Finite Group Databasebeta

← #5956430

Subgroups — #5956430

Order 512.

5710 conjugacy classes of subgroups.

The subgroup lattice has 674 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×C22⋊Q8C211yesmaximal, characteristic
2C2×#2317C211yesmaximal, characteristic, complemented
3C2×#2310C211yesmaximal, characteristic, complemented
4C2×#2310C211yesmaximal, characteristic, complemented
5C2×#2280C211yesmaximal, characteristic
6C2×#2280C211yesmaximal, characteristic
7#15236C214yesmaximal, complemented
8#15182C214yesmaximal, complemented
9#15179C218yesmaximal, complemented
10#15073C214yesmaximal, complemented
11#14918C214yesmaximal
12#13795C211yesmaximal, characteristic, complemented
13C24×Q8C2211yescharacteristic
14C25×C4C2211yescharacteristic
15C22×C22⋊Q8—28no—
16C22×C22⋊Q8—28no—
17C22×C22⋊Q8—28no—
18C22×C22⋊Q8C2214yes—
19C22×C22⋊Q8C2214yes—
20C22×#117C2211yescharacteristic
21C22×#117C2211yescharacteristic
22C22×#117C2211yescharacteristic
23C22×#117C2211yescharacteristic
24C23×C4⋊C4C2212yes—
25C23×C4⋊C4C2211yescharacteristic
26C23×C22⋊C4C2212yes—
27#2317—28no—
28#2310—28no—
29#2310—28no—
30C2×#391C2214yescomplemented
31C2×#391C2214yescomplemented
32C2×#391C2212yescomplemented
33C2×#391C2214yescomplemented
34C2×#391C2212yescomplemented
35C2×#389C2214yescomplemented
36C2×#385C2214yes—
37C2×#384C2212yescomplemented
38C2×#382C2212yescomplemented
39#2280—28no—
40#2280—28no—
41C2×#381C2214yes—
42C2×#381C2214yes—
43C2×#381C2212yes—
44C2×#381C2212yes—
45C2×#380C2214yes—
46C2×#380C2212yes—
47C2×#380C2212yes—
48C2×#379C2212yes—
49C2×#379C2212yes—
50C2×C4×C22⋊C4C2214yes—
51C2×C4×C22⋊C4C2214yes—
52#1442C22116yescomplemented
53#1441C22116yescomplemented
54#1440C22116yes—
55#1439C22116yes—
56C4⋊#95C2214yescomplemented
57C4⋊#95C2214yescomplemented
58C22⋊#95C2214yes—
59C26C2311yescharacteristic
60C23×Q8—24no—
61C23×Q8—24no—
62C23×Q8—24no—
63C23×Q8—24no—
64C23×Q8—24no—
65C23×Q8—24no—
66C23×Q8D418yes—
67C23×Q8C2312yes—
68C23×Q8C2312yes—
69C24×C4—24no—
70C24×C4—24no—
71C24×C4—24no—
72C24×C4—24no—
73C24×C4—24no—
74C24×C4—24no—
75C24×C4D418yes—
76C24×C4C2311yescharacteristic
77C24×C4C2312yes—
78C24×C4C2312yes—
79C24×C4C2312yes—
80C24×C4C2312yes—
81C24×C4C2311yescharacteristic
82C2×C22⋊Q8—2128no—
83C2×C22⋊Q8—216no—
84C2×C22⋊Q8—216no—
85C2×C22⋊Q8—216no—
86C2×C22⋊Q8—216no—
87C2×C22⋊Q8—216no—
88C2×C22⋊Q8—216no—
89C2×#117—24no—
90C2×#117—24no—
91C2×#117—24no—
92C2×#117—24no—
93C2×#117—24no—
94C2×#117—24no—
95C2×#117—24no—
96C2×#117—24no—
97C2×#117—24no—
98C2×#117—24no—
99C2×#117—24no—
100C2×#117D418yes—
101C22×C4⋊C4—28no—
102C22×C4⋊C4—28no—
103C22×C4⋊C4—28no—
104C22×C4⋊C4—24no—
105C22×C4⋊C4—24no—
106C22×C4⋊C4D418yes—
107C22×C4⋊C4C2318yes—
108C22×C4⋊C4C2312yes—
109C22×C4⋊C4C2312yes—
110C22×C4⋊C4C2311yescharacteristic
111C22×C4⋊C4C2311yescharacteristic
112C22×C4⋊C4C2312yes—
113C22×C4⋊C4C2312yes—
114C22×C4⋊C4C2312yes—
115C22×C4⋊C4C2318yes—
116C22×C22⋊C4—28no—
117C22×C22⋊C4—28no—
118C22×C22⋊C4—28no—
119C22×C22⋊C4C2318yes—
120C22×C22⋊C4C2312yes—
121C22×C22⋊C4C2312yes—
122C22×C22⋊C4C2312yes—
123C22×C22⋊C4C2314yes—
124C22×C22⋊C4C2318yes—
125C22×C22⋊C4C2312yes—
126C22×C22⋊C4C2314yes—
127C22×C42C2312yes—
128C22×C42C2312yes—
129C22×C42C2312yes—
130C22×C42C2312yes—
131#391—216no—
132#391—28no—
133#391—28no—
134#391—216no—
135#391—216no—
136#389—216no—
137#385—216no—
138#384—28no—
139#382—28no—
140#381—216no—
141#381—28no—
142#381—216no—
143#381—28no—
144#380—28no—
145#380—28no—
146#380—216no—
147#379—28no—
148#379—28no—
149C4×C22⋊C4—216no—
150C4×C22⋊C4—216no—
151C2×#95C2318yes—
152C2×#95C23116yes—
153C2×#95C2318yes—
154C2×#95C2318yes—
155C2×#95C2314yes—
156C2×#95C2318yes—
157C2×#95C2314yes—
158C2×#95C23116yes—
159C25—22no—
160C25—22no—
161C25—28no—
162C25—22no—
163C25—22no—
164C25—22no—
165C25—22no—
166C25—44no—
167C25C2×D414yes—
168C25C2412yes—
169C25C2411yescharacteristic
170C22×Q8—232no—
171C22×Q8—232no—
172C22×Q8—232no—
173C22×Q8—24no—
174C22×Q8—24no—
175C22×Q8—24no—
176C22×Q8—24no—
177C22×Q8—24no—
178C22×Q8—24no—
179C22×Q8—24no—
180C22×Q8—24no—
181C22×Q8—24no—
182C22×Q8—24no—
183C22×Q8—24no—
184C22×Q8—24no—
185C22×Q8—416no—
186C22×Q8—416no—
187C22×Q8—416no—
188C22×Q8—416no—
189C22×Q8C2×D418yes—
190C22×Q8C2×D418yes—
191C23×C4—24no—
192C23×C4—24no—
193C23×C4—24no—
194C23×C4—24no—
195C23×C4—22no—
196C23×C4—22no—
197C23×C4—22no—
198C23×C4—24no—
199C23×C4—22no—
200C23×C4—22no—
201C23×C4—22no—
202C23×C4—24no—
203C23×C4—24no—
204C23×C4—24no—
205C23×C4—232no—
206C23×C4—232no—
207C23×C4—232no—
208C23×C4—24no—
209C23×C4—24no—
210C23×C4—24no—
211C23×C4—24no—
212C23×C4—24no—
213C23×C4—24no—
214C23×C4—24no—
215C23×C4—24no—
216C23×C4—24no—
217C23×C4—24no—
218C23×C4—24no—
219C23×C4—24no—
220C23×C4—48no—
221C23×C4—416no—
222C23×C4—416no—
223C23×C4—416no—
224C23×C4—416no—
225C23×C4C2×D418yes—
226C23×C4C2×D414yes—
227C23×C4C2×D418yes—
228C23×C4C2×D418yes—
229C23×C4C2×D418yes—
230C23×C4C2×D414yes—
231C23×C4C2×D414yes—
232C23×C4C2×D414yes—
233C23×C4C2×D414yes—
234C23×C4#4114yes—
235C23×C4#4114yes—
236C23×C4#4114yes—
237C23×C4C2412yes—
238C23×C4C2412yes—
239C23×C4C2414yes—
240C23×C4C2414yes—
241C23×C4C2414yes—
242C23×C4C2414yes—
243C23×C4C2414yes—
244C23×C4C2412yes—
245C23×C4C2411yescharacteristic
246C23×C4C2411yescharacteristic
247C22⋊Q8—2128no—
248C22⋊Q8—2128no—
249#117—232no—
250#117—416no—
251#117—416no—
252#117—416no—
253C2×C4⋊C4—264no—
254C2×C4⋊C4—232no—
255C2×C4⋊C4—216no—
256C2×C4⋊C4—216no—
257C2×C4⋊C4—216no—
258C2×C4⋊C4—24no—
259C2×C4⋊C4—24no—
260C2×C4⋊C4—24no—
261C2×C4⋊C4—24no—
262C2×C4⋊C4—24no—
263C2×C4⋊C4—24no—
264C2×C4⋊C4—22no—
265C2×C4⋊C4—22no—
266C2×C4⋊C4—24no—
267C2×C4⋊C4—24no—
268C2×C4⋊C4—22no—
269C2×C4⋊C4—22no—
270C2×C4⋊C4—22no—
271C2×C4⋊C4—22no—
272C2×C4⋊C4—24no—
273C2×C4⋊C4—24no—
274C2×C4⋊C4—24no—
275C2×C4⋊C4—24no—
276C2×C4⋊C4—216no—
277C2×C4⋊C4—416no—
278C2×C4⋊C4C2×D418yes—
279C2×C4⋊C4C2×D418yes—
280C2×C4⋊C4C2×D418yes—
281C2×C22⋊C4—264no—
282C2×C22⋊C4—216no—
283C2×C22⋊C4—24no—
284C2×C22⋊C4—24no—
285C2×C22⋊C4—216no—
286C2×C22⋊C4—216no—
287C2×C22⋊C4—24no—
288C2×C22⋊C4—24no—
289C2×C22⋊C4—24no—
290C2×C22⋊C4—24no—
291C2×C22⋊C4—24no—
292C2×C22⋊C4—24no—
293C2×C22⋊C4—24no—
294C2×C22⋊C4—28no—
295C2×C22⋊C4—28no—
296C2×C22⋊C4—24no—
297C2×C22⋊C4—24no—
298C2×C22⋊C4—216no—
299C2×C22⋊C4—216no—
300C2×C22⋊C4—216no—
301C2×C22⋊C4—48no—
302C2×C22⋊C4—48no—
303C2×C22⋊C4C2×D418yes—
304C2×C42—24no—
305C2×C42—24no—
306C2×C42—24no—
307C2×C42—24no—
308C2×C42—24no—
309C2×C42—24no—
310C2×C42—24no—
311C2×C42—24no—
312C2×C42—24no—
313C2×C42—24no—
314C2×C42—24no—
315C2×C42C2×D418yes—
316#95—216no—
317#95—28no—
318#95—28no—
319#95—232no—
320#95—216no—
321#95—216no—
322#95—216no—
323#95—416no—
324C24—24no—
325C24—24no—
326C24—216no—
327C24—216no—
328C24—28no—
329C24—216no—
330C24—22no—
331C24—22no—
332C24—22no—
333C24—22no—
334C24—22no—
335C24—22no—
336C24—24no—
337C24—24no—
338C24—24no—
339C24—28no—
340C24—28no—
341C24—48no—
342C24—416no—
343C24—416no—
344C24—416no—
345C24—416no—
346C24—48no—
347C24—48no—
348C24—44no—
349C24—44no—
350C24—44no—
351C24—44no—
352C24C22×D414yes—
353C24C22×D412yes—
354C24C22×D412yes—
355C24C22×D412yes—
356C24C22×D412yes—
357C24C2×#4112yes—
358C24C2×#4112yes—
359C24C2×#4112yes—
360C242−1+418yes—
361C24C2511yescharacteristic
362C2×Q8—216no—
363C2×Q8—216no—
364C2×Q8—216no—
365C2×Q8—216no—
366C2×Q8—216no—
367C2×Q8—216no—
368C2×Q8—48no—
369C2×Q8—48no—
370C2×Q8—48no—
371C2×Q8—48no—
372C2×Q8—48no—
373C2×Q8—48no—
374C2×Q8—48no—
375C2×Q8—48no—
376C2×Q8—4128no—
377C22×C4—216no—
378C22×C4—216no—
379C22×C4—28no—
380C22×C4—28no—
381C22×C4—216no—
382C22×C4—28no—
383C22×C4—216no—
384C22×C4—22no—
385C22×C4—24no—
386C22×C4—24no—
387C22×C4—24no—
388C22×C4—24no—
389C22×C4—24no—
390C22×C4—24no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—24no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—24no—
398C22×C4—24no—
399C22×C4—24no—
400C22×C4—28no—
401C22×C4—216no—
402C22×C4—28no—
403C22×C4—216no—
404C22×C4—216no—
405C22×C4—216no—
406C22×C4—216no—
407C22×C4—216no—
408C22×C4—216no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—24no—
413C22×C4—24no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—24no—
417C22×C4—24no—
418C22×C4—22no—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—28no—
422C22×C4—28no—
423C22×C4—28no—
424C22×C4—28no—
425C22×C4—28no—
426C22×C4—28no—
427C22×C4—416no—
428C22×C4—416no—
429C22×C4—416no—
430C22×C4—44no—
431C22×C4—44no—
432C22×C4—48no—
433C22×C4—48no—
434C22×C4—44no—
435C22×C4—48no—
436C22×C4—44no—
437C22×C4—48no—
438C22×C4—48no—
439C22×C4—48no—
440C22×C4—48no—
441C22×C4—48no—
442C22×C4—48no—
443C22×C4—48no—
444C22×C4—48no—
445C22×C4—48no—
446C22×C4—4128no—
447C22×C4#120116yes—
448C22×C4#121116yes—
449C22×C4#121116yes—
450C22×C4#121116yes—
451C22×C4#123116yes—
452C22×C4#123116yes—
453C22×C4#124116yes—
454C22×C4C22×D418yes—
455C22×C4C22×D418yes—
456C22×C4C22×D414yes—
457C22×C4C22×D418yes—
458C22×C4C22×D412yes—
459C22×C4C22×D412yes—
460C22×C4C22×D412yes—
461C22×C4C22×D412yes—
462C22×C4C22×D412yes—
463C22×C4C22×D412yes—
464C22×C4C22×D412yes—
465C22×C4C22×D412yes—
466C22×C4C2×#4112yes—
467C22×C4C2×#4112yes—
468C22×C4C2×#4112yes—
469C22×C4C2×#4112yes—
470C22×C4C2×#4112yes—
471C22×C4C2×#4112yes—
472C4⋊C4—264no—
473C4⋊C4—216no—
474C4⋊C4—216no—
475C4⋊C4—216no—
476C4⋊C4—44no—
477C4⋊C4—44no—
478C4⋊C4—48no—
479C4⋊C4—48no—
480C4⋊C4—816no—
481C22⋊C4—264no—
482C22⋊C4—216no—
483C22⋊C4—432no—
484C22⋊C4—416no—
485C22⋊C4—48no—
486C22⋊C4—48no—
487C22⋊C4—48no—
488C22⋊C4—416no—
489C42—216no—
490C42—48no—
491C42—48no—
492C42—48no—
493C23—24no—
494C23—24no—
495C23—24no—
496C23—24no—
497C23—24no—
498C23—24no—
499C23—28no—
500C23—28no—
501C23—28no—
502C23—28no—
503C23—232no—
504C23—24no—
505C23—24no—
506C23—24no—
507C23—24no—
508C23—24no—
509C23—24no—
510C23—28no—
511C23—28no—
512C23—28no—
513C23—28no—
514C23—28no—
515C23—28no—
516C23—432no—
517C23—432no—
518C23—432no—
519C23—432no—
520C23—416no—
521C23—432no—
522C23—432no—
523C23—416no—
524C23—42no—
525C23—42no—
526C23—42no—
527C23—42no—
528C23—44no—
529C23—44no—
530C23—44no—
531C23—44no—
532C23—44no—
533C23—44no—
534C23C2×#12014yes—
535C23C2×#12114yes—
536C23C2×#12114yes—
537C23C2×#12114yes—
538C23C2×#12314yes—
539C23C2×#12314yes—
540C23C2×#12414yes—
541C23#53618yes—
542C23C23×D411yescharacteristic
543C23C23×D411yescharacteristic
544C23C23×D411yescharacteristic
545C23C23×D411yescharacteristic
546C23C22×#4111yescharacteristic
547C23C22×#4111yescharacteristic
548C23C22×#4111yescharacteristic
549C23C2×2−1+414yes—
550C23#58514yes—
551Q8—432no—
552Q8—432no—
553C2×C4—28no—
554C2×C4—28no—
555C2×C4—24no—
556C2×C4—28no—
557C2×C4—24no—
558C2×C4—28no—
559C2×C4—24no—
560C2×C4—28no—
561C2×C4—216no—
562C2×C4—216no—
563C2×C4—432no—
564C2×C4—416no—
565C2×C4—432no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—44no—
569C2×C4—44no—
570C2×C4—42no—
571C2×C4—44no—
572C2×C4—42no—
573C2×C4—44no—
574C2×C4—48no—
575C2×C4—44no—
576C2×C4—44no—
577C2×C4—48no—
578C2×C4—48no—
579C2×C4—432no—
580C2×C4—432no—
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—44no—
590C2×C4—44no—
591C2×C4—44no—
592C2×C4—44no—
593C2×C4—44no—
594C2×C4—44no—
595C2×C4—48no—
596C2×C4—48no—
597C2×C4—48no—
598C2×C4—48no—
599C2×C4#394164yes—
600C2×C4C2×#12014yes—
601C2×C4C2×#12014yes—
602C2×C4C2×#12114yes—
603C2×C4C2×#12114yes—
604C2×C4C2×#12114yes—
605C2×C4C2×#12114yes—
606C2×C4C2×#12114yes—
607C2×C4C2×#12114yes—
608C2×C4C2×#12314yes—
609C2×C4C2×#12314yes—
610C2×C4C2×#12314yes—
611C2×C4C2×#12314yes—
612C2×C4C2×#12414yes—
613C2×C4C2×#12414yes—
614C22—28no—
615C22—28no—
616C22—24no—
617C22—28no—
618C22—28no—
619C22—24no—
620C22—28no—
621C22—28no—
622C22—28no—
623C22—28no—
624C22—28no—
625C22—464no—
626C22—48no—
627C22—48no—
628C22—44no—
629C22—48no—
630C22—48no—
631C22—44no—
632C22—44no—
633C22—44no—
634C22C2×#39418yes—
635C22C22×#12011yescharacteristic
636C22C22×#12111yescharacteristic
637C22C22×#12111yescharacteristic
638C22C22×#12111yescharacteristic
639C22C22×#12311yescharacteristic
640C22C22×#12311yescharacteristic
641C22C22×#12411yescharacteristic
642C22C2×#53612yes—
643C22#344912yes—
644C22#348512yes—
645C22#348612yes—
646C22#348612yes—
647C22#348712yes—
648C22#348914yes—
649C22#349514yes—
650C22#349514yes—
651C22#349514yes—
652C4—48no—
653C4—48no—
654C4—44no—
655C4—48no—
656C4—82no—
657C4—84no—
658C4—82no—
659C4—84no—
660C4C2×#39418yes—
661C4C2×#39418yes—
662C2—28no—
663C2—48no—
664C2C22×#39411yescharacteristic
665C2#3514911yescharacteristic
666C2#3524211yescharacteristic
667C2#3530612yes—
668C2#3531112yes—
669C2#3563312yes—
670C2#3564412yes—
671C2#3566112yes—
672C2#3567212yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC26C23
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2317C2semidirect
3. C2×#2310C2semidirect
4. C2×#2310C2semidirect
7. #15236C2semidirect
8. #15182C2semidirect
9. #15179C2semidirect
10. #15073C2semidirect
12. #13795C2semidirect
30. C2×#391C22semidirect
31. C2×#391C22semidirect
32. C2×#391C22semidirect
33. C2×#391C22semidirect
34. C2×#391C22semidirect
35. C2×#389C22semidirect
37. C2×#384C22semidirect
38. C2×#382C22semidirect
52. #1442C22semidirect
53. #1441C22semidirect
56. C4⋊#95C22semidirect
57. C4⋊#95C22semidirect

Maximal subgroups

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

subgroupindex
C23×C22⋊Q82
C2×#23172
C2×#23102
C2×#22802
#152362
#151822
#151792
#150732
#149182
#137952