Finite Group Databasebeta

← #4607021

Subgroups — #4607021

Order 512.

2852 conjugacy classes of subgroups.

The subgroup lattice has 666 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2733C211yesmaximal, characteristic, complemented
2C2×#2545C211yesmaximal, characteristic, complemented
3#17091C214yesmaximal, complemented
4#16551C212yesmaximal, complemented
5#16540C214yesmaximal, complemented
6#16509C212yesmaximal, complemented
7#16291C214yesmaximal, complemented
8#16035C214yesmaximal, complemented
9#15763C212yesmaximal, complemented
10#15761C212yesmaximal, complemented
11#15107C212yesmaximal, complemented
12#14883C212yesmaximal, complemented
13#14668C211yesmaximal, characteristic, complemented
14C22×C4⋊Q8C2211yescharacteristic
15C22×#124C2211yescharacteristic
16C22×#124C2211yescharacteristic
17C22×#124C2212yes—
18C22×C4×Q8C2211yescharacteristic
19#2733—28no—
20#2545—28no—
21C2×#400C2212yes—
22C2×#396C2214yes—
23C2×#394C2212yes—
24C2×#394C2212yes—
25C2×#394C2212yes—
26C2×#394C2214yes—
27C2×#394C2214yes—
28C2×#393C2214yes—
29C2×#392C2214yes—
30C2×#392C2214yes—
31C2×#392C2214yes—
32C2×#392C2212yes—
33C2×#392C2212yes—
34C2×#392C2214yes—
35C2×#391C2211yescharacteristic
36C2×#390C2212yes—
37C2×#390C2211yescharacteristic
38C2×#390C2212yes—
39C2×#388C2214yes—
40C2×#388C2214yes—
41C2×C4×C4⋊C4C2211yescharacteristic
42#1446C2212yes—
43#1445C2212yes—
44#1443C2214yes—
45#1442C2214yes—
46#1441C2212yes—
47#1441C2212yes—
48#1440C2214yes—
49#1440C2214yes—
50#1439C2214yes—
51#1439C2214yes—
52#1439C2214yes—
53#1439C2214yes—
54#1439C2214yes—
55#1438C2212yes—
56#1438C2214yes—
57#1438C2214yes—
58#1438C2214yes—
59#1438C2214yes—
60#1438C2214yes—
61#1437C2214yes—
62#1437C2214yes—
63#1437C2214yes—
64#1437C2212yes—
65#1437C2212yes—
66#1436C2214yes—
67#1434C2212yes—
68#1434C2212yes—
69C23×Q8C2311yescharacteristic
70C23×Q8C2311yescharacteristic
71C23×D4C2311yescharacteristic
72C23×D4C2312yes—
73C23×D4C2312yes—
74C23×D4C2312yes—
75C23×D4C2314yes—
76C23×D4C2314yes—
77C2×C4⋊Q8—24no—
78C2×C4⋊Q8—44no—
79C2×#124—28no—
80C2×#124—24no—
81C2×#124—24no—
82C2×#124—42no—
83C2×#124—42no—
84C2×#124—44no—
85C2×#124—48no—
86C2×C4×Q8—24no—
87C2×C4×Q8—42no—
88C2×C4×Q8—42no—
89C22×C4⋊C4C2312yes—
90C22×C4⋊C4C2312yes—
91C22×C4⋊C4C2314yes—
92C22×C4⋊C4C2312yes—
93C22×C4⋊C4C2312yes—
94C22×C4⋊C4C2311yescharacteristic
95C22×C22⋊C4C2314yes—
96C22×C22⋊C4C2314yes—
97C22×C22⋊C4C2312yes—
98C22×C22⋊C4C2314yes—
99C22×C22⋊C4C2312yes—
100C22×C22⋊C4C2314yes—
101C22×C22⋊C4C2314yes—
102C22×C22⋊C4C2314yes—
103C22×C22⋊C4C2314yes—
104C22×C22⋊C4C2314yes—
105C22×C22⋊C4C2314yes—
106C22×C22⋊C4C2314yes—
107C22×C22⋊C4C2314yes—
108C22×C22⋊C4C2314yes—
109C22×C22⋊C4C2314yes—
110C22×C22⋊C4C2314yes—
111C22×C42C2311yescharacteristic
112C22×C42C2312yes—
113C22×C42C2312yes—
114C22×C42C2311yescharacteristic
115C22×C42C2311yescharacteristic
116#400—44no—
117#396—216no—
118#394—28no—
119#394—48no—
120#394—44no—
121#394—48no—
122#394—44no—
123#393—48no—
124#392—216no—
125#392—48no—
126#392—44no—
127#392—48no—
128#392—44no—
129#392—48no—
130#391D418yescomplemented
131#390—24no—
132#390—28no—
133#390—44no—
134#388—216no—
135#388—216no—
136C4×C4⋊C4—24no—
137C2×#95C2314yes—
138C2×#95C2314yes—
139C2×#95C2312yes—
140C2×#95C2312yes—
141C2×#95C2314yes—
142C2×#95C2312yes—
143C2×#95C2312yes—
144C2×#95C2314yes—
145C2×#95C2314yes—
146C2×#95C2314yes—
147C2×#95C2312yes—
148C2×#95C2312yes—
149C2×#95C2314yes—
150C2×#95C2312yes—
151C2×#95C2314yes—
152C2×#95C2314yes—
153C2×#95C2314yes—
154C2×#95C2314yes—
155C25C2412yes—
156C25C2414yes—
157C22×Q8—22no—
158C22×Q8—24no—
159C22×Q8—24no—
160C22×Q8—24no—
161C22×Q8—41no—
162C22×Q8—41no—
163C22×Q8—41no—
164C22×Q8—41no—
165C22×Q8—42no—
166C22×Q8C2×D414yes—
167C22×D4—22no—
168C22×D4—24no—
169C22×D4—24no—
170C22×D4—24no—
171C22×D4—24no—
172C22×D4—41no—
173C22×D4—41no—
174C22×D4—42no—
175C22×D4—44no—
176C22×D4—44no—
177C22×D4—44no—
178C22×D4—42no—
179C22×D4—42no—
180C22×D4—44no—
181C22×D4—44no—
182C22×D4—44no—
183C22×D4—44no—
184C22×D4—44no—
185C22×D4—44no—
186C22×D4—44no—
187C22×D4—42no—
188C22×D4—42no—
189C22×D4—44no—
190C22×D4—44no—
191C22×D4—44no—
192C22×D4—44no—
193C22×D4—44no—
194C22×D4—44no—
195C22×D4—44no—
196C22×D4—44no—
197C22×D4—42no—
198C22×D4—42no—
199C22×D4—44no—
200C22×D4—44no—
201C23×C4C2414yes—
202C23×C4C2414yes—
203C23×C4C2414yes—
204C23×C4C2412yes—
205C23×C4C2412yes—
206C23×C4C2412yes—
207C23×C4C2411yescharacteristic
208C23×C4C2412yes—
209C23×C4C2412yes—
210C23×C4C2411yescharacteristic
211C23×C4C2411yescharacteristic
212C4⋊Q8—416no—
213#124—416no—
214#124—416no—
215#124—432no—
216C4×Q8—416no—
217C2×C4⋊C4—24no—
218C2×C4⋊C4—28no—
219C2×C4⋊C4—24no—
220C2×C4⋊C4—24no—
221C2×C4⋊C4—28no—
222C2×C4⋊C4—24no—
223C2×C4⋊C4—24no—
224C2×C4⋊C4—42no—
225C2×C4⋊C4—41no—
226C2×C4⋊C4—41no—
227C2×C4⋊C4—42no—
228C2×C4⋊C4—44no—
229C2×C4⋊C4—42no—
230C2×C4⋊C4—42no—
231C2×C4⋊C4—44no—
232C2×C4⋊C4—42no—
233C2×C4⋊C4C2×D414yes—
234C2×C22⋊C4—24no—
235C2×C22⋊C4—28no—
236C2×C22⋊C4—28no—
237C2×C22⋊C4—28no—
238C2×C22⋊C4—24no—
239C2×C22⋊C4—28no—
240C2×C22⋊C4—28no—
241C2×C22⋊C4—28no—
242C2×C22⋊C4—24no—
243C2×C22⋊C4—24no—
244C2×C22⋊C4—28no—
245C2×C22⋊C4—28no—
246C2×C22⋊C4—28no—
247C2×C22⋊C4—28no—
248C2×C22⋊C4—28no—
249C2×C22⋊C4—28no—
250C2×C22⋊C4—28no—
251C2×C22⋊C4—28no—
252C2×C22⋊C4—28no—
253C2×C22⋊C4—28no—
254C2×C22⋊C4—44no—
255C2×C22⋊C4—44no—
256C2×C22⋊C4—44no—
257C2×C22⋊C4—44no—
258C2×C22⋊C4—42no—
259C2×C22⋊C4—42no—
260C2×C22⋊C4—44no—
261C2×C22⋊C4—44no—
262C2×C22⋊C4—44no—
263C2×C22⋊C4—44no—
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⋊C4—44no—
273C2×C22⋊C4—44no—
274C2×C22⋊C4—44no—
275C2×C22⋊C4—44no—
276C2×C22⋊C4—44no—
277C2×C22⋊C4—44no—
278C2×C22⋊C4—44no—
279C2×C22⋊C4—44no—
280C2×C22⋊C4—44no—
281C2×C22⋊C4—44no—
282C2×C42—22no—
283C2×C42—22no—
284C2×C42—22no—
285C2×C42—24no—
286C2×C42—24no—
287C2×C42—22no—
288C2×C42—24no—
289C2×C42—42no—
290C2×C42—42no—
291C2×C42—44no—
292C2×C42—42no—
293C2×C42C2×D414yes—
294#95—24no—
295#95—24no—
296#95—28no—
297#95—28no—
298#95—44no—
299#95—44no—
300#95—44no—
301#95—44no—
302#95—42no—
303#95—42no—
304#95—44no—
305#95—44no—
306#95—42no—
307#95—42no—
308#95—44no—
309#95—44no—
310#95—42no—
311#95C2×D4116yes—
312C24—22no—
313C24—24no—
314C24—24no—
315C24—24no—
316C24—22no—
317C24—22no—
318C24—24no—
319C24—24no—
320C24—24no—
321C24—24no—
322C24—24no—
323C24—24no—
324C24—24no—
325C24—24no—
326C24—24no—
327C24—24no—
328C24—24no—
329C24—24no—
330C24—24no—
331C24—24no—
332C24—24no—
333C24—24no—
334C24—24no—
335C24—24no—
336C24C2511yescharacteristic
337C2×Q8—28no—
338C2×Q8—28no—
339C2×Q8—44no—
340C2×Q8—44no—
341C2×Q8—48no—
342C2×Q8—48no—
343C2×Q8—44no—
344C2×Q8—44no—
345C2×Q8—84no—
346C2×Q8—84no—
347C2×D4—44no—
348C2×D4—44no—
349C2×D4—48no—
350C2×D4—48no—
351C2×D4—48no—
352C2×D4—44no—
353C2×D4—84no—
354C2×D4—84no—
355C2×D4—84no—
356C2×D4—84no—
357C2×D4—84no—
358C2×D4—84no—
359C2×D4—88no—
360C2×D4—84no—
361C2×D4—88no—
362C2×D4—88no—
363C2×D4—84no—
364C2×D4—88no—
365C2×D4—88no—
366C2×D4—88no—
367C2×D4—88no—
368C2×D4—88no—
369C2×D4—88no—
370C2×D4—88no—
371C2×D4—88no—
372C2×D4—88no—
373C2×D4—88no—
374C2×D4—88no—
375C2×D4—88no—
376C2×D4—88no—
377C2×D4—88no—
378C2×D4—88no—
379C2×D4—88no—
380C2×D4—88no—
381C22×C4—21no—
382C22×C4—21no—
383C22×C4—22no—
384C22×C4—21no—
385C22×C4—22no—
386C22×C4—21no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—24no—
393C22×C4—22no—
394C22×C4—24no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—24no—
398C22×C4—21no—
399C22×C4—21no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—24no—
403C22×C4—22no—
404C22×C4—24no—
405C22×C4—22no—
406C22×C4—24no—
407C22×C4—24no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—24no—
411C22×C4—24no—
412C22×C4—22no—
413C22×C4—24no—
414C22×C4—24no—
415C22×C4—24no—
416C22×C4—24no—
417C22×C4—24no—
418C22×C4—24no—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—24no—
422C22×C4—24no—
423C22×C4—24no—
424C22×C4—24no—
425C22×C4—24no—
426C22×C4—24no—
427C22×C4—24no—
428C22×C4—24no—
429C22×C4—24no—
430C22×C4—24no—
431C22×C4—24no—
432C22×C4—24no—
433C22×C4—24no—
434C22×C4C22×D414yes—
435C22×C4C22×D414yes—
436C22×C4C22×D414yes—
437C22×C4C22×D412yes—
438C22×C4H(5,𝔽2)14yes—
439C22×C4H(5,𝔽2)12yes—
440C22×C4H(5,𝔽2)12yes—
441C22×C4H(5,𝔽2)14yes—
442C22×C4H(5,𝔽2)14yes—
443C4⋊C4—44no—
444C4⋊C4—48no—
445C4⋊C4—48no—
446C4⋊C4—416no—
447C4⋊C4—84no—
448C4⋊C4—84no—
449C22⋊C4—416no—
450C22⋊C4—48no—
451C22⋊C4—48no—
452C22⋊C4—416no—
453C22⋊C4—416no—
454C22⋊C4—416no—
455C22⋊C4—416no—
456C22⋊C4—88no—
457C22⋊C4—88no—
458C22⋊C4—88no—
459C22⋊C4—88no—
460C22⋊C4—88no—
461C22⋊C4—88no—
462C22⋊C4—88no—
463C22⋊C4—88no—
464C22⋊C4—88no—
465C42—28no—
466C42—44no—
467C42—48no—
468C42—48no—
469C42—44no—
470C23—44no—
471C23—42no—
472C23—42no—
473C23—42no—
474C23—44no—
475C23—44no—
476C23—44no—
477C23—44no—
478C23—44no—
479C23—44no—
480C23—44no—
481C23—44no—
482C23—44no—
483C23—42no—
484C23—44no—
485C23—44no—
486C23—44no—
487C23—44no—
488C23—44no—
489C23—44no—
490C23—44no—
491C23—44no—
492C23—44no—
493C23—44no—
494C23—44no—
495C23—44no—
496C23—44no—
497C23—44no—
498C23—44no—
499C23—44no—
500C23—42no—
501C23—42no—
502C23—44no—
503C23—44no—
504C23—44no—
505C23—44no—
506C23—44no—
507C23—42no—
508C23—44no—
509C23—44no—
510C23—44no—
511C23—44no—
512C23—44no—
513C23—44no—
514C23—44no—
515C23—44no—
516C23—44no—
517C23—44no—
518C23—44no—
519C23—44no—
520C23—44no—
521C23—44no—
522C23—44no—
523C23—44no—
524C23—44no—
525C23—44no—
526C23C23×D411yescharacteristic
527C23C2×H(5,𝔽2)12yes—
528C23C2×H(5,𝔽2)11yescharacteristic
529C23C2×H(5,𝔽2)11yescharacteristic
530C23C2×H(5,𝔽2)12yes—
531C23C2×H(5,𝔽2)12yes—
532C23#58514yes—
533C23#58512yes—
534Q8—416no—
535Q8—88no—
536D4—88no—
537D4—816no—
538D4—816no—
539D4—816no—
540D4—832no—
541D4—832no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—24no—
545C2×C4—24no—
546C2×C4—24no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—24no—
550C2×C4—24no—
551C2×C4—24no—
552C2×C4—24no—
553C2×C4—24no—
554C2×C4—24no—
555C2×C4—24no—
556C2×C4—28no—
557C2×C4—24no—
558C2×C4—24no—
559C2×C4—24no—
560C2×C4—24no—
561C2×C4—24no—
562C2×C4—28no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—41no—
570C2×C4—42no—
571C2×C4—44no—
572C2×C4—44no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—41no—
577C2×C4—42no—
578C2×C4—44no—
579C2×C4—44no—
580C2×C4—44no—
581C2×C4—44no—
582C2×C4—44no—
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—44no—
596C2×C4—44no—
597C2×C4—44no—
598C2×C4—44no—
599C2×C4—44no—
600C2×C4—44no—
601C2×C4—44no—
602C2×C4—44no—
603C2×C4#53514yes—
604C22—84no—
605C22—84no—
606C22—84no—
607C22—84no—
608C22—84no—
609C22—84no—
610C22—82no—
611C22—84no—
612C22—82no—
613C22—84no—
614C22—84no—
615C22—84no—
616C22—84no—
617C22—84no—
618C22—82no—
619C22—84no—
620C22—84no—
621C22—84no—
622C22—84no—
623C22—84no—
624C22—84no—
625C22—84no—
626C22—84no—
627C22—84no—
628C22C2×#53511yescharacteristic
629C22#348411yescharacteristic
630C22#348411yescharacteristic
631C22#348712yes—
632C22#348812yes—
633C22#352911yescharacteristic
634C22#353712yes—
635C22#353712yes—
636C22#354512yes—
637C22#354614yes—
638C22#354911yescharacteristic
639C22#355112yes—
640C22#355114yes—
641C22#355812yes—
642C22#355814yes—
643C22#356214yes—
644C4—24no—
645C4—44no—
646C4—44no—
647C4—42no—
648C4—42no—
649C4—44no—
650C4—44no—
651C4—44no—
652C4—84no—
653C4—84no—
654C4—84no—
655C2—162no—
656C2—164no—
657C2#3671711yescharacteristic
658C2#4288912yes—
659C2#4622511yescharacteristic
660C2#4955312yes—
661C2#5208311yescharacteristic
662C2#5288612yes—
663C2#5352514yes—
664C2#5390512yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
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×#2733C2semidirect
2. C2×#2545C2semidirect
3. #17091C2semidirect
4. #16551C2semidirect
5. #16540C2semidirect
6. #16509C2semidirect
7. #16291C2semidirect
8. #16035C2semidirect
9. #15763C2semidirect
10. #15761C2semidirect
11. #15107C2semidirect
12. #14883C2semidirect
13. #14668C2semidirect
130. #391D4semidirect

Maximal subgroups

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

subgroupindex
C2×#27332
C2×#25452
#170912
#165512
#165402
#165092
#162912
#160352
#157632
#157612
#151072
#148832
#146682