Finite Group Databasebeta

← #360519

Subgroups — #360519

Order 512.

2596 conjugacy classes of subgroups.

The subgroup lattice has 712 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#410C211yesmaximal, characteristic
2C2×#2339C211yesmaximal, characteristic
3C2×#1799C212yesmaximal, complemented
4C2×#1782C211yesmaximal, characteristic
5#13878C212yesmaximal, complemented
6#11043C214yesmaximal, complemented
7#10761C212yesmaximal, complemented
8#10736C212yesmaximal, complemented
9C23×#41C2211yescharacteristic
10C22×C4×D4C2212yes—
11C22×#117C2212yes—
12C2×#410—28no—
13C2×#410—24no—
14C22×#99C2211yescharacteristic
15C22×#99C2212yes—
16C22×#99C2211yescharacteristic
17C22×#99C2212yes—
18C22×#99C2212yes—
19#2339—24no—
20#2339—22no—
21#2339—22no—
22C2×#391C412yescomplemented
23C2×#386C412yescomplemented
24C2×#386C2212yes—
25C2×#381C411yescharacteristic, complemented
26C2×#381C2212yes—
27C2×#381C2211yescharacteristic
28C2×#380C411yescharacteristic, complemented
29C2×#380C2214yes—
30C2×C4×C22⋊C4C411yescharacteristic, complemented
31C22×#95C411yescharacteristic, complemented
32C22×#95C2211yescharacteristic
33#1799—28no—
34#1782—24no—
35C2×#342C2214yes—
36C2×#342C2214yes—
37#1444—22no—
38#1442—22no—
39#1436—24no—
40C22⋊#95C2214yes—
41C22×#41—24no—
42C22×#41—24no—
43C22×#41—22no—
44C22×#41—22no—
45C22×#41—24no—
46C22×#41—44no—
47C22×#41D414yes—
48C22×#41Q814yes—
49C23×Q8C2×C411yescharacteristic
50C23×D4C2×C411yescharacteristic
51C23×D4C2311yescharacteristic
52C23×D4C2311yescharacteristic
53C24×C4C2×C411yescharacteristic
54C24×C4C2×C411yescharacteristic
55C24×C4C2×C412yes—
56C24×C4C2311yescharacteristic
57C24×C4C2312yes—
58C2×C4×D4—24no—
59C2×C4×D4—24no—
60C2×C4×D4—24no—
61C2×C4×D4—24no—
62C2×C4×D4—44no—
63C2×#117—24no—
64C2×#117—24no—
65C2×#117—28no—
66C2×#117—44no—
67C22×C4⋊C4—22no—
68C22×C4⋊C4—24no—
69C22×C4⋊C4C2×C412yes—
70C22×C22⋊C4C2×C412yes—
71C22×C22⋊C4C2×C412yes—
72C22×C22⋊C4C2×C412yes—
73C22×C22⋊C4C2312yes—
74C22×C22⋊C4C2312yes—
75C22×C22⋊C4C2312yes—
76C22×C22⋊C4C2314yes—
77C22×C42—22no—
78C22×C42C2×C412yes—
79#410—232no—
80C2×#99—24no—
81C2×#99—24no—
82C2×#99—24no—
83C2×#99—28no—
84C2×#99—24no—
85C2×#99—24no—
86C2×#99—22no—
87C2×#99—22no—
88C2×#99—24no—
89C2×#99—44no—
90C2×#99D418yes—
91#391—24no—
92#391—24no—
93#386—24no—
94#386—24no—
95#386—24no—
96#386—24no—
97#381—22no—
98#381—22no—
99#381—22no—
100#381—22no—
101#381—24no—
102#381—24no—
103#380—22no—
104#380—22no—
105#380—28no—
106#380—44no—
107C4×C22⋊C4—21no—
108C4×C22⋊C4—22no—
109C4×C22⋊C4—21no—
110C2×#95—21no—
111C2×#95—22no—
112C2×#95—21no—
113C2×#95—24no—
114C2×#95—22no—
115C2×#95—24no—
116C2×#95—22no—
117C2×#95—24no—
118C2×#95C2×C411yescharacteristic
119C2×#95C2×C412yes—
120C2×#95C2×C411yescharacteristic
121C2×#95C2×C412yes—
122C2×#95C2×C412yes—
123C2×#95C2×C411yescharacteristic
124C2×#95C2×C411yescharacteristic
125C2×#95C2×C412yes—
126C2×#95D414yes—
127C2×#95D412yes—
128C2×#95D412yes—
129#342—28no—
130#342—44no—
131C25C22×C411yescharacteristic
132C25C22×C411yescharacteristic
133C25C2411yescharacteristic
134C2×#41—28no—
135C2×#41—216no—
136C2×#41—28no—
137C2×#41—28no—
138C2×#41—28no—
139C2×#41—28no—
140C2×#41—28no—
141C2×#41—48no—
142C2×#41—48no—
143C2×#41—48no—
144C2×#41—48no—
145C2×#41—416no—
146C2×#41—48no—
147C2×#41—48no—
148C2×#41—416no—
149C22×Q8—22no—
150C22×Q8—22no—
151C22×Q8—24no—
152C22×Q8—42no—
153C22×Q8C4⋊C414yescomplemented
154C22×D4—22no—
155C22×D4—22no—
156C22×D4—24no—
157C22×D4—24no—
158C22×D4—22no—
159C22×D4—22no—
160C22×D4—22no—
161C22×D4—22no—
162C22×D4—24no—
163C22×D4—22no—
164C22×D4—22no—
165C22×D4—42no—
166C22×D4C4⋊C414yescomplemented
167C22×D4C2×D412yes—
168C22×D4C2×D412yes—
169C22×D4C2×D414yes—
170C22×D4C2×Q812yes—
171C22×D4C2×Q812yes—
172C22×D4#4114yes—
173C23×C4—22no—
174C23×C4—22no—
175C23×C4—24no—
176C23×C4—24no—
177C23×C4—22no—
178C23×C4—22no—
179C23×C4—24no—
180C23×C4—21no—
181C23×C4—21no—
182C23×C4—21no—
183C23×C4—21no—
184C23×C4—22no—
185C23×C4—22no—
186C23×C4—24no—
187C23×C4—24no—
188C23×C4—24no—
189C23×C4—22no—
190C23×C4—22no—
191C23×C4—22no—
192C23×C4—22no—
193C23×C4—44no—
194C23×C4—42no—
195C23×C4—42no—
196C23×C4—42no—
197C23×C4—44no—
198C23×C4—82no—
199C23×C4—82no—
200C23×C4C4⋊C414yescomplemented
201C23×C4C4⋊C414yescomplemented
202C23×C4C22×C411yescharacteristic
203C23×C4C22×C412yes—
204C23×C4C22×C411yescharacteristic
205C23×C4C22×C411yescharacteristic
206C23×C4C22×C411yescharacteristic
207C23×C4C22×C412yes—
208C23×C4C22×C412yes—
209C23×C4C22×C412yes—
210C23×C4C2×D414yes—
211C23×C4C2×D412yes—
212C23×C4C2×D412yes—
213C23×C4C2×D412yes—
214C23×C4C2×D414yes—
215C23×C4C2×Q812yes—
216C23×C4#4114yes—
217C4×D4—48no—
218C4×D4—48no—
219C4×D4—48no—
220C4×D4—48no—
221#117—216no—
222#117—416no—
223#117—48no—
224C2×C4⋊C4—24no—
225C2×C4⋊C4—24no—
226C2×C4⋊C4—42no—
227C2×C4⋊C4—48no—
228C2×C4⋊C4—44no—
229C2×C4⋊C4—48no—
230C2×C4⋊C4—48no—
231C2×C4⋊C4—84no—
232C2×C22⋊C4—24no—
233C2×C22⋊C4—24no—
234C2×C22⋊C4—24no—
235C2×C22⋊C4—24no—
236C2×C22⋊C4—24no—
237C2×C22⋊C4—22no—
238C2×C22⋊C4—24no—
239C2×C22⋊C4—22no—
240C2×C22⋊C4—24no—
241C2×C22⋊C4—24no—
242C2×C22⋊C4—24no—
243C2×C22⋊C4—22no—
244C2×C22⋊C4—22no—
245C2×C22⋊C4—24no—
246C2×C22⋊C4—22no—
247C2×C22⋊C4—22no—
248C2×C22⋊C4—22no—
249C2×C22⋊C4—22no—
250C2×C22⋊C4—24no—
251C2×C22⋊C4—24no—
252C2×C22⋊C4—24no—
253C2×C22⋊C4—24no—
254C2×C22⋊C4—42no—
255C2×C22⋊C4—44no—
256C2×C22⋊C4—82no—
257C2×C22⋊C4C2×D418yes—
258C2×C42—22no—
259C2×C42—22no—
260C2×C42—24no—
261C2×C42—42no—
262C2×C42—48no—
263C2×C42—44no—
264#99—216no—
265#99—216no—
266#99—48no—
267#99—44no—
268#99—44no—
269#95—21no—
270#95—21no—
271#95—22no—
272#95—22no—
273#95—21no—
274#95—21no—
275#95—24no—
276#95—24no—
277#95—48no—
278#95—48no—
279#95—44no—
280#95—44no—
281#95—84no—
282#95C22⋊C411yescharacteristic
283#95C22⋊C412yes—
284#95C22⋊C411yescharacteristic
285#95C22⋊C412yes—
286#95C22⋊C412yes—
287#95C22⋊C414yes—
288#95C22⋊C414yes—
289C24—21no—
290C24—21no—
291C24—22no—
292C24—21no—
293C24—21no—
294C24—22no—
295C24—22no—
296C24—42no—
297C24—41no—
298C24—41no—
299C24—42no—
300C24—41no—
301C24—41no—
302C24—42no—
303C24—41no—
304C24—41no—
305C24C2×C4⋊C412yes—
306C24C2×C4⋊C412yes—
307C24C4×D414yes—
308C24C23×C411yescharacteristic
309C24C22×D412yes—
310C24C22×D411yescharacteristic
311C24C22×D412yes—
312C24C22×Q811yescharacteristic
313C24C2×#4112yes—
314C24C2×#4112yes—
315C24H(5,𝔽2)14yes—
316#41—416no—
317#41—416no—
318#41—416no—
319#41—416no—
320#41—432no—
321#41—432no—
322C2×Q8—28no—
323C2×Q8—28no—
324C2×Q8—44no—
325C2×Q8—48no—
326C2×Q8—48no—
327C2×D4—24no—
328C2×D4—24no—
329C2×D4—24no—
330C2×D4—24no—
331C2×D4—28no—
332C2×D4—28no—
333C2×D4—28no—
334C2×D4—28no—
335C2×D4—22no—
336C2×D4—24no—
337C2×D4—24no—
338C2×D4—24no—
339C2×D4—22no—
340C2×D4—24no—
341C2×D4—24no—
342C2×D4—28no—
343C2×D4—28no—
344C2×D4—44no—
345C2×D4—48no—
346C2×D4—48no—
347C2×D4—44no—
348C2×D4—44no—
349C2×D4—44no—
350C2×D4—44no—
351C2×D4#12014yes—
352C2×D4C22⋊Q818yes—
353C2×D4#12314yes—
354C22×C4—21no—
355C22×C4—21no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—24no—
363C22×C4—22no—
364C22×C4—24no—
365C22×C4—24no—
366C22×C4—24no—
367C22×C4—24no—
368C22×C4—28no—
369C22×C4—24no—
370C22×C4—24no—
371C22×C4—28no—
372C22×C4—21no—
373C22×C4—21no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—24no—
379C22×C4—24no—
380C22×C4—28no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—24no—
384C22×C4—24no—
385C22×C4—24no—
386C22×C4—24no—
387C22×C4—24no—
388C22×C4—24no—
389C22×C4—28no—
390C22×C4—28no—
391C22×C4—24no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—28no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—44no—
400C22×C4—44no—
401C22×C4—44no—
402C22×C4—42no—
403C22×C4—44no—
404C22×C4—44no—
405C22×C4—48no—
406C22×C4—44no—
407C22×C4—42no—
408C22×C4—42no—
409C22×C4—44no—
410C22×C4—44no—
411C22×C4—44no—
412C22×C4—48no—
413C22×C4—44no—
414C22×C4—44no—
415C22×C4—44no—
416C22×C4—44no—
417C22×C4—44no—
418C22×C4—44no—
419C22×C4—44no—
420C22×C4—44no—
421C22×C4—44no—
422C22×C4—44no—
423C22×C4—42no—
424C22×C4—42no—
425C22×C4—42no—
426C22×C4—42no—
427C22×C4—44no—
428C22×C4—42no—
429C22×C4—44no—
430C22×C4—42no—
431C22×C4—42no—
432C22×C4—42no—
433C22×C4—42no—
434C22×C4—42no—
435C22×C4—44no—
436C22×C4—44no—
437C22×C4—82no—
438C22×C4—82no—
439C22×C4—84no—
440C22×C4—88no—
441C22×C4—84no—
442C22×C4—88no—
443C22×C4—88no—
444C22×C4C2×C22⋊C412yes—
445C22×C4C2×C22⋊C414yes—
446C22×C4C2×C22⋊C411yescharacteristic
447C22×C4C2×C22⋊C411yescharacteristic
448C22×C4C2×C22⋊C412yes—
449C22×C4C2×C22⋊C412yes—
450C22×C4C2×C4⋊C412yes—
451C22×C4C2×C4⋊C412yes—
452C22×C4C2×C4⋊C412yes—
453C22×C4C2×C4⋊C412yes—
454C22×C4#11712yes—
455C22×C4#11712yes—
456C22×C4C4×D414yes—
457C22×C4#12114yes—
458C22×C4#12114yes—
459C22×C4C22⋊Q814yes—
460C22×C4C22⋊Q814yes—
461C4⋊C4—48no—
462C4⋊C4—416no—
463C4⋊C4—816no—
464C22⋊C4—28no—
465C22⋊C4—28no—
466C22⋊C4—48no—
467C22⋊C4—44no—
468C22⋊C4—44no—
469C22⋊C4—44no—
470C22⋊C4—44no—
471C22⋊C4—44no—
472C22⋊C4—44no—
473C22⋊C4—42no—
474C22⋊C4—42no—
475C22⋊C4—42no—
476C22⋊C4—42no—
477C22⋊C4—44no—
478C22⋊C4—44no—
479C22⋊C4—84no—
480C42—44no—
481C42—44no—
482C42—416no—
483C23—24no—
484C23—22no—
485C23—22no—
486C23—21no—
487C23—22no—
488C23—21no—
489C23—22no—
490C23—24no—
491C23—24no—
492C23—22no—
493C23—22no—
494C23—22no—
495C23—22no—
496C23—42no—
497C23—42no—
498C23—44no—
499C23—42no—
500C23—41no—
501C23—41no—
502C23—41no—
503C23—41no—
504C23—42no—
505C23—42no—
506C23—42no—
507C23—44no—
508C23—44no—
509C23—42no—
510C23—42no—
511C23—44no—
512C23—44no—
513C23—44no—
514C23—42no—
515C23—42no—
516C23—42no—
517C23—42no—
518C23—42no—
519C23—42no—
520C23—42no—
521C23—41no—
522C23—41no—
523C23—42no—
524C23—42no—
525C23—42no—
526C23—41no—
527C23—41no—
528C23—82no—
529C23—82no—
530C23—82no—
531C23—82no—
532C23#38514yes—
533C23C22×C22⋊C411yescharacteristic
534C23C22×C4⋊C411yescharacteristic
535C23C2×#11711yescharacteristic
536C23C2×C4×D412yes—
537C23#51812yes—
538C23C2×#12011yescharacteristic
539C23C2×#12112yes—
540C23C2×C22⋊Q812yes—
541C23C2×C22⋊Q812yes—
542C23C2×#12311yescharacteristic
543C23#53412yes—
544C23#54312yes—
545C23D4214yes—
546C23#54614yes—
547C23#54614yes—
548C23#55114yes—
549Q8—416no—
550D4—28no—
551D4—28no—
552D4—216no—
553D4—416no—
554D4—44no—
555D4—44no—
556D4—48no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—24no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—24no—
565C2×C4—24no—
566C2×C4—24no—
567C2×C4—24no—
568C2×C4—22no—
569C2×C4—24no—
570C2×C4—24no—
571C2×C4—24no—
572C2×C4—24no—
573C2×C4—28no—
574C2×C4—28no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—24no—
581C2×C4—22no—
582C2×C4—24no—
583C2×C4—24no—
584C2×C4—22no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—48no—
588C2×C4—48no—
589C2×C4—48no—
590C2×C4—48no—
591C2×C4—48no—
592C2×C4—48no—
593C2×C4—44no—
594C2×C4—44no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—48no—
598C2×C4—48no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—48no—
606C2×C4—88no—
607C2×C4—82no—
608C2×C4—82no—
609C2×C4—84no—
610C2×C4—84no—
611C2×C4—88no—
612C2×C4—84no—
613C2×C4—84no—
614C2×C4—84no—
615C2×C4—84no—
616C2×C4—82no—
617C2×C4—82no—
618C2×C4—82no—
619C2×C4—82no—
620C2×C4—84no—
621C2×C4—84no—
622C2×C4—84no—
623C2×C4—84no—
624C2×C4—82no—
625C2×C4—82no—
626C2×C4—82no—
627C2×C4—82no—
628C2×C4—82no—
629C2×C4—82no—
630C2×C4#38012yes—
631C2×C4#38012yes—
632C2×C4#38014yes—
633C2×C4#38514yes—
634C2×C4#41014yescomplemented
635C22—24no—
636C22—22no—
637C22—22no—
638C22—24no—
639C22—24no—
640C22—22no—
641C22—22no—
642C22—24no—
643C22—22no—
644C22—21no—
645C22—21no—
646C22—22no—
647C22—22no—
648C22—21no—
649C22—21no—
650C22—22no—
651C22—24no—
652C22—41no—
653C22—42no—
654C22—41no—
655C22—42no—
656C22—42no—
657C22—44no—
658C22—44no—
659C22—44no—
660C22—44no—
661C22—48no—
662C22—44no—
663C22—44no—
664C22—42no—
665C22—42no—
666C22—42no—
667C22—42no—
668C22—42no—
669C22—82no—
670C22—82no—
671C22—84no—
672C22—82no—
673C22—82no—
674C22C2×#38011yescharacteristic
675C22C2×#38511yescharacteristic
676C22#231511yescharacteristic
677C22D4×C22⋊C412yes—
678C22#234212yes—
679C22#241012yes—
680C22#241212yes—
681C22#243714yes—
682C22C2×#41012yes—
683C22#288311yescharacteristic
684C22#288411yescharacteristic
685C4—22no—
686C4—22no—
687C4—24no—
688C4—24no—
689C4—24no—
690C4—42no—
691C4—44no—
692C4—42no—
693C4—88no—
694C4—82no—
695C4—82no—
696C4—162no—
697C4—162no—
698C2—22no—
699C2—22no—
700C2—21no—
701C2—22no—
702C2—21no—
703C2—24no—
704C2—44no—
705C2—82no—
706C2#1387711yescharacteristic
707C2#2063211yescharacteristic
708C2#2063411yescharacteristic
709C2#2240512yes—
710C2#2240612yes—
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC24C23×C4
Frattini, cosocleC25C24
prime radicalC23×#41C22
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×#95 ◁ G
LECSG ▷ C25 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. C2×#1799C2semidirect
5. #13878C2semidirect
6. #11043C2semidirect
7. #10761C2semidirect
8. #10736C2semidirect
22. C2×#391C4semidirect
23. C2×#386C4semidirect
25. C2×#381C4semidirect
28. C2×#380C4semidirect
30. C2×C4×C22⋊C4C4semidirect
31. C22×#95C4semidirect
153. C22×Q8C4⋊C4semidirect
166. C22×D4C4⋊C4semidirect
200. C23×C4C4⋊C4semidirect
201. C23×C4C4⋊C4semidirect
634. C2×C4#410semidirect

Maximal subgroups

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

subgroupindex
C22×#4102
C2×#23392
C2×#17992
C2×#17822
#138782
#110432
#107612
#107362