Finite Group Databasebeta

← #410011

Subgroups — #410011

Order 512.

2344 conjugacy classes of subgroups.

The subgroup lattice has 627 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#481C211yesmaximal, characteristic
2C2×#2854C211yesmaximal, characteristic
3C22×#389C211yesmaximal, characteristic, complemented
4C2×#1874C211yesmaximal, characteristic
5C2×#1824C211yesmaximal, characteristic, complemented
6C2×#1817C211yesmaximal, characteristic, complemented
7#11327C214yesmaximal, complemented
8#11066C214yesmaximal, complemented
9#10730C211yesmaximal, characteristic, complemented
10C22×#123—21no—
11C22×#121C2211yescharacteristic
12C23×C4⋊C4—21no—
13C23×C4⋊C4C2211yescharacteristic
14C23×C4⋊C4C2211yescharacteristic
15C23×C42C2211yescharacteristic
16C2×#481—24no—
17C2×#481—24no—
18C2×#481—24no—
19C22×#106C2212yes—
20C22×#102C2212yes—
21C22×#102C2212yes—
22C22×C22⋊C8C2211yescharacteristic
23C22×C22⋊C8C2211yescharacteristic
24#2854—24no—
25#2854—24no—
26C2×#390C2212yes—
27C2×#389—22no—
28C2×#389—24no—
29C2×#389—24no—
30C2×#389C2212yes—
31C2×#389C2214yes—
32C2×#388—22no—
33C2×#386—21no—
34C2×#380C2211yescharacteristic
35C22×#95—21no—
36#1874—24no—
37#1824—24no—
38#1817—24no—
39C2×#340C2214yes—
40C2×#340C2214yes—
41C2×#340C2212yes—
42C2×#336C2214yes—
43C23×D4—21no—
44C23×D4C2311yescharacteristic
45C24×C4—21no—
46C24×C4—21no—
47C24×C4D411yescharacteristic
48C24×C4D411yescharacteristic
49C24×C4C2311yescharacteristic
50C23×C8C2312yes—
51C23×C8C2312yes—
52C2×#123—28no—
53C2×#123—44no—
54C2×#123—44no—
55C2×#121—22no—
56C2×#121—22no—
57C2×#121—22no—
58C2×#121—22no—
59C2×#121—22no—
60C2×#121D414yes—
61C22×C4⋊C4—21no—
62C22×C4⋊C4—22no—
63C22×C4⋊C4—28no—
64C22×C4⋊C4—22no—
65C22×C4⋊C4—22no—
66C22×C4⋊C4—22no—
67C22×C4⋊C4—22no—
68C22×C4⋊C4—28no—
69C22×C4⋊C4—24no—
70C22×C4⋊C4—21no—
71C22×C4⋊C4—22no—
72C22×C4⋊C4—22no—
73C22×C4⋊C4—48no—
74C22×C4⋊C4D414yes—
75C22×C4⋊C4D414yes—
76C22×C4⋊C4C2312yes—
77C22×C4⋊C4C2311yescharacteristic
78C22×C4⋊C4C2312yes—
79C22×C4⋊C4C2312yes—
80C22×C22⋊C4—21no—
81C22×C22⋊C4—21no—
82C22×C22⋊C4—22no—
83C22×C22⋊C4—22no—
84C22×C42—22no—
85C22×C42—22no—
86C22×C42—22no—
87C22×C42—22no—
88C22×C42—22no—
89C22×C42—22no—
90C22×C42D412yes—
91C22×C42C2312yes—
92#481—232no—
93C2×#106—24no—
94C2×#106—24no—
95C2×#106—24no—
96C2×#102—24no—
97C2×#102—24no—
98C2×#102—24no—
99C2×#102—24no—
100C2×#102—28no—
101C2×C22⋊C8—22no—
102C2×C22⋊C8—22no—
103C2×C22⋊C8—22no—
104C2×C22⋊C8—22no—
105C2×C22⋊C8—22no—
106C2×C22⋊C8D414yes—
107#390—24no—
108#390—24no—
109#389—216no—
110#389—24no—
111#389—24no—
112#389—28no—
113#389—28no—
114#388—216no—
115#386—28no—
116#380—22no—
117#380D414yes—
118C2×#95—28no—
119C2×#95—24no—
120C2×#95—21no—
121C2×#95—22no—
122#340—28no—
123#340—24no—
124#340—28no—
125#336—28no—
126C25—21no—
127C25C2×D411yescharacteristic
128C22×D4—24no—
129C22×D4—21no—
130C22×D4—21no—
131C22×D4—21no—
132C22×D4—21no—
133C22×D4—21no—
134C22×D4—24no—
135C22×D4—24no—
136C22×D4—42no—
137C22×D4—42no—
138C22×D4—42no—
139C22×D4—42no—
140C22×D4—42no—
141C22×D4—42no—
142C22×D4C2×D412yes—
143C23×C4—24no—
144C23×C4—24no—
145C23×C4—28no—
146C23×C4—22no—
147C23×C4—24no—
148C23×C4—22no—
149C23×C4—22no—
150C23×C4—22no—
151C23×C4—22no—
152C23×C4—22no—
153C23×C4—22no—
154C23×C4—22no—
155C23×C4—24no—
156C23×C4—24no—
157C23×C4—22no—
158C23×C4—22no—
159C23×C4—22no—
160C23×C4—21no—
161C23×C4—22no—
162C23×C4—42no—
163C23×C4—44no—
164C23×C4—42no—
165C23×C4—42no—
166C23×C4—44no—
167C23×C4—44no—
168C23×C4—42no—
169C23×C4—42no—
170C23×C4SD1614yes—
171C23×C4C2×D412yes—
172C23×C4C2×D412yes—
173C23×C4C2×D411yescharacteristic
174C23×C4C2×D412yes—
175C23×C4C2×D412yes—
176C23×C4C2×D412yes—
177C23×C4C2×D412yes—
178C23×C4C2×D412yes—
179C23×C4#4112yes—
180C23×C4C2411yescharacteristic
181C22×C8—22no—
182C22×C8—22no—
183C22×C8—24no—
184C22×C8—24no—
185C22×C8—22no—
186C22×C8—22no—
187C22×C8—22no—
188C22×C8—24no—
189C22×C8—24no—
190C22×C8C2×D414yes—
191#123—432no—
192#121—28no—
193#121—28no—
194#121—44no—
195#121—44no—
196C2×C4⋊C4—28no—
197C2×C4⋊C4—216no—
198C2×C4⋊C4—24no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—24no—
201C2×C4⋊C4—216no—
202C2×C4⋊C4—28no—
203C2×C4⋊C4—28no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—24no—
213C2×C4⋊C4—24no—
214C2×C4⋊C4—216no—
215C2×C4⋊C4—28no—
216C2×C4⋊C4—24no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—432no—
221C2×C4⋊C4—48no—
222C2×C4⋊C4—44no—
223C2×C4⋊C4—42no—
224C2×C4⋊C4—44no—
225C2×C4⋊C4—42no—
226C2×C4⋊C4—44no—
227C2×C4⋊C4—44no—
228C2×C4⋊C4—48no—
229C2×C4⋊C4C2×D414yes—
230C2×C4⋊C4C2×D412yes—
231C2×C4⋊C4C2×D414yes—
232C2×C4⋊C4C2×D414yes—
233C2×C4⋊C4#4118yes—
234C2×C22⋊C4—28no—
235C2×C22⋊C4—28no—
236C2×C22⋊C4—24no—
237C2×C22⋊C4—24no—
238C2×C22⋊C4—44no—
239C2×C22⋊C4—44no—
240C2×C22⋊C4—42no—
241C2×C22⋊C4—42no—
242C2×C22⋊C4—42no—
243C2×C22⋊C4—42no—
244C2×C22⋊C4—44no—
245C2×C22⋊C4—44no—
246C2×C42—24no—
247C2×C42—24no—
248C2×C42—24no—
249C2×C42—22no—
250C2×C42—22no—
251C2×C42—22no—
252C2×C42—22no—
253C2×C42—22no—
254C2×C42—22no—
255C2×C42—44no—
256C2×C42—44no—
257C2×C42—44no—
258C2×C42—44no—
259#106—216no—
260#102—216no—
261#102—48no—
262C22⋊C8—28no—
263C22⋊C8—44no—
264#95—216no—
265#95—28no—
266#95—24no—
267C24—22no—
268C24—22no—
269C24—22no—
270C24—22no—
271C24—22no—
272C24—41no—
273C24—41no—
274C24—42no—
275C24—42no—
276C24—41no—
277C24—41no—
278C24—41no—
279C24—42no—
280C24—41no—
281C24—42no—
282C24#12012yes—
283C24#12012yes—
284C24#12112yes—
285C24C2×SD1612yes—
286C24#13512yes—
287C24AGL(1,ℤ/8)12yes—
288C24#13712yes—
289C24C22×D411yescharacteristic
290C2×D4—22no—
291C2×D4—22no—
292C2×D4—28no—
293C2×D4—48no—
294C2×D4—48no—
295C2×D4—48no—
296C2×D4—41no—
297C2×D4—42no—
298C2×D4—42no—
299C2×D4—41no—
300C2×D4—42no—
301C2×D4—42no—
302C2×D4—44no—
303C2×D4—44no—
304C2×D4—44no—
305C2×D4—84no—
306C2×D4—84no—
307C2×D4—84no—
308C2×D4—84no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—24no—
312C22×C4—24no—
313C22×C4—24no—
314C22×C4—24no—
315C22×C4—28no—
316C22×C4—28no—
317C22×C4—28no—
318C22×C4—28no—
319C22×C4—216no—
320C22×C4—28no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—24no—
324C22×C4—22no—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—28no—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—28no—
336C22×C4—28no—
337C22×C4—28no—
338C22×C4—24no—
339C22×C4—22no—
340C22×C4—24no—
341C22×C4—48no—
342C22×C4—48no—
343C22×C4—48no—
344C22×C4—416no—
345C22×C4—416no—
346C22×C4—44no—
347C22×C4—42no—
348C22×C4—44no—
349C22×C4—42no—
350C22×C4—44no—
351C22×C4—44no—
352C22×C4—44no—
353C22×C4—48no—
354C22×C4—44no—
355C22×C4—44no—
356C22×C4—48no—
357C22×C4—48no—
358C22×C4—41no—
359C22×C4—42no—
360C22×C4—41no—
361C22×C4—42no—
362C22×C4—42no—
363C22×C4—42no—
364C22×C4—42no—
365C22×C4—41no—
366C22×C4—41no—
367C22×C4—42no—
368C22×C4—42no—
369C22×C4—44no—
370C22×C4—41no—
371C22×C4—41no—
372C22×C4—42no—
373C22×C4—44no—
374C22×C4—48no—
375C22×C4—88no—
376C22×C4—88no—
377C22×C4#12014yes—
378C22×C4#12012yes—
379C22×C4#12012yes—
380C22×C4#12114yes—
381C22×C4#12114yes—
382C22×C4#12112yes—
383C22×C4#12714yes—
384C22×C4C2×SD1614yes—
385C22×C4AGL(1,ℤ/8)14yes—
386C22×C4C22×D411yescharacteristic
387C22×C4C22×D411yescharacteristic
388C22×C4C22×D411yescharacteristic
389C22×C4C22×D411yescharacteristic
390C22×C4C22×D411yescharacteristic
391C22×C4C2×#4111yescharacteristic
392C22×C4C2×#4114yes—
393C22×C42−1+414yes—
394C2×C8—24no—
395C2×C8—28no—
396C2×C8—42no—
397C2×C8—42no—
398C2×C8—44no—
399C2×C8—42no—
400C2×C8—42no—
401C2×C8—42no—
402C2×C8—44no—
403C2×C8—44no—
404C4⋊C4—216no—
405C4⋊C4—24no—
406C4⋊C4—28no—
407C4⋊C4—24no—
408C4⋊C4—22no—
409C4⋊C4—24no—
410C4⋊C4—22no—
411C4⋊C4—28no—
412C4⋊C4—432no—
413C4⋊C4—416no—
414C4⋊C4—42no—
415C4⋊C4—41no—
416C4⋊C4—42no—
417C4⋊C4—41no—
418C4⋊C4—44no—
419C4⋊C4—48no—
420C4⋊C4#12318yescomplemented
421C22⋊C4—416no—
422C22⋊C4—416no—
423C22⋊C4—48no—
424C22⋊C4—48no—
425C42—42no—
426C42—42no—
427C42—42no—
428C42—42no—
429C42—48no—
430C23—24no—
431C23—22no—
432C23—24no—
433C23—24no—
434C23—24no—
435C23—22no—
436C23—24no—
437C23—24no—
438C23—24no—
439C23—24no—
440C23—24no—
441C23—24no—
442C23—22no—
443C23—22no—
444C23—24no—
445C23—24no—
446C23—42no—
447C23—44no—
448C23—42no—
449C23—42no—
450C23—44no—
451C23—82no—
452C23—81no—
453C23—82no—
454C23—82no—
455C23—82no—
456C23—81no—
457C23—82no—
458C23—82no—
459C23—82no—
460C23—82no—
461C23—82no—
462C23—81no—
463C23—81no—
464C23—81no—
465C23—81no—
466C23—81no—
467C23—81no—
468C23—82no—
469C23#39214yes—
470C23#44814yes—
471C23#44914yes—
472C23#45014yes—
473C23#45214yes—
474C23#46514yes—
475C23#46814yes—
476C23C2×#12011yescharacteristic
477C23C2×#12011yescharacteristic
478C23C2×#12111yescharacteristic
479C23#52512yes—
480C23#53612yes—
481C23C22×SD1611yescharacteristic
482C23C2×#13511yescharacteristic
483C23C2×AGL(1,ℤ/8)12yes—
484C23C2×AGL(1,ℤ/8)11yescharacteristic
485C23C2×#13711yescharacteristic
486C23#57512yes—
487D4—44no—
488D4—44no—
489D4—816no—
490D4—82no—
491D4—82no—
492C2×C4—22no—
493C2×C4—22no—
494C2×C4—24no—
495C2×C4—24no—
496C2×C4—28no—
497C2×C4—28no—
498C2×C4—28no—
499C2×C4—28no—
500C2×C4—28no—
501C2×C4—24no—
502C2×C4—24no—
503C2×C4—24no—
504C2×C4—24no—
505C2×C4—24no—
506C2×C4—24no—
507C2×C4—416no—
508C2×C4—432no—
509C2×C4—48no—
510C2×C4—48no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—44no—
515C2×C4—42no—
516C2×C4—44no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—416no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—44no—
524C2×C4—44no—
525C2×C4—44no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—44no—
529C2×C4—44no—
530C2×C4—44no—
531C2×C4—816no—
532C2×C4—82no—
533C2×C4—82no—
534C2×C4—82no—
535C2×C4—82no—
536C2×C4—81no—
537C2×C4—81no—
538C2×C4—84no—
539C2×C4—81no—
540C2×C4—81no—
541C2×C4—84no—
542C2×C4#39214yes—
543C2×C4C2×#12011yescharacteristic
544C2×C4C2×#12111yescharacteristic
545C2×C4C2×#12111yescharacteristic
546C2×C4C2×#12312yes—
547C2×C4#52512yes—
548C2×C4C2×#12711yescharacteristic
549C2×C4#53612yes—
550C2×C4#53612yes—
551C2×C4#54714yes—
552C2×C4#54714yes—
553C2×C4#54714yes—
554C2×C4#55214yes—
555C8—44no—
556C8—82no—
557C22—28no—
558C22—28no—
559C22—28no—
560C22—24no—
561C22—24no—
562C22—24no—
563C22—24no—
564C22—24no—
565C22—24no—
566C22—28no—
567C22—84no—
568C22—82no—
569C22—82no—
570C22—82no—
571C22—84no—
572C22—162no—
573C22—162no—
574C22—161no—
575C22—161no—
576C22—161no—
577C22—161no—
578C22#200018yes—
579C22C2×#39211yescharacteristic
580C22#240612yes—
581C22#241312yes—
582C22#261411yescharacteristic
583C22#284911yescharacteristic
584C22C2×#45011yescharacteristic
585C22C2×#44811yescharacteristic
586C22C2×#44911yescharacteristic
587C22C2×#45211yescharacteristic
588C22#300311yescharacteristic
589C22#300511yescharacteristic
590C22#300612yes—
591C22C2×#46511yescharacteristic
592C22C2×#46811yescharacteristic
593C22#305512yes—
594C22#310312yes—
595C22#310512yes—
596C22#310912yes—
597C22#311712yes—
598C22#322712yes—
599C22#323212yes—
600C22#323412yes—
601C22#323912yes—
602C4—24no—
603C4—44no—
604C4—44no—
605C4—48no—
606C4—82no—
607C4—84no—
608C4—84no—
609C4#240612yes—
610C4#241312yes—
611C4#248612yes—
612C4#250112yes—
613C2—28no—
614C2—162no—
615C2#1506911yescharacteristic
616C2C2×#200011yescharacteristic
617C2#2339111yescharacteristic
618C2#2406112yes—
619C2#2406712yes—
620C2#2460312yes—
621C2#2460712yes—
622C2#2963911yescharacteristic
623C2#2964011yescharacteristic
624C2#3300411yescharacteristic
625C2#3300511yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC22×#121C22
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C23×C42 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. C22×#389C2semidirect
5. C2×#1824C2semidirect
6. C2×#1817C2semidirect
7. #11327C2semidirect
8. #11066C2semidirect
9. #10730C2semidirect
420. C4⋊C4#123semidirect

Maximal subgroups

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

subgroupindex
C22×#4812
C2×#28542
C22×#3892
C2×#18742
C2×#18242
C2×#18172
#113272
#110662
#107302