Finite Group Databasebeta

← #229234

Subgroups — #229234

Order 512.

594 conjugacy classes of subgroups.

The subgroup lattice has 564 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23234C211yesmaximal, characteristic, complemented
2#22717C211yesmaximal, characteristic, complemented
3#19110C211yesmaximal, characteristic
4#18001C211yesmaximal, characteristic
5#18000C211yesmaximal, characteristic, complemented
6#9857C211yesmaximal, characteristic, complemented
7#9629C211yesmaximal, characteristic, complemented
8#9592C211yesmaximal, characteristic, complemented
9#9591C211yesmaximal, characteristic, complemented
10#8991C211yesmaximal, characteristic
11#8989C211yesmaximal, characteristic, complemented
12#8931C211yesmaximal, characteristic, complemented
13#8930C211yesmaximal, characteristic, complemented
14#8293C211yesmaximal, characteristic, complemented
15C8×#102C211yesmaximal, characteristic, complemented
16#3131C2211yescharacteristic
17#3101C2211yescharacteristic
18#3100C2211yescharacteristic
19#2962C2211yescharacteristic
20#2935C2211yescharacteristic
21D4×M16C2211yescharacteristic
22D4×M16C2211yescharacteristic
23C2×C8×D4C2211yescharacteristic
24#1986C2211yescharacteristic
25#1980C2211yescharacteristic
26#1950C2211yescharacteristic
27#1950C2211yescharacteristic
28#1937—21no—
29#1929—21no—
30#1866—21no—
31#1855—21no—
32#1855—21no—
33#1837C2211yescharacteristic
34#1810—21no—
35C4×#108C2211yescharacteristic
36C8×C4⋊C4C2211yescharacteristic
37#1764—21no—
38C4×#102C2211yescharacteristic
39C2×#341—21no—
40#1683C411yescharacteristic, complemented
41#1683C411yescharacteristic, complemented
42#1681C411yescharacteristic, complemented
43#1681C411yescharacteristic, complemented
44#1679C411yescharacteristic, complemented
45#1679C411yescharacteristic, complemented
46#1677C411yescharacteristic, complemented
47#1677C411yescharacteristic, complemented
48#1594—21no—
49#1592—21no—
50#1588—21no—
51#1586—21no—
52#1586—21no—
53#1585—21no—
54#1580—21no—
55SD16⋊C8—21no—
56#1571C2211yescharacteristic
57#1570C2211yescharacteristic
58#1564C2211yescharacteristic
59#1542C2211yescharacteristic
60#1540C2211yescharacteristic
61#1493C2211yescharacteristic
62#1493C2211yescharacteristic
63#1492C2211yescharacteristic
64#1492C2211yescharacteristic
65C2×D4⋊C8C2211yescharacteristic
66#1451C2211yescharacteristic
67#1451C2211yescharacteristic
68C2×C82C2211yescharacteristic
69#1303C2211yescharacteristic
70#1303C2211yescharacteristic
71#1279C2211yescharacteristic
72#1278C2211yescharacteristic
73#1278C2211yescharacteristic
74#1277C2211yescharacteristic
75C2×#131—21no—
76C22×M16—21no—
77C22×M16—21no—
78C23×C8—21no—
79#533C2311yescharacteristic
80#532C2311yescharacteristic
81C2×C4×D4C2311yescharacteristic
82#488—21no—
83#488—21no—
84#487C2×C411yescharacteristic
85#487C2×C411yescharacteristic
86#486C2×C411yescharacteristic
87#486C2×C411yescharacteristic
88#483—21no—
89#480—21no—
90#476C2×C411yescharacteristic
91#476C2×C411yescharacteristic
92#474C2×C411yescharacteristic
93#474C2×C411yescharacteristic
94#461C2×C411yescharacteristic
95#461C2×C411yescharacteristic
96#459C2×C411yescharacteristic
97#459C2×C411yescharacteristic
98#450—21no—
99#447—21no—
100#446—21no—
101#446—41no—
102C8×Q8—21no—
103C4×SD16—21no—
104C4×SD16—21no—
105C4×D8—21no—
106C4×D8—21no—
107#436—21no—
108#436—21no—
109#436—21no—
110#435—41no—
111#435—41no—
112#435—41no—
113C8×D4—21no—
114C8×D4—21no—
115C8×D4—21no—
116C8×D4—21no—
117C8×D4—21no—
118C8×D4C2×C411yescharacteristic
119C8×D4C2×C411yescharacteristic
120C8×D4C2×C411yescharacteristic
121C8×D4C2×C411yescharacteristic
122#431C2311yescharacteristic
123#431C2311yescharacteristic
124C2×#108—21no—
125C2×#108—21no—
126C2×#108D411yescharacteristic
127C2×#108D411yescharacteristic
128#424—21no—
129#423C2311yescharacteristic
130#423C2311yescharacteristic
131C2×C4⋊C8C2311yescharacteristic
132C2×C4⋊C8C2311yescharacteristic
133C2×#104—21no—
134C2×#104—21no—
135#416—21no—
136#416—21no—
137C2×#102—21no—
138#408—21no—
139#407—21no—
140#407—21no—
141C2×C22⋊C8—21no—
142#405—21no—
143C4×M16C2311yescharacteristic
144C4×M16C2311yescharacteristic
145C2×C4×C8—21no—
146C2×C4×C8C2311yescharacteristic
147C2×C4×C8C2311yescharacteristic
148C2×C4×C8C2311yescharacteristic
149C4×C4⋊C4C2311yescharacteristic
150#341—21no—
151#341—21no—
152#341—21no—
153#341—21no—
154#341D411yescharacteristic
155#341D411yescharacteristic
156#340—21no—
157#339—21no—
158#339—21no—
159#336—21no—
160#335—21no—
161#335—21no—
162#335C2×C411yescharacteristic
163#335C2×C411yescharacteristic
164#334—21no—
165#334—21no—
166#334C2×C411yescharacteristic
167#334C2×C411yescharacteristic
168#333C2×C411yescharacteristic
169#333C2×C411yescharacteristic
170#332C2×C411yescharacteristic
171#332C2×C411yescharacteristic
172#332C2×C411yescharacteristic
173#332C2×C411yescharacteristic
174#331C2×C411yescharacteristic
175#331C2×C411yescharacteristic
176Q8⋊C8—21no—
177Q8⋊C8—21no—
178Q8⋊C8—41no—
179D4⋊C8—21no—
180D4⋊C8—21no—
181D4⋊C8—41no—
182D4⋊C8D411yescharacteristic
183D4⋊C8D411yescharacteristic
184D4⋊C8D411yescharacteristic
185D4⋊C8D411yescharacteristic
186#322—21no—
187#322—21no—
188#322—21no—
189#322—21no—
190C82—21no—
191C82—21no—
192C2×#41—21no—
193C22×D4#4111yescharacteristic
194C23×C4—21no—
195C2×SD16—21no—
196C2×SD16—21no—
197C2×D8—21no—
198C2×D8—21no—
199#131—42no—
200#131—81no—
201C2×M16—21no—
202C2×M16—21no—
203C2×M16—41no—
204C2×M16—41no—
205C2×M16—41no—
206C2×M16—42no—
207C2×M16—42no—
208C2×M16—81no—
209C2×M16—81no—
210C2×M16C2×D411yescharacteristic
211C2×M16C2×D411yescharacteristic
212C2×M16C2×D411yescharacteristic
213C2×M16C2×D411yescharacteristic
214C22×C8—21no—
215C22×C8—21no—
216C22×C8—21no—
217C22×C8—21no—
218C22×C8—21no—
219C22×C8—21no—
220C22×C8—21no—
221C22×C8—21no—
222C22×C8—41no—
223C22×C8—41no—
224C22×C8—41no—
225C22×C8—81no—
226C22×C8—81no—
227C22×C8C2×D411yescharacteristic
228C22×C8C2×D411yescharacteristic
229C4⋊Q8C22×C411yescharacteristic
230C4⋊Q8C22×C411yescharacteristic
231C4⋊Q8C22×C411yescharacteristic
232#127C22×C411yescharacteristic
233#125—21no—
234#124—21no—
235C22⋊Q8—21no—
236C22⋊Q8—21no—
237#121—21no—
238#121—21no—
239C4×Q8—21no—
240C4×Q8—21no—
241C4×D4—21no—
242C4×D4—21no—
243C4×D4—21no—
244C4×D4—21no—
245C4×D4C22×C411yescharacteristic
246C4×D4C22×C411yescharacteristic
247C4×D4C2×D411yescharacteristic
248C4×D4C2×D411yescharacteristic
249#117—21no—
250C2×C4⋊C4—21no—
251C2×C4⋊C4#4111yescharacteristic
252C2×C22⋊C4—21no—
253C2×C42—21no—
254C2×C42C2411yescharacteristic
255#108—41no—
256#108—41no—
257#108—81no—
258#108—81no—
259#107—21no—
260#107—21no—
261#106—21no—
262#106—21no—
263C4⋊C8—21no—
264C4⋊C8—21no—
265C4⋊C8—21no—
266C4⋊C8—21no—
267C4⋊C8—21no—
268C4⋊C8—21no—
269C4⋊C8—41no—
270C4⋊C8—41no—
271C4⋊C8C22×C411yescharacteristic
272C4⋊C8C22×C411yescharacteristic
273C4⋊C8C22×C411yescharacteristic
274C4⋊C8C22×C411yescharacteristic
275C4⋊C8C22×C411yescharacteristic
276C4⋊C8C22×C411yescharacteristic
277C4⋊C8C2×D411yescharacteristic
278C4⋊C8C2×D411yescharacteristic
279#104—42no—
280#104—81no—
281#103—21no—
282#103—21no—
283#103—21no—
284#103—21no—
285#103—21no—
286#103—21no—
287#102—24no—
288#102—21no—
289#102—21no—
290#102—21no—
291#102—21no—
292#102—21no—
293#102—21no—
294#102—21no—
295#102—21no—
296#102—21no—
297#102—21no—
298C22⋊C8—21no—
299C22⋊C8—21no—
300C22⋊C8—21no—
301C22⋊C8—21no—
302C22⋊C8—21no—
303C22⋊C8—21no—
304C22⋊C8—41no—
305C22⋊C8—41no—
306C22⋊C8—41no—
307C22⋊C8—41no—
308C22⋊C8—41no—
309C22⋊C8—41no—
310#97—21no—
311#97—21no—
312#97—41no—
313C4×C8—21no—
314C4×C8—21no—
315C4×C8—21no—
316C4×C8—21no—
317C4×C8—21no—
318C4×C8—21no—
319C4×C8—21no—
320C4×C8—41no—
321C4×C8—81no—
322C4×C8C22×C411yescharacteristic
323C4×C8C22×C411yescharacteristic
324C4×C8C2×D411yescharacteristic
325C4×C8C2×D411yescharacteristic
326C4×C8#4111yescharacteristic
327C4×C8#4111yescharacteristic
328#95—21no—
329C24—21no—
330#41—41no—
331#41—41no—
332#41—41no—
333#41—41no—
334C2×Q8—21no—
335C2×Q8—21no—
336C2×Q8—21no—
337C2×D4—22no—
338C2×D4—21no—
339C2×D4—21no—
340C2×D4—21no—
341C2×D4—21no—
342C2×D4—41no—
343C2×D4—41no—
344C2×D4C4×D411yescharacteristic
345C2×D4C4×D411yescharacteristic
346C22×C4—21no—
347C22×C4—21no—
348C22×C4—21no—
349C22×C4—21no—
350C22×C4—21no—
351C22×C4—41no—
352C22×C4—41no—
353C22×C4—41no—
354C22×C4—41no—
355C22×C4—41no—
356C22×C4—41no—
357C22×C4C22×D411yescharacteristic
358C22×C4C2×#4111yescharacteristic
359C22×C42−1+411yescharacteristic
360SD16—42no—
361SD16—81no—
362D8—42no—
363D8—81no—
364M16—21no—
365M16—21no—
366M16—21no—
367M16—21no—
368M16—21no—
369M16—21no—
370M16—21no—
371M16—21no—
372M16—41no—
373M16—41no—
374M16—41no—
375M16—41no—
376M16—82no—
377M16—82no—
378M16—82no—
379C2×C8—21no—
380C2×C8—21no—
381C2×C8—21no—
382C2×C8—21no—
383C2×C8—21no—
384C2×C8—22no—
385C2×C8—21no—
386C2×C8—21no—
387C2×C8—21no—
388C2×C8—21no—
389C2×C8—21no—
390C2×C8—21no—
391C2×C8—21no—
392C2×C8—21no—
393C2×C8—21no—
394C2×C8—21no—
395C2×C8—21no—
396C2×C8—21no—
397C2×C8—21no—
398C2×C8—42no—
399C2×C8—41no—
400C2×C8—41no—
401C2×C8—41no—
402C2×C8—41no—
403C2×C8—41no—
404C2×C8—41no—
405C2×C8—41no—
406C2×C8—41no—
407C2×C8—81no—
408C2×C8—81no—
409C2×C8—81no—
410C2×C8—81no—
411C2×C8—82no—
412C2×C8—82no—
413C2×C8—82no—
414C2×C8C4×D411yescharacteristic
415C2×C8C4×D411yescharacteristic
416C2×C8C4×D411yescharacteristic
417C2×C8C4×D411yescharacteristic
418C4⋊C4—21no—
419C4⋊C4—22no—
420C4⋊C4—21no—
421C4⋊C4—21no—
422C4⋊C4—21no—
423C4⋊C4—21no—
424C4⋊C4—21no—
425C4⋊C4—21no—
426C4⋊C4—41no—
427C4⋊C4—41no—
428C4⋊C4—41no—
429C4⋊C4C4×D411yescharacteristic
430C4⋊C4C4×D411yescharacteristic
431C22⋊C4—21no—
432C22⋊C4—21no—
433C22⋊C4—21no—
434C22⋊C4—21no—
435C22⋊C4—41no—
436C22⋊C4—41no—
437C22⋊C4—41no—
438C42—41no—
439C42—41no—
440C42—41no—
441C42C23×C411yescharacteristic
442C42C22×D411yescharacteristic
443C42C2×#4111yescharacteristic
444C42H(5,𝔽2)11yescharacteristic
445C23—21no—
446C23—21no—
447C23—42no—
448C23—41no—
449C23—41no—
450C23—41no—
451C23—81no—
452C23#54811yescharacteristic
453Q8—41no—
454Q8—41no—
455Q8—41no—
456Q8—41no—
457Q8—81no—
458D4—21no—
459D4—21no—
460D4—21no—
461D4—42no—
462D4—41no—
463D4—41no—
464D4—41no—
465D4—41no—
466D4—81no—
467D4—81no—
468D4—81no—
469D4#44312yes—
470C2×C4—21no—
471C2×C4—21no—
472C2×C4—21no—
473C2×C4—21no—
474C2×C4—41no—
475C2×C4—41no—
476C2×C4—41no—
477C2×C4—41no—
478C2×C4—41no—
479C2×C4—41no—
480C2×C4—41no—
481C2×C4—41no—
482C2×C4—41no—
483C2×C4—41no—
484C2×C4—41no—
485C2×C4—42no—
486C2×C4—41no—
487C2×C4—41no—
488C2×C4—41no—
489C2×C4—81no—
490C2×C4—81no—
491C2×C4—81no—
492C2×C4—81no—
493C2×C4C2×C4×D411yescharacteristic
494C2×C4C2×C4×D411yescharacteristic
495C2×C4#52011yescharacteristic
496C2×C4D4211yescharacteristic
497C2×C4#54711yescharacteristic
498C2×C4#55311yescharacteristic
499C2×C4#57511yescharacteristic
500C2×C4#57511yescharacteristic
501C2×C4#57611yescharacteristic
502C2×C4#57711yescharacteristic
503C8—22no—
504C8—21no—
505C8—21no—
506C8—21no—
507C8—21no—
508C8—41no—
509C8—42no—
510C8—41no—
511C8—41no—
512C8—41no—
513C8—41no—
514C8—41no—
515C8—41no—
516C8—41no—
517C8—41no—
518C8—41no—
519C8—41no—
520C8—41no—
521C8—41no—
522C8—81no—
523C8—81no—
524C22—21no—
525C22—21no—
526C22—22no—
527C22—41no—
528C22—41no—
529C22—41no—
530C22—41no—
531C22—41no—
532C22—82no—
533C22—82no—
534C22—81no—
535C22—81no—
536C22—81no—
537C22#236111yescharacteristic
538C22#337011yescharacteristic
539C22#337611yescharacteristic
540C4—21no—
541C4—21no—
542C4—21no—
543C4—41no—
544C4—41no—
545C4—82no—
546C4—81no—
547C4—81no—
548C4—81no—
549C4#294311yescharacteristic
550C4#294611yescharacteristic
551C4C2×#44311yescharacteristic
552C4#295711yescharacteristic
553C4#327311yescharacteristic
554C4#327511yescharacteristic
555C2—21no—
556C2—21no—
557C2—42no—
558C2—81no—
559C2—161no—
560C2#2300511yescharacteristic
561C2#2301811yescharacteristic
562C2#2304211yescharacteristic
subgroupG/H
centerC2×C4D42
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
socleC22#2361
prime radical#3100C22
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. #23234C2semidirect
2. #22717C2semidirect
5. #18000C2semidirect
6. #9857C2semidirect
7. #9629C2semidirect
8. #9592C2semidirect
9. #9591C2semidirect
11. #8989C2semidirect
12. #8931C2semidirect
13. #8930C2semidirect
14. #8293C2semidirect
15. C8×#102C2semidirect
40. #1683C4semidirect
41. #1683C4semidirect
42. #1681C4semidirect
43. #1681C4semidirect
44. #1679C4semidirect
45. #1679C4semidirect
46. #1677C4semidirect
47. #1677C4semidirect

Maximal subgroups

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

subgroupindex
#232342
#227172
#191102
#180012
#180002
#98572
#96292
#95922
#95912
#89912
#89892
#89312
#89302
#82932
C8×#1022