Finite Group Databasebeta

← #2736983

Subgroups — #2736983

Order 512.

2188 conjugacy classes of subgroups.

The subgroup lattice has 667 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2552C211yesmaximal, characteristic, complemented
2C2×#2492C211yesmaximal, characteristic
3#17164C214yesmaximal, complemented
4#17100C214yesmaximal
5#17079C212yesmaximal, complemented
6#16608C212yesmaximal, complemented
7#16604C212yesmaximal
8#15195C211yesmaximal, characteristic
9#15193C211yesmaximal, characteristic, complemented
10#15185C211yesmaximal, characteristic, complemented
11#15184C212yesmaximal, complemented
12#15179C212yesmaximal
13#15131C212yesmaximal
14#15128C212yesmaximal, complemented
15#14628C212yesmaximal
16#13933C211yesmaximal, characteristic, complemented
17C4×#399C211yesmaximal, characteristic
18C22×C4⋊Q8C2211yescharacteristic
19C22×#125C2212yes—
20C22×C22⋊Q8C2211yescharacteristic
21#2552—28no—
22#2492—28no—
23C2×#400C2212yes—
24C2×#399C2211yescharacteristic
25C2×#398C2214yes—
26C2×#396C2214yes—
27C2×#395C2214yes—
28C2×#393C2212yes—
29C2×#391C2211yescharacteristic
30C2×#391C2211yescharacteristic
31C2×#391C2212yes—
32C2×#391C2212yes—
33C2×#391C2212yes—
34C2×#389C2212yes—
35C2×#389C2212yes—
36C2×#389C2212yes—
37C2×#389C2212yes—
38C2×#388C2212yes—
39C2×#384C2212yes—
40C2×#384C2211yescharacteristic
41C2×#380C2211yescharacteristic
42C2×#380C2212yes—
43C2×C4×C4⋊C4C2211yescharacteristic
44C2×C4×C4⋊C4C2212yes—
45C2×C4×C4⋊C4C2211yescharacteristic
46C2×C4×C22⋊C4C2211yescharacteristic
47C2×C4×C22⋊C4C2212yes—
48#1446C2214yes—
49#1445C2212yes—
50#1445C2214yes—
51#1445C2214yes—
52#1445C2212yes—
53#1445C2211yescharacteristic
54#1444C2214yes—
55#1444C2211yescharacteristic
56#1444C2212yes—
57#1442C2212yes—
58#1442C2214yes—
59#1442C2214yes—
60#1442C2212yes—
61#1442C2212yes—
62#1442C2212yes—
63#1441C2214yes—
64#1441C2214yes—
65#1441C2214yes—
66#1441C2214yes—
67#1441C2214yes—
68#1441C2214yes—
69#1441C2214yes—
70#1441C2214yes—
71#1440C2214yes—
72#1439C2214yes—
73#1439C2212yes—
74#1439C2212yes—
75#1439C2211yescharacteristic
76#1439C2214yes—
77#1439C2212yes—
78C4⋊#95C2212yes—
79C4⋊#95C2212yes—
80C4⋊#95C2212yes—
81C4⋊#95C2211yescharacteristic
82C4⋊#95C2212yes—
83#1434C2212yes—
84C4×#95C2211yescharacteristic
85C23×Q8C2311yescharacteristic
86C23×Q8C2312yes—
87C24×C4C2311yescharacteristic
88C2×C4⋊Q8—24no—
89C2×C4⋊Q8—44no—
90C2×#125—28no—
91C2×#125—48no—
92C2×C22⋊Q8—24no—
93C2×C22⋊Q8—44no—
94C22×C4⋊C4C2311yescharacteristic
95C22×C4⋊C4C2311yescharacteristic
96C22×C4⋊C4C2312yes—
97C22×C4⋊C4C2312yes—
98C22×C4⋊C4C2312yes—
99C22×C4⋊C4C2314yes—
100C22×C4⋊C4C2314yes—
101C22×C4⋊C4C2311yescharacteristic
102C22×C4⋊C4C2311yescharacteristic
103C22×C4⋊C4C2311yescharacteristic
104C22×C4⋊C4C2314yes—
105C22×C4⋊C4C2312yes—
106C22×C4⋊C4C2314yes—
107C22×C4⋊C4C2312yes—
108C22×C4⋊C4C2312yes—
109C22×C4⋊C4C2312yes—
110C22×C4⋊C4C2314yes—
111C22×C4⋊C4C2312yes—
112C22×C22⋊C4C2311yescharacteristic
113C22×C22⋊C4C2311yescharacteristic
114C22×C22⋊C4C2312yes—
115C22×C22⋊C4C2312yes—
116C22×C22⋊C4C2312yes—
117C22×C22⋊C4C2314yes—
118C22×C22⋊C4C2312yes—
119C22×C42C2311yescharacteristic
120C22×C42C2312yes—
121C22×C42C2312yes—
122C22×C42C2311yescharacteristic
123C22×C42C2312yes—
124#400—28no—
125#399Q818yescomplemented
126#398—216no—
127#396—48no—
128#395—48no—
129#393—28no—
130#391—24no—
131#391—28no—
132#391—24no—
133#391—44no—
134#391—44no—
135#389—28no—
136#389—28no—
137#389—44no—
138#389—44no—
139#388—28no—
140#384—28no—
141#384—24no—
142#380—24no—
143#380—44no—
144C4×C4⋊C4—24no—
145C4×C4⋊C4—44no—
146C4×C4⋊C4D418yes—
147C4×C22⋊C4—24no—
148C4×C22⋊C4—44no—
149C2×#95C2312yes—
150C2×#95C2312yes—
151C2×#95C2312yes—
152C2×#95C2314yes—
153C2×#95C2311yescharacteristic
154C2×#95C2312yes—
155C2×#95C2312yes—
156C2×#95C2311yescharacteristic
157C2×#95C2312yes—
158C2×#95C2311yescharacteristic
159C2×#95C2312yes—
160C2×#95C2314yes—
161C2×#95C2314yes—
162C2×#95C2314yes—
163C2×#95C2314yes—
164C2×#95C2312yes—
165C2×#95C2314yes—
166C2×#95C2314yes—
167C2×#95C2314yes—
168C2×#95C2312yes—
169C2×#95C2314yes—
170C2×#95C2314yes—
171C2×#95C2312yes—
172C2×#95C2314yes—
173C2×#95C2312yes—
174C2×#95C2314yes—
175C2×#95C2314yes—
176C2×#95C2312yes—
177C2×#95C2314yes—
178C2×#95C2312yes—
179C2×#95C2312yes—
180C2×#95C2311yescharacteristic
181C25C2411yescharacteristic
182C22×Q8—22no—
183C22×Q8—24no—
184C22×Q8—24no—
185C22×Q8—22no—
186C22×Q8—24no—
187C22×Q8—22no—
188C22×Q8—24no—
189C22×Q8—44no—
190C22×Q8—42no—
191C22×Q8—44no—
192C23×C4—22no—
193C23×C4—24no—
194C23×C4—22no—
195C23×C4—24no—
196C23×C4—22no—
197C23×C4C2414yes—
198C23×C4C2412yes—
199C23×C4C2414yes—
200C23×C4C2412yes—
201C23×C4C2412yes—
202C23×C4C2411yescharacteristic
203C23×C4C2412yes—
204C23×C4C2412yes—
205C23×C4C2411yescharacteristic
206C23×C4C2411yescharacteristic
207C23×C4C2412yes—
208C23×C4C2412yes—
209C23×C4C2412yes—
210C23×C4C2411yescharacteristic
211C23×C4C2411yescharacteristic
212C23×C4C2411yescharacteristic
213C4⋊Q8—416no—
214#125—432no—
215C22⋊Q8—416no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—24no—
219C2×C4⋊C4—24no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—24no—
224C2×C4⋊C4—24no—
225C2×C4⋊C4—28no—
226C2×C4⋊C4—28no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—24no—
229C2×C4⋊C4—24no—
230C2×C4⋊C4—28no—
231C2×C4⋊C4—28no—
232C2×C4⋊C4—24no—
233C2×C4⋊C4—24no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—24no—
236C2×C4⋊C4—24no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—24no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—28no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—28no—
243C2×C4⋊C4—42no—
244C2×C4⋊C4—44no—
245C2×C4⋊C4—44no—
246C2×C4⋊C4—44no—
247C2×C4⋊C4—44no—
248C2×C4⋊C4—44no—
249C2×C4⋊C4—42no—
250C2×C4⋊C4—42no—
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⋊C4—44no—
259C2×C4⋊C4C2×D414yes—
260C2×C4⋊C4C2×D414yes—
261C2×C4⋊C4C2×Q814yes—
262C2×C4⋊C4C2×Q818yes—
263C2×C22⋊C4—22no—
264C2×C22⋊C4—22no—
265C2×C22⋊C4—24no—
266C2×C22⋊C4—22no—
267C2×C22⋊C4—24no—
268C2×C22⋊C4—22no—
269C2×C22⋊C4—24no—
270C2×C22⋊C4—42no—
271C2×C22⋊C4—41no—
272C2×C22⋊C4—42no—
273C2×C22⋊C4—44no—
274C2×C22⋊C4—42no—
275C2×C22⋊C4—42no—
276C2×C22⋊C4—44no—
277C2×C22⋊C4—44no—
278C2×C22⋊C4—44no—
279C2×C22⋊C4—44no—
280C2×C22⋊C4C2×Q814yes—
281C2×C22⋊C4C2×Q818yes—
282C2×C42—24no—
283C2×C42—22no—
284C2×C42—24no—
285C2×C42—24no—
286C2×C42—24no—
287C2×C42—24no—
288C2×C42—24no—
289C2×C42—24no—
290C2×C42—24no—
291C2×C42—44no—
292C2×C42C2×D414yes—
293C2×C42C2×D418yes—
294#95—22no—
295#95—22no—
296#95—24no—
297#95—24no—
298#95—24no—
299#95—24no—
300#95—28no—
301#95—24no—
302#95—24no—
303#95—22no—
304#95—24no—
305#95—42no—
306#95—44no—
307#95—44no—
308#95—42no—
309#95—42no—
310#95—42no—
311#95—42no—
312#95—42no—
313#95—44no—
314#95—44no—
315#95—44no—
316#95—44no—
317#95—44no—
318#95—44no—
319#95—44no—
320#95—44no—
321#95—44no—
322#95—44no—
323#95—44no—
324#95C2×D418yes—
325#95C2×Q814yes—
326C24—21no—
327C24—22no—
328C24—22no—
329C24—21no—
330C24—21no—
331C24—21no—
332C24—21no—
333C24—22no—
334C24—22no—
335C24—21no—
336C24C22×Q812yes—
337C24C2511yescharacteristic
338C2×Q8—48no—
339C2×Q8—48no—
340C2×Q8—44no—
341C2×Q8—44no—
342C2×Q8—44no—
343C2×Q8—48no—
344C2×Q8—48no—
345C2×Q8—84no—
346C2×Q8—88no—
347C2×Q8—88no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—21no—
351C22×C4—21no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—24no—
366C22×C4—24no—
367C22×C4—24no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—21no—
372C22×C4—21no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—21no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—21no—
386C22×C4—24no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—21no—
390C22×C4—22no—
391C22×C4—24no—
392C22×C4—21no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—24no—
396C22×C4—24no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—24no—
400C22×C4—24no—
401C22×C4—24no—
402C22×C4—24no—
403C22×C4—24no—
404C22×C4—22no—
405C22×C4—24no—
406C22×C4—24no—
407C22×C4—24no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—24no—
411C22×C4—24no—
412C22×C4—22no—
413C22×C4—21no—
414C22×C4—24no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—24no—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—22no—
422C22×C4—24no—
423C22×C4—22no—
424C22×C4—44no—
425C22×C4—44no—
426C22×C4—44no—
427C22×C4—48no—
428C22×C4—48no—
429C22×C4C22×D412yes—
430C22×C4C22×D414yes—
431C22×C4C22×D414yes—
432C22×C4C22×D412yes—
433C22×C4C22×D412yes—
434C22×C4C22×Q814yes—
435C22×C4C22×Q812yes—
436C22×C4C22×Q814yes—
437C22×C4C22×Q812yes—
438C22×C4H(5,𝔽2)14yes—
439C22×C4H(5,𝔽2)14yes—
440C22×C4H(5,𝔽2)12yes—
441C22×C42−1+412yes—
442C22×C42−1+412yes—
443C22×C42−1+414yes—
444C22×C42−1+412yes—
445C4⋊C4—28no—
446C4⋊C4—28no—
447C4⋊C4—28no—
448C4⋊C4—28no—
449C4⋊C4—48no—
450C4⋊C4—48no—
451C4⋊C4—416no—
452C4⋊C4—416no—
453C4⋊C4—48no—
454C4⋊C4—48no—
455C4⋊C4—48no—
456C4⋊C4—44no—
457C4⋊C4—48no—
458C4⋊C4—48no—
459C4⋊C4—88no—
460C4⋊C4—84no—
461C4⋊C4—88no—
462C4⋊C4—88no—
463C22⋊C4—28no—
464C22⋊C4—48no—
465C22⋊C4—44no—
466C22⋊C4—48no—
467C22⋊C4—48no—
468C22⋊C4—84no—
469C22⋊C4—88no—
470C42—28no—
471C42—48no—
472C42—48no—
473C42—44no—
474C42—48no—
475C23—22no—
476C23—24no—
477C23—22no—
478C23—24no—
479C23—22no—
480C23—41no—
481C23—41no—
482C23—41no—
483C23—41no—
484C23—41no—
485C23—41no—
486C23—42no—
487C23—42no—
488C23—42no—
489C23—42no—
490C23—42no—
491C23—42no—
492C23—42no—
493C23—41no—
494C23—42no—
495C23—42no—
496C23—41no—
497C23—42no—
498C23C23×D411yescharacteristic
499C23C23×Q811yescharacteristic
500C23C2×H(5,𝔽2)12yes—
501C23C2×H(5,𝔽2)11yescharacteristic
502C23C2×H(5,𝔽2)12yes—
503C23C2×2−1+412yes—
504C23C2×2−1+411yescharacteristic
505C23C2×2−1+411yescharacteristic
506C23C2×2−1+411yescharacteristic
507C23#58511yescharacteristic
508C23#58512yes—
509Q8—816no—
510Q8—88no—
511C2×C4—22no—
512C2×C4—22no—
513C2×C4—24no—
514C2×C4—24no—
515C2×C4—24no—
516C2×C4—22no—
517C2×C4—24no—
518C2×C4—22no—
519C2×C4—24no—
520C2×C4—24no—
521C2×C4—24no—
522C2×C4—24no—
523C2×C4—22no—
524C2×C4—22no—
525C2×C4—24no—
526C2×C4—22no—
527C2×C4—24no—
528C2×C4—24no—
529C2×C4—24no—
530C2×C4—24no—
531C2×C4—24no—
532C2×C4—24no—
533C2×C4—24no—
534C2×C4—24no—
535C2×C4—24no—
536C2×C4—24no—
537C2×C4—41no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—41no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—44no—
549C2×C4—42no—
550C2×C4—44no—
551C2×C4—42no—
552C2×C4—44no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—44no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—41no—
563C2×C4—42no—
564C2×C4—41no—
565C2×C4—42no—
566C2×C4—44no—
567C2×C4—41no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—44no—
571C2×C4—42no—
572C2×C4—44no—
573C2×C4—42no—
574C2×C4—44no—
575C2×C4—42no—
576C2×C4—44no—
577C2×C4—42no—
578C2×C4—44no—
579C2×C4—44no—
580C2×C4—42no—
581C2×C4—44no—
582C2×C4—44no—
583C2×C4—44no—
584C2×C4—44no—
585C2×C4—42no—
586C2×C4—44no—
587C2×C4—42no—
588C2×C4—44no—
589C2×C4—42no—
590C2×C4—44no—
591C2×C4—42no—
592C2×C4—44no—
593C2×C4—44no—
594C2×C4—44no—
595C2×C4—42no—
596C2×C4—44no—
597C2×C4—88no—
598C2×C4#53514yes—
599C2×C4#53614yes—
600C2×C4#53714yes—
601C2×C4D4×Q814yes—
602C2×C4#56218yes—
603C2×C4#56318yes—
604C22—42no—
605C22—42no—
606C22—44no—
607C22—44no—
608C22—42no—
609C22—82no—
610C22—81no—
611C22—81no—
612C22—81no—
613C22—82no—
614C22—81no—
615C22C2×#53511yescharacteristic
616C22C2×#53611yescharacteristic
617C22C2×#53711yescharacteristic
618C22C2×D4×Q811yescharacteristic
619C22#348411yescharacteristic
620C22#348911yescharacteristic
621C22#350011yescharacteristic
622C22#350112yes—
623C22#350211yescharacteristic
624C22#350312yes—
625C22#350311yescharacteristic
626C22C2×#56212yes—
627C22C2×#56312yes—
628C22#353612yes—
629C22#354212yes—
630C22#354612yes—
631C22#354712yes—
632C22#355212yes—
633C22#355411yescharacteristic
634C22#355412yes—
635C22#355512yes—
636C22#355611yescharacteristic
637C22#356512yes—
638C4—24no—
639C4—42no—
640C4—44no—
641C4—44no—
642C4—42no—
643C4—44no—
644C4—44no—
645C4—44no—
646C4—42no—
647C4—82no—
648C4—84no—
649C4—81no—
650C4—82no—
651C4—84no—
652C4—82no—
653C4#282018yes—
654C2—82no—
655C2C2×#282011yescharacteristic
656C2#3959211yescharacteristic
657C2#3959911yescharacteristic
658C2#3961811yescharacteristic
659C2#4242911yescharacteristic
660C2#4247811yescharacteristic
661C2#4291511yescharacteristic
662C2#4549912yes—
663C2#4550112yes—
664C2#5296512yes—
665C2#5304912yes—
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
1. C2×#2552C2semidirect
3. #17164C2semidirect
5. #17079C2semidirect
6. #16608C2semidirect
9. #15193C2semidirect
10. #15185C2semidirect
11. #15184C2semidirect
14. #15128C2semidirect
16. #13933C2semidirect
125. #399Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#25522
C2×#24922
#171642
#171002
#170792
#166082
#166042
#151952
#151932
#151852
#151842
#151792
#151312
#151282
#146282
#139332
C4×#3992