Subgroups — #147265

Order 512.

542 conjugacy classes of subgroups.

The subgroup lattice has 381 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#2159	C₂	1	1	yes	maximal, characteristic
2	#20455	C₂	1	1	yes	maximal, characteristic
3	#12567	C₂	1	1	yes	maximal, characteristic
4	C₂×#1384	C₂	1	1	yes	maximal, characteristic
5	#12557	C₂	1	1	yes	maximal, characteristic
6	C₂×#1378	C₂	1	1	yes	maximal, characteristic
7	#10871	C₂	1	1	yes	maximal, characteristic
8	C₂²×M₃₂	C₂²	1	1	yes	characteristic
9	C₂²×M₃₂	C₂²	1	1	yes	characteristic
10	C₂×#427	C₂²	1	1	yes	characteristic
11	C₂×#427	C₂²	1	1	yes	characteristic
12	C₂×#427	C₂²	1	1	yes	characteristic
13	C₂²×#107	C₂²	1	1	yes	characteristic
14	C₂×#405	C₂²	1	1	yes	characteristic
15	#2274	—	2	1	no	—
16	#2159	—	2	2	no	—
17	#2159	—	2	2	no	—
18	C₂×#367	C₄	1	2	yes	complemented
19	C₂×#366	C₄	1	2	yes	complemented
20	#1776	—	2	2	no	—
21	C₄×#107	—	2	2	no	—
22	#1740	—	2	1	no	—
23	#1736	—	2	1	no	—
24	#1733	—	2	1	no	—
25	#1386	—	2	2	no	—
26	#1384	C₄	1	4	yes	complemented
27	#1380	—	2	2	no	—
28	#1378	C₄	1	4	yes	complemented
29	C₂²×M₁₆	D₄	1	1	yes	characteristic
30	C₂³×C₈	C₂³	1	1	yes	characteristic
31	C₂×#117	—	2	1	no	—
32	C₂×#117	—	2	1	no	—
33	C₂×#117	—	2	1	no	—
34	C₂×#117	D₄	1	1	yes	characteristic
35	C₂²×C₄⋊C₄	—	2	1	no	—
36	C₂×M₃₂	—	2	1	no	—
37	C₂×M₃₂	—	2	1	no	—
38	C₂×M₃₂	—	2	1	no	—
39	C₂×M₃₂	—	2	1	no	—
40	C₂×M₃₂	—	2	1	no	—
41	C₂×M₃₂	—	2	1	no	—
42	C₂×M₃₂	—	2	1	no	—
43	C₂×M₃₂	—	2	1	no	—
44	C₂×M₃₂	—	2	1	no	—
45	C₂×M₃₂	C₂×C₄	1	2	yes	—
46	C₂×M₃₂	C₂×C₄	1	2	yes	—
47	C₂×M₃₂	C₂×C₄	1	2	yes	—
48	C₂²×C₁₆	—	2	1	no	—
49	C₂²×C₁₆	C₂×C₄	1	2	yes	—
50	#427	—	2	1	no	—
51	#427	—	2	1	no	—
52	#427	—	2	1	no	—
53	#427	—	2	1	no	—
54	#427	—	2	1	no	—
55	#427	—	2	1	no	—
56	#427	—	2	1	no	—
57	#427	—	2	1	no	—
58	#427	—	2	1	no	—
59	#427	C₂×C₄	1	2	yes	—
60	#427	C₂×C₄	1	2	yes	—
61	#427	Q₈	1	2	yes	—
62	C₂×#107	—	2	1	no	—
63	C₂×#107	—	2	1	no	—
64	C₂×#107	—	2	1	no	—
65	C₂×#107	—	2	1	no	—
66	C₂×#107	—	2	1	no	—
67	C₂×#107	C₂×C₄	1	1	yes	characteristic
68	C₂×#107	C₂×C₄	1	1	yes	characteristic
69	C₂×#107	C₂×C₄	1	2	yes	—
70	C₂×#107	D₄	1	2	yes	—
71	C₂×#107	D₄	1	2	yes	—
72	C₂×#106	—	2	1	no	—
73	C₂×#106	—	2	1	no	—
74	C₂×#106	D₄	1	2	yes	—
75	#405	—	2	1	no	—
76	#405	—	2	1	no	—
77	#405	C₂×C₄	1	2	yes	—
78	#405	Q₈	1	2	yes	—
79	C₂×#97	—	2	1	no	—
80	C₂×C₄×C₈	—	2	1	no	—
81	C₄×C₄⋊C₄	—	2	4	no	—
82	C₄×C₄⋊C₄	—	4	2	no	—
83	C₄×C₂²⋊C₄	—	4	2	no	—
84	#367	—	2	4	no	—
85	#366	—	2	4	no	—
86	#342	—	4	2	no	—
87	#340	—	2	4	no	—
88	#339	—	4	2	no	—
89	C₂³×C₄	—	4	1	no	—
90	C₂³×C₄	C₂×D₄	1	1	yes	characteristic
91	C₂×M₁₆	—	2	1	no	—
92	C₂×M₁₆	—	2	1	no	—
93	C₂×M₁₆	—	2	1	no	—
94	C₂×M₁₆	—	2	1	no	—
95	C₂×M₁₆	—	2	1	no	—
96	C₂×M₁₆	C₄⋊C₄	1	2	yes	—
97	C₂²×C₈	—	2	1	no	—
98	C₂²×C₈	C₂²×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	#41	1	1	yes	characteristic
107	C₂²×C₈	#41	1	1	yes	characteristic
108	C₂²×C₈	#41	1	1	yes	characteristic
109	C₂²×C₈	#41	1	1	yes	characteristic
110	C₂²×C₈	#41	1	1	yes	characteristic
111	C₂²×C₈	#41	1	1	yes	characteristic
112	#117	—	2	1	no	—
113	#117	—	2	1	no	—
114	#117	—	2	2	no	—
115	#117	—	2	2	no	—
116	#117	—	2	2	no	—
117	#117	—	4	1	no	—
118	#117	—	4	1	no	—
119	#117	—	4	1	no	—
120	#117	—	4	1	no	—
121	#117	—	4	1	no	—
122	#117	—	4	1	no	—
123	#117	—	4	1	no	—
124	#117	—	4	1	no	—
125	#117	—	4	1	no	—
126	#117	C₄⋊C₄	1	2	yes	—
127	#117	Q₁₆	1	2	yes	—
128	C₂×C₄⋊C₄	—	2	1	no	—
129	C₂×C₄⋊C₄	—	2	2	no	—
130	C₂×C₄⋊C₄	—	2	2	no	—
131	C₂×C₄⋊C₄	—	2	2	no	—
132	C₂×C₄⋊C₄	—	2	2	no	—
133	C₂×C₄⋊C₄	—	2	1	no	—
134	C₂×C₄⋊C₄	—	2	1	no	—
135	C₂×C₄⋊C₄	—	4	1	no	—
136	C₂×C₄⋊C₄	—	4	1	no	—
137	C₂×C₄⋊C₄	—	4	1	no	—
138	C₂×C₄⋊C₄	—	4	1	no	—
139	C₂×C₂²⋊C₄	—	4	1	no	—
140	C₂×C₂²⋊C₄	—	4	1	no	—
141	C₂×C₂²⋊C₄	—	4	1	no	—
142	C₂×C₂²⋊C₄	D₈	1	2	yes	—
143	C₂×C₄²	—	2	1	no	—
144	C₂×C₄²	—	4	1	no	—
145	C₂×C₄²	—	4	1	no	—
146	C₂×C₄²	—	4	1	no	—
147	C₂×C₄²	—	4	4	no	—
148	M₃₂	—	2	2	no	—
149	M₃₂	—	2	2	no	—
150	M₃₂	—	2	2	no	—
151	M₃₂	—	2	2	no	—
152	M₃₂	—	2	2	no	—
153	M₃₂	—	2	2	no	—
154	M₃₂	—	4	1	no	—
155	M₃₂	—	4	1	no	—
156	C₂×C₁₆	—	2	2	no	—
157	C₂×C₁₆	—	2	2	no	—
158	C₂×C₁₆	—	2	2	no	—
159	C₂×C₁₆	—	4	1	no	—
160	C₂×C₁₆	—	4	1	no	—
161	C₂×C₁₆	C₄²	1	4	yes	complemented
162	#107	—	2	2	no	—
163	#107	—	2	2	no	—
164	#107	—	2	2	no	—
165	#107	—	2	2	no	—
166	#107	—	2	2	no	—
167	#107	—	2	1	no	—
168	#107	—	2	1	no	—
169	#107	—	2	2	no	—
170	#107	—	2	1	no	—
171	#107	—	2	1	no	—
172	#107	—	4	1	no	—
173	#107	C₂²⋊C₄	1	4	yes	—
174	#106	—	2	2	no	—
175	#106	—	2	2	no	—
176	#106	—	4	1	no	—
177	#106	—	4	1	no	—
178	#106	—	4	1	no	—
179	#106	C₄⋊C₄	1	4	yes	—
180	#97	—	2	2	no	—
181	#97	—	2	2	no	—
182	C₄×C₈	—	2	2	no	—
183	C₄×C₈	—	2	2	no	—
184	#95	—	4	4	no	—
185	C₂⁴	C₂×D₈	1	1	yes	characteristic
186	C₂²×C₄	—	2	2	no	—
187	C₂²×C₄	—	4	2	no	—
188	C₂²×C₄	—	4	2	no	—
189	C₂²×C₄	—	4	2	no	—
190	C₂²×C₄	—	4	2	no	—
191	C₂²×C₄	—	4	1	no	—
192	C₂²×C₄	—	4	1	no	—
193	C₂²×C₄	—	8	1	no	—
194	C₂²×C₄	—	16	2	no	—
195	C₂²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
196	C₂²×C₄	C₄×D₄	1	1	yes	characteristic
197	C₂²×C₄	C₄×D₄	1	1	yes	characteristic
198	C₂²×C₄	#120	1	1	yes	characteristic
199	C₂²×C₄	#121	1	1	yes	characteristic
200	C₂²×C₄	C₂²⋊Q₈	1	1	yes	characteristic
201	C₂²×C₄	#123	1	1	yes	characteristic
202	C₂²×C₄	C₂×Q₁₆	1	1	yes	characteristic
203	C₂²×C₄	#135	1	1	yes	characteristic
204	C₂²×C₄	#135	1	1	yes	characteristic
205	C₂²×C₄	AGL(1,ℤ/8)	1	1	yes	characteristic
206	C₂²×C₄	AGL(1,ℤ/8)	1	1	yes	characteristic
207	C₂²×C₄	#137	1	1	yes	characteristic
208	C₂²×C₄	#137	1	1	yes	characteristic
209	M₁₆	—	2	2	no	—
210	M₁₆	—	2	2	no	—
211	M₁₆	—	4	1	no	—
212	M₁₆	—	4	1	no	—
213	C₂×C₈	—	2	2	no	—
214	C₂×C₈	—	4	1	no	—
215	C₂×C₈	—	4	1	no	—
216	C₂×C₈	C₂×C₄²	1	1	yes	characteristic
217	C₂×C₈	C₂×C₂²⋊C₄	1	1	yes	characteristic
218	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
219	C₂×C₈	#117	1	1	yes	characteristic
220	C₂×C₈	#117	1	1	yes	characteristic
221	C₂×C₈	#117	1	1	yes	characteristic
222	C₂×C₈	C₄×D₄	1	1	yes	characteristic
223	C₂×C₈	C₄×D₄	1	1	yes	characteristic
224	C₂×C₈	C₄×D₄	1	1	yes	characteristic
225	C₂×C₈	C₄×D₄	1	1	yes	characteristic
226	C₂×C₈	C₄×D₄	1	1	yes	characteristic
227	C₂×C₈	C₄×D₄	1	1	yes	characteristic
228	C₂×C₈	C₄×Q₈	1	1	yes	characteristic
229	C₂×C₈	C₄×Q₈	1	1	yes	characteristic
230	C₂×C₈	C₄×Q₈	1	1	yes	characteristic
231	C₂×C₈	C₄×Q₈	1	1	yes	characteristic
232	C₂×C₈	#121	1	1	yes	characteristic
233	C₂×C₈	C₂²⋊Q₈	1	1	yes	characteristic
234	C₂×C₈	C₂²⋊Q₈	1	1	yes	characteristic
235	C₂×C₈	C₂²⋊Q₈	1	1	yes	characteristic
236	C₂×C₈	#123	1	1	yes	characteristic
237	C₂×C₈	#123	1	1	yes	characteristic
238	C₂×C₈	#124	1	1	yes	characteristic
239	C₂×C₈	#125	1	1	yes	characteristic
240	C₂×C₈	#125	1	1	yes	characteristic
241	C₂×C₈	#126	1	1	yes	characteristic
242	C₂×C₈	#126	1	1	yes	characteristic
243	C₂×C₈	C₄⋊Q₈	1	1	yes	characteristic
244	C₄⋊C₄	—	2	2	no	—
245	C₄⋊C₄	—	2	2	no	—
246	C₄⋊C₄	—	2	4	no	—
247	C₄⋊C₄	—	2	4	no	—
248	C₄⋊C₄	—	4	2	no	—
249	C₄⋊C₄	—	4	2	no	—
250	C₄⋊C₄	—	4	2	no	—
251	C₄⋊C₄	—	4	2	no	—
252	C₄⋊C₄	—	4	1	no	—
253	C₄⋊C₄	—	4	1	no	—
254	C₄⋊C₄	—	4	2	no	—
255	C₄⋊C₄	—	4	1	no	—
256	C₄⋊C₄	—	4	1	no	—
257	C₄⋊C₄	—	8	1	no	—
258	C₂²⋊C₄	—	2	2	no	—
259	C₂²⋊C₄	—	4	2	no	—
260	C₂²⋊C₄	—	4	2	no	—
261	C₂²⋊C₄	—	4	2	no	—
262	C₂²⋊C₄	—	8	1	no	—
263	C₂²⋊C₄	—	8	1	no	—
264	C₂²⋊C₄	—	8	1	no	—
265	C₂²⋊C₄	#107	1	4	yes	—
266	C₄²	—	2	2	no	—
267	C₄²	—	2	2	no	—
268	C₄²	—	4	2	no	—
269	C₄²	—	4	2	no	—
270	C₄²	—	4	2	no	—
271	C₄²	—	8	1	no	—
272	C₄²	—	8	1	no	—
273	C₄²	—	8	1	no	—
274	C₄²	—	16	4	no	—
275	C₁₆	—	2	4	no	—
276	C₁₆	—	4	2	no	—
277	C₂³	—	4	1	no	—
278	C₂³	—	4	1	no	—
279	C₂³	C₂×#107	1	1	yes	characteristic
280	C₂³	C₄×D₈	1	1	yes	characteristic
281	C₂³	C₄×D₈	1	1	yes	characteristic
282	C₂³	#447	1	1	yes	characteristic
283	C₂³	#459	1	1	yes	characteristic
284	C₂³	#474	1	1	yes	characteristic
285	C₂³	#480	1	1	yes	characteristic
286	C₂×C₄	—	2	4	no	—
287	C₂×C₄	—	4	2	no	—
288	C₂×C₄	—	4	2	no	—
289	C₂×C₄	—	4	4	no	—
290	C₂×C₄	—	4	4	no	—
291	C₂×C₄	—	8	2	no	—
292	C₂×C₄	—	8	4	no	—
293	C₂×C₄	—	8	2	no	—
294	C₂×C₄	—	8	1	no	—
295	C₂×C₄	—	8	1	no	—
296	C₂×C₄	—	8	2	no	—
297	C₂×C₄	—	16	2	no	—
298	C₂×C₄	—	16	2	no	—
299	C₂×C₄	—	16	1	no	—
300	C₂×C₄	—	16	1	no	—
301	C₂×C₄	—	16	1	no	—
302	C₂×C₄	—	16	1	no	—
303	C₂×C₄	C₄×C₄⋊C₄	1	1	yes	characteristic
304	C₂×C₄	#385	1	1	yes	characteristic
305	C₂×C₄	#385	1	1	yes	characteristic
306	C₂×C₄	#386	1	1	yes	characteristic
307	C₂×C₄	#387	1	1	yes	characteristic
308	C₂×C₄	#388	1	1	yes	characteristic
309	C₂×C₄	#389	1	1	yes	characteristic
310	C₂×C₄	#427	1	1	yes	characteristic
311	C₂×C₄	#428	1	1	yes	characteristic
312	C₂×C₄	#428	1	1	yes	characteristic
313	C₂×C₄	C₄×Q₁₆	1	1	yes	characteristic
314	C₂×C₄	C₄×Q₁₆	1	1	yes	characteristic
315	C₂×C₄	#440	1	1	yes	characteristic
316	C₂×C₄	#440	1	1	yes	characteristic
317	C₂×C₄	#441	1	1	yes	characteristic
318	C₂×C₄	#442	1	1	yes	characteristic
319	C₂×C₄	#449	1	1	yes	characteristic
320	C₂×C₄	#451	1	1	yes	characteristic
321	C₂×C₄	#452	1	1	yes	characteristic
322	C₂×C₄	#462	1	1	yes	characteristic
323	C₂×C₄	#463	1	1	yes	characteristic
324	C₂×C₄	#464	1	1	yes	characteristic
325	C₂×C₄	#477	1	1	yes	characteristic
326	C₂×C₄	#478	1	1	yes	characteristic
327	C₂×C₄	#479	1	1	yes	characteristic
328	C₂×C₄	#482	1	1	yes	characteristic
329	C₂×C₄	#484	1	1	yes	characteristic
330	C₂×C₄	#485	1	1	yes	characteristic
331	C₈	—	4	2	no	—
332	C₈	C₄×C₂²⋊C₄	1	1	yes	characteristic
333	C₈	C₄×C₄⋊C₄	1	1	yes	characteristic
334	C₈	#387	1	1	yes	characteristic
335	C₈	#387	1	1	yes	characteristic
336	C₈	#387	1	1	yes	characteristic
337	C₈	#388	1	1	yes	characteristic
338	C₈	#389	1	1	yes	characteristic
339	C₈	#391	1	1	yes	characteristic
340	C₂²	—	2	1	no	—
341	C₂²	—	2	1	no	—
342	C₂²	—	2	1	no	—
343	C₂²	—	2	1	no	—
344	C₂²	—	2	1	no	—
345	C₂²	—	2	1	no	—
346	C₂²	—	4	1	no	—
347	C₂²	—	4	1	no	—
348	C₂²	—	4	1	no	—
349	C₂²	—	4	1	no	—
350	C₂²	C₄×#107	1	1	yes	characteristic
351	C₂²	#1852	1	1	yes	characteristic
352	C₂²	#1862	1	1	yes	characteristic
353	C₂²	#1879	1	1	yes	characteristic
354	C₂²	#1918	1	1	yes	characteristic
355	C₂²	#1925	1	1	yes	characteristic
356	C₂²	#1947	1	1	yes	characteristic
357	C₄	—	4	4	no	—
358	C₄	—	8	4	no	—
359	C₄	—	8	4	no	—
360	C₄	—	16	2	no	—
361	C₄	—	16	1	no	—
362	C₄	—	16	1	no	—
363	C₄	#1436	1	1	yes	characteristic
364	C₄	#1776	1	1	yes	characteristic
365	C₄	#1854	1	1	yes	characteristic
366	C₄	#1863	1	1	yes	characteristic
367	C₄	#1878	1	1	yes	characteristic
368	C₄	#1921	1	1	yes	characteristic
369	C₄	#1928	1	1	yes	characteristic
370	C₄	#1949	1	1	yes	characteristic
371	C₂	—	2	1	no	—
372	C₂	—	2	1	no	—
373	C₂	—	2	1	no	—
374	C₂	—	2	1	no	—
375	C₂	—	2	1	no	—
376	C₂	—	2	1	no	—
377	C₂	#10907	1	1	yes	characteristic
378	C₂	#13033	1	1	yes	characteristic
379	C₂	#13034	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	#1879
commutator	C₂×C₈	C₂×C₄²
Frattini, cosocle	C₂³×C₈	C₂³
prime radical	C₂⁴	C₂×D₈
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
18. C₂×#367	C₄	semidirect
19. C₂×#366	C₄	semidirect
26. #1384	C₄	semidirect
28. #1378	C₄	semidirect
161. C₂×C₁₆	C₄²	semidirect

Maximal subgroups

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

subgroup	index
C₂×#2159	2
#20455	2
#12567	2
C₂×#1384	2
#12557	2
C₂×#1378	2
#10871	2