Finite Group Databasebeta

← #6480830

Subgroups — #6480830

Order 512.

1310 conjugacy classes of subgroups.

The subgroup lattice has 680 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35291C211yesmaximal, characteristic
2#35286C211yesmaximal, characteristic
3#33287C211yesmaximal, characteristic, complemented
4#33278C211yesmaximal, characteristic, complemented
5#33263C211yesmaximal, characteristic, complemented
6#33258C211yesmaximal, characteristic, complemented
7#30391C214yesmaximal, complemented
8#29585C214yesmaximal, complemented
9#29193C211yesmaximal, characteristic, complemented
10#29191C211yesmaximal, characteristic, complemented
11#29179C211yesmaximal, characteristic, complemented
12#29178C211yesmaximal, characteristic, complemented
13#27975C211yesmaximal, characteristic
14#27960C211yesmaximal, characteristic
15#27949C211yesmaximal, characteristic
16#27910C211yesmaximal, characteristic
17#24614C214yesmaximal
18#22087C214yesmaximal
19C4×#433C211yesmaximal, characteristic
20C2×#484C2214yes—
21C2×#433C2212yes—
22#2862C2211yescharacteristic
23#2859C2211yescharacteristic
24#2859C2211yescharacteristic
25#2859C2211yescharacteristic
26#2859C2211yescharacteristic
27#2859C2211yescharacteristic
28#2857C2211yescharacteristic
29#2857C2211yescharacteristic
30#2647C2212yes—
31#2647C2212yes—
32#2645C2212yes—
33#2625C2212yes—
34#2571—21no—
35#2571—21no—
36#2552—21no—
37#2552—21no—
38#2514—21no—
39#2509—21no—
40#2508—21no—
41#2495—21no—
42#2494—21no—
43#2489—21no—
44#2453—21no—
45#2442—21no—
46#2442—21no—
47#2442—21no—
48#2442—21no—
49#2421—21no—
50#2323C2212yes—
51#2320C2212yes—
52#2317C2212yes—
53#2316C2212yes—
54C4×C4⋊Q8C2211yescharacteristic
55C4×C4⋊Q8C2211yescharacteristic
56#2285C2211yescharacteristic
57C2×#384C2211yescharacteristic
58C4×#117C2211yescharacteristic
59#2099C2214yes—
60#2095C2214yes—
61#2087C2214yes—
62#2086C2214yes—
63#2085C2214yes—
64#2074C2214yes—
65#2068C2214yes—
66#2066C2214yes—
67#1990C2211yescharacteristic
68#1990C2211yescharacteristic
69#1990C2211yescharacteristic
70#1990C2211yescharacteristic
71#1956C2211yescharacteristic
72#1956C2211yescharacteristic
73#1956C2211yescharacteristic
74#1956C2211yescharacteristic
75#1944C2214yes—
76#1943C2214yes—
77#1942C2214yes—
78#1942C2214yes—
79#1935C2214yes—
80#1935C2214yes—
81#1933C2214yes—
82#1933C2214yes—
83#1921C2214yes—
84#1918C2214yes—
85#1873C2214yes—
86#1865C2214yes—
87#1863C2214yes—
88#1854C2214yes—
89#1845C2211yescharacteristic
90#1845C2211yescharacteristic
91#1845C2211yescharacteristic
92#1844C2211yescharacteristic
93#1834C2211yescharacteristic
94#1834C2211yescharacteristic
95#1834C2211yescharacteristic
96#1834C2211yescharacteristic
97#1823C2214yes—
98#1768C2212yes—
99C4×C4⋊C8C2211yescharacteristic
100C4×C4⋊C8C2211yescharacteristic
101#1750C2212yes—
102C4×C22⋊C8C2211yescharacteristic
103C4×C22⋊C8C2211yescharacteristic
104#1746C2212yes—
105C4×#97C2211yescharacteristic
106C42×C8C2211yescharacteristic
107C2×C4⋊Q8C2311yescharacteristic
108C2×C4⋊Q8C2311yescharacteristic
109C2×#125—21no—
110C2×#125—21no—
111C2×#125—21no—
112C2×#125—21no—
113C2×#125—21no—
114C2×C22⋊Q8—21no—
115C2×C22⋊Q8C2312yes—
116C2×C4×Q8—21no—
117C2×C4×Q8C2312yes—
118C2×#117C2312yes—
119C2×#117C2311yescharacteristic
120C2×#117C2311yescharacteristic
121C22×C4⋊C4—21no—
122C22×C4⋊C4—21no—
123C22×C4⋊C4C2312yes—
124C22×C42C2311yescharacteristic
125#484—216no—
126#433—28no—
127C2×#107C2314yes—
128C2×#107C2314yes—
129C2×#107C2314yes—
130C2×#106C2314yes—
131C2×C4⋊C8C2312yes—
132C2×C4⋊C8C2312yes—
133C2×#103C2314yes—
134C2×#103C2314yes—
135C2×#103C2314yes—
136C2×#103C2314yes—
137C2×C22⋊C8C2312yes—
138C2×C22⋊C8C2312yes—
139C2×#97C2312yes—
140C2×#97C2312yes—
141C2×C4×C8C2311yescharacteristic
142C2×C4×C8C2311yescharacteristic
143C2×C4×C8C2311yescharacteristic
144C2×C4×C8C2311yescharacteristic
145C2×C4×C8C2312yes—
146C2×C4×C8C2312yes—
147#401—22no—
148#400—22no—
149#400—22no—
150#400—22no—
151#400—22no—
152#400—22no—
153#400—22no—
154#400—22no—
155#400—22no—
156#398—22no—
157#398—22no—
158#398—22no—
159#398—22no—
160#398—22no—
161#397—22no—
162#397—22no—
163#391—21no—
164#391—21no—
165#391—21no—
166#391—21no—
167#391C2312yes—
168#391C2312yes—
169#391C2312yes—
170#391C2312yes—
171#389—22no—
172#389—22no—
173#389—22no—
174#389—21no—
175#389—21no—
176#389—21no—
177#389—21no—
178#389—21no—
179#389—21no—
180#389—21no—
181#389—21no—
182#389—21no—
183#389—21no—
184#389—21no—
185#389—21no—
186#389—21no—
187#389—21no—
188#389—21no—
189#389—21no—
190#389C2312yes—
191#389C2312yes—
192#389C2312yes—
193#389C2312yes—
194#389C2312yes—
195#389C2312yes—
196#389C2312yes—
197#389C2312yes—
198#389C2312yes—
199#389C2312yes—
200#389C2312yes—
201#389C2312yes—
202#388—22no—
203#388—22no—
204#387—22no—
205#387—22no—
206#387—22no—
207#385—22no—
208#385—22no—
209#384—22no—
210#384—21no—
211#384—21no—
212#384C2311yescharacteristic
213#384C2311yescharacteristic
214#384C2311yescharacteristic
215#384C2311yescharacteristic
216#384C2311yescharacteristic
217#384C2311yescharacteristic
218#384C2312yes—
219#383—22no—
220#382—22no—
221#382—21no—
222#382—21no—
223#382—21no—
224#382—21no—
225#382—21no—
226#382—21no—
227#382—21no—
228#382—22no—
229#382C2312yes—
230#380—21no—
231#380—21no—
232#380—21no—
233#380—21no—
234#380—21no—
235#380—22no—
236#380—22no—
237#380C2312yes—
238C4×C4⋊C4—21no—
239C4×C4⋊C4—21no—
240C4×C4⋊C4—21no—
241C4×C4⋊C4—21no—
242C4×C4⋊C4—21no—
243C4×C4⋊C4D411yescharacteristic
244C4×C4⋊C4D411yescharacteristic
245C4×C4⋊C4D411yescharacteristic
246C4×C4⋊C4D411yescharacteristic
247C4×C4⋊C4C2312yes—
248C4×C22⋊C4D411yescharacteristic
249C4×C22⋊C4D411yescharacteristic
250C4×C22⋊C4D411yescharacteristic
251C4×C22⋊C4D411yescharacteristic
252#376C2311yescharacteristic
253#376C2311yescharacteristic
254C43C2311yescharacteristic
255C43C2311yescharacteristic
256#340C2314yes—
257#340C2314yes—
258#340C2314yes—
259#340C2314yes—
260#340C2314yes—
261#340C2314yes—
262#340C2314yes—
263#340C2314yes—
264#336C2312yes—
265#336C2312yes—
266#336C2312yes—
267#336C2312yes—
268C22×Q8—21no—
269C22×Q8C2412yes—
270C23×C4—22no—
271C23×C4C2×D411yescharacteristic
272C23×C4C2×D411yescharacteristic
273C23×C4C2411yescharacteristic
274C22×C8C2412yes—
275C22×C8C2412yes—
276C22×C8C2412yes—
277C22×C8C2412yes—
278C4⋊Q8—22no—
279C4⋊Q8—22no—
280C4⋊Q8—22no—
281C4⋊Q8—22no—
282#125—82no—
283#125—82no—
284#125—82no—
285#125—82no—
286#125—82no—
287C22⋊Q8—28no—
288C22⋊Q8—82no—
289C4×Q8—44no—
290C4×Q8—82no—
291#117—24no—
292#117—42no—
293#117—44no—
294C2×C4⋊C4—28no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—21no—
305C2×C4⋊C4—21no—
306C2×C4⋊C4—21no—
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—22no—
314C2×C4⋊C4—22no—
315C2×C4⋊C4—22no—
316C2×C4⋊C4—22no—
317C2×C4⋊C4—22no—
318C2×C4⋊C4—22no—
319C2×C4⋊C4—22no—
320C2×C4⋊C4—22no—
321C2×C4⋊C4—22no—
322C2×C4⋊C4—22no—
323C2×C4⋊C4—22no—
324C2×C4⋊C4—22no—
325C2×C4⋊C4—22no—
326C2×C4⋊C4—22no—
327C2×C4⋊C4—82no—
328C2×C4⋊C4—82no—
329C2×C4⋊C4C2×D411yescharacteristic
330C2×C4⋊C4C2×D411yescharacteristic
331C2×C4⋊C4C2×D411yescharacteristic
332C2×C4⋊C4C2×D411yescharacteristic
333C2×C4⋊C4C2412yes—
334C2×C4⋊C4C2412yes—
335C2×C4⋊C4C2412yes—
336C2×C4⋊C4C2412yes—
337C2×C4⋊C4C2412yes—
338C2×C4⋊C4C2412yes—
339C2×C4⋊C4C2412yes—
340C2×C22⋊C4—22no—
341C2×C22⋊C4—22no—
342C2×C22⋊C4—22no—
343C2×C22⋊C4C2×D411yescharacteristic
344C2×C22⋊C4C2×D411yescharacteristic
345C2×C22⋊C4C2×D411yescharacteristic
346C2×C22⋊C4C2×D411yescharacteristic
347C2×C42—22no—
348C2×C42—22no—
349C2×C42—22no—
350C2×C42—22no—
351C2×C42—22no—
352C2×C42C2×D411yescharacteristic
353C2×C42C2×D411yescharacteristic
354C2×C42C2×D411yescharacteristic
355C2×C42C2×D411yescharacteristic
356C2×C42C2×D411yescharacteristic
357C2×C42C2×D411yescharacteristic
358C2×C42C2×D411yescharacteristic
359C2×C42C2×D411yescharacteristic
360C2×C42C2×D411yescharacteristic
361C2×C42C2×D411yescharacteristic
362C2×C42C2411yescharacteristic
363C2×C42C2411yescharacteristic
364C2×C42C2411yescharacteristic
365C2×C42C2411yescharacteristic
366C2×C42C2411yescharacteristic
367C2×C42C2411yescharacteristic
368#107—28no—
369#107—28no—
370#107—44no—
371#106—44no—
372C4⋊C8—24no—
373C4⋊C8—24no—
374#103—28no—
375#103—28no—
376#103—44no—
377#103—44no—
378C22⋊C8—24no—
379C22⋊C8#4118yes—
380#97—24no—
381#97—42no—
382C4×C8—24no—
383C4×C8—22no—
384C4×C8—22no—
385C4×C8—22no—
386C4×C8—22no—
387C4×C8—42no—
388#95—22no—
389#95—22no—
390#95—22no—
391#95—22no—
392#95—22no—
393#95—22no—
394#95—22no—
395#95—22no—
396#95—22no—
397#95—22no—
398#95—22no—
399#95—22no—
400#95—22no—
401#95—22no—
402#95—22no—
403#95—22no—
404#95—22no—
405#95—22no—
406#95—22no—
407#95—22no—
408#95—22no—
409#95—22no—
410#95—22no—
411#95—22no—
412#95C2×D411yescharacteristic
413#95C2×D411yescharacteristic
414#95C2×D411yescharacteristic
415#95C2×D411yescharacteristic
416#95C2×D411yescharacteristic
417#95C2×D411yescharacteristic
418#95C2×D411yescharacteristic
419#95C2×D411yescharacteristic
420C24C22×D411yescharacteristic
421C2×Q8—24no—
422C2×Q8—22no—
423C2×Q8—22no—
424C2×Q8—24no—
425C2×Q8—42no—
426C2×Q8—42no—
427C2×Q8—42no—
428C22×C4—21no—
429C22×C4—21no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—44no—
446C22×C4—82no—
447C22×C4—82no—
448C22×C4C22×D411yescharacteristic
449C22×C4C22×D411yescharacteristic
450C22×C4C22×D411yescharacteristic
451C22×C4C22×D411yescharacteristic
452C22×C4C22×D411yescharacteristic
453C22×C4C22×D411yescharacteristic
454C22×C4C22×D411yescharacteristic
455C22×C4C22×D411yescharacteristic
456C22×C4C22×D411yescharacteristic
457C22×C4C22×D411yescharacteristic
458C22×C4C22×D411yescharacteristic
459C22×C4C22×D411yescharacteristic
460C22×C4C22×D411yescharacteristic
461C22×C4C2×#4114yes—
462C22×C4C2511yescharacteristic
463C2×C8—22no—
464C2×C8—22no—
465C2×C8—22no—
466C2×C8—22no—
467C2×C8—22no—
468C2×C8—22no—
469C2×C8—22no—
470C2×C8—22no—
471C2×C8—22no—
472C2×C8—22no—
473C2×C8C2×#4114yes—
474C2×C8C2×#4114yes—
475C4⋊C4—22no—
476C4⋊C4—22no—
477C4⋊C4—24no—
478C4⋊C4—24no—
479C4⋊C4—24no—
480C4⋊C4—22no—
481C4⋊C4—22no—
482C4⋊C4—24no—
483C4⋊C4—24no—
484C4⋊C4—24no—
485C4⋊C4—41no—
486C4⋊C4—41no—
487C4⋊C4—44no—
488C4⋊C4—44no—
489C4⋊C4—44no—
490C4⋊C4—42no—
491C4⋊C4—42no—
492C4⋊C4—42no—
493C4⋊C4—42no—
494C4⋊C4—42no—
495C4⋊C4—42no—
496C4⋊C4—42no—
497C4⋊C4—42no—
498C4⋊C4—42no—
499C4⋊C4—42no—
500C4⋊C4—42no—
501C4⋊C4—42no—
502C4⋊C4—42no—
503C4⋊C4—42no—
504C4⋊C4—42no—
505C4⋊C4—82no—
506C4⋊C4—82no—
507C4⋊C4—82no—
508C4⋊C4—82no—
509C4⋊C4—82no—
510C4⋊C4—82no—
511C4⋊C4—82no—
512C4⋊C4—82no—
513C4⋊C4—82no—
514C4⋊C4—82no—
515C4⋊C4—82no—
516C4⋊C4—82no—
517C4⋊C4—82no—
518C4⋊C4—82no—
519C22⋊C4—22no—
520C22⋊C4—22no—
521C22⋊C4—41no—
522C22⋊C4—41no—
523C22⋊C4—44no—
524C22⋊C4—44no—
525C22⋊C4—82no—
526C42—22no—
527C42—22no—
528C42—22no—
529C42—22no—
530C42—22no—
531C42—22no—
532C42—22no—
533C42—22no—
534C42—22no—
535C42—22no—
536C42—22no—
537C42—22no—
538C42—21no—
539C42—21no—
540C42—22no—
541C42—22no—
542C42—44no—
543C42—44no—
544C42—82no—
545C422−1+412yes—
546C422−1+412yes—
547C23—21no—
548C23—21no—
549C23—21no—
550C23—21no—
551C23—21no—
552C23—21no—
553C23#53212yes—
554C23C23×D411yescharacteristic
555Q8—44no—
556Q8—44no—
557Q8—84no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—24no—
561C2×C4—24no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—21no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—21no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—21no—
592C2×C4—44no—
593C2×C4—44no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—164no—
619C2×C4#53212yes—
620C2×C4#53612yes—
621C2×C4#53612yes—
622C2×C4#57612yes—
623C2×C4#57712yes—
624C2×C4#57812yes—
625C2×C4#57812yes—
626C2×C4C22×#4112yes—
627C2×C4C2×2−1+411yescharacteristic
628C2×C4C2×2−1+411yescharacteristic
629C2×C4#58512yes—
630C8—24no—
631C8—24no—
632C8—42no—
633C8—42no—
634C22—22no—
635C22—22no—
636C22—22no—
637C22—41no—
638C22—41no—
639C22—41no—
640C22—41no—
641C22C2×#53211yescharacteristic
642C22C2×#53611yescharacteristic
643C22#344911yescharacteristic
644C22C2×#57611yescharacteristic
645C22C2×#57711yescharacteristic
646C22C2×#57811yescharacteristic
647C22C2×#57811yescharacteristic
648C4—22no—
649C4—41no—
650C4—41no—
651C4—41no—
652C4—41no—
653C4—41no—
654C4—41no—
655C4—41no—
656C4—41no—
657C4—41no—
658C4—41no—
659C4—82no—
660C4—82no—
661C4—82no—
662C4—82no—
663C4—82no—
664C4—82no—
665C4—82no—
666C4—82no—
667C4#324214yes—
668C4#324314yes—
669C4#351512yes—
670C4#351512yes—
671C2—42no—
672C2#3513911yescharacteristic
673C2#6071511yescharacteristic
674C2#6072211yescharacteristic
675C2C2×#324211yescharacteristic
676C2C2×#324311yescharacteristic
677C2#6100811yescharacteristic
678C2#6101411yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC24C22×D4
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
3. #33287C2semidirect
4. #33278C2semidirect
5. #33263C2semidirect
6. #33258C2semidirect
7. #30391C2semidirect
8. #29585C2semidirect
9. #29193C2semidirect
10. #29191C2semidirect
11. #29179C2semidirect
12. #29178C2semidirect

Maximal subgroups

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

subgroupindex
#352912
#352862
#332872
#332782
#332632
#332582
#303912
#295852
#291932
#291912
#291792
#291782
#279752
#279602
#279492
#279102
#246142
#220872
C4×#4332