Finite Group Databasebeta

← #378261

Subgroups — #378261

Order 512.

1152 conjugacy classes of subgroups.

The subgroup lattice has 666 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23638C211yesmaximal, characteristic
2#21053C211yesmaximal, characteristic
3#21050C211yesmaximal, characteristic
4C2×#1922C211yesmaximal, characteristic, complemented
5C2×C4×C22⋊C8C211yesmaximal, characteristic
6#11317C211yesmaximal, characteristic, complemented
7#11291C211yesmaximal, characteristic, complemented
8#11097C211yesmaximal, characteristic
9#10910C211yesmaximal, characteristic, complemented
10#10882C211yesmaximal, characteristic, complemented
11#10882C211yesmaximal, characteristic, complemented
12#10859C211yesmaximal, characteristic, complemented
13#10719C211yesmaximal, characteristic
14#10719C211yesmaximal, characteristic
15#10706C211yesmaximal, characteristic, complemented
16C23×C42C2211yescharacteristic
17C22×#108C2211yescharacteristic
18C2×#427C2211yescharacteristic
19C2×#424C2211yescharacteristic
20C2×#424C2211yescharacteristic
21C2×#407C2211yescharacteristic
22C2×#407C2211yescharacteristic
23C22×C22⋊C8C2211yescharacteristic
24C22×C22⋊C8C2211yescharacteristic
25C2×#405C2211yescharacteristic
26C2×#405C2211yescharacteristic
27C22×C4×C8C2211yescharacteristic
28C22×C4×C8C2211yescharacteristic
29#2280C2211yescharacteristic
30#2280C2211yescharacteristic
31#2077C412yescomplemented
32#2077C412yescomplemented
33#2076C412yescomplemented
34#2075C412yescomplemented
35#1924—22no—
36#1924—22no—
37#1923—22no—
38#1923—22no—
39#1922—24no—
40C4×C22⋊C8—24no—
41C2×#341C2211yescharacteristic
42C2×#341C2211yescharacteristic
43C2×#341C2211yescharacteristic
44#1735C2211yescharacteristic
45#1735C2211yescharacteristic
46#1735C2211yescharacteristic
47C2×#339C2211yescharacteristic
48C2×#339C2211yescharacteristic
49C2×#339C2211yescharacteristic
50C2×#339C2211yescharacteristic
51C2×#339C2211yescharacteristic
52C2×#339C2211yescharacteristic
53C2×#339C2211yescharacteristic
54C2×#339C2211yescharacteristic
55#1729C2211yescharacteristic
56#1729C2211yescharacteristic
57#1729C2211yescharacteristic
58#1729C2211yescharacteristic
59C2×#336C2211yescharacteristic
60C2×#336C2211yescharacteristic
61C24×C4—21no—
62C24×C4C2311yescharacteristic
63C22×M16C2311yescharacteristic
64C22×M16C2311yescharacteristic
65C22×M16C2311yescharacteristic
66C22×M16C2311yescharacteristic
67C23×C8C2311yescharacteristic
68C23×C8C2311yescharacteristic
69C23×C8C2311yescharacteristic
70C23×C8C2311yescharacteristic
71C2×#117C2311yescharacteristic
72C2×#117C2311yescharacteristic
73C2×#117C2311yescharacteristic
74C2×#117C2311yescharacteristic
75C22×C42—21no—
76C22×C42—22no—
77C22×C42—22no—
78C22×C42—22no—
79C22×C42—22no—
80C22×C42—41no—
81C22×C42—41no—
82C22×C42C2311yescharacteristic
83C22×C42C2311yescharacteristic
84C2×#108—22no—
85C2×#108—22no—
86C2×#108—41no—
87#427—22no—
88#427—22no—
89C2×#107C2×C411yescharacteristic, complemented
90C2×#107C2×C411yescharacteristic, complemented
91C2×#106C2×C411yescharacteristic, complemented
92C2×#106C2×C411yescharacteristic, complemented
93#424—22no—
94#424—22no—
95#424—22no—
96#424—41no—
97C2×C4⋊C8—21no—
98C2×C4⋊C8—22no—
99C2×C4⋊C8—21no—
100#407—22no—
101#407—22no—
102#407—41no—
103#407—41no—
104C2×C22⋊C8—22no—
105C2×C22⋊C8—22no—
106C2×C22⋊C8—22no—
107C2×C22⋊C8—22no—
108C2×C22⋊C8—22no—
109C2×C22⋊C8—22no—
110C2×C22⋊C8C2×C412yes—
111C2×C22⋊C8C2×C412yes—
112C2×C22⋊C8D412yes—
113C2×C22⋊C8D412yes—
114#405—22no—
115#405—22no—
116#405—22no—
117#405—41no—
118C2×#97—21no—
119C2×#97—21no—
120C2×C4×C8—22no—
121C2×C4×C8—22no—
122C2×C4×C8—21no—
123C2×C4×C8—21no—
124C2×C4×C8—22no—
125C2×C4×C8—22no—
126C2×C4×C8—41no—
127C2×C4×C8—41no—
128#381—21no—
129#381—21no—
130#380C2×C412yes—
131#380C2×C412yes—
132#380C2×C412yes—
133#380C2×C412yes—
134#379—21no—
135#379—21no—
136C4×C22⋊C4—22no—
137C4×C22⋊C4—22no—
138#341—22no—
139#341—22no—
140#341—41no—
141#340C2×C412yescomplemented
142#340C2×C412yescomplemented
143#340C2×C412yescomplemented
144#340C2×C412yescomplemented
145#340C2×C412yescomplemented
146#340C2×C412yescomplemented
147#339—22no—
148#339—22no—
149#339—22no—
150#339—22no—
151#339—22no—
152#339—22no—
153#339—41no—
154#339—41no—
155#336—21no—
156#336—22no—
157#336—22no—
158#336—22no—
159#336—21no—
160#336—22no—
161#336D414yes—
162C25#4111yescharacteristic
163C23×C4—21no—
164C23×C4—28no—
165C23×C4—24no—
166C23×C4—21no—
167C23×C4—21no—
168C23×C4—21no—
169C23×C4—21no—
170C23×C4—21no—
171C23×C4—21no—
172C23×C4—21no—
173C23×C4—21no—
174C23×C4—21no—
175C23×C4—21no—
176C23×C4—21no—
177C23×C4—21no—
178C23×C4—21no—
179C23×C4—42no—
180C23×C4—42no—
181C23×C4—44no—
182C23×C4C22×C412yes—
183C23×C4C2×D412yes—
184C23×C4#4111yescharacteristic
185C23×C4C2411yescharacteristic
186C2×M16—21no—
187C2×M16—21no—
188C2×M16—21no—
189C2×M16—21no—
190C2×M16—21no—
191C2×M16—21no—
192C2×M16—21no—
193C2×M16—21no—
194C2×M16—21no—
195C2×M16—21no—
196C2×M16—21no—
197C2×M16—21no—
198C2×M16—21no—
199C2×M16—21no—
200C2×M16—21no—
201C2×M16—21no—
202C2×M16—21no—
203C2×M16—21no—
204C2×M16—21no—
205C2×M16—21no—
206C2×M16—21no—
207C2×M16—21no—
208C2×M16#4112yes—
209C2×M16#4112yes—
210C22×C8—21no—
211C22×C8—21no—
212C22×C8—21no—
213C22×C8—21no—
214C22×C8—21no—
215C22×C8—21no—
216C22×C8—21no—
217C22×C8—21no—
218C22×C8—21no—
219C22×C8—21no—
220C22×C8—21no—
221C22×C8—21no—
222C22×C8—21no—
223C22×C8—21no—
224C22×C8—21no—
225C22×C8—21no—
226C22×C8—21no—
227C22×C8—21no—
228C22×C8—21no—
229C22×C8—21no—
230C22×C8—21no—
231C22×C8—21no—
232C22×C8—21no—
233C22×C8—21no—
234C22×C8C22×C412yes—
235C22×C8C22×C411yescharacteristic
236C22×C8C22×C411yescharacteristic
237C22×C8C2×D412yes—
238C22×C8C2×D412yes—
239C22×C8C2×D412yes—
240C22×C8C2×D411yescharacteristic
241C22×C8C2×D411yescharacteristic
242C22×C8#4112yes—
243C22×C8#4112yes—
244#117—21no—
245#117—21no—
246#117—21no—
247#117—21no—
248#117—21no—
249#117—21no—
250#117—21no—
251#117—21no—
252#117—21no—
253#117—21no—
254#117—21no—
255#117—21no—
256#117—21no—
257#117—21no—
258#117—21no—
259#117—21no—
260C2×C4⋊C4C22×C412yes—
261C2×C4⋊C4C22×C412yes—
262C2×C4⋊C4C22×C411yescharacteristic
263C2×C4⋊C4C22×C411yescharacteristic
264C2×C4⋊C4C22×C411yescharacteristic
265C2×C4⋊C4C22×C411yescharacteristic
266C2×C22⋊C4—22no—
267C2×C22⋊C4—22no—
268C2×C22⋊C4—21no—
269C2×C22⋊C4—21no—
270C2×C22⋊C4—21no—
271C2×C22⋊C4—21no—
272C2×C22⋊C4—22no—
273C2×C22⋊C4—22no—
274C2×C42—21no—
275C2×C42—21no—
276C2×C42—21no—
277C2×C42—21no—
278C2×C42—21no—
279C2×C42—21no—
280C2×C42—21no—
281C2×C42—24no—
282C2×C42—21no—
283C2×C42—21no—
284C2×C42—21no—
285C2×C42—21no—
286C2×C42—21no—
287C2×C42—44no—
288C2×C42—44no—
289C2×C42—44no—
290C2×C42—42no—
291C2×C42—42no—
292C2×C42—44no—
293C2×C42C2×D412yes—
294C2×C42#4112yes—
295C2×C42#4112yes—
296C2×C42#4112yes—
297#108—44no—
298#107—22no—
299#107—22no—
300#106—22no—
301#106—22no—
302C4⋊C8—44no—
303C4⋊C8—42no—
304C4⋊C8—42no—
305C22⋊C8—24no—
306C22⋊C8—24no—
307C22⋊C8—24no—
308C22⋊C8—24no—
309C22⋊C8—44no—
310C22⋊C8—44no—
311#97—42no—
312#97—42no—
313C4×C8—44no—
314C4×C8—44no—
315C4×C8—42no—
316C4×C8—42no—
317#95—22no—
318#95—22no—
319#95—22no—
320#95—22no—
321C24—21no—
322C24—21no—
323C24—24no—
324C24—21no—
325C24—21no—
326C24—21no—
327C24—21no—
328C24—42no—
329C24C4×D412yes—
330C24C2×#4111yescharacteristic
331C22×C4—24no—
332C22×C4—24no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—216no—
337C22×C4—24no—
338C22×C4—21no—
339C22×C4—22no—
340C22×C4—21no—
341C22×C4—21no—
342C22×C4—21no—
343C22×C4—21no—
344C22×C4—21no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—21no—
360C22×C4—21no—
361C22×C4—21no—
362C22×C4—21no—
363C22×C4—48no—
364C22×C4—416no—
365C22×C4—416no—
366C22×C4—48no—
367C22×C4—42no—
368C22×C4—41no—
369C22×C4—41no—
370C22×C4—41no—
371C22×C4—41no—
372C22×C4—41no—
373C22×C4—41no—
374C22×C4—41no—
375C22×C4—41no—
376C22×C4—41no—
377C22×C4—41no—
378C22×C4—41no—
379C22×C4—41no—
380C22×C4—41no—
381C22×C4—41no—
382C22×C4—41no—
383C22×C4—41no—
384C22×C4—41no—
385C22×C4—41no—
386C22×C4C4×D412yes—
387C22×C4C23×C411yescharacteristic
388C22×C4C22×D411yescharacteristic
389C22×C4C22×D411yescharacteristic
390C22×C4C2×#4111yescharacteristic
391C22×C4C2×#4111yescharacteristic
392C22×C4C2×#4111yescharacteristic
393C22×C4C2×#4111yescharacteristic
394C22×C4C2×#4111yescharacteristic
395C22×C4C2×#4111yescharacteristic
396C22×C4C2×#4111yescharacteristic
397C22×C4H(5,𝔽2)11yescharacteristic
398C22×C4H(5,𝔽2)11yescharacteristic
399C22×C42−1+411yescharacteristic
400C22×C42−1+411yescharacteristic
401M16—22no—
402M16—22no—
403M16—22no—
404M16—22no—
405M16—41no—
406M16—41no—
407M16—41no—
408M16—41no—
409M16—41no—
410M16—41no—
411M16—41no—
412M16—41no—
413M16—41no—
414M16—41no—
415M16—41no—
416M16—41no—
417C2×C8—21no—
418C2×C8—21no—
419C2×C8—21no—
420C2×C8—21no—
421C2×C8—22no—
422C2×C8—22no—
423C2×C8—22no—
424C2×C8—22no—
425C2×C8—21no—
426C2×C8—21no—
427C2×C8—22no—
428C2×C8—22no—
429C2×C8—21no—
430C2×C8—21no—
431C2×C8—22no—
432C2×C8—22no—
433C2×C8—22no—
434C2×C8—22no—
435C2×C8—22no—
436C2×C8—22no—
437C2×C8—22no—
438C2×C8—22no—
439C2×C8—22no—
440C2×C8—22no—
441C2×C8—41no—
442C2×C8—41no—
443C2×C8—41no—
444C2×C8—41no—
445C2×C8—41no—
446C2×C8—41no—
447C2×C8—41no—
448C2×C8—41no—
449C2×C8—41no—
450C2×C8—41no—
451C2×C8—41no—
452C2×C8—41no—
453C2×C8—41no—
454C2×C8—41no—
455C2×C8—41no—
456C2×C8—41no—
457C2×C8—41no—
458C2×C8—41no—
459C2×C8C4×D412yes—
460C2×C8C4×D412yes—
461C2×C8#12112yes—
462C2×C8#12112yes—
463C4⋊C4—21no—
464C4⋊C4—21no—
465C4⋊C4—21no—
466C4⋊C4—21no—
467C4⋊C4—22no—
468C4⋊C4—22no—
469C4⋊C4—21no—
470C4⋊C4—21no—
471C4⋊C4—21no—
472C4⋊C4—21no—
473C4⋊C4—22no—
474C4⋊C4—22no—
475C22⋊C4—41no—
476C22⋊C4—41no—
477C22⋊C4—41no—
478C22⋊C4—41no—
479C22⋊C4—41no—
480C22⋊C4—41no—
481C22⋊C4—41no—
482C22⋊C4—41no—
483C22⋊C4—44no—
484C22⋊C4—44no—
485C22⋊C4—82no—
486C22⋊C4—82no—
487C42—22no—
488C42—22no—
489C42—22no—
490C42—22no—
491C42—22no—
492C42—48no—
493C42—41no—
494C42—41no—
495C42—41no—
496C42—41no—
497C42—41no—
498C42—41no—
499C42—41no—
500C42—41no—
501C42—41no—
502C42—41no—
503C42#12614yes—
504C23—22no—
505C23—22no—
506C23—24no—
507C23—24no—
508C23—22no—
509C23—22no—
510C23—22no—
511C23—22no—
512C23—22no—
513C23—22no—
514C23—44no—
515C23—44no—
516C23—44no—
517C23—44no—
518C23—42no—
519C23—42no—
520C23—41no—
521C23—41no—
522C23—41no—
523C23—41no—
524C23#44314yes—
525C23C2×C4×D411yescharacteristic
526C23#52511yescharacteristic
527C23#52911yescharacteristic
528C23#55111yescharacteristic
529C23#55111yescharacteristic
530C23#55211yescharacteristic
531C23#55211yescharacteristic
532C2×C4—24no—
533C2×C4—24no—
534C2×C4—28no—
535C2×C4—28no—
536C2×C4—24no—
537C2×C4—24no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—432no—
542C2×C4—42no—
543C2×C4—44no—
544C2×C4—44no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—42no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4C2×C4×D411yescharacteristic
573C2×C4C4×#4111yescharacteristic
574C2×C4C4×#4111yescharacteristic
575C2×C4C4×#4111yescharacteristic
576C2×C4#52011yescharacteristic
577C2×C4#52011yescharacteristic
578C2×C4C2×#12111yescharacteristic
579C2×C4C2×#12611yescharacteristic
580C2×C4#52911yescharacteristic
581C2×C4#52911yescharacteristic
582C2×C4#52911yescharacteristic
583C2×C4#53211yescharacteristic
584C2×C4#53811yescharacteristic
585C2×C4#53911yescharacteristic
586C2×C4#54111yescharacteristic
587C2×C4#54211yescharacteristic
588C2×C4#54611yescharacteristic
589C2×C4#54611yescharacteristic
590C2×C4#54611yescharacteristic
591C2×C4#54611yescharacteristic
592C2×C4#54711yescharacteristic
593C2×C4#54711yescharacteristic
594C2×C4#54711yescharacteristic
595C2×C4#54811yescharacteristic
596C2×C4#55111yescharacteristic
597C2×C4#55211yescharacteristic
598C2×C4#55611yescharacteristic
599C2×C4#55911yescharacteristic
600C8—22no—
601C8—22no—
602C8—22no—
603C8—22no—
604C8—42no—
605C8—42no—
606C8—42no—
607C8—42no—
608C8—81no—
609C8—81no—
610C22—21no—
611C22—21no—
612C22—21no—
613C22—21no—
614C22—22no—
615C22—22no—
616C22—22no—
617C22—22no—
618C22—24no—
619C22—24no—
620C22—24no—
621C22—44no—
622C22—44no—
623C22—44no—
624C22—44no—
625C22—42no—
626C22—42no—
627C22—42no—
628C22—42no—
629C22C4×#12111yescharacteristic
630C22#235911yescharacteristic
631C22#235911yescharacteristic
632C22#247511yescharacteristic
633C22#248011yescharacteristic
634C22#252711yescharacteristic
635C22#257511yescharacteristic
636C22C2×#44312yes—
637C22#295612yes—
638C4—28no—
639C4—44no—
640C4—42no—
641C4—81no—
642C4—81no—
643C4—81no—
644C4—81no—
645C4—82no—
646C4—82no—
647C4C4×#12611yescharacteristic
648C4#233411yescharacteristic
649C4#235911yescharacteristic
650C4#236711yescharacteristic
651C4#242511yescharacteristic
652C4#242711yescharacteristic
653C4#249611yescharacteristic
654C4#255711yescharacteristic
655C2—21no—
656C2—21no—
657C2—21no—
658C2—21no—
659C2—44no—
660C2#1420911yescharacteristic
661C2#2154411yescharacteristic
662C2#2155111yescharacteristic
663C2#2294812yes—
664C2#2294812yes—
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
socleC23C2×C4×D4
prime radicalC25#41
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C22×C22⋊C8 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. C2×#1922C2semidirect
6. #11317C2semidirect
7. #11291C2semidirect
9. #10910C2semidirect
10. #10882C2semidirect
11. #10882C2semidirect
12. #10859C2semidirect
15. #10706C2semidirect
31. #2077C4semidirect
32. #2077C4semidirect
33. #2076C4semidirect
34. #2075C4semidirect
89. C2×#107C2×C4semidirect
90. C2×#107C2×C4semidirect
91. C2×#106C2×C4semidirect
92. C2×#106C2×C4semidirect
141. #340C2×C4semidirect
142. #340C2×C4semidirect
143. #340C2×C4semidirect
144. #340C2×C4semidirect
145. #340C2×C4semidirect
146. #340C2×C4semidirect

Maximal subgroups

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

subgroupindex
#236382
#210532
#210502
C2×#19222
C2×C4×C22⋊C82
#113172
#112912
#110972
#109102
#108822
#108592
#107192
#107062