	1	yes	characteristic
223	C₄²	#106	1	1	yes	characteristic
224	C₄²	#106	1	1	yes	characteristic
225	C₄²	#106	1	1	yes	characteristic
226	C₄²	#106	1	1	yes	characteristic
227	C₄²	#106	1	1	yes	characteristic
228	C₄²	#106	1	1	yes	characteristic
229	C₄²	#106	1	1	yes	characteristic
230	C₄²	C₂×C₄⋊C₄	1	1	yes	characteristic
231	C₄²	C₂×C₄⋊C₄	1	1	yes	characteristic
232	C₄²	C₂×C₄⋊C₄	1	1	yes	characteristic
233	C₄²	C₂×C₄⋊C₄	1	1	yes	characteristic
234	C₄²	C₂×SD₁₆	1	1	yes	characteristic
235	C₄²	C₂×SD₁₆	1	1	yes	characteristic
236	C₄²	C₂×SD₁₆	1	1	yes	characteristic
237	C₄²	C₂×SD₁₆	1	1	yes	characteristic
238	C₁₆	—	2	8	no	—
239	C₂³	—	2	2	no	—
240	C₂³	—	2	1	no	—
241	C₂³	—	2	1	no	—
242	C₂³	—	4	1	no	—
243	C₂³	C₂×#106	1	1	yes	characteristic
244	C₂³	C₂×#106	1	1	yes	characteristic
245	C₂³	C₂²×SD₁₆	1	1	yes	characteristic
246	C₂×C₄	—	2	2	no	—
247	C₂×C₄	—	2	2	no	—
248	C₂×C₄	—	2	1	no	—
249	C₂×C₄	—	2	1	no	—
250	C₂×C₄	—	2	1	no	—
251	C₂×C₄	—	2	1	no	—
252	C₂×C₄	—	2	1	no	—
253	C₂×C₄	—	2	1	no	—
254	C₂×C₄	—	2	2	no	—
255	C₂×C₄	—	2	2	no	—
256	C₂×C₄	—	2	2	no	—
257	C₂×C₄	—	2	1	no	—
258	C₂×C₄	—	2	1	no	—
259	C₂×C₄	—	2	1	no	—
260	C₂×C₄	—	2	1	no	—
261	C₂×C₄	—	4	1	no	—
262	C₂×C₄	—	4	1	no	—
263	C₂×C₄	—	4	1	no	—
264	C₂×C₄	—	4	1	no	—
265	C₂×C₄	—	4	1	no	—
266	C₂×C₄	—	4	1	no	—
267	C₂×C₄	—	4	1	no	—
268	C₂×C₄	—	8	2	no	—
269	C₂×C₄	—	8	2	no	—
270	C₂×C₄	—	8	2	no	—
271	C₂×C₄	—	8	2	no	—
272	C₂×C₄	—	8	2	no	—
273	C₂×C₄	—	8	2	no	—
274	C₂×C₄	C₂×#106	1	1	yes	characteristic
275	C₂×C₄	C₂×#106	1	1	yes	characteristic
276	C₂×C₄	C₂×#106	1	1	yes	characteristic
277	C₂×C₄	C₂×#106	1	1	yes	characteristic
278	C₂×C₄	C₂×#106	1	1	yes	characteristic
279	C₂×C₄	C₂×#106	1	1	yes	characteristic
280	C₂×C₄	C₂×#106	1	1	yes	characteristic
281	C₂×C₄	C₂×#106	1	1	yes	characteristic
282	C₂×C₄	C₂×#106	1	1	yes	characteristic
283	C₂×C₄	C₂×#106	1	1	yes	characteristic
284	C₂×C₄	C₂²×C₄⋊C₄	1	1	yes	characteristic
285	C₂×C₄	#535	1	1	yes	characteristic
286	C₂×C₄	#536	1	1	yes	characteristic
287	C₂×C₄	C₂²×SD₁₆	1	1	yes	characteristic
288	C₂×C₄	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
289	C₂×C₄	C₂×#137	1	1	yes	characteristic
290	C₈	—	2	1	no	—
291	C₈	—	2	1	no	—
292	C₈	—	2	1	no	—
293	C₈	—	2	1	no	—
294	C₈	—	8	4	no	—
295	C₈	#428	1	2	yes	—
296	C₈	#428	1	2	yes	—
297	C₈	#520	1	2	yes	—
298	C₈	#544	1	2	yes	—
299	C₂²	—	2	1	no	—
300	C₂²	—	2	1	no	—
301	C₂²	—	2	1	no	—
302	C₂²	—	2	1	no	—
303	C₂²	—	2	1	no	—
304	C₂²	—	2	1	no	—
305	C₂²	—	2	1	no	—
306	C₂²	—	4	2	no	—
307	C₂²	—	4	2	no	—
308	C₂²	#2155	1	2	yes	—
309	C₂²	C₂²×#106	1	1	yes	characteristic
310	C₂²	#3206	1	1	yes	characteristic
311	C₂²	#3227	1	1	yes	characteristic
312	C₄	—	2	1	no	—
313	C₄	—	2	1	no	—
314	C₄	—	2	1	no	—
315	C₄	—	2	1	no	—
316	C₄	—	2	1	no	—
317	C₄	—	2	1	no	—
318	C₄	—	2	1	no	—
319	C₄	—	2	1	no	—
320	C₄	—	2	1	no	—
321	C₄	—	16	2	no	—
322	C₄	—	16	2	no	—
323	C₄	#2155	1	2	yes	—
324	C₄	#2317	1	1	yes	characteristic
325	C₄	C₂×#428	1	1	yes	characteristic
326	C₄	#3208	1	1	yes	characteristic
327	C₄	#3229	1	1	yes	characteristic
328	C₂	—	2	1	no	—
329	C₂	—	2	1	no	—
330	C₂	—	2	2	no	—
331	C₂	—	4	1	no	—
332	C₂	#21981	1	1	yes	characteristic
333	C₂	C₂×#2155	1	1	yes	characteristic
334	C₂	C₂×#2155	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	C₂²×#106
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂⁴	C₂×SD₁₆
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
4. #13185	C₂	semidirect
6. #12904	C₂	semidirect
7. #12887	C₂	semidirect
13. #2163	C₄	semidirect
14. #2163	C₄	semidirect
15. #2162	C₄	semidirect
16. #2162	C₄	semidirect

Maximal subgroups

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

subgroup	index
#33418	2
#20818	2
C₂×#1848	2
#13185	2
#12928	2
#12904	2
#12887	2