Finite Group Databasebeta

← #636545

Subgroups — #636545

Order 512.

3446 conjugacy classes of subgroups.

The subgroup lattice has 673 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2547C211yesmaximal, characteristic, complemented
2C2×#2542C211yesmaximal, characteristic, complemented
3C2×#2537C211yesmaximal, characteristic
4C2×#2293C211yesmaximal, characteristic, complemented
5#16468C218yesmaximal, complemented
6#15219C212yesmaximal, complemented
7#15215C214yesmaximal, complemented
8#15195C212yesmaximal, complemented
9#15179C214yesmaximal, complemented
10#14714C211yesmaximal, characteristic, complemented
11#14340C212yesmaximal, complemented
12#14257C212yesmaximal, complemented
13#13911C211yesmaximal, characteristic, complemented
14C4×#390C211yesmaximal, characteristic, complemented
15C22×C4⋊Q8C2211yescharacteristic
16C22×#124C2212yes—
17C22×C22⋊Q8C2214yes—
18C22×#121C2212yes—
19C22×#121C2212yes—
20C22×C4×D4C2211yescharacteristic
21C22×C4×D4C2211yescharacteristic
22C22×#117C2212yes—
23#2547—28no—
24#2542—28no—
25#2537—28no—
26C2×#399C2214yes—
27C2×#397C2214yes—
28C2×#394C2214yes—
29C2×#391C2212yes—
30C2×#391C2214yes—
31#2293—28no—
32C2×#390C2212yes—
33C2×#390C2212yes—
34C2×#390C2211yescharacteristic
35C2×#390C2211yescharacteristic
36C2×#389C2212yes—
37C2×#388C2214yes—
38C2×#384C2212yes—
39C2×#384C2211yescharacteristic
40C2×#382C2212yes—
41C2×#380C2212yes—
42C2×#380C2212yes—
43C2×#380C2212yes—
44C2×#380C2212yes—
45C2×C4×C4⋊C4C2211yescharacteristic
46C2×C4×C4⋊C4C2211yescharacteristic
47C2×C4×C22⋊C4C2212yes—
48C2×C4×C22⋊C4C2214yes—
49C2×C43C2211yescharacteristic
50#1442C2218yes—
51#1442C2214yes—
52#1441C2218yes—
53#1440C2218yes—
54#1439C2218yes—
55#1439C2218yes—
56#1439C2218yes—
57#1438C2218yes—
58#1438C2218yes—
59#1437C2214yes—
60C4⋊#95C2211yescharacteristic
61C4⋊#95C2212yes—
62C4⋊#95C2214yes—
63C4⋊#95C2214yes—
64C4⋊#95C2212yes—
65C4⋊#95C2211yescharacteristic
66C4⋊#95C2212yes—
67C23×Q8C2312yes—
68C23×D4C2311yescharacteristic
69C23×D4C2311yescharacteristic
70C23×D4C2314yes—
71C24×C4C2312yes—
72C24×C4C2312yes—
73C2×C4⋊Q8—24no—
74C2×C4⋊Q8—24no—
75C2×C4⋊Q8—24no—
76C2×#124—28no—
77C2×#124—48no—
78C2×C22⋊Q8—216no—
79C2×C22⋊Q8—48no—
80C2×C22⋊Q8—48no—
81C2×#121—28no—
82C2×#121—28no—
83C2×#121—28no—
84C2×#121—44no—
85C2×#121—48no—
86C2×C4×D4—24no—
87C2×C4×D4—24no—
88C2×C4×D4—24no—
89C2×C4×D4—24no—
90C2×C4×D4—24no—
91C2×C4×D4D418yes—
92C2×#117—28no—
93C2×#117—28no—
94C2×#117—44no—
95C22×C4⋊C4C2312yes—
96C22×C4⋊C4C2311yescharacteristic
97C22×C4⋊C4C2312yes—
98C22×C4⋊C4C2312yes—
99C22×C4⋊C4C2312yes—
100C22×C4⋊C4C2311yescharacteristic
101C22×C4⋊C4C2311yescharacteristic
102C22×C4⋊C4C2314yes—
103C22×C4⋊C4C2311yescharacteristic
104C22×C4⋊C4C2312yes—
105C22×C4⋊C4C2312yes—
106C22×C4⋊C4C2318yes—
107C22×C22⋊C4C2312yes—
108C22×C22⋊C4C2314yes—
109C22×C22⋊C4C2314yes—
110C22×C22⋊C4C2314yes—
111C22×C22⋊C4C2314yes—
112C22×C22⋊C4C2312yes—
113C22×C22⋊C4C2314yes—
114C22×C22⋊C4C2314yes—
115C22×C22⋊C4C2314yes—
116C22×C22⋊C4C2318yes—
117C22×C22⋊C4C2314yes—
118C22×C42C2312yes—
119C22×C42C2311yescharacteristic
120C22×C42C2312yes—
121C22×C42C2312yes—
122C22×C42C2312yes—
123C22×C42C2311yescharacteristic
124C22×C42C2311yescharacteristic
125#399—216no—
126#397—216no—
127#394—216no—
128#391—28no—
129#391—48no—
130#390—24no—
131#390—28no—
132#390—28no—
133#390—24no—
134#389—28no—
135#388—216no—
136#384—24no—
137#384—28no—
138#382—44no—
139#380—28no—
140#380—28no—
141#380—28no—
142#380—28no—
143C4×C4⋊C4—24no—
144C4×C4⋊C4D418yescomplemented
145C4×C22⋊C4—28no—
146C4×C22⋊C4—48no—
147C2×#95C2314yes—
148C2×#95C2312yes—
149C2×#95C2314yes—
150C2×#95C2314yes—
151C2×#95C2314yes—
152C2×#95C2318yes—
153C2×#95C2318yes—
154C2×#95C2318yes—
155C2×#95C2314yes—
156C2×#95C2314yes—
157C2×#95C2318yes—
158C2×#95C2312yes—
159C43—24no—
160C25C2412yes—
161C25C2412yes—
162C22×Q8—24no—
163C22×Q8—24no—
164C22×Q8—24no—
165C22×Q8—28no—
166C22×Q8—28no—
167C22×D4—28no—
168C22×D4—28no—
169C22×D4—22no—
170C22×D4—22no—
171C22×D4—22no—
172C22×D4—22no—
173C22×D4—22no—
174C22×D4—22no—
175C22×D4—24no—
176C22×D4—22no—
177C22×D4—24no—
178C22×D4—22no—
179C22×D4—44no—
180C22×D4—48no—
181C22×D4—48no—
182C22×D4C2×D414yes—
183C22×D4#4114yes—
184C23×C4—24no—
185C23×C4—24no—
186C23×C4—24no—
187C23×C4—24no—
188C23×C4—24no—
189C23×C4—24no—
190C23×C4—24no—
191C23×C4—24no—
192C23×C4—24no—
193C23×C4—28no—
194C23×C4—28no—
195C23×C4C2×D418yes—
196C23×C4C2414yes—
197C23×C4C2412yes—
198C23×C4C2412yes—
199C23×C4C2412yes—
200C23×C4C2414yes—
201C23×C4C2414yes—
202C23×C4C2412yes—
203C23×C4C2412yes—
204C23×C4C2412yes—
205C23×C4C2411yescharacteristic
206C23×C4C2411yescharacteristic
207C23×C4C2411yescharacteristic
208C4⋊Q8—416no—
209#124—432no—
210C22⋊Q8—464no—
211#121—432no—
212#121—432no—
213C4×D4—232no—
214C4×D4—416no—
215#117—432no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—24no—
219C2×C4⋊C4—24no—
220C2×C4⋊C4—28no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—24no—
224C2×C4⋊C4—24no—
225C2×C4⋊C4—216no—
226C2×C4⋊C4—24no—
227C2×C4⋊C4—24no—
228C2×C4⋊C4—24no—
229C2×C4⋊C4—24no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—28no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—24no—
236C2×C4⋊C4—24no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—28no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—44no—
245C2×C4⋊C4—48no—
246C2×C4⋊C4—48no—
247C2×C4⋊C4—44no—
248C2×C4⋊C4C2×D414yes—
249C2×C4⋊C4C2×D418yes—
250C2×C22⋊C4—24no—
251C2×C22⋊C4—24no—
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—24no—
259C2×C22⋊C4—24no—
260C2×C22⋊C4—28no—
261C2×C22⋊C4—216no—
262C2×C22⋊C4—28no—
263C2×C22⋊C4—28no—
264C2×C22⋊C4—24no—
265C2×C22⋊C4—28no—
266C2×C22⋊C4—28no—
267C2×C22⋊C4—28no—
268C2×C22⋊C4—44no—
269C2×C22⋊C4—44no—
270C2×C22⋊C4—44no—
271C2×C22⋊C4—44no—
272C2×C22⋊C4—48no—
273C2×C22⋊C4—48no—
274C2×C22⋊C4—48no—
275C2×C22⋊C4—48no—
276C2×C22⋊C4—48no—
277C2×C22⋊C4—44no—
278C2×C22⋊C4—44no—
279C2×C22⋊C4C2×D418yes—
280C2×C42—22no—
281C2×C42—24no—
282C2×C42—22no—
283C2×C42—24no—
284C2×C42—24no—
285C2×C42—22no—
286C2×C42—24no—
287C2×C42—24no—
288C2×C42—22no—
289C2×C42—28no—
290C2×C42—24no—
291C2×C42—24no—
292C2×C42—24no—
293C2×C42—22no—
294C2×C42—42no—
295C2×C42C2×D418yes—
296C2×C42C2×D414yes—
297C2×C42C2×D414yes—
298C2×C42#4114yes—
299C2×C42#4114yes—
300#95—28no—
301#95—24no—
302#95—28no—
303#95—28no—
304#95—28no—
305#95—48no—
306#95—44no—
307#95—44no—
308#95—48no—
309#95—48no—
310#95—48no—
311#95C2×D418yes—
312C24—22no—
313C24—22no—
314C24—22no—
315C24—22no—
316C24—22no—
317C24—22no—
318C24—22no—
319C24—22no—
320C24—24no—
321C24—24no—
322C24—22no—
323C24—24no—
324C24—24no—
325C24—22no—
326C24—24no—
327C24—24no—
328C24—22no—
329C24—22no—
330C24—24no—
331C24—24no—
332C24C22×D414yes—
333C24C2×#4114yes—
334C24C2511yescharacteristic
335C2×Q8—48no—
336C2×Q8—416no—
337C2×Q8—48no—
338C2×Q8—48no—
339C2×Q8—416no—
340C2×D4—28no—
341C2×D4—28no—
342C2×D4—28no—
343C2×D4—28no—
344C2×D4—28no—
345C2×D4—28no—
346C2×D4—416no—
347C2×D4—416no—
348C2×D4—44no—
349C2×D4—48no—
350C2×D4—48no—
351C2×D4—44no—
352C2×D4—44no—
353C2×D4—44no—
354C2×D4—88no—
355C2×D4—816no—
356C2×D4—816no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—24no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—24no—
372C22×C4—22no—
373C22×C4—24no—
374C22×C4—24no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—24no—
379C22×C4—22no—
380C22×C4—24no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—216no—
384C22×C4—216no—
385C22×C4—216no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—24no—
390C22×C4—24no—
391C22×C4—24no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—24no—
397C22×C4—24no—
398C22×C4—22no—
399C22×C4—28no—
400C22×C4—24no—
401C22×C4—28no—
402C22×C4—24no—
403C22×C4—24no—
404C22×C4—24no—
405C22×C4—28no—
406C22×C4—24no—
407C22×C4—24no—
408C22×C4—22no—
409C22×C4—24no—
410C22×C4—28no—
411C22×C4—48no—
412C22×C4—48no—
413C22×C4—48no—
414C22×C4—48no—
415C22×C4—48no—
416C22×C4—48no—
417C22×C4—416no—
418C22×C4—48no—
419C22×C4—416no—
420C22×C4C22×D414yes—
421C22×C4C22×D418yes—
422C22×C4C22×D412yes—
423C22×C4C22×D412yes—
424C22×C4C22×D412yes—
425C22×C4C22×D412yes—
426C22×C4C22×D412yes—
427C22×C4C22×D412yes—
428C22×C4C2×#4114yes—
429C22×C4C2×#4112yes—
430C22×C4C2×#4112yes—
431C22×C4C2×#4112yes—
432C22×C4C2×#4112yes—
433C22×C4C2×#4112yes—
434C22×C4H(5,𝔽2)14yes—
435C22×C4H(5,𝔽2)14yes—
436C22×C4H(5,𝔽2)12yes—
437C22×C42−1+414yes—
438C22×C42−1+412yes—
439C4⋊C4—28no—
440C4⋊C4—216no—
441C4⋊C4—416no—
442C4⋊C4—432no—
443C4⋊C4—44no—
444C4⋊C4—48no—
445C4⋊C4—48no—
446C4⋊C4—48no—
447C4⋊C4—44no—
448C4⋊C4—48no—
449C4⋊C4—44no—
450C4⋊C4—84no—
451C22⋊C4—216no—
452C22⋊C4—416no—
453C22⋊C4—416no—
454C22⋊C4—416no—
455C22⋊C4—416no—
456C22⋊C4—416no—
457C22⋊C4—432no—
458C22⋊C4—416no—
459C22⋊C4—416no—
460C22⋊C4—416no—
461C22⋊C4—48no—
462C42—216no—
463C42—28no—
464C42—48no—
465C42—48no—
466C42—48no—
467C42—44no—
468C42#121116yes—
469C23—24no—
470C23—24no—
471C23—24no—
472C23—24no—
473C23—24no—
474C23—24no—
475C23—24no—
476C23—24no—
477C23—24no—
478C23—24no—
479C23—28no—
480C23—28no—
481C23—42no—
482C23—42no—
483C23—44no—
484C23—42no—
485C23—42no—
486C23—44no—
487C23—44no—
488C23—44no—
489C23—42no—
490C23—42no—
491C23—44no—
492C23—44no—
493C23—42no—
494C23—42no—
495C23—42no—
496C23—42no—
497C23—44no—
498C23—44no—
499C23—42no—
500C23—42no—
501C23—44no—
502C23—44no—
503C23—44no—
504C23—44no—
505C23—44no—
506C23—44no—
507C23—44no—
508C23—44no—
509C23—44no—
510C23—44no—
511C23—44no—
512C23—44no—
513C23—44no—
514C23—44no—
515C23C23×D411yescharacteristic
516C23C23×D411yescharacteristic
517C23C22×#4111yescharacteristic
518C23C22×#4111yescharacteristic
519C23C22×#4111yescharacteristic
520C23C2×H(5,𝔽2)12yes—
521C23C2×H(5,𝔽2)12yes—
522C23C2×H(5,𝔽2)11yescharacteristic
523C23C2×2−1+411yescharacteristic
524C23C2×2−1+412yes—
525C23#58512yes—
526Q8—816no—
527D4—416no—
528D4—416no—
529D4—832no—
530C2×C4—24no—
531C2×C4—24no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—24no—
535C2×C4—28no—
536C2×C4—24no—
537C2×C4—28no—
538C2×C4—24no—
539C2×C4—24no—
540C2×C4—24no—
541C2×C4—28no—
542C2×C4—24no—
543C2×C4—24no—
544C2×C4—24no—
545C2×C4—24no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—44no—
555C2×C4—44no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—44no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—44no—
562C2×C4—432no—
563C2×C4—42no—
564C2×C4—44no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—44no—
569C2×C4—44no—
570C2×C4—48no—
571C2×C4—44no—
572C2×C4—42no—
573C2×C4—44no—
574C2×C4—44no—
575C2×C4—48no—
576C2×C4—44no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—44no—
581C2×C4—42no—
582C2×C4—44no—
583C2×C4—48no—
584C2×C4—44no—
585C2×C4—48no—
586C2×C4—44no—
587C2×C4—44no—
588C2×C4—816no—
589C2×C4C2×#12114yes—
590C2×C4C2×#12114yes—
591C2×C4C2×#12114yes—
592C2×C4#52914yes—
593C2×C4#53514yes—
594C2×C4#53518yes—
595C2×C4#53514yes—
596C2×C4#54018yes—
597C2×C4#54214yes—
598C2×C4#54614yes—
599C2×C4#54614yes—
600C2×C4#54714yes—
601C2×C4#54814yes—
602C22—44no—
603C22—44no—
604C22—44no—
605C22—44no—
606C22—44no—
607C22—44no—
608C22—44no—
609C22—48no—
610C22—44no—
611C22—44no—
612C22—48no—
613C22—44no—
614C22—82no—
615C22—84no—
616C22—84no—
617C22—82no—
618C22—82no—
619C22—84no—
620C22—84no—
621C22—82no—
622C22—84no—
623C22—84no—
624C22C22×#12111yescharacteristic
625C22C2×#52911yescharacteristic
626C22C2×#53511yescharacteristic
627C22C2×#53511yescharacteristic
628C22C2×#53512yes—
629C22C2×#54012yes—
630C22C2×#54211yescharacteristic
631C22#346112yes—
632C22C2×#54611yescharacteristic
633C22C2×#54611yescharacteristic
634C22C2×#54711yescharacteristic
635C22C2×#54811yescharacteristic
636C22#348912yes—
637C22#349911yescharacteristic
638C22#350011yescharacteristic
639C22#350612yes—
640C22#351112yes—
641C22#351412yes—
642C22#352012yes—
643C22#354612yes—
644C22#354712yes—
645C22#354812yes—
646C22#355212yes—
647C4—24no—
648C4—48no—
649C4—44no—
650C4—44no—
651C4—44no—
652C4—82no—
653C4—84no—
654C4—82no—
655C4—84no—
656C4—82no—
657C4#242718yes—
658C4#261918yes—
659C2—84no—
660C2—84no—
661C2C2×#261911yescharacteristic
662C2C2×#242711yescharacteristic
663C2#3519612yes—
664C2#3530811yescharacteristic
665C2#3762712yes—
666C2#3895111yescharacteristic
667C2#3895711yescharacteristic
668C2#3912412yes—
669C2#3947112yes—
670C2#3948811yescharacteristic
671C2#3949111yescharacteristic
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC2×#2537C2
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×#2547C2semidirect
2. C2×#2542C2semidirect
4. C2×#2293C2semidirect
5. #16468C2semidirect
6. #15219C2semidirect
7. #15215C2semidirect
8. #15195C2semidirect
9. #15179C2semidirect
10. #14714C2semidirect
11. #14340C2semidirect
12. #14257C2semidirect
13. #13911C2semidirect
14. C4×#390C2semidirect
144. C4×C4⋊C4D4semidirect

Maximal subgroups

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

subgroupindex
C2×#25472
C2×#25422
C2×#25372
C2×#22932
#164682
#152192
#152152
#151952
#151792
#147142
#143402
#142572
#139112
C4×#3902