Finite Group Databasebeta

← #230429

Subgroups — #230429

Order 512.

606 conjugacy classes of subgroups.

The subgroup lattice has 576 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23113C211yesmaximal, characteristic, complemented
2#23016C211yesmaximal, characteristic, complemented
3#19157C211yesmaximal, characteristic
4#18066C211yesmaximal, characteristic, complemented
5#17832C211yesmaximal, characteristic, complemented
6#9852C211yesmaximal, characteristic, complemented
7#9623C211yesmaximal, characteristic, complemented
8#9582C211yesmaximal, characteristic, complemented
9#9577C211yesmaximal, characteristic, complemented
10#8996C211yesmaximal, characteristic, complemented
11#8995C211yesmaximal, characteristic, complemented
12#8932C211yesmaximal, characteristic, complemented
13#8930C211yesmaximal, characteristic, complemented
14#8279C211yesmaximal, characteristic, complemented
15#8197C211yesmaximal, characteristic, complemented
16#3169C2211yescharacteristic
17#3035C2211yescharacteristic
18#3035C2211yescharacteristic
19#2965C2211yescharacteristic
20C2×#444—21no—
21#2935C2211yescharacteristic
22#2935C2211yescharacteristic
23#2925C2211yescharacteristic
24#2906C2211yescharacteristic
25#1974—21no—
26#1962C2211yescharacteristic
27#1962C2211yescharacteristic
28#1950C2211yescharacteristic
29#1950C2211yescharacteristic
30#1938—21no—
31#1929—21no—
32#1876—21no—
33#1876—21no—
34#1866—21no—
35#1767C2211yescharacteristic
36C4×C4⋊C8C2211yescharacteristic
37#1764—21no—
38C4×#104C2211yescharacteristic
39#1744C2211yescharacteristic
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#1589—21no—
49#1581—21no—
50#1581—21no—
51C8×D8—21no—
52#1574—21no—
53#1573—21no—
54#1571C2211yescharacteristic
55#1570C2211yescharacteristic
56#1569—21no—
57#1566C2211yescharacteristic
58#1565—21no—
59C2×#335C2211yescharacteristic
60C2×#334C2211yescharacteristic
61#1540C2211yescharacteristic
62#1540C2211yescharacteristic
63#1492C2211yescharacteristic
64#1492C2211yescharacteristic
65#1480C2211yescharacteristic
66#1480C2211yescharacteristic
67#1476C2211yescharacteristic
68C8×M16C2211yescharacteristic
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#444—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×#97—21no—
149C2×C4×C8—21no—
150C2×C4×C8C2311yescharacteristic
151C2×C4×C8C2311yescharacteristic
152C43C2311yescharacteristic
153#341—21no—
154#341—21no—
155#339D411yescharacteristic
156#339D411yescharacteristic
157#336—21no—
158#336—21no—
159#336—21no—
160#336—21no—
161#335—21no—
162#335—21no—
163#335D411yescharacteristic
164#335D411yescharacteristic
165#335D411yescharacteristic
166#335D411yescharacteristic
167#334—21no—
168#334—21no—
169#334—21no—
170#334—21no—
171#334C2×C411yescharacteristic
172#334C2×C411yescharacteristic
173#334C2×C411yescharacteristic
174#334C2×C411yescharacteristic
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—
192C82—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—
322#97—42no—
323C4×C8—21no—
324C4×C8—21no—
325C4×C8—21no—
326C4×C8—21no—
327C4×C8—21no—
328C4×C8—41no—
329C4×C8—81no—
330C4×C8C22×C411yescharacteristic
331C4×C8C22×C411yescharacteristic
332C4×C8C2×D411yescharacteristic
333C4×C8C2×D411yescharacteristic
334#41—21no—
335#41—21no—
336#41—21no—
337#41—21no—
338#41—41no—
339#41—41no—
340#41—41no—
341#41—41no—
342#41—41no—
343#41—41no—
344#41—41no—
345#41—41no—
346C2×Q8—21no—
347C2×Q8—21no—
348C2×Q8C4×D411yescharacteristic
349C2×D4—21no—
350C2×D4—21no—
351C2×D4—21no—
352C2×D4—21no—
353C2×D4—21no—
354C2×D4—21no—
355C2×D4—21no—
356C2×D4—41no—
357C2×D4—41no—
358C2×D4C4×D411yescharacteristic
359C22×C4—21no—
360C22×C4—21no—
361C22×C4—21no—
362C22×C4—21no—
363C22×C4—21no—
364C22×C4—41no—
365C22×C4—41no—
366C22×C4C22×D411yescharacteristic
367C22×C4C2×#4111yescharacteristic
368C22×C42−1+411yescharacteristic
369Q16—42no—
370Q16—81no—
371SD16—42no—
372SD16—42no—
373SD16—81no—
374SD16—81no—
375D8—42no—
376D8—42no—
377D8—81no—
378D8—81no—
379M16—21no—
380M16—21no—
381M16—21no—
382M16—42no—
383M16—41no—
384M16—41no—
385M16—41no—
386M16—41no—
387M16—81no—
388M16—82no—
389M16—82no—
390M16—82no—
391M16AGL(1,ℤ/8)12yes—
392C2×C8—21no—
393C2×C8—21no—
394C2×C8—21no—
395C2×C8—21no—
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—41no—
405C2×C8—41no—
406C2×C8—41no—
407C2×C8—41no—
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—82no—
421C2×C8—82no—
422C2×C8—82no—
423C2×C8C4×D411yescharacteristic
424C2×C8C4×D411yescharacteristic
425C2×C8C4×D411yescharacteristic
426C2×C8C4×D411yescharacteristic
427C4⋊C4—21no—
428C4⋊C4—21no—
429C4⋊C4—21no—
430C4⋊C4—21no—
431C4⋊C4—21no—
432C4⋊C4—21no—
433C4⋊C4C4×D411yescharacteristic
434C4⋊C4C4×D411yescharacteristic
435C22⋊C4—21no—
436C22⋊C4—21no—
437C22⋊C4—21no—
438C22⋊C4—41no—
439C22⋊C4—41no—
440C22⋊C4—41no—
441C22⋊C4—41no—
442C42—21no—
443C42—21no—
444C42—21no—
445C42—21no—
446C42—21no—
447C42—21no—
448C42—21no—
449C42—41no—
450C42—41no—
451C42—42no—
452C42C23×C411yescharacteristic
453C42C22×D411yescharacteristic
454C42C2×#4111yescharacteristic
455C42H(5,𝔽2)11yescharacteristic
456C23—21no—
457C23—41no—
458C23—41no—
459C23#54711yescharacteristic
460Q8—21no—
461Q8—21no—
462Q8—41no—
463Q8—41no—
464Q8—41no—
465Q8—41no—
466D4—21no—
467D4—21no—
468D4—41no—
469D4—41no—
470D4—41no—
471D4—41no—
472D4—41no—
473D4—41no—
474D4—41no—
475D4—41no—
476D4—41no—
477D4—41no—
478D4—81no—
479D4—81no—
480D4—81no—
481D4—81no—
482D4—81no—
483D4—81no—
484C2×C4—21no—
485C2×C4—21no—
486C2×C4—22no—
487C2×C4—21no—
488C2×C4—21no—
489C2×C4—21no—
490C2×C4—41no—
491C2×C4—41no—
492C2×C4—41no—
493C2×C4—41no—
494C2×C4—41no—
495C2×C4—41no—
496C2×C4—41no—
497C2×C4—41no—
498C2×C4—41no—
499C2×C4—41no—
500C2×C4—41no—
501C2×C4—41no—
502C2×C4—81no—
503C2×C4—81no—
504C2×C4—81no—
505C2×C4—81no—
506C2×C4—81no—
507C2×C4C2×C4×D411yescharacteristic
508C2×C4C2×C4×D411yescharacteristic
509C2×C4#52011yescharacteristic
510C2×C4D4211yescharacteristic
511C2×C4#54811yescharacteristic
512C2×C4#55311yescharacteristic
513C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
514C2×C4#57511yescharacteristic
515C2×C4#57611yescharacteristic
516C2×C4#57711yescharacteristic
517C8—21no—
518C8—21no—
519C8—21no—
520C8—21no—
521C8—22no—
522C8—41no—
523C8—41no—
524C8—41no—
525C8—41no—
526C8—41no—
527C8—41no—
528C8—41no—
529C8—41no—
530C8—41no—
531C8—41no—
532C8—41no—
533C8—81no—
534C8—81no—
535C8—81no—
536C8—81no—
537C22—21no—
538C22—21no—
539C22—41no—
540C22—41no—
541C22—41no—
542C22—81no—
543C22—81no—
544C22—81no—
545C22—81no—
546C22—81no—
547C22—81no—
548C22#236111yescharacteristic
549C22#335411yescharacteristic
550C22#335511yescharacteristic
551C4—21no—
552C4—21no—
553C4—21no—
554C4—42no—
555C4—41no—
556C4—41no—
557C4—41no—
558C4—81no—
559C4—81no—
560C4—81no—
561C4C4×AGL(1,ℤ/8)11yescharacteristic
562C4#294611yescharacteristic
563C4#295511yescharacteristic
564C4#295711yescharacteristic
565C4#326411yescharacteristic
566C4#327111yescharacteristic
567C2—21no—
568C2—21no—
569C2—81no—
570C2—161no—
571C2—161no—
572C2#2301711yescharacteristic
573C2#2303711yescharacteristic
574C2#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. #23016C2semidirect
4. #18066C2semidirect
5. #17832C2semidirect
6. #9852C2semidirect
7. #9623C2semidirect
8. #9582C2semidirect
9. #9577C2semidirect
10. #8996C2semidirect
11. #8995C2semidirect
12. #8932C2semidirect
13. #8930C2semidirect
14. #8279C2semidirect
15. #8197C2semidirect
40. #1702C4semidirect
41. #1702C4semidirect
42. #1701C4semidirect
43. #1701C4semidirect
44. #1694C4semidirect
45. #1693C4semidirect
46. #1688C4semidirect
47. #1687C4semidirect

Maximal subgroups

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

subgroupindex
#231132
#230162
#191572
#180662
#178322
#98522
#96232
#95822
#95772
#89962
#89952
#89322
#89302
#82792
#81972