Finite Group Databasebeta

← #4310576

Subgroups — #4310576

Order 512.

4676 conjugacy classes of subgroups.

The subgroup lattice has 693 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2854C211yesmaximal, characteristic, complemented
2C2×#2613C212yesmaximal, complemented
3C22×#390C211yesmaximal, characteristic, complemented
4#17069C212yesmaximal, complemented
5#17068C212yesmaximal, complemented
6#17022C214yesmaximal, complemented
7#17011C212yesmaximal, complemented
8#16819C214yesmaximal, complemented
9#16810C212yesmaximal, complemented
10#14820C212yesmaximal, complemented
11#14725C214yesmaximal, complemented
12#14716C212yesmaximal, complemented
13#14685C211yesmaximal, characteristic, complemented
14#14256C212yesmaximal, complemented
15C24×D4C2211yescharacteristic
16C22×#123C2212yes—
17C22×#121C2211yescharacteristic
18C22×#120C2214yes—
19C23×C4⋊C4C2211yescharacteristic
20C23×C22⋊C4C2212yes—
21C23×C22⋊C4C2212yes—
22C23×C42C2211yescharacteristic
23#2854—28no—
24#2613—216no—
25C2×#397C2214yescomplemented
26C2×#394C2214yes—
27C2×#394C2214yes—
28C2×#394C2218yescomplemented
29C2×#392C2214yes—
30C2×#392C2214yes—
31C2×#392C2214yescomplemented
32C2×#390—28no—
33C2×#390C2212yes—
34C2×#390C2212yes—
35C2×#390C2212yes—
36C2×#390C2212yes—
37C2×#390C2211yescharacteristic, complemented
38C2×#390C2211yescharacteristic, complemented
39C2×#389C2211yescharacteristic, complemented
40C2×#389C2211yescharacteristic, complemented
41C2×#388C2212yescomplemented
42C2×#388C2212yescomplemented
43C2×#386C2212yes—
44C2×#386C2214yes—
45C2×#381C2212yes—
46C2×#380C2211yescharacteristic
47C2×#380C2212yes—
48C2×#379C2214yes—
49#1441C2214yescomplemented
50#1441C2214yescomplemented
51#1440C2218yescomplemented
52#1440C2218yescomplemented
53#1439C2214yescomplemented
54#1439C2214yescomplemented
55#1439C2214yescomplemented
56#1439C2218yescomplemented
57#1439C2214yescomplemented
58#1438C2218yescomplemented
59#1438C2218yescomplemented
60C4⋊#95C2212yescomplemented
61C4⋊#95C2212yescomplemented
62#1434C2212yescomplemented
63#1434C2212yescomplemented
64C22⋊#95C2212yes—
65C22⋊#95C2214yes—
66C26C2312yes—
67C23×D4—24no—
68C23×D4—24no—
69C23×D4—44no—
70C23×D4—44no—
71C23×D4D418yescomplemented
72C23×D4C2312yes—
73C23×D4C2311yescharacteristic
74C23×D4C2312yes—
75C23×D4C2312yes—
76C23×D4C2318yes—
77C24×C4C2311yescharacteristic
78C24×C4C2311yescharacteristic
79C24×C4C2311yescharacteristic
80C24×C4C2312yes—
81C2×#123—28no—
82C2×#123—28no—
83C2×#123—28no—
84C2×#121—24no—
85C2×#121—44no—
86C2×#120—216no—
87C2×#120—48no—
88C2×#120—48no—
89C22×C4⋊C4—24no—
90C22×C4⋊C4—44no—
91C22×C4⋊C4C2312yes—
92C22×C4⋊C4C2312yes—
93C22×C4⋊C4C2311yescharacteristic
94C22×C4⋊C4C2314yes—
95C22×C4⋊C4C2312yes—
96C22×C4⋊C4C2312yes—
97C22×C4⋊C4C2311yescharacteristic
98C22×C4⋊C4C2311yescharacteristic
99C22×C22⋊C4—28no—
100C22×C22⋊C4—28no—
101C22×C22⋊C4—48no—
102C22×C22⋊C4—48no—
103C22×C22⋊C4C2312yes—
104C22×C22⋊C4C2312yes—
105C22×C22⋊C4C2314yes—
106C22×C22⋊C4C2312yes—
107C22×C22⋊C4C2314yes—
108C22×C22⋊C4C2314yes—
109C22×C22⋊C4C2312yes—
110C22×C22⋊C4C2318yes—
111C22×C22⋊C4C2318yes—
112C22×C22⋊C4C2318yes—
113C22×C22⋊C4C2314yes—
114C22×C22⋊C4C2314yes—
115C22×C22⋊C4C2312yes—
116C22×C22⋊C4C2312yes—
117C22×C22⋊C4C2318yes—
118C22×C42—44no—
119C22×C42D418yescomplemented
120C22×C42C2311yescharacteristic
121C22×C42C2311yescharacteristic
122C22×C42C2311yescharacteristic
123C22×C42C2311yescharacteristic
124#397—48no—
125#394—216no—
126#394—216no—
127#394—416no—
128#392—216no—
129#392—216no—
130#392—48no—
131#390—24no—
132#390—24no—
133#390—28no—
134#390—28no—
135#390—28no—
136#390—28no—
137#389—24no—
138#389—24no—
139#388—28no—
140#388—28no—
141#386—28no—
142#386—48no—
143#381—28no—
144#380—24no—
145#380—44no—
146#379—216no—
147C2×#95C2314yes—
148C2×#95C2314yes—
149C2×#95C2314yes—
150C2×#95C2312yes—
151C2×#95C2312yes—
152C2×#95C2312yes—
153C2×#95C2318yes—
154C2×#95C2318yes—
155C2×#95C2314yes—
156C2×#95C2314yes—
157C2×#95C2318yes—
158C25—24no—
159C25—24no—
160C25—28no—
161C25—28no—
162C25—44no—
163C25—44no—
164C25—44no—
165C25—44no—
166C25C2×D418yes—
167C25C2412yes—
168C25C2414yes—
169C25C2411yescharacteristic
170C22×D4—24no—
171C22×D4—24no—
172C22×D4—24no—
173C22×D4—28no—
174C22×D4—28no—
175C22×D4—216no—
176C22×D4—232no—
177C22×D4—24no—
178C22×D4—24no—
179C22×D4—22no—
180C22×D4—24no—
181C22×D4—24no—
182C22×D4—24no—
183C22×D4—432no—
184C22×D4—44no—
185C22×D4—44no—
186C22×D4—44no—
187C22×D4—44no—
188C22×D4—416no—
189C22×D4—416no—
190C22×D4—432no—
191C22×D4—48no—
192C22×D4—48no—
193C22×D4—42no—
194C22×D4—42no—
195C22×D4—48no—
196C22×D4—48no—
197C22×D4—48no—
198C22×D4—48no—
199C22×D4C2×D418yes—
200C22×D4C2×D418yes—
201C23×C4—24no—
202C23×C4—24no—
203C23×C4—24no—
204C23×C4—24no—
205C23×C4—24no—
206C23×C4—24no—
207C23×C4—24no—
208C23×C4—28no—
209C23×C4—42no—
210C23×C4—42no—
211C23×C4—44no—
212C23×C4—44no—
213C23×C4C2×D414yes—
214C23×C4C2×D414yes—
215C23×C4C2×D414yes—
216C23×C4C2×D414yes—
217C23×C4#4114yes—
218C23×C4C2412yes—
219C23×C4C2412yes—
220C23×C4C2414yes—
221C23×C4C2412yes—
222C23×C4C2418yes—
223C23×C4C2411yescharacteristic
224C23×C4C2411yescharacteristic
225C23×C4C2411yescharacteristic
226C23×C4C2411yescharacteristic
227C23×C4C2411yescharacteristic
228C23×C4C2411yescharacteristic
229#123—432no—
230#121—416no—
231#120—464no—
232C2×C4⋊C4—24no—
233C2×C4⋊C4—24no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—24no—
236C2×C4⋊C4—24no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—24no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—28no—
245C2×C4⋊C4—44no—
246C2×C4⋊C4—44no—
247C2×C4⋊C4—42no—
248C2×C4⋊C4—42no—
249C2×C4⋊C4—416no—
250C2×C4⋊C4—44no—
251C2×C4⋊C4—44no—
252C2×C22⋊C4—28no—
253C2×C22⋊C4—216no—
254C2×C22⋊C4—24no—
255C2×C22⋊C4—24no—
256C2×C22⋊C4—24no—
257C2×C22⋊C4—28no—
258C2×C22⋊C4—28no—
259C2×C22⋊C4—28no—
260C2×C22⋊C4—28no—
261C2×C22⋊C4—28no—
262C2×C22⋊C4—28no—
263C2×C22⋊C4—24no—
264C2×C22⋊C4—24no—
265C2×C22⋊C4—24no—
266C2×C22⋊C4—24no—
267C2×C22⋊C4—24no—
268C2×C22⋊C4—24no—
269C2×C22⋊C4—24no—
270C2×C22⋊C4—24no—
271C2×C22⋊C4—24no—
272C2×C22⋊C4—28no—
273C2×C22⋊C4—216no—
274C2×C22⋊C4—28no—
275C2×C22⋊C4—28no—
276C2×C22⋊C4—216no—
277C2×C22⋊C4—216no—
278C2×C22⋊C4—44no—
279C2×C22⋊C4—48no—
280C2×C22⋊C4—44no—
281C2×C22⋊C4—44no—
282C2×C22⋊C4—48no—
283C2×C22⋊C4—48no—
284C2×C22⋊C4—44no—
285C2×C22⋊C4—432no—
286C2×C22⋊C4—432no—
287C2×C22⋊C4—48no—
288C2×C22⋊C4—48no—
289C2×C22⋊C4—48no—
290C2×C22⋊C4—48no—
291C2×C22⋊C4—48no—
292C2×C22⋊C4—48no—
293C2×C22⋊C4—48no—
294C2×C42—416no—
295C2×C42—42no—
296C2×C42—42no—
297C2×C42—42no—
298C2×C42—42no—
299C2×C42C2×D414yes—
300C2×C42C2×D414yes—
301C2×C42C2×D414yes—
302C2×C42C2×D414yes—
303#95—24no—
304#95—28no—
305#95—28no—
306#95—24no—
307#95—24no—
308#95—28no—
309#95—44no—
310#95—44no—
311#95—48no—
312#95—48no—
313#95—48no—
314C24—22no—
315C24—22no—
316C24—22no—
317C24—22no—
318C24—216no—
319C24—24no—
320C24—24no—
321C24—24no—
322C24—28no—
323C24—24no—
324C24—28no—
325C24—28no—
326C24—24no—
327C24—28no—
328C24—24no—
329C24—232no—
330C24—22no—
331C24—22no—
332C24—22no—
333C24—24no—
334C24—24no—
335C24—24no—
336C24—24no—
337C24—24no—
338C24—24no—
339C24—416no—
340C24—416no—
341C24—48no—
342C24—48no—
343C24—48no—
344C24—48no—
345C24—416no—
346C24—416no—
347C24—48no—
348C24—48no—
349C24—416no—
350C24—816no—
351C24—816no—
352C24—816no—
353C24—816no—
354C24C22×D418yes—
355C24C22×D412yes—
356C24C22×D412yes—
357C24C2×#4112yes—
358C24H(5,𝔽2)14yes—
359C24H(5,𝔽2)14yes—
360C24C2511yescharacteristic
361C2×D4—216no—
362C2×D4—216no—
363C2×D4—416no—
364C2×D4—416no—
365C2×D4—4128no—
366C2×D4—48no—
367C2×D4—48no—
368C2×D4—48no—
369C2×D4—48no—
370C2×D4—416no—
371C2×D4—416no—
372C2×D4—44no—
373C2×D4—44no—
374C2×D4—432no—
375C2×D4—48no—
376C2×D4—416no—
377C2×D4—48no—
378C2×D4—48no—
379C2×D4—416no—
380C2×D4—44no—
381C2×D4—88no—
382C2×D4—816no—
383C2×D4—816no—
384C2×D4—816no—
385C2×D4—816no—
386C2×D4—816no—
387C2×D4—816no—
388C22×C4—28no—
389C22×C4—216no—
390C22×C4—28no—
391C22×C4—216no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—28no—
401C22×C4—28no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—28no—
409C22×C4—28no—
410C22×C4—28no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—24no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—28no—
422C22×C4—24no—
423C22×C4—24no—
424C22×C4—24no—
425C22×C4—24no—
426C22×C4—28no—
427C22×C4—28no—
428C22×C4—28no—
429C22×C4—28no—
430C22×C4—28no—
431C22×C4—24no—
432C22×C4—24no—
433C22×C4—28no—
434C22×C4—24no—
435C22×C4—24no—
436C22×C4—28no—
437C22×C4—44no—
438C22×C4—44no—
439C22×C4—416no—
440C22×C4—416no—
441C22×C4—48no—
442C22×C4—48no—
443C22×C4—48no—
444C22×C4—816no—
445C22×C4#121116yes—
446C22×C4C22×D412yes—
447C22×C4C22×D412yes—
448C22×C4C22×D412yes—
449C22×C4C22×D412yes—
450C22×C4C22×D412yes—
451C22×C4C22×D412yes—
452C22×C4C22×D412yes—
453C22×C4C22×D412yes—
454C22×C4C2×#4112yes—
455C22×C4C2×#4112yes—
456C22×C4H(5,𝔽2)14yes—
457C22×C4H(5,𝔽2)14yes—
458C4⋊C4—44no—
459C4⋊C4—48no—
460C4⋊C4—48no—
461C4⋊C4—44no—
462C4⋊C4—48no—
463C4⋊C4—48no—
464C4⋊C4—44no—
465C4⋊C4—88no—
466C22⋊C4—48no—
467C22⋊C4—416no—
468C22⋊C4—416no—
469C22⋊C4—48no—
470C22⋊C4—432no—
471C22⋊C4—48no—
472C22⋊C4—416no—
473C22⋊C4—48no—
474C22⋊C4—48no—
475C22⋊C4—48no—
476C22⋊C4—416no—
477C22⋊C4—416no—
478C22⋊C4—816no—
479C22⋊C4—816no—
480C22⋊C4—816no—
481C42—44no—
482C42—44no—
483C42—44no—
484C42—44no—
485C23—22no—
486C23—22no—
487C23—22no—
488C23—22no—
489C23—24no—
490C23—24no—
491C23—24no—
492C23—24no—
493C23—28no—
494C23—28no—
495C23—28no—
496C23—28no—
497C23—28no—
498C23—216no—
499C23—28no—
500C23—216no—
501C23—28no—
502C23—432no—
503C23—432no—
504C23—432no—
505C23—44no—
506C23—44no—
507C23—44no—
508C23—44no—
509C23—44no—
510C23—44no—
511C23—48no—
512C23—44no—
513C23—48no—
514C23—44no—
515C23—48no—
516C23—48no—
517C23—48no—
518C23—48no—
519C23—48no—
520C23—48no—
521C23—48no—
522C23—48no—
523C23—42no—
524C23—42no—
525C23—44no—
526C23—42no—
527C23—42no—
528C23—42no—
529C23—42no—
530C23—42no—
531C23—44no—
532C23—44no—
533C23—44no—
534C23—44no—
535C23—44no—
536C23—48no—
537C23—48no—
538C23—48no—
539C23—48no—
540C23—864no—
541C23—816no—
542C23—816no—
543C23—864no—
544C23—816no—
545C23—816no—
546C23C2×#12114yes—
547C23#53418yes—
548C23#53414yes—
549C23#53514yes—
550C23#53818yes—
551C23C23×D411yescharacteristic
552C23C23×D411yescharacteristic
553C23C22×#4111yescharacteristic
554C23C2×H(5,𝔽2)12yes—
555C23C2×H(5,𝔽2)12yes—
556C23C2×H(5,𝔽2)12yes—
557C23C2×H(5,𝔽2)12yes—
558C23#58512yes—
559C23#58512yes—
560D4—432no—
561D4—432no—
562D4—88no—
563D4—864no—
564D4—816no—
565C2×C4—232no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—24no—
571C2×C4—24no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—24no—
577C2×C4—24no—
578C2×C4—28no—
579C2×C4—28no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—22no—
585C2×C4—22no—
586C2×C4—416no—
587C2×C4—416no—
588C2×C4—44no—
589C2×C4—44no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—48no—
594C2×C4—44no—
595C2×C4—42no—
596C2×C4—44no—
597C2×C4—44no—
598C2×C4—42no—
599C2×C4—44no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—44no—
605C2×C4—44no—
606C2×C4—44no—
607C2×C4—48no—
608C2×C4—48no—
609C2×C4—44no—
610C2×C4—44no—
611C2×C4—48no—
612C2×C4—48no—
613C2×C4—48no—
614C2×C4—48no—
615C2×C4—48no—
616C2×C4—48no—
617C2×C4—816no—
618C2×C4C2×#12114yes—
619C2×C4C2×#12114yes—
620C2×C4#53514yes—
621C2×C4#53514yes—
622C22—216no—
623C22—24no—
624C22—24no—
625C22—216no—
626C22—24no—
627C22—24no—
628C22—24no—
629C22—24no—
630C22—416no—
631C22—416no—
632C22—48no—
633C22—48no—
634C22—48no—
635C22—864no—
636C22—84no—
637C22—84no—
638C22—88no—
639C22—88no—
640C22—84no—
641C22—84no—
642C22—88no—
643C22—82no—
644C22—82no—
645C22—88no—
646C22—82no—
647C22—82no—
648C22—82no—
649C22—82no—
650C22—82no—
651C22#261718yes—
652C22C22×#12111yescharacteristic
653C22C2×#53412yes—
654C22C2×#53411yescharacteristic
655C22C2×#53511yescharacteristic
656C22C2×#53511yescharacteristic
657C22C2×#53511yescharacteristic
658C22#344911yescharacteristic
659C22#344911yescharacteristic
660C22C2×#53812yes—
661C22#349612yes—
662C22#349612yes—
663C22#350612yes—
664C22#350612yes—
665C22#353414yes—
666C22#353414yes—
667C22#354314yes—
668C22#354514yes—
669C4—24no—
670C4—24no—
671C4—24no—
672C4—24no—
673C4—42no—
674C4—42no—
675C4—88no—
676C4—82no—
677C4—82no—
678C4—84no—
679C4—82no—
680C2—28no—
681C2—88no—
682C2—162no—
683C2C2×#261711yescharacteristic
684C2#3516511yescharacteristic
685C2#3516511yescharacteristic
686C2#3712914yes—
687C2#3719814yes—
688C2#3744011yescharacteristic
689C2#3744611yescharacteristic
690C2#3760611yescharacteristic
691C2#3761411yescharacteristic
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#2854C2semidirect
2. C2×#2613C2semidirect
3. C22×#390C2semidirect
4. #17069C2semidirect
5. #17068C2semidirect
6. #17022C2semidirect
7. #17011C2semidirect
8. #16819C2semidirect
9. #16810C2semidirect
10. #14820C2semidirect
11. #14725C2semidirect
12. #14716C2semidirect
13. #14685C2semidirect
14. #14256C2semidirect
25. C2×#397C22semidirect
28. C2×#394C22semidirect
31. C2×#392C22semidirect
37. C2×#390C22semidirect
38. C2×#390C22semidirect
39. C2×#389C22semidirect
40. C2×#389C22semidirect
41. C2×#388C22semidirect
42. C2×#388C22semidirect
49. #1441C22semidirect
50. #1441C22semidirect
51. #1440C22semidirect
52. #1440C22semidirect
53. #1439C22semidirect
54. #1439C22semidirect
55. #1439C22semidirect
56. #1439C22semidirect
57. #1439C22semidirect
58. #1438C22semidirect
59. #1438C22semidirect
60. C4⋊#95C22semidirect
61. C4⋊#95C22semidirect
62. #1434C22semidirect
63. #1434C22semidirect
71. C23×D4D4semidirect
119. C22×C42D4semidirect

Maximal subgroups

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

subgroupindex
C2×#28542
C2×#26132
C22×#3902
#170692
#170682
#170222
#170112
#168192
#168102
#148202
#147252
#147162
#146852
#142562