Finite Group Databasebeta

← #6519929

Subgroups — #6519929

Order 512.

1882 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×#3254C211yesmaximal, characteristic
2C2×#3170C211yesmaximal, characteristic, complemented
3#37640C211yesmaximal, characteristic
4#34374C211yesmaximal, characteristic, complemented
5#28590C214yesmaximal, complemented
6#25449C212yesmaximal, complemented
7#25365C212yesmaximal, complemented
8#25320C212yesmaximal, complemented
9#25275C212yesmaximal, complemented
10#24817C212yesmaximal, complemented
11#24765C214yesmaximal, complemented
12#24733C212yesmaximal, complemented
13#24682C212yesmaximal, complemented
14#24633C212yesmaximal, complemented
15#21894C211yesmaximal, characteristic, complemented
16#21401C211yesmaximal, characteristic
17#21143C211yesmaximal, characteristic, complemented
18C2×#533C2211yescharacteristic
19C2×#533C2211yescharacteristic
20C2×#532C2211yescharacteristic
21#3254—28no—
22#3170—28no—
23C2×#500C2211yescharacteristic
24C2×#494C2211yescharacteristic
25C2×#487C2211yescharacteristic
26C2×#487C2211yescharacteristic
27C2×#477C2212yes—
28C2×#474C2212yes—
29C2×#468C2214yes—
30C2×#467C2212yes—
31C2×#467C2212yes—
32C2×#464C2214yes—
33C2×#460C2212yes—
34C2×#461C2212yes—
35C2×C4×Q16C2212yes—
36C2×C4×Q16C2212yes—
37C2×#424C2212yes—
38C2×#423C2211yescharacteristic
39C22×C4×C8C2211yescharacteristic
40#2820—22no—
41#2624—22no—
42#2620C2214yes—
43#2616—22no—
44#2567C2212yes—
45#2548C2212yes—
46#2547C2212yes—
47#2542C2212yes—
48#2538—21no—
49#2538—21no—
50#2382—22no—
51#2365C2212yes—
52#2365C2212yes—
53#2355—21no—
54#2355C2212yes—
55Q8×C4⋊C4—21no—
56Q8×C4⋊C4C2212yes—
57#2323—22no—
58#2323C2214yes—
59#2314—22no—
60#2285C2211yescharacteristic
61#2285C2211yescharacteristic
62#2083C2214yes—
63#2072C2214yes—
64#2068C2214yes—
65#2067C2214yes—
66#2056C2214yes—
67#2055C2214yes—
68#2042C2214yes—
69#2015C2214yes—
70#1990C2211yescharacteristic
71#1990C2211yescharacteristic
72#1988C2212yes—
73#1983C2214yes—
74#1981C2212yes—
75#1957C2211yescharacteristic
76#1957C2211yescharacteristic
77#1947C2214yes—
78#1939C2212yes—
79#1939C2212yes—
80#1939C2212yes—
81#1939C2212yes—
82#1928C2214yes—
83#1925C2214yes—
84#1894C2214yes—
85#1893C2212yes—
86#1893C2212yes—
87#1880C2214yes—
88#1878C2214yes—
89#1803C2212yes—
90#1803C2212yes—
91#1803C2214yes—
92C4×#102C2212yes—
93C4×#102C2212yes—
94#1729C2212yes—
95C22×#41—21no—
96C22×Q16C2312yes—
97C22×Q16C2312yes—
98C22×SD16C2314yes—
99C22×M16C2312yes—
100C23×C8C2311yescharacteristic
101C23×C8C2311yescharacteristic
102#533—24no—
103#533—24no—
104#533—24no—
105#533D418yes—
106#532—24no—
107#532—24no—
108C2×C4⋊Q8C2311yescharacteristic
109C2×C4⋊Q8C2311yescharacteristic
110C2×C4⋊Q8C2311yescharacteristic
111C2×C4⋊Q8C2311yescharacteristic
112C2×C4⋊Q8C2311yescharacteristic
113C2×#127C2311yescharacteristic
114C2×#125—21no—
115C2×#125—21no—
116C2×#124—21no—
117C2×#124C2314yes—
118C2×C22⋊Q8—21no—
119C2×C22⋊Q8—21no—
120C2×C22⋊Q8—22no—
121C2×C22⋊Q8—22no—
122C2×C22⋊Q8C2312yes—
123C2×C22⋊Q8C2312yes—
124C2×#121—21no—
125C2×#121C2312yes—
126C2×C4×Q8—22no—
127C2×C4×Q8—21no—
128C2×C4×Q8—21no—
129C2×C4×Q8C2312yes—
130C2×C4×Q8C2312yes—
131C2×C4×Q8C2314yes—
132C2×C4×D4—21no—
133C2×C4×D4C2312yes—
134C2×#117—21no—
135C2×#117—21no—
136C2×#117—21no—
137C2×#117C2312yes—
138C22×C42C2311yescharacteristic
139#500—24no—
140#494—24no—
141#487—24no—
142#487—24no—
143#477—28no—
144#474—28no—
145#468—216no—
146#467—28no—
147#467—28no—
148#464—48no—
149#461—28no—
150#460—28no—
151C4×Q16—28no—
152C4×Q16—28no—
153C2×#107C2314yes—
154C2×#107C2312yes—
155C2×#107C2312yes—
156#424—44no—
157#423—24no—
158C2×C4⋊C8C2312yes—
159C2×C4⋊C8C2314yes—
160C2×C4⋊C8C2312yes—
161C2×C4⋊C8C2314yes—
162C2×#103C2314yes—
163C2×#103C2312yes—
164C2×#103C2312yes—
165C2×#103C2312yes—
166C2×#103C2312yes—
167C2×#103C2314yes—
168C2×#103C2314yes—
169C2×#102C2312yes—
170C2×#102C2314yes—
171C2×#102C2312yes—
172C2×#102C2312yes—
173C2×#102C2312yes—
174C2×C4×C8—24no—
175C2×C4×C8C2311yescharacteristic
176C2×C4×C8C2311yescharacteristic
177C2×C4×C8C2311yescharacteristic
178C2×C4×C8C2311yescharacteristic
179#399—24no—
180#397—24no—
181#396—24no—
182#396—24no—
183#395—24no—
184#394—24no—
185#394—24no—
186#394—24no—
187#391—22no—
188#391C2314yes—
189#391C2314yes—
190#390—22no—
191#390—22no—
192#390C2314yes—
193#389—22no—
194#389—22no—
195#389—22no—
196#389—24no—
197#389—24no—
198#389C2314yes—
199#388—24no—
200#388—24no—
201#387—24no—
202#387—24no—
203#387—24no—
204#387—24no—
205#384—22no—
206#384D411yescharacteristic
207#384D411yescharacteristic
208#384C2314yes—
209#384C2311yescharacteristic
210#384C2311yescharacteristic
211#382—22no—
212#382—21no—
213#382D412yes—
214#380—21no—
215#380—22no—
216#380—22no—
217#380D412yes—
218#380D412yes—
219#380C2312yes—
220C4×C4⋊C4—22no—
221C4×C4⋊C4—22no—
222C4×C4⋊C4—22no—
223C4×C4⋊C4—21no—
224C4×C4⋊C4C2314yes—
225C4×C4⋊C4C2312yes—
226C4×C4⋊C4C2312yes—
227C4×C4⋊C4C2312yes—
228#340C2314yes—
229#340C2312yes—
230#340C2314yes—
231#340C2312yes—
232#340C2312yes—
233#340C2312yes—
234#336C2314yes—
235#336C2314yes—
236C2×#41—44no—
237C2×#41—82no—
238C2×#41—82no—
239C22×Q8—21no—
240C22×Q8—21no—
241C22×Q8—21no—
242C22×Q8C2412yes—
243C22×Q8C2412yes—
244C22×D4—22no—
245C22×D4—21no—
246C22×D4—21no—
247C22×D4C2412yes—
248C23×C4—22no—
249C23×C4C2×D411yescharacteristic
250C23×C4C2×D411yescharacteristic
251C23×C4C2411yescharacteristic
252C2×Q16—24no—
253C2×Q16—24no—
254C2×Q16—24no—
255C2×Q16—24no—
256C2×Q16—42no—
257C2×Q16—42no—
258C2×SD16—28no—
259C2×SD16—44no—
260C2×SD16—44no—
261C2×M16—24no—
262C2×M16—42no—
263C2×M16—42no—
264C22×C8—22no—
265C22×C8—22no—
266C22×C8—22no—
267C22×C8—22no—
268C22×C8—41no—
269C22×C8—41no—
270C22×C8C2411yescharacteristic
271C22×C8C2411yescharacteristic
272C22×C8C2414yes—
273C22×C8C2411yescharacteristic
274C22×C8C2411yescharacteristic
275C4⋊Q8—22no—
276C4⋊Q8—22no—
277C4⋊Q8—22no—
278C4⋊Q8—22no—
279C4⋊Q8—22no—
280C4⋊Q8—22no—
281C4⋊Q8—22no—
282C4⋊Q8—22no—
283C4⋊Q8C2×D414yes—
284C4⋊Q8C2×D414yes—
285#127—22no—
286#127—22no—
287#125—28no—
288#125—44no—
289#124—44no—
290#124—44no—
291#124—44no—
292C22⋊Q8—28no—
293C22⋊Q8—24no—
294C22⋊Q8—24no—
295C22⋊Q8—24no—
296C22⋊Q8—44no—
297C22⋊Q8—84no—
298C22⋊Q8—84no—
299C22⋊Q8C2×D418yes—
300#121—24no—
301#121—24no—
302#121—44no—
303C4×Q8—28no—
304C4×Q8—24no—
305C4×Q8—24no—
306C4×Q8—24no—
307C4×Q8—44no—
308C4×Q8—44no—
309C4×Q8—44no—
310C4×Q8—84no—
311C4×Q8C2×D418yes—
312C4×D4—24no—
313C4×D4—24no—
314C4×D4—44no—
315#117—28no—
316#117—42no—
317#117—42no—
318#117—44no—
319#117—82no—
320C2×C4⋊C4—21no—
321C2×C4⋊C4—21no—
322C2×C4⋊C4—21no—
323C2×C4⋊C4—22no—
324C2×C4⋊C4—22no—
325C2×C4⋊C4—21no—
326C2×C4⋊C4—21no—
327C2×C4⋊C4—21no—
328C2×C4⋊C4—21no—
329C2×C4⋊C4—22no—
330C2×C4⋊C4—22no—
331C2×C4⋊C4—21no—
332C2×C4⋊C4—21no—
333C2×C4⋊C4—21no—
334C2×C4⋊C4—21no—
335C2×C4⋊C4—24no—
336C2×C4⋊C4—24no—
337C2×C4⋊C4—24no—
338C2×C4⋊C4—24no—
339C2×C4⋊C4C2×D412yes—
340C2×C4⋊C4C2×D412yes—
341C2×C4⋊C4C2×D412yes—
342C2×C4⋊C4C2×D412yes—
343C2×C4⋊C4C2×D412yes—
344C2×C4⋊C4C2×D412yes—
345C2×C4⋊C4C2412yes—
346C2×C4⋊C4C2412yes—
347C2×C4⋊C4C2412yes—
348C2×C4⋊C4C2412yes—
349C2×C4⋊C4C2412yes—
350C2×C22⋊C4—22no—
351C2×C22⋊C4—22no—
352C2×C22⋊C4—22no—
353C2×C22⋊C4—22no—
354C2×C22⋊C4—22no—
355C2×C22⋊C4—24no—
356C2×C22⋊C4—24no—
357C2×C22⋊C4—24no—
358C2×C22⋊C4—24no—
359C2×C22⋊C4C2×D414yes—
360C2×C42—22no—
361C2×C42—22no—
362C2×C42—22no—
363C2×C42—24no—
364C2×C42—24no—
365C2×C42C2×D411yescharacteristic
366C2×C42C2×D411yescharacteristic
367C2×C42C2×D414yes—
368C2×C42#4114yes—
369C2×C42C2414yes—
370C2×C42C2411yescharacteristic
371C2×C42C2411yescharacteristic
372#107—24no—
373#107—24no—
374#107—28no—
375C4⋊C8—28no—
376C4⋊C8—24no—
377C4⋊C8—24no—
378C4⋊C8—44no—
379#103—24no—
380#103—24no—
381#103—28no—
382#103—24no—
383#103—24no—
384#103—44no—
385#103—44no—
386#102—24no—
387#102—24no—
388#102—28no—
389#102—42no—
390#102—42no—
391C4×C8—22no—
392C4×C8—22no—
393C4×C8—22no—
394C4×C8—22no—
395#95—22no—
396#95—24no—
397#95—24no—
398#95—24no—
399#95—22no—
400#95—24no—
401#95—24no—
402#95—24no—
403#95C2×D414yes—
404#95C2×D414yes—
405C24—22no—
406C24C22×D411yescharacteristic
407#41—816no—
408C2×Q8—24no—
409C2×Q8—22no—
410C2×Q8—22no—
411C2×Q8—22no—
412C2×Q8—22no—
413C2×Q8—22no—
414C2×Q8—22no—
415C2×Q8—22no—
416C2×Q8—22no—
417C2×Q8—22no—
418C2×Q8—22no—
419C2×Q8—22no—
420C2×Q8—42no—
421C2×Q8—42no—
422C2×Q8—42no—
423C2×Q8—42no—
424C2×Q8—42no—
425C2×Q8—42no—
426C2×Q8—81no—
427C2×Q8—81no—
428C2×Q8C22×D414yes—
429C2×D4—22no—
430C2×D4—22no—
431C2×D4—22no—
432C2×D4—22no—
433C2×D4—22no—
434C2×D4—22no—
435C2×D4—44no—
436C2×D4—42no—
437C2×D4—42no—
438C2×D4—42no—
439C2×D4—42no—
440C2×D4—81no—
441C2×D4—81no—
442C2×D4—82no—
443C2×D4—82no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—21no—
449C22×C4—21no—
450C22×C4—21no—
451C22×C4—21no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—44no—
460C22×C4—44no—
461C22×C4—44no—
462C22×C4#12114yes—
463C22×C4#12114yes—
464C22×C4C22×D414yes—
465C22×C4C22×D414yes—
466C22×C4C22×D411yescharacteristic
467C22×C4C22×D411yescharacteristic
468C22×C4C22×D411yescharacteristic
469C22×C4C22×D412yes—
470C22×C4C22×D411yescharacteristic
471C22×C4C22×D411yescharacteristic
472C22×C4C2×#4112yes—
473C22×C4C2511yescharacteristic
474Q16—48no—
475Q16—48no—
476SD16—88no—
477M16—84no—
478C2×C8—21no—
479C2×C8—21no—
480C2×C8—21no—
481C2×C8—21no—
482C2×C8—21no—
483C2×C8—21no—
484C2×C8—24no—
485C2×C8—24no—
486C2×C8—21no—
487C2×C8—21no—
488C2×C8—21no—
489C2×C8—21no—
490C2×C8—24no—
491C2×C8—44no—
492C2×C8—44no—
493C2×C8H(5,𝔽2)12yes—
494C2×C82−1+412yes—
495C4⋊C4—24no—
496C4⋊C4—24no—
497C4⋊C4—24no—
498C4⋊C4—28no—
499C4⋊C4—28no—
500C4⋊C4—24no—
501C4⋊C4—24no—
502C4⋊C4—24no—
503C4⋊C4—24no—
504C4⋊C4—22no—
505C4⋊C4—22no—
506C4⋊C4—22no—
507C4⋊C4—22no—
508C4⋊C4—22no—
509C4⋊C4—22no—
510C4⋊C4—22no—
511C4⋊C4—22no—
512C4⋊C4—22no—
513C4⋊C4—42no—
514C4⋊C4—42no—
515C4⋊C4—42no—
516C4⋊C4—42no—
517C4⋊C4—42no—
518C4⋊C4—44no—
519C4⋊C4—44no—
520C4⋊C4—42no—
521C4⋊C4—42no—
522C4⋊C4—42no—
523C4⋊C4—81no—
524C4⋊C4—81no—
525C4⋊C4—84no—
526C4⋊C4—84no—
527C4⋊C4—84no—
528C4⋊C4—84no—
529C4⋊C4C22×D414yes—
530C22⋊C4—28no—
531C22⋊C4—44no—
532C22⋊C4—44no—
533C22⋊C4—44no—
534C22⋊C4—44no—
535C22⋊C4—84no—
536C22⋊C4—84no—
537C22⋊C4—84no—
538C22⋊C4—84no—
539C22⋊C4—82no—
540C42—28no—
541C42—24no—
542C42—24no—
543C42—44no—
544C42—44no—
545C42—84no—
546C42—84no—
547C42#12114yes—
548C42#12114yes—
549C42C22×D412yes—
550C42C22×D412yes—
551C42C2×#4112yes—
552C42C2×#4112yes—
553C23—21no—
554C23—21no—
555C23—21no—
556C23—21no—
557C23—21no—
558C23—21no—
559C23—42no—
560C23—42no—
561C23—42no—
562C23—42no—
563C23—42no—
564C23—42no—
565C23—42no—
566C23C2×#12112yes—
567C23C23×D411yescharacteristic
568Q8—24no—
569Q8—24no—
570Q8—42no—
571Q8—42no—
572Q8—48no—
573Q8—42no—
574Q8—42no—
575Q8—42no—
576Q8—42no—
577Q8—84no—
578Q8—84no—
579D4—42no—
580D4—42no—
581D4—42no—
582D4—42no—
583D4—84no—
584D4—88no—
585D4—84no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—21no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—21no—
592C2×C4—22no—
593C2×C4—22no—
594C2×C4—22no—
595C2×C4—22no—
596C2×C4—28no—
597C2×C4—28no—
598C2×C4—24no—
599C2×C4—24no—
600C2×C4—24no—
601C2×C4—24no—
602C2×C4—24no—
603C2×C4—24no—
604C2×C4—24no—
605C2×C4—24no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—42no—
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—42no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—42no—
624C2×C4—42no—
625C2×C4—88no—
626C2×C4C2×#12112yes—
627C2×C4C2×#12112yes—
628C2×C4C2×#12112yes—
629C2×C4C2×#12112yes—
630C2×C4C2×#12112yes—
631C2×C4#57612yes—
632C2×C4#57712yes—
633C2×C4#57812yes—
634C2×C4#57812yes—
635C2×C4C23×D411yescharacteristic
636C2×C4C22×#4111yescharacteristic
637C2×C4C2×H(5,𝔽2)11yescharacteristic
638C2×C4C2×2−1+411yescharacteristic
639C2×C4#58511yescharacteristic
640C2×C4#58511yescharacteristic
641C8—22no—
642C8—22no—
643C8—41no—
644C8—44no—
645C8—41no—
646C22—22no—
647C22—22no—
648C22—22no—
649C22—41no—
650C22—41no—
651C22—41no—
652C22—41no—
653C22—82no—
654C22—82no—
655C22—82no—
656C22—82no—
657C22—82no—
658C22—82no—
659C22—82no—
660C22C22×#12111yescharacteristic
661C22#349611yescharacteristic
662C22#349711yescharacteristic
663C22C2×#57611yescharacteristic
664C22C2×#57711yescharacteristic
665C22C2×#57811yescharacteristic
666C22C2×#57811yescharacteristic
667C4—22no—
668C4—22no—
669C4—22no—
670C4—44no—
671C4—44no—
672C4—44no—
673C4—44no—
674C4—82no—
675C4—82no—
676C4—82no—
677C4—82no—
678C4—82no—
679C4#304414yes—
680C4#304514yes—
681C4#348811yescharacteristic
682C4#348911yescharacteristic
683C4#351111yescharacteristic
684C4#351211yescharacteristic
685C2—42no—
686C2—162no—
687C2#3556411yescharacteristic
688C2C2×#304411yescharacteristic
689C2C2×#304511yescharacteristic
690C2#6233211yescharacteristic
691C2#6234311yescharacteristic
692C2#6237511yescharacteristic
693C2#6238211yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#532C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ #2285 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#3170C2semidirect
4. #34374C2semidirect
5. #28590C2semidirect
6. #25449C2semidirect
7. #25365C2semidirect
8. #25320C2semidirect
9. #25275C2semidirect
10. #24817C2semidirect
11. #24765C2semidirect
12. #24733C2semidirect
13. #24682C2semidirect
14. #24633C2semidirect
15. #21894C2semidirect
17. #21143C2semidirect

Maximal subgroups

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

subgroupindex
C2×#32542
C2×#31702
#376402
#343742
#285902
#254492
#253652
#253202
#252752
#248172
#247652
#247332
#246822
#246332
#218942
#214012
#211432