Finite Group Databasebeta

← #705795

Subgroups — #705795

Order 512.

2390 conjugacy classes of subgroups.

The subgroup lattice has 669 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2439C211yesmaximal, characteristic
2C2×#2317C211yesmaximal, characteristic
3#17067C212yesmaximal, complemented
4#16694C212yesmaximal, complemented
5#15555C214yesmaximal
6#15378C214yesmaximal
7#15225C212yesmaximal, complemented
8#15221C212yesmaximal, complemented
9#15185C212yesmaximal, complemented
10#15021C212yesmaximal, complemented
11#14264C211yesmaximal, characteristic
12#14130C211yesmaximal, characteristic
13#14053C212yesmaximal, complemented
14#13954C212yesmaximal, complemented
15#13881C211yesmaximal, characteristic
16#13844C211yesmaximal, characteristic
17#13801C211yesmaximal, characteristic
18C22×C4⋊Q8C2211yescharacteristic
19C22×#117C2211yescharacteristic
20C22×#117C2211yescharacteristic
21C23×C4⋊C4C2211yescharacteristic
22#2439—28no—
23C2×#396C2212yes—
24C2×#393C2212yes—
25#2317—28no—
26C2×#391C2212yes—
27C2×#391C2214yes—
28C2×#389C2212yes—
29C2×#389C2212yes—
30C2×#389C2211yescharacteristic
31C2×#389C2211yescharacteristic
32C2×#389C2212yes—
33C2×#389C2212yes—
34C2×#387C2212yes—
35C2×#385C2214yes—
36C2×#385C2212yes—
37C2×#384C2212yes—
38C2×#382C2212yes—
39C2×#382C2211yescharacteristic
40C2×#382C2212yes—
41C2×#380C2211yescharacteristic
42C2×#380C2211yescharacteristic
43C2×C4×C4⋊C4C2212yes—
44C2×C4×C4⋊C4C2212yes—
45C2×C4×C22⋊C4C2211yescharacteristic
46C2×C4×C22⋊C4C2211yescharacteristic
47#1446C2214yes—
48#1445C2214yes—
49#1445C2214yes—
50#1444C2214yes—
51#1444C2214yes—
52#1444C2211yescharacteristic
53#1444C2211yescharacteristic
54#1444C2212yes—
55#1444C2214yes—
56#1444C2212yes—
57#1442C2214yes—
58#1442C2214yes—
59#1442C2211yescharacteristic
60#1442C2211yescharacteristic
61#1442C2212yes—
62#1442C2212yes—
63#1442C2214yes—
64#1442C2212yes—
65#1442C2212yes—
66#1441C2214yes—
67#1441C2214yes—
68#1441C2214yes—
69#1439C2212yes—
70#1439C2214yes—
71#1439C2212yes—
72#1437C2214yes—
73#1436C2214yes—
74C4⋊#95C2214yes—
75C4⋊#95C2212yes—
76C4⋊#95C2212yes—
77C4⋊#95C2214yes—
78C4⋊#95C2212yes—
79C4⋊#95C2212yes—
80#1434C2212yes—
81C22⋊#95C2214yes—
82C22⋊#95C2212yes—
83C4×#95C2212yes—
84C4×#95C2212yes—
85C23×Q8C2312yes—
86C24×C4C2311yescharacteristic
87C24×C4C2312yes—
88C2×C4⋊Q8—24no—
89C2×C4⋊Q8—28no—
90C2×#117—24no—
91C2×#117—28no—
92C2×#117—44no—
93C2×#117D418yes—
94C22×C4⋊C4—24no—
95C22×C4⋊C4—44no—
96C22×C4⋊C4C2312yes—
97C22×C4⋊C4C2312yes—
98C22×C4⋊C4C2311yescharacteristic
99C22×C4⋊C4C2311yescharacteristic
100C22×C4⋊C4C2312yes—
101C22×C4⋊C4C2311yescharacteristic
102C22×C4⋊C4C2312yes—
103C22×C4⋊C4C2312yes—
104C22×C4⋊C4C2312yes—
105C22×C4⋊C4C2311yescharacteristic
106C22×C4⋊C4C2311yescharacteristic
107C22×C4⋊C4C2314yes—
108C22×C4⋊C4C2314yes—
109C22×C4⋊C4C2314yes—
110C22×C4⋊C4C2314yes—
111C22×C4⋊C4C2312yes—
112C22×C4⋊C4C2312yes—
113C22×C4⋊C4C2311yescharacteristic
114C22×C22⋊C4C2311yescharacteristic
115C22×C22⋊C4C2311yescharacteristic
116C22×C22⋊C4C2312yes—
117C22×C22⋊C4C2312yes—
118C22×C22⋊C4C2312yes—
119C22×C22⋊C4C2312yes—
120C22×C22⋊C4C2312yes—
121C22×C42C2312yes—
122C22×C42C2311yescharacteristic
123C22×C42C2311yescharacteristic
124C22×C42C2312yes—
125C22×C42C2312yes—
126C22×C42C2314yes—
127#396—28no—
128#393—28no—
129#391—28no—
130#391—48no—
131#389—28no—
132#389—28no—
133#389—24no—
134#389—24no—
135#389—44no—
136#389—44no—
137#387—28no—
138#385—216no—
139#385—44no—
140#384—28no—
141#382—24no—
142#382—28no—
143#382—44no—
144#380—24no—
145#380Q818yescomplemented
146C4×C4⋊C4—28no—
147C4×C4⋊C4—44no—
148C4×C22⋊C4—24no—
149C4×C22⋊C4Q818yescomplemented
150C2×#95C2312yes—
151C2×#95C2312yes—
152C2×#95C2312yes—
153C2×#95C2312yes—
154C2×#95C2312yes—
155C2×#95C2312yes—
156C2×#95C2314yes—
157C2×#95C2314yes—
158C2×#95C2314yes—
159C2×#95C2314yes—
160C2×#95C2311yescharacteristic
161C2×#95C2312yes—
162C2×#95C2311yescharacteristic
163C2×#95C2314yes—
164C2×#95C2312yes—
165C2×#95C2312yes—
166C2×#95C2314yes—
167C2×#95C2314yes—
168C2×#95C2314yes—
169C2×#95C2314yes—
170C2×#95C2314yes—
171C2×#95C2314yes—
172C2×#95C2314yes—
173C2×#95C2312yes—
174C2×#95C2312yes—
175C2×#95C2314yes—
176C2×#95C2314yes—
177C2×#95C2314yes—
178C2×#95C2311yescharacteristic
179C2×#95C2312yes—
180C2×#95C2311yescharacteristic
181C25C2411yescharacteristic
182C22×Q8—24no—
183C22×Q8—28no—
184C22×Q8—24no—
185C22×Q8—44no—
186C22×Q8—42no—
187C23×C4—24no—
188C23×C4—24no—
189C23×C4—24no—
190C23×C4—44no—
191C23×C4—42no—
192C23×C4—44no—
193C23×C4—42no—
194C23×C4C2×D414yes—
195C23×C4C2×Q814yes—
196C23×C4C2×Q814yes—
197C23×C4C2411yescharacteristic
198C23×C4C2412yes—
199C23×C4C2412yes—
200C23×C4C2412yes—
201C23×C4C2411yescharacteristic
202C23×C4C2412yes—
203C23×C4C2414yes—
204C23×C4C2412yes—
205C23×C4C2414yes—
206C23×C4C2412yes—
207C23×C4C2412yes—
208C23×C4C2411yescharacteristic
209C23×C4C2411yescharacteristic
210C23×C4C2411yescharacteristic
211C23×C4C2412yes—
212C23×C4C2411yescharacteristic
213C4⋊Q8—416no—
214#117—232no—
215#117—416no—
216C2×C4⋊C4—24no—
217C2×C4⋊C4—24no—
218C2×C4⋊C4—24no—
219C2×C4⋊C4—24no—
220C2×C4⋊C4—24no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—24no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—24no—
227C2×C4⋊C4—28no—
228C2×C4⋊C4—24no—
229C2×C4⋊C4—24no—
230C2×C4⋊C4—24no—
231C2×C4⋊C4—24no—
232C2×C4⋊C4—24no—
233C2×C4⋊C4—28no—
234C2×C4⋊C4—28no—
235C2×C4⋊C4—24no—
236C2×C4⋊C4—44no—
237C2×C4⋊C4—42no—
238C2×C4⋊C4—42no—
239C2×C4⋊C4—42no—
240C2×C4⋊C4—42no—
241C2×C4⋊C4—42no—
242C2×C4⋊C4—42no—
243C2×C4⋊C4—42no—
244C2×C4⋊C4—42no—
245C2×C4⋊C4—42no—
246C2×C4⋊C4—416no—
247C2×C4⋊C4—44no—
248C2×C4⋊C4—44no—
249C2×C4⋊C4—44no—
250C2×C4⋊C4—44no—
251C2×C4⋊C4—44no—
252C2×C4⋊C4—44no—
253C2×C4⋊C4—44no—
254C2×C4⋊C4—44no—
255C2×C4⋊C4—44no—
256C2×C4⋊C4—44no—
257C2×C4⋊C4—44no—
258C2×C4⋊C4C2×D414yes—
259C2×C4⋊C4C2×D414yes—
260C2×C4⋊C4C2×Q818yes—
261C2×C4⋊C4#4118yes—
262C2×C22⋊C4—28no—
263C2×C22⋊C4—24no—
264C2×C22⋊C4—22no—
265C2×C22⋊C4—22no—
266C2×C22⋊C4—22no—
267C2×C22⋊C4—22no—
268C2×C22⋊C4—24no—
269C2×C22⋊C4—42no—
270C2×C22⋊C4—42no—
271C2×C22⋊C4—42no—
272C2×C22⋊C4—41no—
273C2×C22⋊C4—41no—
274C2×C22⋊C4—42no—
275C2×C22⋊C4—42no—
276C2×C22⋊C4—42no—
277C2×C22⋊C4—44no—
278C2×C22⋊C4C2×D418yes—
279C2×C22⋊C4C2×Q818yes—
280C2×C22⋊C4C2×Q818yes—
281C2×C42—28no—
282C2×C42—24no—
283C2×C42—24no—
284C2×C42—24no—
285C2×C42—22no—
286C2×C42—22no—
287C2×C42—24no—
288C2×C42—24no—
289C2×C42—24no—
290C2×C42—24no—
291C2×C42—28no—
292C2×C42—44no—
293C2×C42—44no—
294C2×C42C2×D418yes—
295C2×C42C2×Q818yes—
296#95—22no—
297#95—22no—
298#95—24no—
299#95—28no—
300#95—28no—
301#95—24no—
302#95—24no—
303#95—24no—
304#95—24no—
305#95—28no—
306#95—28no—
307#95—24no—
308#95—24no—
309#95—44no—
310#95—42no—
311#95—44no—
312#95—44no—
313#95—44no—
314#95—44no—
315#95—42no—
316#95—42no—
317#95—42no—
318#95—44no—
319#95—44no—
320#95—44no—
321#95—44no—
322#95—44no—
323#95—44no—
324#95C2×Q814yes—
325#95C2×Q814yes—
326#95C2×Q818yes—
327C24—22no—
328C24—21no—
329C24—22no—
330C24—21no—
331C24—22no—
332C24—21no—
333C24—22no—
334C24—21no—
335C24C22×D412yes—
336C24C22×Q812yes—
337C24C22×Q812yes—
338C24C2511yescharacteristic
339C2×Q8—48no—
340C2×Q8—416no—
341C2×Q8—48no—
342C2×Q8—84no—
343C2×Q8—88no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—24no—
348C22×C4—22no—
349C22×C4—28no—
350C22×C4—28no—
351C22×C4—216no—
352C22×C4—216no—
353C22×C4—21no—
354C22×C4—21no—
355C22×C4—22no—
356C22×C4—24no—
357C22×C4—22no—
358C22×C4—24no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—24no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—24no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—24no—
374C22×C4—24no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—24no—
378C22×C4—24no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—21no—
382C22×C4—21no—
383C22×C4—21no—
384C22×C4—22no—
385C22×C4—21no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—24no—
389C22×C4—22no—
390C22×C4—24no—
391C22×C4—22no—
392C22×C4—24no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—24no—
396C22×C4—22no—
397C22×C4—24no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—24no—
402C22×C4—22no—
403C22×C4—21no—
404C22×C4—22no—
405C22×C4—21no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—24no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—24no—
413C22×C4—24no—
414C22×C4—21no—
415C22×C4—21no—
416C22×C4—24no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—24no—
424C22×C4—48no—
425C22×C4—84no—
426C22×C4—84no—
427C22×C4—88no—
428C22×C4—88no—
429C22×C4C4⋊Q8116yes—
430C22×C4C22×D414yes—
431C22×C4C22×D414yes—
432C22×C4C22×D412yes—
433C22×C4C22×D412yes—
434C22×C4C22×Q814yes—
435C22×C4C22×Q814yes—
436C22×C4C22×Q818yes—
437C22×C4C22×Q812yes—
438C22×C4C22×Q812yes—
439C22×C4C22×Q812yes—
440C22×C4C22×Q812yes—
441C22×C4C2×#4118yes—
442C22×C4C2×#4114yes—
443C22×C4H(5,𝔽2)12yes—
444C22×C4H(5,𝔽2)14yes—
445C22×C42−1+412yes—
446C4⋊C4—28no—
447C4⋊C4—216no—
448C4⋊C4—48no—
449C4⋊C4—48no—
450C4⋊C4—44no—
451C4⋊C4—44no—
452C4⋊C4—48no—
453C4⋊C4—48no—
454C4⋊C4—48no—
455C4⋊C4—44no—
456C4⋊C4—44no—
457C4⋊C4—48no—
458C4⋊C4—88no—
459C4⋊C4—84no—
460C4⋊C4—88no—
461C4⋊C4—88no—
462C4⋊C4—88no—
463C4⋊C4#123116yes—
464C22⋊C4—216no—
465C22⋊C4—48no—
466C22⋊C4—48no—
467C22⋊C4—44no—
468C22⋊C4—44no—
469C22⋊C4—48no—
470C22⋊C4—84no—
471C42—216no—
472C42—216no—
473C42—44no—
474C42—44no—
475C42—48no—
476C42—88no—
477C23—22no—
478C23—22no—
479C23—24no—
480C23—24no—
481C23—24no—
482C23—22no—
483C23—22no—
484C23—22no—
485C23—22no—
486C23—22no—
487C23—22no—
488C23—22no—
489C23—24no—
490C23—22no—
491C23—42no—
492C23—42no—
493C23—42no—
494C23—42no—
495C23—42no—
496C23—42no—
497C23—42no—
498C23—42no—
499C23C2×C4⋊Q814yes—
500C23C23×D411yescharacteristic
501C23C23×Q811yescharacteristic
502C23C23×Q811yescharacteristic
503C23C22×#4112yes—
504C23C2×H(5,𝔽2)12yes—
505C23C2×H(5,𝔽2)11yescharacteristic
506C23C2×2−1+411yescharacteristic
507C23#58512yes—
508C23#58512yes—
509C23#58511yescharacteristic
510C23#58511yescharacteristic
511Q8—816no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—24no—
515C2×C4—24no—
516C2×C4—24no—
517C2×C4—24no—
518C2×C4—28no—
519C2×C4—28no—
520C2×C4—232no—
521C2×C4—24no—
522C2×C4—24no—
523C2×C4—24no—
524C2×C4—24no—
525C2×C4—24no—
526C2×C4—28no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—24no—
530C2×C4—24no—
531C2×C4—28no—
532C2×C4—24no—
533C2×C4—28no—
534C2×C4—24no—
535C2×C4—24no—
536C2×C4—24no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—42no—
542C2×C4—44no—
543C2×C4—42no—
544C2×C4—44no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—44no—
548C2×C4—44no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—44no—
554C2×C4—44no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—44no—
563C2×C4—42no—
564C2×C4—41no—
565C2×C4—42no—
566C2×C4—44no—
567C2×C4—42no—
568C2×C4—41no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—41no—
575C2×C4—44no—
576C2×C4—41no—
577C2×C4—42no—
578C2×C4—44no—
579C2×C4—44no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—44no—
586C2×C4—42no—
587C2×C4—44no—
588C2×C4—44no—
589C2×C4—42no—
590C2×C4—44no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—44no—
594C2×C4—42no—
595C2×C4—44no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—816no—
599C2×C4C2×#12318yes—
600C2×C4C2×#12314yes—
601C2×C4C2×C4⋊Q814yes—
602C2×C4C2×C4⋊Q814yes—
603C2×C4#53514yes—
604C2×C4#55418yes—
605C2×C4#55418yes—
606C22—24no—
607C22—24no—
608C22—24no—
609C22—44no—
610C22—44no—
611C22—44no—
612C22—42no—
613C22—42no—
614C22—44no—
615C22—42no—
616C22—42no—
617C22C22×#12311yescharacteristic
618C22C22×C4⋊Q811yescharacteristic
619C22C2×#53511yescharacteristic
620C22#345812yes—
621C22C2×#55412yes—
622C22C2×#55412yes—
623C22#348811yescharacteristic
624C22#348911yescharacteristic
625C22#350011yescharacteristic
626C22#350011yescharacteristic
627C22#350111yescharacteristic
628C22#350211yescharacteristic
629C22#350311yescharacteristic
630C22#350412yes—
631C22#350411yescharacteristic
632C22#351312yes—
633C22#351612yes—
634C22#352112yes—
635C22#352412yes—
636C22#354012yes—
637C22#354012yes—
638C22#354612yes—
639C22#354712yes—
640C4—28no—
641C4—24no—
642C4—44no—
643C4—48no—
644C4—42no—
645C4—44no—
646C4—42no—
647C4—44no—
648C4—82no—
649C4—82no—
650C4—81no—
651C4—84no—
652C4—82no—
653C4—81no—
654C4—82no—
655C4#250218yes—
656C2—44no—
657C2C2×#250211yescharacteristic
658C2#3533011yescharacteristic
659C2#3556611yescharacteristic
660C2#3597611yescharacteristic
661C2#3598311yescharacteristic
662C2#3810612yes—
663C2#3896012yes—
664C2#4298711yescharacteristic
665C2#4299511yescharacteristic
666C2#4536112yes—
667C2#4541512yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC25C24
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #17067C2semidirect
4. #16694C2semidirect
7. #15225C2semidirect
8. #15221C2semidirect
9. #15185C2semidirect
10. #15021C2semidirect
13. #14053C2semidirect
14. #13954C2semidirect
145. #380Q8semidirect
149. C4×C22⋊C4Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#24392
C2×#23172
#170672
#166942
#155552
#153782
#152252
#152212
#151852
#150212
#142642
#141302
#140532
#139542
#138812
#138442
#138012