Finite Group Databasebeta

← #2034206

Subgroups — #2034206

Order 512.

3140 conjugacy classes of subgroups.

The subgroup lattice has 702 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2804C211yesmaximal, characteristic, complemented
2C2×#2856C211yesmaximal, characteristic
3#17077C212yesmaximal, complemented
4#16253C214yesmaximal, complemented
5#15984C212yesmaximal, complemented
6#15798C212yesmaximal, complemented
7#15674C212yesmaximal, complemented
8#15631C214yesmaximal, complemented
9#15313C212yesmaximal, complemented
10#15300C214yesmaximal, complemented
11#14702C212yesmaximal, complemented
12#14445C211yesmaximal, characteristic, complemented
13#14361C214yesmaximal, complemented
14C22×#127C2211yescharacteristic
15C22×C22⋊Q8C2212yes—
16C22×#121C2214yes—
17C22×#121C2211yescharacteristic
18C22×#121C2211yescharacteristic
19C23×C42C2211yescharacteristic
20#2856—28no—
21#2804—28no—
22C2×#400C2212yes—
23C2×#397C2212yes—
24C2×#394C2214yes—
25C2×#394C2214yes—
26C2×#393C2214yes—
27C2×#393C2214yes—
28C2×#391C2212yes—
29C2×#391C2212yes—
30C2×#390C2214yes—
31C2×#390C2211yescharacteristic
32C2×#390C2212yes—
33C2×#390C2211yescharacteristic
34C2×#388C2214yes—
35C2×#386C2212yes—
36C2×#386C2212yes—
37C2×#386C2214yes—
38C2×#385C2214yes—
39C2×#384C2211yescharacteristic
40C2×#381C2212yes—
41C2×#380C2212yes—
42#1442C2214yes—
43#1441C2214yes—
44#1441C2214yes—
45#1441C2214yes—
46#1440C2212yes—
47#1440C2214yes—
48#1440C2214yes—
49#1439C2214yes—
50#1439C2214yes—
51#1439C2214yes—
52#1439C2214yes—
53#1438C2214yes—
54#1438C2214yes—
55#1438C2214yes—
56#1438C2212yes—
57#1438C2214yes—
58#1438C2212yes—
59#1438C2212yes—
60#1437C2212yes—
61#1436C2214yes—
62#1436C2214yes—
63#1436C2212yes—
64#1436C2214yes—
65C4⋊#95C2212yes—
66C4⋊#95C2212yes—
67C22⋊#95C2214yes—
68C22⋊#95C2214yes—
69C23×Q8C2312yes—
70C23×D4C2314yes—
71C23×D4C2312yes—
72C23×D4C2312yes—
73C23×D4C2311yescharacteristic
74C23×D4C2311yescharacteristic
75C24×C4C2311yescharacteristic
76C24×C4C2312yes—
77C24×C4C2312yes—
78C2×#127—24no—
79C2×#127—44no—
80C2×C22⋊Q8—28no—
81C2×C22⋊Q8—44no—
82C2×C22⋊Q8—44no—
83C2×#121—24no—
84C2×#121—216no—
85C2×#121—24no—
86C2×#121—216no—
87C2×#121—48no—
88C2×#121—42no—
89C2×#121—42no—
90C2×#121—42no—
91C2×#121—42no—
92C22×C4⋊C4C2314yes—
93C22×C4⋊C4C2314yes—
94C22×C4⋊C4C2312yes—
95C22×C4⋊C4C2311yescharacteristic
96C22×C4⋊C4C2312yes—
97C22×C4⋊C4C2312yes—
98C22×C4⋊C4C2312yes—
99C22×C4⋊C4C2311yescharacteristic
100C22×C22⋊C4C2314yes—
101C22×C22⋊C4C2312yes—
102C22×C22⋊C4C2312yes—
103C22×C22⋊C4C2314yes—
104C22×C22⋊C4C2312yes—
105C22×C22⋊C4C2314yes—
106C22×C22⋊C4C2314yes—
107C22×C22⋊C4C2314yes—
108C22×C22⋊C4C2314yes—
109C22×C22⋊C4C2314yes—
110C22×C22⋊C4C2312yes—
111C22×C22⋊C4C2314yes—
112C22×C22⋊C4C2314yes—
113C22×C22⋊C4C2312yes—
114C22×C22⋊C4C2314yes—
115C22×C42—24no—
116C22×C42—44no—
117C22×C42C2311yescharacteristic
118C22×C42C2312yes—
119C22×C42C2312yes—
120C22×C42C2311yescharacteristic
121#400—28no—
122#397—44no—
123#394—48no—
124#394—48no—
125#393—216no—
126#393—48no—
127#391—28no—
128#391—44no—
129#390—216no—
130#390—24no—
131#390—44no—
132#390D418yes—
133#388—48no—
134#386—28no—
135#386—48no—
136#386—44no—
137#385—216no—
138#384—24no—
139#381—44no—
140#380—44no—
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×#95C2312yes—
154C2×#95C2314yes—
155C2×#95C2314yes—
156C2×#95C2314yes—
157C2×#95C2314yes—
158C25C2412yes—
159C25C2412yes—
160C25C2411yescharacteristic
161C22×Q8—24no—
162C22×Q8—24no—
163C22×Q8—24no—
164C22×Q8—42no—
165C22×Q8—42no—
166C22×Q8—42no—
167C22×Q8—42no—
168C22×D4—24no—
169C22×D4—28no—
170C22×D4—28no—
171C22×D4—28no—
172C22×D4—24no—
173C22×D4—24no—
174C22×D4—24no—
175C22×D4—22no—
176C22×D4—28no—
177C22×D4—28no—
178C22×D4—28no—
179C22×D4—24no—
180C22×D4—24no—
181C22×D4—41no—
182C22×D4—41no—
183C22×D4—44no—
184C22×D4—41no—
185C22×D4—41no—
186C22×D4—44no—
187C22×D4—42no—
188C22×D4—44no—
189C22×D4—42no—
190C22×D4—42no—
191C22×D4—44no—
192C22×D4—42no—
193C22×D4—44no—
194C22×D4C2×D414yes—
195C23×C4—22no—
196C23×C4—24no—
197C23×C4—22no—
198C23×C4—22no—
199C23×C4—24no—
200C23×C4—24no—
201C23×C4—24no—
202C23×C4—24no—
203C23×C4—24no—
204C23×C4—42no—
205C23×C4—42no—
206C23×C4—42no—
207C23×C4—44no—
208C23×C4—42no—
209C23×C4—44no—
210C23×C4—42no—
211C23×C4#4118yes—
212C23×C4C2414yes—
213C23×C4C2414yes—
214C23×C4C2414yes—
215C23×C4C2414yes—
216C23×C4C2412yes—
217C23×C4C2412yes—
218C23×C4C2411yescharacteristic
219C23×C4C2412yes—
220C23×C4C2411yescharacteristic
221C23×C4C2412yes—
222#127—416no—
223C22⋊Q8—432no—
224#121—464no—
225#121—416no—
226#121—416no—
227C2×C4⋊C4—28no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—28no—
230C2×C4⋊C4—24no—
231C2×C4⋊C4—28no—
232C2×C4⋊C4—28no—
233C2×C4⋊C4—28no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—24no—
236C2×C4⋊C4—24no—
237C2×C4⋊C4—41no—
238C2×C4⋊C4—41no—
239C2×C4⋊C4—44no—
240C2×C4⋊C4—41no—
241C2×C4⋊C4—41no—
242C2×C4⋊C4—42no—
243C2×C4⋊C4—42no—
244C2×C4⋊C4—42no—
245C2×C4⋊C4—42no—
246C2×C4⋊C4—42no—
247C2×C4⋊C4—42no—
248C2×C4⋊C4—42no—
249C2×C4⋊C4—42no—
250C2×C4⋊C4C2×D414yes—
251C2×C22⋊C4—28no—
252C2×C22⋊C4—28no—
253C2×C22⋊C4—24no—
254C2×C22⋊C4—28no—
255C2×C22⋊C4—24no—
256C2×C22⋊C4—28no—
257C2×C22⋊C4—28no—
258C2×C22⋊C4—24no—
259C2×C22⋊C4—28no—
260C2×C22⋊C4—24no—
261C2×C22⋊C4—28no—
262C2×C22⋊C4—24no—
263C2×C22⋊C4—28no—
264C2×C22⋊C4—42no—
265C2×C22⋊C4—42no—
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—42no—
274C2×C22⋊C4—44no—
275C2×C22⋊C4—44no—
276C2×C22⋊C4—44no—
277C2×C22⋊C4—42no—
278C2×C22⋊C4—44no—
279C2×C22⋊C4—44no—
280C2×C22⋊C4—44no—
281C2×C22⋊C4—44no—
282C2×C22⋊C4—44no—
283C2×C22⋊C4—44no—
284C2×C22⋊C4—44no—
285C2×C22⋊C4—44no—
286C2×C22⋊C4—44no—
287C2×C22⋊C4—44no—
288C2×C22⋊C4—44no—
289C2×C22⋊C4—44no—
290C2×C22⋊C4—44no—
291C2×C22⋊C4—44no—
292C2×C22⋊C4C2×D4116yes—
293C2×C42—22no—
294C2×C42—24no—
295C2×C42—24no—
296C2×C42—24no—
297C2×C42—416no—
298C2×C42—42no—
299C2×C42—42no—
300C2×C42—44no—
301C2×C42—42no—
302C2×C42C2×D414yes—
303#95—28no—
304#95—28no—
305#95—28no—
306#95—24no—
307#95—28no—
308#95—28no—
309#95—28no—
310#95—28no—
311#95—28no—
312#95—28no—
313#95—44no—
314#95—44no—
315#95—44no—
316#95—44no—
317#95—44no—
318#95—44no—
319#95—42no—
320C24—21no—
321C24—22no—
322C24—22no—
323C24—22no—
324C24—21no—
325C24—22no—
326C24—22no—
327C24—24no—
328C24—24no—
329C24—22no—
330C24—24no—
331C24—22no—
332C24—24no—
333C24—24no—
334C24—24no—
335C24—22no—
336C24—24no—
337C24—24no—
338C24—22no—
339C24—24no—
340C24—22no—
341C24—24no—
342C24—24no—
343C24—24no—
344C24C22×D414yes—
345C24C2×#4114yes—
346C24H(5,𝔽2)12yes—
347C24C2511yescharacteristic
348C2×Q8—48no—
349C2×Q8—48no—
350C2×Q8—48no—
351C2×Q8—84no—
352C2×Q8—84no—
353C2×Q8—84no—
354C2×Q8—84no—
355C2×D4—28no—
356C2×D4—44no—
357C2×D4—44no—
358C2×D4—416no—
359C2×D4—416no—
360C2×D4—48no—
361C2×D4—416no—
362C2×D4—416no—
363C2×D4—44no—
364C2×D4—44no—
365C2×D4—416no—
366C2×D4—48no—
367C2×D4—48no—
368C2×D4—48no—
369C2×D4—44no—
370C2×D4—416no—
371C2×D4—88no—
372C2×D4—88no—
373C2×D4—84no—
374C2×D4—84no—
375C2×D4—84no—
376C2×D4—84no—
377C2×D4—88no—
378C2×D4—88no—
379C2×D4—84no—
380C2×D4—88no—
381C2×D4—84no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—21no—
390C22×C4—21no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—28no—
394C22×C4—21no—
395C22×C4—24no—
396C22×C4—22no—
397C22×C4—24no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—21no—
401C22×C4—22no—
402C22×C4—216no—
403C22×C4—216no—
404C22×C4—24no—
405C22×C4—24no—
406C22×C4—22no—
407C22×C4—24no—
408C22×C4—24no—
409C22×C4—24no—
410C22×C4—24no—
411C22×C4—24no—
412C22×C4—24no—
413C22×C4—24no—
414C22×C4—24no—
415C22×C4—24no—
416C22×C4—24no—
417C22×C4—24no—
418C22×C4—24no—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—22no—
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—24no—
431C22×C4—24no—
432C22×C4—24no—
433C22×C4—24no—
434C22×C4—24no—
435C22×C4—24no—
436C22×C4—24no—
437C22×C4—48no—
438C22×C4—48no—
439C22×C4—48no—
440C22×C4—48no—
441C22×C4—48no—
442C22×C4—48no—
443C22×C4—48no—
444C22×C4—44no—
445C22×C4—44no—
446C22×C4—48no—
447C22×C4—84no—
448C22×C4—88no—
449C22×C4—84no—
450C22×C4—88no—
451C22×C4C22×D414yes—
452C22×C4C22×D412yes—
453C22×C4C22×D414yes—
454C22×C4C2×#4114yes—
455C22×C4C2×#4114yes—
456C22×C4H(5,𝔽2)12yes—
457C22×C4H(5,𝔽2)14yes—
458C22×C4H(5,𝔽2)14yes—
459C22×C4H(5,𝔽2)14yes—
460C22×C42−1+414yes—
461C4⋊C4—44no—
462C4⋊C4—416no—
463C4⋊C4—44no—
464C4⋊C4—48no—
465C4⋊C4—416no—
466C4⋊C4—84no—
467C4⋊C4—84no—
468C4⋊C4—84no—
469C22⋊C4—232no—
470C22⋊C4—48no—
471C22⋊C4—48no—
472C22⋊C4—416no—
473C22⋊C4—416no—
474C22⋊C4—84no—
475C22⋊C4—88no—
476C22⋊C4—88no—
477C22⋊C4—88no—
478C22⋊C4—88no—
479C22⋊C4—88no—
480C22⋊C4—84no—
481C22⋊C4—84no—
482C22⋊C4—88no—
483C22⋊C4—88no—
484C42—28no—
485C42—44no—
486C42—48no—
487C42—84no—
488C23—24no—
489C23—24no—
490C23—22no—
491C23—24no—
492C23—24no—
493C23—24no—
494C23—24no—
495C23—24no—
496C23—22no—
497C23—24no—
498C23—24no—
499C23—28no—
500C23—24no—
501C23—24no—
502C23—28no—
503C23—41no—
504C23—41no—
505C23—42no—
506C23—41no—
507C23—42no—
508C23—42no—
509C23—42no—
510C23—42no—
511C23—42no—
512C23—41no—
513C23—44no—
514C23—42no—
515C23—44no—
516C23—44no—
517C23—44no—
518C23—44no—
519C23—42no—
520C23—44no—
521C23—44no—
522C23—44no—
523C23—44no—
524C23—42no—
525C23—44no—
526C23—44no—
527C23—42no—
528C23—44no—
529C23—42no—
530C23—44no—
531C23—44no—
532C23—42no—
533C23—42no—
534C23—42no—
535C23—44no—
536C23—44no—
537C23—44no—
538C23—44no—
539C23—44no—
540C23—42no—
541C23—44no—
542C23—44no—
543C23—44no—
544C23—44no—
545C23—42no—
546C23—44no—
547C23—44no—
548C23—42no—
549C23—44no—
550C23#55914yes—
551C23C23×D411yescharacteristic
552C23C22×#4112yes—
553C23C2×H(5,𝔽2)12yes—
554C23C2×H(5,𝔽2)12yes—
555C23C2×H(5,𝔽2)12yes—
556C23C2×H(5,𝔽2)11yescharacteristic
557C23C2×H(5,𝔽2)11yescharacteristic
558C23C2×2−1+412yes—
559C23#58512yes—
560Q8—816no—
561D4—88no—
562D4—832no—
563D4—88no—
564D4—816no—
565D4—816no—
566C2×C4—24no—
567C2×C4—24no—
568C2×C4—22no—
569C2×C4—24no—
570C2×C4—24no—
571C2×C4—22no—
572C2×C4—24no—
573C2×C4—24no—
574C2×C4—24no—
575C2×C4—24no—
576C2×C4—24no—
577C2×C4—24no—
578C2×C4—24no—
579C2×C4—24no—
580C2×C4—24no—
581C2×C4—24no—
582C2×C4—28no—
583C2×C4—22no—
584C2×C4—24no—
585C2×C4—24no—
586C2×C4—432no—
587C2×C4—416no—
588C2×C4—41no—
589C2×C4—42no—
590C2×C4—41no—
591C2×C4—44no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—44no—
595C2×C4—44no—
596C2×C4—44no—
597C2×C4—44no—
598C2×C4—42no—
599C2×C4—42no—
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—44no—
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—44no—
622C2×C4—44no—
623C2×C4—44no—
624C2×C4—44no—
625C2×C4—44no—
626C2×C4—44no—
627C2×C4—816no—
628C2×C4#53614yes—
629C2×C4#53818yes—
630C2×C4#54618yes—
631C22—24no—
632C22—28no—
633C22—44no—
634C22—44no—
635C22—42no—
636C22—42no—
637C22—44no—
638C22—44no—
639C22—44no—
640C22—44no—
641C22—48no—
642C22—48no—
643C22—44no—
644C22—44no—
645C22—84no—
646C22—84no—
647C22—84no—
648C22—84no—
649C22—82no—
650C22—84no—
651C22—84no—
652C22—84no—
653C22—82no—
654C22—82no—
655C22—84no—
656C22—84no—
657C22—84no—
658C22C2×#53611yescharacteristic
659C22C2×#53812yes—
660C22C2×#54612yes—
661C22C2×#55911yescharacteristic
662C22#348411yescharacteristic
663C22#348411yescharacteristic
664C22#348812yes—
665C22#350512yes—
666C22#350712yes—
667C22#350812yes—
668C22#351112yes—
669C22#352911yescharacteristic
670C22#353412yes—
671C22#353612yes—
672C22#353612yes—
673C22#353712yes—
674C22#354311yescharacteristic
675C22#354712yes—
676C22#354812yes—
677C22#354911yescharacteristic
678C22#355812yes—
679C4—28no—
680C4—28no—
681C4—24no—
682C4—44no—
683C4—42no—
684C4—82no—
685C4—84no—
686C4—84no—
687C4—84no—
688C4—84no—
689C2—44no—
690C2—84no—
691C2—162no—
692C2#3671811yescharacteristic
693C2#3731612yes—
694C2#3769312yes—
695C2#3947012yes—
696C2#3976812yes—
697C2#4580812yes—
698C2#4622511yescharacteristic
699C2#5319012yes—
700C2#5914311yescharacteristic
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC2×#2856C2
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×#2804C2semidirect
3. #17077C2semidirect
4. #16253C2semidirect
5. #15984C2semidirect
6. #15798C2semidirect
7. #15674C2semidirect
8. #15631C2semidirect
9. #15313C2semidirect
10. #15300C2semidirect
11. #14702C2semidirect
12. #14445C2semidirect
13. #14361C2semidirect

Maximal subgroups

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

subgroupindex
C2×#28042
C2×#28562
#170772
#162532
#159842
#157982
#156742
#156312
#153132
#153002
#147022
#144452
#143612