Finite Group Databasebeta

← #6489663

Subgroups — #6489663

Order 512.

3296 conjugacy classes of subgroups.

The subgroup lattice has 681 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#554C211yesmaximal, characteristic, complemented
2C2×#3234C211yesmaximal, characteristic, complemented
3C2×#3100C211yesmaximal, characteristic
4#35249C211yesmaximal, characteristic, complemented
5#33004C212yesmaximal, complemented
6#33002C212yesmaximal, complemented
7#29640C212yesmaximal, complemented
8#29638C212yesmaximal, complemented
9#25946C214yesmaximal, complemented
10#25850C214yesmaximal, complemented
11#24335C218yesmaximal, complemented
12#21836C211yesmaximal, characteristic, complemented
13#21820C211yesmaximal, characteristic, complemented
14#20739C211yesmaximal, characteristic, complemented
15C2×#554—28no—
16C2×#554—22no—
17C2×#554—22no—
18C2×#554—28no—
19C2×#554C2214yes—
20C2×#554C2214yes—
21C2×#532C2211yescharacteristic
22C22×C4⋊Q8C2211yescharacteristic
23C22×#125C2211yescharacteristic
24C22×#125C2211yescharacteristic
25C22×C22⋊Q8—21no—
26C22×C22⋊Q8—21no—
27C22×C22⋊Q8—21no—
28C22×C4×Q8C2211yescharacteristic
29C22×C4×D4C2211yescharacteristic
30C22×C4×D4C2211yescharacteristic
31C22×C4×D4C2211yescharacteristic
32C23×C4⋊C4—21no—
33#3234—28no—
34C2×#501C2212yescomplemented
35C2×#490C2212yescomplemented
36#3100—28no—
37C2×#481C2212yes—
38C2×#480C2212yes—
39C2×#480C2212yes—
40C2×#478C2214yescomplemented
41C2×#474C2212yes—
42C2×#459C2212yes—
43C2×#447C2212yes—
44C2×C4×D8C2214yes—
45C2×#432C2211yescharacteristic
46C2×#431C2211yescharacteristic
47#2854C2212yes—
48#2854C2212yes—
49#2632—22no—
50#2580C2214yescomplemented
51#2567C2212yescomplemented
52#2548C2212yescomplemented
53#2501—21no—
54#2436—21no—
55#2406—21no—
56#2406—21no—
57#2326—22no—
58#2283C2211yescharacteristic
59#2059C2218yescomplemented
60#2057C2218yescomplemented
61#2025C2218yescomplemented
62#2023C2218yescomplemented
63#1991C2214yescomplemented
64#1989C2214yescomplemented
65#1985C2214yescomplemented
66#1984C2214yescomplemented
67#1879C2218yes—
68#1879C2218yes—
69#1864C2218yes—
70#1862C2218yes—
71#1824C2212yes—
72#1823C2212yes—
73#1823C2212yes—
74#1817C2212yes—
75#1817C2212yes—
76#1804C2214yes—
77#1786C2212yes—
78#1762C2214yes—
79C22×#41—21no—
80C23×Q8—21no—
81C23×Q8D411yescharacteristic
82C23×D4C2311yescharacteristic
83C24×C4—21no—
84C24×C4—21no—
85C24×C4—21no—
86C22×D8C2314yes—
87#554—232no—
88#554—232no—
89#554—216no—
90#554—216no—
91#554—216no—
92#554—216no—
93#532—24no—
94#532—24no—
95C2×C4⋊Q8—24no—
96C2×C4⋊Q8—24no—
97C2×C4⋊Q8—22no—
98C2×C4⋊Q8C2311yescharacteristic
99C2×C4⋊Q8C2314yes—
100C2×#127C2311yescharacteristic
101C2×#125—24no—
102C2×#125—24no—
103C2×#125—22no—
104C2×#125—21no—
105C2×#125—21no—
106C2×#125—24no—
107C2×#125D418yes—
108C2×#125C2314yes—
109C2×#125C2312yes—
110C2×#125C2312yes—
111C2×#124—22no—
112C2×#124—21no—
113C2×#123—22no—
114C2×#123—22no—
115C2×C22⋊Q8—216no—
116C2×C22⋊Q8—216no—
117C2×C22⋊Q8—216no—
118C2×C22⋊Q8—24no—
119C2×C22⋊Q8—24no—
120C2×C22⋊Q8—24no—
121C2×C22⋊Q8—24no—
122C2×C22⋊Q8—24no—
123C2×C22⋊Q8—24no—
124C2×#121—21no—
125C2×#121C2312yes—
126C2×C4×Q8—24no—
127C2×C4×Q8—24no—
128C2×C4×Q8D412yes—
129C2×C4×Q8D412yes—
130C2×C4×Q8C2312yes—
131C2×C4×Q8C2312yes—
132C2×C4×D4—24no—
133C2×C4×D4—24no—
134C2×C4×D4—24no—
135C2×C4×D4—24no—
136C2×C4×D4—22no—
137C2×C4×D4—22no—
138C2×C4×D4—21no—
139C2×C4×D4—21no—
140C2×C4×D4—24no—
141C2×C4×D4—21no—
142C2×C4×D4D418yes—
143C2×C4×D4C2312yes—
144C2×C4×D4C2314yes—
145C2×C4×D4C2314yes—
146C2×C4×D4C2312yes—
147C2×C4×D4C2312yes—
148C22×C4⋊C4—216no—
149C22×C4⋊C4—21no—
150C22×C4⋊C4—21no—
151C22×C4⋊C4—21no—
152C22×C4⋊C4—21no—
153C22×C4⋊C4—21no—
154C22×C4⋊C4—24no—
155C22×C4⋊C4—24no—
156C22×C4⋊C4D411yescharacteristic
157C22×C4⋊C4D411yescharacteristic
158C22×C4⋊C4D411yescharacteristic
159C22×C4⋊C4C2311yescharacteristic
160C22×C4⋊C4C2311yescharacteristic
161C22×C4⋊C4C2311yescharacteristic
162C22×C22⋊C4—21no—
163C22×C22⋊C4—21no—
164C22×C22⋊C4—21no—
165C22×C42C2311yescharacteristic
166C22×C42C2311yescharacteristic
167C22×C42C2311yescharacteristic
168#501—28no—
169#490—28no—
170#481—28no—
171#480—28no—
172#480—28no—
173#478—216no—
174#474—28no—
175#459—28no—
176#447—28no—
177C4×D8—216no—
178#432—24no—
179#431—24no—
180C2×#107C2314yes—
181C2×#107C2314yes—
182C2×#106C2314yes—
183C2×C4⋊C8C2312yes—
184C2×C4⋊C8C2312yes—
185C2×#102C2314yes—
186C2×#102C2314yes—
187C2×#102C2318yes—
188C2×#102C2314yes—
189C2×#102C2314yes—
190C2×C22⋊C8C2312yes—
191C2×C22⋊C8C2312yes—
192C2×#97C2312yes—
193C2×C4×C8C2312yes—
194#399—24no—
195#395—22no—
196#395—22no—
197#394—22no—
198#394—22no—
199#393—24no—
200#391—22no—
201#390—22no—
202#390C2314yes—
203#390C2314yes—
204#389C2314yes—
205#389C2314yes—
206#388—24no—
207#388—24no—
208#386—22no—
209#386—22no—
210#385—24no—
211#385—24no—
212#384—22no—
213#383—21no—
214#382C2312yes—
215#381—21no—
216#380C2312yes—
217C4×C4⋊C4C2312yes—
218C4×C22⋊C4—21no—
219#376—21no—
220#340C2318yes—
221#340C2318yes—
222#340C2318yes—
223#336C2318yes—
224#336C2318yes—
225C25—21no—
226C2×#41—44no—
227C2×#41—84no—
228C22×Q8—28no—
229C22×Q8—24no—
230C22×Q8—24no—
231C22×Q8—21no—
232C22×Q8—44no—
233C22×Q8—84no—
234C22×Q8C2×D412yes—
235C22×Q8C2×D412yes—
236C22×D4—28no—
237C22×D4—22no—
238C22×D4—21no—
239C22×D4—22no—
240C22×D4—21no—
241C22×D4—21no—
242C22×D4C2×D414yes—
243C22×D4C2412yes—
244C22×D4C2412yes—
245C23×C4—28no—
246C23×C4—21no—
247C23×C4—21no—
248C23×C4—21no—
249C23×C4—28no—
250C23×C4—24no—
251C23×C4—28no—
252C23×C4—24no—
253C23×C4—22no—
254C23×C4—22no—
255C23×C4—22no—
256C23×C4—48no—
257C23×C4—84no—
258C23×C4—84no—
259C23×C4C2×D411yescharacteristic
260C23×C4C2×D411yescharacteristic
261C23×C4C2×D411yescharacteristic
262C23×C4C2×D411yescharacteristic
263C23×C4C2×D411yescharacteristic
264C23×C4C2×D411yescharacteristic
265C23×C4C2411yescharacteristic
266C2×D8—28no—
267C2×D8—216no—
268C22×C8C2414yes—
269C22×C8C2414yes—
270C4⋊Q8—216no—
271C4⋊Q8—28no—
272C4⋊Q8—28no—
273C4⋊Q8—22no—
274C4⋊Q8—22no—
275#127—22no—
276#127—22no—
277#125—216no—
278#125—28no—
279#125—28no—
280#125—28no—
281#125—28no—
282#125—24no—
283#125—24no—
284#125C2×D418yes—
285#125C2×D418yes—
286#124—48no—
287#124—44no—
288#123—84no—
289#123—84no—
290C22⋊Q8—232no—
291C22⋊Q8—232no—
292C22⋊Q8—232no—
293C22⋊Q8—232no—
294C22⋊Q8—232no—
295C22⋊Q8—232no—
296#121—24no—
297#121—24no—
298#121—44no—
299C4×Q8—28no—
300C4×Q8—28no—
301C4×Q8—24no—
302C4×Q8—24no—
303C4×Q8—24no—
304C4×Q8—24no—
305C4×D4—216no—
306C4×D4—216no—
307C4×D4—28no—
308C4×D4—28no—
309C4×D4—28no—
310C4×D4—28no—
311C4×D4—28no—
312C4×D4—28no—
313C4×D4—24no—
314C4×D4—24no—
315C4×D4—24no—
316C4×D4—24no—
317C4×D4—44no—
318C4×D4C2×D418yes—
319C4×D4C2×D418yes—
320C2×C4⋊C4—28no—
321C2×C4⋊C4—28no—
322C2×C4⋊C4—232no—
323C2×C4⋊C4—232no—
324C2×C4⋊C4—28no—
325C2×C4⋊C4—24no—
326C2×C4⋊C4—28no—
327C2×C4⋊C4—28no—
328C2×C4⋊C4—24no—
329C2×C4⋊C4—24no—
330C2×C4⋊C4—24no—
331C2×C4⋊C4—22no—
332C2×C4⋊C4—22no—
333C2×C4⋊C4—24no—
334C2×C4⋊C4—24no—
335C2×C4⋊C4—22no—
336C2×C4⋊C4—22no—
337C2×C4⋊C4—21no—
338C2×C4⋊C4—22no—
339C2×C4⋊C4—21no—
340C2×C4⋊C4—21no—
341C2×C4⋊C4—22no—
342C2×C4⋊C4—22no—
343C2×C4⋊C4—21no—
344C2×C4⋊C4—24no—
345C2×C4⋊C4—24no—
346C2×C4⋊C4C2×D412yes—
347C2×C4⋊C4C2×D412yes—
348C2×C4⋊C4C2×D412yes—
349C2×C4⋊C4C2×D412yes—
350C2×C4⋊C4C2×D412yes—
351C2×C4⋊C4C2×D412yes—
352C2×C4⋊C4C2×D414yes—
353C2×C4⋊C4C2×D414yes—
354C2×C4⋊C4C2×D414yes—
355C2×C4⋊C4C2412yes—
356C2×C4⋊C4C2414yes—
357C2×C4⋊C4C2412yes—
358C2×C4⋊C4C2412yes—
359C2×C4⋊C4C2412yes—
360C2×C22⋊C4—28no—
361C2×C22⋊C4—28no—
362C2×C22⋊C4—28no—
363C2×C22⋊C4—24no—
364C2×C22⋊C4—24no—
365C2×C22⋊C4—22no—
366C2×C22⋊C4—22no—
367C2×C22⋊C4—22no—
368C2×C22⋊C4—22no—
369C2×C22⋊C4—22no—
370C2×C22⋊C4—24no—
371C2×C22⋊C4—24no—
372C2×C22⋊C4—24no—
373C2×C22⋊C4—24no—
374C2×C42—22no—
375C2×C42—22no—
376C2×C42—22no—
377C2×C42—22no—
378C2×C42—22no—
379C2×C42C2×D412yes—
380C2×C42C2×D412yes—
381C2×C42C2×D411yescharacteristic
382C2×C42C2×D411yescharacteristic
383C2×C42C2×D414yes—
384C2×C42C2×D414yes—
385C2×C42C2412yes—
386C2×C42C2412yes—
387C2×C42C2411yescharacteristic
388C2×C42C2411yescharacteristic
389#107—28no—
390#107—28no—
391#106—28no—
392C4⋊C8—24no—
393C4⋊C8—24no—
394#102—28no—
395#102—216no—
396#102—28no—
397#102—28no—
398#102—28no—
399C22⋊C8—24no—
400C22⋊C8—24no—
401#97—24no—
402C4×C8—24no—
403#95—22no—
404#95—22no—
405#95—22no—
406#95—22no—
407#95—24no—
408#95—24no—
409C24—28no—
410C24—24no—
411C24—22no—
412C24—22no—
413C24—44no—
414C24—44no—
415C24C22×D411yescharacteristic
416#41—816no—
417C2×Q8—216no—
418C2×Q8—24no—
419C2×Q8—24no—
420C2×Q8—24no—
421C2×Q8—24no—
422C2×Q8—42no—
423C2×Q8—42no—
424C2×Q8—416no—
425C2×Q8—42no—
426C2×Q8—88no—
427C2×Q8—82no—
428C2×Q8—816no—
429C2×D4—28no—
430C2×D4—216no—
431C2×D4—216no—
432C2×D4—24no—
433C2×D4—22no—
434C2×D4—22no—
435C2×D4—22no—
436C2×D4—24no—
437C2×D4—28no—
438C2×D4—44no—
439C2×D4—44no—
440C2×D4—44no—
441C2×D4—44no—
442C2×D4—42no—
443C2×D4—42no—
444C2×D4—82no—
445C2×D4C22×D414yes—
446C2×D42−1+418yes—
447C22×C4—22no—
448C22×C4—24no—
449C22×C4—24no—
450C22×C4—24no—
451C22×C4—216no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—24no—
456C22×C4—24no—
457C22×C4—24no—
458C22×C4—24no—
459C22×C4—24no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—216no—
464C22×C4—28no—
465C22×C4—28no—
466C22×C4—24no—
467C22×C4—22no—
468C22×C4—42no—
469C22×C4—42no—
470C22×C4—432no—
471C22×C4—48no—
472C22×C4—44no—
473C22×C4—44no—
474C22×C4—44no—
475C22×C4—48no—
476C22×C4—44no—
477C22×C4—816no—
478C22×C4—84no—
479C22×C4—84no—
480C22×C4—88no—
481C22×C4—82no—
482C22×C4—82no—
483C22×C4—816no—
484C22×C4#12014yes—
485C22×C4#12014yes—
486C22×C4C22×D412yes—
487C22×C4C22×D412yes—
488C22×C4C22×D412yes—
489C22×C4C22×D411yescharacteristic
490C22×C4C22×D411yescharacteristic
491C22×C4C22×D411yescharacteristic
492C22×C4C22×D411yescharacteristic
493C22×C4C22×D411yescharacteristic
494C22×C4C22×D412yes—
495C22×C4C22×D412yes—
496C22×C4C22×D412yes—
497C22×C4C2511yescharacteristic
498D8—416no—
499C2×C8—24no—
500C2×C8—24no—
501C2×C8—28no—
502C2×C8—28no—
503C4⋊C4—216no—
504C4⋊C4—216no—
505C4⋊C4—216no—
506C4⋊C4—24no—
507C4⋊C4—24no—
508C4⋊C4—28no—
509C4⋊C4—28no—
510C4⋊C4—28no—
511C4⋊C4—28no—
512C4⋊C4—24no—
513C4⋊C4—24no—
514C4⋊C4—24no—
515C4⋊C4—24no—
516C4⋊C4—24no—
517C4⋊C4—24no—
518C4⋊C4—24no—
519C4⋊C4—22no—
520C4⋊C4—22no—
521C4⋊C4—22no—
522C4⋊C4—22no—
523C4⋊C4—22no—
524C4⋊C4—28no—
525C4⋊C4—42no—
526C4⋊C4—84no—
527C4⋊C4—84no—
528C4⋊C4C22×D414yes—
529C4⋊C4C22×D414yes—
530C4⋊C4C22×D414yes—
531C22⋊C4—216no—
532C22⋊C4—216no—
533C22⋊C4—28no—
534C22⋊C4—28no—
535C22⋊C4—48no—
536C22⋊C4—44no—
537C22⋊C4—44no—
538C22⋊C4—44no—
539C22⋊C4—48no—
540C22⋊C4—84no—
541C22⋊C4—84no—
542C42—24no—
543C42—24no—
544C42—22no—
545C42—22no—
546C42—22no—
547C42—22no—
548C42—44no—
549C42#12014yes—
550C42#12014yes—
551C42C22×D412yes—
552C42C22×D412yes—
553C42C22×D412yes—
554C42C22×D412yes—
555C23—22no—
556C23—22no—
557C23—216no—
558C23—24no—
559C23—28no—
560C23—48no—
561C23—432no—
562C23—48no—
563C23—44no—
564C23—44no—
565C23—42no—
566C23—44no—
567C23—44no—
568C23—42no—
569C23—42no—
570C23C2×#12012yes—
571C23#57614yes—
572C23C23×D411yescharacteristic
573Q8—48no—
574Q8—48no—
575Q8—84no—
576Q8—816no—
577D4—216no—
578D4—24no—
579D4—28no—
580D4—44no—
581D4—44no—
582D4—84no—
583D4—84no—
584D4—88no—
585D4#53618yes—
586C2×C4—28no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—28no—
590C2×C4—22no—
591C2×C4—22no—
592C2×C4—22no—
593C2×C4—22no—
594C2×C4—22no—
595C2×C4—24no—
596C2×C4—24no—
597C2×C4—22no—
598C2×C4—22no—
599C2×C4—22no—
600C2×C4—24no—
601C2×C4—24no—
602C2×C4—24no—
603C2×C4—22no—
604C2×C4—22no—
605C2×C4—24no—
606C2×C4—24no—
607C2×C4—44no—
608C2×C4—432no—
609C2×C4—48no—
610C2×C4—44no—
611C2×C4—44no—
612C2×C4—44no—
613C2×C4—48no—
614C2×C4—48no—
615C2×C4—44no—
616C2×C4—44no—
617C2×C4—44no—
618C2×C4—42no—
619C2×C4—44no—
620C2×C4—44no—
621C2×C4—44no—
622C2×C4—42no—
623C2×C4—42no—
624C2×C4—88no—
625C2×C4—88no—
626C2×C4—88no—
627C2×C4—816no—
628C2×C4C2×#12012yes—
629C2×C4C2×#12012yes—
630C2×C4C2×#12012yes—
631C2×C4C2×#12012yes—
632C2×C4C2×#12012yes—
633C2×C4#57514yes—
634C2×C4C23×D411yescharacteristic
635C2×C4C23×D411yescharacteristic
636C2×C4C2×2−1+414yes—
637C8—44no—
638C8—44no—
639C22—28no—
640C22—22no—
641C22—22no—
642C22—28no—
643C22—432no—
644C22—416no—
645C22—44no—
646C22—44no—
647C22—84no—
648C22—84no—
649C22—82no—
650C22—82no—
651C22—82no—
652C22#300814yes—
653C22C22×#12011yescharacteristic
654C22#348612yes—
655C22C2×#57512yes—
656C22C2×#57612yes—
657C4—22no—
658C4—24no—
659C4—22no—
660C4—22no—
661C4—22no—
662C4—22no—
663C4—42no—
664C4—44no—
665C4—42no—
666C4—42no—
667C4—44no—
668C4—44no—
669C4—84no—
670C4—82no—
671C4C2×#53612yes—
672C4#348612yes—
673C2—24no—
674C2—48no—
675C2—162no—
676C2#3524211yescharacteristic
677C2#6035911yescharacteristic
678C2C2×#300811yescharacteristic
679C2#6183514yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#3100C2
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×#3234C2semidirect
4. #35249C2semidirect
5. #33004C2semidirect
6. #33002C2semidirect
7. #29640C2semidirect
8. #29638C2semidirect
9. #25946C2semidirect
10. #25850C2semidirect
11. #24335C2semidirect
12. #21836C2semidirect
13. #21820C2semidirect
14. #20739C2semidirect
34. C2×#501C22semidirect
35. C2×#490C22semidirect
40. C2×#478C22semidirect
50. #2580C22semidirect
51. #2567C22semidirect
52. #2548C22semidirect
59. #2059C22semidirect
60. #2057C22semidirect
61. #2025C22semidirect
62. #2023C22semidirect
63. #1991C22semidirect
64. #1989C22semidirect
65. #1985C22semidirect
66. #1984C22semidirect

Maximal subgroups

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

subgroupindex
C22×#5542
C2×#32342
C2×#31002
#352492
#330042
#330022
#296402
#296382
#259462
#258502
#243352
#218362
#218202
#207392