Finite Group Databasebeta

← #399576

Subgroups — #399576

Order 512.

2012 conjugacy classes of subgroups.

The subgroup lattice has 596 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#384C211yesmaximal, characteristic
2C2×#2084C212yesmaximal, complemented
3C2×#1853C212yesmaximal
4#14697C211yesmaximal, characteristic
5#11357C212yesmaximal, complemented
6#11323C212yesmaximal, complemented
7#11207C212yesmaximal, complemented
8#10730C212yesmaximal
9#10712C211yesmaximal, characteristic
10C23×C4⋊C4—21no—
11C23×C4⋊C4—21no—
12C23×C4⋊C4C2212yes—
13C23×C42C2211yescharacteristic
14C22×#106C2214yes—
15C22×C22⋊C8C2212yes—
16C2×#389C2211yescharacteristic
17C2×#389C2211yescharacteristic
18C2×#385—21no—
19C2×#384—21no—
20C2×#384—21no—
21C2×#384—28no—
22C2×#384C2212yes—
23C2×#384C2212yes—
24C2×#380C2211yescharacteristic
25C2×#380C2211yescharacteristic
26#2084—28no—
27#1853—28no—
28C2×#340C2214yes—
29C2×#340C2214yes—
30C2×#340C2214yes—
31C2×#336C2212yes—
32C2×#336C2212yes—
33#1436—21no—
34#1436—21no—
35C4⋊#95C2211yescharacteristic
36C4⋊#95C2211yescharacteristic
37C22⋊#95—21no—
38C24×C4—22no—
39C24×C4D411yescharacteristic
40C24×C4D411yescharacteristic
41C24×C4C2311yescharacteristic
42C23×C8C2314yes—
43C22×C4⋊C4—28no—
44C22×C4⋊C4—28no—
45C22×C4⋊C4—22no—
46C22×C4⋊C4—22no—
47C22×C4⋊C4—22no—
48C22×C4⋊C4—22no—
49C22×C4⋊C4—22no—
50C22×C4⋊C4—28no—
51C22×C4⋊C4—24no—
52C22×C4⋊C4—24no—
53C22×C4⋊C4—21no—
54C22×C4⋊C4—21no—
55C22×C4⋊C4—44no—
56C22×C4⋊C4—44no—
57C22×C4⋊C4—44no—
58C22×C4⋊C4—44no—
59C22×C4⋊C4—44no—
60C22×C4⋊C4C2311yescharacteristic
61C22×C4⋊C4C2311yescharacteristic
62C22×C4⋊C4C2311yescharacteristic
63C22×C4⋊C4C2314yes—
64C22×C4⋊C4C2311yescharacteristic
65C22×C22⋊C4—21no—
66C22×C22⋊C4—21no—
67C22×C42—24no—
68C22×C42—24no—
69C22×C42—24no—
70C22×C42D411yescharacteristic
71C22×C42D411yescharacteristic
72C22×C42C2311yescharacteristic
73C22×C42C2311yescharacteristic
74C2×#106—28no—
75C2×#106—28no—
76C2×#106—44no—
77C2×C22⋊C8—24no—
78C2×C22⋊C8—24no—
79C2×C22⋊C8Q818yes—
80#389—24no—
81#389—24no—
82#385—44no—
83#384—28no—
84#384—28no—
85#384—28no—
86#384—24no—
87#384D418yes—
88#380—22no—
89#380—24no—
90#380D414yes—
91C2×#95—21no—
92C2×#95—21no—
93C2×#95—21no—
94C2×#95—21no—
95C2×#95—21no—
96C2×#95—21no—
97C2×#95—21no—
98C2×#95—21no—
99#340—28no—
100#340—28no—
101#340—44no—
102#336—24no—
103#336—24no—
104C25C2×D411yescharacteristic
105C23×C4—22no—
106C23×C4—22no—
107C23×C4—22no—
108C23×C4—22no—
109C23×C4—22no—
110C23×C4—22no—
111C23×C4—22no—
112C23×C4—22no—
113C23×C4—22no—
114C23×C4—22no—
115C23×C4—22no—
116C23×C4—22no—
117C23×C4—22no—
118C23×C4—22no—
119C23×C4—22no—
120C23×C4—22no—
121C23×C4—28no—
122C23×C4—21no—
123C23×C4—21no—
124C23×C4—24no—
125C23×C4—21no—
126C23×C4—21no—
127C23×C4—44no—
128C23×C4—44no—
129C23×C4—44no—
130C23×C4—44no—
131C23×C4—44no—
132C23×C4—44no—
133C23×C4SD1614yes—
134C23×C4SD1614yes—
135C23×C4C2×D411yescharacteristic
136C23×C4C2×D411yescharacteristic
137C23×C4C2×D411yescharacteristic
138C23×C4C2×D412yes—
139C23×C4C2×D411yescharacteristic
140C23×C4C2×D411yescharacteristic
141C23×C4C2×Q814yes—
142C23×C4#4112yes—
143C23×C4#4114yes—
144C23×C4C2411yescharacteristic
145C22×C8—24no—
146C22×C8—24no—
147C22×C8—24no—
148C22×C8—24no—
149C22×C8—24no—
150C22×C8—24no—
151C22×C8C2×Q818yes—
152C2×C4⋊C4—24no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—24no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—24no—
157C2×C4⋊C4—24no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—28no—
161C2×C4⋊C4—28no—
162C2×C4⋊C4—28no—
163C2×C4⋊C4—28no—
164C2×C4⋊C4—28no—
165C2×C4⋊C4—24no—
166C2×C4⋊C4—21no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—416no—
175C2×C4⋊C4—416no—
176C2×C4⋊C4—432no—
177C2×C4⋊C4—44no—
178C2×C4⋊C4—44no—
179C2×C4⋊C4—44no—
180C2×C4⋊C4—44no—
181C2×C4⋊C4—44no—
182C2×C4⋊C4—432no—
183C2×C4⋊C4—44no—
184C2×C4⋊C4—44no—
185C2×C4⋊C4—44no—
186C2×C4⋊C4—44no—
187C2×C4⋊C4—44no—
188C2×C4⋊C4—42no—
189C2×C4⋊C4—42no—
190C2×C4⋊C4—42no—
191C2×C4⋊C4—42no—
192C2×C4⋊C4—42no—
193C2×C4⋊C4—42no—
194C2×C4⋊C4C2×D418yes—
195C2×C4⋊C4C2×D412yes—
196C2×C4⋊C4C2×D412yes—
197C2×C22⋊C4—24no—
198C2×C22⋊C4—42no—
199C2×C22⋊C4—42no—
200C2×C22⋊C4—42no—
201C2×C22⋊C4—42no—
202C2×C22⋊C4—42no—
203C2×C42—232no—
204C2×C42—22no—
205C2×C42—22no—
206C2×C42—22no—
207C2×C42—22no—
208C2×C42—22no—
209C2×C42—22no—
210C2×C42—22no—
211C2×C42—22no—
212C2×C42—22no—
213C2×C42—22no—
214C2×C42C2×D414yes—
215C2×C42#4114yes—
216#106—416no—
217C22⋊C8—216no—
218#95—24no—
219#95—42no—
220#95—42no—
221#95—42no—
222#95—42no—
223#95—42no—
224#95—42no—
225#95—42no—
226C24—21no—
227C24—21no—
228C24—21no—
229C24—21no—
230C24—21no—
231C24—21no—
232C24—21no—
233C24—21no—
234C24#12112yes—
235C24#12312yes—
236C24C4⋊Q812yes—
237C24C2×SD1612yes—
238C24C2×SD1612yes—
239C24AGL(1,ℤ/8)12yes—
240C24#13712yes—
241C24C22×D411yescharacteristic
242C22×C4—28no—
243C22×C4—28no—
244C22×C4—28no—
245C22×C4—28no—
246C22×C4—28no—
247C22×C4—28no—
248C22×C4—28no—
249C22×C4—28no—
250C22×C4—21no—
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—21no—
256C22×C4—21no—
257C22×C4—21no—
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—21no—
261C22×C4—21no—
262C22×C4—21no—
263C22×C4—21no—
264C22×C4—21no—
265C22×C4—21no—
266C22×C4—21no—
267C22×C4—21no—
268C22×C4—21no—
269C22×C4—21no—
270C22×C4—28no—
271C22×C4—28no—
272C22×C4—28no—
273C22×C4—28no—
274C22×C4—24no—
275C22×C4—24no—
276C22×C4—24no—
277C22×C4—28no—
278C22×C4—24no—
279C22×C4—28no—
280C22×C4—28no—
281C22×C4—28no—
282C22×C4—28no—
283C22×C4—28no—
284C22×C4—28no—
285C22×C4—22no—
286C22×C4—22no—
287C22×C4—416no—
288C22×C4—416no—
289C22×C4—416no—
290C22×C4—41no—
291C22×C4—44no—
292C22×C4—41no—
293C22×C4—44no—
294C22×C4—41no—
295C22×C4—41no—
296C22×C4—41no—
297C22×C4—41no—
298C22×C4—44no—
299C22×C4—41no—
300C22×C4—44no—
301C22×C4—44no—
302C22×C4—44no—
303C22×C4—41no—
304C22×C4—41no—
305C22×C4—41no—
306C22×C4—41no—
307C22×C4—41no—
308C22×C4—41no—
309C22×C4—41no—
310C22×C4—41no—
311C22×C4—41no—
312C22×C4—41no—
313C22×C4—41no—
314C22×C4—41no—
315C22×C4—41no—
316C22×C4—41no—
317C22×C4—41no—
318C22×C4—41no—
319C22×C4—41no—
320C22×C4—41no—
321C22×C4—41no—
322C22×C4—88no—
323C22×C4—88no—
324C22×C4—88no—
325C22×C4—88no—
326C22×C4#12012yes—
327C22×C4#12012yes—
328C22×C4#12112yes—
329C22×C4C22⋊Q818yes—
330C22×C4#12312yes—
331C22×C4#12312yes—
332C22×C4#12312yes—
333C22×C4#12518yes—
334C22×C4C4⋊Q812yes—
335C22×C4C2×SD1612yes—
336C22×C4C2×SD1612yes—
337C22×C4C2×SD1612yes—
338C22×C4C2×SD1612yes—
339C22×C4AGL(1,ℤ/8)12yes—
340C22×C4AGL(1,ℤ/8)12yes—
341C22×C4#13712yes—
342C22×C4#13712yes—
343C22×C4C22×D411yescharacteristic
344C22×C4C22×D412yes—
345C22×C4C22×Q812yes—
346C22×C4C2×#4112yes—
347C22×C4C2×#4112yes—
348C22×C4C2×#4111yescharacteristic
349C22×C4H(5,𝔽2)12yes—
350C22×C42−1+412yes—
351C2×C8—28no—
352C2×C8—28no—
353C2×C8—28no—
354C2×C8—44no—
355C2×C8—44no—
356C2×C8—44no—
357C2×C8—44no—
358C4⋊C4—24no—
359C4⋊C4—28no—
360C4⋊C4—28no—
361C4⋊C4—24no—
362C4⋊C4—416no—
363C4⋊C4—416no—
364C4⋊C4—416no—
365C4⋊C4—416no—
366C4⋊C4—44no—
367C4⋊C4—42no—
368C4⋊C4—44no—
369C4⋊C4—42no—
370C4⋊C4—416no—
371C4⋊C4—44no—
372C4⋊C4—44no—
373C4⋊C4—84no—
374C4⋊C4—84no—
375C22⋊C4—48no—
376C22⋊C4—84no—
377C42—28no—
378C42—28no—
379C42—28no—
380C42—28no—
381C23—22no—
382C23—22no—
383C23—22no—
384C23—22no—
385C23—22no—
386C23—22no—
387C23—22no—
388C23—22no—
389C23—22no—
390C23—22no—
391C23—22no—
392C23—22no—
393C23—22no—
394C23—22no—
395C23—22no—
396C23—22no—
397C23—22no—
398C23—22no—
399C23—22no—
400C23—22no—
401C23—22no—
402C23—22no—
403C23—22no—
404C23—22no—
405C23—22no—
406C23—22no—
407C23—22no—
408C23—22no—
409C23#39814yes—
410C23#46014yes—
411C23#46114yes—
412C23#48114yes—
413C23#48314yes—
414C23#49814yes—
415C23#50114yes—
416C23C2×#12011yescharacteristic
417C23C2×#12111yescharacteristic
418C23C2×#12311yescharacteristic
419C23C2×#12311yescharacteristic
420C23C2×C4⋊Q811yescharacteristic
421C23#53511yescharacteristic
422C23#53611yescharacteristic
423C23C22×SD1611yescharacteristic
424C23C22×SD1611yescharacteristic
425C23C2×AGL(1,ℤ/8)11yescharacteristic
426C23C2×AGL(1,ℤ/8)11yescharacteristic
427C23C2×AGL(1,ℤ/8)11yescharacteristic
428C23C2×#13711yescharacteristic
429C23C2×#13711yescharacteristic
430C23C2×#13711yescharacteristic
431C2×C4—232no—
432C2×C4—232no—
433C2×C4—22no—
434C2×C4—22no—
435C2×C4—22no—
436C2×C4—22no—
437C2×C4—22no—
438C2×C4—22no—
439C2×C4—22no—
440C2×C4—22no—
441C2×C4—22no—
442C2×C4—22no—
443C2×C4—22no—
444C2×C4—22no—
445C2×C4—22no—
446C2×C4—22no—
447C2×C4—22no—
448C2×C4—22no—
449C2×C4—22no—
450C2×C4—22no—
451C2×C4—22no—
452C2×C4—22no—
453C2×C4—22no—
454C2×C4—22no—
455C2×C4—22no—
456C2×C4—22no—
457C2×C4—22no—
458C2×C4—22no—
459C2×C4—22no—
460C2×C4—22no—
461C2×C4—22no—
462C2×C4—22no—
463C2×C4—216no—
464C2×C4—216no—
465C2×C4—48no—
466C2×C4—42no—
467C2×C4—48no—
468C2×C4—42no—
469C2×C4—48no—
470C2×C4—42no—
471C2×C4—42no—
472C2×C4—42no—
473C2×C4—42no—
474C2×C4—832no—
475C2×C4—81no—
476C2×C4—81no—
477C2×C4—81no—
478C2×C4—84no—
479C2×C4—84no—
480C2×C4—81no—
481C2×C4—81no—
482C2×C4—81no—
483C2×C4—81no—
484C2×C4—81no—
485C2×C4—84no—
486C2×C4—84no—
487C2×C4—81no—
488C2×C4—81no—
489C2×C4—81no—
490C2×C4—81no—
491C2×C4—81no—
492C2×C4—81no—
493C2×C4—81no—
494C2×C4—81no—
495C2×C4—81no—
496C2×C4—81no—
497C2×C4—81no—
498C2×C4—81no—
499C2×C4—81no—
500C2×C4—81no—
501C2×C4#39814yes—
502C2×C4#44814yes—
503C2×C4#45014yes—
504C2×C4#48114yes—
505C2×C4#48314yes—
506C2×C4C2×C22⋊Q812yes—
507C2×C4#52514yes—
508C2×C4C2×#12512yes—
509C2×C4#53914yes—
510C2×C4D4212yes—
511C2×C4#54612yes—
512C2×C4#54712yes—
513C2×C4D4×Q814yes—
514C2×C4#55212yes—
515C2×C4#55414yes—
516C8—48no—
517C22—24no—
518C22—24no—
519C22—24no—
520C22—24no—
521C22—24no—
522C22—24no—
523C22—24no—
524C22—24no—
525C22—24no—
526C22—24no—
527C22—24no—
528C22—24no—
529C22—24no—
530C22—24no—
531C22#208418yescomplemented
532C22C2×#39811yescharacteristic
533C22#240811yescharacteristic
534C22#243411yescharacteristic
535C22#248611yescharacteristic
536C22#250211yescharacteristic
537C22#262311yescharacteristic
538C22#284911yescharacteristic
539C22C2×#45011yescharacteristic
540C22C2×#44811yescharacteristic
541C22#300311yescharacteristic
542C22#300411yescharacteristic
543C22C2×#46111yescharacteristic
544C22C2×#46011yescharacteristic
545C22#303811yescharacteristic
546C22#303911yescharacteristic
547C22C2×#48311yescharacteristic
548C22C2×#48311yescharacteristic
549C22C2×#48111yescharacteristic
550C22C2×#48111yescharacteristic
551C22#309211yescharacteristic
552C22#309211yescharacteristic
553C22#309211yescharacteristic
554C22#309211yescharacteristic
555C22C2×#49811yescharacteristic
556C22C2×#50111yescharacteristic
557C22#316511yescharacteristic
558C22#316511yescharacteristic
559C22#320611yescharacteristic
560C22#320811yescharacteristic
561C22#321711yescharacteristic
562C22#321811yescharacteristic
563C22#322711yescharacteristic
564C22#322911yescharacteristic
565C22#323811yescharacteristic
566C22#323911yescharacteristic
567C4—24no—
568C4—24no—
569C4—24no—
570C4—24no—
571C4—24no—
572C4—82no—
573C4—88no—
574C4—82no—
575C4—161no—
576C4—161no—
577C4#244914yes—
578C4#247314yes—
579C2—28no—
580C2#1478611yescharacteristic
581C2C2×#208411yescharacteristic
582C2#2365811yescharacteristic
583C2#2420911yescharacteristic
584C2#2421111yescharacteristic
585C2#2581111yescharacteristic
586C2#2581611yescharacteristic
587C2#2764511yescharacteristic
588C2#2764711yescharacteristic
589C2#2794811yescharacteristic
590C2#2795311yescharacteristic
591C2#2990711yescharacteristic
592C2#2990811yescharacteristic
593C2#3297911yescharacteristic
594C2#3298211yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
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
2. C2×#2084C2semidirect
5. #11357C2semidirect
6. #11323C2semidirect
7. #11207C2semidirect
531. C22#2084semidirect

Maximal subgroups

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

subgroupindex
C22×#3842
C2×#20842
C2×#18532
#146972
#113572
#113232
#112072
#107302
#107122