Finite Group Databasebeta

← #202463

Subgroups — #202463

Order 512.

826 conjugacy classes of subgroups.

The subgroup lattice has 613 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21380C211yesmaximal, characteristic
2#20737C211yesmaximal, characteristic
3#20441C211yesmaximal, characteristic, complemented
4C2×#1573C211yesmaximal, characteristic, complemented
5C2×#1477C211yesmaximal, characteristic
6#9548C211yesmaximal, characteristic, complemented
7#9547C211yesmaximal, characteristic, complemented
8#9546C211yesmaximal, characteristic, complemented
9#9545C211yesmaximal, characteristic, complemented
10#9544C211yesmaximal, characteristic, complemented
11#9543C211yesmaximal, characteristic, complemented
12#9542C211yesmaximal, characteristic, complemented
13#9541C211yesmaximal, characteristic, complemented
14#7900C211yesmaximal, characteristic, complemented
15#7850C211yesmaximal, characteristic, complemented
16C2×C4×#41C2211yescharacteristic
17C2×#432C2211yescharacteristic
18C2×#431C2211yescharacteristic
19C2×#431C2211yescharacteristic
20C22×C4⋊C8C2211yescharacteristic
21C2×#405—21no—
22C2×C4×M16C2211yescharacteristic
23#2285C2211yescharacteristic
24#1984C2211yescharacteristic
25#1984C2211yescharacteristic
26#1954C2211yescharacteristic
27#1954C2211yescharacteristic
28#1870—21no—
29#1870—21no—
30#1825C411yescharacteristic
31#1825C411yescharacteristic
32#1803C411yescharacteristic
33#1803C411yescharacteristic
34#1802C411yescharacteristic
35#1802C411yescharacteristic
36#1788C411yescharacteristic
37#1788C411yescharacteristic
38#1771—21no—
39#1771—21no—
40#1770C2211yescharacteristic
41#1770C2211yescharacteristic
42C8×C4⋊C4C2211yescharacteristic
43C8×C4⋊C4C2211yescharacteristic
44C4×#103C2211yescharacteristic
45C4×#103C2211yescharacteristic
46C4×#102C2211yescharacteristic
47C4×#102C2211yescharacteristic
48#1749—21no—
49#1749—21no—
50#1729—21no—
51#1573—24no—
52C2×#335C2211yescharacteristic
53C2×#335C2211yescharacteristic
54C2×#334C2211yescharacteristic
55C2×#334C2211yescharacteristic
56#1477—24no—
57C2×Q8⋊C8C2211yescharacteristic
58C2×Q8⋊C8C2211yescharacteristic
59C2×D4⋊C8C2211yescharacteristic
60C2×D4⋊C8C2211yescharacteristic
61#1279C2211yescharacteristic
62#1279C2211yescharacteristic
63#1279C2211yescharacteristic
64#1279C2211yescharacteristic
65#1273C2211yescharacteristic
66#1273C2211yescharacteristic
67#1273C2211yescharacteristic
68#1273C2211yescharacteristic
69C22×#41C2×C411yescharacteristic
70C22×M16D411yescharacteristic
71C22×M16D411yescharacteristic
72C23×C8—21no—
73C4×#41—22no—
74C4×#41—22no—
75C4×#41—22no—
76C4×#41—22no—
77C2×C4×Q8C2×C411yescharacteristic
78C2×C4×Q8C2311yescharacteristic
79C2×C4×D4—21no—
80C2×C4×D4—21no—
81C2×C4×D4C2×C411yescharacteristic
82C2×C4×D4C2311yescharacteristic
83C2×#117—21no—
84C2×#117—21no—
85C2×#117C2×C411yescharacteristic
86C22×C42—21no—
87C22×C42C2311yescharacteristic
88#432—22no—
89#432—22no—
90#431—22no—
91#431—22no—
92#431—22no—
93#431—22no—
94C2×C4⋊C8—21no—
95C2×C4⋊C8—22no—
96C2×C4⋊C8—21no—
97C2×C4⋊C8C2×C411yescharacteristic
98C2×C4⋊C8C2×C411yescharacteristic
99C2×C4⋊C8D414yes—
100C2×C4⋊C8C2311yescharacteristic
101C2×C4⋊C8C2311yescharacteristic
102C2×C4⋊C8C2311yescharacteristic
103C2×C4⋊C8C2311yescharacteristic
104C2×#103C2×C411yescharacteristic
105C2×#103C2×C411yescharacteristic
106C2×#103C2×C411yescharacteristic
107C2×#103C2×C411yescharacteristic
108C2×#102C2×C411yescharacteristic
109C2×#102C2×C411yescharacteristic
110C2×#102C2×C411yescharacteristic
111C2×#102C2×C411yescharacteristic
112C2×C22⋊C8—21no—
113C2×C22⋊C8—21no—
114C2×C22⋊C8—21no—
115C2×C22⋊C8—21no—
116C2×C22⋊C8—21no—
117C2×C22⋊C8—21no—
118C2×C22⋊C8C2×C411yescharacteristic
119C2×C22⋊C8C2×C411yescharacteristic
120#405—44no—
121C4×M16—22no—
122C4×M16—22no—
123C2×#97—21no—
124C2×#97—21no—
125C2×#97—21no—
126C2×#97D411yescharacteristic
127C2×#97D411yescharacteristic
128C2×C4×C8—21no—
129C2×C4×C8—21no—
130C2×C4×C8C2311yescharacteristic
131C2×C4×C8C2311yescharacteristic
132C2×C4×C8C2311yescharacteristic
133C2×C4×C8C2311yescharacteristic
134C2×C4×C8C2311yescharacteristic
135C2×C4×C8C2311yescharacteristic
136#384C2×C411yescharacteristic
137#384C2×C411yescharacteristic
138#382—21no—
139#380—21no—
140#380C2×C411yescharacteristic
141#380C2×C411yescharacteristic
142C4×C4⋊C4—21no—
143C4×C4⋊C4C2311yescharacteristic
144C4×C4⋊C4C2311yescharacteristic
145#340C2×C411yescharacteristic
146#340C2×C411yescharacteristic
147#340C2×C411yescharacteristic
148#340C2×C411yescharacteristic
149#340C2×C411yescharacteristic
150#340C2×C411yescharacteristic
151#340C2×C411yescharacteristic
152#340C2×C411yescharacteristic
153#336—21no—
154#336—21no—
155#336—21no—
156#336—21no—
157#336—21no—
158#336—21no—
159#336—21no—
160#336—21no—
161#335—22no—
162#335—22no—
163#334—22no—
164#334—22no—
165Q8⋊C8—22no—
166Q8⋊C8—22no—
167D4⋊C8—22no—
168D4⋊C8—22no—
169C2×#41—22no—
170C2×#41—22no—
171C2×#41—22no—
172C2×#41—22no—
173C2×#41—41no—
174C2×#41—41no—
175C22×Q8C22×C411yescharacteristic
176C22×D4—21no—
177C22×D4C22×C411yescharacteristic
178C23×C4—21no—
179C23×C4—21no—
180C23×C4—21no—
181C23×C4C22×C411yescharacteristic
182C23×C4C2×D411yescharacteristic
183C23×C4#4111yescharacteristic
184C2×M16—22no—
185C2×M16—22no—
186C2×M16—22no—
187C2×M16—22no—
188C2×M16—41no—
189C2×M16—41no—
190C22×C8—24no—
191C22×C8—21no—
192C22×C8—21no—
193C22×C8—21no—
194C22×C8—21no—
195C22×C8—21no—
196C22×C8—21no—
197C22×C8—42no—
198C22×C8—42no—
199C22×C8C22×C411yescharacteristic
200C22×C8C22×C411yescharacteristic
201C22×C8C22×C411yescharacteristic
202C22×C8C22×C411yescharacteristic
203C22×C8C2×D411yescharacteristic
204C22×C8C2×D411yescharacteristic
205C22×C8C2×D411yescharacteristic
206C22×C8C2×D411yescharacteristic
207C4×Q8—22no—
208C4×Q8—22no—
209C4×Q8—21no—
210C4×Q8—21no—
211C4×Q8—21no—
212C4×Q8—21no—
213C4×D4—22no—
214C4×D4—22no—
215C4×D4—21no—
216C4×D4—21no—
217C4×D4—21no—
218C4×D4—21no—
219C4×D4—42no—
220C4×D4—42no—
221C4×D4—42no—
222C4×D4—42no—
223#117—22no—
224#117—22no—
225#117—42no—
226#117—42no—
227#117—42no—
228#117—42no—
229C2×C4⋊C4—21no—
230C2×C4⋊C4—21no—
231C2×C4⋊C4—21no—
232C2×C4⋊C4—21no—
233C2×C4⋊C4—21no—
234C2×C4⋊C4—21no—
235C2×C4⋊C4C22×C411yescharacteristic
236C2×C4⋊C4C22×C411yescharacteristic
237C2×C4⋊C4C22×C411yescharacteristic
238C2×C4⋊C4C22×C411yescharacteristic
239C2×C4⋊C4C22×C411yescharacteristic
240C2×C4⋊C4C22×C411yescharacteristic
241C2×C22⋊C4—21no—
242C2×C22⋊C4—21no—
243C2×C22⋊C4—21no—
244C2×C22⋊C4—21no—
245C2×C22⋊C4—21no—
246C2×C42—21no—
247C2×C42—21no—
248C2×C42—21no—
249C2×C42—21no—
250C2×C42—21no—
251C2×C42—42no—
252C2×C42—42no—
253C2×C42C22×C411yescharacteristic
254C2×C42C2×D412yes—
255C2×C42C2×D411yescharacteristic
256C2×C42#4112yes—
257C2×C42#4112yes—
258C2×C42#4112yes—
259C2×C42#4111yescharacteristic
260C2×C42C2411yescharacteristic
261C4⋊C8—22no—
262C4⋊C8—22no—
263C4⋊C8—24no—
264C4⋊C8—21no—
265C4⋊C8—21no—
266C4⋊C8—21no—
267C4⋊C8—21no—
268C4⋊C8—21no—
269C4⋊C8—21no—
270C4⋊C8—42no—
271C4⋊C8C2×D412yes—
272C4⋊C8C2×D412yes—
273#103—22no—
274#103—22no—
275#103—41no—
276#103—41no—
277#102—22no—
278#102—22no—
279#102—41no—
280#102—41no—
281C22⋊C8—22no—
282C22⋊C8—22no—
283C22⋊C8—42no—
284C22⋊C8—42no—
285C22⋊C8—42no—
286C22⋊C8—42no—
287C22⋊C8—42no—
288C22⋊C8—42no—
289#97—22no—
290#97—22no—
291#97—42no—
292#97—42no—
293#97—42no—
294C4×C8—21no—
295C4×C8—21no—
296C4×C8—21no—
297C4×C8—21no—
298C4×C8—21no—
299C4×C8—21no—
300C4×C8—21no—
301C4×C8—21no—
302C4×C8—21no—
303C4×C8—21no—
304C4×C8—21no—
305C4×C8—21no—
306C4×C8—42no—
307C4×C8—42no—
308#95—21no—
309#95—21no—
310#95—21no—
311#95—21no—
312C24—21no—
313C24C4×D411yescharacteristic
314#41—28no—
315#41—44no—
316#41—44no—
317#41—44no—
318C2×Q8—21no—
319C2×Q8—21no—
320C2×Q8—21no—
321C2×Q8—21no—
322C2×Q8—21no—
323C2×Q8#13112yes—
324C2×D4—21no—
325C2×D4—21no—
326C2×D4—21no—
327C2×D4—21no—
328C2×D4—21no—
329C2×D4—42no—
330C2×D4—42no—
331C2×D4—42no—
332C2×D4#13112yes—
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—21no—
347C22×C4—21no—
348C22×C4—41no—
349C22×C4—41no—
350C22×C4—42no—
351C22×C4—42no—
352C22×C4—42no—
353C22×C4—41no—
354C22×C4—41no—
355C22×C4—41no—
356C22×C4—41no—
357C22×C4—41no—
358C22×C4—41no—
359C22×C4—42no—
360C22×C4—81no—
361C22×C4—81no—
362C22×C4#11712yes—
363C22×C4C4×D412yes—
364C22×C4C4×D411yescharacteristic
365C22×C4C4×D411yescharacteristic
366C22×C4C4×D411yescharacteristic
367C22×C4#12112yes—
368C22×C4#12312yes—
369C22×C4#12412yes—
370C22×C4#12612yes—
371C22×C4C23×C411yescharacteristic
372C22×C4C22×D411yescharacteristic
373C22×C4C2×#4111yescharacteristic
374M16—28no—
375M16—44no—
376C2×C8—21no—
377C2×C8—21no—
378C2×C8—21no—
379C2×C8—21no—
380C2×C8—21no—
381C2×C8—21no—
382C2×C8—21no—
383C2×C8—21no—
384C2×C8—22no—
385C2×C8—22no—
386C2×C8—21no—
387C2×C8—21no—
388C2×C8—21no—
389C2×C8—21no—
390C2×C8—21no—
391C2×C8—21no—
392C2×C8—21no—
393C2×C8—21no—
394C2×C8—21no—
395C2×C8—21no—
396C2×C8—21no—
397C2×C8—21no—
398C2×C8—21no—
399C2×C8—21no—
400C2×C8—48no—
401C2×C8—41no—
402C2×C8—41no—
403C2×C8—41no—
404C2×C8—41no—
405C2×C8—41no—
406C2×C8—41no—
407C2×C8—41no—
408C2×C8—41no—
409C2×C8—41no—
410C2×C8—41no—
411C2×C8—41no—
412C2×C8—41no—
413C2×C8—41no—
414C2×C8—41no—
415C2×C8—41no—
416C2×C8—41no—
417C2×C8AGL(1,ℤ/8)12yes—
418C2×C8#13712yes—
419C4⋊C4—21no—
420C4⋊C4—21no—
421C4⋊C4—21no—
422C4⋊C4—21no—
423C4⋊C4—21no—
424C4⋊C4—21no—
425C4⋊C4—21no—
426C4⋊C4—21no—
427C4⋊C4—21no—
428C4⋊C4—21no—
429C4⋊C4—21no—
430C4⋊C4—21no—
431C4⋊C4—42no—
432C4⋊C4—42no—
433C4⋊C4—41no—
434C4⋊C4—41no—
435C4⋊C4—42no—
436C4⋊C4—81no—
437C4⋊C4—81no—
438C22⋊C4—42no—
439C22⋊C4—42no—
440C22⋊C4—42no—
441C22⋊C4—41no—
442C22⋊C4—41no—
443C22⋊C4—42no—
444C42—41no—
445C42—41no—
446C42—41no—
447C42—41no—
448C42—41no—
449C42—41no—
450C42—42no—
451C42—42no—
452C42#11712yes—
453C42C4×D412yes—
454C42#12112yes—
455C42#12312yes—
456C42#12412yes—
457C42#12612yes—
458C42C22×D411yescharacteristic
459C42C2×#4111yescharacteristic
460C42C2×#4111yescharacteristic
461C42C2×#4111yescharacteristic
462C23—21no—
463C23—21no—
464C23—21no—
465C23—21no—
466C23—21no—
467C23—21no—
468C23—22no—
469C23—41no—
470C23—41no—
471C23—41no—
472C23—41no—
473C23—41no—
474C23—41no—
475C23#38812yes—
476C23C2×C4×D411yescharacteristic
477Q8—22no—
478Q8—22no—
479Q8—41no—
480Q8—41no—
481D4—22no—
482D4—22no—
483D4—48no—
484D4—41no—
485D4—41no—
486C2×C4—21no—
487C2×C4—21no—
488C2×C4—21no—
489C2×C4—21no—
490C2×C4—21no—
491C2×C4—21no—
492C2×C4—21no—
493C2×C4—21no—
494C2×C4—22no—
495C2×C4—22no—
496C2×C4—22no—
497C2×C4—22no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—48no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—84no—
527C2×C4—81no—
528C2×C4—81no—
529C2×C4—81no—
530C2×C4—81no—
531C2×C4#38812yes—
532C2×C4#38812yes—
533C2×C4#38812yes—
534C2×C4C2×#11711yescharacteristic
535C2×C4C2×C4×D411yescharacteristic
536C2×C4C2×#12111yescharacteristic
537C2×C4C2×#12311yescharacteristic
538C2×C4C2×#12411yescharacteristic
539C2×C4C2×#12611yescharacteristic
540C2×C4C2×#13111yescharacteristic
541C2×C4C2×#13111yescharacteristic
542C2×C4#56811yescharacteristic
543C2×C4#56811yescharacteristic
544C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
545C2×C4C2×#13711yescharacteristic
546C2×C4#57511yescharacteristic
547C2×C4#57511yescharacteristic
548C8—22no—
549C8—22no—
550C8—41no—
551C8—41no—
552C8—41no—
553C8—41no—
554C8—41no—
555C8—41no—
556C8—42no—
557C8—42no—
558C8—81no—
559C8—81no—
560C8—81no—
561C8—81no—
562C22—22no—
563C22—22no—
564C22—22no—
565C22—41no—
566C22—41no—
567C22—41no—
568C22—41no—
569C22—42no—
570C22—42no—
571C22—42no—
572C22—81no—
573C22—81no—
574C22—81no—
575C22—81no—
576C22C2×#38811yescharacteristic
577C22#293311yescharacteristic
578C22#293311yescharacteristic
579C22C4×AGL(1,ℤ/8)11yescharacteristic
580C22C4×#13711yescharacteristic
581C22#295711yescharacteristic
582C22#295711yescharacteristic
583C4—22no—
584C4—22no—
585C4—21no—
586C4—22no—
587C4—21no—
588C4—42no—
589C4—81no—
590C4—81no—
591C4—81no—
592C4—81no—
593C4—81no—
594C4—81no—
595C4#292411yescharacteristic
596C4#292411yescharacteristic
597C4#292911yescharacteristic
598C4#293011yescharacteristic
599C4#303611yescharacteristic
600C4#303711yescharacteristic
601C4#309211yescharacteristic
602C4#309311yescharacteristic
603C2—42no—
604C2—82no—
605C2#1800711yescharacteristic
606C2#1800911yescharacteristic
607C2#1801111yescharacteristic
608C2#1801311yescharacteristic
609C2#2105311yescharacteristic
610C2#2105911yescharacteristic
611C2#2107211yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC23C2×C4×D4
prime radicalC22×#41C2×C4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C2×C4×M16 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #20441C2semidirect
4. C2×#1573C2semidirect
6. #9548C2semidirect
7. #9547C2semidirect
8. #9546C2semidirect
9. #9545C2semidirect
10. #9544C2semidirect
11. #9543C2semidirect
12. #9542C2semidirect
13. #9541C2semidirect
14. #7900C2semidirect
15. #7850C2semidirect

Maximal subgroups

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

subgroupindex
#213802
#207372
#204412
C2×#15732
C2×#14772
#95482
#95472
#95462
#95452
#95442
#95432
#95422
#95412
#79002
#78502