Finite Group Databasebeta

← #6797971

Subgroups — #6797971

Order 512.

1669 conjugacy classes of subgroups.

The subgroup lattice has 681 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36164C211yesmaximal, characteristic
2#36134C211yesmaximal, characteristic, complemented
3#33293C211yesmaximal, characteristic, complemented
4#31951C214yesmaximal, complemented
5#31947C214yesmaximal, complemented
6#29854C212yesmaximal, complemented
7#28726C211yesmaximal, characteristic, complemented
8#28522C211yesmaximal, characteristic, complemented
9#27956C212yesmaximal, complemented
10#25603C212yesmaximal, complemented
11#24818C212yesmaximal, complemented
12#24814C212yesmaximal, complemented
13#24122C212yesmaximal, complemented
14#24120C212yesmaximal
15#21866C212yesmaximal, complemented
16C4×#497C211yesmaximal, characteristic, complemented
17#21270C211yesmaximal, characteristic, complemented
18C2×#554C2211yescharacteristic
19C2×#547C2211yescharacteristic
20C2×#533C2212yes—
21C2×#497C2211yescharacteristic
22C2×#489C2212yes—
23C2×#477C2212yes—
24C2×#475C2212yes—
25C2×C4×Q16C2212yes—
26C2×C4×SD16C2212yes—
27C2×#432C2212yes—
28#2868C2211yescharacteristic
29#2862C2211yescharacteristic
30#2857C2212yes—
31#2850C2212yes—
32#2568C2211yescharacteristic
33#2563C2211yescharacteristic
34#2552—21no—
35#2542—21no—
36#2538C2212yes—
37#2504C2212yes—
38#2503C2212yes—
39#2489—21no—
40#2489—21no—
41#2488—22no—
42#2488—22no—
43#2441—22no—
44#2429—22no—
45#2423C2214yes—
46#2365—21no—
47#2354—21no—
48Q8×C4⋊C4C2212yes—
49#2339—21no—
50#2339—21no—
51C4×C4⋊Q8C2211yescharacteristic
52C4×#124C2211yescharacteristic
53C2×#382C2212yes—
54C42×D4C2211yescharacteristic
55#2097C2214yes—
56#2085C2214yes—
57#2085C2214yes—
58#2072C2212yes—
59#2071C2212yes—
60#2067C2214yes—
61#2066C2214yes—
62#2036C2214yes—
63#2034C2214yes—
64#2001C2214yes—
65#2000C2214yes—
66#1993C2212yes—
67#1993C2212yes—
68#1993C2212yes—
69#1990C2212yes—
70#1990C2212yes—
71#1973C2212yes—
72#1971C2212yes—
73#1960C2212yes—
74#1960C2211yescharacteristic
75#1959C2211yescharacteristic
76#1954C2212yes—
77#1939C2212yes—
78#1924C2214yes—
79#1880C2214yes—
80#1878C2214yes—
81#1872C2214yes—
82#1872C2214yes—
83#1853C2214yes—
84#1852C2214yes—
85#1847C2211yescharacteristic
86#1847C2212yes—
87#1836C2214yes—
88#1805C2214yes—
89#1805C2214yes—
90#1803C2212yes—
91#1802C2212yes—
92#1767C2211yescharacteristic
93#1749C2212yes—
94C4×#97C2211yescharacteristic
95C22×#41—21no—
96C22×Q16C2312yes—
97C22×SD16C2312yes—
98#554—44no—
99#547—44no—
100#533—28no—
101#533—44no—
102C2×C4⋊Q8—21no—
103C2×C4⋊Q8—21no—
104C2×C4⋊Q8C2311yescharacteristic
105C2×C4⋊Q8C2311yescharacteristic
106C2×C4⋊Q8C2311yescharacteristic
107C2×C4⋊Q8C2312yes—
108C2×#125—21no—
109C2×#125—22no—
110C2×#124C2311yescharacteristic
111C2×#123—22no—
112C2×C22⋊Q8—22no—
113C2×C22⋊Q8—22no—
114C2×C22⋊Q8—21no—
115C2×C22⋊Q8—22no—
116C2×C22⋊Q8C2314yes—
117C2×C22⋊Q8C2312yes—
118C2×#121C2312yes—
119C2×C4×Q8—22no—
120C2×C4×Q8C2312yes—
121C2×C4×Q8C2311yescharacteristic
122C2×C4×Q8C2314yes—
123C2×C4×D4—21no—
124C2×C4×D4D411yescharacteristic
125C2×C4×D4D411yescharacteristic
126C2×C4×D4C2311yescharacteristic
127C2×C4×D4C2311yescharacteristic
128C2×#117—22no—
129C22×C4⋊C4C2312yes—
130C22×C4⋊C4C2312yes—
131C22×C42C2312yes—
132#497—24no—
133#489—44no—
134#477—44no—
135#475—44no—
136C4×Q16—44no—
137C4×SD16—44no—
138#432—44no—
139C2×#107C2314yes—
140C2×#107C2312yes—
141C2×#106C2314yes—
142C2×#106C2312yes—
143C2×C4⋊C8C2312yes—
144C2×C4⋊C8C2314yes—
145C2×#103C2314yes—
146C2×#103C2314yes—
147C2×#103C2314yes—
148C2×#103C2312yes—
149C2×#103C2312yes—
150C2×#103C2312yes—
151C2×#102C2314yes—
152C2×#102C2312yes—
153C2×C22⋊C8C2314yes—
154C2×C22⋊C8C2314yes—
155C2×#97C2312yes—
156C2×#97C2312yes—
157C2×#97C2311yescharacteristic
158C2×#97C2311yescharacteristic
159C2×C4×C8C2312yes—
160#400—22no—
161#400—22no—
162#400—22no—
163#399—24no—
164#398—24no—
165#397—22no—
166#397—22no—
167#396—24no—
168#393—22no—
169#391—21no—
170#391—21no—
171#391—21no—
172#391—21no—
173#391—22no—
174#391—21no—
175#391—21no—
176#391C2312yes—
177#391C2312yes—
178#391C2311yescharacteristic
179#390—22no—
180#390C2312yes—
181#390C2311yescharacteristic
182#389—22no—
183#389—21no—
184#389—21no—
185#389—22no—
186#389—24no—
187#389C2314yes—
188#388—24no—
189#388—22no—
190#388—22no—
191#388—21no—
192#388—21no—
193#387—24no—
194#386—22no—
195#386—22no—
196#386—22no—
197#386—22no—
198#384—22no—
199#384—21no—
200#384—21no—
201#384C2311yescharacteristic
202#384C2312yes—
203#384C2312yes—
204#382—22no—
205#382—22no—
206#382C2312yes—
207#382C2312yes—
208#382C2312yes—
209#382C2312yes—
210#382C2311yescharacteristic
211#381—22no—
212#381—22no—
213#380—22no—
214#380—21no—
215#380—21no—
216#380C2314yes—
217C4×C4⋊C4—22no—
218C4×C4⋊C4—21no—
219C4×C4⋊C4—21no—
220C4×C4⋊C4D411yescharacteristic
221C4×C4⋊C4D411yescharacteristic
222C4×C4⋊C4C2311yescharacteristic
223C4×C22⋊C4—21no—
224C4×C22⋊C4—21no—
225C4×C22⋊C4D412yes—
226C4×C22⋊C4D412yes—
227C4×C22⋊C4C2312yes—
228C2×#95—22no—
229C2×#95—22no—
230C43C2311yescharacteristic
231#340C2314yes—
232#340C2314yes—
233#340C2314yes—
234#340C2314yes—
235#340C2314yes—
236#340C2314yes—
237#336C2312yes—
238#336C2314yes—
239C2×#41—44no—
240C2×#41—44no—
241C2×#41—44no—
242C22×Q8—22no—
243C22×Q8—21no—
244C22×Q8—21no—
245C22×Q8C2411yescharacteristic
246C22×Q8C2412yes—
247C22×D4—22no—
248C22×D4C2×D411yescharacteristic
249C22×D4C2411yescharacteristic
250C23×C4—22no—
251C23×C4—22no—
252C23×C4—21no—
253C23×C4C2×D412yes—
254C23×C4C2×D412yes—
255C23×C4C2412yes—
256C2×Q16—24no—
257C2×Q16—24no—
258C2×Q16—42no—
259C2×SD16—24no—
260C2×SD16—24no—
261C2×SD16—42no—
262C22×C8C2414yes—
263C22×C8C2412yes—
264C22×C8C2412yes—
265C4⋊Q8—24no—
266C4⋊Q8—22no—
267C4⋊Q8—24no—
268C4⋊Q8—24no—
269C4⋊Q8—24no—
270C4⋊Q8—44no—
271C4⋊Q8—82no—
272C4⋊Q8#4114yescomplemented
273#125—48no—
274#125—44no—
275#124—22no—
276#124—22no—
277#123—48no—
278C22⋊Q8—28no—
279C22⋊Q8—48no—
280C22⋊Q8—44no—
281C22⋊Q8—44no—
282C22⋊Q8—48no—
283C22⋊Q8—44no—
284C22⋊Q8—84no—
285#121—44no—
286C4×Q8—28no—
287C4×Q8—24no—
288C4×Q8—24no—
289C4×Q8—24no—
290C4×Q8—44no—
291C4×Q8—48no—
292C4×D4—24no—
293C4×D4—24no—
294C4×D4—24no—
295C4×D4—42no—
296C4×D4—44no—
297#117—48no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—21no—
304C2×C4⋊C4—24no—
305C2×C4⋊C4—21no—
306C2×C4⋊C4—24no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—21no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—21no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—22no—
313C2×C4⋊C4—21no—
314C2×C4⋊C4—24no—
315C2×C4⋊C4—22no—
316C2×C4⋊C4—22no—
317C2×C4⋊C4—44no—
318C2×C4⋊C4—44no—
319C2×C4⋊C4C2×D411yescharacteristic
320C2×C4⋊C4C2×D411yescharacteristic
321C2×C4⋊C4C2×D411yescharacteristic
322C2×C4⋊C4C2411yescharacteristic
323C2×C4⋊C4C2411yescharacteristic
324C2×C4⋊C4C2412yes—
325C2×C4⋊C4C2412yes—
326C2×C4⋊C4C2414yes—
327C2×C4⋊C4C2412yes—
328C2×C22⋊C4—24no—
329C2×C22⋊C4—22no—
330C2×C22⋊C4—21no—
331C2×C22⋊C4—22no—
332C2×C22⋊C4—21no—
333C2×C22⋊C4—21no—
334C2×C22⋊C4—24no—
335C2×C22⋊C4—21no—
336C2×C22⋊C4—21no—
337C2×C22⋊C4—22no—
338C2×C22⋊C4—22no—
339C2×C22⋊C4C2×D412yes—
340C2×C22⋊C4C2×D412yes—
341C2×C22⋊C4C2×D412yes—
342C2×C42—24no—
343C2×C42—24no—
344C2×C42—24no—
345C2×C42—21no—
346C2×C42—22no—
347C2×C42C2×D411yescharacteristic
348C2×C42C2×D411yescharacteristic
349C2×C42C2×D412yes—
350C2×C42C2×D412yes—
351C2×C42C2×D411yescharacteristic
352C2×C42C2×D411yescharacteristic
353C2×C42C2412yes—
354C2×C42C2411yescharacteristic
355C2×C42C2411yescharacteristic
356C2×C42C2411yescharacteristic
357#107—42no—
358#107—44no—
359#106—42no—
360#106—44no—
361C4⋊C8—24no—
362C4⋊C8—44no—
363#103—42no—
364#103—42no—
365#103—42no—
366#103—44no—
367#103—44no—
368#103—44no—
369#102—42no—
370#102—44no—
371C22⋊C8—28no—
372C22⋊C8—44no—
373#97—24no—
374#97—22no—
375#97—22no—
376#97—24no—
377C4×C8—24no—
378#95—21no—
379#95—21no—
380#95—22no—
381#95—22no—
382#95—22no—
383#95—22no—
384#95—22no—
385#95—22no—
386#95—21no—
387#95—21no—
388#95—24no—
389#95—22no—
390#95—22no—
391#95—24no—
392#95—22no—
393#95—22no—
394#95—22no—
395#95—22no—
396#95C2×D412yes—
397#95C2×D412yes—
398#95C2×D412yes—
399C24—21no—
400C24C22×D412yes—
401#41—816no—
402C2×Q8—22no—
403C2×Q8—21no—
404C2×Q8—22no—
405C2×Q8—22no—
406C2×Q8—21no—
407C2×Q8—22no—
408C2×Q8—22no—
409C2×Q8—22no—
410C2×Q8—22no—
411C2×Q8—44no—
412C2×Q8—44no—
413C2×Q8—42no—
414C2×Q8—42no—
415C2×Q8—44no—
416C2×Q8—42no—
417C2×Q8—42no—
418C2×Q8—81no—
419C2×Q8—81no—
420C2×Q8C2×#4114yes—
421C2×D4—22no—
422C2×D4—22no—
423C2×D4—21no—
424C2×D4—21no—
425C2×D4—22no—
426C2×D4—22no—
427C2×D4—44no—
428C2×D4—44no—
429C2×D4—44no—
430C2×D4#13714yes—
431C22×C4—24no—
432C22×C4—22no—
433C22×C4—24no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—24no—
437C22×C4—24no—
438C22×C4—24no—
439C22×C4—21no—
440C22×C4—22no—
441C22×C4—21no—
442C22×C4—22no—
443C22×C4—21no—
444C22×C4—22no—
445C22×C4—24no—
446C22×C4—42no—
447C22×C4—42no—
448C22×C4—42no—
449C22×C4—44no—
450C22×C4—44no—
451C22×C4—44no—
452C22×C4—44no—
453C22×C4—42no—
454C22×C4—42no—
455C22×C4—42no—
456C22×C4—82no—
457C22×C4—82no—
458C22×C4C22×D412yes—
459C22×C4C22×D411yescharacteristic
460C22×C4C22×D411yescharacteristic
461C22×C4C22×D412yes—
462C22×C4C22×D412yes—
463C22×C4C22×D411yescharacteristic
464C22×C4C22×D411yescharacteristic
465C22×C4C22×D411yescharacteristic
466C22×C4C22×D411yescharacteristic
467C22×C4C2511yescharacteristic
468Q16—48no—
469SD16—48no—
470C2×C8—22no—
471C2×C8—22no—
472C2×C8—22no—
473C2×C8—22no—
474C2×C8—24no—
475C2×C8—24no—
476C2×C8—22no—
477C2×C8—24no—
478C2×C82−1+414yes—
479C4⋊C4—22no—
480C4⋊C4—24no—
481C4⋊C4—24no—
482C4⋊C4—22no—
483C4⋊C4—22no—
484C4⋊C4—24no—
485C4⋊C4—24no—
486C4⋊C4—22no—
487C4⋊C4—22no—
488C4⋊C4—22no—
489C4⋊C4—24no—
490C4⋊C4—22no—
491C4⋊C4—24no—
492C4⋊C4—48no—
493C4⋊C4—48no—
494C4⋊C4—42no—
495C4⋊C4—44no—
496C4⋊C4—44no—
497C4⋊C4—44no—
498C4⋊C4—42no—
499C4⋊C4—44no—
500C4⋊C4—42no—
501C4⋊C4—44no—
502C4⋊C4—42no—
503C4⋊C4—42no—
504C4⋊C4—44no—
505C4⋊C4—44no—
506C4⋊C4—84no—
507C4⋊C4—82no—
508C4⋊C4—82no—
509C22⋊C4—24no—
510C22⋊C4—24no—
511C22⋊C4—42no—
512C22⋊C4—48no—
513C22⋊C4—44no—
514C22⋊C4—44no—
515C22⋊C4—42no—
516C22⋊C4—42no—
517C22⋊C4—42no—
518C22⋊C4—81no—
519C22⋊C4—82no—
520C22⋊C4—81no—
521C22⋊C4—82no—
522C22⋊C4—84no—
523C42—24no—
524C42—24no—
525C42—22no—
526C42—22no—
527C42—22no—
528C42—22no—
529C42—22no—
530C42—41no—
531C42—41no—
532C42—48no—
533C42—42no—
534C42—44no—
535C42C2×#4112yes—
536C42H(5,𝔽2)14yes—
537C42H(5,𝔽2)12yes—
538C23—22no—
539C23—22no—
540C23—22no—
541C23—22no—
542C23—22no—
543C23—22no—
544C23—41no—
545C23—41no—
546C23—41no—
547C23—41no—
548C23—41no—
549C23—41no—
550C23—41no—
551C23C2×#13714yes—
552C23C23×D411yescharacteristic
553Q8—24no—
554Q8—24no—
555Q8—42no—
556Q8—42no—
557Q8—42no—
558Q8—42no—
559Q8—88no—
560Q8—84no—
561Q8—162no—
562D4—28no—
563D4—42no—
564D4—42no—
565D4—88no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—22no—
574C2×C4—21no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—24no—
581C2×C4—21no—
582C2×C4—21no—
583C2×C4—22no—
584C2×C4—21no—
585C2×C4—22no—
586C2×C4—21no—
587C2×C4—22no—
588C2×C4—48no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—44no—
592C2×C4—44no—
593C2×C4—42no—
594C2×C4—44no—
595C2×C4—41no—
596C2×C4—42no—
597C2×C4—41no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—41no—
601C2×C4—41no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—41no—
605C2×C4—42no—
606C2×C4—44no—
607C2×C4—41no—
608C2×C4—41no—
609C2×C4—41no—
610C2×C4—42no—
611C2×C4—44no—
612C2×C4—41no—
613C2×C4—42no—
614C2×C4—84no—
615C2×C4—88no—
616C2×C4—84no—
617C2×C4—164no—
618C2×C4#53512yes—
619C2×C4#53512yes—
620C2×C4#54612yes—
621C2×C4#54612yes—
622C2×C4C2×#13712yes—
623C2×C4#57512yes—
624C2×C4C22×#4111yescharacteristic
625C2×C4C2×H(5,𝔽2)12yes—
626C2×C4C2×H(5,𝔽2)11yescharacteristic
627C2×C4C2×2−1+412yes—
628C8—24no—
629C8—42no—
630C8—44no—
631C22—24no—
632C22—24no—
633C22—24no—
634C22—42no—
635C22—42no—
636C22—42no—
637C22—42no—
638C22—81no—
639C22—81no—
640C22—81no—
641C22—81no—
642C22—81no—
643C22—81no—
644C22—81no—
645C22C2×#53511yescharacteristic
646C22C2×#54611yescharacteristic
647C22#349911yescharacteristic
648C22C22×#13711yescharacteristic
649C22C2×#57511yescharacteristic
650C22#358511yescharacteristic
651C22#358611yescharacteristic
652C4—22no—
653C4—22no—
654C4—22no—
655C4—42no—
656C4—44no—
657C4—41no—
658C4—41no—
659C4—42no—
660C4—42no—
661C4—81no—
662C4—81no—
663C4—82no—
664C4—82no—
665C4—81no—
666C4—82no—
667C4—84no—
668C4#321514yes—
669C4#352012yes—
670C4#354812yes—
671C2—44no—
672C2—161no—
673C2#3895111yescharacteristic
674C2C2×#321511yescharacteristic
675C2#6082911yescharacteristic
676C2#6115711yescharacteristic
677C2#6116011yescharacteristic
678C2#6250411yescharacteristic
679C2#6251111yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#547C22
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. #36134C2semidirect
3. #33293C2semidirect
4. #31951C2semidirect
5. #31947C2semidirect
6. #29854C2semidirect
7. #28726C2semidirect
8. #28522C2semidirect
9. #27956C2semidirect
10. #25603C2semidirect
11. #24818C2semidirect
12. #24814C2semidirect
13. #24122C2semidirect
15. #21866C2semidirect
16. C4×#497C2semidirect
17. #21270C2semidirect
272. C4⋊Q8#41semidirect

Maximal subgroups

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

subgroupindex
#361642
#361342
#332932
#319512
#319472
#298542
#287262
#285222
#279562
#256032
#248182
#248142
#241222
#241202
#218662
C4×#4972
#212702