Finite Group Databasebeta

← #415847

Subgroups — #415847

Order 512.

2114 conjugacy classes of subgroups.

The subgroup lattice has 614 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#460C211yesmaximal, characteristic
2C2×#2582C211yesmaximal, characteristic, complemented
3C2×#2548C211yesmaximal, characteristic
4C2×#1984C211yesmaximal, characteristic, complemented
5C2×#1957C211yesmaximal, characteristic
6C2×C4×#103C211yesmaximal, characteristic, complemented
7#11066C214yesmaximal, complemented
8#11023C214yesmaximal, complemented
9#10849C211yesmaximal, characteristic, complemented
10C23×SD16C2212yes—
11C22×#127C2211yescharacteristic
12C22×#124—21no—
13C22×C4×Q8C2211yescharacteristic
14C22×C4×Q8C2211yescharacteristic
15C2×#460—24no—
16C2×#460—24no—
17C2×#460—24no—
18C22×C4⋊C8C2211yescharacteristic
19C22×#103C2212yes—
20C22×#102C2212yes—
21C22×#102C2214yes—
22C22×C4×C8C2211yescharacteristic
23#2582—24no—
24#2582—24no—
25#2548—24no—
26#2548—24no—
27C2×#398—22no—
28C2×#394—22no—
29C2×#391—21no—
30C2×#391C2212yes—
31C2×#390—21no—
32C2×#390C2214yes—
33C2×#389—21no—
34C2×#389C2212yes—
35C2×#384C2211yescharacteristic
36C2×C4×C4⋊C4C2211yescharacteristic
37#1984—24no—
38#1957—24no—
39C4×#103—24no—
40C2×#340C2212yes—
41C2×#340C2214yes—
42C2×#336C2214yes—
43C23×Q8D411yescharacteristic
44C23×Q8C2311yescharacteristic
45C23×D4—21no—
46C23×D4—21no—
47C23×D4C2312yes—
48C22×SD16—24no—
49C22×SD16—24no—
50C22×SD16—28no—
51C22×SD16—24no—
52C22×SD16—44no—
53C23×C8C2312yes—
54C23×C8C2312yes—
55C2×#127—22no—
56C2×#127—22no—
57C2×#127—22no—
58C2×#127—22no—
59C2×#127—22no—
60C2×#127D414yes—
61C2×#124—28no—
62C2×#124—44no—
63C2×#124—44no—
64C2×C4×Q8—22no—
65C2×C4×Q8—22no—
66C2×C4×Q8—22no—
67C2×C4×Q8—22no—
68C2×C4×Q8—22no—
69C2×C4×Q8—22no—
70C2×C4×Q8—22no—
71C2×C4×Q8—22no—
72C2×C4×Q8—22no—
73C2×C4×Q8—22no—
74C2×C4×Q8—22no—
75C2×C4×Q8D414yes—
76C22×C4⋊C4—21no—
77C22×C4⋊C4—21no—
78C22×C4⋊C4—21no—
79C22×C4⋊C4—22no—
80C22×C4⋊C4D412yes—
81C22×C4⋊C4D411yescharacteristic
82C22×C4⋊C4C2311yescharacteristic
83C22×C4⋊C4C2312yes—
84C22×C4⋊C4C2312yes—
85C22×C22⋊C4—24no—
86C22×C22⋊C4—24no—
87C22×C42—21no—
88C22×C42—21no—
89C22×C42C2312yes—
90C22×C42C2311yescharacteristic
91#460—232no—
92C2×C4⋊C8—22no—
93C2×C4⋊C8—22no—
94C2×C4⋊C8D414yes—
95C2×#103—24no—
96C2×#103—24no—
97C2×#103—24no—
98C2×#102—24no—
99C2×#102—24no—
100C2×#102—24no—
101C2×#102—28no—
102C2×#102—28no—
103C2×#102—44no—
104C2×C4×C8—22no—
105C2×C4×C8—22no—
106C2×C4×C8—22no—
107#398—216no—
108#394—216no—
109#391—28no—
110#391—24no—
111#391—24no—
112#390—28no—
113#390—28no—
114#390—28no—
115#389—28no—
116#389—24no—
117#389—24no—
118#384—22no—
119#384D414yes—
120C4×C4⋊C4—22no—
121C4×C4⋊C4D414yes—
122C2×#95—21no—
123C2×#95—22no—
124C2×#95—22no—
125C2×#95—22no—
126#340—24no—
127#340—44no—
128#336—28no—
129C25—22no—
130C22×Q8—22no—
131C22×Q8—22no—
132C22×Q8—22no—
133C22×Q8—22no—
134C22×Q8—22no—
135C22×Q8—22no—
136C22×Q8—21no—
137C22×Q8—22no—
138C22×Q8—21no—
139C22×Q8—21no—
140C22×Q8—21no—
141C22×Q8—21no—
142C22×Q8—22no—
143C22×Q8—22no—
144C22×Q8SD1614yescomplemented
145C22×Q8C2×D412yes—
146C22×Q8#4114yes—
147C22×D4—24no—
148C22×D4—24no—
149C22×D4—22no—
150C22×D4—22no—
151C22×D4—22no—
152C22×D4—22no—
153C22×D4—22no—
154C22×D4—24no—
155C22×D4—24no—
156C22×D4—24no—
157C22×D4—24no—
158C22×D4—42no—
159C22×D4—42no—
160C22×D4—42no—
161C22×D4—42no—
162C22×D4—42no—
163C22×D4—42no—
164C22×D4—42no—
165C22×D4—42no—
166C22×D4—82no—
167C22×D4—82no—
168C22×D4C2×D414yes—
169C23×C4—21no—
170C23×C4—22no—
171C23×C4—22no—
172C23×C4—21no—
173C23×C4C2×D412yes—
174C23×C4C2×D412yes—
175C23×C4C2×D411yescharacteristic
176C23×C4C2×D411yescharacteristic
177C23×C4C2411yescharacteristic
178C2×SD16—216no—
179C2×SD16—416no—
180C2×SD16—416no—
181C2×SD16—48no—
182C2×SD16—48no—
183C22×C8—22no—
184C22×C8—22no—
185C22×C8—22no—
186C22×C8—22no—
187C22×C8—22no—
188C22×C8—24no—
189C22×C8—22no—
190C22×C8—24no—
191C22×C8—22no—
192C22×C8—22no—
193C22×C8C2×D414yes—
194C22×C8#4114yes—
195#127—28no—
196#127—28no—
197#127—44no—
198#127—44no—
199#124—432no—
200C4×Q8—28no—
201C4×Q8—28no—
202C4×Q8—44no—
203C4×Q8—44no—
204C4×Q8—44no—
205C4×Q8—44no—
206C4×Q8—44no—
207C4×Q8—44no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—24no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—24no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—24no—
219C2×C4⋊C4—24no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—24no—
223C2×C4⋊C4—24no—
224C2×C4⋊C4—24no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—21no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—28no—
230C2×C4⋊C4—24no—
231C2×C4⋊C4—42no—
232C2×C4⋊C4—42no—
233C2×C4⋊C4—42no—
234C2×C4⋊C4—42no—
235C2×C4⋊C4—82no—
236C2×C4⋊C4—84no—
237C2×C4⋊C4C2×D412yes—
238C2×C4⋊C4C2×D414yes—
239C2×C4⋊C4C2×D414yes—
240C2×C22⋊C4—216no—
241C2×C22⋊C4—216no—
242C2×C22⋊C4—48no—
243C2×C22⋊C4—48no—
244C2×C22⋊C4—84no—
245C2×C22⋊C4—84no—
246C2×C42—24no—
247C2×C42—24no—
248C2×C42—22no—
249C2×C42—22no—
250C2×C42—22no—
251C2×C42—22no—
252C2×C42—22no—
253C2×C42—22no—
254C2×C42—42no—
255C2×C42—42no—
256C2×C42—42no—
257C2×C42—42no—
258C2×C42C2×D412yes—
259C2×C42C2×D412yes—
260C2×C42C2×D412yes—
261C2×C42C2×D412yes—
262C2×C42C2×D412yes—
263C2×C42#4112yes—
264C4⋊C8—28no—
265#103—216no—
266#102—216no—
267#102—416no—
268C4×C8—44no—
269#95—28no—
270#95—28no—
271#95—28no—
272#95—24no—
273C24—24no—
274C24—42no—
275C24—42no—
276C24—44no—
277C24—44no—
278C24—42no—
279C24—42no—
280C24—42no—
281C24—42no—
282C24—44no—
283C24—44no—
284C24C22×D411yescharacteristic
285C2×Q8—22no—
286C2×Q8—24no—
287C2×Q8—28no—
288C2×Q8—28no—
289C2×Q8—28no—
290C2×Q8—22no—
291C2×Q8—22no—
292C2×Q8—24no—
293C2×Q8—24no—
294C2×Q8—22no—
295C2×Q8—22no—
296C2×Q8—24no—
297C2×Q8—24no—
298C2×Q8—41no—
299C2×Q8—41no—
300C2×Q8—42no—
301C2×Q8—42no—
302C2×Q8—42no—
303C2×Q8—42no—
304C2×Q8—42no—
305C2×Q8—44no—
306C2×Q8—44no—
307C2×Q8—44no—
308C2×Q8—44no—
309C2×Q8#12414yescomplemented
310C2×D4—24no—
311C2×D4—24no—
312C2×D4—28no—
313C2×D4—28no—
314C2×D4—42no—
315C2×D4—44no—
316C2×D4—42no—
317C2×D4—44no—
318C2×D4—42no—
319C2×D4—42no—
320C2×D4—42no—
321C2×D4—42no—
322C2×D4—44no—
323C2×D4—44no—
324C2×D4—44no—
325C2×D4—44no—
326C2×D4—44no—
327C2×D4—44no—
328C2×D4—48no—
329C2×D4—48no—
330C2×D4—48no—
331C2×D4—48no—
332C2×D4—88no—
333C2×D4—84no—
334C2×D4—84no—
335C2×D4—88no—
336C2×D4—84no—
337C2×D4—84no—
338C2×D4—84no—
339C2×D4—84no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—24no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—24no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—24no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—24no—
357C22×C4—24no—
358C22×C4—41no—
359C22×C4—41no—
360C22×C4—42no—
361C22×C4—41no—
362C22×C4—41no—
363C22×C4—42no—
364C22×C4—42no—
365C22×C4—44no—
366C22×C4—42no—
367C22×C4—42no—
368C22×C4—42no—
369C22×C4—42no—
370C22×C4—42no—
371C22×C4—44no—
372C22×C4—42no—
373C22×C4—42no—
374C22×C4—42no—
375C22×C4#12012yes—
376C22×C4#12012yes—
377C22×C4#12012yes—
378C22×C4#12012yes—
379C22×C4#12112yes—
380C22×C4#12112yes—
381C22×C4C2×SD1614yes—
382C22×C4C2×SD1612yes—
383C22×C4#13512yes—
384C22×C4AGL(1,ℤ/8)12yes—
385C22×C4#13712yes—
386C22×C4C22×D411yescharacteristic
387C22×C4C22×D411yescharacteristic
388C22×C4C22×D411yescharacteristic
389C22×C4C22×D411yescharacteristic
390C22×C4C22×D411yescharacteristic
391C22×C4C2×#4111yescharacteristic
392C22×C4C2×#4112yes—
393C22×C4C2×#4112yes—
394C22×C4H(5,𝔽2)12yes—
395C22×C42−1+412yes—
396SD16—432no—
397C2×C8—24no—
398C2×C8—24no—
399C2×C8—24no—
400C2×C8—28no—
401C2×C8—24no—
402C2×C8—42no—
403C2×C8—42no—
404C2×C8—42no—
405C2×C8—42no—
406C2×C8—42no—
407C2×C8—44no—
408C2×C8—42no—
409C4⋊C4—22no—
410C4⋊C4—22no—
411C4⋊C4—24no—
412C4⋊C4—24no—
413C4⋊C4—28no—
414C4⋊C4—41no—
415C4⋊C4—41no—
416C4⋊C4—42no—
417C4⋊C4—42no—
418C4⋊C4—42no—
419C4⋊C4—42no—
420C4⋊C4—48no—
421C4⋊C4—44no—
422C4⋊C4—48no—
423C4⋊C4—48no—
424C4⋊C4—84no—
425C4⋊C4—88no—
426C22⋊C4—432no—
427C22⋊C4—816no—
428C42—42no—
429C42—42no—
430C42—42no—
431C42—42no—
432C42—48no—
433C42—48no—
434C42#12014yes—
435C42#12114yes—
436C42#12114yes—
437C42#12714yes—
438C23—44no—
439C23—48no—
440C23—44no—
441C23—44no—
442C23—48no—
443C23—84no—
444C23—84no—
445C23—84no—
446C23—84no—
447C23—84no—
448C23—84no—
449C23—84no—
450C23—84no—
451C23—84no—
452C23—82no—
453C23—82no—
454C23—82no—
455C23—82no—
456C23—82no—
457C23—82no—
458C23—82no—
459C23—82no—
460C23—84no—
461C23C2×#12011yescharacteristic
462C23C2×#12011yescharacteristic
463C23C2×#12111yescharacteristic
464C23#54812yes—
465C23#55012yes—
466C23C22×SD1611yescharacteristic
467C23C2×#13511yescharacteristic
468C23C2×AGL(1,ℤ/8)11yescharacteristic
469C23C2×#13711yescharacteristic
470C23#57612yes—
471C23#57712yes—
472Q8—24no—
473Q8—24no—
474Q8—42no—
475Q8—44no—
476Q8—42no—
477Q8—416no—
478Q8—44no—
479D4—48no—
480D4—44no—
481D4—44no—
482D4—816no—
483D4—82no—
484D4—82no—
485D4—816no—
486D4—84no—
487C2×C4—22no—
488C2×C4—22no—
489C2×C4—24no—
490C2×C4—24no—
491C2×C4—24no—
492C2×C4—24no—
493C2×C4—24no—
494C2×C4—24no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—44no—
498C2×C4—44no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—44no—
506C2×C4—44no—
507C2×C4—44no—
508C2×C4—44no—
509C2×C4—42no—
510C2×C4—44no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—44no—
515C2×C4—44no—
516C2×C4—44no—
517C2×C4—82no—
518C2×C4—82no—
519C2×C4—82no—
520C2×C4—84no—
521C2×C4—84no—
522C2×C4—82no—
523C2×C4—82no—
524C2×C4—82no—
525C2×C4#39214yes—
526C2×C4#39214yes—
527C2×C4#44814yes—
528C2×C4#44914yes—
529C2×C4#45014yes—
530C2×C4#45214yes—
531C2×C4#46514yes—
532C2×C4#46814yes—
533C2×C4C2×#12011yescharacteristic
534C2×C4C2×#12111yescharacteristic
535C2×C4C2×#12111yescharacteristic
536C2×C4#52512yes—
537C2×C4C2×#12411yescharacteristic
538C2×C4#52912yes—
539C2×C4C2×#12711yescharacteristic
540C2×C4#53211yescharacteristic
541C2×C4#53511yescharacteristic
542C2×C4#53611yescharacteristic
543C2×C4#53712yes—
544C2×C4#54212yes—
545C2×C4#54612yes—
546C2×C4#54612yes—
547C2×C4#54612yes—
548C2×C4#54712yes—
549C2×C4#54812yes—
550C2×C4#54812yes—
551C8—44no—
552C8—44no—
553C22—84no—
554C22—84no—
555C22—84no—
556C22—88no—
557C22—88no—
558C22—162no—
559C22—162no—
560C22—162no—
561C22—162no—
562C22—164no—
563C22—164no—
564C22C2×#39211yescharacteristic
565C22#240712yes—
566C22#242612yes—
567C22#243111yescharacteristic
568C22#243611yescharacteristic
569C22C2×#45011yescharacteristic
570C22C2×#44811yescharacteristic
571C22C2×#44911yescharacteristic
572C22C2×#45211yescharacteristic
573C22#300811yescharacteristic
574C22#301011yescharacteristic
575C22#301212yes—
576C22C2×#46511yescharacteristic
577C22C2×#46811yescharacteristic
578C22#306212yes—
579C22#336212yes—
580C22#336512yes—
581C22#336812yes—
582C22#337512yes—
583C22#339212yes—
584C22#339612yes—
585C22#339912yes—
586C22#340012yes—
587C4—24no—
588C4—44no—
589C4—44no—
590C4—84no—
591C4—82no—
592C4—82no—
593C4—84no—
594C4#200018yes—
595C4#242712yes—
596C4#243211yescharacteristic
597C4#243511yescharacteristic
598C4#254012yes—
599C4#262011yescharacteristic
600C4#285011yescharacteristic
601C2—164no—
602C2#1521511yescharacteristic
603C2C2×#200011yescharacteristic
604C2#2339311yescharacteristic
605C2#2418812yes—
606C2#2419012yes—
607C2#2529512yes—
608C2#2530312yes—
609C2#2563011yescharacteristic
610C2#2563311yescharacteristic
611C2#2593211yescharacteristic
612C2#2593511yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC22×#127C22
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C22×C4×Q8 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2582C2semidirect
4. C2×#1984C2semidirect
6. C2×C4×#103C2semidirect
7. #11066C2semidirect
8. #11023C2semidirect
9. #10849C2semidirect
144. C22×Q8SD16semidirect
309. C2×Q8#124semidirect

Maximal subgroups

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

subgroupindex
C22×#4602
C2×#25822
C2×#25482
C2×#19842
C2×#19572
C2×C4×#1032
#110662
#110232
#108492