Finite Group Databasebeta

← #6815197

Subgroups — #6815197

Order 512.

1756 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
1C2×#3364C211yesmaximal, characteristic
2C2×#3359C211yesmaximal, characteristic
3#39170C211yesmaximal, characteristic, complemented
4#38954C211yesmaximal, characteristic
5#33242C211yesmaximal, characteristic, complemented
6#31242C214yesmaximal, complemented
7#31215C214yesmaximal
8#29831C211yesmaximal, characteristic, complemented
9#29829C211yesmaximal, characteristic
10#28639C211yesmaximal, characteristic, complemented
11#28639C211yesmaximal, characteristic, complemented
12#26822C214yesmaximal, complemented
13#26817C214yesmaximal
14#25840C211yesmaximal, characteristic, complemented
15#25838C211yesmaximal, characteristic
16#25634C211yesmaximal, characteristic, complemented
17#25632C211yesmaximal, characteristic
18C8×C4⋊Q8C211yesmaximal, characteristic, complemented
19#21274C211yesmaximal, characteristic
20C2×#557C2211yescharacteristic
21C2×#557C2211yescharacteristic
22C2×#556C2211yescharacteristic
23C2×D4×Q8C2211yescharacteristic
24#3364—28no—
25#3359—28no—
26C2×#500C2211yescharacteristic
27C2×#494C2211yescharacteristic
28C2×#488C2212yes—
29C2×#487C2212yes—
30C2×#487C2211yescharacteristic
31C2×#487C2211yescharacteristic
32C2×#479C2212yes—
33C2×#475C2212yes—
34C2×#475C2211yescharacteristic
35C2×#474C2211yescharacteristic
36C2×#464C2212yes—
37C2×#462C2211yescharacteristic
38C2×#461C2211yescharacteristic
39C2×#461C2212yes—
40C2×C8×Q8C2211yescharacteristic
41C2×C8×Q8C2211yescharacteristic
42C2×C4×Q16C2212yes—
43C2×C4×SD16C2212yes—
44C2×C4×SD16C2211yescharacteristic
45#2862C2211yescharacteristic
46#2862C2211yescharacteristic
47#2719—24no—
48#2698—24no—
49#2642—21no—
50#2621—21no—
51#2568C2211yescharacteristic
52#2568C2211yescharacteristic
53#2565—21no—
54#2546—21no—
55#2546C2211yescharacteristic
56#2546C2211yescharacteristic
57#2499—21no—
58#2453C2214yes—
59#2451C2214yes—
60#2447C2214yes—
61#2446C2214yes—
62#2434—21no—
63#2327—21no—
64#2310—21no—
65C4×C4⋊Q8C2211yescharacteristic
66#2295C2211yescharacteristic
67#2293C2211yescharacteristic
68#2068C2214yes—
69#2066C2214yes—
70#2066C2214yes—
71#2066C2214yes—
72#2059C2214yes—
73#2057C2214yes—
74#2056C2214yes—
75#2055C2214yes—
76#2016C2214yes—
77#2015C2214yes—
78#1988C2211yescharacteristic
79#1988C2211yescharacteristic
80#1984C2211yescharacteristic
81#1984C2211yescharacteristic
82#1981C2211yescharacteristic
83#1981C2211yescharacteristic
84#1969C2214yes—
85#1969C2214yes—
86#1968C2212yes—
87#1941C2212yes—
88#1939C2214yes—
89#1934C2212yes—
90#1933C2212yes—
91#1933C2214yes—
92#1933C2214yes—
93#1878C2214yes—
94#1878C2214yes—
95#1863C2214yes—
96#1862C2214yes—
97#1842C2211yescharacteristic
98C4×#106C2211yescharacteristic
99C8×C4⋊C4C2214yes—
100C4×#103C2211yescharacteristic
101C4×#102C2211yescharacteristic
102C42×C8C2211yescharacteristic
103C23×Q8—21no—
104C22×Q16C2312yes—
105C22×Q16C2314yes—
106C22×SD16C2312yes—
107C22×SD16C2311yescharacteristic
108#557—24no—
109#557—24no—
110#557—42no—
111#557—42no—
112#556—24no—
113#556—42no—
114D4×Q8—24no—
115D4×Q8—42no—
116C2×C4⋊Q8—21no—
117C2×C4⋊Q8—21no—
118C2×C4⋊Q8—21no—
119C2×C4⋊Q8D411yescharacteristic
120C2×C4⋊Q8C2311yescharacteristic
121C2×C4⋊Q8C2311yescharacteristic
122C2×C4⋊Q8C2312yes—
123C2×C4⋊Q8C2311yescharacteristic
124C2×C4⋊Q8C2311yescharacteristic
125C2×C4⋊Q8C2312yes—
126C2×#126—22no—
127C2×#125—21no—
128C2×#125—22no—
129C2×#125—22no—
130C2×#125C2312yes—
131C2×#124C2312yes—
132C2×C22⋊Q8—21no—
133C2×C22⋊Q8—21no—
134C2×C22⋊Q8—22no—
135C2×C4×Q8—21no—
136C2×C4×Q8—21no—
137C2×C4×Q8C2312yes—
138C2×C4×Q8C2312yes—
139C2×C4×Q8C2312yes—
140C2×C4×Q8C2311yescharacteristic
141C2×C4×Q8C2311yescharacteristic
142C2×C4×Q8C2311yescharacteristic
143C2×C4×D4C2312yes—
144C2×C4×D4C2311yescharacteristic
145C2×#117—21no—
146C2×#117—21no—
147#500—42no—
148#494—42no—
149#488—28no—
150#487—28no—
151#487—24no—
152#487—24no—
153#479—28no—
154#475—24no—
155#475—28no—
156#474—24no—
157#464—28no—
158#462—24no—
159#461—28no—
160#461—24no—
161C8×Q8—24no—
162C8×Q8—24no—
163C4×Q16—28no—
164C4×SD16—28no—
165C4×SD16D418yes—
166C2×#107C2312yes—
167C2×#107C2314yes—
168C2×#106C2312yes—
169C2×#106C2312yes—
170C2×#106C2311yescharacteristic
171C2×C4⋊C8C2312yes—
172C2×C4⋊C8C2311yescharacteristic
173C2×C4⋊C8C2312yes—
174C2×C4⋊C8C2311yescharacteristic
175C2×C4⋊C8C2314yes—
176C2×#103C2312yes—
177C2×#103C2312yes—
178C2×#103C2312yes—
179C2×#103C2312yes—
180C2×#103C2312yes—
181C2×#103C2312yes—
182C2×#103C2312yes—
183C2×#103C2312yes—
184C2×#103C2314yes—
185C2×#103C2314yes—
186C2×#103C2311yescharacteristic
187C2×#102C2312yes—
188C2×#102C2312yes—
189C2×#102C2311yescharacteristic
190C2×C4×C8C2312yes—
191C2×C4×C8C2312yes—
192C2×C4×C8C2311yescharacteristic
193C2×C4×C8C2312yes—
194C2×C4×C8C2312yes—
195C2×C4×C8C2311yescharacteristic
196#400—22no—
197#400—22no—
198#400—24no—
199#400—24no—
200#398—24no—
201#398—22no—
202#398—24no—
203#398—22no—
204#398—24no—
205#397—24no—
206#397—24no—
207#395—24no—
208#391—21no—
209#391—21no—
210#391—21no—
211#391—21no—
212#391—21no—
213#391—21no—
214#391—21no—
215#391—21no—
216#391—24no—
217#391D411yescharacteristic
218#391D411yescharacteristic
219#391C2314yes—
220#391C2312yes—
221#391C2314yes—
222#391C2314yes—
223#391C2314yes—
224#390C2312yes—
225#389—24no—
226#389D414yes—
227#389C2314yes—
228#389C2314yes—
229#389C2312yes—
230#389C2312yes—
231#388—22no—
232#387—24no—
233#387—24no—
234#387—22no—
235#387—24no—
236#387—24no—
237#385—24no—
238#384D411yescharacteristic
239#382—21no—
240#382—21no—
241#382—22no—
242#382—22no—
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#340C2314yes—
253#340C2312yes—
254#340C2312yes—
255#340C2312yes—
256#336C2314yes—
257#336C2314yes—
258C22×Q8—21no—
259C22×Q8—21no—
260C22×Q8—22no—
261C22×Q8—44no—
262C22×Q8—44no—
263C22×Q8—82no—
264C22×Q8C2×D411yescharacteristic
265C22×Q8C2×D411yescharacteristic
266C22×Q8C2412yes—
267C22×Q8C2412yes—
268C22×Q8C2411yescharacteristic
269C22×D4C2411yescharacteristic
270C23×C4—21no—
271C23×C4—22no—
272C2×Q16—24no—
273C2×Q16—28no—
274C2×Q16—44no—
275C2×Q16—44no—
276C2×Q16—44no—
277C2×SD16—24no—
278C2×SD16—42no—
279C2×SD16—44no—
280C2×SD16C2×D414yes—
281C22×C8C2412yes—
282C22×C8C2412yes—
283C22×C8C2411yescharacteristic
284C22×C8C2411yescharacteristic
285C22×C8C2412yes—
286C4⋊Q8—24no—
287C4⋊Q8—24no—
288C4⋊Q8—24no—
289C4⋊Q8—22no—
290C4⋊Q8—22no—
291C4⋊Q8—22no—
292C4⋊Q8—22no—
293C4⋊Q8—22no—
294C4⋊Q8—22no—
295C4⋊Q8—22no—
296C4⋊Q8—22no—
297C4⋊Q8—42no—
298C4⋊Q8—42no—
299C4⋊Q8—44no—
300C4⋊Q8—44no—
301C4⋊Q8—82no—
302#126—48no—
303#125—24no—
304#125—42no—
305#125—44no—
306#125—48no—
307#125—48no—
308#124—24no—
309#124—42no—
310C22⋊Q8—48no—
311C22⋊Q8—44no—
312C22⋊Q8—44no—
313C4×Q8—22no—
314C4×Q8—22no—
315C4×Q8—22no—
316C4×Q8—22no—
317C4×Q8—22no—
318C4×Q8—24no—
319C4×Q8—24no—
320C4×Q8—24no—
321C4×Q8—42no—
322C4×Q8—42no—
323C4×Q8—42no—
324C4×Q8—44no—
325C4×Q8—44no—
326C4×Q8C2×D414yes—
327C4×D4—24no—
328C4×D4—22no—
329C4×D4—42no—
330C4×D4C2×D414yes—
331#117—44no—
332#117—82no—
333C2×C4⋊C4—21no—
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—21no—
342C2×C4⋊C4—22no—
343C2×C4⋊C4—21no—
344C2×C4⋊C4—21no—
345C2×C4⋊C4—22no—
346C2×C4⋊C4—22no—
347C2×C4⋊C4—21no—
348C2×C4⋊C4—24no—
349C2×C4⋊C4—22no—
350C2×C4⋊C4—22no—
351C2×C4⋊C4—24no—
352C2×C4⋊C4C2×D411yescharacteristic
353C2×C4⋊C4C2×D411yescharacteristic
354C2×C4⋊C4C2×D412yes—
355C2×C4⋊C4C2×D412yes—
356C2×C4⋊C4C2×D414yes—
357C2×C4⋊C4C2×D414yes—
358C2×C4⋊C4C2412yes—
359C2×C4⋊C4C2412yes—
360C2×C4⋊C4C2412yes—
361C2×C4⋊C4C2412yes—
362C2×C4⋊C4C2411yescharacteristic
363C2×C4⋊C4C2411yescharacteristic
364C2×C22⋊C4—21no—
365C2×C22⋊C4—22no—
366C2×C22⋊C4—22no—
367C2×C22⋊C4—22no—
368C2×C22⋊C4—22no—
369C2×C22⋊C4—22no—
370C2×C42—21no—
371C2×C42—22no—
372C2×C42—21no—
373C2×C42—22no—
374C2×C42—22no—
375C2×C42—21no—
376C2×C42C2×D411yescharacteristic
377C2×C42C2×D411yescharacteristic
378C2×C42C2×D412yes—
379C2×C42C2412yes—
380C2×C42C2412yes—
381C2×C42C2412yes—
382C2×C42C2411yescharacteristic
383#107—24no—
384#107—44no—
385#106—24no—
386#106—42no—
387#106C2×D414yes—
388C4⋊C8—24no—
389C4⋊C8—24no—
390C4⋊C8—22no—
391C4⋊C8—22no—
392C4⋊C8—44no—
393#103—24no—
394#103—24no—
395#103—24no—
396#103—24no—
397#103—24no—
398#103—24no—
399#103—24no—
400#103—24no—
401#103—44no—
402#103—44no—
403#103C2×D414yes—
404#102—24no—
405#102—24no—
406#102C2×D414yes—
407C4×C8—24no—
408C4×C8—24no—
409C4×C8—24no—
410C4×C8—22no—
411C4×C8—42no—
412C4×C8C2×D414yes—
413#95—22no—
414#95—22no—
415#95—22no—
416#95—22no—
417#95—21no—
418#95—21no—
419#95—24no—
420#95—24no—
421#95—24no—
422#95—24no—
423#95—24no—
424#95—24no—
425#95—24no—
426#95—24no—
427#95—24no—
428#95—24no—
429#95C2×D414yes—
430#95C2×D414yes—
431C24—21no—
432C2×Q8—22no—
433C2×Q8—22no—
434C2×Q8—22no—
435C2×Q8—22no—
436C2×Q8—22no—
437C2×Q8—21no—
438C2×Q8—22no—
439C2×Q8—22no—
440C2×Q8—22no—
441C2×Q8—22no—
442C2×Q8—22no—
443C2×Q8—24no—
444C2×Q8—24no—
445C2×Q8—22no—
446C2×Q8—24no—
447C2×Q8—24no—
448C2×Q8—44no—
449C2×Q8—42no—
450C2×Q8—42no—
451C2×Q8—42no—
452C2×Q8—44no—
453C2×Q8—42no—
454C2×Q8—42no—
455C2×Q8—42no—
456C2×Q8—82no—
457C2×Q8—816no—
458C2×Q8#13514yes—
459C2×Q8C22×D412yes—
460C2×D4—21no—
461C2×D4—22no—
462C2×D4—22no—
463C2×D4C22×D412yes—
464C22×C4—21no—
465C22×C4—21no—
466C22×C4—21no—
467C22×C4—24no—
468C22×C4—22no—
469C22×C4—22no—
470C22×C4—22no—
471C22×C4—22no—
472C22×C4—22no—
473C22×C4—22no—
474C22×C4—44no—
475C22×C4—42no—
476C22×C4—42no—
477C22×C4—44no—
478C22×C4—82no—
479C22×C4C22×D412yes—
480C22×C4C22×D412yes—
481C22×C4C22×D412yes—
482C22×C4C22×D412yes—
483C22×C4C22×D411yescharacteristic
484C22×C4C22×D412yes—
485C22×C4C22×D412yes—
486C22×C4C22×D411yescharacteristic
487C22×C4C2511yescharacteristic
488Q16—48no—
489Q16—88no—
490SD16—44no—
491SD16—48no—
492C2×C8—22no—
493C2×C8—22no—
494C2×C8—22no—
495C2×C8—22no—
496C2×C8—22no—
497C2×C8—22no—
498C2×C8—24no—
499C2×C8—24no—
500C2×C8C22×D412yes—
501C2×C8C22×D412yes—
502C2×C82−1+414yes—
503C4⋊C4—24no—
504C4⋊C4—22no—
505C4⋊C4—24no—
506C4⋊C4—22no—
507C4⋊C4—24no—
508C4⋊C4—28no—
509C4⋊C4—28no—
510C4⋊C4—21no—
511C4⋊C4—21no—
512C4⋊C4—22no—
513C4⋊C4—22no—
514C4⋊C4—22no—
515C4⋊C4—22no—
516C4⋊C4—22no—
517C4⋊C4—22no—
518C4⋊C4—22no—
519C4⋊C4—22no—
520C4⋊C4—44no—
521C4⋊C4—44no—
522C4⋊C4—44no—
523C4⋊C4—44no—
524C4⋊C4—44no—
525C4⋊C4—44no—
526C4⋊C4—44no—
527C4⋊C4—44no—
528C4⋊C4—42no—
529C4⋊C4—42no—
530C4⋊C4—42no—
531C4⋊C4—42no—
532C4⋊C4—42no—
533C4⋊C4—42no—
534C4⋊C4—48no—
535C4⋊C4—44no—
536C4⋊C4—82no—
537C4⋊C4—84no—
538C4⋊C4C22×D412yes—
539C4⋊C4C22×D412yes—
540C22⋊C4—24no—
541C22⋊C4—44no—
542C22⋊C4—44no—
543C22⋊C4—44no—
544C22⋊C4—44no—
545C22⋊C4—82no—
546C42—24no—
547C42—22no—
548C42—22no—
549C42—24no—
550C42—22no—
551C42—22no—
552C42—22no—
553C42—22no—
554C42—22no—
555C42—22no—
556C42—44no—
557C42—44no—
558C42—44no—
559C42—42no—
560C42—42no—
561C42C22×D412yes—
562C42H(5,𝔽2)12yes—
563C23—22no—
564C23—42no—
565C23—41no—
566C23—41no—
567C23—42no—
568C23C23×D411yescharacteristic
569Q8—28no—
570Q8—22no—
571Q8—22no—
572Q8—44no—
573Q8—42no—
574Q8—42no—
575Q8—42no—
576Q8—42no—
577Q8—42no—
578Q8—44no—
579Q8—44no—
580Q8—88no—
581Q8—84no—
582Q8—84no—
583D4—22no—
584D4—22no—
585D4—42no—
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—22no—
598C2×C4—22no—
599C2×C4—22no—
600C2×C4—22no—
601C2×C4—22no—
602C2×C4—24no—
603C2×C4—24no—
604C2×C4—42no—
605C2×C4—41no—
606C2×C4—41no—
607C2×C4—42no—
608C2×C4—48no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—44no—
616C2×C4—44no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—42no—
624C2×C4—42no—
625C2×C4—88no—
626C2×C4D4212yes—
627C2×C4D4212yes—
628C2×C4C2×#13514yes—
629C2×C4C2×#13512yes—
630C2×C4#57712yes—
631C2×C4#57814yes—
632C2×C4C23×D411yescharacteristic
633C2×C4C2×H(5,𝔽2)11yescharacteristic
634C2×C4C2×2−1+412yes—
635C2×C4#58512yes—
636C8—22no—
637C8—22no—
638C8—24no—
639C8—42no—
640C8—42no—
641C22—42no—
642C22—42no—
643C22—42no—
644C22—82no—
645C22—82no—
646C22C2×D4211yescharacteristic
647C22#349712yes—
648C22C22×#13511yescharacteristic
649C22C2×#57711yescharacteristic
650C22C2×#57812yes—
651C4—24no—
652C4—24no—
653C4—22no—
654C4—42no—
655C4—42no—
656C4—42no—
657C4—42no—
658C4—42no—
659C4—42no—
660C4—42no—
661C4—82no—
662C4—82no—
663C4—82no—
664C4—82no—
665C4—82no—
666C4—82no—
667C4#328214yes—
668C4#344911yescharacteristic
669C4#349911yescharacteristic
670C4#356112yes—
671C2—82no—
672C2#3935611yescharacteristic
673C2C2×#328211yescharacteristic
674C2#6109111yescharacteristic
675C2#6236412yes—
676C2#6238212yes—
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. #39170C2semidirect
5. #33242C2semidirect
6. #31242C2semidirect
8. #29831C2semidirect
10. #28639C2semidirect
11. #28639C2semidirect
12. #26822C2semidirect
14. #25840C2semidirect
16. #25634C2semidirect
18. C8×C4⋊Q8C2semidirect

Maximal subgroups

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

subgroupindex
C2×#33642
C2×#33592
#391702
#389542
#332422
#312422
#312152
#298312
#298292
#286392
#268222
#268172
#258402
#258382
#256342
#256322
C8×C4⋊Q82
#212742