Finite Group Databasebeta

← #359278

Subgroups — #359278

Order 512.

968 conjugacy classes of subgroups.

The subgroup lattice has 614 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2080C211yesmaximal, characteristic
2#20436C211yesmaximal, characteristic, complemented
3#20420C211yesmaximal, characteristic
4C2×#1765C211yesmaximal, characteristic
5C2×#1757C211yesmaximal, characteristic
6#11350C211yesmaximal, characteristic, complemented
7#11321C211yesmaximal, characteristic, complemented
8#11306C211yesmaximal, characteristic, complemented
9#11282C211yesmaximal, characteristic, complemented
10#11176C211yesmaximal, characteristic
11#11151C211yesmaximal, characteristic
12#11085C211yesmaximal, characteristic
13#10711C211yesmaximal, characteristic, complemented
14#10707C211yesmaximal, characteristic, complemented
15#10699C211yesmaximal, characteristic, complemented
16C22×#131C2211yescharacteristic
17C2×#424C2211yescharacteristic
18C2×#421C2211yescharacteristic
19C22×#104C2211yescharacteristic
20#2896C2211yescharacteristic
21C2×#416C2211yescharacteristic
22C2×#413C2211yescharacteristic
23C2×#410C2211yescharacteristic
24C2×#405C2211yescharacteristic
25C2×#405C2211yescharacteristic
26C2×#405C2211yescharacteristic
27C2×#405C2211yescharacteristic
28C2×#405C2211yescharacteristic
29C22×#97C2211yescharacteristic
30#2296C2211yescharacteristic
31#2080C418yescomplemented
32#1771—22no—
33#1770—22no—
34#1765—22no—
35#1765—24no—
36#1764—22no—
37#1757—24no—
38C2×#344C2211yescharacteristic
39C2×#344C2211yescharacteristic
40C2×#344C2211yescharacteristic
41C2×#344C2211yescharacteristic
42#1742C2211yescharacteristic
43#1740C2211yescharacteristic
44#1738C2211yescharacteristic
45C2×#341C2211yescharacteristic
46C2×#341C2211yescharacteristic
47#1735C2211yescharacteristic
48#1733C2211yescharacteristic
49C2×#339C2211yescharacteristic
50C2×#339C2211yescharacteristic
51C2×#339C2211yescharacteristic
52C2×#338C2211yescharacteristic
53C2×H(3,ℤ/4)C2211yescharacteristic
54C2×H(3,ℤ/4)C2211yescharacteristic
55C2×H(3,ℤ/4)C2211yescharacteristic
56#1729C2211yescharacteristic
57#1728C2211yescharacteristic
58C22×#41C2311yescharacteristic
59C2×#131—22no—
60C2×#131—21no—
61C2×#131—21no—
62C2×#131—21no—
63C2×#131—21no—
64C2×#131—22no—
65C2×#131—22no—
66C2×#131D414yes—
67C22×M16—21no—
68C22×M16C2311yescharacteristic
69C22×M16C2311yescharacteristic
70C22×M16C2311yescharacteristic
71C22×M16C2311yescharacteristic
72C22×M16C2311yescharacteristic
73C22×M16C2311yescharacteristic
74C23×C8—21no—
75C23×C8C2311yescharacteristic
76C23×C8C2311yescharacteristic
77C2×#117C2311yescharacteristic
78C2×#117C2311yescharacteristic
79C2×#117C2311yescharacteristic
80C2×#117C2311yescharacteristic
81C2×#117C2311yescharacteristic
82C22×C42C2311yescharacteristic
83#424—22no—
84#424C2×C414yes—
85C2×C4⋊C8—22no—
86#421—22no—
87#421C2×C414yes—
88C2×#104—22no—
89C2×#104—22no—
90C2×#104—22no—
91C2×#104—22no—
92#419—21no—
93#418—21no—
94#417—22no—
95#416—22no—
96#416—22no—
97#416C2×C414yes—
98C2×#103—21no—
99C2×#102—21no—
100#413—22no—
101#413C2×C414yes—
102C2×#101—21no—
103C2×#100—21no—
104#410—22no—
105#410—22no—
106C2×#99—22no—
107#405—22no—
108#405—22no—
109#405—22no—
110#405—22no—
111#405—21no—
112#405—22no—
113#405—21no—
114#405—22no—
115#405—22no—
116#405—22no—
117#405—21no—
118#405—21no—
119C2×#97—22no—
120C2×#97—21no—
121C2×#97—21no—
122C2×#97—21no—
123C2×#97—22no—
124C2×#97—21no—
125C2×#97—21no—
126C2×#97—21no—
127C2×#97D412yes—
128C2×C4×C8—21no—
129C2×C4×C8—21no—
130C2×C4×C8—21no—
131C2×C4×C8—21no—
132C2×C4×C8D412yes—
133#389—22no—
134#385—22no—
135C4×C4⋊C4—22no—
136#344—22no—
137#344—22no—
138#344—22no—
139#344C2×C414yes—
140#343—22no—
141#342—22no—
142#341—22no—
143#341—22no—
144#341C2×C414yes—
145#340—22no—
146#339—22no—
147#339—22no—
148#339—22no—
149#339—22no—
150#338—22no—
151H(3,ℤ/4)—22no—
152H(3,ℤ/4)—22no—
153H(3,ℤ/4)C2×C414yes—
154#336—22no—
155#336—22no—
156C2×#41—21no—
157C2×#41—21no—
158C2×#41—21no—
159C2×#41—21no—
160C2×#41—21no—
161C2×#41—21no—
162C2×#41—21no—
163C2×#41—21no—
164C2×#41—21no—
165C2×#41—21no—
166C2×#41C22×C412yes—
167C2×#41C2×D412yes—
168C22×Q8#4111yescharacteristic
169C22×D4—21no—
170C22×D4#4111yescharacteristic
171C23×C4—21no—
172C23×C4—21no—
173C23×C4C2411yescharacteristic
174#131—24no—
175#131—24no—
176#131—24no—
177#131—24no—
178#131—42no—
179#131—42no—
180#131—42no—
181#131—42no—
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—24no—
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—21no—
206C2×M16—21no—
207C2×M16—21no—
208C2×M16—21no—
209C2×M16—21no—
210C2×M16—21no—
211C2×M16—21no—
212C2×M16—21no—
213C2×M16—21no—
214C2×M16—41no—
215C2×M16—41no—
216C2×M16—42no—
217C2×M16C22×C412yes—
218C2×M16C22×C412yes—
219C2×M16C22×C412yes—
220C2×M16C2×D412yes—
221C2×M16#4112yes—
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×C8—21no—
232C22×C8—21no—
233C22×C8—24no—
234C22×C8—41no—
235C22×C8—41no—
236C22×C8—42no—
237C22×C8C22×C412yes—
238C22×C8C2×D412yes—
239C22×C8C2×D412yes—
240C22×C8C2×D411yescharacteristic
241C22×C8C2×D411yescharacteristic
242C22×C8#4112yes—
243C22×C8#4112yes—
244C22×C8#4112yes—
245C22×C8#4111yescharacteristic
246C22×C8#4111yescharacteristic
247C22×C8#4111yescharacteristic
248C22×C8#4111yescharacteristic
249#117—21no—
250#117—21no—
251#117—21no—
252#117—21no—
253#117—21no—
254#117—21no—
255#117—21no—
256#117—21no—
257#117—21no—
258#117—21no—
259#117—21no—
260#117—21no—
261#117—21no—
262#117—21no—
263#117—21no—
264#117—21no—
265#117C22×C412yes—
266#117C22×C412yes—
267#117#4112yes—
268#117#4112yes—
269C2×C4⋊C4—21no—
270C2×C4⋊C4—21no—
271C2×C4⋊C4—21no—
272C2×C4⋊C4—21no—
273C2×C4⋊C4—24no—
274C2×C4⋊C4#4112yes—
275C2×C22⋊C4—21no—
276C2×C22⋊C4—21no—
277C2×C22⋊C4—21no—
278C2×C22⋊C4—21no—
279C2×C22⋊C4#4112yes—
280C2×C42—21no—
281C2×C42—21no—
282C2×C42—21no—
283C2×C42—21no—
284C2×C42—21no—
285C2×C42—21no—
286C2×C42—21no—
287C2×C42—21no—
288C2×C42—21no—
289C2×C42C2×D412yes—
290C2×C42#4112yes—
291C4⋊C8—28no—
292#104—28no—
293#104—28no—
294#103—24no—
295#103—24no—
296#102—24no—
297#102—24no—
298#101—24no—
299#100—24no—
300#99—28no—
301#97—24no—
302#97—42no—
303#97—42no—
304#97—42no—
305#97—42no—
306#97—42no—
307#97—41no—
308#97—41no—
309C4×C8—24no—
310C4×C8—42no—
311C4×C8—42no—
312C4×C8—42no—
313C4×C8—41no—
314C4×C8—41no—
315#95—24no—
316C24—21no—
317C24C2×#4111yescharacteristic
318#41—22no—
319#41—22no—
320#41—22no—
321#41—22no—
322#41—22no—
323#41—22no—
324#41—22no—
325#41—22no—
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×Q8C4×D412yes—
340C2×D4—22no—
341C2×D4—22no—
342C2×D4—21no—
343C2×D4—21no—
344C2×D4—21no—
345C2×D4—21no—
346C2×D4—21no—
347C2×D4—41no—
348C2×D4—41no—
349C2×D4—41no—
350C2×D4—41no—
351C2×D4C4×D412yes—
352C22×C4—22no—
353C22×C4—21no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—41no—
360C22×C4—41no—
361C22×C4—41no—
362C22×C4—41no—
363C22×C4—41no—
364C22×C4—41no—
365C22×C4—41no—
366C22×C4—41no—
367C22×C4—82no—
368C22×C4C23×C411yescharacteristic
369C22×C4C22×D411yescharacteristic
370C22×C4C22×D411yescharacteristic
371C22×C4C2×#4111yescharacteristic
372C22×C4C2×#4111yescharacteristic
373C22×C4C2×#4111yescharacteristic
374C22×C4C2×#4111yescharacteristic
375C22×C4C2×#4111yescharacteristic
376C22×C4C2×#4111yescharacteristic
377C22×C4C2×#4111yescharacteristic
378C22×C4C2×#4111yescharacteristic
379C22×C4C2×#4111yescharacteristic
380C22×C4C2×#4111yescharacteristic
381C22×C4C2×#4111yescharacteristic
382M16—22no—
383M16—22no—
384M16—22no—
385M16—22no—
386M16—22no—
387M16—22no—
388M16—22no—
389M16—22no—
390M16—22no—
391M16—22no—
392M16—41no—
393M16—41no—
394M16—41no—
395M16—41no—
396M16—41no—
397M16—41no—
398M16—41no—
399M16—41no—
400M16—41no—
401M16—41no—
402M16—41no—
403M16—41no—
404M16—41no—
405M16—41no—
406M16—44no—
407M16—44no—
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—44no—
434C2×C8—44no—
435C2×C8#12112yes—
436C2×C8#12112yes—
437C2×C8#12312yes—
438C2×C8#12312yes—
439C2×C8#12312yes—
440C2×C8#12312yes—
441C2×C8#12312yes—
442C2×C8#12312yes—
443C2×C8#12411yescharacteristic
444C2×C8#12411yescharacteristic
445C2×C8#12411yescharacteristic
446C2×C8#12411yescharacteristic
447C2×C8#12612yes—
448C2×C8#12612yes—
449C2×C8#12612yes—
450C2×C8#12612yes—
451C4⋊C4—22no—
452C4⋊C4—22no—
453C4⋊C4—22no—
454C4⋊C4—22no—
455C4⋊C4—22no—
456C4⋊C4—22no—
457C4⋊C4—41no—
458C4⋊C4—41no—
459C4⋊C4—41no—
460C4⋊C4—41no—
461C4⋊C4—84no—
462C22⋊C4—22no—
463C22⋊C4—22no—
464C22⋊C4—22no—
465C22⋊C4—22no—
466C22⋊C4—22no—
467C22⋊C4—22no—
468C22⋊C4—41no—
469C22⋊C4—41no—
470C22⋊C4—41no—
471C22⋊C4—41no—
472C42—22no—
473C42—22no—
474C42—22no—
475C42—22no—
476C42—22no—
477C42—41no—
478C42—41no—
479C42—41no—
480C42—41no—
481C42—41no—
482C42—41no—
483C42—41no—
484C42—41no—
485C42C4×D414yes—
486C23—22no—
487C23—42no—
488C23—41no—
489C23—41no—
490C23—82no—
491C23C4×#4111yescharacteristic
492C23#52511yescharacteristic
493C23#52511yescharacteristic
494C23#52911yescharacteristic
495C23#52911yescharacteristic
496C23#52911yescharacteristic
497C23#52911yescharacteristic
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—44no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—44no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—82no—
528C2×C4—82no—
529C2×C4—82no—
530C2×C4—82no—
531C2×C4—82no—
532C2×C4C2×C4×D411yescharacteristic
533C2×C4C2×C4×D411yescharacteristic
534C2×C4C4×#4111yescharacteristic
535C2×C4C4×#4111yescharacteristic
536C2×C4C4×#4111yescharacteristic
537C2×C4C4×#4111yescharacteristic
538C2×C4C2×#12111yescharacteristic
539C2×C4C2×#12311yescharacteristic
540C2×C4C2×#12311yescharacteristic
541C2×C4C2×#12311yescharacteristic
542C2×C4#52511yescharacteristic
543C2×C4#52511yescharacteristic
544C2×C4C2×#12411yescharacteristic
545C2×C4C2×#12611yescharacteristic
546C2×C4C2×#12611yescharacteristic
547C2×C4#52911yescharacteristic
548C2×C4#52911yescharacteristic
549C2×C4#52911yescharacteristic
550C2×C4#52911yescharacteristic
551C2×C4#52911yescharacteristic
552C2×C4#52911yescharacteristic
553C2×C4#52911yescharacteristic
554C2×C4#52911yescharacteristic
555C2×C4#52911yescharacteristic
556C2×C4#52911yescharacteristic
557C2×C4#52911yescharacteristic
558C2×C4#53211yescharacteristic
559C2×C4#53211yescharacteristic
560C8—42no—
561C8—42no—
562C8—42no—
563C8—42no—
564C8—42no—
565C8—42no—
566C8—42no—
567C8#39712yes—
568C8#39712yes—
569C8#39712yes—
570C8#39712yes—
571C22—22no—
572C22—22no—
573C22—22no—
574C22—41no—
575C22—41no—
576C22—41no—
577C22—41no—
578C22—42no—
579C22—82no—
580C22—82no—
581C22—82no—
582C22C4×#12311yescharacteristic
583C22C4×#12311yescharacteristic
584C22C4×#12611yescharacteristic
585C22#239211yescharacteristic
586C22#239511yescharacteristic
587C22#239811yescharacteristic
588C22#240111yescharacteristic
589C4—44no—
590C4—42no—
591C4—82no—
592C4—82no—
593C4—82no—
594C4—82no—
595C4—82no—
596C4C4×#12111yescharacteristic
597C4C4×#12311yescharacteristic
598C4C4×#12411yescharacteristic
599C4C4×#12611yescharacteristic
600C4C2×#39711yescharacteristic
601C4#239511yescharacteristic
602C4#239511yescharacteristic
603C4#240111yescharacteristic
604C2—21no—
605C2—21no—
606C2—21no—
607C2—21no—
608C2—21no—
609C2—21no—
610C2—82no—
611C2C4×#39711yescharacteristic
612C2#2383412yes—
subgroupG/H
centerC2×C4C2×#124
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
socleC22#2398
prime radicalC22×#41C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C23×C8 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #20436C2semidirect
6. #11350C2semidirect
7. #11321C2semidirect
8. #11306C2semidirect
9. #11282C2semidirect
13. #10711C2semidirect
14. #10707C2semidirect
15. #10699C2semidirect
31. #2080C4semidirect

Maximal subgroups

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

subgroupindex
C2×#20802
#204362
#204202
C2×#17652
C2×#17572
#113502
#113212
#113062
#112822
#111762
#111512
#110852
#107112
#107072
#106992