Finite Group Databasebeta

← #6815222

Subgroups — #6815222

Order 512.

1722 conjugacy classes of subgroups.

The subgroup lattice has 675 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3375C211yesmaximal, characteristic
2C2×#3374C211yesmaximal, characteristic, complemented
3#39746C211yesmaximal, characteristic
4#39619C211yesmaximal, characteristic, complemented
5#33239C211yesmaximal, characteristic
6#31235C214yesmaximal, complemented
7#31211C214yesmaximal, complemented
8#29830C211yesmaximal, characteristic, complemented
9#29826C211yesmaximal, characteristic
10#28646C211yesmaximal, characteristic, complemented
11#28645C211yesmaximal, characteristic
12#26846C214yesmaximal, complemented
13#26845C214yesmaximal, complemented
14#25837C211yesmaximal, characteristic, complemented
15#25836C211yesmaximal, characteristic, complemented
16#25630C211yesmaximal, characteristic
17#25629C211yesmaximal, characteristic
18#21712C211yesmaximal, characteristic, complemented
19#21276C211yesmaximal, characteristic, complemented
20C2×#559C2211yescharacteristic
21C2×Q82C2211yescharacteristic
22C2×#557C2211yescharacteristic
23C2×#557C2211yescharacteristic
24#3375—28no—
25#3374—28no—
26C2×#498C2211yescharacteristic
27C2×#492C2211yescharacteristic
28C2×#491C2212yes—
29C2×#489C2212yes—
30C2×#487C2211yescharacteristic
31C2×#486C2211yescharacteristic
32C2×#479C2212yes—
33C2×#477C2211yescharacteristic
34C2×#475C2211yescharacteristic
35C2×#474C2212yes—
36C2×#464C2212yes—
37C2×#462C2211yescharacteristic
38C2×#462C2212yes—
39C2×#460C2211yescharacteristic
40C2×#446C2211yescharacteristic
41C2×#446C2211yescharacteristic
42C2×#441C2212yes—
43C2×#440C2212yes—
44C2×C4×Q16C2211yescharacteristic
45#2862C2211yescharacteristic
46#2858C2211yescharacteristic
47#2713—24no—
48#2694—24no—
49#2660—21no—
50#2658—21no—
51#2568—21no—
52#2568C2211yescharacteristic
53#2567C2211yescharacteristic
54#2565—21no—
55#2548C2211yescharacteristic
56#2547—21no—
57#2546C2211yescharacteristic
58#2545—21no—
59#2454C2214yes—
60#2451C2214yes—
61#2448C2214yes—
62#2446C2214yes—
63#2327—21no—
64#2326—21no—
65C4×C4⋊Q8C2211yescharacteristic
66#2295C2211yescharacteristic
67#2295C2211yescharacteristic
68#2068C2214yes—
69#2068C2214yes—
70#2067C2214yes—
71#2065C2214yes—
72#2059C2214yes—
73#2058C2214yes—
74#2056C2214yes—
75#2055C2214yes—
76#2015C2214yes—
77#2013C2214yes—
78#1988C2211yescharacteristic
79#1987C2211yescharacteristic
80#1984C2211yescharacteristic
81#1984C2211yescharacteristic
82#1982C2211yescharacteristic
83#1981C2211yescharacteristic
84#1969C2212yes—
85#1968C2214yes—
86#1968C2214yes—
87#1942C2212yes—
88#1939C2214yes—
89#1936C2214yes—
90#1935C2214yes—
91#1933C2212yes—
92#1933C2212yes—
93#1880C2214yes—
94#1877C2214yes—
95#1865C2214yes—
96#1865C2214yes—
97#1842C2211yescharacteristic
98C4×#107C2211yescharacteristic
99#1770C2214yes—
100C4×#103C2211yescharacteristic
101C4×#103C2211yescharacteristic
102C42×C8C2211yescharacteristic
103C22×Q16C2312yes—
104C22×Q16C2311yescharacteristic
105C22×SD16C2312yes—
106C22×SD16C2314yes—
107#559—24no—
108#559—42no—
109Q82—24no—
110Q82—42no—
111#557—24no—
112#557—24no—
113#557—42no—
114#557—42no—
115C2×C4⋊Q8—21no—
116C2×C4⋊Q8—21no—
117C2×C4⋊Q8—21no—
118C2×C4⋊Q8—22no—
119C2×C4⋊Q8D411yescharacteristic
120C2×C4⋊Q8C2311yescharacteristic
121C2×C4⋊Q8C2312yes—
122C2×C4⋊Q8C2311yescharacteristic
123C2×C4⋊Q8C2311yescharacteristic
124C2×C4⋊Q8C2312yes—
125C2×#127C2311yescharacteristic
126C2×#125—21no—
127C2×#125—22no—
128C2×#125—22no—
129C2×#125C2312yes—
130C2×#124—21no—
131C2×#124—21no—
132C2×#124—21no—
133C2×#124—21no—
134C2×#124C2312yes—
135C2×#123—22no—
136C2×C4×Q8—21no—
137C2×C4×Q8—21no—
138C2×C4×Q8—21no—
139C2×C4×Q8C2312yes—
140C2×C4×Q8C2312yes—
141C2×C4×Q8C2312yes—
142C2×C4×Q8C2311yescharacteristic
143C2×C4×Q8C2311yescharacteristic
144C2×C4×Q8C2311yescharacteristic
145C2×C4×Q8C2311yescharacteristic
146C2×C4×D4—21no—
147C2×C4×D4C2312yes—
148#498—42no—
149#492—42no—
150#491—28no—
151#489—28no—
152#487—24no—
153#486—24no—
154#479—28no—
155#477—24no—
156#475—24no—
157#474—28no—
158#464—28no—
159#462—28no—
160#462—24no—
161#460—24no—
162#446—24no—
163#446—24no—
164#441—28no—
165#440—28no—
166C4×Q16D418yes—
167C2×#107C2312yes—
168C2×#107C2312yes—
169C2×#107C2311yescharacteristic
170C2×#106C2312yes—
171C2×#106C2314yes—
172C2×C4⋊C8C2311yescharacteristic
173C2×C4⋊C8C2311yescharacteristic
174C2×C4⋊C8C2312yes—
175C2×C4⋊C8C2312yes—
176C2×C4⋊C8C2314yes—
177C2×#103C2312yes—
178C2×#103C2312yes—
179C2×#103C2312yes—
180C2×#103C2312yes—
181C2×#103C2312yes—
182C2×#103C2312yes—
183C2×#103C2312yes—
184C2×#103C2314yes—
185C2×#103C2311yescharacteristic
186C2×#103C2311yescharacteristic
187C2×#102C2312yes—
188C2×#102C2312yes—
189C2×#102C2312yes—
190C2×#102C2314yes—
191C2×#97C2312yes—
192C2×#97C2312yes—
193C2×C4×C8C2311yescharacteristic
194C2×C4×C8C2312yes—
195C2×C4×C8C2312yes—
196C2×C4×C8C2311yescharacteristic
197#400—22no—
198#400—24no—
199#399—24no—
200#399—24no—
201#398—24no—
202#398—22no—
203#398—24no—
204#397—24no—
205#396—24no—
206#395—22no—
207#395—24no—
208#394—22no—
209#391—21no—
210#391—21no—
211#391—21no—
212#391—22no—
213#391—24no—
214#391D411yescharacteristic
215#391D411yescharacteristic
216#391C2312yes—
217#391C2312yes—
218#391C2314yes—
219#391C2314yes—
220#390—21no—
221#390—22no—
222#390C2314yes—
223#390C2314yes—
224#389D414yes—
225#389C2314yes—
226#389C2314yes—
227#389C2312yes—
228#389C2312yes—
229#388—24no—
230#388—24no—
231#387—24no—
232#387—24no—
233#387—22no—
234#387—22no—
235#387—24no—
236#385—24no—
237#384—21no—
238#384D411yescharacteristic
239#382—21no—
240#382—21no—
241#382—21no—
242#380—24no—
243C4×C4⋊C4—21no—
244C4×C4⋊C4—21no—
245C4×C4⋊C4C2311yescharacteristic
246C4×C4⋊C4C2311yescharacteristic
247C4×C4⋊C4C2314yes—
248#376—21no—
249#376—21no—
250C43C2311yescharacteristic
251#340C2314yes—
252#340C2314yes—
253#340C2312yes—
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×Q8C2411yescharacteristic
265C22×Q8C2411yescharacteristic
266C22×D4—21no—
267C22×D4—21no—
268C22×D4C2412yes—
269C23×C4—22no—
270C2×Q16—24no—
271C2×Q16—42no—
272C2×Q16—44no—
273C2×Q16C2×D414yes—
274C2×SD16—24no—
275C2×SD16—28no—
276C2×SD16—44no—
277C2×SD16—44no—
278C2×SD16—44no—
279C22×C8C2412yes—
280C22×C8C2412yes—
281C22×C8C2411yescharacteristic
282C22×C8C2411yescharacteristic
283C22×C8C2412yes—
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—42no—
294C4⋊Q8—44no—
295C4⋊Q8—42no—
296C4⋊Q8—44no—
297C4⋊Q8—48no—
298C4⋊Q8—82no—
299#127—22no—
300#127—22no—
301#125—24no—
302#125—42no—
303#125—44no—
304#125—48no—
305#125—48no—
306#124—24no—
307#124—42no—
308#124—44no—
309#124—44no—
310#124—82no—
311#124—82no—
312#123—48no—
313C4×Q8—22no—
314C4×Q8—22no—
315C4×Q8—22no—
316C4×Q8—22no—
317C4×Q8—22no—
318C4×Q8—22no—
319C4×Q8—24no—
320C4×Q8—24no—
321C4×Q8—24no—
322C4×Q8—42no—
323C4×Q8—42no—
324C4×Q8—42no—
325C4×Q8—44no—
326C4×Q8—44no—
327C4×Q8—44no—
328C4×Q8C2×D414yes—
329C4×Q8C2×D414yes—
330C4×D4—24no—
331C4×D4—42no—
332C4×D4—44no—
333C2×C4⋊C4—21no—
334C2×C4⋊C4—21no—
335C2×C4⋊C4—22no—
336C2×C4⋊C4—21no—
337C2×C4⋊C4—22no—
338C2×C4⋊C4—21no—
339C2×C4⋊C4—22no—
340C2×C4⋊C4—22no—
341C2×C4⋊C4—22no—
342C2×C4⋊C4—22no—
343C2×C4⋊C4—22no—
344C2×C4⋊C4—21no—
345C2×C4⋊C4—21no—
346C2×C4⋊C4—22no—
347C2×C4⋊C4—24no—
348C2×C4⋊C4—22no—
349C2×C4⋊C4—22no—
350C2×C4⋊C4—24no—
351C2×C4⋊C4C2×D411yescharacteristic
352C2×C4⋊C4C2×D411yescharacteristic
353C2×C4⋊C4C2×D412yes—
354C2×C4⋊C4C2×D412yes—
355C2×C4⋊C4C2×D414yes—
356C2×C4⋊C4C2×D414yes—
357C2×C4⋊C4C2412yes—
358C2×C4⋊C4C2412yes—
359C2×C4⋊C4C2412yes—
360C2×C4⋊C4C2412yes—
361C2×C4⋊C4C2411yescharacteristic
362C2×C4⋊C4C2411yescharacteristic
363C2×C22⋊C4—22no—
364C2×C22⋊C4—22no—
365C2×C22⋊C4—22no—
366C2×C22⋊C4—24no—
367C2×C22⋊C4—24no—
368C2×C22⋊C4—24no—
369C2×C22⋊C4—24no—
370C2×C42—21no—
371C2×C42—21no—
372C2×C42—22no—
373C2×C42—22no—
374C2×C42—22no—
375C2×C42—21no—
376C2×C42—21no—
377C2×C42C2×D411yescharacteristic
378C2×C42C2×D412yes—
379C2×C42C2×D411yescharacteristic
380C2×C42C2412yes—
381C2×C42C2412yes—
382C2×C42C2412yes—
383C2×C42C2411yescharacteristic
384#107—24no—
385#107—42no—
386#107C2×D414yes—
387#106—24no—
388#106—44no—
389C4⋊C8—24no—
390C4⋊C8—24no—
391C4⋊C8—22no—
392C4⋊C8—22no—
393C4⋊C8—44no—
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#103C2×D414yes—
403#103C2×D414yes—
404#102—24no—
405#102—24no—
406#102—24no—
407#102—44no—
408#97—24no—
409#97—24no—
410C4×C8—24no—
411C4×C8—22no—
412C4×C8—42no—
413C4×C8C2×D414yes—
414#95—22no—
415#95—22no—
416#95—22no—
417#95—22no—
418#95—22no—
419#95—22no—
420#95—21no—
421#95—21no—
422#95—24no—
423#95—24no—
424#95—24no—
425#95—24no—
426#95—24no—
427#95—24no—
428#95C2×D414yes—
429#95C2×D414yes—
430C24—22no—
431C2×Q8—22no—
432C2×Q8—22no—
433C2×Q8—22no—
434C2×Q8—22no—
435C2×Q8—22no—
436C2×Q8—22no—
437C2×Q8—21no—
438C2×Q8—21no—
439C2×Q8—22no—
440C2×Q8—22no—
441C2×Q8—22no—
442C2×Q8—22no—
443C2×Q8—24no—
444C2×Q8—22no—
445C2×Q8—22no—
446C2×Q8—24no—
447C2×Q8—44no—
448C2×Q8—42no—
449C2×Q8—42no—
450C2×Q8—42no—
451C2×Q8—42no—
452C2×Q8—82no—
453C2×Q8—84no—
454C2×Q8C22×D412yes—
455C2×Q8C22×D412yes—
456C2×D4—22no—
457C2×D4—22no—
458C2×D4—24no—
459C2×D4—24no—
460C2×D4—42no—
461C2×D4—42no—
462C2×D4—42no—
463C2×D4—42no—
464C2×D4—82no—
465C22×C4—21no—
466C22×C4—21no—
467C22×C4—21no—
468C22×C4—21no—
469C22×C4—22no—
470C22×C4—22no—
471C22×C4—22no—
472C22×C4—22no—
473C22×C4—22no—
474C22×C4—44no—
475C22×C4—84no—
476C22×C4C22×D412yes—
477C22×C4C22×D412yes—
478C22×C4C22×D412yes—
479C22×C4C22×D412yes—
480C22×C4C22×D412yes—
481C22×C4C22×D411yescharacteristic
482C22×C4C22×D412yes—
483C22×C4C22×D411yescharacteristic
484C22×C4C2511yescharacteristic
485Q16—44no—
486Q16—48no—
487SD16—48no—
488SD16—88no—
489C2×C8—22no—
490C2×C8—22no—
491C2×C8—22no—
492C2×C8—22no—
493C2×C8—22no—
494C2×C8—22no—
495C2×C8—24no—
496C2×C8—24no—
497C2×C8C22×D412yes—
498C2×C8C22×D412yes—
499C2×C82−1+414yes—
500C4⋊C4—24no—
501C4⋊C4—24no—
502C4⋊C4—22no—
503C4⋊C4—22no—
504C4⋊C4—24no—
505C4⋊C4—24no—
506C4⋊C4—28no—
507C4⋊C4—21no—
508C4⋊C4—21no—
509C4⋊C4—22no—
510C4⋊C4—22no—
511C4⋊C4—22no—
512C4⋊C4—22no—
513C4⋊C4—22no—
514C4⋊C4—22no—
515C4⋊C4—22no—
516C4⋊C4—22no—
517C4⋊C4—44no—
518C4⋊C4—44no—
519C4⋊C4—44no—
520C4⋊C4—44no—
521C4⋊C4—44no—
522C4⋊C4—44no—
523C4⋊C4—44no—
524C4⋊C4—44no—
525C4⋊C4—44no—
526C4⋊C4—42no—
527C4⋊C4—42no—
528C4⋊C4—42no—
529C4⋊C4—42no—
530C4⋊C4—82no—
531C4⋊C4—82no—
532C4⋊C4—84no—
533C4⋊C4—84no—
534C4⋊C4C22×D412yes—
535C4⋊C4C22×D412yes—
536C22⋊C4—44no—
537C22⋊C4—44no—
538C22⋊C4—44no—
539C22⋊C4—48no—
540C22⋊C4—84no—
541C22⋊C4—84no—
542C22⋊C4—84no—
543C42—24no—
544C42—24no—
545C42—22no—
546C42—22no—
547C42—22no—
548C42—22no—
549C42—22no—
550C42—22no—
551C42—22no—
552C42—42no—
553C42—44no—
554C42—44no—
555C42—44no—
556C42—42no—
557C42—42no—
558C42#13714yes—
559C42C22×D412yes—
560C42H(5,𝔽2)12yes—
561C23—44no—
562C23—42no—
563C23—44no—
564C23—42no—
565C23—42no—
566C23C23×D411yescharacteristic
567Q8—22no—
568Q8—22no—
569Q8—22no—
570Q8—22no—
571Q8—44no—
572Q8—44no—
573Q8—42no—
574Q8—42no—
575Q8—42no—
576Q8—42no—
577Q8—44no—
578Q8—84no—
579Q8—162no—
580Q8—164no—
581D4—42no—
582D4—42no—
583D4—44no—
584D4—84no—
585D4—162no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—24no—
591C2×C4—24no—
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—22no—
603C2×C4—22no—
604C2×C4—22no—
605C2×C4—22no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—44no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—44no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—164no—
624C2×C4D4212yes—
625C2×C4D4212yes—
626C2×C4C2×#13714yes—
627C2×C4C2×#13712yes—
628C2×C4#57712yes—
629C2×C4C23×D411yescharacteristic
630C2×C4C2×H(5,𝔽2)11yescharacteristic
631C2×C4C2×2−1+412yes—
632C2×C4#58512yes—
633C8—22no—
634C8—22no—
635C8—24no—
636C8—42no—
637C8—42no—
638C22—82no—
639C22—84no—
640C22—82no—
641C22—84no—
642C22—82no—
643C22C2×D4211yescharacteristic
644C22#349712yes—
645C22C22×#13711yescharacteristic
646C22C2×#57711yescharacteristic
647C22#358412yes—
648C4—24no—
649C4—22no—
650C4—42no—
651C4—42no—
652C4—42no—
653C4—42no—
654C4—42no—
655C4—42no—
656C4—42no—
657C4—42no—
658C4—42no—
659C4—82no—
660C4—82no—
661C4—82no—
662C4—82no—
663C4—82no—
664C4#327814yes—
665C4#344911yescharacteristic
666C4#349911yescharacteristic
667C4#356112yes—
668C2—162no—
669C2#3935611yescharacteristic
670C2C2×#327811yescharacteristic
671C2#6109011yescharacteristic
672C2#6236612yes—
673C2#6237812yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#127C23
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
2. C2×#3374C2semidirect
4. #39619C2semidirect
6. #31235C2semidirect
7. #31211C2semidirect
8. #29830C2semidirect
10. #28646C2semidirect
12. #26846C2semidirect
13. #26845C2semidirect
14. #25837C2semidirect
15. #25836C2semidirect
18. #21712C2semidirect
19. #21276C2semidirect

Maximal subgroups

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

subgroupindex
C2×#33752
C2×#33742
#397462
#396192
#332392
#312352
#312112
#298302
#298262
#286462
#286452
#268462
#268452
#258372
#258362
#256302
#256292
#217122
#212762