Finite Group Databasebeta

← #6489666

Subgroups — #6489666

Order 512.

2930 conjugacy classes of subgroups.

The subgroup lattice has 628 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#557C211yesmaximal, characteristic
2C2×#3236C211yesmaximal, characteristic
3C2×#3104C211yesmaximal, characteristic
4#35248C211yesmaximal, characteristic
5#33010C212yesmaximal
6#33008C212yesmaximal
7#29643C212yesmaximal
8#29642C212yesmaximal
9#25943C214yesmaximal, complemented
10#25849C214yesmaximal, complemented
11#24336C218yesmaximal, complemented
12#21836C211yesmaximal, characteristic
13#21820C211yesmaximal, characteristic
14#20741C211yesmaximal, 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#3104—28no—
37C2×#484C2212yes—
38C2×#484C2212yes—
39C2×#483C2212yes—
40C2×#479C2214yes—
41C2×#477C2212yes—
42C2×#462C2212yes—
43C2×#451C2212yes—
44C2×C4×Q16C2214yes—
45C2×#432C2211yescharacteristic
46C2×#431C2211yescharacteristic
47#2855C2212yes—
48#2855C2212yes—
49#2634—22no—
50#2581C2214yes—
51#2568C2212yes—
52#2546C2212yes—
53#2503—21no—
54#2484—21no—
55#2484—21no—
56#2434—21no—
57#2327—22no—
58#2283C2211yescharacteristic
59#2059C2218yes—
60#2057C2218yes—
61#2026C2218yes—
62#2024C2218yes—
63#1991C2214yes—
64#1989C2214yes—
65#1985C2214yes—
66#1984C2214yes—
67#1878C2218yes—
68#1878C2218yes—
69#1865C2218yes—
70#1863C2218yes—
71#1824C2212yes—
72#1824C2212yes—
73#1823C2212yes—
74#1817C2212yes—
75#1817C2212yes—
76#1805C2214yes—
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—22no—
95C2×C4⋊Q8—24no—
96C2×C4⋊Q8—24no—
97C2×C4⋊Q8—22no—
98C2×C4⋊Q8C2311yescharacteristic
99C2×C4⋊Q8C2311yescharacteristic
100C2×C4⋊Q8C2314yes—
101C2×#125—216no—
102C2×#125—216no—
103C2×#125—24no—
104C2×#125—24no—
105C2×#125—24no—
106C2×#125—24no—
107C2×#125—24no—
108C2×#125—24no—
109C2×#125—22no—
110C2×#125—21no—
111C2×#125—21no—
112C2×#125—24no—
113C2×#125—21no—
114C2×#125D418yes—
115C2×#125C2314yes—
116C2×#125C2312yes—
117C2×#125C2312yes—
118C2×C22⋊Q8—21no—
119C2×C22⋊Q8C2312yes—
120C2×C4×Q8—216no—
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—24no—
128C2×C4×Q8—22no—
129C2×C4×Q8—21no—
130C2×C4×Q8—21no—
131C2×C4×Q8—22no—
132C2×C4×Q8—24no—
133C2×C4×Q8—24no—
134C2×C4×Q8—21no—
135C2×C4×Q8D418yes—
136C2×C4×Q8D412yes—
137C2×C4×Q8D412yes—
138C2×C4×Q8C2312yes—
139C2×C4×Q8C2314yes—
140C2×C4×Q8C2314yes—
141C2×C4×Q8C2312yes—
142C2×C4×Q8C2312yes—
143C2×C4×Q8C2312yes—
144C2×C4×Q8C2312yes—
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#484—28no—
170#483—28no—
171#479—216no—
172#477—28no—
173#462—28no—
174#451—28no—
175C4×Q16—216no—
176#432—24no—
177#431—24no—
178C2×#107C2314yes—
179C2×#107C2314yes—
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×C4×C8C2312yes—
192#400—22no—
193#400—22no—
194#400—24no—
195#398—22no—
196#398—22no—
197#395—24no—
198#391—22no—
199#391—22no—
200#391C2314yes—
201#391C2314yes—
202#389C2314yes—
203#389C2314yes—
204#387—24no—
205#387—24no—
206#387—24no—
207#387—24no—
208#385—22no—
209#385—22no—
210#383—21no—
211#382—22no—
212#382C2312yes—
213#381—22no—
214#381—22no—
215#381—21no—
216#380C2312yes—
217C4×C4⋊C4C2312yes—
218C4×C22⋊C4—21no—
219#376—21no—
220#340C2318yes—
221#340C2318yes—
222#340C2318yes—
223#336C2318yes—
224#336C2318yes—
225C22×Q8—28no—
226C22×Q8—28no—
227C22×Q8—24no—
228C22×Q8—22no—
229C22×Q8—21no—
230C22×Q8—24no—
231C22×Q8—21no—
232C22×Q8—44no—
233C22×Q8—48no—
234C22×Q8C2×D414yes—
235C22×Q8C2×D412yes—
236C22×Q8C2×D412yes—
237C22×Q8C2412yes—
238C22×Q8C2412yes—
239C23×C4—21no—
240C23×C4—21no—
241C23×C4—21no—
242C23×C4—21no—
243C23×C4C2×D411yescharacteristic
244C23×C4C2×D411yescharacteristic
245C23×C4C2×D411yescharacteristic
246C23×C4C2×D411yescharacteristic
247C23×C4C2×D411yescharacteristic
248C23×C4C2×D411yescharacteristic
249C23×C4C2411yescharacteristic
250C2×Q16—28no—
251C2×Q16—216no—
252C22×C8C2414yes—
253C22×C8C2414yes—
254C4⋊Q8—232no—
255C4⋊Q8—232no—
256C4⋊Q8—216no—
257C4⋊Q8—28no—
258C4⋊Q8—28no—
259C4⋊Q8—22no—
260C4⋊Q8—22no—
261C4⋊Q8—22no—
262C4⋊Q8—22no—
263C4⋊Q8—48no—
264#125—232no—
265#125—232no—
266#125—232no—
267#125—232no—
268#125—216no—
269#125—28no—
270#125—28no—
271#125—28no—
272#125—28no—
273#125—24no—
274#125—24no—
275#125—44no—
276#125C2×D418yes—
277#125C2×D418yes—
278C22⋊Q8—24no—
279C22⋊Q8—24no—
280C22⋊Q8—44no—
281C4×Q8—216no—
282C4×Q8—232no—
283C4×Q8—232no—
284C4×Q8—216no—
285C4×Q8—28no—
286C4×Q8—28no—
287C4×Q8—28no—
288C4×Q8—28no—
289C4×Q8—28no—
290C4×Q8—28no—
291C4×Q8—28no—
292C4×Q8—28no—
293C4×Q8—24no—
294C4×Q8—24no—
295C4×Q8—24no—
296C4×Q8—24no—
297C4×Q8—24no—
298C4×Q8—24no—
299C4×Q8—24no—
300C4×Q8—24no—
301C4×Q8—44no—
302C4×Q8C2×D418yes—
303C4×Q8C2×D418yes—
304#117—44no—
305C2×C4⋊C4—28no—
306C2×C4⋊C4—28no—
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—24no—
314C2×C4⋊C4—24no—
315C2×C4⋊C4—24no—
316C2×C4⋊C4—24no—
317C2×C4⋊C4—22no—
318C2×C4⋊C4—22no—
319C2×C4⋊C4—24no—
320C2×C4⋊C4—24no—
321C2×C4⋊C4—22no—
322C2×C4⋊C4—22no—
323C2×C4⋊C4—24no—
324C2×C4⋊C4—24no—
325C2×C4⋊C4—22no—
326C2×C4⋊C4—22no—
327C2×C4⋊C4—21no—
328C2×C4⋊C4—22no—
329C2×C4⋊C4—21no—
330C2×C4⋊C4—21no—
331C2×C4⋊C4—22no—
332C2×C4⋊C4—22no—
333C2×C4⋊C4—21no—
334C2×C4⋊C4—21no—
335C2×C4⋊C4—21no—
336C2×C4⋊C4—22no—
337C2×C4⋊C4—22no—
338C2×C4⋊C4—24no—
339C2×C4⋊C4—24no—
340C2×C4⋊C4C2×D412yes—
341C2×C4⋊C4C2×D412yes—
342C2×C4⋊C4C2×D412yes—
343C2×C4⋊C4C2×D412yes—
344C2×C4⋊C4C2×D412yes—
345C2×C4⋊C4C2×D412yes—
346C2×C4⋊C4C2×D414yes—
347C2×C4⋊C4C2×D414yes—
348C2×C4⋊C4C2×D414yes—
349C2×C4⋊C4C2412yes—
350C2×C4⋊C4C2414yes—
351C2×C4⋊C4C2412yes—
352C2×C4⋊C4C2412yes—
353C2×C4⋊C4C2412yes—
354C2×C22⋊C4—22no—
355C2×C22⋊C4—22no—
356C2×C42—28no—
357C2×C42—28no—
358C2×C42—28no—
359C2×C42—22no—
360C2×C42—22no—
361C2×C42—22no—
362C2×C42—22no—
363C2×C42—24no—
364C2×C42—22no—
365C2×C42—22no—
366C2×C42—24no—
367C2×C42—22no—
368C2×C42—22no—
369C2×C42C2×D412yes—
370C2×C42C2×D412yes—
371C2×C42C2×D411yescharacteristic
372C2×C42C2×D411yescharacteristic
373C2×C42C2×D414yes—
374C2×C42C2×D414yes—
375C2×C42C2412yes—
376C2×C42C2412yes—
377C2×C42C2411yescharacteristic
378C2×C42C2411yescharacteristic
379#107—28no—
380#107—28no—
381#106—28no—
382C4⋊C8—24no—
383C4⋊C8—24no—
384#103—28no—
385#103—216no—
386#103—28no—
387#103—28no—
388#103—28no—
389C22⋊C8—24no—
390C22⋊C8—24no—
391#97—24no—
392C4×C8—24no—
393#95—22no—
394#95—22no—
395#95—22no—
396#95—22no—
397#95—24no—
398#95—24no—
399#95—24no—
400#95—24no—
401#95—24no—
402#95—24no—
403C24C22×D411yescharacteristic
404C2×Q8—216no—
405C2×Q8—28no—
406C2×Q8—216no—
407C2×Q8—216no—
408C2×Q8—24no—
409C2×Q8—24no—
410C2×Q8—24no—
411C2×Q8—24no—
412C2×Q8—24no—
413C2×Q8—22no—
414C2×Q8—24no—
415C2×Q8—22no—
416C2×Q8—22no—
417C2×Q8—28no—
418C2×Q8—42no—
419C2×Q8—42no—
420C2×Q8—416no—
421C2×Q8—432no—
422C2×Q8—48no—
423C2×Q8—44no—
424C2×Q8—42no—
425C2×Q8—48no—
426C2×Q8C22×D414yes—
427C2×Q82−1+418yes—
428C22×C4—28no—
429C22×C4—24no—
430C22×C4—22no—
431C22×C4—24no—
432C22×C4—22no—
433C22×C4—24no—
434C22×C4—24no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—24no—
438C22×C4—22no—
439C22×C4—24no—
440C22×C4—24no—
441C22×C4—24no—
442C22×C4—24no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—24no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—42no—
450C22×C4—42no—
451C22×C4—44no—
452C22×C4—44no—
453C22×C4—44no—
454C22×C4#12014yes—
455C22×C4#12014yes—
456C22×C4C22×D412yes—
457C22×C4C22×D412yes—
458C22×C4C22×D412yes—
459C22×C4C22×D411yescharacteristic
460C22×C4C22×D411yescharacteristic
461C22×C4C22×D411yescharacteristic
462C22×C4C22×D411yescharacteristic
463C22×C4C22×D411yescharacteristic
464C22×C4C22×D412yes—
465C22×C4C22×D412yes—
466C22×C4C22×D412yes—
467C22×C4C2511yescharacteristic
468Q16—416no—
469C2×C8—24no—
470C2×C8—24no—
471C2×C8—28no—
472C2×C8—28no—
473C4⋊C4—216no—
474C4⋊C4—216no—
475C4⋊C4—216no—
476C4⋊C4—216no—
477C4⋊C4—216no—
478C4⋊C4—28no—
479C4⋊C4—28no—
480C4⋊C4—28no—
481C4⋊C4—28no—
482C4⋊C4—28no—
483C4⋊C4—28no—
484C4⋊C4—24no—
485C4⋊C4—24no—
486C4⋊C4—24no—
487C4⋊C4—24no—
488C4⋊C4—24no—
489C4⋊C4—24no—
490C4⋊C4—24no—
491C4⋊C4—24no—
492C4⋊C4—24no—
493C4⋊C4—22no—
494C4⋊C4—22no—
495C4⋊C4—22no—
496C4⋊C4—22no—
497C4⋊C4—22no—
498C4⋊C4—28no—
499C4⋊C4—48no—
500C4⋊C4—42no—
501C4⋊C4—42no—
502C4⋊C4—44no—
503C4⋊C4—44no—
504C4⋊C4—42no—
505C4⋊C4—48no—
506C4⋊C4C22×D414yes—
507C4⋊C4C22×D414yes—
508C4⋊C4C22×D414yes—
509C22⋊C4—44no—
510C22⋊C4—44no—
511C42—216no—
512C42—216no—
513C42—28no—
514C42—28no—
515C42—24no—
516C42—24no—
517C42—22no—
518C42—22no—
519C42—22no—
520C42—22no—
521C42—44no—
522C42—44no—
523C42#12014yes—
524C42#12014yes—
525C42C22×D412yes—
526C42C22×D412yes—
527C42C22×D412yes—
528C42C22×D412yes—
529C23—22no—
530C23—22no—
531C23C2×#12012yes—
532C23#57814yes—
533C23C23×D411yescharacteristic
534Q8—216no—
535Q8—24no—
536Q8—28no—
537Q8—48no—
538Q8—48no—
539Q8—432no—
540Q8—44no—
541Q8—44no—
542Q8—84no—
543Q8#53618yes—
544C2×C4—216no—
545C2×C4—28no—
546C2×C4—24no—
547C2×C4—22no—
548C2×C4—28no—
549C2×C4—22no—
550C2×C4—28no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—24no—
557C2×C4—24no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—24no—
561C2×C4—22no—
562C2×C4—24no—
563C2×C4—24no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—24no—
567C2×C4—24no—
568C2×C4—44no—
569C2×C4—44no—
570C2×C4—48no—
571C2×C4—48no—
572C2×C4—44no—
573C2×C4—44no—
574C2×C4—48no—
575C2×C4—44no—
576C2×C4—44no—
577C2×C4—44no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—44no—
581C2×C4—44no—
582C2×C4—44no—
583C2×C4—44no—
584C2×C4—44no—
585C2×C4C2×#12012yes—
586C2×C4C2×#12012yes—
587C2×C4C2×#12012yes—
588C2×C4C2×#12012yes—
589C2×C4C2×#12012yes—
590C2×C4#57514yes—
591C2×C4C23×D411yescharacteristic
592C2×C4C23×D411yescharacteristic
593C2×C4C2×2−1+414yes—
594C8—44no—
595C8—44no—
596C22—28no—
597C22—22no—
598C22—22no—
599C22#301114yes—
600C22C22×#12011yescharacteristic
601C22#348612yes—
602C22C2×#57512yes—
603C22C2×#57812yes—
604C4—28no—
605C4—22no—
606C4—24no—
607C4—22no—
608C4—22no—
609C4—22no—
610C4—22no—
611C4—42no—
612C4—44no—
613C4—44no—
614C4—42no—
615C4—42no—
616C4—44no—
617C4—84no—
618C4—82no—
619C4—82no—
620C4C2×#53612yes—
621C4#348612yes—
622C2—24no—
623C2#3524211yescharacteristic
624C2#6035911yescharacteristic
625C2C2×#301111yescharacteristic
626C2#6184214yes—
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. #25849C2semidirect
11. #24336C2semidirect

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×#31042
#352482
#330102
#330082
#296432
#296422
#259432
#258492
#243362
#218362
#218202
#207412