Finite Group Databasebeta

← #644394

Subgroups — #644394

Order 512.

3654 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×#2856C211yesmaximal, characteristic
2C2×#2639C212yesmaximal, complemented
3C2×C4×C22⋊Q8C211yesmaximal, characteristic, complemented
4#17077C212yesmaximal, complemented
5#17069C212yesmaximal, complemented
6#16802C214yesmaximal, complemented
7#15157C212yesmaximal, complemented
8#15152C212yesmaximal, complemented
9#15147C212yesmaximal, complemented
10#15127C212yesmaximal, complemented
11#14747C214yesmaximal, complemented
12#14271C212yesmaximal, complemented
13#14137C214yesmaximal, complemented
14C2×C4⋊#95C211yesmaximal, characteristic, complemented
15C22×#127C2211yescharacteristic
16C22×C22⋊Q8C2211yescharacteristic
17C22×#121C2212yes—
18C22×#121C2212yes—
19C22×#121C2212yes—
20C22×C4×Q8C2211yescharacteristic
21C23×C4⋊C4C2212yes—
22C23×C42C2211yescharacteristic
23#2856—28no—
24#2639—216no—
25C2×#400C2214yes—
26C2×#397C2214yes—
27C2×#394C2214yes—
28C2×#394C2214yes—
29C2×#393C2214yes—
30C4×C22⋊Q8—28no—
31C2×#391C2211yescharacteristic
32C2×#390C2212yes—
33C2×#390C2212yes—
34C2×#390C2212yes—
35C2×#390C2214yes—
36C2×#390C2214yes—
37C2×#389C2211yescharacteristic
38C2×#387C2212yes—
39C2×#386C2218yes—
40C2×#386C2218yes—
41C2×#385C2212yes—
42C2×#384C2211yescharacteristic
43C2×#380C2211yescharacteristic
44C2×C4×C4⋊C4C2212yes—
45C2×C4×C4⋊C4C2211yescharacteristic
46C2×C4×C22⋊C4C2212yes—
47C22×#95C2212yes—
48C22×#95C2212yes—
49#1445C2214yes—
50#1442C2214yes—
51#1441C2214yes—
52#1440C2218yes—
53#1439C2214yes—
54#1439C2218yes—
55#1439C2214yes—
56#1439C2218yes—
57#1439C2218yes—
58#1438C2214yes—
59#1438C2214yes—
60C4⋊#95C2212yes—
61C4⋊#95C2214yes—
62C4⋊#95C2212yes—
63C4⋊#95C2214yes—
64C4⋊#95C2212yes—
65C4⋊#95C2212yes—
66C23×Q8C2311yescharacteristic
67C23×D4C2314yes—
68C23×D4C2312yes—
69C23×D4C2312yes—
70C23×D4C2312yes—
71C24×C4C2311yescharacteristic
72C24×C4C2311yescharacteristic
73C24×C4C2314yes—
74C24×C4C2311yescharacteristic
75C2×#127—24no—
76C2×#127—44no—
77C2×C22⋊Q8—24no—
78C2×C22⋊Q8—24no—
79C2×C22⋊Q8D418yescomplemented
80C2×#121—28no—
81C2×#121—28no—
82C2×#121—28no—
83C2×#121—28no—
84C2×#121—28no—
85C2×#121—216no—
86C2×#121—48no—
87C2×C4×Q8—24no—
88C2×C4×Q8—44no—
89C22×C4⋊C4—28no—
90C22×C4⋊C4—44no—
91C22×C4⋊C4—44no—
92C22×C4⋊C4C2312yes—
93C22×C4⋊C4C2312yes—
94C22×C4⋊C4C2314yes—
95C22×C4⋊C4C2312yes—
96C22×C4⋊C4C2312yes—
97C22×C4⋊C4C2311yescharacteristic
98C22×C4⋊C4C2314yes—
99C22×C4⋊C4C2311yescharacteristic
100C22×C4⋊C4C2312yes—
101C22×C4⋊C4C2312yes—
102C22×C4⋊C4C2311yescharacteristic
103C22×C22⋊C4C2314yes—
104C22×C22⋊C4C2314yes—
105C22×C22⋊C4C2312yes—
106C22×C22⋊C4C2312yes—
107C22×C22⋊C4C2318yes—
108C22×C22⋊C4C2318yes—
109C22×C22⋊C4C2314yes—
110C22×C22⋊C4C2318yes—
111C22×C22⋊C4C2318yes—
112C22×C42—44no—
113C22×C42D418yes—
114C22×C42C2312yes—
115C22×C42C2312yes—
116C22×C42C2311yescharacteristic
117C22×C42C2311yescharacteristic
118C22×C42C2311yescharacteristic
119C22×C42C2311yescharacteristic
120#400—216no—
121#397—216no—
122#394—216no—
123#394—48no—
124#393—216no—
125#391—24no—
126#390—28no—
127#390—28no—
128#390—28no—
129#390—48no—
130#390—48no—
131#389—24no—
132#387—28no—
133#386—232no—
134#386—416no—
135#385—28no—
136#384—24no—
137#380—24no—
138C4×C4⋊C4—28no—
139C4×C4⋊C4—24no—
140C4×C22⋊C4—28no—
141C2×#95—28no—
142C2×#95—44no—
143C2×#95C2314yes—
144C2×#95C2318yes—
145C2×#95C2314yes—
146C2×#95C2314yes—
147C2×#95C2318yes—
148C2×#95C2312yes—
149C2×#95C2312yes—
150C2×#95C2312yes—
151C2×#95C2312yes—
152C2×#95C2312yes—
153C2×#95C2314yes—
154C2×#95C2318yes—
155C2×#95C2318yes—
156C25C2414yes—
157C25C2411yescharacteristic
158C22×Q8—22no—
159C22×Q8—22no—
160C22×Q8—42no—
161C22×Q8—42no—
162C22×Q8C2×D414yes—
163C22×D4—28no—
164C22×D4—28no—
165C22×D4—28no—
166C22×D4—216no—
167C22×D4—216no—
168C22×D4—24no—
169C22×D4—24no—
170C22×D4—24no—
171C22×D4—24no—
172C22×D4—28no—
173C22×D4—24no—
174C22×D4—28no—
175C22×D4—24no—
176C22×D4—24no—
177C22×D4—24no—
178C22×D4—24no—
179C22×D4—44no—
180C22×D4—44no—
181C22×D4—44no—
182C22×D4—44no—
183C23×C4—22no—
184C23×C4—24no—
185C23×C4—22no—
186C23×C4—24no—
187C23×C4—22no—
188C23×C4—22no—
189C23×C4—28no—
190C23×C4—28no—
191C23×C4—28no—
192C23×C4—42no—
193C23×C4—42no—
194C23×C4—42no—
195C23×C4—44no—
196C23×C4—44no—
197C23×C4—44no—
198C23×C4—44no—
199C23×C4C2×D414yes—
200C23×C4C2×D414yes—
201C23×C4C2×D414yes—
202C23×C4C2×D414yes—
203C23×C4#4118yes—
204C23×C4#4114yes—
205C23×C4C2412yes—
206C23×C4C2412yes—
207C23×C4C2418yes—
208C23×C4C2414yes—
209C23×C4C2412yes—
210C23×C4C2412yes—
211C23×C4C2411yescharacteristic
212C23×C4C2411yescharacteristic
213C23×C4C2411yescharacteristic
214C23×C4C2411yescharacteristic
215C23×C4C2411yescharacteristic
216C23×C4C2411yescharacteristic
217#127—416no—
218C22⋊Q8—232no—
219#121—432no—
220#121—432no—
221#121—432no—
222C4×Q8—416no—
223C2×C4⋊C4—24no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—24no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—24no—
229C2×C4⋊C4—24no—
230C2×C4⋊C4—24no—
231C2×C4⋊C4—24no—
232C2×C4⋊C4—24no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—28no—
236C2×C4⋊C4—28no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—28no—
239C2×C4⋊C4—28no—
240C2×C4⋊C4—28no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—44no—
244C2×C4⋊C4—44no—
245C2×C4⋊C4—42no—
246C2×C4⋊C4—42no—
247C2×C4⋊C4—44no—
248C2×C4⋊C4—44no—
249C2×C4⋊C4—432no—
250C2×C4⋊C4C2×D418yes—
251C2×C4⋊C4C2×D414yes—
252C2×C22⋊C4—24no—
253C2×C22⋊C4—24no—
254C2×C22⋊C4—24no—
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—216no—
262C2×C22⋊C4—216no—
263C2×C22⋊C4—216no—
264C2×C22⋊C4—48no—
265C2×C22⋊C4—48no—
266C2×C22⋊C4—48no—
267C2×C22⋊C4—48no—
268C2×C22⋊C4—48no—
269C2×C22⋊C4—48no—
270C2×C22⋊C4—48no—
271C2×C22⋊C4—48no—
272C2×C22⋊C4—48no—
273C2×C22⋊C4—48no—
274C2×C22⋊C4—48no—
275C2×C22⋊C4C2×D418yes—
276C2×C42—24no—
277C2×C42—24no—
278C2×C42—24no—
279C2×C42—24no—
280C2×C42—28no—
281C2×C42—24no—
282C2×C42—24no—
283C2×C42—416no—
284C2×C42—42no—
285C2×C42—42no—
286C2×C42C2×D414yes—
287C2×C42C2×D414yes—
288C2×C42C2×D414yes—
289C2×C42C2×D414yes—
290#95—28no—
291#95—28no—
292#95—24no—
293#95—24no—
294#95—24no—
295#95—24no—
296#95—24no—
297#95—216no—
298#95—216no—
299#95—28no—
300#95—44no—
301#95—48no—
302#95—48no—
303C24—22no—
304C24—21no—
305C24—21no—
306C24—21no—
307C24—21no—
308C24—22no—
309C24—24no—
310C24—24no—
311C24—24no—
312C24—28no—
313C24—24no—
314C24—24no—
315C24—28no—
316C24—28no—
317C24—24no—
318C24—24no—
319C24—28no—
320C24C22×D412yes—
321C24C22×D412yes—
322C24C2×#4112yes—
323C24C2×#4114yes—
324C24H(5,𝔽2)14yes—
325C24C2511yescharacteristic
326C2×Q8—28no—
327C2×Q8—48no—
328C2×Q8—44no—
329C2×Q8—44no—
330C2×Q8—48no—
331C2×D4—48no—
332C2×D4—48no—
333C2×D4—48no—
334C2×D4—48no—
335C2×D4—416no—
336C2×D4—416no—
337C2×D4—416no—
338C2×D4—432no—
339C2×D4—432no—
340C2×D4—48no—
341C2×D4—48no—
342C2×D4—48no—
343C2×D4—48no—
344C2×D4—48no—
345C2×D4—416no—
346C2×D4—416no—
347C2×D4—88no—
348C2×D4—88no—
349C2×D4—88no—
350C2×D4—88no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—21no—
356C22×C4—22no—
357C22×C4—21no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—28no—
361C22×C4—28no—
362C22×C4—216no—
363C22×C4—216no—
364C22×C4—28no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—21no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—24no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—21no—
378C22×C4—24no—
379C22×C4—28no—
380C22×C4—22no—
381C22×C4—28no—
382C22×C4—24no—
383C22×C4—22no—
384C22×C4—24no—
385C22×C4—24no—
386C22×C4—28no—
387C22×C4—28no—
388C22×C4—28no—
389C22×C4—24no—
390C22×C4—24no—
391C22×C4—28no—
392C22×C4—28no—
393C22×C4—22no—
394C22×C4—24no—
395C22×C4—22no—
396C22×C4—28no—
397C22×C4—28no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—24no—
401C22×C4—28no—
402C22×C4—44no—
403C22×C4—44no—
404C22×C4—44no—
405C22×C4—48no—
406C22×C4—48no—
407C22×C4—48no—
408C22×C4—48no—
409C22×C4—416no—
410C22×C4—416no—
411C22×C4—416no—
412C22×C4—88no—
413C22×C4—88no—
414C22×C4—88no—
415C22×C4—88no—
416C22×C4#121116yes—
417C22×C4#124116yes—
418C22×C4C22×D414yes—
419C22×C4C22×D414yes—
420C22×C4C22×D412yes—
421C22×C4C22×D412yes—
422C22×C4C22×D412yes—
423C22×C4C22×D412yes—
424C22×C4C22×D412yes—
425C22×C4C22×D412yes—
426C22×C4C22×D412yes—
427C22×C4C22×D412yes—
428C22×C4C2×#4112yes—
429C22×C4C2×#4112yes—
430C22×C4C2×#4114yes—
431C22×C4C2×#4114yes—
432C22×C4H(5,𝔽2)12yes—
433C22×C4H(5,𝔽2)14yes—
434C22×C4H(5,𝔽2)12yes—
435C22×C42−1+412yes—
436C22×C42−1+412yes—
437C4⋊C4—28no—
438C4⋊C4—216no—
439C4⋊C4—44no—
440C4⋊C4—416no—
441C4⋊C4—44no—
442C4⋊C4—48no—
443C4⋊C4—48no—
444C4⋊C4—48no—
445C4⋊C4—48no—
446C4⋊C4—48no—
447C4⋊C4—416no—
448C22⋊C4—216no—
449C22⋊C4—416no—
450C22⋊C4—48no—
451C22⋊C4—416no—
452C22⋊C4—416no—
453C22⋊C4—816no—
454C22⋊C4—816no—
455C22⋊C4—816no—
456C22⋊C4—816no—
457C42—28no—
458C42—28no—
459C42—44no—
460C42—44no—
461C42—48no—
462C42—48no—
463C23—22no—
464C23—22no—
465C23—22no—
466C23—22no—
467C23—24no—
468C23—24no—
469C23—24no—
470C23—22no—
471C23—22no—
472C23—24no—
473C23—24no—
474C23—24no—
475C23—22no—
476C23—22no—
477C23—24no—
478C23—24no—
479C23—24no—
480C23—24no—
481C23—44no—
482C23—44no—
483C23—44no—
484C23—44no—
485C23—44no—
486C23—44no—
487C23—44no—
488C23—48no—
489C23—48no—
490C23—48no—
491C23—48no—
492C23—48no—
493C23—44no—
494C23—44no—
495C23—48no—
496C23—48no—
497C23—48no—
498C23—44no—
499C23—44no—
500C23—48no—
501C23—48no—
502C23—48no—
503C23—48no—
504C23C2×#12114yes—
505C23C2×#12414yes—
506C23#53514yes—
507C23#55018yes—
508C23#55918yes—
509C23C23×D411yescharacteristic
510C23C23×D411yescharacteristic
511C23C22×#4111yescharacteristic
512C23C22×#4112yes—
513C23C2×H(5,𝔽2)12yes—
514C23C2×H(5,𝔽2)12yes—
515C23C2×H(5,𝔽2)11yescharacteristic
516C23C2×H(5,𝔽2)11yescharacteristic
517C23C2×2−1+411yescharacteristic
518C23C2×2−1+411yescharacteristic
519C23#58512yes—
520Q8—416no—
521D4—816no—
522D4—816no—
523D4—832no—
524D4—816no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—24no—
534C2×C4—24no—
535C2×C4—24no—
536C2×C4—24no—
537C2×C4—24no—
538C2×C4—24no—
539C2×C4—24no—
540C2×C4—24no—
541C2×C4—232no—
542C2×C4—24no—
543C2×C4—24no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—28no—
548C2×C4—24no—
549C2×C4—24no—
550C2×C4—416no—
551C2×C4—416no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—44no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—44no—
562C2×C4—44no—
563C2×C4—42no—
564C2×C4—44no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—48no—
571C2×C4—48no—
572C2×C4—48no—
573C2×C4—48no—
574C2×C4—48no—
575C2×C4—48no—
576C2×C4—48no—
577C2×C4—44no—
578C2×C4—44no—
579C2×C4—48no—
580C2×C4—44no—
581C2×C4—48no—
582C2×C4—44no—
583C2×C4—832no—
584C2×C4C2×#12114yes—
585C2×C4C2×#12114yes—
586C2×C4C2×#12414yes—
587C2×C4C2×#12414yes—
588C2×C4#53414yes—
589C2×C4#53714yes—
590C2×C4#53714yes—
591C2×C4#53818yes—
592C2×C4#53918yes—
593C22—24no—
594C22—24no—
595C22—28no—
596C22—28no—
597C22—24no—
598C22—24no—
599C22—24no—
600C22—24no—
601C22—28no—
602C22—28no—
603C22—84no—
604C22—84no—
605C22—88no—
606C22—88no—
607C22—84no—
608C22—84no—
609C22—84no—
610C22—84no—
611C22—84no—
612C22—88no—
613C22—88no—
614C22#254818yes—
615C22C22×#12111yescharacteristic
616C22C22×#12411yescharacteristic
617C22C2×#53411yescharacteristic
618C22C2×#53511yescharacteristic
619C22C2×#53711yescharacteristic
620C22C2×#53711yescharacteristic
621C22#344911yescharacteristic
622C22C2×#53812yes—
623C22C2×#53912yes—
624C22C2×#55012yes—
625C22C2×#55912yes—
626C22#349612yes—
627C22#349811yescharacteristic
628C22#349911yescharacteristic
629C22#350512yes—
630C22#350612yes—
631C22#350712yes—
632C22#351111yescharacteristic
633C22#351111yescharacteristic
634C22#353612yes—
635C22#353712yes—
636C22#355812yes—
637C22#355912yes—
638C4—24no—
639C4—24no—
640C4—24no—
641C4—24no—
642C4—42no—
643C4—44no—
644C4—44no—
645C4—82no—
646C4—84no—
647C4—82no—
648C4—88no—
649C4#263918yescomplemented
650C2—28no—
651C2—164no—
652C2C2×#263911yescharacteristic
653C2C2×#254811yescharacteristic
654C2#3500011yescharacteristic
655C2#3518011yescharacteristic
656C2#3527311yescharacteristic
657C2#3743911yescharacteristic
658C2#3755411yescharacteristic
659C2#3938312yes—
660C2#3938412yes—
661C2#3976712yes—
662C2#3976812yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC2×#2856C2
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. C2×#2639C2semidirect
3. C2×C4×C22⋊Q8C2semidirect
4. #17077C2semidirect
5. #17069C2semidirect
6. #16802C2semidirect
7. #15157C2semidirect
8. #15152C2semidirect
9. #15147C2semidirect
10. #15127C2semidirect
11. #14747C2semidirect
12. #14271C2semidirect
13. #14137C2semidirect
14. C2×C4⋊#95C2semidirect
79. C2×C22⋊Q8D4semidirect
649. C4#2639semidirect

Maximal subgroups

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

subgroupindex
C2×#28562
C2×#26392
C2×C4×C22⋊Q82
#170772
#170692
#168022
#151572
#151522
#151472
#151272
#147472
#142712
#141372
C2×C4⋊#952