Finite Group Databasebeta

← #211065

Subgroups — #211065

Order 512.

622 conjugacy classes of subgroups.

The subgroup lattice has 566 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#471C211yesmaximal, characteristic
2#17891C211yesmaximal, characteristic
3#17870C211yesmaximal, characteristic, complemented
4C2×#1583C211yesmaximal, characteristic, complemented
5#17337C211yesmaximal, characteristic, complemented
6#9672C211yesmaximal, characteristic, complemented
7#9609C211yesmaximal, characteristic, complemented
8#9550C211yesmaximal, characteristic, complemented
9#9549C211yesmaximal, characteristic, complemented
10#9098C211yesmaximal, characteristic, complemented
11#9097C211yesmaximal, characteristic
12#8604C211yesmaximal, characteristic, complemented
13#8297C211yesmaximal, characteristic, complemented
14#8166C211yesmaximal, characteristic, complemented
15C8×#101C211yesmaximal, characteristic, complemented
16C2×#471C411yescharacteristic
17#2965C2211yescharacteristic
18#2965C2211yescharacteristic
19C2×C8×Q8C2211yescharacteristic
20C4×#137C2211yescharacteristic
21C4×AGL(1,ℤ/8)C2211yescharacteristic
22C2×C4×SD16C2211yescharacteristic
23C2×#436C2211yescharacteristic
24C4×#131C2211yescharacteristic
25#1977C411yescharacteristic
26#1951C411yescharacteristic
27#1931C411yescharacteristic
28#1930C411yescharacteristic
29#1908C411yescharacteristic
30#1907C411yescharacteristic
31#1815C411yescharacteristic
32C4×#108C2211yescharacteristic
33C4×#101C2211yescharacteristic
34C4×#100C2211yescharacteristic
35#1594—21no—
36#1592—21no—
37#1590—21no—
38#1589—21no—
39#1588—21no—
40#1587—21no—
41#1584—21no—
42#1583—21no—
43#1583—21no—
44#1583—21no—
45#1583—21no—
46#1583—21no—
47#1565C2211yescharacteristic
48#1565C2211yescharacteristic
49#1564C2211yescharacteristic
50C2×#334C2211yescharacteristic
51#1479C2211yescharacteristic
52#1478C2211yescharacteristic
53#1477C2211yescharacteristic
54#1477C2211yescharacteristic
55C2×Q8⋊C8C2211yescharacteristic
56C2×D4⋊C8C2211yescharacteristic
57#1461C2211yescharacteristic
58#1460C2211yescharacteristic
59#1455—21no—
60#1453—21no—
61#1452C2211yescharacteristic
62#1451—21no—
63#1451C2211yescharacteristic
64C8×M16—21no—
65C8×M16C2211yescharacteristic
66C8×M16C2211yescharacteristic
67C8×M16C2211yescharacteristic
68C2×#322C2211yescharacteristic
69#1283C2211yescharacteristic
70#1283C2211yescharacteristic
71#1283C2211yescharacteristic
72#1283C2211yescharacteristic
73#1278C2211yescharacteristic
74#1278C2211yescharacteristic
75C2×#137C2×C411yescharacteristic
76C2×AGL(1,ℤ/8)C2×C411yescharacteristic
77C22×SD16C2×C411yescharacteristic
78C2×#131C2×C411yescharacteristic
79C2×#131C2×C411yescharacteristic
80C22×M16—21no—
81C4×#41C2311yescharacteristic
82C2×C4×Q8C2311yescharacteristic
83C2×C4×D4C2311yescharacteristic
84#471—24no—
85#446—21no—
86#446D411yescharacteristic
87#446D411yescharacteristic
88C8×Q8—21no—
89C8×Q8—21no—
90C8×Q8—21no—
91C8×Q8—21no—
92#442—21no—
93#441—21no—
94#440—21no—
95#440—21no—
96C4×Q16—21no—
97C4×SD16—21no—
98C4×SD16—21no—
99C4×SD16—21no—
100C4×SD16—21no—
101C4×SD16—21no—
102C4×SD16—21no—
103C4×D8—21no—
104#436—21no—
105#436—21no—
106#436—21no—
107#436—21no—
108#436—21no—
109#436D411yescharacteristic
110#436D411yescharacteristic
111#435—21no—
112#435—21no—
113#435—22no—
114#431—21no—
115#431—21no—
116#431C2311yescharacteristic
117#431C2311yescharacteristic
118#431C2311yescharacteristic
119#431C2311yescharacteristic
120C2×#108C2×C411yescharacteristic
121C2×#108C2×C411yescharacteristic
122#428C2×C411yescharacteristic
123#428C2×C411yescharacteristic
124C2×#106C2×C411yescharacteristic
125C2×C4⋊C8—21no—
126C2×C4⋊C8C2311yescharacteristic
127C2×C4⋊C8C2311yescharacteristic
128#419C2×C411yescharacteristic
129#418C2×C411yescharacteristic
130#417C2×C411yescharacteristic
131#417C2×C411yescharacteristic
132C2×#103C2×C411yescharacteristic
133C2×#102C2×C411yescharacteristic
134C2×#101C2×C411yescharacteristic
135C2×#101C2×C411yescharacteristic
136C2×#100C2×C411yescharacteristic
137C2×#100C2×C411yescharacteristic
138C2×C22⋊C8—21no—
139#405—21no—
140#405—21no—
141#405C2×C411yescharacteristic
142#405C2×C411yescharacteristic
143C4×M16—21no—
144C4×M16—21no—
145C4×M16—21no—
146C4×M16C2311yescharacteristic
147C4×M16C2311yescharacteristic
148C4×M16C2311yescharacteristic
149C2×C4×C8—21no—
150C2×C4×C8—21no—
151C2×C4×C8C2311yescharacteristic
152C2×C4×C8C2311yescharacteristic
153C2×C4×C8C2311yescharacteristic
154#343C2×C411yescharacteristic
155#343C2×C411yescharacteristic
156#343C2×C411yescharacteristic
157#343C2×C411yescharacteristic
158#341C2×C411yescharacteristic
159#341C2×C411yescharacteristic
160#335—21no—
161#335—21no—
162#335—21no—
163#334—21no—
164#334—21no—
165#334—21no—
166#334—21no—
167#334—21no—
168Q8⋊C8—21no—
169Q8⋊C8—21no—
170Q8⋊C8—21no—
171Q8⋊C8—21no—
172Q8⋊C8—21no—
173Q8⋊C8—21no—
174D4⋊C8—21no—
175D4⋊C8—21no—
176D4⋊C8—21no—
177D4⋊C8—21no—
178D4⋊C8—21no—
179D4⋊C8—21no—
180#324—22no—
181#323—22no—
182#322—21no—
183#322—21no—
184#322—21no—
185#322—21no—
186#322—21no—
187#322—21no—
188#322—21no—
189#322D411yescharacteristic
190#322D411yescharacteristic
191#322D411yescharacteristic
192#322D411yescharacteristic
193C82—21no—
194C82—21no—
195C82—21no—
196C2×#41C22×C411yescharacteristic
197C22×Q8C22×C411yescharacteristic
198C22×D4C22×C411yescharacteristic
199C23×C4—21no—
200#137—22no—
201#137—41no—
202AGL(1,ℤ/8)—22no—
203AGL(1,ℤ/8)—41no—
204C2×Q16—21no—
205C2×SD16—21no—
206C2×SD16—22no—
207C2×SD16—21no—
208C2×SD16—21no—
209C2×SD16—21no—
210C2×SD16—41no—
211C2×D8—21no—
212#131—22no—
213#131—22no—
214#131—41no—
215#131—41no—
216C2×M16—22no—
217C2×M16—21no—
218C2×M16—21no—
219C2×M16—21no—
220C2×M16—21no—
221C2×M16—41no—
222C2×M16—41no—
223C2×M16—82no—
224C2×M16C22×C411yescharacteristic
225C2×M16C22×C411yescharacteristic
226C2×M16C22×C411yescharacteristic
227C2×M16C22×C411yescharacteristic
228C2×M16C22×C411yescharacteristic
229C2×M16C22×C411yescharacteristic
230C22×C8—21no—
231C22×C8—21no—
232C22×C8—21no—
233C22×C8—21no—
234C22×C8—21no—
235C22×C8C22×C411yescharacteristic
236C22×C8C22×C411yescharacteristic
237C22×C8#4111yescharacteristic
238C22×C8#4111yescharacteristic
239C4×Q8—21no—
240C4×Q8—21no—
241C4×Q8—21no—
242C4×Q8—21no—
243C4×Q8C2×D411yescharacteristic
244C4×Q8#4111yescharacteristic
245C4×D4—21no—
246C4×D4—21no—
247C4×D4—21no—
248C4×D4—21no—
249C4×D4—21no—
250C4×D4—21no—
251C4×D4C2×D411yescharacteristic
252C4×D4#4111yescharacteristic
253#117—21no—
254#117C22×C411yescharacteristic
255C2×C4⋊C4—21no—
256C2×C4⋊C4C22×C411yescharacteristic
257C2×C4⋊C4C22×C411yescharacteristic
258C2×C22⋊C4—21no—
259C2×C42—21no—
260C2×C42—21no—
261C2×C42C2411yescharacteristic
262#108—22no—
263#108—41no—
264#107—21no—
265#107—21no—
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⋊C8C2×D411yescharacteristic
283C4⋊C8C2×D411yescharacteristic
284#103—21no—
285#103—21no—
286#103—21no—
287#103—21no—
288#103—21no—
289#103—21no—
290#102—21no—
291#102—21no—
292#102—21no—
293#102—21no—
294#102—21no—
295#102—21no—
296#101—22no—
297#101M1614yescomplemented
298#100—22no—
299#100—22no—
300C22⋊C8—22no—
301C22⋊C8—21no—
302C22⋊C8—21no—
303C22⋊C8—21no—
304C22⋊C8—21no—
305C22⋊C8—21no—
306C22⋊C8—41no—
307C22⋊C8—41no—
308C22⋊C8#4112yes—
309#97—22no—
310#97—21no—
311#97—21no—
312#97—21no—
313#97—21no—
314#97—21no—
315#97—21no—
316#97—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×C8—21no—
326C4×C8—21no—
327C4×C8—41no—
328C4×C8—41no—
329C4×C8C2×D411yescharacteristic
330C4×C8C2×D411yescharacteristic
331C4×C8C2×D411yescharacteristic
332C4×C8C2×D411yescharacteristic
333C4×C8C2×D411yescharacteristic
334C4×C8C2×D411yescharacteristic
335C4×C8C2×D411yescharacteristic
336C4×C8C2×D411yescharacteristic
337C4×C8#4111yescharacteristic
338C4×C8#4111yescharacteristic
339C24—21no—
340#41—21no—
341#41—21no—
342#41—21no—
343#41—21no—
344C2×Q8—21no—
345C2×Q8—21no—
346C2×Q8—21no—
347C2×Q8—21no—
348C2×Q8—21no—
349C2×Q8C4×D411yescharacteristic
350C2×Q8C2×M1612yes—
351C2×D4—21no—
352C2×D4—21no—
353C2×D4—21no—
354C2×D4—21no—
355C2×D4—21no—
356C2×D4—21no—
357C2×D4—21no—
358C2×D4C4×D411yescharacteristic
359C22×C4—21no—
360C22×C4—21no—
361C22×C4—21no—
362C22×C4—21no—
363C22×C4—21no—
364C22×C4—21no—
365C22×C4—41no—
366C22×C4—41no—
367C22×C4—81no—
368C22×C4—81no—
369C22×C4C23×C411yescharacteristic
370C22×C4C2×#4111yescharacteristic
371C22×C4C2×#4111yescharacteristic
372Q16—42no—
373SD16—42no—
374SD16—42no—
375SD16—42no—
376SD16—42no—
377D8—42no—
378M16—21no—
379M16—21no—
380M16—21no—
381M16—21no—
382M16—21no—
383M16—21no—
384M16—21no—
385M16—21no—
386M16—21no—
387M16—42no—
388M16—42no—
389M16—42no—
390M16—42no—
391M16—82no—
392M16—82no—
393M16—82no—
394M16C2×M1612yes—
395M16C2×M1612yes—
396M16#13112yes—
397C2×C8—21no—
398C2×C8—21no—
399C2×C8—21no—
400C2×C8—21no—
401C2×C8—21no—
402C2×C8—21no—
403C2×C8—21no—
404C2×C8—22no—
405C2×C8—21no—
406C2×C8—21no—
407C2×C8—21no—
408C2×C8—21no—
409C2×C8—21no—
410C2×C8—21no—
411C2×C8—21no—
412C2×C8—22no—
413C2×C8—41no—
414C2×C8—42no—
415C2×C8—42no—
416C2×C8—41no—
417C2×C8—41no—
418C2×C8—41no—
419C2×C8—41no—
420C2×C8—81no—
421C2×C8—81no—
422C2×C8—81no—
423C2×C8C4×D411yescharacteristic
424C2×C8C4×D411yescharacteristic
425C2×C8C4×D411yescharacteristic
426C2×C8C4×D411yescharacteristic
427C2×C8#12111yescharacteristic
428C2×C8#12111yescharacteristic
429C2×C8#12111yescharacteristic
430C2×C8#12111yescharacteristic
431C4⋊C4—21no—
432C4⋊C4—21no—
433C4⋊C4—21no—
434C4⋊C4—21no—
435C4⋊C4—21no—
436C4⋊C4—21no—
437C4⋊C4—41no—
438C4⋊C4—41no—
439C4⋊C4C4×D411yescharacteristic
440C4⋊C4C4×D411yescharacteristic
441C22⋊C4—21no—
442C22⋊C4—21no—
443C22⋊C4—21no—
444C22⋊C4—41no—
445C22⋊C4—41no—
446C42—21no—
447C42—21no—
448C42—21no—
449C42—41no—
450C42—41no—
451C42C22×D411yescharacteristic
452C42C22×D411yescharacteristic
453C42C2×#4111yescharacteristic
454C42C2×#4111yescharacteristic
455C23—21no—
456C23—41no—
457C23—41no—
458C23—41no—
459C23—81no—
460C23—81no—
461C23C4×#4111yescharacteristic
462Q8—21no—
463Q8—22no—
464Q8—22no—
465Q8—21no—
466Q8—41no—
467Q8—41no—
468D4—21no—
469D4—21no—
470D4—41no—
471D4—41no—
472D4—41no—
473D4—41no—
474D4—41no—
475D4—41no—
476C2×C4—21no—
477C2×C4—21no—
478C2×C4—22no—
479C2×C4—21no—
480C2×C4—21no—
481C2×C4—21no—
482C2×C4—41no—
483C2×C4—42no—
484C2×C4—41no—
485C2×C4—41no—
486C2×C4—41no—
487C2×C4—41no—
488C2×C4—41no—
489C2×C4—41no—
490C2×C4—41no—
491C2×C4—41no—
492C2×C4—82no—
493C2×C4—81no—
494C2×C4—81no—
495C2×C4—81no—
496C2×C4—81no—
497C2×C4C2×C4×D411yescharacteristic
498C2×C4C2×C4×D411yescharacteristic
499C2×C4C2×#12111yescharacteristic
500C2×C4#52511yescharacteristic
501C2×C4#52911yescharacteristic
502C2×C4#53211yescharacteristic
503C2×C4C22×M1611yescharacteristic
504C2×C4C2×#13111yescharacteristic
505C2×C4#56811yescharacteristic
506C2×C4#56811yescharacteristic
507C8—21no—
508C8—21no—
509C8—22no—
510C8—21no—
511C8—21no—
512C8—22no—
513C8—22no—
514C8—21no—
515C8—21no—
516C8—21no—
517C8—41no—
518C8—41no—
519C8—42no—
520C8—41no—
521C8—41no—
522C8—41no—
523C8—41no—
524C8—41no—
525C8—81no—
526C8—81no—
527C8—81no—
528C8—81no—
529C8#47112yescomplemented
530C22—21no—
531C22—21no—
532C22—41no—
533C22—41no—
534C22—41no—
535C22—81no—
536C22—81no—
537C22—81no—
538C22—161no—
539C22—161no—
540C22C4×#12111yescharacteristic
541C22#296511yescharacteristic
542C22#296711yescharacteristic
543C4—21no—
544C4—21no—
545C4—21no—
546C4—41no—
547C4—41no—
548C4—41no—
549C4—42no—
550C4—81no—
551C4—81no—
552C4#293311yescharacteristic
553C4D4×M1611yescharacteristic
554C4D4×M1611yescharacteristic
555C4#293511yescharacteristic
556C4C2×#47111yescharacteristic
557C4#306711yescharacteristic
558C2—21no—
559C2—21no—
560C2—81no—
561C2—161no—
562C2#2149011yescharacteristic
563C2C4×#47111yescharacteristic
564C2#2155911yescharacteristic
subgroupG/H
centerC2×C4C2×#121
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC22C4×#121
prime radicalC2×AGL(1,ℤ/8)C2×C4
LCSG ▷ C22×C4 ▷ C2 ▷ 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
3. #17870C2semidirect
4. C2×#1583C2semidirect
5. #17337C2semidirect
6. #9672C2semidirect
7. #9609C2semidirect
8. #9550C2semidirect
9. #9549C2semidirect
10. #9098C2semidirect
12. #8604C2semidirect
13. #8297C2semidirect
14. #8166C2semidirect
15. C8×#101C2semidirect
297. #101M16semidirect
529. C8#471semidirect

Maximal subgroups

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

subgroupindex
C4×#4712
#178912
#178702
C2×#15832
#173372
#96722
#96092
#95502
#95492
#90982
#90972
#86042
#82972
#81662
C8×#1012