Subgroups — #149213

Order 512.

558 conjugacy classes of subgroups.

The subgroup lattice has 390 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	C₂²×#366	C₂	1	1	yes	maximal, characteristic
3	#20803	C₂	1	1	yes	maximal, characteristic
4	C₂×#1387	C₂	1	1	yes	maximal, characteristic
5	C₂×#1382	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×#428	—	2	1	no	—
12	C₂×#427	C₂²	1	1	yes	characteristic
13	C₂²×#107	C₂²	1	1	yes	characteristic
14	C₂²×#107	C₂²	1	1	yes	characteristic
15	C₂×#405	C₂²	1	1	yes	characteristic
16	#2317	—	2	1	no	—
17	#2159	—	2	2	no	—
18	#2159	—	2	2	no	—
19	C₂×#367	—	2	1	no	—
20	C₂×#366	—	2	2	no	—
21	C₂×#366	—	2	2	no	—
22	C₂×#366	—	2	2	no	—
23	C₂×#366	—	2	1	no	—
24	#1847	—	2	2	no	—
25	#1846	—	2	1	no	—
26	#1845	—	2	1	no	—
27	C₂×#343	—	2	1	no	—
28	#1736	—	2	1	no	—
29	#1387	—	2	2	no	—
30	#1382	—	2	2	no	—
31	#1379	C₄	1	4	yes	complemented
32	C₂²×M₁₆	—	2	1	no	—
33	C₂²×M₁₆	D₄	1	1	yes	characteristic
34	C₂³×C₈	C₂³	1	1	yes	characteristic
35	C₂×#117	—	2	1	no	—
36	C₂×#117	D₄	1	1	yes	characteristic
37	C₂²×C₄⋊C₄	—	2	1	no	—
38	C₂²×C₄⋊C₄	—	2	1	no	—
39	C₂×M₃₂	—	2	1	no	—
40	C₂×M₃₂	—	2	1	no	—
41	C₂×M₃₂	—	2	1	no	—
42	C₂×M₃₂	C₂×C₄	1	2	yes	—
43	C₂×M₃₂	D₄	1	2	yes	—
44	C₂×M₃₂	Q₈	1	2	yes	—
45	C₂²×C₁₆	—	2	1	no	—
46	C₂²×C₁₆	—	2	1	no	—
47	C₂²×C₁₆	—	2	1	no	—
48	C₂²×C₁₆	—	2	1	no	—
49	C₂²×C₁₆	—	2	1	no	—
50	C₂²×C₁₆	C₂×C₄	1	2	yes	—
51	C₂²×C₁₆	D₄	1	2	yes	—
52	C₂²×C₁₆	Q₈	1	2	yes	—
53	C₂×#108	—	2	1	no	—
54	C₂×#108	—	2	1	no	—
55	C₂×#108	—	2	1	no	—
56	C₂×#108	—	2	1	no	—
57	C₂×#108	—	2	1	no	—
58	C₂×#108	Q₈	1	2	yes	—
59	#428	—	4	2	no	—
60	#428	—	4	2	no	—
61	#427	—	2	1	no	—
62	#427	—	2	1	no	—
63	#427	—	2	1	no	—
64	#427	—	2	1	no	—
65	C₂×#107	—	2	1	no	—
66	C₂×#107	—	2	1	no	—
67	C₂×#107	—	2	1	no	—
68	C₂×#107	—	2	1	no	—
69	C₂×#107	—	2	1	no	—
70	C₂×#107	—	2	1	no	—
71	C₂×#107	—	2	1	no	—
72	C₂×#107	—	2	1	no	—
73	C₂×#107	—	2	2	no	—
74	C₂×#107	—	2	1	no	—
75	C₂×#107	—	2	1	no	—
76	C₂×#107	D₄	1	1	yes	characteristic
77	C₂×#107	D₄	1	1	yes	characteristic
78	C₂×#107	D₄	1	2	yes	—
79	C₂×#107	D₄	1	2	yes	—
80	C₂×#106	—	2	2	no	—
81	C₂×#106	—	2	1	no	—
82	#405	—	2	1	no	—
83	#405	—	2	1	no	—
84	#405	C₂×C₄	1	2	yes	—
85	#405	Q₈	1	2	yes	—
86	C₂×#97	—	2	1	no	—
87	C₂×C₄×C₈	—	2	1	no	—
88	#389	—	4	2	no	—
89	#385	—	4	2	no	—
90	#384	—	2	2	no	—
91	#382	—	2	2	no	—
92	#367	—	2	4	no	—
93	#366	—	2	8	no	—
94	#366	—	2	4	no	—
95	#343	—	4	2	no	—
96	#340	—	4	2	no	—
97	C₂³×C₄	—	4	1	no	—
98	C₂³×C₄	—	4	1	no	—
99	C₂³×C₄	C₂×D₄	1	1	yes	characteristic
100	C₂×M₁₆	—	2	1	no	—
101	C₂×M₁₆	—	2	1	no	—
102	C₂×M₁₆	—	2	1	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	C₄⋊C₄	1	2	yes	—
110	C₂×M₁₆	D₈	1	2	yes	—
111	C₂×M₁₆	Q₁₆	1	2	yes	—
112	C₂²×C₈	—	2	1	no	—
113	C₂²×C₈	—	4	1	no	—
114	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
115	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
116	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
117	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
118	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
119	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
120	C₂²×C₈	C₂×Q₈	1	1	yes	characteristic
121	C₂²×C₈	C₂×Q₈	1	1	yes	characteristic
122	C₂²×C₈	C₂×Q₈	1	1	yes	characteristic
123	C₂²×C₈	C₂×Q₈	1	1	yes	characteristic
124	C₂²×C₈	#41	1	1	yes	characteristic
125	C₂²×C₈	#41	1	1	yes	characteristic
126	C₂²×C₈	#41	1	1	yes	characteristic
127	C₂²×C₈	#41	1	1	yes	characteristic
128	#117	—	2	1	no	—
129	#117	—	2	1	no	—
130	#117	—	4	1	no	—
131	#117	—	4	1	no	—
132	#117	—	4	1	no	—
133	#117	—	4	1	no	—
134	#117	C₄⋊C₄	1	2	yes	—
135	#117	Q₁₆	1	2	yes	—
136	C₂×C₄⋊C₄	—	2	2	no	—
137	C₂×C₄⋊C₄	—	2	1	no	—
138	C₂×C₄⋊C₄	—	2	1	no	—
139	C₂×C₄⋊C₄	—	2	2	no	—
140	C₂×C₄⋊C₄	—	4	1	no	—
141	C₂×C₄⋊C₄	—	4	1	no	—
142	C₂×C₄⋊C₄	—	4	1	no	—
143	C₂×C₄⋊C₄	—	4	1	no	—
144	C₂×C₄⋊C₄	—	4	1	no	—
145	C₂×C₄⋊C₄	—	4	1	no	—
146	C₂×C₄⋊C₄	—	4	1	no	—
147	C₂×C₄⋊C₄	—	4	1	no	—
148	C₂×C₄⋊C₄	—	4	1	no	—
149	C₂×C₄⋊C₄	—	4	1	no	—
150	C₂×C₄⋊C₄	—	4	4	no	—
151	C₂×C₄⋊C₄	D₈	1	2	yes	—
152	C₂×C₂²⋊C₄	—	2	1	no	—
153	C₂×C₂²⋊C₄	—	4	1	no	—
154	C₂×C₄²	—	2	1	no	—
155	C₂×C₄²	—	4	1	no	—
156	M₃₂	—	2	2	no	—
157	M₃₂	—	2	2	no	—
158	M₃₂	—	2	2	no	—
159	M₃₂	C₄⋊C₄	1	4	yes	complemented
160	C₂×C₁₆	—	2	2	no	—
161	C₂×C₁₆	—	2	2	no	—
162	C₂×C₁₆	—	2	2	no	—
163	C₂×C₁₆	—	2	2	no	—
164	C₂×C₁₆	—	2	2	no	—
165	C₂×C₁₆	—	2	2	no	—
166	C₂×C₁₆	—	2	2	no	—
167	C₂×C₁₆	—	2	2	no	—
168	C₂×C₁₆	—	2	2	no	—
169	C₂×C₁₆	C₄⋊C₄	1	4	yes	complemented
170	#108	—	2	2	no	—
171	#108	—	2	2	no	—
172	#108	—	4	1	no	—
173	#108	—	4	1	no	—
174	#107	—	2	2	no	—
175	#107	—	2	2	no	—
176	#107	—	2	1	no	—
177	#107	—	2	2	no	—
178	#107	—	2	1	no	—
179	#107	—	2	2	no	—
180	#107	—	2	1	no	—
181	#107	—	2	1	no	—
182	#107	—	4	4	no	—
183	#107	—	4	1	no	—
184	#107	—	4	1	no	—
185	#107	—	4	1	no	—
186	#107	—	4	1	no	—
187	#106	—	4	4	no	—
188	#106	—	4	1	no	—
189	#106	—	4	1	no	—
190	#97	—	2	2	no	—
191	#97	—	2	2	no	—
192	C₄×C₈	—	2	2	no	—
193	C₄×C₈	—	2	2	no	—
194	#95	—	4	4	no	—
195	C₂⁴	AGL(1,ℤ/8)	1	1	yes	characteristic
196	C₂²×C₄	—	2	2	no	—
197	C₂²×C₄	—	4	1	no	—
198	C₂²×C₄	—	4	2	no	—
199	C₂²×C₄	—	4	1	no	—
200	C₂²×C₄	—	4	2	no	—
201	C₂²×C₄	—	8	1	no	—
202	C₂²×C₄	—	8	1	no	—
203	C₂²×C₄	—	8	1	no	—
204	C₂²×C₄	—	8	1	no	—
205	C₂²×C₄	—	16	2	no	—
206	C₂²×C₄	—	16	2	no	—
207	C₂²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
208	C₂²×C₄	#120	1	1	yes	characteristic
209	C₂²×C₄	#121	1	1	yes	characteristic
210	C₂²×C₄	C₂²⋊Q₈	1	1	yes	characteristic
211	C₂²×C₄	#123	1	1	yes	characteristic
212	C₂²×C₄	#127	1	1	yes	characteristic
213	C₂²×C₄	C₄⋊Q₈	1	1	yes	characteristic
214	C₂²×C₄	C₂×D₈	1	1	yes	characteristic
215	C₂²×C₄	C₂×D₈	1	1	yes	characteristic
216	C₂²×C₄	C₂×Q₁₆	1	1	yes	characteristic
217	C₂²×C₄	C₂×Q₁₆	1	1	yes	characteristic
218	C₂²×C₄	AGL(1,ℤ/8)	1	1	yes	characteristic
219	C₂²×C₄	#137	1	1	yes	characteristic
220	C₂²×C₄	#137	1	1	yes	characteristic
221	M₁₆	—	2	2	no	—
222	M₁₆	—	2	2	no	—
223	M₁₆	—	2	2	no	—
224	M₁₆	—	4	2	no	—
225	M₁₆	—	4	2	no	—
226	M₁₆	—	8	1	no	—
227	M₁₆	—	8	1	no	—
228	M₁₆	#107	1	4	yes	—
229	C₂×C₈	—	2	2	no	—
230	C₂×C₈	—	2	2	no	—
231	C₂×C₈	—	2	2	no	—
232	C₂×C₈	—	4	2	no	—
233	C₂×C₈	—	8	1	no	—
234	C₂×C₈	—	8	1	no	—
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×D₄	1	1	yes	characteristic
240	C₂×C₈	C₄×Q₈	1	1	yes	characteristic
241	C₂×C₈	#120	1	1	yes	characteristic
242	C₂×C₈	#120	1	1	yes	characteristic
243	C₂×C₈	#121	1	1	yes	characteristic
244	C₂×C₈	#121	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	C₂²⋊Q₈	1	1	yes	characteristic
251	C₂×C₈	C₂²⋊Q₈	1	1	yes	characteristic
252	C₂×C₈	C₂²⋊Q₈	1	1	yes	characteristic
253	C₂×C₈	#123	1	1	yes	characteristic
254	C₂×C₈	#123	1	1	yes	characteristic
255	C₂×C₈	#125	1	1	yes	characteristic
256	C₂×C₈	#125	1	1	yes	characteristic
257	C₂×C₈	#125	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	C₂×C₈	C₄⋊Q₈	1	1	yes	characteristic
263	C₄⋊C₄	—	2	2	no	—
264	C₄⋊C₄	—	4	1	no	—
265	C₄⋊C₄	—	4	1	no	—
266	C₄⋊C₄	—	4	2	no	—
267	C₄⋊C₄	—	4	2	no	—
268	C₄⋊C₄	—	4	1	no	—
269	C₄⋊C₄	—	4	2	no	—
270	C₄⋊C₄	—	4	1	no	—
271	C₄⋊C₄	—	4	2	no	—
272	C₄⋊C₄	—	8	1	no	—
273	C₄⋊C₄	—	8	1	no	—
274	C₄⋊C₄	—	8	1	no	—
275	C₄⋊C₄	—	8	1	no	—
276	C₄⋊C₄	—	16	4	no	—
277	C₄⋊C₄	#107	1	4	yes	—
278	C₂²⋊C₄	—	2	2	no	—
279	C₂²⋊C₄	—	2	2	no	—
280	C₂²⋊C₄	—	8	1	no	—
281	C₂²⋊C₄	—	8	1	no	—
282	C₄²	—	2	2	no	—
283	C₄²	—	2	2	no	—
284	C₄²	—	8	1	no	—
285	C₄²	—	8	1	no	—
286	C₁₆	—	2	4	no	—
287	C₁₆	—	2	4	no	—
288	C₂³	—	2	2	no	—
289	C₂³	—	2	2	no	—
290	C₂³	#428	1	1	yes	characteristic
291	C₂³	#447	1	1	yes	characteristic
292	C₂³	#459	1	1	yes	characteristic
293	C₂³	#477	1	1	yes	characteristic
294	C₂³	#484	1	1	yes	characteristic
295	C₂³	#496	1	1	yes	characteristic
296	C₂³	#501	1	1	yes	characteristic
297	C₂×C₄	—	2	4	no	—
298	C₂×C₄	—	4	2	no	—
299	C₂×C₄	—	4	2	no	—
300	C₂×C₄	—	8	2	no	—
301	C₂×C₄	—	8	4	no	—
302	C₂×C₄	—	8	2	no	—
303	C₂×C₄	—	8	1	no	—
304	C₂×C₄	—	8	1	no	—
305	C₂×C₄	—	8	2	no	—
306	C₂×C₄	—	8	2	no	—
307	C₂×C₄	—	16	2	no	—
308	C₂×C₄	—	16	2	no	—
309	C₂×C₄	—	16	2	no	—
310	C₂×C₄	—	16	2	no	—
311	C₂×C₄	—	16	2	no	—
312	C₂×C₄	#384	1	1	yes	characteristic
313	C₂×C₄	#385	1	1	yes	characteristic
314	C₂×C₄	#389	1	1	yes	characteristic
315	C₂×C₄	#392	1	1	yes	characteristic
316	C₂×C₄	#395	1	1	yes	characteristic
317	C₂×C₄	#398	1	1	yes	characteristic
318	C₂×C₄	#399	1	1	yes	characteristic
319	C₂×C₄	C₂×#107	1	1	yes	characteristic
320	C₂×C₄	C₂×#107	1	1	yes	characteristic
321	C₂×C₄	#428	1	1	yes	characteristic
322	C₂×C₄	#447	1	1	yes	characteristic
323	C₂×C₄	#451	1	1	yes	characteristic
324	C₂×C₄	#451	1	1	yes	characteristic
325	C₂×C₄	#459	1	1	yes	characteristic
326	C₂×C₄	#462	1	1	yes	characteristic
327	C₂×C₄	#462	1	1	yes	characteristic
328	C₂×C₄	#474	1	1	yes	characteristic
329	C₂×C₄	#474	1	1	yes	characteristic
330	C₂×C₄	#477	1	1	yes	characteristic
331	C₂×C₄	#480	1	1	yes	characteristic
332	C₂×C₄	#480	1	1	yes	characteristic
333	C₂×C₄	#484	1	1	yes	characteristic
334	C₂×C₄	#493	1	1	yes	characteristic
335	C₂×C₄	#494	1	1	yes	characteristic
336	C₂×C₄	#497	1	1	yes	characteristic
337	C₂×C₄	#500	1	1	yes	characteristic
338	C₂×C₄	#500	1	1	yes	characteristic
339	C₂×C₄	#501	1	1	yes	characteristic
340	C₈	—	2	4	no	—
341	C₈	—	8	2	no	—
342	C₈	#385	1	1	yes	characteristic
343	C₈	#385	1	1	yes	characteristic
344	C₈	#389	1	1	yes	characteristic
345	C₈	#389	1	1	yes	characteristic
346	C₈	#395	1	1	yes	characteristic
347	C₈	#395	1	1	yes	characteristic
348	C₈	#398	1	1	yes	characteristic
349	C₈	#398	1	1	yes	characteristic
350	C₂²	—	2	1	no	—
351	C₂²	—	2	1	no	—
352	C₂²	—	2	1	no	—
353	C₂²	—	2	1	no	—
354	C₂²	—	2	2	no	—
355	C₂²	—	2	2	no	—
356	C₂²	—	2	2	no	—
357	C₂²	—	2	2	no	—
358	C₂²	#1847	1	1	yes	characteristic
359	C₂²	#1852	1	1	yes	characteristic
360	C₂²	#1947	1	1	yes	characteristic
361	C₂²	#2011	1	1	yes	characteristic
362	C₂²	#2057	1	1	yes	characteristic
363	C₂²	#2083	1	1	yes	characteristic
364	C₂²	#2095	1	1	yes	characteristic
365	C₂²	#2192	1	2	yes	—
366	C₂²	#2195	1	2	yes	—
367	C₄	—	4	4	no	—
368	C₄	—	16	1	no	—
369	C₄	—	16	2	no	—
370	C₄	—	16	2	no	—
371	C₄	—	16	1	no	—
372	C₄	#1444	1	1	yes	characteristic
373	C₄	#1845	1	1	yes	characteristic
374	C₄	#1852	1	1	yes	characteristic
375	C₄	#1947	1	1	yes	characteristic
376	C₄	#2016	1	1	yes	characteristic
377	C₄	#2057	1	1	yes	characteristic
378	C₄	#2083	1	1	yes	characteristic
379	C₄	#2093	1	1	yes	characteristic
380	C₂	—	2	1	no	—
381	C₂	—	2	1	no	—
382	C₂	—	2	1	no	—
383	C₂	—	2	1	no	—
384	C₂	#11324	1	1	yes	characteristic
385	C₂	#12925	1	2	yes	—
386	C₂	#13336	1	1	yes	characteristic
387	C₂	#13339	1	1	yes	characteristic
388	C₂	#13455	1	2	yes	—

	subgroup	G/H
center, socle	C₂³	#447
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
prime radical	C₂⁴	AGL(1,ℤ/8)
LCS	G ▷ C₂²×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
31. #1379	C₄	semidirect
159. M₃₂	C₄⋊C₄	semidirect
169. C₂×C₁₆	C₄⋊C₄	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
C₂²×#366	2
#20803	2
C₂×#1387	2
C₂×#1382	2
C₂×#1379	2
#10870	2