Finite Group Databasebeta

← #7537760

Subgroups — #7537760

Order 512.

1939 conjugacy classes of subgroups.

The subgroup lattice has 680 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3393C212yesmaximal, complemented
2#59821C211yesmaximal, characteristic, complemented
3#36188C211yesmaximal, characteristic, complemented
4#30738C214yesmaximal, complemented
5#28460C212yesmaximal
6#28440C211yesmaximal, characteristic, complemented
7#28438C211yesmaximal, characteristic, complemented
8#26591C214yesmaximal, complemented
9#25632C212yesmaximal, complemented
10#25369C212yesmaximal, complemented
11#25293C212yesmaximal, complemented
12#24729C212yesmaximal, complemented
13#24185C212yesmaximal, complemented
14#23149C211yesmaximal, characteristic, complemented
15#22998C212yesmaximal, complemented
16C4×#461C211yesmaximal, characteristic, complemented
17C4×#459C211yesmaximal, characteristic
18C2×#556C2212yes—
19C2×#556C2212yes—
20#3393—216no—
21C2×#495C2214yes—
22C2×#494C2212yes—
23C2×#492C2212yes—
24C2×#464C2214yes—
25C2×#464C2214yes—
26C2×#462C2212yes—
27C2×#462C2212yes—
28C2×#461C2212yes—
29C2×#461C2212yes—
30C2×#461C2211yescharacteristic
31C2×#459C2211yescharacteristic
32C2×C8×Q8C2212yes—
33C2×C4×Q16C2214yes—
34C2×C4×SD16C2212yes—
35C2×C4×D8C2212yes—
36#2868C2211yescharacteristic
37#2857—21no—
38#2727—22no—
39#2568C2212yes—
40#2548C2212yes—
41#2547C2212yes—
42#2547C2212yes—
43#2542—21no—
44#2538C2212yes—
45#2512—22no—
46#2499—22no—
47#2496—22no—
48#2496—22no—
49#2448C2214yes—
50#2415—22no—
51#2365C2212yes—
52#2365C2212yes—
53#2354—21no—
54Q8×C4⋊C4C2212yes—
55C4×C4⋊Q8C2211yescharacteristic
56C4×C4⋊Q8C2211yescharacteristic
57C4×#127C2211yescharacteristic
58#2285—21no—
59C42×D4C2211yescharacteristic
60#2066C2214yes—
61#2055C2214yes—
62#2042C2214yes—
63#2016C2214yes—
64#2015C2214yes—
65#2015C2214yes—
66#2013C2214yes—
67#2012C2214yes—
68#1984C2212yes—
69#1982C2211yescharacteristic
70#1981C2211yescharacteristic
71#1968C2214yes—
72#1956C2212yes—
73#1956C2212yes—
74#1948C2211yescharacteristic
75#1947C2211yescharacteristic
76#1939C2212yes—
77#1933C2214yes—
78#1927C2212yes—
79#1925C2212yes—
80#1916C2214yes—
81#1880C2214yes—
82#1879C2212yes—
83#1878C2214yes—
84#1878C2214yes—
85#1877C2212yes—
86#1863C2214yes—
87#1802C2211yescharacteristic
88#1802C2211yescharacteristic
89C8×C4⋊C4C2212yes—
90C4×C4⋊C8C2211yescharacteristic
91C4×C4⋊C8C2211yescharacteristic
92C4×#103C2212yes—
93C4×#102C2212yes—
94C22×Q16C2314yes—
95C22×Q16C2314yes—
96C22×SD16C2314yes—
97C22×SD16C2314yes—
98C22×SD16C2312yes—
99C22×D8C2312yes—
100#556—28no—
101#556—28no—
102#556—44no—
103#556—44no—
104C2×C4⋊Q8D411yescharacteristic
105C2×C4⋊Q8C2312yes—
106C2×C4⋊Q8C2312yes—
107C2×C4⋊Q8C2311yescharacteristic
108C2×C4⋊Q8C2312yes—
109C2×#127C2311yescharacteristic
110C2×#126—24no—
111C2×#126—24no—
112C2×#124C2314yes—
113C2×#124C2314yes—
114C2×C22⋊Q8—22no—
115C2×C22⋊Q8—22no—
116C2×C22⋊Q8—22no—
117C2×C4×Q8C2312yes—
118C2×C4×Q8C2314yes—
119C2×C4×Q8C2314yes—
120C2×C4×Q8C2312yes—
121C2×C4×D4—21no—
122C2×C4×D4—21no—
123C2×C4×D4C2312yes—
124C2×C4×D4C2311yescharacteristic
125C2×C4×D4C2311yescharacteristic
126C2×C4×D4C2311yescharacteristic
127C2×#117—22no—
128C2×#117—22no—
129C22×C42—21no—
130#495—216no—
131#494—28no—
132#492—44no—
133#464—216no—
134#464—216no—
135#462—28no—
136#462—44no—
137#461—28no—
138#461—44no—
139#461D418yescomplemented
140#459—24no—
141C8×Q8—28no—
142C4×Q16—216no—
143C4×SD16—44no—
144C4×D8—28no—
145C2×#107C2312yes—
146C2×#106C2312yes—
147C2×C4⋊C8C2314yes—
148C2×C4⋊C8C2311yescharacteristic
149C2×C4⋊C8C2311yescharacteristic
150C2×C4⋊C8C2311yescharacteristic
151C2×C4⋊C8C2311yescharacteristic
152C2×#103C2314yes—
153C2×#103C2314yes—
154C2×#103C2314yes—
155C2×#103C2314yes—
156C2×#103C2312yes—
157C2×#103C2312yes—
158C2×#103C2312yes—
159C2×#102C2312yes—
160C2×#102C2312yes—
161C2×#102C2312yes—
162C2×#102C2312yes—
163C2×#102C2312yes—
164C2×C4×C8C2312yes—
165C2×C4×C8C2312yes—
166C2×C4×C8C2314yes—
167#400—24no—
168#399—24no—
169#397—24no—
170#397—24no—
171#397—24no—
172#396—24no—
173#394—24no—
174#391—22no—
175#391D412yes—
176#391C2312yes—
177#391C2314yes—
178#391C2314yes—
179#390—22no—
180#390—22no—
181#390—21no—
182#390—21no—
183#390C2312yes—
184#390C2314yes—
185#389—22no—
186#389—22no—
187#389—21no—
188#389—21no—
189#389—24no—
190#389D412yes—
191#389D412yes—
192#389C2314yes—
193#388—24no—
194#388—24no—
195#387—24no—
196#387—24no—
197#384—22no—
198#384—21no—
199#384—21no—
200#384—21no—
201#384—21no—
202#384D411yescharacteristic
203#384C2311yescharacteristic
204#384C2311yescharacteristic
205#382—22no—
206#380—22no—
207#380—22no—
208#380—22no—
209#380—22no—
210#380—22no—
211C4×C4⋊C4—22no—
212C4×C4⋊C4—22no—
213C4×C4⋊C4—22no—
214C4×C4⋊C4—21no—
215C4×C4⋊C4—21no—
216C4×C4⋊C4C2311yescharacteristic
217C4×C4⋊C4C2311yescharacteristic
218C4×C4⋊C4C2311yescharacteristic
219C4×C4⋊C4C2312yes—
220C4×C4⋊C4C2312yes—
221C4×C22⋊C4—21no—
222C4×C22⋊C4—21no—
223C4×C22⋊C4—21no—
224C43C2311yescharacteristic
225#340C2312yes—
226#340C2312yes—
227#340C2312yes—
228#340C2312yes—
229#336C2314yes—
230#336C2312yes—
231#336C2314yes—
232#336C2312yes—
233C22×Q8C2×D412yes—
234C22×Q8C2414yes—
235C22×Q8C2412yes—
236C22×D4—21no—
237C22×D4—21no—
238C22×D4C2412yes—
239C22×D4C2411yescharacteristic
240C23×C4—21no—
241C23×C4—22no—
242C23×C4—21no—
243C23×C4—21no—
244C2×Q16—28no—
245C2×Q16—28no—
246C2×Q16—28no—
247C2×Q16—44no—
248C2×Q16—44no—
249C2×Q16—44no—
250C2×SD16—28no—
251C2×SD16—42no—
252C2×SD16—42no—
253C2×SD16—44no—
254C2×SD16—44no—
255C2×SD16—44no—
256C2×SD16—44no—
257C2×SD16—44no—
258C2×SD16C2×D418yes—
259C2×D8—24no—
260C2×D8—24no—
261C2×D8—42no—
262C22×C8C2412yes—
263C22×C8C2412yes—
264C22×C8C2412yes—
265C22×C8C2412yes—
266C4⋊Q8—24no—
267C4⋊Q8—24no—
268C4⋊Q8—24no—
269C4⋊Q8—22no—
270C4⋊Q8—24no—
271C4⋊Q8—24no—
272C4⋊Q8—42no—
273C4⋊Q8—42no—
274C4⋊Q8C2×D414yes—
275#127—22no—
276#127—22no—
277#126—416no—
278#126—416no—
279#124—28no—
280#124—28no—
281#124—44no—
282#124—44no—
283C22⋊Q8—48no—
284C22⋊Q8—48no—
285C22⋊Q8—84no—
286C4×Q8—24no—
287C4×Q8—24no—
288C4×Q8—28no—
289C4×Q8—28no—
290C4×Q8—24no—
291C4×Q8—24no—
292C4×Q8—44no—
293C4×Q8—44no—
294C4×D4—24no—
295C4×D4—22no—
296C4×D4—22no—
297C4×D4—24no—
298C4×D4—22no—
299C4×D4—41no—
300C4×D4—41no—
301C4×D4—44no—
302C4×D4—82no—
303C4×D4C2×D414yes—
304#117—48no—
305#117—48no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—21no—
310C2×C4⋊C4—24no—
311C2×C4⋊C4—24no—
312C2×C4⋊C4—22no—
313C2×C4⋊C4—21no—
314C2×C4⋊C4—22no—
315C2×C4⋊C4—22no—
316C2×C4⋊C4—22no—
317C2×C4⋊C4—22no—
318C2×C4⋊C4—22no—
319C2×C4⋊C4—21no—
320C2×C4⋊C4—24no—
321C2×C4⋊C4—21no—
322C2×C4⋊C4C2×D412yes—
323C2×C4⋊C4C2×D414yes—
324C2×C4⋊C4C2×D412yes—
325C2×C4⋊C4C2×D412yes—
326C2×C4⋊C4C2×D412yes—
327C2×C4⋊C4C2×D412yes—
328C2×C4⋊C4C2412yes—
329C2×C4⋊C4C2411yescharacteristic
330C2×C4⋊C4C2411yescharacteristic
331C2×C4⋊C4C2412yes—
332C2×C4⋊C4C2411yescharacteristic
333C2×C22⋊C4—21no—
334C2×C22⋊C4—24no—
335C2×C22⋊C4—24no—
336C2×C22⋊C4—22no—
337C2×C22⋊C4—22no—
338C2×C22⋊C4—21no—
339C2×C22⋊C4—21no—
340C2×C22⋊C4—24no—
341C2×C22⋊C4—24no—
342C2×C22⋊C4—22no—
343C2×C22⋊C4—24no—
344C2×C22⋊C4—24no—
345C2×C42—24no—
346C2×C42—24no—
347C2×C42—22no—
348C2×C42—21no—
349C2×C42—21no—
350C2×C42—21no—
351C2×C42—44no—
352C2×C42C2×D411yescharacteristic
353C2×C42C2×D411yescharacteristic
354C2×C42C2×D411yescharacteristic
355C2×C42C2×D411yescharacteristic
356C2×C42C2411yescharacteristic
357C2×C42C2414yes—
358C2×C42C2411yescharacteristic
359C2×C42C2411yescharacteristic
360#107—24no—
361#106—42no—
362C4⋊C8—28no—
363C4⋊C8—22no—
364C4⋊C8—22no—
365C4⋊C8C2×D414yes—
366C4⋊C8#4114yes—
367#103—28no—
368#103—28no—
369#103—42no—
370#103—42no—
371#103—44no—
372#103—44no—
373#103C2×D418yes—
374#102—24no—
375#102—24no—
376#102—24no—
377#102—42no—
378#102—42no—
379C4×C8—28no—
380C4×C8—24no—
381C4×C8—24no—
382#95—21no—
383#95—21no—
384#95—21no—
385#95—24no—
386#95—24no—
387#95—24no—
388#95—22no—
389#95—22no—
390#95—24no—
391#95—24no—
392#95C2×D414yes—
393#95C2×D414yes—
394C24—21no—
395C24—22no—
396C2×Q8—24no—
397C2×Q8—24no—
398C2×Q8—24no—
399C2×Q8—22no—
400C2×Q8—24no—
401C2×Q8—22no—
402C2×Q8—24no—
403C2×Q8—22no—
404C2×Q8—24no—
405C2×Q8—24no—
406C2×Q8—22no—
407C2×Q8—22no—
408C2×Q8—24no—
409C2×Q8—24no—
410C2×Q8C22×D414yes—
411C2×D4—22no—
412C2×D4—21no—
413C2×D4—22no—
414C2×D4—21no—
415C2×D4—21no—
416C2×D4—22no—
417C2×D4—22no—
418C2×D4—21no—
419C2×D4—21no—
420C2×D4—22no—
421C2×D4—22no—
422C2×D4—42no—
423C2×D4—42no—
424C2×D4—42no—
425C2×D4—42no—
426C2×D4—81no—
427C2×D4—81no—
428C2×D4C22×D412yes—
429C22×C4—21no—
430C22×C4—22no—
431C22×C4—24no—
432C22×C4—21no—
433C22×C4—21no—
434C22×C4—22no—
435C22×C4—24no—
436C22×C4—22no—
437C22×C4—42no—
438C22×C4—42no—
439C22×C4—42no—
440C22×C4—42no—
441C22×C4—44no—
442C22×C4—42no—
443C22×C4—42no—
444C22×C4—42no—
445C22×C4—44no—
446C22×C4—44no—
447C22×C4—42no—
448C22×C4C22×D411yescharacteristic
449C22×C4C22×D414yes—
450C22×C4C22×D414yes—
451C22×C4C22×D411yescharacteristic
452C22×C4C22×D411yescharacteristic
453C22×C4C22×D411yescharacteristic
454C22×C4C22×D411yescharacteristic
455C22×C4C22×D411yescharacteristic
456C22×C4C2511yescharacteristic
457Q16—416no—
458Q16—88no—
459SD16—48no—
460SD16—88no—
461SD16—88no—
462D8—48no—
463C2×C8—22no—
464C2×C8—22no—
465C2×C8—22no—
466C2×C8—22no—
467C2×C8—22no—
468C2×C8—22no—
469C2×C8—22no—
470C2×C8—22no—
471C2×C8—22no—
472C2×C8—22no—
473C2×C8C22×D414yes—
474C2×C8C2×#4114yes—
475C4⋊C4—24no—
476C4⋊C4—24no—
477C4⋊C4—24no—
478C4⋊C4—28no—
479C4⋊C4—24no—
480C4⋊C4—24no—
481C4⋊C4—24no—
482C4⋊C4—21no—
483C4⋊C4—22no—
484C4⋊C4—21no—
485C4⋊C4—22no—
486C4⋊C4—22no—
487C4⋊C4—21no—
488C4⋊C4—22no—
489C4⋊C4—21no—
490C4⋊C4—21no—
491C4⋊C4—42no—
492C4⋊C4—48no—
493C4⋊C4—48no—
494C4⋊C4—44no—
495C4⋊C4—44no—
496C4⋊C4—44no—
497C4⋊C4—44no—
498C4⋊C4—44no—
499C4⋊C4—44no—
500C4⋊C4—44no—
501C4⋊C4—42no—
502C4⋊C4—42no—
503C4⋊C4—81no—
504C4⋊C4—81no—
505C4⋊C4—84no—
506C4⋊C4C22×D412yes—
507C22⋊C4—24no—
508C22⋊C4—48no—
509C22⋊C4—48no—
510C22⋊C4—48no—
511C22⋊C4—48no—
512C22⋊C4—42no—
513C22⋊C4—44no—
514C22⋊C4—42no—
515C22⋊C4—44no—
516C22⋊C4—84no—
517C22⋊C4—82no—
518C22⋊C4—84no—
519C42—22no—
520C42—22no—
521C42—24no—
522C42—24no—
523C42—21no—
524C42—21no—
525C42—41no—
526C42—48no—
527C42—48no—
528C42—44no—
529C42—42no—
530C42—42no—
531C42—42no—
532C42#13514yes—
533C42C22×D412yes—
534C42C2×#4112yes—
535C42H(5,𝔽2)12yes—
536C42H(5,𝔽2)12yes—
537C23—22no—
538C23—41no—
539C23—41no—
540C23—41no—
541C23—41no—
542C23—41no—
543C23—41no—
544C23—42no—
545C23—42no—
546C23—42no—
547C23—42no—
548C23—42no—
549C23—42no—
550C23—42no—
551C23C23×D411yescharacteristic
552Q8—24no—
553Q8—24no—
554Q8—42no—
555Q8—48no—
556Q8—42no—
557Q8—44no—
558Q8—44no—
559Q8—44no—
560Q8—44no—
561D4—22no—
562D4—22no—
563D4—42no—
564D4—42no—
565D4—41no—
566D4—42no—
567D4—41no—
568D4—42no—
569D4—84no—
570D4—84no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—24no—
578C2×C4—24no—
579C2×C4—24no—
580C2×C4—22no—
581C2×C4—24no—
582C2×C4—21no—
583C2×C4—24no—
584C2×C4—21no—
585C2×C4—24no—
586C2×C4—21no—
587C2×C4—24no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—48no—
593C2×C4—42no—
594C2×C4—41no—
595C2×C4—42no—
596C2×C4—44no—
597C2×C4—42no—
598C2×C4—41no—
599C2×C4—44no—
600C2×C4—41no—
601C2×C4—44no—
602C2×C4—41no—
603C2×C4—41no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—41no—
608C2×C4—84no—
609C2×C4—84no—
610C2×C4—88no—
611C2×C4D4212yes—
612C2×C4D4212yes—
613C2×C4#55012yes—
614C2×C4#55012yes—
615C2×C4C2×#13512yes—
616C2×C4C2×#13512yes—
617C2×C4C2×#13512yes—
618C2×C4#57712yes—
619C2×C4#57812yes—
620C2×C4C23×D411yescharacteristic
621C2×C4C22×#4111yescharacteristic
622C2×C4C2×H(5,𝔽2)11yescharacteristic
623C2×C4C2×H(5,𝔽2)11yescharacteristic
624C2×C4#58511yescharacteristic
625C2×C4#58511yescharacteristic
626C8—24no—
627C8—24no—
628C8—42no—
629C8—42no—
630C22—42no—
631C22—42no—
632C22—42no—
633C22—81no—
634C22—81no—
635C22—81no—
636C22—81no—
637C22—82no—
638C22—82no—
639C22—82no—
640C22—82no—
641C22—82no—
642C22—82no—
643C22—82no—
644C22C2×D4211yescharacteristic
645C22C2×#55011yescharacteristic
646C22#348711yescharacteristic
647C22C22×#13511yescharacteristic
648C22C2×#57711yescharacteristic
649C22C2×#57811yescharacteristic
650C22#358611yescharacteristic
651C4—22no—
652C4—22no—
653C4—22no—
654C4—44no—
655C4—42no—
656C4—44no—
657C4—44no—
658C4—41no—
659C4—82no—
660C4—84no—
661C4—81no—
662C4—81no—
663C4—82no—
664C4#328414yes—
665C4#339314yescomplemented
666C4D4×#4111yescharacteristic
667C4#349611yescharacteristic
668C4#351711yescharacteristic
669C4#354611yescharacteristic
670C2—82no—
671C2—162no—
672C2#5905111yescharacteristic
673C2C2×#328411yescharacteristic
674C2#6109711yescharacteristic
675C2C2×#339311yescharacteristic
676C2#6142011yescharacteristic
677C2#6188211yescharacteristic
678C2#6193411yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#459C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×C4⋊Q8 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#3393C2semidirect
2. #59821C2semidirect
3. #36188C2semidirect
4. #30738C2semidirect
6. #28440C2semidirect
7. #28438C2semidirect
8. #26591C2semidirect
9. #25632C2semidirect
10. #25369C2semidirect
11. #25293C2semidirect
12. #24729C2semidirect
13. #24185C2semidirect
14. #23149C2semidirect
15. #22998C2semidirect
16. C4×#461C2semidirect
139. #461D4semidirect
665. C4#3393semidirect

Maximal subgroups

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

subgroupindex
C2×#33932
#598212
#361882
#307382
#284602
#284402
#284382
#265912
#256322
#253692
#252932
#247292
#241852
#231492
#229982
C4×#4612
C4×#4592