Finite Group Databasebeta

← #6856369

Subgroups — #6856369

Order 512.

1693 conjugacy classes of subgroups.

The subgroup lattice has 701 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36580C211yesmaximal, characteristic
2C2×#2582C211yesmaximal, characteristic, complemented
3#32903C211yesmaximal, characteristic
4#32609C214yesmaximal, complemented
5#32552C214yesmaximal, complemented
6#29719C212yesmaximal, complemented
7#29330C212yesmaximal, complemented
8#29237C212yesmaximal, complemented
9#28187C212yesmaximal, complemented
10#27973C211yesmaximal, characteristic
11#27680C211yesmaximal, characteristic, complemented
12#25295C212yesmaximal, complemented
13#24997C212yesmaximal, complemented
14#24190C212yesmaximal, complemented
15Q8×C22⋊C8C211yesmaximal, characteristic, complemented
16#22041C212yesmaximal, complemented
17C4×#451C211yesmaximal, characteristic, complemented
18C2×#559C2211yescharacteristic
19C22×C4×Q8C2211yescharacteristic
20C22×C4×Q8C2211yescharacteristic
21C2×#460C2212yes—
22C2×#451C2211yescharacteristic
23C2×#448C2212yes—
24C2×C8×Q8C2212yes—
25C2×C4×Q16C2212yes—
26#2856C2211yescharacteristic
27#2850C2212yes—
28#2592C2212yes—
29#2582—21no—
30#2582—21no—
31#2582—22no—
32#2582C2211yescharacteristic
33#2582C2212yes—
34#2582C2214yes—
35#2582C2212yes—
36#2580C2212yes—
37#2548C2211yescharacteristic
38#2458—22no—
39#2448—21no—
40#2443—22no—
41#2436—21no—
42C2×#398—22no—
43#2367—21no—
44#2361—21no—
45#2355C2211yescharacteristic
46C4×C22⋊Q8C2211yescharacteristic
47C4×#121C2212yes—
48C2×#391—21no—
49C2×#391C2212yes—
50C2×#389—21no—
51C2×#389C2212yes—
52C2×#384C2211yescharacteristic
53#2101C2214yes—
54#2099C2212yes—
55#2096C2212yes—
56#2087C2214yes—
57#2074C2214yes—
58#2071C2214yes—
59#2065C2214yes—
60#2056C2214yes—
61#2055C2214yes—
62#2042C2214yes—
63#2015C2214yes—
64#1984C2214yes—
65#1984C2212yes—
66#1968C2214yes—
67#1957C2212yes—
68#1948C2212yes—
69#1934C2214yes—
70#1933C2214yes—
71#1933C2214yes—
72#1927C2214yes—
73#1919C2214yes—
74#1880C2214yes—
75#1878C2212yes—
76#1877C2214yes—
77#1874C2212yes—
78#1874C2212yes—
79#1874C2212yes—
80#1872C2212yes—
81#1872C2211yescharacteristic
82#1863C2212yes—
83#1862C2214yes—
84#1853C2212yes—
85#1852C2211yescharacteristic
86#1834C2212yes—
87#1803C2212yes—
88#1803C2212yes—
89#1797C2212yes—
90#1797C2211yescharacteristic
91#1787C2211yescharacteristic
92C4×#103C2212yes—
93C4×C22⋊C8C2212yes—
94C4×C22⋊C8C2211yescharacteristic
95C23×Q8D411yescharacteristic
96C23×Q8C2311yescharacteristic
97C22×Q16C2312yes—
98C22×SD16C2314yes—
99#559—44no—
100C2×#127C2311yescharacteristic
101C2×#124—21no—
102C2×#124—21no—
103C2×#124C2312yes—
104C2×#123—22no—
105C2×C22⋊Q8C2311yescharacteristic
106C2×#121—21no—
107C2×#121—21no—
108C2×#121C2312yes—
109C2×C4×Q8—21no—
110C2×C4×Q8—21no—
111C2×C4×Q8—44no—
112C2×C4×Q8—44no—
113C2×C4×Q8D411yescharacteristic
114C2×C4×Q8D411yescharacteristic
115C2×C4×Q8D412yes—
116C2×C4×Q8C2311yescharacteristic
117C2×C4×Q8C2312yes—
118C2×C4×Q8C2312yes—
119C2×C4×Q8C2311yescharacteristic
120C2×C4×Q8C2311yescharacteristic
121C2×C4×Q8C2312yes—
122C2×C4×D4—22no—
123C2×C4×D4—21no—
124C2×C4×D4C2312yes—
125C22×C4⋊C4—21no—
126C22×C4⋊C4—21no—
127C22×C4⋊C4—21no—
128C22×C4⋊C4—22no—
129C22×C4⋊C4D411yescharacteristic
130C22×C4⋊C4D412yes—
131C22×C4⋊C4C2311yescharacteristic
132C22×C4⋊C4C2312yes—
133C22×C42—21no—
134C22×C42C2311yescharacteristic
135C22×C42C2312yes—
136#460—44no—
137#451—24no—
138#448—28no—
139C8×Q8—44no—
140C4×Q16—44no—
141C2×#107C2312yes—
142C2×#106C2314yes—
143C2×C4⋊C8C2312yes—
144C2×C4⋊C8C2314yes—
145C2×#103C2312yes—
146C2×#103C2314yes—
147C2×#103C2314yes—
148C2×#103C2312yes—
149C2×#103C2312yes—
150C2×#103C2312yes—
151C2×#102C2314yes—
152C2×#102C2314yes—
153C2×#102C2314yes—
154C2×C22⋊C8C2311yescharacteristic
155C2×C22⋊C8C2312yes—
156C2×C22⋊C8C2311yescharacteristic
157C2×C4×C8C2312yes—
158C2×C4×C8C2314yes—
159#399—22no—
160#398—24no—
161#398—24no—
162#398—24no—
163#398—24no—
164#398—22no—
165#397—22no—
166#394—22no—
167#393—22no—
168#393—22no—
169#391—22no—
170#391—24no—
171#391—22no—
172#391—21no—
173#391—21no—
174#391—22no—
175#391—22no—
176#391C2314yes—
177#391C2312yes—
178#391C2314yes—
179#391C2311yescharacteristic
180#390—21no—
181#390—21no—
182#390—21no—
183#390C2312yes—
184#390C2312yes—
185#390C2312yes—
186#389—22no—
187#389—22no—
188#389—24no—
189#389—21no—
190#389—22no—
191#389—22no—
192#389C2314yes—
193#389C2311yescharacteristic
194#389C2312yes—
195#389C2314yes—
196#388—22no—
197#388—22no—
198#388—22no—
199#388—22no—
200#388—22no—
201#387—22no—
202#387—21no—
203#386—22no—
204#386—22no—
205#386—22no—
206#385—21no—
207#385—22no—
208#385—22no—
209#384—21no—
210#384—21no—
211#384C2311yescharacteristic
212#384C2312yes—
213#384C2312yes—
214#382C2312yes—
215#380C2311yescharacteristic
216C4×C4⋊C4—21no—
217C4×C4⋊C4—21no—
218C4×C4⋊C4C2312yes—
219C4×C4⋊C4C2311yescharacteristic
220C4×C22⋊C4—21no—
221C4×C22⋊C4—22no—
222C2×#95—22no—
223C2×#95—22no—
224C2×#95—22no—
225#340C2314yes—
226#340C2314yes—
227#340C2314yes—
228#340C2314yes—
229#340C2312yes—
230#340C2314yes—
231#340C2312yes—
232#336C2314yes—
233#336C2312yes—
234#336C2312yes—
235#336C2314yes—
236C22×Q8—24no—
237C22×Q8—24no—
238C22×Q8—21no—
239C22×Q8—22no—
240C22×Q8—42no—
241C22×Q8—42no—
242C22×Q8—42no—
243C22×Q8C2×D411yescharacteristic
244C22×Q8C2×D412yes—
245C22×Q8C2×D411yescharacteristic
246C22×Q8#4114yes—
247C22×Q8C2411yescharacteristic
248C22×Q8C2412yes—
249C22×D4—22no—
250C22×D4—21no—
251C22×D4—21no—
252C22×D4C2412yes—
253C23×C4—21no—
254C23×C4—22no—
255C23×C4—21no—
256C23×C4—22no—
257C23×C4C2×D411yescharacteristic
258C23×C4C2×D412yes—
259C23×C4C2×D411yescharacteristic
260C23×C4C2×D412yes—
261C23×C4C2411yescharacteristic
262C2×Q16—24no—
263C2×Q16—42no—
264C2×Q16—42no—
265C2×SD16—28no—
266C2×SD16—44no—
267C2×SD16—44no—
268C22×C8C2414yes—
269C22×C8C2412yes—
270C22×C8C2412yes—
271#127—24no—
272#124—44no—
273#124—44no—
274#124—44no—
275#123—48no—
276C22⋊Q8—22no—
277C22⋊Q8—22no—
278#121—24no—
279#121—24no—
280#121—44no—
281#121—82no—
282C4×Q8—24no—
283C4×Q8—24no—
284C4×Q8—24no—
285C4×Q8—24no—
286C4×Q8—24no—
287C4×Q8—44no—
288C4×Q8—44no—
289C4×Q8—42no—
290C4×Q8—42no—
291C4×Q8—42no—
292C4×Q8—42no—
293C4×Q8—44no—
294C4×Q8—44no—
295C4×D4—44no—
296C4×D4—44no—
297C4×D4—84no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—21no—
305C2×C4⋊C4—21no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—24no—
310C2×C4⋊C4—24no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—21no—
313C2×C4⋊C4—21no—
314C2×C4⋊C4—42no—
315C2×C4⋊C4—44no—
316C2×C4⋊C4—44no—
317C2×C4⋊C4—42no—
318C2×C4⋊C4—44no—
319C2×C4⋊C4—44no—
320C2×C4⋊C4—82no—
321C2×C4⋊C4—84no—
322C2×C4⋊C4C2×D411yescharacteristic
323C2×C4⋊C4C2×D412yes—
324C2×C4⋊C4C2×D411yescharacteristic
325C2×C4⋊C4C2×D412yes—
326C2×C4⋊C4C2×D412yes—
327C2×C4⋊C4C2×D412yes—
328C2×C4⋊C4C2×D412yes—
329C2×C4⋊C4C2412yes—
330C2×C4⋊C4C2414yes—
331C2×C4⋊C4C2411yescharacteristic
332C2×C4⋊C4C2412yes—
333C2×C4⋊C4C2411yescharacteristic
334C2×C4⋊C4C2411yescharacteristic
335C2×C22⋊C4—21no—
336C2×C22⋊C4—21no—
337C2×C22⋊C4—22no—
338C2×C22⋊C4—22no—
339C2×C22⋊C4—22no—
340C2×C22⋊C4—22no—
341C2×C22⋊C4—22no—
342C2×C22⋊C4—22no—
343C2×C22⋊C4—22no—
344C2×C22⋊C4—22no—
345C2×C22⋊C4—22no—
346C2×C22⋊C4—22no—
347C2×C42—24no—
348C2×C42—21no—
349C2×C42—22no—
350C2×C42—22no—
351C2×C42—21no—
352C2×C42—22no—
353C2×C42—44no—
354C2×C42—44no—
355C2×C42C2×D411yescharacteristic
356C2×C42C2×D411yescharacteristic
357C2×C42C2×D412yes—
358C2×C42C2×D412yes—
359C2×C42C2412yes—
360C2×C42C2411yescharacteristic
361C2×C42C2412yes—
362C2×C42C2411yescharacteristic
363#107—42no—
364#106—44no—
365C4⋊C8—24no—
366C4⋊C8—44no—
367#103—28no—
368#103—24no—
369#103—44no—
370#103—42no—
371#103—42no—
372#103—42no—
373#102—44no—
374#102—44no—
375#102—44no—
376C22⋊C8—24no—
377C22⋊C8—22no—
378C22⋊C8—22no—
379C4×C8—24no—
380C4×C8—44no—
381#95—24no—
382#95—24no—
383#95—24no—
384#95—24no—
385#95—24no—
386#95—24no—
387#95—21no—
388#95—21no—
389#95—22no—
390#95—21no—
391#95—22no—
392#95—21no—
393#95—21no—
394#95—22no—
395#95—22no—
396#95—22no—
397#95—22no—
398C24—22no—
399C24C22×D411yescharacteristic
400C2×Q8—22no—
401C2×Q8—22no—
402C2×Q8—24no—
403C2×Q8—22no—
404C2×Q8—24no—
405C2×Q8—22no—
406C2×Q8—24no—
407C2×Q8—22no—
408C2×Q8—22no—
409C2×Q8—21no—
410C2×Q8—22no—
411C2×Q8—22no—
412C2×Q8—22no—
413C2×Q8—21no—
414C2×Q8—22no—
415C2×Q8—22no—
416C2×Q8—22no—
417C2×Q8—42no—
418C2×Q8—48no—
419C2×Q8—416no—
420C2×Q8—42no—
421C2×Q8—42no—
422C2×Q8—48no—
423C2×Q8#13714yes—
424C2×Q8C2×#4114yes—
425C2×D4—22no—
426C2×D4—24no—
427C2×D4—22no—
428C2×D4—24no—
429C2×D4—44no—
430C2×D4—44no—
431C2×D4—44no—
432C2×D4—42no—
433C2×D4—42no—
434C2×D4—42no—
435C2×D4—42no—
436C2×D4—81no—
437C2×D4—81no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—24no—
443C22×C4—24no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—24no—
447C22×C4—22no—
448C22×C4—24no—
449C22×C4—21no—
450C22×C4—21no—
451C22×C4—21no—
452C22×C4—22no—
453C22×C4—24no—
454C22×C4—22no—
455C22×C4—21no—
456C22×C4—42no—
457C22×C4—41no—
458C22×C4—42no—
459C22×C4—42no—
460C22×C4—44no—
461C22×C4—42no—
462C22×C4—42no—
463C22×C4—42no—
464C22×C4—44no—
465C22×C4—44no—
466C22×C4—42no—
467C22×C4—44no—
468C22×C4—42no—
469C22×C4—82no—
470C22×C4—82no—
471C22×C4#13514yes—
472C22×C4C22×D412yes—
473C22×C4C22×D412yes—
474C22×C4C22×D412yes—
475C22×C4C22×D411yescharacteristic
476C22×C4C22×D411yescharacteristic
477C22×C4C22×D411yescharacteristic
478C22×C4C22×D411yescharacteristic
479C22×C4C22×D412yes—
480C22×C4C22×D411yescharacteristic
481C22×C4C2×#4112yes—
482C22×C42−1+412yes—
483C22×C4C2511yescharacteristic
484Q16—48no—
485SD16—88no—
486C2×C8—24no—
487C2×C8—22no—
488C2×C8—22no—
489C2×C8—22no—
490C2×C8—24no—
491C2×C8—22no—
492C2×C8—22no—
493C2×C8—22no—
494C2×C8—24no—
495C4⋊C4—22no—
496C4⋊C4—24no—
497C4⋊C4—22no—
498C4⋊C4—24no—
499C4⋊C4—22no—
500C4⋊C4—21no—
501C4⋊C4—22no—
502C4⋊C4—22no—
503C4⋊C4—22no—
504C4⋊C4—22no—
505C4⋊C4—21no—
506C4⋊C4—24no—
507C4⋊C4—24no—
508C4⋊C4—22no—
509C4⋊C4—42no—
510C4⋊C4—42no—
511C4⋊C4—42no—
512C4⋊C4—44no—
513C4⋊C4—42no—
514C4⋊C4—44no—
515C4⋊C4—44no—
516C4⋊C4—42no—
517C4⋊C4—42no—
518C4⋊C4—44no—
519C4⋊C4—44no—
520C4⋊C4—42no—
521C4⋊C4—81no—
522C4⋊C4—82no—
523C4⋊C4—84no—
524C4⋊C4—82no—
525C4⋊C4—84no—
526C4⋊C4—82no—
527C4⋊C4—81no—
528C22⋊C4—44no—
529C22⋊C4—44no—
530C22⋊C4—44no—
531C22⋊C4—42no—
532C22⋊C4—42no—
533C22⋊C4—44no—
534C22⋊C4—44no—
535C22⋊C4—44no—
536C22⋊C4—44no—
537C22⋊C4—82no—
538C22⋊C4—82no—
539C22⋊C4—82no—
540C42—22no—
541C42—22no—
542C42—24no—
543C42—22no—
544C42—24no—
545C42—42no—
546C42—42no—
547C42—44no—
548C42—44no—
549C42—42no—
550C42—42no—
551C42—82no—
552C42H(5,𝔽2)14yes—
553C23—21no—
554C23—21no—
555C23—21no—
556C23—21no—
557C23—42no—
558C23—42no—
559C23—42no—
560C23—42no—
561C23—42no—
562C23—42no—
563C23—42no—
564C23#54812yes—
565C23C2×#13512yes—
566C23#57712yes—
567C23C23×D411yescharacteristic
568Q8—28no—
569Q8—24no—
570Q8—24no—
571Q8—48no—
572Q8—48no—
573Q8—44no—
574Q8—42no—
575Q8—42no—
576Q8—42no—
577Q8—42no—
578D4—44no—
579D4—44no—
580D4—88no—
581D4—84no—
582D4—162no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—22no—
587C2×C4—21no—
588C2×C4—22no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—28no—
592C2×C4—24no—
593C2×C4—24no—
594C2×C4—22no—
595C2×C4—21no—
596C2×C4—22no—
597C2×C4—21no—
598C2×C4—22no—
599C2×C4—22no—
600C2×C4—22no—
601C2×C4—24no—
602C2×C4—22no—
603C2×C4—22no—
604C2×C4—22no—
605C2×C4—21no—
606C2×C4—22no—
607C2×C4—44no—
608C2×C4—48no—
609C2×C4—48no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—41no—
613C2×C4—42no—
614C2×C4—44no—
615C2×C4—41no—
616C2×C4—42no—
617C2×C4—41no—
618C2×C4—41no—
619C2×C4—41no—
620C2×C4—42no—
621C2×C4—41no—
622C2×C4—44no—
623C2×C4—41no—
624C2×C4—41no—
625C2×C4—42no—
626C2×C4—44no—
627C2×C4—41no—
628C2×C4—41no—
629C2×C4—41no—
630C2×C4—42no—
631C2×C4—41no—
632C2×C4—42no—
633C2×C4—84no—
634C2×C4—84no—
635C2×C4—88no—
636C2×C4—164no—
637C2×C4#53412yes—
638C2×C4#53412yes—
639C2×C4#54812yes—
640C2×C4C2×#13512yes—
641C2×C4C2×#13512yes—
642C2×C4C2×#13714yes—
643C2×C4C2×#13712yes—
644C2×C4C22×#4111yescharacteristic
645C2×C4C2×H(5,𝔽2)12yes—
646C2×C4C2×2−1+411yescharacteristic
647C2×C4#58512yes—
648C8—44no—
649C8—42no—
650C8—42no—
651C22—22no—
652C22—22no—
653C22—22no—
654C22—22no—
655C22—22no—
656C22—22no—
657C22—82no—
658C22—82no—
659C22—82no—
660C22—82no—
661C22—82no—
662C22—82no—
663C22—82no—
664C22#336514yes—
665C22C2×#53411yescharacteristic
666C22C2×#54811yescharacteristic
667C22#348711yescharacteristic
668C22C22×#13511yescharacteristic
669C22C22×#13711yescharacteristic
670C22C2×#57711yescharacteristic
671C22#358511yescharacteristic
672C4—22no—
673C4—22no—
674C4—24no—
675C4—22no—
676C4—44no—
677C4—42no—
678C4—41no—
679C4—42no—
680C4—42no—
681C4—81no—
682C4—82no—
683C4—81no—
684C4—82no—
685C4—81no—
686C4—84no—
687C4—81no—
688C4#319014yes—
689C4#350512yes—
690C4#354112yes—
691C2—24no—
692C2—162no—
693C2#3972311yescharacteristic
694C2C2×#319011yescharacteristic
695C2#6076511yescharacteristic
696C2C2×#336511yescharacteristic
697C2#6131911yescharacteristic
698C2#6188511yescharacteristic
699C2#6190611yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#2856C22
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
2. C2×#2582C2semidirect
4. #32609C2semidirect
5. #32552C2semidirect
6. #29719C2semidirect
7. #29330C2semidirect
8. #29237C2semidirect
9. #28187C2semidirect
11. #27680C2semidirect
12. #25295C2semidirect
13. #24997C2semidirect
14. #24190C2semidirect
15. Q8×C22⋊C8C2semidirect
16. #22041C2semidirect
17. C4×#451C2semidirect

Maximal subgroups

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

subgroupindex
#365802
C2×#25822
#329032
#326092
#325522
#297192
#293302
#292372
#281872
#279732
#276802
#252952
#249972
#241902
Q8×C22⋊C82
#220412
C4×#4512