Finite Group Databasebeta

← #285731

Subgroups — #285731

Order 512.

639 conjugacy classes of subgroups.

The subgroup lattice has 532 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#27972C211yesmaximal, characteristic
2#27648C211yesmaximal, characteristic
3#24838C211yesmaximal, characteristic
4#22952C211yesmaximal, characteristic
5#10657C211yesmaximal, characteristic, complemented
6#10626C211yesmaximal, characteristic, complemented
7#10588C211yesmaximal, characteristic, complemented
8#10486C211yesmaximal, characteristic, complemented
9#9613C211yesmaximal, characteristic, complemented
10#9561C211yesmaximal, characteristic, complemented
11#9220C211yesmaximal, characteristic, complemented
12#9131C211yesmaximal, characteristic
13#8830C211yesmaximal, characteristic
14#8647C211yesmaximal, characteristic, complemented
15C2×#1296C211yesmaximal, characteristic
16C2×#484—21no—
17C2×#475C2211yescharacteristic
18C2×C8×Q8C2211yescharacteristic
19C22×C4⋊C8C2211yescharacteristic
20C22×C4⋊C8C2211yescharacteristic
21#2857C2211yescharacteristic
22C4×C22⋊Q8C2211yescharacteristic
23C2×#384C2211yescharacteristic
24#2088—21no—
25#2083—21no—
26#2072—21no—
27#2072—21no—
28#2072—21no—
29#2066—21no—
30#1988C2211yescharacteristic
31#1984C2211yescharacteristic
32#1956C2211yescharacteristic
33#1956C2211yescharacteristic
34#1948C2211yescharacteristic
35#1947C2211yescharacteristic
36#1947C2211yescharacteristic
37#1939—21no—
38#1939C2211yescharacteristic
39#1916—21no—
40#1892—21no—
41#1872—21no—
42#1872—21no—
43#1850—21no—
44#1834C2211yescharacteristic
45#1834C2211yescharacteristic
46#1824—21no—
47#1807—21no—
48#1803C2211yescharacteristic
49#1803C2211yescharacteristic
50#1803C2211yescharacteristic
51#1797C2211yescharacteristic
52C8×C4⋊C4C2211yescharacteristic
53C8×C22⋊C4—21no—
54C4×C22⋊C8C2211yescharacteristic
55C2×#334C2211yescharacteristic
56C2×#334C2211yescharacteristic
57C2×#332C2211yescharacteristic
58C2×#332C2211yescharacteristic
59#1296C2211yescharacteristic
60#1296C2211yescharacteristic
61#1296C2211yescharacteristic
62#1296C2211yescharacteristic
63#1296C2211yescharacteristic
64#1296C2211yescharacteristic
65#1277C2211yescharacteristic
66#1277C2211yescharacteristic
67C23×C8—21no—
68C23×C8—21no—
69C2×C4⋊Q8C2311yescharacteristic
70C2×C22⋊Q8—21no—
71C2×C22⋊Q8D411yescharacteristic
72C2×C4×Q8C2311yescharacteristic
73C22×C4⋊C4—21no—
74C22×C4⋊C4—21no—
75C22×C4⋊C4—21no—
76C22×C42C2311yescharacteristic
77#484—44no—
78#475—22no—
79#475—41no—
80C8×Q8—22no—
81C8×Q8—41no—
82C2×#107—21no—
83C2×#107—21no—
84C2×#106—21no—
85C2×#106—21no—
86C2×C4⋊C8—22no—
87C2×C4⋊C8—22no—
88C2×C4⋊C8—21no—
89C2×C4⋊C8—21no—
90C2×C4⋊C8—21no—
91C2×C4⋊C8—21no—
92C2×C4⋊C8—41no—
93C2×C4⋊C8—41no—
94C2×C4⋊C8C2311yescharacteristic
95C2×C4⋊C8C2311yescharacteristic
96C2×C4⋊C8C2311yescharacteristic
97C2×C4⋊C8C2311yescharacteristic
98C2×C4⋊C8C2311yescharacteristic
99C2×C4⋊C8C2311yescharacteristic
100C2×#103—21no—
101C2×#103—21no—
102C2×#103—21no—
103C2×#103—21no—
104C2×#103—21no—
105C2×#103—21no—
106C2×#103—21no—
107C2×#103—21no—
108C2×C22⋊C8—21no—
109C2×C22⋊C8D411yescharacteristic
110C2×C22⋊C8D411yescharacteristic
111C2×C4×C8—21no—
112C2×C4×C8—21no—
113C2×C4×C8C2311yescharacteristic
114C2×C4×C8C2311yescharacteristic
115#391—21no—
116#391D411yescharacteristic
117#389—21no—
118#389—21no—
119#389D411yescharacteristic
120#387—21no—
121#385—21no—
122#384—21no—
123#384—21no—
124#384—21no—
125#384C2311yescharacteristic
126#384C2311yescharacteristic
127#384C2311yescharacteristic
128#380—21no—
129#380—21no—
130#380D411yescharacteristic
131C4×C4⋊C4—21no—
132C4×C4⋊C4C2311yescharacteristic
133C4×C22⋊C4—21no—
134#340—21no—
135#340—21no—
136#340—21no—
137#340—21no—
138#340—21no—
139#340—21no—
140#340—21no—
141#340—21no—
142#340—21no—
143#340—21no—
144#340—21no—
145#340—21no—
146#336—21no—
147#336—21no—
148#336—21no—
149#336—21no—
150#336—21no—
151#336D411yescharacteristic
152#336D411yescharacteristic
153#334—22no—
154#334—41no—
155#332—22no—
156#332—41no—
157C22×Q8—21no—
158C22×Q8C2×D411yescharacteristic
159C23×C4—41no—
160C23×C4C2×D411yescharacteristic
161C23×C4C2×D411yescharacteristic
162C23×C4#4111yescharacteristic
163C22×C8—21no—
164C22×C8—21no—
165C22×C8—21no—
166C22×C8—21no—
167C22×C8—21no—
168C22×C8—21no—
169C22×C8—42no—
170C22×C8—42no—
171C22×C8—42no—
172C22×C8—42no—
173C22×C8—81no—
174C22×C8—81no—
175C22×C8C2×D411yescharacteristic
176C22×C8C2×D411yescharacteristic
177C22×C8C2×D411yescharacteristic
178C22×C8C2×D411yescharacteristic
179C4⋊Q8—21no—
180C4⋊Q8—21no—
181C4⋊Q8—21no—
182C4⋊Q8—21no—
183C22⋊Q8—22no—
184C22⋊Q8—42no—
185C22⋊Q8—42no—
186C22⋊Q8SD1614yes—
187C4×Q8—21no—
188C4×Q8—21no—
189C4×Q8—21no—
190C4×Q8—21no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—42no—
206C2×C4⋊C4—42no—
207C2×C4⋊C4—41no—
208C2×C4⋊C4—41no—
209C2×C4⋊C4—41no—
210C2×C4⋊C4—41no—
211C2×C4⋊C4—81no—
212C2×C4⋊C4—81no—
213C2×C4⋊C4—82no—
214C2×C4⋊C4C2×D411yescharacteristic
215C2×C4⋊C4C2×D411yescharacteristic
216C2×C4⋊C4C2×D411yescharacteristic
217C2×C22⋊C4—21no—
218C2×C22⋊C4—21no—
219C2×C22⋊C4—41no—
220C2×C42—21no—
221C2×C42—21no—
222C2×C42—21no—
223C2×C42—21no—
224C2×C42—21no—
225C2×C42—21no—
226C2×C42C2×D411yescharacteristic
227C2×C42C2×D411yescharacteristic
228C2×C42#4111yescharacteristic
229C2×C42C2411yescharacteristic
230#107—42no—
231#107—42no—
232#106—42no—
233#106—81no—
234C4⋊C8—21no—
235C4⋊C8—21no—
236C4⋊C8—21no—
237C4⋊C8—21no—
238C4⋊C8—21no—
239C4⋊C8—21no—
240C4⋊C8—21no—
241C4⋊C8—21no—
242C4⋊C8—21no—
243C4⋊C8—21no—
244C4⋊C8—42no—
245C4⋊C8—42no—
246C4⋊C8—42no—
247C4⋊C8—81no—
248C4⋊C8#4112yes—
249C4⋊C8#4112yes—
250#103—42no—
251#103—42no—
252#103—42no—
253#103—81no—
254#103—81no—
255#103—81no—
256#103—81no—
257#103—81no—
258C22⋊C8—22no—
259C22⋊C8—22no—
260C22⋊C8—42no—
261C4×C8—21no—
262C4×C8—21no—
263C4×C8—21no—
264C4×C8—21no—
265C4×C8—42no—
266C4×C8—81no—
267#95—21no—
268#95—21no—
269#95—21no—
270#95—21no—
271#95—21no—
272#95—41no—
273#95—41no—
274C24#12111yescharacteristic
275C2×Q8—21no—
276C2×Q8—21no—
277C2×Q8—21no—
278C2×Q8—21no—
279C2×Q8—21no—
280C2×Q8—41no—
281C2×Q8—41no—
282C2×Q8—41no—
283C2×Q8—41no—
284C2×Q8—41no—
285C2×Q8—41no—
286C2×Q8C2×SD1612yes—
287C22×C4—21no—
288C22×C4—21no—
289C22×C4—21no—
290C22×C4—21no—
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—21no—
294C22×C4—21no—
295C22×C4—21no—
296C22×C4—21no—
297C22×C4—22no—
298C22×C4—21no—
299C22×C4—21no—
300C22×C4—21no—
301C22×C4—21no—
302C22×C4—41no—
303C22×C4—41no—
304C22×C4—41no—
305C22×C4—41no—
306C22×C4—41no—
307C22×C4—82no—
308C22×C4—82no—
309C22×C4—82no—
310C22×C4#12111yescharacteristic
311C22×C4#12111yescharacteristic
312C22×C4#12111yescharacteristic
313C22×C4C2×SD1612yes—
314C22×C4#13512yes—
315C22×C4#13512yes—
316C22×C4#13712yes—
317C22×C4C22×D411yescharacteristic
318C22×C4C22×D411yescharacteristic
319C22×C4C2×#4111yescharacteristic
320C2×C8—21no—
321C2×C8—21no—
322C2×C8—22no—
323C2×C8—22no—
324C2×C8—21no—
325C2×C8—21no—
326C2×C8—21no—
327C2×C8—21no—
328C2×C8—21no—
329C2×C8—21no—
330C2×C8—21no—
331C2×C8—21no—
332C2×C8—21no—
333C2×C8—21no—
334C2×C8—41no—
335C2×C8—41no—
336C2×C8—41no—
337C2×C8—41no—
338C2×C8—41no—
339C2×C8—41no—
340C2×C8—41no—
341C2×C8—41no—
342C2×C8—41no—
343C2×C8—41no—
344C2×C8—41no—
345C2×C8—41no—
346C2×C8—41no—
347C2×C8—41no—
348C2×C8—41no—
349C2×C8—41no—
350C2×C8—84no—
351C2×C8—84no—
352C4⋊C4—21no—
353C4⋊C4—24no—
354C4⋊C4—21no—
355C4⋊C4—21no—
356C4⋊C4—21no—
357C4⋊C4—21no—
358C4⋊C4—41no—
359C4⋊C4—41no—
360C4⋊C4—41no—
361C4⋊C4—41no—
362C4⋊C4—41no—
363C4⋊C4—41no—
364C4⋊C4—41no—
365C4⋊C4—41no—
366C4⋊C4—41no—
367C4⋊C4—41no—
368C4⋊C4—41no—
369C4⋊C4—41no—
370C4⋊C4—41no—
371C4⋊C4—41no—
372C4⋊C4—41no—
373C4⋊C4—41no—
374C4⋊C4—41no—
375C4⋊C4—41no—
376C4⋊C4—81no—
377C4⋊C4—81no—
378C4⋊C4—81no—
379C4⋊C4—81no—
380C4⋊C4—82no—
381C4⋊C4—82no—
382C4⋊C4—82no—
383C4⋊C4—82no—
384C4⋊C4C2×SD1612yes—
385C22⋊C4—24no—
386C22⋊C4—42no—
387C22⋊C4—82no—
388C42—21no—
389C42—21no—
390C42—21no—
391C42—21no—
392C42—21no—
393C42—21no—
394C42—41no—
395C42—41no—
396C42—41no—
397C42—41no—
398C42C2×#4111yescharacteristic
399C42C2×#4111yescharacteristic
400C42H(5,𝔽2)11yescharacteristic
401C422−1+411yescharacteristic
402C23—21no—
403C23—21no—
404C23—21no—
405C23—21no—
406C23#46412yes—
407C23#46512yes—
408C23#47312yes—
409C23C2×#12111yescharacteristic
410Q8—22no—
411Q8—22no—
412Q8—41no—
413Q8—41no—
414Q8—81no—
415Q8—81no—
416Q8—81no—
417Q8—81no—
418C2×C4—21no—
419C2×C4—21no—
420C2×C4—21no—
421C2×C4—21no—
422C2×C4—21no—
423C2×C4—21no—
424C2×C4—22no—
425C2×C4—22no—
426C2×C4—22no—
427C2×C4—22no—
428C2×C4—22no—
429C2×C4—22no—
430C2×C4—22no—
431C2×C4—22no—
432C2×C4—21no—
433C2×C4—22no—
434C2×C4—22no—
435C2×C4—21no—
436C2×C4—21no—
437C2×C4—42no—
438C2×C4—42no—
439C2×C4—41no—
440C2×C4—41no—
441C2×C4—41no—
442C2×C4—41no—
443C2×C4—41no—
444C2×C4—41no—
445C2×C4—41no—
446C2×C4—41no—
447C2×C4—41no—
448C2×C4—41no—
449C2×C4—81no—
450C2×C4—81no—
451C2×C4—81no—
452C2×C4—81no—
453C2×C4—81no—
454C2×C4—81no—
455C2×C4—81no—
456C2×C4—81no—
457C2×C4—81no—
458C2×C4—81no—
459C2×C4—81no—
460C2×C4—81no—
461C2×C4—164no—
462C2×C4#46412yes—
463C2×C4#46512yes—
464C2×C4#52911yescharacteristic
465C2×C4#53911yescharacteristic
466C2×C4#54611yescharacteristic
467C2×C4#54611yescharacteristic
468C2×C4#54711yescharacteristic
469C2×C4#54711yescharacteristic
470C2×C4C22×SD1611yescharacteristic
471C2×C4C2×#13511yescharacteristic
472C2×C4C2×#13511yescharacteristic
473C2×C4C2×#13711yescharacteristic
474C2×C4#57611yescharacteristic
475C2×C4#57711yescharacteristic
476C2×C4#57711yescharacteristic
477C2×C4#57811yescharacteristic
478C8—41no—
479C8—41no—
480C8—42no—
481C8—41no—
482C8—42no—
483C8—41no—
484C8—81no—
485C8—81no—
486C8—81no—
487C8—81no—
488C22—22no—
489C22—22no—
490C22—22no—
491C22—22no—
492C22—22no—
493C22—22no—
494C22#168014yes—
495C22#242511yescharacteristic
496C22C2×#46411yescharacteristic
497C22#304511yescharacteristic
498C22C2×#46511yescharacteristic
499C22#306311yescharacteristic
500C22C2×#47311yescharacteristic
501C22#306811yescharacteristic
502C4—22no—
503C4—22no—
504C4—21no—
505C4—21no—
506C4—42no—
507C4—41no—
508C4—42no—
509C4—81no—
510C4—81no—
511C4—161no—
512C4—161no—
513C4—161no—
514C4—161no—
515C4#329611yescharacteristic
516C4#329711yescharacteristic
517C4#329811yescharacteristic
518C4#332011yescharacteristic
519C4#333511yescharacteristic
520C4#333611yescharacteristic
521C4#333711yescharacteristic
522C4#334611yescharacteristic
523C2—24no—
524C2C2×#168011yescharacteristic
525C2#1924411yescharacteristic
526C2#1940911yescharacteristic
527C2#1958311yescharacteristic
528C2#2521011yescharacteristic
529C2#2523611yescharacteristic
530C2#2526611yescharacteristic
subgroupG/H
center, socleC23C2×#121
commutator, Frattini, cosocleC2×C42C24
prime radicalC24#121
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
5. #10657C2semidirect
6. #10626C2semidirect
7. #10588C2semidirect
8. #10486C2semidirect
9. #9613C2semidirect
10. #9561C2semidirect
11. #9220C2semidirect
14. #8647C2semidirect

Maximal subgroups

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

subgroupindex
#279722
#276482
#248382
#229522
#106572
#106262
#105882
#104862
#96132
#95612
#92202
#91312
#88302
#86472
C2×#12962