Finite Group Databasebeta

← #357435

Subgroups — #357435

Order 512.

5470 conjugacy classes of subgroups.

The subgroup lattice has 698 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#525C211yesmaximal, characteristic
2C22×#417C212yesmaximal, complemented
3C22×#407C211yesmaximal, characteristic, complemented
4C22×#380C211yesmaximal, characteristic, complemented
5C2×#1736C212yesmaximal, complemented
6#10970C214yesmaximal, complemented
7#10969C214yesmaximal, complemented
8C23×#41C2211yescharacteristic
9C25×C4C2211yescharacteristic
10C23×M16C2212yes—
11C2×#525—28no—
12C2×#525—24no—
13C2×#525—28no—
14C2×#525C418yescomplemented
15C22×#123—21no—
16C22×C22⋊Q8C2212yes—
17C22×#121C2212yes—
18C22×#120—21no—
19C22×C4×D4—22no—
20C22×#117C2211yescharacteristic
21C23×C4⋊C4C2212yes—
22C23×C22⋊C4—21no—
23C2×#417—28no—
24C2×#417—28no—
25C2×#417—28no—
26C22×#103C2214yescomplemented
27C22×#102C2214yescomplemented
28C2×#407—24no—
29C2×#407—28no—
30C22×C22⋊C8C2214yes—
31C2×#380—22no—
32C2×#380—28no—
33C2×#380C2214yes—
34C22×#95—21no—
35#1736—28no—
36C2×#340C2218yescomplemented
37C26D411yescharacteristic
38C22×#41—22no—
39C22×#41—24no—
40C22×#41—24no—
41C22×#41—24no—
42C22×#41—24no—
43C22×#41—24no—
44C22×#41C2×C414yes—
45C22×#41D418yes—
46C23×Q8C2311yescharacteristic
47C23×D4—21no—
48C23×D4—22no—
49C23×D4C2311yescharacteristic
50C24×C4—24no—
51C24×C4—24no—
52C24×C4—24no—
53C24×C4—24no—
54C24×C4—21no—
55C24×C4—24no—
56C24×C4—24no—
57C24×C4—22no—
58C24×C4C2×C414yes—
59C24×C4D412yes—
60C24×C4D411yescharacteristic
61C24×C4D414yes—
62C24×C4C2312yes—
63C24×C4C2311yescharacteristic
64C22×M16—24no—
65C22×M16—24no—
66C22×M16—24no—
67C22×M16—24no—
68C22×M16—24no—
69C22×M16—24no—
70C22×M16D418yescomplemented
71C23×C8C2314yes—
72#525—264no—
73#525—232no—
74#525—232no—
75C2×#123—216no—
76C2×#123—28no—
77C2×C22⋊Q8—28no—
78C2×C22⋊Q8—24no—
79C2×C22⋊Q8—28no—
80C2×C22⋊Q8C2×C418yescomplemented
81C2×#121—28no—
82C2×#121—24no—
83C2×#121—28no—
84C2×#121C2×C418yescomplemented
85C2×#120—216no—
86C2×#120—28no—
87C2×C4×D4—232no—
88C2×C4×D4—216no—
89C2×#117—22no—
90C2×#117—24no—
91C2×#117C2×C414yes—
92C2×#117D418yes—
93C22×C4⋊C4—22no—
94C22×C4⋊C4—24no—
95C22×C4⋊C4—24no—
96C22×C4⋊C4—24no—
97C22×C4⋊C4—24no—
98C22×C4⋊C4—22no—
99C22×C4⋊C4—24no—
100C22×C4⋊C4D418yescomplemented
101C22×C4⋊C4C2314yes—
102C22×C4⋊C4C2312yes—
103C22×C22⋊C4—21no—
104C22×C22⋊C4—22no—
105C22×C22⋊C4—22no—
106C22×C22⋊C4—28no—
107C22×C22⋊C4—24no—
108C22×C22⋊C4—48no—
109C22×C42—21no—
110#417—264no—
111C2×#103—28no—
112C2×#103—28no—
113C2×#103—28no—
114C2×#102—28no—
115C2×#102—28no—
116C2×#102—28no—
117#407—232no—
118C2×C22⋊C8—28no—
119C2×C22⋊C8—216no—
120#380—216no—
121#380—216no—
122C2×#95—28no—
123C2×#95—24no—
124#340—216no—
125C25—22no—
126C25—22no—
127C25—24no—
128C25—24no—
129C25—24no—
130C25—24no—
131C25—21no—
132C25—44no—
133C25C22⋊C414yes—
134C25C2×D412yes—
135C25C2×D411yescharacteristic
136C2×#41—28no—
137C2×#41—216no—
138C2×#41—216no—
139C2×#41—216no—
140C2×#41—216no—
141C2×#41—216no—
142C2×#41—216no—
143C2×#41—216no—
144C2×#41—44no—
145C2×#41—48no—
146C2×#41—48no—
147C2×#41—48no—
148C2×#41—44no—
149C2×#41—48no—
150C2×#41—48no—
151C2×#41C22⋊C4116yescomplemented
152C22×Q8—21no—
153C22×Q8—22no—
154C22×Q8—22no—
155C22×Q8—22no—
156C22×Q8—22no—
157C22×Q8—22no—
158C22×Q8C22×C412yes—
159C22×Q8C2×D414yes—
160C22×D4—216no—
161C22×D4—28no—
162C22×D4—24no—
163C22×D4—28no—
164C22×D4—22no—
165C22×D4—22no—
166C22×D4—22no—
167C22×D4—21no—
168C22×D4—22no—
169C22×D4—22no—
170C22×D4—48no—
171C22×D4—42no—
172C22×D4—44no—
173C22×D4—42no—
174C22×D4—44no—
175C22×D4—44no—
176C22×D4C22×C412yes—
177C22×D4C2×D414yes—
178C23×C4—24no—
179C23×C4—22no—
180C23×C4—24no—
181C23×C4—22no—
182C23×C4—22no—
183C23×C4—21no—
184C23×C4—24no—
185C23×C4—28no—
186C23×C4—28no—
187C23×C4—216no—
188C23×C4—216no—
189C23×C4—28no—
190C23×C4—28no—
191C23×C4—216no—
192C23×C4—24no—
193C23×C4—28no—
194C23×C4—28no—
195C23×C4—216no—
196C23×C4—24no—
197C23×C4—28no—
198C23×C4—24no—
199C23×C4—24no—
200C23×C4—24no—
201C23×C4—24no—
202C23×C4—24no—
203C23×C4—24no—
204C23×C4—28no—
205C23×C4—24no—
206C23×C4—48no—
207C23×C4—42no—
208C23×C4—48no—
209C23×C4—48no—
210C23×C4—48no—
211C23×C4—48no—
212C23×C4—44no—
213C23×C4—48no—
214C23×C4—42no—
215C23×C4—48no—
216C23×C4—48no—
217C23×C4—48no—
218C23×C4—48no—
219C23×C4—44no—
220C23×C4—44no—
221C23×C4C22⋊C418yes—
222C23×C4C22⋊C414yes—
223C23×C4C22×C414yes—
224C23×C4C22×C412yes—
225C23×C4C2×D412yes—
226C23×C4C2×D414yes—
227C23×C4C2×D411yescharacteristic
228C23×C4C2×D414yes—
229C23×C4C2×D414yes—
230C23×C4C2×D412yes—
231C23×C4C2×D414yes—
232C23×C4C2×D414yes—
233C23×C4#4114yes—
234C23×C4#4114yes—
235C23×C4C2411yescharacteristic
236C2×M16—216no—
237C2×M16—216no—
238C2×M16—216no—
239C2×M16—48no—
240C2×M16—48no—
241C2×M16—48no—
242C2×M16—48no—
243C22×C8—24no—
244C22×C8—24no—
245C22×C8—24no—
246C22×C8—24no—
247C22×C8—24no—
248C22×C8—24no—
249C22×C8C2×D418yes—
250#123—264no—
251C22⋊Q8—232no—
252C22⋊Q8—216no—
253C22⋊Q8—216no—
254#121—232no—
255#121—216no—
256#121—216no—
257#120—264no—
258C4×D4—2128no—
259#117—28no—
260#117—216no—
261#117C22⋊C4116yescomplemented
262C2×C4⋊C4—216no—
263C2×C4⋊C4—28no—
264C2×C4⋊C4—24no—
265C2×C4⋊C4—22no—
266C2×C4⋊C4—216no—
267C2×C4⋊C4—216no—
268C2×C4⋊C4—24no—
269C2×C4⋊C4—24no—
270C2×C4⋊C4—24no—
271C2×C4⋊C4—24no—
272C2×C4⋊C4—28no—
273C2×C4⋊C4—24no—
274C2×C4⋊C4—44no—
275C2×C4⋊C4—48no—
276C2×C4⋊C4—44no—
277C2×C4⋊C4—48no—
278C2×C4⋊C4C22×C414yes—
279C2×C4⋊C4C2×D418yes—
280C2×C4⋊C4C2×D418yes—
281C2×C22⋊C4—216no—
282C2×C22⋊C4—216no—
283C2×C22⋊C4—24no—
284C2×C22⋊C4—28no—
285C2×C22⋊C4—28no—
286C2×C22⋊C4—28no—
287C2×C22⋊C4—216no—
288C2×C22⋊C4—432no—
289C2×C22⋊C4—48no—
290C2×C22⋊C4—48no—
291C2×C42—24no—
292C2×C42—28no—
293#103—232no—
294#102—232no—
295C22⋊C8—232no—
296#95—216no—
297C24—22no—
298C24—22no—
299C24—24no—
300C24—28no—
301C24—216no—
302C24—28no—
303C24—22no—
304C24—24no—
305C24—24no—
306C24—22no—
307C24—22no—
308C24—22no—
309C24—24no—
310C24—24no—
311C24—28no—
312C24—216no—
313C24—22no—
314C24—24no—
315C24—28no—
316C24—28no—
317C24—48no—
318C24—416no—
319C24—416no—
320C24—416no—
321C24—416no—
322C24—48no—
323C24—48no—
324C24—48no—
325C24—48no—
326C24—41no—
327C24—42no—
328C24—41no—
329C24—42no—
330C24—42no—
331C24—44no—
332C24C2×C22⋊C414yes—
333C24C2×C22⋊C412yes—
334C24#12014yes—
335C24#12012yes—
336C24#12114yes—
337C24#12312yes—
338C24AGL(1,ℤ/8)14yes—
339C24#13714yes—
340C24C22×D411yescharacteristic
341#41—264no—
342#41—432no—
343#41—432no—
344#41—432no—
345C2×Q8—22no—
346C2×Q8—24no—
347C2×Q8—24no—
348C2×Q8—24no—
349C2×Q8—24no—
350C2×Q8—24no—
351C2×Q8—24no—
352C2×Q8—24no—
353C2×Q8—41no—
354C2×Q8—42no—
355C2×Q8—42no—
356C2×Q8—42no—
357C2×Q8—41no—
358C2×Q8—42no—
359C2×Q8—42no—
360C2×Q8C2×C22⋊C414yescomplemented
361C2×D4—232no—
362C2×D4—216no—
363C2×D4—22no—
364C2×D4—24no—
365C2×D4—24no—
366C2×D4—24no—
367C2×D4—24no—
368C2×D4—24no—
369C2×D4—24no—
370C2×D4—24no—
371C2×D4—416no—
372C2×D4—432no—
373C2×D4—432no—
374C2×D4—48no—
375C2×D4—48no—
376C2×D4—416no—
377C2×D4—416no—
378C2×D4—416no—
379C2×D4—41no—
380C2×D4—42no—
381C2×D4—42no—
382C2×D4—42no—
383C2×D4—41no—
384C2×D4—42no—
385C2×D4—42no—
386C2×D4C2×C22⋊C414yescomplemented
387C22×C4—24no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—28no—
391C22×C4—24no—
392C22×C4—24no—
393C22×C4—24no—
394C22×C4—22no—
395C22×C4—28no—
396C22×C4—28no—
397C22×C4—28no—
398C22×C4—24no—
399C22×C4—24no—
400C22×C4—28no—
401C22×C4—28no—
402C22×C4—24no—
403C22×C4—28no—
404C22×C4—28no—
405C22×C4—24no—
406C22×C4—216no—
407C22×C4—216no—
408C22×C4—216no—
409C22×C4—216no—
410C22×C4—28no—
411C22×C4—41no—
412C22×C4—42no—
413C22×C4—41no—
414C22×C4—42no—
415C22×C4—42no—
416C22×C4—44no—
417C22×C4—48no—
418C22×C4—48no—
419C22×C4—44no—
420C22×C4—44no—
421C22×C4—44no—
422C22×C4—44no—
423C22×C4—44no—
424C22×C4—44no—
425C22×C4—44no—
426C22×C4—44no—
427C22×C4—432no—
428C22×C4—48no—
429C22×C4—48no—
430C22×C4—432no—
431C22×C4—416no—
432C22×C4—432no—
433C22×C4—416no—
434C22×C4—416no—
435C22×C4—432no—
436C22×C4—416no—
437C22×C4—416no—
438C22×C4—416no—
439C22×C4—44no—
440C22×C4—44no—
441C22×C4—48no—
442C22×C4—48no—
443C22×C4—416no—
444C22×C4—816no—
445C22×C4—816no—
446C22×C4C2×C22⋊C414yes—
447C22×C4C2×C22⋊C414yes—
448C22×C4C2×C22⋊C414yes—
449C22×C4C2×C22⋊C412yes—
450C22×C4C4×D418yes—
451C22×C4C4×D418yes—
452C22×C4#12018yes—
453C22×C4#12018yes—
454C22×C4#12014yes—
455C22×C4#12014yes—
456C22×C4#12012yes—
457C22×C4#12118yes—
458C22×C4#12118yes—
459C22×C4#12118yes—
460C22×C4#12118yes—
461C22×C4#12118yes—
462C22×C4#12114yes—
463C22×C4#12318yes—
464C22×C4#12314yes—
465C22×C4#12312yes—
466C22×C4#12418yes—
467C22×C4#12718yes—
468C22×C4C23×C411yescharacteristic
469C22×C4C22×D412yes—
470C22×C4C22×D412yes—
471C22×C4C22×D412yes—
472C22×C4C22×D412yes—
473C22×C4C22×D411yescharacteristic
474C22×C4C2×#4112yes—
475C22×C4C2×#4112yes—
476M16—432no—
477C2×C8—28no—
478C2×C8—28no—
479C2×C8—28no—
480C2×C8—44no—
481C2×C8—44no—
482C2×C8—44no—
483C2×C8—44no—
484C4⋊C4—232no—
485C4⋊C4—24no—
486C4⋊C4—28no—
487C4⋊C4—28no—
488C4⋊C4—28no—
489C4⋊C4—44no—
490C4⋊C4—44no—
491C4⋊C4—416no—
492C4⋊C4C2×C22⋊C418yescomplemented
493C22⋊C4—232no—
494C22⋊C4—232no—
495C22⋊C4—216no—
496C22⋊C4—432no—
497C42—216no—
498C23—24no—
499C23—24no—
500C23—24no—
501C23—24no—
502C23—22no—
503C23—22no—
504C23—24no—
505C23—28no—
506C23—28no—
507C23—28no—
508C23—28no—
509C23—232no—
510C23—24no—
511C23—24no—
512C23—24no—
513C23—24no—
514C23—24no—
515C23—24no—
516C23—24no—
517C23—28no—
518C23—28no—
519C23—24no—
520C23—24no—
521C23—24no—
522C23—28no—
523C23—28no—
524C23—24no—
525C23—432no—
526C23—432no—
527C23—432no—
528C23—432no—
529C23—416no—
530C23—464no—
531C23—44no—
532C23—44no—
533C23—44no—
534C23—416no—
535C23—44no—
536C23—44no—
537C23—44no—
538C23—44no—
539C23—44no—
540C23—44no—
541C23—44no—
542C23—44no—
543C23—44no—
544C23—48no—
545C23—48no—
546C23—42no—
547C23—42no—
548C23—82no—
549C23—82no—
550C23—81no—
551C23—81no—
552C23—82no—
553C23—84no—
554C23—84no—
555C23#37914yes—
556C23#38614yes—
557C23#38614yes—
558C23#39218yes—
559C23#39418yes—
560C23#41718yescomplemented
561C23C22×C22⋊C411yescharacteristic
562C23C2×#12011yescharacteristic
563C23C2×#12012yes—
564C23C2×#12112yes—
565C23C2×#12311yescharacteristic
566C23C2×AGL(1,ℤ/8)12yes—
567C23C2×#13712yes—
568C23#57514yes—
569Q8—28no—
570Q8—44no—
571Q8—44no—
572Q8—44no—
573D4—28no—
574D4—464no—
575D4—432no—
576D4—44no—
577D4—44no—
578D4—44no—
579C2×C4—28no—
580C2×C4—24no—
581C2×C4—24no—
582C2×C4—24no—
583C2×C4—24no—
584C2×C4—44no—
585C2×C4—44no—
586C2×C4—44no—
587C2×C4—44no—
588C2×C4—44no—
589C2×C4—44no—
590C2×C4—416no—
591C2×C4—48no—
592C2×C4—48no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—48no—
596C2×C4—48no—
597C2×C4—432no—
598C2×C4—432no—
599C2×C4—48no—
600C2×C4—48no—
601C2×C4—48no—
602C2×C4—48no—
603C2×C4—48no—
604C2×C4—832no—
605C2×C4—88no—
606C2×C4—88no—
607C2×C4#37914yes—
608C2×C4#38614yes—
609C2×C4#38614yes—
610C2×C4#386116yes—
611C2×C4#390116yes—
612C2×C4#392116yes—
613C2×C4#39218yes—
614C2×C4#394116yes—
615C2×C4#39418yes—
616C2×C4C22×C22⋊C411yescharacteristic
617C2×C4C22×C22⋊C411yescharacteristic
618C2×C4C2×C4×D412yes—
619C2×C4C2×C4×D412yes—
620C2×C4C2×#12011yescharacteristic
621C2×C4C2×#12012yes—
622C2×C4C2×#12012yes—
623C2×C4C2×#12112yes—
624C2×C4C2×#12112yes—
625C2×C4C2×#12112yes—
626C2×C4C2×#12112yes—
627C2×C4C2×#12112yes—
628C2×C4C2×#12311yescharacteristic
629C2×C4C2×#12312yes—
630C2×C4C2×#12412yes—
631C2×C4C2×#12712yes—
632C8—48no—
633C22—28no—
634C22—28no—
635C22—28no—
636C22—24no—
637C22—24no—
638C22—216no—
639C22—28no—
640C22—28no—
641C22—28no—
642C22—28no—
643C22—464no—
644C22—44no—
645C22—48no—
646C22—48no—
647C22—44no—
648C22—44no—
649C22—48no—
650C22—48no—
651C22—44no—
652C22—44no—
653C22—48no—
654C22—84no—
655C22—84no—
656C22—82no—
657C22—82no—
658C22—88no—
659C22—84no—
660C22#143818yes—
661C22C2×#37911yescharacteristic
662C22C2×#38611yescharacteristic
663C22C2×#38611yescharacteristic
664C22C2×#39212yes—
665C22C2×#39412yes—
666C22C2×#41712yes—
667C22#289612yes—
668C22#300311yescharacteristic
669C22#300411yescharacteristic
670C22#300512yes—
671C22#300614yes—
672C22#300714yes—
673C22#305514yes—
674C22#305614yes—
675C22#309212yes—
676C22#309312yes—
677C4—44no—
678C4—44no—
679C4—48no—
680C4—88no—
681C4#143818yes—
682C4C2×#38612yes—
683C4C2×#39012yes—
684C4C2×#39212yes—
685C4C2×#39412yes—
686C2—28no—
687C2—48no—
688C2—84no—
689C2C2×#143811yescharacteristic
690C2#2060112yes—
691C2#2090911yescharacteristic
692C2#2091011yescharacteristic
693C2#2097112yes—
694C2#2339112yes—
695C2#2339212yes—
696C2#2348214yes—
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC22×#525C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C25×C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C22×#417C2semidirect
3. C22×#407C2semidirect
4. C22×#380C2semidirect
5. C2×#1736C2semidirect
6. #10970C2semidirect
7. #10969C2semidirect
14. C2×#525C4semidirect
26. C22×#103C22semidirect
27. C22×#102C22semidirect
36. C2×#340C22semidirect
70. C22×M16D4semidirect
80. C2×C22⋊Q8C2×C4semidirect
84. C2×#121C2×C4semidirect
100. C22×C4⋊C4D4semidirect
151. C2×#41C22⋊C4semidirect
261. #117C22⋊C4semidirect
360. C2×Q8C2×C22⋊C4semidirect
386. C2×D4C2×C22⋊C4semidirect
492. C4⋊C4C2×C22⋊C4semidirect
560. C23#417semidirect

Maximal subgroups

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

subgroupindex
C22×#5252
C22×#4172
C22×#4072
C22×#3802
C2×#17362
#109702
#109692