Finite Group Databasebeta

← #6816573

Subgroups — #6816573

Order 512.

1806 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
1C2×#3384C211yesmaximal, characteristic, complemented
2C2×#3341C211yesmaximal, characteristic
3#39489C211yesmaximal, characteristic, complemented
4#38958C211yesmaximal, characteristic, complemented
5#33314C211yesmaximal, characteristic, complemented
6#31108C214yesmaximal, complemented
7#31106C214yesmaximal, complemented
8#29868C211yesmaximal, characteristic, complemented
9#29865C211yesmaximal, characteristic, complemented
10#28706C211yesmaximal, characteristic, complemented
11#28682C211yesmaximal, characteristic, complemented
12#26070C214yesmaximal, complemented
13#25839C211yesmaximal, characteristic, complemented
14#25834C211yesmaximal, characteristic, complemented
15#25704C211yesmaximal, characteristic, complemented
16#25700C211yesmaximal, characteristic, complemented
17#25042C214yesmaximal, complemented
18#21633C211yesmaximal, characteristic, complemented
19#21277C211yesmaximal, characteristic, complemented
20C2×#555C2211yescharacteristic
21C2×#555C2211yescharacteristic
22C2×#554C2211yescharacteristic
23C2×D4×Q8C2211yescharacteristic
24#3384—28no—
25#3341—28no—
26C2×#501C2212yes—
27C2×#500C2211yescharacteristic
28C2×#500C2211yescharacteristic
29C2×#498C2211yescharacteristic
30C2×#498C2211yescharacteristic
31C2×#482C2212yes—
32C2×#482C2212yes—
33C2×#478C2212yes—
34C2×#476C2211yescharacteristic
35C2×#476C2211yescharacteristic
36C2×#476C2211yescharacteristic
37C2×#476C2212yes—
38C2×#474C2211yescharacteristic
39C2×#466C2212yes—
40C2×#446C2211yescharacteristic
41C2×#442C2212yes—
42C2×C4×D8C2211yescharacteristic
43C2×C8×D4C2211yescharacteristic
44C2×#427C2212yes—
45#2868C2211yescharacteristic
46#2862C2211yescharacteristic
47#2700—24no—
48#2674—24no—
49#2620—21no—
50#2620—21no—
51#2567C2211yescharacteristic
52#2567C2211yescharacteristic
53#2563C2211yescharacteristic
54#2547C2211yescharacteristic
55#2501—21no—
56#2499—21no—
57#2499—21no—
58#2441C2214yes—
59#2439C2214yes—
60#2436—21no—
61#2422C2214yes—
62#2420C2214yes—
63#2317—21no—
64#2310—21no—
65C4×#124C2211yescharacteristic
66#2293C2211yescharacteristic
67#2293C2211yescharacteristic
68#2086C2214yes—
69#2085C2214yes—
70#2084C2214yes—
71#2077C2214yes—
72#2073C2214yes—
73#2071C2214yes—
74#2059C2214yes—
75#2058C2214yes—
76#2034C2214yes—
77#2033C2214yes—
78#1988C2211yescharacteristic
79#1987C2211yescharacteristic
80#1987C2211yescharacteristic
81#1987C2211yescharacteristic
82#1984C2211yescharacteristic
83#1972C2212yes—
84#1954C2211yescharacteristic
85#1944C2214yes—
86#1943C2212yes—
87#1936C2212yes—
88#1936C2212yes—
89#1924C2214yes—
90#1864C2214yes—
91#1862C2214yes—
92#1854C2214yes—
93#1854C2214yes—
94#1847C2214yes—
95#1847C2214yes—
96#1831C2211yescharacteristic
97#1807C2214yes—
98C4×#106C2211yescharacteristic
99C4×#102C2211yescharacteristic
100C4×#102C2211yescharacteristic
101#1752C2214yes—
102C4×#97C2211yescharacteristic
103C23×Q8—21no—
104C22×D8C2311yescharacteristic
105C23×C8C2312yes—
106#555—24no—
107#555—24no—
108#555—42no—
109#555—42no—
110#554—24no—
111#554—42no—
112D4×Q8—24no—
113D4×Q8—42no—
114C2×C4⋊Q8—22no—
115C2×C4⋊Q8C2311yescharacteristic
116C2×C4⋊Q8C2311yescharacteristic
117C2×C4⋊Q8C2311yescharacteristic
118C2×C4⋊Q8C2311yescharacteristic
119C2×C4⋊Q8C2312yes—
120C2×#125C2312yes—
121C2×#124—21no—
122C2×#124—22no—
123C2×#124—22no—
124C2×#124D411yescharacteristic
125C2×#123—24no—
126C2×C22⋊Q8—21no—
127C2×C22⋊Q8—21no—
128C2×C22⋊Q8—22no—
129C2×C22⋊Q8—22no—
130C2×#121—21no—
131C2×#121—21no—
132C2×#121C2312yes—
133C2×#121C2312yes—
134C2×C4×Q8C2311yescharacteristic
135C2×C4×D4C2312yes—
136C2×C4×D4C2312yes—
137C2×C4×D4C2311yescharacteristic
138C2×C4×D4C2311yescharacteristic
139C2×C4×D4C2311yescharacteristic
140C2×#117—21no—
141C2×#117—21no—
142C2×#117—21no—
143C2×#117—21no—
144C2×#117C2312yes—
145C2×#117C2312yes—
146C22×C4⋊C4—21no—
147#501—28no—
148#500—24no—
149#500—42no—
150#498—24no—
151#498—42no—
152#482—28no—
153#482—28no—
154#478—28no—
155#476—24no—
156#476—24no—
157#476—24no—
158#476—28no—
159#474—24no—
160#466—28no—
161#446—24no—
162#442—28no—
163C4×D8D418yes—
164C8×D4—24no—
165#427—28no—
166C2×#107C2312yes—
167C2×#107C2312yes—
168C2×#107C2312yes—
169C2×#107C2314yes—
170C2×#107C2311yescharacteristic
171C2×#106C2312yes—
172C2×#106C2312yes—
173C2×#106C2312yes—
174C2×#106C2312yes—
175C2×#106C2312yes—
176C2×#106C2312yes—
177C2×#106C2314yes—
178C2×C4⋊C8C2312yes—
179C2×C4⋊C8C2311yescharacteristic
180C2×C4⋊C8C2311yescharacteristic
181C2×#102C2312yes—
182C2×#102C2312yes—
183C2×#102C2312yes—
184C2×#102C2312yes—
185C2×#102C2312yes—
186C2×#102C2312yes—
187C2×#102C2311yescharacteristic
188C2×#102C2311yescharacteristic
189C2×C22⋊C8C2312yes—
190C2×C22⋊C8C2314yes—
191C2×#97C2312yes—
192C2×#97C2312yes—
193C2×#97C2312yes—
194C2×C4×C8C2311yescharacteristic
195C2×C4×C8C2311yescharacteristic
196#399—24no—
197#397—22no—
198#397—24no—
199#397—24no—
200#396—24no—
201#396—24no—
202#395—22no—
203#395—22no—
204#394—24no—
205#394—24no—
206#393—22no—
207#393—24no—
208#391—21no—
209#391—21no—
210#391—21no—
211#391—21no—
212#391—21no—
213#391—22no—
214#391—24no—
215#391D411yescharacteristic
216#390—21no—
217#390D411yescharacteristic
218#390C2312yes—
219#390C2312yes—
220#389—24no—
221#389C2314yes—
222#389C2314yes—
223#389C2312yes—
224#389C2312yes—
225#388—22no—
226#388—22no—
227#388D414yes—
228#387—24no—
229#387—24no—
230#386—24no—
231#385—24no—
232#384—21no—
233#384—21no—
234#384—21no—
235#384—21no—
236#384C2314yes—
237#382—21no—
238#382D411yescharacteristic
239#382C2314yes—
240#381—24no—
241#380C2314yes—
242#380C2314yes—
243C4×C4⋊C4C2311yescharacteristic
244C4×C4⋊C4C2311yescharacteristic
245C4×C22⋊C4—21no—
246C4×C22⋊C4—21no—
247C4×C22⋊C4C2314yes—
248C43C2311yescharacteristic
249#340C2312yes—
250#340C2314yes—
251#340C2312yes—
252#340C2314yes—
253#340C2314yes—
254#340C2314yes—
255#340C2312yes—
256#336C2314yes—
257#336C2314yes—
258C22×Q8—21no—
259C22×Q8—22no—
260C22×Q8—21no—
261C22×Q8—21no—
262C22×Q8—21no—
263C22×Q8—44no—
264C22×Q8—44no—
265C22×Q8—82no—
266C22×Q8C2×D411yescharacteristic
267C22×D4—22no—
268C22×D4—22no—
269C22×D4C2×D411yescharacteristic
270C22×D4C2411yescharacteristic
271C22×D4C2411yescharacteristic
272C23×C4—21no—
273C23×C4—21no—
274C23×C4—22no—
275C23×C4—22no—
276C23×C4C2412yes—
277C2×D8—24no—
278C2×D8C2×D414yes—
279C22×C8—24no—
280C22×C8—28no—
281C22×C8C2412yes—
282C22×C8C2412yes—
283C22×C8C2411yescharacteristic
284C22×C8C2412yes—
285C22×C8C2411yescharacteristic
286C4⋊Q8—24no—
287C4⋊Q8—22no—
288C4⋊Q8—22no—
289C4⋊Q8—22no—
290C4⋊Q8—22no—
291C4⋊Q8—22no—
292C4⋊Q8—22no—
293C4⋊Q8—22no—
294C4⋊Q8—22no—
295C4⋊Q8—42no—
296C4⋊Q8—84no—
297#125—24no—
298#125—42no—
299#124—42no—
300#124—48no—
301#124—48no—
302#124—82no—
303#123—88no—
304C22⋊Q8—48no—
305C22⋊Q8—48no—
306C22⋊Q8—44no—
307C22⋊Q8—44no—
308#121—24no—
309#121—24no—
310#121—42no—
311#121—42no—
312#121—44no—
313#121—44no—
314C4×Q8—22no—
315C4×Q8—22no—
316C4×D4—24no—
317C4×D4—24no—
318C4×D4—22no—
319C4×D4—22no—
320C4×D4—22no—
321C4×D4—22no—
322C4×D4—42no—
323C4×D4—42no—
324C4×D4C2×D414yes—
325C4×D4C2×D414yes—
326#117—24no—
327#117—24no—
328#117—44no—
329#117—42no—
330#117—42no—
331#117—44no—
332#117—44no—
333#117—82no—
334C2×C4⋊C4—22no—
335C2×C4⋊C4—21no—
336C2×C4⋊C4—21no—
337C2×C4⋊C4—21no—
338C2×C4⋊C4—22no—
339C2×C4⋊C4—22no—
340C2×C4⋊C4—22no—
341C2×C4⋊C4—21no—
342C2×C4⋊C4—24no—
343C2×C4⋊C4—22no—
344C2×C4⋊C4—24no—
345C2×C4⋊C4—24no—
346C2×C4⋊C4—24no—
347C2×C4⋊C4—44no—
348C2×C4⋊C4C2×D411yescharacteristic
349C2×C4⋊C4C2×D411yescharacteristic
350C2×C4⋊C4C2×D412yes—
351C2×C4⋊C4C2412yes—
352C2×C4⋊C4C2412yes—
353C2×C4⋊C4C2412yes—
354C2×C4⋊C4C2412yes—
355C2×C4⋊C4C2412yes—
356C2×C4⋊C4C2412yes—
357C2×C4⋊C4C2411yescharacteristic
358C2×C4⋊C4C2411yescharacteristic
359C2×C22⋊C4—21no—
360C2×C22⋊C4—21no—
361C2×C22⋊C4—22no—
362C2×C22⋊C4—22no—
363C2×C22⋊C4—22no—
364C2×C22⋊C4—22no—
365C2×C22⋊C4—22no—
366C2×C22⋊C4—22no—
367C2×C22⋊C4—22no—
368C2×C22⋊C4—22no—
369C2×C22⋊C4—24no—
370C2×C22⋊C4—24no—
371C2×C22⋊C4—22no—
372C2×C22⋊C4—22no—
373C2×C22⋊C4C2×D412yes—
374C2×C22⋊C4C2×D414yes—
375C2×C22⋊C4C2×D414yes—
376C2×C42—22no—
377C2×C42—22no—
378C2×C42—21no—
379C2×C42—21no—
380C2×C42C2×D411yescharacteristic
381C2×C42C2×D411yescharacteristic
382C2×C42C2×D412yes—
383C2×C42C2412yes—
384C2×C42C2412yes—
385C2×C42C2411yescharacteristic
386#107—24no—
387#107—24no—
388#107—24no—
389#107—44no—
390#107C2×D414yes—
391#106—24no—
392#106—24no—
393#106—24no—
394#106—24no—
395#106—24no—
396#106—24no—
397#106—44no—
398C4⋊C8—24no—
399C4⋊C8—22no—
400C4⋊C8—22no—
401#102—24no—
402#102—24no—
403#102—24no—
404#102—24no—
405#102—24no—
406#102—24no—
407#102C2×D414yes—
408#102C2×D414yes—
409C22⋊C8—24no—
410C22⋊C8—28no—
411#97—24no—
412#97—24no—
413#97—24no—
414C4×C8—22no—
415C4×C8C2×D414yes—
416#95—22no—
417#95—22no—
418#95—21no—
419#95—21no—
420#95—24no—
421#95—24no—
422#95—24no—
423#95—24no—
424#95—24no—
425#95—24no—
426#95C2×D414yes—
427#95C2×D414yes—
428C24—21no—
429C24—21no—
430C24C22×D412yes—
431C2×Q8—22no—
432C2×Q8—22no—
433C2×Q8—22no—
434C2×Q8—44no—
435C2×Q8—44no—
436C2×Q8—42no—
437C2×Q8—42no—
438C2×Q8—42no—
439C2×Q8—42no—
440C2×Q8—42no—
441C2×Q8—42no—
442C2×Q8—82no—
443C2×Q8—82no—
444C2×Q8—82no—
445C2×Q8—82no—
446C2×Q8—816no—
447C2×D4—22no—
448C2×D4—22no—
449C2×D4—22no—
450C2×D4—21no—
451C2×D4—21no—
452C2×D4—22no—
453C2×D4—22no—
454C2×D4—22no—
455C2×D4—22no—
456C2×D4—44no—
457C2×D4—44no—
458C2×D4—84no—
459C2×D4—84no—
460C2×D4C22×D412yes—
461C2×D4C22×D412yes—
462C22×C4—21no—
463C22×C4—21no—
464C22×C4—24no—
465C22×C4—24no—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—24no—
469C22×C4—24no—
470C22×C4—22no—
471C22×C4—22no—
472C22×C4—22no—
473C22×C4—22no—
474C22×C4—22no—
475C22×C4—22no—
476C22×C4—44no—
477C22×C4—44no—
478C22×C4—44no—
479C22×C4—42no—
480C22×C4—42no—
481C22×C4—42no—
482C22×C4—42no—
483C22×C4—82no—
484C22×C4—84no—
485C22×C4C22×D412yes—
486C22×C4C22×D412yes—
487C22×C4C22×D412yes—
488C22×C4C22×D412yes—
489C22×C4C22×D412yes—
490C22×C4C22×D411yescharacteristic
491C22×C4C22×D411yescharacteristic
492C22×C4C2511yescharacteristic
493D8—28no—
494C2×C8—22no—
495C2×C8—22no—
496C2×C8—22no—
497C2×C8—22no—
498C2×C8—22no—
499C2×C8—24no—
500C2×C8—24no—
501C2×C8—48no—
502C2×C8C22×D412yes—
503C2×C8C22×D412yes—
504C2×C82−1+414yes—
505C2×C82−1+414yes—
506C4⋊C4—22no—
507C4⋊C4—22no—
508C4⋊C4—24no—
509C4⋊C4—21no—
510C4⋊C4—21no—
511C4⋊C4—22no—
512C4⋊C4—22no—
513C4⋊C4—22no—
514C4⋊C4—22no—
515C4⋊C4—22no—
516C4⋊C4—22no—
517C4⋊C4—22no—
518C4⋊C4—22no—
519C4⋊C4—22no—
520C4⋊C4—22no—
521C4⋊C4—22no—
522C4⋊C4—22no—
523C4⋊C4—44no—
524C4⋊C4—44no—
525C4⋊C4—44no—
526C4⋊C4—44no—
527C4⋊C4—48no—
528C4⋊C4—42no—
529C4⋊C4—42no—
530C4⋊C4—42no—
531C4⋊C4—42no—
532C4⋊C4—84no—
533C4⋊C4—82no—
534C4⋊C4—84no—
535C4⋊C4—84no—
536C4⋊C4C22×D412yes—
537C4⋊C4C22×D412yes—
538C22⋊C4—24no—
539C22⋊C4—24no—
540C22⋊C4—24no—
541C22⋊C4—44no—
542C22⋊C4—44no—
543C22⋊C4—44no—
544C22⋊C4—44no—
545C22⋊C4—44no—
546C22⋊C4—44no—
547C22⋊C4—44no—
548C22⋊C4—44no—
549C22⋊C4—44no—
550C22⋊C4—44no—
551C22⋊C4—82no—
552C22⋊C4—82no—
553C22⋊C4—84no—
554C22⋊C4—84no—
555C42—22no—
556C42—22no—
557C42—22no—
558C42—22no—
559C42—22no—
560C42—22no—
561C42—42no—
562C42—44no—
563C42—44no—
564C42—42no—
565C42—42no—
566C42C22×D412yes—
567C42H(5,𝔽2)12yes—
568C23—24no—
569C23—22no—
570C23—22no—
571C23—22no—
572C23—24no—
573C23—22no—
574C23—22no—
575C23—42no—
576C23—42no—
577C23—42no—
578C23—42no—
579C23—41no—
580C23—41no—
581C23—41no—
582C23—41no—
583C23C23×D411yescharacteristic
584Q8—44no—
585Q8—88no—
586Q8—84no—
587Q8—162no—
588Q8—162no—
589Q8—162no—
590D4—22no—
591D4—22no—
592D4—22no—
593D4—22no—
594D4—44no—
595D4—42no—
596D4—42no—
597D4—164no—
598D4—164no—
599C2×C4—22no—
600C2×C4—22no—
601C2×C4—22no—
602C2×C4—24no—
603C2×C4—22no—
604C2×C4—22no—
605C2×C4—22no—
606C2×C4—22no—
607C2×C4—22no—
608C2×C4—22no—
609C2×C4—22no—
610C2×C4—24no—
611C2×C4—22no—
612C2×C4—22no—
613C2×C4—22no—
614C2×C4—22no—
615C2×C4—41no—
616C2×C4—41no—
617C2×C4—41no—
618C2×C4—41no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—48no—
622C2×C4—48no—
623C2×C4—44no—
624C2×C4—42no—
625C2×C4—44no—
626C2×C4—42no—
627C2×C4—44no—
628C2×C4—44no—
629C2×C4—42no—
630C2×C4—42no—
631C2×C4—42no—
632C2×C4—42no—
633C2×C4—42no—
634C2×C4—42no—
635C2×C4—42no—
636C2×C4—42no—
637C2×C4—42no—
638C2×C4—42no—
639C2×C4—88no—
640C2×C4—164no—
641C2×C4D4212yes—
642C2×C4D4212yes—
643C2×C4#57512yes—
644C2×C4#57712yes—
645C2×C4C23×D411yescharacteristic
646C2×C4C2×H(5,𝔽2)11yescharacteristic
647C2×C4C2×2−1+412yes—
648C2×C4C2×2−1+412yes—
649C8—22no—
650C8—24no—
651C8—42no—
652C8—42no—
653C8#53614yes—
654C22—44no—
655C22—44no—
656C22—42no—
657C22—42no—
658C22—42no—
659C22—42no—
660C22—42no—
661C22—42no—
662C22—42no—
663C22—42no—
664C22—42no—
665C22—82no—
666C22—82no—
667C22—82no—
668C22—82no—
669C22C2×D4211yescharacteristic
670C22#349912yes—
671C22C2×#57511yescharacteristic
672C22C2×#57711yescharacteristic
673C22#358512yes—
674C4—22no—
675C4—42no—
676C4—42no—
677C4—42no—
678C4—42no—
679C4—42no—
680C4—42no—
681C4—42no—
682C4—82no—
683C4—82no—
684C4—82no—
685C4—82no—
686C4—82no—
687C4—82no—
688C4#326714yes—
689C4C2×#53611yescharacteristic
690C4#349911yescharacteristic
691C4#355212yes—
692C2—82no—
693C2—82no—
694C2—82no—
695C2#3936011yescharacteristic
696C2C2×#326711yescharacteristic
697C2#6108211yescharacteristic
698C2#6250812yes—
699C2#6252112yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#3341C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×#124 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#3384C2semidirect
3. #39489C2semidirect
4. #38958C2semidirect
5. #33314C2semidirect
6. #31108C2semidirect
7. #31106C2semidirect
8. #29868C2semidirect
9. #29865C2semidirect
10. #28706C2semidirect
11. #28682C2semidirect
12. #26070C2semidirect
13. #25839C2semidirect
14. #25834C2semidirect
15. #25704C2semidirect
16. #25700C2semidirect
17. #25042C2semidirect
18. #21633C2semidirect
19. #21277C2semidirect

Maximal subgroups

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

subgroupindex
C2×#33842
C2×#33412
#394892
#389582
#333142
#311082
#311062
#298682
#298652
#287062
#286822
#260702
#258392
#258342
#257042
#257002
#250422
#216332
#212772