Finite Group Databasebeta

← #829136

Subgroups — #829136

Order 512.

2774 conjugacy classes of subgroups.

The subgroup lattice has 712 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2352C212yesmaximal, complemented
2#16808C212yesmaximal
3#16111C212yesmaximal, complemented
4#16054C214yesmaximal, complemented
5#16047C212yesmaximal, complemented
6#15162C214yesmaximal, complemented
7#15150C211yesmaximal, characteristic, complemented
8#15129C212yesmaximal, complemented
9#14248C212yesmaximal, complemented
10#13982C212yesmaximal, complemented
11#13973C212yesmaximal, complemented
12#13924C211yesmaximal, characteristic, complemented
13#13879C212yesmaximal, complemented
14#13871C212yesmaximal, complemented
15C4×#392C211yesmaximal, characteristic
16C22×C4×D4C2212yes—
17C22×C4×D4C2211yescharacteristic
18C22×#117C2212yes—
19C23×C4⋊C4C2211yescharacteristic
20C2×#397C2212yes—
21C2×#397C2212yes—
22C2×#394C2214yes—
23C2×#392C2211yescharacteristic
24#2352—216no—
25C2×#389C2212yes—
26C2×#388C2214yes—
27C2×#387C2214yes—
28C2×#386C2214yes—
29C2×#384C2211yescharacteristic
30C2×#382C2212yes—
31C2×#380C2212yes—
32C2×#380C2212yes—
33C2×#380C2212yes—
34C2×#380C2211yescharacteristic
35C2×#380C2212yes—
36C2×C4×C4⋊C4C2212yes—
37C2×C4×C22⋊C4C2212yes—
38C2×C4×C22⋊C4C2214yes—
39C2×C4×C22⋊C4C2211yescharacteristic
40#1445C2212yes—
41#1445C2212yes—
42#1445C2212yes—
43#1445C2212yes—
44#1445C2214yes—
45#1444C2211yescharacteristic
46#1443C2214yes—
47#1442C2214yes—
48#1441C2214yes—
49#1440C2214yes—
50#1440C2214yes—
51#1440C2214yes—
52#1439C2211yescharacteristic
53#1439C2212yes—
54#1439C2214yes—
55#1439C2214yes—
56#1439C2214yes—
57#1438C2212yes—
58#1438C2211yescharacteristic
59#1438C2214yes—
60#1437C2214yes—
61#1437C2214yes—
62#1436C2214yes—
63#1436C2214yes—
64#1436C2214yes—
65#1436C2214yes—
66C4⋊#95C2212yes—
67C4⋊#95C2212yes—
68C4⋊#95C2211yescharacteristic
69C4⋊#95C2212yes—
70C4⋊#95C2212yes—
71#1434C2214yes—
72#1434C2214yes—
73C22⋊#95C2212yes—
74C22⋊#95C2212yes—
75C4×#95C2211yescharacteristic
76C23×D4C2312yes—
77C23×D4C2311yescharacteristic
78C24×C4C2311yescharacteristic
79C24×C4C2312yes—
80C24×C4C2312yes—
81C2×C4×D4—28no—
82C2×C4×D4—28no—
83C2×C4×D4—28no—
84C2×C4×D4—28no—
85C2×C4×D4—42no—
86C2×#117—28no—
87C2×#117—44no—
88C2×#117—44no—
89C22×C4⋊C4—42no—
90C22×C4⋊C4—42no—
91C22×C4⋊C4D418yescomplemented
92C22×C4⋊C4C2312yes—
93C22×C4⋊C4C2312yes—
94C22×C4⋊C4C2312yes—
95C22×C4⋊C4C2312yes—
96C22×C4⋊C4C2311yescharacteristic
97C22×C4⋊C4C2312yes—
98C22×C4⋊C4C2314yes—
99C22×C4⋊C4C2312yes—
100C22×C4⋊C4C2312yes—
101C22×C4⋊C4C2312yes—
102C22×C22⋊C4C2314yes—
103C22×C22⋊C4C2312yes—
104C22×C22⋊C4C2312yes—
105C22×C22⋊C4C2312yes—
106C22×C22⋊C4C2311yescharacteristic
107C22×C22⋊C4C2312yes—
108C22×C22⋊C4C2312yes—
109C22×C22⋊C4C2312yes—
110C22×C22⋊C4C2314yes—
111C22×C22⋊C4C2314yes—
112C22×C22⋊C4C2314yes—
113C22×C22⋊C4C2314yes—
114C22×C22⋊C4C2311yescharacteristic
115C22×C42C2311yescharacteristic
116C22×C42C2311yescharacteristic
117C22×C42C2312yes—
118C22×C42C2314yes—
119C22×C42C2314yes—
120#397—28no—
121#397—44no—
122#394—48no—
123#392—24no—
124#389—28no—
125#388—216no—
126#387—216no—
127#386—216no—
128#384—42no—
129#382—28no—
130#380—28no—
131#380—28no—
132#380—28no—
133#380—44no—
134#380—42no—
135C4×C4⋊C4—28no—
136C4×C22⋊C4—216no—
137C4×C22⋊C4—28no—
138C4×C22⋊C4—42no—
139C2×#95C2311yescharacteristic
140C2×#95C2312yes—
141C2×#95C2314yes—
142C2×#95C2312yes—
143C2×#95C2314yes—
144C2×#95C2314yes—
145C2×#95C2314yes—
146C2×#95C2314yes—
147C2×#95C2312yes—
148C2×#95C2314yes—
149C2×#95C2314yes—
150C2×#95C2314yes—
151C2×#95C2314yes—
152C2×#95C2314yes—
153C2×#95C2312yes—
154C2×#95C2311yescharacteristic
155C2×#95C2312yes—
156C2×#95C2314yes—
157C2×#95C2314yes—
158C2×#95C2314yes—
159C2×#95C2314yes—
160C2×#95C2314yes—
161C2×#95C2314yes—
162C2×#95C2311yescharacteristic
163C2×#95C2311yescharacteristic
164C2×#95C2312yes—
165C25C2412yes—
166C25C2411yescharacteristic
167C22×D4—24no—
168C22×D4—24no—
169C22×D4—24no—
170C22×D4—24no—
171C22×D4—24no—
172C22×D4—24no—
173C22×D4—22no—
174C22×D4—41no—
175C22×D4—42no—
176C22×D4—42no—
177C22×D4—41no—
178C22×D4—42no—
179C23×C4—24no—
180C23×C4—24no—
181C23×C4—24no—
182C23×C4—24no—
183C23×C4—24no—
184C23×C4—24no—
185C23×C4—22no—
186C23×C4—22no—
187C23×C4—24no—
188C23×C4—24no—
189C23×C4—42no—
190C23×C4—42no—
191C23×C4—44no—
192C23×C4—44no—
193C23×C4C2×D414yes—
194C23×C4C2×D418yes—
195C23×C4#4118yes—
196C23×C4C2414yes—
197C23×C4C2414yes—
198C23×C4C2414yes—
199C23×C4C2411yescharacteristic
200C23×C4C2411yescharacteristic
201C23×C4C2414yes—
202C23×C4C2411yescharacteristic
203C23×C4C2412yes—
204C23×C4C2412yes—
205C23×C4C2412yes—
206C23×C4C2411yescharacteristic
207C23×C4C2411yescharacteristic
208C23×C4C2411yescharacteristic
209C4×D4—264no—
210C4×D4—416no—
211#117—432no—
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—24no—
224C2×C4⋊C4—24no—
225C2×C4⋊C4—24no—
226C2×C4⋊C4—24no—
227C2×C4⋊C4—28no—
228C2×C4⋊C4—24no—
229C2×C4⋊C4—24no—
230C2×C4⋊C4—416no—
231C2×C4⋊C4—42no—
232C2×C4⋊C4—42no—
233C2×C4⋊C4—41no—
234C2×C4⋊C4—42no—
235C2×C4⋊C4—44no—
236C2×C4⋊C4—42no—
237C2×C4⋊C4—42no—
238C2×C4⋊C4—44no—
239C2×C4⋊C4—42no—
240C2×C4⋊C4C2×D418yes—
241C2×C4⋊C4C2×D418yes—
242C2×C22⋊C4—24no—
243C2×C22⋊C4—24no—
244C2×C22⋊C4—24no—
245C2×C22⋊C4—24no—
246C2×C22⋊C4—24no—
247C2×C22⋊C4—24no—
248C2×C22⋊C4—28no—
249C2×C22⋊C4—24no—
250C2×C22⋊C4—28no—
251C2×C22⋊C4—28no—
252C2×C22⋊C4—22no—
253C2×C22⋊C4—28no—
254C2×C22⋊C4—24no—
255C2×C22⋊C4—24no—
256C2×C22⋊C4—24no—
257C2×C22⋊C4—24no—
258C2×C22⋊C4—24no—
259C2×C22⋊C4—22no—
260C2×C22⋊C4—28no—
261C2×C22⋊C4—24no—
262C2×C22⋊C4—28no—
263C2×C22⋊C4—41no—
264C2×C22⋊C4—42no—
265C2×C22⋊C4—42no—
266C2×C22⋊C4—44no—
267C2×C22⋊C4—44no—
268C2×C22⋊C4—42no—
269C2×C22⋊C4—41no—
270C2×C22⋊C4—41no—
271C2×C22⋊C4—44no—
272C2×C22⋊C4—44no—
273C2×C22⋊C4—44no—
274C2×C22⋊C4—42no—
275C2×C22⋊C4—44no—
276C2×C22⋊C4—42no—
277C2×C22⋊C4—44no—
278C2×C22⋊C4—44no—
279C2×C22⋊C4—44no—
280C2×C22⋊C4#4114yes—
281C2×C42—24no—
282C2×C42—24no—
283C2×C42—28no—
284C2×C42—24no—
285C2×C42—24no—
286C2×C42—22no—
287C2×C42—22no—
288C2×C42—28no—
289C2×C42—44no—
290C2×C42—44no—
291C2×C42—44no—
292C2×C42—44no—
293C2×C42#4118yes—
294#95—28no—
295#95—28no—
296#95—28no—
297#95—28no—
298#95—28no—
299#95—24no—
300#95—24no—
301#95—28no—
302#95—28no—
303#95—22no—
304#95—24no—
305#95—28no—
306#95—41no—
307#95—44no—
308#95—44no—
309#95—44no—
310#95—44no—
311#95—42no—
312#95—44no—
313#95—41no—
314#95—41no—
315#95—44no—
316#95—42no—
317#95—44no—
318#95—42no—
319#95—44no—
320C24—22no—
321C24—22no—
322C24—21no—
323C24—21no—
324C24—21no—
325C24—21no—
326C24—22no—
327C24—22no—
328C24—22no—
329C24—22no—
330C24—22no—
331C24—22no—
332C24—22no—
333C24—22no—
334C24—22no—
335C24—22no—
336C24—22no—
337C24—22no—
338C24—22no—
339C24—22no—
340C24—22no—
341C24—22no—
342C24C22×D412yes—
343C24C2×#4114yes—
344C24C2×#4112yes—
345C24H(5,𝔽2)12yes—
346C24C2511yescharacteristic
347C2×D4—216no—
348C2×D4—48no—
349C2×D4—48no—
350C2×D4—48no—
351C2×D4—44no—
352C2×D4—44no—
353C2×D4—48no—
354C2×D4—48no—
355C2×D4—48no—
356C2×D4—48no—
357C2×D4—44no—
358C2×D4—48no—
359C22×C4—22no—
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—21no—
375C22×C4—21no—
376C22×C4—21no—
377C22×C4—22no—
378C22×C4—21no—
379C22×C4—21no—
380C22×C4—21no—
381C22×C4—22no—
382C22×C4—24no—
383C22×C4—24no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—24no—
391C22×C4—24no—
392C22×C4—24no—
393C22×C4—24no—
394C22×C4—22no—
395C22×C4—21no—
396C22×C4—22no—
397C22×C4—21no—
398C22×C4—22no—
399C22×C4—216no—
400C22×C4—216no—
401C22×C4—28no—
402C22×C4—216no—
403C22×C4—216no—
404C22×C4—24no—
405C22×C4—24no—
406C22×C4—24no—
407C22×C4—24no—
408C22×C4—24no—
409C22×C4—24no—
410C22×C4—24no—
411C22×C4—24no—
412C22×C4—22no—
413C22×C4—21no—
414C22×C4—24no—
415C22×C4—24no—
416C22×C4—24no—
417C22×C4—21no—
418C22×C4—24no—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—21no—
422C22×C4—24no—
423C22×C4—22no—
424C22×C4—24no—
425C22×C4—24no—
426C22×C4—24no—
427C22×C4—24no—
428C22×C4—24no—
429C22×C4—24no—
430C22×C4—48no—
431C22×C4—416no—
432C22×C4—48no—
433C22×C4—48no—
434C22×C4—44no—
435C22×C4—48no—
436C22×C4—48no—
437C22×C4—44no—
438C22×C4—48no—
439C22×C4—84no—
440C22×C4—84no—
441C22×C4—88no—
442C22×C4C22×D418yes—
443C22×C4C22×D412yes—
444C22×C4C22×D412yes—
445C22×C4C2×#4112yes—
446C22×C4C2×#4114yes—
447C22×C4C2×#4112yes—
448C22×C4C2×#4114yes—
449C22×C4C2×#4114yes—
450C22×C4C2×#4114yes—
451C22×C4C2×#4114yes—
452C22×C4H(5,𝔽2)12yes—
453C22×C4H(5,𝔽2)12yes—
454C22×C42−1+412yes—
455C4⋊C4—216no—
456C4⋊C4—216no—
457C4⋊C4—48no—
458C4⋊C4—44no—
459C4⋊C4—48no—
460C4⋊C4—48no—
461C4⋊C4—48no—
462C4⋊C4—88no—
463C4⋊C4—84no—
464C4⋊C4—84no—
465C22⋊C4—216no—
466C22⋊C4—216no—
467C22⋊C4—44no—
468C22⋊C4—48no—
469C22⋊C4—48no—
470C22⋊C4—44no—
471C22⋊C4—48no—
472C22⋊C4—416no—
473C22⋊C4—416no—
474C22⋊C4—88no—
475C22⋊C4—88no—
476C22⋊C4—84no—
477C22⋊C4—88no—
478C42—216no—
479C42—416no—
480C42—44no—
481C42—44no—
482C42—88no—
483C23—24no—
484C23—22no—
485C23—24no—
486C23—24no—
487C23—24no—
488C23—24no—
489C23—24no—
490C23—22no—
491C23—22no—
492C23—22no—
493C23—22no—
494C23—22no—
495C23—24no—
496C23—24no—
497C23—24no—
498C23—24no—
499C23—24no—
500C23—24no—
501C23—24no—
502C23—41no—
503C23—42no—
504C23—41no—
505C23—42no—
506C23—42no—
507C23—42no—
508C23—42no—
509C23—41no—
510C23—41no—
511C23—42no—
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—42no—
529C23—42no—
530C23—42no—
531C23—42no—
532C23—42no—
533C23—42no—
534C23—42no—
535C23—42no—
536C23—42no—
537C23—42no—
538C23—42no—
539C23—42no—
540C23#55014yes—
541C23C23×D411yescharacteristic
542C23C22×#4112yes—
543C23C22×#4112yes—
544C23C22×#4111yescharacteristic
545C23C2×H(5,𝔽2)11yescharacteristic
546C23C2×H(5,𝔽2)11yescharacteristic
547C23C2×H(5,𝔽2)11yescharacteristic
548C23C2×2−1+411yescharacteristic
549C23#58512yes—
550C23#58512yes—
551C23#58511yescharacteristic
552D4—432no—
553D4—416no—
554C2×C4—24no—
555C2×C4—24no—
556C2×C4—24no—
557C2×C4—24no—
558C2×C4—28no—
559C2×C4—22no—
560C2×C4—24no—
561C2×C4—24no—
562C2×C4—24no—
563C2×C4—24no—
564C2×C4—24no—
565C2×C4—28no—
566C2×C4—28no—
567C2×C4—24no—
568C2×C4—24no—
569C2×C4—24no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—24no—
573C2×C4—22no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—42no—
585C2×C4—41no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—41no—
596C2×C4—44no—
597C2×C4—41no—
598C2×C4—41no—
599C2×C4—44no—
600C2×C4—44no—
601C2×C4—44no—
602C2×C4—416no—
603C2×C4—432no—
604C2×C4—44no—
605C2×C4—44no—
606C2×C4—44no—
607C2×C4—44no—
608C2×C4—41no—
609C2×C4—44no—
610C2×C4—44no—
611C2×C4—44no—
612C2×C4—44no—
613C2×C4—44no—
614C2×C4—44no—
615C2×C4—44no—
616C2×C4—44no—
617C2×C4—44no—
618C2×C4—42no—
619C2×C4—44no—
620C2×C4—44no—
621C2×C4—42no—
622C2×C4—44no—
623C2×C4—44no—
624C2×C4—41no—
625C2×C4—44no—
626C2×C4—41no—
627C2×C4—44no—
628C2×C4—44no—
629C2×C4—44no—
630C2×C4—816no—
631C2×C4#52518yes—
632C2×C4#53414yes—
633C2×C4#55114yes—
634C2×C4#55318yes—
635C2×C4#55914yes—
636C22—24no—
637C22—28no—
638C22—42no—
639C22—42no—
640C22—42no—
641C22—44no—
642C22—42no—
643C22—44no—
644C22—44no—
645C22—42no—
646C22—42no—
647C22—44no—
648C22—44no—
649C22—44no—
650C22—44no—
651C22—44no—
652C22—44no—
653C22—44no—
654C22—82no—
655C22—82no—
656C22—82no—
657C22—82no—
658C22—82no—
659C22—82no—
660C22—82no—
661C22—82no—
662C22C2×#52512yes—
663C22C2×#53411yescharacteristic
664C22C2×#55011yescharacteristic
665C22C2×#55111yescharacteristic
666C22C2×#55312yes—
667C22#347512yes—
668C22#347512yes—
669C22C2×#55911yescharacteristic
670C22#348512yes—
671C22#349312yes—
672C22#349412yes—
673C22#349412yes—
674C22#349512yes—
675C22#349711yescharacteristic
676C22#350511yescharacteristic
677C22#350712yes—
678C22#351111yescharacteristic
679C22#351212yes—
680C22#353611yescharacteristic
681C22#354011yescharacteristic
682C22#355111yescharacteristic
683C22#355811yescharacteristic
684C22#356212yes—
685C4—42no—
686C4—42no—
687C4—44no—
688C4—42no—
689C4—44no—
690C4—44no—
691C4—48no—
692C4—42no—
693C4—81no—
694C4—84no—
695C4—84no—
696C4—84no—
697C4#260018yes—
698C2—44no—
699C2—84no—
700C2C2×#260011yescharacteristic
701C2#3575812yes—
702C2#3651712yes—
703C2#3974311yescharacteristic
704C2#4267311yescharacteristic
705C2#4511511yescharacteristic
706C2#4516012yes—
707C2#4543311yescharacteristic
708C2#5920211yescharacteristic
709C2#5930612yes—
710C2#5986511yescharacteristic
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC2×#392C22
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×#2352C2semidirect
3. #16111C2semidirect
4. #16054C2semidirect
5. #16047C2semidirect
6. #15162C2semidirect
7. #15150C2semidirect
8. #15129C2semidirect
9. #14248C2semidirect
10. #13982C2semidirect
11. #13973C2semidirect
12. #13924C2semidirect
13. #13879C2semidirect
14. #13871C2semidirect
91. C22×C4⋊C4D4semidirect

Maximal subgroups

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

subgroupindex
C2×#23522
#168082
#161112
#160542
#160472
#151622
#151502
#151292
#142482
#139822
#139732
#139242
#138792
#138712
C4×#3922