Finite Group Databasebeta

← #619803

Subgroups — #619803

Order 512.

3398 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×#2314C211yesmaximal, characteristic, complemented
2C2×#2280C211yesmaximal, characteristic
3#16076C214yesmaximal, complemented
4#16017C214yesmaximal, complemented
5#15432C214yesmaximal, complemented
6#15284C214yesmaximal
7#15233C212yesmaximal, complemented
8#15135C212yesmaximal, complemented
9#14680C212yesmaximal
10#14343C211yesmaximal, characteristic, complemented
11#14264C212yesmaximal, complemented
12#13994C212yesmaximal, complemented
13#13993C212yesmaximal, complemented
14C25×C4C2211yescharacteristic
15C22×#117C2211yescharacteristic
16C22×#117C2211yescharacteristic
17C2×#399C2214yes—
18#2314—28no—
19C2×#389C2212yescomplemented
20C2×#389C2212yescomplemented
21C2×#388C2214yes—
22C2×#388C2214yes—
23C2×#387C2214yescomplemented
24C2×#384C2211yescharacteristic, complemented
25C2×#382C2211yescharacteristic, complemented
26#2280—28no—
27C2×#381C2212yes—
28C2×#380C2212yes—
29C2×#380C2212yes—
30C2×#380C2212yes—
31C2×#380C2212yes—
32C2×#380C2212yes—
33C2×#380C2212yes—
34C2×#380C2212yes—
35C2×#380C2212yes—
36C2×#379C2212yes—
37C2×#379C2214yes—
38C2×C4×C4⋊C4C2212yescomplemented
39C2×C4×C22⋊C4C2212yes—
40C2×C4×C22⋊C4C2212yes—
41#1445C2214yescomplemented
42#1445C2214yescomplemented
43#1444C2214yescomplemented
44#1444C2214yescomplemented
45#1443C2214yescomplemented
46#1443C2214yescomplemented
47#1443C2214yescomplemented
48#1442C2214yescomplemented
49#1442C2214yescomplemented
50#1440C2214yes—
51#1440C2214yes—
52#1440C2214yes—
53#1440C2214yes—
54#1439C2214yes—
55#1439C2214yes—
56#1439C2214yes—
57#1439C2214yes—
58#1437C2214yes—
59#1436C2214yescomplemented
60C4⋊#95C2212yescomplemented
61C4⋊#95C2211yescharacteristic, complemented
62C4⋊#95C2211yescharacteristic, complemented
63C4⋊#95C2212yescomplemented
64C4⋊#95C2212yescomplemented
65#1434C2214yescomplemented
66C22⋊#95C2214yes—
67C22⋊#95C2214yes—
68C22⋊#95C2214yes—
69C26C2311yescharacteristic
70C24×C4—24no—
71C24×C4—28no—
72C24×C4—28no—
73C24×C4—42no—
74C24×C4—42no—
75C24×C4C2312yes—
76C24×C4C2312yes—
77C24×C4C2311yescharacteristic
78C24×C4C2311yescharacteristic
79C2×#117—28no—
80C2×#117—24no—
81C2×#117—42no—
82C2×#117—42no—
83C2×#117D418yes—
84C22×C4⋊C4C2312yes—
85C22×C4⋊C4C2312yes—
86C22×C4⋊C4C2312yes—
87C22×C4⋊C4C2312yes—
88C22×C4⋊C4C2312yes—
89C22×C4⋊C4C2314yes—
90C22×C4⋊C4C2311yescharacteristic
91C22×C4⋊C4C2311yescharacteristic
92C22×C4⋊C4C2314yes—
93C22×C4⋊C4C2314yes—
94C22×C22⋊C4C2314yes—
95C22×C22⋊C4C2312yes—
96C22×C22⋊C4C2312yes—
97C22×C22⋊C4C2312yes—
98C22×C22⋊C4C2312yes—
99C22×C22⋊C4C2312yes—
100C22×C22⋊C4C2314yes—
101C22×C22⋊C4C2314yes—
102C22×C22⋊C4C2312yes—
103C22×C22⋊C4C2314yes—
104C22×C22⋊C4C2314yes—
105C22×C22⋊C4C2314yes—
106C22×C42C2312yes—
107C22×C42C2311yescharacteristic
108C22×C42C2311yescharacteristic
109C22×C42C2314yes—
110#399—48no—
111#389—44no—
112#389—44no—
113#388—216no—
114#388—48no—
115#387—216no—
116#384—24no—
117#382—24no—
118#381—28no—
119#380—28no—
120#380—28no—
121#380—28no—
122#380—28no—
123#380—28no—
124#380—28no—
125#380—44no—
126#380—44no—
127#379—28no—
128#379—48no—
129C4×C4⋊C4—28no—
130C4×C22⋊C4—28no—
131C4×C22⋊C4—28no—
132C2×#95C2314yes—
133C2×#95C2314yes—
134C2×#95C2314yes—
135C2×#95C2314yes—
136C2×#95C2312yes—
137C2×#95C2312yes—
138C2×#95C2314yes—
139C2×#95C2314yes—
140C2×#95C2312yes—
141C2×#95C2312yes—
142C2×#95C2314yes—
143C2×#95C2314yes—
144C2×#95C2314yes—
145C2×#95C2314yes—
146C2×#95C2314yes—
147C2×#95C2314yes—
148C2×#95C2314yes—
149C2×#95C2314yes—
150C2×#95C2314yes—
151C2×#95C2314yes—
152C2×#95C2314yes—
153C2×#95C2314yes—
154C25—22no—
155C25—24no—
156C25—24no—
157C25—24no—
158C25—24no—
159C25—41no—
160C25—41no—
161C25—42no—
162C25—42no—
163C25C2411yescharacteristic
164C25C2412yes—
165C23×C4—24no—
166C23×C4—22no—
167C23×C4—22no—
168C23×C4—232no—
169C23×C4—22no—
170C23×C4—24no—
171C23×C4—24no—
172C23×C4—24no—
173C23×C4—24no—
174C23×C4—24no—
175C23×C4—24no—
176C23×C4—24no—
177C23×C4—24no—
178C23×C4—24no—
179C23×C4—24no—
180C23×C4—24no—
181C23×C4—24no—
182C23×C4—24no—
183C23×C4—42no—
184C23×C4—42no—
185C23×C4—41no—
186C23×C4—41no—
187C23×C4—432no—
188C23×C4—432no—
189C23×C4—416no—
190C23×C4—416no—
191C23×C4C2×D414yes—
192C23×C4#4118yes—
193C23×C4#4118yes—
194C23×C4C2412yes—
195C23×C4C2414yes—
196C23×C4C2414yes—
197C23×C4C2414yes—
198C23×C4C2414yes—
199C23×C4C2412yes—
200C23×C4C2412yes—
201C23×C4C2412yes—
202C23×C4C2412yes—
203C23×C4C2411yescharacteristic
204C23×C4C2411yescharacteristic
205#117—232no—
206#117—416no—
207C2×C4⋊C4—24no—
208C2×C4⋊C4—24no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—24no—
212C2×C4⋊C4—24no—
213C2×C4⋊C4—24no—
214C2×C4⋊C4—24no—
215C2×C4⋊C4—24no—
216C2×C4⋊C4—24no—
217C2×C4⋊C4—24no—
218C2×C4⋊C4—24no—
219C2×C4⋊C4—24no—
220C2×C4⋊C4—24no—
221C2×C4⋊C4—24no—
222C2×C4⋊C4—24no—
223C2×C4⋊C4—28no—
224C2×C4⋊C4—41no—
225C2×C4⋊C4—41no—
226C2×C4⋊C4—41no—
227C2×C4⋊C4—41no—
228C2×C4⋊C4—44no—
229C2×C4⋊C4—44no—
230C2×C4⋊C4—44no—
231C2×C4⋊C4—44no—
232C2×C4⋊C4—44no—
233C2×C4⋊C4—44no—
234C2×C4⋊C4—44no—
235C2×C4⋊C4—44no—
236C2×C4⋊C4C2×D418yes—
237C2×C22⋊C4—24no—
238C2×C22⋊C4—24no—
239C2×C22⋊C4—24no—
240C2×C22⋊C4—28no—
241C2×C22⋊C4—28no—
242C2×C22⋊C4—24no—
243C2×C22⋊C4—24no—
244C2×C22⋊C4—24no—
245C2×C22⋊C4—24no—
246C2×C22⋊C4—28no—
247C2×C22⋊C4—24no—
248C2×C22⋊C4—24no—
249C2×C22⋊C4—24no—
250C2×C22⋊C4—24no—
251C2×C22⋊C4—28no—
252C2×C22⋊C4—28no—
253C2×C22⋊C4—28no—
254C2×C22⋊C4—28no—
255C2×C22⋊C4—28no—
256C2×C22⋊C4—42no—
257C2×C22⋊C4—42no—
258C2×C22⋊C4—42no—
259C2×C22⋊C4—42no—
260C2×C22⋊C4—42no—
261C2×C22⋊C4—42no—
262C2×C22⋊C4—44no—
263C2×C22⋊C4—44no—
264C2×C22⋊C4—44no—
265C2×C22⋊C4—44no—
266C2×C22⋊C4—44no—
267C2×C22⋊C4—44no—
268C2×C22⋊C4—44no—
269C2×C22⋊C4—44no—
270C2×C22⋊C4—44no—
271C2×C22⋊C4—44no—
272C2×C22⋊C4C2×D418yes—
273C2×C42—24no—
274C2×C42—24no—
275C2×C42—24no—
276C2×C42—24no—
277C2×C42—24no—
278C2×C42—28no—
279C2×C42—44no—
280C2×C42—44no—
281C2×C42C2×D414yes—
282C2×C42C2×D414yes—
283#95—24no—
284#95—24no—
285#95—24no—
286#95—24no—
287#95—28no—
288#95—28no—
289#95—28no—
290#95—28no—
291#95—28no—
292#95—28no—
293#95—44no—
294#95—44no—
295#95—44no—
296#95—44no—
297#95—44no—
298#95—44no—
299#95—44no—
300#95—44no—
301#95—44no—
302#95—44no—
303#95—44no—
304#95—44no—
305C24—22no—
306C24—22no—
307C24—22no—
308C24—22no—
309C24—22no—
310C24—22no—
311C24—22no—
312C24—21no—
313C24—21no—
314C24—22no—
315C24—22no—
316C24—22no—
317C24—22no—
318C24—22no—
319C24—22no—
320C24—22no—
321C24—22no—
322C24—22no—
323C24—22no—
324C24—28no—
325C24—216no—
326C24—48no—
327C24—48no—
328C24—48no—
329C24—48no—
330C24—48no—
331C24—48no—
332C24—48no—
333C24—48no—
334C24—44no—
335C24—44no—
336C24—44no—
337C24—48no—
338C24—48no—
339C24—48no—
340C24—48no—
341C24—44no—
342C24—44no—
343C24—44no—
344C24—48no—
345C24C22×D412yes—
346C24C2×#4114yes—
347C24C2×#4114yes—
348C24H(5,𝔽2)14yes—
349C242−1+414yes—
350C24C2511yescharacteristic
351C22×C4—22no—
352C22×C4—21no—
353C22×C4—21no—
354C22×C4—28no—
355C22×C4—28no—
356C22×C4—216no—
357C22×C4—216no—
358C22×C4—216no—
359C22×C4—28no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—216no—
379C22×C4—216no—
380C22×C4—216no—
381C22×C4—24no—
382C22×C4—22no—
383C22×C4—22no—
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—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—24no—
396C22×C4—24no—
397C22×C4—24no—
398C22×C4—24no—
399C22×C4—24no—
400C22×C4—24no—
401C22×C4—24no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—24no—
405C22×C4—24no—
406C22×C4—22no—
407C22×C4—24no—
408C22×C4—24no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—24no—
412C22×C4—24no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—24no—
417C22×C4—24no—
418C22×C4—24no—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—22no—
422C22×C4—24no—
423C22×C4—48no—
424C22×C4—48no—
425C22×C4—48no—
426C22×C4—48no—
427C22×C4—48no—
428C22×C4—48no—
429C22×C4—48no—
430C22×C4—4128no—
431C22×C4—44no—
432C22×C4—44no—
433C22×C4—48no—
434C22×C4—48no—
435C22×C4—48no—
436C22×C4—48no—
437C22×C4—48no—
438C22×C4#123116yes—
439C22×C4C22×D414yes—
440C22×C4C22×D414yes—
441C22×C4C22×D412yes—
442C22×C4C22×D412yes—
443C22×C4C2×#4114yes—
444C22×C4C2×#4114yes—
445C22×C4C2×#4114yes—
446C22×C4C2×#4114yes—
447C22×C4H(5,𝔽2)12yes—
448C22×C4H(5,𝔽2)12yes—
449C4⋊C4—216no—
450C4⋊C4—44no—
451C4⋊C4—44no—
452C4⋊C4—48no—
453C4⋊C4—48no—
454C4⋊C4—48no—
455C4⋊C4—48no—
456C4⋊C4—88no—
457C4⋊C4—88no—
458C4⋊C4—88no—
459C22⋊C4—216no—
460C22⋊C4—48no—
461C22⋊C4—48no—
462C22⋊C4—48no—
463C22⋊C4—48no—
464C22⋊C4—48no—
465C22⋊C4—416no—
466C22⋊C4—88no—
467C22⋊C4—88no—
468C22⋊C4—88no—
469C22⋊C4—88no—
470C22⋊C4—88no—
471C42—28no—
472C42—28no—
473C42—48no—
474C42—88no—
475C23—22no—
476C23—24no—
477C23—22no—
478C23—24no—
479C23—24no—
480C23—24no—
481C23—24no—
482C23—24no—
483C23—24no—
484C23—24no—
485C23—24no—
486C23—24no—
487C23—24no—
488C23—24no—
489C23—24no—
490C23—24no—
491C23—24no—
492C23—24no—
493C23—24no—
494C23—24no—
495C23—24no—
496C23—24no—
497C23—24no—
498C23—24no—
499C23—24no—
500C23—24no—
501C23—24no—
502C23—24no—
503C23—416no—
504C23—432no—
505C23—432no—
506C23—432no—
507C23—432no—
508C23—416no—
509C23—432no—
510C23—432no—
511C23—416no—
512C23—42no—
513C23—42no—
514C23—42no—
515C23—42no—
516C23—42no—
517C23—42no—
518C23—42no—
519C23—42no—
520C23—42no—
521C23—42no—
522C23—42no—
523C23—42no—
524C23—42no—
525C23—42no—
526C23—42no—
527C23—42no—
528C23—41no—
529C23—41no—
530C23—41no—
531C23—42no—
532C23—42no—
533C23—41no—
534C23—42no—
535C23—42no—
536C23—42no—
537C23—42no—
538C23C2×#12314yes—
539C23#54218yes—
540C23C23×D411yescharacteristic
541C23C22×#4112yes—
542C23C22×#4112yes—
543C23C2×H(5,𝔽2)11yescharacteristic
544C23C2×H(5,𝔽2)11yescharacteristic
545C23C2×H(5,𝔽2)12yes—
546C23C2×2−1+412yes—
547C23#58512yes—
548C23#58512yes—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—24no—
552C2×C4—24no—
553C2×C4—232no—
554C2×C4—24no—
555C2×C4—24no—
556C2×C4—24no—
557C2×C4—24no—
558C2×C4—24no—
559C2×C4—24no—
560C2×C4—24no—
561C2×C4—24no—
562C2×C4—24no—
563C2×C4—432no—
564C2×C4—416no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—432no—
578C2×C4—44no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—44no—
582C2×C4—44no—
583C2×C4—44no—
584C2×C4—44no—
585C2×C4—44no—
586C2×C4—44no—
587C2×C4—44no—
588C2×C4—44no—
589C2×C4—44no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—44no—
593C2×C4—42no—
594C2×C4—42no—
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—42no—
607C2×C4—42no—
608C2×C4—44no—
609C2×C4—42no—
610C2×C4—44no—
611C2×C4—42no—
612C2×C4—44no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—44no—
616C2×C4—44no—
617C2×C4—42no—
618C2×C4—44no—
619C2×C4—42no—
620C2×C4C2×#12314yes—
621C2×C4C2×#12314yes—
622C2×C4#52514yes—
623C2×C4#53514yes—
624C2×C4#55118yes—
625C2×C4#55118yes—
626C22—28no—
627C22—28no—
628C22—28no—
629C22—28no—
630C22—24no—
631C22—464no—
632C22—42no—
633C22—44no—
634C22—44no—
635C22—44no—
636C22—44no—
637C22—44no—
638C22—44no—
639C22—44no—
640C22—44no—
641C22—44no—
642C22—44no—
643C22—44no—
644C22—44no—
645C22—44no—
646C22—44no—
647C22—44no—
648C22—44no—
649C22—42no—
650C22—44no—
651C22C22×#12311yescharacteristic
652C22C2×#52511yescharacteristic
653C22C2×#53511yescharacteristic
654C22C2×#54212yes—
655C22#345512yes—
656C22C2×#55112yes—
657C22C2×#55112yes—
658C22#348812yes—
659C22#348912yes—
660C22#349212yes—
661C22#349312yes—
662C22#349412yes—
663C22#349412yes—
664C22#349512yes—
665C22#349512yes—
666C22#353711yescharacteristic
667C22#353811yescharacteristic
668C22#353911yescharacteristic
669C22#353911yescharacteristic
670C22#354112yes—
671C22#354212yes—
672C4—24no—
673C4—44no—
674C4—44no—
675C4—44no—
676C4—82no—
677C4—84no—
678C4—84no—
679C4—84no—
680C4—84no—
681C4—82no—
682C4#248218yes—
683C2—48no—
684C2—44no—
685C2C2×#248211yescharacteristic
686C2#3526012yes—
687C2#3551612yes—
688C2#3727011yescharacteristic
689C2#3737511yescharacteristic
690C2#3785212yes—
691C2#3785312yes—
692C2#3785612yes—
693C2#3785712yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC26C23
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×#2314C2semidirect
3. #16076C2semidirect
4. #16017C2semidirect
5. #15432C2semidirect
7. #15233C2semidirect
8. #15135C2semidirect
10. #14343C2semidirect
11. #14264C2semidirect
12. #13994C2semidirect
13. #13993C2semidirect
19. C2×#389C22semidirect
20. C2×#389C22semidirect
23. C2×#387C22semidirect
24. C2×#384C22semidirect
25. C2×#382C22semidirect
38. C2×C4×C4⋊C4C22semidirect
41. #1445C22semidirect
42. #1445C22semidirect
43. #1444C22semidirect
44. #1444C22semidirect
45. #1443C22semidirect
46. #1443C22semidirect
47. #1443C22semidirect
48. #1442C22semidirect
49. #1442C22semidirect
59. #1436C22semidirect
60. C4⋊#95C22semidirect
61. C4⋊#95C22semidirect
62. C4⋊#95C22semidirect
63. C4⋊#95C22semidirect
64. C4⋊#95C22semidirect
65. #1434C22semidirect

Maximal subgroups

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

subgroupindex
C2×#23142
C2×#22802
#160762
#160172
#154322
#152842
#152332
#151352
#146802
#143432
#142642
#139942
#139932