Finite Group Databasebeta

← #642482

Subgroups — #642482

Order 512.

2836 conjugacy classes of subgroups.

The subgroup lattice has 724 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2647C211yesmaximal, characteristic, complemented
2C2×#2619C211yesmaximal, characteristic
3#16373C214yesmaximal, complemented
4#16371C214yesmaximal, complemented
5#15138C212yesmaximal, complemented
6#15126C212yesmaximal, complemented
7#15007C214yesmaximal, complemented
8#14947C214yesmaximal, complemented
9#14713C211yesmaximal, characteristic, complemented
10#14340C211yesmaximal, characteristic, complemented
11#14272C211yesmaximal, characteristic, complemented
12#14141C212yesmaximal, complemented
13#13998C212yesmaximal, complemented
14C4×#382C211yesmaximal, characteristic, complemented
15#13761C211yesmaximal, characteristic, complemented
16C22×#127C2211yescharacteristic
17C22×#121C2212yes—
18C22×#117C2211yescharacteristic
19#2647—28no—
20#2619—28no—
21C2×#400C2214yes—
22C2×#398C2214yes—
23C2×#397C2214yes—
24C2×#394C2214yes—
25C2×#394C2214yes—
26C2×#392C2214yes—
27C2×#390C2212yes—
28C2×#390C2212yes—
29C2×#390C2212yes—
30C2×#390C2212yes—
31C2×#390C2212yes—
32C2×#390C2214yes—
33C2×#390C2214yes—
34C2×#390C2214yes—
35C2×#389C2212yes—
36C2×#389C2212yes—
37C2×#388C2214yes—
38C2×#384C2211yescharacteristic
39C2×#382C2211yescharacteristic
40C2×#382C2211yescharacteristic
41C2×#380C2212yes—
42C2×C4×C4⋊C4C2212yes—
43C2×C4×C22⋊C4C2211yescharacteristic
44C2×C4×C22⋊C4C2211yescharacteristic
45C2×#376C2211yescharacteristic
46C2×C43C2211yescharacteristic
47#1441C2214yes—
48#1440C2214yes—
49#1440C2214yes—
50#1440C2214yes—
51#1440C2214yes—
52#1439C2214yes—
53#1439C2214yes—
54#1439C2214yes—
55#1439C2214yes—
56#1439C2214yes—
57#1438C2214yes—
58#1438C2214yes—
59#1438C2214yes—
60#1438C2214yes—
61#1437C2214yes—
62#1437C2214yes—
63#1437C2212yes—
64#1437C2212yes—
65#1436C2212yes—
66#1436C2212yes—
67C4⋊#95C2212yes—
68C4⋊#95C2212yes—
69#1434C2214yes—
70C4×#95C2211yescharacteristic
71C4×#95C2211yescharacteristic
72C4×#95C2214yes—
73C23×D4C2312yes—
74C23×D4C2314yes—
75C23×D4C2314yes—
76C24×C4C2311yescharacteristic
77C2×#127—24no—
78C2×#127—24no—
79C2×#127—24no—
80C2×#121—28no—
81C2×#121—28no—
82C2×#121—44no—
83C2×#117—24no—
84C2×#117—24no—
85C2×#117—24no—
86C22×C4⋊C4C2312yes—
87C22×C4⋊C4C2312yes—
88C22×C4⋊C4C2312yes—
89C22×C4⋊C4C2312yes—
90C22×C4⋊C4C2314yes—
91C22×C4⋊C4C2314yes—
92C22×C4⋊C4C2314yes—
93C22×C22⋊C4C2312yes—
94C22×C22⋊C4C2311yescharacteristic
95C22×C22⋊C4C2314yes—
96C22×C22⋊C4C2314yes—
97C22×C22⋊C4C2312yes—
98C22×C22⋊C4C2314yes—
99C22×C22⋊C4C2314yes—
100C22×C22⋊C4C2314yes—
101C22×C22⋊C4C2314yes—
102C22×C22⋊C4C2311yescharacteristic
103C22×C22⋊C4C2314yes—
104C22×C22⋊C4C2314yes—
105C22×C22⋊C4C2314yes—
106C22×C22⋊C4C2314yes—
107C22×C22⋊C4C2314yes—
108C22×C22⋊C4C2314yes—
109C22×C42C2311yescharacteristic
110C22×C42C2311yescharacteristic
111C22×C42C2311yescharacteristic
112C22×C42C2311yescharacteristic
113C22×C42C2312yes—
114C22×C42C2312yes—
115C22×C42C2311yescharacteristic
116C22×C42C2314yes—
117C22×C42C2311yescharacteristic
118#400—216no—
119#398—216no—
120#397—48no—
121#394—216no—
122#394—48no—
123#392—216no—
124#390—28no—
125#390—28no—
126#390—48no—
127#390—48no—
128#390—48no—
129#390—44no—
130#390—44no—
131#390—44no—
132#389—28no—
133#389—28no—
134#388—48no—
135#384—24no—
136#382—24no—
137#382D418yes—
138#380—44no—
139C4×C4⋊C4—44no—
140C4×C22⋊C4—24no—
141C4×C22⋊C4—24no—
142#376—24no—
143C2×#95C2314yes—
144C2×#95C2312yes—
145C2×#95C2312yes—
146C2×#95C2314yes—
147C2×#95C2312yes—
148C2×#95C2314yes—
149C2×#95C2314yes—
150C2×#95C2312yes—
151C2×#95C2311yescharacteristic
152C2×#95C2311yescharacteristic
153C2×#95C2312yes—
154C2×#95C2314yes—
155C2×#95C2314yes—
156C2×#95C2312yes—
157C2×#95C2312yes—
158C2×#95C2314yes—
159C2×#95C2314yes—
160C2×#95C2314yes—
161C2×#95C2314yes—
162C43—24no—
163C25C2414yes—
164C25C2411yescharacteristic
165C22×D4—28no—
166C22×D4—24no—
167C22×D4—28no—
168C22×D4—28no—
169C22×D4—24no—
170C22×D4—28no—
171C22×D4—24no—
172C22×D4—24no—
173C22×D4—24no—
174C22×D4—28no—
175C22×D4—24no—
176C22×D4—24no—
177C22×D4—44no—
178C22×D4—44no—
179C22×D4—44no—
180C22×D4—44no—
181C22×D4—44no—
182C22×D4—44no—
183C22×D4—44no—
184C22×D4—44no—
185C22×D4—44no—
186C23×C4—22no—
187C23×C4—22no—
188C23×C4—22no—
189C23×C4—22no—
190C23×C4—22no—
191C23×C4—22no—
192C23×C4#4114yes—
193C23×C4C2414yes—
194C23×C4C2411yescharacteristic
195C23×C4C2412yes—
196C23×C4C2412yes—
197C23×C4C2414yes—
198C23×C4C2411yescharacteristic
199C23×C4C2414yes—
200C23×C4C2411yescharacteristic
201C23×C4C2411yescharacteristic
202C23×C4C2412yes—
203C23×C4C2412yes—
204C23×C4C2411yescharacteristic
205C23×C4C2411yescharacteristic
206#127—416no—
207#121—432no—
208#117—416no—
209C2×C4⋊C4—24no—
210C2×C4⋊C4—24no—
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—28no—
221C2×C4⋊C4—28no—
222C2×C4⋊C4—28no—
223C2×C4⋊C4—44no—
224C2×C4⋊C4—44no—
225C2×C4⋊C4—44no—
226C2×C4⋊C4—44no—
227C2×C4⋊C4—44no—
228C2×C4⋊C4—44no—
229C2×C4⋊C4C2×D418yes—
230C2×C22⋊C4—24no—
231C2×C22⋊C4—24no—
232C2×C22⋊C4—28no—
233C2×C22⋊C4—24no—
234C2×C22⋊C4—24no—
235C2×C22⋊C4—28no—
236C2×C22⋊C4—22no—
237C2×C22⋊C4—22no—
238C2×C22⋊C4—28no—
239C2×C22⋊C4—28no—
240C2×C22⋊C4—24no—
241C2×C22⋊C4—28no—
242C2×C22⋊C4—24no—
243C2×C22⋊C4—28no—
244C2×C22⋊C4—28no—
245C2×C22⋊C4—28no—
246C2×C22⋊C4—28no—
247C2×C22⋊C4—28no—
248C2×C22⋊C4—42no—
249C2×C22⋊C4—42no—
250C2×C22⋊C4—44no—
251C2×C22⋊C4—44no—
252C2×C22⋊C4—44no—
253C2×C22⋊C4—44no—
254C2×C22⋊C4—44no—
255C2×C22⋊C4—44no—
256C2×C22⋊C4—44no—
257C2×C22⋊C4—44no—
258C2×C22⋊C4—44no—
259C2×C22⋊C4—44no—
260C2×C22⋊C4—44no—
261C2×C22⋊C4—44no—
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⋊C4—44no—
273C2×C22⋊C4—44no—
274C2×C22⋊C4—44no—
275C2×C22⋊C4—44no—
276C2×C42—22no—
277C2×C42—22no—
278C2×C42—22no—
279C2×C42—22no—
280C2×C42—22no—
281C2×C42—22no—
282C2×C42—22no—
283C2×C42—24no—
284C2×C42—24no—
285C2×C42—22no—
286C2×C42—22no—
287C2×C42—22no—
288C2×C42—22no—
289C2×C42—24no—
290C2×C42—24no—
291C2×C42—22no—
292C2×C42—22no—
293C2×C42—22no—
294C2×C42—28no—
295C2×C42—42no—
296C2×C42—42no—
297C2×C42—44no—
298C2×C42—44no—
299C2×C42C2×D414yes—
300C2×C42#4114yes—
301C2×C42#4114yes—
302C2×C42#4114yes—
303#95—24no—
304#95—24no—
305#95—28no—
306#95—28no—
307#95—22no—
308#95—22no—
309#95—24no—
310#95—28no—
311#95—24no—
312#95—28no—
313#95—24no—
314#95—44no—
315#95—42no—
316#95—42no—
317#95—44no—
318#95—44no—
319#95—44no—
320#95—44no—
321#95C2×D4116yes—
322C24—21no—
323C24—21no—
324C24—21no—
325C24—21no—
326C24—22no—
327C24—22no—
328C24—21no—
329C24—21no—
330C24—22no—
331C24—22no—
332C24—24no—
333C24—24no—
334C24—24no—
335C24—24no—
336C24—24no—
337C24—24no—
338C24—24no—
339C24—24no—
340C24—24no—
341C24—24no—
342C24—24no—
343C24—24no—
344C24—24no—
345C24—24no—
346C24—24no—
347C24C2×#4112yes—
348C24C2511yescharacteristic
349C2×D4—416no—
350C2×D4—416no—
351C2×D4—416no—
352C2×D4—416no—
353C2×D4—416no—
354C2×D4—48no—
355C2×D4—48no—
356C2×D4—48no—
357C2×D4—48no—
358C2×D4—48no—
359C2×D4—48no—
360C2×D4—48no—
361C2×D4—88no—
362C2×D4—88no—
363C2×D4—88no—
364C2×D4—88no—
365C2×D4—88no—
366C2×D4—88no—
367C2×D4—88no—
368C2×D4—88no—
369C2×D4—88no—
370C22×C4—21no—
371C22×C4—21no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—21no—
378C22×C4—21no—
379C22×C4—22no—
380C22×C4—21no—
381C22×C4—22no—
382C22×C4—24no—
383C22×C4—21no—
384C22×C4—21no—
385C22×C4—24no—
386C22×C4—21no—
387C22×C4—21no—
388C22×C4—24no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—24no—
393C22×C4—21no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—24no—
398C22×C4—24no—
399C22×C4—28no—
400C22×C4—28no—
401C22×C4—28no—
402C22×C4—21no—
403C22×C4—22no—
404C22×C4—21no—
405C22×C4—24no—
406C22×C4—22no—
407C22×C4—24no—
408C22×C4—21no—
409C22×C4—21no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—24no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—24no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—24no—
422C22×C4—24no—
423C22×C4—24no—
424C22×C4—24no—
425C22×C4—22no—
426C22×C4—21no—
427C22×C4—21no—
428C22×C4—24no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—24no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—24no—
436C22×C4—22no—
437C22×C4—24no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—44no—
441C22×C4—44no—
442C22×C4—44no—
443C22×C4—44no—
444C22×C4C22×D418yes—
445C22×C4C22×D414yes—
446C22×C4C22×D412yes—
447C22×C4C2×#4112yes—
448C22×C4C2×#4112yes—
449C22×C4C2×#4114yes—
450C22×C4C2×#4114yes—
451C22×C4C2×#4112yes—
452C22×C4C2×#4112yes—
453C22×C4C2×#4112yes—
454C22×C4C2×#4112yes—
455C22×C4C2×#4112yes—
456C22×C4H(5,𝔽2)14yes—
457C22×C4H(5,𝔽2)14yes—
458C22×C4H(5,𝔽2)12yes—
459C22×C4H(5,𝔽2)12yes—
460C4⋊C4—216no—
461C4⋊C4—48no—
462C4⋊C4—48no—
463C4⋊C4—48no—
464C4⋊C4—88no—
465C4⋊C4—88no—
466C4⋊C4—88no—
467C22⋊C4—416no—
468C22⋊C4—48no—
469C22⋊C4—44no—
470C22⋊C4—48no—
471C22⋊C4—44no—
472C22⋊C4—416no—
473C22⋊C4—88no—
474C22⋊C4—88no—
475C22⋊C4—88no—
476C22⋊C4—88no—
477C22⋊C4—88no—
478C22⋊C4—88no—
479C22⋊C4—88no—
480C22⋊C4—88no—
481C22⋊C4—88no—
482C22⋊C4—88no—
483C42—28no—
484C42—48no—
485C42—48no—
486C42—44no—
487C42—44no—
488C42—44no—
489C42—44no—
490C42—88no—
491C42#124116yes—
492C23—22no—
493C23—24no—
494C23—22no—
495C23—22no—
496C23—24no—
497C23—41no—
498C23—41no—
499C23—41no—
500C23—41no—
501C23—42no—
502C23—42no—
503C23—41no—
504C23—41no—
505C23—42no—
506C23—42no—
507C23—42no—
508C23—42no—
509C23—41no—
510C23—41no—
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—44no—
526C23—44no—
527C23—44no—
528C23—44no—
529C23—44no—
530C23—44no—
531C23—44no—
532C23—44no—
533C23—44no—
534C23—44no—
535C23—44no—
536C23—44no—
537C23—44no—
538C23—44no—
539C23—44no—
540C23—44no—
541C23—44no—
542C23—44no—
543C23—44no—
544C23—44no—
545C23—44no—
546C23—44no—
547C23—44no—
548C23—44no—
549C23—44no—
550C23C23×D411yescharacteristic
551C23C22×#4111yescharacteristic
552C23C22×#4111yescharacteristic
553C23C22×#4111yescharacteristic
554C23C22×#4111yescharacteristic
555C23C2×H(5,𝔽2)12yes—
556C23C2×H(5,𝔽2)11yescharacteristic
557C23C2×H(5,𝔽2)11yescharacteristic
558C23C2×H(5,𝔽2)12yes—
559C23#58512yes—
560C23#58512yes—
561D4—832no—
562D4—832no—
563D4—816no—
564C2×C4—24no—
565C2×C4—24no—
566C2×C4—22no—
567C2×C4—24no—
568C2×C4—24no—
569C2×C4—24no—
570C2×C4—24no—
571C2×C4—24no—
572C2×C4—22no—
573C2×C4—24no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—28no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—24no—
581C2×C4—24no—
582C2×C4—28no—
583C2×C4—28no—
584C2×C4—22no—
585C2×C4—24no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—42no—
589C2×C4—44no—
590C2×C4—41no—
591C2×C4—42no—
592C2×C4—41no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—44no—
596C2×C4—42no—
597C2×C4—44no—
598C2×C4—44no—
599C2×C4—42no—
600C2×C4—416no—
601C2×C4—42no—
602C2×C4—41no—
603C2×C4—44no—
604C2×C4—44no—
605C2×C4—41no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—44no—
609C2×C4—44no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—41no—
613C2×C4—41no—
614C2×C4—44no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—44no—
621C2×C4—44no—
622C2×C4—42no—
623C2×C4—42no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—44no—
627C2×C4—41no—
628C2×C4—41no—
629C2×C4—44no—
630C2×C4—44no—
631C2×C4—42no—
632C2×C4—44no—
633C2×C4—42no—
634C2×C4—44no—
635C2×C4—44no—
636C2×C4—42no—
637C2×C4—42no—
638C2×C4—44no—
639C2×C4C2×#12418yes—
640C2×C4C2×#12414yes—
641C2×C4#52914yes—
642C2×C4#54014yes—
643C2×C4#54014yes—
644C2×C4#54614yes—
645C2×C4#55014yes—
646C2×C4#55314yes—
647C2×C4#55314yes—
648C22—44no—
649C22—42no—
650C22—42no—
651C22—42no—
652C22—44no—
653C22—81no—
654C22—81no—
655C22—82no—
656C22—82no—
657C22—81no—
658C22—81no—
659C22—84no—
660C22—84no—
661C22—84no—
662C22—84no—
663C22—84no—
664C22—84no—
665C22—84no—
666C22—84no—
667C22—84no—
668C22—84no—
669C22—84no—
670C22—84no—
671C22—84no—
672C22—84no—
673C22—84no—
674C22C22×#12411yescharacteristic
675C22C2×#52911yescharacteristic
676C22C2×#54011yescharacteristic
677C22C2×#54011yescharacteristic
678C22#345511yescharacteristic
679C22#346012yes—
680C22#346012yes—
681C22#346012yes—
682C22C2×#54611yescharacteristic
683C22C2×#55011yescharacteristic
684C22C2×#55311yescharacteristic
685C22C2×#55311yescharacteristic
686C22#349412yes—
687C22#349412yes—
688C22#349412yes—
689C22#349812yes—
690C22#350012yes—
691C22#350811yescharacteristic
692C22#351211yescharacteristic
693C22#354312yes—
694C22#354612yes—
695C22#355812yes—
696C22#356212yes—
697C4—24no—
698C4—28no—
699C4—42no—
700C4—44no—
701C4—42no—
702C4—48no—
703C4—82no—
704C4—84no—
705C4—81no—
706C4—81no—
707C4—84no—
708C4—82no—
709C4#249018yes—
710C2—82no—
711C2—164no—
712C2#3462111yescharacteristic
713C2C2×#249011yescharacteristic
714C2#3526412yes—
715C2#3528711yescharacteristic
716C2#3758712yes—
717C2#3789311yescharacteristic
718C2#3789811yescharacteristic
719C2#3794612yes—
720C2#3938211yescharacteristic
721C2#3938611yescharacteristic
722C2#3941912yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC2×#2619C2
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×#2647C2semidirect
3. #16373C2semidirect
4. #16371C2semidirect
5. #15138C2semidirect
6. #15126C2semidirect
7. #15007C2semidirect
8. #14947C2semidirect
9. #14713C2semidirect
10. #14340C2semidirect
11. #14272C2semidirect
12. #14141C2semidirect
13. #13998C2semidirect
14. C4×#382C2semidirect
15. #13761C2semidirect

Maximal subgroups

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

subgroupindex
C2×#26472
C2×#26192
#163732
#163712
#151382
#151262
#150072
#149472
#147132
#143402
#142722
#141412
#139982
C4×#3822
#137612