Finite Group Databasebeta

← #6489659

Subgroups — #6489659

Order 512.

3184 conjugacy classes of subgroups.

The subgroup lattice has 673 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#554C211yesmaximal, characteristic
2C2×#3236C211yesmaximal, characteristic, complemented
3C2×#3110C211yesmaximal, characteristic
4#35179C211yesmaximal, characteristic, complemented
5#33009C212yesmaximal
6#33007C212yesmaximal
7#29643C212yesmaximal, complemented
8#29642C212yesmaximal, complemented
9#25945C214yesmaximal, complemented
10#25850C214yesmaximal, complemented
11#24349C218yesmaximal, complemented
12#21838C211yesmaximal, characteristic, complemented
13#21822C211yesmaximal, characteristic, complemented
14#20738C211yesmaximal, characteristic
15C2×#554—28no—
16C2×#554—22no—
17C2×#554—22no—
18C2×#554—28no—
19C2×#554C2214yes—
20C2×#554C2214yes—
21C2×#533C2211yescharacteristic
22C22×C4⋊Q8C2211yescharacteristic
23C22×#125C2211yescharacteristic
24C22×#125C2211yescharacteristic
25C22×C22⋊Q8—21no—
26C22×C22⋊Q8—21no—
27C22×C22⋊Q8—21no—
28C22×C4×Q8C2211yescharacteristic
29C22×C4×D4C2211yescharacteristic
30C22×C4×D4C2211yescharacteristic
31C22×C4×D4C2211yescharacteristic
32C23×C4⋊C4—21no—
33#3236—28no—
34C2×#501C2212yescomplemented
35C2×#491C2212yescomplemented
36#3110—28no—
37C2×#484C2212yes—
38C2×#484C2212yes—
39C2×#483C2212yes—
40C2×#479C2214yescomplemented
41C2×#474C2212yes—
42C2×#461C2212yes—
43C2×#450C2212yes—
44C2×#440C2214yes—
45C2×#432C2211yescharacteristic
46C2×#432C2211yescharacteristic
47#2855C2212yes—
48#2855C2212yes—
49#2627—21no—
50#2627—21no—
51#2582C2214yescomplemented
52#2581—22no—
53#2568C2212yescomplemented
54#2567C2212yescomplemented
55#2565—22no—
56#2317—21no—
57#2310—21no—
58#2285C2211yescharacteristic
59#2059C2218yescomplemented
60#2057C2218yescomplemented
61#2026C2218yescomplemented
62#2024C2218yescomplemented
63#1992C2214yescomplemented
64#1989C2214yescomplemented
65#1985C2214yescomplemented
66#1985C2214yescomplemented
67#1877C2218yes—
68#1877C2218yes—
69#1864C2218yes—
70#1862C2218yes—
71#1824C2212yes—
72#1824C2212yes—
73#1823C2212yes—
74#1817C2212yes—
75#1817C2212yes—
76#1802C2214yes—
77#1786C2212yes—
78#1762C2214yes—
79C23×Q8—21no—
80C23×Q8D411yescharacteristic
81C23×D4C2311yescharacteristic
82C24×C4—21no—
83C24×C4—21no—
84C24×C4—21no—
85C22×SD16C2314yes—
86#554—232no—
87#554—232no—
88#554—216no—
89#554—216no—
90#554—216no—
91#554—216no—
92#533—24no—
93#533—24no—
94C2×C4⋊Q8—24no—
95C2×C4⋊Q8—24no—
96C2×C4⋊Q8—22no—
97C2×C4⋊Q8C2311yescharacteristic
98C2×C4⋊Q8C2311yescharacteristic
99C2×C4⋊Q8C2314yes—
100C2×#125—24no—
101C2×#125—24no—
102C2×#125—22no—
103C2×#125—21no—
104C2×#125—21no—
105C2×#125—24no—
106C2×#125—21no—
107C2×#125D418yes—
108C2×#125C2314yes—
109C2×#125C2312yes—
110C2×#125C2312yes—
111C2×C22⋊Q8—216no—
112C2×C22⋊Q8—216no—
113C2×C22⋊Q8—216no—
114C2×C22⋊Q8—24no—
115C2×C22⋊Q8—24no—
116C2×C22⋊Q8—21no—
117C2×C22⋊Q8—24no—
118C2×C22⋊Q8—24no—
119C2×C22⋊Q8—24no—
120C2×C22⋊Q8—24no—
121C2×C22⋊Q8C2312yes—
122C2×C4×Q8—24no—
123C2×C4×Q8—24no—
124C2×C4×Q8—21no—
125C2×C4×Q8D412yes—
126C2×C4×Q8D412yes—
127C2×C4×Q8C2312yes—
128C2×C4×Q8C2312yes—
129C2×C4×Q8C2312yes—
130C2×C4×D4—24no—
131C2×C4×D4—24no—
132C2×C4×D4—24no—
133C2×C4×D4—24no—
134C2×C4×D4—22no—
135C2×C4×D4—22no—
136C2×C4×D4—21no—
137C2×C4×D4—21no—
138C2×C4×D4—24no—
139C2×C4×D4D418yes—
140C2×C4×D4C2314yes—
141C2×C4×D4C2314yes—
142C2×C4×D4C2312yes—
143C2×C4×D4C2312yes—
144C2×#117—21no—
145C2×#117—21no—
146C22×C4⋊C4—216no—
147C22×C4⋊C4—21no—
148C22×C4⋊C4—21no—
149C22×C4⋊C4—21no—
150C22×C4⋊C4—21no—
151C22×C4⋊C4—21no—
152C22×C4⋊C4—24no—
153C22×C4⋊C4—24no—
154C22×C4⋊C4D411yescharacteristic
155C22×C4⋊C4D411yescharacteristic
156C22×C4⋊C4D411yescharacteristic
157C22×C4⋊C4C2311yescharacteristic
158C22×C4⋊C4C2311yescharacteristic
159C22×C4⋊C4C2311yescharacteristic
160C22×C22⋊C4—21no—
161C22×C22⋊C4—21no—
162C22×C22⋊C4—21no—
163C22×C42C2311yescharacteristic
164C22×C42C2311yescharacteristic
165C22×C42C2311yescharacteristic
166#501—28no—
167#491—28no—
168#484—28no—
169#484—28no—
170#483—28no—
171#479—216no—
172#474—28no—
173#461—28no—
174#450—28no—
175#440—216no—
176#432—24no—
177#432—24no—
178C2×#107C2314yes—
179C2×#107C2314yes—
180C2×#106C2314yes—
181C2×C4⋊C8C2312yes—
182C2×C4⋊C8C2312yes—
183C2×#103C2314yes—
184C2×#103C2314yes—
185C2×#103C2314yes—
186C2×#102C2318yes—
187C2×#102C2314yes—
188C2×C22⋊C8C2312yes—
189C2×C22⋊C8C2312yes—
190C2×#97C2312yes—
191C2×#97C2312yes—
192#400—22no—
193#400—22no—
194#398—24no—
195#398—24no—
196#395—22no—
197#395—22no—
198#391—22no—
199#391—22no—
200#391—24no—
201#391C2314yes—
202#391C2314yes—
203#389—24no—
204#389C2314yes—
205#389C2314yes—
206#387—24no—
207#387—24no—
208#385—22no—
209#385—22no—
210#384—22no—
211#384C2312yes—
212#382—22no—
213#382—21no—
214#381—22no—
215#381—22no—
216#380—21no—
217#380C2312yes—
218C4×C4⋊C4—21no—
219C4×C4⋊C4C2312yes—
220#340C2318yes—
221#340C2318yes—
222#340C2318yes—
223#336C2318yes—
224#336C2318yes—
225C25—21no—
226C22×Q8—28no—
227C22×Q8—24no—
228C22×Q8—24no—
229C22×Q8—24no—
230C22×Q8—21no—
231C22×Q8—42no—
232C22×Q8—84no—
233C22×Q8C2×D412yes—
234C22×Q8C2×D412yes—
235C22×Q8C2412yes—
236C22×D4—28no—
237C22×D4—21no—
238C22×D4—22no—
239C22×D4C2×D414yes—
240C22×D4C2412yes—
241C23×C4—28no—
242C23×C4—21no—
243C23×C4—21no—
244C23×C4—21no—
245C23×C4—28no—
246C23×C4—24no—
247C23×C4—28no—
248C23×C4—24no—
249C23×C4—22no—
250C23×C4—22no—
251C23×C4—44no—
252C23×C4—44no—
253C23×C4—84no—
254C23×C4—84no—
255C23×C4C2×D411yescharacteristic
256C23×C4C2×D411yescharacteristic
257C23×C4C2×D411yescharacteristic
258C23×C4C2×D411yescharacteristic
259C23×C4C2×D411yescharacteristic
260C23×C4C2×D411yescharacteristic
261C23×C4C2411yescharacteristic
262C2×SD16—28no—
263C2×SD16—216no—
264C22×C8C2414yes—
265C22×C8C2414yes—
266C4⋊Q8—216no—
267C4⋊Q8—28no—
268C4⋊Q8—28no—
269C4⋊Q8—22no—
270C4⋊Q8—22no—
271C4⋊Q8—22no—
272C4⋊Q8—22no—
273#125—216no—
274#125—28no—
275#125—28no—
276#125—28no—
277#125—28no—
278#125—24no—
279#125—24no—
280#125—44no—
281#125C2×D418yes—
282#125C2×D418yes—
283C22⋊Q8—232no—
284C22⋊Q8—232no—
285C22⋊Q8—232no—
286C22⋊Q8—232no—
287C22⋊Q8—232no—
288C22⋊Q8—232no—
289C22⋊Q8—24no—
290C22⋊Q8—24no—
291C22⋊Q8—44no—
292C4×Q8—28no—
293C4×Q8—28no—
294C4×Q8—24no—
295C4×Q8—24no—
296C4×Q8—24no—
297C4×Q8—24no—
298C4×Q8—24no—
299C4×Q8—24no—
300C4×Q8—44no—
301C4×D4—216no—
302C4×D4—216no—
303C4×D4—28no—
304C4×D4—28no—
305C4×D4—28no—
306C4×D4—28no—
307C4×D4—28no—
308C4×D4—28no—
309C4×D4—24no—
310C4×D4—24no—
311C4×D4C2×D418yes—
312C4×D4C2×D418yes—
313#117—44no—
314#117—44no—
315C2×C4⋊C4—28no—
316C2×C4⋊C4—28no—
317C2×C4⋊C4—232no—
318C2×C4⋊C4—232no—
319C2×C4⋊C4—28no—
320C2×C4⋊C4—24no—
321C2×C4⋊C4—28no—
322C2×C4⋊C4—28no—
323C2×C4⋊C4—24no—
324C2×C4⋊C4—24no—
325C2×C4⋊C4—24no—
326C2×C4⋊C4—22no—
327C2×C4⋊C4—22no—
328C2×C4⋊C4—24no—
329C2×C4⋊C4—24no—
330C2×C4⋊C4—22no—
331C2×C4⋊C4—22no—
332C2×C4⋊C4—21no—
333C2×C4⋊C4—22no—
334C2×C4⋊C4—21no—
335C2×C4⋊C4—21no—
336C2×C4⋊C4—22no—
337C2×C4⋊C4—22no—
338C2×C4⋊C4—21no—
339C2×C4⋊C4—21no—
340C2×C4⋊C4—22no—
341C2×C4⋊C4—21no—
342C2×C4⋊C4—22no—
343C2×C4⋊C4—22no—
344C2×C4⋊C4—22no—
345C2×C4⋊C4—24no—
346C2×C4⋊C4—24no—
347C2×C4⋊C4C2×D412yes—
348C2×C4⋊C4C2×D412yes—
349C2×C4⋊C4C2×D412yes—
350C2×C4⋊C4C2×D412yes—
351C2×C4⋊C4C2×D412yes—
352C2×C4⋊C4C2×D412yes—
353C2×C4⋊C4C2×D414yes—
354C2×C4⋊C4C2×D414yes—
355C2×C4⋊C4C2×D414yes—
356C2×C4⋊C4C2412yes—
357C2×C4⋊C4C2414yes—
358C2×C4⋊C4C2412yes—
359C2×C4⋊C4C2412yes—
360C2×C4⋊C4C2412yes—
361C2×C22⋊C4—28no—
362C2×C22⋊C4—28no—
363C2×C22⋊C4—28no—
364C2×C22⋊C4—24no—
365C2×C22⋊C4—24no—
366C2×C22⋊C4—22no—
367C2×C22⋊C4—22no—
368C2×C22⋊C4—22no—
369C2×C22⋊C4—22no—
370C2×C42—22no—
371C2×C42—22no—
372C2×C42—22no—
373C2×C42—22no—
374C2×C42—22no—
375C2×C42—22no—
376C2×C42C2×D412yes—
377C2×C42C2×D412yes—
378C2×C42C2×D411yescharacteristic
379C2×C42C2×D411yescharacteristic
380C2×C42C2×D414yes—
381C2×C42C2×D414yes—
382C2×C42C2412yes—
383C2×C42C2412yes—
384C2×C42C2411yescharacteristic
385C2×C42C2411yescharacteristic
386#107—28no—
387#107—28no—
388#106—28no—
389C4⋊C8—24no—
390C4⋊C8—24no—
391#103—28no—
392#103—28no—
393#103—28no—
394#102—28no—
395#102—216no—
396C22⋊C8—24no—
397C22⋊C8—24no—
398#97—24no—
399#97—24no—
400#95—22no—
401#95—22no—
402#95—24no—
403#95—24no—
404#95—24no—
405#95—24no—
406#95—24no—
407#95—24no—
408C24—28no—
409C24—24no—
410C24—22no—
411C24—44no—
412C24—42no—
413C24—42no—
414C24C22×D411yescharacteristic
415C2×Q8—216no—
416C2×Q8—24no—
417C2×Q8—24no—
418C2×Q8—24no—
419C2×Q8—24no—
420C2×Q8—22no—
421C2×Q8—22no—
422C2×Q8—28no—
423C2×Q8—42no—
424C2×Q8—42no—
425C2×Q8—416no—
426C2×Q8—42no—
427C2×Q8—82no—
428C2×Q8—88no—
429C2×Q8—816no—
430C2×D4—28no—
431C2×D4—216no—
432C2×D4—216no—
433C2×D4—24no—
434C2×D4—22no—
435C2×D4—24no—
436C2×D4C22×D414yes—
437C2×D42−1+418yes—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—24no—
443C22×C4—216no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—24no—
448C22×C4—24no—
449C22×C4—24no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—216no—
456C22×C4—28no—
457C22×C4—28no—
458C22×C4—24no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—42no—
462C22×C4—42no—
463C22×C4—42no—
464C22×C4—432no—
465C22×C4—48no—
466C22×C4—44no—
467C22×C4—44no—
468C22×C4—42no—
469C22×C4—42no—
470C22×C4—48no—
471C22×C4—84no—
472C22×C4—84no—
473C22×C4—88no—
474C22×C4—816no—
475C22×C4—816no—
476C22×C4#12014yes—
477C22×C4#12014yes—
478C22×C4#13714yes—
479C22×C4C22×D412yes—
480C22×C4C22×D412yes—
481C22×C4C22×D412yes—
482C22×C4C22×D411yescharacteristic
483C22×C4C22×D411yescharacteristic
484C22×C4C22×D411yescharacteristic
485C22×C4C22×D411yescharacteristic
486C22×C4C22×D411yescharacteristic
487C22×C4C22×D412yes—
488C22×C4C22×D412yes—
489C22×C4C22×D412yes—
490C22×C4C2511yescharacteristic
491SD16—416no—
492C2×C8—24no—
493C2×C8—24no—
494C2×C8—28no—
495C2×C8—28no—
496C4⋊C4—216no—
497C4⋊C4—216no—
498C4⋊C4—216no—
499C4⋊C4—24no—
500C4⋊C4—24no—
501C4⋊C4—28no—
502C4⋊C4—28no—
503C4⋊C4—28no—
504C4⋊C4—28no—
505C4⋊C4—24no—
506C4⋊C4—24no—
507C4⋊C4—24no—
508C4⋊C4—24no—
509C4⋊C4—24no—
510C4⋊C4—24no—
511C4⋊C4—24no—
512C4⋊C4—22no—
513C4⋊C4—22no—
514C4⋊C4—22no—
515C4⋊C4—22no—
516C4⋊C4—22no—
517C4⋊C4—28no—
518C4⋊C4—44no—
519C4⋊C4—42no—
520C4⋊C4—42no—
521C4⋊C4—44no—
522C4⋊C4—44no—
523C4⋊C4—42no—
524C4⋊C4—82no—
525C4⋊C4—84no—
526C4⋊C4—84no—
527C4⋊C4C22×D414yes—
528C4⋊C4C22×D414yes—
529C4⋊C4C22×D414yes—
530C22⋊C4—216no—
531C22⋊C4—216no—
532C22⋊C4—28no—
533C22⋊C4—28no—
534C22⋊C4—44no—
535C22⋊C4—44no—
536C42—24no—
537C42—24no—
538C42—22no—
539C42—22no—
540C42—22no—
541C42—22no—
542C42—44no—
543C42—44no—
544C42#12014yes—
545C42#12014yes—
546C42C22×D412yes—
547C42C22×D412yes—
548C42C22×D412yes—
549C42C22×D412yes—
550C23—22no—
551C23—22no—
552C23—216no—
553C23—24no—
554C23—28no—
555C23—48no—
556C23—416no—
557C23—416no—
558C23—44no—
559C23—48no—
560C23—42no—
561C23—42no—
562C23C2×#12012yes—
563C23C2×#13712yes—
564C23#57512yes—
565C23C23×D411yescharacteristic
566Q8—48no—
567Q8—48no—
568Q8—44no—
569Q8—44no—
570Q8—84no—
571Q8—816no—
572D4—216no—
573D4—24no—
574D4—28no—
575D4#53618yes—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—28no—
581C2×C4—28no—
582C2×C4—21no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—22no—
587C2×C4—24no—
588C2×C4—24no—
589C2×C4—24no—
590C2×C4—24no—
591C2×C4—22no—
592C2×C4—22no—
593C2×C4—22no—
594C2×C4—24no—
595C2×C4—24no—
596C2×C4—24no—
597C2×C4—22no—
598C2×C4—22no—
599C2×C4—44no—
600C2×C4—432no—
601C2×C4—44no—
602C2×C4—48no—
603C2×C4—48no—
604C2×C4—44no—
605C2×C4—44no—
606C2×C4—44no—
607C2×C4—44no—
608C2×C4—48no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—44no—
612C2×C4—44no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—44no—
616C2×C4—44no—
617C2×C4—44no—
618C2×C4—88no—
619C2×C4—88no—
620C2×C4—816no—
621C2×C4C2×#12012yes—
622C2×C4C2×#12012yes—
623C2×C4C2×#12012yes—
624C2×C4C2×#12012yes—
625C2×C4C2×#12012yes—
626C2×C4C2×#13714yes—
627C2×C4C23×D411yescharacteristic
628C2×C4C23×D411yescharacteristic
629C2×C4C2×2−1+414yes—
630C8—44no—
631C8—44no—
632C22—24no—
633C22—24no—
634C22—22no—
635C22—22no—
636C22—28no—
637C22—432no—
638C22—48no—
639C22—48no—
640C22—44no—
641C22—44no—
642C22#300714yes—
643C22C22×#12011yescharacteristic
644C22#348612yes—
645C22C22×#13711yescharacteristic
646C22C2×#57511yescharacteristic
647C22#358512yes—
648C4—22no—
649C4—24no—
650C4—22no—
651C4—22no—
652C4—22no—
653C4—22no—
654C4—42no—
655C4—44no—
656C4—42no—
657C4—42no—
658C4—44no—
659C4—44no—
660C4—84no—
661C4—82no—
662C4—82no—
663C4C2×#53612yes—
664C4#348612yes—
665C2—24no—
666C2—48no—
667C2#3524211yescharacteristic
668C2C2×#300711yescharacteristic
669C2#6036711yescharacteristic
670C2#6182612yes—
671C2#6183112yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC23×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C4×Q8 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#3236C2semidirect
4. #35179C2semidirect
7. #29643C2semidirect
8. #29642C2semidirect
9. #25945C2semidirect
10. #25850C2semidirect
11. #24349C2semidirect
12. #21838C2semidirect
13. #21822C2semidirect
34. C2×#501C22semidirect
35. C2×#491C22semidirect
40. C2×#479C22semidirect
51. #2582C22semidirect
53. #2568C22semidirect
54. #2567C22semidirect
59. #2059C22semidirect
60. #2057C22semidirect
61. #2026C22semidirect
62. #2024C22semidirect
63. #1992C22semidirect
64. #1989C22semidirect
65. #1985C22semidirect
66. #1985C22semidirect

Maximal subgroups

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

subgroupindex
C22×#5542
C2×#32362
C2×#31102
#351792
#330092
#330072
#296432
#296422
#259452
#258502
#243492
#218382
#218222
#207382