Finite Group Databasebeta

← #365713

Subgroups — #365713

Order 512.

1622 conjugacy classes of subgroups.

The subgroup lattice has 653 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2367C211yesmaximal, characteristic
2C2×#1983C211yesmaximal, characteristic
3C2×#1959C211yesmaximal, characteristic
4C2×#1947C211yesmaximal, characteristic, complemented
5#13911C211yesmaximal, characteristic, complemented
6#11124C212yesmaximal, complemented
7#11123C212yesmaximal
8#10835C211yesmaximal, characteristic, complemented
9#10789C212yesmaximal, complemented
10#10788C212yesmaximal
11#10703C211yesmaximal, characteristic, complemented
12C22×#124C2211yescharacteristic
13C22×C4×Q8C2211yescharacteristic
14C22×C4×D4C2211yescharacteristic
15C22×#107C2212yes—
16C22×C4⋊C8C2211yescharacteristic
17C22×#103C2212yes—
18C22×#103C2212yes—
19C22×#102C2212yes—
20C22×#102C2212yes—
21C22×#97C2211yescharacteristic
22#2367—24no—
23#2367—24no—
24C2×#391C2211yescharacteristic
25C2×#390C2211yescharacteristic
26C2×#388—21no—
27C2×#388—21no—
28C2×#385—21no—
29C2×#384C2211yescharacteristic
30C2×#382—21no—
31C2×C4×C4⋊C4C2211yescharacteristic
32C2×C4×C4⋊C4C2211yescharacteristic
33C2×C4×C4⋊C4C2211yescharacteristic
34C2×C4×C22⋊C4—21no—
35#1983C418yescomplemented
36#1959—24no—
37#1947—24no—
38C2×#340C2212yes—
39C2×#340C2212yes—
40C2×#340C2212yes—
41C2×#336C2212yes—
42C2×#336C2212yes—
43#1442—21no—
44#1442—21no—
45C4⋊#95—21no—
46C4⋊#95C2212yes—
47C4⋊#95C2212yes—
48C23×Q8C2311yescharacteristic
49C23×D4C2311yescharacteristic
50C24×C4—21no—
51C23×C8C2312yes—
52C23×C8C2312yes—
53C2×#124—22no—
54C2×#124—22no—
55C2×#124—22no—
56C2×#124C2×C414yes—
57C2×#124D414yes—
58C2×#124Q814yes—
59C2×C4×Q8—22no—
60C2×C4×Q8—22no—
61C2×C4×Q8—24no—
62C2×C4×Q8—42no—
63C2×C4×D4—22no—
64C2×C4×D4—22no—
65C2×C4×D4—24no—
66C2×C4×D4—42no—
67C22×C4⋊C4—21no—
68C22×C4⋊C4—21no—
69C22×C4⋊C4—21no—
70C22×C4⋊C4D411yescharacteristic
71C22×C4⋊C4D411yescharacteristic
72C22×C4⋊C4C2312yes—
73C22×C4⋊C4C2312yes—
74C22×C4⋊C4C2311yescharacteristic
75C22×C4⋊C4C2311yescharacteristic
76C22×C22⋊C4—21no—
77C22×C22⋊C4—21no—
78C22×C22⋊C4—21no—
79C22×C22⋊C4—21no—
80C22×C22⋊C4—21no—
81C22×C42—21no—
82C22×C42—21no—
83C22×C42—21no—
84C22×C42C2311yescharacteristic
85C22×C42C2311yescharacteristic
86C22×C42C2311yescharacteristic
87C2×#107—24no—
88C2×#107—42no—
89C2×#107—42no—
90C2×C4⋊C8—22no—
91C2×C4⋊C8—22no—
92C2×C4⋊C8C2×C414yescomplemented
93C2×#103—24no—
94C2×#103—24no—
95C2×#103—42no—
96C2×#103—42no—
97C2×#103—42no—
98C2×#103C2×C418yescomplemented
99C2×#102—24no—
100C2×#102—24no—
101C2×#102—42no—
102C2×#102—42no—
103C2×#102—42no—
104C2×#102C2×C418yes—
105C2×#97—22no—
106C2×#97—22no—
107C2×#97—22no—
108#391—22no—
109#391—22no—
110#390—22no—
111#390—22no—
112#388—28no—
113#388—28no—
114#385—28no—
115#384—22no—
116#384—22no—
117#382—44no—
118C4×C4⋊C4—22no—
119C4×C4⋊C4—22no—
120C4×C4⋊C4—22no—
121C4×C4⋊C4—22no—
122C4×C4⋊C4C2×C414yescomplemented
123C4×C4⋊C4D414yes—
124C4×C22⋊C4—28no—
125C2×#95—21no—
126C2×#95—21no—
127C2×#95—21no—
128C2×#95—21no—
129C2×#95—22no—
130C2×#95—22no—
131C2×#95—22no—
132C2×#95—22no—
133C2×#95D411yescharacteristic
134C2×#95D411yescharacteristic
135#340—24no—
136#340—24no—
137#340—24no—
138#336—42no—
139#336—42no—
140C25—21no—
141C22×Q8—21no—
142C22×Q8—21no—
143C22×Q8—21no—
144C22×Q8—22no—
145C22×Q8—22no—
146C22×Q8—22no—
147C22×Q8—22no—
148C22×Q8C22×C412yes—
149C22×Q8C2×D412yes—
150C22×Q8C2×Q812yes—
151C22×D4—21no—
152C22×D4—22no—
153C22×D4—22no—
154C22×D4—21no—
155C22×D4—21no—
156C22×D4—22no—
157C22×D4—22no—
158C22×D4C22×C412yes—
159C22×D4C2×D412yes—
160C22×D4C2×Q812yes—
161C23×C4—21no—
162C23×C4—21no—
163C23×C4—21no—
164C23×C4—22no—
165C23×C4—24no—
166C23×C4—22no—
167C23×C4—44no—
168C23×C4—42no—
169C23×C4—42no—
170C23×C4—82no—
171C23×C4C2×D411yescharacteristic
172C23×C4C2×D411yescharacteristic
173C23×C4C2×D411yescharacteristic
174C23×C4C2×D411yescharacteristic
175C23×C4C2×D411yescharacteristic
176C23×C4C2×D411yescharacteristic
177C23×C4C2411yescharacteristic
178C22×C8—22no—
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×C8—22no—
190C22×C8C22×C414yes—
191C22×C8#4114yes—
192#124—28no—
193#124—28no—
194#124—28no—
195#124C4⋊C4116yes—
196C4×Q8—48no—
197C4×Q8—44no—
198C4×Q8—44no—
199C4×D4—48no—
200C4×D4—44no—
201C4×D4—44no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—24no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—41no—
224C2×C4⋊C4—41no—
225C2×C4⋊C4—44no—
226C2×C4⋊C4—42no—
227C2×C4⋊C4—42no—
228C2×C4⋊C4—42no—
229C2×C4⋊C4—42no—
230C2×C4⋊C4—82no—
231C2×C4⋊C4C22×C414yes—
232C2×C4⋊C4C2×D412yes—
233C2×C4⋊C4C2×D412yes—
234C2×C4⋊C4#4114yes—
235C2×C22⋊C4—24no—
236C2×C22⋊C4—24no—
237C2×C22⋊C4—24no—
238C2×C22⋊C4—24no—
239C2×C22⋊C4—24no—
240C2×C22⋊C4—24no—
241C2×C22⋊C4—24no—
242C2×C22⋊C4—24no—
243C2×C22⋊C4—24no—
244C2×C22⋊C4—44no—
245C2×C22⋊C4—42no—
246C2×C22⋊C4—42no—
247C2×C22⋊C4—82no—
248C2×C42—24no—
249C2×C42—24no—
250C2×C42—21no—
251C2×C42—21no—
252C2×C42—21no—
253C2×C42—21no—
254C2×C42—22no—
255C2×C42—22no—
256C2×C42—22no—
257C2×C42—22no—
258C2×C42—24no—
259C2×C42—44no—
260C2×C42—42no—
261C2×C42—42no—
262C2×C42—82no—
263C2×C42C22×C412yes—
264C2×C42C2×D412yes—
265C2×C42C2×D412yes—
266C2×C42C2×Q812yes—
267C2×C42#4112yes—
268C2×C42#4112yes—
269#107—48no—
270C4⋊C8—28no—
271#103—48no—
272#103—48no—
273#102—48no—
274#102—48no—
275#97—44no—
276#95—24no—
277#95—24no—
278#95—24no—
279#95—24no—
280#95—22no—
281#95—22no—
282#95—44no—
283#95—44no—
284#95—82no—
285#95—82no—
286C24—22no—
287C24—22no—
288C24—22no—
289C24—42no—
290C24—42no—
291C24—41no—
292C24—41no—
293C24—41no—
294C24—41no—
295C24—42no—
296C24—42no—
297C24C22×D411yescharacteristic
298C2×Q8—22no—
299C2×Q8—22no—
300C2×Q8—22no—
301C2×Q8—24no—
302C2×Q8—24no—
303C2×Q8—24no—
304C2×Q8—24no—
305C2×Q8—24no—
306C2×Q8—24no—
307C2×Q8—41no—
308C2×Q8—42no—
309C2×Q8—41no—
310C2×Q8—41no—
311C2×Q8—41no—
312C2×Q8—42no—
313C2×Q8—42no—
314C2×Q8—42no—
315C2×Q8C2×C4⋊C414yes—
316C2×D4—22no—
317C2×D4—24no—
318C2×D4—24no—
319C2×D4—22no—
320C2×D4—22no—
321C2×D4—24no—
322C2×D4—24no—
323C2×D4—24no—
324C2×D4—24no—
325C2×D4—41no—
326C2×D4—42no—
327C2×D4—41no—
328C2×D4—41no—
329C2×D4—41no—
330C2×D4—42no—
331C2×D4—42no—
332C2×D4—42no—
333C2×D4C2×C4⋊C414yes—
334C22×C4—21no—
335C22×C4—21no—
336C22×C4—21no—
337C22×C4—21no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—21no—
342C22×C4—22no—
343C22×C4—21no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—24no—
349C22×C4—21no—
350C22×C4—21no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—24no—
354C22×C4—21no—
355C22×C4—22no—
356C22×C4—21no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—21no—
360C22×C4—21no—
361C22×C4—21no—
362C22×C4—21no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—48no—
366C22×C4—48no—
367C22×C4—42no—
368C22×C4—42no—
369C22×C4—41no—
370C22×C4—41no—
371C22×C4—42no—
372C22×C4—42no—
373C22×C4—42no—
374C22×C4—42no—
375C22×C4—42no—
376C22×C4—41no—
377C22×C4—42no—
378C22×C4—41no—
379C22×C4—42no—
380C22×C4—42no—
381C22×C4—42no—
382C22×C4—42no—
383C22×C4—42no—
384C22×C4—42no—
385C22×C4—42no—
386C22×C4—42no—
387C22×C4—42no—
388C22×C4—42no—
389C22×C4—88no—
390C22×C4—84no—
391C22×C4—88no—
392C22×C4C4×D412yes—
393C22×C4C4×D412yes—
394C22×C4#12012yes—
395C22×C4#12012yes—
396C22×C4C22⋊Q812yes—
397C22×C4C22⋊Q812yes—
398C22×C4AGL(1,ℤ/8)12yes—
399C22×C4#13712yes—
400C22×C4C23×C411yescharacteristic
401C22×C4C22×D411yescharacteristic
402C22×C4C22×D411yescharacteristic
403C22×C4C22×Q811yescharacteristic
404C22×C4C2×#4111yescharacteristic
405C22×C4C2×#4112yes—
406C22×C4C2×#4112yes—
407C22×C4C2×#4111yescharacteristic
408C22×C4H(5,𝔽2)12yes—
409C22×C4H(5,𝔽2)12yes—
410C2×C8—24no—
411C2×C8—24no—
412C2×C8—24no—
413C2×C8—24no—
414C2×C8—24no—
415C2×C8—24no—
416C2×C8—42no—
417C2×C8—42no—
418C2×C8—42no—
419C2×C8—42no—
420C2×C8—42no—
421C2×C8—42no—
422C2×C8—42no—
423C2×C8—42no—
424C4⋊C4—22no—
425C4⋊C4—24no—
426C4⋊C4—24no—
427C4⋊C4—24no—
428C4⋊C4—22no—
429C4⋊C4—22no—
430C4⋊C4—44no—
431C4⋊C4—44no—
432C4⋊C4—48no—
433C4⋊C4—42no—
434C4⋊C4—42no—
435C4⋊C4—42no—
436C4⋊C4—42no—
437C4⋊C4—42no—
438C4⋊C4—42no—
439C4⋊C4—42no—
440C4⋊C4—84no—
441C4⋊C4—84no—
442C4⋊C4#12414yes—
443C22⋊C4—216no—
444C22⋊C4—216no—
445C22⋊C4—48no—
446C22⋊C4—48no—
447C22⋊C4—84no—
448C42—48no—
449C42—48no—
450C42—41no—
451C42—41no—
452C42—41no—
453C42—41no—
454C42—42no—
455C42—42no—
456C42—84no—
457C42C2×C4⋊C414yes—
458C42C4×D414yes—
459C42C22⋊Q814yes—
460C42#12314yes—
461C23—24no—
462C23—42no—
463C23—42no—
464C23—44no—
465C23—44no—
466C23—42no—
467C23—42no—
468C23—42no—
469C23—42no—
470C23—44no—
471C23—44no—
472C23—42no—
473C23—42no—
474C23—42no—
475C23—42no—
476C23—82no—
477C23—82no—
478C23—82no—
479C23—82no—
480C23—82no—
481C23—82no—
482C23—82no—
483C23—82no—
484C23C2×C4×D411yescharacteristic
485C23C2×#12011yescharacteristic
486C23C2×C22⋊Q811yescharacteristic
487C23#54612yes—
488C23#55012yes—
489C23C2×AGL(1,ℤ/8)11yescharacteristic
490C23C2×#13711yescharacteristic
491C23#57511yescharacteristic
492C23#57511yescharacteristic
493C23#57612yes—
494C23#57812yes—
495Q8—28no—
496Q8—42no—
497Q8—42no—
498Q8—42no—
499Q8—44no—
500Q8—42no—
501D4—28no—
502D4—42no—
503D4—42no—
504D4—42no—
505D4—44no—
506D4—42no—
507C2×C4—22no—
508C2×C4—22no—
509C2×C4—24no—
510C2×C4—22no—
511C2×C4—22no—
512C2×C4—24no—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—24no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—24no—
521C2×C4—24no—
522C2×C4—24no—
523C2×C4—44no—
524C2×C4—42no—
525C2×C4—41no—
526C2×C4—42no—
527C2×C4—41no—
528C2×C4—44no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—42no—
534C2×C4—44no—
535C2×C4—42no—
536C2×C4—44no—
537C2×C4—44no—
538C2×C4—42no—
539C2×C4—44no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—44no—
543C2×C4—42no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—42no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—816no—
554C2×C4—82no—
555C2×C4—82no—
556C2×C4—82no—
557C2×C4—82no—
558C2×C4—82no—
559C2×C4—82no—
560C2×C4—82no—
561C2×C4—82no—
562C2×C4—82no—
563C2×C4—82no—
564C2×C4—82no—
565C2×C4—82no—
566C2×C4—82no—
567C2×C4#38514yes—
568C2×C4#38514yes—
569C2×C4C22×C4⋊C411yescharacteristic
570C2×C4C2×C4×D411yescharacteristic
571C2×C4C4×#4112yes—
572C2×C4#51812yes—
573C2×C4C2×C22⋊Q811yescharacteristic
574C2×C4C2×#12311yescharacteristic
575C2×C4C2×#12411yescharacteristic
576C2×C4#52912yes—
577C2×C4#53211yescharacteristic
578C2×C4#53511yescharacteristic
579C2×C4#53511yescharacteristic
580C2×C4#54012yes—
581C2×C4#54612yes—
582C2×C4#54612yes—
583C2×C4#55312yes—
584C2×C4#55412yes—
585C2×C4#55412yes—
586C2×C4#55512yes—
587C8—44no—
588C8—44no—
589C22—44no—
590C22—44no—
591C22—44no—
592C22—82no—
593C22—82no—
594C22—82no—
595C22—82no—
596C22—84no—
597C22—84no—
598C22—84no—
599C22—84no—
600C22C2×#38511yescharacteristic
601C22#236312yes—
602C22#243111yescharacteristic
603C22#243511yescharacteristic
604C22#245412yes—
605C22C4×AGL(1,ℤ/8)11yescharacteristic
606C22C4×#13711yescharacteristic
607C22#295112yes—
608C22#300611yescharacteristic
609C22#300711yescharacteristic
610C22#300811yescharacteristic
611C22#301111yescharacteristic
612C22#307911yescharacteristic
613C22#308011yescharacteristic
614C22#308312yes—
615C22#330812yes—
616C22#331312yes—
617C22#331412yes—
618C22#331912yes—
619C22#339812yes—
620C22#339912yes—
621C22#340212yes—
622C22#340312yes—
623C4—24no—
624C4—24no—
625C4—42no—
626C4—42no—
627C4—44no—
628C4—44no—
629C4—81no—
630C4—81no—
631C4—82no—
632C4—84no—
633C4—162no—
634C4C4×#12411yescharacteristic
635C4#232811yescharacteristic
636C4#234912yes—
637C4#242012yes—
638C4#256311yescharacteristic
639C4#256611yescharacteristic
640C2—84no—
641C2#1397911yescharacteristic
642C2#2087511yescharacteristic
643C2#2088111yescharacteristic
644C2#2310512yes—
645C2#2311112yes—
646C2#2564211yescharacteristic
647C2#2564311yescharacteristic
648C2#2589611yescharacteristic
649C2#2589711yescharacteristic
650C2#2654212yes—
651C2#2654512yes—
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
4. C2×#1947C2semidirect
5. #13911C2semidirect
6. #11124C2semidirect
8. #10835C2semidirect
9. #10789C2semidirect
11. #10703C2semidirect
35. #1983C4semidirect
92. C2×C4⋊C8C2×C4semidirect
98. C2×#103C2×C4semidirect
122. C4×C4⋊C4C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#23672
C2×#19832
C2×#19592
C2×#19472
#139112
#111242
#111232
#108352
#107892
#107882
#107032