Finite Group Databasebeta

← #505194

Subgroups — #505194

Order 512.

879 conjugacy classes of subgroups.

The subgroup lattice has 585 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2240C211yesmaximal, characteristic, complemented
2C22×#357C211yesmaximal, characteristic, complemented
3#26065C211yesmaximal, characteristic, complemented
4#25779C211yesmaximal, characteristic
5#13480C211yesmaximal, characteristic, complemented
6#13407C211yesmaximal, characteristic, complemented
7#13407C211yesmaximal, characteristic, complemented
8#13363C211yesmaximal, characteristic, complemented
9#13202C211yesmaximal, characteristic, complemented
10#13147C211yesmaximal, characteristic, complemented
11#13147C211yesmaximal, characteristic, complemented
12#13059C211yesmaximal, characteristic, complemented
13#12991C211yesmaximal, characteristic, complemented
14#12731C211yesmaximal, characteristic, complemented
15C16×C22⋊C4C211yesmaximal, characteristic, complemented
16C23×D8C2211yescharacteristic
17C23×C16C2211yescharacteristic
18C2×#500C2211yescharacteristic, complemented
19C2×#493C2211yescharacteristic
20C2×#493C2211yescharacteristic
21C2×#466C2211yescharacteristic
22C2×#447—21no—
23C22×#107C2211yescharacteristic
24#2440—21no—
25#2433—21no—
26#2240—24no—
27C2×#357—24no—
28C2×#357C2211yescharacteristic
29C2×#357C2211yescharacteristic
30C2×#357C2211yescharacteristic
31C2×#357C2211yescharacteristic
32C2×#357C2211yescharacteristic
33C2×#357C2211yescharacteristic
34C2×#357C2211yescharacteristic
35C2×#357C2211yescharacteristic
36C2×#357C2211yescharacteristic, complemented
37C2×#357C2211yescharacteristic, complemented
38C2×#357C2211yescharacteristic, complemented
39C2×#357C2211yescharacteristic, complemented
40C2×C22⋊C16C2211yescharacteristic
41C2×C4×C16C2211yescharacteristic, complemented
42C2×C4×C16C2211yescharacteristic, complemented
43#2083—21no—
44#2057—21no—
45#2011—21no—
46#2011—21no—
47#1970C2211yescharacteristic
48#1970C2211yescharacteristic
49#1942C2211yescharacteristic, complemented
50#1942C2211yescharacteristic, complemented
51#1852—21no—
52#1845C2211yescharacteristic, complemented
53#1808C2211yescharacteristic
54C8×C22⋊C4C2211yescharacteristic
55#1380C2211yescharacteristic, complemented
56#1380C2211yescharacteristic, complemented
57#1380C2211yescharacteristic, complemented
58#1380C2211yescharacteristic, complemented
59#1374C2211yescharacteristic, complemented
60#1374C2211yescharacteristic, complemented
61C23×D4—21no—
62C22×D8—22no—
63C22×D8—22no—
64C22×D8—21no—
65C22×D8—21no—
66C22×D8—21no—
67C22×D8—22no—
68C22×D8—22no—
69C22×D8—41no—
70C22×D8—41no—
71C22×D8C2311yescharacteristic
72C22×D8C2311yescharacteristic
73C22×D8C2311yescharacteristic
74C23×C8C2311yescharacteristic
75C2×C4⋊Q8—21no—
76C2×#127—21no—
77C2×#127—21no—
78C2×C22⋊Q8—41no—
79C2×#121—21no—
80C2×#120—41no—
81C2×#120—41no—
82C22×C4⋊C4—21no—
83C22×C16—22no—
84C22×C16—22no—
85C22×C16—41no—
86C22×C16C2311yescharacteristic
87C22×C16C2311yescharacteristic
88C22×C16C2311yescharacteristic
89C22×C16C2311yescharacteristic
90#500—22no—
91#500—41no—
92#493—22no—
93#493—22no—
94#493—41no—
95#493—41no—
96#466—22no—
97#466—22no—
98#447—44no—
99C2×#107—22no—
100C2×#107—21no—
101C2×#107—21no—
102C2×#107—21no—
103C2×#107—41no—
104C2×#107C2311yescharacteristic
105C2×#107C2311yescharacteristic
106C2×#107C2311yescharacteristic
107C2×#107C2311yescharacteristic
108C2×#107C2311yescharacteristic
109C2×#102—21no—
110C2×#102—21no—
111C2×#102—21no—
112C2×C22⋊C8D411yescharacteristic
113C2×C4×C8C2311yescharacteristic
114C2×C4×C8C2311yescharacteristic
115#398—41no—
116#395—41no—
117#392—41no—
118#392—41no—
119#390—21no—
120#390—21no—
121#389—21no—
122#389—21no—
123#385—41no—
124#384—21no—
125#380—21no—
126C4×C22⋊C4D411yescharacteristic
127#357—22no—
128#357—22no—
129#357—22no—
130#357—22no—
131#357—22no—
132#357—22no—
133#357—22no—
134#357—41no—
135#357—41no—
136#357—41no—
137#357—41no—
138#357D414yescomplemented
139C22⋊C16—22no—
140C4×C16—22no—
141C4×C16—41no—
142#340—21no—
143#340—21no—
144#340—21no—
145#336D411yescharacteristic
146#336D411yescharacteristic
147C25—41no—
148C22×Q8—41no—
149C22×D4—21no—
150C22×D4—21no—
151C22×D4—21no—
152C22×D4—42no—
153C22×D4—42no—
154C22×D4—41no—
155C22×D4—42no—
156C22×D4—41no—
157C22×D4—41no—
158C22×D4—41no—
159C22×D4—41no—
160C22×D4—41no—
161C22×D4—42no—
162C22×D4—81no—
163C22×D4—81no—
164C23×C4—41no—
165C23×C4C2×D411yescharacteristic
166C2×D8—28no—
167C2×D8—21no—
168C2×D8—21no—
169C2×D8—21no—
170C2×D8—21no—
171C2×D8—21no—
172C2×D8—21no—
173C2×D8—21no—
174C2×D8—21no—
175C2×D8—21no—
176C2×D8—21no—
177C2×D8—21no—
178C2×D8—21no—
179C2×D8—21no—
180C2×D8—21no—
181C2×D8—21no—
182C2×D8—21no—
183C2×D8—44no—
184C2×D8—44no—
185C2×D8—42no—
186C2×D8—42no—
187C2×D8—44no—
188C2×D8—42no—
189C2×D8—42no—
190C2×D8—42no—
191C2×D8—42no—
192C2×D8—42no—
193C2×D8—81no—
194C2×D8—81no—
195C2×D8C2×D412yes—
196C2×D8#4112yes—
197C22×C8—21no—
198C22×C8—21no—
199C22×C8—21no—
200C22×C8—21no—
201C22×C8C2×D411yescharacteristic
202C22×C8C2×D411yescharacteristic
203C22×C8C2×D411yescharacteristic
204C22×C8#4112yes—
205C22×C8#4112yes—
206C22×C8C2411yescharacteristic
207C4⋊Q8—42no—
208C4⋊Q8—81no—
209#127—42no—
210#127—42no—
211#127—81no—
212#127—81no—
213C22⋊Q8—84no—
214#121—42no—
215#121—42no—
216#120—84no—
217#120—84no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—21no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4—42no—
224C2×C4⋊C4—41no—
225C2×C4⋊C4—41no—
226C2×C4⋊C4—41no—
227C2×C4⋊C4—41no—
228C2×C4⋊C4—41no—
229C2×C4⋊C4—41no—
230C2×C4⋊C4—81no—
231C2×C22⋊C4—41no—
232C2×C22⋊C4—41no—
233C2×C22⋊C4—41no—
234C2×C22⋊C4—41no—
235C2×C22⋊C4—41no—
236C2×C22⋊C4—41no—
237C2×C22⋊C4D811yescharacteristic
238C2×C22⋊C4D811yescharacteristic
239C2×C42C2×D411yescharacteristic
240C2×C42C2×D411yescharacteristic
241C2×C16—21no—
242C2×C16—21no—
243C2×C16—21no—
244C2×C16—21no—
245C2×C16—21no—
246C2×C16—21no—
247C2×C16—21no—
248C2×C16—21no—
249C2×C16—21no—
250C2×C16—21no—
251C2×C16—21no—
252C2×C16—44no—
253C2×C16C2×D412yes—
254#107—21no—
255#107—21no—
256#107—21no—
257#107—21no—
258#107—21no—
259#107—21no—
260#107—21no—
261#107—21no—
262#107—21no—
263#107—42no—
264#107—42no—
265#107—81no—
266#107C2×D412yes—
267#102—42no—
268#102—42no—
269#102—42no—
270C22⋊C8SD1614yes—
271C4×C8—21no—
272C4×C8—21no—
273C4×C8—21no—
274C4×C8—21no—
275#95—41no—
276#95—41no—
277#95—41no—
278#95D811yescharacteristic
279#95D811yescharacteristic
280C24—41no—
281C24—41no—
282C24—41no—
283C24—82no—
284C24—82no—
285C24—82no—
286C24—82no—
287C24—82no—
288C24—161no—
289C24—161no—
290C24C2×D811yescharacteristic
291C2×Q8—82no—
292C2×Q8—82no—
293C2×Q8—161no—
294C2×D4—22no—
295C2×D4—22no—
296C2×D4—48no—
297C2×D4—41no—
298C2×D4—41no—
299C2×D4—41no—
300C2×D4—41no—
301C2×D4—41no—
302C2×D4—41no—
303C2×D4—41no—
304C2×D4—41no—
305C2×D4—41no—
306C2×D4—41no—
307C2×D4—41no—
308C2×D4—41no—
309C2×D4—41no—
310C2×D4—41no—
311C2×D4—41no—
312C2×D4—41no—
313C2×D4—84no—
314C2×D4—84no—
315C2×D4—82no—
316C2×D4—82no—
317C2×D4—84no—
318C2×D4—82no—
319C2×D4—82no—
320C2×D4—82no—
321C2×D4—82no—
322C2×D4—82no—
323C2×D4—82no—
324C2×D4—82no—
325C2×D4—82no—
326C2×D4—82no—
327C2×D4—82no—
328C2×D4—82no—
329C2×D4—82no—
330C2×D4—161no—
331C2×D4—161no—
332C2×D4—161no—
333C2×D4—161no—
334C22×C4—21no—
335C22×C4—21no—
336C22×C4—21no—
337C22×C4—21no—
338C22×C4—41no—
339C22×C4—41no—
340C22×C4—41no—
341C22×C4—41no—
342C22×C4—41no—
343C22×C4—82no—
344C22×C4—82no—
345C22×C4—161no—
346C22×C4#12412yes—
347C22×C4C2×D811yescharacteristic
348C22×C4C2×D811yescharacteristic
349C22×C4C2×D811yescharacteristic
350C22×C4C2×D811yescharacteristic
351C22×C4C2×D811yescharacteristic
352C22×C4C2×SD1612yes—
353C22×C4C22×D411yescharacteristic
354D8—22no—
355D8—22no—
356D8—22no—
357D8—22no—
358D8—48no—
359D8—41no—
360D8—41no—
361D8—41no—
362D8—41no—
363D8—41no—
364D8—41no—
365D8—41no—
366D8—41no—
367D8—84no—
368D8—84no—
369C2×C8—21no—
370C2×C8—21no—
371C2×C8—21no—
372C2×C8—21no—
373C2×C8—21no—
374C2×C8—21no—
375C2×C8—22no—
376C2×C8—22no—
377C2×C8—22no—
378C2×C8—22no—
379C2×C8#12412yes—
380C2×C8C2×SD1612yes—
381C2×C8C2×SD1612yes—
382C2×C8C22×D411yescharacteristic
383C2×C8C2×#4111yescharacteristic
384C2×C8C2×#4111yescharacteristic
385C2×C8C2×#4111yescharacteristic
386C2×C8H(5,𝔽2)11yescharacteristic
387C2×C8H(5,𝔽2)11yescharacteristic
388C4⋊C4—22no—
389C4⋊C4—41no—
390C4⋊C4—41no—
391C4⋊C4—41no—
392C4⋊C4—41no—
393C4⋊C4—41no—
394C4⋊C4—41no—
395C4⋊C4—41no—
396C4⋊C4—41no—
397C4⋊C4—41no—
398C4⋊C4—82no—
399C4⋊C4—82no—
400C4⋊C4—82no—
401C4⋊C4—82no—
402C4⋊C4—82no—
403C4⋊C4—161no—
404C22⋊C4—82no—
405C22⋊C4—82no—
406C22⋊C4—82no—
407C22⋊C4—82no—
408C22⋊C4—82no—
409C22⋊C4—82no—
410C22⋊C4D1614yes—
411C22⋊C4SD3214yes—
412C42—21no—
413C42—21no—
414C42—21no—
415C42—21no—
416C16—22no—
417C16—41no—
418C16—41no—
419C16—41no—
420C23—21no—
421C23—21no—
422C23—21no—
423C23—21no—
424C23—42no—
425C23—42no—
426C23—88no—
427C23—81no—
428C23—81no—
429C23—81no—
430C23—81no—
431C23—81no—
432C23—81no—
433C23—81no—
434C23—81no—
435C23—81no—
436C23—81no—
437C23—81no—
438C23—81no—
439C23—81no—
440C23—81no—
441C23—81no—
442C23—81no—
443C23—81no—
444C23—81no—
445C23—81no—
446C23—164no—
447C23—164no—
448C23—164no—
449C23—164no—
450C23—164no—
451C23—164no—
452C23#48612yes—
453C23C2×D1612yes—
454C23C2×SD3212yes—
455C23C22×D811yescharacteristic
456Q8—164no—
457D4—42no—
458D4—42no—
459D4—42no—
460D4—42no—
461D4—88no—
462D4—81no—
463D4—81no—
464D4—81no—
465D4—81no—
466D4—81no—
467D4—81no—
468D4—81no—
469D4—81no—
470D4—164no—
471D4—164no—
472D4—164no—
473D4—164no—
474D4—164no—
475C2×C4—21no—
476C2×C4—21no—
477C2×C4—21no—
478C2×C4—21no—
479C2×C4—21no—
480C2×C4—21no—
481C2×C4—21no—
482C2×C4—21no—
483C2×C4—21no—
484C2×C4—21no—
485C2×C4—22no—
486C2×C4—22no—
487C2×C4—22no—
488C2×C4—22no—
489C2×C4—42no—
490C2×C4—81no—
491C2×C4—81no—
492C2×C4—81no—
493C2×C4—81no—
494C2×C4—81no—
495C2×C4—81no—
496C2×C4—81no—
497C2×C4—81no—
498C2×C4—81no—
499C2×C4—81no—
500C2×C4—81no—
501C2×C4—81no—
502C2×C4—81no—
503C2×C4—81no—
504C2×C4—164no—
505C2×C4#48612yes—
506C2×C4C2×D1612yes—
507C2×C4C2×D1612yes—
508C2×C4C2×SD3212yes—
509C2×C4C2×SD3212yes—
510C2×C4C2×#12411yescharacteristic
511C2×C4D4211yescharacteristic
512C2×C4#54611yescharacteristic
513C2×C4C22×SD1611yescharacteristic
514C2×C4#57611yescharacteristic
515C2×C4#57711yescharacteristic
516C8—22no—
517C8—22no—
518C8—22no—
519C8#52511yescharacteristic
520C8#53811yescharacteristic
521C8#54611yescharacteristic
522C8#55111yescharacteristic
523C22—22no—
524C22—22no—
525C22—22no—
526C22—22no—
527C22—22no—
528C22—22no—
529C22—82no—
530C22—82no—
531C22—82no—
532C22—82no—
533C22—82no—
534C22—82no—
535C22—1616no—
536C22—161no—
537C22—161no—
538C22—161no—
539C22—161no—
540C22—161no—
541C22—161no—
542C22—161no—
543C22—161no—
544C22—161no—
545C22—161no—
546C22—161no—
547C22—161no—
548C22—161no—
549C22—161no—
550C22—161no—
551C22#224014yescomplemented
552C22C2×#48611yescharacteristic
553C22D4×D811yescharacteristic
554C22#329411yescharacteristic
555C22C22×D1611yescharacteristic
556C22C22×SD3211yescharacteristic
557C22#341511yescharacteristic
558C22#341611yescharacteristic
559C4—22no—
560C4—22no—
561C4—22no—
562C4—22no—
563C4—22no—
564C4—82no—
565C4—161no—
566C4—161no—
567C4—161no—
568C4—161no—
569C4#247211yescharacteristic
570C4#313611yescharacteristic
571C4D4×SD1611yescharacteristic
572C4#329511yescharacteristic
573C2—24no—
574C2—162no—
575C2—162no—
576C2—321no—
577C2#2725411yescharacteristic
578C2C2×#224011yescharacteristic
579C2#3370211yescharacteristic
580C2D4×D1611yescharacteristic
581C2D4×SD3211yescharacteristic
582C2#3386111yescharacteristic
583C2#3386211yescharacteristic
subgroupG/H
center, socleC23C22×D8
commutator, Frattini, cosocleC22×C8C24
prime radical#25779C2
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ C8×C22⋊C4 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#2240C2semidirect
2. C22×#357C2semidirect
3. #26065C2semidirect
5. #13480C2semidirect
6. #13407C2semidirect
7. #13407C2semidirect
8. #13363C2semidirect
9. #13202C2semidirect
10. #13147C2semidirect
11. #13147C2semidirect
12. #13059C2semidirect
13. #12991C2semidirect
14. #12731C2semidirect
15. C16×C22⋊C4C2semidirect
18. C2×#500C22semidirect
36. C2×#357C22semidirect
37. C2×#357C22semidirect
38. C2×#357C22semidirect
39. C2×#357C22semidirect
41. C2×C4×C16C22semidirect
42. C2×C4×C16C22semidirect
49. #1942C22semidirect
50. #1942C22semidirect
52. #1845C22semidirect
55. #1380C22semidirect
56. #1380C22semidirect
57. #1380C22semidirect
58. #1380C22semidirect
59. #1374C22semidirect
60. #1374C22semidirect
138. #357D4semidirect
551. C22#2240semidirect

Maximal subgroups

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

subgroupindex
C2×#22402
C22×#3572
#260652
#257792
#134802
#134072
#133632
#132022
#131472
#130592
#129912
#127312
C16×C22⋊C42