Finite Group Databasebeta

← #6489669

Subgroups — #6489669

Order 512.

3227 conjugacy classes of subgroups.

The subgroup lattice has 659 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#554C211yesmaximal, characteristic, complemented
2C2×#3238C211yesmaximal, characteristic, complemented
3C2×#3105C211yesmaximal, characteristic
4#35247C211yesmaximal, characteristic, complemented
5#33007C212yesmaximal, complemented
6#33002C212yesmaximal
7#29643C212yesmaximal, complemented
8#29640C212yesmaximal, complemented
9#25940C214yesmaximal, complemented
10#25845C214yesmaximal, complemented
11#24331C214yesmaximal, complemented
12#24327C214yesmaximal, complemented
13#21833C211yesmaximal, characteristic, complemented
14#21820C211yesmaximal, characteristic, complemented
15#20738C211yesmaximal, characteristic, complemented
16C2×#554—28no—
17C2×#554—22no—
18C2×#554—22no—
19C2×#554—28no—
20C2×#554C2214yes—
21C2×#554C2214yes—
22C22×C4⋊Q8C2211yescharacteristic
23C2×#529C2211yescharacteristic
24C22×#125C2212yes—
25C22×C22⋊Q8—21no—
26C22×C22⋊Q8—22no—
27C22×C4×Q8C2211yescharacteristic
28C22×C4×D4C2212yes—
29C22×C4×D4C2211yescharacteristic
30C23×C4⋊C4—21no—
31#3238—28no—
32C2×#501C2212yescomplemented
33C2×#489C2212yescomplemented
34#3105—28no—
35C2×#484C2211yescharacteristic
36C2×#481C2211yescharacteristic
37C2×#483C2212yes—
38C2×#480C2212yes—
39C2×#477C2212yescomplemented
40C2×#478C2212yes—
41C2×#476C2212yescomplemented
42C2×#463C2212yes—
43C2×#450C2211yescharacteristic
44C2×#447C2211yescharacteristic
45C2×C4×SD16C2212yes—
46C2×C4×D8C2212yes—
47C2×#432C2211yescharacteristic
48C2×#431C2211yescharacteristic
49#2855C2212yes—
50#2854C2212yes—
51#2637—22no—
52#2592C2212yescomplemented
53#2584C2212yescomplemented
54#2582C2212yescomplemented
55#2538C2212yescomplemented
56#2486—21no—
57#2413—22no—
58#2409—21no—
59#2323—22no—
60#2285C2211yescharacteristic
61#2059C2218yescomplemented
62#2059C2218yescomplemented
63#2026C2218yescomplemented
64#2025C2218yescomplemented
65#1990C2214yescomplemented
66#1988C2214yescomplemented
67#1985C2214yescomplemented
68#1984C2214yescomplemented
69#1879C2218yes—
70#1877C2218yes—
71#1864C2218yes—
72#1862C2218yes—
73#1824C2212yes—
74#1823C2212yes—
75#1823C2212yes—
76#1817C2212yes—
77#1817C2212yes—
78#1802C2214yes—
79#1786C2212yes—
80#1762C2214yes—
81C23×Q8—21no—
82C23×Q8D411yescharacteristic
83C23×D4C2311yescharacteristic
84C24×C4—21no—
85C24×C4—22no—
86C22×SD16C2312yes—
87C22×D8C2312yes—
88#554—232no—
89#554—232no—
90#554—216no—
91#554—216no—
92#554—216no—
93#554—216no—
94C2×C4⋊Q8—21no—
95C2×C4⋊Q8—21no—
96C2×C4⋊Q8—24no—
97C2×C4⋊Q8D418yes—
98C2×C4⋊Q8C2312yes—
99C2×C4⋊Q8C2312yes—
100#529—24no—
101#529—24no—
102C2×#126—21no—
103C2×#125—28no—
104C2×#125—28no—
105C2×#125—24no—
106C2×#125C2311yescharacteristic
107C2×#125C2318yes—
108C2×#124C2311yescharacteristic
109C2×#123—21no—
110C2×#123—22no—
111C2×C22⋊Q8—216no—
112C2×C22⋊Q8—232no—
113C2×C22⋊Q8—24no—
114C2×C22⋊Q8—24no—
115C2×C22⋊Q8—22no—
116C2×C22⋊Q8—28no—
117C2×C22⋊Q8—28no—
118C2×C22⋊Q8C2311yescharacteristic
119C2×#121C2311yescharacteristic
120C2×C4×Q8—24no—
121C2×C4×Q8—24no—
122C2×C4×Q8D412yes—
123C2×C4×Q8D412yes—
124C2×C4×Q8C2311yescharacteristic
125C2×C4×Q8C2312yes—
126C2×C4×Q8C2312yes—
127C2×C4×D4—28no—
128C2×C4×D4—21no—
129C2×C4×D4—28no—
130C2×C4×D4—24no—
131C2×C4×D4—21no—
132C2×C4×D4—21no—
133C2×C4×D4—24no—
134C2×C4×D4D418yes—
135C2×C4×D4C2318yes—
136C2×C4×D4C2311yescharacteristic
137C2×C4×D4C2312yes—
138C2×C4×D4C2312yes—
139C2×#117—21no—
140C2×#117—21no—
141C22×C4⋊C4—216no—
142C22×C4⋊C4—22no—
143C22×C4⋊C4—21no—
144C22×C4⋊C4—22no—
145C22×C4⋊C4—24no—
146C22×C4⋊C4—24no—
147C22×C4⋊C4D411yescharacteristic
148C22×C4⋊C4D412yes—
149C22×C4⋊C4C2312yes—
150C22×C4⋊C4C2311yescharacteristic
151C22×C22⋊C4—22no—
152C22×C22⋊C4—21no—
153C22×C42C2312yes—
154C22×C42C2311yescharacteristic
155#501—28no—
156#489—28no—
157#484—24no—
158#483—28no—
159#481—24no—
160#480—28no—
161#478—28no—
162#477—28no—
163#476—28no—
164#463—28no—
165#450—24no—
166#447—24no—
167C4×SD16—28no—
168C4×D8—28no—
169#432—24no—
170#431—24no—
171C2×#107C2312yes—
172C2×#107C2314yes—
173C2×#106C2312yes—
174C2×#106C2314yes—
175C2×C4⋊C8C2312yes—
176C2×C4⋊C8C2312yes—
177C2×#103C2314yes—
178C2×#103C2312yes—
179C2×#102C2312yes—
180C2×#102C2312yes—
181C2×#102C2318yes—
182C2×#102C2314yes—
183C2×#102C2312yes—
184C2×C22⋊C8C2312yes—
185C2×C22⋊C8C2311yescharacteristic
186C2×C22⋊C8C2311yescharacteristic
187C2×#97C2312yes—
188C2×C4×C8C2312yes—
189#398—24no—
190#398—22no—
191#397—22no—
192#396—24no—
193#395—22no—
194#393—22no—
195#391—22no—
196#391C2314yes—
197#390C2314yes—
198#389—22no—
199#389—24no—
200#389C2314yes—
201#389C2314yes—
202#388—24no—
203#388—24no—
204#387—24no—
205#386—22no—
206#385—22no—
207#384C2312yes—
208#382—21no—
209#381—22no—
210#380—21no—
211#380—22no—
212#380C2312yes—
213C4×C4⋊C4—21no—
214C4×C4⋊C4C2312yes—
215#340C2318yes—
216#340C2318yes—
217#340C2318yes—
218#336C2318yes—
219#336C2318yes—
220C25—21no—
221C22×Q8—24no—
222C22×Q8—28no—
223C22×Q8—22no—
224C22×Q8—22no—
225C22×Q8—44no—
226C22×Q8—82no—
227C22×Q8—82no—
228C22×Q8C2×D412yes—
229C22×Q8C2×D412yes—
230C22×Q8C2411yescharacteristic
231C22×D4—28no—
232C22×D4—21no—
233C22×D4—21no—
234C22×D4—22no—
235C22×D4C2×D414yes—
236C22×D4C2411yescharacteristic
237C22×D4C2412yes—
238C23×C4—21no—
239C23×C4—22no—
240C23×C4—216no—
241C23×C4—28no—
242C23×C4—22no—
243C23×C4—22no—
244C23×C4—28no—
245C23×C4—21no—
246C23×C4—48no—
247C23×C4—84no—
248C23×C4—84no—
249C23×C4C2×D412yes—
250C23×C4C2×D411yescharacteristic
251C23×C4C2×D411yescharacteristic
252C23×C4C2×D412yes—
253C23×C4C2411yescharacteristic
254C2×SD16—24no—
255C2×SD16—28no—
256C2×D8—24no—
257C2×D8—28no—
258C22×C8C2412yes—
259C22×C8C2412yes—
260C22×C8C2414yes—
261C4⋊Q8—28no—
262C4⋊Q8—28no—
263C4⋊Q8—24no—
264C4⋊Q8—24no—
265C4⋊Q8C2×D418yes—
266C4⋊Q8C2×D418yes—
267#126—44no—
268#125—232no—
269#125—216no—
270#125—216no—
271#125—22no—
272#125—22no—
273#124—22no—
274#124—22no—
275#123—44no—
276#123—84no—
277C22⋊Q8—232no—
278C22⋊Q8—232no—
279C22⋊Q8—264no—
280C22⋊Q8—264no—
281C22⋊Q8—22no—
282C22⋊Q8—22no—
283C22⋊Q8—48no—
284#121—22no—
285#121—22no—
286C4×Q8—28no—
287C4×Q8—28no—
288C4×Q8—24no—
289C4×Q8—24no—
290C4×Q8—24no—
291C4×Q8—24no—
292C4×Q8—22no—
293C4×Q8—22no—
294C4×D4—232no—
295C4×D4—216no—
296C4×D4—216no—
297C4×D4—28no—
298C4×D4—28no—
299C4×D4—24no—
300C4×D4—24no—
301C4×D4—22no—
302C4×D4—22no—
303C4×D4—44no—
304C4×D4C2×D418yes—
305C4×D4C2×D418yes—
306#117—44no—
307#117—82no—
308C2×C4⋊C4—216no—
309C2×C4⋊C4—232no—
310C2×C4⋊C4—232no—
311C2×C4⋊C4—216no—
312C2×C4⋊C4—24no—
313C2×C4⋊C4—28no—
314C2×C4⋊C4—28no—
315C2×C4⋊C4—28no—
316C2×C4⋊C4—22no—
317C2×C4⋊C4—22no—
318C2×C4⋊C4—28no—
319C2×C4⋊C4—21no—
320C2×C4⋊C4—22no—
321C2×C4⋊C4—22no—
322C2×C4⋊C4—24no—
323C2×C4⋊C4—21no—
324C2×C4⋊C4—21no—
325C2×C4⋊C4—21no—
326C2×C4⋊C4—21no—
327C2×C4⋊C4—22no—
328C2×C4⋊C4—24no—
329C2×C4⋊C4—22no—
330C2×C4⋊C4—24no—
331C2×C4⋊C4C2×D412yes—
332C2×C4⋊C4C2×D412yes—
333C2×C4⋊C4C2×D414yes—
334C2×C4⋊C4C2×D414yes—
335C2×C4⋊C4C2×D418yes—
336C2×C4⋊C4C2×D414yes—
337C2×C4⋊C4C2411yescharacteristic
338C2×C4⋊C4C2411yescharacteristic
339C2×C4⋊C4C2414yes—
340C2×C4⋊C4C2412yes—
341C2×C4⋊C4C2414yes—
342C2×C22⋊C4—216no—
343C2×C22⋊C4—28no—
344C2×C22⋊C4—28no—
345C2×C22⋊C4—22no—
346C2×C22⋊C4—22no—
347C2×C22⋊C4—22no—
348C2×C22⋊C4—21no—
349C2×C22⋊C4—21no—
350C2×C22⋊C4—21no—
351C2×C22⋊C4—21no—
352C2×C22⋊C4—21no—
353C2×C22⋊C4—24no—
354C2×C22⋊C4—24no—
355C2×C42—24no—
356C2×C42—24no—
357C2×C42—21no—
358C2×C42—22no—
359C2×C42C2×D414yes—
360C2×C42C2×D411yescharacteristic
361C2×C42C2×D411yescharacteristic
362C2×C42C2×D414yes—
363C2×C42C2×D414yes—
364C2×C42C2414yes—
365C2×C42C2411yescharacteristic
366C2×C42C2411yescharacteristic
367#107—24no—
368#107—28no—
369#106—24no—
370#106—28no—
371C4⋊C8—24no—
372C4⋊C8—24no—
373#103—24no—
374#103—28no—
375#102—24no—
376#102—24no—
377#102—216no—
378#102—24no—
379#102—28no—
380C22⋊C8—22no—
381C22⋊C8—22no—
382C22⋊C8—24no—
383#97—24no—
384C4×C8—24no—
385#95—22no—
386#95—22no—
387#95—24no—
388#95—24no—
389#95—24no—
390#95—24no—
391C24—28no—
392C24—24no—
393C24—22no—
394C24—21no—
395C24—44no—
396C24—42no—
397C24—42no—
398C24C22×D411yescharacteristic
399C2×Q8—24no—
400C2×Q8—24no—
401C2×Q8—28no—
402C2×Q8—28no—
403C2×Q8—24no—
404C2×Q8—24no—
405C2×Q8—21no—
406C2×Q8—21no—
407C2×Q8—24no—
408C2×Q8—44no—
409C2×Q8—416no—
410C2×Q8—816no—
411C2×Q8—82no—
412C2×Q8—82no—
413C2×Q8—82no—
414C2×Q8—82no—
415C2×D4—28no—
416C2×D4—216no—
417C2×D4—216no—
418C2×D4—24no—
419C2×D4—22no—
420C2×D4—21no—
421C2×D4—21no—
422C2×D4—24no—
423C2×D4—24no—
424C2×D4—42no—
425C2×D4—42no—
426C2×D4—42no—
427C2×D4C22×D414yes—
428C2×D42−1+418yes—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—24no—
432C22×C4—24no—
433C22×C4—24no—
434C22×C4—24no—
435C22×C4—24no—
436C22×C4—28no—
437C22×C4—24no—
438C22×C4—24no—
439C22×C4—22no—
440C22×C4—24no—
441C22×C4—232no—
442C22×C4—28no—
443C22×C4—28no—
444C22×C4—21no—
445C22×C4—24no—
446C22×C4—21no—
447C22×C4—21no—
448C22×C4—44no—
449C22×C4—48no—
450C22×C4—432no—
451C22×C4—44no—
452C22×C4—44no—
453C22×C4—44no—
454C22×C4—48no—
455C22×C4—42no—
456C22×C4—44no—
457C22×C4—88no—
458C22×C4—88no—
459C22×C4—832no—
460C22×C4#12014yes—
461C22×C4#12014yes—
462C22×C4C22×D414yes—
463C22×C4C22×D414yes—
464C22×C4C22×D411yescharacteristic
465C22×C4C22×D411yescharacteristic
466C22×C4C22×D411yescharacteristic
467C22×C4C22×D411yescharacteristic
468C22×C4C22×D412yes—
469C22×C4C22×D412yes—
470C22×C4C22×D411yescharacteristic
471C22×C4C2511yescharacteristic
472SD16—48no—
473D8—48no—
474C2×C8—22no—
475C2×C8—22no—
476C2×C8—24no—
477C2×C8—28no—
478C2×C8—24no—
479C2×C8—24no—
480C4⋊C4—232no—
481C4⋊C4—232no—
482C4⋊C4—24no—
483C4⋊C4—24no—
484C4⋊C4—28no—
485C4⋊C4—28no—
486C4⋊C4—24no—
487C4⋊C4—28no—
488C4⋊C4—28no—
489C4⋊C4—28no—
490C4⋊C4—22no—
491C4⋊C4—24no—
492C4⋊C4—21no—
493C4⋊C4—21no—
494C4⋊C4—21no—
495C4⋊C4—21no—
496C4⋊C4—28no—
497C4⋊C4—48no—
498C4⋊C4—42no—
499C4⋊C4—42no—
500C4⋊C4—42no—
501C4⋊C4—42no—
502C4⋊C4—44no—
503C4⋊C4—82no—
504C4⋊C4—84no—
505C4⋊C4C22×D418yes—
506C4⋊C4C22×D414yes—
507C22⋊C4—232no—
508C22⋊C4—28no—
509C22⋊C4—28no—
510C22⋊C4—48no—
511C22⋊C4—42no—
512C22⋊C4—44no—
513C22⋊C4—42no—
514C22⋊C4—42no—
515C22⋊C4—42no—
516C22⋊C4—42no—
517C22⋊C4—84no—
518C42—28no—
519C42—24no—
520C42—24no—
521C42—42no—
522C42—82no—
523C42#12014yes—
524C42#12014yes—
525C42C22×D412yes—
526C42C22×D412yes—
527C42C22×D412yes—
528C42C22×D412yes—
529C23—22no—
530C23—22no—
531C23—216no—
532C23—24no—
533C23—28no—
534C23—48no—
535C23—416no—
536C23—416no—
537C23—48no—
538C23—44no—
539C23—42no—
540C23—42no—
541C23—41no—
542C23—42no—
543C23—42no—
544C23—41no—
545C23—41no—
546C23C2×#12012yes—
547C23#57612yes—
548C23#57712yes—
549C23C23×D411yescharacteristic
550Q8—48no—
551Q8—48no—
552Q8—42no—
553Q8—42no—
554Q8—88no—
555Q8—88no—
556D4—216no—
557D4—24no—
558D4—28no—
559D4—42no—
560D4—42no—
561D4—84no—
562D4#53618yes—
563C2×C4—22no—
564C2×C4—28no—
565C2×C4—22no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—24no—
571C2×C4—24no—
572C2×C4—28no—
573C2×C4—24no—
574C2×C4—24no—
575C2×C4—24no—
576C2×C4—28no—
577C2×C4—24no—
578C2×C4—24no—
579C2×C4—24no—
580C2×C4—24no—
581C2×C4—416no—
582C2×C4—48no—
583C2×C4—48no—
584C2×C4—48no—
585C2×C4—416no—
586C2×C4—44no—
587C2×C4—44no—
588C2×C4—44no—
589C2×C4—416no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—44no—
594C2×C4—42no—
595C2×C4—44no—
596C2×C4—42no—
597C2×C4—44no—
598C2×C4—41no—
599C2×C4—41no—
600C2×C4—832no—
601C2×C4—84no—
602C2×C4C2×#12012yes—
603C2×C4C2×#12012yes—
604C2×C4C2×#12012yes—
605C2×C4C2×#12012yes—
606C2×C4C2×#12012yes—
607C2×C4#57514yes—
608C2×C4C23×D411yescharacteristic
609C2×C4C23×D411yescharacteristic
610C2×C4C2×2−1+414yes—
611C8—42no—
612C8—42no—
613C8—44no—
614C22—24no—
615C22—24no—
616C22—22no—
617C22—22no—
618C22—28no—
619C22—432no—
620C22—48no—
621C22—48no—
622C22—44no—
623C22—44no—
624C22—82no—
625C22—82no—
626C22—81no—
627C22—81no—
628C22—81no—
629C22#301214yes—
630C22C22×#12011yescharacteristic
631C22#348612yes—
632C22C2×#57512yes—
633C22C2×#57611yescharacteristic
634C22C2×#57711yescharacteristic
635C4—22no—
636C4—22no—
637C4—22no—
638C4—22no—
639C4—44no—
640C4—44no—
641C4—44no—
642C4—48no—
643C4—84no—
644C4—81no—
645C4—81no—
646C4—81no—
647C4#300514yes—
648C4C2×#53612yes—
649C4#348612yes—
650C2—24no—
651C2—48no—
652C2—161no—
653C2#3524211yescharacteristic
654C2C2×#300511yescharacteristic
655C2C2×#301211yescharacteristic
656C2#6183512yes—
657C2#6184112yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#447C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C4×Q8 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C22×#554C2semidirect
2. C2×#3238C2semidirect
4. #35247C2semidirect
5. #33007C2semidirect
7. #29643C2semidirect
8. #29640C2semidirect
9. #25940C2semidirect
10. #25845C2semidirect
11. #24331C2semidirect
12. #24327C2semidirect
13. #21833C2semidirect
14. #21820C2semidirect
15. #20738C2semidirect
32. C2×#501C22semidirect
33. C2×#489C22semidirect
39. C2×#477C22semidirect
41. C2×#476C22semidirect
52. #2592C22semidirect
53. #2584C22semidirect
54. #2582C22semidirect
55. #2538C22semidirect
61. #2059C22semidirect
62. #2059C22semidirect
63. #2026C22semidirect
64. #2025C22semidirect
65. #1990C22semidirect
66. #1988C22semidirect
67. #1985C22semidirect
68. #1984C22semidirect

Maximal subgroups

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

subgroupindex
C22×#5542
C2×#32382
C2×#31052
#352472
#330072
#330022
#296432
#296402
#259402
#258452
#243312
#243272
#218332
#218202
#207382