Finite Group Databasebeta

← #374369

Subgroups — #374369

Order 512.

1934 conjugacy classes of subgroups.

The subgroup lattice has 662 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#464C211yesmaximal, characteristic
2C2×#2351C211yesmaximal, characteristic, complemented
3C2×#1958C211yesmaximal, characteristic
4C2×#1880C212yesmaximal
5C2×#1878C212yesmaximal, complemented
6C2×C4×#103C211yesmaximal, characteristic, complemented
7#14153C211yesmaximal, characteristic
8#11067C212yesmaximal, complemented
9#11066C212yesmaximal
10#10850C211yesmaximal, characteristic, complemented
11#10834C211yesmaximal, characteristic, complemented
12C23×Q16C2211yescharacteristic
13C23×SD16C2211yescharacteristic
14C22×#124C2211yescharacteristic
15C22×C4×Q8C2212yes—
16C22×C4×Q8C2211yescharacteristic
17C2×#464—24no—
18C2×#464—24no—
19C2×#464C418yescomplemented
20C22×C4⋊C8C2211yescharacteristic
21C22×#103C2212yes—
22C22×#103C2211yescharacteristic
23C22×#103C2212yes—
24C22×#102C2212yes—
25C22×#102C2211yescharacteristic
26C22×C4×C8C2211yescharacteristic
27#2351—24no—
28#2351—24no—
29C2×#391C2212yes—
30C2×#390C2212yes—
31C2×#389—22no—
32C2×#389C2212yes—
33C2×#382—21no—
34C2×#382C2211yescharacteristic
35C2×C4×C4⋊C4—21no—
36C2×C4×C4⋊C4C2211yescharacteristic
37C2×C4×C4⋊C4C2211yescharacteristic
38#1958—24no—
39#1880—28no—
40#1878—28no—
41C4×#103—24no—
42C2×#340C2212yes—
43C2×#340C2212yes—
44C2×#340C2212yes—
45C2×#336C2212yes—
46C2×#336C2212yes—
47#1439—22no—
48#1437—21no—
49#1437—21no—
50C23×Q8C2311yescharacteristic
51C23×Q8C2311yescharacteristic
52C23×D4C2311yescharacteristic
53C22×Q16—24no—
54C22×Q16—22no—
55C22×Q16—22no—
56C22×Q16—24no—
57C22×Q16C2×C414yes—
58C22×SD16—24no—
59C22×SD16—22no—
60C22×SD16—22no—
61C22×SD16—24no—
62C22×SD16C2×C414yes—
63C23×C8C2311yescharacteristic
64C23×C8C2311yescharacteristic
65C23×C8C2312yes—
66C2×#124—22no—
67C2×#124—22no—
68C2×#124—22no—
69C2×#124—22no—
70C2×#124—22no—
71C2×#124C2×C414yes—
72C2×C4×Q8—24no—
73C2×C4×Q8—24no—
74C2×C4×Q8—22no—
75C2×C4×Q8—22no—
76C2×C4×Q8—22no—
77C2×C4×Q8—24no—
78C2×C4×Q8—24no—
79C2×C4×Q8—42no—
80C2×C4×Q8—42no—
81C2×C4×Q8C2×C414yes—
82C2×C4×Q8D414yes—
83C2×C4×Q8D414yes—
84C22×C4⋊C4—21no—
85C22×C4⋊C4—22no—
86C22×C4⋊C4—22no—
87C22×C4⋊C4D412yes—
88C22×C4⋊C4C2312yes—
89C22×C4⋊C4C2312yes—
90C22×C4⋊C4C2311yescharacteristic
91C22×C22⋊C4—21no—
92C22×C22⋊C4—21no—
93C22×C22⋊C4—22no—
94C22×C22⋊C4—22no—
95C22×C42—21no—
96C22×C42—22no—
97C22×C42—21no—
98C22×C42C2312yes—
99C22×C42C2311yescharacteristic
100#464—232no—
101C2×C4⋊C8—22no—
102C2×C4⋊C8—22no—
103C2×C4⋊C8C2×C414yes—
104C2×#103—24no—
105C2×#103—24no—
106C2×#103—22no—
107C2×#103—22no—
108C2×#103—24no—
109C2×#103—24no—
110C2×#103—42no—
111C2×#103C2×C414yes—
112C2×#103D418yes—
113C2×#102—22no—
114C2×#102—22no—
115C2×#102—24no—
116C2×#102—24no—
117C2×#102—42no—
118C2×#102C2×C414yes—
119C2×C4×C8—22no—
120C2×C4×C8—22no—
121C2×C4×C8—22no—
122#391—24no—
123#391—42no—
124#390—24no—
125#390—42no—
126#389—216no—
127#389—24no—
128#389—24no—
129#382—28no—
130#382—22no—
131#382D414yes—
132C4×C4⋊C4—28no—
133C4×C4⋊C4—22no—
134C4×C4⋊C4—22no—
135C4×C4⋊C4—22no—
136C4×C4⋊C4—22no—
137C2×#95—21no—
138C2×#95—22no—
139C2×#95—21no—
140C2×#95—22no—
141C2×#95—22no—
142C2×#95—21no—
143C2×#95D411yescharacteristic
144C2×#95D411yescharacteristic
145#340—24no—
146#340—24no—
147#340—24no—
148#336—24no—
149#336—24no—
150C25—21no—
151C22×Q8—21no—
152C22×Q8—21no—
153C22×Q8—21no—
154C22×Q8—21no—
155C22×Q8—22no—
156C22×Q8—22no—
157C22×Q8—21no—
158C22×Q8—21no—
159C22×Q8—21no—
160C22×Q8—22no—
161C22×Q8—21no—
162C22×Q8—22no—
163C22×Q8—22no—
164C22×Q8—22no—
165C22×Q8C22×C412yes—
166C22×Q8C22×C412yes—
167C22×Q8C2×D412yes—
168C22×Q8C2×D412yes—
169C22×Q8C2×D414yes—
170C22×Q8#4114yes—
171C22×D4—21no—
172C22×D4—21no—
173C22×D4—21no—
174C22×D4—22no—
175C22×D4—21no—
176C22×D4—22no—
177C22×D4—21no—
178C22×D4—22no—
179C22×D4—22no—
180C22×D4C22×C412yes—
181C23×C4—21no—
182C23×C4—21no—
183C23×C4—22no—
184C23×C4—21no—
185C23×C4—22no—
186C23×C4C2×D412yes—
187C23×C4C2×D412yes—
188C23×C4C2×D411yescharacteristic
189C23×C4C2×D411yescharacteristic
190C23×C4C2411yescharacteristic
191C2×Q16—28no—
192C2×Q16—216no—
193C2×Q16—48no—
194C2×Q16—44no—
195C2×Q16—44no—
196C2×SD16—28no—
197C2×SD16—216no—
198C2×SD16—48no—
199C2×SD16—44no—
200C2×SD16—44no—
201C22×C8—22no—
202C22×C8—21no—
203C22×C8—21no—
204C22×C8—22no—
205C22×C8—21no—
206C22×C8—21no—
207C22×C8—22no—
208C22×C8—22no—
209C22×C8—22no—
210C22×C8—22no—
211C22×C8—22no—
212C22×C8—22no—
213C22×C8—22no—
214C22×C8—22no—
215C22×C8C22×C412yes—
216C22×C8C22×C412yes—
217C22×C8C2×D414yes—
218#124—28no—
219#124—28no—
220#124—44no—
221#124—44no—
222C4×Q8—28no—
223C4×Q8—28no—
224C4×Q8—28no—
225C4×Q8—48no—
226C4×Q8—48no—
227C4×Q8—48no—
228C4×Q8—48no—
229C4×Q8C22⋊C4116yescomplemented
230C2×C4⋊C4—24no—
231C2×C4⋊C4—24no—
232C2×C4⋊C4—24no—
233C2×C4⋊C4—21no—
234C2×C4⋊C4—21no—
235C2×C4⋊C4—21no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—28no—
245C2×C4⋊C4—28no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—42no—
251C2×C4⋊C4—44no—
252C2×C4⋊C4—44no—
253C2×C4⋊C4—44no—
254C2×C4⋊C4—44no—
255C2×C4⋊C4C22×C412yes—
256C2×C4⋊C4C2×D414yes—
257C2×C4⋊C4C2×D414yes—
258C2×C4⋊C4C2×D412yes—
259C2×C4⋊C4C2×D412yes—
260C2×C22⋊C4—24no—
261C2×C22⋊C4—24no—
262C2×C22⋊C4—42no—
263C2×C22⋊C4—42no—
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—82no—
271C2×C22⋊C4—82no—
272C2×C42—24no—
273C2×C42—24no—
274C2×C42—24no—
275C2×C42—28no—
276C2×C42—22no—
277C2×C42—22no—
278C2×C42—22no—
279C2×C42—22no—
280C2×C42—22no—
281C2×C42—22no—
282C2×C42—44no—
283C2×C42—44no—
284C2×C42—42no—
285C2×C42—82no—
286C2×C42C22×C412yes—
287C2×C42C2×D412yes—
288C2×C42C2×D412yes—
289C2×C42C2×D412yes—
290C2×C42#4112yes—
291C2×C42#4112yes—
292C4⋊C8—28no—
293#103—216no—
294#103—28no—
295#103—48no—
296#102—28no—
297#102—48no—
298C4×C8—44no—
299#95—28no—
300#95—24no—
301#95—24no—
302#95—22no—
303#95—22no—
304#95—44no—
305#95—44no—
306#95—42no—
307C24—22no—
308C24—41no—
309C24—41no—
310C24—42no—
311C24—42no—
312C24—42no—
313C24—42no—
314C24—42no—
315C24—42no—
316C24C22×D411yescharacteristic
317C2×Q8—22no—
318C2×Q8—24no—
319C2×Q8—22no—
320C2×Q8—22no—
321C2×Q8—22no—
322C2×Q8—22no—
323C2×Q8—24no—
324C2×Q8—24no—
325C2×Q8—24no—
326C2×Q8—24no—
327C2×Q8—24no—
328C2×Q8—24no—
329C2×Q8—24no—
330C2×Q8—22no—
331C2×Q8—22no—
332C2×Q8—24no—
333C2×Q8—24no—
334C2×Q8—41no—
335C2×Q8—41no—
336C2×Q8—42no—
337C2×Q8—42no—
338C2×Q8—42no—
339C2×Q8—41no—
340C2×Q8—41no—
341C2×Q8—41no—
342C2×Q8—41no—
343C2×Q8—42no—
344C2×Q8—42no—
345C2×Q8—42no—
346C2×Q8—42no—
347C2×Q8—42no—
348C2×Q8C2×C22⋊C414yes—
349C2×Q8C4×D418yes—
350C2×Q8#12118yes—
351C2×Q8#12414yes—
352C2×Q8#12714yes—
353C2×D4—22no—
354C2×D4—22no—
355C2×D4—24no—
356C2×D4—24no—
357C2×D4—42no—
358C2×D4—41no—
359C2×D4—41no—
360C2×D4—41no—
361C2×D4—41no—
362C2×D4—41no—
363C2×D4—41no—
364C2×D4—42no—
365C2×D4—42no—
366C2×D4—42no—
367C2×D4—42no—
368C2×D4—42no—
369C2×D4—42no—
370C2×D4—42no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—24no—
375C22×C4—24no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—24no—
392C22×C4—22no—
393C22×C4—24no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—42no—
400C22×C4—42no—
401C22×C4—41no—
402C22×C4—42no—
403C22×C4—41no—
404C22×C4—42no—
405C22×C4—42no—
406C22×C4—42no—
407C22×C4—42no—
408C22×C4—42no—
409C22×C4—42no—
410C22×C4—42no—
411C22×C4—42no—
412C22×C4—42no—
413C22×C4—42no—
414C22×C4—42no—
415C22×C4—42no—
416C22×C4—42no—
417C22×C4C4×D412yes—
418C22×C4C4×D412yes—
419C22×C4#12012yes—
420C22×C4#12012yes—
421C22×C4#12112yes—
422C22×C4#12112yes—
423C22×C4#13514yes—
424C22×C4C23×C411yescharacteristic
425C22×C4C22×D411yescharacteristic
426C22×C4C22×D411yescharacteristic
427C22×C4C22×D412yes—
428C22×C4C22×D411yescharacteristic
429C22×C4C2×#4112yes—
430C22×C4C2×#4111yescharacteristic
431C22×C4C2×#4111yescharacteristic
432C22×C4H(5,𝔽2)12yes—
433C22×C42−1+412yes—
434Q16—416no—
435SD16—416no—
436C2×C8—22no—
437C2×C8—22no—
438C2×C8—24no—
439C2×C8—24no—
440C2×C8—24no—
441C2×C8—24no—
442C2×C8—24no—
443C2×C8—42no—
444C2×C8—42no—
445C2×C8—42no—
446C2×C8—41no—
447C2×C8—41no—
448C2×C8—41no—
449C2×C8—41no—
450C2×C8—42no—
451C2×C8—42no—
452C2×C8—42no—
453C4⋊C4—22no—
454C4⋊C4—22no—
455C4⋊C4—22no—
456C4⋊C4—216no—
457C4⋊C4—24no—
458C4⋊C4—24no—
459C4⋊C4—24no—
460C4⋊C4—24no—
461C4⋊C4—48no—
462C4⋊C4—416no—
463C4⋊C4—416no—
464C4⋊C4—42no—
465C4⋊C4—42no—
466C4⋊C4—42no—
467C4⋊C4—42no—
468C4⋊C4C2×C22⋊C414yes—
469C22⋊C4—48no—
470C22⋊C4—48no—
471C22⋊C4—88no—
472C22⋊C4—88no—
473C42—216no—
474C42—416no—
475C42—42no—
476C42—42no—
477C42—42no—
478C42—42no—
479C42—84no—
480C42C2×C22⋊C414yes—
481C42C4×D414yes—
482C42#12114yes—
483C42#12314yes—
484C23—44no—
485C23—42no—
486C23—42no—
487C23—44no—
488C23—42no—
489C23—82no—
490C23—82no—
491C23—82no—
492C23—82no—
493C23—82no—
494C23—81no—
495C23—81no—
496C23—82no—
497C23—82no—
498C23—82no—
499C23—82no—
500C23—82no—
501C23—82no—
502C23—82no—
503C23—81no—
504C23—81no—
505C23C2×C4×D411yescharacteristic
506C23C2×#12011yescharacteristic
507C23C2×#12111yescharacteristic
508C23D4212yes—
509C23#54812yes—
510C23C2×#13512yes—
511C23#57512yes—
512C23#57712yes—
513C23#57812yes—
514Q8—24no—
515Q8—28no—
516Q8—28no—
517Q8—24no—
518Q8—24no—
519Q8—42no—
520Q8—44no—
521Q8—42no—
522Q8—82no—
523Q8—81no—
524Q8—81no—
525Q8#39018yes—
526D4—42no—
527D4—44no—
528D4—42no—
529D4—82no—
530D4—81no—
531D4—81no—
532C2×C4—28no—
533C2×C4—28no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—24no—
537C2×C4—24no—
538C2×C4—24no—
539C2×C4—24no—
540C2×C4—24no—
541C2×C4—24no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—24no—
545C2×C4—24no—
546C2×C4—24no—
547C2×C4—24no—
548C2×C4—24no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—44no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—44no—
557C2×C4—44no—
558C2×C4—44no—
559C2×C4—44no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—44no—
563C2×C4—42no—
564C2×C4—44no—
565C2×C4—42no—
566C2×C4—44no—
567C2×C4—42no—
568C2×C4—44no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—82no—
573C2×C4—82no—
574C2×C4—82no—
575C2×C4—81no—
576C2×C4—81no—
577C2×C4—82no—
578C2×C4—82no—
579C2×C4—82no—
580C2×C4—82no—
581C2×C4—82no—
582C2×C4—82no—
583C2×C4#38614yes—
584C2×C4#38614yes—
585C2×C4C22×C22⋊C411yescharacteristic
586C2×C4C2×C4×D411yescharacteristic
587C2×C4C2×C4×D412yes—
588C2×C4#52012yes—
589C2×C4C2×#12111yescharacteristic
590C2×C4C2×#12112yes—
591C2×C4C2×#12311yescharacteristic
592C2×C4C2×#12411yescharacteristic
593C2×C4C2×#12711yescharacteristic
594C2×C4#53511yescharacteristic
595C2×C4#53611yescharacteristic
596C2×C4#53712yes—
597C2×C4D4212yes—
598C2×C4#54712yes—
599C2×C4#54712yes—
600C2×C4#54812yes—
601C2×C4#55312yes—
602C2×C4#55312yes—
603C8—44no—
604C8—42no—
605C8—42no—
606C22—82no—
607C22—82no—
608C22—82no—
609C22—84no—
610C22—84no—
611C22—162no—
612C22—162no—
613C22—162no—
614C22—162no—
615C22C2×#38611yescharacteristic
616C22#236112yes—
617C22#241612yes—
618C22#243311yescharacteristic
619C22#243611yescharacteristic
620C22C4×#13511yescharacteristic
621C22#294311yescharacteristic
622C22#294712yes—
623C22#300212yes—
624C22#301312yes—
625C22#303512yes—
626C22#304311yescharacteristic
627C22#304411yescharacteristic
628C22#327512yes—
629C22#327712yes—
630C22#328212yes—
631C22#328512yes—
632C22#336412yes—
633C22#336512yes—
634C22#337112yes—
635C22#337712yes—
636C4—28no—
637C4—24no—
638C4—24no—
639C4—44no—
640C4—44no—
641C4—82no—
642C4—84no—
643C4—82no—
644C4—84no—
645C4C2×#39011yescharacteristic
646C4#231111yescharacteristic
647C4#236112yes—
648C4#241612yes—
649C4#250111yescharacteristic
650C4#250411yescharacteristic
651C2—162no—
652C2#1413711yescharacteristic
653C2#2092411yescharacteristic
654C2#2093011yescharacteristic
655C2#2300012yes—
656C2#2300612yes—
657C2#2473412yes—
658C2#2473612yes—
659C2#2576112yes—
660C2#2593712yes—
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC23×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2351C2semidirect
5. C2×#1878C2semidirect
6. C2×C4×#103C2semidirect
8. #11067C2semidirect
10. #10850C2semidirect
11. #10834C2semidirect
19. C2×#464C4semidirect
229. C4×Q8C22⋊C4semidirect

Maximal subgroups

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

subgroupindex
C22×#4642
C2×#23512
C2×#19582
C2×#18802
C2×#18782
C2×C4×#1032
#141532
#110672
#110662
#108502
#108342