Finite Group Databasebeta

← #4463904

Subgroups — #4463904

Order 512.

2110 conjugacy classes of subgroups.

The subgroup lattice has 647 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#17305C212yesmaximal, complemented
2#17298C212yesmaximal, complemented
3#17293C214yesmaximal, complemented
4#17234C212yesmaximal
5#17174C214yesmaximal, complemented
6#17159C212yesmaximal
7#15182C212yesmaximal, complemented
8#15180C211yesmaximal, characteristic, complemented
9#15179C212yesmaximal, complemented
10#15178C212yesmaximal, complemented
11#15177C212yesmaximal
12#15152C211yesmaximal, characteristic, complemented
13#15132C212yesmaximal, complemented
14#14261C212yesmaximal, complemented
15#13847C211yesmaximal, characteristic
16C22×C4⋊Q8C2211yescharacteristic
17C22×C22⋊Q8C2212yes—
18C22×C22⋊Q8C2212yes—
19C22×C4×D4C2211yescharacteristic
20C2×#400C2212yes—
21C2×#397C2214yes—
22C2×#391C2212yes—
23C2×#391C2212yes—
24C2×#391C2212yes—
25C2×#391C2212yes—
26C2×#389C2212yes—
27C2×#384C2211yescharacteristic
28C2×#380C2212yes—
29C2×#380C2212yes—
30C2×C4×C4⋊C4C2211yescharacteristic
31C2×C4×C22⋊C4C2212yes—
32C2×C4×C22⋊C4C2212yes—
33C2×#376C2211yescharacteristic
34#1445C2211yescharacteristic
35#1445C2211yescharacteristic
36#1445C2212yes—
37#1444C2214yes—
38#1444C2214yes—
39#1444C2214yes—
40#1443C2214yes—
41#1443C2214yes—
42#1443C2214yes—
43#1442C2212yes—
44#1442C2212yes—
45#1442C2214yes—
46#1442C2214yes—
47#1442C2214yes—
48#1442C2214yes—
49#1442C2212yes—
50#1441C2212yes—
51#1441C2214yes—
52#1441C2214yes—
53#1441C2212yes—
54#1441C2212yes—
55#1441C2214yes—
56#1440C2214yes—
57#1440C2214yes—
58#1439C2214yes—
59#1439C2214yes—
60#1439C2214yes—
61#1439C2212yes—
62#1439C2214yes—
63#1439C2214yes—
64#1439C2214yes—
65#1439C2212yes—
66#1439C2214yes—
67#1438C2211yescharacteristic
68#1438C2214yes—
69#1438C2211yescharacteristic
70C4⋊#95C2211yescharacteristic
71C4⋊#95C2212yes—
72C4⋊#95C2212yes—
73C4⋊#95C2212yes—
74C4⋊#95C2211yescharacteristic
75C23×Q8C2312yes—
76C23×D4C2311yescharacteristic
77C24×C4C2312yes—
78C2×C4⋊Q8—44no—
79C2×C4⋊Q8—42no—
80C2×C22⋊Q8—44no—
81C2×C22⋊Q8—44no—
82C2×C22⋊Q8—44no—
83C2×C22⋊Q8—44no—
84C2×C22⋊Q8—44no—
85C2×C22⋊Q8—44no—
86C2×C4×D4—24no—
87C2×C4×D4—42no—
88C2×C4×D4—42no—
89C22×C4⋊C4C2312yes—
90C22×C4⋊C4C2312yes—
91C22×C4⋊C4C2312yes—
92C22×C4⋊C4C2311yescharacteristic
93C22×C4⋊C4C2311yescharacteristic
94C22×C4⋊C4C2312yes—
95C22×C4⋊C4C2311yescharacteristic
96C22×C4⋊C4C2314yes—
97C22×C4⋊C4C2314yes—
98C22×C4⋊C4C2314yes—
99C22×C4⋊C4C2312yes—
100C22×C4⋊C4C2312yes—
101C22×C4⋊C4C2312yes—
102C22×C22⋊C4C2312yes—
103C22×C22⋊C4C2312yes—
104C22×C22⋊C4C2314yes—
105C22×C22⋊C4C2312yes—
106C22×C22⋊C4C2312yes—
107C22×C22⋊C4C2312yes—
108C22×C22⋊C4C2314yes—
109C22×C22⋊C4C2314yes—
110C22×C22⋊C4C2314yes—
111C22×C42C2311yescharacteristic
112C22×C42C2312yes—
113C22×C42C2311yescharacteristic
114C22×C42C2311yescharacteristic
115#400—44no—
116#397—48no—
117#391—28no—
118#391—44no—
119#391—44no—
120#391—44no—
121#389—44no—
122#384—42no—
123#380—44no—
124#380—44no—
125C4×C4⋊C4—42no—
126C4×C22⋊C4—44no—
127C4×C22⋊C4—44no—
128#376—42no—
129C2×#95C2312yes—
130C2×#95C2314yes—
131C2×#95C2312yes—
132C2×#95C2312yes—
133C2×#95C2314yes—
134C2×#95C2314yes—
135C2×#95C2314yes—
136C2×#95C2314yes—
137C2×#95C2312yes—
138C2×#95C2314yes—
139C2×#95C2314yes—
140C2×#95C2314yes—
141C2×#95C2311yescharacteristic
142C2×#95C2311yescharacteristic
143C2×#95C2311yescharacteristic
144C2×#95C2311yescharacteristic
145C2×#95C2314yes—
146C2×#95C2314yes—
147C2×#95C2314yes—
148C2×#95C2314yes—
149C2×#95C2312yes—
150C2×#95C2314yes—
151C2×#95C2314yes—
152C2×#95C2312yes—
153C2×#95C2312yes—
154C2×#95C2314yes—
155C2×#95C2314yes—
156C2×#95C2312yes—
157C2×#95C2312yes—
158C2×#95C2314yes—
159C25C2412yes—
160C22×Q8—24no—
161C22×Q8—24no—
162C22×Q8—24no—
163C22×Q8—24no—
164C22×Q8—24no—
165C22×Q8—24no—
166C22×Q8—24no—
167C22×D4—24no—
168C22×D4—22no—
169C22×D4—41no—
170C22×D4—41no—
171C22×D4—42no—
172C23×C4—24no—
173C23×C4—24no—
174C23×C4—24no—
175C23×C4—24no—
176C23×C4—24no—
177C23×C4—24no—
178C23×C4—24no—
179C23×C4C2414yes—
180C23×C4C2414yes—
181C23×C4C2414yes—
182C23×C4C2414yes—
183C23×C4C2412yes—
184C23×C4C2412yes—
185C23×C4C2412yes—
186C23×C4C2411yescharacteristic
187C23×C4C2411yescharacteristic
188C23×C4C2411yescharacteristic
189C23×C4C2411yescharacteristic
190C23×C4C2411yescharacteristic
191C23×C4C2411yescharacteristic
192C23×C4C2411yescharacteristic
193C4⋊Q8—416no—
194C22⋊Q8—432no—
195C22⋊Q8—432no—
196C4×D4—416no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—24no—
199C2×C4⋊C4—24no—
200C2×C4⋊C4—24no—
201C2×C4⋊C4—24no—
202C2×C4⋊C4—24no—
203C2×C4⋊C4—24no—
204C2×C4⋊C4—24no—
205C2×C4⋊C4—24no—
206C2×C4⋊C4—24no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—24no—
212C2×C4⋊C4—41no—
213C2×C4⋊C4—41no—
214C2×C4⋊C4—44no—
215C2×C4⋊C4—44no—
216C2×C4⋊C4—44no—
217C2×C4⋊C4—44no—
218C2×C4⋊C4—44no—
219C2×C4⋊C4—44no—
220C2×C4⋊C4—44no—
221C2×C4⋊C4—44no—
222C2×C4⋊C4—42no—
223C2×C4⋊C4—42no—
224C2×C4⋊C4—41no—
225C2×C4⋊C4—41no—
226C2×C4⋊C4—44no—
227C2×C4⋊C4—44no—
228C2×C4⋊C4—42no—
229C2×C4⋊C4—42no—
230C2×C4⋊C4—44no—
231C2×C4⋊C4—44no—
232C2×C4⋊C4—42no—
233C2×C22⋊C4—24no—
234C2×C22⋊C4—24no—
235C2×C22⋊C4—24no—
236C2×C22⋊C4—24no—
237C2×C22⋊C4—24no—
238C2×C22⋊C4—24no—
239C2×C22⋊C4—24no—
240C2×C22⋊C4—28no—
241C2×C22⋊C4—42no—
242C2×C22⋊C4—42no—
243C2×C22⋊C4—42no—
244C2×C22⋊C4—42no—
245C2×C22⋊C4—44no—
246C2×C22⋊C4—44no—
247C2×C22⋊C4—44no—
248C2×C22⋊C4—44no—
249C2×C22⋊C4—44no—
250C2×C22⋊C4—44no—
251C2×C22⋊C4—42no—
252C2×C22⋊C4—42no—
253C2×C22⋊C4—42no—
254C2×C22⋊C4—42no—
255C2×C22⋊C4—44no—
256C2×C22⋊C4—44no—
257C2×C22⋊C4—44no—
258C2×C22⋊C4—44no—
259C2×C22⋊C4—44no—
260C2×C42—22no—
261C2×C42—22no—
262C2×C42—24no—
263C2×C42—22no—
264C2×C42—24no—
265C2×C42—24no—
266C2×C42—24no—
267C2×C42—22no—
268C2×C42—24no—
269C2×C42#4114yes—
270#95—28no—
271#95—28no—
272#95—24no—
273#95—28no—
274#95—24no—
275#95—24no—
276#95—44no—
277#95—44no—
278#95—44no—
279#95—42no—
280#95—41no—
281#95—41no—
282#95—42no—
283#95—42no—
284#95—42no—
285#95—42no—
286#95—41no—
287#95—41no—
288#95—44no—
289#95—44no—
290#95—44no—
291#95—44no—
292#95—44no—
293#95—44no—
294#95—44no—
295#95—44no—
296#95—44no—
297#95—44no—
298#95—44no—
299#95—42no—
300C24—22no—
301C24—22no—
302C24—22no—
303C24—22no—
304C24—22no—
305C24—22no—
306C24—22no—
307C24—22no—
308C24—22no—
309C24—22no—
310C24—22no—
311C24—22no—
312C24—22no—
313C24—22no—
314C242−1+414yes—
315C24C2511yescharacteristic
316C2×Q8—48no—
317C2×Q8—48no—
318C2×Q8—48no—
319C2×Q8—48no—
320C2×Q8—48no—
321C2×Q8—48no—
322C2×Q8—48no—
323C2×D4—48no—
324C2×D4—44no—
325C2×D4—44no—
326C2×D4—48no—
327C2×D4—44no—
328C22×C4—21no—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—21no—
332C22×C4—21no—
333C22×C4—21no—
334C22×C4—21no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—21no—
338C22×C4—21no—
339C22×C4—21no—
340C22×C4—21no—
341C22×C4—21no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—21no—
345C22×C4—21no—
346C22×C4—21no—
347C22×C4—21no—
348C22×C4—21no—
349C22×C4—21no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—24no—
355C22×C4—24no—
356C22×C4—22no—
357C22×C4—24no—
358C22×C4—24no—
359C22×C4—24no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—24no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—24no—
370C22×C4—24no—
371C22×C4—24no—
372C22×C4—24no—
373C22×C4—24no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—24no—
377C22×C4—24no—
378C22×C4—24no—
379C22×C4—24no—
380C22×C4—24no—
381C22×C4—24no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—24no—
385C22×C4—24no—
386C22×C4—22no—
387C22×C4—24no—
388C22×C4—24no—
389C22×C4—24no—
390C22×C4—22no—
391C22×C4—24no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—24no—
397C22×C4—22no—
398C22×C4—24no—
399C22×C4—24no—
400C22×C4—24no—
401C22×C4—24no—
402C22×C4—22no—
403C22×C4—48no—
404C22×C4—48no—
405C22×C4—48no—
406C22×C4—48no—
407C22×C4—48no—
408C22×C4—48no—
409C22×C4—48no—
410C22×C4C2×#4112yes—
411C22×C4C2×#4112yes—
412C22×C4C2×#4112yes—
413C22×C4H(5,𝔽2)12yes—
414C22×C4H(5,𝔽2)12yes—
415C22×C4H(5,𝔽2)12yes—
416C22×C4H(5,𝔽2)14yes—
417C22×C4H(5,𝔽2)14yes—
418C22×C42−1+412yes—
419C22×C42−1+412yes—
420C22×C42−1+412yes—
421C4⋊C4—44no—
422C4⋊C4—44no—
423C4⋊C4—44no—
424C4⋊C4—416no—
425C4⋊C4—416no—
426C4⋊C4—48no—
427C4⋊C4—48no—
428C4⋊C4—48no—
429C4⋊C4—48no—
430C4⋊C4—48no—
431C4⋊C4—84no—
432C4⋊C4—84no—
433C4⋊C4—88no—
434C22⋊C4—48no—
435C22⋊C4—48no—
436C22⋊C4—48no—
437C22⋊C4—48no—
438C22⋊C4—48no—
439C22⋊C4—416no—
440C22⋊C4—416no—
441C22⋊C4—88no—
442C22⋊C4—88no—
443C42—28no—
444C42—44no—
445C42—48no—
446C42—44no—
447C23—24no—
448C23—24no—
449C23—24no—
450C23—24no—
451C23—24no—
452C23—24no—
453C23—24no—
454C23—42no—
455C23—42no—
456C23—42no—
457C23—42no—
458C23—42no—
459C23—42no—
460C23—42no—
461C23—42no—
462C23—42no—
463C23—42no—
464C23—42no—
465C23—42no—
466C23—42no—
467C23—42no—
468C23—42no—
469C23—42no—
470C23—42no—
471C23—42no—
472C23—42no—
473C23—42no—
474C23—42no—
475C23—42no—
476C23—42no—
477C23—42no—
478C23—42no—
479C23—42no—
480C23—42no—
481C23—42no—
482C23C22×#4111yescharacteristic
483C23C2×H(5,𝔽2)11yescharacteristic
484C23C2×H(5,𝔽2)11yescharacteristic
485C23C2×H(5,𝔽2)12yes—
486C23C2×H(5,𝔽2)12yes—
487C23C2×H(5,𝔽2)11yescharacteristic
488C23C2×2−1+412yes—
489C23C2×2−1+411yescharacteristic
490C23C2×2−1+411yescharacteristic
491C23C2×2−1+411yescharacteristic
492C23#58512yes—
493Q8—816no—
494D4—416no—
495C2×C4—22no—
496C2×C4—22no—
497C2×C4—24no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—24no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—24no—
504C2×C4—22no—
505C2×C4—22no—
506C2×C4—24no—
507C2×C4—24no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—42no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—42no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—42no—
528C2×C4—44no—
529C2×C4—44no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—44no—
533C2×C4—44no—
534C2×C4—44no—
535C2×C4—44no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—44no—
539C2×C4—44no—
540C2×C4—42no—
541C2×C4—44no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—44no—
545C2×C4—44no—
546C2×C4—44no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—44no—
550C2×C4—44no—
551C2×C4—44no—
552C2×C4—42no—
553C2×C4—44no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—44no—
558C2×C4—44no—
559C2×C4—42no—
560C2×C4—44no—
561C2×C4—44no—
562C2×C4—44no—
563C2×C4—42no—
564C2×C4—44no—
565C2×C4—44no—
566C2×C4—44no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—44no—
571C2×C4—42no—
572C2×C4—44no—
573C2×C4—44no—
574C2×C4—44no—
575C2×C4—42no—
576C2×C4—816no—
577C2×C4#53814yes—
578C2×C4#54018yes—
579C2×C4#56014yes—
580C2×C4#56018yes—
581C2×C4H(3,𝔽4)14yes—
582C22—44no—
583C22—44no—
584C22—44no—
585C22—44no—
586C22—44no—
587C22—44no—
588C22—44no—
589C22—82no—
590C22—82no—
591C22—82no—
592C22—82no—
593C22—82no—
594C22—82no—
595C22—82no—
596C22—82no—
597C22C2×#53811yescharacteristic
598C22C2×#54012yes—
599C22#349512yes—
600C22#350911yescharacteristic
601C22#351111yescharacteristic
602C22C2×#56011yescharacteristic
603C22C2×#56012yes—
604C22C2×H(3,𝔽4)11yescharacteristic
605C22#353611yescharacteristic
606C22#353612yes—
607C22#353811yescharacteristic
608C22#353812yes—
609C22#353811yescharacteristic
610C22#354011yescharacteristic
611C22#354112yes—
612C22#354112yes—
613C22#354112yes—
614C22#354612yes—
615C22#354811yescharacteristic
616C22#355011yescharacteristic
617C22#355212yes—
618C22#355212yes—
619C22#355412yes—
620C4—42no—
621C4—42no—
622C4—42no—
623C4—42no—
624C4—42no—
625C4—42no—
626C4—82no—
627C4—84no—
628C4—84no—
629C4—82no—
630C4—84no—
631C4—82no—
632C4—84no—
633C4#282818yes—
634C2—84no—
635C2C2×#282811yescharacteristic
636C2#3732411yescharacteristic
637C2#3734112yes—
638C2#3738111yescharacteristic
639C2#3739912yes—
640C2#5299311yescharacteristic
641C2#5303211yescharacteristic
642C2#5308011yescharacteristic
643C2#5309512yes—
644C2#5313612yes—
645C2#5313711yescharacteristic
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC23×D4C23
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #17305C2semidirect
2. #17298C2semidirect
3. #17293C2semidirect
5. #17174C2semidirect
7. #15182C2semidirect
8. #15180C2semidirect
9. #15179C2semidirect
10. #15178C2semidirect
12. #15152C2semidirect
13. #15132C2semidirect
14. #14261C2semidirect

Maximal subgroups

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

subgroupindex
#173052
#172982
#172932
#172342
#171742
#171592
#151822
#151802
#151792
#151782
#151772
#151522
#151322
#142612
#138472