Finite Group Databasebeta

← #375148

Subgroups — #375148

Order 512.

2032 conjugacy classes of subgroups.

The subgroup lattice has 650 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#465C211yesmaximal, characteristic
2C2×#1874C211yesmaximal, characteristic
3C2×#1872C211yesmaximal, characteristic, complemented
4C2×C8×C22⋊C4C211yesmaximal, characteristic, complemented
5C22×#340C211yesmaximal, characteristic, complemented
6#14139C211yesmaximal, characteristic, complemented
7#14137C211yesmaximal, characteristic
8#11378C212yesmaximal, complemented
9#11023C212yesmaximal, complemented
10#10926C212yesmaximal, complemented
11#10925C212yesmaximal, complemented
12C23×SD16C2211yescharacteristic
13C24×C8C2211yescharacteristic
14C22×C22⋊Q8C2211yescharacteristic
15C22×#121C2211yescharacteristic
16C23×C4⋊C4C2211yescharacteristic
17C23×C4⋊C4C2211yescharacteristic
18C2×#465—24no—
19C2×#465—24no—
20C2×#465C418yescomplemented
21C22×#106C2211yescharacteristic
22C22×#103C2212yes—
23C22×#103C2211yescharacteristic
24C22×#102C2212yes—
25C22×#102C2211yescharacteristic
26C22×C22⋊C8C2211yescharacteristic
27C22×C4×C8C2212yes—
28C2×#391C2212yes—
29C2×#390C2212yes—
30C2×#386—22no—
31C2×C4×C22⋊C4C2211yescharacteristic
32#1874—24no—
33#1872—24no—
34C8×C22⋊C4—24no—
35C2×#340—24no—
36C2×#340C2212yes—
37C2×#340C2212yes—
38C2×#340C2212yes—
39C2×#340C2212yes—
40C2×#336C2212yes—
41#1441—22no—
42#1439—22no—
43C4⋊#95C2212yes—
44C4⋊#95C2212yes—
45C22⋊#95—22no—
46C23×Q8C2311yescharacteristic
47C23×D4—21no—
48C23×D4C2311yescharacteristic
49C24×C4—21no—
50C24×C4—21no—
51C24×C4C2311yescharacteristic
52C22×SD16—22no—
53C22×SD16—22no—
54C22×SD16—42no—
55C22×SD16—42no—
56C22×SD16C2×C414yescomplemented
57C23×C8—42no—
58C23×C8—42no—
59C23×C8C2×C414yes—
60C23×C8D414yes—
61C23×C8D414yes—
62C23×C8C2312yes—
63C23×C8C2311yescharacteristic
64C23×C8C2311yescharacteristic
65C2×C22⋊Q8—22no—
66C2×C22⋊Q8—22no—
67C2×C22⋊Q8—22no—
68C2×C22⋊Q8—22no—
69C2×C22⋊Q8—22no—
70C2×C22⋊Q8C2×C414yes—
71C2×#121—22no—
72C2×#121—22no—
73C2×#121—22no—
74C2×#121—22no—
75C2×#121—22no—
76C2×#121C2×C414yes—
77C22×C4⋊C4—22no—
78C22×C4⋊C4—22no—
79C22×C4⋊C4—24no—
80C22×C4⋊C4—24no—
81C22×C4⋊C4—21no—
82C22×C4⋊C4—22no—
83C22×C4⋊C4—22no—
84C22×C4⋊C4—21no—
85C22×C4⋊C4—21no—
86C22×C4⋊C4—42no—
87C22×C4⋊C4—42no—
88C22×C4⋊C4C2311yescharacteristic
89C22×C4⋊C4C2311yescharacteristic
90C22×C4⋊C4C2312yes—
91C22×C4⋊C4C2312yes—
92C22×C22⋊C4—21no—
93C22×C22⋊C4—21no—
94C22×C22⋊C4—24no—
95C22×C22⋊C4D412yes—
96C22×C42C2312yes—
97#465—232no—
98C2×#106—22no—
99C2×#106—22no—
100C2×#106C2×C414yescomplemented
101C2×#103—22no—
102C2×#103—22no—
103C2×#103—24no—
104C2×#103—24no—
105C2×#103—42no—
106C2×#103C2×C414yescomplemented
107C2×#102—22no—
108C2×#102—22no—
109C2×#102—24no—
110C2×#102—24no—
111C2×#102—42no—
112C2×#102C2×C414yescomplemented
113C2×C22⋊C8—22no—
114C2×C22⋊C8—22no—
115C2×C22⋊C8D414yes—
116C2×C4×C8—24no—
117C2×C4×C8—42no—
118C2×C4×C8—42no—
119#391—24no—
120#391—42no—
121#390—24no—
122#390—42no—
123#386—48no—
124C4×C22⋊C4—22no—
125C4×C22⋊C4—22no—
126C2×#95—24no—
127C2×#95—24no—
128C2×#95—24no—
129C2×#95D412yes—
130#340—24no—
131#340—24no—
132#340—24no—
133#340—24no—
134#336—24no—
135C25—21no—
136C25C2×D411yescharacteristic
137C22×Q8—21no—
138C22×Q8—21no—
139C22×Q8—21no—
140C22×Q8—21no—
141C22×Q8—21no—
142C22×Q8—22no—
143C22×Q8—22no—
144C22×Q8—22no—
145C22×Q8—22no—
146C22×Q8C22×C412yes—
147C22×D4—24no—
148C22×D4—21no—
149C22×D4—21no—
150C22×D4—21no—
151C22×D4—21no—
152C22×D4—22no—
153C22×D4—22no—
154C22×D4—21no—
155C22×D4—22no—
156C22×D4—22no—
157C22×D4—42no—
158C22×D4—42no—
159C22×D4—44no—
160C22×D4—44no—
161C22×D4C22×C412yes—
162C23×C4—22no—
163C23×C4—22no—
164C23×C4—22no—
165C23×C4—22no—
166C23×C4—22no—
167C23×C4—21no—
168C23×C4—21no—
169C23×C4—21no—
170C23×C4—22no—
171C23×C4—44no—
172C23×C4—44no—
173C23×C4—42no—
174C23×C4—42no—
175C23×C4—44no—
176C23×C4—44no—
177C23×C4—82no—
178C23×C4—82no—
179C23×C4C22×C412yes—
180C23×C4C2×D414yes—
181C23×C4C2×D412yes—
182C23×C4C2×D412yes—
183C23×C4C2×D411yescharacteristic
184C23×C4C2×D412yes—
185C23×C4#4112yes—
186C23×C4#4112yes—
187C23×C4C2411yescharacteristic
188C2×SD16—28no—
189C2×SD16—48no—
190C2×SD16—44no—
191C2×SD16—44no—
192C2×SD16—48no—
193C22×C8—22no—
194C22×C8—22no—
195C22×C8—22no—
196C22×C8—22no—
197C22×C8—22no—
198C22×C8—22no—
199C22×C8—22no—
200C22×C8—22no—
201C22×C8—22no—
202C22×C8—22no—
203C22×C8—48no—
204C22×C8—48no—
205C22×C8—44no—
206C22×C8—44no—
207C22×C8C22⋊C4116yescomplemented
208C22×C8C22×C412yes—
209C22×C8C22×C412yes—
210C22×C8C2×D414yes—
211C22×C8C2×D412yes—
212C22×C8C2×D412yes—
213C22×C8C2×D412yes—
214C22×C8C2×D412yes—
215C22⋊Q8—28no—
216C22⋊Q8—28no—
217C22⋊Q8—44no—
218C22⋊Q8—44no—
219#121—28no—
220#121—28no—
221#121—44no—
222#121—44no—
223C2×C4⋊C4—28no—
224C2×C4⋊C4—28no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—21no—
230C2×C4⋊C4—21no—
231C2×C4⋊C4—21no—
232C2×C4⋊C4—21no—
233C2×C4⋊C4—21no—
234C2×C4⋊C4—21no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—21no—
238C2×C4⋊C4—21no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—21no—
243C2×C4⋊C4—21no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—48no—
247C2×C4⋊C4—48no—
248C2×C4⋊C4—44no—
249C2×C4⋊C4—44no—
250C2×C4⋊C4—44no—
251C2×C4⋊C4—44no—
252C2×C4⋊C4—42no—
253C2×C4⋊C4—42no—
254C2×C4⋊C4—42no—
255C2×C4⋊C4—42no—
256C2×C4⋊C4C22×C412yes—
257C2×C4⋊C4C22×C412yes—
258C2×C4⋊C4#4114yes—
259C2×C4⋊C4#4114yes—
260C2×C22⋊C4—24no—
261C2×C22⋊C4—24no—
262C2×C22⋊C4—24no—
263C2×C22⋊C4—24no—
264C2×C22⋊C4—42no—
265C2×C22⋊C4—42no—
266C2×C22⋊C4—42no—
267C2×C22⋊C4—42no—
268C2×C22⋊C4—48no—
269C2×C22⋊C4—48no—
270C2×C22⋊C4—84no—
271C2×C22⋊C4SD1618yescomplemented
272C2×C42—22no—
273C2×C42—22no—
274C2×C42—22no—
275C2×C42—22no—
276C2×C42—22no—
277C2×C42—22no—
278#106—28no—
279#103—28no—
280#103—48no—
281#102—28no—
282#102—48no—
283C22⋊C8—28no—
284C4×C8—48no—
285#95—24no—
286#95—48no—
287#95—48no—
288#95—84no—
289C24—22no—
290C24—22no—
291C24—22no—
292C24—22no—
293C24—22no—
294C24—41no—
295C24—41no—
296C24—42no—
297C24—42no—
298C24—42no—
299C24—42no—
300C24—42no—
301C24—42no—
302C24C4×D412yes—
303C24#12112yes—
304C24#12712yes—
305C24C2×SD1614yes—
306C24#13514yes—
307C24C22×D411yescharacteristic
308C2×Q8—22no—
309C2×Q8—22no—
310C2×Q8—24no—
311C2×Q8—24no—
312C2×Q8—41no—
313C2×Q8—41no—
314C2×Q8—42no—
315C2×Q8—42no—
316C2×Q8—41no—
317C2×Q8—41no—
318C2×Q8—42no—
319C2×Q8—41no—
320C2×Q8—41no—
321C2×Q8—42no—
322C2×Q8—42no—
323C2×Q8—42no—
324C2×Q8—42no—
325C2×Q8—42no—
326C2×D4—22no—
327C2×D4—22no—
328C2×D4—24no—
329C2×D4—24no—
330C2×D4—48no—
331C2×D4—416no—
332C2×D4—41no—
333C2×D4—41no—
334C2×D4—41no—
335C2×D4—41no—
336C2×D4—42no—
337C2×D4—41no—
338C2×D4—41no—
339C2×D4—42no—
340C2×D4—42no—
341C2×D4—42no—
342C2×D4—42no—
343C2×D4—42no—
344C2×D4—42no—
345C2×D4—42no—
346C2×D4—88no—
347C2×D4—84no—
348C2×D4—84no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—24no—
352C22×C4—24no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—24no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—24no—
359C22×C4—24no—
360C22×C4—24no—
361C22×C4—24no—
362C22×C4—24no—
363C22×C4—24no—
364C22×C4—24no—
365C22×C4—24no—
366C22×C4—24no—
367C22×C4—24no—
368C22×C4—24no—
369C22×C4—24no—
370C22×C4—24no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—24no—
375C22×C4—24no—
376C22×C4—48no—
377C22×C4—48no—
378C22×C4—41no—
379C22×C4—41no—
380C22×C4—41no—
381C22×C4—41no—
382C22×C4—41no—
383C22×C4—41no—
384C22×C4—42no—
385C22×C4—42no—
386C22×C4—42no—
387C22×C4—42no—
388C22×C4—42no—
389C22×C4—42no—
390C22×C4—42no—
391C22×C4—44no—
392C22×C4—44no—
393C22×C4—42no—
394C22×C4—42no—
395C22×C4—44no—
396C22×C4—44no—
397C22×C4—42no—
398C22×C4—816no—
399C22×C4—816no—
400C22×C4—84no—
401C22×C4—84no—
402C22×C4—84no—
403C22×C4—84no—
404C22×C4C2×C22⋊C414yes—
405C22×C4C4×D414yes—
406C22×C4C4×D412yes—
407C22×C4#12114yes—
408C22×C4#12112yes—
409C22×C4#12414yes—
410C22×C4#12712yes—
411C22×C4C2×SD1618yes—
412C22×C4C23×C411yescharacteristic
413C22×C4C22×D411yescharacteristic
414C22×C4C22×D411yescharacteristic
415C22×C4C22×D411yescharacteristic
416C22×C4C2×#4111yescharacteristic
417C22×C4C2×#4112yes—
418C22×C4C2×#4112yes—
419C22×C4C2×#4111yescharacteristic
420C22×C4H(5,𝔽2)12yes—
421C22×C4H(5,𝔽2)12yes—
422SD16—416no—
423C2×C8—24no—
424C2×C8—24no—
425C2×C8—24no—
426C2×C8—22no—
427C2×C8—22no—
428C2×C8—22no—
429C2×C8—22no—
430C2×C8—24no—
431C2×C8—24no—
432C2×C8—24no—
433C2×C8—24no—
434C2×C8—416no—
435C2×C8—42no—
436C2×C8—42no—
437C2×C8—42no—
438C2×C8—42no—
439C2×C8C2×C22⋊C414yes—
440C2×C8C2×C22⋊C414yes—
441C4⋊C4—22no—
442C4⋊C4—22no—
443C4⋊C4—22no—
444C4⋊C4—22no—
445C4⋊C4—24no—
446C4⋊C4—24no—
447C4⋊C4—24no—
448C4⋊C4—24no—
449C4⋊C4—48no—
450C4⋊C4—48no—
451C4⋊C4—41no—
452C4⋊C4—41no—
453C4⋊C4—41no—
454C4⋊C4—41no—
455C4⋊C4—42no—
456C4⋊C4—42no—
457C4⋊C4—42no—
458C4⋊C4—42no—
459C4⋊C4—48no—
460C22⋊C4—216no—
461C22⋊C4—48no—
462C22⋊C4—48no—
463C22⋊C4—816no—
464C42—42no—
465C42—42no—
466C42—42no—
467C42—42no—
468C23—24no—
469C23—22no—
470C23—22no—
471C23—24no—
472C23—28no—
473C23—28no—
474C23—28no—
475C23—28no—
476C23—22no—
477C23—22no—
478C23—24no—
479C23—24no—
480C23—44no—
481C23—44no—
482C23—44no—
483C23—42no—
484C23—81no—
485C23—81no—
486C23—81no—
487C23—81no—
488C23—84no—
489C23—84no—
490C23—84no—
491C23—84no—
492C23—82no—
493C23—81no—
494C23—81no—
495C23—82no—
496C23—82no—
497C23#39014yes—
498C23C4×SD1618yes—
499C23#46518yes—
500C23#49214yes—
501C23#49514yes—
502C23C2×C4×D411yescharacteristic
503C23C2×#12111yescharacteristic
504C23C2×#12711yescharacteristic
505C23#54612yes—
506C23#54612yes—
507C23C22×SD1612yes—
508C23C2×#13512yes—
509C23#57612yes—
510C23#57812yes—
511Q8—42no—
512Q8—44no—
513Q8—42no—
514Q8—81no—
515Q8—81no—
516Q8—82no—
517D4—42no—
518D4—44no—
519D4—42no—
520D4—816no—
521D4—81no—
522D4—81no—
523D4—82no—
524C2×C4—24no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—24no—
528C2×C4—24no—
529C2×C4—24no—
530C2×C4—24no—
531C2×C4—28no—
532C2×C4—28no—
533C2×C4—24no—
534C2×C4—28no—
535C2×C4—28no—
536C2×C4—28no—
537C2×C4—44no—
538C2×C4—44no—
539C2×C4—44no—
540C2×C4—44no—
541C2×C4—44no—
542C2×C4—44no—
543C2×C4—44no—
544C2×C4—48no—
545C2×C4—48no—
546C2×C4—44no—
547C2×C4—44no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—816no—
552C2×C4—816no—
553C2×C4—82no—
554C2×C4—81no—
555C2×C4—82no—
556C2×C4—81no—
557C2×C4—81no—
558C2×C4—82no—
559C2×C4—82no—
560C2×C4—81no—
561C2×C4—84no—
562C2×C4—84no—
563C2×C4—82no—
564C2×C4—81no—
565C2×C4—81no—
566C2×C4—82no—
567C2×C4—82no—
568C2×C4#39014yes—
569C2×C4C22×C22⋊C411yescharacteristic
570C2×C4C2×C4×D411yescharacteristic
571C2×C4C4×#4112yes—
572C2×C4#51812yes—
573C2×C4C2×#12111yescharacteristic
574C2×C4#52511yescharacteristic
575C2×C4#52511yescharacteristic
576C2×C4C2×#12411yescharacteristic
577C2×C4#53212yes—
578C2×C4#53411yescharacteristic
579C2×C4#53411yescharacteristic
580C2×C4#53412yes—
581C2×C4#54612yes—
582C2×C4#54612yes—
583C2×C4#55312yes—
584C2×C4#55312yes—
585C2×C4#55312yes—
586C2×C4#55312yes—
587C8—24no—
588C8—24no—
589C8—44no—
590C22—24no—
591C22—216no—
592C22—216no—
593C22—24no—
594C22—24no—
595C22—24no—
596C22—24no—
597C22—24no—
598C22—82no—
599C22—88no—
600C22—82no—
601C22—82no—
602C22—164no—
603C22—161no—
604C22—161no—
605C22—161no—
606C22—161no—
607C22#196818yes—
608C22C2×#39011yescharacteristic
609C22#236412yes—
610C22#253911yescharacteristic
611C22#253911yescharacteristic
612C22#254312yes—
613C22C2×C4×SD1612yes—
614C22#295112yes—
615C22C2×#46512yes—
616C22#306112yes—
617C22C2×#49211yescharacteristic
618C22C2×#49511yescharacteristic
619C22#315511yescharacteristic
620C22#315611yescharacteristic
621C22#329514yes—
622C22#329714yes—
623C22#329814yes—
624C22#330014yes—
625C4—24no—
626C4—48no—
627C4—82no—
628C4—84no—
629C4—84no—
630C4—82no—
631C4—82no—
632C4#229711yescharacteristic
633C4#230811yescharacteristic
634C4#236412yes—
635C4#254312yes—
636C4#259111yescharacteristic
637C4#259111yescharacteristic
638C2—28no—
639C2—162no—
640C2#1415211yescharacteristic
641C2C2×#196811yescharacteristic
642C2#2123011yescharacteristic
643C2#2312614yes—
644C2#2833111yescharacteristic
645C2#2833211yescharacteristic
646C2#2833511yescharacteristic
647C2#2833611yescharacteristic
648C2#2855514yes—
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC22×#121C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. C2×#1872C2semidirect
4. C2×C8×C22⋊C4C2semidirect
5. C22×#340C2semidirect
6. #14139C2semidirect
8. #11378C2semidirect
9. #11023C2semidirect
10. #10926C2semidirect
11. #10925C2semidirect
20. C2×#465C4semidirect
56. C22×SD16C2×C4semidirect
100. C2×#106C2×C4semidirect
106. C2×#103C2×C4semidirect
112. C2×#102C2×C4semidirect
207. C22×C8C22⋊C4semidirect
271. C2×C22⋊C4SD16semidirect

Maximal subgroups

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

subgroupindex
C22×#4652
C2×#18742
C2×#18722
C2×C8×C22⋊C42
C22×#3402
#141392
#141372
#113782
#110232
#109262
#109252