Finite Group Databasebeta

← #6526855

Subgroups — #6526855

Order 512.

2263 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×#3238C211yesmaximal, characteristic, complemented
2C2×#3226C211yesmaximal, characteristic, complemented
3C2×#3112C211yesmaximal, characteristic, complemented
4C2×#3108C211yesmaximal, characteristic, complemented
5#35238C211yesmaximal, characteristic
6C2×#2567C211yesmaximal, characteristic, complemented
7#28543C214yesmaximal, complemented
8#28539C212yesmaximal, complemented
9#28531C212yesmaximal, complemented
10#28428C214yesmaximal, complemented
11#28418C212yesmaximal, complemented
12#28404C212yesmaximal, complemented
13#28314C212yesmaximal, complemented
14#28306C212yesmaximal, complemented
15#27914C212yesmaximal, complemented
16#27909C212yesmaximal, complemented
17#20535C211yesmaximal, characteristic, complemented
18C2×#532C2211yescharacteristic
19C22×C4⋊Q8C2211yescharacteristic
20C2×#529C2211yescharacteristic
21C22×C4×Q8C2211yescharacteristic
22#3238—28no—
23#3226—28no—
24C2×#501C2212yes—
25C2×#500C2212yes—
26C2×#489C2212yes—
27C2×#486C2212yes—
28#3112—28no—
29#3108—28no—
30C2×#483C2211yescharacteristic
31C2×#483C2212yes—
32C2×#480C2211yescharacteristic
33C2×#480C2212yes—
34C2×#480C2212yes—
35C2×#480C2212yes—
36C2×#480C2212yes—
37C2×#479C2212yes—
38C2×#477C2212yes—
39C2×#477C2212yes—
40C2×#476C2212yes—
41C2×#474C2214yes—
42C2×#464C2212yes—
43C2×#460C2212yes—
44C2×#451C2211yescharacteristic
45C2×#448C2211yescharacteristic
46C2×#448C2212yes—
47C2×#440C2214yes—
48C2×C4×Q16C2212yes—
49C2×C4×SD16C2212yes—
50C2×#432C2211yescharacteristic
51C2×#431C2211yescharacteristic
52#2856C2212yes—
53#2850C2212yes—
54#2662—22no—
55#2567—22no—
56#2567—22no—
57#2567C2212yes—
58#2567C2214yes—
59#2567C2211yescharacteristic
60#2567C2212yes—
61#2566C2212yes—
62#2563C2211yescharacteristic
63#2548C2211yescharacteristic
64#2544C2212yes—
65#2541C2211yescharacteristic
66#2430—22no—
67#2416—22no—
68C2×#395—22no—
69#2324—22no—
70C2×#391C2212yes—
71C2×#389—22no—
72C2×#384C2212yes—
73C2×C4×C4⋊C4C2211yescharacteristic
74#2097C2214yes—
75#2095C2214yes—
76#2095C2214yes—
77#2094C2214yes—
78#2093C2214yes—
79#2093C2214yes—
80#2015C2214yes—
81#2014C2214yes—
82#1973C2214yes—
83#1973C2214yes—
84#1969C2214yes—
85#1968C2214yes—
86#1959C2214yes—
87#1957C2214yes—
88#1847C2214yes—
89#1845C2214yes—
90#1771C2214yes—
91#1770C2212yes—
92C8×C4⋊C4C2212yes—
93#1749C2212yes—
94C4×C22⋊C8C2212yes—
95C22×#41—21no—
96C23×Q8—21no—
97C23×Q8C2311yescharacteristic
98C22×Q16C2312yes—
99C22×SD16C2312yes—
100C22×SD16C2314yes—
101#532—24no—
102#532—24no—
103C2×C4⋊Q8—21no—
104C2×C4⋊Q8—21no—
105C2×C4⋊Q8—24no—
106C2×C4⋊Q8—24no—
107C2×C4⋊Q8C2311yescharacteristic
108C2×C4⋊Q8C2312yes—
109C2×C4⋊Q8C2312yes—
110C2×#127C2311yescharacteristic
111C2×#127C2312yes—
112#529—24no—
113#529—24no—
114C2×#126—21no—
115C2×#125C2311yescharacteristic
116C2×#124—21no—
117C2×#124C2311yescharacteristic
118C2×#124C2312yes—
119C2×#123—21no—
120C2×C22⋊Q8C2311yescharacteristic
121C2×#121—21no—
122C2×#121—24no—
123C2×#121—22no—
124C2×#121C2312yes—
125C2×#121C2311yescharacteristic
126C2×C4×Q8—21no—
127C2×C4×Q8—21no—
128C2×C4×Q8—24no—
129C2×C4×Q8—24no—
130C2×C4×Q8C2311yescharacteristic
131C2×C4×Q8C2312yes—
132C2×C4×Q8C2312yes—
133C2×C4×D4—21no—
134C2×C4×D4—21no—
135C2×C4×D4C2311yescharacteristic
136C2×C4×D4C2312yes—
137C2×#117—21no—
138C22×C4⋊C4—21no—
139C22×C4⋊C4—21no—
140C22×C4⋊C4—24no—
141C22×C4⋊C4D412yes—
142C22×C4⋊C4C2311yescharacteristic
143C22×C4⋊C4C2312yes—
144C22×C42—21no—
145C22×C42C2312yes—
146C22×C42C2311yescharacteristic
147#501—28no—
148#500—28no—
149#489—28no—
150#486—28no—
151#483—24no—
152#483—28no—
153#480—24no—
154#480—28no—
155#480—28no—
156#480—28no—
157#480—28no—
158#479—28no—
159#477—28no—
160#477—28no—
161#476—28no—
162#474—216no—
163#464—28no—
164#460—28no—
165#451—24no—
166#448—24no—
167#448—28no—
168#440—216no—
169C4×Q16—28no—
170C4×SD16—28no—
171#432D418yescomplemented
172#431D418yescomplemented
173C2×#107C2314yes—
174C2×#107C2314yes—
175C2×#107C2312yes—
176C2×#107C2314yes—
177C2×#107C2314yes—
178C2×#106C2312yes—
179C2×#106C2314yes—
180C2×C4⋊C8C2312yes—
181C2×C4⋊C8C2312yes—
182C2×#103C2314yes—
183C2×#103C2314yes—
184C2×#103C2312yes—
185C2×#103C2312yes—
186C2×#103C2312yes—
187C2×#102C2312yes—
188C2×#102C2314yes—
189C2×#102C2314yes—
190C2×#102C2314yes—
191C2×#102C2314yes—
192C2×C22⋊C8C2311yescharacteristic
193C2×C22⋊C8C2311yescharacteristic
194C2×C22⋊C8C2312yes—
195C2×#97C2314yes—
196C2×#97C2312yes—
197C2×C4×C8C2314yes—
198C2×C4×C8C2312yes—
199#396—24no—
200#395—28no—
201#395—24no—
202#395—24no—
203#395—22no—
204#394—24no—
205#394—22no—
206#393—22no—
207#392—22no—
208#391—24no—
209#391C2314yes—
210#391C2314yes—
211#391C2312yes—
212#390—22no—
213#390—24no—
214#390C2312yes—
215#390C2312yes—
216#390C2312yes—
217#389—24no—
218#389—28no—
219#389—24no—
220#389—22no—
221#388—24no—
222#386—24no—
223#386—24no—
224#385—24no—
225#384—24no—
226#384C2312yes—
227#384C2314yes—
228#384C2314yes—
229#382C2312yes—
230#380—22no—
231C4×C4⋊C4D412yes—
232C4×C4⋊C4D412yes—
233C4×C4⋊C4C2311yescharacteristic
234C4×C4⋊C4C2312yes—
235C4×C4⋊C4C2311yescharacteristic
236C2×#95—24no—
237C2×#95D412yes—
238#336C2314yes—
239#336C2314yes—
240C2×#41—44no—
241C2×#41—84no—
242C22×Q8—24no—
243C22×Q8—21no—
244C22×Q8—21no—
245C22×Q8—22no—
246C22×Q8—22no—
247C22×Q8—22no—
248C22×Q8—22no—
249C22×Q8—42no—
250C22×Q8—44no—
251C22×Q8—84no—
252C22×Q8C2411yescharacteristic
253C22×Q8C2412yes—
254C22×D4—21no—
255C22×D4—22no—
256C22×D4—21no—
257C22×D4—24no—
258C22×D4—21no—
259C22×D4C2411yescharacteristic
260C22×D4C2412yes—
261C23×C4—21no—
262C23×C4—21no—
263C23×C4—22no—
264C23×C4—21no—
265C23×C4—22no—
266C23×C4C2×D414yes—
267C23×C4C2×D411yescharacteristic
268C23×C4C2×D411yescharacteristic
269C23×C4C2411yescharacteristic
270C2×Q16—24no—
271C2×Q16—24no—
272C2×Q16—42no—
273C2×SD16—24no—
274C2×SD16—28no—
275C2×SD16—24no—
276C2×SD16—28no—
277C2×SD16—44no—
278C2×SD16—42no—
279C22×C8C2414yes—
280C22×C8C2412yes—
281C22×C8C2412yes—
282C4⋊Q8—24no—
283C4⋊Q8—24no—
284C4⋊Q8—24no—
285C4⋊Q8—24no—
286C4⋊Q8—22no—
287C4⋊Q8—22no—
288C4⋊Q8—44no—
289C4⋊Q8—44no—
290#127—22no—
291#127—22no—
292#127—44no—
293#126—44no—
294#125—22no—
295#125—22no—
296#124—22no—
297#124—22no—
298#124—44no—
299#124—44no—
300#123—44no—
301C22⋊Q8—22no—
302C22⋊Q8—22no—
303#121—22no—
304#121—22no—
305#121—24no—
306#121—24no—
307#121—44no—
308#121—48no—
309#121—88no—
310C4×Q8—24no—
311C4×Q8—24no—
312C4×Q8—24no—
313C4×Q8—24no—
314C4×Q8—22no—
315C4×Q8—22no—
316C4×Q8—44no—
317C4×Q8—44no—
318C4×D4—22no—
319C4×D4—22no—
320C4×D4—24no—
321C4×D4—24no—
322C4×D4—44no—
323C4×D4—44no—
324#117—44no—
325C2×C4⋊C4—28no—
326C2×C4⋊C4—22no—
327C2×C4⋊C4—22no—
328C2×C4⋊C4—24no—
329C2×C4⋊C4—24no—
330C2×C4⋊C4—22no—
331C2×C4⋊C4—22no—
332C2×C4⋊C4—22no—
333C2×C4⋊C4—22no—
334C2×C4⋊C4—21no—
335C2×C4⋊C4—22no—
336C2×C4⋊C4—21no—
337C2×C4⋊C4—21no—
338C2×C4⋊C4—21no—
339C2×C4⋊C4—21no—
340C2×C4⋊C4—21no—
341C2×C4⋊C4—28no—
342C2×C4⋊C4—28no—
343C2×C4⋊C4—24no—
344C2×C4⋊C4—44no—
345C2×C4⋊C4—44no—
346C2×C4⋊C4—88no—
347C2×C4⋊C4C2×D412yes—
348C2×C4⋊C4C2×D414yes—
349C2×C4⋊C4C2×D412yes—
350C2×C4⋊C4C2412yes—
351C2×C4⋊C4C2414yes—
352C2×C4⋊C4C2412yes—
353C2×C4⋊C4C2411yescharacteristic
354C2×C4⋊C4C2411yescharacteristic
355C2×C22⋊C4—21no—
356C2×C22⋊C4—21no—
357C2×C22⋊C4—21no—
358C2×C22⋊C4—22no—
359C2×C22⋊C4—21no—
360C2×C22⋊C4—22no—
361C2×C22⋊C4—21no—
362C2×C22⋊C4—24no—
363C2×C22⋊C4—24no—
364C2×C22⋊C4—24no—
365C2×C22⋊C4—24no—
366C2×C42—21no—
367C2×C42—22no—
368C2×C42—22no—
369C2×C42—44no—
370C2×C42—44no—
371C2×C42C2×D414yes—
372C2×C42C2×D411yescharacteristic
373C2×C42C2×D411yescharacteristic
374C2×C42C2×D414yes—
375C2×C42C2×D414yes—
376C2×C42C2412yes—
377C2×C42C2412yes—
378C2×C42C2411yescharacteristic
379C2×C42C2411yescharacteristic
380#107—24no—
381#107—28no—
382#107—28no—
383#107—28no—
384#107—28no—
385#106—24no—
386#106—28no—
387C4⋊C8C2×D418yes—
388C4⋊C8C2×D418yes—
389#103—24no—
390#103—24no—
391#103—24no—
392#103—28no—
393#103—28no—
394#102—24no—
395#102—28no—
396#102—28no—
397#102—28no—
398#102—28no—
399C22⋊C8C2×D414yes—
400C22⋊C8C2×D414yes—
401C22⋊C8C2×D418yes—
402#97—44no—
403#97C2×D418yes—
404C4×C8—44no—
405C4×C8C2×D418yes—
406#95—28no—
407#95—28no—
408#95—24no—
409#95—24no—
410#95C2×D414yes—
411#95C2×D412yes—
412#95C2×D412yes—
413C24—21no—
414C24—22no—
415C24C22×D411yescharacteristic
416#41—816no—
417C2×Q8—24no—
418C2×Q8—22no—
419C2×Q8—22no—
420C2×Q8—22no—
421C2×Q8—21no—
422C2×Q8—21no—
423C2×Q8—22no—
424C2×Q8—22no—
425C2×Q8—22no—
426C2×Q8—22no—
427C2×Q8—48no—
428C2×Q8—48no—
429C2×Q8—42no—
430C2×Q8—42no—
431C2×Q8—42no—
432C2×Q8—44no—
433C2×Q8—44no—
434C2×Q8—84no—
435C2×Q8—84no—
436C2×Q8—82no—
437C2×Q8—816no—
438C2×Q82−1+414yes—
439C2×D4—21no—
440C2×D4—22no—
441C2×D4—21no—
442C2×D4—22no—
443C2×D4—22no—
444C2×D4—24no—
445C2×D4—24no—
446C2×D4—22no—
447C2×D4—42no—
448C2×D4—44no—
449C2×D4—44no—
450C2×D4—48no—
451C2×D4—42no—
452C2×D4—42no—
453C2×D4—44no—
454C2×D4—48no—
455C2×D4—82no—
456C2×D4—84no—
457C2×D4—84no—
458C22×C4—24no—
459C22×C4—24no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—24no—
463C22×C4—28no—
464C22×C4—22no—
465C22×C4—28no—
466C22×C4—21no—
467C22×C4—21no—
468C22×C4—22no—
469C22×C4—24no—
470C22×C4—22no—
471C22×C4—21no—
472C22×C4—44no—
473C22×C4—42no—
474C22×C4—42no—
475C22×C4—42no—
476C22×C4—44no—
477C22×C4—44no—
478C22×C4—48no—
479C22×C4—42no—
480C22×C4—44no—
481C22×C4—44no—
482C22×C4—84no—
483C22×C4#12714yes—
484C22×C4#12714yes—
485C22×C4C22×D414yes—
486C22×C4C22×D411yescharacteristic
487C22×C4C22×D411yescharacteristic
488C22×C4C22×D411yescharacteristic
489C22×C4C22×D411yescharacteristic
490C22×C4C22×D412yes—
491C22×C4C22×D412yes—
492C22×C4C22×D414yes—
493C22×C4C22×D411yescharacteristic
494C22×C4C2511yescharacteristic
495Q16—48no—
496SD16—416no—
497SD16—48no—
498C2×C8—24no—
499C2×C8—22no—
500C2×C8—22no—
501C2×C8—22no—
502C2×C8—24no—
503C2×C8—22no—
504C2×C8C22×D414yes—
505C2×C8C22×D414yes—
506C2×C8C22×D418yes—
507C4⋊C4—28no—
508C4⋊C4—24no—
509C4⋊C4—22no—
510C4⋊C4—24no—
511C4⋊C4—22no—
512C4⋊C4—24no—
513C4⋊C4—21no—
514C4⋊C4—22no—
515C4⋊C4—21no—
516C4⋊C4—21no—
517C4⋊C4—22no—
518C4⋊C4—21no—
519C4⋊C4—42no—
520C4⋊C4—44no—
521C4⋊C4—44no—
522C4⋊C4—44no—
523C4⋊C4—44no—
524C4⋊C4—44no—
525C4⋊C4—42no—
526C4⋊C4—42no—
527C4⋊C4—42no—
528C4⋊C4—42no—
529C4⋊C4—42no—
530C4⋊C4—88no—
531C4⋊C4—88no—
532C4⋊C4—84no—
533C4⋊C4#13718yes—
534C22⋊C4—42no—
535C22⋊C4—42no—
536C22⋊C4—42no—
537C22⋊C4—44no—
538C22⋊C4—42no—
539C22⋊C4—44no—
540C22⋊C4—42no—
541C22⋊C4—48no—
542C22⋊C4—84no—
543C22⋊C4—84no—
544C22⋊C4—84no—
545C42—24no—
546C42—24no—
547C42—44no—
548C42—44no—
549C42—44no—
550C42—42no—
551C42#12714yes—
552C42#12714yes—
553C42C22×D412yes—
554C42C22×D412yes—
555C42C22×D412yes—
556C42C22×D412yes—
557C23—21no—
558C23—21no—
559C23—22no—
560C23—41no—
561C23—42no—
562C23—42no—
563C23—41no—
564C23—41no—
565C23—42no—
566C23—44no—
567C23—42no—
568C23—42no—
569C23—44no—
570C23C2×#12712yes—
571C23#57614yes—
572C23C23×D411yescharacteristic
573Q8—24no—
574Q8—24no—
575Q8—48no—
576Q8—42no—
577Q8—42no—
578Q8—42no—
579Q8—42no—
580Q8—88no—
581Q8—88no—
582Q8—84no—
583D4—42no—
584D4—44no—
585D4—42no—
586D4—44no—
587D4—84no—
588D4—84no—
589D4—88no—
590D4—84no—
591D4—168no—
592C2×C4—28no—
593C2×C4—24no—
594C2×C4—21no—
595C2×C4—21no—
596C2×C4—22no—
597C2×C4—24no—
598C2×C4—22no—
599C2×C4—22no—
600C2×C4—28no—
601C2×C4—24no—
602C2×C4—24no—
603C2×C4—24no—
604C2×C4—24no—
605C2×C4—24no—
606C2×C4—24no—
607C2×C4—416no—
608C2×C4—48no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—41no—
613C2×C4—41no—
614C2×C4—41no—
615C2×C4—42no—
616C2×C4—44no—
617C2×C4—41no—
618C2×C4—41no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—44no—
622C2×C4—44no—
623C2×C4—48no—
624C2×C4—88no—
625C2×C4—84no—
626C2×C4—88no—
627C2×C4—84no—
628C2×C4C2×#12712yes—
629C2×C4C2×#12712yes—
630C2×C4C2×#12712yes—
631C2×C4C2×#12712yes—
632C2×C4C2×#12712yes—
633C2×C4C2×#13718yes—
634C2×C4C2×#13714yes—
635C2×C4C23×D411yescharacteristic
636C2×C4C23×D411yescharacteristic
637C2×C4C2×2−1+412yes—
638C2×C4#58512yes—
639C8—28no—
640C8—24no—
641C8—24no—
642C22—24no—
643C22—22no—
644C22—24no—
645C22—22no—
646C22—82no—
647C22—84no—
648C22—81no—
649C22—82no—
650C22—84no—
651C22—82no—
652C22—81no—
653C22—82no—
654C22—82no—
655C22—81no—
656C22#315514yes—
657C22C22×#12711yescharacteristic
658C22#348511yescharacteristic
659C22#348611yescharacteristic
660C22C22×#13712yes—
661C22C2×#57612yes—
662C4—28no—
663C4—22no—
664C4—22no—
665C4—22no—
666C4—22no—
667C4—44no—
668C4—44no—
669C4—81no—
670C4—81no—
671C4—82no—
672C4—82no—
673C4—84no—
674C4—81no—
675C4#315214yes—
676C4#348912yes—
677C4#348912yes—
678C2—24no—
679C2—161no—
680C2—162no—
681C2#3563411yescharacteristic
682C2C2×#315211yescharacteristic
683C2C2×#315511yescharacteristic
684C2#6172612yes—
685C2#6184412yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#35238C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#3238C2semidirect
2. C2×#3226C2semidirect
3. C2×#3112C2semidirect
4. C2×#3108C2semidirect
6. C2×#2567C2semidirect
7. #28543C2semidirect
8. #28539C2semidirect
9. #28531C2semidirect
10. #28428C2semidirect
11. #28418C2semidirect
12. #28404C2semidirect
13. #28314C2semidirect
14. #28306C2semidirect
15. #27914C2semidirect
16. #27909C2semidirect
17. #20535C2semidirect
171. #432D4semidirect
172. #431D4semidirect

Maximal subgroups

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

subgroupindex
C2×#32382
C2×#32262
C2×#31122
C2×#31082
#352382
C2×#25672
#285432
#285392
#285312
#284282
#284182
#284042
#283142
#283062
#279142
#279092
#205352