Finite Group Databasebeta

← #225452

Subgroups — #225452

Order 512.

789 conjugacy classes of subgroups.

The subgroup lattice has 589 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22700C211yesmaximal, characteristic, complemented
2#22650C211yesmaximal, characteristic, complemented
3C4×#442C211yesmaximal, characteristic
4C2×#1594C211yesmaximal, characteristic
5C2×#1589C211yesmaximal, characteristic
6#9596C211yesmaximal, characteristic, complemented
7#9593C211yesmaximal, characteristic, complemented
8#8849C211yesmaximal, characteristic, complemented
9#8849C211yesmaximal, characteristic, complemented
10#8847C211yesmaximal, characteristic, complemented
11#8847C211yesmaximal, characteristic, complemented
12#8808C211yesmaximal, characteristic, complemented
13#8565C211yesmaximal, characteristic, complemented
14#8563C211yesmaximal, characteristic, complemented
15#8262C211yesmaximal, characteristic, complemented
16C2×#442C411yescharacteristic
17C2×#442C411yescharacteristic
18C2×C4×D8C2211yescharacteristic
19C2×#436C2211yescharacteristic
20C2×#436C2211yescharacteristic
21C2×#436C2211yescharacteristic
22C2×#436C2211yescharacteristic
23C2×#423—21no—
24C42×D4C2211yescharacteristic
25C42×D4C2211yescharacteristic
26#1939C2211yescharacteristic
27#1939C2211yescharacteristic
28#1939C2211yescharacteristic
29#1939C2211yescharacteristic
30#1851—21no—
31#1851—21no—
32#1851—21no—
33#1851—21no—
34#1842C2211yescharacteristic
35#1831C2211yescharacteristic
36#1797—21no—
37#1797—21no—
38#1776C411yescharacteristic
39#1776C411yescharacteristic
40C4×#106C2211yescharacteristic
41C4×C4⋊C8C2211yescharacteristic
42C4×C4⋊C8C2211yescharacteristic
43#1762C411yescharacteristic
44#1762C411yescharacteristic
45#1762C411yescharacteristic
46#1762C411yescharacteristic
47C4×#102C2211yescharacteristic
48C4×#102C2211yescharacteristic
49#1728—21no—
50C4×#97C2211yescharacteristic
51#1594—24no—
52#1589—24no—
53C2×#335C2211yescharacteristic
54C2×#335C2211yescharacteristic
55C2×#334C2211yescharacteristic
56C2×#334C2211yescharacteristic
57C2×D4⋊C8C2211yescharacteristic
58C2×D4⋊C8C2211yescharacteristic
59C2×D4⋊C8C2211yescharacteristic
60C2×D4⋊C8C2211yescharacteristic
61C2×#322C2211yescharacteristic
62C2×C82C2211yescharacteristic
63#1279C2211yescharacteristic
64#1279C2211yescharacteristic
65#1279C2211yescharacteristic
66#1279C2211yescharacteristic
67#1273C2211yescharacteristic
68#1273C2211yescharacteristic
69C22×D8C2×C411yescharacteristic
70C22×M16—21no—
71C22×M16—21no—
72C22×M16—21no—
73C22×M16—21no—
74C2×C4×D4C2×C411yescharacteristic
75C2×C4×D4C2×C411yescharacteristic
76C2×C4×D4C2×C411yescharacteristic
77C2×C4×D4C2×C411yescharacteristic
78C2×C4×D4C2311yescharacteristic
79C2×C4×D4C2311yescharacteristic
80C22×C42—21no—
81C22×C42—21no—
82#442—24no—
83#442—24no—
84C4×D8D414yes—
85C4×D8Q814yes—
86#436—22no—
87#436—22no—
88#436—22no—
89#436—22no—
90#436—41no—
91#436—41no—
92#436—41no—
93#436—41no—
94C2×#107C2×C411yescharacteristic
95C2×#106C2×C411yescharacteristic
96C2×#106C2×C411yescharacteristic
97#423—44no—
98C2×C4⋊C8—21no—
99C2×C4⋊C8—21no—
100C2×C4⋊C8—21no—
101C2×C4⋊C8—21no—
102C2×C4⋊C8—21no—
103C2×C4⋊C8—21no—
104C2×C4⋊C8—21no—
105C2×C4⋊C8D411yescharacteristic
106C2×C4⋊C8D411yescharacteristic
107C2×C4⋊C8C2311yescharacteristic
108C2×C4⋊C8C2311yescharacteristic
109C2×C4⋊C8C2311yescharacteristic
110C2×C4⋊C8C2311yescharacteristic
111C2×#102C2×C411yescharacteristic
112C2×#102C2×C411yescharacteristic
113C2×#102C2×C411yescharacteristic
114C2×#102C2×C411yescharacteristic
115C2×#102C2×C411yescharacteristic
116C2×#102C2×C411yescharacteristic
117C2×C22⋊C8—21no—
118C2×C22⋊C8—21no—
119C2×C22⋊C8—21no—
120C2×C22⋊C8—21no—
121C2×#97C2×C411yescharacteristic
122C2×#97C2×C411yescharacteristic
123C2×#97C2×C411yescharacteristic
124C2×#97C2×C411yescharacteristic
125C2×C4×C8C2311yescharacteristic
126C2×C4×C8C2311yescharacteristic
127C2×C4×C8C2311yescharacteristic
128C2×C4×C8C2311yescharacteristic
129C2×C4×C8C2311yescharacteristic
130C2×C4×C8C2311yescharacteristic
131C4×C4⋊C4C2×C411yescharacteristic
132C4×C4⋊C4C2×C411yescharacteristic
133C4×C4⋊C4C2×C411yescharacteristic
134C4×C4⋊C4C2×C411yescharacteristic
135C4×C4⋊C4C2311yescharacteristic
136C4×C4⋊C4C2311yescharacteristic
137C4×C22⋊C4—21no—
138C4×C22⋊C4—21no—
139C4×C22⋊C4—21no—
140C4×C22⋊C4—21no—
141C4×C22⋊C4—21no—
142C4×C22⋊C4—21no—
143C43C2311yescharacteristic
144#340C2×C411yescharacteristic
145#340C2×C411yescharacteristic
146#340C2×C411yescharacteristic
147#340C2×C411yescharacteristic
148#340C2×C411yescharacteristic
149#340C2×C411yescharacteristic
150#336—21no—
151#336—21no—
152#336—21no—
153#336—21no—
154#336—21no—
155#336—21no—
156#336—21no—
157#336D411yescharacteristic
158#336D411yescharacteristic
159#335—22no—
160#335—41no—
161#334—22no—
162#334—41no—
163D4⋊C8—22no—
164D4⋊C8—22no—
165D4⋊C8—22no—
166D4⋊C8—22no—
167#322—22no—
168C82—22no—
169C22×D4C22×C411yescharacteristic
170C22×D4C22×C411yescharacteristic
171C23×C4—21no—
172C23×C4—21no—
173C23×C4—21no—
174C23×C4—21no—
175C23×C4—21no—
176C23×C4—21no—
177C2×D8—22no—
178C2×D8—22no—
179C2×D8C4⋊C414yes—
180C2×M16—42no—
181C2×M16—42no—
182C2×M16—42no—
183C2×M16—42no—
184C2×M16—42no—
185C2×M16—42no—
186C2×M16—42no—
187C2×M16—42no—
188C2×M16—81no—
189C2×M16—81no—
190C2×M16—81no—
191C2×M16—81no—
192C22×C8—21no—
193C22×C8—21no—
194C22×C8—21no—
195C22×C8—21no—
196C22×C8—21no—
197C22×C8—21no—
198C22×C8C22×C411yescharacteristic
199C22×C8C22×C411yescharacteristic
200C22×C8C22×C411yescharacteristic
201C22×C8C22×C411yescharacteristic
202C22×C8C2×D411yescharacteristic
203C22×C8C2×D411yescharacteristic
204C22×C8C2×D411yescharacteristic
205C22×C8C2×D411yescharacteristic
206C4×D4—22no—
207C4×D4—22no—
208C4×D4—22no—
209C4×D4—22no—
210C4×D4—22no—
211C4×D4—22no—
212C4×D4—21no—
213C4×D4—21no—
214C4×D4—21no—
215C4×D4—21no—
216C4×D4—41no—
217C4×D4—41no—
218C4×D4C2×D412yes—
219C4×D4C2×D412yes—
220C4×D4C2×Q812yes—
221C4×D4C2×Q812yes—
222C2×C4⋊C4C22×C411yescharacteristic
223C2×C4⋊C4C22×C411yescharacteristic
224C2×C4⋊C4C22×C411yescharacteristic
225C2×C4⋊C4C22×C411yescharacteristic
226C2×C4⋊C4C22×C411yescharacteristic
227C2×C4⋊C4C22×C411yescharacteristic
228C2×C22⋊C4—21no—
229C2×C22⋊C4—21no—
230C2×C22⋊C4—21no—
231C2×C22⋊C4—21no—
232C2×C22⋊C4—21no—
233C2×C22⋊C4—21no—
234C2×C42—21no—
235C2×C42—21no—
236C2×C42—21no—
237C2×C42—21no—
238C2×C42—21no—
239C2×C42—21no—
240C2×C42—42no—
241C2×C42—42no—
242C2×C42—81no—
243C2×C42—81no—
244C2×C42C22×C411yescharacteristic
245C2×C42C22×C411yescharacteristic
246C2×C42C2×D411yescharacteristic
247C2×C42C2×D411yescharacteristic
248C2×C42#4111yescharacteristic
249C2×C42#4111yescharacteristic
250C2×C42C2411yescharacteristic
251#107C4⋊C414yes—
252#106—22no—
253#106M1614yescomplemented
254C4⋊C8—22no—
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—41no—
264C4⋊C8—42no—
265C4⋊C8—42no—
266C4⋊C8—42no—
267C4⋊C8—42no—
268C4⋊C8—81no—
269C4⋊C8—81no—
270C4⋊C8—81no—
271#102—22no—
272#102—22no—
273#102—22no—
274#102—22no—
275#102C4⋊C414yes—
276#102C4⋊C414yes—
277C22⋊C8—42no—
278C22⋊C8—42no—
279C22⋊C8—42no—
280C22⋊C8—42no—
281#97—22no—
282#97—22no—
283#97—41no—
284#97M1614yescomplemented
285C4×C8—21no—
286C4×C8—21no—
287C4×C8—21no—
288C4×C8—21no—
289C4×C8—21no—
290C4×C8—21no—
291C4×C8—21no—
292C4×C8—21no—
293C4×C8—21no—
294C4×C8—21no—
295C4×C8C2×D412yes—
296C4×C8C2×Q812yes—
297#95—21no—
298#95—21no—
299#95—21no—
300#95—21no—
301#95—21no—
302#95—21no—
303C24—21no—
304C24—21no—
305C2×D4—21no—
306C2×D4—21no—
307C2×D4—21no—
308C2×D4—21no—
309C2×D4—21no—
310C2×D4—21no—
311C2×D4—21no—
312C2×D4—21no—
313C2×D4—21no—
314C2×D4—21no—
315C2×D4C2×C4⋊C412yes—
316C2×D4C2×C4⋊C412yes—
317C22×C4—21no—
318C22×C4—21no—
319C22×C4—21no—
320C22×C4—21no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—22no—
325C22×C4—21no—
326C22×C4—21no—
327C22×C4—42no—
328C22×C4—42no—
329C22×C4—42no—
330C22×C4—42no—
331C22×C4—42no—
332C22×C4—42no—
333C22×C4—41no—
334C22×C4—41no—
335C22×C4—41no—
336C22×C4—41no—
337C22×C4—41no—
338C22×C4—41no—
339C22×C4—41no—
340C22×C4—41no—
341C22×C4—42no—
342C22×C4—42no—
343C22×C4—81no—
344C22×C4—81no—
345C22×C4—81no—
346C22×C4—81no—
347C22×C4C4×D411yescharacteristic
348C22×C4C4×D411yescharacteristic
349C22×C4C4×D411yescharacteristic
350C22×C4C4×D411yescharacteristic
351C22×C4C23×C411yescharacteristic
352C22×C4C22×D411yescharacteristic
353C22×C4C2×#4111yescharacteristic
354D8—28no—
355M16—84no—
356M16—84no—
357M16—84no—
358M16—84no—
359C2×C8—21no—
360C2×C8—21no—
361C2×C8—21no—
362C2×C8—21no—
363C2×C8—21no—
364C2×C8—21no—
365C2×C8—22no—
366C2×C8—21no—
367C2×C8—21no—
368C2×C8—21no—
369C2×C8—21no—
370C2×C8—21no—
371C2×C8—21no—
372C2×C8—21no—
373C2×C8—21no—
374C2×C8—21no—
375C2×C8—21no—
376C2×C8—21no—
377C2×C8—21no—
378C2×C8—41no—
379C2×C8—41no—
380C2×C8—41no—
381C2×C8—41no—
382C2×C8—41no—
383C2×C8—41no—
384C2×C8—41no—
385C2×C8—41no—
386C2×C8—41no—
387C2×C8—41no—
388C2×C8—41no—
389C2×C8—41no—
390C2×C8—41no—
391C2×C8—41no—
392C2×C8—41no—
393C2×C8—41no—
394C2×C8—41no—
395C2×C8C2×C4⋊C412yes—
396C2×C8C2×C4⋊C412yes—
397C2×C8C2×M1612yes—
398C2×C8C2×M1612yes—
399C2×C8C2×M1612yes—
400C2×C8AGL(1,ℤ/8)12yes—
401C4⋊C4—21no—
402C4⋊C4—21no—
403C4⋊C4—21no—
404C4⋊C4—21no—
405C4⋊C4—21no—
406C4⋊C4—21no—
407C4⋊C4—21no—
408C4⋊C4—21no—
409C4⋊C4C2×C4⋊C412yes—
410C4⋊C4C2×C4⋊C412yes—
411C4⋊C4C2×M1612yes—
412C4⋊C4C2×M1612yes—
413C22⋊C4—22no—
414C22⋊C4—22no—
415C22⋊C4—22no—
416C22⋊C4—22no—
417C22⋊C4—42no—
418C22⋊C4—42no—
419C22⋊C4—42no—
420C22⋊C4—42no—
421C42—21no—
422C42—21no—
423C42—21no—
424C42—21no—
425C42—21no—
426C42—21no—
427C42—41no—
428C42—41no—
429C42—42no—
430C42—42no—
431C42—41no—
432C42—41no—
433C42—41no—
434C42—41no—
435C42—81no—
436C42—81no—
437C42C2×M1612yes—
438C42#13112yes—
439C42C22×D411yescharacteristic
440C42C22×Q811yescharacteristic
441C42H(5,𝔽2)11yescharacteristic
442C422−1+411yescharacteristic
443C23—22no—
444C23—22no—
445C23—41no—
446C23—41no—
447C23—41no—
448C23—41no—
449C23—41no—
450C23—41no—
451C23—41no—
452C23—41no—
453C23—41no—
454C23—41no—
455C23—41no—
456C23—41no—
457C23C2×C4×D411yescharacteristic
458D4—22no—
459D4—22no—
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—21no—
472C2×C4—22no—
473C2×C4—22no—
474C2×C4—21no—
475C2×C4—21no—
476C2×C4—22no—
477C2×C4—22no—
478C2×C4—21no—
479C2×C4—21no—
480C2×C4—21no—
481C2×C4—21no—
482C2×C4—41no—
483C2×C4—41no—
484C2×C4—41no—
485C2×C4—41no—
486C2×C4—41no—
487C2×C4—41no—
488C2×C4—41no—
489C2×C4—41no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—41no—
499C2×C4—41no—
500C2×C4—41no—
501C2×C4—41no—
502C2×C4—41no—
503C2×C4—41no—
504C2×C4—84no—
505C2×C4—84no—
506C2×C4—84no—
507C2×C4—84no—
508C2×C4#43612yes—
509C2×C4#43612yes—
510C2×C4C22×C4⋊C411yescharacteristic
511C2×C4#52011yescharacteristic
512C2×C4D4211yescharacteristic
513C2×C4#54711yescharacteristic
514C2×C4D4×Q811yescharacteristic
515C2×C4#55411yescharacteristic
516C2×C4C22×M1611yescharacteristic
517C2×C4C22×M1611yescharacteristic
518C2×C4C2×#13111yescharacteristic
519C2×C4#56811yescharacteristic
520C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
521C2×C4#57511yescharacteristic
522C2×C4#57611yescharacteristic
523C2×C4#57711yescharacteristic
524C8—22no—
525C8—22no—
526C8—22no—
527C8—42no—
528C8—41no—
529C8—41no—
530C8—41no—
531C8—41no—
532C8—81no—
533C8—81no—
534C8—81no—
535C8—81no—
536C8—81no—
537C8#42314yescomplemented
538C22—42no—
539C22—42no—
540C22—42no—
541C22—42no—
542C22—42no—
543C22—42no—
544C22—81no—
545C22—81no—
546C22—81no—
547C22—81no—
548C22—81no—
549C22—81no—
550C22—81no—
551C22—81no—
552C22D4×C4⋊C411yescharacteristic
553C22C2×#43611yescharacteristic
554C22D4×M1611yescharacteristic
555C22C4×AGL(1,ℤ/8)11yescharacteristic
556C22#294611yescharacteristic
557C22#295511yescharacteristic
558C22#295611yescharacteristic
559C4—21no—
560C4—21no—
561C4—22no—
562C4—22no—
563C4—41no—
564C4—42no—
565C4—41no—
566C4—42no—
567C4—42no—
568C4—42no—
569C4—81no—
570C4—81no—
571C4C2×#42311yescharacteristic
572C4#290611yescharacteristic
573C4#298911yescharacteristic
574C4#299111yescharacteristic
575C4#326411yescharacteristic
576C4#326711yescharacteristic
577C4#338411yescharacteristic
578C4#338911yescharacteristic
579C2—82no—
580C2—82no—
581C2#1789111yescharacteristic
582C2#1789311yescharacteristic
583C2#1804811yescharacteristic
584C2#1805211yescharacteristic
585C2#2265011yescharacteristic
586C2#2267611yescharacteristic
587C2#2268711yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC23C2×C4×D4
prime radicalC22×D8C2×C4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C4×C4⋊C8 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. #22700C2semidirect
2. #22650C2semidirect
6. #9596C2semidirect
7. #9593C2semidirect
8. #8849C2semidirect
9. #8849C2semidirect
10. #8847C2semidirect
11. #8847C2semidirect
12. #8808C2semidirect
13. #8565C2semidirect
14. #8563C2semidirect
15. #8262C2semidirect
253. #106M16semidirect
284. #97M16semidirect
537. C8#423semidirect

Maximal subgroups

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

subgroupindex
#227002
#226502
C4×#4422
C2×#15942
C2×#15892
#95962
#95932
#88492
#88472
#88082
#85652
#85632
#82622