Finite Group Databasebeta

← #368456

Subgroups — #368456

Order 512.

1880 conjugacy classes of subgroups.

The subgroup lattice has 645 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2296C211yesmaximal, characteristic, complemented
2C2×#1936C212yesmaximal, complemented
3C2×#1875C211yesmaximal, characteristic
4C2×#1875C211yesmaximal, characteristic
5C2×#1749C211yesmaximal, characteristic, complemented
6#14698C211yesmaximal, characteristic
7#11292C212yesmaximal, complemented
8#11066C212yesmaximal, complemented
9#10881C212yesmaximal, complemented
10#10731C211yesmaximal, characteristic, complemented
11#10731C211yesmaximal, characteristic, complemented
12C22×#121C2211yescharacteristic
13C22×#117C2211yescharacteristic
14C22×#117C2211yescharacteristic
15C23×C42C2211yescharacteristic
16C22×#102C2212yes—
17C22×#102C2212yes—
18C22×C22⋊C8C2211yescharacteristic
19C22×C22⋊C8C2211yescharacteristic
20C22×#97C2212yes—
21#2296—28no—
22C2×#390C2211yescharacteristic
23C2×#390C2211yescharacteristic
24C2×#389C2212yes—
25C2×#389C2212yes—
26C2×#386—22no—
27C2×#385—22no—
28C2×#380C2211yescharacteristic
29C2×C4×C4⋊C4—22no—
30#1936—28no—
31#1875—24no—
32#1875C418yescomplemented
33#1749—24no—
34C2×#340C2212yes—
35C2×#340C2212yes—
36C2×#340C2212yes—
37C2×#340C2212yes—
38C2×#340C2212yes—
39C2×#340C2212yes—
40C2×#336C2212yes—
41#1437—22no—
42C4⋊#95C2211yescharacteristic
43C4⋊#95C2211yescharacteristic
44C23×D4—21no—
45C23×D4C2311yescharacteristic
46C24×C4D412yes—
47C24×C4C2311yescharacteristic
48C23×C8C2312yes—
49C23×C8C2312yes—
50C2×#121—22no—
51C2×#121—22no—
52C2×#121—22no—
53C2×#121—22no—
54C2×#121—22no—
55C2×#121C2×C414yes—
56C2×#117—22no—
57C2×#117—24no—
58C2×#117—24no—
59C2×#117—24no—
60C2×#117—22no—
61C2×#117—22no—
62C2×#117—42no—
63C2×#117C2×C414yes—
64C22×C4⋊C4—22no—
65C22×C4⋊C4—22no—
66C22×C4⋊C4C2311yescharacteristic
67C22×C4⋊C4C2311yescharacteristic
68C22×C4⋊C4C2311yescharacteristic
69C22×C4⋊C4C2312yes—
70C22×C4⋊C4C2312yes—
71C22×C22⋊C4—21no—
72C22×C22⋊C4—21no—
73C22×C22⋊C4—21no—
74C22×C22⋊C4—21no—
75C22×C22⋊C4—22no—
76C22×C22⋊C4—22no—
77C22×C42—21no—
78C22×C42—22no—
79C22×C42—21no—
80C22×C42—22no—
81C22×C42—24no—
82C22×C42—42no—
83C22×C42D412yes—
84C22×C42C2311yescharacteristic
85C22×C42C2311yescharacteristic
86C2×#102—24no—
87C2×#102—24no—
88C2×#102—24no—
89C2×#102—24no—
90C2×#102—42no—
91C2×#102C2×C418yescomplemented
92C2×C22⋊C8—22no—
93C2×C22⋊C8—22no—
94C2×C22⋊C8—22no—
95C2×C22⋊C8—22no—
96C2×C22⋊C8C2×C414yes—
97C2×C22⋊C8D414yes—
98C2×#97—24no—
99C2×#97—24no—
100C2×#97—42no—
101#390—24no—
102#390—42no—
103#389—24no—
104#389—24no—
105#389—24no—
106#389D418yescomplemented
107#386—48no—
108#385—216no—
109#380—22no—
110#380—22no—
111C4×C4⋊C4—216no—
112C2×#95—22no—
113C2×#95—22no—
114C2×#95—22no—
115C2×#95—22no—
116C2×#95—21no—
117#340—24no—
118#340—24no—
119#340—24no—
120#340—24no—
121#340—42no—
122#340C2×C418yescomplemented
123#336—24no—
124C25—21no—
125C25C2×D411yescharacteristic
126C22×D4—24no—
127C22×D4—21no—
128C22×D4—22no—
129C22×D4—21no—
130C22×D4—21no—
131C22×D4—21no—
132C22×D4—22no—
133C22×D4—21no—
134C22×D4—22no—
135C22×D4—22no—
136C22×D4—44no—
137C22×D4—44no—
138C22×D4—42no—
139C22×D4—42no—
140C22×D4C22×C412yes—
141C23×C4—22no—
142C23×C4—22no—
143C23×C4—22no—
144C23×C4—22no—
145C23×C4—22no—
146C23×C4—24no—
147C23×C4—22no—
148C23×C4—21no—
149C23×C4—24no—
150C23×C4—24no—
151C23×C4—21no—
152C23×C4—21no—
153C23×C4—42no—
154C23×C4—42no—
155C23×C4—42no—
156C23×C4—42no—
157C23×C4C22×C412yes—
158C23×C4C2×D412yes—
159C23×C4C2×D412yes—
160C23×C4C2×D411yescharacteristic
161C23×C4C2×D412yes—
162C23×C4#4112yes—
163C23×C4#4112yes—
164C23×C4#4112yes—
165C23×C4#4112yes—
166C23×C4C2411yescharacteristic
167C22×C8—22no—
168C22×C8—22no—
169C22×C8—22no—
170C22×C8—22no—
171C22×C8—22no—
172C22×C8—22no—
173C22×C8—22no—
174C22×C8—22no—
175C22×C8—22no—
176C22×C8—22no—
177C22×C8—22no—
178C22×C8—22no—
179C22×C8C22×C414yes—
180C22×C8C2×D414yes—
181#121—28no—
182#121—28no—
183#121—44no—
184#121—44no—
185#117—216no—
186#117—216no—
187#117—48no—
188#117—48no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—28no—
201C2×C4⋊C4—28no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—44no—
213C2×C4⋊C4—44no—
214C2×C4⋊C4—44no—
215C2×C4⋊C4—44no—
216C2×C4⋊C4C22×C412yes—
217C2×C4⋊C4C22×C414yes—
218C2×C4⋊C4C2×D414yes—
219C2×C4⋊C4C2×D414yes—
220C2×C4⋊C4#4114yes—
221C2×C22⋊C4—24no—
222C2×C22⋊C4—24no—
223C2×C22⋊C4—24no—
224C2×C22⋊C4—24no—
225C2×C22⋊C4—24no—
226C2×C22⋊C4—42no—
227C2×C22⋊C4—42no—
228C2×C22⋊C4—42no—
229C2×C22⋊C4—42no—
230C2×C22⋊C4—44no—
231C2×C22⋊C4—44no—
232C2×C22⋊C4—44no—
233C2×C22⋊C4—44no—
234C2×C22⋊C4—44no—
235C2×C22⋊C4—42no—
236C2×C22⋊C4—44no—
237C2×C22⋊C4—44no—
238C2×C42—24no—
239C2×C42—24no—
240C2×C42—24no—
241C2×C42—21no—
242C2×C42—21no—
243C2×C42—22no—
244C2×C42—24no—
245C2×C42—24no—
246C2×C42—24no—
247C2×C42—21no—
248C2×C42—22no—
249C2×C42—21no—
250C2×C42—22no—
251C2×C42—48no—
252C2×C42—48no—
253C2×C42—42no—
254C2×C42—42no—
255C2×C42—42no—
256C2×C42C2×D414yes—
257#102—216no—
258#102—48no—
259C22⋊C8—28no—
260C22⋊C8—28no—
261#97—48no—
262#95—28no—
263#95—28no—
264#95—44no—
265#95—42no—
266#95—44no—
267C24—21no—
268C24—21no—
269C24—21no—
270C24—21no—
271C24—22no—
272C24—22no—
273C24—22no—
274C24—22no—
275C24—22no—
276C24—41no—
277C24—41no—
278C24—41no—
279C24—41no—
280C24—41no—
281C24—41no—
282C24—42no—
283C24—42no—
284C24—42no—
285C24—42no—
286C24C4×D412yes—
287C24#12112yes—
288C24#12412yes—
289C24C22×D411yescharacteristic
290C2×D4—22no—
291C2×D4—24no—
292C2×D4—22no—
293C2×D4—24no—
294C2×D4—416no—
295C2×D4—48no—
296C2×D4—41no—
297C2×D4—42no—
298C2×D4—41no—
299C2×D4—42no—
300C2×D4—42no—
301C2×D4—42no—
302C2×D4—42no—
303C2×D4—42no—
304C2×D4—42no—
305C2×D4—42no—
306C2×D4—42no—
307C2×D4—42no—
308C2×D4—84no—
309C2×D4—84no—
310C2×D4—88no—
311C22×C4—21no—
312C22×C4—21no—
313C22×C4—21no—
314C22×C4—21no—
315C22×C4—24no—
316C22×C4—24no—
317C22×C4—24no—
318C22×C4—24no—
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—22no—
329C22×C4—24no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—24no—
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—24no—
344C22×C4—48no—
345C22×C4—48no—
346C22×C4—48no—
347C22×C4—48no—
348C22×C4—48no—
349C22×C4—48no—
350C22×C4—48no—
351C22×C4—42no—
352C22×C4—42no—
353C22×C4—42no—
354C22×C4—42no—
355C22×C4—42no—
356C22×C4—41no—
357C22×C4—41no—
358C22×C4—42no—
359C22×C4—42no—
360C22×C4—42no—
361C22×C4—42no—
362C22×C4—42no—
363C22×C4—42no—
364C22×C4—42no—
365C22×C4—41no—
366C22×C4—41no—
367C22×C4—41no—
368C22×C4—41no—
369C22×C4—41no—
370C22×C4—41no—
371C22×C4—42no—
372C22×C4—42no—
373C22×C4—42no—
374C22×C4—42no—
375C22×C4#11714yes—
376C22×C4C4×D414yes—
377C22×C4C4×D412yes—
378C22×C4#12014yes—
379C22×C4#12112yes—
380C22×C4#12314yes—
381C22×C4#12412yes—
382C22×C4#12614yes—
383C22×C4AGL(1,ℤ/8)14yes—
384C22×C4C23×C411yescharacteristic
385C22×C4C22×D411yescharacteristic
386C22×C4C22×D412yes—
387C22×C4C2×#4111yescharacteristic
388C22×C4C2×#4111yescharacteristic
389C22×C4C2×#4112yes—
390C22×C4C2×#4111yescharacteristic
391C22×C4C2×#4111yescharacteristic
392C22×C4H(5,𝔽2)12yes—
393C22×C42−1+412yes—
394C2×C8—24no—
395C2×C8—24no—
396C2×C8—24no—
397C2×C8—24no—
398C2×C8—24no—
399C2×C8—24no—
400C2×C8—42no—
401C2×C8—42no—
402C2×C8—42no—
403C2×C8—42no—
404C2×C8—42no—
405C2×C8—42no—
406C2×C8—42no—
407C2×C8—42no—
408C4⋊C4—24no—
409C4⋊C4—24no—
410C4⋊C4—24no—
411C4⋊C4—22no—
412C4⋊C4—22no—
413C4⋊C4—24no—
414C4⋊C4—24no—
415C4⋊C4—416no—
416C4⋊C4—416no—
417C4⋊C4—41no—
418C4⋊C4—41no—
419C4⋊C4—42no—
420C4⋊C4—42no—
421C4⋊C4—42no—
422C4⋊C4—42no—
423C4⋊C4—42no—
424C4⋊C4—42no—
425C4⋊C4C4×D418yes—
426C22⋊C4—216no—
427C22⋊C4—48no—
428C22⋊C4—48no—
429C22⋊C4—88no—
430C22⋊C4—88no—
431C22⋊C4—84no—
432C42—216no—
433C42—24no—
434C42—24no—
435C42—48no—
436C42—48no—
437C42—42no—
438C42—42no—
439C23—22no—
440C23—22no—
441C23—22no—
442C23—22no—
443C23—22no—
444C23—22no—
445C23—22no—
446C23—22no—
447C23—24no—
448C23—24no—
449C23—22no—
450C23—24no—
451C23—24no—
452C23—22no—
453C23—42no—
454C23—42no—
455C23—42no—
456C23—42no—
457C23—42no—
458C23—42no—
459C23—42no—
460C23—42no—
461C23—42no—
462C23—44no—
463C23—44no—
464C23—42no—
465C23—42no—
466C23—81no—
467C23—82no—
468C23—82no—
469C23—81no—
470C23—81no—
471C23—81no—
472C23—81no—
473C23—81no—
474C23—81no—
475C23—82no—
476C23—82no—
477C23—81no—
478C23—82no—
479C23—82no—
480C23—82no—
481C23—82no—
482C23—82no—
483C23—82no—
484C23#38814yes—
485C23C2×C4×D411yescharacteristic
486C23C2×#12011yescharacteristic
487C23C2×#12111yescharacteristic
488C23#52511yescharacteristic
489C23C2×#12411yescharacteristic
490C23#53411yescharacteristic
491C23#53611yescharacteristic
492C23C2×AGL(1,ℤ/8)12yes—
493C23#57512yes—
494C23#57512yes—
495C23#57512yes—
496D4—44no—
497D4—44no—
498D4—816no—
499D4—82no—
500D4—82no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—22no—
506C2×C4—22no—
507C2×C4—28no—
508C2×C4—28no—
509C2×C4—28no—
510C2×C4—28no—
511C2×C4—24no—
512C2×C4—24no—
513C2×C4—24no—
514C2×C4—24no—
515C2×C4—24no—
516C2×C4—24no—
517C2×C4—28no—
518C2×C4—432no—
519C2×C4—44no—
520C2×C4—44no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—44no—
526C2×C4—44no—
527C2×C4—44no—
528C2×C4—42no—
529C2×C4—44no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—44no—
534C2×C4—44no—
535C2×C4—44no—
536C2×C4—44no—
537C2×C4—42no—
538C2×C4—81no—
539C2×C4—81no—
540C2×C4—82no—
541C2×C4—82no—
542C2×C4—82no—
543C2×C4—82no—
544C2×C4—82no—
545C2×C4—81no—
546C2×C4—82no—
547C2×C4—81no—
548C2×C4—81no—
549C2×C4—81no—
550C2×C4—81no—
551C2×C4—82no—
552C2×C4—81no—
553C2×C4—82no—
554C2×C4—82no—
555C2×C4#38814yes—
556C2×C4C2×#11711yescharacteristic
557C2×C4C2×C4×D412yes—
558C2×C4C2×C4×D411yescharacteristic
559C2×C4#52012yes—
560C2×C4C2×#12311yescharacteristic
561C2×C4#52512yes—
562C2×C4C2×#12611yescharacteristic
563C2×C4#53912yes—
564C2×C4D4212yes—
565C2×C4#54612yes—
566C2×C4#54712yes—
567C2×C4#54812yes—
568C2×C4#54812yes—
569C2×C4#55212yes—
570C2×C4#55312yes—
571C2×C4#55312yes—
572C8—44no—
573C8—44no—
574C22—24no—
575C22—24no—
576C22—24no—
577C22—44no—
578C22—42no—
579C22—42no—
580C22—44no—
581C22—44no—
582C22—42no—
583C22—42no—
584C22—44no—
585C22—84no—
586C22—82no—
587C22—82no—
588C22—82no—
589C22—84no—
590C22—162no—
591C22—161no—
592C22—161no—
593C22—162no—
594C22—161no—
595C22—161no—
596C22C2×#38811yescharacteristic
597C22C4×#12011yescharacteristic
598C22#232211yescharacteristic
599C22#241011yescharacteristic
600C22#241311yescharacteristic
601C22#243611yescharacteristic
602C22#259111yescharacteristic
603C22#294312yes—
604C22C4×AGL(1,ℤ/8)12yes—
605C22#300512yes—
606C22#300612yes—
607C22#305412yes—
608C22#305512yes—
609C22#311112yes—
610C22#311512yes—
611C22#313511yescharacteristic
612C22#313511yescharacteristic
613C22#313811yescharacteristic
614C22#314011yescharacteristic
615C22#319412yes—
616C22#319712yes—
617C22#323512yes—
618C22#324012yes—
619C4—24no—
620C4—44no—
621C4—44no—
622C4—48no—
623C4—84no—
624C4—82no—
625C4—161no—
626C4—161no—
627C4#235212yes—
628C4#236112yes—
629C4#244312yes—
630C4#248112yes—
631C2—44no—
632C2—162no—
633C2#1401911yescharacteristic
634C2#2108611yescharacteristic
635C2#2109511yescharacteristic
636C2#2145612yes—
637C2#2208612yes—
638C2#2438812yes—
639C2#2461012yes—
640C2#2590911yescharacteristic
641C2#2591111yescharacteristic
642C2#2933711yescharacteristic
643C2#2933911yescharacteristic
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
1. C2×#2296C2semidirect
2. C2×#1936C2semidirect
5. C2×#1749C2semidirect
7. #11292C2semidirect
8. #11066C2semidirect
9. #10881C2semidirect
10. #10731C2semidirect
11. #10731C2semidirect
32. #1875C4semidirect
91. C2×#102C2×C4semidirect
106. #389D4semidirect
122. #340C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#22962
C2×#19362
C2×#18752
C2×#17492
#146982
#112922
#110662
#108812
#107312