Finite Group Databasebeta

← #6585570

Subgroups — #6585570

Order 512.

1880 conjugacy classes of subgroups.

The subgroup lattice has 697 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3356C211yesmaximal, characteristic, complemented
2C2×#3342C211yesmaximal, characteristic, complemented
3#36594C211yesmaximal, characteristic
4#36534C211yesmaximal, characteristic
5#33314C211yesmaximal, characteristic, complemented
6#33202C211yesmaximal, characteristic, complemented
7#33196C211yesmaximal, characteristic, complemented
8#28712C211yesmaximal, characteristic, complemented
9#28706C211yesmaximal, characteristic, complemented
10#28238C214yesmaximal, complemented
11#25705C211yesmaximal, characteristic, complemented
12#25705C211yesmaximal, characteristic, complemented
13#25702C211yesmaximal, characteristic, complemented
14#25701C211yesmaximal, characteristic, complemented
15#25029C214yesmaximal, complemented
16#25025C214yesmaximal, complemented
17#22662C214yesmaximal, complemented
18#21682C211yesmaximal, characteristic
19C4×#441C211yesmaximal, characteristic, complemented
20C2×#556C2211yescharacteristic
21C2×#556C2211yescharacteristic
22C2×#556C2211yescharacteristic
23C2×#554C2211yescharacteristic
24#3356—28no—
25#3342—28no—
26C2×#500C2211yescharacteristic
27C2×#500C2211yescharacteristic
28C2×#498C2211yescharacteristic
29C2×#498C2211yescharacteristic
30C2×#485C2212yes—
31C2×#485C2212yes—
32C2×#485C2212yes—
33C2×#483C2212yes—
34C2×#477C2211yescharacteristic
35C2×#475C2211yescharacteristic
36C2×#475C2211yescharacteristic
37C2×#475C2211yescharacteristic
38C2×#470C2212yes—
39C2×#467C2212yes—
40C2×#441C2212yes—
41C2×C4×Q16C2211yescharacteristic
42C2×#436C2211yescharacteristic
43C2×#435C2214yes—
44C2×C8×D4C2211yescharacteristic
45C2×#428C2212yes—
46C2×#427C2212yes—
47#2868C2211yescharacteristic
48#2862C2211yescharacteristic
49#2861—21no—
50#2858—21no—
51#2565—21no—
52#2563C2211yescharacteristic
53#2560—24no—
54#2547C2211yescharacteristic
55#2547C2211yescharacteristic
56#2547C2211yescharacteristic
57#2545—21no—
58#2545—21no—
59#2544—21no—
60#2538C2214yes—
61#2536C2214yes—
62#2512—24no—
63#2354C2214yes—
64#2352C2214yes—
65C4×C4⋊Q8—21no—
66C4×#127C2211yescharacteristic
67C4×#124—21no—
68C42×Q8C2211yescharacteristic
69C42×Q8C2211yescharacteristic
70#2077C2214yes—
71#2076C2214yes—
72#2036C2214yes—
73#2036C2214yes—
74#2035C2214yes—
75#2035C2214yes—
76#1988C2211yescharacteristic
77#1988C2211yescharacteristic
78#1988C2211yescharacteristic
79#1987C2211yescharacteristic
80#1955C2214yes—
81#1954C2211yescharacteristic
82#1954C2211yescharacteristic
83#1949C2214yes—
84#1948C2214yes—
85#1854C2214yes—
86#1854C2214yes—
87#1854C2214yes—
88#1853C2214yes—
89#1847C2214yes—
90#1847C2214yes—
91#1842C2211yescharacteristic
92#1809C2214yes—
93#1807C2214yes—
94#1805C2214yes—
95#1803C2214yes—
96#1776C2212yes—
97C4×#106C2211yescharacteristic
98#1763C2212yes—
99#1763C2212yes—
100C4×#103C2211yescharacteristic
101C4×#103C2211yescharacteristic
102C4×#97C2211yescharacteristic
103C22×Q16C2311yescharacteristic
104C22×M16C2312yes—
105C23×C8C2312yes—
106#556—24no—
107#556—24no—
108#556—24no—
109#556—42no—
110#556—42no—
111#556—42no—
112#554—24no—
113#554—42no—
114C2×C4⋊Q8—21no—
115C2×C4⋊Q8—21no—
116C2×C4⋊Q8C2311yescharacteristic
117C2×C4⋊Q8C2311yescharacteristic
118C2×C4⋊Q8C2311yescharacteristic
119C2×C4⋊Q8C2311yescharacteristic
120C2×#127D411yescharacteristic
121C2×#126—22no—
122C2×#126—22no—
123C2×#126—22no—
124C2×#125—21no—
125C2×#124—21no—
126C2×#124—21no—
127C2×#124—21no—
128C2×#124—21no—
129C2×#124—21no—
130C2×#124—21no—
131C2×C22⋊Q8—22no—
132C2×C22⋊Q8C2312yes—
133C2×C22⋊Q8C2312yes—
134C2×C22⋊Q8C2312yes—
135C2×C22⋊Q8C2312yes—
136C2×C4×Q8—21no—
137C2×C4×Q8—21no—
138C2×C4×Q8—21no—
139C2×C4×Q8C2312yes—
140C2×C4×Q8C2312yes—
141C2×C4×Q8C2311yescharacteristic
142C2×C4×Q8C2311yescharacteristic
143C2×C4×D4—21no—
144C2×C4×D4C2311yescharacteristic
145C2×C4×D4C2314yes—
146C2×C4×D4C2311yescharacteristic
147C2×#117C2312yes—
148C2×#117C2312yes—
149C2×#117C2312yes—
150C22×C4⋊C4C2312yes—
151#500—24no—
152#500—42no—
153#498—24no—
154#498—42no—
155#485—28no—
156#485—28no—
157#485—28no—
158#483—28no—
159#477—24no—
160#475—24no—
161#475—24no—
162#475—24no—
163#470—28no—
164#467—28no—
165#441—28no—
166C4×Q16D418yescomplemented
167#436—24no—
168#435—216no—
169C8×D4—24no—
170#428—28no—
171#427—28no—
172C2×#107C2312yes—
173C2×#107C2312yes—
174C2×#107C2312yes—
175C2×#107C2314yes—
176C2×#107C2311yescharacteristic
177C2×#106C2312yes—
178C2×#106C2312yes—
179C2×#106C2312yes—
180C2×#106C2312yes—
181C2×#106C2312yes—
182C2×#106C2312yes—
183C2×#106C2314yes—
184C2×C4⋊C8C2311yescharacteristic
185C2×C4⋊C8C2311yescharacteristic
186C2×C4⋊C8C2314yes—
187C2×#103C2312yes—
188C2×#103C2312yes—
189C2×#103C2312yes—
190C2×#103C2312yes—
191C2×#103C2312yes—
192C2×#103C2312yes—
193C2×#103C2311yescharacteristic
194C2×#103C2311yescharacteristic
195C2×C22⋊C8C2312yes—
196C2×C22⋊C8C2312yes—
197C2×C22⋊C8C2314yes—
198C2×C22⋊C8C2314yes—
199C2×#97C2312yes—
200C2×#97C2312yes—
201C2×C4×C8C2311yescharacteristic
202C2×C4×C8C2311yescharacteristic
203#398—24no—
204#397—22no—
205#397—22no—
206#397—22no—
207#396—24no—
208#396—24no—
209#394—24no—
210#393—22no—
211#391—21no—
212#391—21no—
213#391—21no—
214#391—22no—
215#391—21no—
216#391—21no—
217#390—21no—
218#390—24no—
219#390—24no—
220#390D414yes—
221#390D411yescharacteristic
222#390D411yescharacteristic
223#389—24no—
224#388—24no—
225#388—24no—
226#388—24no—
227#388—24no—
228#388—24no—
229#387—24no—
230#386—24no—
231#384—22no—
232#384D411yescharacteristic
233#384C2314yes—
234#382—21no—
235#382C2314yes—
236#380C2314yes—
237#380C2314yes—
238#380C2314yes—
239#380C2314yes—
240C4×C4⋊C4—24no—
241C4×C4⋊C4—22no—
242C4×C4⋊C4—21no—
243C4×C4⋊C4—22no—
244C4×C4⋊C4—21no—
245C4×C4⋊C4C2311yescharacteristic
246C4×C4⋊C4C2312yes—
247C4×C4⋊C4C2312yes—
248C4×C4⋊C4C2311yescharacteristic
249C4×C22⋊C4—24no—
250C43—21no—
251C43—21no—
252C43C2311yescharacteristic
253#340C2314yes—
254#340C2314yes—
255#340C2314yes—
256#340C2314yes—
257#340C2312yes—
258#340C2312yes—
259#340C2312yes—
260C22×Q8—21no—
261C22×Q8—21no—
262C22×Q8—21no—
263C22×Q8—21no—
264C22×Q8C2411yescharacteristic
265C22×Q8C2411yescharacteristic
266C22×D4C2×D411yescharacteristic
267C22×D4C2×D411yescharacteristic
268C22×D4C2×D414yes—
269C23×C4—22no—
270C23×C4C2412yes—
271C23×C4C2412yes—
272C2×Q16—24no—
273C2×Q16C2×D414yes—
274C2×M16—24no—
275C2×M16—28no—
276C22×C8—24no—
277C22×C8—28no—
278C22×C8C2412yes—
279C22×C8C2412yes—
280C22×C8C2411yescharacteristic
281C22×C8C2412yes—
282C22×C8C2411yescharacteristic
283C4⋊Q8—22no—
284C4⋊Q8—22no—
285C4⋊Q8—22no—
286C4⋊Q8—22no—
287C4⋊Q8—22no—
288C4⋊Q8—22no—
289C4⋊Q8—22no—
290C4⋊Q8—22no—
291C4⋊Q8—44no—
292C4⋊Q8—82no—
293#127—42no—
294#126—48no—
295#126—48no—
296#126—48no—
297#125—44no—
298#124—44no—
299#124—44no—
300#124—44no—
301#124—44no—
302#124—82no—
303#124—82no—
304C22⋊Q8—24no—
305C22⋊Q8—24no—
306C22⋊Q8—24no—
307C22⋊Q8—24no—
308C22⋊Q8—48no—
309C22⋊Q8—42no—
310C22⋊Q8—42no—
311C22⋊Q8—42no—
312C22⋊Q8—42no—
313C4×Q8—22no—
314C4×Q8—22no—
315C4×Q8—24no—
316C4×Q8—24no—
317C4×Q8—24no—
318C4×Q8—24no—
319C4×Q8—44no—
320C4×Q8—44no—
321C4×Q8—44no—
322C4×Q8C2×D414yes—
323C4×Q8C2×D414yes—
324C4×D4—22no—
325C4×D4—22no—
326C4×D4—28no—
327C4×D4—22no—
328C4×D4—22no—
329C4×D4—28no—
330C4×D4—44no—
331#117—24no—
332#117—24no—
333#117—24no—
334#117—42no—
335#117—42no—
336#117—42no—
337C2×C4⋊C4—21no—
338C2×C4⋊C4—21no—
339C2×C4⋊C4—24no—
340C2×C4⋊C4—22no—
341C2×C4⋊C4—21no—
342C2×C4⋊C4—22no—
343C2×C4⋊C4—22no—
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—21no—
351C2×C4⋊C4—24no—
352C2×C4⋊C4—24no—
353C2×C4⋊C4—42no—
354C2×C4⋊C4C2×D411yescharacteristic
355C2×C4⋊C4C2×D411yescharacteristic
356C2×C4⋊C4C2×D414yes—
357C2×C4⋊C4C2412yes—
358C2×C4⋊C4C2412yes—
359C2×C4⋊C4C2412yes—
360C2×C4⋊C4C2412yes—
361C2×C4⋊C4C2412yes—
362C2×C4⋊C4C2412yes—
363C2×C4⋊C4C2411yescharacteristic
364C2×C4⋊C4C2411yescharacteristic
365C2×C22⋊C4—22no—
366C2×C22⋊C4—22no—
367C2×C22⋊C4—22no—
368C2×C22⋊C4—22no—
369C2×C22⋊C4—22no—
370C2×C22⋊C4—22no—
371C2×C22⋊C4—22no—
372C2×C22⋊C4—24no—
373C2×C22⋊C4—24no—
374C2×C22⋊C4—24no—
375C2×C22⋊C4—24no—
376C2×C22⋊C4C2×D412yes—
377C2×C22⋊C4C2×D412yes—
378C2×C22⋊C4C2×D414yes—
379C2×C22⋊C4C2×D414yes—
380C2×C42—21no—
381C2×C42—21no—
382C2×C42—22no—
383C2×C42—22no—
384C2×C42—22no—
385C2×C42—22no—
386C2×C42—22no—
387C2×C42—21no—
388C2×C42—21no—
389C2×C42—22no—
390C2×C42—22no—
391C2×C42C2×D411yescharacteristic
392C2×C42C2×D411yescharacteristic
393C2×C42C2×D412yes—
394C2×C42C2411yescharacteristic
395C2×C42C2412yes—
396#107—24no—
397#107—24no—
398#107—24no—
399#107—44no—
400#107C2×D414yes—
401#106—24no—
402#106—24no—
403#106—24no—
404#106—24no—
405#106—24no—
406#106—24no—
407#106—44no—
408C4⋊C8—22no—
409C4⋊C8—22no—
410C4⋊C8—28no—
411#103—24no—
412#103—24no—
413#103—24no—
414#103—24no—
415#103—24no—
416#103—24no—
417#103C2×D414yes—
418#103C2×D414yes—
419C22⋊C8—24no—
420C22⋊C8—24no—
421C22⋊C8—28no—
422C22⋊C8—28no—
423#97—24no—
424#97#4118yes—
425C4×C8—22no—
426C4×C8C2×D414yes—
427#95—24no—
428#95—24no—
429#95—22no—
430#95—22no—
431#95—21no—
432#95—21no—
433#95—24no—
434#95—24no—
435#95—24no—
436#95—24no—
437C24C22×D412yes—
438C24C22×D412yes—
439C2×Q8—21no—
440C2×Q8—21no—
441C2×Q8—22no—
442C2×Q8—22no—
443C2×Q8—22no—
444C2×Q8—22no—
445C2×Q8—42no—
446C2×Q8—42no—
447C2×Q8—42no—
448C2×Q8—42no—
449C2×Q8—42no—
450C2×Q8—42no—
451C2×Q8—42no—
452C2×Q8—42no—
453C2×Q8—42no—
454C2×Q8—42no—
455C2×Q8—82no—
456C2×Q8C22×D412yes—
457C2×Q8C22×D412yes—
458C2×D4—22no—
459C2×D4—22no—
460C2×D4—28no—
461C2×D4—22no—
462C2×D4—22no—
463C2×D4—22no—
464C2×D4—22no—
465C2×D4—44no—
466C2×D4—44no—
467C22×C4—21no—
468C22×C4—21no—
469C22×C4—21no—
470C22×C4—21no—
471C22×C4—22no—
472C22×C4—22no—
473C22×C4—24no—
474C22×C4—24no—
475C22×C4—22no—
476C22×C4—22no—
477C22×C4—24no—
478C22×C4—24no—
479C22×C4—22no—
480C22×C4—22no—
481C22×C4—22no—
482C22×C4—22no—
483C22×C4—22no—
484C22×C4—22no—
485C22×C4—44no—
486C22×C4—84no—
487C22×C4C22×D412yes—
488C22×C4C22×D412yes—
489C22×C4C22×D411yescharacteristic
490C22×C4C22×D412yes—
491C22×C4C22×D412yes—
492C22×C4C22×D411yescharacteristic
493C22×C4C2511yescharacteristic
494Q16—28no—
495M16—48no—
496C2×C8—22no—
497C2×C8—22no—
498C2×C8—22no—
499C2×C8—22no—
500C2×C8—22no—
501C2×C8—24no—
502C2×C8—24no—
503C2×C8—48no—
504C2×C8C22×D412yes—
505C2×C8C22×D412yes—
506C2×C8C2×#4114yes—
507C2×C8C2×#4114yes—
508C4⋊C4—24no—
509C4⋊C4—24no—
510C4⋊C4—22no—
511C4⋊C4—22no—
512C4⋊C4—28no—
513C4⋊C4—21no—
514C4⋊C4—21no—
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—22no—
524C4⋊C4—22no—
525C4⋊C4—22no—
526C4⋊C4—22no—
527C4⋊C4—44no—
528C4⋊C4—44no—
529C4⋊C4—44no—
530C4⋊C4—44no—
531C4⋊C4—44no—
532C4⋊C4—48no—
533C4⋊C4—42no—
534C4⋊C4—42no—
535C4⋊C4—42no—
536C4⋊C4—42no—
537C4⋊C4—44no—
538C4⋊C4—44no—
539C4⋊C4—84no—
540C4⋊C4C22×D412yes—
541C4⋊C4C22×D412yes—
542C22⋊C4—24no—
543C22⋊C4—24no—
544C22⋊C4—28no—
545C22⋊C4—28no—
546C22⋊C4—44no—
547C22⋊C4—44no—
548C22⋊C4—44no—
549C22⋊C4—44no—
550C22⋊C4—44no—
551C22⋊C4—44no—
552C22⋊C4—44no—
553C22⋊C4—48no—
554C22⋊C4—84no—
555C22⋊C4—84no—
556C22⋊C4—84no—
557C42—24no—
558C42—24no—
559C42—22no—
560C42—22no—
561C42—42no—
562C42—44no—
563C42—44no—
564C42—44no—
565C42—44no—
566C42—44no—
567C42—44no—
568C42—42no—
569C42—42no—
570C42—82no—
571C42C22×D412yes—
572C42C2×#4114yes—
573C42H(5,𝔽2)12yes—
574C422−1+414yes—
575C23—24no—
576C23—24no—
577C23—22no—
578C23—22no—
579C23—24no—
580C23—24no—
581C23—22no—
582C23—22no—
583C23—22no—
584C23—22no—
585C23C23×D411yescharacteristic
586Q8—22no—
587Q8—22no—
588Q8—22no—
589Q8—22no—
590Q8—42no—
591Q8—42no—
592Q8—84no—
593Q8—84no—
594Q8—84no—
595Q8—162no—
596D4—44no—
597D4—44no—
598D4—88no—
599C2×C4—22no—
600C2×C4—22no—
601C2×C4—22no—
602C2×C4—22no—
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—24no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—44no—
621C2×C4—44no—
622C2×C4—42no—
623C2×C4—42no—
624C2×C4—44no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—42no—
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—164no—
640C2×C4D4212yes—
641C2×C4D4212yes—
642C2×C4#54714yes—
643C2×C4#54714yes—
644C2×C4#57512yes—
645C2×C4#57712yes—
646C2×C4C23×D411yescharacteristic
647C2×C4C22×#4112yes—
648C2×C4C2×H(5,𝔽2)11yescharacteristic
649C2×C4C2×2−1+412yes—
650C8—22no—
651C8—24no—
652C8—42no—
653C8—42no—
654C8#53214yes—
655C22—42no—
656C22—42no—
657C22—44no—
658C22—44no—
659C22—44no—
660C22—44no—
661C22—42no—
662C22—42no—
663C22—42no—
664C22—42no—
665C22C2×D4211yescharacteristic
666C22C2×#54712yes—
667C22C2×#57511yescharacteristic
668C22C2×#57711yescharacteristic
669C22#358612yes—
670C4—24no—
671C4—24no—
672C4—22no—
673C4—42no—
674C4—42no—
675C4—42no—
676C4—42no—
677C4—42no—
678C4—42no—
679C4—82no—
680C4—82no—
681C4—82no—
682C4—82no—
683C4—82no—
684C4—82no—
685C4#327114yes—
686C4C2×#53211yescharacteristic
687C4#349911yescharacteristic
688C4#351112yes—
689C2—82no—
690C2—82no—
691C2#3612711yescharacteristic
692C2C2×#327111yescharacteristic
693C2#6108311yescharacteristic
694C2#6125712yes—
695C2#6128112yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#127D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×#127 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#3356C2semidirect
2. C2×#3342C2semidirect
5. #33314C2semidirect
6. #33202C2semidirect
7. #33196C2semidirect
8. #28712C2semidirect
9. #28706C2semidirect
10. #28238C2semidirect
11. #25705C2semidirect
12. #25705C2semidirect
13. #25702C2semidirect
14. #25701C2semidirect
15. #25029C2semidirect
16. #25025C2semidirect
17. #22662C2semidirect
19. C4×#441C2semidirect
166. C4×Q16D4semidirect

Maximal subgroups

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

subgroupindex
C2×#33562
C2×#33422
#365942
#365342
#333142
#332022
#331962
#287122
#287062
#282382
#257052
#257022
#257012
#250292
#250252
#226622
#216822
C4×#4412