Finite Group Databasebeta

← #4353736

Subgroups — #4353736

Order 512.

2632 conjugacy classes of subgroups.

The subgroup lattice has 681 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2543C211yesmaximal, characteristic
2C2×#2487C211yesmaximal, characteristic, complemented
3#17013C212yesmaximal, complemented
4#16932C212yesmaximal, complemented
5#16884C212yesmaximal, complemented
6#16362C212yesmaximal, complemented
7#15579C212yesmaximal, complemented
8#15420C212yesmaximal, complemented
9#15399C214yesmaximal, complemented
10#15019C212yesmaximal, complemented
11#14894C212yesmaximal, complemented
12#14808C212yesmaximal, complemented
13#14776C212yesmaximal, complemented
14#14363C214yesmaximal, complemented
15#13785C211yesmaximal, characteristic, complemented
16C22×#121C2212yes—
17C22×#121C2211yescharacteristic
18C22×C4×D4C2211yescharacteristic
19C23×C4⋊C4C2211yescharacteristic
20#2543—28no—
21#2487—28no—
22C2×#400C2212yes—
23C2×#399C2212yes—
24C2×#398C2211yescharacteristic
25C2×#397C2211yescharacteristic
26C2×#397C2212yes—
27C2×#397C2212yes—
28C2×#394C2211yescharacteristic
29C2×#392C2211yescharacteristic
30C2×#390C2212yes—
31C2×#389C2212yes—
32C2×#388C2212yes—
33C2×#388C2214yes—
34C2×#386C2212yes—
35C2×#386C2212yes—
36C2×#386C2212yes—
37C2×#386C2214yes—
38C2×#385C2212yes—
39C2×#381C2214yes—
40C2×#380C2212yes—
41C2×#380C2211yescharacteristic
42C2×C4×C4⋊C4C2211yescharacteristic
43C2×#376C2211yescharacteristic
44C22×#95C2211yescharacteristic
45#1445C2212yes—
46#1445C2212yes—
47#1443C2214yes—
48#1443C2214yes—
49#1443C2214yes—
50#1442C2214yes—
51#1442C2214yes—
52#1442C2212yes—
53#1442C2214yes—
54#1442C2212yes—
55#1441C2214yes—
56#1440C2214yes—
57#1440C2214yes—
58#1439C2214yes—
59#1439C2214yes—
60#1439C2214yes—
61#1439C2212yes—
62#1439C2212yes—
63#1438C2214yes—
64#1438C2214yes—
65#1438C2214yes—
66#1438C2214yes—
67#1437C2214yes—
68#1437C2212yes—
69#1437C2212yes—
70#1436C2214yes—
71#1436C2214yes—
72C4⋊#95C2212yes—
73C22⋊#95C2214yes—
74C22⋊#95C2214yes—
75C22⋊#95C2212yes—
76C22⋊#95C2212yes—
77C23×D4C2312yes—
78C23×D4C2312yes—
79C23×D4C2311yescharacteristic
80C23×D4C2311yescharacteristic
81C24×C4C2312yes—
82C24×C4C2311yescharacteristic
83C24×C4C2311yescharacteristic
84C24×C4C2312yes—
85C2×#121—28no—
86C2×#121—216no—
87C2×#121—24no—
88C2×#121—42no—
89C2×#121—42no—
90C2×C4×D4—42no—
91C2×C4×D4—42no—
92C2×C4×D4D418yes—
93C22×C4⋊C4—24no—
94C22×C4⋊C4—42no—
95C22×C4⋊C4—42no—
96C22×C4⋊C4C2312yes—
97C22×C4⋊C4C2312yes—
98C22×C4⋊C4C2311yescharacteristic
99C22×C4⋊C4C2312yes—
100C22×C4⋊C4C2314yes—
101C22×C4⋊C4C2312yes—
102C22×C4⋊C4C2312yes—
103C22×C4⋊C4C2311yescharacteristic
104C22×C22⋊C4C2312yes—
105C22×C22⋊C4C2312yes—
106C22×C22⋊C4C2312yes—
107C22×C22⋊C4C2312yes—
108C22×C22⋊C4C2312yes—
109C22×C22⋊C4C2314yes—
110C22×C22⋊C4C2314yes—
111C22×C22⋊C4C2314yes—
112C22×C22⋊C4C2312yes—
113C22×C22⋊C4C2314yes—
114C22×C22⋊C4C2314yes—
115C22×C22⋊C4C2314yes—
116C22×C22⋊C4C2314yes—
117C22×C22⋊C4C2311yescharacteristic
118C22×C22⋊C4C2311yescharacteristic
119C22×C42C2311yescharacteristic
120C22×C42C2312yes—
121C22×C42C2312yes—
122#400—28no—
123#399—28no—
124#398—24no—
125#397—24no—
126#397—44no—
127#397—44no—
128#394—24no—
129#392—24no—
130#390—28no—
131#389—28no—
132#388—28no—
133#388—48no—
134#386—28no—
135#386—28no—
136#386—28no—
137#386—48no—
138#385—28no—
139#381—48no—
140#380—28no—
141#380—24no—
142C4×C4⋊C4—42no—
143#376—42no—
144C2×#95—24no—
145C2×#95C2311yescharacteristic
146C2×#95C2311yescharacteristic
147C2×#95C2312yes—
148C2×#95C2312yes—
149C2×#95C2312yes—
150C2×#95C2312yes—
151C2×#95C2312yes—
152C2×#95C2314yes—
153C2×#95C2314yes—
154C2×#95C2314yes—
155C2×#95C2314yes—
156C2×#95C2314yes—
157C2×#95C2312yes—
158C2×#95C2314yes—
159C2×#95C2314yes—
160C2×#95C2314yes—
161C2×#95C2314yes—
162C2×#95C2314yes—
163C2×#95C2314yes—
164C2×#95C2314yes—
165C2×#95C2314yes—
166C2×#95C2314yes—
167C2×#95C2314yes—
168C2×#95C2312yes—
169C2×#95C2314yes—
170C25C2412yes—
171C25C2411yescharacteristic
172C25C2411yescharacteristic
173C22×D4—24no—
174C22×D4—24no—
175C22×D4—28no—
176C22×D4—28no—
177C22×D4—22no—
178C22×D4—24no—
179C22×D4—41no—
180C22×D4—41no—
181C22×D4—42no—
182C22×D4—41no—
183C22×D4—41no—
184C22×D4—42no—
185C22×D4—42no—
186C22×D4—44no—
187C22×D4—42no—
188C22×D4—44no—
189C22×D4—44no—
190C22×D4C2×D414yes—
191C22×D4#4118yes—
192C23×C4—22no—
193C23×C4—22no—
194C23×C4—22no—
195C23×C4—22no—
196C23×C4—24no—
197C23×C4—28no—
198C23×C4—24no—
199C23×C4—24no—
200C23×C4—44no—
201C23×C4—42no—
202C23×C4—42no—
203C23×C4—44no—
204C23×C4—44no—
205C23×C4—42no—
206C23×C4—44no—
207C23×C4—44no—
208C23×C4C2×D414yes—
209C23×C4C2×D414yes—
210C23×C4C2414yes—
211C23×C4C2414yes—
212C23×C4C2414yes—
213C23×C4C2414yes—
214C23×C4C2412yes—
215C23×C4C2412yes—
216C23×C4C2411yescharacteristic
217C23×C4C2411yescharacteristic
218C23×C4C2411yescharacteristic
219C23×C4C2412yes—
220C23×C4C2411yescharacteristic
221C23×C4C2411yescharacteristic
222#121—416no—
223#121—432no—
224C4×D4—416no—
225C2×C4⋊C4—24no—
226C2×C4⋊C4—24no—
227C2×C4⋊C4—24no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—24no—
230C2×C4⋊C4—28no—
231C2×C4⋊C4—24no—
232C2×C4⋊C4—24no—
233C2×C4⋊C4—24no—
234C2×C4⋊C4—28no—
235C2×C4⋊C4—42no—
236C2×C4⋊C4—42no—
237C2×C4⋊C4—42no—
238C2×C4⋊C4—44no—
239C2×C4⋊C4—41no—
240C2×C4⋊C4—41no—
241C2×C4⋊C4—41no—
242C2×C4⋊C4—41no—
243C2×C4⋊C4—42no—
244C2×C4⋊C4—416no—
245C2×C4⋊C4—44no—
246C2×C4⋊C4—44no—
247C2×C4⋊C4C2×D414yes—
248C2×C22⋊C4—24no—
249C2×C22⋊C4—24no—
250C2×C22⋊C4—28no—
251C2×C22⋊C4—24no—
252C2×C22⋊C4—24no—
253C2×C22⋊C4—24no—
254C2×C22⋊C4—28no—
255C2×C22⋊C4—24no—
256C2×C22⋊C4—24no—
257C2×C22⋊C4—24no—
258C2×C22⋊C4—28no—
259C2×C22⋊C4—28no—
260C2×C22⋊C4—28no—
261C2×C22⋊C4—24no—
262C2×C22⋊C4—42no—
263C2×C22⋊C4—44no—
264C2×C22⋊C4—44no—
265C2×C22⋊C4—44no—
266C2×C22⋊C4—41no—
267C2×C22⋊C4—41no—
268C2×C22⋊C4—41no—
269C2×C22⋊C4—41no—
270C2×C22⋊C4—44no—
271C2×C22⋊C4—44no—
272C2×C22⋊C4—44no—
273C2×C22⋊C4—44no—
274C2×C22⋊C4—44no—
275C2×C22⋊C4—44no—
276C2×C22⋊C4—44no—
277C2×C22⋊C4—44no—
278C2×C22⋊C4—44no—
279C2×C22⋊C4—44no—
280C2×C22⋊C4—44no—
281C2×C22⋊C4—44no—
282C2×C22⋊C4—44no—
283C2×C22⋊C4—44no—
284C2×C22⋊C4—44no—
285C2×C22⋊C4—44no—
286C2×C22⋊C4—44no—
287C2×C22⋊C4C2×D414yes—
288C2×C22⋊C4C2×D414yes—
289C2×C42—22no—
290C2×C42—22no—
291C2×C42—24no—
292C2×C42—42no—
293C2×C42—42no—
294C2×C42—42no—
295C2×C42—44no—
296C2×C42C2×D414yes—
297#95—24no—
298#95—22no—
299#95—22no—
300#95—28no—
301#95—24no—
302#95—24no—
303#95—24no—
304#95—28no—
305#95—24no—
306#95—24no—
307#95—28no—
308#95—28no—
309#95—28no—
310#95—44no—
311#95—44no—
312#95—44no—
313#95—44no—
314#95—44no—
315#95—44no—
316#95—44no—
317#95—42no—
318#95—44no—
319#95—44no—
320#95—44no—
321#95—44no—
322C24—22no—
323C24—22no—
324C24—21no—
325C24—21no—
326C24—21no—
327C24—21no—
328C24—22no—
329C24—22no—
330C24—24no—
331C24—24no—
332C24—22no—
333C24—22no—
334C24—22no—
335C24—24no—
336C24—22no—
337C24—22no—
338C24—22no—
339C24—22no—
340C24—22no—
341C24—24no—
342C24—24no—
343C24—22no—
344C24—24no—
345C24C22×D412yes—
346C24C22×D412yes—
347C24C2×#4114yes—
348C24C2×#4114yes—
349C24C2511yescharacteristic
350C2×D4—44no—
351C2×D4—44no—
352C2×D4—48no—
353C2×D4—48no—
354C2×D4—48no—
355C2×D4—416no—
356C2×D4—44no—
357C2×D4—48no—
358C2×D4—48no—
359C2×D4—48no—
360C2×D4—48no—
361C2×D4—84no—
362C2×D4—84no—
363C2×D4—88no—
364C2×D4—88no—
365C2×D4—88no—
366C2×D4—84no—
367C2×D4—84no—
368C2×D4#124116yes—
369C22×C4—21no—
370C22×C4—21no—
371C22×C4—22no—
372C22×C4—24no—
373C22×C4—22no—
374C22×C4—24no—
375C22×C4—21no—
376C22×C4—21no—
377C22×C4—22no—
378C22×C4—21no—
379C22×C4—21no—
380C22×C4—21no—
381C22×C4—21no—
382C22×C4—24no—
383C22×C4—24no—
384C22×C4—24no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—28no—
388C22×C4—28no—
389C22×C4—22no—
390C22×C4—24no—
391C22×C4—22no—
392C22×C4—24no—
393C22×C4—24no—
394C22×C4—24no—
395C22×C4—24no—
396C22×C4—24no—
397C22×C4—24no—
398C22×C4—24no—
399C22×C4—24no—
400C22×C4—24no—
401C22×C4—24no—
402C22×C4—24no—
403C22×C4—24no—
404C22×C4—24no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—24no—
409C22×C4—24no—
410C22×C4—24no—
411C22×C4—24no—
412C22×C4—24no—
413C22×C4—22no—
414C22×C4—24no—
415C22×C4—24no—
416C22×C4—24no—
417C22×C4—24no—
418C22×C4—24no—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—24no—
422C22×C4—24no—
423C22×C4—24no—
424C22×C4—24no—
425C22×C4—24no—
426C22×C4—24no—
427C22×C4—48no—
428C22×C4—48no—
429C22×C4—48no—
430C22×C4—48no—
431C22×C4—48no—
432C22×C4—416no—
433C22×C4—48no—
434C22×C4—48no—
435C22×C4—84no—
436C22×C4—84no—
437C22×C4—84no—
438C22×C4—88no—
439C22×C4—88no—
440C22×C4—88no—
441C22×C4—84no—
442C22×C4—84no—
443C22×C4C22×D412yes—
444C22×C4C22×D412yes—
445C22×C4C22×D412yes—
446C22×C4C22×D412yes—
447C22×C4C22×D412yes—
448C22×C4C2×#4114yes—
449C22×C4H(5,𝔽2)14yes—
450C22×C4H(5,𝔽2)12yes—
451C22×C4H(5,𝔽2)12yes—
452C4⋊C4—44no—
453C4⋊C4—48no—
454C4⋊C4—48no—
455C4⋊C4—48no—
456C4⋊C4—44no—
457C4⋊C4—48no—
458C4⋊C4—48no—
459C4⋊C4—88no—
460C22⋊C4—44no—
461C22⋊C4—44no—
462C22⋊C4—48no—
463C22⋊C4—48no—
464C22⋊C4—416no—
465C22⋊C4—48no—
466C22⋊C4—48no—
467C22⋊C4—48no—
468C22⋊C4—84no—
469C22⋊C4—88no—
470C22⋊C4—88no—
471C22⋊C4—88no—
472C22⋊C4—88no—
473C22⋊C4—88no—
474C22⋊C4—88no—
475C42—28no—
476C42—48no—
477C42—84no—
478C23—22no—
479C23—22no—
480C23—22no—
481C23—22no—
482C23—24no—
483C23—24no—
484C23—24no—
485C23—24no—
486C23—24no—
487C23—24no—
488C23—24no—
489C23—24no—
490C23—28no—
491C23—28no—
492C23—28no—
493C23—28no—
494C23—42no—
495C23—42no—
496C23—42no—
497C23—42no—
498C23—44no—
499C23—44no—
500C23—42no—
501C23—42no—
502C23—41no—
503C23—41no—
504C23—41no—
505C23—41no—
506C23—44no—
507C23—44no—
508C23—42no—
509C23—42no—
510C23—42no—
511C23—42no—
512C23—42no—
513C23—42no—
514C23—42no—
515C23—44no—
516C23—44no—
517C23—44no—
518C23—44no—
519C23—44no—
520C23—44no—
521C23—44no—
522C23—44no—
523C23—44no—
524C23—44no—
525C23—42no—
526C23—42no—
527C23—42no—
528C23—42no—
529C23—44no—
530C23—44no—
531C23C2×#12414yes—
532C23C2×#12414yes—
533C23C23×D411yescharacteristic
534C23C22×#4112yes—
535C23C2×H(5,𝔽2)11yescharacteristic
536C23C2×H(5,𝔽2)11yescharacteristic
537C23C2×H(5,𝔽2)12yes—
538C23#58512yes—
539C23#58514yes—
540C23#58512yes—
541D4—416no—
542D4—816no—
543D4—88no—
544D4—816no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—24no—
548C2×C4—24no—
549C2×C4—24no—
550C2×C4—24no—
551C2×C4—24no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—24no—
556C2×C4—24no—
557C2×C4—24no—
558C2×C4—24no—
559C2×C4—28no—
560C2×C4—416no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—44no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—44no—
575C2×C4—44no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—44no—
579C2×C4—44no—
580C2×C4—44no—
581C2×C4—44no—
582C2×C4—42no—
583C2×C4—44no—
584C2×C4—44no—
585C2×C4—44no—
586C2×C4—44no—
587C2×C4—44no—
588C2×C4—44no—
589C2×C4—44no—
590C2×C4—44no—
591C2×C4—44no—
592C2×C4—44no—
593C2×C4—44no—
594C2×C4—44no—
595C2×C4—44no—
596C2×C4—44no—
597C2×C4—44no—
598C2×C4—44no—
599C2×C4—44no—
600C2×C4—44no—
601C2×C4—44no—
602C2×C4—44no—
603C2×C4—44no—
604C2×C4—44no—
605C2×C4—44no—
606C2×C4—44no—
607C2×C4—44no—
608C2×C4—44no—
609C2×C4—88no—
610C2×C4—816no—
611C2×C4—816no—
612C2×C4C2×#12414yes—
613C2×C4#53414yes—
614C22—28no—
615C22—28no—
616C22—24no—
617C22—24no—
618C22—44no—
619C22—44no—
620C22—42no—
621C22—42no—
622C22—48no—
623C22—48no—
624C22—42no—
625C22—42no—
626C22—44no—
627C22—44no—
628C22—44no—
629C22—44no—
630C22—82no—
631C22—82no—
632C22—82no—
633C22—82no—
634C22—82no—
635C22—84no—
636C22—84no—
637C22—82no—
638C22—84no—
639C22—84no—
640C22—82no—
641C22C22×#12411yescharacteristic
642C22C2×#53411yescharacteristic
643C22#344911yescharacteristic
644C22#348512yes—
645C22#348812yes—
646C22#350512yes—
647C22#350512yes—
648C22#351314yes—
649C22#352114yes—
650C22#353312yes—
651C22#353312yes—
652C22#353914yes—
653C22#354612yes—
654C22#354911yescharacteristic
655C22#355111yescharacteristic
656C22#356614yes—
657C4—24no—
658C4—24no—
659C4—44no—
660C4—42no—
661C4—42no—
662C4—42no—
663C4—84no—
664C4—82no—
665C4—84no—
666C4—84no—
667C4—84no—
668C4—82no—
669C2—44no—
670C2—44no—
671C2—162no—
672C2#3501911yescharacteristic
673C2#3584712yes—
674C2#3696511yescharacteristic
675C2#3727311yescharacteristic
676C2#4199012yes—
677C2#4753512yes—
678C2#4754512yes—
679C2#5185514yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC2×#2543C2
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2487C2semidirect
3. #17013C2semidirect
4. #16932C2semidirect
5. #16884C2semidirect
6. #16362C2semidirect
7. #15579C2semidirect
8. #15420C2semidirect
9. #15399C2semidirect
10. #15019C2semidirect
11. #14894C2semidirect
12. #14808C2semidirect
13. #14776C2semidirect
14. #14363C2semidirect
15. #13785C2semidirect

Maximal subgroups

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

subgroupindex
C2×#25432
C2×#24872
#170132
#169322
#168842
#163622
#155792
#154202
#153992
#150192
#148942
#148082
#147762
#143632
#137852