Finite Group Databasebeta

← #229599

Subgroups — #229599

Order 512.

623 conjugacy classes of subgroups.

The subgroup lattice has 578 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23113C211yesmaximal, characteristic, complemented
2#23015C211yesmaximal, characteristic, complemented
3#19157C211yesmaximal, characteristic
4#18073C211yesmaximal, characteristic, complemented
5#17830C211yesmaximal, characteristic, complemented
6#9844C211yesmaximal, characteristic, complemented
7#9626C211yesmaximal, characteristic, complemented
8#9578C211yesmaximal, characteristic, complemented
9#9577C211yesmaximal, characteristic, complemented
10#8996C211yesmaximal, characteristic, complemented
11#8995C211yesmaximal, characteristic, complemented
12#8932C211yesmaximal, characteristic, complemented
13#8930C211yesmaximal, characteristic, complemented
14C4×#334C211yesmaximal, characteristic, complemented
15#8194C211yesmaximal, characteristic, complemented
16#3169C2211yescharacteristic
17#3035C2211yescharacteristic
18#3035C2211yescharacteristic
19#2965C2211yescharacteristic
20C2×#443—21no—
21#2935C2211yescharacteristic
22#2935C2211yescharacteristic
23#2925C2211yescharacteristic
24#2906C2211yescharacteristic
25#1974—21no—
26#1962C2211yescharacteristic
27#1962C2211yescharacteristic
28#1950C2211yescharacteristic
29#1950C2211yescharacteristic
30#1937—21no—
31#1929—21no—
32#1876—21no—
33#1876—21no—
34#1866—21no—
35C4×C4⋊C8C2211yescharacteristic
36C4×C4⋊C8C2211yescharacteristic
37#1764—21no—
38C4×#104C2211yescharacteristic
39C42×C8C2211yescharacteristic
40#1702C411yescharacteristic, complemented
41#1702C411yescharacteristic, complemented
42#1701C411yescharacteristic, complemented
43#1701C411yescharacteristic, complemented
44#1694C411yescharacteristic, complemented
45#1693C411yescharacteristic, complemented
46#1688C411yescharacteristic, complemented
47#1687C411yescharacteristic, complemented
48#1594—21no—
49#1586—21no—
50#1586—21no—
51#1580—21no—
52#1574—21no—
53#1573—21no—
54#1571C2211yescharacteristic
55#1570C2211yescharacteristic
56#1569—21no—
57#1567C2211yescharacteristic
58#1565—21no—
59C2×#334C2211yescharacteristic
60C2×#334C2211yescharacteristic
61#1540C2211yescharacteristic
62#1540C2211yescharacteristic
63#1492C2211yescharacteristic
64#1492C2211yescharacteristic
65#1480C2211yescharacteristic
66#1480C2211yescharacteristic
67#1476C2211yescharacteristic
68#1451C2211yescharacteristic
69#1301C2211yescharacteristic
70#1301C2211yescharacteristic
71#1278C2211yescharacteristic
72#1277C2211yescharacteristic
73#1277C2211yescharacteristic
74#1276C2211yescharacteristic
75C2×#135—21no—
76C2×#135—21no—
77C2×#131—21no—
78C2×#131—21no—
79C2×#131—21no—
80#532C2311yescharacteristic
81#532C2311yescharacteristic
82C4×#41C2311yescharacteristic
83#500C2×C411yescharacteristic
84#493C2×C411yescharacteristic
85#486C2×C411yescharacteristic
86#486C2×C411yescharacteristic
87#466—21no—
88#466—21no—
89#464—21no—
90#464—21no—
91#463—21no—
92#463—21no—
93#462C2×C411yescharacteristic
94#462C2×C411yescharacteristic
95#461C2×C411yescharacteristic
96#461C2×C411yescharacteristic
97#460C2×C411yescharacteristic
98#460C2×C411yescharacteristic
99#459C2×C411yescharacteristic
100#459C2×C411yescharacteristic
101#446C2×C411yescharacteristic
102#446C2×C411yescharacteristic
103#443—44no—
104C4×D8—21no—
105C4×D8—21no—
106#436—21no—
107#436—21no—
108#436C2×C411yescharacteristic
109#436C2×C411yescharacteristic
110#435—21no—
111#435—21no—
112#435—41no—
113#435—41no—
114C8×D4—21no—
115C8×D4—21no—
116#433—41no—
117#432—21no—
118#431—21no—
119#431—21no—
120#431C2311yescharacteristic
121#431C2311yescharacteristic
122C2×#108—21no—
123C2×#108—21no—
124#424—21no—
125#424—41no—
126#423—21no—
127#423C2311yescharacteristic
128#423C2311yescharacteristic
129C2×C4⋊C8—21no—
130C2×C4⋊C8C2311yescharacteristic
131C2×C4⋊C8C2311yescharacteristic
132C2×C4⋊C8C2311yescharacteristic
133C2×C4⋊C8C2311yescharacteristic
134C2×#104—21no—
135C2×#104—21no—
136C2×#104—21no—
137C2×#104—21no—
138C2×#104D411yescharacteristic
139C2×#104D411yescharacteristic
140#416—21no—
141#416—21no—
142#408—21no—
143#408—21no—
144#408—21no—
145#405—21no—
146C4×M16—21no—
147C4×M16C2311yescharacteristic
148C2×C4×C8—21no—
149C2×C4×C8—21no—
150C2×C4×C8—21no—
151C2×C4×C8—21no—
152C2×C4×C8C2311yescharacteristic
153C2×C4×C8C2311yescharacteristic
154C43C2311yescharacteristic
155#341—21no—
156#341—21no—
157#339D411yescharacteristic
158#339D411yescharacteristic
159#336—21no—
160#336—21no—
161#335—21no—
162#335—21no—
163#335—21no—
164#335—21no—
165#334—21no—
166#334—21no—
167#334C2×C411yescharacteristic
168#334C2×C411yescharacteristic
169#334C2×C411yescharacteristic
170#334C2×C411yescharacteristic
171#334D411yescharacteristic
172#334D411yescharacteristic
173#334D411yescharacteristic
174#334D411yescharacteristic
175#332C2×C411yescharacteristic
176#332C2×C411yescharacteristic
177#332C2×C411yescharacteristic
178#332C2×C411yescharacteristic
179#331C2×C411yescharacteristic
180#331C2×C411yescharacteristic
181#331C2×C411yescharacteristic
182#331C2×C411yescharacteristic
183Q8⋊C8—21no—
184D4⋊C8—21no—
185D4⋊C8—21no—
186D4⋊C8—21no—
187D4⋊C8—21no—
188D4⋊C8—21no—
189D4⋊C8—41no—
190D4⋊C8—41no—
191#322—21no—
192#322—21no—
193C2×#41—21no—
194C2×#41—21no—
195C2×#41C2×D411yescharacteristic
196#135—42no—
197#135—42no—
198#135—81no—
199#135—81no—
200C2×Q16—21no—
201C2×Q16—21no—
202C2×SD16—21no—
203C2×SD16—21no—
204C2×SD16—21no—
205C2×SD16—21no—
206C2×D8—21no—
207C2×D8—21no—
208C2×D8—21no—
209C2×D8—21no—
210#131—42no—
211#131—42no—
212#131—42no—
213#131—81no—
214#131—81no—
215#131—81no—
216C2×M16—21no—
217C2×M16—21no—
218C2×M16—21no—
219C2×M16—21no—
220C2×M16—41no—
221C2×M16—41no—
222C2×M16—41no—
223C2×M16C2×D411yescharacteristic
224C2×M16C2×D411yescharacteristic
225C22×C8—21no—
226C22×C8—21no—
227C22×C8—21no—
228C22×C8—21no—
229C22×C8—21no—
230C22×C8—41no—
231C22×C8—41no—
232C22×C8—41no—
233C22×C8#4111yescharacteristic
234C22×C8#4111yescharacteristic
235C4⋊Q8C22×C411yescharacteristic
236C4⋊Q8C22×C411yescharacteristic
237#127C22×C411yescharacteristic
238#127C22×C411yescharacteristic
239#124—21no—
240#124—21no—
241#121—21no—
242#121—21no—
243#121—21no—
244#121—21no—
245C4×Q8C22×C411yescharacteristic
246C4×Q8#4111yescharacteristic
247C4×D4—21no—
248C4×D4—21no—
249C4×D4—21no—
250C4×D4—21no—
251C4×D4—21no—
252C4×D4—21no—
253C4×D4C22×C411yescharacteristic
254C4×D4#4111yescharacteristic
255#117—21no—
256#117C2×D411yescharacteristic
257C2×C42—21no—
258C2×C42—21no—
259C2×C42—21no—
260C2×C42C2×D411yescharacteristic
261C2×C42C2×D411yescharacteristic
262C2×C42C2411yescharacteristic
263#108—42no—
264#108—81no—
265#107—21no—
266#107—21no—
267C4⋊C8—21no—
268C4⋊C8—21no—
269C4⋊C8—21no—
270C4⋊C8—21no—
271C4⋊C8—21no—
272C4⋊C8—21no—
273C4⋊C8—21no—
274C4⋊C8—21no—
275C4⋊C8—21no—
276C4⋊C8—21no—
277C4⋊C8—21no—
278C4⋊C8—41no—
279C4⋊C8—41no—
280C4⋊C8C22×C411yescharacteristic
281C4⋊C8C22×C411yescharacteristic
282C4⋊C8C22×C411yescharacteristic
283C4⋊C8C22×C411yescharacteristic
284C4⋊C8C22×C411yescharacteristic
285C4⋊C8C22×C411yescharacteristic
286C4⋊C8C2×D411yescharacteristic
287C4⋊C8C2×D411yescharacteristic
288C4⋊C8C2×D411yescharacteristic
289C4⋊C8C2×D411yescharacteristic
290#104—22no—
291#104—41no—
292#104—42no—
293#104—42no—
294#104—81no—
295#104—81no—
296#103—21no—
297#103—21no—
298#103—21no—
299#103—21no—
300#102—21no—
301#102—21no—
302#102—21no—
303#102—21no—
304#102—21no—
305#102—21no—
306#102—21no—
307#102—21no—
308C22⋊C8—21no—
309C22⋊C8—21no—
310C22⋊C8—21no—
311C22⋊C8—21no—
312C22⋊C8—41no—
313C22⋊C8—41no—
314C22⋊C8—41no—
315C22⋊C8—41no—
316C22⋊C8—41no—
317C22⋊C8—41no—
318#97—21no—
319#97—21no—
320#97—21no—
321#97—41no—
322C4×C8—24no—
323C4×C8—21no—
324C4×C8—21no—
325C4×C8—21no—
326C4×C8—21no—
327C4×C8—21no—
328C4×C8—41no—
329C4×C8—42no—
330C4×C8—42no—
331C4×C8—81no—
332C4×C8C22×C411yescharacteristic
333C4×C8C22×C411yescharacteristic
334C4×C8C2×D411yescharacteristic
335C4×C8C2×D411yescharacteristic
336#41—21no—
337#41—21no—
338#41—21no—
339#41—21no—
340#41—41no—
341#41—41no—
342#41—41no—
343#41—41no—
344#41—41no—
345#41—41no—
346#41—41no—
347#41—41no—
348C2×Q8—21no—
349C2×Q8—21no—
350C2×Q8C4×D411yescharacteristic
351C2×D4—21no—
352C2×D4—21no—
353C2×D4—21no—
354C2×D4—21no—
355C2×D4—21no—
356C2×D4—21no—
357C2×D4—21no—
358C2×D4—41no—
359C2×D4—41no—
360C2×D4C4×D411yescharacteristic
361C22×C4—21no—
362C22×C4—21no—
363C22×C4—21no—
364C22×C4—21no—
365C22×C4—21no—
366C22×C4—41no—
367C22×C4—41no—
368C22×C4C22×D411yescharacteristic
369C22×C4C2×#4111yescharacteristic
370C22×C42−1+411yescharacteristic
371Q16—42no—
372Q16—81no—
373SD16—42no—
374SD16—42no—
375SD16—81no—
376SD16—81no—
377D8—42no—
378D8—42no—
379D8—81no—
380D8—81no—
381M16—21no—
382M16—21no—
383M16—21no—
384M16—21no—
385M16—42no—
386M16—41no—
387M16—41no—
388M16—41no—
389M16—41no—
390M16—81no—
391M16—82no—
392M16—82no—
393M16—82no—
394C2×C8—22no—
395C2×C8—22no—
396C2×C8—21no—
397C2×C8—21no—
398C2×C8—21no—
399C2×C8—21no—
400C2×C8—21no—
401C2×C8—21no—
402C2×C8—21no—
403C2×C8—21no—
404C2×C8—21no—
405C2×C8—21no—
406C2×C8—21no—
407C2×C8—21no—
408C2×C8—41no—
409C2×C8—41no—
410C2×C8—41no—
411C2×C8—41no—
412C2×C8—41no—
413C2×C8—41no—
414C2×C8—41no—
415C2×C8—41no—
416C2×C8—41no—
417C2×C8—41no—
418C2×C8—41no—
419C2×C8—41no—
420C2×C8—41no—
421C2×C8—41no—
422C2×C8—82no—
423C2×C8—82no—
424C2×C8—82no—
425C2×C8C4×D411yescharacteristic
426C2×C8C4×D411yescharacteristic
427C2×C8C4×D411yescharacteristic
428C2×C8C4×D411yescharacteristic
429C4⋊C4—21no—
430C4⋊C4—21no—
431C4⋊C4—21no—
432C4⋊C4—21no—
433C4⋊C4—21no—
434C4⋊C4—21no—
435C4⋊C4C4×D411yescharacteristic
436C4⋊C4C4×D411yescharacteristic
437C22⋊C4—21no—
438C22⋊C4—21no—
439C22⋊C4—21no—
440C22⋊C4—41no—
441C22⋊C4—41no—
442C22⋊C4—41no—
443C22⋊C4—41no—
444C42—22no—
445C42—22no—
446C42—21no—
447C42—21no—
448C42—21no—
449C42—21no—
450C42—21no—
451C42—21no—
452C42—42no—
453C42AGL(1,ℤ/8)12yes—
454C42C23×C411yescharacteristic
455C42C22×D411yescharacteristic
456C42C2×#4111yescharacteristic
457C42H(5,𝔽2)11yescharacteristic
458C23—21no—
459C23—41no—
460C23—41no—
461C23#54711yescharacteristic
462Q8—21no—
463Q8—21no—
464Q8—41no—
465Q8—41no—
466Q8—41no—
467Q8—41no—
468D4—21no—
469D4—21no—
470D4—41no—
471D4—41no—
472D4—41no—
473D4—41no—
474D4—41no—
475D4—41no—
476D4—41no—
477D4—41no—
478D4—41no—
479D4—41no—
480D4—81no—
481D4—81no—
482D4—81no—
483D4—81no—
484D4—81no—
485D4—81no—
486C2×C4—22no—
487C2×C4—22no—
488C2×C4—22no—
489C2×C4—21no—
490C2×C4—21no—
491C2×C4—22no—
492C2×C4—21no—
493C2×C4—21no—
494C2×C4—21no—
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—81no—
505C2×C4—81no—
506C2×C4—81no—
507C2×C4—81no—
508C2×C4—81no—
509C2×C4C2×C4×D411yescharacteristic
510C2×C4C2×C4×D411yescharacteristic
511C2×C4#52011yescharacteristic
512C2×C4D4211yescharacteristic
513C2×C4#54811yescharacteristic
514C2×C4#55311yescharacteristic
515C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
516C2×C4#57511yescharacteristic
517C2×C4#57611yescharacteristic
518C2×C4#57711yescharacteristic
519C8—21no—
520C8—21no—
521C8—21no—
522C8—42no—
523C8—41no—
524C8—41no—
525C8—41no—
526C8—41no—
527C8—41no—
528C8—41no—
529C8—41no—
530C8—41no—
531C8—41no—
532C8—41no—
533C8—41no—
534C8—41no—
535C8—81no—
536C8—81no—
537C8—81no—
538C8#44312yescomplemented
539C22—21no—
540C22—21no—
541C22—41no—
542C22—41no—
543C22—41no—
544C22—81no—
545C22—81no—
546C22—81no—
547C22—81no—
548C22—81no—
549C22—81no—
550C22#236111yescharacteristic
551C22#334511yescharacteristic
552C22#334711yescharacteristic
553C4—24no—
554C4—21no—
555C4—21no—
556C4—21no—
557C4—42no—
558C4—41no—
559C4—41no—
560C4—41no—
561C4—81no—
562C4—81no—
563C4C4×AGL(1,ℤ/8)11yescharacteristic
564C4#294611yescharacteristic
565C4C2×#44311yescharacteristic
566C4#295611yescharacteristic
567C4#327411yescharacteristic
568C4#327511yescharacteristic
569C2—21no—
570C2—21no—
571C2—81no—
572C2—161no—
573C2—161no—
574C2#2301511yescharacteristic
575C2#2303911yescharacteristic
576C2#2304211yescharacteristic
subgroupG/H
centerC2×C4D42
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
socleC22#2361
prime radical#19157C2
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #23113C2semidirect
2. #23015C2semidirect
4. #18073C2semidirect
5. #17830C2semidirect
6. #9844C2semidirect
7. #9626C2semidirect
8. #9578C2semidirect
9. #9577C2semidirect
10. #8996C2semidirect
11. #8995C2semidirect
12. #8932C2semidirect
13. #8930C2semidirect
14. C4×#334C2semidirect
15. #8194C2semidirect
40. #1702C4semidirect
41. #1702C4semidirect
42. #1701C4semidirect
43. #1701C4semidirect
44. #1694C4semidirect
45. #1693C4semidirect
46. #1688C4semidirect
47. #1687C4semidirect
538. C8#443semidirect

Maximal subgroups

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

subgroupindex
#231132
#230152
#191572
#180732
#178302
#98442
#96262
#95782
#95772
#89962
#89952
#89322
#89302
C4×#3342
#81942