Finite Group Databasebeta

← #615863

Subgroups — #615863

Order 512.

4429 conjugacy classes of subgroups.

The subgroup lattice has 675 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2849C211yesmaximal, characteristic
2C22×C4×C22⋊C4C211yesmaximal, characteristic, complemented
3#15043C214yesmaximal, complemented
4#15026C211yesmaximal, characteristic, complemented
5#14964C214yesmaximal, complemented
6#14955C211yesmaximal, characteristic, complemented
7#14946C211yesmaximal, characteristic, complemented
8#14945C211yesmaximal, characteristic, complemented
9#14696C212yesmaximal, complemented
10#14695C212yesmaximal, complemented
11#14134C212yesmaximal
12#14047C214yesmaximal, complemented
13#13999C214yesmaximal, complemented
14#13877C212yesmaximal, complemented
15C2×C22⋊#95C211yesmaximal, characteristic, complemented
16C25×C4C2211yescharacteristic
17C22×#123C2212yes—
18C22×#120C2211yescharacteristic
19C23×C22⋊C4C2212yes—
20C23×C22⋊C4C2212yes—
21C23×C42C2212yes—
22#2849—28no—
23C2×#388C2214yescomplemented
24C2×#387C2214yescomplemented
25C2×#386C2211yescharacteristic
26C2×#386C2211yescharacteristic
27C2×#386C2212yes—
28C2×#386C2212yes—
29C2×#386C2214yes—
30C2×#386C2212yes—
31C2×#385C2211yescharacteristic
32C2×#385C2211yescharacteristic
33C2×#385C2212yes—
34C2×#385C2212yes—
35C2×#385C2212yes—
36C2×#381C2212yes—
37C2×#380C2214yes—
38C2×#379C2211yescharacteristic
39C2×C4×C22⋊C4—28no—
40C2×C4×C22⋊C4C2214yes—
41C2×C4×C22⋊C4C2214yes—
42C22×#95C2211yescharacteristic
43C22×#95C2211yescharacteristic
44C22×#95C2214yes—
45#1445C2214yescomplemented
46#1443C2214yescomplemented
47#1443C2214yescomplemented
48#1442C2214yescomplemented
49#1442C2214yescomplemented
50#1441C2214yescomplemented
51#1440C2214yes—
52#1440C2214yescomplemented
53#1439C2214yes—
54#1439C2214yes—
55#1439C2214yescomplemented
56#1438C2214yes—
57#1437C2214yescomplemented
58#1436C2214yescomplemented
59#1436C2214yescomplemented
60#1436C2214yescomplemented
61#1434C2214yescomplemented
62#1434C2214yescomplemented
63C22⋊#95C2212yes—
64C22⋊#95C2212yes—
65C22⋊#95C2214yes—
66C22⋊#95C2212yes—
67C22⋊#95C2212yes—
68C22⋊#95C2214yes—
69C22⋊#95C2211yescharacteristic
70C22⋊#95C2211yescharacteristic
71C22⋊#95C2212yes—
72C22⋊#95C2214yes—
73C26C2311yescharacteristic
74C23×D4C2311yescharacteristic
75C23×D4C2311yescharacteristic
76C23×D4C2311yescharacteristic
77C24×C4—28no—
78C24×C4—48no—
79C24×C4D418yes—
80C24×C4C2311yescharacteristic
81C24×C4C2312yes—
82C24×C4C2312yes—
83C24×C4C2314yes—
84C24×C4C2312yes—
85C24×C4C2312yes—
86C24×C4C2311yescharacteristic
87C2×#123—28no—
88C2×#123—48no—
89C2×#120—24no—
90C2×#120—28no—
91C22×C4⋊C4C2312yes—
92C22×C4⋊C4C2314yes—
93C22×C4⋊C4C2312yes—
94C22×C4⋊C4C2314yes—
95C22×C22⋊C4—28no—
96C22×C22⋊C4—216no—
97C22×C22⋊C4—28no—
98C22×C22⋊C4—28no—
99C22×C22⋊C4—44no—
100C22×C22⋊C4C2312yes—
101C22×C22⋊C4C2314yes—
102C22×C22⋊C4C2311yescharacteristic
103C22×C22⋊C4C2311yescharacteristic
104C22×C22⋊C4C2311yescharacteristic
105C22×C22⋊C4C2312yes—
106C22×C22⋊C4C2312yes—
107C22×C22⋊C4C2311yescharacteristic
108C22×C22⋊C4C2311yescharacteristic
109C22×C22⋊C4C2314yes—
110C22×C22⋊C4C2314yes—
111C22×C22⋊C4C2312yes—
112C22×C22⋊C4C2314yes—
113C22×C22⋊C4C2312yes—
114C22×C22⋊C4C2312yes—
115C22×C22⋊C4C2314yes—
116C22×C42—28no—
117C22×C42—48no—
118C22×C42C2314yes—
119C22×C42C2314yes—
120#388—48no—
121#387—48no—
122#386—24no—
123#386—28no—
124#386—24no—
125#386—28no—
126#386—216no—
127#386—44no—
128#385—24no—
129#385—28no—
130#385—24no—
131#385—28no—
132#385—44no—
133#381—28no—
134#380—216no—
135#379—24no—
136C4×C22⋊C4—216no—
137C4×C22⋊C4—216no—
138C2×#95—24no—
139C2×#95—24no—
140C2×#95—48no—
141C2×#95C2314yes—
142C2×#95C2314yes—
143C2×#95C2314yes—
144C2×#95C2314yes—
145C2×#95C2314yes—
146C2×#95C2314yes—
147C2×#95C2314yes—
148C2×#95C2314yes—
149C2×#95C2312yes—
150C2×#95C2312yes—
151C2×#95C2314yes—
152C2×#95C2314yes—
153C2×#95C2312yes—
154C2×#95C2312yes—
155C2×#95C2314yes—
156C2×#95C2314yes—
157C2×#95C2314yes—
158C2×#95C2314yes—
159C2×#95C2314yes—
160C2×#95C2314yes—
161C2×#95C2314yes—
162C2×#95C2314yes—
163C25—28no—
164C25—24no—
165C25—22no—
166C25—22no—
167C25—44no—
168C25C2×D414yes—
169C25#4118yes—
170C25C2411yescharacteristic
171C25C2411yescharacteristic
172C25C2411yescharacteristic
173C25C2411yescharacteristic
174C22×D4—22no—
175C22×D4—22no—
176C22×D4—22no—
177C22×D4—24no—
178C22×D4—24no—
179C22×D4—28no—
180C22×D4—24no—
181C22×D4—22no—
182C22×D4—22no—
183C22×D4—24no—
184C22×D4—44no—
185C23×C4—24no—
186C23×C4—28no—
187C23×C4—232no—
188C23×C4—24no—
189C23×C4—24no—
190C23×C4—28no—
191C23×C4—28no—
192C23×C4—24no—
193C23×C4—28no—
194C23×C4—28no—
195C23×C4—28no—
196C23×C4—28no—
197C23×C4—24no—
198C23×C4—24no—
199C23×C4—24no—
200C23×C4—24no—
201C23×C4—28no—
202C23×C4—28no—
203C23×C4—28no—
204C23×C4—28no—
205C23×C4—432no—
206C23×C4—44no—
207C23×C4—44no—
208C23×C4—44no—
209C23×C4—44no—
210C23×C4—44no—
211C23×C4—44no—
212C23×C4—44no—
213C23×C4—44no—
214C23×C4—464no—
215C23×C4C2×D414yes—
216C23×C4C2×D418yes—
217C23×C4C2×D418yes—
218C23×C4C2×D414yes—
219C23×C4C2412yes—
220C23×C4C2412yes—
221C23×C4C2414yes—
222C23×C4C2411yescharacteristic
223C23×C4C2411yescharacteristic
224C23×C4C2411yescharacteristic
225C23×C4C2412yes—
226C23×C4C2414yes—
227C23×C4C2414yes—
228C23×C4C2414yes—
229C23×C4C2412yes—
230#123—432no—
231#120—416no—
232C2×C4⋊C4—24no—
233C2×C4⋊C4—24no—
234C2×C4⋊C4—28no—
235C2×C4⋊C4—28no—
236C2×C4⋊C4—28no—
237C2×C4⋊C4—28no—
238C2×C4⋊C4—44no—
239C2×C4⋊C4—44no—
240C2×C4⋊C4—44no—
241C2×C4⋊C4—44no—
242C2×C22⋊C4—264no—
243C2×C22⋊C4—28no—
244C2×C22⋊C4—28no—
245C2×C22⋊C4—28no—
246C2×C22⋊C4—24no—
247C2×C22⋊C4—24no—
248C2×C22⋊C4—28no—
249C2×C22⋊C4—28no—
250C2×C22⋊C4—22no—
251C2×C22⋊C4—22no—
252C2×C22⋊C4—22no—
253C2×C22⋊C4—24no—
254C2×C22⋊C4—24no—
255C2×C22⋊C4—22no—
256C2×C22⋊C4—22no—
257C2×C22⋊C4—28no—
258C2×C22⋊C4—24no—
259C2×C22⋊C4—24no—
260C2×C22⋊C4—24no—
261C2×C22⋊C4—24no—
262C2×C22⋊C4—24no—
263C2×C22⋊C4—28no—
264C2×C22⋊C4—24no—
265C2×C22⋊C4—24no—
266C2×C22⋊C4—22no—
267C2×C22⋊C4—22no—
268C2×C22⋊C4—24no—
269C2×C22⋊C4—24no—
270C2×C22⋊C4—28no—
271C2×C22⋊C4—28no—
272C2×C22⋊C4—28no—
273C2×C22⋊C4—28no—
274C2×C22⋊C4—28no—
275C2×C22⋊C4—432no—
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#41116yes—
283C2×C22⋊C4#4118yes—
284C2×C42—28no—
285C2×C42—28no—
286C2×C42—28no—
287C2×C42—28no—
288C2×C42—44no—
289C2×C42—44no—
290C2×C42—432no—
291#95—28no—
292#95—28no—
293#95—24no—
294#95—24no—
295#95—24no—
296#95—24no—
297#95—28no—
298#95—28no—
299#95—28no—
300#95—28no—
301#95—28no—
302#95—28no—
303#95—28no—
304#95—28no—
305#95—44no—
306#95—44no—
307#95—44no—
308#95—44no—
309#95—44no—
310#95—44no—
311#95—44no—
312#95—44no—
313C24—24no—
314C24—22no—
315C24—24no—
316C24—24no—
317C24—21no—
318C24—21no—
319C24—216no—
320C24—232no—
321C24—28no—
322C24—28no—
323C24—216no—
324C24—216no—
325C24—24no—
326C24—22no—
327C24—21no—
328C24—24no—
329C24—21no—
330C24—21no—
331C24—24no—
332C24—22no—
333C24—22no—
334C24—24no—
335C24—21no—
336C24—21no—
337C24—48no—
338C24—416no—
339C24—416no—
340C24—416no—
341C24—416no—
342C24—416no—
343C24#123116yes—
344C24C22×D412yes—
345C24C22×D412yes—
346C24C22×D412yes—
347C24C2×#4114yes—
348C24C2×#4114yes—
349C24C2×#4114yes—
350C24C2×#4118yes—
351C24C2×#4114yes—
352C24H(5,𝔽2)12yes—
353C242−1+412yes—
354C24C2511yescharacteristic
355C2×D4—48no—
356C2×D4—44no—
357C2×D4—44no—
358C2×D4—44no—
359C2×D4—48no—
360C2×D4—416no—
361C2×D4—44no—
362C2×D4—44no—
363C2×D4—48no—
364C2×D4—48no—
365C2×D4—88no—
366C22×C4—232no—
367C22×C4—28no—
368C22×C4—24no—
369C22×C4—24no—
370C22×C4—216no—
371C22×C4—216no—
372C22×C4—216no—
373C22×C4—216no—
374C22×C4—216no—
375C22×C4—28no—
376C22×C4—24no—
377C22×C4—24no—
378C22×C4—24no—
379C22×C4—24no—
380C22×C4—24no—
381C22×C4—24no—
382C22×C4—24no—
383C22×C4—24no—
384C22×C4—24no—
385C22×C4—24no—
386C22×C4—24no—
387C22×C4—24no—
388C22×C4—24no—
389C22×C4—24no—
390C22×C4—24no—
391C22×C4—24no—
392C22×C4—24no—
393C22×C4—24no—
394C22×C4—24no—
395C22×C4—24no—
396C22×C4—24no—
397C22×C4—216no—
398C22×C4—216no—
399C22×C4—21no—
400C22×C4—21no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—21no—
404C22×C4—22no—
405C22×C4—24no—
406C22×C4—24no—
407C22×C4—22no—
408C22×C4—24no—
409C22×C4—24no—
410C22×C4—24no—
411C22×C4—24no—
412C22×C4—24no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—24no—
418C22×C4—24no—
419C22×C4—21no—
420C22×C4—22no—
421C22×C4—24no—
422C22×C4—24no—
423C22×C4—24no—
424C22×C4—24no—
425C22×C4—22no—
426C22×C4—21no—
427C22×C4—48no—
428C22×C4—48no—
429C22×C4—48no—
430C22×C4—416no—
431C22×C4—416no—
432C22×C4—416no—
433C22×C4—416no—
434C22×C4—432no—
435C22×C4—432no—
436C22×C4—4128no—
437C22×C4—416no—
438C22×C4—416no—
439C22×C4—416no—
440C22×C4—48no—
441C22×C4—48no—
442C22×C4—416no—
443C22×C4—48no—
444C22×C4—48no—
445C22×C4—416no—
446C22×C4—88no—
447C22×C4—88no—
448C22×C4—88no—
449C22×C4C22×D414yes—
450C22×C4C22×D414yes—
451C22×C4C2×#4118yes—
452C22×C4C2×#4118yes—
453C22×C4C2×#4118yes—
454C22×C4H(5,𝔽2)18yes—
455C4⋊C4—416no—
456C4⋊C4—48no—
457C4⋊C4—48no—
458C4⋊C4—88no—
459C22⋊C4—232no—
460C22⋊C4—232no—
461C22⋊C4—216no—
462C22⋊C4—416no—
463C22⋊C4—416no—
464C22⋊C4—48no—
465C22⋊C4—48no—
466C22⋊C4—44no—
467C22⋊C4—44no—
468C22⋊C4—44no—
469C22⋊C4—48no—
470C22⋊C4—44no—
471C22⋊C4—44no—
472C22⋊C4—48no—
473C22⋊C4—48no—
474C22⋊C4—88no—
475C42—416no—
476C42—416no—
477C23—24no—
478C23—28no—
479C23—22no—
480C23—24no—
481C23—24no—
482C23—22no—
483C23—24no—
484C23—24no—
485C23—28no—
486C23—28no—
487C23—28no—
488C23—28no—
489C23—28no—
490C23—28no—
491C23—28no—
492C23—28no—
493C23—28no—
494C23—28no—
495C23—232no—
496C23—24no—
497C23—28no—
498C23—22no—
499C23—24no—
500C23—22no—
501C23—24no—
502C23—24no—
503C23—28no—
504C23—232no—
505C23—416no—
506C23—432no—
507C23—432no—
508C23—464no—
509C23—464no—
510C23—44no—
511C23—44no—
512C23—44no—
513C23—44no—
514C23—41no—
515C23—42no—
516C23—44no—
517C23—41no—
518C23—42no—
519C23—41no—
520C23—42no—
521C23—42no—
522C23—44no—
523C23—44no—
524C23—44no—
525C23—44no—
526C23—44no—
527C23C2×#12314yes—
528C23C2×#12314yes—
529C23C2×#12314yes—
530C23#52518yes—
531C23#529116yes—
532C23#53214yes—
533C23#53714yes—
534C23#55118yes—
535C23#55218yes—
536C23C23×D411yescharacteristic
537C23C22×#4112yes—
538C23C22×#4114yes—
539C23C22×#4112yes—
540C23C2×H(5,𝔽2)14yes—
541C23C2×H(5,𝔽2)11yescharacteristic
542C23C2×2−1+411yescharacteristic
543D4—88no—
544D4—88no—
545D4—88no—
546C2×C4—232no—
547C2×C4—24no—
548C2×C4—24no—
549C2×C4—216no—
550C2×C4—24no—
551C2×C4—28no—
552C2×C4—28no—
553C2×C4—24no—
554C2×C4—24no—
555C2×C4—28no—
556C2×C4—28no—
557C2×C4—28no—
558C2×C4—28no—
559C2×C4—28no—
560C2×C4—28no—
561C2×C4—28no—
562C2×C4—432no—
563C2×C4—432no—
564C2×C4—432no—
565C2×C4—432no—
566C2×C4—416no—
567C2×C4—44no—
568C2×C4—44no—
569C2×C4—44no—
570C2×C4—44no—
571C2×C4—44no—
572C2×C4—44no—
573C2×C4—44no—
574C2×C4—48no—
575C2×C4—44no—
576C2×C4—44no—
577C2×C4—44no—
578C2×C4—44no—
579C2×C4—44no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—44no—
587C2×C4—44no—
588C2×C4—42no—
589C2×C4—41no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—41no—
595C2×C4—44no—
596C2×C4—42no—
597C2×C4—42no—
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—832no—
609C2×C4#53418yes—
610C22—28no—
611C22—28no—
612C22—24no—
613C22—28no—
614C22—28no—
615C22—216no—
616C22—216no—
617C22—216no—
618C22—216no—
619C22—24no—
620C22—24no—
621C22—24no—
622C22—24no—
623C22—24no—
624C22—24no—
625C22—464no—
626C22—44no—
627C22—44no—
628C22—48no—
629C22—48no—
630C22—82no—
631C22—81no—
632C22—82no—
633C22—84no—
634C22—84no—
635C22—81no—
636C22—81no—
637C22#249818yes—
638C22#285318yes—
639C22C22×#12311yescharacteristic
640C22C2×#52512yes—
641C22C2×#52914yes—
642C22C2×#53211yescharacteristic
643C22C2×#53412yes—
644C22C2×#53711yescharacteristic
645C22C2×#55112yes—
646C22C2×#55212yes—
647C22#348114yes—
648C22#348114yes—
649C22#349014yes—
650C22#351914yes—
651C22#352014yes—
652C4—28no—
653C4—48no—
654C4—48no—
655C4—48no—
656C4—44no—
657C4—82no—
658C4—81no—
659C4—82no—
660C4—84no—
661C4—81no—
662C4—81no—
663C2—28no—
664C2—28no—
665C2—44no—
666C2—161no—
667C2C2×#249811yescharacteristic
668C2#3495712yes—
669C2#3499211yescharacteristic
670C2#3507712yes—
671C2#3624414yes—
672C2#3624514yes—
673C2C2×#285311yescharacteristic
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC22×#120C22
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C22×C4×C22⋊C4C2semidirect
3. #15043C2semidirect
4. #15026C2semidirect
5. #14964C2semidirect
6. #14955C2semidirect
7. #14946C2semidirect
8. #14945C2semidirect
9. #14696C2semidirect
10. #14695C2semidirect
12. #14047C2semidirect
13. #13999C2semidirect
14. #13877C2semidirect
15. C2×C22⋊#95C2semidirect
23. C2×#388C22semidirect
24. C2×#387C22semidirect
45. #1445C22semidirect
46. #1443C22semidirect
47. #1443C22semidirect
48. #1442C22semidirect
49. #1442C22semidirect
50. #1441C22semidirect
52. #1440C22semidirect
55. #1439C22semidirect
57. #1437C22semidirect
58. #1436C22semidirect
59. #1436C22semidirect
60. #1436C22semidirect
61. #1434C22semidirect
62. #1434C22semidirect

Maximal subgroups

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

subgroupindex
C2×#28492
C22×C4×C22⋊C42
#150432
#150262
#149642
#149552
#149462
#149452
#146962
#146952
#141342
#140472
#139992
#138772
C2×C22⋊#952