Finite Group Databasebeta

← #876497

Subgroups — #876497

Order 512.

3254 conjugacy classes of subgroups.

The subgroup lattice has 658 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2732C212yesmaximal, complemented
2C2×#2728C212yesmaximal, complemented
3C2×#2536C212yesmaximal, complemented
4#17075C211yesmaximal, characteristic, complemented
5#17068C211yesmaximal, characteristic
6#17015C212yesmaximal, complemented
7#15513C214yesmaximal, complemented
8#15493C214yesmaximal, complemented
9#15163C212yesmaximal, complemented
10#15020C214yesmaximal, complemented
11#14700C212yesmaximal, complemented
12#14189C214yesmaximal, complemented
13C4×#391C211yesmaximal, characteristic, complemented
14C22×C22⋊Q8C2214yes—
15C22×C4×Q8C2211yescharacteristic
16C22×C4×D4C2212yes—
17#2732—216no—
18#2728—216no—
19#2536—216no—
20C2×#400C2214yes—
21C2×#399C2214yes—
22C2×#398C2212yes—
23C2×#398C2212yes—
24C2×#398C2212yes—
25C2×#398C2214yes—
26C2×#397C2214yes—
27C2×#396C2214yes—
28C2×#394C2212yes—
29C2×#394C2212yes—
30C2×#394C2212yes—
31C2×#392C2214yes—
32C2×#391C2211yescharacteristic
33C2×#391C2211yescharacteristic
34C2×#390C2212yes—
35C2×#390C2212yes—
36C2×#390C2212yes—
37C2×#390C2212yes—
38C2×#389C2212yes—
39C2×#388C2214yes—
40C2×#388C2214yes—
41C2×#388C2214yes—
42C2×#386C2214yes—
43C2×#386C2214yes—
44C2×#385C2214yes—
45C2×#385C2214yes—
46C2×#382C2212yes—
47C2×#380C2214yes—
48C2×C4×C4⋊C4C2211yescharacteristic
49C2×C43C2211yescharacteristic
50#1445C2212yes—
51#1445C2214yes—
52#1445C2214yes—
53#1444C2214yes—
54#1439C2214yes—
55#1439C2214yes—
56#1439C2214yes—
57#1439C2214yes—
58#1438C2212yes—
59#1438C2214yes—
60#1437C2214yes—
61#1437C2214yes—
62#1437C2214yes—
63#1436C2214yes—
64C4⋊#95C2212yes—
65C4⋊#95C2212yes—
66C4⋊#95C2212yes—
67C4⋊#95C2211yescharacteristic
68C4⋊#95C2211yescharacteristic
69C4×#95C2214yes—
70C23×Q8C2311yescharacteristic
71C23×D4C2312yes—
72C23×D4C2312yes—
73C23×D4C2312yes—
74C24×C4C2314yes—
75C2×C22⋊Q8—216no—
76C2×C22⋊Q8—216no—
77C2×C22⋊Q8—216no—
78C2×C4×Q8—42no—
79C2×C4×Q8—42no—
80C2×C4×Q8D418yescomplemented
81C2×C4×D4—28no—
82C2×C4×D4—44no—
83C2×C4×D4—44no—
84C22×C4⋊C4C2314yes—
85C22×C4⋊C4C2314yes—
86C22×C4⋊C4C2314yes—
87C22×C4⋊C4C2312yes—
88C22×C4⋊C4C2314yes—
89C22×C4⋊C4C2312yes—
90C22×C4⋊C4C2312yes—
91C22×C4⋊C4C2312yes—
92C22×C4⋊C4C2312yes—
93C22×C4⋊C4C2311yescharacteristic
94C22×C4⋊C4C2311yescharacteristic
95C22×C4⋊C4C2312yes—
96C22×C4⋊C4C2311yescharacteristic
97C22×C22⋊C4C2314yes—
98C22×C22⋊C4C2314yes—
99C22×C22⋊C4C2314yes—
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×C42C2312yes—
109C22×C42C2312yes—
110C22×C42C2311yescharacteristic
111C22×C42C2312yes—
112C22×C42C2312yes—
113C22×C42C2311yescharacteristic
114C22×C42C2311yescharacteristic
115#400—216no—
116#399—216no—
117#398—28no—
118#398—28no—
119#398—28no—
120#398—216no—
121#397—216no—
122#396—216no—
123#394—28no—
124#394—28no—
125#394—28no—
126#392—216no—
127#391D418yescomplemented
128#391D418yescomplemented
129#390—28no—
130#390—28no—
131#390—44no—
132#390—44no—
133#389—28no—
134#388—216no—
135#388—216no—
136#388—216no—
137#386—216no—
138#386—216no—
139#385—216no—
140#385—216no—
141#382—28no—
142#380—216no—
143C4×C4⋊C4—42no—
144C2×#95C2314yes—
145C2×#95C2312yes—
146C2×#95C2312yes—
147C2×#95C2312yes—
148C2×#95C2314yes—
149C2×#95C2312yes—
150C2×#95C2314yes—
151C2×#95C2314yes—
152C2×#95C2314yes—
153C2×#95C2314yes—
154C2×#95C2314yes—
155C2×#95C2314yes—
156C2×#95C2314yes—
157C2×#95C2314yes—
158C2×#95C2312yes—
159C2×#95C2314yes—
160C2×#95C2314yes—
161C43—42no—
162C25C2414yes—
163C22×Q8—41no—
164C22×Q8—41no—
165C22×Q8—41no—
166C22×Q8—41no—
167C22×Q8C2×D414yes—
168C22×Q8C2×D414yes—
169C22×Q8C2×D414yes—
170C22×D4—28no—
171C22×D4—28no—
172C22×D4—24no—
173C22×D4—24no—
174C22×D4—24no—
175C22×D4—24no—
176C22×D4—42no—
177C22×D4—42no—
178C22×D4—44no—
179C22×D4—42no—
180C22×D4—42no—
181C22×D4—42no—
182C22×D4—44no—
183C22×D4—42no—
184C22×D4—42no—
185C22×D4—42no—
186C22×D4#4118yes—
187C23×C4—28no—
188C23×C4—28no—
189C23×C4—28no—
190C23×C4—44no—
191C23×C4—44no—
192C23×C4—44no—
193C23×C4—44no—
194C23×C4C2414yes—
195C23×C4C2414yes—
196C23×C4C2414yes—
197C23×C4C2412yes—
198C23×C4C2411yescharacteristic
199C23×C4C2412yes—
200C23×C4C2412yes—
201C23×C4C2412yes—
202C23×C4C2411yescharacteristic
203C23×C4C2412yes—
204C23×C4C2412yes—
205C23×C4C2411yescharacteristic
206C22⋊Q8—464no—
207C4×Q8—416no—
208C4×D4—432no—
209C2×C4⋊C4—24no—
210C2×C4⋊C4—28no—
211C2×C4⋊C4—28no—
212C2×C4⋊C4—28no—
213C2×C4⋊C4—28no—
214C2×C4⋊C4—28no—
215C2×C4⋊C4—28no—
216C2×C4⋊C4—28no—
217C2×C4⋊C4—28no—
218C2×C4⋊C4—28no—
219C2×C4⋊C4—24no—
220C2×C4⋊C4—24no—
221C2×C4⋊C4—24no—
222C2×C4⋊C4—24no—
223C2×C4⋊C4—24no—
224C2×C4⋊C4—24no—
225C2×C4⋊C4—28no—
226C2×C4⋊C4—41no—
227C2×C4⋊C4—41no—
228C2×C4⋊C4—42no—
229C2×C4⋊C4—42no—
230C2×C4⋊C4—41no—
231C2×C4⋊C4—41no—
232C2×C4⋊C4—42no—
233C2×C4⋊C4—42no—
234C2×C4⋊C4—41no—
235C2×C4⋊C4—41no—
236C2×C4⋊C4—42no—
237C2×C4⋊C4—42no—
238C2×C4⋊C4—42no—
239C2×C4⋊C4—44no—
240C2×C4⋊C4—44no—
241C2×C4⋊C4—42no—
242C2×C4⋊C4—42no—
243C2×C4⋊C4—42no—
244C2×C4⋊C4C2×D418yes—
245C2×C4⋊C4C2×D414yes—
246C2×C4⋊C4C2×D414yes—
247C2×C4⋊C4C2×D414yes—
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—28no—
255C2×C22⋊C4—28no—
256C2×C22⋊C4—28no—
257C2×C22⋊C4—28no—
258C2×C22⋊C4—28no—
259C2×C22⋊C4—28no—
260C2×C22⋊C4—28no—
261C2×C22⋊C4—28no—
262C2×C22⋊C4—28no—
263C2×C22⋊C4—28no—
264C2×C22⋊C4—28no—
265C2×C22⋊C4—28no—
266C2×C22⋊C4—28no—
267C2×C22⋊C4—28no—
268C2×C22⋊C4—28no—
269C2×C22⋊C4—28no—
270C2×C22⋊C4—28no—
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×C42—24no—
282C2×C42—24no—
283C2×C42—22no—
284C2×C42—22no—
285C2×C42—24no—
286C2×C42—24no—
287C2×C42—24no—
288C2×C42—24no—
289C2×C42—41no—
290C2×C42—41no—
291C2×C42—42no—
292C2×C42—41no—
293C2×C42—41no—
294C2×C42—42no—
295C2×C42—42no—
296C2×C42—44no—
297C2×C42C2×D418yes—
298C2×C42C2×D414yes—
299C2×C42C2×D414yes—
300C2×C42C2×D414yes—
301#95—28no—
302#95—28no—
303#95—28no—
304#95—24no—
305#95—24no—
306#95—28no—
307#95—28no—
308#95—28no—
309#95—28no—
310#95—28no—
311#95—24no—
312#95—24no—
313#95—28no—
314#95—44no—
315#95—42no—
316#95C2×D4116yes—
317#95C2×D4116yes—
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—
332C24C2×#4118yes—
333C24C2511yescharacteristic
334C2×Q8—44no—
335C2×Q8—44no—
336C2×Q8—44no—
337C2×Q8—44no—
338C2×Q8—44no—
339C2×Q8—44no—
340C2×Q8#127116yes—
341C2×D4—216no—
342C2×D4—416no—
343C2×D4—48no—
344C2×D4—48no—
345C2×D4—48no—
346C2×D4—416no—
347C2×D4—48no—
348C2×D4—416no—
349C2×D4—48no—
350C2×D4—48no—
351C2×D4—84no—
352C2×D4—84no—
353C2×D4—84no—
354C2×D4—84no—
355C2×D4—84no—
356C2×D4—88no—
357C2×D4—84no—
358C22×C4—22no—
359C22×C4—21no—
360C22×C4—21no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—21no—
369C22×C4—21no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—24no—
373C22×C4—24no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—21no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—21no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—24no—
386C22×C4—24no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—24no—
391C22×C4—24no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—24no—
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—24no—
412C22×C4—24no—
413C22×C4—24no—
414C22×C4—24no—
415C22×C4—416no—
416C22×C4—416no—
417C22×C4—416no—
418C22×C4—88no—
419C22×C4—88no—
420C22×C4—88no—
421C22×C4—88no—
422C22×C4C22×D414yes—
423C22×C4C22×D414yes—
424C22×C4C22×D412yes—
425C22×C4C22×D414yes—
426C22×C4C22×D414yes—
427C22×C4C22×D414yes—
428C22×C4C22×D414yes—
429C22×C4C22×D412yes—
430C22×C4C22×D412yes—
431C22×C4C22×D412yes—
432C22×C4C22×D414yes—
433C22×C4C22×D412yes—
434C22×C4C22×D414yes—
435C22×C4C2×#4114yes—
436C22×C4H(5,𝔽2)12yes—
437C22×C4H(5,𝔽2)12yes—
438C22×C42−1+414yes—
439C4⋊C4—44no—
440C4⋊C4—44no—
441C4⋊C4—48no—
442C4⋊C4—44no—
443C4⋊C4—48no—
444C4⋊C4—416no—
445C4⋊C4—416no—
446C4⋊C4—416no—
447C4⋊C4—48no—
448C4⋊C4—84no—
449C4⋊C4—84no—
450C4⋊C4—84no—
451C4⋊C4—88no—
452C22⋊C4—416no—
453C22⋊C4—416no—
454C22⋊C4—416no—
455C22⋊C4—416no—
456C22⋊C4—416no—
457C22⋊C4—416no—
458C22⋊C4—416no—
459C22⋊C4—416no—
460C22⋊C4—88no—
461C22⋊C4—88no—
462C22⋊C4—88no—
463C42—216no—
464C42—28no—
465C42—44no—
466C42—44no—
467C42—48no—
468C42—48no—
469C42—84no—
470C23—28no—
471C23—28no—
472C23—28no—
473C23—28no—
474C23—28no—
475C23—28no—
476C23—28no—
477C23—44no—
478C23—44no—
479C23—44no—
480C23—44no—
481C23—44no—
482C23—44no—
483C23—44no—
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—44no—
501C23—44no—
502C23—44no—
503C23—44no—
504C23—44no—
505C23C23×D411yescharacteristic
506C23C23×D411yescharacteristic
507C23C23×D411yescharacteristic
508C23C22×#4112yes—
509C23C2×H(5,𝔽2)11yescharacteristic
510C23C2×H(5,𝔽2)11yescharacteristic
511C23C2×2−1+412yes—
512C23#58511yescharacteristic
513C23#58511yescharacteristic
514C23#58512yes—
515C23#58512yes—
516Q8—416no—
517D4—432no—
518D4—816no—
519D4—816no—
520C2×C4—24no—
521C2×C4—22no—
522C2×C4—24no—
523C2×C4—22no—
524C2×C4—24no—
525C2×C4—24no—
526C2×C4—24no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—24no—
532C2×C4—24no—
533C2×C4—24no—
534C2×C4—24no—
535C2×C4—24no—
536C2×C4—24no—
537C2×C4—24no—
538C2×C4—24no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—24no—
542C2×C4—24no—
543C2×C4—22no—
544C2×C4—24no—
545C2×C4—24no—
546C2×C4—28no—
547C2×C4—28no—
548C2×C4—24no—
549C2×C4—24no—
550C2×C4—24no—
551C2×C4—24no—
552C2×C4—28no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—44no—
556C2×C4—41no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—44no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—41no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—44no—
569C2×C4—44no—
570C2×C4—44no—
571C2×C4—44no—
572C2×C4—44no—
573C2×C4—44no—
574C2×C4—44no—
575C2×C4—44no—
576C2×C4—44no—
577C2×C4—44no—
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—832no—
591C2×C4C2×#12714yes—
592C2×C4C2×#12718yes—
593C2×C4#53414yes—
594C2×C4#54818yes—
595C22—48no—
596C22—48no—
597C22—48no—
598C22—48no—
599C22—48no—
600C22—48no—
601C22—48no—
602C22—84no—
603C22—84no—
604C22—84no—
605C22—84no—
606C22—84no—
607C22—84no—
608C22—84no—
609C22—84no—
610C22C22×#12711yescharacteristic
611C22C2×#53411yescharacteristic
612C22#344911yescharacteristic
613C22#345912yes—
614C22C2×#54812yes—
615C22D4×#4112yes—
616C22D4×#4112yes—
617C22#348111yescharacteristic
618C22#349112yes—
619C22#349611yescharacteristic
620C22#349611yescharacteristic
621C22#349611yescharacteristic
622C22#349611yescharacteristic
623C22#349712yes—
624C22#349712yes—
625C22#350012yes—
626C22#351512yes—
627C22#352911yescharacteristic
628C22#353512yes—
629C22#354411yescharacteristic
630C22#354511yescharacteristic
631C22#356412yes—
632C22#356712yes—
633C4—24no—
634C4—28no—
635C4—24no—
636C4—28no—
637C4—44no—
638C4—44no—
639C4—44no—
640C4—42no—
641C4—44no—
642C4—84no—
643C4—84no—
644C4—84no—
645C2—88no—
646C2#3500111yescharacteristic
647C2#3646512yes—
648C2#3719011yescharacteristic
649C2#3843412yes—
650C2#3942212yes—
651C2#3945312yes—
652C2#4623611yescharacteristic
653C2#4631211yescharacteristic
654C2#5216311yescharacteristic
655C2#5966011yescharacteristic
656C2#5983611yescharacteristic
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radical#17068C2
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×#2732C2semidirect
2. C2×#2728C2semidirect
3. C2×#2536C2semidirect
4. #17075C2semidirect
6. #17015C2semidirect
7. #15513C2semidirect
8. #15493C2semidirect
9. #15163C2semidirect
10. #15020C2semidirect
11. #14700C2semidirect
12. #14189C2semidirect
13. C4×#391C2semidirect
80. C2×C4×Q8D4semidirect
127. #391D4semidirect
128. #391D4semidirect

Maximal subgroups

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

subgroupindex
C2×#27322
C2×#27282
C2×#25362
#170752
#170682
#170152
#155132
#154932
#151632
#150202
#147002
#141892
C4×#3912