Finite Group Databasebeta

← #726006

Subgroups — #726006

Order 512.

2418 conjugacy classes of subgroups.

The subgroup lattice has 647 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2399C211yesmaximal, characteristic, complemented
2C2×#2357C211yesmaximal, characteristic
3#16299C214yesmaximal, complemented
4#15741C214yesmaximal, complemented
5#15664C214yesmaximal, complemented
6#15222C212yesmaximal
7#15157C212yesmaximal
8#14775C214yesmaximal, complemented
9#14276C211yesmaximal, characteristic, complemented
10#14042C212yesmaximal, complemented
11#13955C211yesmaximal, characteristic, complemented
12#13925C212yesmaximal, complemented
13#13911C211yesmaximal, characteristic, complemented
14#13906C211yesmaximal, characteristic
15#13785C211yesmaximal, characteristic
16C22×C4×Q8C2212yes—
17C22×C4×D4C2211yescharacteristic
18#2399—28no—
19C2×#398C2212yes—
20C2×#398C2212yes—
21C2×#395C2212yes—
22C2×#395C2212yes—
23C2×#393C2214yes—
24#2357—28no—
25C2×#390C2211yescharacteristic
26C2×#390C2212yes—
27C2×#390C2211yescharacteristic
28C2×#390C2212yes—
29C2×#389C2212yes—
30C2×#388C2214yes—
31C2×#387C2214yes—
32C2×#384C2211yescharacteristic
33C2×#382C2211yescharacteristic
34C2×#382C2212yes—
35C2×#381C2212yes—
36C2×#380C2214yes—
37C2×C4×C4⋊C4C2212yes—
38C2×C4×C4⋊C4C2211yescharacteristic
39C2×C4×C4⋊C4C2211yescharacteristic
40C2×C4×C4⋊C4C2212yes—
41C2×C4×C4⋊C4C2211yescharacteristic
42C2×C4×C22⋊C4C2212yes—
43C2×#376C2211yescharacteristic
44#1446C2214yes—
45#1445C2214yes—
46#1443C2214yes—
47#1442C2212yes—
48#1442C2212yes—
49#1442C2214yes—
50#1441C2214yes—
51#1441C2214yes—
52#1441C2214yes—
53#1439C2212yes—
54#1439C2214yes—
55#1439C2214yes—
56#1439C2214yes—
57#1439C2214yes—
58#1439C2214yes—
59#1439C2212yes—
60#1439C2214yes—
61#1439C2214yes—
62#1437C2214yes—
63#1436C2214yes—
64#1436C2214yes—
65C4⋊#95C2214yes—
66C4⋊#95C2212yes—
67C4⋊#95C2212yes—
68C4⋊#95C2212yes—
69C4⋊#95C2212yes—
70C4⋊#95C2211yescharacteristic
71C4⋊#95C2212yes—
72C4⋊#95C2211yescharacteristic
73#1434C2212yes—
74#1434C2212yes—
75#1434C2214yes—
76C22⋊#95C2214yes—
77C23×Q8C2312yes—
78C23×D4C2311yescharacteristic
79C24×C4C2312yes—
80C2×C4×Q8—28no—
81C2×C4×Q8—28no—
82C2×C4×Q8—44no—
83C2×C4×D4—24no—
84C2×C4×D4—24no—
85C2×C4×D4Q818yescomplemented
86C22×C4⋊C4C2311yescharacteristic
87C22×C4⋊C4C2311yescharacteristic
88C22×C4⋊C4C2312yes—
89C22×C4⋊C4C2312yes—
90C22×C4⋊C4C2312yes—
91C22×C4⋊C4C2312yes—
92C22×C4⋊C4C2312yes—
93C22×C4⋊C4C2312yes—
94C22×C4⋊C4C2312yes—
95C22×C4⋊C4C2311yescharacteristic
96C22×C4⋊C4C2312yes—
97C22×C4⋊C4C2312yes—
98C22×C4⋊C4C2314yes—
99C22×C4⋊C4C2312yes—
100C22×C4⋊C4C2312yes—
101C22×C22⋊C4C2312yes—
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×C42C2311yescharacteristic
112C22×C42C2312yes—
113C22×C42C2314yes—
114C22×C42C2311yescharacteristic
115#398—28no—
116#398—28no—
117#395—28no—
118#395—28no—
119#393—48no—
120#390—28no—
121#390—24no—
122#390—24no—
123#390—44no—
124#389—28no—
125#388—216no—
126#387—48no—
127#384—24no—
128#382—24no—
129#382—44no—
130#381—28no—
131#380—48no—
132C4×C4⋊C4—28no—
133C4×C4⋊C4—28no—
134C4×C4⋊C4—24no—
135C4×C4⋊C4—24no—
136C4×C4⋊C4D418yes—
137C4×C22⋊C4—28no—
138#376—24no—
139C2×#95C2314yes—
140C2×#95C2311yescharacteristic
141C2×#95C2312yes—
142C2×#95C2312yes—
143C2×#95C2312yes—
144C2×#95C2311yescharacteristic
145C2×#95C2312yes—
146C2×#95C2312yes—
147C2×#95C2312yes—
148C2×#95C2314yes—
149C2×#95C2314yes—
150C2×#95C2312yes—
151C2×#95C2314yes—
152C2×#95C2314yes—
153C2×#95C2314yes—
154C2×#95C2314yes—
155C2×#95C2314yes—
156C2×#95C2314yes—
157C2×#95C2314yes—
158C2×#95C2314yes—
159C2×#95C2314yes—
160C2×#95C2314yes—
161C2×#95C2314yes—
162C2×#95C2314yes—
163C2×#95C2314yes—
164C2×#95C2311yescharacteristic
165C2×#95C2311yescharacteristic
166C25C2412yes—
167C22×Q8—24no—
168C22×Q8—24no—
169C22×Q8—24no—
170C22×Q8—42no—
171C22×Q8—42no—
172C22×Q8—42no—
173C22×Q8—42no—
174C22×D4—22no—
175C22×D4—22no—
176C22×D4—22no—
177C22×D4—22no—
178C22×D4—22no—
179C22×D4—22no—
180C22×D4C2×Q814yes—
181C23×C4—24no—
182C23×C4—24no—
183C23×C4—44no—
184C23×C4—44no—
185C23×C4C2×Q818yes—
186C23×C4C2414yes—
187C23×C4C2412yes—
188C23×C4C2414yes—
189C23×C4C2412yes—
190C23×C4C2412yes—
191C23×C4C2412yes—
192C23×C4C2412yes—
193C23×C4C2412yes—
194C23×C4C2412yes—
195C23×C4C2412yes—
196C23×C4C2412yes—
197C23×C4C2411yescharacteristic
198C23×C4C2411yescharacteristic
199C23×C4C2411yescharacteristic
200C4×Q8—432no—
201C4×D4—232no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—24no—
205C2×C4⋊C4—24no—
206C2×C4⋊C4—24no—
207C2×C4⋊C4—24no—
208C2×C4⋊C4—24no—
209C2×C4⋊C4—24no—
210C2×C4⋊C4—24no—
211C2×C4⋊C4—24no—
212C2×C4⋊C4—24no—
213C2×C4⋊C4—24no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—24no—
217C2×C4⋊C4—28no—
218C2×C4⋊C4—24no—
219C2×C4⋊C4—24no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—24no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—24no—
225C2×C4⋊C4—24no—
226C2×C4⋊C4—24no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—24no—
229C2×C4⋊C4—24no—
230C2×C4⋊C4—42no—
231C2×C4⋊C4—42no—
232C2×C4⋊C4—42no—
233C2×C4⋊C4—42no—
234C2×C4⋊C4—42no—
235C2×C4⋊C4—42no—
236C2×C4⋊C4—44no—
237C2×C4⋊C4—42no—
238C2×C4⋊C4—42no—
239C2×C4⋊C4—42no—
240C2×C4⋊C4—44no—
241C2×C4⋊C4—44no—
242C2×C4⋊C4—44no—
243C2×C4⋊C4C2×D418yes—
244C2×C4⋊C4C2×Q814yes—
245C2×C22⋊C4—24no—
246C2×C22⋊C4—24no—
247C2×C22⋊C4—28no—
248C2×C22⋊C4—28no—
249C2×C22⋊C4—28no—
250C2×C22⋊C4—28no—
251C2×C22⋊C4—28no—
252C2×C22⋊C4—44no—
253C2×C22⋊C4—44no—
254C2×C22⋊C4—44no—
255C2×C22⋊C4—44no—
256C2×C22⋊C4—44no—
257C2×C22⋊C4—44no—
258C2×C22⋊C4—44no—
259C2×C22⋊C4—44no—
260C2×C22⋊C4—44no—
261C2×C22⋊C4—44no—
262C2×C22⋊C4—44no—
263C2×C22⋊C4—44no—
264C2×C22⋊C4—44no—
265C2×C22⋊C4C2×Q818yes—
266C2×C42—22no—
267C2×C42—22no—
268C2×C42—24no—
269C2×C42—22no—
270C2×C42—24no—
271C2×C42—22no—
272C2×C42—24no—
273C2×C42—24no—
274C2×C42—24no—
275C2×C42—28no—
276C2×C42—24no—
277C2×C42—44no—
278C2×C42—44no—
279C2×C42—44no—
280C2×C42C2×D418yes—
281C2×C42C2×D414yes—
282C2×C42C2×Q814yes—
283C2×C42#4114yes—
284C2×C42#4114yes—
285C2×C42#4114yes—
286#95—24no—
287#95—24no—
288#95—24no—
289#95—28no—
290#95—24no—
291#95—22no—
292#95—24no—
293#95—22no—
294#95—24no—
295#95—24no—
296#95—28no—
297#95—28no—
298#95—28no—
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—44no—
308#95—44no—
309#95—44no—
310#95—44no—
311#95C2×D414yes—
312#95C2×D414yes—
313C24—22no—
314C24—22no—
315C24—22no—
316C24—22no—
317C24—24no—
318C24—22no—
319C24—22no—
320C24—24no—
321C24—24no—
322C24—24no—
323C24C22×Q814yes—
324C24C2511yescharacteristic
325C2×Q8—48no—
326C2×Q8—48no—
327C2×Q8—48no—
328C2×Q8—48no—
329C2×Q8—48no—
330C2×Q8—48no—
331C2×Q8—48no—
332C2×D4—28no—
333C2×D4—28no—
334C2×D4—28no—
335C2×D4—44no—
336C2×D4—44no—
337C2×D4—44no—
338C2×D4—44no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—24no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—24no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—216no—
378C22×C4—24no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—24no—
383C22×C4—24no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—24no—
388C22×C4—24no—
389C22×C4—24no—
390C22×C4—24no—
391C22×C4—22no—
392C22×C4—24no—
393C22×C4—24no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—24no—
397C22×C4—22no—
398C22×C4—24no—
399C22×C4—24no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—24no—
403C22×C4—24no—
404C22×C4—24no—
405C22×C4—24no—
406C22×C4—24no—
407C22×C4—22no—
408C22×C4—24no—
409C22×C4—416no—
410C22×C4—88no—
411C22×C4—84no—
412C22×C4—84no—
413C22×C4C22×D414yes—
414C22×C4C22×D414yes—
415C22×C4C22×D412yes—
416C22×C4C22×D412yes—
417C22×C4C22×D412yes—
418C22×C4C22×Q814yes—
419C22×C4C22×Q812yes—
420C22×C4C22×Q812yes—
421C22×C4C22×Q812yes—
422C22×C4C2×#4114yes—
423C22×C4C2×#4114yes—
424C22×C4C2×#4112yes—
425C22×C4C2×#4112yes—
426C22×C4C2×#4112yes—
427C22×C4C2×#4112yes—
428C22×C4C2×#4112yes—
429C22×C4H(5,𝔽2)14yes—
430C22×C4H(5,𝔽2)12yes—
431C22×C42−1+412yes—
432C4⋊C4—28no—
433C4⋊C4—48no—
434C4⋊C4—48no—
435C4⋊C4—48no—
436C4⋊C4—48no—
437C4⋊C4—48no—
438C4⋊C4—48no—
439C4⋊C4—48no—
440C4⋊C4—416no—
441C4⋊C4—48no—
442C4⋊C4—44no—
443C4⋊C4—48no—
444C4⋊C4—44no—
445C4⋊C4—84no—
446C4⋊C4—84no—
447C22⋊C4—216no—
448C22⋊C4—416no—
449C22⋊C4—88no—
450C22⋊C4—88no—
451C22⋊C4—88no—
452C22⋊C4—88no—
453C22⋊C4—88no—
454C42—216no—
455C42—28no—
456C42—28no—
457C42—48no—
458C42—84no—
459C42—88no—
460C42C22⋊Q8116yes—
461C23—24no—
462C23—24no—
463C23—24no—
464C23—28no—
465C23—28no—
466C23—44no—
467C23—42no—
468C23—42no—
469C23—42no—
470C23—42no—
471C23—44no—
472C23—44no—
473C23—44no—
474C23—44no—
475C23—42no—
476C23—44no—
477C23—42no—
478C23—42no—
479C23—42no—
480C23—44no—
481C23—44no—
482C23—44no—
483C23—44no—
484C23C23×D411yescharacteristic
485C23C23×Q811yescharacteristic
486C23C22×#4111yescharacteristic
487C23C22×#4111yescharacteristic
488C23C22×#4111yescharacteristic
489C23C2×H(5,𝔽2)11yescharacteristic
490C23C2×H(5,𝔽2)12yes—
491C23C2×2−1+411yescharacteristic
492C23#58512yes—
493C23#58512yes—
494C23#58512yes—
495Q8—432no—
496D4—416no—
497C2×C4—24no—
498C2×C4—24no—
499C2×C4—24no—
500C2×C4—24no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—24no—
506C2×C4—24no—
507C2×C4—24no—
508C2×C4—24no—
509C2×C4—24no—
510C2×C4—24no—
511C2×C4—24no—
512C2×C4—24no—
513C2×C4—24no—
514C2×C4—24no—
515C2×C4—24no—
516C2×C4—24no—
517C2×C4—24no—
518C2×C4—28no—
519C2×C4—24no—
520C2×C4—24no—
521C2×C4—24no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—44no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—44no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—44no—
549C2×C4—42no—
550C2×C4—44no—
551C2×C4—42no—
552C2×C4—44no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—44no—
556C2×C4—42no—
557C2×C4—44no—
558C2×C4—44no—
559C2×C4—44no—
560C2×C4—44no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—44no—
565C2×C4—44no—
566C2×C4—44no—
567C2×C4—42no—
568C2×C4—44no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—44no—
572C2×C4—44no—
573C2×C4—44no—
574C2×C4—44no—
575C2×C4—44no—
576C2×C4—816no—
577C2×C4C2×C22⋊Q814yes—
578C2×C4C2×C22⋊Q814yes—
579C2×C4C2×C22⋊Q814yes—
580C2×C4#53314yes—
581C2×C4#54414yes—
582C2×C4#54614yes—
583C2×C4#54814yes—
584C2×C4#55514yes—
585C2×C4#55614yes—
586C22—44no—
587C22—48no—
588C22—44no—
589C22—44no—
590C22—48no—
591C22—84no—
592C22—82no—
593C22—82no—
594C22—82no—
595C22—82no—
596C22—84no—
597C22C22×C22⋊Q811yescharacteristic
598C22C2×#53311yescharacteristic
599C22#344912yes—
600C22C2×#54411yescharacteristic
601C22C2×#54611yescharacteristic
602C22C2×#54811yescharacteristic
603C22C2×#55511yescharacteristic
604C22C2×#55611yescharacteristic
605C22#348812yes—
606C22#349312yes—
607C22#349412yes—
608C22#350112yes—
609C22#350412yes—
610C22#350612yes—
611C22#350811yescharacteristic
612C22#351011yescharacteristic
613C22#351011yescharacteristic
614C22#351012yes—
615C22#351011yescharacteristic
616C22#353912yes—
617C22#354212yes—
618C22#355412yes—
619C22#355712yes—
620C4—24no—
621C4—24no—
622C4—44no—
623C4—44no—
624C4—44no—
625C4—44no—
626C4—44no—
627C4—44no—
628C4—84no—
629C4—82no—
630C4—82no—
631C4—84no—
632C4—82no—
633C4#243918yes—
634C2—84no—
635C2C2×#243911yescharacteristic
636C2#3533411yescharacteristic
637C2#3551511yescharacteristic
638C2#3595212yes—
639C2#3784612yes—
640C2#3971812yes—
641C2#4249512yes—
642C2#4496311yescharacteristic
643C2#4516311yescharacteristic
644C2#4542811yescharacteristic
645C2#4543011yescharacteristic
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC23×D4C23
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×#2399C2semidirect
3. #16299C2semidirect
4. #15741C2semidirect
5. #15664C2semidirect
8. #14775C2semidirect
9. #14276C2semidirect
10. #14042C2semidirect
11. #13955C2semidirect
12. #13925C2semidirect
13. #13911C2semidirect
85. C2×C4×D4Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#23992
C2×#23572
#162992
#157412
#156642
#152222
#151572
#147752
#142762
#140422
#139552
#139252
#139112
#139062
#137852