Finite Group Databasebeta

← #211892

Subgroups — #211892

Order 512.

627 conjugacy classes of subgroups.

The subgroup lattice has 565 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21627C211yesmaximal, characteristic, complemented
2#21517C211yesmaximal, characteristic
3#19156C211yesmaximal, characteristic
4#17957C211yesmaximal, characteristic, complemented
5#17335C211yesmaximal, characteristic, complemented
6#9722C211yesmaximal, characteristic, complemented
7#9625C211yesmaximal, characteristic, complemented
8#9592C211yesmaximal, characteristic, complemented
9#9591C211yesmaximal, characteristic, complemented
10#9210C211yesmaximal, characteristic
11#9209C211yesmaximal, characteristic, complemented
12#8668C211yesmaximal, characteristic, complemented
13C8×#107C211yesmaximal, characteristic, complemented
14#8252C211yesmaximal, characteristic, complemented
15#8251C211yesmaximal, characteristic, complemented
16#3167C2211yescharacteristic
17#3076C2211yescharacteristic
18#3035C2211yescharacteristic
19#2965C2211yescharacteristic
20#2965C2211yescharacteristic
21#2960C2211yescharacteristic
22C2×#443—21no—
23#2931C2211yescharacteristic
24C4×#131C2211yescharacteristic
25#1986C2211yescharacteristic
26#1980C2211yescharacteristic
27#1950C2211yescharacteristic
28#1950C2211yescharacteristic
29#1929—21no—
30#1922—21no—
31#1876—21no—
32#1866—21no—
33#1806C2211yescharacteristic
34#1806C2211yescharacteristic
35C4×#107C2211yescharacteristic
36#1772—21no—
37C8×C4⋊C4C2211yescharacteristic
38C8×C4⋊C4C2211yescharacteristic
39#1764—21no—
40#1764—21no—
41#1700C411yescharacteristic, complemented
42#1699C411yescharacteristic, complemented
43#1698C411yescharacteristic, complemented
44#1697C411yescharacteristic, complemented
45#1692C411yescharacteristic, complemented
46#1691C411yescharacteristic, complemented
47#1686C411yescharacteristic, complemented
48#1685C411yescharacteristic, complemented
49#1590—21no—
50#1590—21no—
51C8×SD16—21no—
52C8×SD16—21no—
53#1566C2211yescharacteristic
54C2×#335C2211yescharacteristic
55#1542C2211yescharacteristic
56#1540C2211yescharacteristic
57#1503C2211yescharacteristic
58#1502C2211yescharacteristic
59#1493C2211yescharacteristic
60#1492C2211yescharacteristic
61#1455—21no—
62#1453—21no—
63#1451C2211yescharacteristic
64#1451C2211yescharacteristic
65#1450C2211yescharacteristic
66C8×M16—21no—
67C8×M16—21no—
68C2×C82C2211yescharacteristic
69#1303C2211yescharacteristic
70#1303C2211yescharacteristic
71#1301C2211yescharacteristic
72#1301C2211yescharacteristic
73#1279C2211yescharacteristic
74#1279C2211yescharacteristic
75C2×#135—21no—
76C2×#131—21no—
77C2×#131D411yescharacteristic
78C2×#131D411yescharacteristic
79#533C2311yescharacteristic
80#532C2311yescharacteristic
81C4×#41C2311yescharacteristic
82#498C2×C411yescharacteristic
83#492C2×C411yescharacteristic
84#487C2×C411yescharacteristic
85#486C2×C411yescharacteristic
86#479—21no—
87#478—21no—
88#477C2×C411yescharacteristic
89#476C2×C411yescharacteristic
90#475C2×C411yescharacteristic
91#474C2×C411yescharacteristic
92#465—21no—
93#465—21no—
94#464—21no—
95#463—21no—
96#462C2×C411yescharacteristic
97#461C2×C411yescharacteristic
98#460C2×C411yescharacteristic
99#459C2×C411yescharacteristic
100#446—21no—
101#446—21no—
102#446—21no—
103#446C2×C411yescharacteristic
104#446C2×C411yescharacteristic
105C8×Q8—21no—
106C8×Q8—21no—
107#443—44no—
108C4×SD16—21no—
109C4×SD16—21no—
110#436—21no—
111#436—21no—
112#436—21no—
113#436C2×C411yescharacteristic
114#436C2×C411yescharacteristic
115#435—21no—
116#435—21no—
117#435—22no—
118C8×D4—21no—
119C8×D4—21no—
120#431—21no—
121#431—21no—
122#431C2311yescharacteristic
123#431C2311yescharacteristic
124C2×#108—21no—
125C2×#108—21no—
126#427—21no—
127C2×#107—21no—
128#424—21no—
129#423C2311yescharacteristic
130#423C2311yescharacteristic
131C2×C4⋊C8C2311yescharacteristic
132C2×C4⋊C8C2311yescharacteristic
133C2×#104—21no—
134C2×#104—21no—
135C2×#104—21no—
136C2×#104—21no—
137#416—21no—
138#416—21no—
139#408—21no—
140#405—21no—
141#405—21no—
142#405—21no—
143#405D411yescharacteristic
144#405D411yescharacteristic
145C4×M16—21no—
146C4×M16—21no—
147C4×M16—21no—
148C4×M16C2311yescharacteristic
149C2×C4×C8—21no—
150C2×C4×C8—21no—
151C2×C4×C8—21no—
152C2×C4×C8C2311yescharacteristic
153C2×C4×C8C2311yescharacteristic
154C2×C4×C8C2311yescharacteristic
155C4×C4⋊C4C2311yescharacteristic
156C4×C4⋊C4C2311yescharacteristic
157#341—21no—
158#341—21no—
159#340—21no—
160#339—21no—
161#339—21no—
162#339—21no—
163#339—21no—
164#336—21no—
165#336—21no—
166#335C2×C411yescharacteristic
167#335C2×C411yescharacteristic
168#335C2×C411yescharacteristic
169#335C2×C411yescharacteristic
170#334—21no—
171#334—21no—
172#333C2×C411yescharacteristic
173#333C2×C411yescharacteristic
174#332C2×C411yescharacteristic
175#332C2×C411yescharacteristic
176#332C2×C411yescharacteristic
177#332C2×C411yescharacteristic
178#331C2×C411yescharacteristic
179#331C2×C411yescharacteristic
180#328—22no—
181#327—22no—
182Q8⋊C8—21no—
183Q8⋊C8—21no—
184D4⋊C8—21no—
185D4⋊C8—21no—
186#322—21no—
187#322—21no—
188#322—21no—
189#322—21no—
190#322—21no—
191C82—21no—
192C82D411yescharacteristic
193C82D411yescharacteristic
194C82D411yescharacteristic
195C82D411yescharacteristic
196C2×#41—21no—
197C2×#41C2×D411yescharacteristic
198#135—42no—
199#135—42no—
200C2×Q16—21no—
201C2×SD16—21no—
202C2×SD16—21no—
203C2×SD16—21no—
204C2×SD16—21no—
205C2×D8—21no—
206#131—22no—
207#131—22no—
208#131—41no—
209#131—41no—
210#131—42no—
211#131—42no—
212C2×M16—22no—
213C2×M16—21no—
214C2×M16—21no—
215C2×M16—21no—
216C2×M16—21no—
217C2×M16—21no—
218C2×M16—21no—
219C2×M16—41no—
220C2×M16C2×D411yescharacteristic
221C2×M16C2×D411yescharacteristic
222C22×C8—21no—
223C22×C8—21no—
224C22×C8—21no—
225C22×C8—21no—
226C22×C8—21no—
227C22×C8—41no—
228C22×C8C2×D411yescharacteristic
229C22×C8C2×D411yescharacteristic
230C22×C8#4111yescharacteristic
231C22×C8#4111yescharacteristic
232C4⋊Q8C22×C411yescharacteristic
233C4⋊Q8C22×C411yescharacteristic
234C4⋊Q8C22×C411yescharacteristic
235#127C22×C411yescharacteristic
236#125—21no—
237#124—21no—
238C22⋊Q8—21no—
239C22⋊Q8—21no—
240#121—21no—
241#121—21no—
242C4×Q8—21no—
243C4×Q8—21no—
244C4×Q8C22×C411yescharacteristic
245C4×Q8#4111yescharacteristic
246C4×D4—21no—
247C4×D4—21no—
248C4×D4—21no—
249C4×D4—21no—
250C4×D4C22×C411yescharacteristic
251C4×D4#4111yescharacteristic
252#117—21no—
253#117—21no—
254#117C2×D411yescharacteristic
255C2×C4⋊C4—21no—
256C2×C4⋊C4—21no—
257C2×C42—21no—
258C2×C42—21no—
259C2×C42—21no—
260C2×C42C2411yescharacteristic
261#108—42no—
262#108—42no—
263#107—24no—
264#107—21no—
265#107—21no—
266#106—21no—
267#106—21no—
268#106—21no—
269#106—21no—
270C4⋊C8—22no—
271C4⋊C8—21no—
272C4⋊C8—21no—
273C4⋊C8—21no—
274C4⋊C8—21no—
275C4⋊C8—21no—
276C4⋊C8—21no—
277C4⋊C8—21no—
278C4⋊C8—21no—
279C4⋊C8—21no—
280C4⋊C8—41no—
281C4⋊C8—41no—
282C4⋊C8C22×C411yescharacteristic
283C4⋊C8C22×C411yescharacteristic
284C4⋊C8C22×C411yescharacteristic
285C4⋊C8C22×C411yescharacteristic
286C4⋊C8C22×C411yescharacteristic
287C4⋊C8C22×C411yescharacteristic
288#104—42no—
289#104—42no—
290#104—42no—
291#104—42no—
292#103—21no—
293#103—21no—
294#103—21no—
295#103—21no—
296#103—21no—
297#103—21no—
298#102—21no—
299#102—21no—
300#102—21no—
301#102—21no—
302#102—21no—
303#102—21no—
304C22⋊C8—22no—
305C22⋊C8—21no—
306C22⋊C8—21no—
307C22⋊C8—21no—
308C22⋊C8—41no—
309C22⋊C8—41no—
310C22⋊C8#4112yes—
311#97—22no—
312#97—21no—
313#97—21no—
314#97—21no—
315#97—21no—
316#97—21no—
317#97—41no—
318C4×C8—21no—
319C4×C8—21no—
320C4×C8—21no—
321C4×C8—21no—
322C4×C8—21no—
323C4×C8—21no—
324C4×C8—41no—
325C4×C8—42no—
326C4×C8—42no—
327C4×C8C22×C411yescharacteristic
328C4×C8C22×C411yescharacteristic
329C4×C8C2×D411yescharacteristic
330C4×C8C2×D411yescharacteristic
331C4×C8C2×D411yescharacteristic
332C4×C8C2×D411yescharacteristic
333C4×C8C2×D411yescharacteristic
334C4×C8C2×D411yescharacteristic
335C4×C8#4111yescharacteristic
336C4×C8#4111yescharacteristic
337#95—21no—
338#95—21no—
339#41—21no—
340#41—21no—
341#41—21no—
342#41—21no—
343#41—41no—
344#41—41no—
345#41—41no—
346#41—41no—
347C2×Q8—21no—
348C2×Q8—21no—
349C2×Q8—21no—
350C2×Q8C4×D411yescharacteristic
351C2×D4—21no—
352C2×D4—21no—
353C2×D4—21no—
354C2×D4—21no—
355C2×D4—41no—
356C2×D4C4×D411yescharacteristic
357C22×C4—21no—
358C22×C4—21no—
359C22×C4—21no—
360C22×C4—41no—
361C22×C4—41no—
362C22×C4—41no—
363C22×C4C22×D411yescharacteristic
364C22×C4C2×#4111yescharacteristic
365C22×C4C2×#4111yescharacteristic
366Q16—42no—
367SD16—42no—
368SD16—42no—
369SD16—42no—
370SD16—42no—
371D8—42no—
372M16—21no—
373M16—21no—
374M16—21no—
375M16—21no—
376M16—44no—
377M16—42no—
378M16—42no—
379M16—42no—
380M16—42no—
381M16—41no—
382M16—41no—
383M16—41no—
384M16—41no—
385M16—82no—
386C2×C8—21no—
387C2×C8—21no—
388C2×C8—21no—
389C2×C8—22no—
390C2×C8—21no—
391C2×C8—21no—
392C2×C8—21no—
393C2×C8—21no—
394C2×C8—21no—
395C2×C8—21no—
396C2×C8—21no—
397C2×C8—21no—
398C2×C8—21no—
399C2×C8—22no—
400C2×C8—41no—
401C2×C8—42no—
402C2×C8—42no—
403C2×C8—41no—
404C2×C8—41no—
405C2×C8—41no—
406C2×C8—41no—
407C2×C8—42no—
408C2×C8—41no—
409C2×C8—41no—
410C2×C8—82no—
411C2×C8C4×D411yescharacteristic
412C2×C8C4×D411yescharacteristic
413C2×C8C4×D411yescharacteristic
414C2×C8C4×D411yescharacteristic
415C2×C8#12111yescharacteristic
416C2×C8#12111yescharacteristic
417C2×C8#12111yescharacteristic
418C2×C8#12111yescharacteristic
419C2×C8AGL(1,ℤ/8)12yes—
420C2×C8#13712yes—
421C4⋊C4—21no—
422C4⋊C4—21no—
423C4⋊C4—22no—
424C4⋊C4—22no—
425C4⋊C4—21no—
426C4⋊C4—21no—
427C4⋊C4—21no—
428C4⋊C4—21no—
429C4⋊C4—21no—
430C4⋊C4—21no—
431C4⋊C4—41no—
432C4⋊C4—41no—
433C4⋊C4—41no—
434C4⋊C4—41no—
435C4⋊C4C4×D411yescharacteristic
436C4⋊C4C4×D411yescharacteristic
437C22⋊C4—21no—
438C22⋊C4—21no—
439C22⋊C4—21no—
440C22⋊C4—41no—
441C22⋊C4—41no—
442C22⋊C4—41no—
443C42—21no—
444C42—21no—
445C42—21no—
446C42—41no—
447C42—41no—
448C42—41no—
449C42—41no—
450C42—41no—
451C42C23×C411yescharacteristic
452C42C22×D411yescharacteristic
453C42C2×#4111yescharacteristic
454C42C2×#4111yescharacteristic
455C23—21no—
456C23—41no—
457C23#53211yescharacteristic
458Q8—21no—
459Q8—21no—
460Q8—41no—
461Q8—41no—
462Q8—41no—
463Q8—41no—
464Q8—81no—
465D4—21no—
466D4—21no—
467D4—41no—
468D4—41no—
469D4—41no—
470D4—41no—
471D4—41no—
472D4—41no—
473D4—81no—
474D4—81no—
475D4—81no—
476C2×C4—21no—
477C2×C4—21no—
478C2×C4—21no—
479C2×C4—21no—
480C2×C4—21no—
481C2×C4—41no—
482C2×C4—42no—
483C2×C4—41no—
484C2×C4—41no—
485C2×C4—41no—
486C2×C4—41no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—41no—
490C2×C4—41no—
491C2×C4—41no—
492C2×C4—81no—
493C2×C4—81no—
494C2×C4—81no—
495C2×C4—81no—
496C2×C4—81no—
497C2×C4—81no—
498C2×C4C2×C4×D411yescharacteristic
499C2×C4C2×C4×D411yescharacteristic
500C2×C4C4×#4111yescharacteristic
501C2×C4C2×#12111yescharacteristic
502C2×C4#52511yescharacteristic
503C2×C4#52911yescharacteristic
504C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
505C2×C4C2×#13711yescharacteristic
506C2×C4#57511yescharacteristic
507C2×C4#57511yescharacteristic
508C8—22no—
509C8—21no—
510C8—21no—
511C8—21no—
512C8—21no—
513C8—21no—
514C8—41no—
515C8—42no—
516C8—41no—
517C8—41no—
518C8—41no—
519C8—41no—
520C8—41no—
521C8—41no—
522C8—41no—
523C8—41no—
524C8—81no—
525C8—81no—
526C8#44312yescomplemented
527C8#46812yes—
528C8#46812yes—
529C22—21no—
530C22—21no—
531C22—41no—
532C22—41no—
533C22—41no—
534C22—81no—
535C22—81no—
536C22—81no—
537C22C4×#12111yescharacteristic
538C22#317111yescharacteristic
539C22#317211yescharacteristic
540C4—21no—
541C4—21no—
542C4—21no—
543C4—41no—
544C4—41no—
545C4—41no—
546C4—82no—
547C4—82no—
548C4—81no—
549C4—81no—
550C4—81no—
551C4C4×AGL(1,ℤ/8)11yescharacteristic
552C4C4×#13711yescharacteristic
553C4C2×#44311yescharacteristic
554C4#295511yescharacteristic
555C4C2×#46811yescharacteristic
556C4#305411yescharacteristic
557C2—21no—
558C2—21no—
559C2—81no—
560C2—161no—
561C2#2151711yescharacteristic
562C2C4×#46811yescharacteristic
563C2#2155811yescharacteristic
subgroupG/H
centerC2×C8#121
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
socleC22C4×#121
prime radical#3035C22
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C2×C8 ◁ C2×C4×C8 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #21627C2semidirect
4. #17957C2semidirect
5. #17335C2semidirect
6. #9722C2semidirect
7. #9625C2semidirect
8. #9592C2semidirect
9. #9591C2semidirect
11. #9209C2semidirect
12. #8668C2semidirect
13. C8×#107C2semidirect
14. #8252C2semidirect
15. #8251C2semidirect
41. #1700C4semidirect
42. #1699C4semidirect
43. #1698C4semidirect
44. #1697C4semidirect
45. #1692C4semidirect
46. #1691C4semidirect
47. #1686C4semidirect
48. #1685C4semidirect
526. C8#443semidirect

Maximal subgroups

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

subgroupindex
#216272
#215172
#191562
#179572
#173352
#97222
#96252
#95922
#95912
#92102
#92092
#86682
C8×#1072
#82522
#82512