Finite Group Databasebeta

← #6494678

Subgroups — #6494678

Order 512.

1517 conjugacy classes of subgroups.

The subgroup lattice has 744 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35288C211yesmaximal, characteristic
2#35283C211yesmaximal, characteristic, complemented
3#33316C211yesmaximal, characteristic, complemented
4#33252C211yesmaximal, characteristic, complemented
5#33212C211yesmaximal, characteristic, complemented
6#33174C211yesmaximal, characteristic
7#30206C211yesmaximal, characteristic, complemented
8#30203C211yesmaximal, characteristic, complemented
9#29845C211yesmaximal, characteristic, complemented
10#29842C211yesmaximal, characteristic
11#28018C214yesmaximal, complemented
12#27795C214yesmaximal, complemented
13#26487C214yesmaximal, complemented
14#25037C214yesmaximal, complemented
15#22221C211yesmaximal, characteristic, complemented
16#22219C211yesmaximal, characteristic, complemented
17C4×#500C211yesmaximal, characteristic, complemented
18C4×#492C211yesmaximal, characteristic, complemented
19#20472C211yesmaximal, characteristic, complemented
20C2×#501C2214yes—
21C2×#500C2211yescharacteristic
22C2×#492C2211yescharacteristic
23C2×#487C2211yescharacteristic
24C2×#486C2211yescharacteristic
25C2×#468C2214yes—
26C2×C4×SD16C2214yes—
27C2×#431C2211yescharacteristic
28C2×C4×M16C2211yescharacteristic
29#2868C2211yescharacteristic
30#2862C2211yescharacteristic
31#2859C2211yescharacteristic
32#2858C2211yescharacteristic
33#2857C2211yescharacteristic
34#2857C2211yescharacteristic
35#2857C2211yescharacteristic
36#2856C2211yescharacteristic
37#2647—21no—
38#2644—21no—
39#2628—21no—
40#2624—21no—
41#2513C2212yes—
42#2511—21no—
43#2511C2212yes—
44#2492—21no—
45#2492C2212yes—
46#2490C2212yes—
47#2454C2212yes—
48#2451—21no—
49#2451C2212yes—
50#2448—21no—
51#2447—21no—
52#2427—21no—
53#2426—21no—
54#2423—21no—
55#2423C2212yes—
56#2420C2212yes—
57#2331—21no—
58#2324—21no—
59#2319—21no—
60#2314—21no—
61C4×C4⋊Q8C2211yescharacteristic
62C4×C4⋊Q8C2211yescharacteristic
63C4×C4⋊Q8C2211yescharacteristic
64C4×#127C2211yescharacteristic
65C4×#117C2211yescharacteristic
66#2074C2214yes—
67#2071C2214yes—
68#2067C2214yes—
69#2066C2214yes—
70#2035C2214yes—
71#2033C2214yes—
72#2026C2214yes—
73#2023C2214yes—
74#1990C2211yescharacteristic
75#1990C2211yescharacteristic
76#1990C2211yescharacteristic
77#1990C2211yescharacteristic
78#1973C2214yes—
79#1973C2214yes—
80#1973C2214yes—
81#1968C2214yes—
82#1957C2211yescharacteristic
83#1957C2211yescharacteristic
84#1956C2211yescharacteristic
85#1956C2211yescharacteristic
86#1944C2214yes—
87#1942C2214yes—
88#1928C2214yes—
89#1926C2214yes—
90#1864C2214yes—
91#1863C2214yes—
92#1847C2214yes—
93#1845C2211yescharacteristic
94#1845C2211yescharacteristic
95#1844C2211yescharacteristic
96#1844C2211yescharacteristic
97#1809C2214yes—
98C4×#107C2214yes—
99#1771C2212yes—
100#1771C2212yes—
101#1753C2212yes—
102#1753C2212yes—
103#1744C2211yescharacteristic
104#1744C2211yescharacteristic
105C4×#97C2211yescharacteristic
106C42×C8C2211yescharacteristic
107C22×SD16C2314yes—
108C22×M16C2312yes—
109C2×C4⋊Q8—21no—
110C2×C4⋊Q8—21no—
111C2×C4⋊Q8C2311yescharacteristic
112C2×C4⋊Q8C2311yescharacteristic
113C2×C4⋊Q8C2311yescharacteristic
114C2×C4⋊Q8C2312yes—
115C2×#127C2311yescharacteristic
116C2×#125C2312yes—
117C2×#124—21no—
118C2×#124—21no—
119C2×C22⋊Q8—21no—
120C2×C22⋊Q8—21no—
121C2×C22⋊Q8C2312yes—
122C2×#121—21no—
123C2×#121—21no—
124C2×#121C2312yes—
125C2×C4×Q8—21no—
126C2×C4×Q8—21no—
127C2×C4×Q8C2312yes—
128C2×C4×D4—21no—
129C2×C4×D4—21no—
130C2×C4×D4C2312yes—
131C2×#117D411yescharacteristic
132C2×#117D411yescharacteristic
133C22×C42C2311yescharacteristic
134#501—48no—
135#500—24no—
136#492—24no—
137#487—24no—
138#486—24no—
139#468—48no—
140C4×SD16—48no—
141#431—24no—
142C2×#107C2314yes—
143C2×#107C2314yes—
144C2×#107C2314yes—
145C2×#106C2314yes—
146C2×#106C2314yes—
147C2×C4⋊C8C2312yes—
148C2×#103C2314yes—
149C2×#103C2314yes—
150C2×#103C2314yes—
151C2×#102C2314yes—
152C2×#102C2314yes—
153C2×#102C2314yes—
154C2×C22⋊C8C2312yes—
155C4×M16—24no—
156C2×#97C2312yes—
157C2×#97C2312yes—
158C2×#97C2312yes—
159C2×C4×C8C2311yescharacteristic
160C2×C4×C8C2311yescharacteristic
161C2×C4×C8C2312yes—
162C2×C4×C8C2312yes—
163C2×C4×C8C2311yescharacteristic
164C2×C4×C8C2311yescharacteristic
165#400—22no—
166#400—22no—
167#400—22no—
168#399—22no—
169#399—22no—
170#398—22no—
171#398—22no—
172#397—22no—
173#397—22no—
174#397—22no—
175#397—22no—
176#396—22no—
177#395—22no—
178#394—22no—
179#394—22no—
180#393—22no—
181#391—21no—
182#391—21no—
183#391—21no—
184#391—21no—
185#391—21no—
186#391—21no—
187#391—22no—
188#391—22no—
189#391C2312yes—
190#391C2312yes—
191#391C2312yes—
192#391C2312yes—
193#390—22no—
194#390—21no—
195#390—21no—
196#390—21no—
197#390—21no—
198#390—21no—
199#390—21no—
200#390C2312yes—
201#390C2312yes—
202#390C2312yes—
203#390C2312yes—
204#389—21no—
205#389—21no—
206#389—21no—
207#389—21no—
208#389—21no—
209#389—21no—
210#389—21no—
211#389—21no—
212#389—21no—
213#389—21no—
214#389—21no—
215#389—21no—
216#389C2312yes—
217#389C2312yes—
218#389C2312yes—
219#389C2312yes—
220#388—22no—
221#388—22no—
222#388—22no—
223#388—22no—
224#388—22no—
225#387—22no—
226#387—22no—
227#387—22no—
228#387—22no—
229#386—22no—
230#385—22no—
231#384—21no—
232#384—21no—
233#384C2311yescharacteristic
234#384C2311yescharacteristic
235#384C2311yescharacteristic
236#384C2311yescharacteristic
237#384C2312yes—
238#382—21no—
239#382—21no—
240#382C2312yes—
241#381—22no—
242#380—21no—
243#380—21no—
244#380—21no—
245#380—21no—
246#380—22no—
247#380C2312yes—
248#380C2312yes—
249C4×C4⋊C4—21no—
250C4×C4⋊C4—21no—
251C4×C4⋊C4—21no—
252C4×C4⋊C4—21no—
253C4×C4⋊C4D411yescharacteristic
254C4×C4⋊C4D411yescharacteristic
255C4×C4⋊C4C2312yes—
256C4×C4⋊C4C2312yes—
257C4×C4⋊C4C2311yescharacteristic
258C4×C4⋊C4C2311yescharacteristic
259C4×C22⋊C4D411yescharacteristic
260C4×C22⋊C4D411yescharacteristic
261C4×C22⋊C4C2311yescharacteristic
262C4×C22⋊C4C2311yescharacteristic
263#376D411yescharacteristic
264#376D411yescharacteristic
265C43C2311yescharacteristic
266C43C2311yescharacteristic
267#340C2314yes—
268#340C2314yes—
269#340C2314yes—
270#340C2314yes—
271#336C2312yes—
272#336C2312yes—
273#336C2312yes—
274#336C2312yes—
275C22×Q8—21no—
276C22×Q8—21no—
277C22×Q8—22no—
278C22×Q8C2412yes—
279C22×D4—22no—
280C22×D4—21no—
281C22×D4—21no—
282C22×D4C2412yes—
283C23×C4—22no—
284C23×C4C2×D411yescharacteristic
285C23×C4C2×D411yescharacteristic
286C23×C4C2411yescharacteristic
287C2×SD16—28no—
288C2×SD16—28no—
289C2×SD16—28no—
290C2×M16—24no—
291C2×M16—24no—
292C2×M16—24no—
293C22×C8C2412yes—
294C22×C8C2412yes—
295C22×C8C2412yes—
296C22×C8C2412yes—
297C4⋊Q8—22no—
298C4⋊Q8—22no—
299C4⋊Q8—22no—
300C4⋊Q8—22no—
301C4⋊Q8—22no—
302C4⋊Q8—22no—
303C4⋊Q8—44no—
304C4⋊Q8—44no—
305C4⋊Q8—82no—
306#127—22no—
307#127—22no—
308#125—44no—
309#124—82no—
310#124—82no—
311C22⋊Q8—44no—
312C22⋊Q8—82no—
313C22⋊Q8—82no—
314#121—44no—
315#121—44no—
316#121—82no—
317C4×Q8—44no—
318C4×Q8—82no—
319C4×Q8—82no—
320C4×D4—44no—
321C4×D4—44no—
322C4×D4—82no—
323#117—24no—
324#117—42no—
325C2×C4⋊C4—21no—
326C2×C4⋊C4—21no—
327C2×C4⋊C4—21no—
328C2×C4⋊C4—21no—
329C2×C4⋊C4—21no—
330C2×C4⋊C4—21no—
331C2×C4⋊C4—21no—
332C2×C4⋊C4—21no—
333C2×C4⋊C4—22no—
334C2×C4⋊C4—21no—
335C2×C4⋊C4—21no—
336C2×C4⋊C4—21no—
337C2×C4⋊C4—21no—
338C2×C4⋊C4—22no—
339C2×C4⋊C4—22no—
340C2×C4⋊C4—22no—
341C2×C4⋊C4—22no—
342C2×C4⋊C4—22no—
343C2×C4⋊C4—22no—
344C2×C4⋊C4—22no—
345C2×C4⋊C4—22no—
346C2×C4⋊C4—22no—
347C2×C4⋊C4—22no—
348C2×C4⋊C4C2×D411yescharacteristic
349C2×C4⋊C4C2×D411yescharacteristic
350C2×C4⋊C4C2×D411yescharacteristic
351C2×C4⋊C4C2×D411yescharacteristic
352C2×C4⋊C4C2412yes—
353C2×C4⋊C4C2412yes—
354C2×C4⋊C4C2412yes—
355C2×C4⋊C4C2412yes—
356C2×C4⋊C4C2412yes—
357C2×C4⋊C4C2412yes—
358C2×C22⋊C4—22no—
359C2×C22⋊C4—22no—
360C2×C22⋊C4—22no—
361C2×C22⋊C4—22no—
362C2×C22⋊C4—22no—
363C2×C22⋊C4—22no—
364C2×C22⋊C4—22no—
365C2×C22⋊C4—22no—
366C2×C22⋊C4—22no—
367C2×C22⋊C4—22no—
368C2×C22⋊C4—22no—
369C2×C22⋊C4—22no—
370C2×C22⋊C4—22no—
371C2×C22⋊C4C2×D411yescharacteristic
372C2×C22⋊C4C2×D411yescharacteristic
373C2×C22⋊C4C2×D411yescharacteristic
374C2×C22⋊C4C2×D411yescharacteristic
375C2×C42—22no—
376C2×C42—22no—
377C2×C42—22no—
378C2×C42—22no—
379C2×C42—22no—
380C2×C42C2×D411yescharacteristic
381C2×C42C2×D411yescharacteristic
382C2×C42C2×D411yescharacteristic
383C2×C42C2×D411yescharacteristic
384C2×C42C2×D411yescharacteristic
385C2×C42C2×D411yescharacteristic
386C2×C42C2×D411yescharacteristic
387C2×C42C2×D411yescharacteristic
388C2×C42C2×D411yescharacteristic
389C2×C42C2×D411yescharacteristic
390C2×C42C2411yescharacteristic
391C2×C42C2411yescharacteristic
392C2×C42C2411yescharacteristic
393C2×C42C2411yescharacteristic
394C2×C42C2411yescharacteristic
395C2×C42C2411yescharacteristic
396#107—28no—
397#107—28no—
398#107—44no—
399#106—28no—
400#106—44no—
401C4⋊C8—24no—
402#103—28no—
403#103—44no—
404#103—44no—
405#102—28no—
406#102—44no—
407#102—44no—
408C22⋊C8—24no—
409#97—24no—
410#97—24no—
411#97—24no—
412C4×C8—24no—
413C4×C8—24no—
414C4×C8—22no—
415C4×C8—22no—
416C4×C8#4114yes—
417C4×C8#4114yes—
418#95—22no—
419#95—22no—
420#95—22no—
421#95—22no—
422#95—22no—
423#95—22no—
424#95—22no—
425#95—22no—
426#95—22no—
427#95—22no—
428#95—22no—
429#95—22no—
430#95—22no—
431#95—22no—
432#95—22no—
433#95—22no—
434#95—22no—
435#95—22no—
436#95C2×D411yescharacteristic
437#95C2×D411yescharacteristic
438#95C2×D411yescharacteristic
439#95C2×D411yescharacteristic
440#95C2×D411yescharacteristic
441#95C2×D411yescharacteristic
442#95C2×D411yescharacteristic
443#95C2×D411yescharacteristic
444C24—22no—
445C24C22×D411yescharacteristic
446C2×Q8—22no—
447C2×Q8—22no—
448C2×Q8—24no—
449C2×Q8—24no—
450C2×Q8—44no—
451C2×Q8—42no—
452C2×Q8—42no—
453C2×Q8—42no—
454C2×Q8—42no—
455C2×Q8—42no—
456C2×Q8—42no—
457C2×Q8—82no—
458C2×Q8—82no—
459C2×D4—22no—
460C2×D4—22no—
461C2×D4—24no—
462C2×D4—24no—
463C2×D4—44no—
464C2×D4—42no—
465C2×D4—42no—
466C2×D4—42no—
467C2×D4—42no—
468C2×D4—42no—
469C2×D4—42no—
470C2×D4—82no—
471C2×D4—82no—
472C22×C4—22no—
473C22×C4—22no—
474C22×C4—22no—
475C22×C4—22no—
476C22×C4—22no—
477C22×C4—22no—
478C22×C4—21no—
479C22×C4—21no—
480C22×C4—22no—
481C22×C4—22no—
482C22×C4—22no—
483C22×C4—22no—
484C22×C4—22no—
485C22×C4—22no—
486C22×C4—22no—
487C22×C4—22no—
488C22×C4—22no—
489C22×C4—44no—
490C22×C4—82no—
491C22×C4—82no—
492C22×C4C22×D411yescharacteristic
493C22×C4C22×D411yescharacteristic
494C22×C4C22×D411yescharacteristic
495C22×C4C22×D411yescharacteristic
496C22×C4C22×D411yescharacteristic
497C22×C4C22×D411yescharacteristic
498C22×C4C22×D411yescharacteristic
499C22×C4C22×D411yescharacteristic
500C22×C4C22×D411yescharacteristic
501C22×C4C22×D411yescharacteristic
502C22×C4C22×D411yescharacteristic
503C22×C4C22×D411yescharacteristic
504C22×C4C22×D411yescharacteristic
505C22×C4C2511yescharacteristic
506SD16—416no—
507M16—48no—
508C2×C8—22no—
509C2×C8—22no—
510C2×C8—22no—
511C2×C8—22no—
512C2×C8—22no—
513C2×C8—22no—
514C2×C8—22no—
515C2×C8—22no—
516C2×C8C2×#4114yes—
517C2×C8C2×#4114yes—
518C2×C8H(5,𝔽2)14yes—
519C2×C82−1+414yes—
520C4⋊C4—22no—
521C4⋊C4—22no—
522C4⋊C4—22no—
523C4⋊C4—22no—
524C4⋊C4—22no—
525C4⋊C4—22no—
526C4⋊C4—24no—
527C4⋊C4—24no—
528C4⋊C4—24no—
529C4⋊C4—24no—
530C4⋊C4—41no—
531C4⋊C4—41no—
532C4⋊C4—44no—
533C4⋊C4—42no—
534C4⋊C4—42no—
535C4⋊C4—42no—
536C4⋊C4—42no—
537C4⋊C4—42no—
538C4⋊C4—42no—
539C4⋊C4—42no—
540C4⋊C4—42no—
541C4⋊C4—42no—
542C4⋊C4—42no—
543C4⋊C4—42no—
544C4⋊C4—42no—
545C4⋊C4—82no—
546C4⋊C4—82no—
547C4⋊C4—82no—
548C4⋊C4—82no—
549C4⋊C4—82no—
550C4⋊C4—82no—
551C4⋊C4—82no—
552C4⋊C4—82no—
553C4⋊C4—82no—
554C4⋊C4—82no—
555C22⋊C4—22no—
556C22⋊C4—22no—
557C22⋊C4—41no—
558C22⋊C4—41no—
559C22⋊C4—44no—
560C22⋊C4—44no—
561C22⋊C4—82no—
562C22⋊C4—82no—
563C22⋊C4—82no—
564C22⋊C4—82no—
565C22⋊C4—82no—
566C22⋊C4—82no—
567C22⋊C4—82no—
568C22⋊C4—82no—
569C22⋊C4—82no—
570C22⋊C4—82no—
571C22⋊C4—82no—
572C42—22no—
573C42—22no—
574C42—22no—
575C42—22no—
576C42—22no—
577C42—22no—
578C42—22no—
579C42—22no—
580C42—22no—
581C42—21no—
582C42—21no—
583C42—22no—
584C42—22no—
585C42—22no—
586C42—22no—
587C42—82no—
588C42—82no—
589C42—82no—
590C42#13714yes—
591C42C2×#4112yes—
592C42C2×#4112yes—
593C23—21no—
594C23—21no—
595C23—21no—
596C23—21no—
597C23—21no—
598C23—21no—
599C23—21no—
600C23—42no—
601C23—42no—
602C23—42no—
603C23—42no—
604C23—42no—
605C23—42no—
606C23—42no—
607C23C23×D411yescharacteristic
608Q8—44no—
609Q8—44no—
610Q8—84no—
611Q8—84no—
612Q8—164no—
613D4—44no—
614D4—44no—
615D4—84no—
616D4—84no—
617D4—164no—
618C2×C4—21no—
619C2×C4—21no—
620C2×C4—21no—
621C2×C4—21no—
622C2×C4—21no—
623C2×C4—21no—
624C2×C4—21no—
625C2×C4—21no—
626C2×C4—21no—
627C2×C4—21no—
628C2×C4—21no—
629C2×C4—21no—
630C2×C4—21no—
631C2×C4—21no—
632C2×C4—21no—
633C2×C4—21no—
634C2×C4—21no—
635C2×C4—21no—
636C2×C4—21no—
637C2×C4—21no—
638C2×C4—21no—
639C2×C4—21no—
640C2×C4—21no—
641C2×C4—21no—
642C2×C4—21no—
643C2×C4—21no—
644C2×C4—21no—
645C2×C4—21no—
646C2×C4—21no—
647C2×C4—21no—
648C2×C4—21no—
649C2×C4—44no—
650C2×C4—44no—
651C2×C4—42no—
652C2×C4—42no—
653C2×C4—42no—
654C2×C4—42no—
655C2×C4—42no—
656C2×C4—42no—
657C2×C4—42no—
658C2×C4—42no—
659C2×C4—42no—
660C2×C4—42no—
661C2×C4—42no—
662C2×C4—42no—
663C2×C4—42no—
664C2×C4—42no—
665C2×C4—42no—
666C2×C4—42no—
667C2×C4—42no—
668C2×C4—42no—
669C2×C4—42no—
670C2×C4—42no—
671C2×C4—42no—
672C2×C4—42no—
673C2×C4—42no—
674C2×C4—164no—
675C2×C4#53212yes—
676C2×C4#53212yes—
677C2×C4C2×#13712yes—
678C2×C4C2×#13712yes—
679C2×C4C2×#13712yes—
680C2×C4#57512yes—
681C2×C4#57612yes—
682C2×C4#57712yes—
683C2×C4C22×#4111yescharacteristic
684C2×C4C22×#4111yescharacteristic
685C2×C4C2×H(5,𝔽2)12yes—
686C2×C4C2×2−1+412yes—
687C8—24no—
688C8—42no—
689C8—42no—
690C8#54218yes—
691C22—41no—
692C22—41no—
693C22—41no—
694C22—41no—
695C22—41no—
696C22—41no—
697C22—41no—
698C22—82no—
699C22—82no—
700C22—82no—
701C22—82no—
702C22—82no—
703C22—82no—
704C22—82no—
705C22C2×#53211yescharacteristic
706C22#349811yescharacteristic
707C22#349911yescharacteristic
708C22C22×#13711yescharacteristic
709C22C2×#57511yescharacteristic
710C22C2×#57611yescharacteristic
711C22C2×#57711yescharacteristic
712C4—22no—
713C4—22no—
714C4—41no—
715C4—41no—
716C4—41no—
717C4—41no—
718C4—41no—
719C4—41no—
720C4—41no—
721C4—41no—
722C4—41no—
723C4—82no—
724C4—82no—
725C4—82no—
726C4—82no—
727C4—82no—
728C4—82no—
729C4—82no—
730C4#317214yes—
731C4#317914yes—
732C4C2×#54212yes—
733C4#351112yes—
734C2—81no—
735C2—162no—
736C2#3527711yescharacteristic
737C2C2×#317211yescharacteristic
738C2C2×#317911yescharacteristic
739C2#6242211yescharacteristic
740C2#6242611yescharacteristic
741C2#6248011yescharacteristic
742C2#6248311yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#2856C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #35283C2semidirect
3. #33316C2semidirect
4. #33252C2semidirect
5. #33212C2semidirect
7. #30206C2semidirect
8. #30203C2semidirect
9. #29845C2semidirect
11. #28018C2semidirect
12. #27795C2semidirect
13. #26487C2semidirect
14. #25037C2semidirect
15. #22221C2semidirect
16. #22219C2semidirect
17. C4×#500C2semidirect
18. C4×#492C2semidirect
19. #20472C2semidirect

Maximal subgroups

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

subgroupindex
#352882
#352832
#333162
#332522
#332122
#331742
#302062
#302032
#298452
#298422
#280182
#277952
#264872
#250372
#222212
#222192
C4×#5002
C4×#4922
#204722