Finite Group Databasebeta

← #226021

Subgroups — #226021

Order 512.

558 conjugacy classes of subgroups.

The subgroup lattice has 541 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21137C211yesmaximal, characteristic
2#17898C211yesmaximal, characteristic
3#17881C211yesmaximal, characteristic
4#17812C211yesmaximal, characteristic, complemented
5#17799C211yesmaximal, characteristic, complemented
6#9610C211yesmaximal, characteristic, complemented
7#9608C211yesmaximal, characteristic, complemented
8#8841C211yesmaximal, characteristic, complemented
9#8841C211yesmaximal, characteristic, complemented
10#8841C211yesmaximal, characteristic, complemented
11#8839C211yesmaximal, characteristic, complemented
12#8810C211yesmaximal, characteristic, complemented
13#8606C211yesmaximal, characteristic, complemented
14#8605C211yesmaximal, characteristic, complemented
15#8270C211yesmaximal, characteristic, complemented
16#2989C2211yescharacteristic
17#2989C2211yescharacteristic
18#2989C2211yescharacteristic
19#2988C2211yescharacteristic
20C4×#135C2211yescharacteristic
21#2922C2211yescharacteristic
22C4×#131C2211yescharacteristic
23#1929C411yescharacteristic
24#1929C411yescharacteristic
25#1922C411yescharacteristic
26#1922C411yescharacteristic
27#1876C411yescharacteristic
28#1876C411yescharacteristic
29#1866C411yescharacteristic
30#1866C411yescharacteristic
31#1842C2211yescharacteristic
32#1842C2211yescharacteristic
33C4×#108C2211yescharacteristic
34C4×#104C2211yescharacteristic
35C4×#104C2211yescharacteristic
36#1744C2211yescharacteristic
37#1592—21no—
38#1592—21no—
39#1591—21no—
40#1589—21no—
41#1588—21no—
42#1585—21no—
43#1584—21no—
44#1581—21no—
45#1574—21no—
46#1573—21no—
47#1572—21no—
48#1571—21no—
49#1569—21no—
50#1569—21no—
51#1568—21no—
52#1567—21no—
53#1565C2211yescharacteristic
54#1565C2211yescharacteristic
55#1565C2211yescharacteristic
56#1565C2211yescharacteristic
57#1564C2211yescharacteristic
58#1564C2211yescharacteristic
59C2×#335C2211yescharacteristic
60C2×#335C2211yescharacteristic
61C2×#335C2211yescharacteristic
62C2×#335C2211yescharacteristic
63#1477C2211yescharacteristic
64#1477C2211yescharacteristic
65#1477C2211yescharacteristic
66#1477C2211yescharacteristic
67#1452C2211yescharacteristic
68#1450C2211yescharacteristic
69#1283C2211yescharacteristic
70#1283C2211yescharacteristic
71#1283C2211yescharacteristic
72#1283C2211yescharacteristic
73#1276C2211yescharacteristic
74#1276C2211yescharacteristic
75C2×#135C2×C411yescharacteristic
76C2×#131C2×C411yescharacteristic
77C2×#131C2×C411yescharacteristic
78C4×#41C2311yescharacteristic
79C4×#41C2311yescharacteristic
80#446—21no—
81#446—21no—
82#446—21no—
83C8×Q8—21no—
84C4×Q16D411yescharacteristic
85C4×Q16Q811yescharacteristic
86C4×SD16D411yescharacteristic
87C4×SD16D411yescharacteristic
88C4×SD16Q811yescharacteristic
89C4×SD16Q811yescharacteristic
90C4×D8D411yescharacteristic
91C4×D8Q811yescharacteristic
92#436—21no—
93#436—21no—
94#436—21no—
95#435—41no—
96#435—41no—
97#435—41no—
98C8×D4—21no—
99C8×D4—41no—
100#433—21no—
101#433—21no—
102#432—21no—
103#432—21no—
104#432—21no—
105#431—21no—
106#431—21no—
107#431—21no—
108#431C2311yescharacteristic
109#431C2311yescharacteristic
110#431C2311yescharacteristic
111#431C2311yescharacteristic
112C2×#108C2×C411yescharacteristic
113C2×#108C2×C411yescharacteristic
114#427C2×C411yescharacteristic
115#424C2×C411yescharacteristic
116#424C2×C411yescharacteristic
117#423—21no—
118#423—21no—
119#423—21no—
120C2×C4⋊C8—21no—
121C2×C4⋊C8—41no—
122C2×C4⋊C8—41no—
123C2×C4⋊C8C2311yescharacteristic
124C2×C4⋊C8C2311yescharacteristic
125C2×C4⋊C8C2311yescharacteristic
126C2×C4⋊C8C2311yescharacteristic
127C2×#104C2×C411yescharacteristic
128C2×#104C2×C411yescharacteristic
129C2×#104C2×C411yescharacteristic
130C2×#104C2×C411yescharacteristic
131#416C2×C411yescharacteristic
132#416C2×C411yescharacteristic
133#408C2×C411yescharacteristic
134#408C2×C411yescharacteristic
135#405—21no—
136#405—21no—
137#405C2×C411yescharacteristic
138#405C2×C411yescharacteristic
139C4×M16—21no—
140C4×M16C2311yescharacteristic
141C4×M16C2311yescharacteristic
142C2×C4×C8—21no—
143C2×C4×C8C2311yescharacteristic
144C2×C4×C8C2311yescharacteristic
145C43C2311yescharacteristic
146#341C2×C411yescharacteristic
147#341C2×C411yescharacteristic
148#339C2×C411yescharacteristic
149#339C2×C411yescharacteristic
150#339C2×C411yescharacteristic
151#339C2×C411yescharacteristic
152#336C2×C411yescharacteristic
153#336C2×C411yescharacteristic
154#336C2×C411yescharacteristic
155#336C2×C411yescharacteristic
156#335—21no—
157#335—21no—
158#335—21no—
159#335—21no—
160#335—21no—
161#335—21no—
162#335—21no—
163#335—21no—
164#335—21no—
165#335—21no—
166#335—21no—
167#335—21no—
168#335—21no—
169#335D411yescharacteristic
170#335D411yescharacteristic
171#334—21no—
172#334—21no—
173#334—21no—
174#334—21no—
175#334—21no—
176#334D411yescharacteristic
177#334D411yescharacteristic
178Q8⋊C8—21no—
179Q8⋊C8—21no—
180Q8⋊C8—21no—
181Q8⋊C8—21no—
182D4⋊C8—21no—
183D4⋊C8—21no—
184D4⋊C8—21no—
185D4⋊C8—21no—
186#322—21no—
187#322—21no—
188#322—21no—
189C82—21no—
190C2×#41C22×C411yescharacteristic
191C2×#41C22×C411yescharacteristic
192#135—41no—
193#135—41no—
194C2×Q16C4⋊C411yescharacteristic
195C2×SD16C4⋊C411yescharacteristic
196C2×SD16C4⋊C411yescharacteristic
197C2×D8C4⋊C411yescharacteristic
198#131—22no—
199#131—41no—
200#131—41no—
201#131—41no—
202C2×M16—21no—
203C2×M16—21no—
204C2×M16—41no—
205C2×M16—41no—
206C2×M16—41no—
207C2×M16C22×C411yescharacteristic
208C2×M16C22×C411yescharacteristic
209C2×M16C22×C411yescharacteristic
210C2×M16C22×C411yescharacteristic
211C22×C8—21no—
212C22×C8—21no—
213C22×C8—21no—
214C22×C8—21no—
215C22×C8—21no—
216C22×C8—21no—
217C22×C8—41no—
218C22×C8C22×C411yescharacteristic
219C22×C8C22×C411yescharacteristic
220C22×C8C22×C411yescharacteristic
221C22×C8C22×C411yescharacteristic
222C4×Q8C2×D411yescharacteristic
223C4×Q8C2×D411yescharacteristic
224C4×Q8C2×Q811yescharacteristic
225C4×Q8C2×Q811yescharacteristic
226C4×D4—21no—
227C4×D4—21no—
228C4×D4—21no—
229C4×D4—21no—
230C4×D4C2×D411yescharacteristic
231C4×D4C2×D411yescharacteristic
232C4×D4C2×Q811yescharacteristic
233C4×D4C2×Q811yescharacteristic
234#117—21no—
235#117—21no—
236#117C22×C411yescharacteristic
237#117C22×C411yescharacteristic
238C2×C42—21no—
239C2×C42—21no—
240C2×C42—21no—
241C2×C42—21no—
242C2×C42C22×C411yescharacteristic
243C2×C42C22×C411yescharacteristic
244C2×C42C2411yescharacteristic
245#108—22no—
246#108—41no—
247#107C4⋊C411yescharacteristic
248#107C4⋊C411yescharacteristic
249#106C4⋊C411yescharacteristic
250#106C4⋊C411yescharacteristic
251C4⋊C8—21no—
252C4⋊C8—21no—
253C4⋊C8—21no—
254C4⋊C8—21no—
255C4⋊C8—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—41no—
266C4⋊C8—41no—
267C4⋊C8—41no—
268C4⋊C8—41no—
269C4⋊C8—82no—
270C4⋊C8—82no—
271C4⋊C8C2×D411yescharacteristic
272C4⋊C8C2×D411yescharacteristic
273C4⋊C8C2×D411yescharacteristic
274C4⋊C8C2×D411yescharacteristic
275#104—22no—
276#104—41no—
277#104—41no—
278#104—41no—
279#103C4⋊C411yescharacteristic
280#103C4⋊C411yescharacteristic
281#103C4⋊C411yescharacteristic
282#103C4⋊C411yescharacteristic
283#102C4⋊C411yescharacteristic
284#102C4⋊C411yescharacteristic
285#102C4⋊C411yescharacteristic
286#102C4⋊C411yescharacteristic
287C22⋊C8—21no—
288C22⋊C8—21no—
289C22⋊C8—21no—
290C22⋊C8—21no—
291C22⋊C8—21no—
292C22⋊C8—21no—
293C22⋊C8—21no—
294C22⋊C8—21no—
295C22⋊C8—41no—
296C22⋊C8—41no—
297C22⋊C8—41no—
298C22⋊C8—41no—
299#97—21no—
300#97—21no—
301#97—21no—
302#97—21no—
303#97—41no—
304#97—41no—
305#97—41no—
306C4×C8—21no—
307C4×C8—21no—
308C4×C8—21no—
309C4×C8—21no—
310C4×C8—21no—
311C4×C8—21no—
312C4×C8—21no—
313C4×C8—21no—
314C4×C8—41no—
315C4×C8C2×D411yescharacteristic
316C4×C8C2×D411yescharacteristic
317C4×C8C2×D411yescharacteristic
318C4×C8C2×D411yescharacteristic
319C4×C8C2×Q811yescharacteristic
320C4×C8C2×Q811yescharacteristic
321C4×C8#4111yescharacteristic
322C4×C8#4111yescharacteristic
323#41—21no—
324#41—21no—
325#41—21no—
326#41—21no—
327#41—21no—
328#41—21no—
329#41—21no—
330#41—21no—
331C2×Q8C2×C4⋊C411yescharacteristic
332C2×Q8C2×C4⋊C411yescharacteristic
333C2×D4—21no—
334C2×D4—21no—
335C2×D4C2×C4⋊C411yescharacteristic
336C2×D4C2×C4⋊C411yescharacteristic
337C22×C4—21no—
338C22×C4—21no—
339C22×C4—21no—
340C22×C4—21no—
341C22×C4—21no—
342C22×C4—21no—
343C22×C4—21no—
344C22×C4—21no—
345C22×C4C23×C411yescharacteristic
346C22×C4H(5,𝔽2)11yescharacteristic
347C22×C42−1+411yescharacteristic
348Q16—41no—
349SD16—41no—
350SD16—41no—
351D8—41no—
352M16—21no—
353M16—21no—
354M16—21no—
355M16—21no—
356M16—21no—
357M16—21no—
358M16—21no—
359M16—42no—
360M16—42no—
361M16—161no—
362M16—161no—
363M16—161no—
364M16#13112yes—
365C2×C8—21no—
366C2×C8—21no—
367C2×C8—21no—
368C2×C8—21no—
369C2×C8—21no—
370C2×C8—21no—
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—42no—
382C2×C8—42no—
383C2×C8—41no—
384C2×C8—41no—
385C2×C8—41no—
386C2×C8—41no—
387C2×C8—41no—
388C2×C8—41no—
389C2×C8—81no—
390C2×C8—81no—
391C2×C8—81no—
392C2×C8—81no—
393C2×C8—161no—
394C2×C8C2×C4⋊C411yescharacteristic
395C2×C8C2×C4⋊C411yescharacteristic
396C2×C8C2×C4⋊C411yescharacteristic
397C2×C8C2×C4⋊C411yescharacteristic
398C2×C8C4×D411yescharacteristic
399C2×C8C4×D411yescharacteristic
400C2×C8C4×D411yescharacteristic
401C2×C8C4×D411yescharacteristic
402C4⋊C4—21no—
403C4⋊C4—21no—
404C4⋊C4—21no—
405C4⋊C4—21no—
406C4⋊C4C2×C4⋊C411yescharacteristic
407C4⋊C4C2×C4⋊C411yescharacteristic
408C4⋊C4C2×C4⋊C411yescharacteristic
409C4⋊C4C2×C4⋊C411yescharacteristic
410C22⋊C4—21no—
411C22⋊C4—21no—
412C22⋊C4—21no—
413C22⋊C4—21no—
414C22⋊C4—21no—
415C22⋊C4—21no—
416C42—22no—
417C42—22no—
418C42—21no—
419C42—21no—
420C42—21no—
421C42—21no—
422C42—21no—
423C42—21no—
424C42—21no—
425C42—21no—
426C42—21no—
427C42—21no—
428C42—81no—
429C42C22×D411yescharacteristic
430C42C22×D411yescharacteristic
431C42C22×Q811yescharacteristic
432C42C2×#4111yescharacteristic
433C23—21no—
434C23—21no—
435C23#52011yescharacteristic
436Q8—21no—
437Q8—21no—
438Q8—21no—
439Q8—21no—
440D4—21no—
441D4—21no—
442D4—21no—
443D4—21no—
444D4—41no—
445D4—41no—
446D4—41no—
447D4—41no—
448C2×C4—21no—
449C2×C4—21no—
450C2×C4—22no—
451C2×C4—22no—
452C2×C4—21no—
453C2×C4—21no—
454C2×C4—21no—
455C2×C4—21no—
456C2×C4—21no—
457C2×C4—21no—
458C2×C4—41no—
459C2×C4—41no—
460C2×C4—41no—
461C2×C4—41no—
462C2×C4—41no—
463C2×C4—41no—
464C2×C4—41no—
465C2×C4—41no—
466C2×C4—41no—
467C2×C4—41no—
468C2×C4—41no—
469C2×C4—41no—
470C2×C4—41no—
471C2×C4—81no—
472C2×C4—81no—
473C2×C4C22×C4⋊C411yescharacteristic
474C2×C4C2×C4×D411yescharacteristic
475C2×C4D4211yescharacteristic
476C2×C4#54711yescharacteristic
477C2×C4D4×Q811yescharacteristic
478C2×C4#55411yescharacteristic
479C2×C4C2×#13111yescharacteristic
480C2×C4#56811yescharacteristic
481C2×C4#56811yescharacteristic
482C2×C4#56811yescharacteristic
483C8—21no—
484C8—21no—
485C8—21no—
486C8—21no—
487C8—21no—
488C8—21no—
489C8—21no—
490C8—21no—
491C8—22no—
492C8—41no—
493C8—41no—
494C8—41no—
495C8—41no—
496C8—41no—
497C8—41no—
498C8—41no—
499C8—81no—
500C8—81no—
501C8—81no—
502C8—81no—
503C8—81no—
504C8—81no—
505C22—21no—
506C22—21no—
507C22—41no—
508C22—41no—
509C22—41no—
510C22—41no—
511C22—41no—
512C22—41no—
513C22D4×C4⋊C411yescharacteristic
514C22#299311yescharacteristic
515C22#299511yescharacteristic
516C4—21no—
517C4—21no—
518C4—21no—
519C4—42no—
520C4—42no—
521C4—41no—
522C4—41no—
523C4—41no—
524C4—41no—
525C4—41no—
526C4—41no—
527C4#290511yescharacteristic
528C4#290611yescharacteristic
529C4#293011yescharacteristic
530C4#293311yescharacteristic
531C4#329211yescharacteristic
532C4#329311yescharacteristic
533C2—21no—
534C2—21no—
535C2—81no—
536C2—81no—
537C2#2264911yescharacteristic
538C2#2269211yescharacteristic
539C2#2269511yescharacteristic
subgroupG/H
centerC2×C4D42
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC22D4×C4⋊C4
prime radicalC2×#135C2×C4
LCSG ▷ C22×C4 ▷ C2 ▷ 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
4. #17812C2semidirect
5. #17799C2semidirect
6. #9610C2semidirect
7. #9608C2semidirect
8. #8841C2semidirect
9. #8841C2semidirect
10. #8841C2semidirect
11. #8839C2semidirect
12. #8810C2semidirect
13. #8606C2semidirect
14. #8605C2semidirect
15. #8270C2semidirect

Maximal subgroups

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

subgroupindex
#211372
#178982
#178812
#178122
#177992
#96102
#96082
#88412
#88392
#88102
#86062
#86052
#82702