Finite Group Databasebeta

← #6489657

Subgroups — #6489657

Order 512.

2924 conjugacy classes of subgroups.

The subgroup lattice has 645 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#557C211yesmaximal, characteristic
2C2×#3236C211yesmaximal, characteristic
3C2×#3113C211yesmaximal, characteristic
4#35179C211yesmaximal, characteristic
5#33010C212yesmaximal
6#33008C212yesmaximal
7#29643C212yesmaximal
8#29641C212yesmaximal
9#25943C214yesmaximal, complemented
10#25848C214yesmaximal, complemented
11#24348C218yesmaximal, complemented
12#21838C211yesmaximal, characteristic
13#21822C211yesmaximal, characteristic
14#20740C211yesmaximal, characteristic
15C2×#557—28no—
16C2×#557—28no—
17C2×#557—22no—
18C2×#557—22no—
19C2×#557C2214yes—
20C2×#557C2214yes—
21C2×#533C2211yescharacteristic
22C22×C4⋊Q8—21no—
23C22×C4⋊Q8C2211yescharacteristic
24C22×#125—21no—
25C22×#125—21no—
26C22×#125C2211yescharacteristic
27C22×#125C2211yescharacteristic
28C22×C4×Q8—21no—
29C22×C4×Q8C2211yescharacteristic
30C22×C4×Q8C2211yescharacteristic
31C22×C4×Q8C2211yescharacteristic
32C22×C4×Q8C2211yescharacteristic
33#3236—28no—
34C2×#501C2212yes—
35C2×#491C2212yes—
36#3113—28no—
37C2×#484C2212yes—
38C2×#483C2212yes—
39C2×#483C2212yes—
40C2×#479C2214yes—
41C2×#475C2212yes—
42C2×#462C2212yes—
43C2×#451C2212yes—
44C2×#441C2214yes—
45C2×#432C2211yescharacteristic
46C2×#432C2211yescharacteristic
47#2855C2212yes—
48#2855C2212yes—
49#2627—21no—
50#2627—21no—
51#2582C2214yes—
52#2581—22no—
53#2568C2212yes—
54#2567C2212yes—
55#2565—22no—
56#2317—21no—
57#2310—21no—
58#2285C2211yescharacteristic
59#2059C2218yes—
60#2058C2218yes—
61#2026C2218yes—
62#2024C2218yes—
63#1992C2214yes—
64#1989C2214yes—
65#1985C2214yes—
66#1985C2214yes—
67#1878C2218yes—
68#1878C2218yes—
69#1865C2218yes—
70#1863C2218yes—
71#1824C2212yes—
72#1823C2212yes—
73#1823C2212yes—
74#1817C2212yes—
75#1817C2212yes—
76#1803C2214yes—
77#1787C2212yes—
78#1763C2214yes—
79C23×Q8—21no—
80C23×Q8D411yescharacteristic
81C23×Q8C2311yescharacteristic
82C22×Q16C2314yes—
83#557—232no—
84#557—232no—
85#557—216no—
86#557—216no—
87#557—216no—
88#557—216no—
89#533—24no—
90#533—24no—
91C2×C4⋊Q8—216no—
92C2×C4⋊Q8—24no—
93C2×C4⋊Q8—24no—
94C2×C4⋊Q8—24no—
95C2×C4⋊Q8—24no—
96C2×C4⋊Q8—22no—
97C2×C4⋊Q8C2311yescharacteristic
98C2×C4⋊Q8C2311yescharacteristic
99C2×C4⋊Q8C2314yes—
100C2×#125—216no—
101C2×#125—216no—
102C2×#125—24no—
103C2×#125—24no—
104C2×#125—24no—
105C2×#125—24no—
106C2×#125—24no—
107C2×#125—24no—
108C2×#125—22no—
109C2×#125—21no—
110C2×#125—21no—
111C2×#125—24no—
112C2×#125—21no—
113C2×#125D418yes—
114C2×#125C2314yes—
115C2×#125C2312yes—
116C2×#125C2312yes—
117C2×C22⋊Q8—21no—
118C2×C22⋊Q8C2312yes—
119C2×C4×Q8—216no—
120C2×C4×Q8—24no—
121C2×C4×Q8—24no—
122C2×C4×Q8—24no—
123C2×C4×Q8—24no—
124C2×C4×Q8—24no—
125C2×C4×Q8—24no—
126C2×C4×Q8—24no—
127C2×C4×Q8—22no—
128C2×C4×Q8—21no—
129C2×C4×Q8—21no—
130C2×C4×Q8—22no—
131C2×C4×Q8—24no—
132C2×C4×Q8—24no—
133C2×C4×Q8—21no—
134C2×C4×Q8D418yes—
135C2×C4×Q8D412yes—
136C2×C4×Q8D412yes—
137C2×C4×Q8C2312yes—
138C2×C4×Q8C2314yes—
139C2×C4×Q8C2314yes—
140C2×C4×Q8C2312yes—
141C2×C4×Q8C2312yes—
142C2×C4×Q8C2312yes—
143C2×C4×Q8C2312yes—
144C2×#117—21no—
145C2×#117—21no—
146C22×C4⋊C4—21no—
147C22×C4⋊C4—21no—
148C22×C4⋊C4—21no—
149C22×C4⋊C4—21no—
150C22×C4⋊C4—21no—
151C22×C4⋊C4—21no—
152C22×C4⋊C4—21no—
153C22×C4⋊C4—21no—
154C22×C4⋊C4D411yescharacteristic
155C22×C4⋊C4D411yescharacteristic
156C22×C4⋊C4D411yescharacteristic
157C22×C4⋊C4C2311yescharacteristic
158C22×C4⋊C4C2311yescharacteristic
159C22×C4⋊C4C2311yescharacteristic
160C22×C42—21no—
161C22×C42—21no—
162C22×C42—21no—
163C22×C42C2311yescharacteristic
164C22×C42C2311yescharacteristic
165C22×C42C2311yescharacteristic
166#501—28no—
167#491—28no—
168#484—28no—
169#483—28no—
170#483—28no—
171#479—216no—
172#475—28no—
173#462—28no—
174#451—28no—
175#441—216no—
176#432—24no—
177#432—24no—
178C2×#107C2314yes—
179C2×#106C2314yes—
180C2×#106C2314yes—
181C2×C4⋊C8C2312yes—
182C2×C4⋊C8C2312yes—
183C2×#103C2314yes—
184C2×#103C2314yes—
185C2×#103C2314yes—
186C2×#103C2318yes—
187C2×#103C2314yes—
188C2×C22⋊C8C2312yes—
189C2×C22⋊C8C2312yes—
190C2×#97C2312yes—
191C2×#97C2312yes—
192#400—22no—
193#400—22no—
194#398—24no—
195#398—24no—
196#395—22no—
197#395—22no—
198#391—22no—
199#391—22no—
200#391—24no—
201#391C2314yes—
202#391C2314yes—
203#389—24no—
204#389C2314yes—
205#389C2314yes—
206#387—24no—
207#387—24no—
208#385—22no—
209#385—22no—
210#384—22no—
211#384C2312yes—
212#382—22no—
213#382—21no—
214#381—22no—
215#381—22no—
216#380—21no—
217#380C2312yes—
218C4×C4⋊C4—21no—
219C4×C4⋊C4C2312yes—
220#340C2318yes—
221#340C2318yes—
222#340C2318yes—
223#336C2318yes—
224#336C2318yes—
225C22×Q8—28no—
226C22×Q8—28no—
227C22×Q8—24no—
228C22×Q8—24no—
229C22×Q8—22no—
230C22×Q8—21no—
231C22×Q8—24no—
232C22×Q8—21no—
233C22×Q8—42no—
234C22×Q8—44no—
235C22×Q8—44no—
236C22×Q8C2×D414yes—
237C22×Q8C2×D412yes—
238C22×Q8C2×D412yes—
239C22×Q8C2412yes—
240C22×Q8C2412yes—
241C23×C4—21no—
242C23×C4—21no—
243C23×C4—21no—
244C23×C4—21no—
245C23×C4C2×D411yescharacteristic
246C23×C4C2×D411yescharacteristic
247C23×C4C2×D411yescharacteristic
248C23×C4C2×D411yescharacteristic
249C23×C4C2×D411yescharacteristic
250C23×C4C2×D411yescharacteristic
251C23×C4C2411yescharacteristic
252C2×Q16—28no—
253C2×Q16—216no—
254C22×C8C2414yes—
255C22×C8C2414yes—
256C4⋊Q8—232no—
257C4⋊Q8—232no—
258C4⋊Q8—216no—
259C4⋊Q8—28no—
260C4⋊Q8—28no—
261C4⋊Q8—22no—
262C4⋊Q8—22no—
263C4⋊Q8—22no—
264C4⋊Q8—22no—
265#125—232no—
266#125—232no—
267#125—232no—
268#125—232no—
269#125—216no—
270#125—28no—
271#125—28no—
272#125—28no—
273#125—28no—
274#125—24no—
275#125—24no—
276#125—44no—
277#125C2×D418yes—
278#125C2×D418yes—
279C22⋊Q8—24no—
280C22⋊Q8—24no—
281C22⋊Q8—44no—
282C4×Q8—216no—
283C4×Q8—232no—
284C4×Q8—232no—
285C4×Q8—216no—
286C4×Q8—28no—
287C4×Q8—28no—
288C4×Q8—28no—
289C4×Q8—28no—
290C4×Q8—28no—
291C4×Q8—28no—
292C4×Q8—28no—
293C4×Q8—28no—
294C4×Q8—24no—
295C4×Q8—24no—
296C4×Q8—24no—
297C4×Q8—24no—
298C4×Q8—24no—
299C4×Q8—24no—
300C4×Q8—24no—
301C4×Q8—24no—
302C4×Q8—44no—
303C4×Q8C2×D418yes—
304C4×Q8C2×D418yes—
305#117—44no—
306#117—44no—
307C2×C4⋊C4—28no—
308C2×C4⋊C4—28no—
309C2×C4⋊C4—28no—
310C2×C4⋊C4—28no—
311C2×C4⋊C4—28no—
312C2×C4⋊C4—28no—
313C2×C4⋊C4—28no—
314C2×C4⋊C4—28no—
315C2×C4⋊C4—24no—
316C2×C4⋊C4—24no—
317C2×C4⋊C4—24no—
318C2×C4⋊C4—24no—
319C2×C4⋊C4—22no—
320C2×C4⋊C4—22no—
321C2×C4⋊C4—24no—
322C2×C4⋊C4—24no—
323C2×C4⋊C4—22no—
324C2×C4⋊C4—22no—
325C2×C4⋊C4—24no—
326C2×C4⋊C4—24no—
327C2×C4⋊C4—22no—
328C2×C4⋊C4—22no—
329C2×C4⋊C4—21no—
330C2×C4⋊C4—22no—
331C2×C4⋊C4—21no—
332C2×C4⋊C4—21no—
333C2×C4⋊C4—22no—
334C2×C4⋊C4—22no—
335C2×C4⋊C4—21no—
336C2×C4⋊C4—21no—
337C2×C4⋊C4—22no—
338C2×C4⋊C4—21no—
339C2×C4⋊C4—22no—
340C2×C4⋊C4—22no—
341C2×C4⋊C4—22no—
342C2×C4⋊C4—24no—
343C2×C4⋊C4—24no—
344C2×C4⋊C4C2×D412yes—
345C2×C4⋊C4C2×D412yes—
346C2×C4⋊C4C2×D412yes—
347C2×C4⋊C4C2×D412yes—
348C2×C4⋊C4C2×D412yes—
349C2×C4⋊C4C2×D412yes—
350C2×C4⋊C4C2×D414yes—
351C2×C4⋊C4C2×D414yes—
352C2×C4⋊C4C2×D414yes—
353C2×C4⋊C4C2412yes—
354C2×C4⋊C4C2414yes—
355C2×C4⋊C4C2412yes—
356C2×C4⋊C4C2412yes—
357C2×C4⋊C4C2412yes—
358C2×C22⋊C4—22no—
359C2×C22⋊C4—22no—
360C2×C42—28no—
361C2×C42—28no—
362C2×C42—28no—
363C2×C42—22no—
364C2×C42—22no—
365C2×C42—22no—
366C2×C42—22no—
367C2×C42—24no—
368C2×C42—22no—
369C2×C42—22no—
370C2×C42—24no—
371C2×C42—22no—
372C2×C42—22no—
373C2×C42C2×D412yes—
374C2×C42C2×D412yes—
375C2×C42C2×D411yescharacteristic
376C2×C42C2×D411yescharacteristic
377C2×C42C2×D414yes—
378C2×C42C2×D414yes—
379C2×C42C2412yes—
380C2×C42C2412yes—
381C2×C42C2411yescharacteristic
382C2×C42C2411yescharacteristic
383#107—28no—
384#106—28no—
385#106—28no—
386C4⋊C8—24no—
387C4⋊C8—24no—
388#103—28no—
389#103—216no—
390#103—28no—
391#103—28no—
392#103—28no—
393C22⋊C8—24no—
394C22⋊C8—24no—
395#97—24no—
396#97—24no—
397#95—22no—
398#95—22no—
399#95—24no—
400#95—24no—
401#95—24no—
402#95—24no—
403#95—24no—
404#95—24no—
405C24C22×D411yescharacteristic
406C2×Q8—216no—
407C2×Q8—28no—
408C2×Q8—216no—
409C2×Q8—216no—
410C2×Q8—24no—
411C2×Q8—24no—
412C2×Q8—24no—
413C2×Q8—24no—
414C2×Q8—24no—
415C2×Q8—22no—
416C2×Q8—24no—
417C2×Q8—22no—
418C2×Q8—22no—
419C2×Q8—28no—
420C2×Q8—42no—
421C2×Q8—42no—
422C2×Q8—416no—
423C2×Q8—432no—
424C2×Q8—48no—
425C2×Q8—42no—
426C2×Q8—48no—
427C2×Q8—82no—
428C2×Q8C22×D414yes—
429C2×Q82−1+418yes—
430C22×C4—28no—
431C22×C4—24no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—24no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—24no—
441C22×C4—22no—
442C22×C4—24no—
443C22×C4—24no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—24no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—42no—
453C22×C4—42no—
454C22×C4—42no—
455C22×C4—44no—
456C22×C4—42no—
457C22×C4—42no—
458C22×C4—42no—
459C22×C4—42no—
460C22×C4#12014yes—
461C22×C4#12014yes—
462C22×C4#13714yes—
463C22×C4C22×D412yes—
464C22×C4C22×D412yes—
465C22×C4C22×D412yes—
466C22×C4C22×D411yescharacteristic
467C22×C4C22×D411yescharacteristic
468C22×C4C22×D411yescharacteristic
469C22×C4C22×D411yescharacteristic
470C22×C4C22×D411yescharacteristic
471C22×C4C22×D412yes—
472C22×C4C22×D412yes—
473C22×C4C22×D412yes—
474C22×C4C2511yescharacteristic
475Q16—416no—
476C2×C8—24no—
477C2×C8—24no—
478C2×C8—28no—
479C2×C8—28no—
480C4⋊C4—216no—
481C4⋊C4—216no—
482C4⋊C4—216no—
483C4⋊C4—216no—
484C4⋊C4—216no—
485C4⋊C4—28no—
486C4⋊C4—28no—
487C4⋊C4—28no—
488C4⋊C4—28no—
489C4⋊C4—28no—
490C4⋊C4—28no—
491C4⋊C4—24no—
492C4⋊C4—24no—
493C4⋊C4—24no—
494C4⋊C4—24no—
495C4⋊C4—24no—
496C4⋊C4—24no—
497C4⋊C4—24no—
498C4⋊C4—24no—
499C4⋊C4—24no—
500C4⋊C4—22no—
501C4⋊C4—22no—
502C4⋊C4—22no—
503C4⋊C4—22no—
504C4⋊C4—22no—
505C4⋊C4—28no—
506C4⋊C4—44no—
507C4⋊C4—42no—
508C4⋊C4—42no—
509C4⋊C4—44no—
510C4⋊C4—44no—
511C4⋊C4—42no—
512C4⋊C4—82no—
513C4⋊C4—84no—
514C4⋊C4—84no—
515C4⋊C4C22×D414yes—
516C4⋊C4C22×D414yes—
517C4⋊C4C22×D414yes—
518C22⋊C4—44no—
519C22⋊C4—44no—
520C42—216no—
521C42—216no—
522C42—28no—
523C42—28no—
524C42—24no—
525C42—24no—
526C42—22no—
527C42—22no—
528C42—22no—
529C42—22no—
530C42—44no—
531C42—44no—
532C42#12014yes—
533C42#12014yes—
534C42C22×D412yes—
535C42C22×D412yes—
536C42C22×D412yes—
537C42C22×D412yes—
538C23—22no—
539C23—22no—
540C23C2×#12012yes—
541C23C2×#13712yes—
542C23#57512yes—
543C23C23×D411yescharacteristic
544Q8—216no—
545Q8—24no—
546Q8—28no—
547Q8—48no—
548Q8—48no—
549Q8—432no—
550Q8—44no—
551Q8—44no—
552Q8—84no—
553Q8#53618yes—
554C2×C4—216no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—24no—
558C2×C4—22no—
559C2×C4—28no—
560C2×C4—22no—
561C2×C4—28no—
562C2×C4—28no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—22no—
568C2×C4—24no—
569C2×C4—24no—
570C2×C4—24no—
571C2×C4—24no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—24no—
575C2×C4—22no—
576C2×C4—24no—
577C2×C4—24no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—44no—
581C2×C4—44no—
582C2×C4—44no—
583C2×C4—48no—
584C2×C4—44no—
585C2×C4—44no—
586C2×C4—48no—
587C2×C4—48no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—44no—
593C2×C4—44no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—44no—
597C2×C4—44no—
598C2×C4—44no—
599C2×C4C2×#12012yes—
600C2×C4C2×#12012yes—
601C2×C4C2×#12012yes—
602C2×C4C2×#12012yes—
603C2×C4C2×#12012yes—
604C2×C4C2×#13714yes—
605C2×C4C23×D411yescharacteristic
606C2×C4C23×D411yescharacteristic
607C2×C4C2×2−1+414yes—
608C8—44no—
609C8—44no—
610C22—24no—
611C22—24no—
612C22—22no—
613C22—22no—
614C22#300714yes—
615C22C22×#12011yescharacteristic
616C22#348612yes—
617C22C22×#13711yescharacteristic
618C22C2×#57511yescharacteristic
619C22#358612yes—
620C4—28no—
621C4—22no—
622C4—22no—
623C4—22no—
624C4—24no—
625C4—22no—
626C4—22no—
627C4—42no—
628C4—44no—
629C4—44no—
630C4—42no—
631C4—42no—
632C4—44no—
633C4—84no—
634C4—82no—
635C4—82no—
636C4C2×#53612yes—
637C4#348612yes—
638C2—24no—
639C2#3524211yescharacteristic
640C2C2×#300711yescharacteristic
641C2#6036811yescharacteristic
642C2#6182912yes—
643C2#6183212yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC24C22×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C4×Q8 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
9. #25943C2semidirect
10. #25848C2semidirect
11. #24348C2semidirect

Maximal subgroups

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

subgroupindex
C22×#5572
C2×#32362
C2×#31132
#351792
#330102
#330082
#296432
#296412
#259432
#258482
#243482
#218382
#218222
#207402