Finite Group Databasebeta

← #281929

Subgroups — #281929

Order 512.

725 conjugacy classes of subgroups.

The subgroup lattice has 570 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#25813C211yesmaximal, characteristic
2#24711C211yesmaximal, characteristic
3#24302C211yesmaximal, characteristic
4#23051C211yesmaximal, characteristic
5C2×#1676C211yesmaximal, characteristic, complemented
6C2×#1558C211yesmaximal, characteristic
7#10601C211yesmaximal, characteristic, complemented
8#10543C211yesmaximal, characteristic, complemented
9#10482C211yesmaximal, characteristic, complemented
10#10279C211yesmaximal, characteristic, complemented
11#9739C211yesmaximal, characteristic, complemented
12#9575C211yesmaximal, characteristic, complemented
13#9216C211yesmaximal, characteristic, complemented
14#9136C211yesmaximal, characteristic
15#8834C211yesmaximal, characteristic
16C22×C4⋊Q8C2211yescharacteristic
17C2×#500C2211yescharacteristic
18C2×#500C2211yescharacteristic
19C2×#477C2211yescharacteristic
20C2×#477C2211yescharacteristic
21C2×#477C2211yescharacteristic
22C2×C8×Q8C2211yescharacteristic
23C2×#432C2211yescharacteristic
24C2×#432C2211yescharacteristic
25#2857C2211yescharacteristic
26C4×C22⋊Q8C2211yescharacteristic
27#2075—21no—
28#2057—21no—
29#2057—21no—
30#2057—21no—
31#2035—21no—
32#2024—21no—
33#1990C2211yescharacteristic
34#1990C2211yescharacteristic
35#1987C2211yescharacteristic
36#1987C2211yescharacteristic
37#1987C2211yescharacteristic
38#1984C2211yescharacteristic
39#1948C2211yescharacteristic
40#1944—21no—
41#1939C2211yescharacteristic
42#1935—21no—
43#1924—21no—
44#1923—21no—
45#1917—21no—
46#1872—21no—
47#1852—21no—
48#1852—21no—
49#1852—21no—
50#1803C2211yescharacteristic
51#1797C2211yescharacteristic
52#1771—21no—
53#1770C2211yescharacteristic
54C4×C22⋊C8C2211yescharacteristic
55#1676—24no—
56C2×#335C2211yescharacteristic
57C2×#334C2211yescharacteristic
58#1558—24no—
59C2×#333C2211yescharacteristic
60C2×#333C2211yescharacteristic
61C2×#333C2211yescharacteristic
62C2×#333C2211yescharacteristic
63C2×#333C2211yescharacteristic
64C2×#333C2211yescharacteristic
65#1296C2211yescharacteristic
66#1296C2211yescharacteristic
67#1277C2211yescharacteristic
68#1277C2211yescharacteristic
69C23×Q8—21no—
70C2×C4⋊Q8—22no—
71C2×C4⋊Q8—22no—
72C2×C4⋊Q8—21no—
73C2×C4⋊Q8—21no—
74C2×C4⋊Q8—21no—
75C2×C4⋊Q8—41no—
76C2×C4⋊Q8—41no—
77C2×C4⋊Q8C2311yescharacteristic
78C2×C4⋊Q8C2311yescharacteristic
79C2×C4⋊Q8C2311yescharacteristic
80C2×C22⋊Q8—21no—
81C2×C22⋊Q8D411yescharacteristic
82C2×C4×Q8C2311yescharacteristic
83C22×C4⋊C4—21no—
84C22×C4⋊C4—21no—
85C22×C42C2311yescharacteristic
86#500—22no—
87#500—22no—
88#500—41no—
89#500—41no—
90#477—22no—
91#477—22no—
92#477—22no—
93#477—41no—
94#477—41no—
95#477—41no—
96C8×Q8—22no—
97C8×Q8—41no—
98#432—22no—
99#432—22no—
100#432—41no—
101#432—41no—
102C2×#107—21no—
103C2×#107—21no—
104C2×#107—21no—
105C2×#107—21no—
106C2×#107—21no—
107C2×#107—21no—
108C2×#107—21no—
109C2×#106—21no—
110C2×C4⋊C8—21no—
111C2×C4⋊C8—21no—
112C2×C4⋊C8C2311yescharacteristic
113C2×C4⋊C8C2311yescharacteristic
114C2×C4⋊C8C2311yescharacteristic
115C2×C4⋊C8C2311yescharacteristic
116C2×C4⋊C8C2311yescharacteristic
117C2×C4⋊C8C2311yescharacteristic
118C2×#103—21no—
119C2×#103—21no—
120C2×#103—21no—
121C2×#103—21no—
122C2×C22⋊C8—21no—
123C2×C22⋊C8—21no—
124C2×C22⋊C8—21no—
125C2×C22⋊C8D411yescharacteristic
126C2×C22⋊C8D411yescharacteristic
127C2×#97—21no—
128C2×#97—21no—
129C2×#97—21no—
130C2×C4×C8—21no—
131C2×C4×C8C2311yescharacteristic
132C2×C4×C8C2311yescharacteristic
133#391—21no—
134#391D411yescharacteristic
135#389—21no—
136#389—21no—
137#389D411yescharacteristic
138#387—21no—
139#385—21no—
140#384C2311yescharacteristic
141#380—21no—
142#380—21no—
143#380D411yescharacteristic
144C4×C4⋊C4—21no—
145C4×C4⋊C4C2311yescharacteristic
146C4×C22⋊C4—21no—
147#340—21no—
148#340—21no—
149#340—21no—
150#340—21no—
151#340—21no—
152#340—21no—
153#340—21no—
154#340—21no—
155#340—21no—
156#340—21no—
157#340—21no—
158#340—21no—
159#336—21no—
160#336—21no—
161#336—21no—
162#336—21no—
163#336—21no—
164#336D411yescharacteristic
165#336D411yescharacteristic
166#335—22no—
167#334—41no—
168#333—22no—
169#333—22no—
170#333—22no—
171#333—22no—
172#333—41no—
173#333D414yes—
174C22×Q8—24no—
175C22×Q8—21no—
176C22×Q8—21no—
177C22×Q8—21no—
178C22×Q8—42no—
179C22×Q8—41no—
180C22×Q8—82no—
181C22×Q8—82no—
182C22×Q8C2×D411yescharacteristic
183C23×C4—41no—
184C23×C4C2×D411yescharacteristic
185C23×C4C2×D411yescharacteristic
186C23×C4#4111yescharacteristic
187C22×C8—21no—
188C22×C8—21no—
189C22×C8—21no—
190C22×C8—21no—
191C22×C8—21no—
192C22×C8—21no—
193C22×C8C2×D411yescharacteristic
194C22×C8C2×D411yescharacteristic
195C22×C8C2×D411yescharacteristic
196C22×C8C2×D411yescharacteristic
197C4⋊Q8—24no—
198C4⋊Q8—21no—
199C4⋊Q8—21no—
200C4⋊Q8—21no—
201C4⋊Q8—21no—
202C4⋊Q8—21no—
203C4⋊Q8—21no—
204C4⋊Q8—21no—
205C4⋊Q8—21no—
206C4⋊Q8—21no—
207C4⋊Q8—21no—
208C4⋊Q8—42no—
209C4⋊Q8—42no—
210C4⋊Q8—42no—
211C4⋊Q8—42no—
212C4⋊Q8—42no—
213C4⋊Q8C2×D412yes—
214C4⋊Q8#4112yes—
215C22⋊Q8—22no—
216C22⋊Q8—41no—
217C22⋊Q8—82no—
218C4×Q8—21no—
219C4×Q8—21no—
220C4×Q8—21no—
221C4×Q8—21no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4—21no—
224C2×C4⋊C4—21no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4—21no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4—21no—
229C2×C4⋊C4—21no—
230C2×C4⋊C4—21no—
231C2×C4⋊C4—21no—
232C2×C4⋊C4—21no—
233C2×C4⋊C4—21no—
234C2×C4⋊C4—42no—
235C2×C4⋊C4—42no—
236C2×C4⋊C4—42no—
237C2×C4⋊C4—42no—
238C2×C4⋊C4—41no—
239C2×C4⋊C4—41no—
240C2×C4⋊C4—41no—
241C2×C4⋊C4C2×D411yescharacteristic
242C2×C4⋊C4C2×D411yescharacteristic
243C2×C4⋊C4C2×D411yescharacteristic
244C2×C22⋊C4—21no—
245C2×C22⋊C4—21no—
246C2×C22⋊C4—41no—
247C2×C42—21no—
248C2×C42—21no—
249C2×C42—21no—
250C2×C42—21no—
251C2×C42—21no—
252C2×C42—21no—
253C2×C42C2×D411yescharacteristic
254C2×C42C2×D411yescharacteristic
255C2×C42#4111yescharacteristic
256C2×C42C2411yescharacteristic
257#107—42no—
258#107—42no—
259#107—42no—
260#107—42no—
261#107—42no—
262#107—42no—
263#107—42no—
264#106—42no—
265C4⋊C8—21no—
266C4⋊C8—21no—
267C4⋊C8—21no—
268C4⋊C8—21no—
269C4⋊C8—21no—
270C4⋊C8—21no—
271C4⋊C8—21no—
272C4⋊C8—21no—
273C4⋊C8—21no—
274C4⋊C8—21no—
275C4⋊C8—42no—
276C4⋊C8—42no—
277C4⋊C8C2×D412yes—
278C4⋊C8C2×D412yes—
279#103—42no—
280#103—42no—
281#103—42no—
282#103—42no—
283C22⋊C8—22no—
284C22⋊C8—22no—
285C22⋊C8—42no—
286C22⋊C8—42no—
287C22⋊C8—42no—
288#97—42no—
289#97—42no—
290#97—42no—
291C4×C8—21no—
292C4×C8—21no—
293C4×C8—21no—
294C4×C8—21no—
295C4×C8—42no—
296#95—21no—
297#95—21no—
298#95—21no—
299#95—21no—
300#95—21no—
301#95—41no—
302#95—41no—
303C24#12111yescharacteristic
304C2×Q8—21no—
305C2×Q8—22no—
306C2×Q8—22no—
307C2×Q8—21no—
308C2×Q8—21no—
309C2×Q8—21no—
310C2×Q8—21no—
311C2×Q8—44no—
312C2×Q8—41no—
313C2×Q8—41no—
314C2×Q8—41no—
315C2×Q8—41no—
316C2×Q8—82no—
317C2×Q8—82no—
318C2×Q8—81no—
319C2×Q8—81no—
320C2×Q8—81no—
321C2×Q8—81no—
322C2×Q8—81no—
323C2×Q8—81no—
324C2×Q8—88no—
325C2×Q8—88no—
326C2×Q8#13712yes—
327C22×C4—21no—
328C22×C4—22no—
329C22×C4—21no—
330C22×C4—21no—
331C22×C4—21no—
332C22×C4—21no—
333C22×C4—21no—
334C22×C4—21no—
335C22×C4—21no—
336C22×C4—21no—
337C22×C4—21no—
338C22×C4—21no—
339C22×C4—21no—
340C22×C4—21no—
341C22×C4—21no—
342C22×C4—21no—
343C22×C4—21no—
344C22×C4—21no—
345C22×C4—21no—
346C22×C4—41no—
347C22×C4—44no—
348C22×C4—41no—
349C22×C4—41no—
350C22×C4—41no—
351C22×C4—41no—
352C22×C4—82no—
353C22×C4—82no—
354C22×C4#12111yescharacteristic
355C22×C4#12111yescharacteristic
356C22×C4#12111yescharacteristic
357C22×C4C22×D411yescharacteristic
358C22×C4C22×D411yescharacteristic
359C22×C4C2×#4111yescharacteristic
360C2×C8—21no—
361C2×C8—21no—
362C2×C8—21no—
363C2×C8—21no—
364C2×C8—21no—
365C2×C8—22no—
366C2×C8—22no—
367C2×C8—21no—
368C2×C8—21no—
369C2×C8—21no—
370C2×C8—21no—
371C2×C8—21no—
372C2×C8—41no—
373C2×C8—41no—
374C2×C8—41no—
375C2×C8—41no—
376C2×C8—41no—
377C2×C8—41no—
378C2×C8—41no—
379C2×C8—41no—
380C2×C8—41no—
381C2×C8—41no—
382C2×C8—41no—
383C2×C8—41no—
384C2×C8—41no—
385C2×C8—41no—
386C2×C8—41no—
387C2×C8—41no—
388C2×C8#13712yes—
389C2×C8#13712yes—
390C4⋊C4—22no—
391C4⋊C4—21no—
392C4⋊C4—22no—
393C4⋊C4—22no—
394C4⋊C4—22no—
395C4⋊C4—21no—
396C4⋊C4—21no—
397C4⋊C4—21no—
398C4⋊C4—21no—
399C4⋊C4—21no—
400C4⋊C4—42no—
401C4⋊C4—44no—
402C4⋊C4—44no—
403C4⋊C4—41no—
404C4⋊C4—41no—
405C4⋊C4—41no—
406C4⋊C4—41no—
407C4⋊C4—41no—
408C4⋊C4—41no—
409C4⋊C4—41no—
410C4⋊C4—41no—
411C4⋊C4—41no—
412C4⋊C4—41no—
413C4⋊C4—41no—
414C4⋊C4—41no—
415C4⋊C4—41no—
416C4⋊C4—41no—
417C4⋊C4—81no—
418C4⋊C4—81no—
419C4⋊C4—82no—
420C22⋊C4—42no—
421C22⋊C4—42no—
422C22⋊C4—82no—
423C42—21no—
424C42—21no—
425C42—21no—
426C42—21no—
427C42—21no—
428C42—21no—
429C42—41no—
430C42—41no—
431C42—41no—
432C42—41no—
433C42C22×D411yescharacteristic
434C42C2×#4111yescharacteristic
435C42H(5,𝔽2)11yescharacteristic
436C42H(5,𝔽2)11yescharacteristic
437C23—21no—
438C23—21no—
439C23—21no—
440C23—21no—
441C23—21no—
442C23—21no—
443C23#47212yes—
444C23C2×#12111yescharacteristic
445Q8—22no—
446Q8—22no—
447Q8—41no—
448Q8—41no—
449Q8—82no—
450Q8—82no—
451Q8—82no—
452Q8—82no—
453Q8—88no—
454Q8—161no—
455Q8—161no—
456C2×C4—21no—
457C2×C4—21no—
458C2×C4—21no—
459C2×C4—21no—
460C2×C4—21no—
461C2×C4—21no—
462C2×C4—21no—
463C2×C4—22no—
464C2×C4—21no—
465C2×C4—21no—
466C2×C4—21no—
467C2×C4—21no—
468C2×C4—42no—
469C2×C4—42no—
470C2×C4—41no—
471C2×C4—41no—
472C2×C4—41no—
473C2×C4—41no—
474C2×C4—41no—
475C2×C4—41no—
476C2×C4—41no—
477C2×C4—41no—
478C2×C4—41no—
479C2×C4—41no—
480C2×C4—41no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—41no—
484C2×C4—41no—
485C2×C4—41no—
486C2×C4—41no—
487C2×C4—41no—
488C2×C4—41no—
489C2×C4—41no—
490C2×C4—41no—
491C2×C4—88no—
492C2×C4—81no—
493C2×C4—81no—
494C2×C4—81no—
495C2×C4—81no—
496C2×C4—81no—
497C2×C4—81no—
498C2×C4—81no—
499C2×C4—81no—
500C2×C4—81no—
501C2×C4—81no—
502C2×C4#52511yescharacteristic
503C2×C4#53811yescharacteristic
504C2×C4D4211yescharacteristic
505C2×C4D4211yescharacteristic
506C2×C4#54611yescharacteristic
507C2×C4#54611yescharacteristic
508C2×C4C2×#13711yescharacteristic
509C2×C4C2×#13711yescharacteristic
510C2×C4C2×#13711yescharacteristic
511C2×C4#57511yescharacteristic
512C2×C4#57811yescharacteristic
513C2×C4#57811yescharacteristic
514C2×C4#57811yescharacteristic
515C2×C4#57811yescharacteristic
516C8—22no—
517C8—22no—
518C8—41no—
519C8—42no—
520C8—41no—
521C8—42no—
522C8—81no—
523C8—81no—
524C8—81no—
525C8—81no—
526C22—22no—
527C22—22no—
528C22—22no—
529C22—41no—
530C22—41no—
531C22—41no—
532C22—41no—
533C22#242411yescharacteristic
534C22#303911yescharacteristic
535C22#304411yescharacteristic
536C22#305611yescharacteristic
537C22#305911yescharacteristic
538C22C2×#47211yescharacteristic
539C22#306811yescharacteristic
540C4—21no—
541C4—21no—
542C4—41no—
543C4—41no—
544C4—42no—
545C4—41no—
546C4—81no—
547C4—82no—
548C4—82no—
549C4—81no—
550C4—81no—
551C4—161no—
552C4—161no—
553C4#326911yescharacteristic
554C4#327011yescharacteristic
555C4#327011yescharacteristic
556C4#327611yescharacteristic
557C4#331211yescharacteristic
558C4#331211yescharacteristic
559C4#331311yescharacteristic
560C4#332811yescharacteristic
561C2—42no—
562C2#1932311yescharacteristic
563C2#1935911yescharacteristic
564C2#1947711yescharacteristic
565C2#1953111yescharacteristic
566C2#2509511yescharacteristic
567C2#2511811yescharacteristic
568C2#2514011yescharacteristic
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. C2×#1676C2semidirect
7. #10601C2semidirect
8. #10543C2semidirect
9. #10482C2semidirect
10. #10279C2semidirect
11. #9739C2semidirect
12. #9575C2semidirect
13. #9216C2semidirect

Maximal subgroups

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

subgroupindex
#258132
#247112
#243022
#230512
C2×#16762
C2×#15582
#106012
#105432
#104822
#102792
#97392
#95752
#92162
#91362
#88342