Finite Group Databasebeta

← #211105

Subgroups — #211105

Order 512.

596 conjugacy classes of subgroups.

The subgroup lattice has 557 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23718C211yesmaximal, characteristic
2#17891C211yesmaximal, characteristic
3#17870C211yesmaximal, characteristic, complemented
4C2×#1583C211yesmaximal, characteristic, complemented
5#17331C211yesmaximal, characteristic, complemented
6#9667C211yesmaximal, characteristic, complemented
7#9607C211yesmaximal, characteristic, complemented
8#9552C211yesmaximal, characteristic, complemented
9#9551C211yesmaximal, characteristic, complemented
10#9096C211yesmaximal, characteristic, complemented
11#9095C211yesmaximal, characteristic
12#8607C211yesmaximal, characteristic, complemented
13#8297C211yesmaximal, characteristic, complemented
14#8166C211yesmaximal, characteristic, complemented
15C8×#101C211yesmaximal, characteristic, complemented
16#2965C2211yescharacteristic
17C8×#41C2211yescharacteristic
18C2×C8×Q8C2211yescharacteristic
19C4×#137C2211yescharacteristic
20C4×AGL(1,ℤ/8)C2211yescharacteristic
21C2×C4×SD16C2211yescharacteristic
22C2×#436C2211yescharacteristic
23#2922C2211yescharacteristic
24#2090C411yescharacteristic
25#2079C411yescharacteristic
26#2064C411yescharacteristic
27#2063C411yescharacteristic
28#2052C411yescharacteristic
29#2051C411yescharacteristic
30#2046C411yescharacteristic
31#2019C411yescharacteristic
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#1454—21no—
60#1453—21no—
61#1452—21no—
62#1451C2211yescharacteristic
63#1451C2211yescharacteristic
64#1450C2211yescharacteristic
65C8×M16—21no—
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
78C22×M16—21no—
79C4×#41C2311yescharacteristic
80C2×C4×Q8C2311yescharacteristic
81C2×C4×D4C2311yescharacteristic
82#446D411yescharacteristic
83#446D411yescharacteristic
84C8×Q8—21no—
85C8×Q8—21no—
86C8×Q8—21no—
87C8×Q8—21no—
88C8×Q8—21no—
89#442—21no—
90#441—21no—
91#440—21no—
92#440—21no—
93C4×Q16—21no—
94C4×SD16—21no—
95C4×SD16—21no—
96C4×SD16—21no—
97C4×SD16—21no—
98C4×SD16—21no—
99C4×SD16—21no—
100C4×D8—21no—
101#436—21no—
102#436—21no—
103#436—21no—
104#436—21no—
105#436D411yescharacteristic
106#436D411yescharacteristic
107#435—21no—
108#435—21no—
109C8×D4—21no—
110C8×D4—22no—
111#433—21no—
112#433—21no—
113#431—21no—
114#431—21no—
115#431—21no—
116#431—21no—
117#431C2311yescharacteristic
118#431C2311yescharacteristic
119#431C2311yescharacteristic
120#431C2311yescharacteristic
121C2×#108C2×C411yescharacteristic
122C2×#108C2×C411yescharacteristic
123#428C2×C411yescharacteristic
124#428C2×C411yescharacteristic
125C2×#106C2×C411yescharacteristic
126#424C2×C411yescharacteristic
127#424C2×C411yescharacteristic
128C2×C4⋊C8—21no—
129C2×C4⋊C8C2311yescharacteristic
130C2×C4⋊C8C2311yescharacteristic
131#419C2×C411yescharacteristic
132#418C2×C411yescharacteristic
133#417C2×C411yescharacteristic
134#417C2×C411yescharacteristic
135C2×#103C2×C411yescharacteristic
136C2×#102C2×C411yescharacteristic
137C2×#101C2×C411yescharacteristic
138C2×#101C2×C411yescharacteristic
139C2×#100C2×C411yescharacteristic
140C2×#100C2×C411yescharacteristic
141#408C2×C411yescharacteristic
142#408C2×C411yescharacteristic
143C2×C22⋊C8—21no—
144C4×M16—21no—
145C4×M16C2311yescharacteristic
146C4×M16C2311yescharacteristic
147C4×M16C2311yescharacteristic
148C2×C4×C8—21no—
149C2×C4×C8—21no—
150C2×C4×C8C2311yescharacteristic
151C2×C4×C8C2311yescharacteristic
152C2×C4×C8C2311yescharacteristic
153#343C2×C411yescharacteristic
154#343C2×C411yescharacteristic
155#343C2×C411yescharacteristic
156#343C2×C411yescharacteristic
157#341C2×C411yescharacteristic
158#341C2×C411yescharacteristic
159#335—21no—
160#335—21no—
161#335—21no—
162#334—21no—
163#334—21no—
164#334—21no—
165#334—21no—
166#334—21no—
167Q8⋊C8—21no—
168Q8⋊C8—21no—
169Q8⋊C8—21no—
170Q8⋊C8—21no—
171Q8⋊C8—21no—
172Q8⋊C8—21no—
173D4⋊C8—21no—
174D4⋊C8—21no—
175D4⋊C8—21no—
176D4⋊C8—21no—
177D4⋊C8—21no—
178D4⋊C8—21no—
179#324—22no—
180#323—22no—
181#322—21no—
182#322—21no—
183#322—21no—
184#322—21no—
185#322—21no—
186#322—21no—
187#322—21no—
188#322D411yescharacteristic
189#322D411yescharacteristic
190#322D411yescharacteristic
191#322D411yescharacteristic
192C82—21no—
193C82—21no—
194C82—21no—
195C2×#41C22×C411yescharacteristic
196C22×Q8C22×C411yescharacteristic
197C22×D4C22×C411yescharacteristic
198C23×C4—21no—
199#137—22no—
200#137—41no—
201AGL(1,ℤ/8)—22no—
202AGL(1,ℤ/8)—41no—
203C2×Q16—21no—
204C2×SD16—21no—
205C2×SD16—22no—
206C2×SD16—21no—
207C2×SD16—21no—
208C2×SD16—21no—
209C2×SD16—41no—
210C2×D8—21no—
211C2×M16—21no—
212C2×M16—21no—
213C2×M16—41no—
214C2×M16—41no—
215C2×M16—82no—
216C2×M16C22×C411yescharacteristic
217C2×M16C22×C411yescharacteristic
218C2×M16C22×C411yescharacteristic
219C2×M16C22×C411yescharacteristic
220C2×M16C22×C411yescharacteristic
221C2×M16C22×C411yescharacteristic
222C22×C8—22no—
223C22×C8—21no—
224C22×C8—21no—
225C22×C8—21no—
226C22×C8C22×C411yescharacteristic
227C22×C8C22×C411yescharacteristic
228C22×C8#4111yescharacteristic
229C22×C8#4111yescharacteristic
230C4×Q8—21no—
231C4×Q8—21no—
232C4×Q8—21no—
233C4×Q8—21no—
234C4×Q8C2×D411yescharacteristic
235C4×Q8#4111yescharacteristic
236C4×D4—21no—
237C4×D4—21no—
238C4×D4—21no—
239C4×D4—21no—
240C4×D4—21no—
241C4×D4—21no—
242C4×D4C2×D411yescharacteristic
243C4×D4#4111yescharacteristic
244#117—21no—
245#117C22×C411yescharacteristic
246C2×C4⋊C4—21no—
247C2×C4⋊C4C22×C411yescharacteristic
248C2×C4⋊C4C22×C411yescharacteristic
249C2×C22⋊C4—21no—
250C2×C42—21no—
251C2×C42—21no—
252C2×C42C2411yescharacteristic
253#108—22no—
254#108—41no—
255#107—21no—
256#107—21no—
257#106—21no—
258#106—21no—
259#106—21no—
260#106—21no—
261C4⋊C8—22no—
262C4⋊C8—21no—
263C4⋊C8—21no—
264C4⋊C8—21no—
265C4⋊C8—21no—
266C4⋊C8—21no—
267C4⋊C8—21no—
268C4⋊C8—21no—
269C4⋊C8—21no—
270C4⋊C8—21no—
271C4⋊C8—21no—
272C4⋊C8—21no—
273C4⋊C8—21no—
274C4⋊C8—21no—
275C4⋊C8—41no—
276C4⋊C8—41no—
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—22no—
293#100—41no—
294#100—41no—
295C22⋊C8—22no—
296C22⋊C8—21no—
297C22⋊C8—21no—
298C22⋊C8—21no—
299C22⋊C8—21no—
300C22⋊C8—21no—
301C22⋊C8—21no—
302C22⋊C8—21no—
303C22⋊C8—21no—
304C22⋊C8—21no—
305C22⋊C8—41no—
306C22⋊C8—41no—
307C22⋊C8#4112yes—
308#97—21no—
309#97—21no—
310#97—21no—
311#97—21no—
312#97—21no—
313C4×C8—22no—
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—41no—
323C4×C8—41no—
324C4×C8C2×D411yescharacteristic
325C4×C8C2×D411yescharacteristic
326C4×C8C2×D411yescharacteristic
327C4×C8C2×D411yescharacteristic
328C4×C8C2×D411yescharacteristic
329C4×C8C2×D411yescharacteristic
330C4×C8C2×D411yescharacteristic
331C4×C8C2×D411yescharacteristic
332C4×C8#4111yescharacteristic
333C4×C8#4111yescharacteristic
334C24—21no—
335#41—21no—
336#41—21no—
337#41—21no—
338#41—21no—
339C2×Q8—21no—
340C2×Q8—21no—
341C2×Q8—21no—
342C2×Q8—21no—
343C2×Q8—21no—
344C2×Q8C4×D411yescharacteristic
345C2×Q8#13112yes—
346C2×D4—21no—
347C2×D4—21no—
348C2×D4—21no—
349C2×D4—21no—
350C2×D4—21no—
351C2×D4—21no—
352C2×D4—21no—
353C2×D4C4×D411yescharacteristic
354C22×C4—21no—
355C22×C4—21no—
356C22×C4—21no—
357C22×C4—21no—
358C22×C4—21no—
359C22×C4—21no—
360C22×C4—41no—
361C22×C4—41no—
362C22×C4—81no—
363C22×C4—81no—
364C22×C4C23×C411yescharacteristic
365C22×C4C2×#4111yescharacteristic
366C22×C4C2×#4111yescharacteristic
367Q16—42no—
368SD16—42no—
369SD16—42no—
370SD16—42no—
371SD16—42no—
372D8—42no—
373M16—21no—
374M16—21no—
375M16—21no—
376M16—21no—
377M16—21no—
378M16—21no—
379M16—21no—
380M16—21no—
381M16—21no—
382M16—21no—
383M16—21no—
384M16—42no—
385M16—42no—
386M16—82no—
387M16—82no—
388M16#13112yes—
389C2×C8—21no—
390C2×C8—21no—
391C2×C8—21no—
392C2×C8—21no—
393C2×C8—21no—
394C2×C8—22no—
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—22no—
404C2×C8—21no—
405C2×C8—41no—
406C2×C8—41no—
407C2×C8—41no—
408C2×C8—41no—
409C2×C8—41no—
410C2×C8—81no—
411C2×C8—82no—
412C2×C8—81no—
413C2×C8—81no—
414C2×C8C4×D411yescharacteristic
415C2×C8C4×D411yescharacteristic
416C2×C8C4×D411yescharacteristic
417C2×C8C4×D411yescharacteristic
418C2×C8#12111yescharacteristic
419C2×C8#12111yescharacteristic
420C2×C8#12111yescharacteristic
421C2×C8#12111yescharacteristic
422C4⋊C4—21no—
423C4⋊C4—21no—
424C4⋊C4—21no—
425C4⋊C4—21no—
426C4⋊C4—21no—
427C4⋊C4—21no—
428C4⋊C4—41no—
429C4⋊C4—41no—
430C4⋊C4C4×D411yescharacteristic
431C4⋊C4C4×D411yescharacteristic
432C22⋊C4—21no—
433C22⋊C4—21no—
434C22⋊C4—21no—
435C22⋊C4—41no—
436C22⋊C4—41no—
437C42—21no—
438C42—21no—
439C42—21no—
440C42—41no—
441C42—41no—
442C42C22×D411yescharacteristic
443C42C22×D411yescharacteristic
444C42C2×#4111yescharacteristic
445C42C2×#4111yescharacteristic
446C23—21no—
447C23—41no—
448C23—41no—
449C23—41no—
450C23—81no—
451C23—81no—
452C23C4×#4111yescharacteristic
453Q8—22no—
454Q8—21no—
455Q8—22no—
456Q8—21no—
457Q8—41no—
458Q8—41no—
459D4—21no—
460D4—21no—
461D4—41no—
462D4—41no—
463D4—41no—
464D4—41no—
465D4—41no—
466D4—41no—
467C2×C4—21no—
468C2×C4—21no—
469C2×C4—22no—
470C2×C4—21no—
471C2×C4—21no—
472C2×C4—21no—
473C2×C4—41no—
474C2×C4—42no—
475C2×C4—41no—
476C2×C4—41no—
477C2×C4—41no—
478C2×C4—41no—
479C2×C4—41no—
480C2×C4—41no—
481C2×C4—41no—
482C2×C4—41no—
483C2×C4—82no—
484C2×C4—81no—
485C2×C4—81no—
486C2×C4—81no—
487C2×C4—81no—
488C2×C4C2×C4×D411yescharacteristic
489C2×C4C2×C4×D411yescharacteristic
490C2×C4C2×#12111yescharacteristic
491C2×C4#52511yescharacteristic
492C2×C4#52911yescharacteristic
493C2×C4#53211yescharacteristic
494C2×C4C2×#13111yescharacteristic
495C2×C4C2×#13111yescharacteristic
496C2×C4#56811yescharacteristic
497C2×C4#56811yescharacteristic
498C8—21no—
499C8—21no—
500C8—22no—
501C8—21no—
502C8—21no—
503C8—21no—
504C8—21no—
505C8—21no—
506C8—42no—
507C8—41no—
508C8—41no—
509C8—41no—
510C8—41no—
511C8—41no—
512C8—41no—
513C8—41no—
514C8—41no—
515C8—41no—
516C8—81no—
517C8—81no—
518C8—81no—
519C8—81no—
520C8#47112yes—
521C22—21no—
522C22—21no—
523C22—41no—
524C22—41no—
525C22—41no—
526C22—81no—
527C22—81no—
528C22—81no—
529C22—161no—
530C22—161no—
531C22C4×#12111yescharacteristic
532C22#296611yescharacteristic
533C22#296711yescharacteristic
534C4—21no—
535C4—21no—
536C4—21no—
537C4—41no—
538C4—41no—
539C4—41no—
540C4—42no—
541C4—81no—
542C4—81no—
543C4#293311yescharacteristic
544C4#293311yescharacteristic
545C4#293311yescharacteristic
546C4#293511yescharacteristic
547C4C2×#47111yescharacteristic
548C4#306711yescharacteristic
549C2—21no—
550C2—21no—
551C2—81no—
552C2—161no—
553C2#2149611yescharacteristic
554C2C4×#47111yescharacteristic
555C2#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. #17331C2semidirect
6. #9667C2semidirect
7. #9607C2semidirect
8. #9552C2semidirect
9. #9551C2semidirect
10. #9096C2semidirect
12. #8607C2semidirect
13. #8297C2semidirect
14. #8166C2semidirect
15. C8×#101C2semidirect

Maximal subgroups

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

subgroupindex
#237182
#178912
#178702
C2×#15832
#173312
#96672
#96072
#95522
#95512
#90962
#90952
#86072
#82972
#81662
C8×#1012