Subgroups — #147253

Order 512.

554 conjugacy classes of subgroups.

The subgroup lattice has 369 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#367	C₂	1	1	yes	maximal, characteristic
2	#20455	C₂	1	1	yes	maximal, characteristic
3	C₂×#1387	C₂	1	1	yes	maximal, characteristic
4	C₂×#1382	C₂	1	1	yes	maximal, characteristic
5	#12557	C₂	1	1	yes	maximal, characteristic
6	C₂×#1379	C₂	1	1	yes	maximal, characteristic
7	#10870	C₂	1	1	yes	maximal, characteristic
8	C₂²×M₃₂	C₂²	1	1	yes	characteristic
9	C₂³×C₁₆	C₂²	1	1	yes	characteristic
10	C₂²×#108	C₂²	1	1	yes	characteristic
11	C₂×#427	C₂²	1	1	yes	characteristic
12	C₂²×#107	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	C₂×#367	—	2	2	no	—
17	C₂×#367	—	2	2	no	—
18	C₂×#367	C₄	1	4	yes	complemented
19	#1776	—	2	2	no	—
20	C₄×#107	—	2	2	no	—
21	C₂×#343	—	2	1	no	—
22	#1736	—	2	1	no	—
23	#1736	—	2	1	no	—
24	#1387	—	2	2	no	—
25	#1382	C₄	1	4	yes	complemented
26	#1380	—	2	2	no	—
27	#1379	C₄	1	4	yes	complemented
28	C₂²×M₁₆	—	2	1	no	—
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	D₄	1	1	yes	characteristic
33	C₂²×C₄⋊C₄	—	2	1	no	—
34	C₂²×C₄⋊C₄	—	2	1	no	—
35	C₂×M₃₂	—	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	C₂×C₄	1	2	yes	—
41	C₂²×C₁₆	—	2	1	no	—
42	C₂²×C₁₆	—	2	1	no	—
43	C₂²×C₁₆	—	2	1	no	—
44	C₂²×C₁₆	—	2	1	no	—
45	C₂²×C₁₆	—	2	1	no	—
46	C₂²×C₁₆	C₂×C₄	1	2	yes	—
47	C₂²×C₁₆	C₂×C₄	1	2	yes	—
48	C₂²×C₁₆	C₂×C₄	1	2	yes	—
49	C₂×#108	—	2	1	no	—
50	C₂×#108	—	2	1	no	—
51	C₂×#108	—	2	1	no	—
52	C₂×#108	—	2	1	no	—
53	C₂×#108	—	2	1	no	—
54	C₂×#108	C₂×C₄	1	2	yes	—
55	#427	—	2	1	no	—
56	#427	—	2	1	no	—
57	#427	C₂×C₄	1	2	yes	—
58	#427	Q₈	1	2	yes	—
59	C₂×#107	—	2	1	no	—
60	C₂×#107	—	2	1	no	—
61	C₂×#107	—	2	1	no	—
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	2	yes	—
68	C₂×#107	C₂×C₄	1	2	yes	—
69	C₂×#107	D₄	1	2	yes	—
70	C₂×#107	D₄	1	2	yes	—
71	C₂×#107	D₄	1	2	yes	—
72	C₂×#106	—	2	1	no	—
73	#405	—	2	1	no	—
74	#405	—	2	1	no	—
75	#405	C₂×C₄	1	2	yes	—
76	#405	Q₈	1	2	yes	—
77	C₂×#97	—	2	1	no	—
78	C₂×C₄×C₈	—	2	1	no	—
79	C₄×C₄⋊C₄	—	2	4	no	—
80	C₄×C₄⋊C₄	—	4	2	no	—
81	C₄×C₂²⋊C₄	—	4	2	no	—
82	#367	—	2	8	no	—
83	#343	—	4	2	no	—
84	#340	—	2	4	no	—
85	#340	—	4	2	no	—
86	C₂³×C₄	—	4	1	no	—
87	C₂³×C₄	—	4	1	no	—
88	C₂³×C₄	C₂×D₄	1	1	yes	characteristic
89	C₂×M₁₆	—	2	1	no	—
90	C₂×M₁₆	—	2	1	no	—
91	C₂×M₁₆	—	2	1	no	—
92	C₂×M₁₆	—	2	1	no	—
93	C₂×M₁₆	—	2	1	no	—
94	C₂×M₁₆	—	2	2	no	—
95	C₂×M₁₆	—	4	1	no	—
96	C₂×M₁₆	—	4	1	no	—
97	C₂×M₁₆	—	4	1	no	—
98	C₂×M₁₆	—	4	1	no	—
99	C₂×M₁₆	—	4	1	no	—
100	C₂×M₁₆	C₄⋊C₄	1	2	yes	—
101	C₂²×C₈	—	2	1	no	—
102	C₂²×C₈	—	4	1	no	—
103	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
104	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
105	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
106	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
107	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
108	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
109	C₂²×C₈	C₂×Q₈	1	1	yes	characteristic
110	C₂²×C₈	C₂×Q₈	1	1	yes	characteristic
111	C₂²×C₈	#41	1	1	yes	characteristic
112	C₂²×C₈	#41	1	1	yes	characteristic
113	C₂²×C₈	#41	1	1	yes	characteristic
114	C₂²×C₈	#41	1	1	yes	characteristic
115	C₂²×C₈	#41	1	1	yes	characteristic
116	C₂²×C₈	#41	1	1	yes	characteristic
117	#117	—	2	1	no	—
118	#117	—	2	1	no	—
119	#117	—	2	2	no	—
120	#117	—	2	2	no	—
121	#117	—	4	1	no	—
122	#117	—	4	1	no	—
123	#117	C₄⋊C₄	1	2	yes	—
124	#117	Q₁₆	1	2	yes	—
125	C₂×C₄⋊C₄	—	2	2	no	—
126	C₂×C₄⋊C₄	—	2	2	no	—
127	C₂×C₄⋊C₄	—	2	2	no	—
128	C₂×C₄⋊C₄	—	2	2	no	—
129	C₂×C₄⋊C₄	—	2	2	no	—
130	C₂×C₄⋊C₄	—	4	1	no	—
131	C₂×C₄⋊C₄	—	4	1	no	—
132	C₂×C₄⋊C₄	—	4	1	no	—
133	C₂×C₄⋊C₄	—	4	1	no	—
134	C₂×C₄⋊C₄	—	4	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	D₈	1	2	yes	—
139	C₂×C₂²⋊C₄	—	2	1	no	—
140	C₂×C₂²⋊C₄	—	4	1	no	—
141	C₂×C₄²	—	2	1	no	—
142	C₂×C₄²	—	4	1	no	—
143	C₂×C₄²	—	4	4	no	—
144	M₃₂	—	2	2	no	—
145	M₃₂	—	2	2	no	—
146	M₃₂	—	4	1	no	—
147	M₃₂	—	4	1	no	—
148	C₂×C₁₆	—	2	2	no	—
149	C₂×C₁₆	—	2	2	no	—
150	C₂×C₁₆	—	2	2	no	—
151	C₂×C₁₆	—	2	2	no	—
152	C₂×C₁₆	—	2	2	no	—
153	C₂×C₁₆	—	2	2	no	—
154	C₂×C₁₆	—	2	2	no	—
155	C₂×C₁₆	—	4	1	no	—
156	C₂×C₁₆	—	4	1	no	—
157	C₂×C₁₆	C₄²	1	4	yes	complemented
158	#108	—	2	2	no	—
159	#108	—	2	2	no	—
160	#108	—	4	1	no	—
161	#108	—	4	1	no	—
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	2	no	—
168	#107	—	4	1	no	—
169	#107	—	4	1	no	—
170	#107	C₂²⋊C₄	1	4	yes	—
171	#107	C₄⋊C₄	1	4	yes	—
172	#106	—	2	2	no	—
173	#106	—	2	2	no	—
174	#97	—	2	2	no	—
175	#97	—	2	2	no	—
176	C₄×C₈	—	2	2	no	—
177	C₄×C₈	—	2	2	no	—
178	#95	—	4	4	no	—
179	C₂⁴	AGL(1,ℤ/8)	1	1	yes	characteristic
180	C₂²×C₄	—	2	2	no	—
181	C₂²×C₄	—	4	2	no	—
182	C₂²×C₄	—	4	2	no	—
183	C₂²×C₄	—	4	2	no	—
184	C₂²×C₄	—	4	2	no	—
185	C₂²×C₄	—	4	2	no	—
186	C₂²×C₄	—	8	1	no	—
187	C₂²×C₄	—	8	1	no	—
188	C₂²×C₄	—	16	2	no	—
189	C₂²×C₄	—	16	2	no	—
190	C₂²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
191	C₂²×C₄	C₄×D₄	1	1	yes	characteristic
192	C₂²×C₄	C₄×D₄	1	1	yes	characteristic
193	C₂²×C₄	#120	1	1	yes	characteristic
194	C₂²×C₄	#121	1	1	yes	characteristic
195	C₂²×C₄	C₂²⋊Q₈	1	1	yes	characteristic
196	C₂²×C₄	#123	1	1	yes	characteristic
197	C₂²×C₄	C₂×D₈	1	1	yes	characteristic
198	C₂²×C₄	C₂×Q₁₆	1	1	yes	characteristic
199	C₂²×C₄	#135	1	1	yes	characteristic
200	C₂²×C₄	#135	1	1	yes	characteristic
201	C₂²×C₄	AGL(1,ℤ/8)	1	1	yes	characteristic
202	C₂²×C₄	#137	1	1	yes	characteristic
203	C₂²×C₄	#137	1	1	yes	characteristic
204	M₁₆	—	2	2	no	—
205	M₁₆	—	2	2	no	—
206	M₁₆	—	4	1	no	—
207	M₁₆	—	4	1	no	—
208	M₁₆	—	4	2	no	—
209	M₁₆	—	4	2	no	—
210	M₁₆	—	8	1	no	—
211	M₁₆	—	8	1	no	—
212	C₂×C₈	—	2	2	no	—
213	C₂×C₈	—	4	1	no	—
214	C₂×C₈	—	4	1	no	—
215	C₂×C₈	—	4	2	no	—
216	C₂×C₈	—	8	1	no	—
217	C₂×C₈	—	8	1	no	—
218	C₂×C₈	C₂×C₄²	1	1	yes	characteristic
219	C₂×C₈	C₂×C₂²⋊C₄	1	1	yes	characteristic
220	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
221	C₂×C₈	#117	1	1	yes	characteristic
222	C₂×C₈	#117	1	1	yes	characteristic
223	C₂×C₈	#117	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×D₄	1	1	yes	characteristic
229	C₂×C₈	C₄×D₄	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	C₄×Q₈	1	1	yes	characteristic
233	C₂×C₈	C₄×Q₈	1	1	yes	characteristic
234	C₂×C₈	#121	1	1	yes	characteristic
235	C₂×C₈	C₂²⋊Q₈	1	1	yes	characteristic
236	C₂×C₈	C₂²⋊Q₈	1	1	yes	characteristic
237	C₂×C₈	C₂²⋊Q₈	1	1	yes	characteristic
238	C₂×C₈	#123	1	1	yes	characteristic
239	C₂×C₈	#123	1	1	yes	characteristic
240	C₂×C₈	#124	1	1	yes	characteristic
241	C₂×C₈	#125	1	1	yes	characteristic
242	C₂×C₈	#125	1	1	yes	characteristic
243	C₂×C₈	#126	1	1	yes	characteristic
244	C₂×C₈	#126	1	1	yes	characteristic
245	C₂×C₈	C₄⋊Q₈	1	1	yes	characteristic
246	C₄⋊C₄	—	2	2	no	—
247	C₄⋊C₄	—	2	4	no	—
248	C₄⋊C₄	—	2	4	no	—
249	C₄⋊C₄	—	4	2	no	—
250	C₄⋊C₄	—	4	2	no	—
251	C₄⋊C₄	—	4	2	no	—
252	C₄⋊C₄	—	4	2	no	—
253	C₄⋊C₄	—	4	2	no	—
254	C₄⋊C₄	—	4	2	no	—
255	C₄⋊C₄	—	8	1	no	—
256	C₄⋊C₄	—	8	1	no	—
257	C₄⋊C₄	#107	1	4	yes	—
258	C₂²⋊C₄	—	2	2	no	—
259	C₂²⋊C₄	—	2	2	no	—
260	C₂²⋊C₄	—	4	2	no	—
261	C₂²⋊C₄	—	4	2	no	—
262	C₄²	—	2	2	no	—
263	C₄²	—	2	2	no	—
264	C₄²	—	4	2	no	—
265	C₄²	—	4	2	no	—
266	C₄²	—	16	4	no	—
267	C₁₆	—	2	4	no	—
268	C₁₆	—	4	2	no	—
269	C₂³	—	2	2	no	—
270	C₂³	—	2	2	no	—
271	C₂³	#428	1	1	yes	characteristic
272	C₂³	#440	1	1	yes	characteristic
273	C₂³	#442	1	1	yes	characteristic
274	C₂³	#447	1	1	yes	characteristic
275	C₂³	#463	1	1	yes	characteristic
276	C₂³	#478	1	1	yes	characteristic
277	C₂³	#484	1	1	yes	characteristic
278	C₂×C₄	—	2	4	no	—
279	C₂×C₄	—	4	2	no	—
280	C₂×C₄	—	4	2	no	—
281	C₂×C₄	—	4	4	no	—
282	C₂×C₄	—	4	4	no	—
283	C₂×C₄	—	8	2	no	—
284	C₂×C₄	—	8	2	no	—
285	C₂×C₄	—	8	4	no	—
286	C₂×C₄	—	8	2	no	—
287	C₂×C₄	—	16	2	no	—
288	C₂×C₄	—	16	2	no	—
289	C₂×C₄	—	16	2	no	—
290	C₂×C₄	—	16	2	no	—
291	C₂×C₄	—	16	2	no	—
292	C₂×C₄	C₄×C₄⋊C₄	1	1	yes	characteristic
293	C₂×C₄	#385	1	1	yes	characteristic
294	C₂×C₄	#385	1	1	yes	characteristic
295	C₂×C₄	#386	1	1	yes	characteristic
296	C₂×C₄	#387	1	1	yes	characteristic
297	C₂×C₄	#388	1	1	yes	characteristic
298	C₂×C₄	#389	1	1	yes	characteristic
299	C₂×C₄	C₂×#107	1	1	yes	characteristic
300	C₂×C₄	#427	1	1	yes	characteristic
301	C₂×C₄	#428	1	1	yes	characteristic
302	C₂×C₄	C₄×D₈	1	1	yes	characteristic
303	C₂×C₄	C₄×D₈	1	1	yes	characteristic
304	C₂×C₄	C₄×Q₁₆	1	1	yes	characteristic
305	C₂×C₄	C₄×Q₁₆	1	1	yes	characteristic
306	C₂×C₄	#440	1	1	yes	characteristic
307	C₂×C₄	#441	1	1	yes	characteristic
308	C₂×C₄	#449	1	1	yes	characteristic
309	C₂×C₄	#451	1	1	yes	characteristic
310	C₂×C₄	#452	1	1	yes	characteristic
311	C₂×C₄	#459	1	1	yes	characteristic
312	C₂×C₄	#462	1	1	yes	characteristic
313	C₂×C₄	#464	1	1	yes	characteristic
314	C₂×C₄	#474	1	1	yes	characteristic
315	C₂×C₄	#477	1	1	yes	characteristic
316	C₂×C₄	#479	1	1	yes	characteristic
317	C₂×C₄	#480	1	1	yes	characteristic
318	C₂×C₄	#482	1	1	yes	characteristic
319	C₂×C₄	#485	1	1	yes	characteristic
320	C₈	—	4	2	no	—
321	C₈	—	8	2	no	—
322	C₈	C₄×C₂²⋊C₄	1	1	yes	characteristic
323	C₈	C₄×C₄⋊C₄	1	1	yes	characteristic
324	C₈	#387	1	1	yes	characteristic
325	C₈	#387	1	1	yes	characteristic
326	C₈	#387	1	1	yes	characteristic
327	C₈	#388	1	1	yes	characteristic
328	C₈	#389	1	1	yes	characteristic
329	C₈	#391	1	1	yes	characteristic
330	C₂²	—	2	2	no	—
331	C₂²	—	2	2	no	—
332	C₂²	—	2	2	no	—
333	C₂²	—	2	2	no	—
334	C₂²	—	2	1	no	—
335	C₂²	—	2	1	no	—
336	C₂²	—	2	1	no	—
337	C₂²	—	2	1	no	—
338	C₂²	#1776	1	1	yes	characteristic
339	C₂²	#1852	1	1	yes	characteristic
340	C₂²	#1862	1	1	yes	characteristic
341	C₂²	#1879	1	1	yes	characteristic
342	C₂²	#1921	1	1	yes	characteristic
343	C₂²	#1928	1	1	yes	characteristic
344	C₂²	#1949	1	1	yes	characteristic
345	C₂²	#2193	1	2	yes	—
346	C₂²	#2194	1	2	yes	—
347	C₄	—	4	4	no	—
348	C₄	—	8	4	no	—
349	C₄	—	8	4	no	—
350	C₄	—	16	2	no	—
351	C₄	#1436	1	1	yes	characteristic
352	C₄	C₄×#107	1	1	yes	characteristic
353	C₄	#1854	1	1	yes	characteristic
354	C₄	#1863	1	1	yes	characteristic
355	C₄	#1878	1	1	yes	characteristic
356	C₄	#1918	1	1	yes	characteristic
357	C₄	#1925	1	1	yes	characteristic
358	C₄	#1947	1	1	yes	characteristic
359	C₂	—	2	1	no	—
360	C₂	—	2	1	no	—
361	C₂	—	2	1	no	—
362	C₂	—	2	1	no	—
363	C₂	#10907	1	1	yes	characteristic
364	C₂	#12926	1	2	yes	—
365	C₂	#12946	1	2	yes	—
366	C₂	#13018	1	1	yes	characteristic
367	C₂	#13019	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³	#447
commutator	C₂×C₈	C₂×C₄²
Frattini, cosocle	C₂³×C₈	C₂³
prime radical	C₂⁴	AGL(1,ℤ/8)
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ 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
25. #1382	C₄	semidirect
27. #1379	C₄	semidirect
157. 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
C₂²×#367	2
#20455	2
C₂×#1387	2
C₂×#1382	2
#12557	2
C₂×#1379	2
#10870	2