Finite Group Databasebeta

← #371625

Subgroups — #371625

Order 512.

937 conjugacy classes of subgroups.

The subgroup lattice has 645 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21147C211yesmaximal, characteristic, complemented
2C2×#1938C211yesmaximal, characteristic
3#21083C211yesmaximal, characteristic
4#21052C211yesmaximal, characteristic
5C2×#1770C211yesmaximal, characteristic, complemented
6#11317C211yesmaximal, characteristic, complemented
7#11226C211yesmaximal, characteristic, complemented
8#11075C211yesmaximal, characteristic
9#11068C211yesmaximal, characteristic
10#10934C211yesmaximal, characteristic
11#10930C211yesmaximal, characteristic
12#10887C211yesmaximal, characteristic, complemented
13#10883C211yesmaximal, characteristic, complemented
14#10822C211yesmaximal, characteristic, complemented
15#10707C211yesmaximal, characteristic, complemented
16C2×#424C2211yescharacteristic
17C2×#424C2211yescharacteristic
18C22×C4⋊C8C2211yescharacteristic
19C22×#104C2211yescharacteristic
20C22×#104C2211yescharacteristic
21C2×#416C2211yescharacteristic
22C2×#416C2211yescharacteristic
23C2×#408C2211yescharacteristic
24C2×#408C2211yescharacteristic
25C2×#405C2211yescharacteristic
26C2×#405C2211yescharacteristic
27C22×#97C2211yescharacteristic
28C22×#97C2211yescharacteristic
29C22×C4×C8C2211yescharacteristic
30#2297C2211yescharacteristic
31#2296C2211yescharacteristic
32C2×C4×C4⋊C4C2211yescharacteristic
33#1940—22no—
34#1939—22no—
35#1938—24no—
36#1934C412yescomplemented
37#1934C412yescomplemented
38#1933C412yescomplemented
39#1933C412yescomplemented
40#1924—22no—
41#1923—22no—
42#1770—24no—
43C2×#341C2211yescharacteristic
44C2×#341C2211yescharacteristic
45C2×#341C2211yescharacteristic
46C2×#341C2211yescharacteristic
47#1735C2211yescharacteristic
48#1735C2211yescharacteristic
49C2×#339C2211yescharacteristic
50C2×#339C2211yescharacteristic
51C2×#339C2211yescharacteristic
52C2×#339C2211yescharacteristic
53C2×#339C2211yescharacteristic
54C2×#339C2211yescharacteristic
55#1729C2211yescharacteristic
56#1729C2211yescharacteristic
57#1728C2211yescharacteristic
58#1728C2211yescharacteristic
59C2×#336C2211yescharacteristic
60C2×#336C2211yescharacteristic
61C22×#41C2311yescharacteristic
62C22×M16C2311yescharacteristic
63C22×M16C2311yescharacteristic
64C22×M16C2311yescharacteristic
65C22×M16C2311yescharacteristic
66C23×C8C2311yescharacteristic
67C23×C8C2311yescharacteristic
68C23×C8C2311yescharacteristic
69C23×C8C2311yescharacteristic
70C2×#117C2311yescharacteristic
71C2×#117C2311yescharacteristic
72C2×#117C2311yescharacteristic
73C22×C4⋊C4—21no—
74C22×C42C2311yescharacteristic
75C22×C42C2311yescharacteristic
76C22×C42C2311yescharacteristic
77#424—22no—
78#424—22no—
79#424—22no—
80#424—41no—
81C2×C4⋊C8—22no—
82C2×C4⋊C8—22no—
83C2×C4⋊C8—21no—
84C2×C4⋊C8—22no—
85C2×C4⋊C8—21no—
86C2×C4⋊C8D412yes—
87C2×C4⋊C8D412yes—
88C2×#104—22no—
89C2×#104—22no—
90C2×#104—22no—
91C2×#104—41no—
92C2×#104—41no—
93C2×#104—41no—
94#416—22no—
95#416—22no—
96#416—22no—
97#416—22no—
98C2×#103C2×C411yescharacteristic
99C2×#103C2×C411yescharacteristic
100C2×#103C2×C412yes—
101C2×#102C2×C411yescharacteristic
102C2×#102C2×C411yescharacteristic
103C2×#102C2×C412yes—
104#408—22no—
105#408—22no—
106#408—22no—
107#408—41no—
108C2×C22⋊C8—21no—
109C2×C22⋊C8—22no—
110C2×C22⋊C8—21no—
111#405—22no—
112#405—22no—
113#405—41no—
114#405—41no—
115C2×#97—22no—
116C2×#97—21no—
117C2×#97—21no—
118C2×#97—41no—
119C2×#97—41no—
120C2×#97—41no—
121C2×#97—41no—
122C2×#97D414yes—
123C2×C4×C8—22no—
124C2×C4×C8—22no—
125C2×C4×C8—21no—
126C2×C4×C8—21no—
127C2×C4×C8C2×C412yes—
128C2×C4×C8C2×C412yes—
129#391C2×C412yes—
130#390C2×C412yes—
131#389C2×C412yes—
132#389C2×C412yes—
133#386—21no—
134#386—21no—
135#385—21no—
136#385—21no—
137C4×C4⋊C4—22no—
138C4×C4⋊C4—22no—
139C4×C4⋊C4—22no—
140C4×C22⋊C4—22no—
141C2×#95—21no—
142#341—22no—
143#341—22no—
144#341—22no—
145#341—22no—
146#340C2×C412yes—
147#340C2×C412yes—
148#340C2×C412yes—
149#340C2×C412yes—
150#339—22no—
151#339—22no—
152#339—22no—
153#339—22no—
154#339—22no—
155#339—41no—
156#336—22no—
157#336—22no—
158#336—22no—
159#336—22no—
160#336—22no—
161#336—22no—
162C2×#41—21no—
163C2×#41—21no—
164C2×#41—21no—
165C2×#41—21no—
166C2×#41—21no—
167C2×#41—21no—
168C2×#41—21no—
169C2×#41—21no—
170C2×#41—21no—
171C2×#41—21no—
172C2×#41—21no—
173C2×#41—21no—
174C22×Q8C22×C411yescharacteristic
175C22×D4—21no—
176C22×D4C22×C411yescharacteristic
177C23×C4—21no—
178C23×C4—21no—
179C23×C4—21no—
180C23×C4#4111yescharacteristic
181C23×C4#4111yescharacteristic
182C23×C4C2411yescharacteristic
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—
198C2×M16—21no—
199C2×M16—21no—
200C2×M16—21no—
201C2×M16—21no—
202C2×M16—21no—
203C2×M16—21no—
204C2×M16—21no—
205C2×M16#4112yes—
206C2×M16#4112yes—
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×C8—21no—
225C22×C8—21no—
226C22×C8—21no—
227C22×C8—21no—
228C22×C8—21no—
229C22×C8—21no—
230C22×C8—21no—
231C22×C8C22×C412yes—
232C22×C8C22×C411yescharacteristic
233C22×C8C22×C411yescharacteristic
234C22×C8C2×D412yes—
235C22×C8C2×D412yes—
236C22×C8C2×D412yes—
237C22×C8C2×D411yescharacteristic
238C22×C8C2×D411yescharacteristic
239C22×C8#4112yes—
240C22×C8#4112yes—
241#117—21no—
242#117—21no—
243#117—21no—
244#117—21no—
245#117—21no—
246#117—21no—
247#117—21no—
248#117—21no—
249#117—21no—
250#117—21no—
251#117#4112yes—
252#117#4112yes—
253C2×C4⋊C4—24no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—44no—
257C2×C4⋊C4C22×C412yes—
258C2×C4⋊C4C22×C411yescharacteristic
259C2×C4⋊C4C22×C411yescharacteristic
260C2×C4⋊C4C22×C412yes—
261C2×C22⋊C4—21no—
262C2×C22⋊C4—21no—
263C2×C22⋊C4—21no—
264C2×C22⋊C4—22no—
265C2×C22⋊C4—22no—
266C2×C42—21no—
267C2×C42—21no—
268C2×C42—21no—
269C2×C42—21no—
270C2×C42—21no—
271C2×C42—21no—
272C2×C42—21no—
273C2×C42—21no—
274C2×C42—21no—
275C2×C42—21no—
276C2×C42—21no—
277C2×C42—21no—
278C2×C42—21no—
279C2×C42—21no—
280C2×C42—21no—
281C2×C42—21no—
282C2×C42—21no—
283C2×C42C22×C412yes—
284C2×C42C2×D412yes—
285C2×C42C2×D412yes—
286C2×C42#4112yes—
287C4⋊C8—24no—
288C4⋊C8—24no—
289C4⋊C8—44no—
290C4⋊C8—42no—
291C4⋊C8—42no—
292#104—44no—
293#104—44no—
294#103—22no—
295#103—22no—
296#103—24no—
297#102—22no—
298#102—22no—
299#102—24no—
300C22⋊C8—44no—
301C22⋊C8—42no—
302C22⋊C8—42no—
303#97—44no—
304#97—42no—
305#97—42no—
306#97—44no—
307C4×C8—24no—
308C4×C8—24no—
309C4×C8—42no—
310C4×C8—42no—
311#95—24no—
312#95—22no—
313#95—22no—
314#95—22no—
315#95—22no—
316C24—21no—
317C24C2×#4111yescharacteristic
318#41—41no—
319#41—41no—
320#41—41no—
321#41—41no—
322#41—41no—
323#41—41no—
324#41—41no—
325#41—41no—
326#41—41no—
327#41—41no—
328#41—41no—
329#41—41no—
330#41—41no—
331#41—41no—
332#41—41no—
333#41—41no—
334C2×Q8—21no—
335C2×Q8—21no—
336C2×Q8—21no—
337C2×Q8—21no—
338C2×Q8—21no—
339C2×Q8—21no—
340C2×D4—21no—
341C2×D4—21no—
342C2×D4—21no—
343C2×D4—21no—
344C2×D4—21no—
345C2×D4—21no—
346C2×D4—41no—
347C2×D4—41no—
348C2×D4—41no—
349C2×D4—41no—
350C2×D4—41no—
351C2×D4—41no—
352C22×C4—21no—
353C22×C4—21no—
354C22×C4—24no—
355C22×C4—22no—
356C22×C4—21no—
357C22×C4—21no—
358C22×C4—21no—
359C22×C4—22no—
360C22×C4—21no—
361C22×C4—21no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—42no—
365C22×C4—42no—
366C22×C4—42no—
367C22×C4—42no—
368C22×C4—42no—
369C22×C4—41no—
370C22×C4—41no—
371C22×C4—41no—
372C22×C4—41no—
373C22×C4—41no—
374C22×C4—41no—
375C22×C4—41no—
376C22×C4—41no—
377C22×C4—41no—
378C22×C4—41no—
379C22×C4C4×D412yes—
380C22×C4C4×D412yes—
381C22×C4C23×C411yescharacteristic
382C22×C4C22×D411yescharacteristic
383C22×C4C22×D411yescharacteristic
384C22×C4C2×#4111yescharacteristic
385C22×C4C2×#4111yescharacteristic
386C22×C4C2×#4111yescharacteristic
387C22×C4C2×#4111yescharacteristic
388C22×C4C2×#4111yescharacteristic
389C22×C4C2×#4111yescharacteristic
390C22×C4C2×#4111yescharacteristic
391C22×C4H(5,𝔽2)11yescharacteristic
392C22×C4H(5,𝔽2)11yescharacteristic
393C22×C42−1+411yescharacteristic
394C22×C42−1+411yescharacteristic
395M16—22no—
396M16—22no—
397M16—22no—
398M16—22no—
399M16—41no—
400M16—41no—
401M16—41no—
402M16—41no—
403M16—41no—
404M16—41no—
405M16—41no—
406M16—41no—
407M16—41no—
408M16—41no—
409M16—41no—
410M16—41no—
411C2×C8—21no—
412C2×C8—21no—
413C2×C8—21no—
414C2×C8—21no—
415C2×C8—22no—
416C2×C8—22no—
417C2×C8—22no—
418C2×C8—22no—
419C2×C8—22no—
420C2×C8—22no—
421C2×C8—21no—
422C2×C8—21no—
423C2×C8—21no—
424C2×C8—21no—
425C2×C8—22no—
426C2×C8—22no—
427C2×C8—22no—
428C2×C8—22no—
429C2×C8—22no—
430C2×C8—22no—
431C2×C8—22no—
432C2×C8—22no—
433C2×C8—22no—
434C2×C8—22no—
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—41no—
444C2×C8—41no—
445C2×C8—41no—
446C2×C8—41no—
447C2×C8—41no—
448C2×C8—41no—
449C2×C8—41no—
450C2×C8—41no—
451C2×C8—41no—
452C2×C8—41no—
453C2×C8C4×D412yes—
454C2×C8C4×D412yes—
455C2×C8#12112yes—
456C2×C8#12112yes—
457C4⋊C4—22no—
458C4⋊C4—21no—
459C4⋊C4—21no—
460C4⋊C4—22no—
461C4⋊C4—21no—
462C4⋊C4—21no—
463C4⋊C4—22no—
464C4⋊C4—48no—
465C4⋊C4—44no—
466C4⋊C4—82no—
467C4⋊C4#11714yes—
468C22⋊C4—22no—
469C22⋊C4—22no—
470C22⋊C4—41no—
471C22⋊C4—41no—
472C22⋊C4—41no—
473C22⋊C4—41no—
474C22⋊C4—44no—
475C22⋊C4—82no—
476C42—22no—
477C42—22no—
478C42—22no—
479C42—22no—
480C42—22no—
481C42—22no—
482C42—22no—
483C42—22no—
484C42—22no—
485C42—22no—
486C42—41no—
487C42—41no—
488C42—41no—
489C42—41no—
490C42—41no—
491C42—41no—
492C42—41no—
493C42—41no—
494C23—22no—
495C23—22no—
496C23—41no—
497C23—41no—
498C23—42no—
499C23—42no—
500C23—41no—
501C23C2×C4×D411yescharacteristic
502C23#52511yescharacteristic
503C23#52911yescharacteristic
504C23#55211yescharacteristic
505C23#55311yescharacteristic
506C23#55611yescharacteristic
507C23#55911yescharacteristic
508Q8—41no—
509Q8—41no—
510Q8—41no—
511Q8—41no—
512D4—41no—
513D4—41no—
514D4—41no—
515D4—41no—
516D4—81no—
517D4—81no—
518D4—81no—
519D4—81no—
520C2×C4—24no—
521C2×C4—24no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—24no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—22no—
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—41no—
539C2×C4—42no—
540C2×C4—41no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—82no—
549C2×C4—82no—
550C2×C4—82no—
551C2×C4—82no—
552C2×C4—81no—
553C2×C4—81no—
554C2×C4—81no—
555C2×C4—81no—
556C2×C4#44414yes—
557C2×C4C2×#11711yescharacteristic
558C2×C4C2×C4×D411yescharacteristic
559C2×C4C4×#4111yescharacteristic
560C2×C4C4×#4111yescharacteristic
561C2×C4#52011yescharacteristic
562C2×C4#52011yescharacteristic
563C2×C4C2×#12111yescharacteristic
564C2×C4#52911yescharacteristic
565C2×C4#52911yescharacteristic
566C2×C4#53211yescharacteristic
567C2×C4#53211yescharacteristic
568C2×C4#53211yescharacteristic
569C2×C4#53411yescharacteristic
570C2×C4#53611yescharacteristic
571C2×C4#53911yescharacteristic
572C2×C4#53911yescharacteristic
573C2×C4#54611yescharacteristic
574C2×C4#54611yescharacteristic
575C2×C4#54811yescharacteristic
576C2×C4#54811yescharacteristic
577C2×C4#55111yescharacteristic
578C2×C4#55211yescharacteristic
579C2×C4#55311yescharacteristic
580C2×C4#55311yescharacteristic
581C2×C4#55311yescharacteristic
582C2×C4#55611yescharacteristic
583C2×C4#55611yescharacteristic
584C2×C4#55611yescharacteristic
585C8—22no—
586C8—22no—
587C8—22no—
588C8—22no—
589C8—42no—
590C8—42no—
591C8—42no—
592C8—42no—
593C8—81no—
594C8—81no—
595C22—21no—
596C22—21no—
597C22—21no—
598C22—21no—
599C22—22no—
600C22—22no—
601C22—22no—
602C22—22no—
603C22—81no—
604C22—81no—
605C22—82no—
606C22—81no—
607C22—82no—
608C22C4×#12111yescharacteristic
609C22#235211yescharacteristic
610C22#236711yescharacteristic
611C22#259411yescharacteristic
612C22#259611yescharacteristic
613C22#260211yescharacteristic
614C22#260411yescharacteristic
615C22C2×#44412yes—
616C22#295712yes—
617C4—24no—
618C4—42no—
619C4—44no—
620C4—82no—
621C4—82no—
622C4—81no—
623C4—81no—
624C4—82no—
625C4—81no—
626C4#229311yescharacteristic
627C4#232211yescharacteristic
628C4#234711yescharacteristic
629C4#237311yescharacteristic
630C4#254311yescharacteristic
631C4#254511yescharacteristic
632C4#255811yescharacteristic
633C4#256011yescharacteristic
634C2—21no—
635C2—21no—
636C2—21no—
637C2—21no—
638C2—161no—
639C2#1407711yescharacteristic
640C2#2154611yescharacteristic
641C2#2154711yescharacteristic
642C2#2278212yes—
643C2#2323512yes—
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 ◁ C22×C4⋊C8 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #21147C2semidirect
5. C2×#1770C2semidirect
6. #11317C2semidirect
7. #11226C2semidirect
12. #10887C2semidirect
13. #10883C2semidirect
14. #10822C2semidirect
15. #10707C2semidirect
36. #1934C4semidirect
37. #1934C4semidirect
38. #1933C4semidirect
39. #1933C4semidirect

Maximal subgroups

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

subgroupindex
#211472
C2×#19382
#210832
#210522
C2×#17702
#113172
#112262
#110752
#110682
#109342
#109302
#108872
#108832
#108222
#107072