Finite Group Databasebeta

← #6683909

Subgroups — #6683909

Order 512.

1720 conjugacy classes of subgroups.

The subgroup lattice has 671 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3383C211yesmaximal, characteristic, complemented
2C2×#3375C211yesmaximal, characteristic
3#39746C211yesmaximal, characteristic
4#36474C211yesmaximal, characteristic, complemented
5#33025C211yesmaximal, characteristic
6#30950C214yesmaximal, complemented
7#30830C214yesmaximal, complemented
8#30278C211yesmaximal, characteristic
9#29867C211yesmaximal, characteristic, complemented
10#28683C211yesmaximal, characteristic, complemented
11#28640C211yesmaximal, characteristic
12#26399C214yesmaximal, complemented
13#26173C214yesmaximal, complemented
14#25935C211yesmaximal, characteristic
15#25837C211yesmaximal, characteristic, complemented
16#25836C211yesmaximal, characteristic, complemented
17#25652C211yesmaximal, characteristic
18#22240C211yesmaximal, characteristic, complemented
19#21116C211yesmaximal, characteristic, complemented
20C2×#559C2211yescharacteristic
21C2×Q82C2211yescharacteristic
22C2×#557C2211yescharacteristic
23C2×#557C2211yescharacteristic
24#3383—28no—
25#3375—28no—
26C2×#500C2211yescharacteristic
27C2×#499C2212yes—
28C2×#498C2211yescharacteristic
29C2×#489C2212yes—
30C2×#486C2211yescharacteristic
31C2×#486C2211yescharacteristic
32C2×#479C2212yes—
33C2×#479C2212yes—
34C2×#477C2211yescharacteristic
35C2×#475C2211yescharacteristic
36C2×#475C2211yescharacteristic
37C2×#474C2212yes—
38C2×#464C2212yes—
39C2×#460C2211yescharacteristic
40C2×#446C2211yescharacteristic
41C2×C8×Q8C2211yescharacteristic
42C2×#440C2212yes—
43C2×C4×Q16C2212yes—
44C2×C4×Q16C2211yescharacteristic
45#2862—21no—
46#2862C2211yescharacteristic
47#2858C2211yescharacteristic
48#2694—24no—
49#2658—21no—
50#2567—21no—
51#2567C2211yescharacteristic
52#2550—24no—
53#2550C2214yes—
54#2548C2211yescharacteristic
55#2547—21no—
56#2546—21no—
57#2546C2211yescharacteristic
58#2546C2211yescharacteristic
59#2545—21no—
60#2454C2214yes—
61#2448C2214yes—
62#2351C2214yes—
63#2326—21no—
64C4×C4⋊Q8—21no—
65C4×C4⋊Q8C2211yescharacteristic
66#2295C2211yescharacteristic
67C42×Q8C2211yescharacteristic
68#2092C2214yes—
69#2092C2214yes—
70#2087C2214yes—
71#2085C2214yes—
72#2074C2214yes—
73#2065C2214yes—
74#2059C2214yes—
75#2058C2214yes—
76#2042C2214yes—
77#2025C2214yes—
78#1988C2211yescharacteristic
79#1988C2211yescharacteristic
80#1987C2211yescharacteristic
81#1984C2211yescharacteristic
82#1984C2211yescharacteristic
83#1982C2211yescharacteristic
84#1944C2214yes—
85#1934C2214yes—
86#1927C2212yes—
87#1926C2214yes—
88#1918C2212yes—
89#1917C2214yes—
90#1916C2214yes—
91#1880C2214yes—
92#1880C2214yes—
93#1878C2212yes—
94#1865C2212yes—
95#1865C2214yes—
96#1865C2214yes—
97#1864C2214yes—
98#1831C2211yescharacteristic
99C4×#106C2211yescharacteristic
100C4×#103C2211yescharacteristic
101C4×#103C2211yescharacteristic
102#1744C2211yescharacteristic
103C22×Q16C2311yescharacteristic
104C22×SD16C2312yes—
105#559—24no—
106#559—42no—
107Q82—24no—
108Q82—42no—
109#557—24no—
110#557—24no—
111#557—42no—
112#557—42no—
113C2×C4⋊Q8—21no—
114C2×C4⋊Q8—21no—
115C2×C4⋊Q8—21no—
116C2×C4⋊Q8—21no—
117C2×C4⋊Q8—21no—
118C2×C4⋊Q8—22no—
119C2×C4⋊Q8D411yescharacteristic
120C2×C4⋊Q8C2311yescharacteristic
121C2×C4⋊Q8C2311yescharacteristic
122C2×C4⋊Q8C2311yescharacteristic
123C2×C4⋊Q8C2312yes—
124C2×#127C2311yescharacteristic
125C2×#125—22no—
126C2×#125—22no—
127C2×#125C2312yes—
128C2×#125C2312yes—
129C2×#124—21no—
130C2×#124—21no—
131C2×#124—21no—
132C2×#124—21no—
133C2×#124C2312yes—
134C2×#123—22no—
135C2×C4×Q8—21no—
136C2×C4×Q8—21no—
137C2×C4×Q8—21no—
138C2×C4×Q8C2312yes—
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#500—24no—
149#499—28no—
150#498—42no—
151#489—28no—
152#486—24no—
153#486—42no—
154#479—28no—
155#479—28no—
156#477—24no—
157#475—24no—
158#475—24no—
159#474—28no—
160#464—28no—
161#460—24no—
162#446—24no—
163C8×Q8—24no—
164#440—28no—
165C4×Q16—28no—
166C4×Q16D418yes—
167C2×#107C2312yes—
168C2×#107C2312yes—
169C2×#107C2312yes—
170C2×#107C2311yescharacteristic
171C2×#106C2312yes—
172C2×#106C2312yes—
173C2×#106C2312yes—
174C2×#106C2314yes—
175C2×C4⋊C8C2311yescharacteristic
176C2×C4⋊C8C2311yescharacteristic
177C2×C4⋊C8C2312yes—
178C2×C4⋊C8C2312yes—
179C2×#103C2312yes—
180C2×#103C2312yes—
181C2×#103C2312yes—
182C2×#103C2312yes—
183C2×#103C2312yes—
184C2×#103C2311yescharacteristic
185C2×#103C2311yescharacteristic
186C2×#102C2312yes—
187C2×#102C2314yes—
188C2×#102C2312yes—
189C2×#102C2312yes—
190C2×#102C2314yes—
191C2×#97C2312yes—
192C2×C4×C8C2311yescharacteristic
193C2×C4×C8C2312yes—
194C2×C4×C8C2311yescharacteristic
195#399—24no—
196#399—24no—
197#398—22no—
198#398—24no—
199#398—22no—
200#397—24no—
201#396—24no—
202#395—24no—
203#395—22no—
204#394—22no—
205#391—21no—
206#391—21no—
207#391—21no—
208#391—21no—
209#391—21no—
210#391—22no—
211#391—24no—
212#391D411yescharacteristic
213#391D411yescharacteristic
214#391C2312yes—
215#390—21no—
216#390—22no—
217#390C2314yes—
218#390C2314yes—
219#389—24no—
220#389—24no—
221#389—24no—
222#389D414yes—
223#389C2314yes—
224#389C2314yes—
225#389C2312yes—
226#389C2314yes—
227#388—24no—
228#388—24no—
229#387—22no—
230#385—24no—
231#384—21no—
232#384D411yescharacteristic
233#382—24no—
234#382C2314yes—
235#380—24no—
236C4×C4⋊C4—21no—
237C4×C4⋊C4—22no—
238C4×C4⋊C4—21no—
239C4×C4⋊C4—24no—
240C4×C4⋊C4C2311yescharacteristic
241C4×C4⋊C4C2312yes—
242C4×C4⋊C4C2311yescharacteristic
243C4×C4⋊C4C2314yes—
244#376—21no—
245C43—21no—
246C43C2311yescharacteristic
247#340C2312yes—
248#340C2314yes—
249#340C2312yes—
250#340C2314yes—
251#340C2314yes—
252#340C2312yes—
253#336C2314yes—
254#336C2314yes—
255#336C2314yes—
256#336C2312yes—
257#336C2312yes—
258C22×Q8—21no—
259C22×Q8—21no—
260C22×Q8—22no—
261C22×Q8—21no—
262C22×Q8C2×D411yescharacteristic
263C22×Q8C2×D411yescharacteristic
264C22×Q8C2411yescharacteristic
265C22×Q8C2411yescharacteristic
266C22×D4—21no—
267C22×D4—21no—
268C22×D4C2412yes—
269C23×C4—22no—
270C2×Q16—24no—
271C2×Q16C2×D414yes—
272C2×SD16—24no—
273C2×SD16—28no—
274C22×C8C2412yes—
275C22×C8C2412yes—
276C22×C8C2412yes—
277C22×C8C2411yescharacteristic
278C22×C8C2411yescharacteristic
279C4⋊Q8—24no—
280C4⋊Q8—22no—
281C4⋊Q8—22no—
282C4⋊Q8—22no—
283C4⋊Q8—22no—
284C4⋊Q8—22no—
285C4⋊Q8—22no—
286C4⋊Q8—42no—
287C4⋊Q8—44no—
288C4⋊Q8—42no—
289C4⋊Q8—44no—
290C4⋊Q8—44no—
291C4⋊Q8—48no—
292C4⋊Q8—44no—
293C4⋊Q8—82no—
294#127—22no—
295#127—22no—
296#125—24no—
297#125—24no—
298#125—42no—
299#125—42no—
300#125—48no—
301#125—48no—
302#124—24no—
303#124—42no—
304#124—44no—
305#124—44no—
306#124—82no—
307#124—82no—
308#123—48no—
309C4×Q8—24no—
310C4×Q8—24no—
311C4×Q8—22no—
312C4×Q8—22no—
313C4×Q8—22no—
314C4×Q8—22no—
315C4×Q8—22no—
316C4×Q8—22no—
317C4×Q8—24no—
318C4×Q8—24no—
319C4×Q8—24no—
320C4×Q8—42no—
321C4×Q8—42no—
322C4×Q8—42no—
323C4×Q8—44no—
324C4×Q8—44no—
325C4×Q8—44no—
326C4×Q8C2×D414yes—
327C4×Q8C2×D414yes—
328C4×D4—24no—
329C4×D4—42no—
330C4×D4—44no—
331C2×C4⋊C4—21no—
332C2×C4⋊C4—21no—
333C2×C4⋊C4—22no—
334C2×C4⋊C4—22no—
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—21no—
343C2×C4⋊C4—22no—
344C2×C4⋊C4—22no—
345C2×C4⋊C4—24no—
346C2×C4⋊C4—24no—
347C2×C4⋊C4—22no—
348C2×C4⋊C4—24no—
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×C4⋊C4C2412yes—
362C2×C22⋊C4—22no—
363C2×C22⋊C4—22no—
364C2×C22⋊C4—22no—
365C2×C22⋊C4—24no—
366C2×C22⋊C4—24no—
367C2×C22⋊C4—24no—
368C2×C22⋊C4—24no—
369C2×C42—21no—
370C2×C42—21no—
371C2×C42—22no—
372C2×C42—22no—
373C2×C42—22no—
374C2×C42—22no—
375C2×C42—21no—
376C2×C42—21no—
377C2×C42—22no—
378C2×C42C2×D411yescharacteristic
379C2×C42C2×D411yescharacteristic
380C2×C42C2×D412yes—
381C2×C42C2412yes—
382C2×C42C2412yes—
383C2×C42C2412yes—
384C2×C42C2411yescharacteristic
385#107—24no—
386#107—24no—
387#107—24no—
388#107C2×D414yes—
389#106—24no—
390#106—24no—
391#106—24no—
392#106—44no—
393C4⋊C8—24no—
394C4⋊C8—24no—
395C4⋊C8—22no—
396C4⋊C8—22no—
397#103—24no—
398#103—24no—
399#103—24no—
400#103—24no—
401#103—24no—
402#103C2×D414yes—
403#103C2×D414yes—
404#102—24no—
405#102—24no—
406#102—24no—
407#102—28no—
408#102—44no—
409#97—24no—
410C4×C8—24no—
411C4×C8—22no—
412C4×C8C2×D414yes—
413#95—22no—
414#95—22no—
415#95—21no—
416#95—22no—
417#95—24no—
418#95—22no—
419#95—21no—
420#95—24no—
421#95—24no—
422#95—24no—
423#95—24no—
424#95C2×D414yes—
425#95C2×D414yes—
426C24—22no—
427C2×Q8—22no—
428C2×Q8—22no—
429C2×Q8—22no—
430C2×Q8—22no—
431C2×Q8—22no—
432C2×Q8—22no—
433C2×Q8—21no—
434C2×Q8—21no—
435C2×Q8—22no—
436C2×Q8—22no—
437C2×Q8—22no—
438C2×Q8—44no—
439C2×Q8—42no—
440C2×Q8—42no—
441C2×Q8—42no—
442C2×Q8—42no—
443C2×Q8—42no—
444C2×Q8—42no—
445C2×Q8—42no—
446C2×Q8—82no—
447C2×Q8—84no—
448C2×Q8C22×D412yes—
449C2×Q8C22×D412yes—
450C2×Q82−1+414yes—
451C2×D4—22no—
452C2×D4—22no—
453C2×D4—24no—
454C2×D4—24no—
455C2×D4—42no—
456C2×D4—42no—
457C2×D4—42no—
458C2×D4—42no—
459C2×D4—82no—
460C22×C4—21no—
461C22×C4—22no—
462C22×C4—21no—
463C22×C4—21no—
464C22×C4—21no—
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—22no—
469C22×C4—44no—
470C22×C4—84no—
471C22×C4C22×D412yes—
472C22×C4C22×D412yes—
473C22×C4C22×D412yes—
474C22×C4C22×D412yes—
475C22×C4C22×D412yes—
476C22×C4C22×D412yes—
477C22×C4C22×D411yescharacteristic
478C22×C4C22×D411yescharacteristic
479C22×C4C2511yescharacteristic
480Q16—28no—
481SD16—48no—
482C2×C8—22no—
483C2×C8—22no—
484C2×C8—22no—
485C2×C8—22no—
486C2×C8—22no—
487C2×C8—24no—
488C2×C8—22no—
489C2×C8—22no—
490C2×C8—24no—
491C2×C8C22×D412yes—
492C2×C8C22×D412yes—
493C4⋊C4—28no—
494C4⋊C4—24no—
495C4⋊C4—24no—
496C4⋊C4—22no—
497C4⋊C4—22no—
498C4⋊C4—24no—
499C4⋊C4—24no—
500C4⋊C4—21no—
501C4⋊C4—21no—
502C4⋊C4—22no—
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—44no—
521C4⋊C4—48no—
522C4⋊C4—42no—
523C4⋊C4—42no—
524C4⋊C4—42no—
525C4⋊C4—82no—
526C4⋊C4—84no—
527C4⋊C4—84no—
528C4⋊C4C22×D412yes—
529C4⋊C4C22×D412yes—
530C4⋊C42−1+414yes—
531C22⋊C4—44no—
532C22⋊C4—44no—
533C22⋊C4—44no—
534C22⋊C4—48no—
535C22⋊C4—84no—
536C22⋊C4—84no—
537C22⋊C4—84no—
538C42—28no—
539C42—24no—
540C42—24no—
541C42—22no—
542C42—22no—
543C42—22no—
544C42—22no—
545C42—22no—
546C42—22no—
547C42—22no—
548C42—22no—
549C42—42no—
550C42—44no—
551C42—44no—
552C42—44no—
553C42—42no—
554C42—42no—
555C42—82no—
556C42C22×D412yes—
557C42H(5,𝔽2)12yes—
558C23—44no—
559C23—42no—
560C23—44no—
561C23—42no—
562C23—42no—
563C23C23×D411yescharacteristic
564Q8—22no—
565Q8—24no—
566Q8—22no—
567Q8—22no—
568Q8—44no—
569Q8—44no—
570Q8—42no—
571Q8—84no—
572Q8—84no—
573Q8—162no—
574Q8—164no—
575Q8#53614yes—
576D4—42no—
577D4—42no—
578D4—44no—
579D4—84no—
580D4—162no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—24no—
584C2×C4—22no—
585C2×C4—22no—
586C2×C4—24no—
587C2×C4—24no—
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—24no—
598C2×C4—22no—
599C2×C4—22no—
600C2×C4—22no—
601C2×C4—22no—
602C2×C4—22no—
603C2×C4—22no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—44no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—164no—
621C2×C4D4212yes—
622C2×C4D4212yes—
623C2×C4#57512yes—
624C2×C4#57712yes—
625C2×C4C23×D411yescharacteristic
626C2×C4C2×H(5,𝔽2)11yescharacteristic
627C2×C4C2×2−1+412yes—
628C2×C4C2×2−1+412yes—
629C8—22no—
630C8—22no—
631C8—42no—
632C8—42no—
633C8—42no—
634C22—82no—
635C22—82no—
636C22—84no—
637C22—82no—
638C22—84no—
639C22C2×D4211yescharacteristic
640C22#348612yes—
641C22C2×#57511yescharacteristic
642C22C2×#57711yescharacteristic
643C22#358412yes—
644C4—24no—
645C4—22no—
646C4—42no—
647C4—42no—
648C4—42no—
649C4—42no—
650C4—44no—
651C4—42no—
652C4—42no—
653C4—42no—
654C4—42no—
655C4—42no—
656C4—82no—
657C4—82no—
658C4—82no—
659C4—82no—
660C4#327114yes—
661C4C2×#53611yescharacteristic
662C4#349911yescharacteristic
663C4#355212yes—
664C2—162no—
665C2#3767511yescharacteristic
666C2C2×#327111yescharacteristic
667C2#6109011yescharacteristic
668C2#6184512yes—
669C2#6186512yes—
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
1. C2×#3383C2semidirect
4. #36474C2semidirect
6. #30950C2semidirect
7. #30830C2semidirect
9. #29867C2semidirect
10. #28683C2semidirect
12. #26399C2semidirect
13. #26173C2semidirect
15. #25837C2semidirect
16. #25836C2semidirect
18. #22240C2semidirect
19. #21116C2semidirect

Maximal subgroups

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

subgroupindex
C2×#33832
C2×#33752
#397462
#364742
#330252
#309502
#308302
#302782
#298672
#286832
#286402
#263992
#261732
#259352
#258372
#258362
#256522
#222402
#211162