Finite Group Databasebeta

← #5021373

Subgroups — #5021373

Order 512.

2732 conjugacy classes of subgroups.

The subgroup lattice has 642 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2858C211yesmaximal, characteristic
2C2×#2379C211yesmaximal, characteristic, complemented
3#16094C212yesmaximal, complemented
4#15784C212yesmaximal, complemented
5#15783C212yesmaximal, complemented
6#15445C214yesmaximal, complemented
7#15348C214yesmaximal, complemented
8#15329C214yesmaximal, complemented
9#15271C214yesmaximal, complemented
10#15261C212yesmaximal
11#14707C212yesmaximal, complemented
12#14662C212yesmaximal, complemented
13#14662C211yesmaximal, characteristic, complemented
14C22×C4⋊Q8C2211yescharacteristic
15C22×#127C2211yescharacteristic
16C22×#124C2211yescharacteristic
17C22×#124C2211yescharacteristic
18C22×#124C2211yescharacteristic
19C22×#124C2211yescharacteristic
20C22×#123C2212yes—
21#2858—28no—
22C2×#400C2212yes—
23C2×#398C2212yes—
24C2×#397C2214yes—
25C2×#394C2214yes—
26#2379—28no—
27C2×#391C2212yes—
28C2×#390C2214yes—
29C2×#390C2214yes—
30C2×#389C2212yes—
31C2×#389C2212yes—
32C2×#388C2214yes—
33C2×#388C2214yes—
34C2×#386C2212yes—
35C2×#386C2212yes—
36C2×#385C2214yes—
37C2×#382C2212yes—
38C2×#382C2212yes—
39C2×C43C2211yescharacteristic
40#1446C2214yes—
41#1446C2212yes—
42#1442C2214yes—
43#1441C2214yes—
44#1440C2214yes—
45#1440C2214yes—
46#1439C2214yes—
47#1439C2214yes—
48#1439C2214yes—
49#1439C2214yes—
50#1439C2214yes—
51#1439C2214yes—
52#1439C2214yes—
53#1439C2214yes—
54#1439C2214yes—
55#1438C2212yes—
56#1438C2214yes—
57#1438C2212yes—
58#1438C2214yes—
59#1438C2212yes—
60#1436C2212yes—
61#1436C2214yes—
62#1436C2214yes—
63#1436C2214yes—
64#1436C2214yes—
65#1436C2212yes—
66C4⋊#95C2212yes—
67C4⋊#95C2212yes—
68C22⋊#95C2214yes—
69C23×Q8C2311yescharacteristic
70C23×Q8C2311yescharacteristic
71C23×D4C2312yes—
72C23×D4C2312yes—
73C23×D4C2311yescharacteristic
74C23×D4C2311yescharacteristic
75C24×C4C2312yes—
76C2×C4⋊Q8—24no—
77C2×C4⋊Q8—42no—
78C2×C4⋊Q8—42no—
79C2×#127—24no—
80C2×#127—42no—
81C2×#127—42no—
82C2×#124—24no—
83C2×#124—24no—
84C2×#124—24no—
85C2×#124—28no—
86C2×#124—28no—
87C2×#124—44no—
88C2×#124—44no—
89C2×#124D418yes—
90C2×#123—48no—
91C2×#123—44no—
92C22×C4⋊C4C2314yes—
93C22×C4⋊C4C2312yes—
94C22×C4⋊C4C2312yes—
95C22×C4⋊C4C2314yes—
96C22×C4⋊C4C2312yes—
97C22×C4⋊C4C2312yes—
98C22×C4⋊C4C2312yes—
99C22×C4⋊C4C2312yes—
100C22×C4⋊C4C2312yes—
101C22×C22⋊C4C2314yes—
102C22×C22⋊C4C2314yes—
103C22×C22⋊C4C2314yes—
104C22×C22⋊C4C2314yes—
105C22×C22⋊C4C2314yes—
106C22×C22⋊C4C2312yes—
107C22×C22⋊C4C2314yes—
108C22×C22⋊C4C2314yes—
109C22×C22⋊C4C2314yes—
110C22×C22⋊C4C2314yes—
111C22×C22⋊C4C2312yes—
112C22×C22⋊C4C2314yes—
113C22×C22⋊C4C2312yes—
114C22×C42C2311yescharacteristic
115C22×C42C2311yescharacteristic
116C22×C42C2312yes—
117C22×C42C2312yes—
118C22×C42C2312yes—
119C22×C42C2311yescharacteristic
120#400—44no—
121#398—44no—
122#397—48no—
123#394—48no—
124#391—28no—
125#390—216no—
126#390—216no—
127#389—44no—
128#389—44no—
129#388—216no—
130#388—48no—
131#386—28no—
132#386—44no—
133#385—216no—
134#382—28no—
135#382—28no—
136C2×#95C2314yes—
137C2×#95C2314yes—
138C2×#95C2314yes—
139C2×#95C2314yes—
140C2×#95C2314yes—
141C2×#95C2314yes—
142C2×#95C2314yes—
143C2×#95C2314yes—
144C2×#95C2314yes—
145C2×#95C2314yes—
146C2×#95C2314yes—
147C2×#95C2314yes—
148C2×#95C2312yes—
149C2×#95C2314yes—
150C2×#95C2312yes—
151C2×#95C2314yes—
152C2×#95C2314yes—
153C2×#95C2314yes—
154C43—24no—
155C25C2412yes—
156C25C2412yes—
157C22×Q8—22no—
158C22×Q8—24no—
159C22×Q8—24no—
160C22×Q8—42no—
161C22×Q8—42no—
162C22×Q8—41no—
163C22×Q8—41no—
164C22×Q8—41no—
165C22×Q8—41no—
166C22×Q8C2×D414yes—
167C22×D4—24no—
168C22×D4—24no—
169C22×D4—22no—
170C22×D4—24no—
171C22×D4—24no—
172C22×D4—24no—
173C22×D4—28no—
174C22×D4—24no—
175C22×D4—28no—
176C22×D4—42no—
177C22×D4—42no—
178C22×D4—42no—
179C22×D4—42no—
180C22×D4—41no—
181C22×D4—41no—
182C22×D4—41no—
183C22×D4—41no—
184C22×D4—44no—
185C22×D4—44no—
186C22×D4C2×D414yes—
187C23×C4—24no—
188C23×C4—42no—
189C23×C4—44no—
190C23×C4—44no—
191C23×C4—42no—
192C23×C4C2412yes—
193C23×C4C2414yes—
194C23×C4C2414yes—
195C23×C4C2414yes—
196C23×C4C2414yes—
197C23×C4C2412yes—
198C23×C4C2412yes—
199C23×C4C2412yes—
200C23×C4C2411yescharacteristic
201C23×C4C2411yescharacteristic
202C23×C4C2411yescharacteristic
203C4⋊Q8—416no—
204#127—416no—
205#124—232no—
206#124—416no—
207#124—416no—
208#124—416no—
209#123—432no—
210C2×C4⋊C4—24no—
211C2×C4⋊C4—24no—
212C2×C4⋊C4—24no—
213C2×C4⋊C4—24no—
214C2×C4⋊C4—28no—
215C2×C4⋊C4—28no—
216C2×C4⋊C4—28no—
217C2×C4⋊C4—24no—
218C2×C4⋊C4—24no—
219C2×C4⋊C4—24no—
220C2×C4⋊C4—24no—
221C2×C4⋊C4—24no—
222C2×C4⋊C4—28no—
223C2×C4⋊C4—42no—
224C2×C4⋊C4—42no—
225C2×C4⋊C4—42no—
226C2×C4⋊C4—42no—
227C2×C4⋊C4—42no—
228C2×C4⋊C4—42no—
229C2×C4⋊C4—42no—
230C2×C4⋊C4—42no—
231C2×C4⋊C4—44no—
232C2×C4⋊C4—42no—
233C2×C4⋊C4—42no—
234C2×C4⋊C4—44no—
235C2×C4⋊C4—42no—
236C2×C4⋊C4—42no—
237C2×C22⋊C4—28no—
238C2×C22⋊C4—28no—
239C2×C22⋊C4—28no—
240C2×C22⋊C4—28no—
241C2×C22⋊C4—28no—
242C2×C22⋊C4—28no—
243C2×C22⋊C4—28no—
244C2×C22⋊C4—28no—
245C2×C22⋊C4—28no—
246C2×C22⋊C4—28no—
247C2×C22⋊C4—24no—
248C2×C22⋊C4—44no—
249C2×C22⋊C4—44no—
250C2×C22⋊C4—42no—
251C2×C22⋊C4—44no—
252C2×C22⋊C4—44no—
253C2×C22⋊C4—44no—
254C2×C22⋊C4—44no—
255C2×C22⋊C4—44no—
256C2×C22⋊C4—44no—
257C2×C22⋊C4—44no—
258C2×C22⋊C4—44no—
259C2×C22⋊C4—44no—
260C2×C22⋊C4—44no—
261C2×C22⋊C4—44no—
262C2×C22⋊C4—44no—
263C2×C22⋊C4—42no—
264C2×C22⋊C4—44no—
265C2×C22⋊C4—44no—
266C2×C22⋊C4—44no—
267C2×C22⋊C4—44no—
268C2×C22⋊C4—44no—
269C2×C22⋊C4—44no—
270C2×C22⋊C4—44no—
271C2×C22⋊C4—44no—
272C2×C22⋊C4C2×D4116yes—
273C2×C42—22no—
274C2×C42—22no—
275C2×C42—24no—
276C2×C42—22no—
277C2×C42—22no—
278C2×C42—24no—
279C2×C42—24no—
280C2×C42—24no—
281C2×C42—24no—
282C2×C42—24no—
283C2×C42—42no—
284C2×C42—42no—
285C2×C42—44no—
286C2×C42C2×D414yes—
287C2×C42#4118yes—
288#95—28no—
289#95—28no—
290#95—28no—
291#95—28no—
292#95—28no—
293#95—28no—
294#95—28no—
295#95—28no—
296#95—28no—
297#95—44no—
298#95—44no—
299#95—44no—
300#95—44no—
301#95—44no—
302#95—42no—
303#95—44no—
304#95—42no—
305#95—44no—
306C24—24no—
307C24—24no—
308C24—24no—
309C24—22no—
310C24—22no—
311C24—24no—
312C24—24no—
313C24—24no—
314C24—22no—
315C24—24no—
316C24—24no—
317C24—24no—
318C24—22no—
319C24—22no—
320C24—24no—
321C24—24no—
322C24—24no—
323C24C22×D414yes—
324C24C2511yescharacteristic
325C2×Q8—28no—
326C2×Q8—44no—
327C2×Q8—44no—
328C2×Q8—44no—
329C2×Q8—44no—
330C2×Q8—44no—
331C2×Q8—84no—
332C2×Q8—84no—
333C2×Q8—84no—
334C2×Q8—84no—
335C2×D4—28no—
336C2×D4—44no—
337C2×D4—44no—
338C2×D4—48no—
339C2×D4—48no—
340C2×D4—416no—
341C2×D4—416no—
342C2×D4—44no—
343C2×D4—44no—
344C2×D4—44no—
345C2×D4—48no—
346C2×D4—48no—
347C2×D4—84no—
348C2×D4—84no—
349C2×D4—84no—
350C2×D4—84no—
351C2×D4—84no—
352C2×D4—84no—
353C2×D4—88no—
354C2×D4—88no—
355C22×C4—21no—
356C22×C4—21no—
357C22×C4—22no—
358C22×C4—21no—
359C22×C4—21no—
360C22×C4—21no—
361C22×C4—21no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—24no—
370C22×C4—22no—
371C22×C4—24no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—24no—
379C22×C4—24no—
380C22×C4—22no—
381C22×C4—24no—
382C22×C4—24no—
383C22×C4—24no—
384C22×C4—24no—
385C22×C4—24no—
386C22×C4—24no—
387C22×C4—24no—
388C22×C4—24no—
389C22×C4—24no—
390C22×C4—24no—
391C22×C4—24no—
392C22×C4—24no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—24no—
396C22×C4—24no—
397C22×C4—24no—
398C22×C4—24no—
399C22×C4—24no—
400C22×C4—24no—
401C22×C4—24no—
402C22×C4—24no—
403C22×C4—24no—
404C22×C4—24no—
405C22×C4—24no—
406C22×C4—24no—
407C22×C4—24no—
408C22×C4—24no—
409C22×C4—24no—
410C22×C4—24no—
411C22×C4—48no—
412C22×C4—84no—
413C22×C4—88no—
414C22×C4—84no—
415C22×C4—88no—
416C22×C4C22×D414yes—
417C22×C4C22×D412yes—
418C22×C4C22×D414yes—
419C22×C4C2×#4114yes—
420C22×C4C2×#4114yes—
421C22×C4C2×#4114yes—
422C22×C4H(5,𝔽2)14yes—
423C22×C4H(5,𝔽2)14yes—
424C22×C4H(5,𝔽2)12yes—
425C22×C4H(5,𝔽2)12yes—
426C22×C42−1+414yes—
427C4⋊C4—48no—
428C4⋊C4—48no—
429C4⋊C4—48no—
430C4⋊C4—48no—
431C4⋊C4—416no—
432C4⋊C4—84no—
433C4⋊C4—84no—
434C4⋊C4—84no—
435C4⋊C4—88no—
436C22⋊C4—232no—
437C22⋊C4—416no—
438C22⋊C4—416no—
439C22⋊C4—416no—
440C22⋊C4—48no—
441C22⋊C4—416no—
442C22⋊C4—416no—
443C22⋊C4—84no—
444C22⋊C4—88no—
445C22⋊C4—84no—
446C22⋊C4—88no—
447C22⋊C4—88no—
448C22⋊C4—88no—
449C42—28no—
450C42—28no—
451C42—48no—
452C42—48no—
453C42—84no—
454C42#123116yescomplemented
455C23—24no—
456C23—24no—
457C23—24no—
458C23—28no—
459C23—28no—
460C23—44no—
461C23—44no—
462C23—44no—
463C23—44no—
464C23—44no—
465C23—44no—
466C23—42no—
467C23—42no—
468C23—44no—
469C23—42no—
470C23—42no—
471C23—44no—
472C23—44no—
473C23—44no—
474C23—44no—
475C23—44no—
476C23—42no—
477C23—42no—
478C23—42no—
479C23—44no—
480C23—44no—
481C23—44no—
482C23—44no—
483C23—44no—
484C23—44no—
485C23—44no—
486C23—44no—
487C23—42no—
488C23—42no—
489C23—44no—
490C23—42no—
491C23—42no—
492C23—44no—
493C23—44no—
494C23—44no—
495C23—44no—
496C23—44no—
497C23C23×D411yescharacteristic
498C23C22×#4112yes—
499C23C2×H(5,𝔽2)11yescharacteristic
500C23C2×H(5,𝔽2)11yescharacteristic
501C23C2×H(5,𝔽2)12yes—
502C23C2×H(5,𝔽2)12yes—
503C23C2×2−1+412yes—
504C23#58512yes—
505C23#58512yes—
506Q8—88no—
507Q8—88no—
508D4—88no—
509D4—816no—
510D4—816no—
511D4—88no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—22no—
516C2×C4—24no—
517C2×C4—24no—
518C2×C4—24no—
519C2×C4—24no—
520C2×C4—24no—
521C2×C4—24no—
522C2×C4—24no—
523C2×C4—24no—
524C2×C4—24no—
525C2×C4—24no—
526C2×C4—22no—
527C2×C4—22no—
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—42no—
537C2×C4—42no—
538C2×C4—44no—
539C2×C4—44no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—44no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—44no—
547C2×C4—44no—
548C2×C4—44no—
549C2×C4—44no—
550C2×C4—44no—
551C2×C4—44no—
552C2×C4—44no—
553C2×C4—44no—
554C2×C4—44no—
555C2×C4—44no—
556C2×C4—44no—
557C2×C4—44no—
558C2×C4—44no—
559C2×C4—44no—
560C2×C4—44no—
561C2×C4—44no—
562C2×C4—44no—
563C2×C4—44no—
564C2×C4—44no—
565C2×C4—44no—
566C2×C4—44no—
567C2×C4—44no—
568C2×C4—44no—
569C2×C4—44no—
570C2×C4—44no—
571C2×C4—44no—
572C2×C4—44no—
573C2×C4—44no—
574C2×C4—44no—
575C2×C4—816no—
576C2×C4C2×#12318yes—
577C2×C4C2×#12314yes—
578C2×C4H(3,𝔽4)14yes—
579C2×C4#56314yes—
580C22—44no—
581C22—48no—
582C22—44no—
583C22—48no—
584C22—44no—
585C22—84no—
586C22—84no—
587C22—84no—
588C22—84no—
589C22—82no—
590C22—84no—
591C22—82no—
592C22—84no—
593C22—84no—
594C22—84no—
595C22—84no—
596C22—84no—
597C22—82no—
598C22C22×#12311yescharacteristic
599C22#348411yescharacteristic
600C22#348411yescharacteristic
601C22#348812yes—
602C22#348912yes—
603C22#350612yes—
604C22#350712yes—
605C22#350812yes—
606C22#350812yes—
607C22#351112yes—
608C22#351212yes—
609C22C2×H(3,𝔽4)11yescharacteristic
610C22C2×#56311yescharacteristic
611C22#352912yes—
612C22#354511yescharacteristic
613C22#354511yescharacteristic
614C22#354812yes—
615C22#355812yes—
616C22#355812yes—
617C22#355912yes—
618C22#356112yes—
619C4—24no—
620C4—24no—
621C4—28no—
622C4—24no—
623C4—44no—
624C4—44no—
625C4—84no—
626C4—84no—
627C4—84no—
628C4—84no—
629C4—82no—
630C2—84no—
631C2—162no—
632C2#3551712yes—
633C2#3555411yescharacteristic
634C2#4597711yescharacteristic
635C2#4601412yes—
636C2#4601612yes—
637C2#4601712yes—
638C2#4601812yes—
639C2#4613611yescharacteristic
640C2#4622412yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC22×#127C22
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×#2379C2semidirect
3. #16094C2semidirect
4. #15784C2semidirect
5. #15783C2semidirect
6. #15445C2semidirect
7. #15348C2semidirect
8. #15329C2semidirect
9. #15271C2semidirect
11. #14707C2semidirect
12. #14662C2semidirect
13. #14662C2semidirect
454. C42#123semidirect

Maximal subgroups

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

subgroupindex
C2×#28582
C2×#23792
#160942
#157842
#157832
#154452
#153482
#153292
#152712
#152612
#147072
#146622