Finite Group Databasebeta

← #6481518

Subgroups — #6481518

Order 512.

2882 conjugacy classes of subgroups.

The subgroup lattice has 650 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#541C211yesmaximal, characteristic
2C2×#3236C211yesmaximal, characteristic, complemented
3C2×#3209C211yesmaximal, characteristic, complemented
4#35325C211yesmaximal, characteristic
5#33010C212yesmaximal, complemented
6#33008C212yesmaximal, complemented
7#29599C212yesmaximal
8#29597C212yesmaximal
9#25686C214yesmaximal, complemented
10#25609C214yesmaximal, complemented
11#24165C218yesmaximal, complemented
12#21836C211yesmaximal, characteristic, complemented
13#21811C211yesmaximal, characteristic, complemented
14#20717C211yesmaximal, characteristic
15C2×#541—28no—
16C2×#541—28no—
17C2×#541—22no—
18C2×#541—22no—
19C2×#541C2214yes—
20C2×#541C2214yes—
21C2×#533C2211yescharacteristic
22C22×C4⋊Q8C2211yescharacteristic
23C22×#126—21no—
24C22×#126—21no—
25C22×#125—21no—
26C22×#125—21no—
27C22×#125C2211yescharacteristic
28C22×C22⋊Q8C2211yescharacteristic
29C22×C22⋊Q8C2211yescharacteristic
30C22×C4×Q8C2211yescharacteristic
31C22×C4×Q8C2211yescharacteristic
32C22×#117C2211yescharacteristic
33#3236—28no—
34#3209—28no—
35C2×#501C2212yescomplemented
36C2×#494C2212yescomplemented
37C2×#491C2212yescomplemented
38C2×#487C2212yescomplemented
39C2×#484C2212yes—
40C2×#483C2212yes—
41C2×#479C2214yescomplemented
42C2×#462C2214yescomplemented
43C2×#451C2212yes—
44C2×#451C2212yes—
45C2×#432C2211yescharacteristic
46C2×#431C2211yescharacteristic
47#2855C2212yes—
48#2848C2212yes—
49#2628C2212yes—
50#2621C2212yes—
51#2611—22no—
52#2599—22no—
53#2583—21no—
54#2561—21no—
55#2484—21no—
56#2482—21no—
57#2323C2214yes—
58#2283C2211yescharacteristic
59#2036C2218yes—
60#2035C2218yes—
61#2001C2218yes—
62#2001C2218yes—
63#1991C2214yescomplemented
64#1989C2214yescomplemented
65#1981C2214yescomplemented
66#1960C2214yescomplemented
67#1878C2218yescomplemented
68#1878C2218yescomplemented
69#1865C2218yescomplemented
70#1863C2218yescomplemented
71#1824C2212yes—
72#1823C2212yes—
73#1817C2212yes—
74#1798C2214yes—
75#1787C2212yes—
76#1787C2212yes—
77#1779C2212yes—
78#1750C2214yes—
79C23×Q8C2311yescharacteristic
80C23×Q8C2311yescharacteristic
81C24×C4C2311yescharacteristic
82C22×Q16C2314yes—
83C22×Q16C2314yes—
84#541—232no—
85#541—232no—
86#541—216no—
87#541—216no—
88#541—216no—
89#541—216no—
90#533—24no—
91#533—24no—
92C2×C4⋊Q8—21no—
93C2×C4⋊Q8—21no—
94C2×C4⋊Q8—24no—
95C2×C4⋊Q8D418yes—
96C2×C4⋊Q8C2311yescharacteristic
97C2×C4⋊Q8C2312yes—
98C2×C4⋊Q8C2312yes—
99C2×C4⋊Q8C2311yescharacteristic
100C2×#126—216no—
101C2×#126—216no—
102C2×#126—24no—
103C2×#126—24no—
104C2×#126—24no—
105C2×#126—24no—
106C2×#125—216no—
107C2×#125—216no—
108C2×#125—24no—
109C2×#125—24no—
110C2×#125—24no—
111C2×#125—24no—
112C2×#125—21no—
113C2×#125—21no—
114C2×#125—24no—
115C2×#125—21no—
116C2×#125D418yes—
117C2×#125C2312yes—
118C2×#125C2312yes—
119C2×C22⋊Q8—24no—
120C2×C22⋊Q8—24no—
121C2×C22⋊Q8—22no—
122C2×C22⋊Q8—22no—
123C2×C22⋊Q8—21no—
124C2×C22⋊Q8—24no—
125C2×C22⋊Q8—24no—
126C2×C22⋊Q8C2312yes—
127C2×C22⋊Q8C2314yes—
128C2×C22⋊Q8C2314yes—
129C2×C22⋊Q8C2314yes—
130C2×C4×Q8—24no—
131C2×C4×Q8—24no—
132C2×C4×Q8—24no—
133C2×C4×Q8—24no—
134C2×C4×Q8—22no—
135C2×C4×Q8—22no—
136C2×C4×Q8—21no—
137C2×C4×Q8C2312yes—
138C2×C4×Q8C2314yes—
139C2×C4×Q8C2314yes—
140C2×#117—21no—
141C2×#117—24no—
142C2×#117—24no—
143C2×#117D412yes—
144C2×#117D412yes—
145C2×#117C2312yes—
146C2×#117C2312yes—
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⋊C4—21no—
155C22×C4⋊C4D411yescharacteristic
156C22×C4⋊C4D411yescharacteristic
157C22×C4⋊C4C2311yescharacteristic
158C22×C4⋊C4C2311yescharacteristic
159C22×C22⋊C4—21no—
160C22×C22⋊C4—21no—
161C22×C22⋊C4D411yescharacteristic
162C22×C22⋊C4D411yescharacteristic
163C22×C42—21no—
164C22×C42—21no—
165C22×C42C2311yescharacteristic
166C22×C42C2311yescharacteristic
167#501—28no—
168#494—28no—
169#491—28no—
170#487—28no—
171#484—28no—
172#483—28no—
173#479—216no—
174#462—216no—
175#451—28no—
176#451—28no—
177#432—24no—
178#431—24no—
179C2×#107C2314yes—
180C2×#106C2314yes—
181C2×C4⋊C8C2312yes—
182C2×C4⋊C8C2312yes—
183C2×#103C2314yes—
184C2×#103C2314yes—
185C2×#103C2314yes—
186C2×#103C2314yes—
187C2×#103C2318yes—
188C2×#103C2318yes—
189C2×C22⋊C8C2312yes—
190C2×C22⋊C8C2312yes—
191C2×C22⋊C8C2318yes—
192C2×#97C2312yes—
193C2×C4×C8C2312yes—
194#400—22no—
195#399—22no—
196#398—22no—
197#398—22no—
198#397—22no—
199#397—22no—
200#395—22no—
201#393—22no—
202#391C2314yes—
203#389—24no—
204#389C2314yes—
205#388—24no—
206#388—24no—
207#387—24no—
208#387—24no—
209#385—22no—
210#385—22no—
211#385—24no—
212#383—21no—
213#382C2312yes—
214#381—22no—
215#381—21no—
216#380C2314yes—
217#380C2312yes—
218#379—22no—
219C4×C4⋊C4C2312yes—
220C4×C22⋊C4—21no—
221#376—21no—
222#340C2318yes—
223#340C2318yes—
224#336C2318yes—
225C25C2×D411yescharacteristic
226C22×Q8—22no—
227C22×Q8—22no—
228C22×Q8—21no—
229C22×Q8—21no—
230C22×Q8—21no—
231C22×Q8—44no—
232C22×Q8—44no—
233C22×Q8C2×D414yes—
234C22×Q8C2×D414yes—
235C22×Q8C2412yes—
236C22×Q8C2412yes—
237C22×Q8C2412yes—
238C23×C4—21no—
239C23×C4—21no—
240C23×C4—21no—
241C23×C4—21no—
242C23×C4—22no—
243C23×C4—22no—
244C23×C4—44no—
245C23×C4C2×D411yescharacteristic
246C23×C4C2×D411yescharacteristic
247C23×C4C2×D411yescharacteristic
248C23×C4C2×D411yescharacteristic
249C23×C4C2×D412yes—
250C23×C4C2×D411yescharacteristic
251C23×C4C2412yes—
252C23×C4C2411yescharacteristic
253C2×Q16—28no—
254C2×Q16—28no—
255C2×Q16—48no—
256C2×Q16—48no—
257C22×C8C2414yes—
258C22×C8C2414yes—
259C4⋊Q8—28no—
260C4⋊Q8—28no—
261C4⋊Q8—24no—
262C4⋊Q8—24no—
263C4⋊Q8—22no—
264C4⋊Q8—22no—
265C4⋊Q8—22no—
266C4⋊Q8—22no—
267C4⋊Q8C2×D418yes—
268C4⋊Q8C2×D418yes—
269#126—232no—
270#126—232no—
271#126—232no—
272#126—232no—
273#125—232no—
274#125—232no—
275#125—232no—
276#125—232no—
277#125—28no—
278#125—28no—
279#125—24no—
280#125—24no—
281#125—44no—
282#125C2×D418yes—
283#125C2×D418yes—
284C22⋊Q8—216no—
285C22⋊Q8—216no—
286C22⋊Q8—28no—
287C22⋊Q8—28no—
288C22⋊Q8—28no—
289C22⋊Q8—28no—
290C22⋊Q8—24no—
291C22⋊Q8—24no—
292C22⋊Q8—48no—
293C22⋊Q8—44no—
294C4×Q8—216no—
295C4×Q8—216no—
296C4×Q8—28no—
297C4×Q8—28no—
298C4×Q8—28no—
299C4×Q8—28no—
300C4×Q8—24no—
301C4×Q8—24no—
302C4×Q8—44no—
303#117—28no—
304#117—28no—
305#117—24no—
306#117—24no—
307#117—24no—
308#117—24no—
309#117—44no—
310C2×C4⋊C4—28no—
311C2×C4⋊C4—28no—
312C2×C4⋊C4—28no—
313C2×C4⋊C4—28no—
314C2×C4⋊C4—28no—
315C2×C4⋊C4—28no—
316C2×C4⋊C4—28no—
317C2×C4⋊C4—28no—
318C2×C4⋊C4—24no—
319C2×C4⋊C4—24no—
320C2×C4⋊C4—24no—
321C2×C4⋊C4—24no—
322C2×C4⋊C4—24no—
323C2×C4⋊C4—24no—
324C2×C4⋊C4—22no—
325C2×C4⋊C4—22no—
326C2×C4⋊C4—22no—
327C2×C4⋊C4—22no—
328C2×C4⋊C4—22no—
329C2×C4⋊C4—22no—
330C2×C4⋊C4—22no—
331C2×C4⋊C4—22no—
332C2×C4⋊C4—21no—
333C2×C4⋊C4—21no—
334C2×C4⋊C4—22no—
335C2×C4⋊C4—22no—
336C2×C4⋊C4—21no—
337C2×C4⋊C4—21no—
338C2×C4⋊C4—21no—
339C2×C4⋊C4—22no—
340C2×C4⋊C4—22no—
341C2×C4⋊C4—24no—
342C2×C4⋊C4—24no—
343C2×C4⋊C4C2×D412yes—
344C2×C4⋊C4C2×D412yes—
345C2×C4⋊C4C2×D412yes—
346C2×C4⋊C4C2×D412yes—
347C2×C4⋊C4C2×D414yes—
348C2×C4⋊C4C2×D414yes—
349C2×C4⋊C4C2412yes—
350C2×C4⋊C4C2414yes—
351C2×C4⋊C4C2412yes—
352C2×C4⋊C4C2412yes—
353C2×C22⋊C4—28no—
354C2×C22⋊C4—28no—
355C2×C22⋊C4—24no—
356C2×C22⋊C4—24no—
357C2×C22⋊C4—24no—
358C2×C22⋊C4—24no—
359C2×C22⋊C4—22no—
360C2×C22⋊C4—24no—
361C2×C22⋊C4—22no—
362C2×C22⋊C4—24no—
363C2×C22⋊C4C2×D412yes—
364C2×C22⋊C4C2×D412yes—
365C2×C22⋊C4C2×D412yes—
366C2×C22⋊C4C2×D412yes—
367C2×C42—28no—
368C2×C42—28no—
369C2×C42—22no—
370C2×C42—22no—
371C2×C42—24no—
372C2×C42—24no—
373C2×C42—22no—
374C2×C42—22no—
375C2×C42C2×D412yes—
376C2×C42C2×D411yescharacteristic
377C2×C42C2×D411yescharacteristic
378C2×C42C2×D414yes—
379C2×C42C2×D414yes—
380C2×C42C2412yes—
381C2×C42C2411yescharacteristic
382C2×C42C2411yescharacteristic
383#107—28no—
384#106—28no—
385C4⋊C8—24no—
386C4⋊C8—24no—
387#103—28no—
388#103—28no—
389#103—28no—
390#103—28no—
391#103—48no—
392#103—48no—
393C22⋊C8—24no—
394C22⋊C8—24no—
395C22⋊C8—48no—
396#97—24no—
397C4×C8—24no—
398#95—22no—
399#95—22no—
400#95—22no—
401#95—22no—
402#95—24no—
403#95—24no—
404#95—24no—
405#95—24no—
406C24—22no—
407C24—24no—
408C24—42no—
409C24—42no—
410C24C22×D412yes—
411C24C22×D411yescharacteristic
412C2×Q8—24no—
413C2×Q8—24no—
414C2×Q8—22no—
415C2×Q8—22no—
416C2×Q8—22no—
417C2×Q8—22no—
418C2×Q8—28no—
419C2×Q8—28no—
420C2×Q8—28no—
421C2×Q8—48no—
422C2×Q8—48no—
423C2×Q8—44no—
424C2×Q8—44no—
425C2×Q8—48no—
426C2×Q8—48no—
427C2×Q8—42no—
428C2×Q8—82no—
429C2×Q8C22×D414yes—
430C2×Q8C22×D414yes—
431C22×C4—24no—
432C22×C4—24no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—24no—
436C22×C4—24no—
437C22×C4—24no—
438C22×C4—24no—
439C22×C4—22no—
440C22×C4—24no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—24no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—24no—
449C22×C4—24no—
450C22×C4—24no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—42no—
454C22×C4—42no—
455C22×C4—42no—
456C22×C4—42no—
457C22×C4—42no—
458C22×C4—48no—
459C22×C4—48no—
460C22×C4—44no—
461C22×C4—44no—
462C22×C4—44no—
463C22×C4—44no—
464C22×C4#12014yes—
465C22×C4#12014yes—
466C22×C4C22×D412yes—
467C22×C4C22×D412yes—
468C22×C4C22×D412yes—
469C22×C4C22×D411yescharacteristic
470C22×C4C22×D411yescharacteristic
471C22×C4C22×D411yescharacteristic
472C22×C4C22×D411yescharacteristic
473C22×C4C22×D411yescharacteristic
474C22×C4C22×D412yes—
475C22×C4C22×D412yes—
476C22×C4C2511yescharacteristic
477Q16—88no—
478Q16—88no—
479C2×C8—24no—
480C2×C8—24no—
481C2×C8—28no—
482C2×C8—28no—
483C4⋊C4—216no—
484C4⋊C4—216no—
485C4⋊C4—216no—
486C4⋊C4—216no—
487C4⋊C4—28no—
488C4⋊C4—28no—
489C4⋊C4—28no—
490C4⋊C4—28no—
491C4⋊C4—28no—
492C4⋊C4—28no—
493C4⋊C4—28no—
494C4⋊C4—28no—
495C4⋊C4—24no—
496C4⋊C4—24no—
497C4⋊C4—24no—
498C4⋊C4—24no—
499C4⋊C4—24no—
500C4⋊C4—24no—
501C4⋊C4—22no—
502C4⋊C4—22no—
503C4⋊C4—22no—
504C4⋊C4—22no—
505C4⋊C4—28no—
506C4⋊C4—48no—
507C4⋊C4—42no—
508C4⋊C4—42no—
509C4⋊C4—44no—
510C4⋊C4—44no—
511C4⋊C4—42no—
512C4⋊C4—84no—
513C4⋊C4C22×D414yes—
514C4⋊C4C22×D414yes—
515C22⋊C4—216no—
516C22⋊C4—216no—
517C22⋊C4—24no—
518C22⋊C4—24no—
519C22⋊C4—24no—
520C22⋊C4—24no—
521C22⋊C4—48no—
522C22⋊C4—44no—
523C22⋊C4—44no—
524C22⋊C4—84no—
525C42—216no—
526C42—216no—
527C42—24no—
528C42—22no—
529C42—22no—
530C42—44no—
531C42—82no—
532C42#12014yes—
533C42#12014yes—
534C42C22×D412yes—
535C42C22×D412yes—
536C42C22×D412yes—
537C42C22×D412yes—
538C23—22no—
539C23—22no—
540C23—24no—
541C23—22no—
542C23—22no—
543C23—22no—
544C23—24no—
545C23—44no—
546C23—44no—
547C23—44no—
548C23—88no—
549C23C2×#12012yes—
550C23#57814yes—
551C23C23×D411yescharacteristic
552Q8—28no—
553Q8—28no—
554Q8—48no—
555Q8—48no—
556Q8—44no—
557Q8—44no—
558Q8—44no—
559Q8—44no—
560Q8—84no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—28no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—24no—
568C2×C4—24no—
569C2×C4—24no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—24no—
573C2×C4—24no—
574C2×C4—24no—
575C2×C4—24no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—24no—
580C2×C4—44no—
581C2×C4—48no—
582C2×C4—44no—
583C2×C4—48no—
584C2×C4—44no—
585C2×C4—48no—
586C2×C4—48no—
587C2×C4—44no—
588C2×C4—44no—
589C2×C4—44no—
590C2×C4—44no—
591C2×C4—48no—
592C2×C4—44no—
593C2×C4—44no—
594C2×C4—44no—
595C2×C4—44no—
596C2×C4—44no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—44no—
600C2×C4—44no—
601C2×C4—44no—
602C2×C4C2×#12012yes—
603C2×C4C2×#12012yes—
604C2×C4C2×#12012yes—
605C2×C4C2×#12012yes—
606C2×C4C2×#12012yes—
607C2×C4C23×D411yescharacteristic
608C2×C4C23×D411yescharacteristic
609C2×C4#58514yes—
610C8—44no—
611C8—44no—
612C22—28no—
613C22—22no—
614C22—22no—
615C22—44no—
616C22—44no—
617C22—42no—
618C22—42no—
619C22—42no—
620C22—88no—
621C22#301114yes—
622C22C22×#12011yescharacteristic
623C22#344912yes—
624C22C2×#57812yes—
625C22#358612yes—
626C4—22no—
627C4—22no—
628C4—22no—
629C4—22no—
630C4—22no—
631C4—42no—
632C4—44no—
633C4—44no—
634C4—42no—
635C4—44no—
636C4—42no—
637C4—44no—
638C4—84no—
639C4—82no—
640C4—82no—
641C4#348512yes—
642C4#348712yes—
643C2—24no—
644C2—82no—
645C2#3514911yescharacteristic
646C2C2×#301111yescharacteristic
647C2#6036811yescharacteristic
648C2#6098314yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#3236C2semidirect
3. C2×#3209C2semidirect
5. #33010C2semidirect
6. #33008C2semidirect
9. #25686C2semidirect
10. #25609C2semidirect
11. #24165C2semidirect
12. #21836C2semidirect
13. #21811C2semidirect
35. C2×#501C22semidirect
36. C2×#494C22semidirect
37. C2×#491C22semidirect
38. C2×#487C22semidirect
41. C2×#479C22semidirect
42. C2×#462C22semidirect
63. #1991C22semidirect
64. #1989C22semidirect
65. #1981C22semidirect
66. #1960C22semidirect
67. #1878C22semidirect
68. #1878C22semidirect
69. #1865C22semidirect
70. #1863C22semidirect

Maximal subgroups

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

subgroupindex
C22×#5412
C2×#32362
C2×#32092
#353252
#330102
#330082
#295992
#295972
#256862
#256092
#241652
#218362
#218112
#207172