Finite Group Databasebeta

← #229593

Subgroups — #229593

Order 512.

568 conjugacy classes of subgroups.

The subgroup lattice has 546 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23113C211yesmaximal, characteristic
2#23080C211yesmaximal, characteristic, complemented
3#19124C211yesmaximal, characteristic
4#18071C211yesmaximal, characteristic, complemented
5#17829C211yesmaximal, characteristic, complemented
6#9851C211yesmaximal, characteristic, complemented
7#9626C211yesmaximal, characteristic, complemented
8#9579C211yesmaximal, characteristic, complemented
9#9578C211yesmaximal, characteristic, complemented
10#8996C211yesmaximal, characteristic
11#8994C211yesmaximal, characteristic, complemented
12#8932C211yesmaximal, characteristic, complemented
13#8931C211yesmaximal, characteristic, complemented
14#8275C211yesmaximal, characteristic, complemented
15#8192C211yesmaximal, characteristic, complemented
16#3167C2211yescharacteristic
17#3076C2211yescharacteristic
18#3035C2211yescharacteristic
19C8×#41C2211yescharacteristic
20#2962C2211yescharacteristic
21#2935C2211yescharacteristic
22#2925C2211yescharacteristic
23#2925C2211yescharacteristic
24#1994C2211yescharacteristic
25#1974—21no—
26#1962C2211yescharacteristic
27#1950C2211yescharacteristic
28#1950C2211yescharacteristic
29#1937—21no—
30#1929—21no—
31#1929—21no—
32#1922—21no—
33#1876—21no—
34#1772—21no—
35C4×C4⋊C8C2211yescharacteristic
36C4×C4⋊C8C2211yescharacteristic
37#1764—21no—
38C4×#104C2211yescharacteristic
39C4×#97C2211yescharacteristic
40#1680C411yescharacteristic, complemented
41#1679C411yescharacteristic, complemented
42#1678C411yescharacteristic, complemented
43#1677C411yescharacteristic, complemented
44#1672C411yescharacteristic, complemented
45#1671C411yescharacteristic, complemented
46#1666C411yescharacteristic, complemented
47#1665C411yescharacteristic, complemented
48#1592—21no—
49#1588—21no—
50#1585—21no—
51SD16⋊C8—21no—
52#1572—21no—
53#1572—21no—
54#1570C2211yescharacteristic
55#1570C2211yescharacteristic
56#1568—21no—
57#1567C2211yescharacteristic
58#1564—21no—
59C2×#335C2211yescharacteristic
60C2×#334C2211yescharacteristic
61#1542C2211yescharacteristic
62#1540C2211yescharacteristic
63#1493C2211yescharacteristic
64#1492C2211yescharacteristic
65#1481C2211yescharacteristic
66#1480C2211yescharacteristic
67#1476C2211yescharacteristic
68#1451C2211yescharacteristic
69#1301C2211yescharacteristic
70#1301C2211yescharacteristic
71#1278C2211yescharacteristic
72#1277C2211yescharacteristic
73#1277C2211yescharacteristic
74#1276C2211yescharacteristic
75C2×#135—21no—
76C2×#131—21no—
77#533C2311yescharacteristic
78#532C2311yescharacteristic
79C4×#41C2311yescharacteristic
80#498C2×C411yescharacteristic
81#492C2×C411yescharacteristic
82#487C2×C411yescharacteristic
83#486C2×C411yescharacteristic
84#479—21no—
85#478—21no—
86#477C2×C411yescharacteristic
87#476C2×C411yescharacteristic
88#475C2×C411yescharacteristic
89#474C2×C411yescharacteristic
90#465—21no—
91#465—21no—
92#464—21no—
93#463—21no—
94#462C2×C411yescharacteristic
95#461C2×C411yescharacteristic
96#460C2×C411yescharacteristic
97#459C2×C411yescharacteristic
98#446—21no—
99#446—41no—
100C8×Q8—21no—
101C8×Q8C2×C411yescharacteristic
102C8×Q8C2×C411yescharacteristic
103C4×SD16—21no—
104C4×SD16—21no—
105#436—21no—
106#435—41no—
107C8×D4—21no—
108C8×D4—21no—
109C8×D4—21no—
110C8×D4C2×C411yescharacteristic
111C8×D4C2×C411yescharacteristic
112#433—41no—
113#433—41no—
114#432—21no—
115#432—21no—
116#431—21no—
117#431—21no—
118#431—21no—
119#431C2311yescharacteristic
120#431C2311yescharacteristic
121C2×#108—21no—
122C2×#108—21no—
123#427—21no—
124#424—21no—
125#423C2311yescharacteristic
126#423C2311yescharacteristic
127C2×C4⋊C8C2311yescharacteristic
128C2×C4⋊C8C2311yescharacteristic
129C2×C4⋊C8C2311yescharacteristic
130C2×C4⋊C8C2311yescharacteristic
131C2×#104—21no—
132C2×#104—21no—
133C2×#104D411yescharacteristic
134C2×#104D411yescharacteristic
135#416—21no—
136#416—21no—
137#408—21no—
138#408—21no—
139#408—21no—
140#405—21no—
141#405—21no—
142#405—21no—
143C4×M16C2311yescharacteristic
144C2×#97—21no—
145C2×#97—21no—
146C2×C4×C8—21no—
147C2×C4×C8—21no—
148C2×C4×C8—21no—
149C2×C4×C8C2311yescharacteristic
150C2×C4×C8C2311yescharacteristic
151C43C2311yescharacteristic
152#341—21no—
153#341—21no—
154#339—21no—
155#339—21no—
156#339D411yescharacteristic
157#339D411yescharacteristic
158#336—21no—
159#336—21no—
160#335—21no—
161#335—21no—
162#335D411yescharacteristic
163#335D411yescharacteristic
164#335D411yescharacteristic
165#335D411yescharacteristic
166#334—21no—
167#334—21no—
168#334—21no—
169#334—21no—
170#334C2×C411yescharacteristic
171#334C2×C411yescharacteristic
172#334C2×C411yescharacteristic
173#334C2×C411yescharacteristic
174#333C2×C411yescharacteristic
175#333C2×C411yescharacteristic
176#332C2×C411yescharacteristic
177#332C2×C411yescharacteristic
178#332C2×C411yescharacteristic
179#332C2×C411yescharacteristic
180#331C2×C411yescharacteristic
181#331C2×C411yescharacteristic
182Q8⋊C8—21no—
183Q8⋊C8—21no—
184Q8⋊C8—21no—
185Q8⋊C8—41no—
186D4⋊C8—21no—
187D4⋊C8—21no—
188D4⋊C8—21no—
189D4⋊C8—41no—
190#322—21no—
191#322—21no—
192C2×#41—21no—
193C2×#41C2×D411yescharacteristic
194#135—81no—
195#135—81no—
196C2×Q16—21no—
197C2×SD16—21no—
198C2×SD16—21no—
199C2×SD16—21no—
200C2×SD16—21no—
201C2×D8—21no—
202#131—42no—
203#131—81no—
204C2×M16—21no—
205C2×M16—21no—
206C2×M16—41no—
207C2×M16C2×D411yescharacteristic
208C2×M16C2×D411yescharacteristic
209C22×C8—21no—
210C22×C8—21no—
211C22×C8—21no—
212C22×C8—21no—
213C22×C8—21no—
214C22×C8—21no—
215C22×C8—21no—
216C22×C8—41no—
217C22×C8#4111yescharacteristic
218C22×C8#4111yescharacteristic
219C4⋊Q8C22×C411yescharacteristic
220C4⋊Q8C22×C411yescharacteristic
221C4⋊Q8C22×C411yescharacteristic
222#127C22×C411yescharacteristic
223#125—21no—
224#124—21no—
225C22⋊Q8—21no—
226C22⋊Q8—21no—
227#121—21no—
228#121—21no—
229C4×Q8—21no—
230C4×Q8—21no—
231C4×Q8C22×C411yescharacteristic
232C4×Q8#4111yescharacteristic
233C4×D4—21no—
234C4×D4—21no—
235C4×D4—21no—
236C4×D4—21no—
237C4×D4C22×C411yescharacteristic
238C4×D4#4111yescharacteristic
239#117—21no—
240#117—21no—
241#117C2×D411yescharacteristic
242C2×C42—21no—
243C2×C42—21no—
244C2×C42—21no—
245C2×C42C2×D411yescharacteristic
246C2×C42C2×D411yescharacteristic
247C2×C42C2411yescharacteristic
248#108—42no—
249#108—81no—
250#107—21no—
251#107—21no—
252#106—21no—
253#106—21no—
254#106—21no—
255#106—21no—
256C4⋊C8—21no—
257C4⋊C8—21no—
258C4⋊C8—21no—
259C4⋊C8—21no—
260C4⋊C8—21no—
261C4⋊C8—21no—
262C4⋊C8—21no—
263C4⋊C8—21no—
264C4⋊C8—21no—
265C4⋊C8—21no—
266C4⋊C8—21no—
267C4⋊C8—41no—
268C4⋊C8—41no—
269C4⋊C8C22×C411yescharacteristic
270C4⋊C8C22×C411yescharacteristic
271C4⋊C8C22×C411yescharacteristic
272C4⋊C8C22×C411yescharacteristic
273C4⋊C8C22×C411yescharacteristic
274C4⋊C8C22×C411yescharacteristic
275C4⋊C8C2×D411yescharacteristic
276C4⋊C8C2×D411yescharacteristic
277C4⋊C8C2×D411yescharacteristic
278C4⋊C8C2×D411yescharacteristic
279#104—41no—
280#104—41no—
281#104—42no—
282#104—81no—
283#103—21no—
284#103—21no—
285#103—21no—
286#103—21no—
287#103—21no—
288#103—21no—
289#102—21no—
290#102—21no—
291#102—21no—
292#102—21no—
293#102—21no—
294#102—21no—
295C22⋊C8—21no—
296C22⋊C8—21no—
297C22⋊C8—21no—
298C22⋊C8—21no—
299C22⋊C8—41no—
300C22⋊C8—41no—
301C22⋊C8—41no—
302C22⋊C8—41no—
303C22⋊C8—41no—
304C22⋊C8—41no—
305#97—24no—
306#97—21no—
307#97—41no—
308#97—41no—
309#97—42no—
310#97—41no—
311C4×C8—21no—
312C4×C8—21no—
313C4×C8—21no—
314C4×C8—21no—
315C4×C8—21no—
316C4×C8—21no—
317C4×C8—21no—
318C4×C8—41no—
319C4×C8—41no—
320C4×C8—42no—
321C4×C8—41no—
322C4×C8—81no—
323C4×C8C22×C411yescharacteristic
324C4×C8C22×C411yescharacteristic
325C4×C8C2×D411yescharacteristic
326C4×C8C2×D411yescharacteristic
327#41—21no—
328#41—21no—
329#41—21no—
330#41—21no—
331#41—41no—
332#41—41no—
333#41—41no—
334#41—41no—
335C2×Q8—21no—
336C2×Q8—21no—
337C2×Q8—21no—
338C2×Q8C4×D411yescharacteristic
339C2×D4—21no—
340C2×D4—21no—
341C2×D4—21no—
342C2×D4—21no—
343C2×D4—41no—
344C2×D4C4×D411yescharacteristic
345C22×C4—21no—
346C22×C4—21no—
347C22×C4—21no—
348C22×C4—21no—
349C22×C4—21no—
350C22×C4—41no—
351C22×C4C22×D411yescharacteristic
352C22×C4C2×#4111yescharacteristic
353C22×C42−1+411yescharacteristic
354Q16—81no—
355SD16—42no—
356SD16—81no—
357SD16—81no—
358SD16—81no—
359D8—81no—
360M16—21no—
361M16—21no—
362M16—21no—
363M16—21no—
364M16—41no—
365M16—41no—
366M16—41no—
367M16—41no—
368M16—82no—
369C2×C8—22no—
370C2×C8—22no—
371C2×C8—21no—
372C2×C8—21no—
373C2×C8—21no—
374C2×C8—21no—
375C2×C8—21no—
376C2×C8—21no—
377C2×C8—21no—
378C2×C8—21no—
379C2×C8—21no—
380C2×C8—21no—
381C2×C8—21no—
382C2×C8—21no—
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—42no—
395C2×C8—41no—
396C2×C8—41no—
397C2×C8—41no—
398C2×C8—81no—
399C2×C8—82no—
400C2×C8C4×D411yescharacteristic
401C2×C8C4×D411yescharacteristic
402C2×C8C4×D411yescharacteristic
403C2×C8C4×D411yescharacteristic
404C4⋊C4—21no—
405C4⋊C4—21no—
406C4⋊C4—21no—
407C4⋊C4—21no—
408C4⋊C4—21no—
409C4⋊C4—21no—
410C4⋊C4—21no—
411C4⋊C4—21no—
412C4⋊C4—41no—
413C4⋊C4—41no—
414C4⋊C4C4×D411yescharacteristic
415C4⋊C4C4×D411yescharacteristic
416C22⋊C4—21no—
417C22⋊C4—21no—
418C22⋊C4—21no—
419C22⋊C4—41no—
420C22⋊C4—41no—
421C22⋊C4—41no—
422C42—22no—
423C42—22no—
424C42—21no—
425C42—21no—
426C42—21no—
427C42—21no—
428C42—21no—
429C42—21no—
430C42—21no—
431C42—41no—
432C42—81no—
433C42C23×C411yescharacteristic
434C42C22×D411yescharacteristic
435C42C2×#4111yescharacteristic
436C42H(5,𝔽2)11yescharacteristic
437C23—21no—
438C23—41no—
439C23#54711yescharacteristic
440Q8—21no—
441Q8—21no—
442Q8—41no—
443Q8—41no—
444Q8—41no—
445Q8—41no—
446Q8—81no—
447D4—21no—
448D4—21no—
449D4—41no—
450D4—41no—
451D4—41no—
452D4—41no—
453D4—41no—
454D4—41no—
455D4—81no—
456D4—81no—
457D4—81no—
458C2×C4—21no—
459C2×C4—21no—
460C2×C4—22no—
461C2×C4—22no—
462C2×C4—21no—
463C2×C4—21no—
464C2×C4—21no—
465C2×C4—41no—
466C2×C4—41no—
467C2×C4—41no—
468C2×C4—41no—
469C2×C4—41no—
470C2×C4—41no—
471C2×C4—41no—
472C2×C4—41no—
473C2×C4—41no—
474C2×C4—41no—
475C2×C4—41no—
476C2×C4—41no—
477C2×C4—81no—
478C2×C4—81no—
479C2×C4—81no—
480C2×C4C2×C4×D411yescharacteristic
481C2×C4C2×C4×D411yescharacteristic
482C2×C4#52011yescharacteristic
483C2×C4D4211yescharacteristic
484C2×C4#54811yescharacteristic
485C2×C4#55311yescharacteristic
486C2×C4#57511yescharacteristic
487C2×C4#57511yescharacteristic
488C2×C4#57711yescharacteristic
489C2×C4#57711yescharacteristic
490C8—21no—
491C8—21no—
492C8—21no—
493C8—42no—
494C8—41no—
495C8—41no—
496C8—41no—
497C8—41no—
498C8—41no—
499C8—41no—
500C8—41no—
501C8—41no—
502C8—41no—
503C8—41no—
504C8—41no—
505C8—41no—
506C8—81no—
507C8—81no—
508C8—81no—
509C8#44312yes—
510C22—21no—
511C22—21no—
512C22—41no—
513C22—41no—
514C22—41no—
515C22—81no—
516C22—81no—
517C22—81no—
518C22#236111yescharacteristic
519C22#334811yescharacteristic
520C22#335011yescharacteristic
521C4—21no—
522C4—21no—
523C4—21no—
524C4—42no—
525C4—42no—
526C4—41no—
527C4—41no—
528C4—41no—
529C4—81no—
530C4—81no—
531C4—81no—
532C4#294311yescharacteristic
533C4#294911yescharacteristic
534C4C2×#44311yescharacteristic
535C4#295611yescharacteristic
536C4#326711yescharacteristic
537C4#327111yescharacteristic
538C2—21no—
539C2—21no—
540C2—81no—
541C2—161no—
542C2#2301511yescharacteristic
543C2#2303311yescharacteristic
544C2#2304311yescharacteristic
subgroupG/H
centerC2×C4D42
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
socleC22#2361
prime radical#3035C22
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
2. #23080C2semidirect
4. #18071C2semidirect
5. #17829C2semidirect
6. #9851C2semidirect
7. #9626C2semidirect
8. #9579C2semidirect
9. #9578C2semidirect
11. #8994C2semidirect
12. #8932C2semidirect
13. #8931C2semidirect
14. #8275C2semidirect
15. #8192C2semidirect
40. #1680C4semidirect
41. #1679C4semidirect
42. #1678C4semidirect
43. #1677C4semidirect
44. #1672C4semidirect
45. #1671C4semidirect
46. #1666C4semidirect
47. #1665C4semidirect

Maximal subgroups

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

subgroupindex
#231132
#230802
#191242
#180712
#178292
#98512
#96262
#95792
#95782
#89962
#89942
#89322
#89312
#82752
#81922