Finite Group Databasebeta

← #644354

Subgroups — #644354

Order 512.

4340 conjugacy classes of subgroups.

The subgroup lattice has 700 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2847C211yesmaximal, characteristic
2C22×#380C211yesmaximal, characteristic, complemented
3#17058C212yesmaximal, complemented
4#16884C214yesmaximal, complemented
5#16362C212yesmaximal, complemented
6#15091C212yesmaximal, complemented
7#15015C212yesmaximal, complemented
8#14913C212yesmaximal, complemented
9#14776C212yesmaximal, complemented
10#14735C214yesmaximal, complemented
11#14347C212yesmaximal, complemented
12#14271C212yesmaximal, complemented
13#14134C214yesmaximal, complemented
14C2×C22⋊#95C211yesmaximal, characteristic, complemented
15C25×C4C2211yescharacteristic
16C22×#121C2211yescharacteristic
17C22×#121C2211yescharacteristic
18C22×#121C2211yescharacteristic
19C22×#120C2212yes—
20C23×C4⋊C4C2211yescharacteristic
21C23×C4⋊C4C2211yescharacteristic
22C23×C22⋊C4C2212yes—
23C23×C22⋊C4C2212yes—
24#2847—28no—
25C2×#397C2214yescomplemented
26C2×#386C2212yes—
27C2×#386C2212yes—
28C2×#386C2212yes—
29C2×#386C2214yes—
30C2×#386C2214yes—
31C2×#386C2218yes—
32C2×#380—28no—
33C2×#380C2211yescharacteristic
34C2×#380C2211yescharacteristic
35C2×#380C2211yescharacteristic
36C2×#380C2212yes—
37C2×#380C2212yes—
38C2×#380C2212yes—
39C2×#380C2212yes—
40C22×#95C2212yes—
41C22×#95C2212yes—
42C22×#95C2212yes—
43#1445C2214yescomplemented
44#1445C2214yescomplemented
45#1443C2218yescomplemented
46#1442C2214yescomplemented
47#1442C2214yescomplemented
48#1442C2214yescomplemented
49#1442C2214yescomplemented
50#1442C2214yescomplemented
51#1440C2218yes—
52#1440C2218yescomplemented
53#1439C2214yescomplemented
54#1439C2214yescomplemented
55#1439C2214yescomplemented
56#1438C2218yes—
57#1438C2214yescomplemented
58C22⋊#95C2218yes—
59C22⋊#95C2214yes—
60C22⋊#95C2214yes—
61C22⋊#95C2212yes—
62C22⋊#95C2212yes—
63C22⋊#95C2212yes—
64C22⋊#95C2212yes—
65C26C2311yescharacteristic
66C23×D4C2312yes—
67C23×D4C2312yes—
68C23×D4C2312yes—
69C23×D4C2311yescharacteristic
70C24×C4—28no—
71C24×C4—44no—
72C24×C4—44no—
73C24×C4D418yes—
74C24×C4C2311yescharacteristic
75C24×C4C2311yescharacteristic
76C24×C4C2311yescharacteristic
77C24×C4C2311yescharacteristic
78C24×C4C2312yes—
79C24×C4C2312yes—
80C24×C4C2314yes—
81C24×C4C2311yescharacteristic
82C24×C4C2311yescharacteristic
83C2×#121—24no—
84C2×#121—24no—
85C2×#121—24no—
86C2×#121—28no—
87C2×#121—44no—
88C2×#121—44no—
89C2×#120—28no—
90C2×#120—216no—
91C22×C4⋊C4—24no—
92C22×C4⋊C4—24no—
93C22×C4⋊C4—44no—
94C22×C4⋊C4—44no—
95C22×C4⋊C4C2312yes—
96C22×C4⋊C4C2311yescharacteristic
97C22×C4⋊C4C2312yes—
98C22×C4⋊C4C2312yes—
99C22×C4⋊C4C2312yes—
100C22×C4⋊C4C2311yescharacteristic
101C22×C4⋊C4C2311yescharacteristic
102C22×C22⋊C4—28no—
103C22×C22⋊C4—28no—
104C22×C22⋊C4—44no—
105C22×C22⋊C4—48no—
106C22×C22⋊C4—44no—
107C22×C22⋊C4C2312yes—
108C22×C22⋊C4C2314yes—
109C22×C22⋊C4C2314yes—
110C22×C22⋊C4C2312yes—
111C22×C22⋊C4C2312yes—
112C22×C22⋊C4C2312yes—
113C22×C22⋊C4C2312yes—
114C22×C22⋊C4C2314yes—
115C22×C22⋊C4C2314yes—
116C22×C22⋊C4C2312yes—
117C22×C22⋊C4C2318yes—
118C22×C22⋊C4C2318yes—
119#397—48no—
120#386—28no—
121#386—28no—
122#386—28no—
123#386—48no—
124#386—48no—
125#386—416no—
126#380—24no—
127#380—24no—
128#380—24no—
129#380—28no—
130#380—28no—
131#380—28no—
132#380—28no—
133C2×#95—28no—
134C2×#95—28no—
135C2×#95—44no—
136C2×#95C2314yes—
137C2×#95C2314yes—
138C2×#95C2314yes—
139C2×#95C2314yes—
140C2×#95C2314yes—
141C2×#95C2312yes—
142C2×#95C2312yes—
143C2×#95C2312yes—
144C2×#95C2318yes—
145C2×#95C2318yes—
146C2×#95C2314yes—
147C2×#95C2318yes—
148C2×#95C2318yes—
149C2×#95C2318yes—
150C2×#95C2318yes—
151C25—24no—
152C25—24no—
153C25—22no—
154C25—24no—
155C25—22no—
156C25—42no—
157C25—42no—
158C25C2×D414yes—
159C25#4118yes—
160C25C2412yes—
161C25C2411yescharacteristic
162C25C2411yescharacteristic
163C25C2411yescharacteristic
164C22×D4—24no—
165C22×D4—24no—
166C22×D4—24no—
167C22×D4—28no—
168C22×D4—28no—
169C22×D4—28no—
170C22×D4—28no—
171C22×D4—28no—
172C22×D4—24no—
173C22×D4—24no—
174C22×D4—28no—
175C22×D4—22no—
176C22×D4—24no—
177C22×D4—42no—
178C22×D4—42no—
179C22×D4—44no—
180C22×D4—44no—
181C23×C4—24no—
182C23×C4—24no—
183C23×C4—24no—
184C23×C4—24no—
185C23×C4—24no—
186C23×C4—232no—
187C23×C4—24no—
188C23×C4—24no—
189C23×C4—24no—
190C23×C4—24no—
191C23×C4—24no—
192C23×C4—24no—
193C23×C4—24no—
194C23×C4—28no—
195C23×C4—28no—
196C23×C4—28no—
197C23×C4—24no—
198C23×C4—28no—
199C23×C4—24no—
200C23×C4—416no—
201C23×C4—416no—
202C23×C4—42no—
203C23×C4—42no—
204C23×C4—42no—
205C23×C4—42no—
206C23×C4—42no—
207C23×C4—42no—
208C23×C4—42no—
209C23×C4—44no—
210C23×C4—44no—
211C23×C4—44no—
212C23×C4—44no—
213C23×C4—44no—
214C23×C4—44no—
215C23×C4—44no—
216C23×C4—44no—
217C23×C4—464no—
218C23×C4C2×D414yes—
219C23×C4C2×D414yes—
220C23×C4C2×D414yes—
221C23×C4C2×D414yes—
222C23×C4C2×D414yes—
223C23×C4C2×D414yes—
224C23×C4#4118yes—
225C23×C4C2414yes—
226C23×C4C2414yes—
227C23×C4C2418yes—
228C23×C4C2412yes—
229C23×C4C2412yes—
230C23×C4C2412yes—
231C23×C4C2411yescharacteristic
232C23×C4C2411yescharacteristic
233C23×C4C2411yescharacteristic
234C23×C4C2411yescharacteristic
235#121—416no—
236#121—416no—
237#121—416no—
238#120—432no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—28no—
248C2×C4⋊C4—44no—
249C2×C4⋊C4—44no—
250C2×C4⋊C4—44no—
251C2×C4⋊C4—42no—
252C2×C4⋊C4—42no—
253C2×C4⋊C4—416no—
254C2×C4⋊C4—416no—
255C2×C22⋊C4—28no—
256C2×C22⋊C4—28no—
257C2×C22⋊C4—24no—
258C2×C22⋊C4—24no—
259C2×C22⋊C4—24no—
260C2×C22⋊C4—28no—
261C2×C22⋊C4—28no—
262C2×C22⋊C4—28no—
263C2×C22⋊C4—28no—
264C2×C22⋊C4—28no—
265C2×C22⋊C4—216no—
266C2×C22⋊C4—28no—
267C2×C22⋊C4—28no—
268C2×C22⋊C4—24no—
269C2×C22⋊C4—24no—
270C2×C22⋊C4—24no—
271C2×C22⋊C4—28no—
272C2×C22⋊C4—28no—
273C2×C22⋊C4—44no—
274C2×C22⋊C4—48no—
275C2×C22⋊C4—44no—
276C2×C22⋊C4—44no—
277C2×C22⋊C4—432no—
278C2×C22⋊C4—432no—
279C2×C22⋊C4—48no—
280C2×C22⋊C4—48no—
281C2×C22⋊C4—48no—
282C2×C22⋊C4—48no—
283C2×C22⋊C4—48no—
284C2×C22⋊C4—48no—
285#95—28no—
286#95—28no—
287#95—24no—
288#95—24no—
289#95—24no—
290#95—28no—
291#95—28no—
292#95—28no—
293#95—44no—
294#95—48no—
295#95—48no—
296#95—48no—
297#95—48no—
298#95—48no—
299#95—48no—
300C24—22no—
301C24—22no—
302C24—22no—
303C24—22no—
304C24—22no—
305C24—21no—
306C24—21no—
307C24—22no—
308C24—22no—
309C24—28no—
310C24—28no—
311C24—28no—
312C24—28no—
313C24—216no—
314C24—216no—
315C24—232no—
316C24—22no—
317C24—22no—
318C24—21no—
319C24—22no—
320C24—22no—
321C24—21no—
322C24—22no—
323C24—22no—
324C24—24no—
325C24—24no—
326C24—24no—
327C24—24no—
328C24—22no—
329C24—24no—
330C24—22no—
331C24—24no—
332C24—44no—
333C24—44no—
334C24—416no—
335C24—416no—
336C24—48no—
337C24—48no—
338C24—416no—
339C24—48no—
340C24—48no—
341C24#124116yes—
342C24C22×D412yes—
343C24C22×D412yes—
344C24C22×D412yes—
345C24C2×#4114yes—
346C24C2×#4114yes—
347C24C2×#4114yes—
348C24C2×#4114yes—
349C24H(5,𝔽2)14yes—
350C24H(5,𝔽2)14yes—
351C24H(5,𝔽2)12yes—
352C242−1+412yes—
353C24C2511yescharacteristic
354C2×D4—44no—
355C2×D4—44no—
356C2×D4—48no—
357C2×D4—48no—
358C2×D4—48no—
359C2×D4—48no—
360C2×D4—48no—
361C2×D4—48no—
362C2×D4—48no—
363C2×D4—48no—
364C2×D4—48no—
365C2×D4—416no—
366C2×D4—416no—
367C2×D4—432no—
368C2×D4—44no—
369C2×D4—88no—
370C2×D4—816no—
371C22×C4—232no—
372C22×C4—28no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—28no—
380C22×C4—28no—
381C22×C4—28no—
382C22×C4—28no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—28no—
387C22×C4—28no—
388C22×C4—28no—
389C22×C4—28no—
390C22×C4—28no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—24no—
395C22×C4—24no—
396C22×C4—28no—
397C22×C4—28no—
398C22×C4—24no—
399C22×C4—24no—
400C22×C4—24no—
401C22×C4—24no—
402C22×C4—28no—
403C22×C4—24no—
404C22×C4—22no—
405C22×C4—24no—
406C22×C4—24no—
407C22×C4—28no—
408C22×C4—24no—
409C22×C4—28no—
410C22×C4—28no—
411C22×C4—28no—
412C22×C4—28no—
413C22×C4—28no—
414C22×C4—28no—
415C22×C4—24no—
416C22×C4—22no—
417C22×C4—24no—
418C22×C4—24no—
419C22×C4—28no—
420C22×C4—24no—
421C22×C4—44no—
422C22×C4—44no—
423C22×C4—44no—
424C22×C4—44no—
425C22×C4—44no—
426C22×C4—44no—
427C22×C4—44no—
428C22×C4—416no—
429C22×C4—416no—
430C22×C4—416no—
431C22×C4—416no—
432C22×C4—4128no—
433C22×C4—416no—
434C22×C4—48no—
435C22×C4—48no—
436C22×C4—416no—
437C22×C4—48no—
438C22×C4—48no—
439C22×C4—416no—
440C22×C4—48no—
441C22×C4—48no—
442C22×C4—416no—
443C22×C4—88no—
444C22×C4—88no—
445C22×C4—88no—
446C22×C4—816no—
447C22×C4—88no—
448C22×C4—88no—
449C22×C4—88no—
450C22×C4#123116yes—
451C22×C4C22×D412yes—
452C22×C4C22×D412yes—
453C22×C4C22×D412yes—
454C22×C4C22×D412yes—
455C22×C4C2×#4114yes—
456C22×C4C2×#4114yes—
457C22×C4H(5,𝔽2)14yes—
458C4⋊C4—48no—
459C4⋊C4—48no—
460C4⋊C4—48no—
461C4⋊C4—44no—
462C4⋊C4—44no—
463C4⋊C4—44no—
464C4⋊C4—48no—
465C22⋊C4—48no—
466C22⋊C4—416no—
467C22⋊C4—416no—
468C22⋊C4—48no—
469C22⋊C4—48no—
470C22⋊C4—48no—
471C22⋊C4—416no—
472C22⋊C4—416no—
473C22⋊C4—48no—
474C22⋊C4—48no—
475C22⋊C4—816no—
476C22⋊C4—816no—
477C23—22no—
478C23—22no—
479C23—22no—
480C23—22no—
481C23—22no—
482C23—22no—
483C23—22no—
484C23—22no—
485C23—24no—
486C23—24no—
487C23—28no—
488C23—28no—
489C23—28no—
490C23—24no—
491C23—24no—
492C23—24no—
493C23—24no—
494C23—28no—
495C23—28no—
496C23—28no—
497C23—28no—
498C23—24no—
499C23—24no—
500C23—24no—
501C23—24no—
502C23—232no—
503C23—22no—
504C23—22no—
505C23—22no—
506C23—22no—
507C23—22no—
508C23—22no—
509C23—24no—
510C23—24no—
511C23—28no—
512C23—28no—
513C23—232no—
514C23—416no—
515C23—416no—
516C23—464no—
517C23—464no—
518C23—416no—
519C23—416no—
520C23—416no—
521C23—44no—
522C23—42no—
523C23—42no—
524C23—44no—
525C23—42no—
526C23—42no—
527C23—42no—
528C23—42no—
529C23—42no—
530C23—42no—
531C23—42no—
532C23—42no—
533C23—42no—
534C23—44no—
535C23—44no—
536C23—48no—
537C23—48no—
538C23—44no—
539C23—44no—
540C23—48no—
541C23—48no—
542C23—44no—
543C23—44no—
544C23C2×#12314yes—
545C23C2×#12414yes—
546C23C2×#12414yes—
547C23C2×#12414yes—
548C23#53414yes—
549C23#53514yes—
550C23#53714yes—
551C23#53818yes—
552C23#53918yes—
553C23#553116yes—
554C23C23×D411yescharacteristic
555C23C22×#4112yes—
556C23C22×#4112yes—
557C23C2×H(5,𝔽2)12yes—
558C23C2×H(5,𝔽2)12yes—
559C23C2×H(5,𝔽2)12yes—
560C23C2×H(5,𝔽2)11yescharacteristic
561C23C2×2−1+411yescharacteristic
562C23#58512yes—
563D4—88no—
564D4—816no—
565D4—816no—
566D4—816no—
567C2×C4—232no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—28no—
575C2×C4—28no—
576C2×C4—28no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—416no—
581C2×C4—416no—
582C2×C4—416no—
583C2×C4—416no—
584C2×C4—416no—
585C2×C4—44no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—44no—
593C2×C4—44no—
594C2×C4—44no—
595C2×C4—44no—
596C2×C4—42no—
597C2×C4—44no—
598C2×C4—42no—
599C2×C4—44no—
600C2×C4—42no—
601C2×C4—48no—
602C2×C4—44no—
603C2×C4—44no—
604C2×C4—48no—
605C2×C4—48no—
606C2×C4—48no—
607C2×C4—48no—
608C2×C4—48no—
609C2×C4—48no—
610C2×C4—48no—
611C2×C4—48no—
612C2×C4—48no—
613C2×C4—48no—
614C2×C4—48no—
615C2×C4—48no—
616C2×C4—832no—
617C2×C4—816no—
618C2×C4—816no—
619C2×C4C2×#12314yes—
620C2×C4C2×#12314yes—
621C2×C4#53414yes—
622C22—216no—
623C22—216no—
624C22—24no—
625C22—216no—
626C22—216no—
627C22—24no—
628C22—24no—
629C22—24no—
630C22—24no—
631C22—24no—
632C22—24no—
633C22—24no—
634C22—24no—
635C22—24no—
636C22—24no—
637C22—24no—
638C22—24no—
639C22—24no—
640C22—24no—
641C22—464no—
642C22—48no—
643C22—48no—
644C22—42no—
645C22—42no—
646C22—42no—
647C22—42no—
648C22—88no—
649C22—82no—
650C22—88no—
651C22—82no—
652C22—82no—
653C22—82no—
654C22—82no—
655C22—82no—
656C22—82no—
657C22#250418yes—
658C22#263918yes—
659C22C22×#12311yescharacteristic
660C22C22×#12411yescharacteristic
661C22C2×#53411yescharacteristic
662C22C2×#53411yescharacteristic
663C22C2×#53511yescharacteristic
664C22C2×#53711yescharacteristic
665C22#344911yescharacteristic
666C22C2×#53812yes—
667C22C2×#53912yes—
668C22C2×#55314yes—
669C22#349014yes—
670C22#349412yes—
671C22#349512yes—
672C22#350514yes—
673C22#353414yes—
674C22#353612yes—
675C22#353712yes—
676C4—24no—
677C4—24no—
678C4—24no—
679C4—42no—
680C4—82no—
681C4—88no—
682C4—84no—
683C4—84no—
684C4—82no—
685C4—82no—
686C2—28no—
687C2—28no—
688C2—44no—
689C2—162no—
690C2C2×#263911yescharacteristic
691C2C2×#250411yescharacteristic
692C2#3499211yescharacteristic
693C2#3501911yescharacteristic
694C2#3526311yescharacteristic
695C2#3743911yescharacteristic
696C2#3744611yescharacteristic
697C2#3789014yes—
698C2#3789114yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC2×#2847C2
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C22×#380C2semidirect
3. #17058C2semidirect
4. #16884C2semidirect
5. #16362C2semidirect
6. #15091C2semidirect
7. #15015C2semidirect
8. #14913C2semidirect
9. #14776C2semidirect
10. #14735C2semidirect
11. #14347C2semidirect
12. #14271C2semidirect
13. #14134C2semidirect
14. C2×C22⋊#95C2semidirect
25. C2×#397C22semidirect
43. #1445C22semidirect
44. #1445C22semidirect
45. #1443C22semidirect
46. #1442C22semidirect
47. #1442C22semidirect
48. #1442C22semidirect
49. #1442C22semidirect
50. #1442C22semidirect
52. #1440C22semidirect
53. #1439C22semidirect
54. #1439C22semidirect
55. #1439C22semidirect
57. #1438C22semidirect

Maximal subgroups

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

subgroupindex
C2×#28472
C22×#3802
#170582
#168842
#163622
#150912
#150152
#149132
#147762
#147352
#143472
#142712
#141342
C2×C22⋊#952