Finite Group Databasebeta

← #367791

Subgroups — #367791

Order 512.

941 conjugacy classes of subgroups.

The subgroup lattice has 629 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21148C211yesmaximal, characteristic, complemented
2C2×#1937C211yesmaximal, characteristic
3#21083C211yesmaximal, characteristic
4#21051C211yesmaximal, characteristic
5C2×#1837C211yesmaximal, characteristic, complemented
6#11291C211yesmaximal, characteristic, complemented
7#11209C211yesmaximal, characteristic, complemented
8#11068C211yesmaximal, characteristic
9#11053C211yesmaximal, characteristic
10#10934C211yesmaximal, characteristic
11#10929C211yesmaximal, characteristic
12#10882C211yesmaximal, characteristic, complemented
13#10809C211yesmaximal, characteristic, complemented
14#10706C211yesmaximal, characteristic, complemented
15C8×#95C211yesmaximal, characteristic, complemented
16C2×#424C2211yescharacteristic
17C2×#424C2211yescharacteristic
18C2×#423C2211yescharacteristic
19C2×#423C2211yescharacteristic
20C22×#104C2211yescharacteristic
21C22×#104C2211yescharacteristic
22C2×#416C2211yescharacteristic
23C2×#416C2211yescharacteristic
24C2×#408C2211yescharacteristic
25C2×#408C2211yescharacteristic
26C2×#405C2211yescharacteristic
27C2×#405C2211yescharacteristic
28C22×C4×C8C2211yescharacteristic
29C22×C4×C8C2211yescharacteristic
30C22×C4×C8C2211yescharacteristic
31#2297C2211yescharacteristic
32#2296C2211yescharacteristic
33#1940—22no—
34#1939—22no—
35#1937—24no—
36#1934C412yescomplemented
37#1934C412yescomplemented
38#1933C412yescomplemented
39#1933C412yescomplemented
40#1924—22no—
41#1923—22no—
42#1837—24no—
43C2×#341C2211yescharacteristic
44C2×#341C2211yescharacteristic
45C2×#341C2211yescharacteristic
46C2×#341C2211yescharacteristic
47#1735C2211yescharacteristic
48#1735C2211yescharacteristic
49C2×#339C2211yescharacteristic
50C2×#339C2211yescharacteristic
51C2×#339C2211yescharacteristic
52C2×#339C2211yescharacteristic
53C2×#339C2211yescharacteristic
54C2×#339C2211yescharacteristic
55#1729C2211yescharacteristic
56#1729C2211yescharacteristic
57C2×#336C2211yescharacteristic
58C2×#336C2211yescharacteristic
59C2×#336C2211yescharacteristic
60C4×#95C2211yescharacteristic
61C22×#41C2311yescharacteristic
62C22×M16C2311yescharacteristic
63C22×M16C2311yescharacteristic
64C22×M16C2311yescharacteristic
65C22×M16C2311yescharacteristic
66C23×C8C2311yescharacteristic
67C23×C8C2311yescharacteristic
68C23×C8C2311yescharacteristic
69C23×C8C2311yescharacteristic
70C2×#117C2311yescharacteristic
71C2×#117C2311yescharacteristic
72C2×#117C2311yescharacteristic
73C22×C42C2311yescharacteristic
74C22×C42C2311yescharacteristic
75C22×C42C2311yescharacteristic
76#424—22no—
77#424—22no—
78#424—41no—
79#424—41no—
80#423—22no—
81#423—22no—
82#423—41no—
83#423—41no—
84C2×C4⋊C8—22no—
85C2×C4⋊C8—22no—
86C2×C4⋊C8—22no—
87C2×C4⋊C8—22no—
88C2×#104—22no—
89C2×#104—22no—
90C2×#104—22no—
91C2×#104—41no—
92C2×#104—41no—
93C2×#104—41no—
94#416—22no—
95#416—22no—
96#416—22no—
97#416—22no—
98C2×#103C2×C412yes—
99C2×#103C2×C412yes—
100C2×#102C2×C412yes—
101C2×#102C2×C412yes—
102#408—22no—
103#408—22no—
104#408—22no—
105#408—41no—
106C2×C22⋊C8—21no—
107C2×C22⋊C8—22no—
108C2×C22⋊C8—21no—
109#405—22no—
110#405—22no—
111#405—22no—
112#405—41no—
113C2×#97—21no—
114C2×#97—21no—
115C2×C4×C8—22no—
116C2×C4×C8—22no—
117C2×C4×C8—22no—
118C2×C4×C8—21no—
119C2×C4×C8—21no—
120C2×C4×C8—22no—
121C2×C4×C8—22no—
122C2×C4×C8—22no—
123C2×C4×C8—41no—
124C2×C4×C8—41no—
125C2×C4×C8C2×C412yes—
126C2×C4×C8C2×C412yes—
127#391C2×C411yescharacteristic
128#391C2×C411yescharacteristic
129#390C2×C411yescharacteristic
130#390C2×C411yescharacteristic
131#389C2×C412yes—
132#389C2×C412yes—
133#386—22no—
134#385—22no—
135C4×C4⋊C4—22no—
136C4×C22⋊C4—22no—
137C2×#95—22no—
138#341—22no—
139#341—22no—
140#341—22no—
141#341D414yes—
142#340C2×C412yes—
143#340C2×C412yes—
144#340C2×C412yes—
145#340C2×C412yes—
146#339—22no—
147#339—22no—
148#339—22no—
149#339—22no—
150#339—22no—
151#339—41no—
152#336—22no—
153#336—21no—
154#336—21no—
155#336—22no—
156#336—22no—
157#336D412yes—
158#336D412yes—
159C2×#41—21no—
160C2×#41—21no—
161C2×#41—21no—
162C2×#41—21no—
163C2×#41—21no—
164C2×#41—21no—
165C2×#41—21no—
166C2×#41—21no—
167C2×#41—21no—
168C2×#41—21no—
169C2×#41—21no—
170C2×#41—21no—
171C22×Q8C22×C411yescharacteristic
172C22×D4—21no—
173C22×D4C22×C411yescharacteristic
174C23×C4—21no—
175C23×C4—21no—
176C23×C4—21no—
177C23×C4#4112yes—
178C23×C4C2411yescharacteristic
179C2×M16—21no—
180C2×M16—21no—
181C2×M16—21no—
182C2×M16—21no—
183C2×M16—21no—
184C2×M16—21no—
185C2×M16—21no—
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×M16C2×D412yes—
202C2×M16C2×D412yes—
203C22×C8—21no—
204C22×C8—21no—
205C22×C8—21no—
206C22×C8—21no—
207C22×C8—21no—
208C22×C8—21no—
209C22×C8—21no—
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×C8C22×C412yes—
228C22×C8C22×C412yes—
229C22×C8C2×D412yes—
230C22×C8C2×D412yes—
231C22×C8C2×D411yescharacteristic
232C22×C8C2×D411yescharacteristic
233C22×C8#4112yes—
234C22×C8#4112yes—
235C22×C8#4112yes—
236#117—21no—
237#117—21no—
238#117—21no—
239#117—21no—
240#117—21no—
241#117—21no—
242#117—21no—
243#117—21no—
244#117—21no—
245#117—21no—
246#117#4112yes—
247#117#4112yes—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4C22×C411yescharacteristic
251C2×C4⋊C4C22×C411yescharacteristic
252C2×C4⋊C4C22×C412yes—
253C2×C4⋊C4C22×C412yes—
254C2×C22⋊C4—21no—
255C2×C22⋊C4—21no—
256C2×C22⋊C4—21no—
257C2×C22⋊C4—22no—
258C2×C22⋊C4—22no—
259C2×C42—21no—
260C2×C42—21no—
261C2×C42—21no—
262C2×C42—21no—
263C2×C42—21no—
264C2×C42—21no—
265C2×C42—21no—
266C2×C42—21no—
267C2×C42—21no—
268C2×C42—21no—
269C2×C42—21no—
270C2×C42—21no—
271C2×C42—21no—
272C2×C42—21no—
273C2×C42—21no—
274C2×C42—21no—
275C2×C42—21no—
276C2×C42C22×C411yescharacteristic
277C2×C42C22×C411yescharacteristic
278C2×C42C2×D412yes—
279C2×C42#4112yes—
280C2×C42#4112yes—
281C4⋊C8—44no—
282C4⋊C8—44no—
283C4⋊C8—44no—
284C4⋊C8—44no—
285#104—44no—
286#104—44no—
287#103—24no—
288#103—24no—
289#102—24no—
290#102—24no—
291C22⋊C8—44no—
292C22⋊C8—42no—
293C22⋊C8—42no—
294#97—42no—
295#97—42no—
296C4×C8—24no—
297C4×C8—24no—
298C4×C8—44no—
299C4×C8—42no—
300C4×C8—44no—
301C4×C8—42no—
302#95—28no—
303#95—22no—
304#95—22no—
305#95—22no—
306#95—22no—
307C24—21no—
308C24C2×#4111yescharacteristic
309#41—41no—
310#41—41no—
311#41—41no—
312#41—41no—
313#41—41no—
314#41—41no—
315#41—41no—
316#41—41no—
317#41—41no—
318#41—41no—
319#41—41no—
320#41—41no—
321#41—41no—
322#41—41no—
323#41—41no—
324#41—41no—
325C2×Q8—21no—
326C2×Q8—21no—
327C2×Q8—21no—
328C2×Q8—21no—
329C2×Q8—21no—
330C2×Q8—21no—
331C2×D4—21no—
332C2×D4—21no—
333C2×D4—21no—
334C2×D4—21no—
335C2×D4—21no—
336C2×D4—21no—
337C2×D4—41no—
338C2×D4—41no—
339C2×D4—41no—
340C2×D4—41no—
341C2×D4—41no—
342C2×D4—41no—
343C22×C4—24no—
344C22×C4—24no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—24no—
351C22×C4—21no—
352C22×C4—21no—
353C22×C4—21no—
354C22×C4—22no—
355C22×C4—42no—
356C22×C4—42no—
357C22×C4—42no—
358C22×C4—41no—
359C22×C4—41no—
360C22×C4—41no—
361C22×C4—41no—
362C22×C4—41no—
363C22×C4—41no—
364C22×C4—41no—
365C22×C4—41no—
366C22×C4—41no—
367C22×C4—41no—
368C22×C4C4×D412yes—
369C22×C4C4×D412yes—
370C22×C4C23×C411yescharacteristic
371C22×C4C22×D411yescharacteristic
372C22×C4C22×D411yescharacteristic
373C22×C4C2×#4111yescharacteristic
374C22×C4C2×#4111yescharacteristic
375C22×C4C2×#4111yescharacteristic
376C22×C4C2×#4111yescharacteristic
377C22×C4C2×#4111yescharacteristic
378C22×C4C2×#4111yescharacteristic
379C22×C4C2×#4111yescharacteristic
380C22×C4H(5,𝔽2)11yescharacteristic
381C22×C4H(5,𝔽2)11yescharacteristic
382C22×C42−1+411yescharacteristic
383C22×C42−1+411yescharacteristic
384M16—22no—
385M16—22no—
386M16—22no—
387M16—22no—
388M16—41no—
389M16—41no—
390M16—41no—
391M16—41no—
392M16—41no—
393M16—41no—
394M16—41no—
395M16—41no—
396M16—41no—
397M16—41no—
398M16—41no—
399M16—41no—
400C2×C8—22no—
401C2×C8—22no—
402C2×C8—22no—
403C2×C8—22no—
404C2×C8—22no—
405C2×C8—21no—
406C2×C8—21no—
407C2×C8—22no—
408C2×C8—21no—
409C2×C8—21no—
410C2×C8—22no—
411C2×C8—22no—
412C2×C8—22no—
413C2×C8—22no—
414C2×C8—22no—
415C2×C8—22no—
416C2×C8—22no—
417C2×C8—22no—
418C2×C8—22no—
419C2×C8—22no—
420C2×C8—22no—
421C2×C8—22no—
422C2×C8—41no—
423C2×C8—41no—
424C2×C8—41no—
425C2×C8—41no—
426C2×C8—41no—
427C2×C8—41no—
428C2×C8—41no—
429C2×C8—41no—
430C2×C8—41no—
431C2×C8—41no—
432C2×C8—41no—
433C2×C8—41no—
434C2×C8—41no—
435C2×C8—41no—
436C2×C8—41no—
437C2×C8—41no—
438C2×C8—41no—
439C2×C8—41no—
440C2×C8C4×D414yes—
441C2×C8#12412yes—
442C2×C8#12412yes—
443C4⋊C4—22no—
444C4⋊C4—22no—
445C4⋊C4—21no—
446C4⋊C4—21no—
447C4⋊C4—21no—
448C4⋊C4—21no—
449C4⋊C4—22no—
450C4⋊C4—44no—
451C4⋊C4—82no—
452C4⋊C4#11714yes—
453C22⋊C4—22no—
454C22⋊C4—22no—
455C22⋊C4—41no—
456C22⋊C4—41no—
457C22⋊C4—41no—
458C22⋊C4—41no—
459C22⋊C4—44no—
460C22⋊C4—82no—
461C42—21no—
462C42—21no—
463C42—21no—
464C42—21no—
465C42—22no—
466C42—22no—
467C42—22no—
468C42—22no—
469C42—22no—
470C42—22no—
471C42—22no—
472C42—22no—
473C42—41no—
474C42—41no—
475C42—41no—
476C42—41no—
477C42—41no—
478C42—41no—
479C42—41no—
480C42—41no—
481C23—22no—
482C23—22no—
483C23—41no—
484C23—41no—
485C23—42no—
486C23—42no—
487C23—41no—
488C23C2×C4×D411yescharacteristic
489C23#52911yescharacteristic
490C23#52911yescharacteristic
491C23#54811yescharacteristic
492C23#55211yescharacteristic
493C23#55311yescharacteristic
494C23#55311yescharacteristic
495Q8—41no—
496Q8—41no—
497Q8—41no—
498Q8—41no—
499D4—41no—
500D4—41no—
501D4—41no—
502D4—41no—
503D4—81no—
504D4—81no—
505D4—81no—
506D4—81no—
507C2×C4—22no—
508C2×C4—22no—
509C2×C4—24no—
510C2×C4—24no—
511C2×C4—24no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—44no—
516C2×C4—44no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—44no—
520C2×C4—44no—
521C2×C4—44no—
522C2×C4—44no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—41no—
528C2×C4—42no—
529C2×C4—41no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—81no—
538C2×C4—81no—
539C2×C4—81no—
540C2×C4—81no—
541C2×C4#44314yes—
542C2×C4C2×#11711yescharacteristic
543C2×C4C2×C4×D411yescharacteristic
544C2×C4C4×#4111yescharacteristic
545C2×C4C4×#4111yescharacteristic
546C2×C4#52011yescharacteristic
547C2×C4#52011yescharacteristic
548C2×C4C2×#12411yescharacteristic
549C2×C4#52911yescharacteristic
550C2×C4#52911yescharacteristic
551C2×C4#53211yescharacteristic
552C2×C4#53211yescharacteristic
553C2×C4#53211yescharacteristic
554C2×C4#53811yescharacteristic
555C2×C4#53911yescharacteristic
556C2×C4#54111yescharacteristic
557C2×C4#54211yescharacteristic
558C2×C4D4211yescharacteristic
559C2×C4#54611yescharacteristic
560C2×C4#54611yescharacteristic
561C2×C4#54711yescharacteristic
562C2×C4#54711yescharacteristic
563C2×C4#54811yescharacteristic
564C2×C4#55111yescharacteristic
565C2×C4#55211yescharacteristic
566C2×C4#55211yescharacteristic
567C2×C4#55211yescharacteristic
568C2×C4#55511yescharacteristic
569C2×C4#55511yescharacteristic
570C8—24no—
571C8—22no—
572C8—22no—
573C8—42no—
574C8—42no—
575C8—42no—
576C8—42no—
577C8—81no—
578C8—81no—
579C22—21no—
580C22—21no—
581C22—21no—
582C22—21no—
583C22—22no—
584C22—22no—
585C22—22no—
586C22—22no—
587C22—81no—
588C22—81no—
589C22—81no—
590C22—82no—
591C22—82no—
592C22C4×#12411yescharacteristic
593C22#235211yescharacteristic
594C22#236111yescharacteristic
595C22#251811yescharacteristic
596C22#251911yescharacteristic
597C22#252911yescharacteristic
598C22#255811yescharacteristic
599C22C2×#44312yes—
600C22#295712yes—
601C4—24no—
602C4—44no—
603C4—44no—
604C4—42no—
605C4—44no—
606C4—81no—
607C4—81no—
608C4—81no—
609C4—82no—
610C4#229311yescharacteristic
611C4#233411yescharacteristic
612C4#234611yescharacteristic
613C4#235911yescharacteristic
614C4#248811yescharacteristic
615C4#249111yescharacteristic
616C4#251911yescharacteristic
617C4#255511yescharacteristic
618C2—21no—
619C2—21no—
620C2—21no—
621C2—21no—
622C2—161no—
623C2#1401011yescharacteristic
624C2#2165811yescharacteristic
625C2#2165911yescharacteristic
626C2#2278312yes—
627C2#2301812yes—
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
socleC23C2×C4×D4
prime radicalC22×#41C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C2×#336 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #21148C2semidirect
5. C2×#1837C2semidirect
6. #11291C2semidirect
7. #11209C2semidirect
12. #10882C2semidirect
13. #10809C2semidirect
14. #10706C2semidirect
15. C8×#95C2semidirect
36. #1934C4semidirect
37. #1934C4semidirect
38. #1933C4semidirect
39. #1933C4semidirect

Maximal subgroups

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

subgroupindex
#211482
C2×#19372
#210832
#210512
C2×#18372
#112912
#112092
#110682
#110532
#109342
#109292
#108822
#108092
#107062
C8×#952