Finite Group Databasebeta

← #230050

Subgroups — #230050

Order 512.

591 conjugacy classes of subgroups.

The subgroup lattice has 555 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#471C211yesmaximal, characteristic
2#17894C211yesmaximal, characteristic, complemented
3#17879C211yesmaximal, characteristic, complemented
4#17873C211yesmaximal, characteristic
5#17798C211yesmaximal, characteristic, complemented
6#9661C211yesmaximal, characteristic, complemented
7#9609C211yesmaximal, characteristic, complemented
8#9551C211yesmaximal, characteristic, complemented
9#9550C211yesmaximal, characteristic, complemented
10#8956C211yesmaximal, characteristic
11#8955C211yesmaximal, characteristic, complemented
12#8838C211yesmaximal, characteristic, complemented
13#8835C211yesmaximal, characteristic, complemented
14#8296C211yesmaximal, characteristic, complemented
15#8186C211yesmaximal, characteristic, complemented
16C2×#471C411yescharacteristic
17#2989C2211yescharacteristic
18#2989C2211yescharacteristic
19#2983C2211yescharacteristic
20#2972C2211yescharacteristic
21C4×#137C2211yescharacteristic
22C4×AGL(1,ℤ/8)C2211yescharacteristic
23C2×C4×SD16C2211yescharacteristic
24C4×#131C2211yescharacteristic
25#1984C2211yescharacteristic
26#1977C411yescharacteristic
27#1954C2211yescharacteristic
28#1951C411yescharacteristic
29#1931C411yescharacteristic
30#1930C411yescharacteristic
31#1908C411yescharacteristic
32#1907C411yescharacteristic
33#1815C411yescharacteristic
34C4×#108C2211yescharacteristic
35C4×#101C2211yescharacteristic
36C4×#100C2211yescharacteristic
37#1593—21no—
38#1592—21no—
39#1591—21no—
40#1590—21no—
41#1586—21no—
42#1585—21no—
43#1585—21no—
44#1585—21no—
45#1582—21no—
46#1582—21no—
47#1582—21no—
48#1581—21no—
49#1569—21no—
50#1569—21no—
51#1568—21no—
52#1566—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#1452C2211yescharacteristic
66#1451C2211yescharacteristic
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
78C2×#131C2×C411yescharacteristic
79C2×#131C2×C411yescharacteristic
80C4×#41C2311yescharacteristic
81C2×C4×Q8C2311yescharacteristic
82C2×C4×D4C2311yescharacteristic
83#471—24no—
84#446—21no—
85#446—21no—
86#446—21no—
87#446—21no—
88#446—21no—
89#446—21no—
90#442—21no—
91#441—21no—
92#440—21no—
93#440—21no—
94C4×Q16—21no—
95C4×SD16—21no—
96C4×SD16—21no—
97C4×SD16—21no—
98C4×SD16—21no—
99C4×SD16—21no—
100C4×SD16—21no—
101C4×D8—21no—
102#436—21no—
103#436—21no—
104#435—41no—
105#435—41no—
106C8×D4—21no—
107C8×D4—21no—
108C8×D4—21no—
109C8×D4—21no—
110#432—21no—
111#432—21no—
112#432—41no—
113#431C2311yescharacteristic
114#431C2311yescharacteristic
115#431C2311yescharacteristic
116#431C2311yescharacteristic
117C2×#108C2×C411yescharacteristic
118C2×#108C2×C411yescharacteristic
119#428C2×C411yescharacteristic
120#428C2×C411yescharacteristic
121C2×#106C2×C411yescharacteristic
122#423—21no—
123#423—21no—
124C2×C4⋊C8C2311yescharacteristic
125C2×C4⋊C8C2311yescharacteristic
126#419C2×C411yescharacteristic
127#418C2×C411yescharacteristic
128#417C2×C411yescharacteristic
129#417C2×C411yescharacteristic
130C2×#103C2×C411yescharacteristic
131C2×#102C2×C411yescharacteristic
132C2×#101C2×C411yescharacteristic
133C2×#101C2×C411yescharacteristic
134C2×#100C2×C411yescharacteristic
135C2×#100C2×C411yescharacteristic
136C2×C22⋊C8—41no—
137C2×C22⋊C8—41no—
138#405—21no—
139#405—21no—
140#405C2×C411yescharacteristic
141#405C2×C411yescharacteristic
142C4×M16—21no—
143C4×M16C2311yescharacteristic
144C4×M16C2311yescharacteristic
145C4×M16C2311yescharacteristic
146C2×C4×C8—21no—
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#336—41no—
157#335—21no—
158#335—21no—
159#335—21no—
160#335—21no—
161#335D411yescharacteristic
162#335D411yescharacteristic
163#334—21no—
164#334—21no—
165#334—21no—
166#334—21no—
167#334—21no—
168#334—21no—
169#334D411yescharacteristic
170#334D411yescharacteristic
171Q8⋊C8—21no—
172Q8⋊C8—21no—
173Q8⋊C8—21no—
174Q8⋊C8—21no—
175Q8⋊C8—21no—
176Q8⋊C8D411yescharacteristic
177Q8⋊C8D411yescharacteristic
178D4⋊C8—21no—
179D4⋊C8—21no—
180D4⋊C8—21no—
181D4⋊C8—21no—
182D4⋊C8—21no—
183D4⋊C8D411yescharacteristic
184D4⋊C8D411yescharacteristic
185#322—21no—
186#322—21no—
187#322—21no—
188#322—21no—
189#322—21no—
190#322—21no—
191#322—21no—
192C82—21no—
193C2×#41C22×C411yescharacteristic
194C22×Q8C22×C411yescharacteristic
195C22×D4C22×C411yescharacteristic
196C23×C4—21no—
197#137—22no—
198#137—41no—
199AGL(1,ℤ/8)—22no—
200AGL(1,ℤ/8)—41no—
201C2×Q16—21no—
202C2×SD16—21no—
203C2×SD16—21no—
204C2×SD16—22no—
205C2×SD16—21no—
206C2×SD16—21no—
207C2×SD16—41no—
208C2×D8—21no—
209#131—22no—
210#131—41no—
211#131—41no—
212#131—41no—
213C2×M16—21no—
214C2×M16—21no—
215C2×M16—41no—
216C2×M16—41no—
217C2×M16C22×C411yescharacteristic
218C2×M16C22×C411yescharacteristic
219C2×M16C22×C411yescharacteristic
220C2×M16C22×C411yescharacteristic
221C2×M16C22×C411yescharacteristic
222C2×M16C22×C411yescharacteristic
223C22×C8—21no—
224C22×C8—21no—
225C22×C8—21no—
226C22×C8—21no—
227C22×C8—21no—
228C22×C8—21no—
229C22×C8—41no—
230C22×C8—41no—
231C22×C8C22×C411yescharacteristic
232C22×C8C22×C411yescharacteristic
233C4×Q8—21no—
234C4×Q8—21no—
235C4×Q8—21no—
236C4×Q8—21no—
237C4×Q8C2×D411yescharacteristic
238C4×Q8#4111yescharacteristic
239C4×D4—21no—
240C4×D4—21no—
241C4×D4—21no—
242C4×D4—21no—
243C4×D4—21no—
244C4×D4—21no—
245C4×D4C2×D411yescharacteristic
246C4×D4#4111yescharacteristic
247#117—21no—
248#117C22×C411yescharacteristic
249C2×C4⋊C4—21no—
250C2×C4⋊C4C22×C411yescharacteristic
251C2×C4⋊C4C22×C411yescharacteristic
252C2×C22⋊C4—21no—
253C2×C42—21no—
254C2×C42—21no—
255C2×C42C2411yescharacteristic
256#108—22no—
257#108—41no—
258#107—21no—
259#107—21no—
260#106—21no—
261#106—21no—
262#106—21no—
263#106—21no—
264C4⋊C8—21no—
265C4⋊C8—21no—
266C4⋊C8—21no—
267C4⋊C8—21no—
268C4⋊C8—21no—
269C4⋊C8—41no—
270C4⋊C8—41no—
271C4⋊C8—41no—
272C4⋊C8—41no—
273C4⋊C8C2×D411yescharacteristic
274C4⋊C8C2×D411yescharacteristic
275C4⋊C8C2×D411yescharacteristic
276C4⋊C8C2×D411yescharacteristic
277C4⋊C8C2×D411yescharacteristic
278C4⋊C8C2×D411yescharacteristic
279#103—21no—
280#103—21no—
281#103—21no—
282#103—21no—
283#103—21no—
284#103—21no—
285#102—21no—
286#102—21no—
287#102—21no—
288#102—21no—
289#102—21no—
290#102—21no—
291#101—22no—
292#101—41no—
293#100—41no—
294#100M1614yescomplemented
295C22⋊C8—21no—
296C22⋊C8—21no—
297C22⋊C8—21no—
298C22⋊C8—21no—
299C22⋊C8—41no—
300C22⋊C8—41no—
301C22⋊C8—41no—
302C22⋊C8—41no—
303C22⋊C8—82no—
304#97—21no—
305#97—21no—
306#97—21no—
307#97—21no—
308#97—21no—
309#97—41no—
310#97—41no—
311#97—41no—
312#97—41no—
313C4×C8—21no—
314C4×C8—21no—
315C4×C8—21no—
316C4×C8—21no—
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×C8C2×D411yescharacteristic
326C4×C8C2×D411yescharacteristic
327C4×C8C2×D411yescharacteristic
328C4×C8C2×D411yescharacteristic
329C4×C8#4111yescharacteristic
330C4×C8#4111yescharacteristic
331C24—21no—
332#41—21no—
333#41—21no—
334#41—21no—
335#41—21no—
336C2×Q8—21no—
337C2×Q8—21no—
338C2×Q8—21no—
339C2×Q8—21no—
340C2×Q8—21no—
341C2×Q8—21no—
342C2×Q8C4×D411yescharacteristic
343C2×D4—21no—
344C2×D4—21no—
345C2×D4—21no—
346C2×D4—21no—
347C2×D4—21no—
348C2×D4—21no—
349C2×D4C4×D411yescharacteristic
350C2×D4C2×M1612yes—
351C22×C4—21no—
352C22×C4—21no—
353C22×C4—21no—
354C22×C4—21no—
355C22×C4—21no—
356C22×C4—21no—
357C22×C4—41no—
358C22×C4—41no—
359C22×C4—81no—
360C22×C4—81no—
361C22×C4C23×C411yescharacteristic
362C22×C4H(5,𝔽2)11yescharacteristic
363C22×C42−1+411yescharacteristic
364Q16—42no—
365SD16—42no—
366SD16—42no—
367SD16—42no—
368SD16—42no—
369D8—42no—
370M16—21no—
371M16—21no—
372M16—21no—
373M16—21no—
374M16—21no—
375M16—21no—
376M16—21no—
377M16—21no—
378M16—21no—
379M16—21no—
380M16—42no—
381M16—42no—
382M16—161no—
383M16—161no—
384M16C2×M1612yes—
385M16C2×M1612yes—
386C2×C8—21no—
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—42no—
406C2×C8—42no—
407C2×C8—41no—
408C2×C8—41no—
409C2×C8—41no—
410C2×C8—41no—
411C2×C8—41no—
412C2×C8—81no—
413C2×C8—81no—
414C2×C8—81no—
415C2×C8—161no—
416C2×C8—161no—
417C2×C8C4×D411yescharacteristic
418C2×C8C4×D411yescharacteristic
419C2×C8C4×D411yescharacteristic
420C2×C8C4×D411yescharacteristic
421C4⋊C4—21no—
422C4⋊C4—21no—
423C4⋊C4—21no—
424C4⋊C4—21no—
425C4⋊C4—21no—
426C4⋊C4—21no—
427C4⋊C4—41no—
428C4⋊C4—41no—
429C4⋊C4C4×D411yescharacteristic
430C4⋊C4C4×D411yescharacteristic
431C22⋊C4—21no—
432C22⋊C4—21no—
433C22⋊C4—21no—
434C22⋊C4—41no—
435C22⋊C4—41no—
436C42—21no—
437C42—21no—
438C42—21no—
439C42—41no—
440C42—41no—
441C42C22×D411yescharacteristic
442C42C22×D411yescharacteristic
443C42C2×#4111yescharacteristic
444C42C2×#4111yescharacteristic
445C23—21no—
446C23—22no—
447C23—41no—
448C23—41no—
449C23—81no—
450C23—81no—
451C23#52011yescharacteristic
452Q8—21no—
453Q8—21no—
454Q8—41no—
455Q8—41no—
456Q8—41no—
457Q8—41no—
458D4—21no—
459D4—21no—
460D4—22no—
461D4—22no—
462D4—41no—
463D4—41no—
464D4—41no—
465D4—41no—
466C2×C4—21no—
467C2×C4—21no—
468C2×C4—21no—
469C2×C4—21no—
470C2×C4—21no—
471C2×C4—41no—
472C2×C4—41no—
473C2×C4—41no—
474C2×C4—41no—
475C2×C4—41no—
476C2×C4—41no—
477C2×C4—41no—
478C2×C4—41no—
479C2×C4—41no—
480C2×C4—41no—
481C2×C4—81no—
482C2×C4—81no—
483C2×C4—81no—
484C2×C4—81no—
485C2×C4—81no—
486C2×C4—161no—
487C2×C4C2×C4×D411yescharacteristic
488C2×C4C2×C4×D411yescharacteristic
489C2×C4D4211yescharacteristic
490C2×C4#54711yescharacteristic
491C2×C4#54811yescharacteristic
492C2×C4#55311yescharacteristic
493C2×C4C22×M1611yescharacteristic
494C2×C4#56811yescharacteristic
495C2×C4#56811yescharacteristic
496C2×C4#56811yescharacteristic
497C8—21no—
498C8—21no—
499C8—21no—
500C8—21no—
501C8—22no—
502C8—22no—
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—81no—
517C8—81no—
518C8—81no—
519C22—21no—
520C22—21no—
521C22—41no—
522C22—41no—
523C22—41no—
524C22—42no—
525C22—82no—
526C22—81no—
527C22—81no—
528C22—161no—
529C22#236111yescharacteristic
530C22#298911yescharacteristic
531C22#299511yescharacteristic
532C4—21no—
533C4—21no—
534C4—21no—
535C4—41no—
536C4—41no—
537C4—41no—
538C4—81no—
539C4—81no—
540C4—81no—
541C4#293011yescharacteristic
542C4#293211yescharacteristic
543C4D4×M1611yescharacteristic
544C4D4×M1611yescharacteristic
545C4#329011yescharacteristic
546C4#329111yescharacteristic
547C2—21no—
548C2—21no—
549C2—81no—
550C2—82no—
551C2#2298811yescharacteristic
552C2#2304311yescharacteristic
553C2#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. #17894C2semidirect
3. #17879C2semidirect
5. #17798C2semidirect
6. #9661C2semidirect
7. #9609C2semidirect
8. #9551C2semidirect
9. #9550C2semidirect
11. #8955C2semidirect
12. #8838C2semidirect
13. #8835C2semidirect
14. #8296C2semidirect
15. #8186C2semidirect
294. #100M16semidirect

Maximal subgroups

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

subgroupindex
C4×#4712
#178942
#178792
#178732
#177982
#96612
#96092
#95512
#95502
#89562
#89552
#88382
#88352
#82962
#81862