Finite Group Databasebeta

← #230085

Subgroups — #230085

Order 512.

578 conjugacy classes of subgroups.

The subgroup lattice has 561 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23716C211yesmaximal, characteristic
2#17895C211yesmaximal, characteristic, complemented
3#17893C211yesmaximal, characteristic, complemented
4#17875C211yesmaximal, characteristic, complemented
5#17812C211yesmaximal, characteristic, complemented
6#9663C211yesmaximal, characteristic, complemented
7#9608C211yesmaximal, characteristic, complemented
8#9552C211yesmaximal, characteristic, complemented
9#9550C211yesmaximal, characteristic, complemented
10#8956C211yesmaximal, characteristic, complemented
11#8954C211yesmaximal, characteristic, complemented
12#8837C211yesmaximal, characteristic, complemented
13#8837C211yesmaximal, characteristic, complemented
14#8296C211yesmaximal, characteristic, complemented
15#8189C211yesmaximal, characteristic, complemented
16#2989C2211yescharacteristic
17#2988C2211yescharacteristic
18#2975C2211yescharacteristic
19#2975C2211yescharacteristic
20C4×AGL(1,ℤ/8)C2211yescharacteristic
21C4×AGL(1,ℤ/8)C2211yescharacteristic
22C2×C4×D8C2211yescharacteristic
23#2922C2211yescharacteristic
24#2089C411yescharacteristic
25#2078C411yescharacteristic
26#2063C411yescharacteristic
27#2063C411yescharacteristic
28#2051C411yescharacteristic
29#2051C411yescharacteristic
30#2045C411yescharacteristic
31#2018C411yescharacteristic
32#1954C2211yescharacteristic
33#1954C2211yescharacteristic
34C4×#108C2211yescharacteristic
35C4×#100C2211yescharacteristic
36C4×#100C2211yescharacteristic
37#1594—21no—
38#1594—21no—
39#1594—21no—
40#1592—21no—
41#1590—21no—
42#1589—21no—
43#1589—21no—
44#1589—21no—
45#1588—21no—
46#1586—21no—
47#1583—21no—
48#1581—21no—
49#1574—21no—
50#1573—21no—
51#1572—21no—
52#1571—21no—
53#1565C2211yescharacteristic
54#1565C2211yescharacteristic
55#1565C2211yescharacteristic
56#1565C2211yescharacteristic
57#1564C2211yescharacteristic
58C2×#335C2211yescharacteristic
59#1478C2211yescharacteristic
60#1478C2211yescharacteristic
61#1476C2211yescharacteristic
62#1476C2211yescharacteristic
63C2×D4⋊C8C2211yescharacteristic
64C2×D4⋊C8C2211yescharacteristic
65#1451C2211yescharacteristic
66#1450C2211yescharacteristic
67C8×M16C2211yescharacteristic
68C2×#322C2211yescharacteristic
69#1283C2211yescharacteristic
70#1283C2211yescharacteristic
71#1278C2211yescharacteristic
72#1278C2211yescharacteristic
73#1278C2211yescharacteristic
74#1278C2211yescharacteristic
75C2×AGL(1,ℤ/8)C2×C411yescharacteristic
76C2×AGL(1,ℤ/8)C2×C411yescharacteristic
77C22×D8C2×C411yescharacteristic
78C4×#41C2311yescharacteristic
79C2×C4×D4C2311yescharacteristic
80C2×C4×D4C2311yescharacteristic
81#446—21no—
82C8×Q8—21no—
83#442—21no—
84#442—21no—
85#440—21no—
86#440—21no—
87C4×SD16—21no—
88C4×SD16—21no—
89C4×D8—21no—
90C4×D8—21no—
91C4×D8—21no—
92C4×D8—21no—
93C4×D8—21no—
94C4×D8—21no—
95#436—21no—
96#436—21no—
97#436—21no—
98#436—21no—
99#436—21no—
100#436—21no—
101#436—21no—
102#436—21no—
103#436—21no—
104#435—41no—
105C8×D4—21no—
106C8×D4—41no—
107#433—21no—
108#433—21no—
109#432—21no—
110#431—21no—
111#431—21no—
112#431—21no—
113#431C2311yescharacteristic
114#431C2311yescharacteristic
115#431C2311yescharacteristic
116#431C2311yescharacteristic
117C2×#108C2×C411yescharacteristic
118C2×#108C2×C411yescharacteristic
119#428C2×C411yescharacteristic
120#428C2×C411yescharacteristic
121C2×#107C2×C411yescharacteristic
122#424C2×C411yescharacteristic
123#424C2×C411yescharacteristic
124#423—21no—
125C2×C4⋊C8—21no—
126C2×C4⋊C8C2311yescharacteristic
127C2×C4⋊C8C2311yescharacteristic
128#418C2×C411yescharacteristic
129#418C2×C411yescharacteristic
130#417C2×C411yescharacteristic
131#417C2×C411yescharacteristic
132C2×#102C2×C411yescharacteristic
133C2×#102C2×C411yescharacteristic
134C2×#100C2×C411yescharacteristic
135C2×#100C2×C411yescharacteristic
136C2×#100C2×C411yescharacteristic
137C2×#100C2×C411yescharacteristic
138#408C2×C411yescharacteristic
139#408C2×C411yescharacteristic
140#407—41no—
141#407—41no—
142C2×C22⋊C8—41no—
143C2×C22⋊C8—41no—
144C4×M16C2311yescharacteristic
145C4×M16C2311yescharacteristic
146C4×M16C2311yescharacteristic
147C2×C4×C8C2311yescharacteristic
148C2×C4×C8C2311yescharacteristic
149C2×C4×C8C2311yescharacteristic
150#343C2×C411yescharacteristic
151#343C2×C411yescharacteristic
152#343C2×C411yescharacteristic
153#343C2×C411yescharacteristic
154#341C2×C411yescharacteristic
155#341C2×C411yescharacteristic
156#335—21no—
157#335—21no—
158#335—21no—
159#335—21no—
160#335—21no—
161#335—21no—
162#335D411yescharacteristic
163#335D411yescharacteristic
164#334—21no—
165#334—21no—
166#334—21no—
167#334—21no—
168#334D411yescharacteristic
169#334D411yescharacteristic
170Q8⋊C8—21no—
171Q8⋊C8D411yescharacteristic
172Q8⋊C8D411yescharacteristic
173D4⋊C8—21no—
174D4⋊C8—21no—
175D4⋊C8—21no—
176D4⋊C8—21no—
177D4⋊C8—21no—
178D4⋊C8—21no—
179D4⋊C8—21no—
180D4⋊C8—21no—
181D4⋊C8—21no—
182D4⋊C8D411yescharacteristic
183D4⋊C8D411yescharacteristic
184#322—21no—
185#322—21no—
186#322—21no—
187#322—21no—
188#322—21no—
189#322—21no—
190C82—21no—
191C82—21no—
192C2×#41C22×C411yescharacteristic
193C22×D4C22×C411yescharacteristic
194C22×D4C22×C411yescharacteristic
195C23×C4—21no—
196C23×C4—21no—
197AGL(1,ℤ/8)—22no—
198AGL(1,ℤ/8)—41no—
199AGL(1,ℤ/8)—41no—
200AGL(1,ℤ/8)—41no—
201C2×SD16—21no—
202C2×SD16—21no—
203C2×D8—21no—
204C2×D8—21no—
205C2×D8—21no—
206C2×D8—21no—
207C2×D8—41no—
208C2×D8—41no—
209C2×M16—41no—
210C2×M16—41no—
211C2×M16—41no—
212C2×M16—41no—
213C2×M16—41no—
214C2×M16C22×C411yescharacteristic
215C2×M16C22×C411yescharacteristic
216C2×M16C22×C411yescharacteristic
217C2×M16C22×C411yescharacteristic
218C2×M16C22×C411yescharacteristic
219C2×M16C22×C411yescharacteristic
220C22×C8—21no—
221C22×C8—21no—
222C22×C8—21no—
223C22×C8—21no—
224C22×C8—41no—
225C22×C8C22×C411yescharacteristic
226C22×C8C22×C411yescharacteristic
227C4×Q8C2×D411yescharacteristic
228C4×Q8#4111yescharacteristic
229C4×D4—21no—
230C4×D4—21no—
231C4×D4—21no—
232C4×D4—21no—
233C4×D4—21no—
234C4×D4—21no—
235C4×D4—21no—
236C4×D4—21no—
237C4×D4—21no—
238C4×D4—21no—
239C4×D4C2×D411yescharacteristic
240C4×D4#4111yescharacteristic
241#117—21no—
242#117C22×C411yescharacteristic
243C2×C4⋊C4C22×C411yescharacteristic
244C2×C4⋊C4C22×C411yescharacteristic
245C2×C22⋊C4—21no—
246C2×C22⋊C4—21no—
247C2×C42—21no—
248C2×C42C2411yescharacteristic
249#108—22no—
250#108—41no—
251#107—21no—
252#107—21no—
253#107—21no—
254#107—21no—
255#106—21no—
256#106—21no—
257C4⋊C8—21no—
258C4⋊C8—21no—
259C4⋊C8—21no—
260C4⋊C8—21no—
261C4⋊C8—21no—
262C4⋊C8—21no—
263C4⋊C8—21no—
264C4⋊C8—21no—
265C4⋊C8—21no—
266C4⋊C8—41no—
267C4⋊C8—41no—
268C4⋊C8C2×D411yescharacteristic
269C4⋊C8C2×D411yescharacteristic
270C4⋊C8C2×D411yescharacteristic
271C4⋊C8C2×D411yescharacteristic
272C4⋊C8C2×D411yescharacteristic
273C4⋊C8C2×D411yescharacteristic
274#103—21no—
275#103—21no—
276#102—21no—
277#102—21no—
278#102—21no—
279#102—21no—
280#102—21no—
281#102—21no—
282#102—21no—
283#102—21no—
284#102—21no—
285#102—21no—
286#100—22no—
287#100—41no—
288#100—41no—
289#100—41no—
290C22⋊C8—21no—
291C22⋊C8—21no—
292C22⋊C8—21no—
293C22⋊C8—21no—
294C22⋊C8—21no—
295C22⋊C8—21no—
296C22⋊C8—21no—
297C22⋊C8—21no—
298C22⋊C8—41no—
299C22⋊C8—41no—
300C22⋊C8—41no—
301C22⋊C8—41no—
302C22⋊C8—41no—
303C22⋊C8—41no—
304C22⋊C8—82no—
305C22⋊C8—82no—
306#97—21no—
307#97—21no—
308#97—21no—
309#97—41no—
310C4×C8—21no—
311C4×C8—21no—
312C4×C8—21no—
313C4×C8—21no—
314C4×C8—21no—
315C4×C8—21no—
316C4×C8—21no—
317C4×C8—21no—
318C4×C8—21no—
319C4×C8—21no—
320C4×C8—41no—
321C4×C8C2×D411yescharacteristic
322C4×C8C2×D411yescharacteristic
323C4×C8C2×D411yescharacteristic
324C4×C8C2×D411yescharacteristic
325C4×C8#4111yescharacteristic
326C4×C8#4111yescharacteristic
327C24—21no—
328C24—21no—
329#41—21no—
330#41—21no—
331#41—21no—
332#41—21no—
333C2×Q8C4×D411yescharacteristic
334C2×D4—21no—
335C2×D4—21no—
336C2×D4—21no—
337C2×D4—21no—
338C2×D4—21no—
339C2×D4—21no—
340C2×D4—21no—
341C2×D4—21no—
342C2×D4—21no—
343C2×D4—21no—
344C2×D4—21no—
345C2×D4—21no—
346C2×D4—21no—
347C2×D4C4×D411yescharacteristic
348C22×C4—21no—
349C22×C4—21no—
350C22×C4—21no—
351C22×C4—21no—
352C22×C4—21no—
353C22×C4—41no—
354C22×C4—41no—
355C22×C4—41no—
356C22×C4—41no—
357C22×C4—81no—
358C22×C4—81no—
359C22×C4—81no—
360C22×C4—81no—
361C22×C4C23×C411yescharacteristic
362C22×C4H(5,𝔽2)11yescharacteristic
363C22×C42−1+411yescharacteristic
364SD16—42no—
365SD16—42no—
366D8—42no—
367D8—42no—
368D8—42no—
369D8—42no—
370M16—21no—
371M16—21no—
372M16—21no—
373M16—21no—
374M16—21no—
375M16—21no—
376M16—21no—
377M16—21no—
378M16—21no—
379M16—21no—
380M16—21no—
381M16—161no—
382M16—161no—
383M16—161no—
384M16—161no—
385M16—161no—
386M16#13112yes—
387C2×C8—21no—
388C2×C8—21no—
389C2×C8—21no—
390C2×C8—21no—
391C2×C8—21no—
392C2×C8—22no—
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—21no—
405C2×C8—21no—
406C2×C8—41no—
407C2×C8—41no—
408C2×C8—41no—
409C2×C8—41no—
410C2×C8—41no—
411C2×C8—81no—
412C2×C8—81no—
413C2×C8—81no—
414C2×C8—161no—
415C2×C8C4×D411yescharacteristic
416C2×C8C4×D411yescharacteristic
417C2×C8C4×D411yescharacteristic
418C2×C8C4×D411yescharacteristic
419C4⋊C4—21no—
420C4⋊C4—21no—
421C4⋊C4—21no—
422C4⋊C4—21no—
423C4⋊C4—21no—
424C4⋊C4—21no—
425C4⋊C4C4×D411yescharacteristic
426C4⋊C4C4×D411yescharacteristic
427C22⋊C4—21no—
428C22⋊C4—21no—
429C22⋊C4—21no—
430C22⋊C4—41no—
431C22⋊C4—41no—
432C22⋊C4—41no—
433C22⋊C4—41no—
434C42—21no—
435C42—21no—
436C42—21no—
437C42C22×D411yescharacteristic
438C42C22×D411yescharacteristic
439C42C2×#4111yescharacteristic
440C42C2×#4111yescharacteristic
441C23—21no—
442C23—41no—
443C23—41no—
444C23—41no—
445C23—41no—
446C23—41no—
447C23—41no—
448C23—81no—
449C23—81no—
450C23—81no—
451C23—81no—
452C23#52011yescharacteristic
453Q8—21no—
454Q8—21no—
455D4—21no—
456D4—21no—
457D4—41no—
458D4—41no—
459D4—41no—
460D4—41no—
461D4—41no—
462D4—41no—
463D4—41no—
464D4—41no—
465D4—41no—
466D4—41no—
467C2×C4—21no—
468C2×C4—21no—
469C2×C4—21no—
470C2×C4—21no—
471C2×C4—21no—
472C2×C4—41no—
473C2×C4—41no—
474C2×C4—41no—
475C2×C4—41no—
476C2×C4—41no—
477C2×C4—41no—
478C2×C4—41no—
479C2×C4—81no—
480C2×C4—81no—
481C2×C4—81no—
482C2×C4—81no—
483C2×C4—81no—
484C2×C4—81no—
485C2×C4—81no—
486C2×C4—161no—
487C2×C4—161no—
488C2×C4C2×C4×D411yescharacteristic
489C2×C4C2×C4×D411yescharacteristic
490C2×C4D4211yescharacteristic
491C2×C4#54711yescharacteristic
492C2×C4#54811yescharacteristic
493C2×C4#55311yescharacteristic
494C2×C4C2×#13111yescharacteristic
495C2×C4#56811yescharacteristic
496C2×C4#56811yescharacteristic
497C2×C4#56811yescharacteristic
498C8—21no—
499C8—21no—
500C8—22no—
501C8—21no—
502C8—21no—
503C8—42no—
504C8—41no—
505C8—41no—
506C8—41no—
507C8—41no—
508C8—41no—
509C8—41no—
510C8—41no—
511C8—41no—
512C8—41no—
513C8—41no—
514C8—41no—
515C8—41no—
516C8—41no—
517C8—81no—
518C8—81no—
519C8—81no—
520C22—21no—
521C22—21no—
522C22—41no—
523C22—41no—
524C22—41no—
525C22—82no—
526C22—82no—
527C22—81no—
528C22—81no—
529C22—81no—
530C22—81no—
531C22—81no—
532C22—81no—
533C22—161no—
534C22—161no—
535C22#236111yescharacteristic
536C22#299311yescharacteristic
537C22#299511yescharacteristic
538C4—21no—
539C4—21no—
540C4—21no—
541C4—41no—
542C4—41no—
543C4—41no—
544C4—81no—
545C4—81no—
546C4#293011yescharacteristic
547C4#293011yescharacteristic
548C4#293311yescharacteristic
549C4#293511yescharacteristic
550C4#329011yescharacteristic
551C4#329211yescharacteristic
552C2—21no—
553C2—21no—
554C2—81no—
555C2—161no—
556C2—161no—
557C2#2299411yescharacteristic
558C2#2304311yescharacteristic
559C2#2304611yescharacteristic
subgroupG/H
centerC2×C4D42
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC22#2361
prime radical#2018C4
LCSG ▷ C22×C4 ▷ C2 ▷ 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
2. #17895C2semidirect
3. #17893C2semidirect
4. #17875C2semidirect
5. #17812C2semidirect
6. #9663C2semidirect
7. #9608C2semidirect
8. #9552C2semidirect
9. #9550C2semidirect
10. #8956C2semidirect
11. #8954C2semidirect
12. #8837C2semidirect
13. #8837C2semidirect
14. #8296C2semidirect
15. #8189C2semidirect

Maximal subgroups

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

subgroupindex
#237162
#178952
#178932
#178752
#178122
#96632
#96082
#95522
#95502
#89562
#89542
#88372
#82962
#81892