Subgroups — #149272

Order 512.

499 conjugacy classes of subgroups.

The subgroup lattice has 353 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#2161	C₂	1	1	yes	maximal, characteristic
2	C₂×#1953	C₂	1	1	yes	maximal, characteristic
3	C₂×#1953	C₂	1	1	yes	maximal, characteristic
4	C₂×#1388	C₂	1	1	yes	maximal, characteristic
5	C₂×#1388	C₂	1	1	yes	maximal, characteristic
6	#12559	C₂	1	1	yes	maximal, characteristic
7	#12551	C₂	1	1	yes	maximal, characteristic
8	C₂²×M₃₂	C₂²	1	1	yes	characteristic
9	C₂²×M₃₂	C₂²	1	1	yes	characteristic
10	C₂×#430	—	2	1	no	—
11	C₂×#430	—	2	1	no	—
12	C₂²×#108	C₂²	1	1	yes	characteristic
13	C₂²×#108	C₂²	1	1	yes	characteristic
14	C₂²×#108	C₂²	1	1	yes	characteristic
15	C₂²×#108	C₂²	1	1	yes	characteristic
16	C₂×#405	C₂²	1	1	yes	characteristic
17	#2161	—	2	2	no	—
18	#2161	—	2	2	no	—
19	C₂×#368	—	2	2	no	—
20	#1953	—	2	2	no	—
21	#1953	—	2	2	no	—
22	#1953	—	2	2	no	—
23	#1953	—	2	2	no	—
24	C₂×#344	—	2	1	no	—
25	C₂×#344	—	2	1	no	—
26	#1388	—	2	2	no	—
27	#1388	C₄	1	4	yes	complemented
28	#1381	—	2	2	no	—
29	#1377	—	2	2	no	—
30	C₂²×M₁₆	—	2	1	no	—
31	C₂²×M₁₆	—	2	1	no	—
32	C₂²×M₁₆	—	2	1	no	—
33	C₂²×M₁₆	—	2	1	no	—
34	C₂²×M₁₆	D₄	1	1	yes	characteristic
35	C₂³×C₈	C₂³	1	1	yes	characteristic
36	C₂×#117	D₄	1	1	yes	characteristic
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	D₄	1	2	yes	—
47	C₂×M₃₂	Q₈	1	2	yes	—
48	C₂×M₃₂	Q₈	1	2	yes	—
49	C₂²×C₁₆	—	2	1	no	—
50	C₂²×C₁₆	—	2	1	no	—
51	#430	—	2	4	no	—
52	#430	—	2	4	no	—
53	#430	—	4	2	no	—
54	#430	—	4	2	no	—
55	C₂×#108	—	2	1	no	—
56	C₂×#108	—	2	1	no	—
57	C₂×#108	—	2	1	no	—
58	C₂×#108	—	2	1	no	—
59	C₂×#108	—	2	1	no	—
60	C₂×#108	—	2	1	no	—
61	C₂×#108	—	2	1	no	—
62	C₂×#108	—	2	1	no	—
63	C₂×#108	—	2	1	no	—
64	C₂×#108	—	2	1	no	—
65	C₂×#108	—	2	1	no	—
66	C₂×#108	—	2	1	no	—
67	C₂×#108	—	2	1	no	—
68	C₂×#108	—	2	1	no	—
69	C₂×#108	—	2	1	no	—
70	C₂×#108	—	2	1	no	—
71	C₂×#108	—	4	2	no	—
72	C₂×#108	—	4	2	no	—
73	C₂×#108	C₂×C₄	1	2	yes	—
74	C₂×#108	C₂×C₄	1	2	yes	—
75	C₂×#108	D₄	1	2	yes	—
76	C₂×#108	D₄	1	2	yes	—
77	C₂×#108	D₄	1	2	yes	—
78	C₂×#108	D₄	1	2	yes	—
79	C₂×#108	Q₈	1	2	yes	—
80	C₂×#108	Q₈	1	2	yes	—
81	#405	—	2	1	no	—
82	#405	—	2	1	no	—
83	#405	—	2	1	no	—
84	#405	—	2	1	no	—
85	C₂×#97	—	2	1	no	—
86	C₂×C₄×C₈	—	2	1	no	—
87	#368	—	2	8	no	—
88	#344	—	2	4	no	—
89	#344	—	2	4	no	—
90	C₂³×C₄	C₂×D₄	1	1	yes	characteristic
91	C₂×M₁₆	—	2	1	no	—
92	C₂×M₁₆	—	2	2	no	—
93	C₂×M₁₆	—	2	2	no	—
94	C₂×M₁₆	—	2	2	no	—
95	C₂×M₁₆	—	2	2	no	—
96	C₂×M₁₆	—	2	1	no	—
97	C₂×M₁₆	—	2	1	no	—
98	C₂×M₁₆	—	2	2	no	—
99	C₂×M₁₆	—	2	2	no	—
100	C₂×M₁₆	—	2	1	no	—
101	C₂×M₁₆	—	2	1	no	—
102	C₂×M₁₆	—	2	2	no	—
103	C₂×M₁₆	—	2	2	no	—
104	C₂×M₁₆	—	4	1	no	—
105	C₂×M₁₆	—	4	1	no	—
106	C₂×M₁₆	—	4	1	no	—
107	C₂×M₁₆	—	4	1	no	—
108	C₂×M₁₆	—	4	1	no	—
109	C₂×M₁₆	—	4	1	no	—
110	C₂×M₁₆	—	4	1	no	—
111	C₂×M₁₆	—	4	1	no	—
112	C₂×M₁₆	—	4	1	no	—
113	C₂×M₁₆	—	4	1	no	—
114	C₂×M₁₆	—	4	1	no	—
115	C₂×M₁₆	—	4	1	no	—
116	C₂×M₁₆	—	4	1	no	—
117	C₂×M₁₆	—	4	1	no	—
118	C₂×M₁₆	—	4	1	no	—
119	C₂×M₁₆	—	4	1	no	—
120	C₂×M₁₆	D₈	1	2	yes	—
121	C₂²×C₈	—	2	1	no	—
122	C₂²×C₈	—	4	1	no	—
123	C₂²×C₈	—	4	1	no	—
124	C₂²×C₈	—	4	1	no	—
125	C₂²×C₈	—	4	1	no	—
126	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
127	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
128	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
129	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
130	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
131	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
132	C₂²×C₈	C₂×Q₈	1	1	yes	characteristic
133	C₂²×C₈	C₂×Q₈	1	1	yes	characteristic
134	C₂²×C₈	C₂×Q₈	1	1	yes	characteristic
135	C₂²×C₈	C₂×Q₈	1	1	yes	characteristic
136	C₂²×C₈	#41	1	1	yes	characteristic
137	C₂²×C₈	#41	1	1	yes	characteristic
138	C₂²×C₈	#41	1	1	yes	characteristic
139	C₂²×C₈	#41	1	1	yes	characteristic
140	#117	—	2	1	no	—
141	#117	—	2	1	no	—
142	#117	—	2	1	no	—
143	#117	—	2	1	no	—
144	C₂×C₄⋊C₄	—	2	1	no	—
145	C₂×C₂²⋊C₄	Q₁₆	1	2	yes	—
146	C₂×C₄²	—	2	1	no	—
147	M₃₂	—	2	2	no	—
148	M₃₂	—	2	2	no	—
149	M₃₂	—	2	2	no	—
150	M₃₂	—	2	2	no	—
151	M₃₂	—	2	2	no	—
152	M₃₂	—	4	1	no	—
153	M₃₂	—	4	1	no	—
154	M₃₂	C₄⋊C₄	1	4	yes	—
155	C₂×C₁₆	—	2	2	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	#108	—	2	2	no	—
162	#108	—	2	2	no	—
163	#108	—	2	2	no	—
164	#108	—	2	2	no	—
165	#108	—	2	2	no	—
166	#108	—	2	2	no	—
167	#108	—	2	2	no	—
168	#108	—	2	2	no	—
169	#108	—	2	2	no	—
170	#108	—	2	2	no	—
171	#108	—	4	8	no	—
172	#108	—	4	8	no	—
173	#108	—	4	1	no	—
174	#108	—	4	1	no	—
175	#108	—	4	1	no	—
176	#108	—	4	1	no	—
177	#108	C₄⋊C₄	1	4	yes	—
178	#108	C₄⋊C₄	1	4	yes	—
179	#97	—	4	1	no	—
180	#97	—	4	1	no	—
181	C₄×C₈	—	4	1	no	—
182	C₄×C₈	—	4	1	no	—
183	C₂⁴	C₂×Q₁₆	1	1	yes	characteristic
184	C₂²×C₄	—	2	2	no	—
185	C₂²×C₄	C₄×D₄	1	1	yes	characteristic
186	C₂²×C₄	#120	1	1	yes	characteristic
187	C₂²×C₄	#120	1	1	yes	characteristic
188	C₂²×C₄	#121	1	1	yes	characteristic
189	C₂²×C₄	C₂²⋊Q₈	1	1	yes	characteristic
190	C₂²×C₄	C₂²⋊Q₈	1	1	yes	characteristic
191	C₂²×C₄	C₄⋊Q₈	1	1	yes	characteristic
192	C₂²×C₄	C₂×D₈	1	1	yes	characteristic
193	C₂²×C₄	#135	1	1	yes	characteristic
194	C₂²×C₄	#135	1	1	yes	characteristic
195	C₂²×C₄	AGL(1,ℤ/8)	1	1	yes	characteristic
196	C₂²×C₄	AGL(1,ℤ/8)	1	1	yes	characteristic
197	C₂²×C₄	#137	1	1	yes	characteristic
198	C₂²×C₄	#137	1	1	yes	characteristic
199	M₁₆	—	2	4	no	—
200	M₁₆	—	2	4	no	—
201	M₁₆	—	2	2	no	—
202	M₁₆	—	2	2	no	—
203	M₁₆	—	4	2	no	—
204	M₁₆	—	4	2	no	—
205	M₁₆	—	4	1	no	—
206	M₁₆	—	4	1	no	—
207	M₁₆	—	4	2	no	—
208	M₁₆	—	4	2	no	—
209	M₁₆	—	4	2	no	—
210	M₁₆	—	4	2	no	—
211	M₁₆	—	4	2	no	—
212	M₁₆	—	4	2	no	—
213	M₁₆	—	4	2	no	—
214	M₁₆	—	4	2	no	—
215	M₁₆	—	8	1	no	—
216	M₁₆	—	8	1	no	—
217	M₁₆	—	8	1	no	—
218	M₁₆	—	8	1	no	—
219	C₂×C₈	—	2	2	no	—
220	C₂×C₈	—	4	1	no	—
221	C₂×C₈	—	4	2	no	—
222	C₂×C₈	—	4	2	no	—
223	C₂×C₈	—	4	2	no	—
224	C₂×C₈	—	4	2	no	—
225	C₂×C₈	—	4	1	no	—
226	C₂×C₈	—	4	2	no	—
227	C₂×C₈	—	4	2	no	—
228	C₂×C₈	—	4	2	no	—
229	C₂×C₈	—	4	2	no	—
230	C₂×C₈	—	8	1	no	—
231	C₂×C₈	—	8	1	no	—
232	C₂×C₈	—	8	1	no	—
233	C₂×C₈	—	8	1	no	—
234	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
235	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
236	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
237	C₂×C₈	C₄×D₄	1	1	yes	characteristic
238	C₂×C₈	C₄×D₄	1	1	yes	characteristic
239	C₂×C₈	C₄×Q₈	1	1	yes	characteristic
240	C₂×C₈	#120	1	1	yes	characteristic
241	C₂×C₈	#121	1	1	yes	characteristic
242	C₂×C₈	#121	1	1	yes	characteristic
243	C₂×C₈	C₂²⋊Q₈	1	1	yes	characteristic
244	C₂×C₈	C₂²⋊Q₈	1	1	yes	characteristic
245	C₂×C₈	C₂²⋊Q₈	1	1	yes	characteristic
246	C₂×C₈	C₂²⋊Q₈	1	1	yes	characteristic
247	C₂×C₈	C₂²⋊Q₈	1	1	yes	characteristic
248	C₂×C₈	C₂²⋊Q₈	1	1	yes	characteristic
249	C₂×C₈	C₂²⋊Q₈	1	1	yes	characteristic
250	C₂×C₈	#123	1	1	yes	characteristic
251	C₂×C₈	#123	1	1	yes	characteristic
252	C₂×C₈	#123	1	1	yes	characteristic
253	C₂×C₈	#125	1	1	yes	characteristic
254	C₂×C₈	#125	1	1	yes	characteristic
255	C₂×C₈	#125	1	1	yes	characteristic
256	C₂×C₈	#127	1	1	yes	characteristic
257	C₂×C₈	C₄⋊Q₈	1	1	yes	characteristic
258	C₂×C₈	C₄⋊Q₈	1	1	yes	characteristic
259	C₂×C₈	C₄⋊Q₈	1	1	yes	characteristic
260	C₂×C₈	C₄⋊Q₈	1	1	yes	characteristic
261	C₂×C₈	C₄⋊Q₈	1	1	yes	characteristic
262	C₄⋊C₄	—	4	1	no	—
263	C₄⋊C₄	—	4	1	no	—
264	C₂²⋊C₄	—	2	2	no	—
265	C₂²⋊C₄	—	2	2	no	—
266	C₄²	—	4	1	no	—
267	C₄²	—	4	1	no	—
268	C₁₆	—	2	4	no	—
269	C₁₆	—	4	2	no	—
270	C₂³	—	8	1	no	—
271	C₂³	C₄×Q₁₆	1	1	yes	characteristic
272	C₂³	#451	1	1	yes	characteristic
273	C₂³	#451	1	1	yes	characteristic
274	C₂³	#467	1	1	yes	characteristic
275	C₂³	#477	1	1	yes	characteristic
276	C₂³	#477	1	1	yes	characteristic
277	C₂³	#500	1	1	yes	characteristic
278	C₂×C₄	—	4	2	no	—
279	C₂×C₄	—	8	2	no	—
280	C₂×C₄	#385	1	1	yes	characteristic
281	C₂×C₄	#385	1	1	yes	characteristic
282	C₂×C₄	#389	1	1	yes	characteristic
283	C₂×C₄	#392	1	1	yes	characteristic
284	C₂×C₄	#395	1	1	yes	characteristic
285	C₂×C₄	#395	1	1	yes	characteristic
286	C₂×C₄	#398	1	1	yes	characteristic
287	C₂×C₄	C₄×D₈	1	1	yes	characteristic
288	C₂×C₄	#440	1	1	yes	characteristic
289	C₂×C₄	#440	1	1	yes	characteristic
290	C₂×C₄	#447	1	1	yes	characteristic
291	C₂×C₄	#447	1	1	yes	characteristic
292	C₂×C₄	#449	1	1	yes	characteristic
293	C₂×C₄	#449	1	1	yes	characteristic
294	C₂×C₄	#452	1	1	yes	characteristic
295	C₂×C₄	#452	1	1	yes	characteristic
296	C₂×C₄	#466	1	1	yes	characteristic
297	C₂×C₄	#469	1	1	yes	characteristic
298	C₂×C₄	#470	1	1	yes	characteristic
299	C₂×C₄	#474	1	1	yes	characteristic
300	C₂×C₄	#474	1	1	yes	characteristic
301	C₂×C₄	#478	1	1	yes	characteristic
302	C₂×C₄	#478	1	1	yes	characteristic
303	C₂×C₄	#479	1	1	yes	characteristic
304	C₂×C₄	#479	1	1	yes	characteristic
305	C₂×C₄	#499	1	1	yes	characteristic
306	C₂×C₄	#501	1	1	yes	characteristic
307	C₂×C₄	#501	1	1	yes	characteristic
308	C₈	—	4	4	no	—
309	C₈	—	4	4	no	—
310	C₈	—	4	2	no	—
311	C₈	—	8	2	no	—
312	C₈	—	8	2	no	—
313	C₈	#384	1	1	yes	characteristic
314	C₈	#385	1	1	yes	characteristic
315	C₈	#389	1	1	yes	characteristic
316	C₈	#389	1	1	yes	characteristic
317	C₈	#395	1	1	yes	characteristic
318	C₈	#398	1	1	yes	characteristic
319	C₈	#398	1	1	yes	characteristic
320	C₈	#399	1	1	yes	characteristic
321	C₂²	—	4	1	no	—
322	C₂²	—	4	1	no	—
323	C₂²	—	4	1	no	—
324	C₂²	—	4	1	no	—
325	C₂²	—	4	1	no	—
326	C₂²	—	4	1	no	—
327	C₂²	—	4	1	no	—
328	C₂²	#1863	1	1	yes	characteristic
329	C₂²	#1863	1	1	yes	characteristic
330	C₂²	#1942	1	1	yes	characteristic
331	C₂²	#2001	1	1	yes	characteristic
332	C₂²	#2057	1	1	yes	characteristic
333	C₂²	#2057	1	1	yes	characteristic
334	C₂²	#2087	1	1	yes	characteristic
335	C₄	—	8	2	no	—
336	C₄	#1444	1	1	yes	characteristic
337	C₄	#1862	1	1	yes	characteristic
338	C₄	#1862	1	1	yes	characteristic
339	C₄	#1944	1	1	yes	characteristic
340	C₄	#1999	1	1	yes	characteristic
341	C₄	#2059	1	1	yes	characteristic
342	C₄	#2059	1	1	yes	characteristic
343	C₄	#2086	1	1	yes	characteristic
344	C₂	—	2	1	no	—
345	C₂	—	2	1	no	—
346	C₂	—	2	1	no	—
347	C₂	—	2	1	no	—
348	C₂	—	2	1	no	—
349	C₂	—	2	1	no	—
350	C₂	—	2	1	no	—
351	C₂	#11328	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#1999
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂	#11328
prime radical	C₂⁴	C₂×Q₁₆
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ 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
27. #1388	C₄	semidirect

Maximal subgroups

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

subgroup	index
C₂×#2161	2
C₂×#1953	2
C₂×#1388	2
#12559	2
#12551	2