Finite Group Databasebeta

← #211887

Subgroups — #211887

Order 512.

623 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#21518C211yesmaximal, characteristic
3#19124C211yesmaximal, characteristic
4#17979C211yesmaximal, characteristic, complemented
5#17320C211yesmaximal, characteristic, complemented
6#9722C211yesmaximal, characteristic, complemented
7#9626C211yesmaximal, characteristic, complemented
8#9592C211yesmaximal, characteristic, complemented
9#9591C211yesmaximal, characteristic, complemented
10#9208C211yesmaximal, characteristic
11#9207C211yesmaximal, characteristic, complemented
12#8667C211yesmaximal, characteristic, complemented
13#8288C211yesmaximal, characteristic, complemented
14#8257C211yesmaximal, characteristic, complemented
15#8254C211yesmaximal, characteristic, complemented
16#3167C2211yescharacteristic
17#3076C2211yescharacteristic
18#3035C2211yescharacteristic
19C8×#41C2211yescharacteristic
20C8×#41C2211yescharacteristic
21#2960C2211yescharacteristic
22C2×#444—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—
37#1770C2211yescharacteristic
38C8×C4⋊C4C2211yescharacteristic
39#1765—21no—
40#1764—21no—
41#1680C411yescharacteristic, complemented
42#1679C411yescharacteristic, complemented
43#1678C411yescharacteristic, complemented
44#1677C411yescharacteristic, complemented
45#1672C411yescharacteristic, complemented
46#1671C411yescharacteristic, complemented
47#1666C411yescharacteristic, complemented
48#1665C411yescharacteristic, complemented
49#1588—21no—
50#1588—21no—
51#1585—21no—
52#1585—21no—
53#1567C2211yescharacteristic
54C2×#334C2211yescharacteristic
55#1542C2211yescharacteristic
56#1540C2211yescharacteristic
57#1503C2211yescharacteristic
58#1502C2211yescharacteristic
59#1493C2211yescharacteristic
60#1492C2211yescharacteristic
61#1452—21no—
62#1452C2211yescharacteristic
63#1451—21no—
64#1451—21no—
65#1451C2211yescharacteristic
66#1451C2211yescharacteristic
67C2×#322—21no—
68C2×#322C2211yescharacteristic
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—
102C8×Q8—21no—
103C8×Q8—21no—
104C8×Q8—21no—
105C8×Q8C2×C411yescharacteristic
106C8×Q8C2×C411yescharacteristic
107#444—44no—
108C4×SD16—21no—
109C4×SD16—21no—
110#436—21no—
111#436—21no—
112C8×D4—21no—
113C8×D4—21no—
114C8×D4—21no—
115C8×D4—22no—
116C8×D4—21no—
117C8×D4—21no—
118C8×D4C2×C411yescharacteristic
119C8×D4C2×C411yescharacteristic
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×M16C2311yescharacteristic
147C2×#97—21no—
148C2×C4×C8—21no—
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#335—21no—
167#335—21no—
168#334C2×C411yescharacteristic
169#334C2×C411yescharacteristic
170#334C2×C411yescharacteristic
171#334C2×C411yescharacteristic
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—
191#322—21no—
192#322D411yescharacteristic
193#322D411yescharacteristic
194#322D411yescharacteristic
195#322D411yescharacteristic
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—21no—
213C2×M16—21no—
214C2×M16—21no—
215C2×M16—21no—
216C2×M16—41no—
217C2×M16C2×D411yescharacteristic
218C2×M16C2×D411yescharacteristic
219C22×C8—22no—
220C22×C8—21no—
221C22×C8—21no—
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—21no—
264#107—21no—
265#107—42no—
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—21no—
312#97—21no—
313#97—21no—
314#97—42no—
315#97—41no—
316C4×C8—22no—
317C4×C8—21no—
318C4×C8—21no—
319C4×C8—21no—
320C4×C8—21no—
321C4×C8—21no—
322C4×C8—21no—
323C4×C8—21no—
324C4×C8—21no—
325C4×C8—42no—
326C4×C8—41no—
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—41no—
342#41—41no—
343#41—41no—
344#41—41no—
345#41AGL(1,ℤ/8)12yes—
346#41#13712yes—
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—81no—
370SD16—81no—
371D8—42no—
372M16—21no—
373M16—21no—
374M16—21no—
375M16—21no—
376M16—42no—
377M16—42no—
378M16—41no—
379M16—41no—
380M16—41no—
381M16—41no—
382M16—82no—
383C2×C8—21no—
384C2×C8—21no—
385C2×C8—21no—
386C2×C8—21no—
387C2×C8—21no—
388C2×C8—21no—
389C2×C8—21no—
390C2×C8—21no—
391C2×C8—21no—
392C2×C8—21no—
393C2×C8—21no—
394C2×C8—21no—
395C2×C8—21no—
396C2×C8—22no—
397C2×C8—21no—
398C2×C8—41no—
399C2×C8—41no—
400C2×C8—42no—
401C2×C8—42no—
402C2×C8—42no—
403C2×C8—44no—
404C2×C8—41no—
405C2×C8—41no—
406C2×C8—41no—
407C2×C8—41no—
408C2×C8—41no—
409C2×C8—41no—
410C2×C8—81no—
411C2×C8—81no—
412C2×C8—82no—
413C2×C8C4×D411yescharacteristic
414C2×C8C4×D411yescharacteristic
415C2×C8C4×D411yescharacteristic
416C2×C8C4×D411yescharacteristic
417C2×C8#12111yescharacteristic
418C2×C8#12111yescharacteristic
419C2×C8#12111yescharacteristic
420C2×C8#12111yescharacteristic
421C4⋊C4—21no—
422C4⋊C4—21no—
423C4⋊C4—22no—
424C4⋊C4—21no—
425C4⋊C4—21no—
426C4⋊C4—21no—
427C4⋊C4—21no—
428C4⋊C4—21no—
429C4⋊C4—21no—
430C4⋊C4—41no—
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—41no—
460Q8—41no—
461Q8—41no—
462Q8—41no—
463Q8—81no—
464Q8#44012yes—
465D4—22no—
466D4—22no—
467D4—21no—
468D4—41no—
469D4—41no—
470D4—41no—
471D4—41no—
472D4—81no—
473D4—81no—
474D4—81no—
475D4#44012yes—
476C2×C4—21no—
477C2×C4—21no—
478C2×C4—21no—
479C2×C4—22no—
480C2×C4—22no—
481C2×C4—21no—
482C2×C4—21no—
483C2×C4—41no—
484C2×C4—42no—
485C2×C4—41no—
486C2×C4—41no—
487C2×C4—42no—
488C2×C4—41no—
489C2×C4—41no—
490C2×C4—41no—
491C2×C4—81no—
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—21no—
515C8—21no—
516C8—21no—
517C8—42no—
518C8—41no—
519C8—41no—
520C8—41no—
521C8—41no—
522C8—41no—
523C8—41no—
524C8—41no—
525C8—41no—
526C8—41no—
527C8—81no—
528C8—81no—
529C22—21no—
530C22—21no—
531C22—22no—
532C22—41no—
533C22—41no—
534C22—81no—
535C22—81no—
536C22—81no—
537C22C4×#12111yescharacteristic
538C22#318011yescharacteristic
539C22#318111yescharacteristic
540C4—21no—
541C4—22no—
542C4—21no—
543C4—21no—
544C4—41no—
545C4—41no—
546C4—82no—
547C4—81no—
548C4—81no—
549C4—81no—
550C4—161no—
551C4C2×#44011yescharacteristic
552C4#294311yescharacteristic
553C4#295611yescharacteristic
554C4#295711yescharacteristic
555C4#305511yescharacteristic
556C4#305611yescharacteristic
557C2—21no—
558C2—21no—
559C2—42no—
560C2—161no—
561C2#2151911yescharacteristic
562C2#2152811yescharacteristic
563C2#2155811yescharacteristic
subgroupG/H
centerC2×C4C2×#121
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
socleC22C4×#121
prime radical#3035C22
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ 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. #17979C2semidirect
5. #17320C2semidirect
6. #9722C2semidirect
7. #9626C2semidirect
8. #9592C2semidirect
9. #9591C2semidirect
11. #9207C2semidirect
12. #8667C2semidirect
13. #8288C2semidirect
14. #8257C2semidirect
15. #8254C2semidirect
41. #1680C4semidirect
42. #1679C4semidirect
43. #1678C4semidirect
44. #1677C4semidirect
45. #1672C4semidirect
46. #1671C4semidirect
47. #1666C4semidirect
48. #1665C4semidirect

Maximal subgroups

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

subgroupindex
#216272
#215182
#191242
#179792
#173202
#97222
#96262
#95922
#95912
#92082
#92072
#86672
#82882
#82572
#82542