Finite Group Databasebeta

← #6817315

Subgroups — #6817315

Order 512.

1834 conjugacy classes of subgroups.

The subgroup lattice has 700 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3392C211yesmaximal, characteristic
2C2×#3381C211yesmaximal, characteristic, complemented
3#39170C211yesmaximal, characteristic, complemented
4#38952C211yesmaximal, characteristic
5#33239C211yesmaximal, characteristic, complemented
6#31158C214yesmaximal, complemented
7#31157C214yesmaximal, complemented
8#29829C211yesmaximal, characteristic, complemented
9#29827C211yesmaximal, characteristic, complemented
10#28738C211yesmaximal, characteristic, complemented
11#28677C211yesmaximal, characteristic, complemented
12#26481C214yesmaximal, complemented
13#26250C214yesmaximal, complemented
14#25935C211yesmaximal, characteristic, complemented
15#25934C211yesmaximal, characteristic, complemented
16#25651C211yesmaximal, characteristic, complemented
17#25645C211yesmaximal, characteristic
18C8×C4⋊Q8C211yesmaximal, characteristic, complemented
19#21274C211yesmaximal, characteristic, complemented
20C2×#557C2211yescharacteristic
21C2×#557C2211yescharacteristic
22C2×#554C2211yescharacteristic
23C2×#550C2211yescharacteristic
24#3392—28no—
25#3381—28no—
26C2×#499C2212yes—
27C2×#498C2211yescharacteristic
28C2×#492C2211yescharacteristic
29C2×#492C2212yes—
30C2×#487C2211yescharacteristic
31C2×#486C2211yescharacteristic
32C2×#479C2212yes—
33C2×#477C2211yescharacteristic
34C2×#478C2212yes—
35C2×#474C2211yescharacteristic
36C2×#460C2211yescharacteristic
37C2×#460C2212yes—
38C2×#461C2211yescharacteristic
39C2×#461C2212yes—
40C2×C8×Q8C2211yescharacteristic
41C2×C8×Q8C2211yescharacteristic
42C2×C4×SD16C2212yes—
43C2×C4×SD16C2212yes—
44C2×C4×SD16C2211yescharacteristic
45#2862C2211yescharacteristic
46#2858C2211yescharacteristic
47#2719—24no—
48#2672—24no—
49#2642—21no—
50#2620—21no—
51#2568C2211yescharacteristic
52#2567C2211yescharacteristic
53#2565—21no—
54#2548C2211yescharacteristic
55#2546—21no—
56#2546C2211yescharacteristic
57#2501—21no—
58#2454C2214yes—
59#2453C2214yes—
60#2446C2214yes—
61#2445C2214yes—
62#2432—21no—
63#2327—21no—
64#2311—21no—
65C4×C4⋊Q8C2211yescharacteristic
66#2295C2211yescharacteristic
67#2293C2211yescharacteristic
68#2088C2214yes—
69#2068C2214yes—
70#2067C2214yes—
71#2055C2214yes—
72#2055C2214yes—
73#2055C2214yes—
74#2043C2214yes—
75#2041C2214yes—
76#2026C2214yes—
77#2025C2214yes—
78#1988C2211yescharacteristic
79#1987C2211yescharacteristic
80#1984C2211yescharacteristic
81#1984C2211yescharacteristic
82#1982C2211yescharacteristic
83#1981C2211yescharacteristic
84#1968C2212yes—
85#1968C2214yes—
86#1968C2214yes—
87#1968C2214yes—
88#1941C2214yes—
89#1941C2212yes—
90#1939C2214yes—
91#1934C2212yes—
92#1933C2212yes—
93#1880C2214yes—
94#1877C2214yes—
95#1863C2214yes—
96#1862C2214yes—
97#1842C2211yescharacteristic
98C4×#106C2211yescharacteristic
99C8×C4⋊C4C2214yes—
100C4×#103C2211yescharacteristic
101C4×#102C2211yescharacteristic
102C42×C8C2211yescharacteristic
103C22×#41—21no—
104C22×SD16C2312yes—
105C22×SD16C2311yescharacteristic
106C22×SD16C2312yes—
107C22×SD16C2312yes—
108C22×SD16C2312yes—
109#557—24no—
110#557—24no—
111#557—42no—
112#557—42no—
113#554—24no—
114#554—42no—
115#550—24no—
116#550—42no—
117C2×C4⋊Q8—21no—
118C2×C4⋊Q8—21no—
119C2×C4⋊Q8D411yescharacteristic
120C2×C4⋊Q8C2311yescharacteristic
121C2×C4⋊Q8C2312yes—
122C2×C4⋊Q8C2311yescharacteristic
123C2×C4⋊Q8C2311yescharacteristic
124C2×#127C2311yescharacteristic
125C2×#127C2312yes—
126C2×#125—21no—
127C2×#125—22no—
128C2×#125—22no—
129C2×#125C2312yes—
130C2×#125C2312yes—
131C2×#124—21no—
132C2×#124—22no—
133C2×#123—24no—
134C2×C22⋊Q8—21no—
135C2×C22⋊Q8—22no—
136C2×#121—21no—
137C2×#121—22no—
138C2×C4×Q8—21no—
139C2×C4×Q8—21no—
140C2×C4×Q8C2312yes—
141C2×C4×Q8C2312yes—
142C2×C4×Q8C2311yescharacteristic
143C2×C4×Q8C2311yescharacteristic
144C2×C4×Q8C2311yescharacteristic
145C2×C4×D4C2312yes—
146C2×C4×D4C2312yes—
147C2×C4×D4C2311yescharacteristic
148C2×#117—21no—
149C22×C4⋊C4—21no—
150#499—28no—
151#498—24no—
152#492—28no—
153#492—24no—
154#487—42no—
155#486—42no—
156#479—28no—
157#478—28no—
158#477—24no—
159#474—24no—
160#461—24no—
161#461—28no—
162#460—24no—
163#460—28no—
164C8×Q8—24no—
165C8×Q8—24no—
166C4×SD16—28no—
167C4×SD16—28no—
168C4×SD16D418yes—
169C2×#107C2312yes—
170C2×#107C2312yes—
171C2×#106C2312yes—
172C2×#106C2312yes—
173C2×#106C2312yes—
174C2×#106C2311yescharacteristic
175C2×C4⋊C8C2312yes—
176C2×C4⋊C8C2311yescharacteristic
177C2×C4⋊C8C2312yes—
178C2×C4⋊C8C2311yescharacteristic
179C2×C4⋊C8C2314yes—
180C2×#103C2312yes—
181C2×#103C2312yes—
182C2×#103C2314yes—
183C2×#103C2314yes—
184C2×#103C2312yes—
185C2×#103C2311yescharacteristic
186C2×#102C2312yes—
187C2×#102C2312yes—
188C2×#102C2314yes—
189C2×#102C2314yes—
190C2×#102C2312yes—
191C2×#102C2311yescharacteristic
192C2×C4×C8C2311yescharacteristic
193C2×C4×C8C2312yes—
194C2×C4×C8C2312yes—
195C2×C4×C8C2312yes—
196C2×C4×C8C2312yes—
197C2×C4×C8C2311yescharacteristic
198#400—22no—
199#400—24no—
200#400—24no—
201#399—24no—
202#398—24no—
203#398—24no—
204#398—22no—
205#396—24no—
206#395—22no—
207#394—22no—
208#394—24no—
209#394—24no—
210#391—21no—
211#391—21no—
212#391—21no—
213#391—21no—
214#391—21no—
215#391D411yescharacteristic
216#391D411yescharacteristic
217#391C2312yes—
218#391C2314yes—
219#391C2314yes—
220#390—21no—
221#390—21no—
222#390—21no—
223#390C2312yes—
224#390C2314yes—
225#390C2314yes—
226#389—24no—
227#389D414yes—
228#389C2314yes—
229#389C2312yes—
230#389C2312yes—
231#389C2314yes—
232#388—22no—
233#387—24no—
234#387—22no—
235#387—24no—
236#387—24no—
237#386—24no—
238#385—24no—
239#384—21no—
240#384D411yescharacteristic
241#382—21no—
242#382—22no—
243#380—24no—
244C4×C4⋊C4—21no—
245C4×C4⋊C4C2311yescharacteristic
246C4×C4⋊C4C2311yescharacteristic
247C4×C4⋊C4C2314yes—
248C4×C22⋊C4—21no—
249#376—21no—
250C43C2311yescharacteristic
251#340C2312yes—
252#340C2314yes—
253#340C2314yes—
254#340C2312yes—
255#340C2312yes—
256#336C2314yes—
257#336C2314yes—
258C2×#41—44no—
259C2×#41—44no—
260C2×#41—82no—
261C22×Q8—21no—
262C22×Q8—21no—
263C22×Q8C2×D411yescharacteristic
264C22×Q8C2×D411yescharacteristic
265C22×Q8C2412yes—
266C22×Q8C2411yescharacteristic
267C22×D4—21no—
268C22×D4—22no—
269C22×D4—22no—
270C22×D4—21no—
271C22×D4C2412yes—
272C22×D4C2411yescharacteristic
273C23×C4—21no—
274C23×C4—22no—
275C23×C4—22no—
276C2×SD16—24no—
277C2×SD16—24no—
278C2×SD16—24no—
279C2×SD16—24no—
280C2×SD16—24no—
281C2×SD16—28no—
282C2×SD16—44no—
283C2×SD16—44no—
284C2×SD16—44no—
285C2×SD16C2×D414yes—
286C22×C8C2412yes—
287C22×C8C2412yes—
288C22×C8C2412yes—
289C22×C8C2411yescharacteristic
290C22×C8C2411yescharacteristic
291C4⋊Q8—24no—
292C4⋊Q8—24no—
293C4⋊Q8—22no—
294C4⋊Q8—22no—
295C4⋊Q8—22no—
296C4⋊Q8—22no—
297C4⋊Q8—22no—
298C4⋊Q8—22no—
299C4⋊Q8—42no—
300C4⋊Q8—44no—
301C4⋊Q8—44no—
302#127—24no—
303#127—22no—
304#127—22no—
305#127—42no—
306#125—24no—
307#125—24no—
308#125—42no—
309#125—42no—
310#125—44no—
311#125—48no—
312#125—48no—
313#124—84no—
314#124—82no—
315#123—88no—
316C22⋊Q8—48no—
317C22⋊Q8—44no—
318#121—48no—
319#121—44no—
320C4×Q8—22no—
321C4×Q8—22no—
322C4×Q8—22no—
323C4×Q8—22no—
324C4×Q8—22no—
325C4×Q8—24no—
326C4×Q8—24no—
327C4×Q8—42no—
328C4×Q8—42no—
329C4×Q8—44no—
330C4×Q8—44no—
331C4×Q8C2×D414yes—
332C4×D4—24no—
333C4×D4—24no—
334C4×D4—22no—
335C4×D4—42no—
336C4×D4—42no—
337C4×D4C2×D414yes—
338#117—82no—
339C2×C4⋊C4—21no—
340C2×C4⋊C4—22no—
341C2×C4⋊C4—21no—
342C2×C4⋊C4—22no—
343C2×C4⋊C4—21no—
344C2×C4⋊C4—22no—
345C2×C4⋊C4—22no—
346C2×C4⋊C4—22no—
347C2×C4⋊C4—22no—
348C2×C4⋊C4—21no—
349C2×C4⋊C4—21no—
350C2×C4⋊C4—22no—
351C2×C4⋊C4—22no—
352C2×C4⋊C4—24no—
353C2×C4⋊C4—22no—
354C2×C4⋊C4—22no—
355C2×C4⋊C4—24no—
356C2×C4⋊C4—44no—
357C2×C4⋊C4C2×D411yescharacteristic
358C2×C4⋊C4C2×D411yescharacteristic
359C2×C4⋊C4C2×D412yes—
360C2×C4⋊C4C2×D412yes—
361C2×C4⋊C4C2×D414yes—
362C2×C4⋊C4C2×D414yes—
363C2×C4⋊C4C2412yes—
364C2×C4⋊C4C2412yes—
365C2×C4⋊C4C2412yes—
366C2×C4⋊C4C2412yes—
367C2×C4⋊C4C2411yescharacteristic
368C2×C4⋊C4C2411yescharacteristic
369C2×C22⋊C4—21no—
370C2×C22⋊C4—22no—
371C2×C22⋊C4—22no—
372C2×C22⋊C4—22no—
373C2×C22⋊C4—24no—
374C2×C22⋊C4—22no—
375C2×C22⋊C4—24no—
376C2×C22⋊C4—24no—
377C2×C22⋊C4—24no—
378C2×C42—21no—
379C2×C42—21no—
380C2×C42—22no—
381C2×C42—22no—
382C2×C42—21no—
383C2×C42C2×D411yescharacteristic
384C2×C42C2×D412yes—
385C2×C42C2×D411yescharacteristic
386C2×C42C2412yes—
387C2×C42C2412yes—
388C2×C42C2412yes—
389C2×C42C2411yescharacteristic
390#107—24no—
391#107—24no—
392#106—24no—
393#106—24no—
394#106—24no—
395#106C2×D414yes—
396C4⋊C8—24no—
397C4⋊C8—24no—
398C4⋊C8—22no—
399C4⋊C8—22no—
400C4⋊C8—44no—
401#103—24no—
402#103—24no—
403#103—42no—
404#103—44no—
405#103—44no—
406#103C2×D414yes—
407#102—24no—
408#102—24no—
409#102—42no—
410#102—44no—
411#102—44no—
412#102C2×D414yes—
413C4×C8—24no—
414C4×C8—24no—
415C4×C8—24no—
416C4×C8—22no—
417C4×C8—42no—
418C4×C8C2×D414yes—
419#95—22no—
420#95—22no—
421#95—22no—
422#95—22no—
423#95—21no—
424#95—21no—
425#95—24no—
426#95—24no—
427#95—24no—
428#95—24no—
429#95—24no—
430#95—24no—
431#95C2×D414yes—
432#95C2×D414yes—
433C24—21no—
434C24—22no—
435#41—816no—
436C2×Q8—22no—
437C2×Q8—22no—
438C2×Q8—22no—
439C2×Q8—22no—
440C2×Q8—22no—
441C2×Q8—21no—
442C2×Q8—22no—
443C2×Q8—22no—
444C2×Q8—22no—
445C2×Q8—24no—
446C2×Q8—22no—
447C2×Q8—24no—
448C2×Q8—42no—
449C2×Q8—42no—
450C2×Q8—42no—
451C2×Q8—42no—
452C2×Q8—82no—
453C2×Q8#13514yes—
454C2×Q8C22×D412yes—
455C2×D4—22no—
456C2×D4—21no—
457C2×D4—22no—
458C2×D4—22no—
459C2×D4—24no—
460C2×D4—22no—
461C2×D4—24no—
462C2×D4—44no—
463C2×D4—44no—
464C2×D4—42no—
465C2×D4—42no—
466C2×D4—44no—
467C2×D4—42no—
468C2×D4—42no—
469C2×D4—82no—
470C2×D4—84no—
471C2×D4—82no—
472C2×D4C22×D412yes—
473C22×C4—21no—
474C22×C4—21no—
475C22×C4—21no—
476C22×C4—24no—
477C22×C4—22no—
478C22×C4—22no—
479C22×C4—22no—
480C22×C4—22no—
481C22×C4—22no—
482C22×C4—44no—
483C22×C4—44no—
484C22×C4—42no—
485C22×C4—42no—
486C22×C4—44no—
487C22×C4—82no—
488C22×C4—84no—
489C22×C4C22×D412yes—
490C22×C4C22×D412yes—
491C22×C4C22×D412yes—
492C22×C4C22×D412yes—
493C22×C4C22×D411yescharacteristic
494C22×C4C22×D412yes—
495C22×C4C22×D412yes—
496C22×C4C22×D411yescharacteristic
497C22×C4C2511yescharacteristic
498SD16—28no—
499SD16—48no—
500SD16—84no—
501SD16—84no—
502SD16—84no—
503C2×C8—22no—
504C2×C8—22no—
505C2×C8—22no—
506C2×C8—22no—
507C2×C8—22no—
508C2×C8—22no—
509C2×C8—24no—
510C2×C8—24no—
511C2×C8C22×D412yes—
512C2×C8C22×D412yes—
513C2×C82−1+414yes—
514C4⋊C4—24no—
515C4⋊C4—22no—
516C4⋊C4—24no—
517C4⋊C4—22no—
518C4⋊C4—24no—
519C4⋊C4—28no—
520C4⋊C4—28no—
521C4⋊C4—21no—
522C4⋊C4—21no—
523C4⋊C4—22no—
524C4⋊C4—22no—
525C4⋊C4—22no—
526C4⋊C4—22no—
527C4⋊C4—22no—
528C4⋊C4—22no—
529C4⋊C4—22no—
530C4⋊C4—22no—
531C4⋊C4—44no—
532C4⋊C4—44no—
533C4⋊C4—44no—
534C4⋊C4—44no—
535C4⋊C4—44no—
536C4⋊C4—44no—
537C4⋊C4—44no—
538C4⋊C4—44no—
539C4⋊C4—42no—
540C4⋊C4—42no—
541C4⋊C4—42no—
542C4⋊C4—42no—
543C4⋊C4—48no—
544C4⋊C4—44no—
545C4⋊C4—82no—
546C4⋊C4—84no—
547C4⋊C4C22×D412yes—
548C4⋊C4C22×D412yes—
549C22⋊C4—24no—
550C22⋊C4—44no—
551C22⋊C4—44no—
552C22⋊C4—44no—
553C22⋊C4—48no—
554C22⋊C4—82no—
555C22⋊C4—84no—
556C22⋊C4—84no—
557C22⋊C4—84no—
558C42—24no—
559C42—22no—
560C42—22no—
561C42—24no—
562C42—22no—
563C42—22no—
564C42—22no—
565C42—22no—
566C42—22no—
567C42—22no—
568C42—44no—
569C42—44no—
570C42—42no—
571C42—42no—
572C42C22×D412yes—
573C42H(5,𝔽2)12yes—
574C23—22no—
575C23—42no—
576C23—41no—
577C23—42no—
578C23—41no—
579C23—42no—
580C23—44no—
581C23—42no—
582C23—44no—
583C23—42no—
584C23C23×D411yescharacteristic
585Q8—28no—
586Q8—22no—
587Q8—22no—
588Q8—44no—
589Q8—42no—
590Q8—42no—
591Q8—42no—
592Q8—44no—
593Q8—84no—
594Q8—162no—
595D4—22no—
596D4—22no—
597D4—42no—
598D4—42no—
599D4—42no—
600D4—44no—
601D4—88no—
602D4—84no—
603D4—162no—
604D4—164no—
605C2×C4—22no—
606C2×C4—22no—
607C2×C4—22no—
608C2×C4—22no—
609C2×C4—22no—
610C2×C4—22no—
611C2×C4—22no—
612C2×C4—22no—
613C2×C4—22no—
614C2×C4—22no—
615C2×C4—22no—
616C2×C4—22no—
617C2×C4—22no—
618C2×C4—22no—
619C2×C4—22no—
620C2×C4—22no—
621C2×C4—24no—
622C2×C4—24no—
623C2×C4—42no—
624C2×C4—41no—
625C2×C4—42no—
626C2×C4—41no—
627C2×C4—48no—
628C2×C4—42no—
629C2×C4—42no—
630C2×C4—42no—
631C2×C4—42no—
632C2×C4—42no—
633C2×C4—44no—
634C2×C4—42no—
635C2×C4—42no—
636C2×C4—42no—
637C2×C4—42no—
638C2×C4—44no—
639C2×C4—42no—
640C2×C4—42no—
641C2×C4—88no—
642C2×C4—164no—
643C2×C4D4212yes—
644C2×C4D4212yes—
645C2×C4C2×#13514yes—
646C2×C4C2×#13512yes—
647C2×C4#57714yes—
648C2×C4#57712yes—
649C2×C4C23×D411yescharacteristic
650C2×C4C2×H(5,𝔽2)11yescharacteristic
651C2×C4C2×2−1+412yes—
652C2×C4#58512yes—
653C8—22no—
654C8—42no—
655C8—42no—
656C8—42no—
657C8#53614yes—
658C22—42no—
659C22—42no—
660C22—42no—
661C22—82no—
662C22—82no—
663C22—82no—
664C22—84no—
665C22—82no—
666C22—84no—
667C22—82no—
668C22C2×D4211yescharacteristic
669C22#349712yes—
670C22C22×#13511yescharacteristic
671C22C2×#57711yescharacteristic
672C22C2×#57712yes—
673C4—24no—
674C4—24no—
675C4—22no—
676C4—42no—
677C4—42no—
678C4—42no—
679C4—42no—
680C4—42no—
681C4—42no—
682C4—42no—
683C4—82no—
684C4—82no—
685C4—82no—
686C4—82no—
687C4—82no—
688C4#328214yes—
689C4C2×#53611yescharacteristic
690C4#350011yescharacteristic
691C4#355412yes—
692C2—82no—
693C2—162no—
694C2#3936311yescharacteristic
695C2C2×#328211yescharacteristic
696C2#6108711yescharacteristic
697C2#6236212yes—
698C2#6237112yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#550C22
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
2. C2×#3381C2semidirect
3. #39170C2semidirect
5. #33239C2semidirect
6. #31158C2semidirect
7. #31157C2semidirect
8. #29829C2semidirect
9. #29827C2semidirect
10. #28738C2semidirect
11. #28677C2semidirect
12. #26481C2semidirect
13. #26250C2semidirect
14. #25935C2semidirect
15. #25934C2semidirect
16. #25651C2semidirect
18. C8×C4⋊Q8C2semidirect
19. #21274C2semidirect

Maximal subgroups

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

subgroupindex
C2×#33922
C2×#33812
#391702
#389522
#332392
#311582
#311572
#298292
#298272
#287382
#286772
#264812
#262502
#259352
#259342
#256512
#256452
C8×C4⋊Q82
#212742