Finite Group Databasebeta

← #636501

Subgroups — #636501

Order 512.

2886 conjugacy classes of subgroups.

The subgroup lattice has 644 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2857C211yesmaximal, characteristic
2C2×#2567C211yesmaximal, characteristic, complemented
3C2×C4×C22⋊Q8C211yesmaximal, characteristic
4C2×#2285C211yesmaximal, characteristic
5#16565C214yesmaximal, complemented
6#16264C214yesmaximal
7#15237C212yesmaximal, complemented
8#15183C214yesmaximal
9#15151C212yesmaximal, complemented
10#15133C214yesmaximal, complemented
11#14340C212yesmaximal
12#14258C212yesmaximal
13#14161C211yesmaximal, characteristic, complemented
14#13926C211yesmaximal, characteristic, complemented
15#13911C211yesmaximal, characteristic, complemented
16C22×C4⋊Q8C2211yescharacteristic
17C22×C22⋊Q8C2211yescharacteristic
18C22×C22⋊Q8C2212yes—
19C22×C4×Q8C2211yescharacteristic
20C22×#117C2212yes—
21C23×C42C2211yescharacteristic
22#2857—28no—
23#2567—28no—
24C2×#398C2218yes—
25C2×#395C2214yes—
26C4×C22⋊Q8—28no—
27C2×#391C2212yes—
28C2×#391C2212yes—
29C2×#391C2211yescharacteristic
30C2×#391C2212yes—
31C2×#389C2212yes—
32C2×#389C2212yes—
33C2×#389C2214yes—
34C2×#389C2211yescharacteristic
35C2×#387C2212yes—
36C2×#385C2212yes—
37#2285—28no—
38C2×#384C2212yes—
39C2×#384C2211yescharacteristic
40C2×#384C2212yes—
41C2×#382C2212yes—
42C2×#382C2212yes—
43C2×#380C2211yescharacteristic
44C2×#380C2212yes—
45C2×#380C2212yes—
46C2×#380C2212yes—
47C2×#380C2212yes—
48C2×C4×C4⋊C4C2212yes—
49C2×C4×C4⋊C4C2212yes—
50C2×C4×C4⋊C4C2211yescharacteristic
51C2×C4×C4⋊C4C2211yescharacteristic
52C2×C4×C4⋊C4C2211yescharacteristic
53C2×C4×C22⋊C4C2212yes—
54#1446C2214yes—
55#1445C2218yes—
56#1443C2218yes—
57#1442C2214yes—
58#1441C2218yes—
59#1441C2218yes—
60#1441C2218yes—
61#1441C2214yes—
62#1440C2218yes—
63#1439C2218yes—
64C4⋊#95C2212yes—
65C4⋊#95C2212yes—
66C4⋊#95C2214yes—
67C4⋊#95C2214yes—
68C4⋊#95C2214yes—
69C4⋊#95C2212yes—
70C4⋊#95C2212yes—
71C23×Q8C2311yescharacteristic
72C23×Q8C2312yes—
73C24×C4C2311yescharacteristic
74C24×C4C2311yescharacteristic
75C24×C4C2311yescharacteristic
76C2×C4⋊Q8—24no—
77C2×C4⋊Q8—24no—
78C2×C4⋊Q8—24no—
79C2×C22⋊Q8—24no—
80C2×C22⋊Q8—24no—
81C2×C22⋊Q8—28no—
82C2×C22⋊Q8—48no—
83C2×C22⋊Q8Q818yescomplemented
84C2×C4×Q8—24no—
85C2×C4×Q8—24no—
86C2×C4×Q8—24no—
87C2×#117—28no—
88C2×#117—28no—
89C2×#117—44no—
90C22×C4⋊C4C2312yes—
91C22×C4⋊C4C2312yes—
92C22×C4⋊C4C2314yes—
93C22×C4⋊C4C2311yescharacteristic
94C22×C4⋊C4C2312yes—
95C22×C4⋊C4C2312yes—
96C22×C4⋊C4C2312yes—
97C22×C4⋊C4C2312yes—
98C22×C4⋊C4C2311yescharacteristic
99C22×C4⋊C4C2312yes—
100C22×C4⋊C4C2314yes—
101C22×C4⋊C4C2314yes—
102C22×C4⋊C4C2312yes—
103C22×C4⋊C4C2311yescharacteristic
104C22×C4⋊C4C2312yes—
105C22×C4⋊C4C2318yes—
106C22×C4⋊C4C2312yes—
107C22×C22⋊C4C2314yes—
108C22×C22⋊C4C2312yes—
109C22×C22⋊C4C2314yes—
110C22×C22⋊C4C2312yes—
111C22×C42—24no—
112C22×C42—24no—
113C22×C42D418yes—
114C22×C42C2312yes—
115C22×C42C2312yes—
116C22×C42C2312yes—
117C22×C42C2311yescharacteristic
118C22×C42C2311yescharacteristic
119C22×C42C2311yescharacteristic
120C22×C42C2312yes—
121C22×C42C2311yescharacteristic
122#398—416no—
123#395—216no—
124#391—28no—
125#391—28no—
126#391—24no—
127#391—44no—
128#389—216no—
129#389—28no—
130#389—28no—
131#389—24no—
132#387—28no—
133#385—28no—
134#384—28no—
135#384—28no—
136#384—24no—
137#382—28no—
138#382—44no—
139#380—28no—
140#380—28no—
141#380—24no—
142#380—28no—
143#380—28no—
144C4×C4⋊C4—28no—
145C4×C4⋊C4—24no—
146C4×C4⋊C4—28no—
147C4×C4⋊C4—24no—
148C4×C4⋊C4D418yes—
149C4×C22⋊C4—28no—
150C2×#95C2318yes—
151C2×#95C2312yes—
152C2×#95C2312yes—
153C2×#95C2314yes—
154C2×#95C2312yes—
155C2×#95C2312yes—
156C2×#95C2312yes—
157C2×#95C2314yes—
158C2×#95C2314yes—
159C2×#95C2312yes—
160C2×#95C2318yes—
161C2×#95C2318yes—
162C2×#95C2318yes—
163C2×#95C2318yes—
164C2×#95C2318yes—
165C2×#95C2318yes—
166C2×#95C2312yes—
167C25C2411yescharacteristic
168C22×Q8—22no—
169C22×Q8—22no—
170C22×Q8—24no—
171C22×Q8—24no—
172C22×Q8—22no—
173C22×Q8—22no—
174C22×Q8—24no—
175C22×Q8—22no—
176C22×Q8—22no—
177C22×Q8—44no—
178C22×Q8—44no—
179C22×Q8C2×Q814yes—
180C23×C4—22no—
181C23×C4—22no—
182C23×C4—22no—
183C23×C4—22no—
184C23×C4—22no—
185C23×C4—22no—
186C23×C4—22no—
187C23×C4—22no—
188C23×C4—24no—
189C23×C4—24no—
190C23×C4—24no—
191C23×C4—42no—
192C23×C4C2×D414yes—
193C23×C4C2×D414yes—
194C23×C4C2×D414yes—
195C23×C4C2×Q814yes—
196C23×C4#4114yes—
197C23×C4C2412yes—
198C23×C4C2414yes—
199C23×C4C2414yes—
200C23×C4C2414yes—
201C23×C4C2412yes—
202C23×C4C2414yes—
203C23×C4C2411yescharacteristic
204C23×C4C2412yes—
205C23×C4C2412yes—
206C23×C4C2411yescharacteristic
207C23×C4C2411yescharacteristic
208C23×C4C2411yescharacteristic
209C23×C4C2411yescharacteristic
210C23×C4C2411yescharacteristic
211C4⋊Q8—416no—
212C22⋊Q8—232no—
213C22⋊Q8—432no—
214C4×Q8—416no—
215#117—432no—
216C2×C4⋊C4—24no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—24no—
220C2×C4⋊C4—24no—
221C2×C4⋊C4—24no—
222C2×C4⋊C4—28no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—24no—
226C2×C4⋊C4—24no—
227C2×C4⋊C4—24no—
228C2×C4⋊C4—28no—
229C2×C4⋊C4—24no—
230C2×C4⋊C4—24no—
231C2×C4⋊C4—24no—
232C2×C4⋊C4—24no—
233C2×C4⋊C4—28no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—24no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—24no—
239C2×C4⋊C4—28no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—216no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—28no—
248C2×C4⋊C4—28no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—28no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—24no—
254C2×C4⋊C4—216no—
255C2×C4⋊C4—44no—
256C2×C4⋊C4—44no—
257C2×C4⋊C4—44no—
258C2×C4⋊C4—48no—
259C2×C4⋊C4—48no—
260C2×C4⋊C4C2×D418yes—
261C2×C4⋊C4C2×Q814yes—
262C2×C4⋊C4C2×Q818yes—
263C2×C22⋊C4—24no—
264C2×C22⋊C4—24no—
265C2×C22⋊C4—28no—
266C2×C22⋊C4—24no—
267C2×C22⋊C4—28no—
268C2×C22⋊C4—28no—
269C2×C22⋊C4—44no—
270C2×C22⋊C4—44no—
271C2×C22⋊C4—48no—
272C2×C22⋊C4C2×Q818yes—
273C2×C42—22no—
274C2×C42—22no—
275C2×C42—22no—
276C2×C42—24no—
277C2×C42—22no—
278C2×C42—22no—
279C2×C42—22no—
280C2×C42—24no—
281C2×C42—232no—
282C2×C42—24no—
283C2×C42—24no—
284C2×C42—24no—
285C2×C42—24no—
286C2×C42—24no—
287C2×C42—24no—
288C2×C42—24no—
289C2×C42—28no—
290C2×C42C2×D418yes—
291C2×C42C2×D414yes—
292C2×C42C2×D414yes—
293C2×C42C2×D414yes—
294C2×C42C2×D414yes—
295C2×C42C2×D414yes—
296C2×C42#4114yes—
297#95—24no—
298#95—24no—
299#95—28no—
300#95—24no—
301#95—24no—
302#95—24no—
303#95—24no—
304#95—28no—
305#95—28no—
306#95—216no—
307#95—48no—
308#95—48no—
309#95—48no—
310#95—48no—
311#95—48no—
312#95—48no—
313#95C2×D418yes—
314C24—21no—
315C24—21no—
316C24—22no—
317C24—21no—
318C24—22no—
319C24—22no—
320C24—21no—
321C24—22no—
322C24C22×D412yes—
323C24C22×Q812yes—
324C24C2×#4112yes—
325C24C2511yescharacteristic
326C2×Q8—28no—
327C2×Q8—28no—
328C2×Q8—28no—
329C2×Q8—48no—
330C2×Q8—48no—
331C2×Q8—44no—
332C2×Q8—48no—
333C2×Q8—44no—
334C2×Q8—44no—
335C2×Q8—44no—
336C2×Q8—88no—
337C2×Q8—88no—
338C22×C4—21no—
339C22×C4—21no—
340C22×C4—21no—
341C22×C4—21no—
342C22×C4—21no—
343C22×C4—22no—
344C22×C4—21no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—28no—
348C22×C4—28no—
349C22×C4—28no—
350C22×C4—28no—
351C22×C4—28no—
352C22×C4—28no—
353C22×C4—21no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—21no—
358C22×C4—24no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—24no—
363C22×C4—22no—
364C22×C4—24no—
365C22×C4—24no—
366C22×C4—24no—
367C22×C4—24no—
368C22×C4—24no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—24no—
373C22×C4—28no—
374C22×C4—28no—
375C22×C4—28no—
376C22×C4—28no—
377C22×C4—28no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—24no—
381C22×C4—24no—
382C22×C4—24no—
383C22×C4—24no—
384C22×C4—22no—
385C22×C4—24no—
386C22×C4—22no—
387C22×C4—24no—
388C22×C4—22no—
389C22×C4—24no—
390C22×C4—24no—
391C22×C4—24no—
392C22×C4—28no—
393C22×C4—24no—
394C22×C4—28no—
395C22×C4—24no—
396C22×C4—24no—
397C22×C4—24no—
398C22×C4—28no—
399C22×C4—24no—
400C22×C4—22no—
401C22×C4—24no—
402C22×C4—24no—
403C22×C4—28no—
404C22×C4—24no—
405C22×C4—44no—
406C22×C4—48no—
407C22×C4—48no—
408C22×C4—48no—
409C22×C4—48no—
410C22×C4C22⋊Q8116yes—
411C22×C4C22×D412yes—
412C22×C4C22×D418yes—
413C22×C4C22×D412yes—
414C22×C4C22×D412yes—
415C22×C4C22×D412yes—
416C22×C4C22×D412yes—
417C22×C4C22×D412yes—
418C22×C4C22×D412yes—
419C22×C4C22×D412yes—
420C22×C4C22×D412yes—
421C22×C4C22×Q814yes—
422C22×C4C22×Q814yes—
423C22×C4C22×Q812yes—
424C22×C4C22×Q812yes—
425C22×C4C2×#4112yes—
426C22×C4C2×#4112yes—
427C22×C4C2×#4112yes—
428C22×C4C2×#4112yes—
429C22×C4C2×#4112yes—
430C22×C4H(5,𝔽2)14yes—
431C22×C4H(5,𝔽2)12yes—
432C22×C42−1+414yes—
433C22×C42−1+414yes—
434C22×C42−1+412yes—
435C4⋊C4—28no—
436C4⋊C4—216no—
437C4⋊C4—216no—
438C4⋊C4—416no—
439C4⋊C4—416no—
440C4⋊C4—48no—
441C4⋊C4—48no—
442C4⋊C4—44no—
443C4⋊C4—48no—
444C4⋊C4—48no—
445C4⋊C4—48no—
446C4⋊C4—416no—
447C4⋊C4—48no—
448C4⋊C4—48no—
449C4⋊C4—44no—
450C4⋊C4—48no—
451C4⋊C4—816no—
452C22⋊C4—216no—
453C22⋊C4—416no—
454C22⋊C4—48no—
455C22⋊C4—416no—
456C42—28no—
457C42—28no—
458C42—28no—
459C42—216no—
460C42—48no—
461C42—48no—
462C42—48no—
463C42#121116yes—
464C23—22no—
465C23—22no—
466C23—22no—
467C23—22no—
468C23—22no—
469C23—22no—
470C23—22no—
471C23—22no—
472C23—22no—
473C23—22no—
474C23—24no—
475C23—24no—
476C23—24no—
477C23—24no—
478C23—42no—
479C23—42no—
480C23—42no—
481C23—42no—
482C23—42no—
483C23—42no—
484C23—42no—
485C23—42no—
486C23C2×C22⋊Q814yes—
487C23C23×D411yescharacteristic
488C23C23×D411yescharacteristic
489C23C23×Q811yescharacteristic
490C23C22×#4111yescharacteristic
491C23C22×#4111yescharacteristic
492C23C2×H(5,𝔽2)12yes—
493C23C2×H(5,𝔽2)11yescharacteristic
494C23C2×2−1+412yes—
495C23C2×2−1+412yes—
496C23C2×2−1+411yescharacteristic
497C23#58512yes—
498Q8—416no—
499Q8—816no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—22no—
506C2×C4—22no—
507C2×C4—22no—
508C2×C4—22no—
509C2×C4—22no—
510C2×C4—24no—
511C2×C4—24no—
512C2×C4—24no—
513C2×C4—24no—
514C2×C4—232no—
515C2×C4—22no—
516C2×C4—24no—
517C2×C4—24no—
518C2×C4—28no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—28no—
522C2×C4—24no—
523C2×C4—24no—
524C2×C4—28no—
525C2×C4—28no—
526C2×C4—416no—
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—44no—
536C2×C4—44no—
537C2×C4—44no—
538C2×C4—44no—
539C2×C4—44no—
540C2×C4—416no—
541C2×C4—42no—
542C2×C4—44no—
543C2×C4—44no—
544C2×C4—44no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—44no—
548C2×C4—44no—
549C2×C4—42no—
550C2×C4—44no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—44no—
554C2×C4—44no—
555C2×C4—44no—
556C2×C4—44no—
557C2×C4—48no—
558C2×C4—44no—
559C2×C4—48no—
560C2×C4—44no—
561C2×C4—44no—
562C2×C4—44no—
563C2×C4—44no—
564C2×C4—48no—
565C2×C4—44no—
566C2×C4—48no—
567C2×C4—44no—
568C2×C4C2×#12114yes—
569C2×C4C2×#12114yes—
570C2×C4C2×#12114yes—
571C2×C4C2×C22⋊Q814yes—
572C2×C4C2×C22⋊Q814yes—
573C2×C4#53514yes—
574C2×C4#53614yes—
575C2×C4#53618yes—
576C2×C4#53614yes—
577C2×C4#53714yes—
578C2×C4#54118yes—
579C2×C4#54614yes—
580C2×C4D4×Q814yes—
581C2×C4#55214yes—
582C2×C4#55414yes—
583C22—24no—
584C22—24no—
585C22—24no—
586C22—42no—
587C22—42no—
588C22—44no—
589C22—44no—
590C22—42no—
591C22—44no—
592C22—42no—
593C22—44no—
594C22C22×#12111yescharacteristic
595C22C22×C22⋊Q811yescharacteristic
596C22C2×#53511yescharacteristic
597C22C2×#53611yescharacteristic
598C22C2×#53611yescharacteristic
599C22C2×#53612yes—
600C22C2×#53711yescharacteristic
601C22#344911yescharacteristic
602C22C2×#54112yes—
603C22C2×#54611yescharacteristic
604C22C2×D4×Q811yescharacteristic
605C22C2×#55211yescharacteristic
606C22C2×#55411yescharacteristic
607C22#348812yes—
608C22#349412yes—
609C22#350212yes—
610C22#350312yes—
611C22#350712yes—
612C22#351112yes—
613C22#354712yes—
614C22#354812yes—
615C22#355012yes—
616C22#355512yes—
617C4—24no—
618C4—24no—
619C4—24no—
620C4—48no—
621C4—42no—
622C4—44no—
623C4—44no—
624C4—82no—
625C4—84no—
626C4—84no—
627C4—82no—
628C4—84no—
629C4#241718yes—
630C4#262118yes—
631C2—44no—
632C2C2×#262111yescharacteristic
633C2C2×#241711yescharacteristic
634C2#3517911yescharacteristic
635C2#3525211yescharacteristic
636C2#3527911yescharacteristic
637C2#3763311yescharacteristic
638C2#3764011yescharacteristic
639C2#3947012yes—
640C2#3948012yes—
641C2#3959712yes—
642C2#3960712yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC25C24
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×#2567C2semidirect
5. #16565C2semidirect
7. #15237C2semidirect
9. #15151C2semidirect
10. #15133C2semidirect
13. #14161C2semidirect
14. #13926C2semidirect
15. #13911C2semidirect
83. C2×C22⋊Q8Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#28572
C2×#25672
C2×C4×C22⋊Q82
C2×#22852
#165652
#162642
#152372
#151832
#151512
#151332
#143402
#142582
#141612
#139262
#139112