Finite Group Databasebeta

← #211112

Subgroups — #211112

Order 512.

618 conjugacy classes of subgroups.

The subgroup lattice has 572 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23716C211yesmaximal, characteristic
2#17891C211yesmaximal, characteristic, complemented
3#17869C211yesmaximal, characteristic, complemented
4C2×#1586C211yesmaximal, characteristic, complemented
5#17330C211yesmaximal, characteristic, complemented
6#9668C211yesmaximal, characteristic, complemented
7#9607C211yesmaximal, characteristic, complemented
8#9552C211yesmaximal, characteristic, complemented
9#9552C211yesmaximal, characteristic, complemented
10#9095C211yesmaximal, characteristic, complemented
11#9095C211yesmaximal, characteristic, complemented
12#8607C211yesmaximal, characteristic, complemented
13#8296C211yesmaximal, characteristic, complemented
14#8166C211yesmaximal, characteristic, complemented
15C8×#100C211yesmaximal, characteristic, complemented
16#2965C2211yescharacteristic
17C8×#41C2211yescharacteristic
18C4×AGL(1,ℤ/8)C2211yescharacteristic
19C4×AGL(1,ℤ/8)C2211yescharacteristic
20C2×C4×D8C2211yescharacteristic
21C2×#436C2211yescharacteristic
22C2×C8×D4C2211yescharacteristic
23#2922C2211yescharacteristic
24#2089C411yescharacteristic
25#2078C411yescharacteristic
26#2063C411yescharacteristic
27#2063C411yescharacteristic
28#2051C411yescharacteristic
29#2051C411yescharacteristic
30#2045C411yescharacteristic
31#2018C411yescharacteristic
32C4×#108C2211yescharacteristic
33C4×#100C2211yescharacteristic
34C4×#100C2211yescharacteristic
35#1594—21no—
36#1592—21no—
37#1590—21no—
38#1589—21no—
39#1586—21no—
40#1586—21no—
41#1586—21no—
42#1586—21no—
43#1586—21no—
44#1585—21no—
45#1582—21no—
46#1581—21no—
47#1565C2211yescharacteristic
48#1565C2211yescharacteristic
49#1564C2211yescharacteristic
50C2×#334C2211yescharacteristic
51#1478C2211yescharacteristic
52#1478C2211yescharacteristic
53#1477C2211yescharacteristic
54#1477C2211yescharacteristic
55C2×D4⋊C8C2211yescharacteristic
56C2×D4⋊C8C2211yescharacteristic
57#1460C2211yescharacteristic
58#1460C2211yescharacteristic
59#1454—21no—
60#1453—21no—
61#1451—21no—
62#1451C2211yescharacteristic
63#1451C2211yescharacteristic
64#1450—21no—
65#1450C2211yescharacteristic
66C8×M16C2211yescharacteristic
67C8×M16C2211yescharacteristic
68C2×#322C2211yescharacteristic
69#1283C2211yescharacteristic
70#1283C2211yescharacteristic
71#1283C2211yescharacteristic
72#1283C2211yescharacteristic
73#1278C2211yescharacteristic
74#1278C2211yescharacteristic
75C2×AGL(1,ℤ/8)C2×C411yescharacteristic
76C2×AGL(1,ℤ/8)C2×C411yescharacteristic
77C22×D8C2×C411yescharacteristic
78C22×M16—21no—
79C23×C8—21no—
80C4×#41C2311yescharacteristic
81C2×C4×D4C2311yescharacteristic
82C2×C4×D4C2311yescharacteristic
83#446D411yescharacteristic
84#446D411yescharacteristic
85C8×Q8—21no—
86#442—21no—
87#442—21no—
88#440—21no—
89#440—21no—
90C4×SD16—21no—
91C4×SD16—21no—
92C4×D8—21no—
93C4×D8—21no—
94C4×D8—21no—
95C4×D8—21no—
96C4×D8—21no—
97C4×D8—21no—
98#436—21no—
99#436—21no—
100#436—21no—
101#436—21no—
102#436D411yescharacteristic
103#436D411yescharacteristic
104#435—21no—
105#435—21no—
106C8×D4—21no—
107C8×D4—22no—
108C8×D4—21no—
109C8×D4—21no—
110C8×D4—21no—
111C8×D4—21no—
112#433—21no—
113#433—21no—
114#431—21no—
115#431—21no—
116#431—21no—
117#431—21no—
118#431C2311yescharacteristic
119#431C2311yescharacteristic
120#431C2311yescharacteristic
121#431C2311yescharacteristic
122C2×#108C2×C411yescharacteristic
123C2×#108C2×C411yescharacteristic
124#428C2×C411yescharacteristic
125#428C2×C411yescharacteristic
126C2×#107C2×C411yescharacteristic
127#424C2×C411yescharacteristic
128#424C2×C411yescharacteristic
129C2×C4⋊C8C2311yescharacteristic
130C2×C4⋊C8C2311yescharacteristic
131#418C2×C411yescharacteristic
132#418C2×C411yescharacteristic
133#417C2×C411yescharacteristic
134#417C2×C411yescharacteristic
135C2×#102C2×C411yescharacteristic
136C2×#102C2×C411yescharacteristic
137C2×#100C2×C411yescharacteristic
138C2×#100C2×C411yescharacteristic
139C2×#100C2×C411yescharacteristic
140C2×#100C2×C411yescharacteristic
141#408C2×C411yescharacteristic
142#408C2×C411yescharacteristic
143C2×C22⋊C8—21no—
144C2×C22⋊C8—21no—
145C4×M16—21no—
146C4×M16C2311yescharacteristic
147C4×M16C2311yescharacteristic
148C4×M16C2311yescharacteristic
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—
169D4⋊C8—21no—
170D4⋊C8—21no—
171D4⋊C8—21no—
172D4⋊C8—21no—
173D4⋊C8—21no—
174D4⋊C8—21no—
175D4⋊C8—21no—
176D4⋊C8—21no—
177D4⋊C8—21no—
178D4⋊C8—21no—
179#323—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×D4C22×C411yescharacteristic
197C22×D4C22×C411yescharacteristic
198C23×C4—21no—
199C23×C4—21no—
200AGL(1,ℤ/8)—22no—
201AGL(1,ℤ/8)—22no—
202AGL(1,ℤ/8)—41no—
203AGL(1,ℤ/8)—41no—
204C2×SD16—21no—
205C2×SD16—21no—
206C2×D8—21no—
207C2×D8—21no—
208C2×D8—22no—
209C2×D8—21no—
210C2×D8—21no—
211C2×D8—41no—
212C2×M16—21no—
213C2×M16—21no—
214C2×M16—41no—
215C2×M16—41no—
216C2×M16—82no—
217C2×M16C22×C411yescharacteristic
218C2×M16C22×C411yescharacteristic
219C2×M16C22×C411yescharacteristic
220C2×M16C22×C411yescharacteristic
221C2×M16C22×C411yescharacteristic
222C2×M16C22×C411yescharacteristic
223C22×C8—22no—
224C22×C8—21no—
225C22×C8—21no—
226C22×C8—21no—
227C22×C8—41no—
228C22×C8—41no—
229C22×C8—82no—
230C22×C8C22×C411yescharacteristic
231C22×C8C22×C411yescharacteristic
232C22×C8#4111yescharacteristic
233C22×C8#4111yescharacteristic
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×D4—21no—
243C4×D4—21no—
244C4×D4—21no—
245C4×D4—21no—
246C4×D4C2×D411yescharacteristic
247C4×D4#4111yescharacteristic
248#117—21no—
249#117C22×C411yescharacteristic
250C2×C4⋊C4C22×C411yescharacteristic
251C2×C4⋊C4C22×C411yescharacteristic
252C2×C22⋊C4—21no—
253C2×C22⋊C4—21no—
254C2×C42—21no—
255C2×C42C2411yescharacteristic
256#108—22no—
257#108—41no—
258#107—21no—
259#107—21no—
260#107—21no—
261#107—21no—
262#106—21no—
263#106—21no—
264C4⋊C8—22no—
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—21no—
276C4⋊C8—21no—
277C4⋊C8—21no—
278C4⋊C8C2×D411yescharacteristic
279C4⋊C8C2×D411yescharacteristic
280#103—21no—
281#103—21no—
282#102—21no—
283#102—21no—
284#102—21no—
285#102—21no—
286#102—21no—
287#102—21no—
288#102—21no—
289#102—21no—
290#102—21no—
291#102—21no—
292#100—22no—
293#100—22no—
294#100—41no—
295#100—41no—
296C22⋊C8—22no—
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—21no—
306C22⋊C8—41no—
307C22⋊C8—41no—
308C22⋊C8—41no—
309C22⋊C8—41no—
310C22⋊C8#4112yes—
311#97—21no—
312#97—21no—
313#97—21no—
314#97—21no—
315#97—21no—
316C4×C8—22no—
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×C8C2×D411yescharacteristic
330C4×C8C2×D411yescharacteristic
331C4×C8C2×D411yescharacteristic
332C4×C8C2×D411yescharacteristic
333C4×C8#4111yescharacteristic
334C4×C8#4111yescharacteristic
335C24—21no—
336C24—21no—
337#41—21no—
338#41—21no—
339#41—21no—
340#41—21no—
341C2×Q8C4×D411yescharacteristic
342C2×D4—21no—
343C2×D4—21no—
344C2×D4—21no—
345C2×D4—21no—
346C2×D4—21no—
347C2×D4—21no—
348C2×D4—21no—
349C2×D4—21no—
350C2×D4—21no—
351C2×D4—21no—
352C2×D4—21no—
353C2×D4—21no—
354C2×D4C4×D411yescharacteristic
355C2×D4#13112yes—
356C22×C4—21no—
357C22×C4—21no—
358C22×C4—21no—
359C22×C4—21no—
360C22×C4—21no—
361C22×C4—41no—
362C22×C4—41no—
363C22×C4—41no—
364C22×C4—41no—
365C22×C4—81no—
366C22×C4—81no—
367C22×C4—81no—
368C22×C4—81no—
369C22×C4C23×C411yescharacteristic
370C22×C4C2×#4111yescharacteristic
371C22×C4C2×#4111yescharacteristic
372SD16—42no—
373SD16—42no—
374D8—42no—
375D8—42no—
376D8—42no—
377D8—42no—
378M16—21no—
379M16—21no—
380M16—21no—
381M16—21no—
382M16—21no—
383M16—21no—
384M16—21no—
385M16—21no—
386M16—21no—
387M16—21no—
388M16—21no—
389M16—81no—
390M16—81no—
391M16—82no—
392M16—82no—
393M16#13112yes—
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—21no—
407C2×C8—22no—
408C2×C8—21no—
409C2×C8—41no—
410C2×C8—41no—
411C2×C8—42no—
412C2×C8—44no—
413C2×C8—41no—
414C2×C8—41no—
415C2×C8—41no—
416C2×C8—41no—
417C2×C8—82no—
418C2×C8—82no—
419C2×C8—81no—
420C2×C8—81no—
421C2×C8C4×D411yescharacteristic
422C2×C8C4×D411yescharacteristic
423C2×C8C4×D411yescharacteristic
424C2×C8C4×D411yescharacteristic
425C2×C8#12111yescharacteristic
426C2×C8#12111yescharacteristic
427C2×C8#12111yescharacteristic
428C2×C8#12111yescharacteristic
429C4⋊C4—21no—
430C4⋊C4—21no—
431C4⋊C4—21no—
432C4⋊C4—21no—
433C4⋊C4—21no—
434C4⋊C4—21no—
435C4⋊C4C4×D411yescharacteristic
436C4⋊C4C4×D411yescharacteristic
437C22⋊C4—21no—
438C22⋊C4—21no—
439C22⋊C4—21no—
440C22⋊C4—41no—
441C22⋊C4—41no—
442C22⋊C4—41no—
443C22⋊C4—41no—
444C42—21no—
445C42—21no—
446C42—21no—
447C42C22×D411yescharacteristic
448C42C22×D411yescharacteristic
449C42C2×#4111yescharacteristic
450C42C2×#4111yescharacteristic
451C23—21no—
452C23—22no—
453C23—41no—
454C23—41no—
455C23—41no—
456C23—41no—
457C23—41no—
458C23—81no—
459C23—81no—
460C23—81no—
461C23—81no—
462C23C4×#4111yescharacteristic
463Q8—21no—
464Q8—21no—
465D4—22no—
466D4—21no—
467D4—21no—
468D4—22no—
469D4—41no—
470D4—41no—
471D4—41no—
472D4—41no—
473D4—41no—
474D4—41no—
475D4—41no—
476D4—41no—
477C2×C4—21no—
478C2×C4—21no—
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—82no—
491C2×C4—82no—
492C2×C4—81no—
493C2×C4—81no—
494C2×C4—81no—
495C2×C4—81no—
496C2×C4—81no—
497C2×C4—81no—
498C2×C4C2×C4×D411yescharacteristic
499C2×C4C2×C4×D411yescharacteristic
500C2×C4C2×#12111yescharacteristic
501C2×C4#52511yescharacteristic
502C2×C4#52911yescharacteristic
503C2×C4#53211yescharacteristic
504C2×C4C2×#13111yescharacteristic
505C2×C4C2×#13111yescharacteristic
506C2×C4#56811yescharacteristic
507C2×C4#56811yescharacteristic
508C8—22no—
509C8—21no—
510C8—21no—
511C8—22no—
512C8—21no—
513C8—21no—
514C8—21no—
515C8—21no—
516C8—21no—
517C8—21no—
518C8—42no—
519C8—41no—
520C8—41no—
521C8—41no—
522C8—41no—
523C8—41no—
524C8—41no—
525C8—41no—
526C8—41no—
527C8—81no—
528C8—81no—
529C8—81no—
530C8—81no—
531C22—21no—
532C22—21no—
533C22—41no—
534C22—41no—
535C22—41no—
536C22—42no—
537C22—82no—
538C22—82no—
539C22—81no—
540C22—81no—
541C22—81no—
542C22—81no—
543C22—81no—
544C22—161no—
545C22—161no—
546C22C4×#12111yescharacteristic
547C22#296611yescharacteristic
548C22#296711yescharacteristic
549C4—21no—
550C4—21no—
551C4—21no—
552C4—41no—
553C4—41no—
554C4—41no—
555C4—81no—
556C4—81no—
557C4#293311yescharacteristic
558C4#293311yescharacteristic
559C4#293311yescharacteristic
560C4#293511yescharacteristic
561C4#306811yescharacteristic
562C4#306811yescharacteristic
563C2—21no—
564C2—21no—
565C2—81no—
566C2—82no—
567C2—161no—
568C2#2149611yescharacteristic
569C2#2155511yescharacteristic
570C2#2155811yescharacteristic
subgroupG/H
centerC2×C4C2×#121
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC22C4×#121
prime radical#2018C4
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
2. #17891C2semidirect
3. #17869C2semidirect
4. C2×#1586C2semidirect
5. #17330C2semidirect
6. #9668C2semidirect
7. #9607C2semidirect
8. #9552C2semidirect
9. #9552C2semidirect
10. #9095C2semidirect
11. #9095C2semidirect
12. #8607C2semidirect
13. #8296C2semidirect
14. #8166C2semidirect
15. C8×#100C2semidirect

Maximal subgroups

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

subgroupindex
#237162
#178912
#178692
C2×#15862
#173302
#96682
#96072
#95522
#90952
#86072
#82962
#81662
C8×#1002