Finite Group Databasebeta

← #4311789

Subgroups — #4311789

Order 512.

2970 conjugacy classes of subgroups.

The subgroup lattice has 664 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2804C211yesmaximal, characteristic, complemented
2C2×#2311C211yesmaximal, characteristic
3#16891C212yesmaximal, complemented
4#16653C212yesmaximal, complemented
5#16299C214yesmaximal, complemented
6#15958C214yesmaximal, complemented
7#15506C214yesmaximal, complemented
8#14887C212yesmaximal, complemented
9#14822C212yesmaximal, complemented
10#14767C214yesmaximal, complemented
11#14513C212yesmaximal, complemented
12#14486C211yesmaximal, characteristic, complemented
13#13956C212yesmaximal
14C23×#41C2211yescharacteristic
15C22×#117C2211yescharacteristic
16C23×C4⋊C4C2211yescharacteristic
17#2804—28no—
18C2×#400C2212yes—
19C2×#400C2212yes—
20C2×#395C2212yes—
21C2×#393C2214yes—
22C2×#393C2212yes—
23#2311—28no—
24C2×#391C2211yescharacteristic
25C2×#391C2212yes—
26C2×#391C2211yescharacteristic
27C2×#390C2211yescharacteristic
28C2×#390C2211yescharacteristic
29C2×#386C2212yes—
30C2×#386C2214yes—
31C2×#386C2212yes—
32C2×#386C2212yes—
33C2×#386C2212yes—
34C2×#385C2214yes—
35C2×#381C2212yes—
36C2×#380C2214yes—
37C2×#380C2212yes—
38C22×#95C2212yes—
39#1446C2214yes—
40#1446C2214yes—
41#1445C2214yes—
42#1445C2214yes—
43#1443C2214yes—
44#1443C2214yes—
45#1443C2214yes—
46#1442C2214yes—
47#1442C2214yes—
48#1442C2214yes—
49#1442C2214yes—
50#1441C2214yes—
51#1441C2212yes—
52#1441C2212yes—
53#1441C2212yes—
54#1439C2212yes—
55#1439C2214yes—
56#1439C2214yes—
57#1439C2214yes—
58#1439C2214yes—
59#1438C2214yes—
60#1436C2214yes—
61#1434C2212yes—
62C22⋊#95C2214yes—
63C22⋊#95C2214yes—
64C22⋊#95C2214yes—
65C22⋊#95C2214yes—
66C22⋊#95C2212yes—
67C22⋊#95C2214yes—
68C22⋊#95C2214yes—
69C22×#41—24no—
70C22×#41—24no—
71C22×#41—42no—
72C22×#41—42no—
73C22×#41—42no—
74C22×#41—42no—
75C22×#41Q818yescomplemented
76C23×Q8C2311yescharacteristic
77C23×Q8C2312yes—
78C23×D4C2312yes—
79C23×D4C2311yescharacteristic
80C24×C4C2311yescharacteristic
81C24×C4C2312yes—
82C24×C4C2312yes—
83C24×C4C2312yes—
84C24×C4C2312yes—
85C2×#117—24no—
86C2×#117—42no—
87C2×#117—42no—
88C22×C4⋊C4—24no—
89C22×C4⋊C4—42no—
90C22×C4⋊C4—42no—
91C22×C4⋊C4C2312yes—
92C22×C4⋊C4C2312yes—
93C22×C4⋊C4C2312yes—
94C22×C4⋊C4C2311yescharacteristic
95C22×C4⋊C4C2312yes—
96C22×C4⋊C4C2314yes—
97C22×C4⋊C4C2312yes—
98C22×C4⋊C4C2311yescharacteristic
99C22×C22⋊C4C2312yes—
100C22×C22⋊C4C2312yes—
101C22×C22⋊C4C2314yes—
102C22×C22⋊C4C2314yes—
103C22×C22⋊C4C2314yes—
104C22×C22⋊C4C2314yes—
105C22×C22⋊C4C2314yes—
106C22×C22⋊C4C2312yes—
107C22×C22⋊C4C2314yes—
108C22×C42C2311yescharacteristic
109C22×C42C2311yescharacteristic
110#400—28no—
111#400—44no—
112#395—44no—
113#393—216no—
114#393—44no—
115#391—24no—
116#391—24no—
117#391—28no—
118#390—24no—
119#390D418yes—
120#386—28no—
121#386—28no—
122#386—216no—
123#386—44no—
124#386—44no—
125#385—216no—
126#381—44no—
127#380—216no—
128#380—44no—
129C2×#95—44no—
130C2×#95C2312yes—
131C2×#95C2314yes—
132C2×#95C2314yes—
133C2×#95C2314yes—
134C2×#95C2314yes—
135C2×#95C2314yes—
136C2×#95C2314yes—
137C2×#95C2314yes—
138C2×#95C2314yes—
139C2×#95C2312yes—
140C2×#95C2314yes—
141C2×#95C2312yes—
142C2×#95C2312yes—
143C2×#95C2314yes—
144C2×#95C2314yes—
145C2×#95C2314yes—
146C2×#95C2314yes—
147C2×#95C2314yes—
148C2×#95C2314yes—
149C2×#95C2314yes—
150C2×#95C2314yes—
151C2×#95C2314yes—
152C2×#95C2314yes—
153C2×#95C2314yes—
154C2×#95C2314yes—
155C25C2412yes—
156C25C2411yescharacteristic
157C2×#41—232no—
158C2×#41—416no—
159C2×#41—416no—
160C2×#41—416no—
161C2×#41—416no—
162C2×#41—416no—
163C2×#41—416no—
164C22×Q8—22no—
165C22×Q8—24no—
166C22×Q8—22no—
167C22×Q8—41no—
168C22×Q8—41no—
169C22×Q8—41no—
170C22×Q8—41no—
171C22×Q8—44no—
172C22×Q8—44no—
173C22×Q8—42no—
174C22×Q8—42no—
175C22×Q8C2×Q814yes—
176C22×D4—24no—
177C22×D4—24no—
178C22×D4—24no—
179C22×D4—24no—
180C22×D4—41no—
181C22×D4—41no—
182C22×D4—41no—
183C22×D4—41no—
184C22×D4—42no—
185C22×D4—42no—
186C22×D4C2×D414yes—
187C22×D4C2×Q818yes—
188C22×D4C2×Q814yes—
189C22×D4#4114yes—
190C23×C4—22no—
191C23×C4—22no—
192C23×C4—22no—
193C23×C4—22no—
194C23×C4—22no—
195C23×C4—22no—
196C23×C4—24no—
197C23×C4—24no—
198C23×C4—24no—
199C23×C4—24no—
200C23×C4—24no—
201C23×C4—24no—
202C23×C4—24no—
203C23×C4—24no—
204C23×C4—42no—
205C23×C4—42no—
206C23×C4—42no—
207C23×C4—44no—
208C23×C4—44no—
209C23×C4—44no—
210C23×C4—42no—
211C23×C4—42no—
212C23×C4—44no—
213C23×C4—44no—
214C23×C4—44no—
215C23×C4—42no—
216C23×C4—42no—
217C23×C4C2×Q814yes—
218C23×C4C2×Q818yes—
219C23×C4C2412yes—
220C23×C4C2414yes—
221C23×C4C2414yes—
222C23×C4C2414yes—
223C23×C4C2412yes—
224C23×C4C2414yes—
225C23×C4C2412yes—
226C23×C4C2412yes—
227C23×C4C2412yes—
228C23×C4C2411yescharacteristic
229C23×C4C2411yescharacteristic
230#117—416no—
231C2×C4⋊C4—24no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—24no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—24no—
236C2×C4⋊C4—24no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—41no—
239C2×C4⋊C4—41no—
240C2×C4⋊C4—42no—
241C2×C4⋊C4—41no—
242C2×C4⋊C4—41no—
243C2×C4⋊C4—42no—
244C2×C4⋊C4—416no—
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—42no—
253C2×C4⋊C4C2×D414yes—
254C2×C22⋊C4—24no—
255C2×C22⋊C4—24no—
256C2×C22⋊C4—28no—
257C2×C22⋊C4—28no—
258C2×C22⋊C4—24no—
259C2×C22⋊C4—24no—
260C2×C22⋊C4—28no—
261C2×C22⋊C4—28no—
262C2×C22⋊C4—28no—
263C2×C22⋊C4—44no—
264C2×C22⋊C4—44no—
265C2×C22⋊C4—42no—
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⋊C4C2×D4116yes—
281C2×C42—22no—
282C2×C42—41no—
283C2×C42—41no—
284C2×C42—41no—
285C2×C42—41no—
286C2×C42C2×D414yes—
287#95—28no—
288#95—28no—
289#95—28no—
290#95—28no—
291#95—28no—
292#95—24no—
293#95—28no—
294#95—28no—
295#95—28no—
296#95—44no—
297#95—44no—
298#95—44no—
299#95—44no—
300#95—44no—
301#95—44no—
302#95—44no—
303#95—44no—
304#95—44no—
305#95—44no—
306#95—44no—
307#95—42no—
308#95—44no—
309#95—42no—
310#95—42no—
311#95—44no—
312C24—21no—
313C24—21no—
314C24—21no—
315C24—22no—
316C24—22no—
317C24—22no—
318C24—21no—
319C24—22no—
320C24—24no—
321C24—24no—
322C24—24no—
323C24—24no—
324C24—24no—
325C24—24no—
326C24C22×D414yes—
327C24C22×Q814yes—
328C24C22×Q812yes—
329C24C2×#4114yes—
330C24H(5,𝔽2)12yes—
331C24H(5,𝔽2)12yes—
332C24C2511yescharacteristic
333#41—4128no—
334C2×Q8—28no—
335C2×Q8—44no—
336C2×Q8—44no—
337C2×Q8—44no—
338C2×Q8—44no—
339C2×Q8—44no—
340C2×Q8—44no—
341C2×Q8—48no—
342C2×Q8—84no—
343C2×Q8—84no—
344C2×Q8—88no—
345C2×Q8—88no—
346C2×D4—216no—
347C2×D4—216no—
348C2×D4—44no—
349C2×D4—44no—
350C2×D4—44no—
351C2×D4—44no—
352C2×D4—48no—
353C2×D4—44no—
354C2×D4—44no—
355C2×D4—48no—
356C2×D4—48no—
357C2×D4—48no—
358C2×D4—48no—
359C2×D4C22⋊Q8116yes—
360C22×C4—28no—
361C22×C4—28no—
362C22×C4—28no—
363C22×C4—28no—
364C22×C4—28no—
365C22×C4—21no—
366C22×C4—21no—
367C22×C4—21no—
368C22×C4—21no—
369C22×C4—21no—
370C22×C4—21no—
371C22×C4—21no—
372C22×C4—21no—
373C22×C4—28no—
374C22×C4—28no—
375C22×C4—216no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
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—24no—
388C22×C4—24no—
389C22×C4—24no—
390C22×C4—216no—
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—22no—
403C22×C4—24no—
404C22×C4—24no—
405C22×C4—24no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—24no—
410C22×C4—24no—
411C22×C4—24no—
412C22×C4—24no—
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—48no—
430C22×C4—48no—
431C22×C4—48no—
432C22×C4—48no—
433C22×C4—48no—
434C22×C4—48no—
435C22×C4—48no—
436C22×C4—48no—
437C22×C4—48no—
438C22×C4—88no—
439C22×C4—84no—
440C22×C4—84no—
441C22×C4—84no—
442C22×C4—88no—
443C22×C4—88no—
444C22×C4—88no—
445C22×C4—84no—
446C22×C4—84no—
447C22×C4—88no—
448C22×C4—88no—
449C22×C4C22×D414yes—
450C22×C4C22×D414yes—
451C22×C4C22×D412yes—
452C22×C4C22×Q814yes—
453C22×C4C22×Q812yes—
454C22×C4C22×Q812yes—
455C22×C4C2×#4112yes—
456C22×C4H(5,𝔽2)12yes—
457C22×C4H(5,𝔽2)12yes—
458C4⋊C4—44no—
459C4⋊C4—48no—
460C4⋊C4—48no—
461C4⋊C4—44no—
462C4⋊C4—84no—
463C4⋊C4—84no—
464C4⋊C4—84no—
465C4⋊C4—88no—
466C22⋊C4—416no—
467C22⋊C4—48no—
468C22⋊C4—84no—
469C22⋊C4—88no—
470C22⋊C4—84no—
471C22⋊C4—88no—
472C22⋊C4—88no—
473C22⋊C4—88no—
474C22⋊C4—88no—
475C42—44no—
476C42—44no—
477C23—22no—
478C23—22no—
479C23—24no—
480C23—24no—
481C23—22no—
482C23—24no—
483C23—22no—
484C23—24no—
485C23—28no—
486C23—24no—
487C23—22no—
488C23—22no—
489C23—24no—
490C23—28no—
491C23—28no—
492C23—28no—
493C23—28no—
494C23—28no—
495C23—28no—
496C23—24no—
497C23—24no—
498C23—28no—
499C23—42no—
500C23—42no—
501C23—44no—
502C23—44no—
503C23—42no—
504C23—42no—
505C23—42no—
506C23—42no—
507C23—42no—
508C23—42no—
509C23—44no—
510C23—44no—
511C23—44no—
512C23—44no—
513C23—44no—
514C23—44no—
515C23C2×C22⋊Q818yes—
516C23#54314yes—
517C23C23×D411yescharacteristic
518C23C23×Q811yescharacteristic
519C23C22×#4111yescharacteristic
520C23C2×H(5,𝔽2)11yescharacteristic
521C23C2×H(5,𝔽2)11yescharacteristic
522C23C2×H(5,𝔽2)11yescharacteristic
523C23C2×H(5,𝔽2)11yescharacteristic
524C23#58514yes—
525C23#58514yes—
526Q8—416no—
527Q8—816no—
528D4—416no—
529D4—432no—
530C2×C4—232no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—24no—
544C2×C4—24no—
545C2×C4—24no—
546C2×C4—24no—
547C2×C4—24no—
548C2×C4—24no—
549C2×C4—28no—
550C2×C4—24no—
551C2×C4—432no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—44no—
559C2×C4—44no—
560C2×C4—44no—
561C2×C4—42no—
562C2×C4—44no—
563C2×C4—42no—
564C2×C4—44no—
565C2×C4—42no—
566C2×C4—44no—
567C2×C4—44no—
568C2×C4—44no—
569C2×C4—44no—
570C2×C4—44no—
571C2×C4—44no—
572C2×C4—44no—
573C2×C4—42no—
574C2×C4—44no—
575C2×C4—44no—
576C2×C4—44no—
577C2×C4—44no—
578C2×C4—44no—
579C2×C4—42no—
580C2×C4—42no—
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—816no—
600C2×C4—816no—
601C2×C4—816no—
602C2×C4C2×C22⋊Q814yes—
603C2×C4#54314yes—
604C22—24no—
605C22—28no—
606C22—216no—
607C22—28no—
608C22—44no—
609C22—42no—
610C22—48no—
611C22—44no—
612C22—44no—
613C22—44no—
614C22—44no—
615C22—42no—
616C22—44no—
617C22—44no—
618C22—48no—
619C22—44no—
620C22—44no—
621C22—44no—
622C22—44no—
623C22—48no—
624C22C22×C22⋊Q811yescharacteristic
625C22C2×#54311yescharacteristic
626C22C2×#54311yescharacteristic
627C22#348411yescharacteristic
628C22#348411yescharacteristic
629C22#348512yes—
630C22#349011yescharacteristic
631C22#349011yescharacteristic
632C22#350012yes—
633C22#350414yes—
634C22#352114yes—
635C22#353312yes—
636C22#354012yes—
637C22#354012yes—
638C22#354512yes—
639C22#355114yes—
640C22#356014yes—
641C4—24no—
642C4—24no—
643C4—44no—
644C4—44no—
645C4—44no—
646C4—84no—
647C4—84no—
648C4—82no—
649C4—82no—
650C4—84no—
651C4—84no—
652C2—44no—
653C2—48no—
654C2#3498211yescharacteristic
655C2#3603112yes—
656C2#3742612yes—
657C2#4218012yes—
658C2#4642712yes—
659C2#4766512yes—
660C2#5174911yescharacteristic
661C2#5180312yes—
662C2#5205411yescharacteristic
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC23×#41C22
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×#2804C2semidirect
3. #16891C2semidirect
4. #16653C2semidirect
5. #16299C2semidirect
6. #15958C2semidirect
7. #15506C2semidirect
8. #14887C2semidirect
9. #14822C2semidirect
10. #14767C2semidirect
11. #14513C2semidirect
12. #14486C2semidirect
75. C22×#41Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#28042
C2×#23112
#168912
#166532
#162992
#159582
#155062
#148872
#148222
#147672
#145132
#144862
#139562