Finite Group Databasebeta

← #379291

Subgroups — #379291

Order 512.

1013 conjugacy classes of subgroups.

The subgroup lattice has 639 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23451C211yesmaximal, characteristic
2C2×#1994C211yesmaximal, characteristic, complemented
3#21304C211yesmaximal, characteristic, complemented
4C2×#1937C211yesmaximal, characteristic
5#21051C211yesmaximal, characteristic
6#11291C211yesmaximal, characteristic, complemented
7#11269C211yesmaximal, characteristic, complemented
8#11191C211yesmaximal, characteristic
9#10934C211yesmaximal, characteristic
10#10929C211yesmaximal, characteristic
11#10923C211yesmaximal, characteristic
12#10818C211yesmaximal, characteristic, complemented
13#10809C211yesmaximal, characteristic, complemented
14#10706C211yesmaximal, characteristic, complemented
15C4×#336C211yesmaximal, characteristic, complemented
16C2×#533C2211yescharacteristic
17C2×#424C2211yescharacteristic
18C2×#423C2211yescharacteristic
19C22×#104C2211yescharacteristic
20C22×#104C2211yescharacteristic
21C2×#416C2211yescharacteristic
22C2×#416C2211yescharacteristic
23C2×#408C2211yescharacteristic
24C2×#408C2211yescharacteristic
25C2×#405C2211yescharacteristic
26C2×#405C2211yescharacteristic
27C22×C4×C8C2211yescharacteristic
28C22×C4×C8C2211yescharacteristic
29#2297C2211yescharacteristic
30C2×C43C2211yescharacteristic
31#2026C412yescomplemented
32#2025C412yescomplemented
33#2024C412yescomplemented
34#2023C412yescomplemented
35#1994—24no—
36#1985—22no—
37#1984—22no—
38#1937—24no—
39#1924—22no—
40#1923—22no—
41C2×#341C2211yescharacteristic
42C2×#341C2211yescharacteristic
43C2×#341C2211yescharacteristic
44C2×#341C2211yescharacteristic
45#1735C2211yescharacteristic
46#1735C2211yescharacteristic
47C2×#339C2211yescharacteristic
48C2×#339C2211yescharacteristic
49C2×#339C2211yescharacteristic
50C2×#339C2211yescharacteristic
51C2×#339C2211yescharacteristic
52C2×#339C2211yescharacteristic
53#1729C2211yescharacteristic
54#1729C2211yescharacteristic
55#1729C2211yescharacteristic
56#1728C2211yescharacteristic
57C2×#336C2211yescharacteristic
58C2×#336C2211yescharacteristic
59C2×#336C2211yescharacteristic
60C2×#336C2211yescharacteristic
61C22×#41C2311yescharacteristic
62C22×M16C2311yescharacteristic
63C22×M16C2311yescharacteristic
64C22×M16C2311yescharacteristic
65C22×M16C2311yescharacteristic
66C23×C8C2311yescharacteristic
67C23×C8C2311yescharacteristic
68C23×C8C2311yescharacteristic
69C23×C8C2311yescharacteristic
70#533—22no—
71#533—22no—
72#533—41no—
73#533—41no—
74C2×C4⋊Q8C2×C412yes—
75C2×#125C2×C412yes—
76C2×C22⋊Q8—22no—
77C2×C4×Q8—22no—
78C2×#117C2311yescharacteristic
79C2×#117C2311yescharacteristic
80C2×#117C2311yescharacteristic
81C22×C42—22no—
82C22×C42C2311yescharacteristic
83C22×C42C2311yescharacteristic
84C22×C42C2311yescharacteristic
85#424—22no—
86#424—41no—
87#423—22no—
88#423—41no—
89C2×C4⋊C8—22no—
90C2×C4⋊C8—22no—
91C2×#104—22no—
92C2×#104—22no—
93C2×#104—22no—
94C2×#104—22no—
95C2×#104—22no—
96C2×#104—41no—
97#416—22no—
98#416—22no—
99#416—41no—
100#416—41no—
101C2×#103C2×C412yes—
102C2×#103C2×C412yes—
103C2×#102C2×C412yes—
104C2×#102C2×C412yes—
105#408—22no—
106#408—22no—
107#408—22no—
108#408—41no—
109C2×C22⋊C8—21no—
110C2×C22⋊C8—22no—
111C2×C22⋊C8—21no—
112#405—22no—
113#405—22no—
114#405—22no—
115#405—41no—
116C2×#97—21no—
117C2×#97—21no—
118C2×C4×C8—21no—
119C2×C4×C8—21no—
120C2×C4×C8—22no—
121C2×C4×C8—22no—
122C2×C4×C8—22no—
123C2×C4×C8—22no—
124C2×C4×C8—22no—
125C2×C4×C8—41no—
126#391C2×C411yescharacteristic
127#391C2×C411yescharacteristic
128#390C2×C411yescharacteristic
129#390C2×C411yescharacteristic
130#386—22no—
131C4×C22⋊C4—22no—
132C43—22no—
133C43—41no—
134#341—22no—
135#341—22no—
136#341—22no—
137#341—41no—
138#340C2×C412yes—
139#340C2×C412yes—
140#340C2×C412yes—
141#340C2×C412yes—
142#339—22no—
143#339—22no—
144#339—22no—
145#339—22no—
146#339—22no—
147#339D414yes—
148#336—22no—
149#336—21no—
150#336—22no—
151#336—21no—
152#336—22no—
153#336—22no—
154#336—22no—
155#336C2×C412yes—
156#336C2×C412yes—
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×Q8—22no—
172C22×Q8C22×C411yescharacteristic
173C22×D4—21no—
174C22×D4C22×C411yescharacteristic
175C23×C4—21no—
176C23×C4—21no—
177C23×C4—21no—
178C23×C4#4112yes—
179C23×C4C2411yescharacteristic
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×M16—21no—
202C2×M16C2×D412yes—
203C2×M16#4112yes—
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×C8—21no—
228C22×C8—21no—
229C22×C8—21no—
230C22×C8C22×C412yes—
231C22×C8C22×C412yes—
232C22×C8C2×D412yes—
233C22×C8C2×D411yescharacteristic
234C22×C8C2×D411yescharacteristic
235C22×C8#4112yes—
236C22×C8#4112yes—
237C4⋊Q8—24no—
238C4⋊Q8—42no—
239#125—24no—
240#125—42no—
241C22⋊Q8—44no—
242C22⋊Q8—44no—
243C4×Q8—44no—
244C4×Q8—44no—
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#117C2×D412yes—
256#117#4112yes—
257C2×C4⋊C4—24no—
258C2×C4⋊C4—22no—
259C2×C4⋊C4C22×C412yes—
260C2×C4⋊C4C22×C411yescharacteristic
261C2×C4⋊C4C22×C411yescharacteristic
262C2×C4⋊C4C22×C412yes—
263C2×C22⋊C4—21no—
264C2×C22⋊C4—21no—
265C2×C22⋊C4—21no—
266C2×C22⋊C4—22no—
267C2×C22⋊C4—22no—
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×C42—21no—
277C2×C42—21no—
278C2×C42—21no—
279C2×C42—21no—
280C2×C42—28no—
281C2×C42—21no—
282C2×C42—21no—
283C2×C42—21no—
284C2×C42—44no—
285C2×C42—44no—
286C2×C42C22×C411yescharacteristic
287C2×C42C22×C411yescharacteristic
288C2×C42C2×D412yes—
289C2×C42C2×D412yes—
290C2×C42#4112yes—
291C2×C42#4112yes—
292C2×C42#4112yes—
293C4⋊C8—44no—
294C4⋊C8—44no—
295#104—44no—
296#104—44no—
297#103—24no—
298#103—42no—
299#102—24no—
300#102—42no—
301C22⋊C8—44no—
302C22⋊C8—42no—
303C22⋊C8—42no—
304#97—42no—
305#97—42no—
306C4×C8—44no—
307C4×C8—42no—
308C4×C8—42no—
309C4×C8—44no—
310#95—22no—
311#95—22no—
312C24—21no—
313C24C2×#4111yescharacteristic
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—
325#41—41no—
326#41—41no—
327#41—41no—
328#41—41no—
329#41—41no—
330C2×Q8—21no—
331C2×Q8—21no—
332C2×Q8—21no—
333C2×Q8—21no—
334C2×Q8—21no—
335C2×Q8—21no—
336C2×Q8—44no—
337C2×Q8—44no—
338C2×Q8—82no—
339C2×D4—21no—
340C2×D4—21no—
341C2×D4—21no—
342C2×D4—21no—
343C2×D4—21no—
344C2×D4—21no—
345C2×D4—41no—
346C2×D4—41no—
347C2×D4—41no—
348C2×D4—41no—
349C2×D4—41no—
350C2×D4—41no—
351C22×C4—24no—
352C22×C4—24no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—24no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—24no—
362C22×C4—21no—
363C22×C4—21no—
364C22×C4—21no—
365C22×C4—22no—
366C22×C4—42no—
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×C4C4×D412yes—
377C22×C4C4×D412yes—
378C22×C4C23×C411yescharacteristic
379C22×C4C22×D411yescharacteristic
380C22×C4C22×D411yescharacteristic
381C22×C4C2×#4111yescharacteristic
382C22×C4C2×#4111yescharacteristic
383C22×C4C2×#4111yescharacteristic
384C22×C4C2×#4111yescharacteristic
385C22×C4C2×#4111yescharacteristic
386C22×C4C2×#4111yescharacteristic
387C22×C4C2×#4111yescharacteristic
388C22×C4H(5,𝔽2)11yescharacteristic
389C22×C4H(5,𝔽2)11yescharacteristic
390C22×C42−1+411yescharacteristic
391C22×C42−1+411yescharacteristic
392M16—22no—
393M16—22no—
394M16—22no—
395M16—22no—
396M16—41no—
397M16—41no—
398M16—41no—
399M16—41no—
400M16—41no—
401M16—41no—
402M16—41no—
403M16—41no—
404M16—41no—
405M16—41no—
406M16—41no—
407M16—41no—
408C2×C8—22no—
409C2×C8—22no—
410C2×C8—21no—
411C2×C8—21no—
412C2×C8—21no—
413C2×C8—21no—
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—22no—
423C2×C8—22no—
424C2×C8—22no—
425C2×C8—22no—
426C2×C8—22no—
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×C8—41no—
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#12412yes—
450C2×C8#12412yes—
451C4⋊C4—22no—
452C4⋊C4—22no—
453C4⋊C4—21no—
454C4⋊C4—21no—
455C4⋊C4—21no—
456C4⋊C4—21no—
457C4⋊C4—22no—
458C4⋊C4—48no—
459C4⋊C4—44no—
460C4⋊C4C4×D414yes—
461C22⋊C4—22no—
462C22⋊C4—22no—
463C22⋊C4—41no—
464C22⋊C4—41no—
465C22⋊C4—41no—
466C22⋊C4—41no—
467C22⋊C4—44no—
468C22⋊C4—82no—
469C42—21no—
470C42—21no—
471C42—21no—
472C42—21no—
473C42—22no—
474C42—22no—
475C42—22no—
476C42—22no—
477C42—22no—
478C42—22no—
479C42—22no—
480C42—22no—
481C42—22no—
482C42—416no—
483C42—41no—
484C42—41no—
485C42—41no—
486C42—41no—
487C42—41no—
488C42—41no—
489C42#12414yes—
490C23—22no—
491C23—22no—
492C23—41no—
493C23—41no—
494C23—42no—
495C23—42no—
496C23—41no—
497C23C2×C4×D411yescharacteristic
498C23#52911yescharacteristic
499C23#52911yescharacteristic
500C23#54611yescharacteristic
501C23#55111yescharacteristic
502C23#55211yescharacteristic
503C23#55611yescharacteristic
504Q8—41no—
505Q8—41no—
506Q8—41no—
507Q8—41no—
508Q8—88no—
509D4—41no—
510D4—41no—
511D4—41no—
512D4—41no—
513D4—81no—
514D4—81no—
515D4—81no—
516D4—81no—
517C2×C4—28no—
518C2×C4—28no—
519C2×C4—24no—
520C2×C4—24no—
521C2×C4—22no—
522C2×C4—22no—
523C2×C4—24no—
524C2×C4—24no—
525C2×C4—22no—
526C2×C4—24no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—44no—
530C2×C4—44no—
531C2×C4—44no—
532C2×C4—44no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—81no—
548C2×C4—81no—
549C2×C4—81no—
550C2×C4—81no—
551C2×C4#44314yes—
552C2×C4C2×C4×D411yescharacteristic
553C2×C4C4×#4111yescharacteristic
554C2×C4C4×#4111yescharacteristic
555C2×C4C4×#4111yescharacteristic
556C2×C4#51811yescharacteristic
557C2×C4#51911yescharacteristic
558C2×C4C2×#12411yescharacteristic
559C2×C4C2×#12411yescharacteristic
560C2×C4#52911yescharacteristic
561C2×C4#52911yescharacteristic
562C2×C4#53211yescharacteristic
563C2×C4#53211yescharacteristic
564C2×C4#53911yescharacteristic
565C2×C4#53911yescharacteristic
566C2×C4#54211yescharacteristic
567C2×C4#54211yescharacteristic
568C2×C4D4211yescharacteristic
569C2×C4#54611yescharacteristic
570C2×C4#54711yescharacteristic
571C2×C4#54711yescharacteristic
572C2×C4#54811yescharacteristic
573C2×C4#54811yescharacteristic
574C2×C4#55111yescharacteristic
575C2×C4#55111yescharacteristic
576C2×C4#55211yescharacteristic
577C2×C4#55511yescharacteristic
578C2×C4#55511yescharacteristic
579C2×C4#55611yescharacteristic
580C8—22no—
581C8—22no—
582C8—42no—
583C8—42no—
584C8—42no—
585C8—42no—
586C8—42no—
587C8—81no—
588C8—81no—
589C22—22no—
590C22—22no—
591C22—22no—
592C22—22no—
593C22—21no—
594C22—21no—
595C22—21no—
596C22—21no—
597C22—82no—
598C22—82no—
599C22—81no—
600C22—81no—
601C22—81no—
602C22C4×#12411yescharacteristic
603C22#235811yescharacteristic
604C22#236211yescharacteristic
605C22#248811yescharacteristic
606C22#253211yescharacteristic
607C22#253211yescharacteristic
608C22#256011yescharacteristic
609C22C2×#44312yes—
610C22#295612yes—
611C4—28no—
612C4—24no—
613C4—44no—
614C4—42no—
615C4—44no—
616C4—82no—
617C4—81no—
618C4—81no—
619C4—81no—
620C4C4×#12411yescharacteristic
621C4#233211yescharacteristic
622C4#236211yescharacteristic
623C4#237211yescharacteristic
624C4#248811yescharacteristic
625C4#249111yescharacteristic
626C4#249111yescharacteristic
627C4#256011yescharacteristic
628C2—21no—
629C2—21no—
630C2—21no—
631C2—21no—
632C2—161no—
633C2#1423211yescharacteristic
634C2#2165811yescharacteristic
635C2#2166211yescharacteristic
636C2#2288812yes—
637C2#2308012yes—
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
2. C2×#1994C2semidirect
3. #21304C2semidirect
6. #11291C2semidirect
7. #11269C2semidirect
12. #10818C2semidirect
13. #10809C2semidirect
14. #10706C2semidirect
15. C4×#336C2semidirect
31. #2026C4semidirect
32. #2025C4semidirect
33. #2024C4semidirect
34. #2023C4semidirect

Maximal subgroups

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

subgroupindex
#234512
C2×#19942
#213042
C2×#19372
#210512
#112912
#112692
#111912
#109342
#109292
#109232
#108182
#108092
#107062
C4×#3362