Finite Group Databasebeta

← #206057

Subgroups — #206057

Order 512.

1212 conjugacy classes of subgroups.

The subgroup lattice has 677 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3188C211yesmaximal, characteristic
2#21948C211yesmaximal, characteristic, complemented
3#20523C211yesmaximal, characteristic, complemented
4#17542C211yesmaximal, characteristic, complemented
5C2×#1507C211yesmaximal, characteristic, complemented
6#17381C211yesmaximal, characteristic, complemented
7C2×#1471C211yesmaximal, characteristic, complemented
8#9815C214yesmaximal, complemented
9#9813C214yesmaximal, complemented
10C2×#525C2211yescharacteristic
11C22×#121C2211yescharacteristic
12#3188—22no—
13#3188—22no—
14#3188—22no—
15#3188—22no—
16C2×#484C411yescharacteristic, complemented
17C2×#481C411yescharacteristic, complemented
18C2×#482C412yescomplemented
19C2×#466C2212yes—
20C2×#465C2212yes—
21C2×#449C412yescomplemented
22C2×#450C411yescharacteristic, complemented
23C2×#447C411yescharacteristic, complemented
24C2×#407C2211yescharacteristic
25C2×#407C2211yescharacteristic
26C22×C22⋊C8C2211yescharacteristic
27C22×C22⋊C8C2211yescharacteristic
28C2×C4×C22⋊C4C2211yescharacteristic
29#1855—22no—
30#1855—22no—
31#1810C2214yescomplemented
32C8×C22⋊C4—22no—
33C8×C22⋊C4C2214yescomplemented
34C4×C22⋊C8—22no—
35#1508C2212yescomplemented
36#1507—22no—
37#1507C2214yes—
38#1507C2212yescomplemented
39#1504—22no—
40#1501—22no—
41C2×#327—22no—
42#1472C2212yescomplemented
43#1471—22no—
44#1471C2214yes—
45#1471C2212yescomplemented
46#1468—22no—
47#1459—22no—
48C2×#323—22no—
49C22×#41C2×C411yescharacteristic
50C23×D4—21no—
51C23×D4C2×C411yescharacteristic
52C24×C4C2311yescharacteristic
53C22×SD16C2×C412yescomplemented
54C22×D8C2×C412yescomplemented
55C22×M16—21no—
56C22×M16—21no—
57C23×C8—21no—
58C23×C8C2312yes—
59#525—21no—
60#525—21no—
61#525—21no—
62#525—21no—
63#525—21no—
64#525—21no—
65#525—21no—
66#525—21no—
67C2×#123—21no—
68C2×C22⋊Q8C2×C411yescharacteristic
69C2×C22⋊Q8C2311yescharacteristic
70C2×#121—21no—
71C2×#121—21no—
72C2×#121—21no—
73C2×#121—21no—
74C2×#121—21no—
75C2×#121—21no—
76C2×#121—21no—
77C2×#121—21no—
78C2×#121—21no—
79C2×#121—21no—
80C2×#121C2×C412yes—
81C2×#121C2×C411yescharacteristic
82C2×#121C2312yes—
83C2×#121C2311yescharacteristic
84C2×#120—21no—
85C2×C4×D4—21no—
86C2×C4×D4—21no—
87C2×#117C2×C411yescharacteristic
88C22×C4⋊C4C2×C411yescharacteristic
89C22×C22⋊C4—21no—
90C22×C22⋊C4—21no—
91C22×C42—21no—
92#484—22no—
93#484—22no—
94#482—24no—
95#482—24no—
96#481—22no—
97#481—22no—
98#466—24no—
99#466—24no—
100#465—24no—
101#465—24no—
102#450—22no—
103#450—22no—
104#449—24no—
105#449—24no—
106#447—22no—
107#447—22no—
108C2×#108—24no—
109C2×#107C2×C412yescomplemented
110C2×#106C2×C412yescomplemented
111C2×#103C2×C412yescomplemented
112C2×#102C2×C412yescomplemented
113C2×#102C2×C412yescomplemented
114C2×#102C2×C412yescomplemented
115#407—21no—
116#407—21no—
117#407—21no—
118#407—21no—
119#407—21no—
120#407—21no—
121#407C2312yes—
122#407C2312yes—
123C2×C22⋊C8—21no—
124C2×C22⋊C8—21no—
125C2×C22⋊C8—21no—
126C2×C22⋊C8—21no—
127C2×C22⋊C8—21no—
128C2×C22⋊C8—21no—
129C2×C22⋊C8—21no—
130C2×C22⋊C8C2×C412yes—
131C2×C22⋊C8C2×C411yescharacteristic
132C2×C22⋊C8C2×C411yescharacteristic
133C2×C22⋊C8D412yes—
134C2×C22⋊C8D412yes—
135C2×C22⋊C8D412yes—
136C2×C22⋊C8D412yes—
137C2×C22⋊C8D412yes—
138C2×C22⋊C8D412yes—
139C2×C22⋊C8C2312yes—
140C2×C4×C8—24no—
141C4×C22⋊C4—21no—
142C4×C22⋊C4—21no—
143C4×C22⋊C4D412yes—
144C4×C22⋊C4C2312yes—
145C2×#95D412yes—
146#341—24no—
147#336—24no—
148#328—24no—
149#327—28no—
150#327—24no—
151#324—24no—
152#323—28no—
153#323—24no—
154C25—21no—
155C25#4111yescharacteristic
156C2×#41—21no—
157C2×#41—21no—
158C2×#41—21no—
159C2×#41—21no—
160C2×#41—82no—
161C22×Q8C22×C411yescharacteristic
162C22×D4—21no—
163C22×D4—21no—
164C22×D4—22no—
165C22×D4—21no—
166C22×D4—22no—
167C22×D4—21no—
168C22×D4—21no—
169C22×D4—21no—
170C22×D4—21no—
171C22×D4—21no—
172C22×D4—41no—
173C22×D4—41no—
174C22×D4—41no—
175C22×D4—41no—
176C22×D4—82no—
177C22×D4—84no—
178C22×D4C22×C411yescharacteristic
179C22×D4C22×C412yes—
180C23×C4—22no—
181C23×C4—21no—
182C23×C4—21no—
183C23×C4—82no—
184C23×C4C22×C411yescharacteristic
185C23×C4C22×C411yescharacteristic
186C23×C4C2×D411yescharacteristic
187C23×C4C2×D411yescharacteristic
188C23×C4C2×D411yescharacteristic
189C23×C4C2×D411yescharacteristic
190C23×C4C2×D411yescharacteristic
191C23×C4C2×D411yescharacteristic
192C23×C4C2×D411yescharacteristic
193C23×C4C2×D411yescharacteristic
194C23×C4#4111yescharacteristic
195C23×C4#4111yescharacteristic
196C23×C4#4111yescharacteristic
197C23×C4C2411yescharacteristic
198C2×SD16—24no—
199C2×SD16—82no—
200C2×D8—24no—
201C2×D8—82no—
202C2×M16—22no—
203C2×M16—22no—
204C2×M16—41no—
205C2×M16—41no—
206C2×M16—82no—
207C2×M16—82no—
208C22×C8—22no—
209C22×C8—22no—
210C22×C8—22no—
211C22×C8—41no—
212C22×C8—41no—
213C22×C8—82no—
214C22×C8—82no—
215C22×C8C22×C412yes—
216C22×C8C22×C412yes—
217C22×C8C2×D414yes—
218C22×C8C2×D414yes—
219#123—41no—
220#123—41no—
221#123—41no—
222#123—41no—
223C22⋊Q8—21no—
224C22⋊Q8—21no—
225C22⋊Q8—21no—
226C22⋊Q8—21no—
227C22⋊Q8—21no—
228C22⋊Q8—21no—
229C22⋊Q8—21no—
230C22⋊Q8—21no—
231#121—22no—
232#121—22no—
233#121—21no—
234#121—21no—
235#121—22no—
236#121—22no—
237#121—21no—
238#121—21no—
239#121—22no—
240#121—22no—
241#121—22no—
242#121—21no—
243#121—21no—
244#121—21no—
245#121—22no—
246#121—21no—
247#121—41no—
248#121—41no—
249#121—41no—
250#121—41no—
251#121—41no—
252#121—41no—
253#121—41no—
254#121—41no—
255#120—41no—
256#120—41no—
257#120—41no—
258#120—41no—
259C4×D4—41no—
260C4×D4—41no—
261C4×D4—41no—
262C4×D4—41no—
263C4×D4—41no—
264C4×D4—41no—
265C4×D4—41no—
266C4×D4—41no—
267#117—21no—
268#117—21no—
269#117—21no—
270#117—21no—
271C2×C4⋊C4—21no—
272C2×C4⋊C4—21no—
273C2×C4⋊C4—21no—
274C2×C4⋊C4—21no—
275C2×C4⋊C4—21no—
276C2×C4⋊C4—21no—
277C2×C4⋊C4—21no—
278C2×C4⋊C4C22×C412yes—
279C2×C4⋊C4C22×C411yescharacteristic
280C2×C4⋊C4C22×C411yescharacteristic
281C2×C22⋊C4—28no—
282C2×C22⋊C4—22no—
283C2×C22⋊C4—22no—
284C2×C22⋊C4—21no—
285C2×C22⋊C4—21no—
286C2×C22⋊C4—21no—
287C2×C22⋊C4—22no—
288C2×C22⋊C4—21no—
289C2×C22⋊C4—22no—
290C2×C22⋊C4—21no—
291C2×C22⋊C4—41no—
292C2×C22⋊C4—41no—
293C2×C22⋊C4—41no—
294C2×C22⋊C4—41no—
295C2×C42—21no—
296C2×C42—22no—
297C2×C42—22no—
298C2×C42—82no—
299#108—84no—
300#107—24no—
301#106—24no—
302#103—24no—
303#102—24no—
304#102—24no—
305#102—24no—
306C22⋊C8—22no—
307C22⋊C8—22no—
308C22⋊C8—22no—
309C22⋊C8—22no—
310C22⋊C8—22no—
311C22⋊C8—22no—
312C22⋊C8—22no—
313C22⋊C8—41no—
314C22⋊C8—41no—
315C22⋊C8—41no—
316C22⋊C8—41no—
317C22⋊C8C22⋊C412yes—
318C22⋊C8C22⋊C412yes—
319C22⋊C8C22⋊C414yes—
320C22⋊C8C22⋊C414yes—
321C22⋊C8C22⋊C412yes—
322C22⋊C8C22⋊C412yes—
323C22⋊C8C2×D414yes—
324C4×C8—84no—
325#95—22no—
326#95C2×D414yes—
327C24—24no—
328C24—21no—
329C24—22no—
330C24—42no—
331C24—41no—
332C24—41no—
333C24—44no—
334C24—81no—
335C24—82no—
336C24—82no—
337C24C4×D411yescharacteristic
338C24C4×D411yescharacteristic
339C24#12111yescharacteristic
340C24#12111yescharacteristic
341C24#12411yescharacteristic
342C24#12411yescharacteristic
343C24C2×#4111yescharacteristic
344#41—82no—
345#41—82no—
346#41—82no—
347#41—82no—
348C2×Q8—21no—
349C2×Q8—21no—
350C2×Q8—42no—
351C2×D4—21no—
352C2×D4—22no—
353C2×D4—22no—
354C2×D4—21no—
355C2×D4—42no—
356C2×D4—42no—
357C2×D4—42no—
358C2×D4—41no—
359C2×D4—42no—
360C2×D4—41no—
361C2×D4—42no—
362C2×D4—41no—
363C2×D4—41no—
364C2×D4—41no—
365C2×D4—41no—
366C2×D4—42no—
367C2×D4—42no—
368C2×D4—44no—
369C2×D4—41no—
370C2×D4—41no—
371C2×D4—82no—
372C2×D4—82no—
373C2×D4—82no—
374C2×D4—82no—
375C2×D4—82no—
376C2×D4—84no—
377C2×D4—84no—
378C2×D4—84no—
379C2×D4—84no—
380C2×D4—82no—
381C2×D4—84no—
382C2×D4—84no—
383C2×D4—82no—
384C22×C4—24no—
385C22×C4—28no—
386C22×C4—21no—
387C22×C4—21no—
388C22×C4—21no—
389C22×C4—22no—
390C22×C4—42no—
391C22×C4—42no—
392C22×C4—42no—
393C22×C4—42no—
394C22×C4—42no—
395C22×C4—42no—
396C22×C4—42no—
397C22×C4—41no—
398C22×C4—41no—
399C22×C4—41no—
400C22×C4—41no—
401C22×C4—81no—
402C22×C4—81no—
403C22×C4—81no—
404C22×C4—81no—
405C22×C4—81no—
406C22×C4—81no—
407C22×C4—81no—
408C22×C4—82no—
409C22×C4—82no—
410C22×C4—82no—
411C22×C4—82no—
412C22×C4C2×C22⋊C411yescharacteristic
413C22×C4C2×C22⋊C411yescharacteristic
414C22×C4C2×C22⋊C411yescharacteristic
415C22×C4C2×C22⋊C411yescharacteristic
416C22×C4C4×D411yescharacteristic
417C22×C4C4×D411yescharacteristic
418C22×C4C4×D411yescharacteristic
419C22×C4C4×D411yescharacteristic
420C22×C4C4×D411yescharacteristic
421C22×C4C4×D411yescharacteristic
422C22×C4#12111yescharacteristic
423C22×C4#12111yescharacteristic
424C22×C4#12111yescharacteristic
425C22×C4#12111yescharacteristic
426C22×C4#12111yescharacteristic
427C22×C4#12111yescharacteristic
428C22×C4#12411yescharacteristic
429C22×C4#12411yescharacteristic
430C22×C4#12711yescharacteristic
431C22×C4#12711yescharacteristic
432C22×C4#12711yescharacteristic
433C22×C4#12711yescharacteristic
434C22×C4C23×C411yescharacteristic
435C22×C4C22×D411yescharacteristic
436C22×C4C22×D411yescharacteristic
437C22×C4C22×D411yescharacteristic
438C22×C4C22×D411yescharacteristic
439C22×C4C2×#4111yescharacteristic
440SD16—84no—
441D8—84no—
442M16—44no—
443M16—44no—
444M16—82no—
445M16—82no—
446C2×C8—24no—
447C2×C8—44no—
448C2×C8—44no—
449C2×C8—42no—
450C2×C8—42no—
451C2×C8—44no—
452C2×C8—42no—
453C2×C8—42no—
454C2×C8—82no—
455C2×C8—82no—
456C2×C8—82no—
457C2×C8—82no—
458C2×C8C2×C22⋊C414yes—
459C2×C8C2×C22⋊C414yes—
460C4⋊C4—21no—
461C4⋊C4—21no—
462C4⋊C4—22no—
463C4⋊C4—22no—
464C4⋊C4—21no—
465C4⋊C4—21no—
466C4⋊C4—41no—
467C4⋊C4—41no—
468C4⋊C4—41no—
469C4⋊C4—41no—
470C4⋊C4—41no—
471C4⋊C4—41no—
472C22⋊C4—28no—
473C22⋊C4—28no—
474C22⋊C4—41no—
475C22⋊C4—41no—
476C22⋊C4—41no—
477C22⋊C4—41no—
478C22⋊C4—41no—
479C22⋊C4—41no—
480C22⋊C4—42no—
481C22⋊C4—41no—
482C22⋊C4—42no—
483C22⋊C4—41no—
484C22⋊C4—42no—
485C22⋊C4—42no—
486C22⋊C4—41no—
487C22⋊C4—41no—
488C42—44no—
489C42—41no—
490C42—41no—
491C42—82no—
492C23—22no—
493C23—22no—
494C23—22no—
495C23—22no—
496C23—22no—
497C23—22no—
498C23—22no—
499C23—22no—
500C23—22no—
501C23—41no—
502C23—41no—
503C23—41no—
504C23—41no—
505C23—41no—
506C23—41no—
507C23—41no—
508C23—41no—
509C23—41no—
510C23—41no—
511C23—41no—
512C23—41no—
513C23—42no—
514C23—42no—
515C23—44no—
516C23—82no—
517C23—82no—
518C23—82no—
519C23—82no—
520C23—81no—
521C23—82no—
522C23—82no—
523C23—81no—
524C23—88no—
525C23—82no—
526C23—82no—
527C23—168no—
528C23#39011yescharacteristic
529C23#39011yescharacteristic
530C23#39011yescharacteristic
531C23#39011yescharacteristic
532C23#44314yes—
533C23C2×C4×D411yescharacteristic
534C23C2×#12111yescharacteristic
535C23C2×#12411yescharacteristic
536Q8—42no—
537Q8—81no—
538D4—42no—
539D4—42no—
540D4—42no—
541D4—82no—
542D4—81no—
543D4—82no—
544D4—82no—
545D4—82no—
546D4—84no—
547D4—84no—
548D4—84no—
549D4—82no—
550D4—82no—
551D4—82no—
552D4—82no—
553D4—82no—
554D4—84no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—28no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—44no—
585C2×C4—44no—
586C2×C4—42no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—42no—
590C2×C4—41no—
591C2×C4—82no—
592C2×C4—82no—
593C2×C4—88no—
594C2×C4—82no—
595C2×C4—82no—
596C2×C4—82no—
597C2×C4—81no—
598C2×C4—81no—
599C2×C4#39011yescharacteristic
600C2×C4#39011yescharacteristic
601C2×C4#39011yescharacteristic
602C2×C4#39011yescharacteristic
603C2×C4C22×C22⋊C411yescharacteristic
604C2×C4C2×C4×D411yescharacteristic
605C2×C4C2×#12111yescharacteristic
606C2×C4C2×#12711yescharacteristic
607C8—24no—
608C8—24no—
609C8—44no—
610C8—82no—
611C8—82no—
612C22—21no—
613C22—21no—
614C22—21no—
615C22—21no—
616C22—21no—
617C22—21no—
618C22—21no—
619C22—21no—
620C22—21no—
621C22—21no—
622C22—21no—
623C22—21no—
624C22—21no—
625C22—21no—
626C22—21no—
627C22—21no—
628C22—21no—
629C22—44no—
630C22—44no—
631C22—44no—
632C22—44no—
633C22—82no—
634C22—82no—
635C22—82no—
636C22—82no—
637C22—81no—
638C22—81no—
639C22—82no—
640C22—81no—
641C22—82no—
642C22—162no—
643C22—164no—
644C22—164no—
645C22—164no—
646C22#197412yes—
647C22#197412yes—
648C22C2×#39011yescharacteristic
649C22C2×#44312yes—
650C22#306812yes—
651C4—21no—
652C4—21no—
653C4—21no—
654C4—21no—
655C4—21no—
656C4—21no—
657C4—21no—
658C4—48no—
659C4—82no—
660C4—81no—
661C4—81no—
662C4—81no—
663C4#288312yes—
664C2—22no—
665C2—22no—
666C2—22no—
667C2—22no—
668C2—41no—
669C2—41no—
670C2—41no—
671C2—161no—
672C2—162no—
673C2#2121311yescharacteristic
674C2C2×#197411yescharacteristic
675C2#2124911yescharacteristic
subgroupG/H
center, socleC22C2×#390
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC2×#3188C2
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C24×C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #21948C2semidirect
3. #20523C2semidirect
4. #17542C2semidirect
5. C2×#1507C2semidirect
6. #17381C2semidirect
7. C2×#1471C2semidirect
8. #9815C2semidirect
9. #9813C2semidirect
16. C2×#484C4semidirect
17. C2×#481C4semidirect
18. C2×#482C4semidirect
21. C2×#449C4semidirect
22. C2×#450C4semidirect
23. C2×#447C4semidirect
31. #1810C22semidirect
33. C8×C22⋊C4C22semidirect
35. #1508C22semidirect
38. #1507C22semidirect
42. #1472C22semidirect
45. #1471C22semidirect
53. C22×SD16C2×C4semidirect
54. C22×D8C2×C4semidirect
109. C2×#107C2×C4semidirect
110. C2×#106C2×C4semidirect
111. C2×#103C2×C4semidirect
112. C2×#102C2×C4semidirect
113. C2×#102C2×C4semidirect
114. C2×#102C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#31882
#219482
#205232
#175422
C2×#15072
#173812
C2×#14712
#98152
#98132