Finite Group Databasebeta

← #6433212

Subgroups — #6433212

Order 512.

2339 conjugacy classes of subgroups.

The subgroup lattice has 670 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3092C211yesmaximal, characteristic, complemented
2C22×#481C211yesmaximal, characteristic, complemented
3#35252C211yesmaximal, characteristic
4C2×#2316C211yesmaximal, characteristic, complemented
5#29768C214yesmaximal, complemented
6#29763C214yesmaximal, complemented
7#29761C214yesmaximal, complemented
8#29755C214yesmaximal, complemented
9#23661C212yesmaximal, complemented
10#23614C212yesmaximal, complemented
11#23610C212yesmaximal, complemented
12#23511C212yesmaximal, complemented
13#20912C211yesmaximal, characteristic, complemented
14#20908C211yesmaximal, characteristic, complemented
15#20628C211yesmaximal, characteristic, complemented
16C2×#535C2211yescharacteristic
17C2×#525C2211yescharacteristic
18C22×#121C2211yescharacteristic
19C22×#117C2211yescharacteristic
20C22×#117C2211yescharacteristic
21C23×C4⋊C4C2211yescharacteristic
22#3092—28no—
23C2×#484C2212yescomplemented
24C2×#481—28no—
25C2×#481C2214yes—
26C2×#481C2214yes—
27C2×#481C2212yescomplemented
28C2×#483C2212yescomplemented
29C2×#480C2212yescomplemented
30C2×#428C2212yes—
31C22×#106C2212yes—
32C2×#424C2212yes—
33C2×#417C2212yes—
34C2×#418C2212yes—
35C2×#416C2212yes—
36C22×#102C2212yes—
37C2×#407C2211yescharacteristic
38C22×C22⋊C8C2211yescharacteristic
39#2856C2212yes—
40#2850C2212yes—
41#2620C2212yes—
42#2619C2212yes—
43#2416—22no—
44#2414—22no—
45#2411—22no—
46#2410—22no—
47#2317C2211yescharacteristic
48#2316—22no—
49#2316—22no—
50#2316C2212yes—
51#2316C2212yes—
52#2316C2214yescomplemented
53#2311C2211yescharacteristic
54#2309C2211yescharacteristic
55C2×#389—22no—
56C2×#389C2214yescomplemented
57C2×#380—22no—
58#2085C2218yescomplemented
59#2084C2218yescomplemented
60#2067C2218yescomplemented
61#2066C2218yescomplemented
62#2065C2218yescomplemented
63#2065C2218yescomplemented
64#2042C2218yescomplemented
65#2041C2218yescomplemented
66#1875C2214yes—
67#1875C2212yescomplemented
68#1874C2214yes—
69#1874C2212yescomplemented
70#1873C2212yescomplemented
71#1872C2212yescomplemented
72#1798C2214yescomplemented
73#1797C2214yescomplemented
74#1736C2212yes—
75#1735C2212yes—
76#1729C2212yes—
77C22×#41C2311yescharacteristic
78C23×D4—21no—
79C23×D4C2311yescharacteristic
80C24×C4—21no—
81C24×C4C2311yescharacteristic
82C22×M16C2312yes—
83C23×C8C2312yes—
84#535—44no—
85C2×#127C2312yes—
86C2×#124C2312yes—
87#525—24no—
88#525—24no—
89C2×#123—21no—
90C2×C22⋊Q8C2311yescharacteristic
91C2×C22⋊Q8C2311yescharacteristic
92C2×#121—21no—
93C2×#121—21no—
94C2×#121—24no—
95C2×#121—22no—
96C2×#121—24no—
97C2×#121—24no—
98C2×#121C2311yescharacteristic
99C2×#121C2312yes—
100C2×#121C2312yes—
101C2×#121C2311yescharacteristic
102C2×#120—21no—
103C2×#120—22no—
104C2×C4×D4—21no—
105C2×C4×D4—21no—
106C2×#117—21no—
107C2×#117—21no—
108C2×#117—44no—
109C2×#117—44no—
110C2×#117D412yes—
111C2×#117D412yes—
112C2×#117C2311yescharacteristic
113C2×#117C2312yes—
114C2×#117C2312yes—
115C2×#117C2311yescharacteristic
116C2×#117C2311yescharacteristic
117C2×#117C2312yes—
118C22×C4⋊C4—21no—
119C22×C4⋊C4—21no—
120C22×C4⋊C4—21no—
121C22×C4⋊C4—24no—
122C22×C4⋊C4—24no—
123C22×C4⋊C4D412yes—
124C22×C4⋊C4D418yescomplemented
125C22×C4⋊C4C2311yescharacteristic
126C22×C4⋊C4C2312yes—
127C22×C4⋊C4C2312yes—
128C22×C4⋊C4C2312yes—
129C22×C4⋊C4C2312yes—
130C22×C22⋊C4—21no—
131C22×C22⋊C4—21no—
132C22×C22⋊C4D412yes—
133C22×C42—21no—
134C22×C42C2312yes—
135#484—28no—
136#483—28no—
137#481—216no—
138#481—216no—
139#481—28no—
140#480—28no—
141#428—28no—
142C2×#107C2314yes—
143C2×#106—28no—
144C2×#106C2314yes—
145C2×#106C2318yes—
146#424—44no—
147C2×C4⋊C8C2318yes—
148#418—44no—
149#417—28no—
150#416—44no—
151C2×#103C2314yes—
152C2×#103C2314yes—
153C2×#102—28no—
154C2×#102C2314yes—
155C2×#102C2318yes—
156C2×#102C2314yes—
157C2×#102C2318yes—
158#407—24no—
159C2×C22⋊C8—24no—
160C2×C22⋊C8C2312yes—
161C2×C22⋊C8C2312yes—
162C2×C22⋊C8C2312yes—
163C2×C22⋊C8C2312yes—
164#396—24no—
165#394—24no—
166#394—24no—
167#392—24no—
168#391C2312yes—
169#390—22no—
170#390—24no—
171#390C2312yes—
172#390C2312yes—
173#390C2312yes—
174#389—24no—
175#389—24no—
176#389—28no—
177#389—22no—
178#389—24no—
179#389—24no—
180#389C2318yes—
181#389C2318yes—
182#388—24no—
183#386—24no—
184#386—22no—
185#385—22no—
186#384C2312yes—
187#382C2312yes—
188#380—22no—
189#380—24no—
190#380—24no—
191#380—28no—
192#379—22no—
193C2×#95—24no—
194#340C2314yes—
195#340C2318yes—
196#340C2314yes—
197#336C2318yes—
198C25—21no—
199C25C2×D411yescharacteristic
200C2×#41—22no—
201C2×#41—22no—
202C2×#41—24no—
203C2×#41—24no—
204C22×Q8C2411yescharacteristic
205C22×D4—21no—
206C22×D4—21no—
207C22×D4—22no—
208C22×D4—21no—
209C22×D4—22no—
210C22×D4—22no—
211C22×D4—21no—
212C22×D4—24no—
213C22×D4—22no—
214C22×D4—24no—
215C22×D4—24no—
216C22×D4—44no—
217C22×D4—84no—
218C22×D4C2412yes—
219C22×D4C2411yescharacteristic
220C23×C4—24no—
221C23×C4—24no—
222C23×C4—22no—
223C23×C4—28no—
224C23×C4—22no—
225C23×C4—22no—
226C23×C4—22no—
227C23×C4—21no—
228C23×C4—21no—
229C23×C4—84no—
230C23×C4C2×D411yescharacteristic
231C23×C4C2×D411yescharacteristic
232C23×C4C2×D411yescharacteristic
233C23×C4C2×D414yes—
234C23×C4C2×D414yes—
235C23×C4#4114yes—
236C23×C4C2411yescharacteristic
237C23×C4C2411yescharacteristic
238C23×C4C2411yescharacteristic
239C2×M16—24no—
240C2×M16—24no—
241C2×M16—24no—
242C22×C8—24no—
243C22×C8—24no—
244C22×C8—24no—
245C22×C8C2414yes—
246C22×C8C2414yes—
247#127—44no—
248#124—44no—
249#123—44no—
250C22⋊Q8—22no—
251C22⋊Q8—22no—
252C22⋊Q8—22no—
253C22⋊Q8—22no—
254#121—24no—
255#121—24no—
256#121—22no—
257#121—22no—
258#121—24no—
259#121—24no—
260#121—22no—
261#121—22no—
262#121—48no—
263#121—44no—
264#121—44no—
265#121—88no—
266#120—48no—
267#120—44no—
268C4×D4—44no—
269C4×D4—44no—
270#117—24no—
271#117—24no—
272#117—22no—
273#117—22no—
274#117—24no—
275#117—24no—
276#117—44no—
277#117—44no—
278#117—44no—
279#117—44no—
280#117—44no—
281#117—44no—
282C2×C4⋊C4—28no—
283C2×C4⋊C4—28no—
284C2×C4⋊C4—28no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—24no—
289C2×C4⋊C4—24no—
290C2×C4⋊C4—24no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—28no—
298C2×C4⋊C4—28no—
299C2×C4⋊C4—24no—
300C2×C4⋊C4—44no—
301C2×C4⋊C4—44no—
302C2×C4⋊C4—88no—
303C2×C4⋊C4C2×D412yes—
304C2×C4⋊C4C2×D412yes—
305C2×C4⋊C4C2×D418yes—
306C2×C4⋊C4C2×D418yes—
307C2×C4⋊C4C2×D418yes—
308C2×C4⋊C4C2×D414yes—
309C2×C4⋊C4C2414yes—
310C2×C4⋊C4C2412yes—
311C2×C4⋊C4C2411yescharacteristic
312C2×C4⋊C4C2411yescharacteristic
313C2×C4⋊C4C2414yes—
314C2×C22⋊C4—22no—
315C2×C22⋊C4—22no—
316C2×C22⋊C4—21no—
317C2×C22⋊C4—21no—
318C2×C22⋊C4—21no—
319C2×C22⋊C4—22no—
320C2×C22⋊C4—22no—
321C2×C22⋊C4—21no—
322C2×C22⋊C4—21no—
323C2×C22⋊C4—24no—
324C2×C22⋊C4—24no—
325C2×C22⋊C4—24no—
326C2×C22⋊C4—24no—
327C2×C22⋊C4—44no—
328C2×C22⋊C4—44no—
329C2×C22⋊C4—44no—
330C2×C22⋊C4C2×D412yes—
331C2×C22⋊C4C2×D412yes—
332C2×C22⋊C4C2×D414yes—
333C2×C42—22no—
334C2×C42—22no—
335C2×C42—21no—
336C2×C42—44no—
337C2×C42—44no—
338C2×C42C2×D414yes—
339C2×C42C2412yes—
340C2×C42C2412yes—
341#107—28no—
342#106—216no—
343#106—28no—
344C4⋊C8—48no—
345#103—28no—
346#103—44no—
347#102—216no—
348#102—28no—
349#102—44no—
350#102—48no—
351C22⋊C8—24no—
352C22⋊C8—24no—
353C22⋊C8—24no—
354C22⋊C8—24no—
355#95—28no—
356#95—28no—
357#95—24no—
358#95—24no—
359C24—22no—
360C24—22no—
361C24—24no—
362C24—22no—
363C24—21no—
364C24—42no—
365C24—42no—
366C24—42no—
367C24—42no—
368C24—82no—
369C24—82no—
370C24#12114yes—
371C24C22×D411yescharacteristic
372C24C22×D411yescharacteristic
373C24C22×D411yescharacteristic
374#41—48no—
375#41—48no—
376C2×Q8—21no—
377C2×Q8—21no—
378C2×Q8—22no—
379C2×Q8—22no—
380C2×D4—22no—
381C2×D4—21no—
382C2×D4—22no—
383C2×D4—21no—
384C2×D4—24no—
385C2×D4—22no—
386C2×D4—24no—
387C2×D4—22no—
388C2×D4—48no—
389C2×D4—42no—
390C2×D4—42no—
391C2×D4—42no—
392C2×D4—44no—
393C2×D4—44no—
394C2×D4—42no—
395C2×D4—42no—
396C2×D4—44no—
397C2×D4—48no—
398C2×D4—42no—
399C2×D4—42no—
400C2×D4—42no—
401C2×D4—48no—
402C2×D4—48no—
403C2×D4—84no—
404C2×D4—84no—
405C2×D4—82no—
406C2×D4—84no—
407C2×D4—816no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—216no—
415C22×C4—216no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—28no—
421C22×C4—24no—
422C22×C4—28no—
423C22×C4—28no—
424C22×C4—28no—
425C22×C4—24no—
426C22×C4—21no—
427C22×C4—21no—
428C22×C4—22no—
429C22×C4—21no—
430C22×C4—44no—
431C22×C4—44no—
432C22×C4—48no—
433C22×C4—44no—
434C22×C4—44no—
435C22×C4—42no—
436C22×C4—44no—
437C22×C4—42no—
438C22×C4—42no—
439C22×C4—42no—
440C22×C4—84no—
441C22×C4—84no—
442C22×C4—82no—
443C22×C4—816no—
444C22×C4#12114yes—
445C22×C4#12114yes—
446C22×C4#12114yes—
447C22×C4C22×D414yes—
448C22×C4C22×D414yes—
449C22×C4C22×D411yescharacteristic
450C22×C4C22×D411yescharacteristic
451C22×C4C22×D412yes—
452C22×C4C22×D412yes—
453C22×C4C22×D411yescharacteristic
454C22×C4C22×D412yes—
455C22×C4C2×#4112yes—
456C22×C4C2×#4112yes—
457C22×C4C2×#4112yes—
458C22×C4H(5,𝔽2)14yes—
459C22×C42−1+414yes—
460C22×C4C2511yescharacteristic
461M16—48no—
462C2×C8—24no—
463C2×C8—24no—
464C2×C8—24no—
465C2×C8—24no—
466C2×C8—24no—
467C2×C8—24no—
468C2×C8—48no—
469C4⋊C4—28no—
470C4⋊C4—28no—
471C4⋊C4—28no—
472C4⋊C4—24no—
473C4⋊C4—24no—
474C4⋊C4—24no—
475C4⋊C4—22no—
476C4⋊C4—21no—
477C4⋊C4—21no—
478C4⋊C4—22no—
479C4⋊C4—21no—
480C4⋊C4—21no—
481C4⋊C4—24no—
482C4⋊C4—24no—
483C4⋊C4—44no—
484C4⋊C4—44no—
485C4⋊C4—42no—
486C4⋊C4—42no—
487C4⋊C4—42no—
488C4⋊C4—88no—
489C4⋊C4—88no—
490C4⋊C4—84no—
491C4⋊C4C22×D418yes—
492C22⋊C4—24no—
493C22⋊C4—24no—
494C22⋊C4—44no—
495C22⋊C4—48no—
496C22⋊C4—44no—
497C22⋊C4—42no—
498C22⋊C4—42no—
499C22⋊C4—42no—
500C22⋊C4—44no—
501C22⋊C4—44no—
502C22⋊C4—44no—
503C22⋊C4—42no—
504C22⋊C4—42no—
505C22⋊C4—84no—
506C22⋊C4—84no—
507C22⋊C4—84no—
508C42—24no—
509C42—24no—
510C42—44no—
511C42—44no—
512C42—42no—
513C42—44no—
514C23—22no—
515C23—22no—
516C23—21no—
517C23—21no—
518C23—22no—
519C23—22no—
520C23—28no—
521C23—21no—
522C23—21no—
523C23—28no—
524C23—22no—
525C23—44no—
526C23—48no—
527C23—48no—
528C23—44no—
529C23—42no—
530C23—41no—
531C23—42no—
532C23—42no—
533C23—41no—
534C23—41no—
535C23—42no—
536C23—42no—
537C23—44no—
538C23—84no—
539C23—84no—
540C23—84no—
541C23—164no—
542C23—164no—
543C23C2×#12112yes—
544C23C2×#12112yes—
545C23C2×#12112yes—
546C23#53412yes—
547C23#53612yes—
548C23#57714yes—
549C23C23×D411yescharacteristic
550Q8—42no—
551Q8—42no—
552D4—44no—
553D4—42no—
554D4—44no—
555D4—42no—
556D4—84no—
557D4—88no—
558D4—84no—
559D4—84no—
560D4—88no—
561D4—84no—
562D4—168no—
563C2×C4—28no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—28no—
569C2×C4—28no—
570C2×C4—24no—
571C2×C4—24no—
572C2×C4—24no—
573C2×C4—24no—
574C2×C4—24no—
575C2×C4—24no—
576C2×C4—28no—
577C2×C4—24no—
578C2×C4—24no—
579C2×C4—24no—
580C2×C4—24no—
581C2×C4—416no—
582C2×C4—48no—
583C2×C4—42no—
584C2×C4—44no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—42no—
588C2×C4—41no—
589C2×C4—44no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—42no—
593C2×C4—44no—
594C2×C4—44no—
595C2×C4—88no—
596C2×C4—88no—
597C2×C4—88no—
598C2×C4—88no—
599C2×C4—84no—
600C2×C4—84no—
601C2×C4C2×#12112yes—
602C2×C4C2×#12112yes—
603C2×C4C2×#12112yes—
604C2×C4#53412yes—
605C2×C4#53612yes—
606C2×C4#53718yes—
607C2×C4#53918yes—
608C2×C4C23×D411yescharacteristic
609C2×C4C22×#4111yescharacteristic
610C2×C4C2×H(5,𝔽2)12yes—
611C2×C4C2×2−1+412yes—
612C8—44no—
613C8—44no—
614C22—22no—
615C22—22no—
616C22—22no—
617C22—24no—
618C22—24no—
619C22—24no—
620C22—24no—
621C22—22no—
622C22—22no—
623C22—22no—
624C22—22no—
625C22—416no—
626C22—42no—
627C22—42no—
628C22—82no—
629C22—81no—
630C22—81no—
631C22—82no—
632C22—82no—
633C22—84no—
634C22—82no—
635C22—84no—
636C22—81no—
637C22—82no—
638C22—168no—
639C22#261814yes—
640C22#304314yes—
641C22C22×#12111yescharacteristic
642C22C2×#53411yescharacteristic
643C22C2×#53611yescharacteristic
644C22C2×#57712yes—
645C22#358512yes—
646C4—22no—
647C4—22no—
648C4—48no—
649C4—44no—
650C4—44no—
651C4—82no—
652C4—81no—
653C4—81no—
654C4—84no—
655C4—81no—
656C4#261814yes—
657C4C2×#53712yes—
658C4C2×#53912yes—
659C2—24no—
660C2—24no—
661C2—42no—
662C2—161no—
663C2—162no—
664C2C2×#261811yescharacteristic
665C2C2×#304311yescharacteristic
666C2#6042311yescharacteristic
667C2#6076512yes—
668C2#6087312yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#35252C2
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
1. C2×#3092C2semidirect
2. C22×#481C2semidirect
4. C2×#2316C2semidirect
5. #29768C2semidirect
6. #29763C2semidirect
7. #29761C2semidirect
8. #29755C2semidirect
9. #23661C2semidirect
10. #23614C2semidirect
11. #23610C2semidirect
12. #23511C2semidirect
13. #20912C2semidirect
14. #20908C2semidirect
15. #20628C2semidirect
23. C2×#484C22semidirect
27. C2×#481C22semidirect
28. C2×#483C22semidirect
29. C2×#480C22semidirect
52. #2316C22semidirect
56. C2×#389C22semidirect
58. #2085C22semidirect
59. #2084C22semidirect
60. #2067C22semidirect
61. #2066C22semidirect
62. #2065C22semidirect
63. #2065C22semidirect
64. #2042C22semidirect
65. #2041C22semidirect
67. #1875C22semidirect
69. #1874C22semidirect
70. #1873C22semidirect
71. #1872C22semidirect
72. #1798C22semidirect
73. #1797C22semidirect
124. C22×C4⋊C4D4semidirect

Maximal subgroups

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

subgroupindex
C2×#30922
C22×#4812
#352522
C2×#23162
#297682
#297632
#297612
#297552
#236612
#236142
#236102
#235112
#209122
#209082
#206282