Finite Group Databasebeta

← #6514223

Subgroups — #6514223

Order 512.

1896 conjugacy classes of subgroups.

The subgroup lattice has 676 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3254C211yesmaximal, characteristic
2C2×#3104C211yesmaximal, characteristic, complemented
3#35249C211yesmaximal, characteristic
4#34373C211yesmaximal, characteristic, complemented
5#28288C212yesmaximal, complemented
6#28258C212yesmaximal, complemented
7#28211C212yesmaximal, complemented
8#28168C212yesmaximal, complemented
9#25217C214yesmaximal, complemented
10#25107C214yesmaximal, complemented
11#24817C212yesmaximal, complemented
12#24733C212yesmaximal, complemented
13#24682C212yesmaximal, complemented
14#24633C212yesmaximal, complemented
15#21975C211yesmaximal, characteristic, complemented
16#21863C211yesmaximal, characteristic
17#20647C211yesmaximal, characteristic, complemented
18C2×#533C2211yescharacteristic
19C2×#532C2211yescharacteristic
20C22×C4×Q8C2211yescharacteristic
21#3254—28no—
22#3104—28no—
23C2×#484C2212yes—
24C2×#477C2211yescharacteristic
25C2×#477C2212yes—
26C2×#477C2211yescharacteristic
27C2×#474C2212yes—
28C2×#468C2214yes—
29C2×#464C2214yes—
30C2×#462C2211yescharacteristic
31C2×#462C2211yescharacteristic
32C2×#460C2212yes—
33C2×#461C2212yes—
34C2×#451C2212yes—
35C2×C4×Q16C2212yes—
36C2×C4×Q16C2212yes—
37C2×#431C2211yescharacteristic
38C2×#423C2211yescharacteristic
39C2×#419C2212yes—
40#2854—22no—
41#2632—22no—
42#2580—22no—
43#2567C2211yescharacteristic
44#2567C2211yescharacteristic
45#2548C2211yescharacteristic
46#2548C2211yescharacteristic
47#2501C2212yes—
48#2436C2212yes—
49#2406—22no—
50#2378—22no—
51#2362C2214yes—
52#2355—22no—
53#2355C2212yes—
54Q8×C4⋊C4C2212yes—
55#2336—22no—
56#2331C2212yes—
57#2327C2212yes—
58#2326C2212yes—
59#2326C2212yes—
60#2314—22no—
61#2285C2211yescharacteristic
62#2283C2211yescharacteristic
63#2083C2214yes—
64#2072C2214yes—
65#2068C2214yes—
66#2067C2214yes—
67#2056C2214yes—
68#2055C2214yes—
69#2042C2214yes—
70#2015C2214yes—
71#1990C2212yes—
72#1983C2214yes—
73#1957C2212yes—
74#1947C2214yes—
75#1939C2212yes—
76#1939C2212yes—
77#1928C2214yes—
78#1925C2214yes—
79#1880C2214yes—
80#1878C2214yes—
81#1875C2214yes—
82#1851C2214yes—
83#1843C2212yes—
84#1832C2212yes—
85#1805C2212yes—
86#1805C2212yes—
87#1804C2212yes—
88#1804C2212yes—
89#1803C2212yes—
90#1803C2214yes—
91#1803C2212yes—
92#1736C2212yes—
93C22×#41—21no—
94C23×Q8C2311yescharacteristic
95C22×Q16C2312yes—
96C22×Q16C2312yes—
97C22×SD16C2314yes—
98C22×M16C2312yes—
99#533—24no—
100#533D418yes—
101#532—24no—
102#532—24no—
103C2×C4⋊Q8C2311yescharacteristic
104C2×C4⋊Q8C2311yescharacteristic
105C2×C4⋊Q8C2311yescharacteristic
106C2×#127C2311yescharacteristic
107C2×#125—21no—
108C2×#124—21no—
109C2×#124C2312yes—
110C2×#124C2312yes—
111C2×#123—24no—
112C2×C22⋊Q8—21no—
113C2×C22⋊Q8C2312yes—
114C2×#121—21no—
115C2×#121C2312yes—
116C2×C4×Q8—21no—
117C2×C4×Q8—21no—
118C2×C4×Q8—21no—
119C2×C4×Q8—24no—
120C2×C4×Q8—24no—
121C2×C4×Q8C2311yescharacteristic
122C2×C4×Q8C2311yescharacteristic
123C2×C4×Q8C2311yescharacteristic
124C2×C4×Q8C2311yescharacteristic
125C2×C4×Q8C2312yes—
126C2×C4×Q8C2314yes—
127C2×C4×D4—21no—
128C2×C4×D4C2312yes—
129C2×#117—21no—
130C2×#117C2312yes—
131C22×C4⋊C4—21no—
132C22×C4⋊C4C2311yescharacteristic
133C22×C42—21no—
134C22×C42C2311yescharacteristic
135#484—28no—
136#477—24no—
137#477—24no—
138#477—28no—
139#474—28no—
140#468—216no—
141#464—48no—
142#462—24no—
143#462—24no—
144#461—28no—
145#460—28no—
146#451—28no—
147C4×Q16—28no—
148C4×Q16—28no—
149#431—24no—
150C2×#107C2312yes—
151C2×#107C2312yes—
152C2×#107C2314yes—
153#423—24no—
154C2×C4⋊C8C2312yes—
155C2×C4⋊C8C2311yescharacteristic
156C2×C4⋊C8C2311yescharacteristic
157C2×C4⋊C8C2312yes—
158C2×C4⋊C8C2314yes—
159C2×C4⋊C8C2314yes—
160#419—44no—
161C2×#103C2314yes—
162C2×#103C2312yes—
163C2×#103C2312yes—
164C2×#103C2312yes—
165C2×#103C2314yes—
166C2×#103C2312yes—
167C2×#103C2314yes—
168C2×#102C2314yes—
169C2×#102C2314yes—
170C2×#102C2314yes—
171C2×C22⋊C8C2312yes—
172C2×C4×C8C2312yes—
173#399—24no—
174#395—22no—
175#395—22no—
176#394—22no—
177#394—22no—
178#393—24no—
179#391—24no—
180#391C2314yes—
181#391C2312yes—
182#391C2312yes—
183#390—22no—
184#390—22no—
185#390C2312yes—
186#390C2312yes—
187#389—22no—
188#389—22no—
189#389—24no—
190#389C2314yes—
191#388—24no—
192#388—24no—
193#388—24no—
194#388—24no—
195#387—24no—
196#387—24no—
197#387—24no—
198#387—24no—
199#387—24no—
200#386—24no—
201#385—24no—
202#385—24no—
203#384—22no—
204#384C2312yes—
205#384C2312yes—
206#384C2312yes—
207#383D412yes—
208#382—22no—
209#382D412yes—
210#382C2312yes—
211#381D412yes—
212#380—21no—
213#380—21no—
214#380D412yes—
215C4×C4⋊C4—22no—
216C4×C4⋊C4—22no—
217C4×C4⋊C4—21no—
218C4×C4⋊C4—21no—
219C4×C4⋊C4—24no—
220C4×C4⋊C4C2314yes—
221C4×C4⋊C4C2311yescharacteristic
222C4×C4⋊C4C2311yescharacteristic
223C4×C22⋊C4C2312yes—
224#376C2312yes—
225#340C2314yes—
226#340C2314yes—
227#340C2314yes—
228#340C2314yes—
229#336C2314yes—
230#336C2314yes—
231C2×#41—44no—
232C2×#41—84no—
233C22×Q8—21no—
234C22×Q8—21no—
235C22×Q8—21no—
236C22×Q8—22no—
237C22×Q8—22no—
238C22×Q8—22no—
239C22×Q8—22no—
240C22×Q8—41no—
241C22×Q8—41no—
242C22×Q8C2412yes—
243C22×Q8C2411yescharacteristic
244C22×Q8C2411yescharacteristic
245C22×D4—22no—
246C22×D4—21no—
247C22×D4—21no—
248C22×D4C2412yes—
249C23×C4—21no—
250C23×C4—21no—
251C23×C4—22no—
252C23×C4C2×D411yescharacteristic
253C23×C4C2×D411yescharacteristic
254C23×C4C2411yescharacteristic
255C2×Q16—24no—
256C2×Q16—24no—
257C2×Q16—24no—
258C2×Q16—24no—
259C2×Q16—42no—
260C2×Q16—42no—
261C2×SD16—28no—
262C2×SD16—44no—
263C2×SD16—44no—
264C2×M16—24no—
265C2×M16—42no—
266C2×M16—42no—
267C22×C8C2414yes—
268C22×C8C2412yes—
269C22×C8C2412yes—
270C4⋊Q8—22no—
271C4⋊Q8—22no—
272C4⋊Q8—22no—
273C4⋊Q8—22no—
274C4⋊Q8C2×D414yes—
275C4⋊Q8C2×D414yes—
276#127—22no—
277#127—22no—
278#125—28no—
279#124—24no—
280#124—24no—
281#124—42no—
282#124—42no—
283#124—44no—
284#123—88no—
285C22⋊Q8—28no—
286C22⋊Q8—24no—
287C22⋊Q8C2×D418yes—
288#121—24no—
289#121—24no—
290#121—44no—
291C4×Q8—28no—
292C4×Q8—24no—
293C4×Q8—22no—
294C4×Q8—22no—
295C4×Q8—22no—
296C4×Q8—22no—
297C4×Q8—22no—
298C4×Q8—22no—
299C4×Q8—22no—
300C4×Q8—22no—
301C4×Q8—44no—
302C4×Q8—44no—
303C4×Q8—44no—
304C4×Q8—44no—
305C4×Q8C2×D418yes—
306C4×D4—24no—
307C4×D4—24no—
308C4×D4—44no—
309#117—28no—
310#117—42no—
311#117—42no—
312C2×C4⋊C4—21no—
313C2×C4⋊C4—21no—
314C2×C4⋊C4—21no—
315C2×C4⋊C4—22no—
316C2×C4⋊C4—22no—
317C2×C4⋊C4—22no—
318C2×C4⋊C4—22no—
319C2×C4⋊C4—21no—
320C2×C4⋊C4—21no—
321C2×C4⋊C4—21no—
322C2×C4⋊C4—21no—
323C2×C4⋊C4—21no—
324C2×C4⋊C4—21no—
325C2×C4⋊C4—22no—
326C2×C4⋊C4—24no—
327C2×C4⋊C4—24no—
328C2×C4⋊C4—24no—
329C2×C4⋊C4—22no—
330C2×C4⋊C4—24no—
331C2×C4⋊C4—44no—
332C2×C4⋊C4C2×D412yes—
333C2×C4⋊C4C2×D412yes—
334C2×C4⋊C4C2411yescharacteristic
335C2×C4⋊C4C2411yescharacteristic
336C2×C4⋊C4C2414yes—
337C2×C4⋊C4C2412yes—
338C2×C4⋊C4C2412yes—
339C2×C22⋊C4—22no—
340C2×C22⋊C4—22no—
341C2×C22⋊C4—22no—
342C2×C22⋊C4—24no—
343C2×C22⋊C4—24no—
344C2×C22⋊C4—22no—
345C2×C22⋊C4—24no—
346C2×C22⋊C4—24no—
347C2×C22⋊C4C2×D414yes—
348C2×C42—22no—
349C2×C42—21no—
350C2×C42—21no—
351C2×C42—21no—
352C2×C42—21no—
353C2×C42—22no—
354C2×C42—24no—
355C2×C42—24no—
356C2×C42—44no—
357C2×C42C2×D411yescharacteristic
358C2×C42C2×D411yescharacteristic
359C2×C42C2×D414yes—
360C2×C42#4114yes—
361C2×C42C2412yes—
362C2×C42C2412yes—
363C2×C42C2411yescharacteristic
364C2×C42C2411yescharacteristic
365#107—24no—
366#107—24no—
367#107—28no—
368C4⋊C8—24no—
369C4⋊C8—24no—
370C4⋊C8—22no—
371C4⋊C8—22no—
372C4⋊C8—44no—
373C4⋊C8—44no—
374#103—24no—
375#103—24no—
376#103—28no—
377#103—28no—
378#103—24no—
379#103—24no—
380#103—44no—
381#102—28no—
382#102—44no—
383#102—44no—
384C22⋊C8—24no—
385C4×C8—24no—
386#95—22no—
387#95—22no—
388#95—24no—
389#95—24no—
390#95—24no—
391#95—24no—
392#95—24no—
393#95—24no—
394#95C2×D414yes—
395#95C2×D414yes—
396#95C2×D414yes—
397#95C2×D412yes—
398#95C2×D412yes—
399C24—22no—
400C24C22×D411yescharacteristic
401#41—816no—
402C2×Q8—24no—
403C2×Q8—22no—
404C2×Q8—21no—
405C2×Q8—21no—
406C2×Q8—28no—
407C2×Q8—21no—
408C2×Q8—21no—
409C2×Q8—21no—
410C2×Q8—21no—
411C2×Q8—21no—
412C2×Q8—22no—
413C2×Q8—22no—
414C2×Q8—21no—
415C2×Q8—22no—
416C2×Q8—22no—
417C2×Q8—21no—
418C2×Q8—21no—
419C2×Q8—42no—
420C2×Q8—42no—
421C2×Q8—44no—
422C2×Q8—44no—
423C2×Q8—44no—
424C2×Q8—42no—
425C2×Q8—42no—
426C2×Q8—82no—
427C2×Q8—82no—
428C2×Q8C22×D414yes—
429C2×Q8H(5,𝔽2)12yes—
430C2×Q82−1+412yes—
431C2×D4—22no—
432C2×D4—22no—
433C2×D4—22no—
434C2×D4—22no—
435C2×D4—22no—
436C2×D4—22no—
437C2×D4—48no—
438C2×D4—44no—
439C2×D4—42no—
440C2×D4—42no—
441C2×D4—44no—
442C2×D4—82no—
443C22×C4—24no—
444C22×C4—21no—
445C22×C4—21no—
446C22×C4—21no—
447C22×C4—21no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—21no—
451C22×C4—21no—
452C22×C4—24no—
453C22×C4—22no—
454C22×C4—21no—
455C22×C4—21no—
456C22×C4—42no—
457C22×C4—42no—
458C22×C4—42no—
459C22×C4—42no—
460C22×C4—44no—
461C22×C4—42no—
462C22×C4—44no—
463C22×C4—84no—
464C22×C4#12114yes—
465C22×C4#12114yes—
466C22×C4C22×D414yes—
467C22×C4C22×D414yes—
468C22×C4C22×D411yescharacteristic
469C22×C4C22×D411yescharacteristic
470C22×C4C22×D411yescharacteristic
471C22×C4C22×D411yescharacteristic
472C22×C4C22×D412yes—
473C22×C4C22×D411yescharacteristic
474C22×C4C2×#4112yes—
475C22×C4C2511yescharacteristic
476Q16—48no—
477Q16—48no—
478SD16—88no—
479M16—84no—
480C2×C8—22no—
481C2×C8—22no—
482C2×C8—24no—
483C2×C8—22no—
484C2×C8—24no—
485C2×C8—22no—
486C2×C8—22no—
487C2×C8—22no—
488C2×C8—24no—
489C4⋊C4—24no—
490C4⋊C4—24no—
491C4⋊C4—24no—
492C4⋊C4—28no—
493C4⋊C4—28no—
494C4⋊C4—24no—
495C4⋊C4—24no—
496C4⋊C4—22no—
497C4⋊C4—21no—
498C4⋊C4—21no—
499C4⋊C4—22no—
500C4⋊C4—21no—
501C4⋊C4—21no—
502C4⋊C4—22no—
503C4⋊C4—24no—
504C4⋊C4—24no—
505C4⋊C4—42no—
506C4⋊C4—42no—
507C4⋊C4—42no—
508C4⋊C4—42no—
509C4⋊C4—42no—
510C4⋊C4—42no—
511C4⋊C4—44no—
512C4⋊C4—82no—
513C4⋊C4—84no—
514C4⋊C4—84no—
515C4⋊C4—84no—
516C4⋊C4—84no—
517C4⋊C4C22×D414yes—
518C22⋊C4—28no—
519C22⋊C4—44no—
520C22⋊C4—48no—
521C22⋊C4—48no—
522C22⋊C4—44no—
523C22⋊C4—44no—
524C22⋊C4—84no—
525C22⋊C4—84no—
526C22⋊C4—82no—
527C42—28no—
528C42—22no—
529C42—22no—
530C42—22no—
531C42—22no—
532C42—42no—
533C42—42no—
534C42—42no—
535C42—42no—
536C42—84no—
537C42—82no—
538C42—84no—
539C42#12114yes—
540C42#12114yes—
541C42C22×D412yes—
542C42C22×D412yes—
543C42C2×#4112yes—
544C42C2×#4112yes—
545C23—22no—
546C23—21no—
547C23—21no—
548C23—22no—
549C23—44no—
550C23—42no—
551C23—42no—
552C23—42no—
553C23—44no—
554C23C2×#12112yes—
555C23C23×D411yescharacteristic
556Q8—24no—
557Q8—22no—
558Q8—22no—
559Q8—24no—
560Q8—22no—
561Q8—22no—
562Q8—48no—
563Q8—41no—
564Q8—41no—
565Q8—42no—
566Q8—42no—
567Q8—41no—
568Q8—41no—
569Q8—84no—
570Q8—84no—
571D4—42no—
572D4—42no—
573D4—42no—
574D4—42no—
575D4—84no—
576D4—88no—
577D4—84no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—22no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—28no—
588C2×C4—24no—
589C2×C4—24no—
590C2×C4—24no—
591C2×C4—24no—
592C2×C4—24no—
593C2×C4—22no—
594C2×C4—24no—
595C2×C4—22no—
596C2×C4—24no—
597C2×C4—24no—
598C2×C4—44no—
599C2×C4—48no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—41no—
603C2×C4—41no—
604C2×C4—42no—
605C2×C4—41no—
606C2×C4—41no—
607C2×C4—42no—
608C2×C4—44no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—44no—
612C2×C4—41no—
613C2×C4—41no—
614C2×C4—42no—
615C2×C4—44no—
616C2×C4—44no—
617C2×C4—88no—
618C2×C4—84no—
619C2×C4C2×#12112yes—
620C2×C4C2×#12112yes—
621C2×C4C2×#12112yes—
622C2×C4C2×#12112yes—
623C2×C4C2×#12112yes—
624C2×C4#57814yes—
625C2×C4C23×D411yescharacteristic
626C2×C4C22×#4111yescharacteristic
627C2×C4C2×H(5,𝔽2)11yescharacteristic
628C2×C4C2×2−1+411yescharacteristic
629C2×C4#58511yescharacteristic
630C2×C4#58511yescharacteristic
631C8—44no—
632C8—42no—
633C8—42no—
634C22—22no—
635C22—22no—
636C22—22no—
637C22—42no—
638C22—42no—
639C22—82no—
640C22—82no—
641C22—84no—
642C22—82no—
643C22—84no—
644C22C22×#12111yescharacteristic
645C22#348811yescharacteristic
646C22#348911yescharacteristic
647C22C2×#57812yes—
648C22#358512yes—
649C4—22no—
650C4—22no—
651C4—22no—
652C4—22no—
653C4—44no—
654C4—44no—
655C4—42no—
656C4—42no—
657C4—44no—
658C4—44no—
659C4—81no—
660C4—81no—
661C4—82no—
662C4—84no—
663C4#304414yes—
664C4#348811yescharacteristic
665C4#348911yescharacteristic
666C4#349211yescharacteristic
667C4#349311yescharacteristic
668C2—42no—
669C2—162no—
670C2#3551811yescharacteristic
671C2C2×#304411yescharacteristic
672C2#6042611yescharacteristic
673C2#6204612yes—
674C2#6219312yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#532C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ #2283 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#3104C2semidirect
4. #34373C2semidirect
5. #28288C2semidirect
6. #28258C2semidirect
7. #28211C2semidirect
8. #28168C2semidirect
9. #25217C2semidirect
10. #25107C2semidirect
11. #24817C2semidirect
12. #24733C2semidirect
13. #24682C2semidirect
14. #24633C2semidirect
15. #21975C2semidirect
17. #20647C2semidirect

Maximal subgroups

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

subgroupindex
C2×#32542
C2×#31042
#352492
#343732
#282882
#282582
#282112
#281682
#252172
#251072
#248172
#247332
#246822
#246332
#219752
#218632
#206472