Finite Group Databasebeta

← #498162

Subgroups — #498162

Order 512.

751 conjugacy classes of subgroups.

The subgroup lattice has 636 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33516C211yesmaximal, characteristic
2C2×#2179C211yesmaximal, characteristic, complemented
3C22×#359C211yesmaximal, characteristic, complemented
4#23018C211yesmaximal, characteristic, complemented
5#22454C211yesmaximal, characteristic, complemented
6#13242C211yesmaximal, characteristic, complemented
7#13152C211yesmaximal, characteristic, complemented
8#13134C211yesmaximal, characteristic, complemented
9#13063C211yesmaximal, characteristic, complemented
10#13011C211yesmaximal, characteristic, complemented
11#13011C211yesmaximal, characteristic, complemented
12#12939C211yesmaximal, characteristic, complemented
13#12922C211yesmaximal, characteristic, complemented
14#12642C211yesmaximal, characteristic, complemented
15#12611C211yesmaximal, characteristic, complemented
16C22×#135C2211yescharacteristic
17C2×#508C2211yescharacteristic
18C2×#508C2211yescharacteristic
19C23×C16C2211yescharacteristic
20#3050C2211yescharacteristic
21C2×#443C2211yescharacteristic
22C2×#443C2211yescharacteristic
23C2×#443C2211yescharacteristic, complemented
24#2933—21no—
25C22×#108C2211yescharacteristic
26#2880—21no—
27#2189C411yescharacteristic, complemented
28#2189C411yescharacteristic, complemented
29#2188C411yescharacteristic, complemented
30#2187C411yescharacteristic, complemented
31#2186C411yescharacteristic, complemented
32#2186C411yescharacteristic, complemented
33#2185C411yescharacteristic, complemented
34#2184C411yescharacteristic, complemented
35#2179—24no—
36C2×#368C2211yescharacteristic, complemented
37C2×#368C2211yescharacteristic, complemented
38C2×#359—24no—
39C2×#359C2211yescharacteristic
40C2×#359C2211yescharacteristic
41C2×#359C2211yescharacteristic
42C2×#359C2211yescharacteristic
43C2×#359C2211yescharacteristic, complemented
44C2×#359C2211yescharacteristic, complemented
45#2138C2211yescharacteristic
46#2138C2211yescharacteristic
47C2×C22⋊C16C2211yescharacteristic
48C2×#346C2211yescharacteristic, complemented
49C2×#346C2211yescharacteristic, complemented
50#1974C2211yescharacteristic
51#1974C2211yescharacteristic
52#1974C2211yescharacteristic, complemented
53#1945C2211yescharacteristic, complemented
54#1929—21no—
55#1876—21no—
56#1876—21no—
57#1871—21no—
58#1866—21no—
59#1810C2211yescharacteristic
60#1789—21no—
61#1772C2211yescharacteristic, complemented
62C8×C22⋊C4C2211yescharacteristic
63#1382C2211yescharacteristic, complemented
64#1382C2211yescharacteristic, complemented
65#1381C2211yescharacteristic, complemented
66#1381C2211yescharacteristic, complemented
67#1374C2211yescharacteristic, complemented
68#1374C2211yescharacteristic, complemented
69C22×#41—21no—
70C2×#135—22no—
71C2×#135—22no—
72C2×#135—21no—
73C2×#135—21no—
74C2×#135—41no—
75C2×#135—41no—
76C2×#135C2×C411yescharacteristic
77C2×#135C2×C411yescharacteristic
78C2×#135C2311yescharacteristic
79C2×#135C2311yescharacteristic
80C2×#135C2311yescharacteristic
81C22×Q16C2×C411yescharacteristic
82C22×SD16—21no—
83C22×D8C2×C411yescharacteristic
84C2×#131—21no—
85C2×#131—21no—
86C2×#131—21no—
87C22×M16—21no—
88C23×C8C2311yescharacteristic
89#525—21no—
90#508—22no—
91#508—22no—
92#508—41no—
93#508—41no—
94C2×Q32C2×C411yescharacteristic
95C2×Q32C2×C411yescharacteristic
96C2×SD32C2×C411yescharacteristic
97C2×SD32C2×C411yescharacteristic
98C2×SD32C2×C411yescharacteristic
99C2×SD32C2×C411yescharacteristic
100C2×D16C2×C411yescharacteristic
101C2×D16C2×C411yescharacteristic
102C22×C16—21no—
103C22×C16—21no—
104C22×C16—22no—
105C22×C16—41no—
106C22×C16C2311yescharacteristic
107C22×C16C2311yescharacteristic
108C22×C16C2311yescharacteristic
109C22×C16C2311yescharacteristic
110#467C2×C411yescharacteristic
111#467C2×C411yescharacteristic
112#466C2×C411yescharacteristic
113#466C2×C411yescharacteristic
114#465—21no—
115#465—21no—
116#443—22no—
117#443—22no—
118#443—22no—
119#443—41no—
120#443—41no—
121#443—41no—
122#436—41no—
123#435—41no—
124#435—41no—
125C8×D4—41no—
126C2×#108—22no—
127C2×#108—21no—
128C2×#108—41no—
129C2×#108C2311yescharacteristic
130C2×#108C2311yescharacteristic
131C2×#108C2311yescharacteristic
132C2×#108C2311yescharacteristic
133#427C2311yescharacteristic
134#424—21no—
135C2×#104—21no—
136C2×#104—21no—
137C2×#104—21no—
138C2×#104—21no—
139C2×#104—21no—
140C2×#104—21no—
141#416—21no—
142#408—21no—
143#408—21no—
144#408—21no—
145#408—41no—
146#408—41no—
147#407—21no—
148#407—41no—
149C2×C22⋊C8—41no—
150C2×C22⋊C8D411yescharacteristic
151#405—21no—
152C2×C4×C8C2311yescharacteristic
153C2×C4×C8C2311yescharacteristic
154C4×C22⋊C4D411yescharacteristic
155#368—22no—
156#368—41no—
157#359—22no—
158#359—22no—
159#359—22no—
160#359—22no—
161#359—41no—
162#359D414yescomplemented
163#358C2×C411yescharacteristic
164#358C2×C411yescharacteristic
165#358C2×C411yescharacteristic
166#358C2×C411yescharacteristic
167#357C2×C411yescharacteristic
168#357C2×C411yescharacteristic
169#357C2×C411yescharacteristic
170#357C2×C411yescharacteristic
171C22⋊C16C2×C411yescharacteristic
172C22⋊C16C2×C411yescharacteristic
173C22⋊C16C2×C411yescharacteristic
174C22⋊C16C2×C411yescharacteristic
175#346—22no—
176#346—41no—
177#341—21no—
178#339—21no—
179#339—21no—
180#336D411yescharacteristic
181#336D411yescharacteristic
182C2×#41—21no—
183C2×#41—21no—
184C2×#41—21no—
185C2×#41—21no—
186C2×#41—21no—
187C2×#41—42no—
188C2×#41—42no—
189C2×#41—41no—
190C2×#41—41no—
191C2×#41—81no—
192C2×#41—81no—
193C22×Q8—21no—
194C22×D4—21no—
195C22×D4—41no—
196C23×C4—41no—
197C23×C4C2×D411yescharacteristic
198#135—22no—
199#135—22no—
200#135—21no—
201#135—21no—
202#135—21no—
203#135—21no—
204#135—21no—
205#135—21no—
206#135—21no—
207#135—21no—
208#135—21no—
209#135—21no—
210#135—41no—
211#135—41no—
212#135—42no—
213#135—42no—
214#135—42no—
215#135—42no—
216#135C2×D412yes—
217#135#4112yes—
218C2×Q16—22no—
219C2×Q16—21no—
220C2×Q16—41no—
221C2×Q16C22×C411yescharacteristic
222C2×Q16C22×C411yescharacteristic
223C2×Q16C22×C411yescharacteristic
224C2×SD16—21no—
225C2×SD16—21no—
226C2×SD16—21no—
227C2×SD16—21no—
228C2×SD16—21no—
229C2×SD16—42no—
230C2×SD16—42no—
231C2×D8—22no—
232C2×D8—21no—
233C2×D8—41no—
234C2×D8C22×C411yescharacteristic
235C2×D8C22×C411yescharacteristic
236C2×D8C22×C411yescharacteristic
237#131—42no—
238#131—42no—
239#131—42no—
240#131—81no—
241#131—81no—
242#131—81no—
243C2×M16—21no—
244C2×M16—21no—
245C2×M16—21no—
246C2×M16—21no—
247C2×M16—41no—
248C2×M16—41no—
249C2×M16—41no—
250C2×M16—41no—
251C2×M16—42no—
252C2×M16—81no—
253C22×C8—21no—
254C22×C8—21no—
255C22×C8—21no—
256C22×C8—21no—
257C22×C8—41no—
258C22×C8—41no—
259C22×C8—41no—
260C22×C8—41no—
261C22×C8C22×C411yescharacteristic
262C22×C8C22×C411yescharacteristic
263C22×C8C2×D411yescharacteristic
264C22×C8C2×D411yescharacteristic
265C22×C8C2×D411yescharacteristic
266C22×C8#4111yescharacteristic
267C22×C8#4111yescharacteristic
268C22×C8C2411yescharacteristic
269#123—41no—
270C22⋊Q8—21no—
271C22⋊Q8—21no—
272#121—21no—
273#121—21no—
274#120—41no—
275C4×D4—41no—
276C4×D4—41no—
277#117—21no—
278C2×C22⋊C4—21no—
279C2×C42C2×D411yescharacteristic
280C2×C42C2×D411yescharacteristic
281Q32—22no—
282Q32—41no—
283SD32—22no—
284SD32—22no—
285SD32—41no—
286SD32—41no—
287D16—22no—
288D16—41no—
289C2×C16—21no—
290C2×C16—21no—
291C2×C16—21no—
292C2×C16—21no—
293C2×C16—21no—
294C2×C16—21no—
295C2×C16—21no—
296C2×C16—21no—
297C2×C16—21no—
298C2×C16—42no—
299C2×C16—42no—
300C2×C16C22×C411yescharacteristic
301C2×C16C22×C411yescharacteristic
302C2×C16C22×C411yescharacteristic
303C2×C16C22×C411yescharacteristic
304C2×C16C2×D412yes—
305#108—21no—
306#108—21no—
307#108—21no—
308#108—21no—
309#108—21no—
310#108—21no—
311#108—21no—
312#108—42no—
313#108C2×D412yes—
314#107C22×C411yescharacteristic
315#107C22×C411yescharacteristic
316#106—21no—
317C4⋊C8—41no—
318C4⋊C8—41no—
319#104—42no—
320#104—42no—
321#104—42no—
322#104—81no—
323#104—81no—
324#104—81no—
325#103—21no—
326#103—21no—
327#102—21no—
328#102—21no—
329C22⋊C8—41no—
330C22⋊C8—41no—
331C22⋊C8—41no—
332C22⋊C8—41no—
333C22⋊C8—41no—
334C22⋊C8—41no—
335C22⋊C8—41no—
336C22⋊C8—41no—
337C22⋊C8D811yescharacteristic
338C22⋊C8D811yescharacteristic
339C22⋊C8D811yescharacteristic
340C22⋊C8D811yescharacteristic
341#97—41no—
342C4×C8—21no—
343C4×C8—21no—
344C4×C8—21no—
345C4×C8—21no—
346C4×C8—41no—
347#95—21no—
348C24—41no—
349C24#13511yescharacteristic
350#41—22no—
351#41—22no—
352#41—42no—
353#41—42no—
354#41—41no—
355#41—41no—
356#41—41no—
357#41—41no—
358#41—41no—
359#41—41no—
360#41—41no—
361#41—41no—
362#41—41no—
363#41—41no—
364#41—82no—
365#41—82no—
366#41—81no—
367#41—81no—
368#41—82no—
369#41—82no—
370C2×Q8—21no—
371C2×Q8—21no—
372C2×Q8—21no—
373C2×Q8—42no—
374C2×Q8—41no—
375C2×Q8—81no—
376C2×D4—21no—
377C2×D4—21no—
378C2×D4—21no—
379C2×D4—42no—
380C2×D4—41no—
381C2×D4—41no—
382C2×D4—41no—
383C2×D4—41no—
384C2×D4—41no—
385C2×D4—41no—
386C2×D4—41no—
387C2×D4—41no—
388C2×D4—81no—
389C2×D4—82no—
390C2×D4—82no—
391C22×C4—21no—
392C22×C4—21no—
393C22×C4—21no—
394C22×C4—21no—
395C22×C4—21no—
396C22×C4—21no—
397C22×C4—41no—
398C22×C4—41no—
399C22×C4—41no—
400C22×C4—41no—
401C22×C4—41no—
402C22×C4—41no—
403C22×C4—82no—
404C22×C4—82no—
405C22×C4C4×D411yescharacteristic
406C22×C4C4×D411yescharacteristic
407C22×C4C2×D811yescharacteristic
408C22×C4C2×D811yescharacteristic
409C22×C4#13511yescharacteristic
410C22×C4C22×D411yescharacteristic
411Q16—21no—
412Q16—21no—
413Q16—21no—
414Q16—21no—
415Q16—21no—
416Q16—42no—
417Q16C4×D412yes—
418SD16—22no—
419SD16—22no—
420SD16—42no—
421SD16—42no—
422SD16—41no—
423SD16—41no—
424SD16—41no—
425SD16—41no—
426D8—21no—
427D8—21no—
428D8—21no—
429D8—21no—
430D8—21no—
431D8—42no—
432D8C4×D412yes—
433M16—22no—
434M16—41no—
435M16—41no—
436M16—41no—
437M16—41no—
438M16—41no—
439M16—41no—
440M16—41no—
441M16—82no—
442M16—82no—
443M16—82no—
444M16—82no—
445C2×C8—21no—
446C2×C8—21no—
447C2×C8—21no—
448C2×C8—21no—
449C2×C8—21no—
450C2×C8—21no—
451C2×C8—21no—
452C2×C8—21no—
453C2×C8—21no—
454C2×C8—21no—
455C2×C8—21no—
456C2×C8—21no—
457C2×C8—21no—
458C2×C8—21no—
459C2×C8—41no—
460C2×C8—41no—
461C2×C8—41no—
462C2×C8—41no—
463C2×C8—82no—
464C2×C8—82no—
465C2×C8—82no—
466C2×C8—82no—
467C2×C8C4×D411yescharacteristic
468C2×C8C4×D411yescharacteristic
469C2×C8C2×D811yescharacteristic
470C2×C8C2×D811yescharacteristic
471C2×C8C2×D811yescharacteristic
472C2×C8C2×D811yescharacteristic
473C2×C8C23×C411yescharacteristic
474C2×C8C22×D411yescharacteristic
475C2×C8C2×#4111yescharacteristic
476C2×C8C2×#4111yescharacteristic
477C2×C8H(5,𝔽2)11yescharacteristic
478C2×C8H(5,𝔽2)11yescharacteristic
479C4⋊C4—21no—
480C4⋊C4—21no—
481C4⋊C4—41no—
482C4⋊C4—41no—
483C22⋊C4—42no—
484C22⋊C4—41no—
485C22⋊C4—41no—
486C22⋊C4—41no—
487C22⋊C4—41no—
488C22⋊C4—41no—
489C42—21no—
490C42—21no—
491C42—21no—
492C42—21no—
493C42—41no—
494C16—21no—
495C16—21no—
496C16—21no—
497C16—21no—
498C16—22no—
499C16—41no—
500C23—22no—
501C23—21no—
502C23—21no—
503C23—41no—
504C23—41no—
505C23—41no—
506C23—41no—
507C23—82no—
508C23—82no—
509C23—161no—
510C23C4×D811yescharacteristic
511C23C4×D811yescharacteristic
512C23C2×#13511yescharacteristic
513Q8—22no—
514Q8—41no—
515Q8—41no—
516Q8—41no—
517Q8—41no—
518Q8—41no—
519Q8—82no—
520D4—22no—
521D4—42no—
522D4—42no—
523D4—41no—
524D4—41no—
525D4—41no—
526D4—41no—
527D4—41no—
528D4—82no—
529D4—82no—
530D4—82no—
531D4—82no—
532D4—81no—
533D4—81no—
534D4—81no—
535D4—81no—
536D4—161no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—41no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—82no—
561C2×C4—82no—
562C2×C4—82no—
563C2×C4—81no—
564C2×C4—81no—
565C2×C4—81no—
566C2×C4—81no—
567C2×C4C4×D811yescharacteristic
568C2×C4C4×D811yescharacteristic
569C2×C4C2×C4×D411yescharacteristic
570C2×C4D4211yescharacteristic
571C2×C4#54611yescharacteristic
572C2×C4C22×D811yescharacteristic
573C2×C4#57611yescharacteristic
574C2×C4#57711yescharacteristic
575C8—21no—
576C8—21no—
577C8—21no—
578C8—21no—
579C8—21no—
580C8—21no—
581C8—42no—
582C8—81no—
583C8—81no—
584C8—81no—
585C8C2×C4×D411yescharacteristic
586C8#51811yescharacteristic
587C8#54611yescharacteristic
588C8#55111yescharacteristic
589C22—21no—
590C22—21no—
591C22—21no—
592C22—21no—
593C22—21no—
594C22—21no—
595C22—21no—
596C22—42no—
597C22—42no—
598C22—42no—
599C22—81no—
600C22—81no—
601C22—81no—
602C22—81no—
603C22—81no—
604C22—81no—
605C22—81no—
606C22—81no—
607C22—164no—
608C22#217912yescomplemented
609C22C2×C4×D811yescharacteristic
610C22#328211yescharacteristic
611C22#329711yescharacteristic
612C4—21no—
613C4—21no—
614C4—42no—
615C4—81no—
616C4—81no—
617C4—81no—
618C4—81no—
619C4—81no—
620C4—81no—
621C4#235811yescharacteristic
622C4#294611yescharacteristic
623C4D4×D811yescharacteristic
624C4#329411yescharacteristic
625C2—21no—
626C2—21no—
627C2—22no—
628C2—41no—
629C2—82no—
630C2—161no—
631C2—161no—
632C2#2287011yescharacteristic
633C2C2×#217911yescharacteristic
634C2#3346411yescharacteristic
subgroupG/H
centerC2×C4C22×D8
commutatorC2×C8C23×C4
Frattini, cosocleC22×C8C24
socleC22C2×C4×D8
prime radical#33516C2
LCSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C2×C22⋊C8 ◁ C8×C22⋊C4 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2179C2semidirect
3. C22×#359C2semidirect
4. #23018C2semidirect
5. #22454C2semidirect
6. #13242C2semidirect
7. #13152C2semidirect
8. #13134C2semidirect
9. #13063C2semidirect
10. #13011C2semidirect
11. #13011C2semidirect
12. #12939C2semidirect
13. #12922C2semidirect
14. #12642C2semidirect
15. #12611C2semidirect
23. C2×#443C22semidirect
27. #2189C4semidirect
28. #2189C4semidirect
29. #2188C4semidirect
30. #2187C4semidirect
31. #2186C4semidirect
32. #2186C4semidirect
33. #2185C4semidirect
34. #2184C4semidirect
36. C2×#368C22semidirect
37. C2×#368C22semidirect
43. C2×#359C22semidirect
44. C2×#359C22semidirect
48. C2×#346C22semidirect
49. C2×#346C22semidirect
52. #1974C22semidirect
53. #1945C22semidirect
61. #1772C22semidirect
63. #1382C22semidirect
64. #1382C22semidirect
65. #1381C22semidirect
66. #1381C22semidirect
67. #1374C22semidirect
68. #1374C22semidirect
162. #359D4semidirect
608. C22#2179semidirect

Maximal subgroups

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

subgroupindex
#335162
C2×#21792
C22×#3592
#230182
#224542
#132422
#131522
#131342
#130632
#130112
#129392
#129222
#126422
#126112