Finite Group Databasebeta

← #230090

Subgroups — #230090

Order 512.

610 conjugacy classes of subgroups.

The subgroup lattice has 567 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23234C211yesmaximal, characteristic, complemented
2D4×#108C211yesmaximal, characteristic, complemented
3#19110C211yesmaximal, characteristic
4#18000C211yesmaximal, characteristic, complemented
5#17994C211yesmaximal, characteristic
6#9858C211yesmaximal, characteristic, complemented
7#9627C211yesmaximal, characteristic, complemented
8#9592C211yesmaximal, characteristic, complemented
9#9589C211yesmaximal, characteristic, complemented
10#8991C211yesmaximal, characteristic
11#8989C211yesmaximal, characteristic, complemented
12#8933C211yesmaximal, characteristic, complemented
13#8930C211yesmaximal, characteristic, complemented
14#8296C211yesmaximal, characteristic, complemented
15#8200C211yesmaximal, characteristic, complemented
16#3131C2211yescharacteristic
17#3101C2211yescharacteristic
18#3100C2211yescharacteristic
19#2962C2211yescharacteristic
20#2935C2211yescharacteristic
21D4×M16C2211yescharacteristic
22D4×M16C2211yescharacteristic
23C2×C8×D4C2211yescharacteristic
24C22×#108—21no—
25#1986C2211yescharacteristic
26#1980C2211yescharacteristic
27#1950C2211yescharacteristic
28#1950C2211yescharacteristic
29#1937—21no—
30#1929—21no—
31#1866—21no—
32#1855—21no—
33#1855—21no—
34#1848C2211yescharacteristic
35#1810—21no—
36C4×#108C2211yescharacteristic
37C8×C4⋊C4C2211yescharacteristic
38#1764—21no—
39C4×#102C2211yescharacteristic
40#1683C411yescharacteristic, complemented
41#1683C411yescharacteristic, complemented
42#1681C411yescharacteristic, complemented
43#1681C411yescharacteristic, complemented
44#1679C411yescharacteristic, complemented
45#1679C411yescharacteristic, complemented
46#1677C411yescharacteristic, complemented
47#1677C411yescharacteristic, complemented
48#1592—21no—
49#1589—21no—
50#1588—21no—
51#1585—21no—
52#1581—21no—
53#1581—21no—
54SD16⋊C8—21no—
55C8×D8—21no—
56#1571C2211yescharacteristic
57#1571C2211yescharacteristic
58#1564C2211yescharacteristic
59#1542C2211yescharacteristic
60#1540C2211yescharacteristic
61#1493C2211yescharacteristic
62#1493C2211yescharacteristic
63#1492C2211yescharacteristic
64#1492C2211yescharacteristic
65C2×D4⋊C8C2211yescharacteristic
66#1451C2211yescharacteristic
67C8×M16C2211yescharacteristic
68C2×#322C2211yescharacteristic
69#1303C2211yescharacteristic
70#1303C2211yescharacteristic
71#1279C2211yescharacteristic
72#1278C2211yescharacteristic
73#1278C2211yescharacteristic
74#1277C2211yescharacteristic
75C2×#131—21no—
76C22×M16—21no—
77C22×M16—21no—
78C23×C8—21no—
79#533C2311yescharacteristic
80#532C2311yescharacteristic
81C2×C4×D4C2311yescharacteristic
82#488—21no—
83#488—21no—
84#487C2×C411yescharacteristic
85#487C2×C411yescharacteristic
86#486C2×C411yescharacteristic
87#486C2×C411yescharacteristic
88#483—21no—
89#480—21no—
90#476C2×C411yescharacteristic
91#476C2×C411yescharacteristic
92#474C2×C411yescharacteristic
93#474C2×C411yescharacteristic
94#461C2×C411yescharacteristic
95#461C2×C411yescharacteristic
96#459C2×C411yescharacteristic
97#459C2×C411yescharacteristic
98#450—21no—
99#447—21no—
100#446—21no—
101#446—41no—
102C8×Q8—21no—
103C4×SD16—21no—
104C4×SD16—21no—
105C4×D8—21no—
106C4×D8—21no—
107#436—21no—
108#436—21no—
109#436—21no—
110#435—41no—
111#435—41no—
112#435—41no—
113C8×D4—21no—
114C8×D4—21no—
115C8×D4—21no—
116C8×D4—21no—
117C8×D4—21no—
118C8×D4C2×C411yescharacteristic
119C8×D4C2×C411yescharacteristic
120C8×D4C2×C411yescharacteristic
121C8×D4C2×C411yescharacteristic
122#431C2311yescharacteristic
123#431C2311yescharacteristic
124C2×#108—21no—
125C2×#108—21no—
126C2×#108—21no—
127C2×#108—21no—
128C2×#108—44no—
129C2×#108D411yescharacteristic
130C2×#108D411yescharacteristic
131#424—21no—
132#423C2311yescharacteristic
133#423C2311yescharacteristic
134C2×C4⋊C8C2311yescharacteristic
135C2×C4⋊C8C2311yescharacteristic
136C2×#104—21no—
137C2×#104—21no—
138#416—21no—
139#416—21no—
140C2×#102—21no—
141#408—21no—
142#407—21no—
143#407—21no—
144C2×C22⋊C8—21no—
145#405—21no—
146C4×M16C2311yescharacteristic
147C4×M16C2311yescharacteristic
148C2×C4×C8—21no—
149C2×C4×C8C2311yescharacteristic
150C2×C4×C8C2311yescharacteristic
151C2×C4×C8C2311yescharacteristic
152C4×C4⋊C4C2311yescharacteristic
153#341—21no—
154#341—21no—
155#341D411yescharacteristic
156#341D411yescharacteristic
157#340—21no—
158#339—21no—
159#339—21no—
160#336—21no—
161#335—21no—
162#335—21no—
163#335—21no—
164#335—21no—
165#335C2×C411yescharacteristic
166#335C2×C411yescharacteristic
167#334C2×C411yescharacteristic
168#334C2×C411yescharacteristic
169#333C2×C411yescharacteristic
170#333C2×C411yescharacteristic
171#332C2×C411yescharacteristic
172#332C2×C411yescharacteristic
173#332C2×C411yescharacteristic
174#332C2×C411yescharacteristic
175#331C2×C411yescharacteristic
176#331C2×C411yescharacteristic
177Q8⋊C8—21no—
178Q8⋊C8—21no—
179Q8⋊C8—41no—
180D4⋊C8—21no—
181D4⋊C8—21no—
182D4⋊C8—41no—
183D4⋊C8D411yescharacteristic
184D4⋊C8D411yescharacteristic
185D4⋊C8D411yescharacteristic
186D4⋊C8D411yescharacteristic
187#322—21no—
188#322—21no—
189#322—21no—
190#322—21no—
191#322—21no—
192C82—21no—
193C2×#41—21no—
194C22×D4#4111yescharacteristic
195C23×C4—21no—
196C2×SD16—21no—
197C2×SD16—21no—
198C2×D8—21no—
199C2×D8—21no—
200#131—42no—
201#131—81no—
202C2×M16—21no—
203C2×M16—21no—
204C2×M16—41no—
205C2×M16—41no—
206C2×M16—41no—
207C2×M16—42no—
208C2×M16—42no—
209C2×M16—81no—
210C2×M16—81no—
211C2×M16C2×D411yescharacteristic
212C2×M16C2×D411yescharacteristic
213C2×M16C2×D411yescharacteristic
214C2×M16C2×D411yescharacteristic
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—41no—
224C22×C8—42no—
225C22×C8—42no—
226C22×C8—41no—
227C22×C8—41no—
228C22×C8C2×D411yescharacteristic
229C22×C8C2×D411yescharacteristic
230C4⋊Q8C22×C411yescharacteristic
231C4⋊Q8C22×C411yescharacteristic
232C4⋊Q8C22×C411yescharacteristic
233#127C22×C411yescharacteristic
234#125—21no—
235#124—21no—
236C22⋊Q8—21no—
237C22⋊Q8—21no—
238#121—21no—
239#121—21no—
240C4×Q8—21no—
241C4×Q8—21no—
242C4×D4—21no—
243C4×D4—21no—
244C4×D4—21no—
245C4×D4—21no—
246C4×D4C22×C411yescharacteristic
247C4×D4C22×C411yescharacteristic
248C4×D4C2×D411yescharacteristic
249C4×D4C2×D411yescharacteristic
250#117—21no—
251C2×C4⋊C4—21no—
252C2×C4⋊C4#4111yescharacteristic
253C2×C22⋊C4—21no—
254C2×C42—21no—
255C2×C42C2411yescharacteristic
256#108—22no—
257#108—41no—
258#108—42no—
259#108—42no—
260#108—42no—
261#108—81no—
262#107—21no—
263#107—21no—
264#106—21no—
265#106—21no—
266C4⋊C8—21no—
267C4⋊C8—21no—
268C4⋊C8—21no—
269C4⋊C8—21no—
270C4⋊C8—21no—
271C4⋊C8—21no—
272C4⋊C8—41no—
273C4⋊C8—41no—
274C4⋊C8C22×C411yescharacteristic
275C4⋊C8C22×C411yescharacteristic
276C4⋊C8C22×C411yescharacteristic
277C4⋊C8C22×C411yescharacteristic
278C4⋊C8C22×C411yescharacteristic
279C4⋊C8C22×C411yescharacteristic
280C4⋊C8C2×D411yescharacteristic
281C4⋊C8C2×D411yescharacteristic
282#104—42no—
283#104—81no—
284#103—21no—
285#103—21no—
286#103—21no—
287#103—21no—
288#103—21no—
289#103—21no—
290#102—24no—
291#102—21no—
292#102—21no—
293#102—21no—
294#102—21no—
295#102—21no—
296#102—21no—
297#102—21no—
298#102—21no—
299#102—21no—
300#102—21no—
301C22⋊C8—21no—
302C22⋊C8—21no—
303C22⋊C8—21no—
304C22⋊C8—21no—
305C22⋊C8—21no—
306C22⋊C8—21no—
307C22⋊C8—41no—
308C22⋊C8—41no—
309C22⋊C8—41no—
310C22⋊C8—41no—
311C22⋊C8—41no—
312C22⋊C8—41no—
313#97—21no—
314#97—21no—
315#97—41no—
316C4×C8—21no—
317C4×C8—21no—
318C4×C8—21no—
319C4×C8—21no—
320C4×C8—21no—
321C4×C8—21no—
322C4×C8—21no—
323C4×C8—41no—
324C4×C8—81no—
325C4×C8C22×C411yescharacteristic
326C4×C8C22×C411yescharacteristic
327C4×C8C2×D411yescharacteristic
328C4×C8C2×D411yescharacteristic
329C4×C8#4111yescharacteristic
330C4×C8#4111yescharacteristic
331#95—21no—
332C24—21no—
333#41—41no—
334#41—41no—
335#41—41no—
336#41—41no—
337C2×Q8—21no—
338C2×Q8—21no—
339C2×Q8—21no—
340C2×D4—22no—
341C2×D4—21no—
342C2×D4—21no—
343C2×D4—21no—
344C2×D4—21no—
345C2×D4—41no—
346C2×D4—41no—
347C2×D4C4×D411yescharacteristic
348C2×D4C4×D411yescharacteristic
349C22×C4—21no—
350C22×C4—21no—
351C22×C4—21no—
352C22×C4—21no—
353C22×C4—21no—
354C22×C4—41no—
355C22×C4—41no—
356C22×C4—41no—
357C22×C4—41no—
358C22×C4—41no—
359C22×C4—41no—
360C22×C4C22×D411yescharacteristic
361C22×C4C2×#4111yescharacteristic
362C22×C42−1+411yescharacteristic
363SD16—42no—
364SD16—81no—
365D8—42no—
366D8—81no—
367M16—21no—
368M16—21no—
369M16—21no—
370M16—21no—
371M16—21no—
372M16—21no—
373M16—21no—
374M16—41no—
375M16—41no—
376M16—41no—
377M16—41no—
378M16—82no—
379M16—82no—
380M16—82no—
381M16AGL(1,ℤ/8)12yes—
382C2×C8—21no—
383C2×C8—21no—
384C2×C8—21no—
385C2×C8—21no—
386C2×C8—21no—
387C2×C8—22no—
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—21no—
397C2×C8—21no—
398C2×C8—21no—
399C2×C8—21no—
400C2×C8—21no—
401C2×C8—42no—
402C2×C8—42no—
403C2×C8—42no—
404C2×C8—42no—
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—82no—
414C2×C8—82no—
415C2×C8—81no—
416C2×C8—82no—
417C2×C8C4×D411yescharacteristic
418C2×C8C4×D411yescharacteristic
419C2×C8C4×D411yescharacteristic
420C2×C8C4×D411yescharacteristic
421C4⋊C4—21no—
422C4⋊C4—22no—
423C4⋊C4—21no—
424C4⋊C4—21no—
425C4⋊C4—21no—
426C4⋊C4—21no—
427C4⋊C4—21no—
428C4⋊C4—21no—
429C4⋊C4—41no—
430C4⋊C4—41no—
431C4⋊C4—41no—
432C4⋊C4C4×D411yescharacteristic
433C4⋊C4C4×D411yescharacteristic
434C22⋊C4—21no—
435C22⋊C4—21no—
436C22⋊C4—21no—
437C22⋊C4—21no—
438C22⋊C4—41no—
439C22⋊C4—41no—
440C22⋊C4—41no—
441C42—41no—
442C42—41no—
443C42—41no—
444C42C23×C411yescharacteristic
445C42C22×D411yescharacteristic
446C42C2×#4111yescharacteristic
447C42H(5,𝔽2)11yescharacteristic
448C23—21no—
449C23—21no—
450C23—42no—
451C23—41no—
452C23—41no—
453C23—41no—
454C23—81no—
455C23#54811yescharacteristic
456Q8—41no—
457Q8—41no—
458Q8—41no—
459Q8—41no—
460Q8—81no—
461D4—21no—
462D4—21no—
463D4—21no—
464D4—42no—
465D4—41no—
466D4—41no—
467D4—41no—
468D4—41no—
469D4—81no—
470D4—81no—
471D4—81no—
472D4#44312yes—
473C2×C4—21no—
474C2×C4—21no—
475C2×C4—21no—
476C2×C4—21no—
477C2×C4—41no—
478C2×C4—41no—
479C2×C4—41no—
480C2×C4—41no—
481C2×C4—41no—
482C2×C4—41no—
483C2×C4—41no—
484C2×C4—41no—
485C2×C4—41no—
486C2×C4—41no—
487C2×C4—41no—
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×C4C2×C4×D411yescharacteristic
497C2×C4C2×C4×D411yescharacteristic
498C2×C4#52011yescharacteristic
499C2×C4D4211yescharacteristic
500C2×C4#54711yescharacteristic
501C2×C4#55311yescharacteristic
502C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
503C2×C4#57511yescharacteristic
504C2×C4#57611yescharacteristic
505C2×C4#57811yescharacteristic
506C8—21no—
507C8—21no—
508C8—21no—
509C8—21no—
510C8—22no—
511C8—42no—
512C8—41no—
513C8—41no—
514C8—41no—
515C8—41no—
516C8—41no—
517C8—41no—
518C8—41no—
519C8—41no—
520C8—41no—
521C8—41no—
522C8—41no—
523C8—41no—
524C8—41no—
525C8—81no—
526C8—81no—
527C22—21no—
528C22—21no—
529C22—22no—
530C22—41no—
531C22—41no—
532C22—41no—
533C22—41no—
534C22—41no—
535C22—82no—
536C22—82no—
537C22—81no—
538C22—81no—
539C22—81no—
540C22#236111yescharacteristic
541C22#336711yescharacteristic
542C22#337011yescharacteristic
543C4—21no—
544C4—21no—
545C4—21no—
546C4—41no—
547C4—41no—
548C4—82no—
549C4—81no—
550C4—81no—
551C4—81no—
552C4C4×AGL(1,ℤ/8)11yescharacteristic
553C4#295111yescharacteristic
554C4C2×#44311yescharacteristic
555C4#295711yescharacteristic
556C4#326411yescharacteristic
557C4#326911yescharacteristic
558C2—21no—
559C2—21no—
560C2—42no—
561C2—81no—
562C2—161no—
563C2#2301111yescharacteristic
564C2#2301811yescharacteristic
565C2#2304311yescharacteristic
subgroupG/H
centerC2×C4D42
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
socleC22#2361
prime radical#3100C22
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. #23234C2semidirect
2. D4×#108C2semidirect
4. #18000C2semidirect
6. #9858C2semidirect
7. #9627C2semidirect
8. #9592C2semidirect
9. #9589C2semidirect
11. #8989C2semidirect
12. #8933C2semidirect
13. #8930C2semidirect
14. #8296C2semidirect
15. #8200C2semidirect
40. #1683C4semidirect
41. #1683C4semidirect
42. #1681C4semidirect
43. #1681C4semidirect
44. #1679C4semidirect
45. #1679C4semidirect
46. #1677C4semidirect
47. #1677C4semidirect

Maximal subgroups

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

subgroupindex
#232342
D4×#1082
#191102
#180002
#179942
#98582
#96272
#95922
#95892
#89912
#89892
#89332
#89302
#82962
#82002