Finite Group Databasebeta

← #285727

Subgroups — #285727

Order 512.

643 conjugacy classes of subgroups.

The subgroup lattice has 533 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#27961C211yesmaximal, characteristic
2#27680C211yesmaximal, characteristic
3#24752C211yesmaximal, characteristic
4#22952C211yesmaximal, characteristic
5#10657C211yesmaximal, characteristic, complemented
6#10626C211yesmaximal, characteristic, complemented
7#10598C211yesmaximal, characteristic, complemented
8#10479C211yesmaximal, characteristic, complemented
9#9613C211yesmaximal, characteristic, complemented
10#9565C211yesmaximal, characteristic, complemented
11#9218C211yesmaximal, characteristic, complemented
12#9129C211yesmaximal, characteristic
13#8830C211yesmaximal, characteristic
14#8647C211yesmaximal, characteristic, complemented
15C2×#1296C211yesmaximal, characteristic
16C2×#467—21no—
17C2×#462C2211yescharacteristic
18C2×C8×Q8C2211yescharacteristic
19C22×C4⋊C8C2211yescharacteristic
20C22×C4⋊C8C2211yescharacteristic
21#2857C2211yescharacteristic
22C4×C22⋊Q8C2211yescharacteristic
23C2×#384C2211yescharacteristic
24#2100—21no—
25#2100—21no—
26#2099—21no—
27#2096—21no—
28#2068—21no—
29#2056—21no—
30#1990C2211yescharacteristic
31#1990C2211yescharacteristic
32#1984C2211yescharacteristic
33#1981C2211yescharacteristic
34#1948C2211yescharacteristic
35#1948C2211yescharacteristic
36#1947C2211yescharacteristic
37#1939—21no—
38#1939C2211yescharacteristic
39#1916—21no—
40#1892—21no—
41#1872—21no—
42#1853—21no—
43#1850—21no—
44#1834C2211yescharacteristic
45#1834C2211yescharacteristic
46#1824—21no—
47#1824—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
67C22×Q16—21no—
68C23×C8—21no—
69C23×C8—21no—
70C2×C4⋊Q8C2311yescharacteristic
71C2×C22⋊Q8—21no—
72C2×C22⋊Q8D411yescharacteristic
73C2×C4×Q8C2311yescharacteristic
74C22×C4⋊C4—21no—
75C22×C4⋊C4—21no—
76C22×C4⋊C4—21no—
77C22×C42C2311yescharacteristic
78#467—44no—
79#462—22no—
80#462—41no—
81C8×Q8—22no—
82C8×Q8—41no—
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×C22⋊C8—21no—
107C2×C22⋊C8D411yescharacteristic
108C2×C22⋊C8D411yescharacteristic
109C2×C4×C8—21no—
110C2×C4×C8—21no—
111C2×C4×C8C2311yescharacteristic
112C2×C4×C8C2311yescharacteristic
113#391—21no—
114#391D411yescharacteristic
115#389—21no—
116#389—21no—
117#389D411yescharacteristic
118#387—21no—
119#385—21no—
120#384—21no—
121#384—21no—
122#384—21no—
123#384C2311yescharacteristic
124#384C2311yescharacteristic
125#384C2311yescharacteristic
126#380—21no—
127#380—21no—
128#380D411yescharacteristic
129C4×C4⋊C4—21no—
130C4×C4⋊C4C2311yescharacteristic
131C4×C22⋊C4—21no—
132#340—21no—
133#340—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
163C2×Q16—42no—
164C2×Q16—42no—
165C2×Q16—81no—
166C22×C8—21no—
167C22×C8—21no—
168C22×C8—21no—
169C22×C8—21no—
170C22×C8—21no—
171C22×C8—21no—
172C22×C8—42no—
173C22×C8—42no—
174C22×C8—42no—
175C22×C8—42no—
176C22×C8—81no—
177C22×C8—81no—
178C22×C8C2×D411yescharacteristic
179C22×C8C2×D411yescharacteristic
180C22×C8C2×D411yescharacteristic
181C22×C8C2×D411yescharacteristic
182C4⋊Q8—21no—
183C4⋊Q8—21no—
184C4⋊Q8—21no—
185C4⋊Q8—21no—
186C22⋊Q8—22no—
187C22⋊Q8—42no—
188C22⋊Q8—42no—
189C22⋊Q8SD1614yes—
190C4×Q8—21no—
191C4×Q8—21no—
192C4×Q8—21no—
193C4×Q8—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—21no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—42no—
209C2×C4⋊C4—42no—
210C2×C4⋊C4—41no—
211C2×C4⋊C4—41no—
212C2×C4⋊C4—41no—
213C2×C4⋊C4—41no—
214C2×C4⋊C4—81no—
215C2×C4⋊C4—81no—
216C2×C4⋊C4—82no—
217C2×C4⋊C4C2×D411yescharacteristic
218C2×C4⋊C4C2×D411yescharacteristic
219C2×C4⋊C4C2×D411yescharacteristic
220C2×C22⋊C4—21no—
221C2×C22⋊C4—21no—
222C2×C22⋊C4—41no—
223C2×C42—21no—
224C2×C42—21no—
225C2×C42—21no—
226C2×C42—21no—
227C2×C42—21no—
228C2×C42—21no—
229C2×C42C2×D411yescharacteristic
230C2×C42C2×D411yescharacteristic
231C2×C42#4111yescharacteristic
232C2×C42C2411yescharacteristic
233#107—42no—
234#106—81no—
235#106—81no—
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—21no—
245C4⋊C8—21no—
246C4⋊C8—42no—
247C4⋊C8—42no—
248C4⋊C8—42no—
249C4⋊C8—81no—
250C4⋊C8#4112yes—
251C4⋊C8#4112yes—
252#103—42no—
253#103—42no—
254#103—42no—
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
320Q16—84no—
321C2×C8—21no—
322C2×C8—21no—
323C2×C8—22no—
324C2×C8—22no—
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—21no—
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—41no—
351C2×C8—84no—
352C2×C8—84no—
353C4⋊C4—21no—
354C4⋊C4—24no—
355C4⋊C4—21no—
356C4⋊C4—21no—
357C4⋊C4—21no—
358C4⋊C4—21no—
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—41no—
377C4⋊C4—81no—
378C4⋊C4—81no—
379C4⋊C4—81no—
380C4⋊C4—81no—
381C4⋊C4—82no—
382C4⋊C4—82no—
383C4⋊C4—82no—
384C4⋊C4—82no—
385C4⋊C4C2×SD1612yes—
386C22⋊C4—24no—
387C22⋊C4—42no—
388C22⋊C4—82no—
389C42—21no—
390C42—21no—
391C42—21no—
392C42—21no—
393C42—21no—
394C42—21no—
395C42—41no—
396C42—41no—
397C42—41no—
398C42—41no—
399C42C2×#4111yescharacteristic
400C42C2×#4111yescharacteristic
401C42H(5,𝔽2)11yescharacteristic
402C422−1+411yescharacteristic
403C23—21no—
404C23—21no—
405C23—21no—
406C23—21no—
407C23#46412yes—
408C23#46512yes—
409C23#47312yes—
410C23C2×#12111yescharacteristic
411Q8—22no—
412Q8—22no—
413Q8—41no—
414Q8—41no—
415Q8—81no—
416Q8—81no—
417Q8—81no—
418Q8—81no—
419C2×C4—21no—
420C2×C4—21no—
421C2×C4—21no—
422C2×C4—21no—
423C2×C4—21no—
424C2×C4—21no—
425C2×C4—22no—
426C2×C4—22no—
427C2×C4—22no—
428C2×C4—22no—
429C2×C4—22no—
430C2×C4—22no—
431C2×C4—22no—
432C2×C4—22no—
433C2×C4—21no—
434C2×C4—22no—
435C2×C4—22no—
436C2×C4—21no—
437C2×C4—21no—
438C2×C4—42no—
439C2×C4—42no—
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—41no—
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—81no—
462C2×C4—164no—
463C2×C4#46412yes—
464C2×C4#46512yes—
465C2×C4#52911yescharacteristic
466C2×C4#53911yescharacteristic
467C2×C4#54611yescharacteristic
468C2×C4#54611yescharacteristic
469C2×C4#54711yescharacteristic
470C2×C4#54711yescharacteristic
471C2×C4C22×SD1611yescharacteristic
472C2×C4C2×#13511yescharacteristic
473C2×C4C2×#13511yescharacteristic
474C2×C4C2×#13711yescharacteristic
475C2×C4#57611yescharacteristic
476C2×C4#57711yescharacteristic
477C2×C4#57711yescharacteristic
478C2×C4#57811yescharacteristic
479C8—41no—
480C8—41no—
481C8—42no—
482C8—41no—
483C8—42no—
484C8—41no—
485C8—81no—
486C8—81no—
487C8—81no—
488C8—81no—
489C22—22no—
490C22—22no—
491C22—22no—
492C22—22no—
493C22—22no—
494C22—22no—
495C22#168014yes—
496C22#242511yescharacteristic
497C22C2×#46411yescharacteristic
498C22#304611yescharacteristic
499C22C2×#46511yescharacteristic
500C22#306211yescharacteristic
501C22C2×#47311yescharacteristic
502C22#306811yescharacteristic
503C4—22no—
504C4—22no—
505C4—21no—
506C4—21no—
507C4—42no—
508C4—41no—
509C4—42no—
510C4—81no—
511C4—81no—
512C4—161no—
513C4—161no—
514C4—161no—
515C4—161no—
516C4#329511yescharacteristic
517C4#329611yescharacteristic
518C4#330111yescharacteristic
519C4#331211yescharacteristic
520C4#333511yescharacteristic
521C4#333611yescharacteristic
522C4#333911yescharacteristic
523C4#334411yescharacteristic
524C2—24no—
525C2C2×#168011yescharacteristic
526C2#1924211yescharacteristic
527C2#1941411yescharacteristic
528C2#1959011yescharacteristic
529C2#2520811yescharacteristic
530C2#2523811yescharacteristic
531C2#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. #10598C2semidirect
8. #10479C2semidirect
9. #9613C2semidirect
10. #9565C2semidirect
11. #9218C2semidirect
14. #8647C2semidirect

Maximal subgroups

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

subgroupindex
#279612
#276802
#247522
#229522
#106572
#106262
#105982
#104792
#96132
#95652
#92182
#91292
#88302
#86472
C2×#12962