Finite Group Databasebeta

← #366772

Subgroups — #366772

Order 512.

2000 conjugacy classes of subgroups.

The subgroup lattice has 649 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#480C211yesmaximal, characteristic
2C2×#1874C211yesmaximal, characteristic
3C2×#1853C211yesmaximal, characteristic, complemented
4C2×#1823C211yesmaximal, characteristic, complemented
5C2×#1749C211yesmaximal, characteristic, complemented
6#14698C211yesmaximal, characteristic
7C2×C4⋊#95C211yesmaximal, characteristic, complemented
8#11377C212yesmaximal, complemented
9#11121C212yesmaximal, complemented
10#10927C212yesmaximal, complemented
11#10863C212yesmaximal, complemented
12C22×#121C2211yescharacteristic
13C23×C4⋊C4C2211yescharacteristic
14C23×C4⋊C4C2211yescharacteristic
15C23×C42C2211yescharacteristic
16C2×#480—24no—
17C2×#480—24no—
18C2×#480C418yescomplemented
19C22×#107C2212yes—
20C22×#106C2212yes—
21C22×#102C2212yes—
22C22×#102C2212yes—
23C22×C22⋊C8C2211yescharacteristic
24C22×C22⋊C8C2211yescharacteristic
25C22×#97C2212yes—
26C2×#390C2211yescharacteristic
27C2×#390C2211yescharacteristic
28C2×#386—22no—
29C2×#380C2211yescharacteristic
30C22×#95—22no—
31#1874—24no—
32#1853—24no—
33#1823—24no—
34#1749—24no—
35C2×#340C2212yes—
36C2×#340C2212yes—
37C2×#340C2212yes—
38C2×#340C2212yes—
39C2×#336C2212yes—
40#1437—22no—
41C4⋊#95—22no—
42C4⋊#95C2211yescharacteristic
43C4⋊#95C2211yescharacteristic
44C4⋊#95C2212yes—
45C4⋊#95C2212yes—
46C23×D4—21no—
47C23×D4C2311yescharacteristic
48C24×C4—21no—
49C24×C4—21no—
50C24×C4D412yes—
51C24×C4C2311yescharacteristic
52C23×C8C2312yes—
53C23×C8C2312yes—
54C2×#121—22no—
55C2×#121—22no—
56C2×#121—22no—
57C2×#121—22no—
58C2×#121—22no—
59C2×#121C2×C414yes—
60C22×C4⋊C4—22no—
61C22×C4⋊C4—22no—
62C22×C4⋊C4—22no—
63C22×C4⋊C4—22no—
64C22×C4⋊C4—21no—
65C22×C4⋊C4—21no—
66C22×C4⋊C4—22no—
67C22×C4⋊C4—24no—
68C22×C4⋊C4—22no—
69C22×C4⋊C4—22no—
70C22×C4⋊C4—42no—
71C22×C4⋊C4C2×C414yes—
72C22×C4⋊C4C2311yescharacteristic
73C22×C4⋊C4C2311yescharacteristic
74C22×C4⋊C4C2312yes—
75C22×C4⋊C4C2311yescharacteristic
76C22×C4⋊C4C2312yes—
77C22×C22⋊C4—21no—
78C22×C22⋊C4—21no—
79C22×C22⋊C4—22no—
80C22×C22⋊C4—22no—
81C22×C42—22no—
82C22×C42—22no—
83C22×C42—24no—
84C22×C42—42no—
85C22×C42D412yes—
86C22×C42C2311yescharacteristic
87C22×C42C2311yescharacteristic
88#480—232no—
89C2×#107—24no—
90C2×#107—24no—
91C2×#107C2×C418yescomplemented
92C2×#106—24no—
93C2×#106—24no—
94C2×#106—42no—
95C2×#102—24no—
96C2×#102—24no—
97C2×#102—24no—
98C2×#102—24no—
99C2×#102—42no—
100C2×#102C2×C418yescomplemented
101C2×C22⋊C8—22no—
102C2×C22⋊C8—22no—
103C2×C22⋊C8C2×C414yes—
104C2×C22⋊C8D414yes—
105C2×C22⋊C8D414yes—
106C2×C22⋊C8D414yes—
107C2×#97—24no—
108C2×#97—24no—
109C2×#97—42no—
110#390—24no—
111#390—42no—
112#386—48no—
113#380—22no—
114#380—22no—
115C2×#95—24no—
116C2×#95—24no—
117C2×#95—22no—
118C2×#95—22no—
119C2×#95—21no—
120C2×#95—48no—
121#340—24no—
122#340—24no—
123#340—24no—
124#340—24no—
125#336—24no—
126C25—21no—
127C25C2×D411yescharacteristic
128C22×D4—24no—
129C22×D4—21no—
130C22×D4—21no—
131C22×D4—21no—
132C22×D4—21no—
133C22×D4—22no—
134C22×D4—22no—
135C22×D4—21no—
136C22×D4—22no—
137C22×D4—22no—
138C22×D4—42no—
139C22×D4—42no—
140C22×D4—44no—
141C22×D4—44no—
142C22×D4C22×C412yes—
143C23×C4—22no—
144C23×C4—22no—
145C23×C4—22no—
146C23×C4—22no—
147C23×C4—24no—
148C23×C4—24no—
149C23×C4—24no—
150C23×C4—24no—
151C23×C4—24no—
152C23×C4—24no—
153C23×C4—24no—
154C23×C4—21no—
155C23×C4—22no—
156C23×C4—22no—
157C23×C4—21no—
158C23×C4—21no—
159C23×C4—42no—
160C23×C4—42no—
161C23×C4—42no—
162C23×C4—44no—
163C23×C4—42no—
164C23×C4—42no—
165C23×C4—42no—
166C23×C4—44no—
167C23×C4—44no—
168C23×C4—44no—
169C23×C4C22×C412yes—
170C23×C4C2×D412yes—
171C23×C4C2×D412yes—
172C23×C4C2×D412yes—
173C23×C4C2×D412yes—
174C23×C4C2×D411yescharacteristic
175C23×C4C2×D412yes—
176C23×C4#4112yes—
177C23×C4#4112yes—
178C23×C4C2411yescharacteristic
179C22×C8—22no—
180C22×C8—22no—
181C22×C8—22no—
182C22×C8—22no—
183C22×C8—22no—
184C22×C8—22no—
185C22×C8—22no—
186C22×C8—22no—
187C22×C8—22no—
188C22×C8—22no—
189C22×C8C22×C414yes—
190C22×C8C2×D414yes—
191C22×C8C2×D414yes—
192C22×C8C2×D414yes—
193#121—28no—
194#121—28no—
195#121—44no—
196#121—44no—
197C2×C4⋊C4—28no—
198C2×C4⋊C4—28no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—24no—
212C2×C4⋊C4—24no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—48no—
225C2×C4⋊C4—48no—
226C2×C4⋊C4—42no—
227C2×C4⋊C4—44no—
228C2×C4⋊C4—42no—
229C2×C4⋊C4—44no—
230C2×C4⋊C4—42no—
231C2×C4⋊C4—42no—
232C2×C4⋊C4C22×C414yes—
233C2×C4⋊C4C22×C412yes—
234C2×C4⋊C4#4114yes—
235C2×C22⋊C4—24no—
236C2×C22⋊C4—42no—
237C2×C22⋊C4—42no—
238C2×C22⋊C4—44no—
239C2×C22⋊C4—44no—
240C2×C22⋊C4—44no—
241C2×C22⋊C4—44no—
242C2×C22⋊C4—42no—
243C2×C22⋊C4—44no—
244C2×C22⋊C4—44no—
245C2×C22⋊C4—44no—
246C2×C42—21no—
247C2×C42—21no—
248C2×C42—22no—
249C2×C42—24no—
250C2×C42—24no—
251C2×C42—21no—
252C2×C42—22no—
253C2×C42—21no—
254C2×C42—22no—
255C2×C42—48no—
256C2×C42—48no—
257C2×C42—42no—
258C2×C42#4114yes—
259#107—216no—
260#106—48no—
261#102—216no—
262#102—48no—
263C22⋊C8—28no—
264C22⋊C8C22⋊C4116yescomplemented
265#97—48no—
266#95—48no—
267#95—48no—
268#95—44no—
269#95—44no—
270#95—42no—
271C24—21no—
272C24—21no—
273C24—21no—
274C24—21no—
275C24—22no—
276C24—22no—
277C24—22no—
278C24—41no—
279C24—41no—
280C24—41no—
281C24—41no—
282C24—41no—
283C24—41no—
284C24—42no—
285C24—42no—
286C24—42no—
287C24—42no—
288C24C4×D412yes—
289C24#12112yes—
290C24#12712yes—
291C24AGL(1,ℤ/8)14yes—
292C24AGL(1,ℤ/8)14yes—
293C24C22×D411yescharacteristic
294C2×D4—22no—
295C2×D4—22no—
296C2×D4—24no—
297C2×D4—24no—
298C2×D4—48no—
299C2×D4—416no—
300C2×D4—41no—
301C2×D4—41no—
302C2×D4—42no—
303C2×D4—42no—
304C2×D4—42no—
305C2×D4—42no—
306C2×D4—42no—
307C2×D4—42no—
308C2×D4—42no—
309C2×D4—42no—
310C2×D4—42no—
311C2×D4—42no—
312C2×D4—88no—
313C2×D4—84no—
314C2×D4—84no—
315C22×C4—21no—
316C22×C4—21no—
317C22×C4—21no—
318C22×C4—21no—
319C22×C4—24no—
320C22×C4—24no—
321C22×C4—24no—
322C22×C4—24no—
323C22×C4—24no—
324C22×C4—24no—
325C22×C4—24no—
326C22×C4—24no—
327C22×C4—24no—
328C22×C4—24no—
329C22×C4—24no—
330C22×C4—24no—
331C22×C4—216no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—24no—
346C22×C4—24no—
347C22×C4—48no—
348C22×C4—48no—
349C22×C4—48no—
350C22×C4—48no—
351C22×C4—48no—
352C22×C4—48no—
353C22×C4—48no—
354C22×C4—416no—
355C22×C4—41no—
356C22×C4—42no—
357C22×C4—41no—
358C22×C4—42no—
359C22×C4—42no—
360C22×C4—42no—
361C22×C4—44no—
362C22×C4—44no—
363C22×C4—42no—
364C22×C4—44no—
365C22×C4—42no—
366C22×C4—44no—
367C22×C4—48no—
368C22×C4—416no—
369C22×C4—41no—
370C22×C4—41no—
371C22×C4—41no—
372C22×C4—41no—
373C22×C4—41no—
374C22×C4—41no—
375C22×C4—42no—
376C22×C4—42no—
377C22×C4—42no—
378C22×C4—42no—
379C22×C4—84no—
380C22×C4—88no—
381C22×C4—88no—
382C22×C4—84no—
383C22×C4—84no—
384C22×C4—84no—
385C22×C4C2×C22⋊C414yes—
386C22×C4C4×D414yes—
387C22×C4C4×D412yes—
388C22×C4#12114yes—
389C22×C4#12112yes—
390C22×C4#12314yes—
391C22×C4#12414yes—
392C22×C4#12712yes—
393C22×C4#13714yes—
394C22×C4C23×C411yescharacteristic
395C22×C4C22×D411yescharacteristic
396C22×C4C22×D411yescharacteristic
397C22×C4C22×D411yescharacteristic
398C22×C4C2×#4111yescharacteristic
399C22×C4C2×#4112yes—
400C22×C4C2×#4112yes—
401C22×C4C2×#4111yescharacteristic
402C22×C4H(5,𝔽2)12yes—
403C22×C42−1+412yes—
404C2×C8—24no—
405C2×C8—24no—
406C2×C8—24no—
407C2×C8—24no—
408C2×C8—24no—
409C2×C8—24no—
410C2×C8—24no—
411C2×C8—24no—
412C2×C8—24no—
413C2×C8—42no—
414C2×C8—42no—
415C2×C8—42no—
416C2×C8—42no—
417C2×C8C2×C22⋊C418yes—
418C4⋊C4—24no—
419C4⋊C4—22no—
420C4⋊C4—24no—
421C4⋊C4—22no—
422C4⋊C4—24no—
423C4⋊C4—24no—
424C4⋊C4—48no—
425C4⋊C4—42no—
426C4⋊C4—41no—
427C4⋊C4—42no—
428C4⋊C4—41no—
429C4⋊C4—42no—
430C4⋊C4—42no—
431C4⋊C4—48no—
432C4⋊C4—42no—
433C4⋊C4—42no—
434C4⋊C4—42no—
435C4⋊C4—42no—
436C22⋊C4—48no—
437C22⋊C4—88no—
438C22⋊C4—88no—
439C22⋊C4—84no—
440C42—24no—
441C42—24no—
442C42—48no—
443C42—42no—
444C42—42no—
445C23—22no—
446C23—24no—
447C23—24no—
448C23—24no—
449C23—24no—
450C23—22no—
451C23—22no—
452C23—22no—
453C23—22no—
454C23—24no—
455C23—24no—
456C23—24no—
457C23—24no—
458C23—24no—
459C23—22no—
460C23—22no—
461C23—24no—
462C23—22no—
463C23—42no—
464C23—44no—
465C23—44no—
466C23—42no—
467C23—42no—
468C23—82no—
469C23—82no—
470C23—82no—
471C23—82no—
472C23—81no—
473C23—81no—
474C23—81no—
475C23—81no—
476C23—81no—
477C23—81no—
478C23—82no—
479C23—82no—
480C23—82no—
481C23—82no—
482C23—81no—
483C23—81no—
484C23—82no—
485C23—82no—
486C23#39014yes—
487C23#44218yes—
488C23#46918yes—
489C23#49614yes—
490C23#49614yes—
491C23C2×C4×D411yescharacteristic
492C23C2×#12111yescharacteristic
493C23C2×#12311yescharacteristic
494C23#52511yescharacteristic
495C23C2×#12711yescharacteristic
496C23#53411yescharacteristic
497C23#53611yescharacteristic
498C23C2×AGL(1,ℤ/8)12yes—
499C23C2×AGL(1,ℤ/8)12yes—
500C23C2×#13712yes—
501C23#57512yes—
502D4—44no—
503D4—44no—
504D4—816no—
505D4—82no—
506D4—82no—
507C2×C4—22no—
508C2×C4—22no—
509C2×C4—22no—
510C2×C4—22no—
511C2×C4—22no—
512C2×C4—22no—
513C2×C4—28no—
514C2×C4—28no—
515C2×C4—28no—
516C2×C4—28no—
517C2×C4—24no—
518C2×C4—24no—
519C2×C4—24no—
520C2×C4—24no—
521C2×C4—24no—
522C2×C4—24no—
523C2×C4—432no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—48no—
533C2×C4—48no—
534C2×C4—44no—
535C2×C4—44no—
536C2×C4—42no—
537C2×C4—44no—
538C2×C4—816no—
539C2×C4—82no—
540C2×C4—82no—
541C2×C4—82no—
542C2×C4—81no—
543C2×C4—81no—
544C2×C4—82no—
545C2×C4—84no—
546C2×C4—816no—
547C2×C4—84no—
548C2×C4—82no—
549C2×C4—82no—
550C2×C4—82no—
551C2×C4—81no—
552C2×C4—81no—
553C2×C4—81no—
554C2×C4—81no—
555C2×C4—82no—
556C2×C4—82no—
557C2×C4—81no—
558C2×C4—81no—
559C2×C4#39014yes—
560C2×C4C22×C22⋊C411yescharacteristic
561C2×C4C2×C4×D411yescharacteristic
562C2×C4C4×#4112yes—
563C2×C4#52012yes—
564C2×C4C2×#12111yescharacteristic
565C2×C4C2×#12411yescharacteristic
566C2×C4#52912yes—
567C2×C4#53912yes—
568C2×C4#54612yes—
569C2×C4#54612yes—
570C2×C4#54612yes—
571C2×C4#54712yes—
572C2×C4#54712yes—
573C2×C4#54712yes—
574C2×C4#55112yes—
575C2×C4#55212yes—
576C8—28no—
577C8—44no—
578C22—24no—
579C22—24no—
580C22—28no—
581C22—28no—
582C22—28no—
583C22—24no—
584C22—24no—
585C22—24no—
586C22—28no—
587C22—24no—
588C22—84no—
589C22—84no—
590C22—82no—
591C22—82no—
592C22—82no—
593C22—161no—
594C22—161no—
595C22—162no—
596C22—161no—
597C22—161no—
598C22—162no—
599C22#197218yes—
600C22C2×#39011yescharacteristic
601C22C4×#12311yescharacteristic
602C22#232211yescharacteristic
603C22#241211yescharacteristic
604C22#241311yescharacteristic
605C22#250111yescharacteristic
606C22#253911yescharacteristic
607C22C2×#44212yes—
608C22C4×#13712yes—
609C22C2×#46912yes—
610C22#305612yes—
611C22#309212yes—
612C22#309412yes—
613C22#311212yes—
614C22#311412yes—
615C22C2×#49611yescharacteristic
616C22C2×#49611yescharacteristic
617C22#315211yescharacteristic
618C22#315311yescharacteristic
619C22#319312yes—
620C22#319412yes—
621C22#323412yes—
622C22#323812yes—
623C4—24no—
624C4—44no—
625C4—44no—
626C4—84no—
627C4—84no—
628C4—82no—
629C4—161no—
630C4—161no—
631C4#233912yes—
632C4#235912yes—
633C4#242512yes—
634C4#248812yes—
635C2—28no—
636C2—162no—
637C2#1399611yescharacteristic
638C2C2×#197211yescharacteristic
639C2#2123711yescharacteristic
640C2#2157812yes—
641C2#2208112yes—
642C2#2455812yes—
643C2#2460812yes—
644C2#2791411yescharacteristic
645C2#2791811yescharacteristic
646C2#2831211yescharacteristic
647C2#2831411yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC22×#121C22
LCSG ▷ C22×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. C2×#1853C2semidirect
4. C2×#1823C2semidirect
5. C2×#1749C2semidirect
7. C2×C4⋊#95C2semidirect
8. #11377C2semidirect
9. #11121C2semidirect
10. #10927C2semidirect
11. #10863C2semidirect
18. C2×#480C4semidirect
91. C2×#107C2×C4semidirect
100. C2×#102C2×C4semidirect
264. C22⋊C8C22⋊C4semidirect

Maximal subgroups

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

subgroupindex
C22×#4802
C2×#18742
C2×#18532
C2×#18232
C2×#17492
#146982
C2×C4⋊#952
#113772
#111212
#109272
#108632