Finite Group Databasebeta

← #202679

Subgroups — #202679

Order 512.

749 conjugacy classes of subgroups.

The subgroup lattice has 602 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21951C211yesmaximal, characteristic, complemented
2#21850C211yesmaximal, characteristic
3#21388C211yesmaximal, characteristic
4C2×#1482C211yesmaximal, characteristic
5C2×#1454C211yesmaximal, characteristic, complemented
6#9560C211yesmaximal, characteristic, complemented
7#9559C211yesmaximal, characteristic, complemented
8#9558C211yesmaximal, characteristic, complemented
9#9557C211yesmaximal, characteristic, complemented
10#9556C211yesmaximal, characteristic, complemented
11#9555C211yesmaximal, characteristic, complemented
12#9554C211yesmaximal, characteristic, complemented
13#9553C211yesmaximal, characteristic, complemented
14#7901C211yesmaximal, characteristic, complemented
15#7880C211yesmaximal, characteristic, complemented
16C2×#529C2211yescharacteristic
17C2×#431C2211yescharacteristic
18C2×#431C2211yescharacteristic
19C2×#431C2211yescharacteristic
20C2×#431C2211yescharacteristic
21C2×#423C2211yescharacteristic
22C2×C4×C4⋊C4C2211yescharacteristic
23#1989C411yescharacteristic
24#1989C411yescharacteristic
25#1984C2211yescharacteristic
26#1984C2211yescharacteristic
27#1983C411yescharacteristic
28#1983C411yescharacteristic
29#1954C2211yescharacteristic
30#1954C2211yescharacteristic
31#1939—21no—
32#1939—21no—
33#1939C2211yescharacteristic
34#1939C2211yescharacteristic
35#1939C2211yescharacteristic
36#1939C2211yescharacteristic
37#1874C411yescharacteristic
38#1874C411yescharacteristic
39#1872C411yescharacteristic
40#1872C411yescharacteristic
41#1870—21no—
42#1870—21no—
43#1798—21no—
44#1798—21no—
45#1797—21no—
46#1797—21no—
47C4×#103C2211yescharacteristic
48C4×#103C2211yescharacteristic
49C4×#102C2211yescharacteristic
50C4×#102C2211yescharacteristic
51#1482—24no—
52C2×Q8⋊C8C2211yescharacteristic
53C2×Q8⋊C8C2211yescharacteristic
54C2×D4⋊C8C2211yescharacteristic
55C2×D4⋊C8C2211yescharacteristic
56#1454—24no—
57C2×#322C2211yescharacteristic
58C2×#322C2211yescharacteristic
59C2×#322C2211yescharacteristic
60C2×C82C2211yescharacteristic
61#1279C2211yescharacteristic
62#1279C2211yescharacteristic
63#1279C2211yescharacteristic
64#1279C2211yescharacteristic
65#1277C2211yescharacteristic
66#1277C2211yescharacteristic
67#1277C2211yescharacteristic
68#1277C2211yescharacteristic
69C22×M16—21no—
70#529—22no—
71#529—22no—
72#529—22no—
73#529—22no—
74C2×#126—21no—
75C2×#125C2×C411yescharacteristic
76C2×#124C2×C411yescharacteristic
77C2×#123—21no—
78C2×C22⋊Q8C2×C411yescharacteristic
79C2×#121C2×C411yescharacteristic
80C2×C4×Q8C2311yescharacteristic
81C2×C4×D4—21no—
82C2×C4×D4C2311yescharacteristic
83C2×#117—21no—
84C22×C4⋊C4C2×C411yescharacteristic
85C22×C4⋊C4C2×C411yescharacteristic
86C22×C42—21no—
87C22×C42C2311yescharacteristic
88#431—22no—
89#431—22no—
90#431—22no—
91#431—22no—
92#431—22no—
93#431—22no—
94#431—22no—
95#431D414yes—
96#423—22no—
97#423—22no—
98C2×C4⋊C8—21no—
99C2×C4⋊C8—21no—
100C2×C4⋊C8—21no—
101C2×C4⋊C8—21no—
102C2×C4⋊C8—21no—
103C2×C4⋊C8—21no—
104C2×C4⋊C8C2×C411yescharacteristic
105C2×C4⋊C8C2×C411yescharacteristic
106C2×C4⋊C8D411yescharacteristic
107C2×C4⋊C8D411yescharacteristic
108C2×C4⋊C8C2311yescharacteristic
109C2×C4⋊C8C2311yescharacteristic
110C2×C4⋊C8C2311yescharacteristic
111C2×C4⋊C8C2311yescharacteristic
112C2×#103C2×C411yescharacteristic
113C2×#103C2×C411yescharacteristic
114C2×#103C2×C411yescharacteristic
115C2×#103C2×C411yescharacteristic
116C2×#102C2×C411yescharacteristic
117C2×#102C2×C411yescharacteristic
118C2×#102C2×C411yescharacteristic
119C2×#102C2×C411yescharacteristic
120C2×C22⋊C8—21no—
121C2×C22⋊C8—21no—
122C2×C22⋊C8—21no—
123C2×C22⋊C8—21no—
124C2×C22⋊C8—21no—
125C2×C22⋊C8—21no—
126C2×C22⋊C8C2×C411yescharacteristic
127C2×C22⋊C8C2×C411yescharacteristic
128C2×C22⋊C8D411yescharacteristic
129C2×C22⋊C8D411yescharacteristic
130C2×C4×C8—21no—
131C2×C4×C8C2311yescharacteristic
132C2×C4×C8C2311yescharacteristic
133C2×C4×C8C2311yescharacteristic
134C2×C4×C8C2311yescharacteristic
135C2×C4×C8C2311yescharacteristic
136C2×C4×C8C2311yescharacteristic
137C4×C4⋊C4—21no—
138C4×C4⋊C4—21no—
139C4×C4⋊C4C2×C411yescharacteristic
140C4×C4⋊C4C2×C411yescharacteristic
141C4×C4⋊C4C2311yescharacteristic
142C4×C4⋊C4C2311yescharacteristic
143C2×#95—21no—
144#340C2×C411yescharacteristic
145#340C2×C411yescharacteristic
146#340C2×C411yescharacteristic
147#340C2×C411yescharacteristic
148#340C2×C411yescharacteristic
149#340C2×C411yescharacteristic
150#340C2×C411yescharacteristic
151#340C2×C411yescharacteristic
152#336—21no—
153#336—21no—
154#336—21no—
155#336—21no—
156#336—21no—
157#336—21no—
158#336—21no—
159#336—21no—
160Q8⋊C8—22no—
161Q8⋊C8—22no—
162D4⋊C8—22no—
163D4⋊C8—22no—
164#322—22no—
165#322—22no—
166#322—22no—
167C82—22no—
168C22×Q8C22×C411yescharacteristic
169C22×D4—21no—
170C22×D4C22×C411yescharacteristic
171C23×C4—21no—
172C23×C4—21no—
173C23×C4—21no—
174C23×C4C22×C411yescharacteristic
175C23×C4C2×D411yescharacteristic
176C23×C4#4111yescharacteristic
177C2×M16—42no—
178C2×M16—81no—
179C2×M16—81no—
180C22×C8—21no—
181C22×C8—21no—
182C22×C8—21no—
183C22×C8—21no—
184C22×C8—21no—
185C22×C8—21no—
186C22×C8C22×C411yescharacteristic
187C22×C8C22×C411yescharacteristic
188C22×C8C22×C411yescharacteristic
189C22×C8C22×C411yescharacteristic
190C22×C8C2×D411yescharacteristic
191C22×C8C2×D411yescharacteristic
192C22×C8C2×D411yescharacteristic
193C22×C8C2×D411yescharacteristic
194#126—42no—
195#126—42no—
196#125—22no—
197#125—22no—
198#124—22no—
199#124—22no—
200#123—42no—
201#123—42no—
202C22⋊Q8—22no—
203C22⋊Q8—22no—
204#121—22no—
205#121—22no—
206C4×Q8—21no—
207C4×Q8—21no—
208C4×Q8—21no—
209C4×Q8—21no—
210C4×D4—21no—
211C4×D4—21no—
212C4×D4—21no—
213C4×D4—21no—
214C4×D4—42no—
215C4×D4—42no—
216#117—42no—
217#117—42no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4—21no—
224C2×C4⋊C4—21no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4—41no—
227C2×C4⋊C4—41no—
228C2×C4⋊C4C22×C411yescharacteristic
229C2×C4⋊C4C22×C411yescharacteristic
230C2×C4⋊C4C22×C411yescharacteristic
231C2×C4⋊C4C22×C411yescharacteristic
232C2×C4⋊C4C22×C411yescharacteristic
233C2×C4⋊C4C22×C411yescharacteristic
234C2×C22⋊C4—21no—
235C2×C22⋊C4—21no—
236C2×C22⋊C4—21no—
237C2×C22⋊C4—21no—
238C2×C22⋊C4—21no—
239C2×C42—21no—
240C2×C42—21no—
241C2×C42—21no—
242C2×C42—21no—
243C2×C42—21no—
244C2×C42—81no—
245C2×C42—81no—
246C2×C42C22×C411yescharacteristic
247C2×C42C2×D412yes—
248C2×C42C2×D411yescharacteristic
249C2×C42#4112yes—
250C2×C42#4112yes—
251C2×C42#4112yes—
252C2×C42#4111yescharacteristic
253C2×C42C2411yescharacteristic
254C4⋊C8—22no—
255C4⋊C8—22no—
256C4⋊C8—22no—
257C4⋊C8—22no—
258C4⋊C8—24no—
259C4⋊C8—21no—
260C4⋊C8—21no—
261C4⋊C8—21no—
262C4⋊C8—21no—
263C4⋊C8—21no—
264C4⋊C8—21no—
265C4⋊C8—21no—
266C4⋊C8—21no—
267C4⋊C8—42no—
268C4⋊C8—81no—
269C4⋊C8—81no—
270C4⋊C8—81no—
271C4⋊C8—81no—
272#103—22no—
273#103—22no—
274#103—41no—
275#103—41no—
276#102—22no—
277#102—22no—
278#102—41no—
279#102—41no—
280C22⋊C8—22no—
281C22⋊C8—22no—
282C22⋊C8—22no—
283C22⋊C8—22no—
284C22⋊C8—24no—
285C22⋊C8—42no—
286C22⋊C8—81no—
287C22⋊C8—81no—
288C22⋊C8—81no—
289C22⋊C8—81no—
290C4×C8—21no—
291C4×C8—21no—
292C4×C8—21no—
293C4×C8—21no—
294C4×C8—21no—
295C4×C8—21no—
296C4×C8—21no—
297C4×C8—21no—
298C4×C8—21no—
299C4×C8—21no—
300C4×C8—42no—
301C4×C8C2×D412yes—
302C4×C8C2×D412yes—
303#95—21no—
304#95—21no—
305#95—21no—
306#95—21no—
307C24—21no—
308C24C4×D411yescharacteristic
309C2×Q8—21no—
310C2×Q8—21no—
311C2×Q8—21no—
312C2×Q8—21no—
313C2×Q8—21no—
314C2×Q8—21no—
315C2×D4—21no—
316C2×D4—21no—
317C2×D4—21no—
318C2×D4—21no—
319C2×D4—21no—
320C2×D4—21no—
321C2×D4—42no—
322C2×D4—81no—
323C2×D4—81no—
324C22×C4—21no—
325C22×C4—21no—
326C22×C4—21no—
327C22×C4—21no—
328C22×C4—21no—
329C22×C4—21no—
330C22×C4—21no—
331C22×C4—21no—
332C22×C4—21no—
333C22×C4—21no—
334C22×C4—24no—
335C22×C4—21no—
336C22×C4—21no—
337C22×C4—21no—
338C22×C4—21no—
339C22×C4—21no—
340C22×C4—41no—
341C22×C4—41no—
342C22×C4—42no—
343C22×C4—41no—
344C22×C4—41no—
345C22×C4—41no—
346C22×C4—41no—
347C22×C4—41no—
348C22×C4—41no—
349C22×C4—81no—
350C22×C4—81no—
351C22×C4—81no—
352C22×C4—81no—
353C22×C4#11712yes—
354C22×C4C4×D412yes—
355C22×C4C4×D411yescharacteristic
356C22×C4C4×D411yescharacteristic
357C22×C4C4×D411yescharacteristic
358C22×C4#12112yes—
359C22×C4#12312yes—
360C22×C4#12412yes—
361C22×C4#12612yes—
362C22×C4C23×C411yescharacteristic
363C22×C4C22×D411yescharacteristic
364C22×C4C2×#4111yescharacteristic
365M16—84no—
366C2×C8—21no—
367C2×C8—21no—
368C2×C8—21no—
369C2×C8—21no—
370C2×C8—21no—
371C2×C8—21no—
372C2×C8—21no—
373C2×C8—21no—
374C2×C8—22no—
375C2×C8—22no—
376C2×C8—21no—
377C2×C8—21no—
378C2×C8—21no—
379C2×C8—21no—
380C2×C8—21no—
381C2×C8—21no—
382C2×C8—21no—
383C2×C8—21no—
384C2×C8—21no—
385C2×C8—21no—
386C2×C8—21no—
387C2×C8—21no—
388C2×C8—21no—
389C2×C8—21no—
390C2×C8—21no—
391C2×C8—21no—
392C2×C8—41no—
393C2×C8—41no—
394C2×C8—41no—
395C2×C8—41no—
396C2×C8—41no—
397C2×C8—41no—
398C2×C8—41no—
399C2×C8—41no—
400C2×C8—41no—
401C2×C8—41no—
402C2×C8—41no—
403C2×C8—41no—
404C2×C8—41no—
405C2×C8—41no—
406C2×C8—41no—
407C2×C8—41no—
408C4⋊C4—24no—
409C4⋊C4—21no—
410C4⋊C4—21no—
411C4⋊C4—21no—
412C4⋊C4—21no—
413C4⋊C4—21no—
414C4⋊C4—21no—
415C4⋊C4—21no—
416C4⋊C4—21no—
417C4⋊C4—21no—
418C4⋊C4—21no—
419C4⋊C4—42no—
420C4⋊C4—41no—
421C4⋊C4—41no—
422C4⋊C4—42no—
423C4⋊C4—42no—
424C4⋊C4—42no—
425C4⋊C4#13112yes—
426C4⋊C4#13112yes—
427C22⋊C4—42no—
428C22⋊C4—41no—
429C22⋊C4—41no—
430C22⋊C4—42no—
431C22⋊C4—42no—
432C22⋊C4—42no—
433C42—41no—
434C42—41no—
435C42—41no—
436C42—41no—
437C42—41no—
438C42—41no—
439C42—81no—
440C42—81no—
441C42#11712yes—
442C42C4×D412yes—
443C42#12112yes—
444C42#12312yes—
445C42#12412yes—
446C42#12612yes—
447C42C22×D411yescharacteristic
448C42C2×#4111yescharacteristic
449C42C2×#4111yescharacteristic
450C42C2×#4111yescharacteristic
451C23—21no—
452C23—21no—
453C23—21no—
454C23—21no—
455C23—21no—
456C23—21no—
457C23—41no—
458C23—41no—
459C23—41no—
460C23—41no—
461C23—41no—
462C23—41no—
463C23—41no—
464C23#38812yes—
465C23C2×C4×D411yescharacteristic
466Q8—41no—
467Q8—41no—
468Q8—41no—
469Q8—41no—
470D4—41no—
471D4—41no—
472D4—41no—
473D4—41no—
474D4—84no—
475C2×C4—21no—
476C2×C4—21no—
477C2×C4—21no—
478C2×C4—21no—
479C2×C4—21no—
480C2×C4—21no—
481C2×C4—21no—
482C2×C4—21no—
483C2×C4—22no—
484C2×C4—22no—
485C2×C4—22no—
486C2×C4—22no—
487C2×C4—22no—
488C2×C4—22no—
489C2×C4—22no—
490C2×C4—22no—
491C2×C4—22no—
492C2×C4—22no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—41no—
496C2×C4—41no—
497C2×C4—41no—
498C2×C4—41no—
499C2×C4—41no—
500C2×C4—41no—
501C2×C4—41no—
502C2×C4—41no—
503C2×C4—41no—
504C2×C4—41no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—84no—
515C2×C4—84no—
516C2×C4—81no—
517C2×C4—81no—
518C2×C4—81no—
519C2×C4—81no—
520C2×C4#38812yes—
521C2×C4#38812yes—
522C2×C4#38812yes—
523C2×C4C2×#11711yescharacteristic
524C2×C4C2×C4×D411yescharacteristic
525C2×C4C2×#12111yescharacteristic
526C2×C4C2×#12311yescharacteristic
527C2×C4C2×#12411yescharacteristic
528C2×C4C2×#12611yescharacteristic
529C2×C4C2×#13111yescharacteristic
530C2×C4C2×#13111yescharacteristic
531C2×C4#56811yescharacteristic
532C2×C4#56811yescharacteristic
533C2×C4#57611yescharacteristic
534C2×C4#57711yescharacteristic
535C2×C4#57711yescharacteristic
536C2×C4#57811yescharacteristic
537C8—42no—
538C8—42no—
539C8—41no—
540C8—41no—
541C8—41no—
542C8—41no—
543C8—41no—
544C8—41no—
545C8—41no—
546C8—41no—
547C8—81no—
548C8—81no—
549C8—81no—
550C8—81no—
551C22—22no—
552C22—22no—
553C22—22no—
554C22—41no—
555C22—41no—
556C22—41no—
557C22—41no—
558C22—81no—
559C22—81no—
560C22—81no—
561C22—81no—
562C22—81no—
563C22—81no—
564C22—81no—
565C22C2×#38811yescharacteristic
566C22#292911yescharacteristic
567C22#293011yescharacteristic
568C22#294611yescharacteristic
569C22#294711yescharacteristic
570C22#295711yescharacteristic
571C22#295711yescharacteristic
572C4—22no—
573C4—22no—
574C4—21no—
575C4—21no—
576C4—22no—
577C4—42no—
578C4—42no—
579C4—81no—
580C4—81no—
581C4—81no—
582C4—81no—
583C4—81no—
584C4#292511yescharacteristic
585C4#292511yescharacteristic
586C4#293511yescharacteristic
587C4#293511yescharacteristic
588C4#306011yescharacteristic
589C4#306111yescharacteristic
590C4#313611yescharacteristic
591C4#313711yescharacteristic
592C2—42no—
593C2—161no—
594C2#1791811yescharacteristic
595C2#1791911yescharacteristic
596C2#1794611yescharacteristic
597C2#1794711yescharacteristic
598C2#2105211yescharacteristic
599C2#2109211yescharacteristic
600C2#2109811yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC23C2×C4×D4
prime radicalC2×#121C2×C4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C2×#431 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #21951C2semidirect
5. C2×#1454C2semidirect
6. #9560C2semidirect
7. #9559C2semidirect
8. #9558C2semidirect
9. #9557C2semidirect
10. #9556C2semidirect
11. #9555C2semidirect
12. #9554C2semidirect
13. #9553C2semidirect
14. #7901C2semidirect
15. #7880C2semidirect

Maximal subgroups

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

subgroupindex
#219512
#218502
#213882
C2×#14822
C2×#14542
#95602
#95592
#95582
#95572
#95562
#95552
#95542
#95532
#79012
#78802