Finite Group Databasebeta

← #6795519

Subgroups — #6795519

Order 512.

1563 conjugacy classes of subgroups.

The subgroup lattice has 682 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36187C211yesmaximal, characteristic
2#34945C211yesmaximal, characteristic, complemented
3#33198C211yesmaximal, characteristic, complemented
4#32293C214yesmaximal, complemented
5#31944C214yesmaximal, complemented
6#29925C212yesmaximal, complemented
7#28687C211yesmaximal, characteristic, complemented
8#28529C212yesmaximal, complemented
9#28279C211yesmaximal, characteristic, complemented
10#27728C212yesmaximal, complemented
11#27682C212yesmaximal, complemented
12#25700C212yesmaximal, complemented
13#24653C212yesmaximal, complemented
14#24121C212yesmaximal, complemented
15#22795C211yesmaximal, characteristic, complemented
16#21985C212yesmaximal, complemented
17C4×#460C211yesmaximal, characteristic, complemented
18C2×#555C2212yes—
19C2×#498C2212yes—
20C2×#477C2212yes—
21C2×#474C2212yes—
22C2×#460C2211yescharacteristic
23C2×C4×SD16C2212yes—
24C2×#431C2212yes—
25#2868C2211yescharacteristic
26#2856C2212yes—
27#2573—22no—
28#2571C2211yescharacteristic
29#2568C2211yescharacteristic
30#2563C2212yes—
31#2547C2211yescharacteristic
32#2538C2211yescharacteristic
33#2537—21no—
34#2537—21no—
35#2514—21no—
36#2500C2214yes—
37#2494C2212yes—
38#2491—22no—
39#2491—22no—
40#2447—21no—
41#2421C2212yes—
42#2401—22no—
43#2354—21no—
44#2354—21no—
45C4×C4⋊Q8C2211yescharacteristic
46C4×#127C2211yescharacteristic
47C4×#124—21no—
48C4×C22⋊Q8C2212yes—
49#2295—21no—
50#2285C2212yes—
51#2283C2212yes—
52C42×Q8C2211yescharacteristic
53C42×D4C2211yescharacteristic
54#2101C2214yes—
55#2097C2214yes—
56#2092C2214yes—
57#2084C2214yes—
58#2073C2214yes—
59#2067C2214yes—
60#2034C2214yes—
61#2034C2214yes—
62#2033C2214yes—
63#2000C2214yes—
64#1990C2212yes—
65#1987C2212yes—
66#1982C2212yes—
67#1982C2211yescharacteristic
68#1968C2214yes—
69#1954C2212yes—
70#1948C2212yes—
71#1948C2212yes—
72#1948C2211yescharacteristic
73#1947C2212yes—
74#1947C2212yes—
75#1947C2212yes—
76#1934C2214yes—
77#1926C2212yes—
78#1923C2214yes—
79#1880C2212yes—
80#1873C2214yes—
81#1854C2214yes—
82#1854C2214yes—
83#1854C2214yes—
84#1846C2214yes—
85#1842C2211yescharacteristic
86#1835C2214yes—
87#1803C2211yescharacteristic
88#1803C2212yes—
89#1803C2212yes—
90#1802C2212yes—
91#1797C2212yes—
92C8×C4⋊C4C2212yes—
93C4×C4⋊C8C2211yescharacteristic
94C4×#102C2212yes—
95C22×SD16C2312yes—
96#555—44no—
97#555—44no—
98C2×C4⋊Q8—21no—
99C2×C4⋊Q8—21no—
100C2×C4⋊Q8C2312yes—
101C2×C4⋊Q8C2311yescharacteristic
102C2×#127C2311yescharacteristic
103C2×#124—21no—
104C2×#124—24no—
105C2×C22⋊Q8C2312yes—
106C2×#121—22no—
107C2×#121—22no—
108C2×#121—22no—
109C2×#121C2314yes—
110C2×C4×Q8—21no—
111C2×C4×Q8C2311yescharacteristic
112C2×C4×Q8C2311yescharacteristic
113C2×C4×Q8C2311yescharacteristic
114C2×C4×D4—21no—
115C2×C4×D4—21no—
116C2×C4×D4D411yescharacteristic
117C2×C4×D4D411yescharacteristic
118C2×C4×D4C2312yes—
119C2×C4×D4C2311yescharacteristic
120C2×#117—22no—
121C2×#117C2314yes—
122C22×C42C2312yes—
123#498—44no—
124#477—44no—
125#474—44no—
126#460—24no—
127C4×SD16—44no—
128#431—28no—
129C2×#107C2314yes—
130C2×#107C2314yes—
131C2×#106C2314yes—
132C2×#106C2314yes—
133C2×#106C2312yes—
134C2×C4⋊C8C2312yes—
135C2×C4⋊C8C2311yescharacteristic
136C2×C4⋊C8C2312yes—
137C2×C4⋊C8C2312yes—
138C2×C4⋊C8C2311yescharacteristic
139C2×#103C2314yes—
140C2×#103C2312yes—
141C2×#102C2312yes—
142C2×#102C2314yes—
143C2×#102C2314yes—
144C2×#102C2312yes—
145C2×#102C2312yes—
146C2×C22⋊C8C2314yes—
147C2×C22⋊C8C2314yes—
148C2×C4×C8C2314yes—
149C2×C4×C8C2312yes—
150#401—22no—
151#400—22no—
152#400—22no—
153#400—22no—
154#397—22no—
155#397—22no—
156#397—22no—
157#396—22no—
158#394—24no—
159#394—24no—
160#393—24no—
161#391—21no—
162#391—21no—
163#391—21no—
164#391—21no—
165#391C2312yes—
166#390—21no—
167#390—21no—
168#390—21no—
169#390—22no—
170#390—22no—
171#390C2312yes—
172#390C2314yes—
173#389—21no—
174#389—21no—
175#389—21no—
176#389C2312yes—
177#389C2312yes—
178#388—22no—
179#388—22no—
180#388—24no—
181#388—24no—
182#387—22no—
183#387—22no—
184#387—22no—
185#385—22no—
186#384—21no—
187#384—21no—
188#384C2311yescharacteristic
189#384C2311yescharacteristic
190#384C2312yes—
191#384C2312yes—
192#383—22no—
193#382—21no—
194#382C2314yes—
195#382C2312yes—
196#382C2312yes—
197#381—22no—
198#380—22no—
199#380—22no—
200#380—22no—
201#380—22no—
202#380C2312yes—
203#380C2312yes—
204#380C2312yes—
205C4×C4⋊C4—21no—
206C4×C4⋊C4—21no—
207C4×C4⋊C4—22no—
208C4×C4⋊C4—22no—
209C4×C4⋊C4—21no—
210C4×C4⋊C4—22no—
211C4×C4⋊C4—21no—
212C4×C4⋊C4D411yescharacteristic
213C4×C4⋊C4D411yescharacteristic
214C4×C4⋊C4C2312yes—
215C4×C4⋊C4C2311yescharacteristic
216C4×C4⋊C4C2311yescharacteristic
217C4×C4⋊C4C2311yescharacteristic
218C4×C4⋊C4C2311yescharacteristic
219C4×C22⋊C4—22no—
220C4×C22⋊C4—22no—
221C4×C22⋊C4D412yes—
222C4×C22⋊C4D412yes—
223C4×C22⋊C4C2312yes—
224#376—22no—
225C43—21no—
226C43C2311yescharacteristic
227#340C2314yes—
228#340C2314yes—
229#340C2312yes—
230#340C2314yes—
231#340C2314yes—
232#340C2314yes—
233#340C2312yes—
234#336C2314yes—
235#336C2312yes—
236C22×Q8—21no—
237C22×Q8—22no—
238C22×Q8C2411yescharacteristic
239C22×D4—24no—
240C22×D4—21no—
241C22×D4—21no—
242C22×D4C2×D411yescharacteristic
243C22×D4C2412yes—
244C23×C4—22no—
245C23×C4C2×D412yes—
246C23×C4C2×D412yes—
247C23×C4C2412yes—
248C2×SD16—24no—
249C2×SD16—24no—
250C2×SD16—42no—
251C22×C8C2412yes—
252C22×C8C2414yes—
253C22×C8C2412yes—
254C4⋊Q8—24no—
255C4⋊Q8—24no—
256C4⋊Q8—22no—
257C4⋊Q8—22no—
258C4⋊Q8—82no—
259C4⋊Q8—82no—
260#127—22no—
261#127—22no—
262#124—416no—
263#124—44no—
264C22⋊Q8—24no—
265C22⋊Q8—42no—
266#121—44no—
267#121—44no—
268#121—48no—
269#121—84no—
270#121—84no—
271C4×Q8—22no—
272C4×Q8—22no—
273C4×Q8—22no—
274C4×Q8—22no—
275C4×Q8—22no—
276C4×Q8—41no—
277C4×Q8—44no—
278C4×D4—24no—
279C4×D4—22no—
280C4×D4—22no—
281C4×D4—24no—
282C4×D4—24no—
283C4×D4—42no—
284C4×D4—44no—
285C4×D4—44no—
286#117—44no—
287#117—44no—
288#117—48no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—21no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4C2×D411yescharacteristic
306C2×C4⋊C4C2×D411yescharacteristic
307C2×C4⋊C4C2×D411yescharacteristic
308C2×C4⋊C4C2412yes—
309C2×C4⋊C4C2414yes—
310C2×C4⋊C4C2411yescharacteristic
311C2×C4⋊C4C2411yescharacteristic
312C2×C4⋊C4C2411yescharacteristic
313C2×C4⋊C4C2412yes—
314C2×C4⋊C4C2412yes—
315C2×C22⋊C4—24no—
316C2×C22⋊C4—22no—
317C2×C22⋊C4—22no—
318C2×C22⋊C4—22no—
319C2×C22⋊C4—24no—
320C2×C22⋊C4—22no—
321C2×C22⋊C4—22no—
322C2×C22⋊C4—22no—
323C2×C22⋊C4—22no—
324C2×C22⋊C4—24no—
325C2×C22⋊C4—24no—
326C2×C22⋊C4C2×D412yes—
327C2×C22⋊C4C2×D412yes—
328C2×C22⋊C4C2×D412yes—
329C2×C42—24no—
330C2×C42—24no—
331C2×C42—24no—
332C2×C42—22no—
333C2×C42—21no—
334C2×C42—21no—
335C2×C42—21no—
336C2×C42—21no—
337C2×C42—21no—
338C2×C42—21no—
339C2×C42C2×D411yescharacteristic
340C2×C42C2×D411yescharacteristic
341C2×C42C2×D412yes—
342C2×C42C2×D412yes—
343C2×C42C2×D411yescharacteristic
344C2×C42C2×D411yescharacteristic
345C2×C42C2411yescharacteristic
346C2×C42C2412yes—
347C2×C42C2411yescharacteristic
348C2×C42C2411yescharacteristic
349#107—44no—
350#107—44no—
351#106—42no—
352#106—44no—
353#106—44no—
354C4⋊C8—24no—
355C4⋊C8—24no—
356C4⋊C8—22no—
357C4⋊C8—42no—
358C4⋊C8#4114yes—
359#103—42no—
360#103—44no—
361#102—24no—
362#102—42no—
363#102—44no—
364#102—42no—
365#102—44no—
366C22⋊C8—28no—
367C22⋊C8—44no—
368C4×C8—28no—
369C4×C8—24no—
370#95—22no—
371#95—22no—
372#95—22no—
373#95—22no—
374#95—22no—
375#95—22no—
376#95—22no—
377#95—22no—
378#95—21no—
379#95—21no—
380#95—21no—
381#95—24no—
382#95—24no—
383#95—22no—
384#95—22no—
385#95—22no—
386#95—22no—
387#95C2×D412yes—
388#95C2×D412yes—
389#95C2×D412yes—
390C24—22no—
391C24C22×D412yes—
392C2×Q8—21no—
393C2×Q8—21no—
394C2×Q8—21no—
395C2×Q8—21no—
396C2×Q8—21no—
397C2×Q8—42no—
398C2×Q8—44no—
399C2×Q8—42no—
400C2×Q8—42no—
401C2×Q8—82no—
402C2×Q8—82no—
403C2×Q82−1+412yes—
404C2×D4—22no—
405C2×D4—22no—
406C2×D4—22no—
407C2×D4—22no—
408C2×D4—22no—
409C2×D4—22no—
410C2×D4—22no—
411C2×D4—22no—
412C2×D4—22no—
413C2×D4—42no—
414C2×D4—42no—
415C2×D4—48no—
416C2×D4—42no—
417C2×D4—42no—
418C2×D4—42no—
419C2×D4—42no—
420C2×D4—84no—
421C2×D4—84no—
422C22×C4—21no—
423C22×C4—24no—
424C22×C4—24no—
425C22×C4—24no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—24no—
433C22×C4—21no—
434C22×C4—21no—
435C22×C4—22no—
436C22×C4—21no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—24no—
440C22×C4—42no—
441C22×C4—42no—
442C22×C4—44no—
443C22×C4—44no—
444C22×C4—44no—
445C22×C4C22×D412yes—
446C22×C4C22×D412yes—
447C22×C4C22×D412yes—
448C22×C4C22×D411yescharacteristic
449C22×C4C22×D411yescharacteristic
450C22×C4C22×D411yescharacteristic
451C22×C4C22×D411yescharacteristic
452C22×C4C22×D411yescharacteristic
453C22×C4C22×D411yescharacteristic
454C22×C4C2511yescharacteristic
455SD16—48no—
456C2×C8—22no—
457C2×C8—22no—
458C2×C8—24no—
459C2×C8—22no—
460C2×C8—24no—
461C2×C8—22no—
462C2×C8—22no—
463C2×C8—24no—
464C2×C8C2×#4114yes—
465C4⋊C4—22no—
466C4⋊C4—22no—
467C4⋊C4—22no—
468C4⋊C4—21no—
469C4⋊C4—21no—
470C4⋊C4—22no—
471C4⋊C4—22no—
472C4⋊C4—24no—
473C4⋊C4—22no—
474C4⋊C4—24no—
475C4⋊C4—21no—
476C4⋊C4—22no—
477C4⋊C4—21no—
478C4⋊C4—21no—
479C4⋊C4—22no—
480C4⋊C4—21no—
481C4⋊C4—22no—
482C4⋊C4—42no—
483C4⋊C4—44no—
484C4⋊C4—42no—
485C4⋊C4—44no—
486C4⋊C4—44no—
487C4⋊C4—42no—
488C4⋊C4—42no—
489C4⋊C4—42no—
490C4⋊C4—42no—
491C4⋊C4—42no—
492C4⋊C4—44no—
493C4⋊C4—44no—
494C4⋊C4—81no—
495C4⋊C4—82no—
496C4⋊C4—82no—
497C4⋊C4—82no—
498C22⋊C4—24no—
499C22⋊C4—24no—
500C22⋊C4—42no—
501C22⋊C4—44no—
502C22⋊C4—48no—
503C22⋊C4—44no—
504C22⋊C4—48no—
505C22⋊C4—44no—
506C22⋊C4—44no—
507C22⋊C4—44no—
508C22⋊C4—44no—
509C22⋊C4—84no—
510C22⋊C4—84no—
511C22⋊C4—82no—
512C42—24no—
513C42—24no—
514C42—22no—
515C42—22no—
516C42—22no—
517C42—22no—
518C42—21no—
519C42—21no—
520C42—41no—
521C42—42no—
522C42—48no—
523C42—42no—
524C42—42no—
525C42—44no—
526C42—42no—
527C42—42no—
528C42—81no—
529C42#13514yes—
530C42C2×#4112yes—
531C42H(5,𝔽2)12yes—
532C42H(5,𝔽2)12yes—
533C422−1+412yes—
534C23—22no—
535C23—22no—
536C23—22no—
537C23—22no—
538C23—22no—
539C23—22no—
540C23—22no—
541C23—42no—
542C23—42no—
543C23—42no—
544C23—42no—
545C23—42no—
546C23—42no—
547C23—42no—
548C23C23×D411yescharacteristic
549Q8—22no—
550Q8—22no—
551Q8—41no—
552Q8—41no—
553Q8—84no—
554Q8—164no—
555D4—44no—
556D4—42no—
557D4—42no—
558D4—42no—
559D4—42no—
560D4—84no—
561D4—84no—
562D4—168no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—21no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—21no—
579C2×C4—22no—
580C2×C4—21no—
581C2×C4—22no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—48no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—44no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—44no—
595C2×C4—41no—
596C2×C4—42no—
597C2×C4—41no—
598C2×C4—41no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—41no—
604C2×C4—41no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—41no—
608C2×C4—41no—
609C2×C4—44no—
610C2×C4—84no—
611C2×C4—88no—
612C2×C4#53512yes—
613C2×C4#53512yes—
614C2×C4#54712yes—
615C2×C4#54712yes—
616C2×C4C2×#13512yes—
617C2×C4C2×#13512yes—
618C2×C4C2×#13512yes—
619C2×C4#57512yes—
620C2×C4#57712yes—
621C2×C4C22×#4111yescharacteristic
622C2×C4C2×H(5,𝔽2)11yescharacteristic
623C2×C4C2×H(5,𝔽2)11yescharacteristic
624C2×C4C2×2−1+411yescharacteristic
625C2×C4C2×2−1+411yescharacteristic
626C2×C4#58511yescharacteristic
627C8—24no—
628C8—44no—
629C8—42no—
630C22—42no—
631C22—42no—
632C22—42no—
633C22—42no—
634C22—42no—
635C22—42no—
636C22—42no—
637C22—82no—
638C22—82no—
639C22—82no—
640C22—82no—
641C22—82no—
642C22—82no—
643C22—82no—
644C22C2×#53511yescharacteristic
645C22C2×#54711yescharacteristic
646C22#348911yescharacteristic
647C22C22×#13511yescharacteristic
648C22C2×#57511yescharacteristic
649C22C2×#57711yescharacteristic
650C22#358511yescharacteristic
651C4—22no—
652C4—22no—
653C4—22no—
654C4—42no—
655C4—42no—
656C4—42no—
657C4—42no—
658C4—41no—
659C4—81no—
660C4—82no—
661C4—82no—
662C4—82no—
663C4—81no—
664C4—84no—
665C4—81no—
666C4#321214yes—
667C4#334014yes—
668C4#351711yescharacteristic
669C4#352011yescharacteristic
670C4#354711yescharacteristic
671C4#355211yescharacteristic
672C2—82no—
673C2—162no—
674C2#3892011yescharacteristic
675C2C2×#321211yescharacteristic
676C2#6083311yescharacteristic
677C2C2×#334011yescharacteristic
678C2#6125611yescharacteristic
679C2#6210211yescharacteristic
680C2#6214511yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#36187C2
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. #34945C2semidirect
3. #33198C2semidirect
4. #32293C2semidirect
5. #31944C2semidirect
6. #29925C2semidirect
7. #28687C2semidirect
8. #28529C2semidirect
9. #28279C2semidirect
10. #27728C2semidirect
11. #27682C2semidirect
12. #25700C2semidirect
13. #24653C2semidirect
14. #24121C2semidirect
15. #22795C2semidirect
16. #21985C2semidirect
17. C4×#460C2semidirect

Maximal subgroups

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

subgroupindex
#361872
#349452
#331982
#322932
#319442
#299252
#286872
#285292
#282792
#277282
#276822
#257002
#246532
#241212
#227952
#219852
C4×#4602