Finite Group Databasebeta

← #6519548

Subgroups — #6519548

Order 512.

1590 conjugacy classes of subgroups.

The subgroup lattice has 678 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35281C211yesmaximal, characteristic
2#34978C211yesmaximal, characteristic
3#29355C212yesmaximal, complemented
4#29349C212yesmaximal, complemented
5#28755C212yesmaximal, complemented
6#28748C212yesmaximal, complemented
7#27777C212yesmaximal, complemented
8#27769C212yesmaximal, complemented
9#27757C212yesmaximal, complemented
10#27749C212yesmaximal, complemented
11#27283C214yesmaximal
12#27277C214yesmaximal
13#21894C211yesmaximal, characteristic
14#21804C211yesmaximal, characteristic
15#21389C211yesmaximal, characteristic
16#21259C211yesmaximal, characteristic
17#20550C211yesmaximal, characteristic
18C2×#533C2211yescharacteristic
19C22×C4⋊Q8C2211yescharacteristic
20C2×#491C2214yes—
21C2×#487C2214yes—
22C2×#431C2211yescharacteristic
23C2×#423C2211yescharacteristic
24C2×#419C2212yes—
25C2×#419C2212yes—
26#2868C2212yes—
27#2862C2212yes—
28#2660—22no—
29#2612—22no—
30#2571C2211yescharacteristic
31#2567—21no—
32#2567—21no—
33#2552C2211yescharacteristic
34#2552C2211yescharacteristic
35#2551C2211yescharacteristic
36#2546C2212yes—
37#2546C2212yes—
38#2502—22no—
39#2441—21no—
40#2441C2212yes—
41#2439—21no—
42#2439C2212yes—
43#2434C2214yes—
44#2396—22no—
45#2317—21no—
46#2317—21no—
47C4×C4⋊Q8—21no—
48C4×C4⋊Q8C2212yes—
49C4×#125—21no—
50C4×#125C2212yes—
51#2285C2211yescharacteristic
52C4×#117C2211yescharacteristic
53C4×#117C2211yescharacteristic
54#2074C2214yes—
55#2072C2214yes—
56#2043C2214yes—
57#2043C2214yes—
58#2043C2214yes—
59#2043C2214yes—
60#2026C2214yes—
61#2024C2214yes—
62#1988C2212yes—
63#1988C2211yescharacteristic
64#1988C2211yescharacteristic
65#1981C2212yes—
66#1981C2212yes—
67#1981C2212yes—
68#1981C2212yes—
69#1981C2212yes—
70#1981C2212yes—
71#1981C2211yescharacteristic
72#1981C2212yes—
73#1981C2211yescharacteristic
74#1960C2214yes—
75#1956C2212yes—
76#1956C2212yes—
77#1956C2214yes—
78#1935C2214yes—
79#1935C2214yes—
80#1933C2214yes—
81#1933C2214yes—
82#1894C2214yes—
83#1873C2212yes—
84#1873C2212yes—
85#1803C2212yes—
86#1803C2212yes—
87#1787C2214yes—
88#1770C2212yes—
89C8×C4⋊C4C2212yes—
90#1767C2212yes—
91C4×C4⋊C8C2212yes—
92C4×#103C2212yes—
93C4×#103C2212yes—
94#1753C2214yes—
95C23×Q8C2312yes—
96C22×M16C2312yes—
97#533—24no—
98#533—24no—
99C2×C4⋊Q8—21no—
100C2×C4⋊Q8—21no—
101C2×C4⋊Q8—21no—
102C2×C4⋊Q8—24no—
103C2×C4⋊Q8—24no—
104C2×C4⋊Q8C2312yes—
105C2×C4⋊Q8C2311yescharacteristic
106C2×C4⋊Q8C2312yes—
107C2×C4⋊Q8C2314yes—
108C2×C4⋊Q8C2314yes—
109C2×C4⋊Q8C2311yescharacteristic
110C2×C4⋊Q8C2312yes—
111C2×#125—21no—
112C2×#125—21no—
113C2×#125C2312yes—
114C2×C22⋊Q8—21no—
115C2×C22⋊Q8C2312yes—
116C2×C4×Q8—21no—
117C2×C4×Q8—22no—
118C2×C4×Q8C2312yes—
119C2×#117—21no—
120C2×#117—21no—
121C2×#117D411yescharacteristic
122C2×#117D411yescharacteristic
123C2×#117C2311yescharacteristic
124C2×#117C2311yescharacteristic
125C22×C4⋊C4—21no—
126C22×C4⋊C4—21no—
127C22×C42C2311yescharacteristic
128#491—48no—
129#487—48no—
130#431—24no—
131#423—24no—
132C2×C4⋊C8C2312yes—
133C2×C4⋊C8C2312yes—
134C2×C4⋊C8C2311yescharacteristic
135C2×C4⋊C8C2311yescharacteristic
136#419—44no—
137#419—44no—
138C2×#103C2312yes—
139C2×#103C2314yes—
140C2×#103C2312yes—
141C2×#103C2314yes—
142C2×#103C2314yes—
143C2×#103C2314yes—
144C2×#103C2314yes—
145C2×#103C2314yes—
146C2×#103C2314yes—
147C2×#103C2314yes—
148C2×#103C2312yes—
149C2×#103C2312yes—
150C2×#103C2314yes—
151C2×#103C2314yes—
152C2×C22⋊C8C2312yes—
153C2×#97C2314yes—
154C2×C4×C8C2312yes—
155C2×C4×C8C2314yes—
156#400—22no—
157#399—22no—
158#398—22no—
159#398—22no—
160#398—24no—
161#398—24no—
162#396—22no—
163#396—22no—
164#395—24no—
165#395—24no—
166#395—22no—
167#393—22no—
168#391—22no—
169#391—22no—
170#391—22no—
171#391—22no—
172#391C2314yes—
173#391C2314yes—
174#389—21no—
175#389—21no—
176#389—21no—
177#389—21no—
178#389—22no—
179#389—22no—
180#389—22no—
181#389—22no—
182#389—22no—
183#389—22no—
184#389C2312yes—
185#389C2312yes—
186#389C2312yes—
187#389C2312yes—
188#387—22no—
189#387—22no—
190#387—22no—
191#387—22no—
192#385—24no—
193#385—24no—
194#384—21no—
195#384—22no—
196#384—22no—
197#384—22no—
198#384—22no—
199#384C2312yes—
200#384C2311yescharacteristic
201#384C2311yescharacteristic
202#382—21no—
203#382—22no—
204#382—22no—
205#382—21no—
206#382C2312yes—
207#380—21no—
208#380C2312yes—
209C4×C4⋊C4—22no—
210C4×C4⋊C4—21no—
211C4×C4⋊C4—22no—
212C4×C4⋊C4—22no—
213C4×C4⋊C4D411yescharacteristic
214C4×C4⋊C4D411yescharacteristic
215C4×C4⋊C4C2311yescharacteristic
216C4×C4⋊C4C2312yes—
217C4×C4⋊C4C2311yescharacteristic
218C4×C4⋊C4C2311yescharacteristic
219C4×C4⋊C4C2311yescharacteristic
220C4×C4⋊C4C2311yescharacteristic
221C4×C4⋊C4C2311yescharacteristic
222C4×C22⋊C4—21no—
223C4×C22⋊C4—21no—
224C4×C22⋊C4D412yes—
225C4×C22⋊C4C2312yes—
226#376—21no—
227#376D412yes—
228C43—21no—
229C43C2312yes—
230#340C2314yes—
231#340C2314yes—
232#340C2312yes—
233#340C2312yes—
234#340C2312yes—
235#340C2312yes—
236#336C2314yes—
237#336C2314yes—
238C22×Q8—21no—
239C22×Q8—22no—
240C22×Q8—22no—
241C22×Q8—24no—
242C22×Q8—24no—
243C22×Q8—42no—
244C22×Q8—42no—
245C22×Q8—42no—
246C22×Q8—42no—
247C22×Q8C2412yes—
248C22×Q8C2414yes—
249C23×C4—22no—
250C23×C4C2×D411yescharacteristic
251C23×C4C2×D411yescharacteristic
252C23×C4C2411yescharacteristic
253C2×M16—24no—
254C2×M16—42no—
255C2×M16—42no—
256C22×C8C2414yes—
257C22×C8C2412yes—
258C22×C8C2412yes—
259C4⋊Q8—24no—
260C4⋊Q8—22no—
261C4⋊Q8—22no—
262C4⋊Q8—24no—
263C4⋊Q8—24no—
264C4⋊Q8—24no—
265C4⋊Q8—24no—
266C4⋊Q8—22no—
267C4⋊Q8—22no—
268C4⋊Q8—42no—
269C4⋊Q8—44no—
270C4⋊Q8—44no—
271C4⋊Q8—44no—
272C4⋊Q8—44no—
273C4⋊Q8—44no—
274C4⋊Q8—44no—
275C4⋊Q8—44no—
276#125—24no—
277#125—42no—
278#125—44no—
279#125—44no—
280C22⋊Q8—24no—
281C22⋊Q8—24no—
282C22⋊Q8—44no—
283C4×Q8—24no—
284C4×Q8—24no—
285C4×Q8—44no—
286C4×Q8—84no—
287#117—24no—
288#117—24no—
289#117—22no—
290#117—22no—
291#117—41no—
292#117—41no—
293#117—44no—
294#117—82no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—21no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—21no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—21no—
313C2×C4⋊C4—21no—
314C2×C4⋊C4—22no—
315C2×C4⋊C4—22no—
316C2×C4⋊C4—22no—
317C2×C4⋊C4—24no—
318C2×C4⋊C4—24no—
319C2×C4⋊C4—24no—
320C2×C4⋊C4—24no—
321C2×C4⋊C4—44no—
322C2×C4⋊C4—44no—
323C2×C4⋊C4C2×D411yescharacteristic
324C2×C4⋊C4C2×D411yescharacteristic
325C2×C4⋊C4C2×D411yescharacteristic
326C2×C4⋊C4C2×D411yescharacteristic
327C2×C4⋊C4C2411yescharacteristic
328C2×C4⋊C4C2411yescharacteristic
329C2×C4⋊C4C2412yes—
330C2×C4⋊C4C2412yes—
331C2×C4⋊C4C2412yes—
332C2×C4⋊C4C2411yescharacteristic
333C2×C4⋊C4C2411yescharacteristic
334C2×C22⋊C4—22no—
335C2×C22⋊C4—21no—
336C2×C22⋊C4—21no—
337C2×C22⋊C4—22no—
338C2×C22⋊C4C2×D412yes—
339C2×C22⋊C4C2×D412yes—
340C2×C42—22no—
341C2×C42—22no—
342C2×C42—21no—
343C2×C42—21no—
344C2×C42—21no—
345C2×C42—21no—
346C2×C42—21no—
347C2×C42—21no—
348C2×C42C2×D412yes—
349C2×C42C2×D412yes—
350C2×C42C2×D412yes—
351C2×C42C2×D412yes—
352C2×C42C2×D411yescharacteristic
353C2×C42C2×D411yescharacteristic
354C2×C42#4114yes—
355C2×C42#4114yes—
356C2×C42C2412yes—
357C2×C42C2412yes—
358C2×C42C2411yescharacteristic
359C2×C42C2411yescharacteristic
360C4⋊C8—24no—
361C4⋊C8—24no—
362C4⋊C8—22no—
363C4⋊C8—22no—
364#103—28no—
365#103—28no—
366#103—24no—
367#103—24no—
368#103—42no—
369#103—42no—
370#103—44no—
371#103—44no—
372#103—44no—
373#103—44no—
374#103—44no—
375#103—44no—
376#103—44no—
377#103—44no—
378C22⋊C8—24no—
379#97—44no—
380C4×C8—24no—
381C4×C8—44no—
382#95—21no—
383#95—21no—
384#95—21no—
385#95—21no—
386#95—22no—
387#95—22no—
388#95—22no—
389#95—22no—
390#95—22no—
391#95—22no—
392#95—22no—
393#95—22no—
394#95—24no—
395#95—24no—
396#95—24no—
397#95—24no—
398#95C2×D412yes—
399#95C2×D412yes—
400#95C2×D412yes—
401#95C2×D412yes—
402C24C22×D411yescharacteristic
403C2×Q8—22no—
404C2×Q8—22no—
405C2×Q8—24no—
406C2×Q8—24no—
407C2×Q8—22no—
408C2×Q8—22no—
409C2×Q8—24no—
410C2×Q8—24no—
411C2×Q8—22no—
412C2×Q8—22no—
413C2×Q8—24no—
414C2×Q8—24no—
415C2×Q8—44no—
416C2×Q8—48no—
417C2×Q8—48no—
418C2×Q8—42no—
419C2×Q8—44no—
420C2×Q8—44no—
421C2×Q8—42no—
422C2×Q8—42no—
423C2×Q8—82no—
424C2×Q8—82no—
425C2×Q8—82no—
426C2×Q8—84no—
427C2×Q8—84no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—21no—
433C22×C4—21no—
434C22×C4—21no—
435C22×C4—21no—
436C22×C4—21no—
437C22×C4—21no—
438C22×C4—21no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—21no—
442C22×C4—22no—
443C22×C4—24no—
444C22×C4—22no—
445C22×C4—44no—
446C22×C4—82no—
447C22×C4—82no—
448C22×C4#12414yes—
449C22×C4#12414yes—
450C22×C4C22×D412yes—
451C22×C4C22×D412yes—
452C22×C4C22×D411yescharacteristic
453C22×C4C22×D411yescharacteristic
454C22×C4C22×D411yescharacteristic
455C22×C4C22×D412yes—
456C22×C4C22×D412yes—
457C22×C4C22×D411yescharacteristic
458C22×C4C22×D411yescharacteristic
459C22×C4C2×#4112yes—
460C22×C4C2×#4112yes—
461C22×C4C2511yescharacteristic
462M16—84no—
463C2×C8—22no—
464C2×C8—22no—
465C2×C8—24no—
466C2×C8—22no—
467C2×C8—24no—
468C2×C8—22no—
469C2×C8—24no—
470C2×C8—22no—
471C2×C8—22no—
472C4⋊C4—22no—
473C4⋊C4—22no—
474C4⋊C4—21no—
475C4⋊C4—21no—
476C4⋊C4—22no—
477C4⋊C4—22no—
478C4⋊C4—21no—
479C4⋊C4—21no—
480C4⋊C4—22no—
481C4⋊C4—22no—
482C4⋊C4—21no—
483C4⋊C4—21no—
484C4⋊C4—22no—
485C4⋊C4—22no—
486C4⋊C4—44no—
487C4⋊C4—44no—
488C4⋊C4—44no—
489C4⋊C4—44no—
490C4⋊C4—44no—
491C4⋊C4—44no—
492C4⋊C4—44no—
493C4⋊C4—42no—
494C4⋊C4—42no—
495C4⋊C4—44no—
496C4⋊C4—44no—
497C4⋊C4—42no—
498C4⋊C4—44no—
499C4⋊C4—44no—
500C4⋊C4—42no—
501C4⋊C4—44no—
502C4⋊C4—42no—
503C4⋊C4—81no—
504C4⋊C4—81no—
505C4⋊C4—82no—
506C4⋊C4—82no—
507C4⋊C4—82no—
508C4⋊C4—84no—
509C4⋊C4—84no—
510C4⋊C4—84no—
511C4⋊C4—84no—
512C4⋊C4#13714yes—
513C4⋊C4#13714yes—
514C4⋊C4H(5,𝔽2)12yes—
515C4⋊C4H(5,𝔽2)12yes—
516C22⋊C4—24no—
517C22⋊C4—24no—
518C22⋊C4—42no—
519C22⋊C4—42no—
520C22⋊C4—44no—
521C22⋊C4—82no—
522C42—24no—
523C42—24no—
524C42—24no—
525C42—24no—
526C42—22no—
527C42—22no—
528C42—22no—
529C42—22no—
530C42—42no—
531C42—42no—
532C42—44no—
533C42—42no—
534C42—42no—
535C42—44no—
536C42—42no—
537C42—42no—
538C42#12414yes—
539C42#12414yes—
540C42C2×#4112yes—
541C42C2×#4112yes—
542C42C2×#4112yes—
543C42C2×#4112yes—
544C23—21no—
545C23—21no—
546C23—21no—
547C23—21no—
548C23—21no—
549C23—21no—
550C23C2×#12412yes—
551C23C23×D411yescharacteristic
552Q8—42no—
553Q8—42no—
554Q8—44no—
555Q8—44no—
556Q8—42no—
557Q8—44no—
558Q8—42no—
559Q8—44no—
560Q8—88no—
561Q8—88no—
562Q8—84no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—22no—
570C2×C4—21no—
571C2×C4—22no—
572C2×C4—21no—
573C2×C4—28no—
574C2×C4—28no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—22no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—44no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—44no—
601C2×C4—44no—
602C2×C4—41no—
603C2×C4—41no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—41no—
609C2×C4—42no—
610C2×C4—41no—
611C2×C4—42no—
612C2×C4—88no—
613C2×C4C2×#12412yes—
614C2×C4C2×#12412yes—
615C2×C4C2×#12412yes—
616C2×C4C2×#12412yes—
617C2×C4C2×#12412yes—
618C2×C4C2×#13714yes—
619C2×C4C2×#13714yes—
620C2×C4C2×#13712yes—
621C2×C4C2×#13712yes—
622C2×C4#57712yes—
623C2×C4#57812yes—
624C2×C4C22×#4111yescharacteristic
625C2×C4C22×#4111yescharacteristic
626C2×C4C2×H(5,𝔽2)11yescharacteristic
627C2×C4C2×H(5,𝔽2)11yescharacteristic
628C2×C4#58511yescharacteristic
629C2×C4#58511yescharacteristic
630C8—42no—
631C8—42no—
632C8—44no—
633C22—22no—
634C22—22no—
635C22—22no—
636C22—41no—
637C22—41no—
638C22—41no—
639C22—41no—
640C22C22×#12411yescharacteristic
641C22#348811yescharacteristic
642C22#348811yescharacteristic
643C22C22×#13711yescharacteristic
644C22C22×#13711yescharacteristic
645C22C2×#57711yescharacteristic
646C22C2×#57811yescharacteristic
647C4—24no—
648C4—24no—
649C4—22no—
650C4—22no—
651C4—22no—
652C4—42no—
653C4—42no—
654C4—42no—
655C4—42no—
656C4—82no—
657C4—81no—
658C4—81no—
659C4—82no—
660C4—82no—
661C4—84no—
662C4—82no—
663C4#313714yes—
664C4#313914yes—
665C4#350611yescharacteristic
666C4#350611yescharacteristic
667C4#350811yescharacteristic
668C4#351011yescharacteristic
669C2—42no—
670C2#3555611yescharacteristic
671C2C2×#313711yescharacteristic
672C2C2×#313911yescharacteristic
673C2#6198011yescharacteristic
674C2#6198411yescharacteristic
675C2#6200011yescharacteristic
676C2#6200211yescharacteristic
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. #29355C2semidirect
4. #29349C2semidirect
5. #28755C2semidirect
6. #28748C2semidirect
7. #27777C2semidirect
8. #27769C2semidirect
9. #27757C2semidirect
10. #27749C2semidirect

Maximal subgroups

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

subgroupindex
#352812
#349782
#293552
#293492
#287552
#287482
#277772
#277692
#277572
#277492
#272832
#272772
#218942
#218042
#213892
#212592
#205502