Finite Group Databasebeta

← #6594928

Subgroups — #6594928

Order 512.

1538 conjugacy classes of subgroups.

The subgroup lattice has 762 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36211C211yesmaximal, characteristic
2#36189C211yesmaximal, characteristic, complemented
3#33263C211yesmaximal, characteristic, complemented
4#33048C211yesmaximal, characteristic, complemented
5#33042C211yesmaximal, characteristic, complemented
6#29192C211yesmaximal, characteristic, complemented
7#29178C211yesmaximal, characteristic, complemented
8#28854C214yesmaximal, complemented
9#28041C214yesmaximal, complemented
10#27949C211yesmaximal, characteristic, complemented
11#27909C211yesmaximal, characteristic, complemented
12#27890C211yesmaximal, characteristic, complemented
13#27878C211yesmaximal, characteristic, complemented
14#25240C214yesmaximal, complemented
15#22941C214yesmaximal, complemented
16#21646C211yesmaximal, characteristic, complemented
17#20817C211yesmaximal, characteristic, complemented
18#20764C211yesmaximal, characteristic, complemented
19C4×#435C211yesmaximal, characteristic, complemented
20C2×#486C2211yescharacteristic
21C2×#480C2214yes—
22C2×#435C2212yes—
23C2×C4×M16C2211yescharacteristic
24C22×C4×C8C2211yescharacteristic
25#2859C2211yescharacteristic
26#2859C2211yescharacteristic
27#2857C2211yescharacteristic
28#2857C2211yescharacteristic
29#2856C2211yescharacteristic
30#2856C2211yescharacteristic
31#2570—21no—
32#2535—21no—
33#2513—21no—
34#2513C2212yes—
35#2511—21no—
36#2507C2212yes—
37#2497C2212yes—
38#2490—21no—
39#2490—21no—
40#2487C2212yes—
41#2442—21no—
42#2441—21no—
43#2423—21no—
44#2420—21no—
45#2420C2212yes—
46#2417—21no—
47#2414—21no—
48#2414C2212yes—
49#2363—21no—
50#2363C2212yes—
51#2360—21no—
52#2357—21no—
53#2352—21no—
54#2345C2212yes—
55C4×C4⋊Q8C2211yescharacteristic
56C4×#127C2211yescharacteristic
57#2285C2211yescharacteristic
58#2285C2211yescharacteristic
59#2285C2211yescharacteristic
60C2×#384C2211yescharacteristic
61C42×D4C2211yescharacteristic
62#2093C2214yes—
63#2086C2214yes—
64#2085C2214yes—
65#2076C2214yes—
66#2067C2214yes—
67#2034C2214yes—
68#1990C2211yescharacteristic
69#1990C2211yescharacteristic
70#1990C2211yescharacteristic
71#1957C2211yescharacteristic
72#1957C2211yescharacteristic
73#1957C2211yescharacteristic
74#1944C2214yes—
75#1943C2214yes—
76#1936C2214yes—
77#1936C2214yes—
78#1934C2214yes—
79#1926C2214yes—
80#1921C2214yes—
81#1918C2214yes—
82#1895C2214yes—
83#1893C2214yes—
84#1862C2214yes—
85#1854C2214yes—
86#1845C2211yescharacteristic
87#1845C2211yescharacteristic
88#1845C2211yescharacteristic
89#1844C2211yescharacteristic
90#1836C2214yes—
91#1834C2211yescharacteristic
92#1834C2211yescharacteristic
93#1834C2211yescharacteristic
94#1834C2211yescharacteristic
95#1834C2211yescharacteristic
96#1808C2214yes—
97C4×#107C2214yes—
98#1771C2212yes—
99C4×C4⋊C8C2211yescharacteristic
100#1762C2214yes—
101#1753C2212yes—
102#1752C2212yes—
103C4×C22⋊C8C2211yescharacteristic
104C4×C22⋊C8C2211yescharacteristic
105#1736C2214yes—
106C4×#97C2211yescharacteristic
107C22×M16C2312yes—
108C23×C8C2312yes—
109C2×C4⋊Q8C2311yescharacteristic
110C2×#127C2311yescharacteristic
111C2×#125—21no—
112C2×#125—21no—
113C2×C22⋊Q8—21no—
114C2×C22⋊Q8—21no—
115C2×#121—21no—
116C2×#121—21no—
117C2×#121—21no—
118C2×#121—21no—
119C2×#121C2312yes—
120C2×#121C2312yes—
121C2×C4×Q8—21no—
122C2×C4×D4—21no—
123C2×C4×D4—21no—
124C2×C4×D4D411yescharacteristic
125C2×C4×D4D411yescharacteristic
126C2×C4×D4C2312yes—
127C2×C4×D4C2311yescharacteristic
128C2×#117—21no—
129C2×#117—21no—
130C2×#117C2312yes—
131C22×C4⋊C4—21no—
132C22×C4⋊C4—21no—
133C22×C4⋊C4C2312yes—
134C22×C42C2311yescharacteristic
135C22×C42C2311yescharacteristic
136#486—24no—
137#480—216no—
138#435—28no—
139C2×#107C2314yes—
140C2×#107C2314yes—
141C2×#107C2314yes—
142C2×#106C2314yes—
143C2×C4⋊C8C2312yes—
144C2×#102C2314yes—
145C2×#102C2314yes—
146C2×#102C2314yes—
147C2×#102C2314yes—
148C2×C22⋊C8C2312yes—
149C2×C22⋊C8C2312yes—
150C4×M16—24no—
151C2×#97C2312yes—
152C2×#97C2312yes—
153C2×#97C2312yes—
154C2×C4×C8—24no—
155C2×C4×C8C2311yescharacteristic
156C2×C4×C8C2311yescharacteristic
157C2×C4×C8C2311yescharacteristic
158C2×C4×C8C2312yes—
159C2×C4×C8C2311yescharacteristic
160#400—22no—
161#400—22no—
162#399—22no—
163#398—22no—
164#398—22no—
165#397—22no—
166#397—22no—
167#397—22no—
168#396—22no—
169#394—22no—
170#394—22no—
171#392—22no—
172#391—21no—
173#391—21no—
174#391—21no—
175#390—22no—
176#390—22no—
177#390—21no—
178#390—21no—
179#390—21no—
180#390—21no—
181#390—21no—
182#390—21no—
183#390C2312yes—
184#390C2312yes—
185#390C2312yes—
186#389—21no—
187#389—21no—
188#389—21no—
189#389—21no—
190#389—21no—
191#389—21no—
192#389—21no—
193#389—22no—
194#389C2312yes—
195#389C2312yes—
196#389C2312yes—
197#389C2312yes—
198#389C2312yes—
199#389C2312yes—
200#389C2312yes—
201#388—22no—
202#388—22no—
203#388—22no—
204#388—22no—
205#388—22no—
206#387—22no—
207#387—22no—
208#387—22no—
209#387—22no—
210#386—22no—
211#386—22no—
212#386—22no—
213#385—22no—
214#384—21no—
215#384—21no—
216#384C2311yescharacteristic
217#384C2311yescharacteristic
218#384C2311yescharacteristic
219#384C2311yescharacteristic
220#384C2311yescharacteristic
221#384C2311yescharacteristic
222#384C2312yes—
223#382—21no—
224#382—21no—
225#382—21no—
226#382—21no—
227#382C2312yes—
228#381—22no—
229#380—21no—
230#380—21no—
231#380—21no—
232#380—21no—
233#380—21no—
234#380—21no—
235#380—22no—
236#380C2312yes—
237#380C2312yes—
238#380C2312yes—
239#380C2312yes—
240C4×C4⋊C4—21no—
241C4×C4⋊C4—21no—
242C4×C4⋊C4—21no—
243C4×C4⋊C4—21no—
244C4×C4⋊C4—21no—
245C4×C4⋊C4D411yescharacteristic
246C4×C4⋊C4D411yescharacteristic
247C4×C4⋊C4C2312yes—
248C4×C4⋊C4C2312yes—
249C4×C4⋊C4C2312yes—
250C4×C4⋊C4C2311yescharacteristic
251C4×C22⋊C4—22no—
252C4×C22⋊C4—22no—
253C4×C22⋊C4D411yescharacteristic
254C4×C22⋊C4D411yescharacteristic
255C4×C22⋊C4D411yescharacteristic
256C4×C22⋊C4D411yescharacteristic
257C4×C22⋊C4C2311yescharacteristic
258C4×C22⋊C4C2311yescharacteristic
259C43C2311yescharacteristic
260#340C2314yes—
261#340C2314yes—
262#340C2314yes—
263#340C2314yes—
264#340C2314yes—
265#340C2314yes—
266#340C2314yes—
267#340C2314yes—
268#336C2312yes—
269#336C2312yes—
270#336C2312yes—
271C22×Q8—21no—
272C22×D4—22no—
273C22×D4—22no—
274C22×D4—21no—
275C22×D4—21no—
276C22×D4C2×D411yescharacteristic
277C22×D4C2412yes—
278C23×C4—22no—
279C23×C4—22no—
280C23×C4C2×D411yescharacteristic
281C23×C4C2×D411yescharacteristic
282C23×C4C2×D411yescharacteristic
283C23×C4C2×D411yescharacteristic
284C23×C4C2411yescharacteristic
285C23×C4C2411yescharacteristic
286C2×M16—24no—
287C2×M16—24no—
288C2×M16—42no—
289C22×C8—24no—
290C22×C8—24no—
291C22×C8—42no—
292C22×C8C2412yes—
293C22×C8C2412yes—
294C22×C8C2412yes—
295C22×C8C2412yes—
296C4⋊Q8—22no—
297C4⋊Q8—22no—
298#127—22no—
299#127—22no—
300#125—82no—
301#125—82no—
302C22⋊Q8—44no—
303C22⋊Q8—82no—
304#121—28no—
305#121—44no—
306#121—44no—
307#121—82no—
308#121—82no—
309#121—82no—
310C4×Q8—82no—
311C4×D4—24no—
312C4×D4—24no—
313C4×D4—42no—
314C4×D4—44no—
315C4×D4—44no—
316C4×D4—82no—
317#117—44no—
318#117—82no—
319#117—82no—
320C2×C4⋊C4—28no—
321C2×C4⋊C4—21no—
322C2×C4⋊C4—21no—
323C2×C4⋊C4—21no—
324C2×C4⋊C4—21no—
325C2×C4⋊C4—22no—
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—21no—
334C2×C4⋊C4—22no—
335C2×C4⋊C4—21no—
336C2×C4⋊C4—22no—
337C2×C4⋊C4—22no—
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—82no—
345C2×C4⋊C4—82no—
346C2×C4⋊C4C2×D411yescharacteristic
347C2×C4⋊C4C2×D411yescharacteristic
348C2×C4⋊C4C2×D411yescharacteristic
349C2×C4⋊C4C2412yes—
350C2×C4⋊C4C2412yes—
351C2×C4⋊C4C2412yes—
352C2×C4⋊C4C2412yes—
353C2×C4⋊C4C2412yes—
354C2×C4⋊C4C2412yes—
355C2×C4⋊C4C2412yes—
356C2×C22⋊C4—22no—
357C2×C22⋊C4—22no—
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⋊C4C2×D411yescharacteristic
371C2×C22⋊C4C2×D411yescharacteristic
372C2×C22⋊C4C2×D411yescharacteristic
373C2×C22⋊C4C2×D411yescharacteristic
374C2×C22⋊C4C2×D411yescharacteristic
375C2×C22⋊C4C2×D411yescharacteristic
376C2×C42—22no—
377C2×C42—22no—
378C2×C42—22no—
379C2×C42—22no—
380C2×C42—22no—
381C2×C42—22no—
382C2×C42—22no—
383C2×C42—22no—
384C2×C42C2×D411yescharacteristic
385C2×C42C2×D411yescharacteristic
386C2×C42C2×D411yescharacteristic
387C2×C42C2×D411yescharacteristic
388C2×C42C2×D411yescharacteristic
389C2×C42C2×D411yescharacteristic
390C2×C42C2×D411yescharacteristic
391C2×C42C2×D411yescharacteristic
392C2×C42C2411yescharacteristic
393C2×C42C2411yescharacteristic
394C2×C42C2411yescharacteristic
395C2×C42C2411yescharacteristic
396C2×C42C2411yescharacteristic
397#107—28no—
398#107—28no—
399#107—44no—
400#106—44no—
401C4⋊C8—24no—
402#102—28no—
403#102—28no—
404#102—28no—
405#102—44no—
406C22⋊C8—24no—
407C22⋊C8#4118yes—
408#97—24no—
409#97—24no—
410#97—42no—
411C4×C8—24no—
412C4×C8—22no—
413C4×C8—22no—
414C4×C8—22no—
415C4×C8#4114yes—
416#95—22no—
417#95—22no—
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#95C2×D411yescharacteristic
433#95C2×D411yescharacteristic
434#95C2×D411yescharacteristic
435#95C2×D411yescharacteristic
436#95C2×D411yescharacteristic
437#95C2×D411yescharacteristic
438C24—22no—
439C24C22×D411yescharacteristic
440C24C22×D411yescharacteristic
441C2×Q8—42no—
442C2×Q8—42no—
443C2×Q8—42no—
444C2×D4—22no—
445C2×D4—22no—
446C2×D4—24no—
447C2×D4—22no—
448C2×D4—22no—
449C2×D4—24no—
450C2×D4—41no—
451C2×D4—44no—
452C2×D4—44no—
453C2×D4—42no—
454C2×D4—42no—
455C2×D4—42no—
456C2×D4—42no—
457C2×D4—42no—
458C2×D4—42no—
459C2×D4—82no—
460C2×D4—82no—
461C2×D4—82no—
462C2×D4—82no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—21no—
466C22×C4—21no—
467C22×C4—22no—
468C22×C4—22no—
469C22×C4—22no—
470C22×C4—22no—
471C22×C4—22no—
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—44no—
489C22×C4—44no—
490C22×C4—82no—
491C22×C4—82no—
492C22×C4—82no—
493C22×C4—82no—
494C22×C4#13714yes—
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×C4C22×D411yescharacteristic
506C22×C4C22×D411yescharacteristic
507C22×C4C2×#4114yes—
508C22×C4C2511yescharacteristic
509M16—48no—
510C2×C8—22no—
511C2×C8—22no—
512C2×C8—22no—
513C2×C8—22no—
514C2×C8—22no—
515C2×C8—22no—
516C2×C8—22no—
517C2×C8—22no—
518C2×C8—22no—
519C2×C8—48no—
520C2×C8C2×#4114yes—
521C2×C8C2×#4114yes—
522C2×C8C2×#4114yes—
523C4⋊C4—22no—
524C4⋊C4—22no—
525C4⋊C4—22no—
526C4⋊C4—24no—
527C4⋊C4—24no—
528C4⋊C4—24no—
529C4⋊C4—22no—
530C4⋊C4—24no—
531C4⋊C4—24no—
532C4⋊C4—24no—
533C4⋊C4—41no—
534C4⋊C4—44no—
535C4⋊C4—44no—
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—42no—
546C4⋊C4—42no—
547C4⋊C4—42no—
548C4⋊C4—42no—
549C4⋊C4—82no—
550C4⋊C4—82no—
551C4⋊C4—82no—
552C4⋊C4—82no—
553C4⋊C4—82no—
554C4⋊C4—82no—
555C4⋊C4—82no—
556C4⋊C4—82no—
557C22⋊C4—22no—
558C22⋊C4—22no—
559C22⋊C4—22no—
560C22⋊C4—22no—
561C22⋊C4—41no—
562C22⋊C4—41no—
563C22⋊C4—44no—
564C22⋊C4—44no—
565C22⋊C4—44no—
566C22⋊C4—44no—
567C22⋊C4—82no—
568C22⋊C4—82no—
569C22⋊C4—82no—
570C22⋊C4—82no—
571C22⋊C4—82no—
572C22⋊C4—82no—
573C22⋊C4—82no—
574C22⋊C4—82no—
575C22⋊C4—82no—
576C22⋊C4—82no—
577C42—22no—
578C42—22no—
579C42—22no—
580C42—22no—
581C42—22no—
582C42—22no—
583C42—22no—
584C42—22no—
585C42—21no—
586C42—21no—
587C42—22no—
588C42—21no—
589C42—21no—
590C42—22no—
591C42—82no—
592C42—82no—
593C42—82no—
594C42—82no—
595C42C2×#4112yes—
596C422−1+412yes—
597C23—21no—
598C23—21no—
599C23—21no—
600C23—21no—
601C23—21no—
602C23—21no—
603C23—21no—
604C23—21no—
605C23—21no—
606C23—21no—
607C23—21no—
608C23—21no—
609C23—42no—
610C23—42no—
611C23—42no—
612C23—42no—
613C23—42no—
614C23—42no—
615C23—42no—
616C23#53212yes—
617C23C2×#13712yes—
618C23C23×D411yescharacteristic
619Q8—84no—
620D4—44no—
621D4—44no—
622D4—44no—
623D4—84no—
624D4—84no—
625D4—164no—
626D4—164no—
627C2×C4—21no—
628C2×C4—21no—
629C2×C4—21no—
630C2×C4—21no—
631C2×C4—28no—
632C2×C4—24no—
633C2×C4—24no—
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—21no—
650C2×C4—21no—
651C2×C4—21no—
652C2×C4—21no—
653C2×C4—21no—
654C2×C4—21no—
655C2×C4—21no—
656C2×C4—21no—
657C2×C4—21no—
658C2×C4—44no—
659C2×C4—44no—
660C2×C4—44no—
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—42no—
675C2×C4—42no—
676C2×C4—42no—
677C2×C4—42no—
678C2×C4—42no—
679C2×C4—42no—
680C2×C4—42no—
681C2×C4—42no—
682C2×C4—42no—
683C2×C4—42no—
684C2×C4—164no—
685C2×C4—164no—
686C2×C4#53212yes—
687C2×C4#54712yes—
688C2×C4#54712yes—
689C2×C4C2×#13712yes—
690C2×C4C2×#13712yes—
691C2×C4#57512yes—
692C2×C4#57612yes—
693C2×C4#57612yes—
694C2×C4C22×#4111yescharacteristic
695C2×C4C22×#4112yes—
696C2×C4C2×2−1+411yescharacteristic
697C2×C4#58512yes—
698C8—24no—
699C8—24no—
700C8—24no—
701C8—42no—
702C22—22no—
703C22—22no—
704C22—22no—
705C22—22no—
706C22—22no—
707C22—22no—
708C22—41no—
709C22—41no—
710C22—41no—
711C22—41no—
712C22—41no—
713C22—41no—
714C22—41no—
715C22—41no—
716C22—82no—
717C22—82no—
718C22—82no—
719C22—82no—
720C22—82no—
721C22—82no—
722C22—82no—
723C22C2×#53211yescharacteristic
724C22C2×#54711yescharacteristic
725C22#350011yescharacteristic
726C22C22×#13711yescharacteristic
727C22C2×#57511yescharacteristic
728C22C2×#57611yescharacteristic
729C22C2×#57611yescharacteristic
730C4—22no—
731C4—22no—
732C4—41no—
733C4—41no—
734C4—41no—
735C4—41no—
736C4—41no—
737C4—41no—
738C4—41no—
739C4—41no—
740C4—82no—
741C4—82no—
742C4—82no—
743C4—82no—
744C4—82no—
745C4—82no—
746C4—82no—
747C4#334614yes—
748C4#334714yes—
749C4#347512yes—
750C4#351512yes—
751C2—42no—
752C2—42no—
753C2—162no—
754C2#3618411yescharacteristic
755C2#6070111yescharacteristic
756C2#6071411yescharacteristic
757C2C2×#334611yescharacteristic
758C2C2×#334711yescharacteristic
759C2#6253711yescharacteristic
760C2#6254611yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#36211C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C42×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #36189C2semidirect
3. #33263C2semidirect
4. #33048C2semidirect
5. #33042C2semidirect
6. #29192C2semidirect
7. #29178C2semidirect
8. #28854C2semidirect
9. #28041C2semidirect
10. #27949C2semidirect
11. #27909C2semidirect
12. #27890C2semidirect
13. #27878C2semidirect
14. #25240C2semidirect
15. #22941C2semidirect
16. #21646C2semidirect
17. #20817C2semidirect
18. #20764C2semidirect
19. C4×#435C2semidirect

Maximal subgroups

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

subgroupindex
#362112
#361892
#332632
#330482
#330422
#291922
#291782
#288542
#280412
#279492
#279092
#278902
#278782
#252402
#229412
#216462
#208172
#207642
C4×#4352