Finite Group Databasebeta

← #6683860

Subgroups — #6683860

Order 512.

1826 conjugacy classes of subgroups.

The subgroup lattice has 695 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3385C211yesmaximal, characteristic, complemented
2C2×#3373C211yesmaximal, characteristic
3#39168C211yesmaximal, characteristic, complemented
4#36276C211yesmaximal, characteristic, complemented
5#33030C211yesmaximal, characteristic
6#30974C214yesmaximal, complemented
7#30830C214yesmaximal, complemented
8#30278C211yesmaximal, characteristic, complemented
9#29867C211yesmaximal, characteristic, complemented
10#28682C211yesmaximal, characteristic, complemented
11#28648C211yesmaximal, characteristic, complemented
12#26388C214yesmaximal, complemented
13#26184C214yesmaximal, complemented
14#25930C211yesmaximal, characteristic
15#25837C211yesmaximal, characteristic, complemented
16#25834C211yesmaximal, characteristic, complemented
17#25648C211yesmaximal, characteristic, complemented
18#22242C211yesmaximal, characteristic, complemented
19#21117C211yesmaximal, characteristic, complemented
20C2×#559C2211yescharacteristic
21C2×Q82C2211yescharacteristic
22C2×#554C2211yescharacteristic
23C2×D4×Q8C2211yescharacteristic
24#3385—28no—
25#3373—28no—
26C2×#501C2212yes—
27C2×#500C2211yescharacteristic
28C2×#498C2211yescharacteristic
29C2×#489C2212yes—
30C2×#486C2211yescharacteristic
31C2×#486C2211yescharacteristic
32C2×#479C2212yes—
33C2×#477C2211yescharacteristic
34C2×#475C2211yescharacteristic
35C2×#476C2211yescharacteristic
36C2×#476C2212yes—
37C2×#474C2212yes—
38C2×#463C2212yes—
39C2×#459C2211yescharacteristic
40C2×#446C2211yescharacteristic
41C2×#446C2211yescharacteristic
42C2×#442C2212yes—
43C2×#440C2212yes—
44C2×C4×SD16C2211yescharacteristic
45#2862C2211yescharacteristic
46#2858C2211yescharacteristic
47#2857—21no—
48#2674—24no—
49#2640—21no—
50#2567C2211yescharacteristic
51#2567C2211yescharacteristic
52#2550C2214yes—
53#2549—24no—
54#2548C2211yescharacteristic
55#2546C2211yescharacteristic
56#2545—21no—
57#2542—21no—
58#2502—21no—
59#2452C2214yes—
60#2448C2214yes—
61#2434—21no—
62#2349C2214yes—
63#2326—21no—
64C4×C4⋊Q8C2211yescharacteristic
65#2295C2211yescharacteristic
66C2×#391—21no—
67C42×D4C2211yescharacteristic
68#2092C2214yes—
69#2092C2214yes—
70#2086C2214yes—
71#2085C2214yes—
72#2074C2214yes—
73#2065C2214yes—
74#2058C2214yes—
75#2058C2214yes—
76#2042C2214yes—
77#2025C2214yes—
78#1988C2211yescharacteristic
79#1987C2211yescharacteristic
80#1987C2211yescharacteristic
81#1984C2211yescharacteristic
82#1984C2211yescharacteristic
83#1982C2211yescharacteristic
84#1944C2214yes—
85#1936C2214yes—
86#1927C2212yes—
87#1926C2214yes—
88#1920C2212yes—
89#1917C2214yes—
90#1917C2214yes—
91#1879C2214yes—
92#1879C2214yes—
93#1877C2212yes—
94#1865C2214yes—
95#1864C2212yes—
96#1864C2214yes—
97#1862C2214yes—
98#1831C2211yescharacteristic
99C4×#106C2211yescharacteristic
100C4×#103C2211yescharacteristic
101C4×#102C2211yescharacteristic
102#1744C2211yescharacteristic
103C23×Q8—21no—
104C22×SD16C2311yescharacteristic
105C22×D8C2312yes—
106#559—24no—
107#559—42no—
108Q82—24no—
109Q82—42no—
110#554—24no—
111#554—42no—
112D4×Q8—24no—
113D4×Q8—42no—
114C2×C4⋊Q8—21no—
115C2×C4⋊Q8—21no—
116C2×C4⋊Q8—22no—
117C2×C4⋊Q8D411yescharacteristic
118C2×C4⋊Q8C2311yescharacteristic
119C2×C4⋊Q8C2311yescharacteristic
120C2×C4⋊Q8C2311yescharacteristic
121C2×C4⋊Q8C2312yes—
122C2×C4⋊Q8C2312yes—
123C2×#127C2311yescharacteristic
124C2×#125C2312yes—
125C2×#124—21no—
126C2×#124—21no—
127C2×#124—22no—
128C2×#124C2312yes—
129C2×#123—22no—
130C2×C22⋊Q8—21no—
131C2×C22⋊Q8—21no—
132C2×C22⋊Q8—22no—
133C2×C22⋊Q8—22no—
134C2×C22⋊Q8—24no—
135C2×C4×Q8—21no—
136C2×C4×Q8C2312yes—
137C2×C4×Q8C2311yescharacteristic
138C2×C4×Q8C2311yescharacteristic
139C2×C4×Q8C2311yescharacteristic
140C2×C4×D4—21no—
141C2×C4×D4C2312yes—
142C2×C4×D4C2312yes—
143C2×C4×D4C2312yes—
144C2×C4×D4C2312yes—
145C2×C4×D4C2311yescharacteristic
146C22×C4⋊C4—21no—
147C22×C42—21no—
148#501—28no—
149#500—24no—
150#498—42no—
151#489—28no—
152#486—24no—
153#486—42no—
154#479—28no—
155#477—24no—
156#476—24no—
157#476—28no—
158#475—24no—
159#474—28no—
160#463—28no—
161#459—24no—
162#446—24no—
163#446—24no—
164#442—28no—
165#440—28no—
166C4×SD16D418yes—
167C2×#107C2312yes—
168C2×#107C2312yes—
169C2×#107C2312yes—
170C2×#106C2312yes—
171C2×#106C2312yes—
172C2×#106C2312yes—
173C2×#106C2314yes—
174C2×#106C2311yescharacteristic
175C2×C4⋊C8C2311yescharacteristic
176C2×C4⋊C8C2311yescharacteristic
177C2×C4⋊C8C2312yes—
178C2×C4⋊C8C2312yes—
179C2×#103C2312yes—
180C2×#103C2312yes—
181C2×#103C2312yes—
182C2×#103C2311yescharacteristic
183C2×#102C2312yes—
184C2×#102C2314yes—
185C2×#102C2312yes—
186C2×#102C2312yes—
187C2×#102C2312yes—
188C2×#102C2312yes—
189C2×#102C2314yes—
190C2×#102C2311yescharacteristic
191C2×#97C2312yes—
192C2×#97C2312yes—
193C2×C4×C8C2311yescharacteristic
194C2×C4×C8C2311yescharacteristic
195#399—24no—
196#398—24no—
197#398—22no—
198#397—24no—
199#396—24no—
200#395—22no—
201#395—22no—
202#395—24no—
203#394—22no—
204#393—24no—
205#391—21no—
206#391—21no—
207#391—21no—
208#391—21no—
209#391—21no—
210#391—22no—
211#391—24no—
212#391D411yescharacteristic
213#391D411yescharacteristic
214#391C2312yes—
215#390—21no—
216#390—21no—
217#390—21no—
218#390C2314yes—
219#390C2314yes—
220#389—24no—
221#389D414yes—
222#389C2314yes—
223#389C2314yes—
224#389C2314yes—
225#389C2312yes—
226#388—24no—
227#388—24no—
228#387—22no—
229#385—24no—
230#385—24no—
231#385—24no—
232#384—21no—
233#384—21no—
234#384D411yescharacteristic
235#382C2314yes—
236#380—24no—
237C4×C4⋊C4—21no—
238C4×C4⋊C4C2311yescharacteristic
239C4×C4⋊C4C2311yescharacteristic
240C4×C4⋊C4C2312yes—
241C4×C4⋊C4C2314yes—
242C4×C22⋊C4—21no—
243C4×C22⋊C4—22no—
244#376—21no—
245C2×#95—24no—
246C43C2311yescharacteristic
247#340C2312yes—
248#340C2312yes—
249#340C2314yes—
250#340C2314yes—
251#340C2314yes—
252#340C2312yes—
253#336C2314yes—
254#336C2314yes—
255#336C2314yes—
256#336C2312yes—
257#336C2312yes—
258C22×Q8—21no—
259C22×Q8—22no—
260C22×Q8—22no—
261C22×Q8—21no—
262C22×Q8—21no—
263C22×Q8—44no—
264C22×Q8—44no—
265C22×Q8—82no—
266C22×Q8C2×D411yescharacteristic
267C22×Q8C2×D411yescharacteristic
268C22×Q8C2411yescharacteristic
269C22×D4—21no—
270C22×D4—21no—
271C22×D4C2412yes—
272C22×D4C2411yescharacteristic
273C23×C4—21no—
274C23×C4—22no—
275C23×C4—22no—
276C23×C4—22no—
277C23×C4—22no—
278C2×SD16—24no—
279C2×SD16C2×D414yes—
280C2×D8—24no—
281C2×D8—28no—
282C22×C8C2412yes—
283C22×C8C2412yes—
284C22×C8C2412yes—
285C22×C8C2411yescharacteristic
286C22×C8C2411yescharacteristic
287C4⋊Q8—24no—
288C4⋊Q8—24no—
289C4⋊Q8—22no—
290C4⋊Q8—22no—
291C4⋊Q8—22no—
292C4⋊Q8—22no—
293C4⋊Q8—22no—
294C4⋊Q8—22no—
295C4⋊Q8—42no—
296C4⋊Q8—42no—
297C4⋊Q8—44no—
298C4⋊Q8—42no—
299C4⋊Q8—48no—
300C4⋊Q8—82no—
301#127—22no—
302#127—22no—
303#125—24no—
304#125—42no—
305#124—24no—
306#124—42no—
307#124—44no—
308#124—44no—
309#124—84no—
310#123—48no—
311C22⋊Q8—48no—
312C22⋊Q8—48no—
313C22⋊Q8—44no—
314C22⋊Q8—44no—
315C22⋊Q8—88no—
316C4×Q8—22no—
317C4×Q8—22no—
318C4×Q8—22no—
319C4×Q8—22no—
320C4×Q8—22no—
321C4×Q8—24no—
322C4×Q8—42no—
323C4×Q8—44no—
324C4×Q8C2×D414yes—
325C4×D4—24no—
326C4×D4—24no—
327C4×D4—24no—
328C4×D4—24no—
329C4×D4—24no—
330C4×D4—22no—
331C4×D4—42no—
332C4×D4—42no—
333C4×D4—42no—
334C4×D4—44no—
335C4×D4C2×D414yes—
336C2×C4⋊C4—21no—
337C2×C4⋊C4—22no—
338C2×C4⋊C4—21no—
339C2×C4⋊C4—21no—
340C2×C4⋊C4—22no—
341C2×C4⋊C4—22no—
342C2×C4⋊C4—22no—
343C2×C4⋊C4—22no—
344C2×C4⋊C4—21no—
345C2×C4⋊C4—22no—
346C2×C4⋊C4—24no—
347C2×C4⋊C4—24no—
348C2×C4⋊C4—24no—
349C2×C4⋊C4—44no—
350C2×C4⋊C4C2×D411yescharacteristic
351C2×C4⋊C4C2×D411yescharacteristic
352C2×C4⋊C4C2×D412yes—
353C2×C4⋊C4C2×D412yes—
354C2×C4⋊C4C2×D414yes—
355C2×C4⋊C4C2×D414yes—
356C2×C4⋊C4C2412yes—
357C2×C4⋊C4C2412yes—
358C2×C4⋊C4C2412yes—
359C2×C4⋊C4C2412yes—
360C2×C4⋊C4C2411yescharacteristic
361C2×C4⋊C4C2411yescharacteristic
362C2×C4⋊C4C2412yes—
363C2×C22⋊C4—21no—
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—22no—
371C2×C22⋊C4—24no—
372C2×C22⋊C4—24no—
373C2×C22⋊C4—24no—
374C2×C42—21no—
375C2×C42—22no—
376C2×C42—21no—
377C2×C42—21no—
378C2×C42—22no—
379C2×C42—82no—
380C2×C42C2×D411yescharacteristic
381C2×C42C2×D411yescharacteristic
382C2×C42C2×D412yes—
383C2×C42C2412yes—
384C2×C42C2412yes—
385C2×C42C2412yes—
386C2×C42C2411yescharacteristic
387#107—24no—
388#107—24no—
389#107—24no—
390#106—24no—
391#106—24no—
392#106—24no—
393#106—44no—
394#106C2×D414yes—
395C4⋊C8—24no—
396C4⋊C8—24no—
397C4⋊C8—22no—
398C4⋊C8—22no—
399#103—24no—
400#103—24no—
401#103—24no—
402#103C2×D414yes—
403#102—24no—
404#102—24no—
405#102—24no—
406#102—24no—
407#102—24no—
408#102—28no—
409#102—44no—
410#102C2×D414yes—
411#97—24no—
412#97—24no—
413C4×C8—22no—
414C4×C8C2×D414yes—
415#95—24no—
416#95—24no—
417#95—24no—
418#95—22no—
419#95—22no—
420#95—21no—
421#95—22no—
422#95—22no—
423#95—21no—
424#95—24no—
425#95—24no—
426#95C2×D414yes—
427#95C2×D414yes—
428C24—21no—
429C24—22no—
430C2×Q8—22no—
431C2×Q8—22no—
432C2×Q8—22no—
433C2×Q8—22no—
434C2×Q8—22no—
435C2×Q8—22no—
436C2×Q8—21no—
437C2×Q8—22no—
438C2×Q8—22no—
439C2×Q8—44no—
440C2×Q8—44no—
441C2×Q8—44no—
442C2×Q8—42no—
443C2×Q8—42no—
444C2×Q8—42no—
445C2×Q8—42no—
446C2×Q8—42no—
447C2×Q8—82no—
448C2×Q8—84no—
449C2×Q8—82no—
450C2×Q8—82no—
451C2×Q8—816no—
452C2×Q8C22×D412yes—
453C2×D4—22no—
454C2×D4—21no—
455C2×D4—22no—
456C2×D4—22no—
457C2×D4—24no—
458C2×D4—24no—
459C2×D4—42no—
460C2×D4—42no—
461C2×D4—42no—
462C2×D4—42no—
463C2×D4—82no—
464C2×D4C22×D412yes—
465C2×D42−1+414yes—
466C22×C4—22no—
467C22×C4—21no—
468C22×C4—21no—
469C22×C4—21no—
470C22×C4—28no—
471C22×C4—24no—
472C22×C4—22no—
473C22×C4—22no—
474C22×C4—22no—
475C22×C4—22no—
476C22×C4—44no—
477C22×C4—44no—
478C22×C4—44no—
479C22×C4—44no—
480C22×C4—42no—
481C22×C4—42no—
482C22×C4—82no—
483C22×C4—82no—
484C22×C4—82no—
485C22×C4—84no—
486C22×C4—84no—
487C22×C4C22×D412yes—
488C22×C4C22×D412yes—
489C22×C4C22×D412yes—
490C22×C4C22×D412yes—
491C22×C4C22×D412yes—
492C22×C4C22×D412yes—
493C22×C4C22×D411yescharacteristic
494C22×C4C22×D411yescharacteristic
495C22×C4C2511yescharacteristic
496SD16—28no—
497D8—48no—
498C2×C8—22no—
499C2×C8—22no—
500C2×C8—22no—
501C2×C8—22no—
502C2×C8—22no—
503C2×C8—24no—
504C2×C8—22no—
505C2×C8—22no—
506C2×C8—24no—
507C2×C8C22×D412yes—
508C2×C8C22×D412yes—
509C4⋊C4—24no—
510C4⋊C4—22no—
511C4⋊C4—22no—
512C4⋊C4—24no—
513C4⋊C4—24no—
514C4⋊C4—21no—
515C4⋊C4—21no—
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—44no—
526C4⋊C4—44no—
527C4⋊C4—44no—
528C4⋊C4—44no—
529C4⋊C4—44no—
530C4⋊C4—44no—
531C4⋊C4—44no—
532C4⋊C4—44no—
533C4⋊C4—42no—
534C4⋊C4—42no—
535C4⋊C4—42no—
536C4⋊C4—84no—
537C4⋊C4—84no—
538C4⋊C4—84no—
539C4⋊C4C22×D412yes—
540C4⋊C4C22×D412yes—
541C4⋊C42−1+414yes—
542C22⋊C4—28no—
543C22⋊C4—24no—
544C22⋊C4—44no—
545C22⋊C4—44no—
546C22⋊C4—44no—
547C22⋊C4—44no—
548C22⋊C4—44no—
549C22⋊C4—48no—
550C22⋊C4—48no—
551C22⋊C4—84no—
552C22⋊C4—84no—
553C42—24no—
554C42—22no—
555C42—22no—
556C42—22no—
557C42—22no—
558C42—22no—
559C42—22no—
560C42—22no—
561C42—22no—
562C42—42no—
563C42—44no—
564C42—42no—
565C42—42no—
566C42—82no—
567C42C22×D412yes—
568C42H(5,𝔽2)12yes—
569C23—22no—
570C23—24no—
571C23—42no—
572C23—41no—
573C23—41no—
574C23—44no—
575C23—42no—
576C23—44no—
577C23—42no—
578C23—42no—
579C23C23×D411yescharacteristic
580Q8—22no—
581Q8—22no—
582Q8—44no—
583Q8—44no—
584Q8—42no—
585Q8—88no—
586Q8—84no—
587Q8—162no—
588Q8—162no—
589Q8—164no—
590D4—22no—
591D4—24no—
592D4—42no—
593D4—42no—
594D4—44no—
595D4—84no—
596D4—162no—
597D4#53614yes—
598C2×C4—22no—
599C2×C4—22no—
600C2×C4—22no—
601C2×C4—24no—
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—22no—
611C2×C4—22no—
612C2×C4—24no—
613C2×C4—22no—
614C2×C4—22no—
615C2×C4—22no—
616C2×C4—22no—
617C2×C4—22no—
618C2×C4—22no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—41no—
623C2×C4—42no—
624C2×C4—41no—
625C2×C4—48no—
626C2×C4—44no—
627C2×C4—42no—
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—88no—
637C2×C4—88no—
638C2×C4—164no—
639C2×C4—164no—
640C2×C4D4212yes—
641C2×C4D4212yes—
642C2×C4#57512yes—
643C2×C4#57712yes—
644C2×C4C23×D411yescharacteristic
645C2×C4C2×H(5,𝔽2)11yescharacteristic
646C2×C4C2×2−1+412yes—
647C2×C4C2×2−1+412yes—
648C8—22no—
649C8—22no—
650C8—42no—
651C8—42no—
652C8—42no—
653C22—24no—
654C22—42no—
655C22—44no—
656C22—42no—
657C22—44no—
658C22—82no—
659C22—82no—
660C22—84no—
661C22—82no—
662C22—84no—
663C22C2×D4211yescharacteristic
664C22#348612yes—
665C22C2×#57511yescharacteristic
666C22C2×#57711yescharacteristic
667C22#358512yes—
668C4—22no—
669C4—42no—
670C4—42no—
671C4—42no—
672C4—42no—
673C4—42no—
674C4—42no—
675C4—42no—
676C4—42no—
677C4—42no—
678C4—44no—
679C4—82no—
680C4—82no—
681C4—82no—
682C4—82no—
683C4#327514yes—
684C4C2×#53611yescharacteristic
685C4#349911yescharacteristic
686C4#355212yes—
687C2—44no—
688C2—162no—
689C2#3767511yescharacteristic
690C2C2×#327511yescharacteristic
691C2#6108211yescharacteristic
692C2#6183312yes—
693C2#6185012yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#459C22
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
1. C2×#3385C2semidirect
3. #39168C2semidirect
4. #36276C2semidirect
6. #30974C2semidirect
7. #30830C2semidirect
8. #30278C2semidirect
9. #29867C2semidirect
10. #28682C2semidirect
11. #28648C2semidirect
12. #26388C2semidirect
13. #26184C2semidirect
15. #25837C2semidirect
16. #25834C2semidirect
17. #25648C2semidirect
18. #22242C2semidirect
19. #21117C2semidirect

Maximal subgroups

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

subgroupindex
C2×#33852
C2×#33732
#391682
#362762
#330302
#309742
#308302
#302782
#298672
#286822
#286482
#263882
#261842
#259302
#258372
#258342
#256482
#222422
#211172