Finite Group Databasebeta

← #726480

Subgroups — #726480

Order 512.

4858 conjugacy classes of subgroups.

The subgroup lattice has 636 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#547C211yesmaximal, characteristic
2C2×#2562C211yesmaximal, characteristic, complemented
3C2×#2566C211yesmaximal, characteristic, complemented
4C2×#2660C211yesmaximal, characteristic, complemented
5#16601C214yesmaximal, complemented
6#15612C218yesmaximal, complemented
7#15219C212yesmaximal, complemented
8#15130C212yesmaximal, complemented
9#14889C214yesmaximal, complemented
10#14161C211yesmaximal, characteristic, complemented
11#13881C212yesmaximal, complemented
12#13880C212yesmaximal, complemented
13#13847C211yesmaximal, characteristic, complemented
14#13785C211yesmaximal, characteristic, complemented
15C23×#41C2212yes—
16C2×#547—28no—
17C2×#547—28no—
18C2×#547—28no—
19C22×C4⋊Q8C2211yescharacteristic
20C22×#125C2211yescharacteristic
21C22×#123C2214yes—
22C22×C22⋊Q8C2212yes—
23C22×#121C2212yes—
24C22×C4×D4C2211yescharacteristic
25C22×C4×D4C2211yescharacteristic
26C23×C4⋊C4C2212yes—
27#2660—28no—
28#2566—28no—
29#2562—28no—
30C2×#400C2212yes—
31C2×#398C2214yes—
32C2×#396C2214yes—
33C2×#395C2214yes—
34C2×#394C2212yes—
35C2×#394C2218yes—
36C2×#390C2211yescharacteristic
37C2×#390C2212yes—
38C2×#390C2211yescharacteristic
39C2×#388C2214yes—
40C2×#385C2214yes—
41C2×#385C2218yes—
42C2×#384C2212yes—
43C2×#384C2211yescharacteristic
44C2×#382C2211yescharacteristic
45C2×#382C2212yes—
46C2×#380C2212yes—
47C2×#380C2212yes—
48C2×C4×C4⋊C4C2211yescharacteristic
49C2×C4×C4⋊C4C2211yescharacteristic
50C2×C4×C22⋊C4C2212yes—
51C2×C4×C22⋊C4C2212yes—
52C2×#376C2211yescharacteristic
53#1446C2214yes—
54#1445C2212yes—
55#1444C2218yes—
56#1442C2218yes—
57#1440C2218yes—
58#1439C2214yes—
59#1439C2218yes—
60#1438C2212yes—
61#1438C2218yes—
62#1437C2214yes—
63C4⋊#95C2212yes—
64C4⋊#95C2212yes—
65C4⋊#95C2212yes—
66C4⋊#95C2214yes—
67C22⋊#95C2218yes—
68C22⋊#95C2214yes—
69C22×#41—28no—
70C22×#41—28no—
71C22×#41—28no—
72C22×#41—416no—
73C23×Q8C2312yes—
74C23×D4C2311yescharacteristic
75C23×D4C2311yescharacteristic
76C23×D4C2314yes—
77C24×C4C2312yes—
78C24×C4C2314yes—
79C24×C4C2312yes—
80#547—2128no—
81C2×C4⋊Q8—24no—
82C2×C4⋊Q8—24no—
83C2×C4⋊Q8D418yes—
84C2×#125—24no—
85C2×#125—24no—
86C2×#125—24no—
87C2×#123—216no—
88C2×#123—216no—
89C2×#123—216no—
90C2×C22⋊Q8—28no—
91C2×C22⋊Q8—28no—
92C2×C22⋊Q8—28no—
93C2×#121—28no—
94C2×#121—28no—
95C2×#121—28no—
96C2×C4×D4—24no—
97C2×C4×D4—24no—
98C2×C4×D4—24no—
99C2×C4×D4—24no—
100C2×C4×D4D418yes—
101C2×C4×D4D418yes—
102C22×C4⋊C4—28no—
103C22×C4⋊C4—28no—
104C22×C4⋊C4—28no—
105C22×C4⋊C4C2312yes—
106C22×C4⋊C4C2312yes—
107C22×C4⋊C4C2314yes—
108C22×C4⋊C4C2311yescharacteristic
109C22×C4⋊C4C2311yescharacteristic
110C22×C4⋊C4C2312yes—
111C22×C4⋊C4C2312yes—
112C22×C4⋊C4C2312yes—
113C22×C4⋊C4C2311yescharacteristic
114C22×C4⋊C4C2311yescharacteristic
115C22×C4⋊C4C2312yes—
116C22×C4⋊C4C2318yes—
117C22×C4⋊C4C2314yes—
118C22×C22⋊C4C2314yes—
119C22×C22⋊C4C2314yes—
120C22×C22⋊C4C2314yes—
121C22×C22⋊C4C2314yes—
122C22×C22⋊C4C2314yes—
123C22×C22⋊C4C2312yes—
124C22×C22⋊C4C2312yes—
125C22×C22⋊C4C2314yes—
126C22×C22⋊C4C2314yes—
127C22×C22⋊C4C2318yes—
128C22×C42C2311yescharacteristic
129C22×C42C2311yescharacteristic
130C22×C42C2312yes—
131C22×C42C2312yes—
132C22×C42C2311yescharacteristic
133#400—28no—
134#398—216no—
135#396—216no—
136#395—216no—
137#394—28no—
138#394—416no—
139#390—24no—
140#390—24no—
141#390—28no—
142#388—216no—
143#385—216no—
144#385—416no—
145#384—24no—
146#384—28no—
147#382—24no—
148#382—28no—
149#380—28no—
150#380—28no—
151C4×C4⋊C4—24no—
152C4×C4⋊C4—24no—
153C4×C22⋊C4—28no—
154C4×C22⋊C4—28no—
155#376—24no—
156C2×#95C2312yes—
157C2×#95C2312yes—
158C2×#95C2318yes—
159C2×#95C2314yes—
160C2×#95C2312yes—
161C2×#95C2314yes—
162C2×#95C2314yes—
163C2×#95C2312yes—
164C2×#95C2314yes—
165C2×#95C2314yes—
166C2×#95C2318yes—
167C2×#95C2318yes—
168C2×#95C2318yes—
169C25C2412yes—
170C25C2412yes—
171C2×#41—264no—
172C2×#41—464no—
173C2×#41—464no—
174C2×#41—464no—
175C22×Q8—24no—
176C22×Q8—24no—
177C22×Q8—48no—
178C22×Q8C2×D418yes—
179C22×D4—28no—
180C22×D4—28no—
181C22×D4—28no—
182C22×D4—216no—
183C22×D4—22no—
184C22×D4—22no—
185C22×D4—22no—
186C22×D4—22no—
187C22×D4—28no—
188C22×D4—24no—
189C22×D4—24no—
190C22×D4—48no—
191C22×D4C2×D414yes—
192C22×D4C2×D414yes—
193C23×C4—28no—
194C23×C4—24no—
195C23×C4—24no—
196C23×C4—28no—
197C23×C4—28no—
198C23×C4—24no—
199C23×C4—24no—
200C23×C4—24no—
201C23×C4—24no—
202C23×C4—24no—
203C23×C4—24no—
204C23×C4—216no—
205C23×C4—48no—
206C23×C4—48no—
207C23×C4C2×D418yes—
208C23×C4C2×D418yes—
209C23×C4C2412yes—
210C23×C4C2412yes—
211C23×C4C2412yes—
212C23×C4C2414yes—
213C23×C4C2412yes—
214C23×C4C2412yes—
215C23×C4C2414yes—
216C23×C4C2414yes—
217C23×C4C2412yes—
218C23×C4C2411yescharacteristic
219C23×C4C2411yescharacteristic
220C23×C4C2411yescharacteristic
221C4⋊Q8—232no—
222#125—416no—
223#123—2128no—
224C22⋊Q8—264no—
225#121—264no—
226C4×D4—232no—
227C4×D4—232no—
228C2×C4⋊C4—264no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—24no—
232C2×C4⋊C4—24no—
233C2×C4⋊C4—28no—
234C2×C4⋊C4—28no—
235C2×C4⋊C4—28no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—24no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—24no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—24no—
256C2×C4⋊C4—24no—
257C2×C4⋊C4—28no—
258C2×C4⋊C4—216no—
259C2×C4⋊C4—42no—
260C2×C4⋊C4—48no—
261C2×C4⋊C4—48no—
262C2×C4⋊C4—48no—
263C2×C4⋊C4C2×D414yes—
264C2×C4⋊C4C2×D418yes—
265C2×C4⋊C4C2×D414yes—
266C2×C4⋊C4C2×D418yes—
267C2×C22⋊C4—28no—
268C2×C22⋊C4—28no—
269C2×C22⋊C4—24no—
270C2×C22⋊C4—24no—
271C2×C22⋊C4—24no—
272C2×C22⋊C4—24no—
273C2×C22⋊C4—28no—
274C2×C22⋊C4—28no—
275C2×C22⋊C4—28no—
276C2×C22⋊C4—28no—
277C2×C22⋊C4—28no—
278C2×C22⋊C4—28no—
279C2×C22⋊C4—216no—
280C2×C22⋊C4—28no—
281C2×C22⋊C4—28no—
282C2×C22⋊C4—28no—
283C2×C22⋊C4—28no—
284C2×C22⋊C4—28no—
285C2×C22⋊C4—216no—
286C2×C22⋊C4—44no—
287C2×C22⋊C4—44no—
288C2×C22⋊C4—48no—
289C2×C22⋊C4—48no—
290C2×C22⋊C4—48no—
291C2×C22⋊C4—48no—
292C2×C22⋊C4C2×D418yes—
293C2×C22⋊C4C2×D418yes—
294C2×C42—24no—
295C2×C42—24no—
296C2×C42—22no—
297C2×C42—22no—
298C2×C42—24no—
299C2×C42—24no—
300C2×C42—24no—
301C2×C42—44no—
302C2×C42C2×D414yes—
303C2×C42C2×D414yes—
304C2×C42C2×D414yes—
305C2×C42#4118yescomplemented
306#95—24no—
307#95—24no—
308#95—24no—
309#95—24no—
310#95—28no—
311#95—28no—
312#95—28no—
313#95—216no—
314#95—28no—
315#95—28no—
316#95—48no—
317#95—48no—
318#95—48no—
319C24—22no—
320C24—22no—
321C24—22no—
322C24—22no—
323C24—24no—
324C24—24no—
325C24—22no—
326C24—22no—
327C24—24no—
328C24—22no—
329C24—22no—
330C24—24no—
331C24—28no—
332C24—24no—
333C24—24no—
334C24—24no—
335C24C22×D414yes—
336C24C22×D414yes—
337C24H(5,𝔽2)18yes—
338C24C2511yescharacteristic
339#41—4256no—
340C2×Q8—216no—
341C2×Q8—416no—
342C2×Q8—416no—
343C2×Q8—416no—
344C2×D4—216no—
345C2×D4—216no—
346C2×D4—216no—
347C2×D4—232no—
348C2×D4—28no—
349C2×D4—28no—
350C2×D4—28no—
351C2×D4—28no—
352C2×D4—432no—
353C2×D4—432no—
354C2×D4—416no—
355C2×D4—416no—
356C2×D4—48no—
357C2×D4—48no—
358C22×C4—24no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—232no—
362C22×C4—24no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—24no—
369C22×C4—24no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—24no—
373C22×C4—24no—
374C22×C4—24no—
375C22×C4—28no—
376C22×C4—28no—
377C22×C4—28no—
378C22×C4—24no—
379C22×C4—24no—
380C22×C4—24no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—24no—
384C22×C4—22no—
385C22×C4—216no—
386C22×C4—216no—
387C22×C4—216no—
388C22×C4—216no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—24no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—24no—
398C22×C4—24no—
399C22×C4—28no—
400C22×C4—24no—
401C22×C4—28no—
402C22×C4—24no—
403C22×C4—24no—
404C22×C4—22no—
405C22×C4—28no—
406C22×C4—24no—
407C22×C4—22no—
408C22×C4—416no—
409C22×C4—432no—
410C22×C4—432no—
411C22×C4—416no—
412C22×C4—416no—
413C22×C4—416no—
414C22×C4—416no—
415C22×C4—48no—
416C22×C4—48no—
417C22×C4—48no—
418C22×C4—48no—
419C22×C4C22×D414yes—
420C22×C4C22×D418yes—
421C22×C4C22×D414yes—
422C22×C4C22×D412yes—
423C22×C4C22×D412yes—
424C22×C4C22×D412yes—
425C22×C4C22×D412yes—
426C22×C4C22×D412yes—
427C22×C4C22×D412yes—
428C22×C4C22×D412yes—
429C22×C4C22×D412yes—
430C22×C4C22×D412yes—
431C22×C4C2×#4118yes—
432C22×C4C2×#4114yes—
433C22×C42−1+414yes—
434C22×C42−1+414yes—
435C4⋊C4—232no—
436C4⋊C4—28no—
437C4⋊C4—216no—
438C4⋊C4—28no—
439C4⋊C4—216no—
440C4⋊C4—48no—
441C4⋊C4—48no—
442C4⋊C4—48no—
443C4⋊C4—44no—
444C4⋊C4—44no—
445C4⋊C4—48no—
446C4⋊C4—88no—
447C4⋊C4—816no—
448C22⋊C4—232no—
449C22⋊C4—232no—
450C22⋊C4—216no—
451C22⋊C4—216no—
452C22⋊C4—416no—
453C22⋊C4—416no—
454C22⋊C4—416no—
455C22⋊C4—816no—
456C22⋊C4—88no—
457C22⋊C4—88no—
458C42—216no—
459C42—44no—
460C42—44no—
461C42—84no—
462C42#120116yes—
463C23—24no—
464C23—24no—
465C23—24no—
466C23—24no—
467C23—28no—
468C23—28no—
469C23—28no—
470C23—28no—
471C23—24no—
472C23—24no—
473C23—28no—
474C23—28no—
475C23—24no—
476C23—24no—
477C23—24no—
478C23—28no—
479C23—28no—
480C23—42no—
481C23—42no—
482C23—42no—
483C23—42no—
484C23—44no—
485C23—44no—
486C23—44no—
487C23—44no—
488C23—44no—
489C23—44no—
490C23—44no—
491C23—44no—
492C23—44no—
493C23—44no—
494C23—48no—
495C23—48no—
496C23—44no—
497C23—42no—
498C23—42no—
499C23—48no—
500C23—48no—
501C23#53518yes—
502C23C23×D411yescharacteristic
503C23C23×D411yescharacteristic
504C23C23×D411yescharacteristic
505C23C22×#4112yes—
506C23C2×H(5,𝔽2)14yes—
507C23C2×2−1+412yes—
508C23C2×2−1+412yes—
509C23#58512yes—
510Q8—432no—
511D4—232no—
512D4—464no—
513D4—416no—
514C2×C4—24no—
515C2×C4—24no—
516C2×C4—24no—
517C2×C4—24no—
518C2×C4—24no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—24no—
522C2×C4—24no—
523C2×C4—28no—
524C2×C4—24no—
525C2×C4—24no—
526C2×C4—28no—
527C2×C4—28no—
528C2×C4—216no—
529C2×C4—24no—
530C2×C4—28no—
531C2×C4—28no—
532C2×C4—28no—
533C2×C4—432no—
534C2×C4—464no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—44no—
538C2×C4—44no—
539C2×C4—44no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—44no—
544C2×C4—42no—
545C2×C4—44no—
546C2×C4—48no—
547C2×C4—44no—
548C2×C4—44no—
549C2×C4—44no—
550C2×C4—44no—
551C2×C4—432no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—44no—
556C2×C4—44no—
557C2×C4—44no—
558C2×C4—42no—
559C2×C4—44no—
560C2×C4—44no—
561C2×C4—44no—
562C2×C4—42no—
563C2×C4—44no—
564C2×C4—44no—
565C2×C4—48no—
566C2×C4—44no—
567C2×C4—42no—
568C2×C4—44no—
569C2×C4—42no—
570C2×C4—48no—
571C2×C4C2×#12014yes—
572C2×C4C2×#12014yes—
573C2×C4C2×#12014yes—
574C2×C4#53214yes—
575C2×C4#53614yes—
576C2×C4#54718yes—
577C2×C4#54818yes—
578C22—28no—
579C22—28no—
580C22—28no—
581C22—44no—
582C22—48no—
583C22—48no—
584C22—48no—
585C22—44no—
586C22—44no—
587C22—44no—
588C22—48no—
589C22—44no—
590C22—48no—
591C22—44no—
592C22—44no—
593C22—48no—
594C22—88no—
595C22—84no—
596C22—84no—
597C22C22×#12011yescharacteristic
598C22C2×#53211yescharacteristic
599C22C2×#53512yes—
600C22C2×#53611yescharacteristic
601C22C2×#54712yes—
602C22C2×#54812yes—
603C22#348412yes—
604C22#348412yes—
605C22#348912yes—
606C22#349214yes—
607C22#349414yes—
608C22#349712yes—
609C22#349911yescharacteristic
610C22#350011yescharacteristic
611C22#354414yes—
612C22#355914yes—
613C4—24no—
614C4—24no—
615C4—48no—
616C4—44no—
617C4—44no—
618C4—48no—
619C4—44no—
620C4—82no—
621C4—82no—
622C4—82no—
623C4—84no—
624C4#243218yes—
625C2—48no—
626C2—84no—
627C2C2×#243211yescharacteristic
628C2#3522211yescharacteristic
629C2#3566011yescharacteristic
630C2#3592312yes—
631C2#3786312yes—
632C2#3786412yes—
633C2#4248212yes—
634C2#4487814yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC22×#547C2
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×#2562C2semidirect
3. C2×#2566C2semidirect
4. C2×#2660C2semidirect
5. #16601C2semidirect
6. #15612C2semidirect
7. #15219C2semidirect
8. #15130C2semidirect
9. #14889C2semidirect
10. #14161C2semidirect
11. #13881C2semidirect
12. #13880C2semidirect
13. #13847C2semidirect
14. #13785C2semidirect
305. C2×C42#41semidirect

Maximal subgroups

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

subgroupindex
C22×#5472
C2×#25622
C2×#25662
C2×#26602
#166012
#156122
#152192
#151302
#148892
#141612
#138812
#138802
#138472
#137852