Finite Group Databasebeta

← #6529827

Subgroups — #6529827

Order 512.

1950 conjugacy classes of subgroups.

The subgroup lattice has 723 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3112C211yesmaximal, characteristic
2C2×#2984C211yesmaximal, characteristic, complemented
3#35249C211yesmaximal, characteristic
4C2×#2582C211yesmaximal, characteristic, complemented
5#28277C214yesmaximal, complemented
6#28200C214yesmaximal, complemented
7#25480C212yesmaximal, complemented
8#25471C212yesmaximal, complemented
9#25303C212yesmaximal, complemented
10#25295C212yesmaximal, complemented
11#24608C212yesmaximal, complemented
12#24600C212yesmaximal, complemented
13#24509C212yesmaximal, complemented
14#24503C212yesmaximal, complemented
15#21987C211yesmaximal, characteristic, complemented
16#21859C211yesmaximal, characteristic
17#20735C211yesmaximal, characteristic, complemented
18C2×#532C2211yescharacteristic
19C22×C4×Q8C2211yescharacteristic
20C22×C4×Q8C2211yescharacteristic
21#3112—28no—
22C2×#480C2212yes—
23C2×#477C2212yes—
24C2×#460C2212yes—
25C2×#448C2212yes—
26#2984—28no—
27C2×#446C2214yes—
28C2×#446C2212yes—
29C2×C8×Q8C2212yes—
30C2×#440C2214yes—
31C2×#432C2212yes—
32C2×#432C2211yescharacteristic
33C2×#431C2212yes—
34C2×#431C2211yescharacteristic
35#2854C2212yes—
36#2854C2212yes—
37#2632—22no—
38#2582—21no—
39#2582—21no—
40#2582—22no—
41#2582C2214yes—
42#2582C2212yes—
43#2582C2212yes—
44#2580C2212yes—
45#2580C2212yes—
46#2567C2211yescharacteristic
47#2567C2211yescharacteristic
48#2548C2211yescharacteristic
49#2548C2211yescharacteristic
50#2501—21no—
51#2436—21no—
52#2406—22no—
53C2×#398—22no—
54#2326—21no—
55#2326—21no—
56C2×#391—21no—
57C2×#391C2212yes—
58C2×#389—21no—
59C2×#389C2212yes—
60C2×#384C2211yescharacteristic
61#2283C2211yescharacteristic
62#2087C2214yes—
63#2086C2214yes—
64#2068C2214yes—
65#2067C2214yes—
66#2066C2214yes—
67#2065C2214yes—
68#2055C2214yes—
69#2055C2214yes—
70#1990C2212yes—
71#1973C2214yes—
72#1968C2214yes—
73#1957C2212yes—
74#1948C2212yes—
75#1947C2212yes—
76#1944C2214yes—
77#1942C2214yes—
78#1936C2214yes—
79#1935C2214yes—
80#1934C2214yes—
81#1933C2214yes—
82#1874C2212yes—
83#1874C2212yes—
84#1853C2212yes—
85#1852C2212yes—
86#1847C2212yes—
87#1836C2212yes—
88#1823C2211yescharacteristic
89#1823C2211yescharacteristic
90#1805C2212yes—
91#1805C2212yes—
92#1802C2214yes—
93#1787C2211yescharacteristic
94#1787C2211yescharacteristic
95C4×#103C2212yes—
96C4×#103C2212yes—
97C22×#41—21no—
98C23×Q8D411yescharacteristic
99C23×Q8C2311yescharacteristic
100C22×SD16C2314yes—
101#532—24no—
102#532—24no—
103C2×C4⋊Q8C2311yescharacteristic
104C2×#127C2311yescharacteristic
105C2×#124—21no—
106C2×#124—21no—
107C2×#124—21no—
108C2×#123—22no—
109C2×#123—22no—
110C2×#121—21no—
111C2×#121C2312yes—
112C2×C4×Q8—24no—
113C2×C4×Q8—21no—
114C2×C4×Q8—21no—
115C2×C4×Q8—24no—
116C2×C4×Q8—24no—
117C2×C4×Q8—24no—
118C2×C4×Q8D411yescharacteristic
119C2×C4×Q8D411yescharacteristic
120C2×C4×Q8D412yes—
121C2×C4×Q8C2312yes—
122C2×C4×Q8C2312yes—
123C2×C4×Q8C2311yescharacteristic
124C2×C4×Q8C2311yescharacteristic
125C2×C4×Q8C2312yes—
126C2×C4×D4—21no—
127C2×C4×D4C2312yes—
128C22×C4⋊C4—21no—
129C22×C4⋊C4—21no—
130C22×C4⋊C4—21no—
131C22×C4⋊C4—22no—
132C22×C4⋊C4D412yes—
133C22×C4⋊C4D411yescharacteristic
134C22×C4⋊C4C2311yescharacteristic
135C22×C4⋊C4C2312yes—
136C22×C42—21no—
137C22×C42C2312yes—
138C22×C42C2311yescharacteristic
139#480—28no—
140#477—28no—
141#460—28no—
142#448—28no—
143#446—28no—
144#446—216no—
145C8×Q8—28no—
146#440—216no—
147#432—28no—
148#432D418yes—
149#431—24no—
150#431—28no—
151C2×#107C2314yes—
152C2×#107C2314yes—
153C2×#106C2314yes—
154C2×C4⋊C8C2314yes—
155C2×C4⋊C8C2314yes—
156C2×C4⋊C8C2312yes—
157C2×C4⋊C8C2312yes—
158C2×#103C2314yes—
159C2×#103C2312yes—
160C2×#103C2312yes—
161C2×#103C2312yes—
162C2×#103C2312yes—
163C2×#102C2314yes—
164C2×#102C2314yes—
165C2×#102C2314yes—
166C2×C22⋊C8C2311yescharacteristic
167C2×C22⋊C8C2311yescharacteristic
168C2×C22⋊C8C2312yes—
169C2×#97C2314yes—
170C2×#97C2312yes—
171C2×C4×C8C2314yes—
172C2×C4×C8C2312yes—
173#399—22no—
174#399—22no—
175#398—24no—
176#398—24no—
177#398—24no—
178#398—24no—
179#395—22no—
180#395—22no—
181#394—22no—
182#394—22no—
183#393—22no—
184#393—22no—
185#391—21no—
186#391—21no—
187#391—22no—
188#391—24no—
189#391—22no—
190#391—22no—
191#391—22no—
192#391C2314yes—
193#391C2314yes—
194#390—21no—
195#390—21no—
196#390C2312yes—
197#390C2312yes—
198#390C2312yes—
199#390C2312yes—
200#389—22no—
201#389—22no—
202#389—24no—
203#389—22no—
204#389—22no—
205#389C2312yes—
206#389C2312yes—
207#389C2312yes—
208#389C2312yes—
209#389C2314yes—
210#389C2314yes—
211#388—22no—
212#388—22no—
213#388—22no—
214#388—22no—
215#386—22no—
216#386—22no—
217#385—22no—
218#385—22no—
219#385—22no—
220#385—22no—
221#384—21no—
222#384—21no—
223#384—21no—
224#384—21no—
225#384C2312yes—
226#384C2312yes—
227#383—21no—
228#382C2311yescharacteristic
229#382C2311yescharacteristic
230#381—21no—
231#380C2311yescharacteristic
232#380C2311yescharacteristic
233C4×C4⋊C4C2311yescharacteristic
234C4×C4⋊C4C2311yescharacteristic
235C4×C22⋊C4—21no—
236#376—21no—
237C2×#95—22no—
238C2×#95—22no—
239C2×#95—22no—
240#340C2314yes—
241#340C2314yes—
242#340C2314yes—
243#340C2314yes—
244#340C2312yes—
245#340C2312yes—
246#340C2312yes—
247#340C2312yes—
248C2×#41—44no—
249C2×#41—82no—
250C2×#41—82no—
251C22×Q8—24no—
252C22×Q8—24no—
253C22×Q8—24no—
254C22×Q8—21no—
255C22×Q8—21no—
256C22×Q8—22no—
257C22×Q8—22no—
258C22×Q8—42no—
259C22×Q8—42no—
260C22×Q8C2×D411yescharacteristic
261C22×Q8C2×D412yes—
262C22×Q8C2×D411yescharacteristic
263C22×Q8C2412yes—
264C22×D4—22no—
265C22×D4—21no—
266C22×D4—21no—
267C22×D4C2412yes—
268C23×C4—21no—
269C23×C4—22no—
270C23×C4—21no—
271C23×C4—22no—
272C23×C4C2×D412yes—
273C23×C4C2×D412yes—
274C23×C4C2×D411yescharacteristic
275C23×C4C2×D411yescharacteristic
276C23×C4C2411yescharacteristic
277C2×SD16—28no—
278C2×SD16—28no—
279C2×SD16—44no—
280C22×C8C2412yes—
281C22×C8C2412yes—
282C22×C8C2414yes—
283C4⋊Q8—22no—
284C4⋊Q8—22no—
285#127—22no—
286#127—22no—
287#124—44no—
288#124—44no—
289#124—44no—
290#123—84no—
291#123—84no—
292#121—24no—
293#121—24no—
294#121—44no—
295C4×Q8—24no—
296C4×Q8—24no—
297C4×Q8—28no—
298C4×Q8—22no—
299C4×Q8—24no—
300C4×Q8—22no—
301C4×Q8—24no—
302C4×Q8—22no—
303C4×Q8—22no—
304C4×Q8—24no—
305C4×Q8—24no—
306C4×Q8—24no—
307C4×Q8—24no—
308C4×Q8—44no—
309C4×Q8—44no—
310C4×D4—24no—
311C4×D4—24no—
312C4×D4—44no—
313C2×C4⋊C4—24no—
314C2×C4⋊C4—28no—
315C2×C4⋊C4—22no—
316C2×C4⋊C4—22no—
317C2×C4⋊C4—22no—
318C2×C4⋊C4—22no—
319C2×C4⋊C4—22no—
320C2×C4⋊C4—22no—
321C2×C4⋊C4—21no—
322C2×C4⋊C4—22no—
323C2×C4⋊C4—21no—
324C2×C4⋊C4—22no—
325C2×C4⋊C4—22no—
326C2×C4⋊C4—24no—
327C2×C4⋊C4—24no—
328C2×C4⋊C4—22no—
329C2×C4⋊C4—22no—
330C2×C4⋊C4—22no—
331C2×C4⋊C4—22no—
332C2×C4⋊C4—44no—
333C2×C4⋊C4—44no—
334C2×C4⋊C4—44no—
335C2×C4⋊C4—82no—
336C2×C4⋊C4—84no—
337C2×C4⋊C4C2×D411yescharacteristic
338C2×C4⋊C4C2×D411yescharacteristic
339C2×C4⋊C4C2×D412yes—
340C2×C4⋊C4C2×D412yes—
341C2×C4⋊C4C2×D412yes—
342C2×C4⋊C4C2×D412yes—
343C2×C4⋊C4C2×D412yes—
344C2×C4⋊C4C2412yes—
345C2×C4⋊C4C2414yes—
346C2×C4⋊C4C2412yes—
347C2×C4⋊C4C2411yescharacteristic
348C2×C4⋊C4C2411yescharacteristic
349C2×C4⋊C4C2411yescharacteristic
350C2×C4⋊C4C2411yescharacteristic
351C2×C22⋊C4—21no—
352C2×C22⋊C4—21no—
353C2×C22⋊C4—22no—
354C2×C22⋊C4—22no—
355C2×C22⋊C4—22no—
356C2×C22⋊C4—22no—
357C2×C22⋊C4—22no—
358C2×C22⋊C4—22no—
359C2×C22⋊C4—22no—
360C2×C22⋊C4—22no—
361C2×C22⋊C4—22no—
362C2×C22⋊C4—22no—
363C2×C42—24no—
364C2×C42—24no—
365C2×C42—22no—
366C2×C42—22no—
367C2×C42—21no—
368C2×C42—21no—
369C2×C42—44no—
370C2×C42C2×D412yes—
371C2×C42C2×D412yes—
372C2×C42C2×D411yescharacteristic
373C2×C42C2×D411yescharacteristic
374C2×C42C2×D414yes—
375C2×C42#4114yes—
376C2×C42C2412yes—
377C2×C42C2412yes—
378C2×C42C2411yescharacteristic
379C2×C42C2411yescharacteristic
380#107—28no—
381#107—44no—
382#106—44no—
383C4⋊C8—28no—
384C4⋊C8—28no—
385C4⋊C8—24no—
386C4⋊C8C2×D418yes—
387#103—24no—
388#103—24no—
389#103—28no—
390#103—42no—
391#103—42no—
392#102—28no—
393#102—44no—
394#102—44no—
395C22⋊C8—22no—
396C22⋊C8—22no—
397C22⋊C8C2×D418yes—
398#97—28no—
399#97C2×D418yes—
400C4×C8—28no—
401C4×C8—24no—
402#95—24no—
403#95—24no—
404#95—24no—
405#95—24no—
406#95—24no—
407#95—24no—
408#95—21no—
409#95—21no—
410#95—21no—
411#95—21no—
412#95—21no—
413#95—21no—
414#95—21no—
415#95—21no—
416#95—22no—
417#95—22no—
418#95—22no—
419#95—22no—
420C24—22no—
421C24C22×D411yescharacteristic
422#41—816no—
423C2×Q8—24no—
424C2×Q8—22no—
425C2×Q8—24no—
426C2×Q8—22no—
427C2×Q8—24no—
428C2×Q8—22no—
429C2×Q8—22no—
430C2×Q8—22no—
431C2×Q8—22no—
432C2×Q8—22no—
433C2×Q8—22no—
434C2×Q8—22no—
435C2×Q8—22no—
436C2×Q8—42no—
437C2×Q8—48no—
438C2×Q8—48no—
439C2×Q8—416no—
440C2×Q8—42no—
441C2×Q8—42no—
442C2×Q8—42no—
443C2×Q8—81no—
444C2×Q8—81no—
445C2×Q8#13714yes—
446C2×Q82−1+414yes—
447C2×D4—22no—
448C2×D4—22no—
449C2×D4—24no—
450C2×D4—24no—
451C2×D4—44no—
452C2×D4—44no—
453C2×D4—44no—
454C2×D4—42no—
455C2×D4—42no—
456C2×D4—42no—
457C2×D4—42no—
458C2×D4—81no—
459C2×D4—81no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—22no—
466C22×C4—24no—
467C22×C4—24no—
468C22×C4—24no—
469C22×C4—24no—
470C22×C4—24no—
471C22×C4—22no—
472C22×C4—24no—
473C22×C4—21no—
474C22×C4—21no—
475C22×C4—21no—
476C22×C4—21no—
477C22×C4—24no—
478C22×C4—22no—
479C22×C4—22no—
480C22×C4—42no—
481C22×C4—42no—
482C22×C4—42no—
483C22×C4—44no—
484C22×C4—44no—
485C22×C4—42no—
486C22×C4—44no—
487C22×C4—42no—
488C22×C4—42no—
489C22×C4—44no—
490C22×C4—82no—
491C22×C4—82no—
492C22×C4#12114yes—
493C22×C4#12114yes—
494C22×C4C22×D412yes—
495C22×C4C22×D412yes—
496C22×C4C22×D412yes—
497C22×C4C22×D411yescharacteristic
498C22×C4C22×D411yescharacteristic
499C22×C4C22×D411yescharacteristic
500C22×C4C22×D411yescharacteristic
501C22×C4C22×D412yes—
502C22×C4C22×D412yes—
503C22×C4C22×D411yescharacteristic
504C22×C4C2×#4112yes—
505C22×C4C2511yescharacteristic
506SD16—416no—
507C2×C8—24no—
508C2×C8—24no—
509C2×C8—22no—
510C2×C8—22no—
511C2×C8—22no—
512C2×C8—22no—
513C2×C8—22no—
514C2×C8—22no—
515C2×C8C22×D418yes—
516C4⋊C4—24no—
517C4⋊C4—24no—
518C4⋊C4—24no—
519C4⋊C4—22no—
520C4⋊C4—22no—
521C4⋊C4—24no—
522C4⋊C4—24no—
523C4⋊C4—22no—
524C4⋊C4—22no—
525C4⋊C4—22no—
526C4⋊C4—22no—
527C4⋊C4—22no—
528C4⋊C4—22no—
529C4⋊C4—24no—
530C4⋊C4—24no—
531C4⋊C4—22no—
532C4⋊C4—22no—
533C4⋊C4—44no—
534C4⋊C4—44no—
535C4⋊C4—42no—
536C4⋊C4—44no—
537C4⋊C4—44no—
538C4⋊C4—42no—
539C4⋊C4—44no—
540C4⋊C4—82no—
541C4⋊C4—82no—
542C4⋊C4—84no—
543C4⋊C4—84no—
544C4⋊C4—82no—
545C4⋊C4—82no—
546C4⋊C4—82no—
547C4⋊C4—82no—
548C22⋊C4—44no—
549C22⋊C4—44no—
550C22⋊C4—42no—
551C22⋊C4—42no—
552C22⋊C4—44no—
553C22⋊C4—44no—
554C22⋊C4—44no—
555C22⋊C4—44no—
556C22⋊C4—82no—
557C22⋊C4—82no—
558C22⋊C4—82no—
559C22⋊C4—82no—
560C42—24no—
561C42—24no—
562C42—22no—
563C42—22no—
564C42—22no—
565C42—22no—
566C42—44no—
567C42—44no—
568C42—42no—
569C42—42no—
570C42#12114yes—
571C42#12114yes—
572C42C22×D412yes—
573C42C22×D412yes—
574C42C2×#4112yes—
575C42C2×#4112yes—
576C23—21no—
577C23—21no—
578C23—21no—
579C23—21no—
580C23—42no—
581C23—42no—
582C23—42no—
583C23—42no—
584C23—42no—
585C23—42no—
586C23—42no—
587C23C2×#12112yes—
588C23#57612yes—
589C23#57712yes—
590C23C23×D411yescharacteristic
591Q8—28no—
592Q8—24no—
593Q8—24no—
594Q8—48no—
595Q8—48no—
596Q8—42no—
597Q8—42no—
598Q8—44no—
599Q8—84no—
600D4—44no—
601D4—44no—
602D4—84no—
603D4—88no—
604D4—84no—
605C2×C4—22no—
606C2×C4—21no—
607C2×C4—21no—
608C2×C4—28no—
609C2×C4—21no—
610C2×C4—21no—
611C2×C4—22no—
612C2×C4—21no—
613C2×C4—21no—
614C2×C4—21no—
615C2×C4—21no—
616C2×C4—24no—
617C2×C4—24no—
618C2×C4—28no—
619C2×C4—22no—
620C2×C4—22no—
621C2×C4—22no—
622C2×C4—24no—
623C2×C4—22no—
624C2×C4—22no—
625C2×C4—22no—
626C2×C4—22no—
627C2×C4—22no—
628C2×C4—22no—
629C2×C4—48no—
630C2×C4—48no—
631C2×C4—42no—
632C2×C4—42no—
633C2×C4—48no—
634C2×C4—42no—
635C2×C4—42no—
636C2×C4—41no—
637C2×C4—41no—
638C2×C4—44no—
639C2×C4—41no—
640C2×C4—41no—
641C2×C4—41no—
642C2×C4—41no—
643C2×C4—41no—
644C2×C4—41no—
645C2×C4—44no—
646C2×C4—41no—
647C2×C4—41no—
648C2×C4—42no—
649C2×C4—44no—
650C2×C4—42no—
651C2×C4—41no—
652C2×C4—41no—
653C2×C4—42no—
654C2×C4—42no—
655C2×C4—88no—
656C2×C4—88no—
657C2×C4—84no—
658C2×C4C2×#12112yes—
659C2×C4C2×#12112yes—
660C2×C4C2×#12112yes—
661C2×C4C2×#12112yes—
662C2×C4C2×#12112yes—
663C2×C4C2×#13714yes—
664C2×C4C2×#13712yes—
665C2×C4#57512yes—
666C2×C4C23×D411yescharacteristic
667C2×C4C22×#4111yescharacteristic
668C2×C4C2×2−1+412yes—
669C2×C4#58512yes—
670C8—28no—
671C8—42no—
672C8—42no—
673C22—22no—
674C22—22no—
675C22—22no—
676C22—22no—
677C22—22no—
678C22—22no—
679C22—82no—
680C22—82no—
681C22—82no—
682C22—82no—
683C22—82no—
684C22—82no—
685C22—82no—
686C22#306214yes—
687C22C22×#12111yescharacteristic
688C22#348611yescharacteristic
689C22#348711yescharacteristic
690C22C22×#13711yescharacteristic
691C22C2×#57511yescharacteristic
692C22C2×#57611yescharacteristic
693C22C2×#57711yescharacteristic
694C4—22no—
695C4—24no—
696C4—22no—
697C4—22no—
698C4—22no—
699C4—42no—
700C4—42no—
701C4—42no—
702C4—44no—
703C4—82no—
704C4—82no—
705C4—81no—
706C4—81no—
707C4—81no—
708C4—84no—
709C4—81no—
710C4#305614yes—
711C4#348912yes—
712C4#349512yes—
713C2—24no—
714C2—162no—
715C2#3566111yescharacteristic
716C2C2×#305611yescharacteristic
717C2C2×#306211yescharacteristic
718C2#6183611yescharacteristic
719C2#6184411yescharacteristic
720C2#6189811yescharacteristic
721C2#6190611yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#532C22
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×#2984C2semidirect
4. C2×#2582C2semidirect
5. #28277C2semidirect
6. #28200C2semidirect
7. #25480C2semidirect
8. #25471C2semidirect
9. #25303C2semidirect
10. #25295C2semidirect
11. #24608C2semidirect
12. #24600C2semidirect
13. #24509C2semidirect
14. #24503C2semidirect
15. #21987C2semidirect
17. #20735C2semidirect

Maximal subgroups

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

subgroupindex
C2×#31122
C2×#29842
#352492
C2×#25822
#282772
#282002
#254802
#254712
#253032
#252952
#246082
#246002
#245092
#245032
#219872
#218592
#207352