Finite Group Databasebeta

← #6816550

Subgroups — #6816550

Order 512.

1906 conjugacy classes of subgroups.

The subgroup lattice has 705 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3386C211yesmaximal, characteristic, complemented
2C2×#3353C211yesmaximal, characteristic
3#39744C211yesmaximal, characteristic, complemented
4#39360C211yesmaximal, characteristic
5#33312C211yesmaximal, characteristic, complemented
6#31084C214yesmaximal, complemented
7#31081C214yesmaximal, complemented
8#29866C211yesmaximal, characteristic, complemented
9#29864C211yesmaximal, characteristic, complemented
10#28712C211yesmaximal, characteristic, complemented
11#28683C211yesmaximal, characteristic, complemented
12#26077C214yesmaximal, complemented
13#25840C211yesmaximal, characteristic, complemented
14#25838C211yesmaximal, characteristic, complemented
15#25705C211yesmaximal, characteristic, complemented
16#25703C211yesmaximal, characteristic
17#25045C214yesmaximal, complemented
18#21631C211yesmaximal, characteristic, complemented
19#21274C211yesmaximal, characteristic, complemented
20C2×#557C2211yescharacteristic
21C2×#556C2211yescharacteristic
22C2×D4×Q8C2211yescharacteristic
23C2×#547C2211yescharacteristic
24C2×#536—21no—
25#3386—28no—
26#3353—28no—
27C2×#501C2212yes—
28C2×#500C2211yescharacteristic
29C2×#500C2211yescharacteristic
30C2×#500C2211yescharacteristic
31C2×#500C2211yescharacteristic
32C2×#484C2212yes—
33C2×#482C2212yes—
34C2×#477C2212yes—
35C2×#475C2211yescharacteristic
36C2×#475C2211yescharacteristic
37C2×#478C2212yes—
38C2×#474C2211yescharacteristic
39C2×#474C2211yescharacteristic
40C2×#468C2212yes—
41C2×#446C2211yescharacteristic
42C2×#440C2212yes—
43C2×C4×SD16C2211yescharacteristic
44C2×#436C2211yescharacteristic
45C2×#428C2212yes—
46#2868C2211yescharacteristic
47#2868C2211yescharacteristic
48#2721—24no—
49#2680—24no—
50#2658—21no—
51#2621—21no—
52#2547C2211yescharacteristic
53#2547C2211yescharacteristic
54#2546—21no—
55#2546C2211yescharacteristic
56#2546C2211yescharacteristic
57#2545—21no—
58#2441C2214yes—
59#2440C2214yes—
60#2434—21no—
61#2432—21no—
62#2423C2214yes—
63#2422C2214yes—
64#2327—21no—
65C4×#124C2211yescharacteristic
66#2295C2211yescharacteristic
67#2293C2211yescharacteristic
68#2087C2214yes—
69#2084C2214yes—
70#2083C2214yes—
71#2076C2214yes—
72#2072C2214yes—
73#2071C2214yes—
74#2059C2214yes—
75#2057C2214yes—
76#2036C2214yes—
77#2033C2214yes—
78#1988C2211yescharacteristic
79#1988C2211yescharacteristic
80#1988C2211yescharacteristic
81#1988C2211yescharacteristic
82#1984C2211yescharacteristic
83#1968C2212yes—
84#1954C2211yescharacteristic
85#1944C2214yes—
86#1941C2212yes—
87#1934C2212yes—
88#1933C2212yes—
89#1923C2214yes—
90#1863C2214yes—
91#1862C2214yes—
92#1854C2214yes—
93#1852C2214yes—
94#1845C2214yes—
95#1845C2214yes—
96#1831C2211yescharacteristic
97#1809C2214yes—
98C4×#106C2211yescharacteristic
99C4×#103C2211yescharacteristic
100C4×#102C2211yescharacteristic
101#1753C2214yes—
102C42×C8C2211yescharacteristic
103C22×#41—21no—
104C23×Q8—21no—
105C22×SD16C2311yescharacteristic
106C22×M16C2312yes—
107#557—24no—
108#557—42no—
109#556—24no—
110#556—42no—
111D4×Q8—24no—
112D4×Q8—42no—
113#547—24no—
114#547—42no—
115#536—48no—
116C2×C4⋊Q8—21no—
117C2×C4⋊Q8—21no—
118C2×C4⋊Q8—21no—
119C2×C4⋊Q8—21no—
120C2×C4⋊Q8—21no—
121C2×C4⋊Q8—21no—
122C2×C4⋊Q8—21no—
123C2×C4⋊Q8C2311yescharacteristic
124C2×C4⋊Q8C2311yescharacteristic
125C2×C4⋊Q8C2311yescharacteristic
126C2×C4⋊Q8C2311yescharacteristic
127C2×C4⋊Q8C2312yes—
128C2×#126—22no—
129C2×#125—22no—
130C2×#125C2312yes—
131C2×#124—21no—
132C2×#124—21no—
133C2×#124—21no—
134C2×#124D411yescharacteristic
135C2×#123—24no—
136C2×#123—22no—
137C2×C22⋊Q8—21no—
138C2×C22⋊Q8—21no—
139C2×C22⋊Q8—22no—
140C2×C22⋊Q8—24no—
141C2×C22⋊Q8C2312yes—
142C2×#121C2312yes—
143C2×C4×Q8—21no—
144C2×C4×Q8—21no—
145C2×C4×Q8C2312yes—
146C2×C4×Q8C2311yescharacteristic
147C2×C4×Q8C2311yescharacteristic
148C2×C4×D4C2312yes—
149C2×C4×D4C2311yescharacteristic
150C2×C4×D4C2311yescharacteristic
151C2×#117—21no—
152C2×#117C2312yes—
153C22×C4⋊C4C2312yes—
154#501—28no—
155#500—24no—
156#500—24no—
157#500—42no—
158#500—42no—
159#484—28no—
160#482—28no—
161#478—28no—
162#477—28no—
163#475—24no—
164#475—24no—
165#474—24no—
166#474—24no—
167#468—28no—
168#446—24no—
169#440—28no—
170C4×SD16D418yes—
171#436—24no—
172#428—28no—
173C2×#107C2312yes—
174C2×#107C2312yes—
175C2×#107C2312yes—
176C2×#107C2312yes—
177C2×#107C2312yes—
178C2×#107C2312yes—
179C2×#107C2314yes—
180C2×#107C2314yes—
181C2×#106C2312yes—
182C2×#106C2312yes—
183C2×#106C2312yes—
184C2×#106C2311yescharacteristic
185C2×C4⋊C8C2312yes—
186C2×C4⋊C8C2311yescharacteristic
187C2×C4⋊C8C2311yescharacteristic
188C2×#103C2312yes—
189C2×#103C2312yes—
190C2×#103C2312yes—
191C2×#103C2311yescharacteristic
192C2×#102C2312yes—
193C2×#102C2312yes—
194C2×#102C2312yes—
195C2×#102C2311yescharacteristic
196C2×C22⋊C8C2312yes—
197C2×C22⋊C8C2314yes—
198C2×#97C2312yes—
199C2×C4×C8C2312yes—
200C2×C4×C8C2312yes—
201C2×C4×C8C2311yescharacteristic
202C2×C4×C8C2311yescharacteristic
203#399—24no—
204#399—24no—
205#398—22no—
206#398—22no—
207#398—24no—
208#398—24no—
209#397—22no—
210#397—22no—
211#397—24no—
212#396—24no—
213#395—24no—
214#394—24no—
215#391—21no—
216#391—21no—
217#391—21no—
218#391—21no—
219#391D411yescharacteristic
220#391C2312yes—
221#390D411yescharacteristic
222#390C2312yes—
223#389—24no—
224#389C2314yes—
225#389C2314yes—
226#389C2312yes—
227#389C2312yes—
228#388—24no—
229#388—22no—
230#388D414yes—
231#387—22no—
232#387—24no—
233#387—24no—
234#385—24no—
235#385—24no—
236#384—22no—
237#384—22no—
238#384C2314yes—
239#384C2314yes—
240#382D411yescharacteristic
241#380C2314yes—
242#380C2314yes—
243C4×C4⋊C4—21no—
244C4×C4⋊C4C2311yescharacteristic
245C4×C4⋊C4C2311yescharacteristic
246C4×C22⋊C4—21no—
247C4×C22⋊C4C2314yes—
248#376—21no—
249C43C2311yescharacteristic
250#340C2312yes—
251#340C2314yes—
252#340C2312yes—
253#340C2314yes—
254#340C2314yes—
255#340C2314yes—
256#340C2312yes—
257#336C2314yes—
258#336C2314yes—
259C2×#41—44no—
260C2×#41—44no—
261C2×#41—82no—
262C22×Q8—21no—
263C22×Q8—21no—
264C22×Q8—21no—
265C22×Q8—21no—
266C22×Q8—22no—
267C22×Q8—44no—
268C22×Q8—44no—
269C22×Q8—82no—
270C22×Q8C2×D411yescharacteristic
271C22×Q8C2411yescharacteristic
272C22×D4—22no—
273C22×D4C2×D411yescharacteristic
274C22×D4C2411yescharacteristic
275C23×C4—21no—
276C23×C4—22no—
277C23×C4—22no—
278C23×C4C2412yes—
279C2×SD16—24no—
280C2×SD16C2×D414yes—
281C2×M16—24no—
282C2×M16—28no—
283C22×C8C2412yes—
284C22×C8C2412yes—
285C22×C8C2411yescharacteristic
286C22×C8C2412yes—
287C22×C8C2411yescharacteristic
288C4⋊Q8—24no—
289C4⋊Q8—22no—
290C4⋊Q8—22no—
291C4⋊Q8—22no—
292C4⋊Q8—22no—
293C4⋊Q8—22no—
294C4⋊Q8—22no—
295C4⋊Q8—22no—
296C4⋊Q8—22no—
297C4⋊Q8—42no—
298C4⋊Q8—44no—
299C4⋊Q8—44no—
300C4⋊Q8—44no—
301C4⋊Q8—44no—
302C4⋊Q8—82no—
303C4⋊Q8—82no—
304C4⋊Q8—82no—
305#126—48no—
306#125—24no—
307#125—42no—
308#125—48no—
309#124—24no—
310#124—44no—
311#124—44no—
312#124—82no—
313#123—48no—
314#123—416no—
315C22⋊Q8—24no—
316C22⋊Q8—48no—
317C22⋊Q8—416no—
318C22⋊Q8—42no—
319C22⋊Q8—44no—
320C22⋊Q8—44no—
321#121—24no—
322#121—42no—
323C4×Q8—22no—
324C4×Q8—22no—
325C4×Q8—22no—
326C4×Q8—24no—
327C4×Q8—42no—
328C4×Q8—44no—
329C4×Q8—44no—
330C4×Q8C2×D414yes—
331C4×D4—24no—
332C4×D4—22no—
333C4×D4—22no—
334C4×D4—22no—
335C4×D4—42no—
336C4×D4C2×D414yes—
337#117—24no—
338#117—42no—
339#117—44no—
340C2×C4⋊C4—21no—
341C2×C4⋊C4—24no—
342C2×C4⋊C4—22no—
343C2×C4⋊C4—22no—
344C2×C4⋊C4—22no—
345C2×C4⋊C4—22no—
346C2×C4⋊C4—22no—
347C2×C4⋊C4—21no—
348C2×C4⋊C4—22no—
349C2×C4⋊C4—24no—
350C2×C4⋊C4—24no—
351C2×C4⋊C4—21no—
352C2×C4⋊C4—21no—
353C2×C4⋊C4—22no—
354C2×C4⋊C4—21no—
355C2×C4⋊C4—24no—
356C2×C4⋊C4—22no—
357C2×C4⋊C4—24no—
358C2×C4⋊C4—24no—
359C2×C4⋊C4—24no—
360C2×C4⋊C4—42no—
361C2×C4⋊C4C2×D411yescharacteristic
362C2×C4⋊C4C2×D411yescharacteristic
363C2×C4⋊C4C2×D412yes—
364C2×C4⋊C4C2412yes—
365C2×C4⋊C4C2412yes—
366C2×C4⋊C4C2412yes—
367C2×C4⋊C4C2412yes—
368C2×C4⋊C4C2412yes—
369C2×C4⋊C4C2412yes—
370C2×C4⋊C4C2411yescharacteristic
371C2×C4⋊C4C2411yescharacteristic
372C2×C22⋊C4—21no—
373C2×C22⋊C4—22no—
374C2×C22⋊C4—24no—
375C2×C22⋊C4—22no—
376C2×C22⋊C4—22no—
377C2×C22⋊C4—22no—
378C2×C22⋊C4—22no—
379C2×C22⋊C4—22no—
380C2×C22⋊C4—24no—
381C2×C22⋊C4C2×D412yes—
382C2×C22⋊C4C2×D414yes—
383C2×C22⋊C4C2×D414yes—
384C2×C42—21no—
385C2×C42—21no—
386C2×C42—22no—
387C2×C42—22no—
388C2×C42—21no—
389C2×C42C2×D411yescharacteristic
390C2×C42C2×D412yes—
391C2×C42C2×D411yescharacteristic
392C2×C42C2412yes—
393C2×C42C2412yes—
394C2×C42C2411yescharacteristic
395#107—24no—
396#107—24no—
397#107—24no—
398#107—24no—
399#107—24no—
400#107—24no—
401#107—44no—
402#107—44no—
403#106—24no—
404#106—24no—
405#106—24no—
406#106C2×D414yes—
407C4⋊C8—24no—
408C4⋊C8—22no—
409C4⋊C8—22no—
410#103—24no—
411#103—24no—
412#103—24no—
413#103C2×D414yes—
414#102—24no—
415#102—24no—
416#102—24no—
417#102C2×D414yes—
418C22⋊C8—24no—
419C22⋊C8—28no—
420#97—24no—
421C4×C8—24no—
422C4×C8—24no—
423C4×C8—22no—
424C4×C8C2×D414yes—
425#95—22no—
426#95—22no—
427#95—22no—
428#95—22no—
429#95—21no—
430#95—21no—
431#95—24no—
432#95—24no—
433#95—24no—
434#95—24no—
435#95C2×D414yes—
436#95C2×D414yes—
437C24—21no—
438C24C22×D412yes—
439#41—816no—
440C2×Q8—22no—
441C2×Q8—22no—
442C2×Q8—22no—
443C2×Q8—21no—
444C2×Q8—22no—
445C2×Q8—22no—
446C2×Q8—44no—
447C2×Q8—44no—
448C2×Q8—42no—
449C2×Q8—42no—
450C2×Q8—42no—
451C2×Q8—42no—
452C2×Q8—42no—
453C2×Q8—42no—
454C2×Q8—42no—
455C2×Q8—42no—
456C2×Q8—42no—
457C2×Q8—42no—
458C2×Q8—42no—
459C2×Q8—42no—
460C2×Q8—82no—
461C2×Q8—816no—
462C2×Q8C22×D412yes—
463C2×D4—22no—
464C2×D4—22no—
465C2×D4—22no—
466C2×D4—21no—
467C2×D4—22no—
468C2×D4—22no—
469C2×D4—44no—
470C2×D4—44no—
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—24no—
480C22×C4—24no—
481C22×C4—22no—
482C22×C4—22no—
483C22×C4—22no—
484C22×C4—22no—
485C22×C4—22no—
486C22×C4—22no—
487C22×C4—44no—
488C22×C4—44no—
489C22×C4—44no—
490C22×C4—42no—
491C22×C4—42no—
492C22×C4—44no—
493C22×C4—82no—
494C22×C4—82no—
495C22×C4C22×D412yes—
496C22×C4C22×D412yes—
497C22×C4C22×D412yes—
498C22×C4C22×D411yescharacteristic
499C22×C4C22×D412yes—
500C22×C4C22×D412yes—
501C22×C4C22×D411yescharacteristic
502C22×C4C2511yescharacteristic
503SD16—28no—
504M16—48no—
505C2×C8—22no—
506C2×C8—22no—
507C2×C8—22no—
508C2×C8—22no—
509C2×C8—22no—
510C2×C8—24no—
511C2×C8—24no—
512C2×C8C22×D412yes—
513C2×C8C22×D412yes—
514C2×C82−1+414yes—
515C2×C82−1+414yes—
516C4⋊C4—24no—
517C4⋊C4—22no—
518C4⋊C4—22no—
519C4⋊C4—24no—
520C4⋊C4—21no—
521C4⋊C4—21no—
522C4⋊C4—22no—
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—22no—
532C4⋊C4—22no—
533C4⋊C4—22no—
534C4⋊C4—44no—
535C4⋊C4—44no—
536C4⋊C4—44no—
537C4⋊C4—44no—
538C4⋊C4—44no—
539C4⋊C4—44no—
540C4⋊C4—48no—
541C4⋊C4—42no—
542C4⋊C4—42no—
543C4⋊C4—42no—
544C4⋊C4—42no—
545C4⋊C4—44no—
546C4⋊C4—48no—
547C4⋊C4—48no—
548C4⋊C4—44no—
549C4⋊C4—84no—
550C4⋊C4—84no—
551C4⋊C4—84no—
552C4⋊C4C22×D412yes—
553C4⋊C4C22×D412yes—
554C22⋊C4—24no—
555C22⋊C4—24no—
556C22⋊C4—28no—
557C22⋊C4—28no—
558C22⋊C4—44no—
559C22⋊C4—44no—
560C22⋊C4—44no—
561C22⋊C4—44no—
562C22⋊C4—48no—
563C22⋊C4—44no—
564C22⋊C4—44no—
565C22⋊C4—84no—
566C42—24no—
567C42—22no—
568C42—24no—
569C42—22no—
570C42—22no—
571C42—22no—
572C42—22no—
573C42—44no—
574C42—44no—
575C42—42no—
576C42—42no—
577C42AGL(1,ℤ/8)14yes—
578C42C22×D412yes—
579C42H(5,𝔽2)12yes—
580C23—22no—
581C23—22no—
582C23—24no—
583C23—22no—
584C23—22no—
585C23—24no—
586C23—42no—
587C23—41no—
588C23—41no—
589C23—42no—
590C23C23×D411yescharacteristic
591Q8—22no—
592Q8—22no—
593Q8—44no—
594Q8—42no—
595Q8—88no—
596Q8—84no—
597Q8—84no—
598Q8—84no—
599Q8—84no—
600D4—22no—
601D4—22no—
602D4—44no—
603D4—42no—
604D4—88no—
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—24no—
617C2×C4—22no—
618C2×C4—22no—
619C2×C4—24no—
620C2×C4—42no—
621C2×C4—41no—
622C2×C4—41no—
623C2×C4—42no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—48no—
627C2×C4—44no—
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—42no—
640C2×C4—42no—
641C2×C4—42no—
642C2×C4—42no—
643C2×C4—42no—
644C2×C4—42no—
645C2×C4—42no—
646C2×C4—88no—
647C2×C4—88no—
648C2×C4D4212yes—
649C2×C4D4212yes—
650C2×C4C2×AGL(1,ℤ/8)14yes—
651C2×C4C2×AGL(1,ℤ/8)12yes—
652C2×C4#57612yes—
653C2×C4#57814yes—
654C2×C4C23×D411yescharacteristic
655C2×C4C2×H(5,𝔽2)11yescharacteristic
656C2×C4C2×2−1+412yes—
657C2×C4C2×2−1+412yes—
658C8—22no—
659C8—24no—
660C8—42no—
661C8—42no—
662C8#53614yescomplemented
663C22—44no—
664C22—44no—
665C22—42no—
666C22—42no—
667C22—42no—
668C22—42no—
669C22—42no—
670C22—42no—
671C22—82no—
672C22—82no—
673C22C2×D4211yescharacteristic
674C22#349912yes—
675C22C22×AGL(1,ℤ/8)11yescharacteristic
676C22C2×#57611yescharacteristic
677C22C2×#57812yes—
678C4—24no—
679C4—24no—
680C4—22no—
681C4—42no—
682C4—42no—
683C4—42no—
684C4—42no—
685C4—42no—
686C4—42no—
687C4—82no—
688C4—82no—
689C4—82no—
690C4—82no—
691C4—82no—
692C4—82no—
693C4#327414yes—
694C4C2×#53611yescharacteristic
695C4#349911yescharacteristic
696C4#355212yes—
697C2—82no—
698C2—82no—
699C2#3936011yescharacteristic
700C2C2×#327411yescharacteristic
701C2#6109111yescharacteristic
702C2#6247912yes—
703C2#6248812yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#547C22
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×#3386C2semidirect
3. #39744C2semidirect
5. #33312C2semidirect
6. #31084C2semidirect
7. #31081C2semidirect
8. #29866C2semidirect
9. #29864C2semidirect
10. #28712C2semidirect
11. #28683C2semidirect
12. #26077C2semidirect
13. #25840C2semidirect
14. #25838C2semidirect
15. #25705C2semidirect
17. #25045C2semidirect
18. #21631C2semidirect
19. #21274C2semidirect
662. C8#536semidirect

Maximal subgroups

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

subgroupindex
C2×#33862
C2×#33532
#397442
#393602
#333122
#310842
#310812
#298662
#298642
#287122
#286832
#260772
#258402
#258382
#257052
#257032
#250452
#216312
#212742