Finite Group Databasebeta

← #6492359

Subgroups — #6492359

Order 512.

2194 conjugacy classes of subgroups.

The subgroup lattice has 687 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3245C211yesmaximal, characteristic, complemented
2C2×#3213C211yesmaximal, characteristic, complemented
3#37536C211yesmaximal, characteristic, complemented
4#34236C211yesmaximal, characteristic, complemented
5#33166C212yesmaximal, complemented
6#33036C212yesmaximal, complemented
7#29839C212yesmaximal, complemented
8#29802C212yesmaximal, complemented
9#25376C214yesmaximal, complemented
10#25285C214yesmaximal, complemented
11#25090C214yesmaximal, complemented
12#25076C214yesmaximal, complemented
13#21977C211yesmaximal, characteristic, complemented
14#21395C211yesmaximal, characteristic, complemented
15C2×#1983C211yesmaximal, characteristic, complemented
16C2×#532C2211yescharacteristic
17C2×#532C2211yescharacteristic
18C22×#124C2211yescharacteristic
19#3245—28no—
20#3213—28no—
21C2×#496C2212yes—
22C2×#489C2212yes—
23C2×#474C2212yescomplemented
24C2×#474C2212yescomplemented
25C2×#466C2214yes—
26C2×#463C2212yes—
27C2×#463C2212yes—
28C2×#463C2214yescomplemented
29C2×#459C2214yes—
30C2×#459C2212yescomplemented
31C2×#459C2212yescomplemented
32C2×#448C2212yes—
33C2×#447C2212yes—
34C2×#432C2211yescharacteristic
35C22×C4⋊C8C2211yescharacteristic
36C22×#103C2212yes—
37C22×#102C2212yes—
38#2819—21no—
39#2819—21no—
40#2657C2212yes—
41#2640C2212yes—
42#2631—22no—
43#2617—22no—
44#2544C2214yes—
45#2436C2212yes—
46#2433C2212yes—
47#2404—22no—
48#2369C2212yescomplemented
49#2366C2212yescomplemented
50#2361C2214yes—
51#2357—21no—
52#2349—21no—
53#2339—22no—
54#2337—22no—
55#2319C2212yescomplemented
56C4×#121—22no—
57#2283C2211yescharacteristic
58#2275C2212yescomplemented
59C2×C4×C4⋊C4C2211yescharacteristic
60#2068C2214yescomplemented
61#2067C2214yes—
62#2055C2214yescomplemented
63#2054C2214yes—
64#2042C2214yescomplemented
65#2041C2214yes—
66#2012C2214yescomplemented
67#2011C2214yes—
68#1987C2211yescharacteristic, complemented
69#1987C2211yescharacteristic, complemented
70#1983C2212yes—
71#1983C2214yes—
72#1983C2212yes—
73#1983C2214yescomplemented
74#1982C2214yes—
75#1982C2211yescharacteristic, complemented
76#1982C2211yescharacteristic, complemented
77#1940C2214yescomplemented
78#1928C2214yescomplemented
79#1925C2214yes—
80#1879C2214yes—
81#1877C2214yescomplemented
82#1874C2214yes—
83#1850C2214yes—
84#1843C2212yescomplemented
85#1832C2212yescomplemented
86#1762C2214yescomplemented
87C22×#41—21no—
88C22×#41—21no—
89C23×Q8C2311yescharacteristic
90C23×D4C2311yescharacteristic
91C22×SD16C2312yes—
92C22×SD16C2312yes—
93C22×D8C2312yes—
94C22×D8C2312yes—
95C22×D8C2314yes—
96C23×C8C2312yes—
97#532—24no—
98#532—24no—
99#532—24no—
100#532D418yescomplemented
101C2×C4⋊Q8C2311yescharacteristic
102C2×C4⋊Q8C2311yescharacteristic
103C2×#127C2311yescharacteristic
104C2×#127C2311yescharacteristic
105C2×#127C2314yes—
106C2×#124—21no—
107C2×#124—21no—
108C2×#124—21no—
109C2×#124—21no—
110C2×#124—24no—
111C2×#124—24no—
112C2×#124C2312yes—
113C2×#124C2312yes—
114C2×#124C2314yes—
115C2×#121—21no—
116C2×#121—21no—
117C2×#121C2312yes—
118C2×#121C2312yes—
119C2×#120—24no—
120C2×C4×D4—22no—
121C2×C4×D4—24no—
122C2×C4×D4—21no—
123C2×C4×D4—21no—
124C2×C4×D4C2312yes—
125C2×C4×D4C2314yes—
126C2×C4×D4C2312yes—
127C22×C4⋊C4C2312yes—
128C22×C22⋊C4—21no—
129C22×C22⋊C4—21no—
130C22×C42—21no—
131C22×C42C2311yescharacteristic
132#496—28no—
133#489—28no—
134#474—28no—
135#474—28no—
136#466—216no—
137#463—28no—
138#463—28no—
139#463—48no—
140#459—28no—
141#459—28no—
142#459—48no—
143#448—28no—
144#447—28no—
145#432—24no—
146C2×#107C2314yes—
147C2×C4⋊C8—24no—
148C2×C4⋊C8C2311yescharacteristic
149C2×C4⋊C8C2312yes—
150C2×C4⋊C8C2311yescharacteristic
151C2×C4⋊C8C2312yes—
152C2×C4⋊C8C2314yes—
153C2×C4⋊C8C2314yes—
154C2×#103—44no—
155C2×#103C2312yes—
156C2×#103C2312yes—
157C2×#103C2314yes—
158C2×#103C2314yes—
159C2×#102—44no—
160C2×#102C2314yes—
161C2×#102C2312yes—
162C2×#102C2312yes—
163C2×#102C2314yes—
164C2×#102C2314yes—
165C2×#102C2314yes—
166C2×#102C2314yes—
167C2×#102C2314yes—
168C2×C22⋊C8C2312yes—
169C2×#97C2312yes—
170#397—24no—
171#394—24no—
172#394—24no—
173#393—22no—
174#393—22no—
175#393—22no—
176#393—22no—
177#392—22no—
178#392—22no—
179#392—22no—
180#392—22no—
181#392—24no—
182#391—24no—
183#391C2314yes—
184#390—22no—
185#390C2314yes—
186#390C2314yes—
187#389—22no—
188#389C2314yes—
189#388—24no—
190#388—24no—
191#386—24no—
192#386—24no—
193#386—24no—
194#386—24no—
195#386—24no—
196#385—24no—
197#383D412yes—
198#382D411yescharacteristic
199#382D411yescharacteristic
200#381—22no—
201#381D412yes—
202#380—22no—
203#380D412yes—
204C4×C4⋊C4—22no—
205C4×C4⋊C4—21no—
206C4×C4⋊C4—21no—
207C4×C4⋊C4C2311yescharacteristic
208C4×C4⋊C4C2314yes—
209C4×C4⋊C4C2312yes—
210C4×C4⋊C4C2311yescharacteristic
211C4×C4⋊C4C2312yes—
212C4×C22⋊C4—24no—
213C4×C22⋊C4—22no—
214C4×C22⋊C4C2312yes—
215#376C2312yes—
216C2×#95—21no—
217C2×#95—22no—
218#340C2314yes—
219#340C2314yes—
220#336C2314yes—
221#336C2314yes—
222C25—21no—
223C2×#41—28no—
224C2×#41—44no—
225C2×#41—84no—
226C2×#41—84no—
227C22×Q8—21no—
228C22×Q8—21no—
229C22×Q8—21no—
230C22×Q8—22no—
231C22×Q8—22no—
232C22×Q8—42no—
233C22×Q8—42no—
234C22×Q8C2412yes—
235C22×D4—22no—
236C22×D4—22no—
237C22×D4—21no—
238C22×D4—21no—
239C22×D4—22no—
240C22×D4—21no—
241C22×D4—21no—
242C22×D4—21no—
243C22×D4—24no—
244C22×D4—24no—
245C22×D4—22no—
246C22×D4—42no—
247C22×D4—42no—
248C22×D4C2412yes—
249C22×D4C2412yes—
250C22×D4C2412yes—
251C23×C4—22no—
252C23×C4—22no—
253C23×C4—21no—
254C23×C4—24no—
255C23×C4—22no—
256C23×C4C2×D411yescharacteristic
257C23×C4C2×D411yescharacteristic
258C23×C4C2411yescharacteristic
259C2×SD16—24no—
260C2×SD16—24no—
261C2×SD16—44no—
262C2×SD16—44no—
263C2×D8—24no—
264C2×D8—24no—
265C2×D8—28no—
266C2×D8—44no—
267C2×D8—44no—
268C2×D8—44no—
269C2×D8—44no—
270C22×C8—24no—
271C22×C8—42no—
272C22×C8—42no—
273C22×C8C2414yes—
274C22×C8C2412yes—
275C22×C8C2412yes—
276C4⋊Q8—22no—
277C4⋊Q8—22no—
278C4⋊Q8—22no—
279C4⋊Q8C2×D414yes—
280#127—22no—
281#127—22no—
282#127—22no—
283#127—44no—
284#127—44no—
285#127C2×D414yes—
286#124—28no—
287#124—24no—
288#124—24no—
289#124—24no—
290#124—24no—
291#124—44no—
292#124—44no—
293#124—44no—
294#124—44no—
295#124—44no—
296#121—28no—
297#121—24no—
298#121—24no—
299#121—24no—
300#121—44no—
301#121C2×D418yes—
302#120—88no—
303C4×D4—28no—
304C4×D4—28no—
305C4×D4—24no—
306C4×D4—24no—
307C4×D4—24no—
308C4×D4—44no—
309C4×D4—44no—
310C4×D4—84no—
311C4×D4—88no—
312C4×D4C2×D418yes—
313C2×C4⋊C4—21no—
314C2×C4⋊C4—24no—
315C2×C4⋊C4—22no—
316C2×C4⋊C4—21no—
317C2×C4⋊C4—24no—
318C2×C4⋊C4—42no—
319C2×C4⋊C4C2×D412yes—
320C2×C4⋊C4C2×D412yes—
321C2×C4⋊C4C2412yes—
322C2×C4⋊C4C2412yes—
323C2×C4⋊C4C2412yes—
324C2×C4⋊C4C2412yes—
325C2×C22⋊C4—22no—
326C2×C22⋊C4—22no—
327C2×C22⋊C4—22no—
328C2×C22⋊C4—22no—
329C2×C22⋊C4—22no—
330C2×C22⋊C4—22no—
331C2×C22⋊C4—22no—
332C2×C22⋊C4—22no—
333C2×C22⋊C4—24no—
334C2×C22⋊C4—24no—
335C2×C22⋊C4—24no—
336C2×C22⋊C4—24no—
337C2×C22⋊C4—24no—
338C2×C22⋊C4—24no—
339C2×C22⋊C4—24no—
340C2×C22⋊C4—44no—
341C2×C22⋊C4—44no—
342C2×C22⋊C4C2×D414yes—
343C2×C42—22no—
344C2×C42—24no—
345C2×C42—21no—
346C2×C42—21no—
347C2×C42—82no—
348C2×C42C2×D411yescharacteristic
349C2×C42C2×D411yescharacteristic
350C2×C42C2×D414yes—
351C2×C42#4114yes—
352C2×C42C2414yes—
353C2×C42C2411yescharacteristic
354C2×C42C2411yescharacteristic
355#107—28no—
356C4⋊C8—24no—
357C4⋊C8—24no—
358C4⋊C8—22no—
359C4⋊C8—22no—
360C4⋊C8—44no—
361C4⋊C8—44no—
362#103—24no—
363#103—24no—
364#103—28no—
365#103—44no—
366#102—28no—
367#102—24no—
368#102—24no—
369#102—28no—
370#102—28no—
371#102—28no—
372#102—44no—
373#102—44no—
374C22⋊C8—24no—
375#97—24no—
376#95—21no—
377#95—24no—
378#95—21no—
379#95—22no—
380#95—24no—
381#95—24no—
382#95—24no—
383#95C2×D412yes—
384#95C2×D414yes—
385#95C2×D412yes—
386#95C2×D412yes—
387#95C2×D412yes—
388#95C2×D414yes—
389C24—22no—
390C24—22no—
391C24—22no—
392C24—42no—
393C24—44no—
394C24—82no—
395C24—81no—
396C24—81no—
397C24C22×D411yescharacteristic
398#41—816no—
399#41—816no—
400C2×Q8—24no—
401C2×Q8—22no—
402C2×Q8—22no—
403C2×Q8—24no—
404C2×Q8—24no—
405C2×Q8—48no—
406C2×Q8—42no—
407C2×Q8—42no—
408C2×Q8—82no—
409C2×Q8—82no—
410C2×Q8—82no—
411C2×Q8—82no—
412C2×Q8—82no—
413C2×D4—28no—
414C2×D4—24no—
415C2×D4—24no—
416C2×D4—22no—
417C2×D4—22no—
418C2×D4—22no—
419C2×D4—22no—
420C2×D4—22no—
421C2×D4—24no—
422C2×D4—24no—
423C2×D4—24no—
424C2×D4—24no—
425C2×D4—24no—
426C2×D4—24no—
427C2×D4—42no—
428C2×D4—42no—
429C2×D4—48no—
430C2×D4—48no—
431C2×D4—44no—
432C2×D4—48no—
433C2×D4—42no—
434C2×D4—42no—
435C2×D4—42no—
436C2×D4—82no—
437C2×D4—82no—
438C2×D4—81no—
439C2×D4—81no—
440C2×D4—82no—
441C2×D4—82no—
442C2×D4—82no—
443C2×D4—84no—
444C2×D4—84no—
445C2×D4—84no—
446C2×D4—84no—
447C2×D4—84no—
448C2×D4—84no—
449C2×D4C22×D414yes—
450C22×C4—24no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—28no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—41no—
460C22×C4—41no—
461C22×C4—48no—
462C22×C4—44no—
463C22×C4—42no—
464C22×C4—44no—
465C22×C4—44no—
466C22×C4—84no—
467C22×C4—84no—
468C22×C4—84no—
469C22×C4—84no—
470C22×C4—82no—
471C22×C4—82no—
472C22×C4#12114yes—
473C22×C4#12114yes—
474C22×C4C22×D414yes—
475C22×C4C22×D414yes—
476C22×C4C22×D411yescharacteristic
477C22×C4C22×D411yescharacteristic
478C22×C4C22×D411yescharacteristic
479C22×C4C22×D411yescharacteristic
480C22×C4C22×D411yescharacteristic
481C22×C4C22×D412yes—
482C22×C4C2×#4112yes—
483C22×C4C2511yescharacteristic
484SD16—84no—
485SD16—84no—
486D8—88no—
487D8—84no—
488D8—84no—
489C2×C8—22no—
490C2×C8—22no—
491C2×C8—24no—
492C2×C8—24no—
493C2×C8—24no—
494C2×C8—24no—
495C2×C8—24no—
496C2×C8—84no—
497C4⋊C4—24no—
498C4⋊C4—22no—
499C4⋊C4—22no—
500C4⋊C4—22no—
501C4⋊C4—22no—
502C4⋊C4—22no—
503C4⋊C4—22no—
504C4⋊C4—42no—
505C4⋊C4—42no—
506C4⋊C4—44no—
507C4⋊C4—42no—
508C4⋊C4—84no—
509C4⋊C4C22×D414yes—
510C4⋊C4H(5,𝔽2)14yes—
511C22⋊C4—28no—
512C22⋊C4—28no—
513C22⋊C4—28no—
514C22⋊C4—48no—
515C22⋊C4—44no—
516C22⋊C4—44no—
517C22⋊C4—44no—
518C22⋊C4—44no—
519C22⋊C4—44no—
520C22⋊C4—44no—
521C22⋊C4—84no—
522C22⋊C4—84no—
523C22⋊C4—84no—
524C22⋊C4—84no—
525C22⋊C4—84no—
526C22⋊C4—84no—
527C42—24no—
528C42—24no—
529C42—44no—
530C42—81no—
531C42—81no—
532C42—84no—
533C42#12114yes—
534C42#12114yes—
535C42C22×D412yes—
536C42C22×D412yes—
537C42C2×#4112yes—
538C42C2×#4112yes—
539C23—21no—
540C23—21no—
541C23—22no—
542C23—24no—
543C23—42no—
544C23—42no—
545C23—44no—
546C23—44no—
547C23—42no—
548C23—42no—
549C23—42no—
550C23—42no—
551C23—42no—
552C23—42no—
553C23—44no—
554C23—44no—
555C23—44no—
556C23—44no—
557C23—84no—
558C23—84no—
559C23—84no—
560C23—164no—
561C23—164no—
562C23C2×#12112yes—
563C23#57614yes—
564C23C23×D411yescharacteristic
565Q8—44no—
566Q8—42no—
567Q8—42no—
568Q8—84no—
569Q8—84no—
570Q8—84no—
571D4—24no—
572D4—24no—
573D4—48no—
574D4—416no—
575D4—44no—
576D4—44no—
577D4—44no—
578D4—42no—
579D4—42no—
580D4—42no—
581D4—42no—
582D4—84no—
583D4—84no—
584D4—84no—
585D4—84no—
586D4—88no—
587D4—168no—
588D4—168no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—21no—
592C2×C4—21no—
593C2×C4—22no—
594C2×C4—28no—
595C2×C4—22no—
596C2×C4—22no—
597C2×C4—24no—
598C2×C4—22no—
599C2×C4—22no—
600C2×C4—24no—
601C2×C4—24no—
602C2×C4—24no—
603C2×C4—24no—
604C2×C4—24no—
605C2×C4—24no—
606C2×C4—24no—
607C2×C4—24no—
608C2×C4—44no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—44no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—44no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—44no—
619C2×C4—88no—
620C2×C4—88no—
621C2×C4—88no—
622C2×C4—84no—
623C2×C4—168no—
624C2×C4C2×#12112yes—
625C2×C4C2×#12112yes—
626C2×C4C2×#12112yes—
627C2×C4C2×#12112yes—
628C2×C4C2×#12112yes—
629C2×C4C23×D411yescharacteristic
630C2×C4C22×#4111yescharacteristic
631C2×C4C2×H(5,𝔽2)12yes—
632C2×C4#58512yes—
633C8—44no—
634C8—42no—
635C8—42no—
636C22—22no—
637C22—24no—
638C22—24no—
639C22—22no—
640C22—44no—
641C22—44no—
642C22—44no—
643C22—84no—
644C22—84no—
645C22—82no—
646C22—82no—
647C22—82no—
648C22—82no—
649C22—84no—
650C22—84no—
651C22—84no—
652C22—84no—
653C22—82no—
654C22—82no—
655C22—168no—
656C22#304114yes—
657C22C22×#12111yescharacteristic
658C22#348812yes—
659C22C2×#57612yes—
660C22#358511yescharacteristic
661C22#358611yescharacteristic
662C4—22no—
663C4—22no—
664C4—22no—
665C4—22no—
666C4—22no—
667C4—44no—
668C4—44no—
669C4—44no—
670C4—44no—
671C4—82no—
672C4—82no—
673C4—82no—
674C4#346012yes—
675C4#349811yescharacteristic
676C4#350011yescharacteristic
677C2—24no—
678C2—84no—
679C2—162no—
680C2—162no—
681C2#3525611yescharacteristic
682C2C2×#304111yescharacteristic
683C2#6042511yescharacteristic
684C2#6201512yes—
685C2#6202212yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ #2283 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#3245C2semidirect
2. C2×#3213C2semidirect
3. #37536C2semidirect
4. #34236C2semidirect
5. #33166C2semidirect
6. #33036C2semidirect
7. #29839C2semidirect
8. #29802C2semidirect
9. #25376C2semidirect
10. #25285C2semidirect
11. #25090C2semidirect
12. #25076C2semidirect
13. #21977C2semidirect
14. #21395C2semidirect
15. C2×#1983C2semidirect
23. C2×#474C22semidirect
24. C2×#474C22semidirect
28. C2×#463C22semidirect
30. C2×#459C22semidirect
31. C2×#459C22semidirect
48. #2369C22semidirect
49. #2366C22semidirect
55. #2319C22semidirect
58. #2275C22semidirect
60. #2068C22semidirect
62. #2055C22semidirect
64. #2042C22semidirect
66. #2012C22semidirect
68. #1987C22semidirect
69. #1987C22semidirect
73. #1983C22semidirect
75. #1982C22semidirect
76. #1982C22semidirect
77. #1940C22semidirect
78. #1928C22semidirect
81. #1877C22semidirect
84. #1843C22semidirect
85. #1832C22semidirect
86. #1762C22semidirect
100. #532D4semidirect

Maximal subgroups

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

subgroupindex
C2×#32452
C2×#32132
#375362
#342362
#331662
#330362
#298392
#298022
#253762
#252852
#250902
#250762
#219772
#213952
C2×#19832