Finite Group Databasebeta

← #360567

Subgroups — #360567

Order 512.

2450 conjugacy classes of subgroups.

The subgroup lattice has 692 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#103C211yesmaximal, characteristic
2C2×Q8×C22⋊C4C211yesmaximal, characteristic
3C2×#1850C211yesmaximal, characteristic
4C2×#1805C212yesmaximal, complemented
5C2×#1787C211yesmaximal, characteristic
6#13881C211yesmaximal, characteristic
7#11077C212yesmaximal, complemented
8#11052C212yesmaximal, complemented
9#10789C212yesmaximal, complemented
10#10730C211yesmaximal, characteristic
11#10721C211yesmaximal, characteristic
12C24×Q8C2211yescharacteristic
13C24×C8C2211yescharacteristic
14C22×C4×Q8C2212yes—
15C23×C4⋊C4C2211yescharacteristic
16C22×C4⋊C8C2212yes—
17C22×#103—28no—
18C22×#103—24no—
19C22×#103C2211yescharacteristic
20C22×#103C2212yes—
21C22×#103C2211yescharacteristic
22C22×#103C2212yes—
23C22×#103C2211yescharacteristic
24C22×#103C2211yescharacteristic
25C22×C22⋊C8C2211yescharacteristic
26Q8×C22⋊C4—24no—
27Q8×C22⋊C4C418yescomplemented
28C2×#391—22no—
29C2×#391C2212yes—
30C2×#382C2212yes—
31C2×#380—21no—
32C2×#380C2211yescharacteristic
33C2×#380C2211yescharacteristic
34C2×C4×C22⋊C4—21no—
35C2×C4×C22⋊C4C2211yescharacteristic
36#1850—24no—
37#1805—28no—
38#1787—24no—
39C2×#340C2212yes—
40C2×#340C2212yes—
41C2×#340C2212yes—
42C2×#340C2212yes—
43C2×#336C2212yes—
44#1442—22no—
45C4⋊#95C2212yes—
46C22⋊#95—21no—
47C22⋊#95—21no—
48C23×Q8—22no—
49C23×Q8—21no—
50C23×Q8—22no—
51C23×Q8—22no—
52C23×Q8—24no—
53C23×Q8—24no—
54C23×Q8—42no—
55C23×Q8—42no—
56C23×Q8C2×C414yes—
57C23×Q8D414yes—
58C23×Q8Q814yes—
59C23×Q8C2311yescharacteristic
60C23×Q8C2311yescharacteristic
61C24×C4—21no—
62C24×C4—21no—
63C24×C4C2311yescharacteristic
64C23×C8—22no—
65C23×C8—24no—
66C23×C8—22no—
67C23×C8—22no—
68C23×C8—42no—
69C23×C8C2311yescharacteristic
70C23×C8C2311yescharacteristic
71C23×C8C2312yes—
72C2×C4×Q8—24no—
73C2×C4×Q8—24no—
74C2×C4×Q8—24no—
75C2×C4×Q8—42no—
76C2×C4×Q8—42no—
77C2×C4×Q8C2×C418yescomplemented
78C22×C4⋊C4—22no—
79C22×C4⋊C4—24no—
80C22×C4⋊C4—22no—
81C22×C4⋊C4—22no—
82C22×C4⋊C4—21no—
83C22×C4⋊C4—42no—
84C22×C4⋊C4C2311yescharacteristic
85C22×C4⋊C4C2311yescharacteristic
86C22×C4⋊C4C2312yes—
87C22×C4⋊C4C2312yes—
88C22×C22⋊C4—21no—
89C22×C22⋊C4—21no—
90C22×C22⋊C4D411yescharacteristic
91C22×C22⋊C4D411yescharacteristic
92C22×C42—22no—
93C22×C42C2312yes—
94C2×C4⋊C8—24no—
95C2×C4⋊C8—24no—
96C2×C4⋊C8—42no—
97C2×#103—232no—
98C2×#103—24no—
99C2×#103—24no—
100C2×#103—24no—
101C2×#103—24no—
102C2×#103—24no—
103C2×#103—24no—
104C2×#103—22no—
105C2×#103—22no—
106C2×#103—22no—
107C2×#103—22no—
108C2×#103—24no—
109C2×#103—42no—
110C2×#103—42no—
111C2×#103D418yes—
112C2×C22⋊C8—22no—
113C2×C22⋊C8—22no—
114C2×C22⋊C8—22no—
115#391—24no—
116#391—216no—
117#391C2×C418yescomplemented
118#382—24no—
119#382—42no—
120#380—22no—
121#380—28no—
122#380—22no—
123#380—22no—
124#380C2×C414yes—
125C4×C22⋊C4—28no—
126C4×C22⋊C4C2×C414yes—
127C4×C22⋊C4D414yes—
128C2×#95—22no—
129C2×#95—22no—
130C2×#95—21no—
131C2×#95—22no—
132C2×#95—22no—
133C2×#95—21no—
134C2×#95—22no—
135C2×#95—22no—
136C2×#95D412yes—
137#340—24no—
138#340—24no—
139#340—42no—
140#340—42no—
141#336—24no—
142C25C2×D411yescharacteristic
143C22×Q8—24no—
144C22×Q8—28no—
145C22×Q8—24no—
146C22×Q8—28no—
147C22×Q8—24no—
148C22×Q8—28no—
149C22×Q8—21no—
150C22×Q8—21no—
151C22×Q8—21no—
152C22×Q8—21no—
153C22×Q8—21no—
154C22×Q8—21no—
155C22×Q8—22no—
156C22×Q8—22no—
157C22×Q8—216no—
158C22×Q8—28no—
159C22×Q8—216no—
160C22×Q8—28no—
161C22×Q8—22no—
162C22×Q8—22no—
163C22×Q8—22no—
164C22×Q8—22no—
165C22×Q8—44no—
166C22×Q8—44no—
167C22×Q8—48no—
168C22×Q8—44no—
169C22×Q8—44no—
170C22×Q8—48no—
171C22×Q8—48no—
172C22×Q8—48no—
173C22×Q8—48no—
174C22×Q8—48no—
175C22×Q8—48no—
176C22×Q8—48no—
177C22×Q8C4⋊C4116yes—
178C22×Q8C22×C412yes—
179C22×Q8C22×C412yes—
180C22×Q8C2×D412yes—
181C22×Q8C2×D412yes—
182C22×Q8C2×D414yes—
183C22×Q8C2×Q812yes—
184C22×Q8C2×Q812yes—
185C22×Q8#4114yes—
186C23×C4—21no—
187C23×C4—22no—
188C23×C4—22no—
189C23×C4—22no—
190C23×C4—22no—
191C23×C4—24no—
192C23×C4—24no—
193C23×C4—24no—
194C23×C4—21no—
195C23×C4—22no—
196C23×C4—22no—
197C23×C4—21no—
198C23×C4—21no—
199C23×C4—44no—
200C23×C4—44no—
201C23×C4—44no—
202C23×C4—42no—
203C23×C4—42no—
204C23×C4—42no—
205C23×C4—82no—
206C23×C4C22×C412yes—
207C23×C4C2×D412yes—
208C23×C4C2×D412yes—
209C23×C4C2×D411yescharacteristic
210C23×C4C2×D412yes—
211C23×C4C2×D412yes—
212C23×C4C2×Q812yes—
213C23×C4#4112yes—
214C23×C4#4112yes—
215C23×C4C2411yescharacteristic
216C22×C8—28no—
217C22×C8—22no—
218C22×C8—22no—
219C22×C8—22no—
220C22×C8—22no—
221C22×C8—22no—
222C22×C8—22no—
223C22×C8—21no—
224C22×C8—22no—
225C22×C8—21no—
226C22×C8—21no—
227C22×C8—21no—
228C22×C8—21no—
229C22×C8—21no—
230C22×C8—22no—
231C22×C8—22no—
232C22×C8—22no—
233C22×C8—48no—
234C22×C8—48no—
235C22×C8—44no—
236C22×C8—44no—
237C22×C8C2×D414yes—
238C4×Q8—216no—
239C4×Q8—48no—
240C4×Q8—48no—
241C4×Q8—48no—
242C2×C4⋊C4—28no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—28no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—21no—
256C2×C4⋊C4—21no—
257C2×C4⋊C4—21no—
258C2×C4⋊C4—21no—
259C2×C4⋊C4—22no—
260C2×C4⋊C4—22no—
261C2×C4⋊C4—22no—
262C2×C4⋊C4—48no—
263C2×C4⋊C4—44no—
264C2×C4⋊C4—44no—
265C2×C4⋊C4—44no—
266C2×C4⋊C4—44no—
267C2×C4⋊C4—42no—
268C2×C4⋊C4C22×C414yes—
269C2×C4⋊C4C2×D414yes—
270C2×C22⋊C4—22no—
271C2×C22⋊C4—22no—
272C2×C22⋊C4—22no—
273C2×C22⋊C4—22no—
274C2×C22⋊C4—24no—
275C2×C22⋊C4—42no—
276C2×C22⋊C4—42no—
277C2×C22⋊C4—42no—
278C2×C22⋊C4—42no—
279C2×C22⋊C4—42no—
280C2×C22⋊C4C22⋊C414yes—
281C2×C22⋊C4C22⋊C414yes—
282C2×C42—22no—
283C2×C42—22no—
284C2×C42—22no—
285C2×C42—22no—
286C2×C42—28no—
287C2×C42—44no—
288C2×C42—44no—
289C2×C42C22×C414yes—
290C2×C42C2×D414yes—
291C4⋊C8—48no—
292#103—216no—
293#103—216no—
294#103—44no—
295#103—44no—
296#103—44no—
297#103—44no—
298C22⋊C8—44no—
299#95—28no—
300#95—28no—
301#95—24no—
302#95—44no—
303#95—44no—
304#95—42no—
305#95—82no—
306#95—82no—
307#95C22⋊C418yes—
308C24—22no—
309C24—22no—
310C24—22no—
311C24—22no—
312C24C2×C22⋊C412yes—
313C24#12112yes—
314C24C22⋊Q812yes—
315C24#13512yes—
316C24#13512yes—
317C24#13712yes—
318C24#13712yes—
319C24C22×D411yescharacteristic
320C2×Q8—22no—
321C2×Q8—22no—
322C2×Q8—24no—
323C2×Q8—22no—
324C2×Q8—22no—
325C2×Q8—22no—
326C2×Q8—22no—
327C2×Q8—22no—
328C2×Q8—22no—
329C2×Q8—24no—
330C2×Q8—24no—
331C2×Q8—24no—
332C2×Q8—24no—
333C2×Q8—24no—
334C2×Q8—24no—
335C2×Q8—24no—
336C2×Q8—216no—
337C2×Q8—232no—
338C2×Q8—24no—
339C2×Q8—24no—
340C2×Q8—22no—
341C2×Q8—24no—
342C2×Q8—22no—
343C2×Q8—22no—
344C2×Q8—22no—
345C2×Q8—24no—
346C2×Q8—416no—
347C2×Q8—48no—
348C2×Q8—48no—
349C2×Q8—416no—
350C2×Q8—416no—
351C2×Q8—432no—
352C2×Q8—416no—
353C2×Q8—416no—
354C2×Q8—432no—
355C2×Q8—42no—
356C2×Q8—42no—
357C2×Q8—42no—
358C2×Q8—42no—
359C2×Q8—42no—
360C2×Q8—42no—
361C2×Q8C2×C4⋊C414yes—
362C2×Q8C2×C4⋊C414yes—
363C2×Q8C4×D418yes—
364C2×Q8#12014yes—
365C2×Q8C22⋊Q818yes—
366C2×Q8#12314yes—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—24no—
371C22×C4—22no—
372C22×C4—24no—
373C22×C4—22no—
374C22×C4—24no—
375C22×C4—24no—
376C22×C4—22no—
377C22×C4—24no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—24no—
382C22×C4—24no—
383C22×C4—24no—
384C22×C4—24no—
385C22×C4—24no—
386C22×C4—24no—
387C22×C4—24no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—216no—
404C22×C4—24no—
405C22×C4—24no—
406C22×C4—416no—
407C22×C4—48no—
408C22×C4—48no—
409C22×C4—416no—
410C22×C4—42no—
411C22×C4—42no—
412C22×C4—48no—
413C22×C4—42no—
414C22×C4—42no—
415C22×C4—42no—
416C22×C4—42no—
417C22×C4—42no—
418C22×C4—42no—
419C22×C4—42no—
420C22×C4—42no—
421C22×C4—42no—
422C22×C4—42no—
423C22×C4—42no—
424C22×C4—42no—
425C22×C4—41no—
426C22×C4—41no—
427C22×C4—42no—
428C22×C4—42no—
429C22×C4—41no—
430C22×C4—41no—
431C22×C4—41no—
432C22×C4—41no—
433C22×C4—42no—
434C22×C4—42no—
435C22×C4—88no—
436C22×C4—88no—
437C22×C4—84no—
438C22×C4—84no—
439C22×C4C2×C22⋊C414yes—
440C22×C4C2×C22⋊C414yes—
441C22×C4C2×C22⋊C412yes—
442C22×C4C2×C4⋊C414yes—
443C22×C4#11714yes—
444C22×C4#12114yes—
445C22×C4#12112yes—
446C22×C4C22⋊Q814yes—
447C22×C4C22⋊Q812yes—
448C22×C4C23×C411yescharacteristic
449C22×C4C22×D411yescharacteristic
450C22×C4C22×D412yes—
451C22×C4C22×D411yescharacteristic
452C22×C4C22×Q811yescharacteristic
453C22×C4C2×#4111yescharacteristic
454C22×C4C2×#4111yescharacteristic
455C22×C4C2×#4112yes—
456C22×C4H(5,𝔽2)12yes—
457C22×C4H(5,𝔽2)12yes—
458C2×C8—24no—
459C2×C8—24no—
460C2×C8—24no—
461C2×C8—416no—
462C2×C8—42no—
463C2×C8—42no—
464C2×C8—42no—
465C2×C8—42no—
466C2×C8—41no—
467C2×C8—41no—
468C2×C8—41no—
469C2×C8—41no—
470C2×C8—41no—
471C2×C8—41no—
472C2×C8—42no—
473C2×C8—42no—
474C2×C8—42no—
475C2×C8—42no—
476C4⋊C4—24no—
477C4⋊C4—24no—
478C4⋊C4—24no—
479C4⋊C4—24no—
480C4⋊C4—416no—
481C4⋊C4—48no—
482C4⋊C4—42no—
483C4⋊C4—42no—
484C4⋊C4—42no—
485C4⋊C4—42no—
486C4⋊C4—41no—
487C4⋊C4—41no—
488C4⋊C4—41no—
489C4⋊C4—41no—
490C4⋊C4—42no—
491C4⋊C4—42no—
492C22⋊C4—28no—
493C22⋊C4—28no—
494C22⋊C4—48no—
495C22⋊C4—84no—
496C42—24no—
497C42—24no—
498C42—24no—
499C42—24no—
500C42—416no—
501C23—24no—
502C23—24no—
503C23—24no—
504C23—22no—
505C23—22no—
506C23—24no—
507C23—24no—
508C23—24no—
509C23—24no—
510C23—24no—
511C23—22no—
512C23—22no—
513C23—24no—
514C23—24no—
515C23—24no—
516C23—24no—
517C23#38014yes—
518C23#41614yes—
519C23#41914yes—
520C23#46414yes—
521C23#46414yes—
522C23#47914yes—
523C23#47914yes—
524C23C22×C22⋊C411yescharacteristic
525C23C2×#12111yescharacteristic
526C23C2×C22⋊Q811yescharacteristic
527C23D4212yes—
528C23#54612yes—
529C23C2×#13511yescharacteristic
530C23C2×#13511yescharacteristic
531C23C2×#13711yescharacteristic
532C23C2×#13711yescharacteristic
533C23#57712yes—
534C23#57812yes—
535Q8—24no—
536Q8—28no—
537Q8—24no—
538Q8—24no—
539Q8—28no—
540Q8—24no—
541Q8—24no—
542Q8—432no—
543Q8—44no—
544Q8—42no—
545Q8—42no—
546Q8—44no—
547Q8#38518yes—
548C2×C4—28no—
549C2×C4—24no—
550C2×C4—22no—
551C2×C4—24no—
552C2×C4—22no—
553C2×C4—24no—
554C2×C4—24no—
555C2×C4—28no—
556C2×C4—24no—
557C2×C4—24no—
558C2×C4—28no—
559C2×C4—24no—
560C2×C4—24no—
561C2×C4—28no—
562C2×C4—24no—
563C2×C4—24no—
564C2×C4—24no—
565C2×C4—24no—
566C2×C4—24no—
567C2×C4—24no—
568C2×C4—24no—
569C2×C4—24no—
570C2×C4—432no—
571C2×C4—44no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—44no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—44no—
584C2×C4—44no—
585C2×C4—44no—
586C2×C4—44no—
587C2×C4—44no—
588C2×C4—44no—
589C2×C4—816no—
590C2×C4—82no—
591C2×C4—82no—
592C2×C4—82no—
593C2×C4—82no—
594C2×C4—82no—
595C2×C4—82no—
596C2×C4—82no—
597C2×C4—82no—
598C2×C4—81no—
599C2×C4—81no—
600C2×C4—82no—
601C2×C4—82no—
602C2×C4—81no—
603C2×C4—81no—
604C2×C4—81no—
605C2×C4—81no—
606C2×C4—82no—
607C2×C4—82no—
608C2×C4#38014yes—
609C2×C4C22×C4⋊C411yescharacteristic
610C2×C4C2×#11711yescharacteristic
611C2×C4C2×C4×D412yes—
612C2×C4#51812yes—
613C2×C4C2×#12011yescharacteristic
614C2×C4C2×#12111yescharacteristic
615C2×C4C2×C22⋊Q811yescharacteristic
616C2×C4C2×C22⋊Q812yes—
617C2×C4C2×#12311yescharacteristic
618C2×C4#53411yescharacteristic
619C2×C4#53411yescharacteristic
620C2×C4#54312yes—
621C2×C4D4212yes—
622C2×C4#54612yes—
623C2×C4#54612yes—
624C2×C4#54612yes—
625C2×C4#55112yes—
626C2×C4#55112yes—
627C8—44no—
628C8—81no—
629C8—81no—
630C22—24no—
631C22—24no—
632C22—24no—
633C22—28no—
634C22—28no—
635C22—28no—
636C22—28no—
637C22—24no—
638C22—24no—
639C22—24no—
640C22#180518yescomplemented
641C22C2×#38011yescharacteristic
642C22D4×C22⋊C412yes—
643C22#241011yescharacteristic
644C22#241211yescharacteristic
645C22#243712yes—
646C22C2×#41611yescharacteristic
647C22C2×#41911yescharacteristic
648C22#289812yes—
649C22C2×#46411yescharacteristic
650C22C2×#46411yescharacteristic
651C22#304311yescharacteristic
652C22#304411yescharacteristic
653C22C2×#47911yescharacteristic
654C22C2×#47911yescharacteristic
655C22#308212yes—
656C22#326612yes—
657C22#327012yes—
658C22#328212yes—
659C22#328512yes—
660C22#330012yes—
661C22#330112yes—
662C22#331212yes—
663C22#332012yes—
664C4—24no—
665C4—28no—
666C4—44no—
667C4—44no—
668C4—44no—
669C4—84no—
670C4—84no—
671C4—161no—
672C4—161no—
673C4C2×#38511yescharacteristic
674C4#231511yescharacteristic
675C4#234212yes—
676C4#241011yescharacteristic
677C4#241211yescharacteristic
678C4#243712yes—
679C2—28no—
680C2#1387711yescharacteristic
681C2C2×#180511yescharacteristic
682C2#2065111yescharacteristic
683C2#2241612yes—
684C2#2242512yes—
685C2#2435711yescharacteristic
686C2#2436111yescharacteristic
687C2#2452911yescharacteristic
688C2#2453311yescharacteristic
689C2#2595712yes—
690C2#2596512yes—
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
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
4. C2×#1805C2semidirect
7. #11077C2semidirect
8. #11052C2semidirect
9. #10789C2semidirect
27. Q8×C22⋊C4C4semidirect
77. C2×C4×Q8C2×C4semidirect
117. #391C2×C4semidirect
640. C22#1805semidirect

Maximal subgroups

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

subgroupindex
C23×#1032
C2×Q8×C22⋊C42
C2×#18502
C2×#18052
C2×#17872
#138812
#110772
#110522
#107892
#107302
#107212