Finite Group Databasebeta

← #643908

Subgroups — #643908

Order 512.

2746 conjugacy classes of subgroups.

The subgroup lattice has 630 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2647C211yesmaximal, characteristic, complemented
2C22×#384C211yesmaximal, characteristic
3#16570C214yesmaximal, complemented
4#16523C214yesmaximal, complemented
5#15146C212yesmaximal, complemented
6#15137C212yesmaximal, complemented
7#14822C214yesmaximal
8#14728C214yesmaximal
9#14682C211yesmaximal, characteristic
10#14348C211yesmaximal, characteristic, complemented
11#14277C211yesmaximal, characteristic, complemented
12#14276C211yesmaximal, characteristic, complemented
13#14130C212yesmaximal
14#13907C212yesmaximal
15#13753C211yesmaximal, characteristic
16C23×C4⋊C4C2212yes—
17C23×C4⋊C4C2214yes—
18C23×C42C2211yescharacteristic
19#2647—28no—
20C2×#400C2214yes—
21C2×#400C2214yes—
22C2×#398C2214yes—
23C2×#398C2214yes—
24C2×#389C2212yes—
25C2×#389C2212yes—
26C2×#389C2212yes—
27C2×#389C2212yes—
28C2×#385C2214yes—
29C2×#384—28no—
30C2×#384C2211yescharacteristic
31C2×#384C2212yes—
32C2×#384C2214yes—
33C2×#384C2211yescharacteristic
34C2×#382C2211yescharacteristic
35C2×#382C2211yescharacteristic
36C2×#380C2212yes—
37C2×#380C2214yes—
38C2×C4×C22⋊C4C2211yescharacteristic
39C2×C4×C22⋊C4C2211yescharacteristic
40C2×#376C2211yescharacteristic
41C2×#376C2211yescharacteristic
42#1444C2214yes—
43#1444C2214yes—
44#1444C2214yes—
45#1442C2214yes—
46#1442C2214yes—
47#1442C2214yes—
48#1442C2214yes—
49#1442C2214yes—
50#1442C2214yes—
51#1442C2214yes—
52#1441C2214yes—
53#1441C2214yes—
54#1441C2214yes—
55#1441C2214yes—
56#1441C2214yes—
57#1441C2214yes—
58C4⋊#95C2212yes—
59C4⋊#95C2212yes—
60C4⋊#95C2212yes—
61C4⋊#95C2212yes—
62C4⋊#95C2212yes—
63#1434C2214yes—
64#1434C2212yes—
65#1434C2212yes—
66C22⋊#95C2214yes—
67C22⋊#95C2214yes—
68C22⋊#95C2214yes—
69C22⋊#95C2212yes—
70C22⋊#95C2212yes—
71C4×#95C2211yescharacteristic
72C4×#95C2211yescharacteristic
73C24×C4C2312yes—
74C24×C4C2314yes—
75C24×C4C2311yescharacteristic
76C22×C4⋊C4—28no—
77C22×C4⋊C4—28no—
78C22×C4⋊C4—216no—
79C22×C4⋊C4—44no—
80C22×C4⋊C4—48no—
81C22×C4⋊C4—48no—
82C22×C4⋊C4C2312yes—
83C22×C4⋊C4C2312yes—
84C22×C4⋊C4C2312yes—
85C22×C4⋊C4C2314yes—
86C22×C4⋊C4C2314yes—
87C22×C4⋊C4C2314yes—
88C22×C4⋊C4C2312yes—
89C22×C4⋊C4C2314yes—
90C22×C4⋊C4C2314yes—
91C22×C4⋊C4C2314yes—
92C22×C4⋊C4C2314yes—
93C22×C4⋊C4C2314yes—
94C22×C4⋊C4C2314yes—
95C22×C22⋊C4C2311yescharacteristic
96C22×C22⋊C4C2312yes—
97C22×C22⋊C4C2314yes—
98C22×C22⋊C4C2311yescharacteristic
99C22×C42—24no—
100C22×C42—24no—
101C22×C42Q818yes—
102C22×C42C2311yescharacteristic
103C22×C42C2311yescharacteristic
104C22×C42C2311yescharacteristic
105C22×C42C2311yescharacteristic
106C22×C42C2311yescharacteristic
107C22×C42C2312yes—
108C22×C42C2311yescharacteristic
109#400—216no—
110#400—48no—
111#398—216no—
112#398—48no—
113#389—28no—
114#389—28no—
115#389—44no—
116#389—44no—
117#385—48no—
118#384—28no—
119#384—216no—
120#384—24no—
121#384D418yes—
122#382—24no—
123#382—24no—
124#380—28no—
125#380—48no—
126C4×C22⋊C4—24no—
127C4×C22⋊C4—24no—
128#376—24no—
129#376—24no—
130C2×#95C2312yes—
131C2×#95C2312yes—
132C2×#95C2314yes—
133C2×#95C2314yes—
134C2×#95C2314yes—
135C2×#95C2312yes—
136C2×#95C2312yes—
137C2×#95C2314yes—
138C2×#95C2312yes—
139C2×#95C2312yes—
140C2×#95C2314yes—
141C2×#95C2312yes—
142C2×#95C2312yes—
143C2×#95C2314yes—
144C2×#95C2312yes—
145C2×#95C2314yes—
146C2×#95C2311yescharacteristic
147C2×#95C2311yescharacteristic
148C2×#95C2314yes—
149C2×#95C2314yes—
150C2×#95C2314yes—
151C2×#95C2314yes—
152C2×#95C2314yes—
153C2×#95C2314yes—
154C2×#95C2314yes—
155C2×#95C2314yes—
156C2×#95C2314yes—
157C2×#95C2314yes—
158C25C2411yescharacteristic
159C23×C4—24no—
160C23×C4—24no—
161C23×C4—24no—
162C23×C4—22no—
163C23×C4—24no—
164C23×C4—22no—
165C23×C4—24no—
166C23×C4—22no—
167C23×C4—24no—
168C23×C4—22no—
169C23×C4—28no—
170C23×C4—28no—
171C23×C4—28no—
172C23×C4—28no—
173C23×C4—44no—
174C23×C4—44no—
175C23×C4—44no—
176C23×C4C2×Q818yes—
177C23×C4C2×Q814yes—
178C23×C4#4114yes—
179C23×C4#4114yes—
180C23×C4C2411yescharacteristic
181C23×C4C2414yes—
182C23×C4C2414yes—
183C23×C4C2412yes—
184C23×C4C2412yes—
185C23×C4C2411yescharacteristic
186C23×C4C2411yescharacteristic
187C23×C4C2411yescharacteristic
188C23×C4C2414yes—
189C23×C4C2414yes—
190C23×C4C2412yes—
191C23×C4C2411yescharacteristic
192C23×C4C2412yes—
193C23×C4C2411yescharacteristic
194C2×C4⋊C4—24no—
195C2×C4⋊C4—24no—
196C2×C4⋊C4—28no—
197C2×C4⋊C4—28no—
198C2×C4⋊C4—28no—
199C2×C4⋊C4—28no—
200C2×C4⋊C4—24no—
201C2×C4⋊C4—24no—
202C2×C4⋊C4—28no—
203C2×C4⋊C4—24no—
204C2×C4⋊C4—24no—
205C2×C4⋊C4—24no—
206C2×C4⋊C4—24no—
207C2×C4⋊C4—24no—
208C2×C4⋊C4—28no—
209C2×C4⋊C4—28no—
210C2×C4⋊C4—24no—
211C2×C4⋊C4—24no—
212C2×C4⋊C4—28no—
213C2×C4⋊C4—28no—
214C2×C4⋊C4—432no—
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—464no—
223C2×C4⋊C4—44no—
224C2×C4⋊C4—44no—
225C2×C4⋊C4—44no—
226C2×C4⋊C4—44no—
227C2×C4⋊C4—44no—
228C2×C4⋊C4—44no—
229C2×C4⋊C4—44no—
230C2×C4⋊C4—44no—
231C2×C4⋊C4—44no—
232C2×C4⋊C4—44no—
233C2×C4⋊C4C2×D418yes—
234C2×C4⋊C4C2×D4116yes—
235C2×C22⋊C4—22no—
236C2×C22⋊C4—22no—
237C2×C22⋊C4—24no—
238C2×C22⋊C4—28no—
239C2×C22⋊C4—24no—
240C2×C22⋊C4—28no—
241C2×C22⋊C4—24no—
242C2×C22⋊C4—24no—
243C2×C22⋊C4—24no—
244C2×C22⋊C4—28no—
245C2×C42—24no—
246C2×C42—22no—
247C2×C42—24no—
248C2×C42—22no—
249C2×C42—232no—
250C2×C42—22no—
251C2×C42—22no—
252C2×C42—22no—
253C2×C42—22no—
254C2×C42—22no—
255C2×C42—22no—
256C2×C42—22no—
257C2×C42—22no—
258C2×C42—22no—
259C2×C42—22no—
260C2×C42—22no—
261C2×C42—22no—
262C2×C42C2×D414yes—
263C2×C42C2×Q818yes—
264C2×C42C2×Q814yes—
265C2×C42C2×Q814yes—
266C2×C42#4114yes—
267#95—24no—
268#95—24no—
269#95—28no—
270#95—28no—
271#95—28no—
272#95—24no—
273#95—24no—
274#95—24no—
275#95—22no—
276#95—22no—
277#95—28no—
278#95—28no—
279#95—24no—
280#95—28no—
281#95—24no—
282#95—24no—
283#95—24no—
284#95—28no—
285#95—44no—
286#95—44no—
287#95—44no—
288#95—44no—
289#95—44no—
290#95—44no—
291#95—44no—
292#95—44no—
293#95—44no—
294#95—44no—
295C24—21no—
296C24—21no—
297C24—21no—
298C24—22no—
299C24—21no—
300C24—22no—
301C24C22×Q812yes—
302C24C2×#4112yes—
303C24C2×#4112yes—
304C24H(5,𝔽2)14yes—
305C242−1+414yes—
306C24C2511yescharacteristic
307C22×C4—216no—
308C22×C4—216no—
309C22×C4—216no—
310C22×C4—216no—
311C22×C4—216no—
312C22×C4—21no—
313C22×C4—22no—
314C22×C4—21no—
315C22×C4—22no—
316C22×C4—21no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—21no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—216no—
325C22×C4—28no—
326C22×C4—216no—
327C22×C4—28no—
328C22×C4—28no—
329C22×C4—28no—
330C22×C4—28no—
331C22×C4—28no—
332C22×C4—24no—
333C22×C4—24no—
334C22×C4—24no—
335C22×C4—24no—
336C22×C4—24no—
337C22×C4—24no—
338C22×C4—21no—
339C22×C4—22no—
340C22×C4—24no—
341C22×C4—21no—
342C22×C4—24no—
343C22×C4—21no—
344C22×C4—21no—
345C22×C4—22no—
346C22×C4—24no—
347C22×C4—21no—
348C22×C4—21no—
349C22×C4—21no—
350C22×C4—21no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—24no—
354C22×C4—24no—
355C22×C4—24no—
356C22×C4—22no—
357C22×C4—24no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—24no—
361C22×C4—22no—
362C22×C4—21no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—24no—
367C22×C4—22no—
368C22×C4—24no—
369C22×C4—24no—
370C22×C4—24no—
371C22×C4—24no—
372C22×C4—24no—
373C22×C4—24no—
374C22×C4—24no—
375C22×C4—21no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—24no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—24no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—21no—
385C22×C4—21no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—24no—
389C22×C4—22no—
390C22×C4—24no—
391C22×C4—22no—
392C22×C4—24no—
393C22×C4—416no—
394C22×C4—416no—
395C22×C4—416no—
396C22×C4—416no—
397C22×C4—88no—
398C22×C4—88no—
399C22×C4—88no—
400C22×C4#125116yes—
401C22×C4C22×D418yes—
402C22×C4C22×D414yes—
403C22×C4C22×D412yes—
404C22×C4C22×Q814yes—
405C22×C4C22×Q812yes—
406C22×C4C22×Q814yes—
407C22×C4C22×Q812yes—
408C22×C4C2×#4114yes—
409C22×C4C2×#4112yes—
410C22×C4C2×#4112yes—
411C22×C4C2×#4112yes—
412C22×C4C2×#4112yes—
413C22×C4C2×#4112yes—
414C22×C4H(5,𝔽2)14yes—
415C22×C4H(5,𝔽2)12yes—
416C22×C42−1+414yes—
417C22×C42−1+412yes—
418C4⋊C4—216no—
419C4⋊C4—416no—
420C4⋊C4—416no—
421C4⋊C4—416no—
422C4⋊C4—416no—
423C4⋊C4—416no—
424C4⋊C4—48no—
425C4⋊C4—48no—
426C4⋊C4—48no—
427C4⋊C4—88no—
428C4⋊C4—88no—
429C4⋊C4—88no—
430C4⋊C4—88no—
431C22⋊C4—44no—
432C22⋊C4—44no—
433C22⋊C4—48no—
434C22⋊C4—416no—
435C42—216no—
436C42—28no—
437C42—44no—
438C42—44no—
439C42—44no—
440C42—44no—
441C42C22⋊Q8116yes—
442C23—22no—
443C23—22no—
444C23—24no—
445C23—22no—
446C23—22no—
447C23—22no—
448C23—22no—
449C23—24no—
450C23—24no—
451C23—24no—
452C23—24no—
453C23—22no—
454C23—22no—
455C23—24no—
456C23—24no—
457C23—24no—
458C23—24no—
459C23—24no—
460C23C2×#12514yes—
461C23#54014yes—
462C23#54114yes—
463C23#54218yes—
464C23#54418yes—
465C23C23×D411yescharacteristic
466C23C23×Q811yescharacteristic
467C23C22×#4111yescharacteristic
468C23C22×#4111yescharacteristic
469C23C22×#4111yescharacteristic
470C23C2×H(5,𝔽2)12yes—
471C23C2×H(5,𝔽2)11yescharacteristic
472C23C2×H(5,𝔽2)12yes—
473C23C2×2−1+412yes—
474C23C2×2−1+411yescharacteristic
475C23C2×2−1+412yes—
476C2×C4—264no—
477C2×C4—22no—
478C2×C4—24no—
479C2×C4—22no—
480C2×C4—24no—
481C2×C4—24no—
482C2×C4—24no—
483C2×C4—22no—
484C2×C4—22no—
485C2×C4—22no—
486C2×C4—22no—
487C2×C4—24no—
488C2×C4—24no—
489C2×C4—24no—
490C2×C4—24no—
491C2×C4—24no—
492C2×C4—24no—
493C2×C4—24no—
494C2×C4—24no—
495C2×C4—232no—
496C2×C4—28no—
497C2×C4—28no—
498C2×C4—28no—
499C2×C4—44no—
500C2×C4—44no—
501C2×C4—44no—
502C2×C4—44no—
503C2×C4—41no—
504C2×C4—41no—
505C2×C4—44no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—42no—
509C2×C4—44no—
510C2×C4—42no—
511C2×C4—44no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—44no—
515C2×C4—42no—
516C2×C4—44no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—42no—
520C2×C4—44no—
521C2×C4—44no—
522C2×C4—44no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—44no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—44no—
529C2×C4—42no—
530C2×C4—44no—
531C2×C4—42no—
532C2×C4—44no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—44no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—42no—
541C2×C4—44no—
542C2×C4—42no—
543C2×C4—44no—
544C2×C4—42no—
545C2×C4—44no—
546C2×C4—42no—
547C2×C4—44no—
548C2×C4—44no—
549C2×C4—42no—
550C2×C4—44no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—44no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—44no—
557C2×C4—42no—
558C2×C4—832no—
559C2×C4C2×C22⋊Q818yes—
560C2×C4C2×C22⋊Q814yes—
561C2×C4C2×#12514yes—
562C2×C4C2×#12514yes—
563C2×C4#54418yes—
564C2×C4#54414yes—
565C2×C4#54614yes—
566C2×C4#54814yes—
567C2×C4#55214yes—
568C2×C4#55314yes—
569C22—28no—
570C22—24no—
571C22—24no—
572C22—28no—
573C22—24no—
574C22—28no—
575C22—24no—
576C22—28no—
577C22—24no—
578C22—24no—
579C22#264718yescomplemented
580C22C22×C22⋊Q811yescharacteristic
581C22C22×#12511yescharacteristic
582C22C2×#54011yescharacteristic
583C22C2×#54111yescharacteristic
584C22C2×#54212yes—
585C22C2×#54412yes—
586C22C2×#54411yescharacteristic
587C22C2×#54412yes—
588C22C2×#54611yescharacteristic
589C22C2×#54811yescharacteristic
590C22C2×#55211yescharacteristic
591C22C2×#55311yescharacteristic
592C22#348412yes—
593C22#348612yes—
594C22#349212yes—
595C22#349212yes—
596C22#350811yescharacteristic
597C22#350911yescharacteristic
598C22#351112yes—
599C22#354312yes—
600C22#354412yes—
601C22#355212yes—
602C22#355212yes—
603C4—24no—
604C4—28no—
605C4—28no—
606C4—48no—
607C4—82no—
608C4—84no—
609C4—81no—
610C4—84no—
611C4—81no—
612C4—82no—
613C4—84no—
614C4—81no—
615C4—81no—
616C4#244218yes—
617C2—28no—
618C2C2×#264711yescharacteristic
619C2C2×#244211yescharacteristic
620C2#3522412yes—
621C2#3528611yescharacteristic
622C2#3762312yes—
623C2#3769411yescharacteristic
624C2#3769511yescharacteristic
625C2#3770412yes—
626C2#3788211yescharacteristic
627C2#3788311yescharacteristic
628C2#3789212yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC25C24
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×#2647C2semidirect
3. #16570C2semidirect
4. #16523C2semidirect
5. #15146C2semidirect
6. #15137C2semidirect
10. #14348C2semidirect
11. #14277C2semidirect
12. #14276C2semidirect
579. C22#2647semidirect

Maximal subgroups

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

subgroupindex
C2×#26472
C22×#3842
#165702
#165232
#151462
#151372
#148222
#147282
#146822
#143482
#142772
#142762
#141302
#139072
#137532