Finite Group Databasebeta

← #359393

Subgroups — #359393

Order 512.

1056 conjugacy classes of subgroups.

The subgroup lattice has 648 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23601C211yesmaximal, characteristic
2#20444C211yesmaximal, characteristic, complemented
3C2×#1773C211yesmaximal, characteristic, complemented
4#20436C211yesmaximal, characteristic, complemented
5#20420C211yesmaximal, characteristic
6C2×#1765C211yesmaximal, characteristic
7C2×#1757C211yesmaximal, characteristic
8#11304C211yesmaximal, characteristic, complemented
9#11250C211yesmaximal, characteristic, complemented
10#11241C211yesmaximal, characteristic, complemented
11#11215C211yesmaximal, characteristic, complemented
12#11146C211yesmaximal, characteristic
13#11126C211yesmaximal, characteristic
14#11060C211yesmaximal, characteristic
15#10711C211yesmaximal, characteristic, complemented
16C22×#131C2211yescharacteristic
17C2×#430C2211yescharacteristic
18C22×#108C2211yescharacteristic
19#2911C2211yescharacteristic
20C2×#427C2211yescharacteristic
21C2×#424C2211yescharacteristic
22C2×#421C2211yescharacteristic
23C22×#104C2211yescharacteristic
24#2896C2211yescharacteristic
25C2×#416C2211yescharacteristic
26C2×#413C2211yescharacteristic
27C2×#410C2211yescharacteristic
28C2×#405C2211yescharacteristic
29C2×#405C2211yescharacteristic
30C2×#405C2211yescharacteristic
31C2×#405C2211yescharacteristic
32C2×#405C2211yescharacteristic
33C22×#97C2211yescharacteristic
34#2296C2211yescharacteristic
35#2064C414yescomplemented
36#2063C414yescomplemented
37#1773—24no—
38#1773—22no—
39#1772—22no—
40#1771—22no—
41#1770—22no—
42#1765—22no—
43#1765—22no—
44#1765—22no—
45#1764—22no—
46#1757—24no—
47C2×#344C2211yescharacteristic
48C2×#344C2211yescharacteristic
49C2×#344C2211yescharacteristic
50C2×#344C2211yescharacteristic
51#1742C2211yescharacteristic
52#1740C2211yescharacteristic
53#1738C2211yescharacteristic
54C2×#341C2211yescharacteristic
55#1733C2211yescharacteristic
56#1733C2211yescharacteristic
57#1733C2211yescharacteristic
58C2×#339C2211yescharacteristic
59C2×#339C2211yescharacteristic
60C2×#339C2211yescharacteristic
61#1729C2211yescharacteristic
62#1728C2211yescharacteristic
63C22×#41C2311yescharacteristic
64C2×#131—22no—
65C2×#131—21no—
66C2×#131—21no—
67C2×#131—21no—
68C2×#131—21no—
69C2×#131—22no—
70C2×#131—22no—
71C2×#131Q814yes—
72C22×M16—21no—
73C22×M16C2311yescharacteristic
74C22×M16C2311yescharacteristic
75C22×M16C2311yescharacteristic
76C22×M16C2311yescharacteristic
77C22×M16C2311yescharacteristic
78C22×M16C2311yescharacteristic
79C23×C8—21no—
80C23×C8C2311yescharacteristic
81C23×C8C2311yescharacteristic
82C2×#117C2311yescharacteristic
83C2×#117C2311yescharacteristic
84C2×#117C2311yescharacteristic
85C2×#117C2311yescharacteristic
86C2×#117C2311yescharacteristic
87C22×C42C2311yescharacteristic
88#430—22no—
89#430—22no—
90C2×#108—22no—
91C2×#108—22no—
92C2×#108—22no—
93C2×#108C2×C414yes—
94#428—22no—
95#428C2×C414yes—
96#427—22no—
97#427—22no—
98#427—22no—
99C2×#107—21no—
100C2×#106—21no—
101#424—21no—
102#424—21no—
103#424C2×C414yes—
104C2×C4⋊C8—22no—
105#421—22no—
106#421—22no—
107C2×#104—22no—
108C2×#104—22no—
109C2×#104—22no—
110C2×#104—22no—
111#419C2×C412yes—
112#418C2×C412yes—
113#417—22no—
114#416—22no—
115#416—21no—
116#416—21no—
117#416—22no—
118C2×#103C2×C412yes—
119C2×#102C2×C412yes—
120#413—22no—
121#413—22no—
122C2×#101C2×C412yes—
123C2×#100C2×C412yes—
124#410—22no—
125#410—22no—
126C2×#99—22no—
127#405—21no—
128#405—21no—
129#405—22no—
130#405—22no—
131#405—21no—
132#405—21no—
133#405—21no—
134#405—21no—
135#405—22no—
136#405—22no—
137#405—22no—
138#405D414yes—
139#405D412yes—
140#405D412yes—
141C2×#97—22no—
142C2×#97—21no—
143C2×#97—21no—
144C2×#97—22no—
145C2×#97—21no—
146C2×#97—21no—
147C2×#97—21no—
148C2×#97D412yes—
149C2×#97D412yes—
150C2×C4×C8—21no—
151C2×C4×C8—21no—
152C2×C4×C8—21no—
153C2×C4×C8—21no—
154C2×C4×C8—21no—
155#389—22no—
156#385—22no—
157C4×C4⋊C4—22no—
158#344—22no—
159#344—22no—
160#344—22no—
161#344—22no—
162#343C2×C414yes—
163#342—22no—
164#341—22no—
165#341—22no—
166#339—22no—
167#339—22no—
168#339—22no—
169#339—22no—
170#339—22no—
171#339—22no—
172#336—22no—
173#336—22no—
174C2×#41—21no—
175C2×#41—21no—
176C2×#41—21no—
177C2×#41—21no—
178C2×#41—21no—
179C2×#41—21no—
180C2×#41—21no—
181C2×#41—21no—
182C2×#41—21no—
183C2×#41—21no—
184C2×#41C2×Q812yes—
185C2×#41#4112yes—
186C22×Q8C22×C411yescharacteristic
187C22×D4—21no—
188C22×D4C22×C411yescharacteristic
189C23×C4—21no—
190C23×C4—21no—
191C23×C4C2411yescharacteristic
192#131—24no—
193#131—24no—
194#131—24no—
195#131—24no—
196#131—42no—
197#131—42no—
198#131—42no—
199#131—42no—
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—24no—
209C2×M16—21no—
210C2×M16—21no—
211C2×M16—21no—
212C2×M16—21no—
213C2×M16—21no—
214C2×M16—21no—
215C2×M16—21no—
216C2×M16—21no—
217C2×M16—21no—
218C2×M16—21no—
219C2×M16—21no—
220C2×M16—21no—
221C2×M16—21no—
222C2×M16—21no—
223C2×M16—21no—
224C2×M16—21no—
225C2×M16—21no—
226C2×M16—21no—
227C2×M16—21no—
228C2×M16—21no—
229C2×M16—41no—
230C2×M16—41no—
231C2×M16—42no—
232C2×M16C22×C412yes—
233C2×M16C22×C412yes—
234C2×M16C22×C412yes—
235C2×M16C2×D412yes—
236C2×M16C2×D412yes—
237C2×M16C2×Q812yes—
238C2×M16#4112yes—
239C2×M16#4112yes—
240C22×C8—21no—
241C22×C8—21no—
242C22×C8—21no—
243C22×C8—21no—
244C22×C8—21no—
245C22×C8—21no—
246C22×C8—21no—
247C22×C8—21no—
248C22×C8—21no—
249C22×C8—21no—
250C22×C8—24no—
251C22×C8—41no—
252C22×C8—41no—
253C22×C8—42no—
254C22×C8C22×C412yes—
255C22×C8C2×D412yes—
256C22×C8C2×D411yescharacteristic
257C22×C8C2×D411yescharacteristic
258C22×C8C2×D411yescharacteristic
259C22×C8C2×D411yescharacteristic
260C22×C8C2×D412yes—
261C22×C8C2×Q812yes—
262C22×C8#4111yescharacteristic
263C22×C8#4111yescharacteristic
264C22×C8#4111yescharacteristic
265C22×C8#4111yescharacteristic
266C22×C8#4112yes—
267C22×C8#4112yes—
268#117—21no—
269#117—21no—
270#117—21no—
271#117—21no—
272#117—21no—
273#117—21no—
274#117—21no—
275#117—21no—
276#117—21no—
277#117—21no—
278#117—21no—
279#117—21no—
280#117—21no—
281#117—21no—
282#117—21no—
283#117—21no—
284#117C22×C412yes—
285#117C2×D412yes—
286#117C2×D412yes—
287#117#4112yes—
288C2×C4⋊C4—21no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—24no—
292C2×C4⋊C4C22×C412yes—
293C2×C4⋊C4#4112yes—
294C2×C22⋊C4—21no—
295C2×C22⋊C4—21no—
296C2×C22⋊C4—21no—
297C2×C22⋊C4—21no—
298C2×C22⋊C4#4112yes—
299C2×C42—21no—
300C2×C42—21no—
301C2×C42—21no—
302C2×C42—21no—
303C2×C42—21no—
304C2×C42—21no—
305C2×C42—21no—
306C2×C42—21no—
307C2×C42—21no—
308C2×C42—21no—
309C2×C42C2×D412yes—
310#108—28no—
311#108—28no—
312#107—24no—
313#107—24no—
314#106—24no—
315#106—24no—
316C4⋊C8—24no—
317C4⋊C8—24no—
318#104—28no—
319#104—28no—
320#103—22no—
321#103—22no—
322#103—24no—
323#102—22no—
324#102—22no—
325#102—24no—
326#101—24no—
327#100—24no—
328#99—28no—
329#97—24no—
330#97—24no—
331#97—22no—
332#97—22no—
333#97—24no—
334#97—42no—
335#97—42no—
336#97—41no—
337#97—41no—
338C4×C8—22no—
339C4×C8—22no—
340C4×C8—24no—
341C4×C8—42no—
342C4×C8—42no—
343C4×C8—41no—
344C4×C8—41no—
345#95—24no—
346C24—21no—
347C24C2×#4111yescharacteristic
348#41—22no—
349#41—22no—
350#41—22no—
351#41—22no—
352#41—22no—
353#41—22no—
354#41—22no—
355#41—22no—
356#41—41no—
357#41—41no—
358#41—41no—
359#41—41no—
360#41—41no—
361#41—41no—
362#41—41no—
363#41—41no—
364C2×Q8—21no—
365C2×Q8—21no—
366C2×Q8—21no—
367C2×Q8—21no—
368C2×Q8—21no—
369C2×Q8C4×Q812yes—
370C2×D4—22no—
371C2×D4—22no—
372C2×D4—21no—
373C2×D4—21no—
374C2×D4—21no—
375C2×D4—21no—
376C2×D4—21no—
377C2×D4—41no—
378C2×D4—41no—
379C2×D4—41no—
380C2×D4—41no—
381C2×D4C4×Q812yes—
382C22×C4—21no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—41no—
390C22×C4—41no—
391C22×C4—41no—
392C22×C4—41no—
393C22×C4—41no—
394C22×C4—41no—
395C22×C4—41no—
396C22×C4—41no—
397C22×C4—82no—
398C22×C4C23×C411yescharacteristic
399C22×C4C22×D411yescharacteristic
400C22×C4C22×D411yescharacteristic
401C22×C4C22×D411yescharacteristic
402C22×C4C22×Q811yescharacteristic
403C22×C4C2×#4111yescharacteristic
404C22×C4C2×#4111yescharacteristic
405C22×C4C2×#4111yescharacteristic
406C22×C4C2×#4111yescharacteristic
407C22×C4C2×#4111yescharacteristic
408C22×C4C2×#4111yescharacteristic
409C22×C4C2×#4111yescharacteristic
410C22×C4C2×#4111yescharacteristic
411C22×C4C2×#4111yescharacteristic
412M16—22no—
413M16—22no—
414M16—22no—
415M16—22no—
416M16—22no—
417M16—22no—
418M16—22no—
419M16—22no—
420M16—22no—
421M16—22no—
422M16—22no—
423M16—22no—
424M16—22no—
425M16—41no—
426M16—41no—
427M16—41no—
428M16—41no—
429M16—41no—
430M16—41no—
431M16—41no—
432M16—41no—
433M16—41no—
434M16—41no—
435M16—44no—
436M16—44no—
437M16C4×D414yes—
438C2×C8—22no—
439C2×C8—22no—
440C2×C8—22no—
441C2×C8—22no—
442C2×C8—22no—
443C2×C8—21no—
444C2×C8—21no—
445C2×C8—22no—
446C2×C8—21no—
447C2×C8—21no—
448C2×C8—22no—
449C2×C8—22no—
450C2×C8—22no—
451C2×C8—22no—
452C2×C8—22no—
453C2×C8—22no—
454C2×C8—41no—
455C2×C8—41no—
456C2×C8—41no—
457C2×C8—41no—
458C2×C8—41no—
459C2×C8—41no—
460C2×C8—41no—
461C2×C8—41no—
462C2×C8—41no—
463C2×C8—41no—
464C2×C8—44no—
465C2×C8—44no—
466C2×C8C4×D412yes—
467C2×C8C4×D412yes—
468C2×C8#12112yes—
469C2×C8#12112yes—
470C2×C8#12112yes—
471C2×C8#12112yes—
472C2×C8#12111yescharacteristic
473C2×C8#12111yescharacteristic
474C2×C8#12111yescharacteristic
475C2×C8#12111yescharacteristic
476C2×C8C22⋊Q812yes—
477C2×C8C22⋊Q812yes—
478C2×C8C22⋊Q812yes—
479C2×C8C22⋊Q812yes—
480C2×C8C22⋊Q812yes—
481C2×C8C22⋊Q812yes—
482C2×C8#12612yes—
483C2×C8#12612yes—
484C4⋊C4—22no—
485C4⋊C4—22no—
486C4⋊C4—22no—
487C4⋊C4—22no—
488C4⋊C4—22no—
489C4⋊C4—41no—
490C4⋊C4—41no—
491C4⋊C4—41no—
492C4⋊C4—41no—
493C4⋊C4—84no—
494C4⋊C4C4×D414yes—
495C22⋊C4—22no—
496C22⋊C4—22no—
497C22⋊C4—22no—
498C22⋊C4—22no—
499C22⋊C4—22no—
500C22⋊C4—22no—
501C22⋊C4—41no—
502C22⋊C4—41no—
503C22⋊C4—41no—
504C22⋊C4—41no—
505C42—22no—
506C42—22no—
507C42—22no—
508C42—22no—
509C42—22no—
510C42—22no—
511C42—41no—
512C42—41no—
513C42—41no—
514C42—41no—
515C42—41no—
516C42—41no—
517C42—41no—
518C42—41no—
519C23—22no—
520C23—42no—
521C23—41no—
522C23—41no—
523C23—82no—
524C23C4×#4111yescharacteristic
525C23#52511yescharacteristic
526C23#52511yescharacteristic
527C23#52911yescharacteristic
528C23#52911yescharacteristic
529C23#53211yescharacteristic
530C23#53311yescharacteristic
531Q8—22no—
532Q8—22no—
533Q8—41no—
534Q8—41no—
535D4—22no—
536D4—22no—
537D4—42no—
538D4—42no—
539D4—42no—
540D4—42no—
541D4—41no—
542D4—41no—
543C2×C4—24no—
544C2×C4—22no—
545C2×C4—44no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—44no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—82no—
561C2×C4—82no—
562C2×C4—82no—
563C2×C4—82no—
564C2×C4—82no—
565C2×C4C2×C4×D411yescharacteristic
566C2×C4C2×C4×D411yescharacteristic
567C2×C4C2×C4×D411yescharacteristic
568C2×C4C2×C4×Q811yescharacteristic
569C2×C4C4×#4111yescharacteristic
570C2×C4C4×#4111yescharacteristic
571C2×C4C2×#12111yescharacteristic
572C2×C4C2×#12111yescharacteristic
573C2×C4C2×#12111yescharacteristic
574C2×C4C2×C22⋊Q811yescharacteristic
575C2×C4C2×C22⋊Q811yescharacteristic
576C2×C4C2×C22⋊Q811yescharacteristic
577C2×C4#52511yescharacteristic
578C2×C4#52511yescharacteristic
579C2×C4#52511yescharacteristic
580C2×C4#52511yescharacteristic
581C2×C4C2×#12611yescharacteristic
582C2×C4#52911yescharacteristic
583C2×C4#52911yescharacteristic
584C2×C4#52911yescharacteristic
585C2×C4#52911yescharacteristic
586C2×C4#52911yescharacteristic
587C2×C4#52911yescharacteristic
588C2×C4#52911yescharacteristic
589C2×C4#53211yescharacteristic
590C2×C4#53211yescharacteristic
591C2×C4#53311yescharacteristic
592C2×C4#53311yescharacteristic
593C8—22no—
594C8—22no—
595C8—24no—
596C8—42no—
597C8—42no—
598C8—42no—
599C8—42no—
600C8—42no—
601C8#39612yes—
602C8#39612yes—
603C8#39612yes—
604C8#39612yes—
605C22—22no—
606C22—22no—
607C22—22no—
608C22—41no—
609C22—41no—
610C22—41no—
611C22—41no—
612C22—42no—
613C22—82no—
614C22—82no—
615C22—82no—
616C22C4×#12111yescharacteristic
617C22C4×C22⋊Q811yescharacteristic
618C22C4×#12611yescharacteristic
619C22#239211yescharacteristic
620C22#239211yescharacteristic
621C22#239611yescharacteristic
622C22#239611yescharacteristic
623C4—42no—
624C4—44no—
625C4—82no—
626C4—82no—
627C4—82no—
628C4—82no—
629C4—82no—
630C4C4×#12111yescharacteristic
631C4C4×#12111yescharacteristic
632C4C4×C22⋊Q811yescharacteristic
633C4C4×C22⋊Q811yescharacteristic
634C4C2×#39611yescharacteristic
635C4#239211yescharacteristic
636C4#239611yescharacteristic
637C4#240111yescharacteristic
638C2—21no—
639C2—21no—
640C2—21no—
641C2—21no—
642C2—21no—
643C2—21no—
644C2—82no—
645C2C4×#39611yescharacteristic
646C2#2155912yes—
subgroupG/H
centerC2×C4C2×#121
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
socleC22C4×#121
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. #20444C2semidirect
3. C2×#1773C2semidirect
4. #20436C2semidirect
8. #11304C2semidirect
9. #11250C2semidirect
10. #11241C2semidirect
11. #11215C2semidirect
15. #10711C2semidirect
35. #2064C4semidirect
36. #2063C4semidirect

Maximal subgroups

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

subgroupindex
#236012
#204442
C2×#17732
#204362
#204202
C2×#17652
C2×#17572
#113042
#112502
#112412
#112152
#111462
#111262
#110602
#107112