Finite Group Databasebeta

← #227696

Subgroups — #227696

Order 512.

856 conjugacy classes of subgroups.

The subgroup lattice has 636 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23054C211yesmaximal, characteristic, complemented
2#22469C211yesmaximal, characteristic
3C4×#452C211yesmaximal, characteristic
4C2×#1585C211yesmaximal, characteristic, complemented
5C2×#1476C211yesmaximal, characteristic
6#9664C211yesmaximal, characteristic, complemented
7#9559C211yesmaximal, characteristic, complemented
8#9548C211yesmaximal, characteristic, complemented
9#9071C211yesmaximal, characteristic, complemented
10#8950C211yesmaximal, characteristic, complemented
11#8945C211yesmaximal, characteristic, complemented
12#8845C211yesmaximal, characteristic, complemented
13#8831C211yesmaximal, characteristic, complemented
14#8204C211yesmaximal, characteristic, complemented
15#8144C211yesmaximal, characteristic, complemented
16C2×C4×#41C2211yescharacteristic
17C2×#452C411yescharacteristic
18C2×#446C2211yescharacteristic
19C2×C4×Q16C2211yescharacteristic
20C2×C4×SD16C2211yescharacteristic
21C2×C8×D4C2211yescharacteristic
22C2×C8×D4C2211yescharacteristic
23C2×#431C2211yescharacteristic
24C22×C4×C8C2211yescharacteristic
25C4×C22⋊Q8C2211yescharacteristic
26#1984C2211yescharacteristic
27#1984C2211yescharacteristic
28#1984C2211yescharacteristic
29#1939C2211yescharacteristic
30#1924—21no—
31#1924—21no—
32#1916—21no—
33#1878C411yescharacteristic
34#1877C411yescharacteristic
35#1873C411yescharacteristic
36#1869—21no—
37#1869—21no—
38#1865C411yescharacteristic
39#1862C411yescharacteristic
40#1854C411yescharacteristic
41#1817—21no—
42#1797C2211yescharacteristic
43#1788C411yescharacteristic
44#1771—21no—
45#1770C2211yescharacteristic
46C4×#103C2211yescharacteristic
47C4×#102C2211yescharacteristic
48C4×C22⋊C8C2211yescharacteristic
49C4×C22⋊C8C2211yescharacteristic
50C2×#336—21no—
51#1585—24no—
52C2×#335C2211yescharacteristic
53C2×#334C2211yescharacteristic
54#1476—24no—
55C2×Q8⋊C8C2211yescharacteristic
56C2×Q8⋊C8C2211yescharacteristic
57C2×Q8⋊C8C2211yescharacteristic
58C2×Q8⋊C8C2211yescharacteristic
59C2×D4⋊C8C2211yescharacteristic
60C2×D4⋊C8C2211yescharacteristic
61C2×#322C2211yescharacteristic
62C2×#322C2211yescharacteristic
63#1279C2211yescharacteristic
64#1279C2211yescharacteristic
65#1277C2211yescharacteristic
66#1277C2211yescharacteristic
67#1273C2211yescharacteristic
68#1273C2211yescharacteristic
69C22×#41C2×C411yescharacteristic
70C22×Q16C2×C411yescharacteristic
71C22×SD16C2×C411yescharacteristic
72C23×C8—21no—
73C23×C8—21no—
74C23×C8—21no—
75C2×C22⋊Q8C2×C411yescharacteristic
76C4×#41—22no—
77C4×#41—22no—
78C4×#41—41no—
79C4×#41—41no—
80C2×C4×Q8C2×C411yescharacteristic
81C2×C4×Q8C2311yescharacteristic
82C2×C4×Q8C2311yescharacteristic
83C2×C4×D4—21no—
84C2×C4×D4—21no—
85C2×C4×D4C2×C411yescharacteristic
86C2×C4×D4C2311yescharacteristic
87C2×#117—21no—
88C2×#117C2×C411yescharacteristic
89C22×C42—21no—
90C22×C42C2311yescharacteristic
91#452—24no—
92#446—22no—
93#446—41no—
94C4×Q16—41no—
95C4×Q16—41no—
96C4×SD16—22no—
97C4×SD16—22no—
98C8×D4—22no—
99C8×D4—22no—
100C8×D4—41no—
101C8×D4—41no—
102#431—22no—
103#431—41no—
104C2×#107C2×C411yescharacteristic
105C2×#106C2×C411yescharacteristic
106C2×C4⋊C8—21no—
107C2×C4⋊C8—21no—
108C2×C4⋊C8C2311yescharacteristic
109C2×C4⋊C8C2311yescharacteristic
110C2×C4⋊C8C2311yescharacteristic
111C2×C4⋊C8C2311yescharacteristic
112C2×#103C2×C411yescharacteristic
113C2×#103C2×C411yescharacteristic
114C2×#103C2×C411yescharacteristic
115C2×#103C2×C411yescharacteristic
116C2×#103C2×C411yescharacteristic
117C2×#102C2×C411yescharacteristic
118C2×#102C2×C411yescharacteristic
119C2×#102C2×C411yescharacteristic
120C2×C22⋊C8—21no—
121C2×C22⋊C8—21no—
122C2×C22⋊C8—21no—
123C2×C22⋊C8—21no—
124C2×C22⋊C8C2×C411yescharacteristic
125C2×C22⋊C8C2×C411yescharacteristic
126C2×C22⋊C8D411yescharacteristic
127C2×C22⋊C8D411yescharacteristic
128C2×#97—21no—
129C2×#97—21no—
130C2×C4×C8—22no—
131C2×C4×C8—21no—
132C2×C4×C8—21no—
133C2×C4×C8—41no—
134C2×C4×C8C2311yescharacteristic
135C2×C4×C8C2311yescharacteristic
136C2×C4×C8C2311yescharacteristic
137C2×C4×C8C2311yescharacteristic
138C2×C4×C8C2311yescharacteristic
139C2×C4×C8C2311yescharacteristic
140#391C2×C411yescharacteristic
141#389C2×C411yescharacteristic
142#387—21no—
143#385—21no—
144#380C2×C411yescharacteristic
145C4×C4⋊C4—21no—
146C4×C4⋊C4C2311yescharacteristic
147C4×C22⋊C4—21no—
148#340C2×C411yescharacteristic
149#340C2×C411yescharacteristic
150#340C2×C411yescharacteristic
151#340C2×C411yescharacteristic
152#336—21no—
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#336C2×C411yescharacteristic
162#336C2×C411yescharacteristic
163#336D411yescharacteristic
164#336D411yescharacteristic
165#335—22no—
166#334—41no—
167Q8⋊C8—22no—
168Q8⋊C8—22no—
169Q8⋊C8—22no—
170Q8⋊C8—41no—
171D4⋊C8—22no—
172D4⋊C8D414yes—
173#322—22no—
174#322—41no—
175C2×#41—22no—
176C2×#41—22no—
177C2×#41—22no—
178C2×#41—41no—
179C2×#41—41no—
180C2×#41M1614yes—
181C22×Q8C22×C411yescharacteristic
182C22×Q8C22×C411yescharacteristic
183C22×D4—21no—
184C22×D4C22×C411yescharacteristic
185C23×C4—21no—
186C23×C4—21no—
187C23×C4—21no—
188C23×C4C22×C411yescharacteristic
189C23×C4C2×D411yescharacteristic
190C23×C4#4111yescharacteristic
191C2×Q16—22no—
192C2×Q16—41no—
193C2×Q16—41no—
194C2×SD16—22no—
195C2×SD16—22no—
196C2×SD16—22no—
197C22×C8—21no—
198C22×C8—24no—
199C22×C8—21no—
200C22×C8—21no—
201C22×C8—21no—
202C22×C8—21no—
203C22×C8—21no—
204C22×C8—42no—
205C22×C8—42no—
206C22×C8—42no—
207C22×C8—42no—
208C22×C8—42no—
209C22×C8—42no—
210C22×C8—81no—
211C22×C8—81no—
212C22×C8C22×C411yescharacteristic
213C22×C8C22×C411yescharacteristic
214C22×C8C22×C411yescharacteristic
215C22×C8C22×C411yescharacteristic
216C22×C8C2×D411yescharacteristic
217C22×C8C2×D411yescharacteristic
218C22×C8C2×D411yescharacteristic
219C22×C8C2×D411yescharacteristic
220C22⋊Q8—22no—
221C22⋊Q8—22no—
222C4×Q8—22no—
223C4×Q8—21no—
224C4×Q8—21no—
225C4×Q8—21no—
226C4×Q8—21no—
227C4×Q8—21no—
228C4×Q8—21no—
229C4×Q8—21no—
230C4×Q8—21no—
231C4×Q8—41no—
232C4×D4—22no—
233C4×D4—21no—
234C4×D4—21no—
235C4×D4—41no—
236C4×D4—42no—
237C4×D4—42no—
238C4×D4—42no—
239C4×D4—42no—
240C4×D4C2×D412yes—
241C4×D4#4112yes—
242#117—22no—
243#117—42no—
244#117—42no—
245#117—41no—
246C2×C4⋊C4—21no—
247C2×C4⋊C4—21no—
248C2×C4⋊C4—21no—
249C2×C4⋊C4—21no—
250C2×C4⋊C4—21no—
251C2×C4⋊C4C22×C411yescharacteristic
252C2×C4⋊C4C22×C411yescharacteristic
253C2×C4⋊C4C22×C411yescharacteristic
254C2×C4⋊C4C22×C411yescharacteristic
255C2×C4⋊C4C22×C411yescharacteristic
256C2×C22⋊C4—21no—
257C2×C22⋊C4—21no—
258C2×C22⋊C4—21no—
259C2×C22⋊C4—21no—
260C2×C22⋊C4—21no—
261C2×C42—21no—
262C2×C42—21no—
263C2×C42—21no—
264C2×C42—21no—
265C2×C42—21no—
266C2×C42—21no—
267C2×C42—21no—
268C2×C42—21no—
269C2×C42—21no—
270C2×C42—42no—
271C2×C42—42no—
272C2×C42C22×C411yescharacteristic
273C2×C42C2×D411yescharacteristic
274C2×C42#4111yescharacteristic
275C2×C42C2411yescharacteristic
276#107—41no—
277#106—22no—
278C4⋊C8—21no—
279C4⋊C8—21no—
280C4⋊C8—21no—
281C4⋊C8—21no—
282C4⋊C8—21no—
283C4⋊C8—21no—
284C4⋊C8—21no—
285C4⋊C8—42no—
286C4⋊C8—42no—
287C4⋊C8C2×D412yes—
288#103—22no—
289#103—22no—
290#103—41no—
291#103—41no—
292#103—41no—
293#102—22no—
294#102—22no—
295#102—22no—
296C22⋊C8—22no—
297C22⋊C8—22no—
298C22⋊C8—41no—
299C22⋊C8—41no—
300C22⋊C8—42no—
301C22⋊C8—42no—
302C22⋊C8—42no—
303C22⋊C8—81no—
304#97—42no—
305#97—42no—
306C4×C8—21no—
307C4×C8—21no—
308C4×C8—21no—
309C4×C8—21no—
310C4×C8—21no—
311C4×C8—21no—
312C4×C8—21no—
313C4×C8—21no—
314C4×C8—21no—
315C4×C8—21no—
316C4×C8—21no—
317C4×C8—42no—
318C4×C8—42no—
319C4×C8C2×D412yes—
320#95—21no—
321#95—21no—
322#95—21no—
323#95—21no—
324#95—21no—
325C24—21no—
326C24C4×D411yescharacteristic
327#41—28no—
328#41—44no—
329#41—44no—
330#41—44no—
331C2×Q8—21no—
332C2×Q8—21no—
333C2×Q8—21no—
334C2×Q8—21no—
335C2×Q8—21no—
336C2×Q8—21no—
337C2×Q8—21no—
338C2×Q8—21no—
339C2×Q8—21no—
340C2×Q8—21no—
341C2×Q8—21no—
342C2×Q8C2×M1612yes—
343C2×D4—24no—
344C2×D4—21no—
345C2×D4—21no—
346C2×D4—21no—
347C2×D4—42no—
348C2×D4—42no—
349C2×D4C4×D412yes—
350C2×D4C2×M1612yes—
351C2×D4#13112yes—
352C22×C4—21no—
353C22×C4—21no—
354C22×C4—21no—
355C22×C4—21no—
356C22×C4—21no—
357C22×C4—21no—
358C22×C4—21no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—21no—
362C22×C4—21no—
363C22×C4—21no—
364C22×C4—21no—
365C22×C4—21no—
366C22×C4—21no—
367C22×C4—21no—
368C22×C4—24no—
369C22×C4—22no—
370C22×C4—21no—
371C22×C4—21no—
372C22×C4—21no—
373C22×C4—21no—
374C22×C4—21no—
375C22×C4—41no—
376C22×C4—42no—
377C22×C4—42no—
378C22×C4—42no—
379C22×C4—41no—
380C22×C4—41no—
381C22×C4—41no—
382C22×C4—41no—
383C22×C4—42no—
384C22×C4—81no—
385C22×C4C4×D411yescharacteristic
386C22×C4C4×D411yescharacteristic
387C22×C4C4×D411yescharacteristic
388C22×C4C2×M1612yes—
389C22×C4#13112yes—
390C22×C4C23×C411yescharacteristic
391C22×C4C22×D411yescharacteristic
392C22×C4C2×#4111yescharacteristic
393Q16—44no—
394SD16—44no—
395C2×C8—22no—
396C2×C8—22no—
397C2×C8—22no—
398C2×C8—22no—
399C2×C8—21no—
400C2×C8—21no—
401C2×C8—21no—
402C2×C8—21no—
403C2×C8—21no—
404C2×C8—21no—
405C2×C8—22no—
406C2×C8—21no—
407C2×C8—21no—
408C2×C8—21no—
409C2×C8—21no—
410C2×C8—21no—
411C2×C8—21no—
412C2×C8—21no—
413C2×C8—21no—
414C2×C8—21no—
415C2×C8—21no—
416C2×C8—21no—
417C2×C8—21no—
418C2×C8—21no—
419C2×C8—21no—
420C2×C8—21no—
421C2×C8—21no—
422C2×C8—21no—
423C2×C8—21no—
424C2×C8—48no—
425C2×C8—41no—
426C2×C8—41no—
427C2×C8—41no—
428C2×C8—41no—
429C2×C8—41no—
430C2×C8—41no—
431C2×C8—41no—
432C2×C8—41no—
433C2×C8—41no—
434C2×C8—41no—
435C2×C8—41no—
436C2×C8—41no—
437C2×C8—41no—
438C2×C8—84no—
439C2×C8—84no—
440C4⋊C4—21no—
441C4⋊C4—21no—
442C4⋊C4—21no—
443C4⋊C4—21no—
444C4⋊C4—21no—
445C4⋊C4—21no—
446C4⋊C4—21no—
447C4⋊C4—21no—
448C4⋊C4—21no—
449C4⋊C4—42no—
450C4⋊C4—41no—
451C4⋊C4—41no—
452C4⋊C4—41no—
453C4⋊C4—41no—
454C4⋊C4—42no—
455C4⋊C4—42no—
456C4⋊C4C4×D412yes—
457C22⋊C4—22no—
458C22⋊C4—22no—
459C22⋊C4—42no—
460C22⋊C4—42no—
461C22⋊C4—42no—
462C22⋊C4—81no—
463C42—21no—
464C42—21no—
465C42—21no—
466C42—21no—
467C42—21no—
468C42—21no—
469C42—42no—
470C42—42no—
471C42—41no—
472C42—41no—
473C42—41no—
474C42—41no—
475C42—41no—
476C42—41no—
477C42C22×D411yescharacteristic
478C42C2×#4111yescharacteristic
479C42H(5,𝔽2)11yescharacteristic
480C42H(5,𝔽2)11yescharacteristic
481C23—22no—
482C23—21no—
483C23—22no—
484C23—21no—
485C23—22no—
486C23—21no—
487C23—21no—
488C23—21no—
489C23—21no—
490C23—41no—
491C23—41no—
492C23—41no—
493C23—41no—
494C23#43512yes—
495C23C2×C4×D411yescharacteristic
496Q8—22no—
497Q8—22no—
498Q8—41no—
499Q8—41no—
500Q8—41no—
501Q8—41no—
502Q8—41no—
503Q8—41no—
504D4—22no—
505D4—22no—
506D4—22no—
507D4—48no—
508D4#43514yes—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—21no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—48no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—42no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—42no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—84no—
555C2×C4#43512yes—
556C2×C4#44312yes—
557C2×C4C2×C4×D411yescharacteristic
558C2×C4#51811yescharacteristic
559C2×C4D4211yescharacteristic
560C2×C4#54611yescharacteristic
561C2×C4#54611yescharacteristic
562C2×C4#55111yescharacteristic
563C2×C4C22×M1611yescharacteristic
564C2×C4C2×#13111yescharacteristic
565C2×C4C2×#13111yescharacteristic
566C2×C4#56811yescharacteristic
567C2×C4#57511yescharacteristic
568C2×C4#57511yescharacteristic
569C2×C4#57711yescharacteristic
570C2×C4#57811yescharacteristic
571C8—24no—
572C8—42no—
573C8—41no—
574C8—41no—
575C8—41no—
576C8—41no—
577C8—41no—
578C8—42no—
579C8—41no—
580C8—41no—
581C8—41no—
582C8—81no—
583C8—81no—
584C8—81no—
585C22—22no—
586C22—24no—
587C22—22no—
588C22—22no—
589C22—41no—
590C22—42no—
591C22—41no—
592C22—42no—
593C22—41no—
594C22—42no—
595C22—41no—
596C22—42no—
597C22—42no—
598C22—42no—
599C22#235811yescharacteristic
600C22C2×#43511yescharacteristic
601C22#293311yescharacteristic
602C22#294311yescharacteristic
603C22#294711yescharacteristic
604C22C2×#44311yescharacteristic
605C22#295611yescharacteristic
606C4—22no—
607C4—21no—
608C4—22no—
609C4—21no—
610C4—42no—
611C4—41no—
612C4—42no—
613C4—81no—
614C4—81no—
615C4—81no—
616C4—81no—
617C4—81no—
618C4C2×#43511yescharacteristic
619C4#293311yescharacteristic
620C4#297311yescharacteristic
621C4#297711yescharacteristic
622C4#326711yescharacteristic
623C4#326911yescharacteristic
624C4#331111yescharacteristic
625C4#331511yescharacteristic
626C2—42no—
627C2—44no—
628C2#1786811yescharacteristic
629C2#1787611yescharacteristic
630C2#1800211yescharacteristic
631C2#1801611yescharacteristic
632C2#2284411yescharacteristic
633C2#2285611yescharacteristic
634C2#2288811yescharacteristic
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 ◁ C4×C22⋊C8 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #23054C2semidirect
4. C2×#1585C2semidirect
6. #9664C2semidirect
7. #9559C2semidirect
8. #9548C2semidirect
9. #9071C2semidirect
10. #8950C2semidirect
11. #8945C2semidirect
12. #8845C2semidirect
13. #8831C2semidirect
14. #8204C2semidirect
15. #8144C2semidirect

Maximal subgroups

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

subgroupindex
#230542
#224692
C4×#4522
C2×#15852
C2×#14762
#96642
#95592
#95482
#90712
#89502
#89452
#88452
#88312
#82042
#81442