Finite Group Databasebeta

← #6817343

Subgroups — #6817343

Order 512.

1692 conjugacy classes of subgroups.

The subgroup lattice has 668 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3398C211yesmaximal, characteristic
2C2×#3385C211yesmaximal, characteristic
3#39619C211yesmaximal, characteristic, complemented
4#39490C211yesmaximal, characteristic
5#33242C211yesmaximal, characteristic, complemented
6#31152C214yesmaximal
7#31149C214yesmaximal, complemented
8#29830C211yesmaximal, characteristic, complemented
9#29828C211yesmaximal, characteristic
10#28744C211yesmaximal, characteristic, complemented
11#28682C211yesmaximal, characteristic, complemented
12#26497C214yesmaximal, complemented
13#26266C214yesmaximal
14#25936C211yesmaximal, characteristic, complemented
15#25934C211yesmaximal, characteristic
16#25651C211yesmaximal, characteristic, complemented
17#25646C211yesmaximal, characteristic
18#21712C211yesmaximal, characteristic, complemented
19#21274C211yesmaximal, characteristic
20C2×Q82C2211yescharacteristic
21C2×#557C2211yescharacteristic
22C2×#556C2211yescharacteristic
23C2×#554C2211yescharacteristic
24#3398—28no—
25#3385—28no—
26C2×#501C2212yes—
27C2×#500C2211yescharacteristic
28C2×#497C2212yes—
29C2×#494C2211yescharacteristic
30C2×#487C2211yescharacteristic
31C2×#487C2211yescharacteristic
32C2×#479C2212yes—
33C2×#477C2211yescharacteristic
34C2×#476C2211yescharacteristic
35C2×#474C2212yes—
36C2×#464C2212yes—
37C2×#462C2211yescharacteristic
38C2×#462C2212yes—
39C2×#461C2211yescharacteristic
40C2×#446C2211yescharacteristic
41C2×#446C2211yescharacteristic
42C2×#441C2212yes—
43C2×#440C2212yes—
44C2×C4×SD16C2211yescharacteristic
45#2862C2211yescharacteristic
46#2862C2211yescharacteristic
47#2713—24no—
48#2689—24no—
49#2660—21no—
50#2621—21no—
51#2568—21no—
52#2568C2211yescharacteristic
53#2568C2211yescharacteristic
54#2567C2211yescharacteristic
55#2567C2211yescharacteristic
56#2565—21no—
57#2503—21no—
58#2499—21no—
59#2453C2214yes—
60#2451C2214yes—
61#2447C2214yes—
62#2446C2214yes—
63#2327—21no—
64#2317—21no—
65C4×C4⋊Q8C2211yescharacteristic
66#2295C2211yescharacteristic
67#2293C2211yescharacteristic
68#2086C2214yes—
69#2068C2214yes—
70#2066C2214yes—
71#2056C2214yes—
72#2056C2214yes—
73#2055C2214yes—
74#2043C2214yes—
75#2042C2214yes—
76#2026C2214yes—
77#2024C2214yes—
78#1987C2211yescharacteristic
79#1987C2211yescharacteristic
80#1984C2211yescharacteristic
81#1984C2211yescharacteristic
82#1981C2211yescharacteristic
83#1981C2211yescharacteristic
84#1971C2214yes—
85#1969C2214yes—
86#1969C2214yes—
87#1968C2212yes—
88#1944C2214yes—
89#1941C2212yes—
90#1939C2214yes—
91#1934C2212yes—
92#1933C2212yes—
93#1878C2214yes—
94#1877C2214yes—
95#1865C2214yes—
96#1863C2214yes—
97#1842C2211yescharacteristic
98C4×#106C2211yescharacteristic
99#1770C2214yes—
100C4×#103C2211yescharacteristic
101C4×#102C2211yescharacteristic
102C42×C8C2211yescharacteristic
103C22×Q16C2312yes—
104C22×Q16C2312yes—
105C22×Q16C2312yes—
106C22×SD16C2311yescharacteristic
107C22×SD16C2312yes—
108Q82—24no—
109Q82—42no—
110#557—24no—
111#557—42no—
112#556—24no—
113#556—42no—
114#554—24no—
115#554—42no—
116C2×C4⋊Q8—21no—
117C2×C4⋊Q8—22no—
118C2×C4⋊Q8D411yescharacteristic
119C2×C4⋊Q8C2311yescharacteristic
120C2×C4⋊Q8C2311yescharacteristic
121C2×C4⋊Q8C2312yes—
122C2×C4⋊Q8C2311yescharacteristic
123C2×C4⋊Q8C2311yescharacteristic
124C2×C4⋊Q8C2312yes—
125C2×#126—22no—
126C2×#125—21no—
127C2×#125—22no—
128C2×#125C2312yes—
129C2×#124C2312yes—
130C2×C22⋊Q8—21no—
131C2×C22⋊Q8—21no—
132C2×C22⋊Q8—22no—
133C2×C4×Q8—21no—
134C2×C4×Q8—21no—
135C2×C4×Q8C2312yes—
136C2×C4×Q8C2312yes—
137C2×C4×Q8C2312yes—
138C2×C4×Q8C2311yescharacteristic
139C2×C4×Q8C2311yescharacteristic
140C2×C4×Q8C2311yescharacteristic
141C2×C4×D4C2312yes—
142C2×C4×D4C2311yescharacteristic
143C2×#117—21no—
144C2×#117—21no—
145C22×C4⋊C4—21no—
146#501—28no—
147#500—24no—
148#497—28no—
149#494—24no—
150#487—42no—
151#487—42no—
152#479—28no—
153#477—24no—
154#476—24no—
155#474—28no—
156#464—28no—
157#462—24no—
158#462—28no—
159#461—24no—
160#446—24no—
161#446—24no—
162#441—28no—
163#440—28no—
164C4×SD16D418yes—
165C2×#107C2312yes—
166C2×#107C2312yes—
167C2×#107C2312yes—
168C2×#106C2312yes—
169C2×#106C2312yes—
170C2×#106C2311yescharacteristic
171C2×C4⋊C8C2311yescharacteristic
172C2×C4⋊C8C2312yes—
173C2×C4⋊C8C2312yes—
174C2×C4⋊C8C2311yescharacteristic
175C2×C4⋊C8C2314yes—
176C2×#103C2312yes—
177C2×#103C2312yes—
178C2×#103C2312yes—
179C2×#103C2314yes—
180C2×#103C2314yes—
181C2×#103C2314yes—
182C2×#103C2314yes—
183C2×#103C2312yes—
184C2×#103C2311yescharacteristic
185C2×#102C2312yes—
186C2×#102C2312yes—
187C2×#102C2311yescharacteristic
188C2×#97C2312yes—
189C2×#97C2312yes—
190C2×C4×C8C2311yescharacteristic
191C2×C4×C8C2312yes—
192C2×C4×C8C2312yes—
193C2×C4×C8C2311yescharacteristic
194#400—22no—
195#400—22no—
196#400—24no—
197#400—24no—
198#398—24no—
199#398—24no—
200#398—24no—
201#397—24no—
202#397—24no—
203#395—22no—
204#395—22no—
205#395—24no—
206#391—21no—
207#391—21no—
208#391—21no—
209#391—21no—
210#391—22no—
211#391—22no—
212#391—24no—
213#391D411yescharacteristic
214#391D411yescharacteristic
215#391C2314yes—
216#391C2312yes—
217#391C2314yes—
218#391C2314yes—
219#391C2314yes—
220#390C2312yes—
221#389—24no—
222#389D414yes—
223#389C2314yes—
224#389C2314yes—
225#389C2312yes—
226#389C2312yes—
227#388—22no—
228#387—24no—
229#387—24no—
230#387—22no—
231#387—24no—
232#387—24no—
233#385—24no—
234#384—21no—
235#384—21no—
236#384D411yescharacteristic
237#382—21no—
238#382—21no—
239#382—21no—
240#382—21no—
241#382—21no—
242#382—21no—
243#381—24no—
244C4×C4⋊C4—21no—
245C4×C4⋊C4C2311yescharacteristic
246C4×C4⋊C4C2311yescharacteristic
247C4×C4⋊C4C2314yes—
248C4×C22⋊C4—21no—
249#376—21no—
250C43C2311yescharacteristic
251#340C2314yes—
252#340C2312yes—
253#340C2314yes—
254#340C2312yes—
255#340C2312yes—
256#336C2314yes—
257#336C2314yes—
258C22×Q8—21no—
259C22×Q8—22no—
260C22×Q8—21no—
261C22×Q8C2×D411yescharacteristic
262C22×Q8C2×D411yescharacteristic
263C22×Q8C2412yes—
264C22×Q8C2412yes—
265C22×Q8C2411yescharacteristic
266C22×D4C2411yescharacteristic
267C23×C4—21no—
268C23×C4—22no—
269C2×Q16—24no—
270C2×Q16—24no—
271C2×Q16—24no—
272C2×Q16—28no—
273C2×Q16—44no—
274C2×Q16—44no—
275C2×SD16—24no—
276C2×SD16—24no—
277C2×SD16—44no—
278C2×SD16C2×D414yes—
279C22×C8C2412yes—
280C22×C8C2412yes—
281C22×C8C2412yes—
282C22×C8C2411yescharacteristic
283C22×C8C2411yescharacteristic
284C4⋊Q8—24no—
285C4⋊Q8—24no—
286C4⋊Q8—24no—
287C4⋊Q8—22no—
288C4⋊Q8—22no—
289C4⋊Q8—22no—
290C4⋊Q8—22no—
291C4⋊Q8—22no—
292C4⋊Q8—22no—
293C4⋊Q8—22no—
294C4⋊Q8—22no—
295C4⋊Q8—42no—
296C4⋊Q8—44no—
297C4⋊Q8—42no—
298C4⋊Q8—48no—
299#126—48no—
300#125—24no—
301#125—42no—
302#125—44no—
303#125—48no—
304#124—24no—
305#124—42no—
306C22⋊Q8—48no—
307C22⋊Q8—44no—
308C22⋊Q8—44no—
309C4×Q8—22no—
310C4×Q8—22no—
311C4×Q8—22no—
312C4×Q8—22no—
313C4×Q8—22no—
314C4×Q8—24no—
315C4×Q8—24no—
316C4×Q8—24no—
317C4×Q8—42no—
318C4×Q8—42no—
319C4×Q8—42no—
320C4×Q8—44no—
321C4×Q8—44no—
322C4×Q8C2×D414yes—
323C4×D4—24no—
324C4×D4—22no—
325C4×D4—42no—
326C4×D4C2×D414yes—
327#117—44no—
328#117—44no—
329C2×C4⋊C4—21no—
330C2×C4⋊C4—22no—
331C2×C4⋊C4—21no—
332C2×C4⋊C4—21no—
333C2×C4⋊C4—22no—
334C2×C4⋊C4—21no—
335C2×C4⋊C4—22no—
336C2×C4⋊C4—21no—
337C2×C4⋊C4—22no—
338C2×C4⋊C4—22no—
339C2×C4⋊C4—22no—
340C2×C4⋊C4—22no—
341C2×C4⋊C4—22no—
342C2×C4⋊C4—22no—
343C2×C4⋊C4—21no—
344C2×C4⋊C4—21no—
345C2×C4⋊C4—22no—
346C2×C4⋊C4—24no—
347C2×C4⋊C4—24no—
348C2×C4⋊C4—44no—
349C2×C4⋊C4C2×D411yescharacteristic
350C2×C4⋊C4C2×D411yescharacteristic
351C2×C4⋊C4C2×D412yes—
352C2×C4⋊C4C2×D412yes—
353C2×C4⋊C4C2×D414yes—
354C2×C4⋊C4C2×D414yes—
355C2×C4⋊C4C2412yes—
356C2×C4⋊C4C2412yes—
357C2×C4⋊C4C2412yes—
358C2×C4⋊C4C2412yes—
359C2×C4⋊C4C2411yescharacteristic
360C2×C4⋊C4C2411yescharacteristic
361C2×C22⋊C4—21no—
362C2×C22⋊C4—22no—
363C2×C22⋊C4—22no—
364C2×C22⋊C4—22no—
365C2×C22⋊C4—22no—
366C2×C22⋊C4—22no—
367C2×C42—21no—
368C2×C42—22no—
369C2×C42—21no—
370C2×C42—22no—
371C2×C42—22no—
372C2×C42—21no—
373C2×C42C2×D411yescharacteristic
374C2×C42C2×D412yes—
375C2×C42C2×D411yescharacteristic
376C2×C42C2412yes—
377C2×C42C2412yes—
378C2×C42C2412yes—
379C2×C42C2411yescharacteristic
380#107—24no—
381#107—24no—
382#107—24no—
383#106—24no—
384#106—24no—
385#106C2×D414yes—
386C4⋊C8—24no—
387C4⋊C8—24no—
388C4⋊C8—22no—
389C4⋊C8—22no—
390C4⋊C8—44no—
391#103—24no—
392#103—24no—
393#103—24no—
394#103—44no—
395#103—42no—
396#103—44no—
397#103—44no—
398#103—44no—
399#103C2×D414yes—
400#102—24no—
401#102—42no—
402#102C2×D414yes—
403#97—24no—
404#97—24no—
405C4×C8—24no—
406C4×C8—22no—
407C4×C8—42no—
408C4×C8C2×D414yes—
409#95—22no—
410#95—22no—
411#95—22no—
412#95—22no—
413#95—21no—
414#95—21no—
415#95—24no—
416#95—24no—
417#95—24no—
418#95—24no—
419#95—24no—
420#95—24no—
421#95—24no—
422#95—24no—
423#95—24no—
424#95—24no—
425#95C2×D414yes—
426#95C2×D414yes—
427C24—21no—
428C2×Q8—22no—
429C2×Q8—22no—
430C2×Q8—22no—
431C2×Q8—22no—
432C2×Q8—22no—
433C2×Q8—22no—
434C2×Q8—21no—
435C2×Q8—22no—
436C2×Q8—22no—
437C2×Q8—22no—
438C2×Q8—22no—
439C2×Q8—22no—
440C2×Q8—24no—
441C2×Q8—24no—
442C2×Q8—22no—
443C2×Q8—24no—
444C2×Q8—24no—
445C2×Q8—44no—
446C2×Q8—42no—
447C2×Q8—42no—
448C2×Q8—82no—
449C2×Q8—82no—
450C2×Q8—84no—
451C2×Q8C22×D412yes—
452C2×D4—21no—
453C2×D4—22no—
454C2×D4—22no—
455C2×D4C22×D412yes—
456C22×C4—21no—
457C22×C4—21no—
458C22×C4—21no—
459C22×C4—24no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—22no—
466C22×C4—44no—
467C22×C4—42no—
468C22×C4—42no—
469C22×C4—84no—
470C22×C4C22×D412yes—
471C22×C4C22×D412yes—
472C22×C4C22×D412yes—
473C22×C4C22×D412yes—
474C22×C4C22×D412yes—
475C22×C4C22×D411yescharacteristic
476C22×C4C22×D412yes—
477C22×C4C22×D411yescharacteristic
478C22×C4C2511yescharacteristic
479Q16—48no—
480Q16—84no—
481Q16—84no—
482SD16—28no—
483SD16—84no—
484C2×C8—22no—
485C2×C8—22no—
486C2×C8—22no—
487C2×C8—22no—
488C2×C8—22no—
489C2×C8—22no—
490C2×C8—24no—
491C2×C8—24no—
492C2×C8C22×D412yes—
493C2×C8C22×D412yes—
494C2×C82−1+414yes—
495C4⋊C4—24no—
496C4⋊C4—22no—
497C4⋊C4—22no—
498C4⋊C4—24no—
499C4⋊C4—24no—
500C4⋊C4—28no—
501C4⋊C4—21no—
502C4⋊C4—21no—
503C4⋊C4—22no—
504C4⋊C4—22no—
505C4⋊C4—22no—
506C4⋊C4—22no—
507C4⋊C4—22no—
508C4⋊C4—22no—
509C4⋊C4—22no—
510C4⋊C4—22no—
511C4⋊C4—44no—
512C4⋊C4—44no—
513C4⋊C4—44no—
514C4⋊C4—44no—
515C4⋊C4—44no—
516C4⋊C4—44no—
517C4⋊C4—44no—
518C4⋊C4—44no—
519C4⋊C4—44no—
520C4⋊C4—42no—
521C4⋊C4—42no—
522C4⋊C4—42no—
523C4⋊C4—42no—
524C4⋊C4—42no—
525C4⋊C4—42no—
526C4⋊C4—44no—
527C4⋊C4—82no—
528C4⋊C4—84no—
529C4⋊C4—84no—
530C4⋊C4C22×D412yes—
531C4⋊C4C22×D412yes—
532C22⋊C4—24no—
533C22⋊C4—44no—
534C22⋊C4—44no—
535C22⋊C4—44no—
536C22⋊C4—44no—
537C22⋊C4—82no—
538C42—24no—
539C42—22no—
540C42—22no—
541C42—22no—
542C42—22no—
543C42—22no—
544C42—22no—
545C42—22no—
546C42—42no—
547C42—44no—
548C42—44no—
549C42—44no—
550C42—42no—
551C42—42no—
552C42#13714yes—
553C42C22×D412yes—
554C42H(5,𝔽2)12yes—
555C23—22no—
556C23—42no—
557C23—42no—
558C23—41no—
559C23—41no—
560C23C23×D411yescharacteristic
561Q8—22no—
562Q8—22no—
563Q8—44no—
564Q8—44no—
565Q8—42no—
566Q8—42no—
567Q8—42no—
568Q8—42no—
569Q8—42no—
570Q8—44no—
571Q8—44no—
572Q8—162no—
573Q8—162no—
574Q8—164no—
575D4—22no—
576D4—22no—
577D4—42no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—24no—
583C2×C4—24no—
584C2×C4—22no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—22no—
591C2×C4—22no—
592C2×C4—22no—
593C2×C4—22no—
594C2×C4—22no—
595C2×C4—22no—
596C2×C4—22no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—41no—
600C2×C4—41no—
601C2×C4—48no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—44no—
605C2×C4—44no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—44no—
613C2×C4—44no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—164no—
617C2×C4D4212yes—
618C2×C4D4212yes—
619C2×C4C2×#13714yes—
620C2×C4C2×#13712yes—
621C2×C4#57812yes—
622C2×C4C23×D411yescharacteristic
623C2×C4C2×H(5,𝔽2)11yescharacteristic
624C2×C4C2×2−1+412yes—
625C2×C4#58512yes—
626C8—22no—
627C8—42no—
628C8—42no—
629C8—42no—
630C8#53614yes—
631C22—42no—
632C22—42no—
633C22—42no—
634C22—82no—
635C22—82no—
636C22C2×D4211yescharacteristic
637C22#349712yes—
638C22C22×#13711yescharacteristic
639C22C2×#57811yescharacteristic
640C22#358412yes—
641C4—24no—
642C4—22no—
643C4—42no—
644C4—42no—
645C4—42no—
646C4—42no—
647C4—42no—
648C4—42no—
649C4—42no—
650C4—42no—
651C4—82no—
652C4—82no—
653C4—82no—
654C4—82no—
655C4—82no—
656C4—82no—
657C4#327614yes—
658C4C2×#53611yescharacteristic
659C4#350011yescharacteristic
660C4#355412yes—
661C2—82no—
662C2#3936311yescharacteristic
663C2C2×#327611yescharacteristic
664C2#6109011yescharacteristic
665C2#6236612yes—
666C2#6238512yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×D4C24
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×C4⋊Q8 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #39619C2semidirect
5. #33242C2semidirect
7. #31149C2semidirect
8. #29830C2semidirect
10. #28744C2semidirect
11. #28682C2semidirect
12. #26497C2semidirect
14. #25936C2semidirect
16. #25651C2semidirect
18. #21712C2semidirect

Maximal subgroups

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

subgroupindex
C2×#33982
C2×#33852
#396192
#394902
#332422
#311522
#311492
#298302
#298282
#287442
#286822
#264972
#262662
#259362
#259342
#256512
#256462
#217122
#212742