Finite Group Databasebeta

← #230084

Subgroups — #230084

Order 512.

561 conjugacy classes of subgroups.

The subgroup lattice has 546 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23718C211yesmaximal, characteristic
2#17896C211yesmaximal, characteristic, complemented
3#17893C211yesmaximal, characteristic
4#17876C211yesmaximal, characteristic, complemented
5#17811C211yesmaximal, characteristic, complemented
6#9661C211yesmaximal, characteristic, complemented
7#9607C211yesmaximal, characteristic, complemented
8#9552C211yesmaximal, characteristic, complemented
9#9549C211yesmaximal, characteristic, complemented
10#8955C211yesmaximal, characteristic, complemented
11#8954C211yesmaximal, characteristic
12#8838C211yesmaximal, characteristic, complemented
13#8837C211yesmaximal, characteristic, complemented
14#8296C211yesmaximal, characteristic, complemented
15#8189C211yesmaximal, characteristic, complemented
16#2989C2211yescharacteristic
17#2988C2211yescharacteristic
18#2983C2211yescharacteristic
19#2975C2211yescharacteristic
20C4×#137C2211yescharacteristic
21C4×AGL(1,ℤ/8)C2211yescharacteristic
22C2×C4×SD16C2211yescharacteristic
23#2922C2211yescharacteristic
24#2090C411yescharacteristic
25#2079C411yescharacteristic
26#2064C411yescharacteristic
27#2063C411yescharacteristic
28#2052C411yescharacteristic
29#2051C411yescharacteristic
30#2046C411yescharacteristic
31#2019C411yescharacteristic
32#1984C2211yescharacteristic
33#1954C2211yescharacteristic
34C4×#108C2211yescharacteristic
35C4×#101C2211yescharacteristic
36C4×#100C2211yescharacteristic
37#1594—21no—
38#1592—21no—
39#1592—21no—
40#1592—21no—
41#1590—21no—
42#1590—21no—
43#1590—21no—
44#1589—21no—
45#1587—21no—
46#1585—21no—
47#1584—21no—
48#1582—21no—
49#1574—21no—
50#1573—21no—
51#1572—21no—
52#1570—21no—
53#1565C2211yescharacteristic
54#1565C2211yescharacteristic
55#1565C2211yescharacteristic
56#1565C2211yescharacteristic
57#1564C2211yescharacteristic
58C2×#334C2211yescharacteristic
59#1479C2211yescharacteristic
60#1478C2211yescharacteristic
61#1476C2211yescharacteristic
62#1476C2211yescharacteristic
63C2×Q8⋊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×#137C2×C411yescharacteristic
76C2×AGL(1,ℤ/8)C2×C411yescharacteristic
77C22×SD16C2×C411yescharacteristic
78C4×#41C2311yescharacteristic
79C2×C4×Q8C2311yescharacteristic
80C2×C4×D4C2311yescharacteristic
81#446—21no—
82#446—21no—
83#446—21no—
84#446—21no—
85#446—21no—
86C8×Q8—21no—
87#442—21no—
88#441—21no—
89#440—21no—
90#440—21no—
91C4×Q16—21no—
92C4×SD16—21no—
93C4×SD16—21no—
94C4×SD16—21no—
95C4×SD16—21no—
96C4×SD16—21no—
97C4×SD16—21no—
98C4×D8—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#432—41no—
111#431—21no—
112#431—21no—
113#431—21no—
114#431C2311yescharacteristic
115#431C2311yescharacteristic
116#431C2311yescharacteristic
117#431C2311yescharacteristic
118C2×#108C2×C411yescharacteristic
119C2×#108C2×C411yescharacteristic
120#428C2×C411yescharacteristic
121#428C2×C411yescharacteristic
122C2×#106C2×C411yescharacteristic
123#424C2×C411yescharacteristic
124#424C2×C411yescharacteristic
125#423—21no—
126C2×C4⋊C8—21no—
127C2×C4⋊C8C2311yescharacteristic
128C2×C4⋊C8C2311yescharacteristic
129#419C2×C411yescharacteristic
130#418C2×C411yescharacteristic
131#417C2×C411yescharacteristic
132#417C2×C411yescharacteristic
133C2×#103C2×C411yescharacteristic
134C2×#102C2×C411yescharacteristic
135C2×#101C2×C411yescharacteristic
136C2×#101C2×C411yescharacteristic
137C2×#100C2×C411yescharacteristic
138C2×#100C2×C411yescharacteristic
139#408C2×C411yescharacteristic
140#408C2×C411yescharacteristic
141#407—41no—
142C2×C22⋊C8—41no—
143C4×M16C2311yescharacteristic
144C4×M16C2311yescharacteristic
145C4×M16C2311yescharacteristic
146C2×C4×C8C2311yescharacteristic
147C2×C4×C8C2311yescharacteristic
148C2×C4×C8C2311yescharacteristic
149#343C2×C411yescharacteristic
150#343C2×C411yescharacteristic
151#343C2×C411yescharacteristic
152#343C2×C411yescharacteristic
153#341C2×C411yescharacteristic
154#341C2×C411yescharacteristic
155#336—41no—
156#335—21no—
157#335—21no—
158#335—21no—
159#335—21no—
160#335D411yescharacteristic
161#335D411yescharacteristic
162#334—21no—
163#334—21no—
164#334—21no—
165#334—21no—
166#334—21no—
167#334—21no—
168#334D411yescharacteristic
169#334D411yescharacteristic
170Q8⋊C8—21no—
171Q8⋊C8—21no—
172Q8⋊C8—21no—
173Q8⋊C8—21no—
174Q8⋊C8—21no—
175Q8⋊C8D411yescharacteristic
176Q8⋊C8D411yescharacteristic
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×Q8C22×C411yescharacteristic
194C22×D4C22×C411yescharacteristic
195C23×C4—21no—
196#137—41no—
197#137—41no—
198AGL(1,ℤ/8)—22no—
199AGL(1,ℤ/8)—41no—
200C2×Q16—21no—
201C2×SD16—21no—
202C2×SD16—21no—
203C2×SD16—21no—
204C2×SD16—21no—
205C2×SD16—41no—
206C2×SD16—41no—
207C2×D8—21no—
208C2×M16—41no—
209C2×M16—41no—
210C2×M16—41no—
211C2×M16C22×C411yescharacteristic
212C2×M16C22×C411yescharacteristic
213C2×M16C22×C411yescharacteristic
214C2×M16C22×C411yescharacteristic
215C2×M16C22×C411yescharacteristic
216C2×M16C22×C411yescharacteristic
217C22×C8—21no—
218C22×C8—21no—
219C22×C8—21no—
220C22×C8—21no—
221C22×C8—41no—
222C22×C8C22×C411yescharacteristic
223C22×C8C22×C411yescharacteristic
224C4×Q8—21no—
225C4×Q8—21no—
226C4×Q8—21no—
227C4×Q8—21no—
228C4×Q8C2×D411yescharacteristic
229C4×Q8#4111yescharacteristic
230C4×D4—21no—
231C4×D4—21no—
232C4×D4—21no—
233C4×D4—21no—
234C4×D4—21no—
235C4×D4—21no—
236C4×D4C2×D411yescharacteristic
237C4×D4#4111yescharacteristic
238#117—21no—
239#117C22×C411yescharacteristic
240C2×C4⋊C4—21no—
241C2×C4⋊C4C22×C411yescharacteristic
242C2×C4⋊C4C22×C411yescharacteristic
243C2×C22⋊C4—21no—
244C2×C42—21no—
245C2×C42—21no—
246C2×C42C2411yescharacteristic
247#108—22no—
248#108—41no—
249#107—21no—
250#107—21no—
251#106—21no—
252#106—21no—
253#106—21no—
254#106—21no—
255C4⋊C8—21no—
256C4⋊C8—21no—
257C4⋊C8—21no—
258C4⋊C8—21no—
259C4⋊C8—21no—
260C4⋊C8—21no—
261C4⋊C8—21no—
262C4⋊C8—21no—
263C4⋊C8—21no—
264C4⋊C8—41no—
265C4⋊C8—41no—
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#103—21no—
277#103—21no—
278#103—21no—
279#103—21no—
280#102—21no—
281#102—21no—
282#102—21no—
283#102—21no—
284#102—21no—
285#102—21no—
286#101—22no—
287#101—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—82no—
303#97—21no—
304#97—21no—
305#97—21no—
306#97—41no—
307#97—41no—
308#97—41no—
309C4×C8—21no—
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—41no—
320C4×C8C2×D411yescharacteristic
321C4×C8C2×D411yescharacteristic
322C4×C8C2×D411yescharacteristic
323C4×C8C2×D411yescharacteristic
324C4×C8#4111yescharacteristic
325C4×C8#4111yescharacteristic
326C24—21no—
327#41—21no—
328#41—21no—
329#41—21no—
330#41—21no—
331C2×Q8—21no—
332C2×Q8—21no—
333C2×Q8—21no—
334C2×Q8—21no—
335C2×Q8—21no—
336C2×Q8—21no—
337C2×Q8C4×D411yescharacteristic
338C2×D4—21no—
339C2×D4—21no—
340C2×D4—21no—
341C2×D4—21no—
342C2×D4—21no—
343C2×D4—21no—
344C2×D4—21no—
345C2×D4C4×D411yescharacteristic
346C22×C4—21no—
347C22×C4—21no—
348C22×C4—21no—
349C22×C4—21no—
350C22×C4—21no—
351C22×C4—21no—
352C22×C4—41no—
353C22×C4—41no—
354C22×C4—81no—
355C22×C4—81no—
356C22×C4C23×C411yescharacteristic
357C22×C4H(5,𝔽2)11yescharacteristic
358C22×C42−1+411yescharacteristic
359Q16—42no—
360SD16—42no—
361SD16—42no—
362SD16—42no—
363SD16—42no—
364D8—42no—
365M16—21no—
366M16—21no—
367M16—21no—
368M16—21no—
369M16—21no—
370M16—21no—
371M16—21no—
372M16—21no—
373M16—21no—
374M16—21no—
375M16—21no—
376M16—161no—
377M16—161no—
378M16—161no—
379M16#13112yes—
380C2×C8—21no—
381C2×C8—21no—
382C2×C8—21no—
383C2×C8—21no—
384C2×C8—21no—
385C2×C8—22no—
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—21no—
397C2×C8—21no—
398C2×C8—21no—
399C2×C8—41no—
400C2×C8—41no—
401C2×C8—41no—
402C2×C8—41no—
403C2×C8—41no—
404C2×C8—81no—
405C2×C8—81no—
406C2×C8—81no—
407C2×C8—161no—
408C2×C8C4×D411yescharacteristic
409C2×C8C4×D411yescharacteristic
410C2×C8C4×D411yescharacteristic
411C2×C8C4×D411yescharacteristic
412C4⋊C4—21no—
413C4⋊C4—21no—
414C4⋊C4—21no—
415C4⋊C4—21no—
416C4⋊C4—21no—
417C4⋊C4—21no—
418C4⋊C4—41no—
419C4⋊C4—41no—
420C4⋊C4C4×D411yescharacteristic
421C4⋊C4C4×D411yescharacteristic
422C22⋊C4—21no—
423C22⋊C4—21no—
424C22⋊C4—21no—
425C22⋊C4—41no—
426C22⋊C4—41no—
427C42—21no—
428C42—21no—
429C42—21no—
430C42—41no—
431C42—41no—
432C42C22×D411yescharacteristic
433C42C22×D411yescharacteristic
434C42C2×#4111yescharacteristic
435C42C2×#4111yescharacteristic
436C23—21no—
437C23—41no—
438C23—41no—
439C23—41no—
440C23—81no—
441C23—81no—
442C23#52011yescharacteristic
443Q8—21no—
444Q8—21no—
445Q8—41no—
446Q8—41no—
447Q8—41no—
448Q8—41no—
449D4—21no—
450D4—21no—
451D4—41no—
452D4—41no—
453D4—41no—
454D4—41no—
455D4—41no—
456D4—41no—
457C2×C4—21no—
458C2×C4—21no—
459C2×C4—21no—
460C2×C4—21no—
461C2×C4—21no—
462C2×C4—41no—
463C2×C4—41no—
464C2×C4—41no—
465C2×C4—41no—
466C2×C4—41no—
467C2×C4—41no—
468C2×C4—41no—
469C2×C4—41no—
470C2×C4—41no—
471C2×C4—41no—
472C2×C4—81no—
473C2×C4—81no—
474C2×C4—81no—
475C2×C4—81no—
476C2×C4—81no—
477C2×C4—161no—
478C2×C4C2×C4×D411yescharacteristic
479C2×C4C2×C4×D411yescharacteristic
480C2×C4D4211yescharacteristic
481C2×C4#54711yescharacteristic
482C2×C4#54811yescharacteristic
483C2×C4#55311yescharacteristic
484C2×C4C2×#13111yescharacteristic
485C2×C4#56811yescharacteristic
486C2×C4#56811yescharacteristic
487C2×C4#56811yescharacteristic
488C8—21no—
489C8—21no—
490C8—22no—
491C8—21no—
492C8—21no—
493C8—42no—
494C8—41no—
495C8—41no—
496C8—41no—
497C8—41no—
498C8—41no—
499C8—41no—
500C8—41no—
501C8—41no—
502C8—41no—
503C8—41no—
504C8—41no—
505C8—41no—
506C8—41no—
507C8—81no—
508C8—81no—
509C8—81no—
510C22—21no—
511C22—21no—
512C22—41no—
513C22—41no—
514C22—41no—
515C22—82no—
516C22—81no—
517C22—81no—
518C22—81no—
519C22—161no—
520C22#236111yescharacteristic
521C22#299311yescharacteristic
522C22#299511yescharacteristic
523C4—21no—
524C4—21no—
525C4—21no—
526C4—41no—
527C4—41no—
528C4—41no—
529C4—81no—
530C4—81no—
531C4—81no—
532C4#293011yescharacteristic
533C4#293011yescharacteristic
534C4#293311yescharacteristic
535C4#293511yescharacteristic
536C4#329011yescharacteristic
537C4#329111yescharacteristic
538C2—21no—
539C2—21no—
540C2—81no—
541C2—161no—
542C2#2299411yescharacteristic
543C2#2304311yescharacteristic
544C2#2304411yescharacteristic
subgroupG/H
centerC2×C4D42
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC22#2361
prime radicalC2×AGL(1,ℤ/8)C2×C4
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. #17896C2semidirect
4. #17876C2semidirect
5. #17811C2semidirect
6. #9661C2semidirect
7. #9607C2semidirect
8. #9552C2semidirect
9. #9549C2semidirect
10. #8955C2semidirect
12. #8838C2semidirect
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
#237182
#178962
#178932
#178762
#178112
#96612
#96072
#95522
#95492
#89552
#89542
#88382
#88372
#82962
#81892