Finite Group Databasebeta

← #230095

Subgroups — #230095

Order 512.

638 conjugacy classes of subgroups.

The subgroup lattice has 598 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23180C211yesmaximal, characteristic, complemented
2#22717C211yesmaximal, characteristic, complemented
3#19101C211yesmaximal, characteristic
4#18001C211yesmaximal, characteristic, complemented
5#17994C211yesmaximal, characteristic, complemented
6#9860C211yesmaximal, characteristic, complemented
7#9632C211yesmaximal, characteristic, complemented
8#9592C211yesmaximal, characteristic, complemented
9#9590C211yesmaximal, characteristic, complemented
10#8991C211yesmaximal, characteristic, complemented
11#8991C211yesmaximal, characteristic, complemented
12#8932C211yesmaximal, characteristic, complemented
13#8930C211yesmaximal, characteristic, complemented
14#8297C211yesmaximal, characteristic, complemented
15#8204C211yesmaximal, characteristic, complemented
16#3147C2211yescharacteristic
17#3100C2211yescharacteristic
18#3100C2211yescharacteristic
19C2×#444—21no—
20#2935C2211yescharacteristic
21#2935C2211yescharacteristic
22D4×M16C2211yescharacteristic
23D4×M16C2211yescharacteristic
24C2×C8×D4C2211yescharacteristic
25#1980C2211yescharacteristic
26#1980C2211yescharacteristic
27#1975—21no—
28#1950C2211yescharacteristic
29#1950C2211yescharacteristic
30#1876—21no—
31#1876—21no—
32#1855—21no—
33#1855—21no—
34#1837C2211yescharacteristic
35#1810—21no—
36C4×#108C2211yescharacteristic
37#1770C2211yescharacteristic
38C4×#102C2211yescharacteristic
39C2×#341—21no—
40#1669C411yescharacteristic, complemented
41#1669C411yescharacteristic, complemented
42#1667C411yescharacteristic, complemented
43#1667C411yescharacteristic, complemented
44#1665C411yescharacteristic, complemented
45#1665C411yescharacteristic, complemented
46#1665C411yescharacteristic, complemented
47#1665C411yescharacteristic, complemented
48#1594—21no—
49#1589—21no—
50#1586—21no—
51#1586—21no—
52#1581—21no—
53#1581—21no—
54#1580—21no—
55C8×D8—21no—
56#1571C2211yescharacteristic
57#1570C2211yescharacteristic
58#1564C2211yescharacteristic
59#1540C2211yescharacteristic
60#1540C2211yescharacteristic
61#1492C2211yescharacteristic
62#1492C2211yescharacteristic
63#1492C2211yescharacteristic
64#1492C2211yescharacteristic
65C2×D4⋊C8C2211yescharacteristic
66#1451C2211yescharacteristic
67C8×M16C2211yescharacteristic
68C2×#322C2211yescharacteristic
69#1303C2211yescharacteristic
70#1303C2211yescharacteristic
71#1279C2211yescharacteristic
72#1278C2211yescharacteristic
73#1278C2211yescharacteristic
74#1277C2211yescharacteristic
75C2×#135—21no—
76C2×#135—21no—
77C2×#131—21no—
78C2×#131—21no—
79C22×M16—21no—
80C22×M16—21no—
81C23×C8—21no—
82#532C2311yescharacteristic
83#532C2311yescharacteristic
84C2×C4×D4C2311yescharacteristic
85#495—21no—
86#495—21no—
87#494C2×C411yescharacteristic
88#493C2×C411yescharacteristic
89#492C2×C411yescharacteristic
90#492C2×C411yescharacteristic
91#480—21no—
92#480—21no—
93#474C2×C411yescharacteristic
94#474C2×C411yescharacteristic
95#474C2×C411yescharacteristic
96#474C2×C411yescharacteristic
97#459C2×C411yescharacteristic
98#459C2×C411yescharacteristic
99#459C2×C411yescharacteristic
100#459C2×C411yescharacteristic
101#447—21no—
102#447—21no—
103#444—44no—
104C4×D8—21no—
105C4×D8—21no—
106C4×D8—21no—
107C4×D8—21no—
108#436—21no—
109#436—21no—
110#436—21no—
111#436—21no—
112#435—41no—
113#435—41no—
114#435—41no—
115#435—41no—
116C8×D4—21no—
117C8×D4—21no—
118C8×D4—21no—
119C8×D4—21no—
120C8×D4—21no—
121C8×D4—21no—
122C8×D4C2×C411yescharacteristic
123C8×D4C2×C411yescharacteristic
124C8×D4C2×C411yescharacteristic
125C8×D4C2×C411yescharacteristic
126#431C2311yescharacteristic
127#431C2311yescharacteristic
128C2×#108—21no—
129C2×#108—21no—
130C2×#108D411yescharacteristic
131C2×#108D411yescharacteristic
132#423C2311yescharacteristic
133#423C2311yescharacteristic
134C2×C4⋊C8C2311yescharacteristic
135C2×C4⋊C8C2311yescharacteristic
136C2×#104—21no—
137C2×#104—21no—
138C2×#104—21no—
139C2×#104—21no—
140C2×#102—21no—
141#408—21no—
142#408—21no—
143#407—21no—
144#407—21no—
145C2×C22⋊C8—21no—
146C4×M16C2311yescharacteristic
147C4×M16C2311yescharacteristic
148C2×#97—21no—
149C2×C4×C8C2311yescharacteristic
150C2×C4×C8C2311yescharacteristic
151C2×C4×C8C2311yescharacteristic
152C4×C4⋊C4C2311yescharacteristic
153#341—21no—
154#341—21no—
155#341—21no—
156#341—21no—
157#341D411yescharacteristic
158#341D411yescharacteristic
159#340—21no—
160#336—21no—
161#335—21no—
162#335—21no—
163#335C2×C411yescharacteristic
164#335C2×C411yescharacteristic
165#334—21no—
166#334—21no—
167#334C2×C411yescharacteristic
168#334C2×C411yescharacteristic
169#332C2×C411yescharacteristic
170#332C2×C411yescharacteristic
171#332C2×C411yescharacteristic
172#332C2×C411yescharacteristic
173#331C2×C411yescharacteristic
174#331C2×C411yescharacteristic
175#331C2×C411yescharacteristic
176#331C2×C411yescharacteristic
177D4⋊C8—21no—
178D4⋊C8—21no—
179D4⋊C8—21no—
180D4⋊C8—21no—
181D4⋊C8—41no—
182D4⋊C8—41no—
183D4⋊C8D411yescharacteristic
184D4⋊C8D411yescharacteristic
185D4⋊C8D411yescharacteristic
186D4⋊C8D411yescharacteristic
187#322—21no—
188#322—21no—
189#322—21no—
190#322—21no—
191#322—21no—
192C82—21no—
193C2×#41—21no—
194C2×#41—21no—
195C22×D4#4111yescharacteristic
196C23×C4—21no—
197#135—42no—
198#135—42no—
199#135—81no—
200#135—81no—
201C2×Q16—21no—
202C2×Q16—21no—
203C2×SD16—21no—
204C2×SD16—21no—
205C2×SD16—21no—
206C2×SD16—21no—
207C2×D8—21no—
208C2×D8—21no—
209C2×D8—21no—
210C2×D8—21no—
211C2×D8—21no—
212C2×D8—21no—
213#131—42no—
214#131—42no—
215#131—81no—
216#131—81no—
217C2×M16—21no—
218C2×M16—21no—
219C2×M16—41no—
220C2×M16—41no—
221C2×M16—41no—
222C2×M16—41no—
223C2×M16—42no—
224C2×M16—42no—
225C2×M16—81no—
226C2×M16—81no—
227C2×M16C2×D411yescharacteristic
228C2×M16C2×D411yescharacteristic
229C2×M16C2×D411yescharacteristic
230C2×M16C2×D411yescharacteristic
231C22×C8—21no—
232C22×C8—21no—
233C22×C8—21no—
234C22×C8—21no—
235C22×C8—21no—
236C22×C8—21no—
237C22×C8—21no—
238C22×C8—21no—
239C22×C8—41no—
240C22×C8—41no—
241C22×C8—41no—
242C22×C8—41no—
243C22×C8—81no—
244C22×C8—81no—
245C22×C8C2×D411yescharacteristic
246C22×C8C2×D411yescharacteristic
247C4⋊Q8C22×C411yescharacteristic
248C4⋊Q8C22×C411yescharacteristic
249#127C22×C411yescharacteristic
250#127C22×C411yescharacteristic
251#124—21no—
252#124—21no—
253#121—21no—
254#121—21no—
255#121—21no—
256#121—21no—
257C4×D4—21no—
258C4×D4—21no—
259C4×D4—21no—
260C4×D4—21no—
261C4×D4—21no—
262C4×D4—21no—
263C4×D4C22×C411yescharacteristic
264C4×D4C22×C411yescharacteristic
265C4×D4C2×D411yescharacteristic
266C4×D4C2×D411yescharacteristic
267C2×C4⋊C4—21no—
268C2×C4⋊C4#4111yescharacteristic
269C2×C22⋊C4—21no—
270C2×C42—21no—
271C2×C42C2411yescharacteristic
272#108—22no—
273#108—41no—
274#108—81no—
275#108—81no—
276#107—21no—
277#107—21no—
278#107—21no—
279#107—21no—
280C4⋊C8—21no—
281C4⋊C8—21no—
282C4⋊C8—21no—
283C4⋊C8—21no—
284C4⋊C8—21no—
285C4⋊C8—21no—
286C4⋊C8C22×C411yescharacteristic
287C4⋊C8C22×C411yescharacteristic
288C4⋊C8C22×C411yescharacteristic
289C4⋊C8C22×C411yescharacteristic
290C4⋊C8C22×C411yescharacteristic
291C4⋊C8C22×C411yescharacteristic
292C4⋊C8C2×D411yescharacteristic
293C4⋊C8C2×D411yescharacteristic
294#104—42no—
295#104—42no—
296#104—81no—
297#104—81no—
298#102—21no—
299#102—21no—
300#102—21no—
301#102—21no—
302#102—21no—
303#102—21no—
304#102—21no—
305#102—21no—
306#102—42no—
307C22⋊C8—21no—
308C22⋊C8—21no—
309C22⋊C8—21no—
310C22⋊C8—21no—
311C22⋊C8—21no—
312C22⋊C8—21no—
313C22⋊C8—41no—
314C22⋊C8—41no—
315C22⋊C8—41no—
316C22⋊C8—41no—
317C22⋊C8—41no—
318C22⋊C8—41no—
319C22⋊C8—41no—
320C22⋊C8—41no—
321#97—21no—
322#97—21no—
323#97—42no—
324C4×C8—21no—
325C4×C8—21no—
326C4×C8—21no—
327C4×C8—21no—
328C4×C8—21no—
329C4×C8—21no—
330C4×C8—21no—
331C4×C8C22×C411yescharacteristic
332C4×C8C22×C411yescharacteristic
333C4×C8C2×D411yescharacteristic
334C4×C8C2×D411yescharacteristic
335C4×C8#4111yescharacteristic
336C4×C8#4111yescharacteristic
337#95—21no—
338C24—21no—
339#41—41no—
340#41—41no—
341#41—41no—
342#41—41no—
343#41—41no—
344#41—41no—
345#41—41no—
346#41—41no—
347C2×Q8—21no—
348C2×Q8—21no—
349C2×D4—22no—
350C2×D4—21no—
351C2×D4—21no—
352C2×D4—21no—
353C2×D4—21no—
354C2×D4—21no—
355C2×D4—21no—
356C2×D4—21no—
357C2×D4—41no—
358C2×D4—41no—
359C2×D4—41no—
360C2×D4C4×D411yescharacteristic
361C2×D4C4×D411yescharacteristic
362C22×C4—21no—
363C22×C4—21no—
364C22×C4—21no—
365C22×C4—21no—
366C22×C4—21no—
367C22×C4—41no—
368C22×C4—41no—
369C22×C4—41no—
370C22×C4—41no—
371C22×C4—41no—
372C22×C4—41no—
373C22×C4—41no—
374C22×C4C22×D411yescharacteristic
375C22×C4C2×#4111yescharacteristic
376C22×C42−1+411yescharacteristic
377Q16—42no—
378Q16—81no—
379SD16—42no—
380SD16—42no—
381SD16—81no—
382SD16—81no—
383D8—42no—
384D8—42no—
385D8—42no—
386D8—81no—
387D8—81no—
388D8—81no—
389M16—21no—
390M16—21no—
391M16—21no—
392M16—21no—
393M16—21no—
394M16—21no—
395M16—21no—
396M16—41no—
397M16—41no—
398M16—41no—
399M16—41no—
400M16—82no—
401M16—82no—
402M16—82no—
403M16—82no—
404M16AGL(1,ℤ/8)12yes—
405C2×C8—21no—
406C2×C8—21no—
407C2×C8—21no—
408C2×C8—21no—
409C2×C8—21no—
410C2×C8—21no—
411C2×C8—21no—
412C2×C8—21no—
413C2×C8—21no—
414C2×C8—21no—
415C2×C8—21no—
416C2×C8—21no—
417C2×C8—21no—
418C2×C8—21no—
419C2×C8—21no—
420C2×C8—21no—
421C2×C8—21no—
422C2×C8—21no—
423C2×C8—42no—
424C2×C8—41no—
425C2×C8—41no—
426C2×C8—41no—
427C2×C8—41no—
428C2×C8—41no—
429C2×C8—41no—
430C2×C8—41no—
431C2×C8—41no—
432C2×C8—41no—
433C2×C8—81no—
434C2×C8—81no—
435C2×C8—81no—
436C2×C8—81no—
437C2×C8—82no—
438C2×C8—82no—
439C2×C8—82no—
440C2×C8—82no—
441C2×C8C4×D411yescharacteristic
442C2×C8C4×D411yescharacteristic
443C2×C8C4×D411yescharacteristic
444C2×C8C4×D411yescharacteristic
445C4⋊C4—21no—
446C4⋊C4—21no—
447C4⋊C4—21no—
448C4⋊C4—21no—
449C4⋊C4—21no—
450C4⋊C4—41no—
451C4⋊C4—41no—
452C4⋊C4C4×D411yescharacteristic
453C4⋊C4C4×D411yescharacteristic
454C22⋊C4—21no—
455C22⋊C4—21no—
456C22⋊C4—21no—
457C22⋊C4—21no—
458C22⋊C4—41no—
459C22⋊C4—41no—
460C22⋊C4—41no—
461C22⋊C4—41no—
462C42—41no—
463C42—41no—
464C42C23×C411yescharacteristic
465C42C22×D411yescharacteristic
466C42C2×#4111yescharacteristic
467C42H(5,𝔽2)11yescharacteristic
468C23—21no—
469C23—21no—
470C23—42no—
471C23—41no—
472C23—41no—
473C23—41no—
474C23—41no—
475C23—81no—
476C23#54811yescharacteristic
477Q8—41no—
478Q8—41no—
479Q8—41no—
480Q8—41no—
481D4—21no—
482D4—21no—
483D4—21no—
484D4—42no—
485D4—41no—
486D4—41no—
487D4—41no—
488D4—41no—
489D4—41no—
490D4—41no—
491D4—41no—
492D4—41no—
493D4—81no—
494D4—81no—
495D4—81no—
496D4—81no—
497D4—81no—
498D4—81no—
499D4#44412yescomplemented
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—41no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—81no—
518C2×C4—81no—
519C2×C4—81no—
520C2×C4—81no—
521C2×C4—81no—
522C2×C4—81no—
523C2×C4—81no—
524C2×C4C2×C4×D411yescharacteristic
525C2×C4C2×C4×D411yescharacteristic
526C2×C4#52011yescharacteristic
527C2×C4D4211yescharacteristic
528C2×C4#54711yescharacteristic
529C2×C4#55311yescharacteristic
530C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
531C2×C4#57511yescharacteristic
532C2×C4#57611yescharacteristic
533C2×C4#57611yescharacteristic
534C8—22no—
535C8—21no—
536C8—21no—
537C8—21no—
538C8—21no—
539C8—22no—
540C8—41no—
541C8—41no—
542C8—41no—
543C8—41no—
544C8—41no—
545C8—41no—
546C8—41no—
547C8—41no—
548C8—41no—
549C8—41no—
550C8—41no—
551C8—41no—
552C8—81no—
553C8—81no—
554C8—81no—
555C22—21no—
556C22—21no—
557C22—22no—
558C22—41no—
559C22—41no—
560C22—41no—
561C22—41no—
562C22—41no—
563C22—82no—
564C22—82no—
565C22—81no—
566C22—81no—
567C22—81no—
568C22—81no—
569C22—81no—
570C22—81no—
571C22#236111yescharacteristic
572C22#336611yescharacteristic
573C22#337011yescharacteristic
574C4—21no—
575C4—21no—
576C4—21no—
577C4—41no—
578C4—41no—
579C4—81no—
580C4—81no—
581C4—161no—
582C4C4×AGL(1,ℤ/8)11yescharacteristic
583C4#294811yescharacteristic
584C4C2×#44411yescharacteristic
585C4#295711yescharacteristic
586C4#326411yescharacteristic
587C4#326811yescharacteristic
588C2—21no—
589C2—21no—
590C2—42no—
591C2—81no—
592C2—161no—
593C2—161no—
594C2#2300811yescharacteristic
595C2#2301911yescharacteristic
596C2#2304211yescharacteristic
subgroupG/H
centerC2×C4D42
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
socleC22#2361
prime radical#19101C2
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #23180C2semidirect
2. #22717C2semidirect
4. #18001C2semidirect
5. #17994C2semidirect
6. #9860C2semidirect
7. #9632C2semidirect
8. #9592C2semidirect
9. #9590C2semidirect
10. #8991C2semidirect
11. #8991C2semidirect
12. #8932C2semidirect
13. #8930C2semidirect
14. #8297C2semidirect
15. #8204C2semidirect
40. #1669C4semidirect
41. #1669C4semidirect
42. #1667C4semidirect
43. #1667C4semidirect
44. #1665C4semidirect
45. #1665C4semidirect
46. #1665C4semidirect
47. #1665C4semidirect
499. D4#444semidirect

Maximal subgroups

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

subgroupindex
#231802
#227172
#191012
#180012
#179942
#98602
#96322
#95922
#95902
#89912
#89322
#89302
#82972
#82042