Finite Group Databasebeta

← #6513547

Subgroups — #6513547

Order 512.

2028 conjugacy classes of subgroups.

The subgroup lattice has 655 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3113C211yesmaximal, characteristic, complemented
2C2×#2981C211yesmaximal, characteristic
3#37640C211yesmaximal, characteristic
4#35178C211yesmaximal, characteristic
5#28238C214yesmaximal, complemented
6#28153C214yesmaximal, complemented
7#25394C212yesmaximal, complemented
8#25386C212yesmaximal, complemented
9#25348C212yesmaximal, complemented
10#25341C212yesmaximal, complemented
11#24566C214yesmaximal, complemented
12#24392C214yesmaximal, complemented
13#21988C211yesmaximal, characteristic, complemented
14#21864C211yesmaximal, characteristic, complemented
15#20734C211yesmaximal, characteristic, complemented
16C2×#536C2212yes—
17C2×#533C2211yescharacteristic
18C2×#532C2211yescharacteristic
19C22×C4×Q8C2211yescharacteristic
20#3113—28no—
21C2×#483C2212yes—
22C2×#475C2212yes—
23C2×#470C2214yes—
24C2×#467C2214yes—
25C2×#462C2212yes—
26C2×#451C2212yes—
27#2981—28no—
28C2×#441C2214yes—
29C2×#436C2212yes—
30C2×#435C2214yes—
31C2×C8×D4C2212yes—
32C2×#432C2211yescharacteristic
33C2×#431C2211yescharacteristic
34C2×#408C2214yes—
35#2854—22no—
36#2820—22no—
37#2624—22no—
38#2620—22no—
39#2618—22no—
40#2616C2214yes—
41#2567—21no—
42#2548—21no—
43#2548—21no—
44#2547C2212yes—
45#2546—21no—
46#2545—22no—
47#2542C2212yes—
48#2538C2212yes—
49#2538C2212yes—
50#2538C2212yes—
51#2538C2212yes—
52#2536C2214yes—
53#2311C2212yes—
54#2285C2211yescharacteristic
55#2285C2211yescharacteristic
56#2077C2214yes—
57#2077C2214yes—
58#2036C2214yes—
59#2036C2214yes—
60#2035C2214yes—
61#2035C2214yes—
62#2001C2214yes—
63#2001C2214yes—
64#1990C2211yescharacteristic
65#1990C2211yescharacteristic
66#1956C2211yescharacteristic
67#1956C2211yescharacteristic
68#1948C2212yes—
69#1947C2212yes—
70#1894C2214yes—
71#1894C2214yes—
72#1892C2214yes—
73#1892C2214yes—
74#1873C2214yes—
75#1854C2214yes—
76#1847C2212yes—
77#1836C2212yes—
78#1824C2212yes—
79#1809C2214yes—
80#1807C2214yes—
81#1805C2214yes—
82#1803C2212yes—
83#1803C2212yes—
84#1787C2212yes—
85C4×#103C2212yes—
86C4×#103C2212yes—
87C22×#41C2312yes—
88C23×Q8C2311yescharacteristic
89C22×Q16C2314yes—
90C22×M16C2314yes—
91C23×C8C2314yes—
92#536—48no—
93#533—24no—
94#533—24no—
95#532—24no—
96#532—24no—
97C2×C4⋊Q8—22no—
98C2×C4⋊Q8D411yescharacteristic
99C2×C4⋊Q8C2311yescharacteristic
100C2×C4⋊Q8C2311yescharacteristic
101C2×#127D411yescharacteristic
102C2×#125—21no—
103C2×#124—22no—
104C2×#124—22no—
105C2×#124D412yes—
106C2×#123—24no—
107C2×C22⋊Q8—21no—
108C2×C22⋊Q8C2312yes—
109C2×C22⋊Q8C2314yes—
110C2×C22⋊Q8C2314yes—
111C2×C22⋊Q8C2314yes—
112C2×C22⋊Q8C2314yes—
113C2×#121D412yes—
114C2×#121D412yes—
115C2×C4×Q8—21no—
116C2×C4×Q8—21no—
117C2×C4×Q8—21no—
118C2×C4×Q8—24no—
119C2×C4×Q8—24no—
120C2×C4×Q8C2312yes—
121C2×C4×Q8C2312yes—
122C2×C4×Q8C2312yes—
123C2×C4×D4C2312yes—
124C2×C4×D4C2312yes—
125C2×#117—21no—
126C2×#117—21no—
127C2×#117C2312yes—
128C22×C4⋊C4—21no—
129C22×C4⋊C4C2311yescharacteristic
130C22×C42—21no—
131C22×C42C2311yescharacteristic
132#483—28no—
133#475—28no—
134#470—48no—
135#467—48no—
136#462—28no—
137#451—28no—
138#441—216no—
139#436—28no—
140#435—216no—
141C8×D4—28no—
142#432D418yes—
143#431—24no—
144C2×#107C2314yes—
145C2×#106C2314yes—
146C2×#106C2314yes—
147C2×C4⋊C8C2312yes—
148C2×C4⋊C8C2312yes—
149C2×#103C2314yes—
150C2×#103C2312yes—
151C2×#103C2312yes—
152C2×#103C2314yes—
153C2×#103C2312yes—
154C2×#103C2312yes—
155C2×#103C2314yes—
156C2×#103C2314yes—
157#408—216no—
158C2×C22⋊C8C2314yes—
159C2×C22⋊C8C2314yes—
160C2×C22⋊C8C2312yes—
161C2×C22⋊C8C2312yes—
162C2×#97C2312yes—
163C2×C4×C8C2311yescharacteristic
164C2×C4×C8C2311yescharacteristic
165#399—24no—
166#398—24no—
167#397—24no—
168#396—24no—
169#396—24no—
170#396—24no—
171#396—24no—
172#395—24no—
173#394—24no—
174#394—24no—
175#394—24no—
176#394—24no—
177#391—22no—
178#391—22no—
179#390—22no—
180#390—22no—
181#390—22no—
182#390—22no—
183#390—22no—
184#390—22no—
185#389—22no—
186#389—22no—
187#389—22no—
188#389—22no—
189#388—24no—
190#388—24no—
191#386—24no—
192#384—22no—
193#384C2311yescharacteristic
194#384C2311yescharacteristic
195#384C2311yescharacteristic
196#384C2311yescharacteristic
197#382—21no—
198#382C2312yes—
199#380—21no—
200#380—21no—
201#380—21no—
202#380C2314yes—
203#380C2314yes—
204#380C2314yes—
205#380C2314yes—
206#380C2312yes—
207C4×C4⋊C4—21no—
208C4×C4⋊C4—21no—
209C4×C4⋊C4—21no—
210C4×C4⋊C4C2312yes—
211#340C2312yes—
212#340C2312yes—
213#340C2312yes—
214#340C2312yes—
215#340C2314yes—
216#340C2314yes—
217#340C2314yes—
218#340C2314yes—
219C2×#41—24no—
220C2×#41—24no—
221C2×#41—44no—
222C2×#41—44no—
223C22×Q8—24no—
224C22×Q8—21no—
225C22×Q8—21no—
226C22×Q8—22no—
227C22×Q8—22no—
228C22×Q8—42no—
229C22×Q8C2×D412yes—
230C22×Q8C2412yes—
231C22×Q8C2412yes—
232C22×D4C2×D412yes—
233C22×D4C2×D412yes—
234C22×D4C2×D414yes—
235C22×D4C2×D412yes—
236C23×C4—21no—
237C23×C4—21no—
238C23×C4C2×D411yescharacteristic
239C23×C4C2×D411yescharacteristic
240C23×C4C2414yes—
241C23×C4C2411yescharacteristic
242C2×Q16—28no—
243C2×Q16—28no—
244C2×Q16—44no—
245C2×M16—28no—
246C2×M16—44no—
247C2×M16—44no—
248C22×C8—28no—
249C22×C8—44no—
250C22×C8—44no—
251C22×C8C2412yes—
252C22×C8C2412yes—
253C22×C8C2414yes—
254C4⋊Q8—24no—
255C4⋊Q8—22no—
256C4⋊Q8—22no—
257C4⋊Q8—22no—
258C4⋊Q8—22no—
259C4⋊Q8—48no—
260#127—24no—
261#125—44no—
262#124—28no—
263#124—48no—
264#124—84no—
265#123—416no—
266C22⋊Q8—28no—
267C22⋊Q8—28no—
268C22⋊Q8—24no—
269C22⋊Q8—24no—
270C22⋊Q8—44no—
271C22⋊Q8—44no—
272C22⋊Q8—44no—
273C22⋊Q8—44no—
274C22⋊Q8—44no—
275C22⋊Q8—44no—
276C22⋊Q8—44no—
277#121—28no—
278#121—28no—
279C4×Q8—24no—
280C4×Q8—24no—
281C4×Q8—24no—
282C4×Q8—24no—
283C4×Q8—24no—
284C4×Q8—24no—
285C4×Q8—44no—
286C4×Q8—44no—
287C4×Q8—44no—
288C4×D4—24no—
289C4×D4—24no—
290C4×D4—24no—
291C4×D4—24no—
292#117—44no—
293#117—44no—
294#117—82no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—24no—
306C2×C4⋊C4—24no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—21no—
309C2×C4⋊C4—21no—
310C2×C4⋊C4—21no—
311C2×C4⋊C4—21no—
312C2×C4⋊C4—21no—
313C2×C4⋊C4—24no—
314C2×C4⋊C4—24no—
315C2×C4⋊C4—21no—
316C2×C4⋊C4—21no—
317C2×C4⋊C4—44no—
318C2×C4⋊C4C2×D412yes—
319C2×C4⋊C4C2×D412yes—
320C2×C4⋊C4C2412yes—
321C2×C4⋊C4C2412yes—
322C2×C4⋊C4C2412yes—
323C2×C4⋊C4C2412yes—
324C2×C4⋊C4C2412yes—
325C2×C4⋊C4C2412yes—
326C2×C22⋊C4—22no—
327C2×C22⋊C4—22no—
328C2×C22⋊C4—24no—
329C2×C22⋊C4—24no—
330C2×C22⋊C4—22no—
331C2×C22⋊C4—24no—
332C2×C22⋊C4—24no—
333C2×C22⋊C4—24no—
334C2×C22⋊C4—24no—
335C2×C22⋊C4—24no—
336C2×C22⋊C4—24no—
337C2×C22⋊C4C2×D414yes—
338C2×C22⋊C4C2×D414yes—
339C2×C42—22no—
340C2×C42—22no—
341C2×C42—22no—
342C2×C42—22no—
343C2×C42—22no—
344C2×C42—44no—
345C2×C42C2×D411yescharacteristic
346C2×C42C2×D411yescharacteristic
347C2×C42C2×D414yes—
348C2×C42#4114yes—
349C2×C42C2411yescharacteristic
350C2×C42C2411yescharacteristic
351#107—44no—
352#106—28no—
353#106—44no—
354C4⋊C8—24no—
355C4⋊C8C2×D418yes—
356#103—24no—
357#103—24no—
358#103—28no—
359#103—28no—
360#103—42no—
361#103—42no—
362#103—44no—
363#103—44no—
364C22⋊C8—24no—
365C22⋊C8—28no—
366C22⋊C8—28no—
367C22⋊C8C2×D418yes—
368#97C2×D418yes—
369C4×C8—22no—
370C4×C8—22no—
371#95—22no—
372#95—22no—
373#95—22no—
374#95—22no—
375#95—24no—
376#95—24no—
377#95—24no—
378#95—24no—
379C24C22×D414yes—
380C24C22×D411yescharacteristic
381#41—88no—
382#41—88no—
383C2×Q8—24no—
384C2×Q8—24no—
385C2×Q8—22no—
386C2×Q8—22no—
387C2×Q8—22no—
388C2×Q8—22no—
389C2×Q8—22no—
390C2×Q8—24no—
391C2×Q8—22no—
392C2×Q8—22no—
393C2×Q8—24no—
394C2×Q8—42no—
395C2×Q8—42no—
396C2×Q8—48no—
397C2×Q8—48no—
398C2×Q8—42no—
399C2×Q8—42no—
400C2×Q8—42no—
401C2×Q8—42no—
402C2×Q8—42no—
403C2×Q8H(5,𝔽2)14yes—
404C2×D4—28no—
405C2×D4—24no—
406C2×D4—24no—
407C2×D4—24no—
408C2×D4—24no—
409C2×D4—24no—
410C2×D4—44no—
411C2×D4—42no—
412C2×D4—42no—
413C2×D4—42no—
414C2×D4—42no—
415C2×D4—44no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—24no—
421C22×C4—22no—
422C22×C4—24no—
423C22×C4—24no—
424C22×C4—24no—
425C22×C4—24no—
426C22×C4—24no—
427C22×C4—24no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—41no—
436C22×C4—41no—
437C22×C4—42no—
438C22×C4—42no—
439C22×C4—42no—
440C22×C4—81no—
441C22×C4—81no—
442C22×C4#12114yes—
443C22×C4#12114yes—
444C22×C4C22×D414yes—
445C22×C4C22×D411yescharacteristic
446C22×C4C22×D411yescharacteristic
447C22×C4C22×D411yescharacteristic
448C22×C4C22×D411yescharacteristic
449C22×C4C22×D412yes—
450C22×C4C22×D411yescharacteristic
451C22×C4C2×#4112yes—
452C22×C4C2511yescharacteristic
453Q16—416no—
454M16—88no—
455C2×C8—24no—
456C2×C8—24no—
457C2×C8—22no—
458C2×C8—22no—
459C2×C8—22no—
460C2×C8—22no—
461C2×C8—22no—
462C2×C8—22no—
463C2×C8—88no—
464C2×C8C22×D418yes—
465C4⋊C4—24no—
466C4⋊C4—24no—
467C4⋊C4—22no—
468C4⋊C4—22no—
469C4⋊C4—22no—
470C4⋊C4—22no—
471C4⋊C4—22no—
472C4⋊C4—22no—
473C4⋊C4—24no—
474C4⋊C4—24no—
475C4⋊C4—22no—
476C4⋊C4—22no—
477C4⋊C4—48no—
478C4⋊C4—48no—
479C4⋊C4—42no—
480C4⋊C4—42no—
481C4⋊C4—42no—
482C4⋊C4—44no—
483C4⋊C4—44no—
484C4⋊C4—42no—
485C4⋊C4—42no—
486C4⋊C4—44no—
487C4⋊C4—44no—
488C4⋊C4—42no—
489C4⋊C4—42no—
490C4⋊C4—42no—
491C4⋊C4—81no—
492C4⋊C4—81no—
493C4⋊C4—81no—
494C4⋊C4—81no—
495C4⋊C4—84no—
496C4⋊C4—84no—
497C22⋊C4—28no—
498C22⋊C4—28no—
499C22⋊C4—48no—
500C22⋊C4—48no—
501C22⋊C4—44no—
502C22⋊C4—44no—
503C22⋊C4—84no—
504C22⋊C4—84no—
505C22⋊C4—82no—
506C22⋊C4—84no—
507C22⋊C4—84no—
508C22⋊C4—84no—
509C22⋊C4—84no—
510C42—44no—
511C42—44no—
512C42—44no—
513C42—44no—
514C42—44no—
515C42#12114yes—
516C42#12114yes—
517C42C22×D412yes—
518C42C22×D412yes—
519C42C2×#4112yes—
520C42C2×#4112yes—
521C23—21no—
522C23—21no—
523C23—21no—
524C23—21no—
525C23—21no—
526C23—21no—
527C23—24no—
528C23—24no—
529C23—24no—
530C23—24no—
531C23—24no—
532C23—24no—
533C23—24no—
534C23C2×#12112yes—
535C23C23×D411yescharacteristic
536Q8—24no—
537Q8—24no—
538Q8—48no—
539Q8—42no—
540Q8—42no—
541Q8—44no—
542Q8—44no—
543Q8—84no—
544Q8—84no—
545D4—48no—
546D4—84no—
547D4—84no—
548D4—88no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—24no—
558C2×C4—24no—
559C2×C4—28no—
560C2×C4—24no—
561C2×C4—24no—
562C2×C4—24no—
563C2×C4—24no—
564C2×C4—44no—
565C2×C4—48no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—44no—
569C2×C4—44no—
570C2×C4—44no—
571C2×C4—44no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—84no—
594C2×C4C2×#12112yes—
595C2×C4C2×#12112yes—
596C2×C4C2×#12112yes—
597C2×C4C2×#12112yes—
598C2×C4C2×#12112yes—
599C2×C4#57712yes—
600C2×C4#57812yes—
601C2×C4C23×D411yescharacteristic
602C2×C4C22×#4111yescharacteristic
603C2×C4C2×H(5,𝔽2)12yes—
604C2×C4#58512yes—
605C8—28no—
606C8—42no—
607C8—42no—
608C22—22no—
609C22—22no—
610C22—22no—
611C22—41no—
612C22—41no—
613C22—41no—
614C22—41no—
615C22—44no—
616C22—44no—
617C22—44no—
618C22—44no—
619C22—44no—
620C22—44no—
621C22—44no—
622C22C22×#12111yescharacteristic
623C22#348411yescharacteristic
624C22#348511yescharacteristic
625C22C2×#57711yescharacteristic
626C22C2×#57811yescharacteristic
627C22#358611yescharacteristic
628C22#358611yescharacteristic
629C4—22no—
630C4—22no—
631C4—22no—
632C4—22no—
633C4—44no—
634C4—44no—
635C4—82no—
636C4—82no—
637C4—82no—
638C4—82no—
639C4—82no—
640C4—82no—
641C4—82no—
642C4#306314yes—
643C4#348812yes—
644C4#349412yes—
645C2—42no—
646C2—84no—
647C2#3551411yescharacteristic
648C2C2×#306311yescharacteristic
649C2#6045511yescharacteristic
650C2#6168311yescharacteristic
651C2#6168611yescharacteristic
652C2#6180611yescharacteristic
653C2#6180911yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#532C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×#532 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#3113C2semidirect
5. #28238C2semidirect
6. #28153C2semidirect
7. #25394C2semidirect
8. #25386C2semidirect
9. #25348C2semidirect
10. #25341C2semidirect
11. #24566C2semidirect
12. #24392C2semidirect
13. #21988C2semidirect
14. #21864C2semidirect
15. #20734C2semidirect

Maximal subgroups

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

subgroupindex
C2×#31132
C2×#29812
#376402
#351782
#282382
#281532
#253942
#253862
#253482
#253412
#245662
#243922
#219882
#218642
#207342