Finite Group Databasebeta

← #362591

Subgroups — #362591

Order 512.

940 conjugacy classes of subgroups.

The subgroup lattice has 633 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20774C211yesmaximal, characteristic, complemented
2#20630C211yesmaximal, characteristic
3#20411C211yesmaximal, characteristic
4#11291C211yesmaximal, characteristic, complemented
5#11291C211yesmaximal, characteristic, complemented
6#11191C211yesmaximal, characteristic
7#11191C211yesmaximal, characteristic
8#10930C211yesmaximal, characteristic
9#10929C211yesmaximal, characteristic
10#10883C211yesmaximal, characteristic, complemented
11#10882C211yesmaximal, characteristic, complemented
12#10823C211yesmaximal, characteristic, complemented
13#10808C211yesmaximal, characteristic, complemented
14#10790C211yesmaximal, characteristic
15#10696C211yesmaximal, characteristic, complemented
16C2×C4×#41C2211yescharacteristic
17C2×#424C2211yescharacteristic
18C2×#424C2211yescharacteristic
19C22×#104C2211yescharacteristic
20C22×#104C2211yescharacteristic
21C2×#416C2211yescharacteristic
22C2×#416C2211yescharacteristic
23C2×#408C2211yescharacteristic
24C2×#408C2211yescharacteristic
25C22×#97C2211yescharacteristic
26C22×#97C2211yescharacteristic
27C22×C4×C8C2211yescharacteristic
28C4×#117C2211yescharacteristic
29#1840—22no—
30#1839—22no—
31#1798—22no—
32#1797—22no—
33#1763C412yescomplemented
34#1763C412yescomplemented
35#1762C412yescomplemented
36#1762C412yescomplemented
37C2×#341C2211yescharacteristic
38C2×#341C2211yescharacteristic
39C2×#341C2211yescharacteristic
40C2×#341C2211yescharacteristic
41#1735C2211yescharacteristic
42#1735C2211yescharacteristic
43C2×#339C2211yescharacteristic
44C2×#339C2211yescharacteristic
45C2×#339C2211yescharacteristic
46C2×#339C2211yescharacteristic
47C2×#339C2211yescharacteristic
48C2×#339C2211yescharacteristic
49#1729C2211yescharacteristic
50#1729C2211yescharacteristic
51#1729C2211yescharacteristic
52#1729C2211yescharacteristic
53#1728C2211yescharacteristic
54#1728C2211yescharacteristic
55C2×#336C2211yescharacteristic
56C2×#336C2211yescharacteristic
57C2×#336C2211yescharacteristic
58C4×#95C2211yescharacteristic
59C22×#41C2311yescharacteristic
60C22×M16C2311yescharacteristic
61C22×M16C2311yescharacteristic
62C22×M16C2311yescharacteristic
63C22×M16C2311yescharacteristic
64C23×C8C2311yescharacteristic
65C23×C8C2311yescharacteristic
66C23×C8C2311yescharacteristic
67C23×C8C2311yescharacteristic
68C4×#41—22no—
69C4×#41—22no—
70C4×#41—41no—
71C4×#41—41no—
72C2×C4×Q8C2×C412yes—
73C2×C4×D4—22no—
74C2×C4×D4C2×C412yes—
75C2×#117—21no—
76C2×#117C2311yescharacteristic
77C2×#117C2311yescharacteristic
78C2×#117C2311yescharacteristic
79C22×C42—21no—
80C22×C42C2311yescharacteristic
81C22×C42C2311yescharacteristic
82C22×C42C2311yescharacteristic
83#424—22no—
84#424—22no—
85#424—41no—
86#424—41no—
87C2×C4⋊C8—22no—
88C2×C4⋊C8—22no—
89C2×#104—22no—
90C2×#104—22no—
91C2×#104—22no—
92C2×#104—22no—
93C2×#104—41no—
94C2×#104—41no—
95#416—22no—
96#416—22no—
97#416—22no—
98#416—22no—
99C2×#103C2×C412yes—
100C2×#103C2×C411yescharacteristic
101C2×#103C2×C411yescharacteristic
102C2×#102C2×C412yes—
103C2×#102C2×C411yescharacteristic
104C2×#102C2×C411yescharacteristic
105#408—22no—
106#408—22no—
107#408—41no—
108#408—41no—
109C2×C22⋊C8—22no—
110C2×C22⋊C8—22no—
111C2×#97—22no—
112C2×#97—41no—
113C2×#97—41no—
114C2×#97—41no—
115C2×#97—41no—
116C2×#97C2×C412yes—
117C2×#97C2×C412yes—
118C2×C4×C8—22no—
119C2×C4×C8—41no—
120C2×C4×C8—41no—
121C4×C4⋊C4C2×C412yes—
122C4×C4⋊C4C2×C412yes—
123C4×C22⋊C4—22no—
124#376—21no—
125C2×#95—22no—
126C43—21no—
127#341—22no—
128#341—22no—
129#341—41no—
130#341—41no—
131#340C2×C412yes—
132#340C2×C412yes—
133#340C2×C412yes—
134#340C2×C412yes—
135#339—22no—
136#339—22no—
137#339—22no—
138#339—22no—
139#339—41no—
140#339—41no—
141#336—22no—
142#336—22no—
143#336—22no—
144#336—22no—
145#336—21no—
146#336—21no—
147#336—21no—
148#336—21no—
149#336—22no—
150#336—22no—
151#336D412yes—
152#336D412yes—
153C2×#41—21no—
154C2×#41—21no—
155C2×#41—21no—
156C2×#41—21no—
157C2×#41—21no—
158C2×#41—21no—
159C2×#41—21no—
160C2×#41—21no—
161C2×#41—21no—
162C2×#41—21no—
163C2×#41#4112yes—
164C2×#41#4112yes—
165C22×Q8C22×C411yescharacteristic
166C22×D4—21no—
167C22×D4C22×C411yescharacteristic
168C23×C4—21no—
169C23×C4—21no—
170C23×C4—21no—
171C23×C4—22no—
172C23×C4#4112yes—
173C23×C4C2411yescharacteristic
174C2×M16—21no—
175C2×M16—21no—
176C2×M16—21no—
177C2×M16—21no—
178C2×M16—21no—
179C2×M16—21no—
180C2×M16—21no—
181C2×M16—21no—
182C2×M16—21no—
183C2×M16—21no—
184C2×M16—21no—
185C2×M16—21no—
186C2×M16—21no—
187C2×M16—21no—
188C2×M16—21no—
189C2×M16—21no—
190C2×M16—21no—
191C2×M16—21no—
192C2×M16—21no—
193C2×M16—21no—
194C2×M16—21no—
195C2×M16—21no—
196C2×M16—21no—
197C2×M16—21no—
198C22×C8—21no—
199C22×C8—21no—
200C22×C8—21no—
201C22×C8—21no—
202C22×C8—21no—
203C22×C8—21no—
204C22×C8—21no—
205C22×C8—21no—
206C22×C8—21no—
207C22×C8—21no—
208C22×C8—21no—
209C22×C8—21no—
210C22×C8—21no—
211C22×C8—21no—
212C22×C8—21no—
213C22×C8—21no—
214C22×C8—21no—
215C22×C8—21no—
216C22×C8—21no—
217C22×C8—21no—
218C22×C8—21no—
219C22×C8—21no—
220C22×C8—21no—
221C22×C8—21no—
222C22×C8—21no—
223C22×C8—21no—
224C22×C8C22×C412yes—
225C22×C8C22×C411yescharacteristic
226C22×C8C22×C411yescharacteristic
227C22×C8C2×D412yes—
228C22×C8C2×D412yes—
229C22×C8#4112yes—
230C22×C8#4112yes—
231C4×Q8—24no—
232C4×Q8—42no—
233C4×D4—24no—
234C4×D4—42no—
235C4×D4—44no—
236C4×D4—44no—
237#117—21no—
238#117—21no—
239#117—21no—
240#117—21no—
241#117—21no—
242#117—21no—
243#117—21no—
244#117—21no—
245#117—21no—
246#117—21no—
247#117—42no—
248#117—42no—
249#117#4112yes—
250#117#4112yes—
251C2×C4⋊C4—22no—
252C2×C4⋊C4C22×C411yescharacteristic
253C2×C4⋊C4C22×C411yescharacteristic
254C2×C4⋊C4C22×C412yes—
255C2×C4⋊C4C22×C412yes—
256C2×C22⋊C4—21no—
257C2×C22⋊C4—21no—
258C2×C22⋊C4—21no—
259C2×C22⋊C4—22no—
260C2×C42—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—21no—
271C2×C42—21no—
272C2×C42—21no—
273C2×C42—22no—
274C2×C42—22no—
275C2×C42—21no—
276C2×C42—21no—
277C2×C42—21no—
278C2×C42—21no—
279C2×C42—22no—
280C2×C42—42no—
281C2×C42—42no—
282C2×C42C22×C412yes—
283C2×C42C2×D411yescharacteristic
284C2×C42C2×D411yescharacteristic
285C2×C42#4112yes—
286C2×C42#4112yes—
287C4⋊C8—44no—
288C4⋊C8—44no—
289#104—44no—
290#104—44no—
291#103—24no—
292#103—22no—
293#103—22no—
294#102—24no—
295#102—22no—
296#102—22no—
297C22⋊C8—44no—
298C22⋊C8—44no—
299#97—42no—
300#97—42no—
301#97—44no—
302C4×C8—44no—
303#95—22no—
304#95—22no—
305#95—44no—
306C24—21no—
307C24C2×#4111yescharacteristic
308#41—22no—
309#41—22no—
310#41—22no—
311#41—22no—
312#41—22no—
313#41—22no—
314#41—22no—
315#41—22no—
316#41—41no—
317#41—41no—
318#41—41no—
319#41—41no—
320#41—41no—
321#41—41no—
322#41—41no—
323#41—41no—
324C2×Q8—21no—
325C2×Q8—21no—
326C2×Q8—21no—
327C2×Q8—21no—
328C2×Q8—21no—
329C2×Q8#11712yes—
330C2×D4—22no—
331C2×D4—22no—
332C2×D4—21no—
333C2×D4—21no—
334C2×D4—21no—
335C2×D4—21no—
336C2×D4—21no—
337C2×D4—41no—
338C2×D4—41no—
339C2×D4—41no—
340C2×D4—41no—
341C2×D4#11712yes—
342C22×C4—24no—
343C22×C4—22no—
344C22×C4—21no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—21no—
352C22×C4—21no—
353C22×C4—22no—
354C22×C4—42no—
355C22×C4—42no—
356C22×C4—44no—
357C22×C4—44no—
358C22×C4—42no—
359C22×C4—41no—
360C22×C4—41no—
361C22×C4—41no—
362C22×C4—41no—
363C22×C4—41no—
364C22×C4—41no—
365C22×C4—42no—
366C22×C4—42no—
367C22×C4—41no—
368C22×C4—41no—
369C22×C4—82no—
370C22×C4C4×D412yes—
371C22×C4C4×D412yes—
372C22×C4C23×C411yescharacteristic
373C22×C4C22×D411yescharacteristic
374C22×C4C2×#4111yescharacteristic
375C22×C4C2×#4111yescharacteristic
376C22×C4C2×#4111yescharacteristic
377C22×C4C2×#4111yescharacteristic
378C22×C4C2×#4111yescharacteristic
379C22×C4C2×#4111yescharacteristic
380C22×C4C2×#4111yescharacteristic
381C22×C4C2×#4111yescharacteristic
382C22×C4H(5,𝔽2)11yescharacteristic
383C22×C4H(5,𝔽2)11yescharacteristic
384C22×C4H(5,𝔽2)11yescharacteristic
385C22×C42−1+411yescharacteristic
386M16—41no—
387M16—41no—
388M16—41no—
389M16—41no—
390M16—41no—
391M16—41no—
392M16—41no—
393M16—41no—
394M16—41no—
395M16—41no—
396M16—41no—
397M16—41no—
398M16—41no—
399M16—41no—
400M16—41no—
401M16—41no—
402C2×C8—21no—
403C2×C8—21no—
404C2×C8—21no—
405C2×C8—21no—
406C2×C8—22no—
407C2×C8—22no—
408C2×C8—22no—
409C2×C8—22no—
410C2×C8—22no—
411C2×C8—22no—
412C2×C8—22no—
413C2×C8—22no—
414C2×C8—22no—
415C2×C8—22no—
416C2×C8—22no—
417C2×C8—22no—
418C2×C8—22no—
419C2×C8—41no—
420C2×C8—41no—
421C2×C8—41no—
422C2×C8—41no—
423C2×C8—41no—
424C2×C8—41no—
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—41no—
439C2×C8—41no—
440C2×C8—41no—
441C2×C8—41no—
442C2×C8—41no—
443C2×C8#11712yes—
444C2×C8#11712yes—
445C4⋊C4—22no—
446C4⋊C4—22no—
447C4⋊C4—21no—
448C4⋊C4—21no—
449C4⋊C4—21no—
450C4⋊C4—21no—
451C4⋊C4—22no—
452C4⋊C4—44no—
453C4⋊C4#11714yes—
454C22⋊C4—22no—
455C22⋊C4—22no—
456C22⋊C4—44no—
457C22⋊C4—41no—
458C22⋊C4—41no—
459C22⋊C4—41no—
460C22⋊C4—41no—
461C42—22no—
462C42—22no—
463C42—22no—
464C42—22no—
465C42—21no—
466C42—21no—
467C42—21no—
468C42—21no—
469C42—22no—
470C42—22no—
471C42—22no—
472C42—22no—
473C42—44no—
474C42—44no—
475C42—41no—
476C42—41no—
477C42—41no—
478C42—41no—
479C42—41no—
480C42—41no—
481C42—41no—
482C42—41no—
483C42—82no—
484C23—22no—
485C23—22no—
486C23—22no—
487C23—41no—
488C23—42no—
489C23—42no—
490C23—41no—
491C23C2×C4×D411yescharacteristic
492C23#52911yescharacteristic
493C23#52911yescharacteristic
494C23#55111yescharacteristic
495C23#55111yescharacteristic
496C23#55211yescharacteristic
497C23#55311yescharacteristic
498Q8—22no—
499Q8—22no—
500Q8—41no—
501Q8—41no—
502D4—22no—
503D4—22no—
504D4—42no—
505D4—42no—
506D4—42no—
507D4—42no—
508D4—41no—
509D4—41no—
510C2×C4—24no—
511C2×C4—22no—
512C2×C4—22no—
513C2×C4—24no—
514C2×C4—22no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—44no—
520C2×C4—42no—
521C2×C4—44no—
522C2×C4—42no—
523C2×C4—41no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—41no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—82no—
542C2×C4—82no—
543C2×C4—88no—
544C2×C4—82no—
545C2×C4—82no—
546C2×C4C2×#11711yescharacteristic
547C2×C4C2×#11711yescharacteristic
548C2×C4C2×#11711yescharacteristic
549C2×C4C4×#4111yescharacteristic
550C2×C4#51811yescharacteristic
551C2×C4#52011yescharacteristic
552C2×C4#52911yescharacteristic
553C2×C4#52911yescharacteristic
554C2×C4#52911yescharacteristic
555C2×C4#52911yescharacteristic
556C2×C4#53211yescharacteristic
557C2×C4#53211yescharacteristic
558C2×C4#53811yescharacteristic
559C2×C4#53811yescharacteristic
560C2×C4#53911yescharacteristic
561C2×C4#53911yescharacteristic
562C2×C4#54611yescharacteristic
563C2×C4#54611yescharacteristic
564C2×C4#54611yescharacteristic
565C2×C4#54811yescharacteristic
566C2×C4#55111yescharacteristic
567C2×C4#55111yescharacteristic
568C2×C4#55311yescharacteristic
569C2×C4#55311yescharacteristic
570C2×C4#55311yescharacteristic
571C2×C4#55311yescharacteristic
572C2×C4#55611yescharacteristic
573C2×C4#55611yescharacteristic
574C8—22no—
575C8—22no—
576C8—42no—
577C8—42no—
578C8—42no—
579C8—81no—
580C8—81no—
581C8—81no—
582C8—81no—
583C22—22no—
584C22—22no—
585C22—21no—
586C22—21no—
587C22—21no—
588C22—21no—
589C22—22no—
590C22—22no—
591C22—44no—
592C22—42no—
593C22—81no—
594C22—81no—
595C22—82no—
596C22#229311yescharacteristic
597C22#234211yescharacteristic
598C22#235211yescharacteristic
599C22#257711yescharacteristic
600C22#257711yescharacteristic
601C22#257811yescharacteristic
602C22#257811yescharacteristic
603C22#295512yes—
604C22#295612yes—
605C4—44no—
606C4—42no—
607C4—42no—
608C4—44no—
609C4—42no—
610C4—82no—
611C4—82no—
612C4—81no—
613C4—81no—
614C4#228311yescharacteristic
615C4#231811yescharacteristic
616C4#234711yescharacteristic
617C4#235311yescharacteristic
618C4#255911yescharacteristic
619C4#255911yescharacteristic
620C4#256011yescharacteristic
621C4#256011yescharacteristic
622C2—21no—
623C2—21no—
624C2—21no—
625C2—21no—
626C2—82no—
627C2#1391911yescharacteristic
628C2#2129611yescharacteristic
629C2#2129711yescharacteristic
630C2#2258812yes—
631C2#2278412yes—
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
socleC23C2×C4×D4
prime radicalC22×#41C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C2×#336 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #20774C2semidirect
4. #11291C2semidirect
5. #11291C2semidirect
10. #10883C2semidirect
11. #10882C2semidirect
12. #10823C2semidirect
13. #10808C2semidirect
15. #10696C2semidirect
33. #1763C4semidirect
34. #1763C4semidirect
35. #1762C4semidirect
36. #1762C4semidirect

Maximal subgroups

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

subgroupindex
#207742
#206302
#204112
#112912
#111912
#109302
#109292
#108832
#108822
#108232
#108082
#107902
#106962