Finite Group Databasebeta

← #368442

Subgroups — #368442

Order 512.

1810 conjugacy classes of subgroups.

The subgroup lattice has 660 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#428C211yesmaximal, characteristic, complemented
2C2×#1925C212yesmaximal, complemented
3C2×#1895C211yesmaximal, characteristic
4C2×#1875C211yesmaximal, characteristic
5#14137C211yesmaximal, characteristic
6#13801C211yesmaximal, characteristic, complemented
7#11257C212yesmaximal, complemented
8#11121C212yesmaximal, complemented
9#10907C212yesmaximal, complemented
10#10730C211yesmaximal, characteristic, complemented
11#10715C211yesmaximal, characteristic, complemented
12C23×M16C2211yescharacteristic
13C22×#121C2211yescharacteristic
14C22×#117C2211yescharacteristic
15C23×C4⋊C4C2211yescharacteristic
16C2×#428—28no—
17C2×#428—24no—
18C22×#107C2212yes—
19C22×#106C2212yes—
20C22×#102C2211yescharacteristic
21C22×#102C2212yes—
22C22×#102C2211yescharacteristic
23C22×C22⋊C8C2211yescharacteristic
24C2×#390C2212yes—
25C2×#386—22no—
26C2×#380C2211yescharacteristic
27C2×C4×C4⋊C4C2212yes—
28C2×C4×C22⋊C4C2211yescharacteristic
29#1925—28no—
30#1895C418yescomplemented
31#1875—24no—
32C2×#340C2212yes—
33C2×#340C2211yescharacteristic
34C2×#340C2211yescharacteristic
35C2×#340C2212yes—
36C2×#340C2212yes—
37C2×#336C2212yes—
38C2×#336C2212yes—
39#1439—21no—
40#1439—21no—
41#1437—22no—
42C4⋊#95C2212yes—
43C4⋊#95C2212yes—
44#1434—21no—
45C4×#95—21no—
46C23×D4—21no—
47C23×D4C2311yescharacteristic
48C24×C4—21no—
49C24×C4C2311yescharacteristic
50C22×M16—22no—
51C22×M16—22no—
52C22×M16—24no—
53C22×M16—24no—
54C22×M16C2×C414yes—
55C23×C8C2311yescharacteristic
56C23×C8C2311yescharacteristic
57C23×C8C2312yes—
58C2×#121—22no—
59C2×#121—22no—
60C2×#121—22no—
61C2×#121—22no—
62C2×#121—22no—
63C2×#121C2×C414yes—
64C2×#117—22no—
65C2×#117—24no—
66C2×#117—42no—
67C2×#117D414yes—
68C22×C4⋊C4—22no—
69C22×C4⋊C4—24no—
70C22×C4⋊C4—22no—
71C22×C4⋊C4—21no—
72C22×C4⋊C4—42no—
73C22×C4⋊C4C2312yes—
74C22×C4⋊C4C2311yescharacteristic
75C22×C4⋊C4C2311yescharacteristic
76C22×C4⋊C4C2312yes—
77C22×C4⋊C4C2311yescharacteristic
78C22×C22⋊C4—21no—
79C22×C22⋊C4—21no—
80C22×C22⋊C4—21no—
81C22×C22⋊C4—22no—
82C22×C22⋊C4—22no—
83C22×C22⋊C4D412yes—
84C22×C42—21no—
85C22×C42—22no—
86C22×C42C2312yes—
87#428—232no—
88C2×#107—24no—
89C2×#107—24no—
90C2×#107D418yescomplemented
91C2×#106—24no—
92C2×#106—24no—
93C2×#106—24no—
94C2×#102—24no—
95C2×#102—24no—
96C2×#102—22no—
97C2×#102—22no—
98C2×#102—24no—
99C2×#102—42no—
100C2×#102C2×C418yescomplemented
101C2×C22⋊C8—22no—
102C2×C22⋊C8—22no—
103C2×C22⋊C8—22no—
104#390—24no—
105#390—42no—
106#386—48no—
107#380—22no—
108#380C2×C414yes—
109C4×C4⋊C4—24no—
110C4×C4⋊C4—42no—
111C4×C22⋊C4—22no—
112C4×C22⋊C4—22no—
113C2×#95—21no—
114C2×#95—22no—
115C2×#95—22no—
116C2×#95—22no—
117C2×#95—22no—
118C2×#95—22no—
119C2×#95D411yescharacteristic
120C2×#95D411yescharacteristic
121#340—22no—
122#340—22no—
123#340—24no—
124#340—42no—
125#340C2×C418yescomplemented
126#336—24no—
127#336—24no—
128C25—21no—
129C25C2×D411yescharacteristic
130C22×D4—24no—
131C22×D4—21no—
132C22×D4—22no—
133C22×D4—21no—
134C22×D4—21no—
135C22×D4—21no—
136C22×D4—22no—
137C22×D4—21no—
138C22×D4—22no—
139C22×D4—22no—
140C22×D4—44no—
141C22×D4—44no—
142C22×D4—42no—
143C22×D4—42no—
144C22×D4C22×C412yes—
145C23×C4—22no—
146C23×C4—22no—
147C23×C4—22no—
148C23×C4—22no—
149C23×C4—21no—
150C23×C4—22no—
151C23×C4—21no—
152C23×C4—22no—
153C23×C4—21no—
154C23×C4—44no—
155C23×C4—42no—
156C23×C4—44no—
157C23×C4—81no—
158C23×C4—81no—
159C23×C4C22×C412yes—
160C23×C4C2×D412yes—
161C23×C4C2×D412yes—
162C23×C4C2×D411yescharacteristic
163C23×C4C2×D412yes—
164C23×C4#4112yes—
165C23×C4#4112yes—
166C23×C4#4112yes—
167C23×C4#4112yes—
168C23×C4C2411yescharacteristic
169C2×M16—216no—
170C2×M16—28no—
171C2×M16—28no—
172C2×M16—28no—
173C2×M16—216no—
174C22×C8—22no—
175C22×C8—22no—
176C22×C8—22no—
177C22×C8—22no—
178C22×C8—22no—
179C22×C8—21no—
180C22×C8—21no—
181C22×C8—22no—
182C22×C8—21no—
183C22×C8—21no—
184C22×C8—21no—
185C22×C8—21no—
186C22×C8—22no—
187C22×C8—22no—
188C22×C8C22×C414yes—
189C22×C8C2×D414yes—
190C22×C8#4114yes—
191#121—28no—
192#121—28no—
193#121—44no—
194#121—44no—
195#117—28no—
196#117—44no—
197#117—48no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—28no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—21no—
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—22no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—48no—
220C2×C4⋊C4—44no—
221C2×C4⋊C4—44no—
222C2×C4⋊C4—42no—
223C2×C4⋊C4C22×C412yes—
224C2×C4⋊C4C22×C414yes—
225C2×C4⋊C4C2×D414yes—
226C2×C4⋊C4C2×D412yes—
227C2×C4⋊C4C2×D414yes—
228C2×C4⋊C4C2×D412yes—
229C2×C22⋊C4—24no—
230C2×C22⋊C4—24no—
231C2×C22⋊C4—24no—
232C2×C22⋊C4—24no—
233C2×C22⋊C4—24no—
234C2×C22⋊C4—42no—
235C2×C22⋊C4—44no—
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—82no—
243C2×C22⋊C4—82no—
244C2×C22⋊C4—81no—
245C2×C22⋊C4—81no—
246C2×C42—24no—
247C2×C42—22no—
248C2×C42—22no—
249C2×C42—22no—
250C2×C42—22no—
251C2×C42—22no—
252C2×C42—22no—
253C2×C42—44no—
254C2×C42—44no—
255C2×C42—44no—
256C2×C42—82no—
257#107—216no—
258#106—216no—
259#102—216no—
260#102—44no—
261#102—44no—
262C22⋊C8—44no—
263#95—24no—
264#95—22no—
265#95—22no—
266#95—44no—
267#95—44no—
268#95—44no—
269#95—44no—
270#95—82no—
271C24—22no—
272C24—22no—
273C24—22no—
274C24—22no—
275C24—21no—
276C24—21no—
277C24—21no—
278C24—21no—
279C24—22no—
280C24—41no—
281C24—41no—
282C24—42no—
283C24—42no—
284C24—42no—
285C24—42no—
286C24—41no—
287C24—41no—
288C24—41no—
289C24—41no—
290C24C4×D412yes—
291C24#12112yes—
292C24#12312yes—
293C24C22×D411yescharacteristic
294C2×D4—22no—
295C2×D4—24no—
296C2×D4—22no—
297C2×D4—24no—
298C2×D4—416no—
299C2×D4—48no—
300C2×D4—41no—
301C2×D4—41no—
302C2×D4—42no—
303C2×D4—41no—
304C2×D4—41no—
305C2×D4—41no—
306C2×D4—41no—
307C2×D4—42no—
308C2×D4—42no—
309C2×D4—42no—
310C2×D4—42no—
311C2×D4—42no—
312C2×D4—42no—
313C2×D4—42no—
314C2×D4—88no—
315C2×D4—84no—
316C2×D4—84no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—24no—
320C22×C4—24no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—24no—
324C22×C4—22no—
325C22×C4—22no—
326C22×C4—24no—
327C22×C4—24no—
328C22×C4—24no—
329C22×C4—24no—
330C22×C4—24no—
331C22×C4—24no—
332C22×C4—24no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—24no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—24no—
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—22no—
351C22×C4—22no—
352C22×C4—42no—
353C22×C4—41no—
354C22×C4—41no—
355C22×C4—48no—
356C22×C4—41no—
357C22×C4—42no—
358C22×C4—41no—
359C22×C4—42no—
360C22×C4—42no—
361C22×C4—42no—
362C22×C4—42no—
363C22×C4—42no—
364C22×C4—42no—
365C22×C4—42no—
366C22×C4—42no—
367C22×C4—42no—
368C22×C4—42no—
369C22×C4—42no—
370C22×C4—42no—
371C22×C4—42no—
372C22×C4—41no—
373C22×C4—42no—
374C22×C4—42no—
375C22×C4—41no—
376C22×C4—88no—
377C22×C4—88no—
378C22×C4—84no—
379C22×C4—84no—
380C22×C4#11714yes—
381C22×C4C4×D414yes—
382C22×C4C4×D412yes—
383C22×C4#12112yes—
384C22×C4#12312yes—
385C22×C4#12414yes—
386C22×C4#12614yes—
387C22×C4AGL(1,ℤ/8)14yes—
388C22×C4C23×C411yescharacteristic
389C22×C4C22×D411yescharacteristic
390C22×C4C22×D412yes—
391C22×C4C2×#4111yescharacteristic
392C22×C4C2×#4111yescharacteristic
393C22×C4C2×#4111yescharacteristic
394C22×C4C2×#4112yes—
395C22×C4C2×#4111yescharacteristic
396C22×C4H(5,𝔽2)12yes—
397C22×C42−1+412yes—
398M16—232no—
399C2×C8—24no—
400C2×C8—24no—
401C2×C8—24no—
402C2×C8—24no—
403C2×C8—24no—
404C2×C8—24no—
405C2×C8—41no—
406C2×C8—41no—
407C2×C8—41no—
408C2×C8—41no—
409C2×C8—41no—
410C2×C8—41no—
411C2×C8—42no—
412C2×C8—42no—
413C2×C8—42no—
414C2×C8—42no—
415C2×C8C4×D418yes—
416C4⋊C4—22no—
417C4⋊C4—24no—
418C4⋊C4—22no—
419C4⋊C4—24no—
420C4⋊C4—22no—
421C4⋊C4—24no—
422C4⋊C4—22no—
423C4⋊C4—24no—
424C4⋊C4—22no—
425C4⋊C4—24no—
426C4⋊C4—48no—
427C4⋊C4—41no—
428C4⋊C4—41no—
429C4⋊C4—42no—
430C4⋊C4—42no—
431C4⋊C4—42no—
432C4⋊C4—42no—
433C4⋊C4—42no—
434C4⋊C4#12014yes—
435C22⋊C4—48no—
436C22⋊C4—48no—
437C22⋊C4—48no—
438C22⋊C4—88no—
439C22⋊C4—84no—
440C22⋊C4—88no—
441C42—48no—
442C42—42no—
443C42—42no—
444C42—42no—
445C42—42no—
446C42—88no—
447C23—24no—
448C23—22no—
449C23—22no—
450C23—24no—
451C23—22no—
452C23—22no—
453C23—24no—
454C23—22no—
455C23—22no—
456C23—24no—
457C23—22no—
458C23—22no—
459C23—22no—
460C23—22no—
461C23—42no—
462C23—42no—
463C23—42no—
464C23—42no—
465C23—42no—
466C23—42no—
467C23—42no—
468C23—42no—
469C23—44no—
470C23—44no—
471C23—42no—
472C23—42no—
473C23—42no—
474C23—82no—
475C23—82no—
476C23—82no—
477C23—81no—
478C23—82no—
479C23—81no—
480C23—82no—
481C23—81no—
482C23—81no—
483C23—82no—
484C23—82no—
485C23—82no—
486C23—82no—
487C23—82no—
488C23—81no—
489C23—81no—
490C23—81no—
491C23—81no—
492C23#38814yes—
493C23C2×C4×D411yescharacteristic
494C23C2×#12111yescharacteristic
495C23C2×#12311yescharacteristic
496C23D4212yes—
497C23#54712yes—
498C23C2×AGL(1,ℤ/8)12yes—
499C23#57512yes—
500C23#57611yescharacteristic
501C23#57711yescharacteristic
502C23#57711yescharacteristic
503C23#57811yescharacteristic
504D4—42no—
505D4—44no—
506D4—42no—
507D4—816no—
508D4—81no—
509D4—81no—
510D4—82no—
511C2×C4—24no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—24no—
515C2×C4—24no—
516C2×C4—24no—
517C2×C4—28no—
518C2×C4—24no—
519C2×C4—28no—
520C2×C4—24no—
521C2×C4—24no—
522C2×C4—24no—
523C2×C4—24no—
524C2×C4—24no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—44no—
529C2×C4—44no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—44no—
538C2×C4—42no—
539C2×C4—44no—
540C2×C4—44no—
541C2×C4—44no—
542C2×C4—44no—
543C2×C4—42no—
544C2×C4—44no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—44no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—816no—
552C2×C4—82no—
553C2×C4—82no—
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—81no—
563C2×C4—81no—
564C2×C4—82no—
565C2×C4#38814yes—
566C2×C4C2×#11711yescharacteristic
567C2×C4C2×C4×D411yescharacteristic
568C2×C4C2×C4×D412yes—
569C2×C4#52012yes—
570C2×C4C2×#12011yescharacteristic
571C2×C4#52511yescharacteristic
572C2×C4#52512yes—
573C2×C4C2×#12411yescharacteristic
574C2×C4C2×#12611yescharacteristic
575C2×C4#53411yescharacteristic
576C2×C4#53611yescharacteristic
577C2×C4#53912yes—
578C2×C4#54612yes—
579C2×C4#54812yes—
580C2×C4#54812yes—
581C2×C4#55212yes—
582C2×C4#55312yes—
583C2×C4#55312yes—
584C8—28no—
585C8—81no—
586C8—81no—
587C22—24no—
588C22—24no—
589C22—24no—
590C22—44no—
591C22—42no—
592C22—42no—
593C22—44no—
594C22—44no—
595C22—42no—
596C22—42no—
597C22—44no—
598C22—84no—
599C22—82no—
600C22—82no—
601C22—82no—
602C22—84no—
603C22—161no—
604C22—161no—
605C22—162no—
606C22—162no—
607C22—161no—
608C22—161no—
609C22C2×#38811yescharacteristic
610C22#236112yes—
611C22#241011yescharacteristic
612C22#241311yescharacteristic
613C22#244312yes—
614C22C4×AGL(1,ℤ/8)12yes—
615C22#294611yescharacteristic
616C22#294711yescharacteristic
617C22#303811yescharacteristic
618C22#304011yescharacteristic
619C22#304511yescharacteristic
620C22#304611yescharacteristic
621C22#309412yes—
622C22#309811yescharacteristic
623C22#309911yescharacteristic
624C22#326412yes—
625C22#327112yes—
626C22#327212yes—
627C22#327712yes—
628C22#334312yes—
629C22#335012yes—
630C22#335412yes—
631C22#335512yes—
632C4—24no—
633C4—44no—
634C4—44no—
635C4—82no—
636C4—84no—
637C4—82no—
638C4—82no—
639C4—84no—
640C4C4×#12011yescharacteristic
641C4#232211yescharacteristic
642C4#235212yes—
643C4#243611yescharacteristic
644C4#248112yes—
645C4#259111yescharacteristic
646C2—44no—
647C2—162no—
648C2#1401911yescharacteristic
649C2#2106011yescharacteristic
650C2#2106211yescharacteristic
651C2#2303712yes—
652C2#2304012yes—
653C2#2437211yescharacteristic
654C2#2437511yescharacteristic
655C2#2459411yescharacteristic
656C2#2459711yescharacteristic
657C2#2614712yes—
658C2#2615312yes—
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC22×#121C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C22×#428C2semidirect
2. C2×#1925C2semidirect
6. #13801C2semidirect
7. #11257C2semidirect
8. #11121C2semidirect
9. #10907C2semidirect
10. #10730C2semidirect
11. #10715C2semidirect
30. #1895C4semidirect
90. C2×#107D4semidirect
100. C2×#102C2×C4semidirect
125. #340C2×C4semidirect

Maximal subgroups

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

subgroupindex
C22×#4282
C2×#19252
C2×#18952
C2×#18752
#141372
#138012
#112572
#111212
#109072
#107302
#107152