Finite Group Databasebeta

← #4397574

Subgroups — #4397574

Order 512.

2816 conjugacy classes of subgroups.

The subgroup lattice has 665 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2607C211yesmaximal, characteristic, complemented
2C2×#2436C211yesmaximal, characteristic
3#16891C212yesmaximal, complemented
4#16559C212yesmaximal, complemented
5#15636C214yesmaximal, complemented
6#15468C214yesmaximal, complemented
7#15459C214yesmaximal, complemented
8#15399C214yesmaximal, complemented
9#15053C212yesmaximal, complemented
10#14932C212yesmaximal, complemented
11#14659C211yesmaximal, characteristic
12#14336C211yesmaximal, characteristic, complemented
13#14290C211yesmaximal, characteristic
14#13922C211yesmaximal, characteristic, complemented
15#13780C211yesmaximal, characteristic, complemented
16C24×Q8C2211yescharacteristic
17C22×#124C2211yescharacteristic
18C22×#124C2211yescharacteristic
19C22×#121C2211yescharacteristic
20#2607—28no—
21#2436—28no—
22C2×#399C2212yes—
23C2×#393C2214yes—
24C2×#393C2212yes—
25C2×#393C2214yes—
26C2×#391C2212yes—
27C2×#391C2212yes—
28C2×#391C2211yescharacteristic
29C2×#391C2211yescharacteristic
30C2×#391C2212yes—
31C2×#391C2212yes—
32C2×#388C2214yes—
33C2×#387C2212yes—
34C2×#387C2212yes—
35C2×#386C2214yes—
36C2×#381C2211yescharacteristic
37C2×#381C2211yescharacteristic
38C2×#381C2212yes—
39C2×#381C2212yes—
40C2×#380C2211yescharacteristic
41C2×C4×C22⋊C4C2211yescharacteristic
42C2×C4×C22⋊C4C2211yescharacteristic
43#1446C2214yes—
44#1446C2212yes—
45#1446C2212yes—
46#1446C2214yes—
47#1446C2212yes—
48#1446C2212yes—
49#1445C2212yes—
50#1445C2212yes—
51#1443C2214yes—
52#1443C2214yes—
53#1443C2214yes—
54#1442C2214yes—
55#1442C2214yes—
56#1440C2214yes—
57#1440C2214yes—
58#1440C2214yes—
59#1439C2214yes—
60#1439C2214yes—
61#1439C2214yes—
62#1438C2212yes—
63#1438C2214yes—
64#1438C2212yes—
65#1438C2212yes—
66#1438C2212yes—
67#1437C2212yes—
68#1437C2214yes—
69#1437C2214yes—
70#1437C2212yes—
71#1436C2214yes—
72#1436C2214yes—
73#1434C2214yes—
74#1434C2212yes—
75C22⋊#95C2214yes—
76C4×#95C2212yes—
77C23×Q8—24no—
78C23×Q8—24no—
79C23×Q8—24no—
80C23×Q8—44no—
81C23×Q8—44no—
82C23×Q8C2312yes—
83C23×Q8C2311yescharacteristic
84C23×Q8C2311yescharacteristic
85C23×Q8C2312yes—
86C23×D4C2311yescharacteristic
87C23×D4C2312yes—
88C24×C4C2312yes—
89C24×C4C2311yescharacteristic
90C2×#124—24no—
91C2×#124—24no—
92C2×#124—44no—
93C2×#124—44no—
94C2×#121—24no—
95C2×#121—24no—
96C2×#121—24no—
97C22×C4⋊C4C2311yescharacteristic
98C22×C4⋊C4C2312yes—
99C22×C4⋊C4C2312yes—
100C22×C4⋊C4C2312yes—
101C22×C4⋊C4C2312yes—
102C22×C22⋊C4C2312yes—
103C22×C22⋊C4C2314yes—
104C22×C22⋊C4C2312yes—
105C22×C22⋊C4C2314yes—
106C22×C22⋊C4C2311yescharacteristic
107C22×C22⋊C4C2314yes—
108C22×C22⋊C4C2311yescharacteristic
109C22×C22⋊C4C2312yes—
110C22×C22⋊C4C2312yes—
111C22×C22⋊C4C2314yes—
112C22×C22⋊C4C2314yes—
113C22×C22⋊C4C2314yes—
114C22×C22⋊C4C2312yes—
115C22×C42C2311yescharacteristic
116C22×C42C2311yescharacteristic
117C22×C42C2314yes—
118C22×C42C2311yescharacteristic
119C22×C42C2311yescharacteristic
120#399—28no—
121#393—28no—
122#393—216no—
123#393—48no—
124#391—28no—
125#391—24no—
126#391—24no—
127#391—28no—
128#391—44no—
129#391—44no—
130#388—48no—
131#387—28no—
132#387—28no—
133#386—216no—
134#381—24no—
135#381—24no—
136#381—44no—
137#381—44no—
138#380—24no—
139C4×C22⋊C4—24no—
140C4×C22⋊C4D418yes—
141C2×#95C2314yes—
142C2×#95C2312yes—
143C2×#95C2314yes—
144C2×#95C2314yes—
145C2×#95C2312yes—
146C2×#95C2312yes—
147C2×#95C2314yes—
148C2×#95C2312yes—
149C2×#95C2312yes—
150C2×#95C2312yes—
151C2×#95C2314yes—
152C2×#95C2312yes—
153C2×#95C2312yes—
154C2×#95C2312yes—
155C2×#95C2312yes—
156C2×#95C2314yes—
157C2×#95C2314yes—
158C2×#95C2314yes—
159C2×#95C2314yes—
160C2×#95C2314yes—
161C2×#95C2314yes—
162C2×#95C2314yes—
163C2×#95C2314yes—
164C2×#95C2314yes—
165C2×#95C2314yes—
166C2×#95C2314yes—
167C2×#95C2314yes—
168C2×#95C2312yes—
169C25C2412yes—
170C25C2411yescharacteristic
171C22×Q8—22no—
172C22×Q8—22no—
173C22×Q8—24no—
174C22×Q8—24no—
175C22×Q8—24no—
176C22×Q8—22no—
177C22×Q8—22no—
178C22×Q8—22no—
179C22×Q8—22no—
180C22×Q8—24no—
181C22×Q8—24no—
182C22×Q8—24no—
183C22×Q8—24no—
184C22×Q8—24no—
185C22×Q8—24no—
186C22×Q8—42no—
187C22×Q8—42no—
188C22×Q8—42no—
189C22×Q8—42no—
190C22×Q8—42no—
191C22×Q8—44no—
192C22×Q8—44no—
193C22×Q8—432no—
194C22×Q8—416no—
195C22×Q8—416no—
196C22×Q8—432no—
197C22×Q8—416no—
198C22×D4—24no—
199C22×D4—24no—
200C22×D4—24no—
201C22×D4—22no—
202C22×D4—22no—
203C22×D4—22no—
204C22×D4—42no—
205C22×D4—42no—
206C22×D4—44no—
207C22×D4—44no—
208C23×C4—24no—
209C23×C4—24no—
210C23×C4—24no—
211C23×C4—24no—
212C23×C4—24no—
213C23×C4—42no—
214C23×C4—42no—
215C23×C4—42no—
216C23×C4—42no—
217C23×C4C2×D414yes—
218C23×C4#4114yes—
219C23×C4#4114yes—
220C23×C4C2412yes—
221C23×C4C2412yes—
222C23×C4C2414yes—
223C23×C4C2412yes—
224C23×C4C2414yes—
225C23×C4C2412yes—
226C23×C4C2412yes—
227C23×C4C2412yes—
228C23×C4C2412yes—
229C23×C4C2412yes—
230C23×C4C2412yes—
231C23×C4C2411yescharacteristic
232C23×C4C2411yescharacteristic
233#124—416no—
234#124—416no—
235#121—416no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—24no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—42no—
245C2×C4⋊C4—44no—
246C2×C4⋊C4—44no—
247C2×C4⋊C4—42no—
248C2×C4⋊C4—42no—
249C2×C22⋊C4—24no—
250C2×C22⋊C4—24no—
251C2×C22⋊C4—28no—
252C2×C22⋊C4—28no—
253C2×C22⋊C4—24no—
254C2×C22⋊C4—24no—
255C2×C22⋊C4—28no—
256C2×C22⋊C4—28no—
257C2×C22⋊C4—24no—
258C2×C22⋊C4—22no—
259C2×C22⋊C4—22no—
260C2×C22⋊C4—24no—
261C2×C22⋊C4—24no—
262C2×C22⋊C4—28no—
263C2×C22⋊C4—44no—
264C2×C22⋊C4—42no—
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—42no—
277C2×C22⋊C4—42no—
278C2×C22⋊C4—42no—
279C2×C22⋊C4—42no—
280C2×C22⋊C4—44no—
281C2×C22⋊C4—44no—
282C2×C22⋊C4—44no—
283C2×C22⋊C4C2×D418yes—
284C2×C42—22no—
285C2×C42—22no—
286C2×C42—42no—
287C2×C42—42no—
288C2×C42—44no—
289C2×C42—42no—
290C2×C42—42no—
291C2×C42—44no—
292C2×C42—44no—
293C2×C42C2×D414yes—
294C2×C42C2×D414yes—
295#95—24no—
296#95—24no—
297#95—24no—
298#95—28no—
299#95—28no—
300#95—24no—
301#95—24no—
302#95—24no—
303#95—24no—
304#95—24no—
305#95—24no—
306#95—24no—
307#95—28no—
308#95—28no—
309#95—28no—
310#95—28no—
311#95—44no—
312#95—44no—
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#95C2×D418yes—
323C24—22no—
324C24—22no—
325C24—22no—
326C24—22no—
327C24—24no—
328C24—22no—
329C24—22no—
330C24—22no—
331C24—24no—
332C24—24no—
333C24—24no—
334C24—24no—
335C24—24no—
336C24—24no—
337C24C22×D412yes—
338C24C2×#4112yes—
339C24C2×#4112yes—
340C24C2511yescharacteristic
341C2×Q8—28no—
342C2×Q8—28no—
343C2×Q8—216no—
344C2×Q8—48no—
345C2×Q8—48no—
346C2×Q8—48no—
347C2×Q8—48no—
348C2×Q8—48no—
349C2×Q8—48no—
350C2×Q8—48no—
351C2×Q8—48no—
352C2×Q8—48no—
353C2×Q8—44no—
354C2×Q8—44no—
355C2×Q8—48no—
356C2×Q8—48no—
357C2×Q8—4128no—
358C2×Q8—48no—
359C2×Q8—44no—
360C2×Q8—44no—
361C2×Q8—84no—
362C2×Q8—88no—
363C2×Q8—88no—
364C2×D4—44no—
365C2×D4—44no—
366C2×D4—416no—
367C2×D4—48no—
368C2×D4—44no—
369C2×D4—84no—
370C2×D4—84no—
371C2×D4—88no—
372C2×D4—84no—
373C2×D4—84no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—216no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—24no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—24no—
392C22×C4—24no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—24no—
399C22×C4—24no—
400C22×C4—22no—
401C22×C4—24no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—24no—
408C22×C4—24no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—24no—
412C22×C4—24no—
413C22×C4—24no—
414C22×C4—24no—
415C22×C4—24no—
416C22×C4—24no—
417C22×C4—24no—
418C22×C4—22no—
419C22×C4—24no—
420C22×C4—22no—
421C22×C4—22no—
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×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—44no—
438C22×C4—44no—
439C22×C4—48no—
440C22×C4—48no—
441C22×C4—48no—
442C22×C4—48no—
443C22×C4—48no—
444C22×C4—48no—
445C22×C4—48no—
446C22×C4—48no—
447C22×C4#124116yes—
448C22×C4C22×D414yes—
449C22×C4C22×D414yes—
450C22×C4C22×D412yes—
451C22×C4C22×D412yes—
452C22×C4C2×#4112yes—
453C22×C4C2×#4112yes—
454C22×C4C2×#4112yes—
455C22×C4C2×#4112yes—
456C22×C4H(5,𝔽2)14yes—
457C22×C4H(5,𝔽2)14yes—
458C22×C42−1+414yes—
459C4⋊C4—48no—
460C4⋊C4—48no—
461C4⋊C4—44no—
462C4⋊C4—84no—
463C4⋊C4—84no—
464C22⋊C4—48no—
465C22⋊C4—416no—
466C22⋊C4—416no—
467C22⋊C4—48no—
468C22⋊C4—416no—
469C22⋊C4—48no—
470C22⋊C4—88no—
471C22⋊C4—82no—
472C22⋊C4—82no—
473C22⋊C4—84no—
474C22⋊C4—84no—
475C22⋊C4—88no—
476C22⋊C4—88no—
477C42—44no—
478C42—44no—
479C42—82no—
480C42—82no—
481C42—88no—
482C23—24no—
483C23—22no—
484C23—22no—
485C23—24no—
486C23—24no—
487C23—24no—
488C23—24no—
489C23—22no—
490C23—22no—
491C23—24no—
492C23—24no—
493C23—44no—
494C23—44no—
495C23—44no—
496C23—44no—
497C23—42no—
498C23—44no—
499C23—42no—
500C23—42no—
501C23—44no—
502C23—44no—
503C23—44no—
504C23—44no—
505C23—42no—
506C23—44no—
507C23—42no—
508C23—42no—
509C23—42no—
510C23—44no—
511C23—44no—
512C23—44no—
513C23—44no—
514C23—44no—
515C23—44no—
516C23—44no—
517C23—44no—
518C23C2×#12414yes—
519C23C23×D411yescharacteristic
520C23C22×#4111yescharacteristic
521C23C22×#4111yescharacteristic
522C23C2×H(5,𝔽2)12yes—
523C23C2×H(5,𝔽2)12yes—
524C23C2×2−1+412yes—
525C23#58514yes—
526C23#58512yes—
527Q8—416no—
528Q8—416no—
529Q8—432no—
530Q8—816no—
531D4—88no—
532D4—816no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—24no—
538C2×C4—24no—
539C2×C4—24no—
540C2×C4—24no—
541C2×C4—24no—
542C2×C4—24no—
543C2×C4—24no—
544C2×C4—24no—
545C2×C4—24no—
546C2×C4—24no—
547C2×C4—24no—
548C2×C4—24no—
549C2×C4—24no—
550C2×C4—24no—
551C2×C4—24no—
552C2×C4—24no—
553C2×C4—24no—
554C2×C4—28no—
555C2×C4—416no—
556C2×C4—432no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—44no—
568C2×C4—44no—
569C2×C4—44no—
570C2×C4—44no—
571C2×C4—44no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—44no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—44no—
584C2×C4—44no—
585C2×C4—44no—
586C2×C4—44no—
587C2×C4—44no—
588C2×C4—44no—
589C2×C4—44no—
590C2×C4—42no—
591C2×C4—42no—
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—44no—
604C2×C4—44no—
605C2×C4—44no—
606C2×C4C2×#12414yes—
607C2×C4C2×#12414yes—
608C22—24no—
609C22—24no—
610C22—24no—
611C22—44no—
612C22—48no—
613C22—44no—
614C22—44no—
615C22—44no—
616C22—82no—
617C22—82no—
618C22—82no—
619C22—84no—
620C22—84no—
621C22—84no—
622C22—84no—
623C22—84no—
624C22—84no—
625C22C22×#12411yescharacteristic
626C22#344912yes—
627C22#348411yescharacteristic
628C22#348411yescharacteristic
629C22#348912yes—
630C22#349012yes—
631C22#349012yes—
632C22#350712yes—
633C22#350911yescharacteristic
634C22#351012yes—
635C22#351012yes—
636C22#351011yescharacteristic
637C22#354014yes—
638C22#355114yes—
639C22#355414yes—
640C22#356114yes—
641C4—24no—
642C4—24no—
643C4—44no—
644C4—44no—
645C4—44no—
646C4—44no—
647C4—44no—
648C4—82no—
649C4—84no—
650C4—82no—
651C4—82no—
652C4—82no—
653C4—84no—
654C2—44no—
655C2—162no—
656C2#3533811yescharacteristic
657C2#3536711yescharacteristic
658C2#3562811yescharacteristic
659C2#4060712yes—
660C2#4216912yes—
661C2#4645512yes—
662C2#4703612yes—
663C2#4801014yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC22×#121C22
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×#2607C2semidirect
3. #16891C2semidirect
4. #16559C2semidirect
5. #15636C2semidirect
6. #15468C2semidirect
7. #15459C2semidirect
8. #15399C2semidirect
9. #15053C2semidirect
10. #14932C2semidirect
12. #14336C2semidirect
14. #13922C2semidirect
15. #13780C2semidirect

Maximal subgroups

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

subgroupindex
C2×#26072
C2×#24362
#168912
#165592
#156362
#154682
#154592
#153992
#150532
#149322
#146592
#143362
#142902
#139222
#137802