Finite Group Databasebeta

← #225515

Subgroups — #225515

Order 512.

543 conjugacy classes of subgroups.

The subgroup lattice has 536 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21293C211yesmaximal, characteristic
2#17882C211yesmaximal, characteristic
3#17881C211yesmaximal, characteristic
4#17812C211yesmaximal, characteristic, complemented
5#17811C211yesmaximal, characteristic, complemented
6#9610C211yesmaximal, characteristic, complemented
7#9609C211yesmaximal, characteristic, complemented
8#8841C211yesmaximal, characteristic, complemented
9#8841C211yesmaximal, characteristic, complemented
10#8840C211yesmaximal, characteristic, complemented
11#8840C211yesmaximal, characteristic, complemented
12#8808C211yesmaximal, characteristic, complemented
13#8607C211yesmaximal, characteristic, complemented
14#8606C211yesmaximal, characteristic, complemented
15#8269C211yesmaximal, characteristic, complemented
16#2989C2211yescharacteristic
17#2989C2211yescharacteristic
18#2988C2211yescharacteristic
19#2988C2211yescharacteristic
20C4×#135C2211yescharacteristic
21#2922C2211yescharacteristic
22#2922C2211yescharacteristic
23#1975C411yescharacteristic
24#1975C411yescharacteristic
25#1946C411yescharacteristic
26#1946C411yescharacteristic
27#1938C411yescharacteristic
28#1938C411yescharacteristic
29#1938C411yescharacteristic
30#1938C411yescharacteristic
31#1842C2211yescharacteristic
32#1831C2211yescharacteristic
33C4×#108C2211yescharacteristic
34C4×#104C2211yescharacteristic
35C4×#104C2211yescharacteristic
36C4×#97C2211yescharacteristic
37#1588—21no—
38#1587—21no—
39#1586—21no—
40#1585—21no—
41#1584—21no—
42#1583—21no—
43#1582—21no—
44#1581—21no—
45#1574—21no—
46#1574—21no—
47#1573—21no—
48#1573—21no—
49#1572—21no—
50#1572—21no—
51#1571—21no—
52#1570—21no—
53#1565C2211yescharacteristic
54#1565C2211yescharacteristic
55#1565C2211yescharacteristic
56#1565C2211yescharacteristic
57#1564C2211yescharacteristic
58#1564C2211yescharacteristic
59C2×#335C2211yescharacteristic
60C2×#335C2211yescharacteristic
61C2×#334C2211yescharacteristic
62C2×#334C2211yescharacteristic
63#1477C2211yescharacteristic
64#1477C2211yescharacteristic
65#1477C2211yescharacteristic
66#1477C2211yescharacteristic
67#1452C2211yescharacteristic
68#1452C2211yescharacteristic
69#1283C2211yescharacteristic
70#1283C2211yescharacteristic
71#1283C2211yescharacteristic
72#1283C2211yescharacteristic
73#1276C2211yescharacteristic
74#1276C2211yescharacteristic
75C2×#135C2×C411yescharacteristic
76C4×#41C2311yescharacteristic
77C4×#41C2311yescharacteristic
78#446—21no—
79#446—21no—
80C8×Q8—21no—
81C8×Q8—21no—
82C4×Q16D411yescharacteristic
83C4×Q16Q811yescharacteristic
84C4×SD16D411yescharacteristic
85C4×SD16D411yescharacteristic
86C4×SD16Q811yescharacteristic
87C4×SD16Q811yescharacteristic
88C4×D8D411yescharacteristic
89C4×D8Q811yescharacteristic
90#436—21no—
91#436—21no—
92#435—41no—
93#435—41no—
94C8×D4—21no—
95C8×D4—21no—
96C8×D4—41no—
97C8×D4—41no—
98#433—21no—
99#433—21no—
100#433—21no—
101#433—21no—
102#432—21no—
103#432—21no—
104#431—21no—
105#431—21no—
106#431—21no—
107#431—21no—
108#431—21no—
109#431—21no—
110#431C2311yescharacteristic
111#431C2311yescharacteristic
112#431C2311yescharacteristic
113#431C2311yescharacteristic
114C2×#108C2×C411yescharacteristic
115C2×#108C2×C411yescharacteristic
116#427C2×C411yescharacteristic
117#424C2×C411yescharacteristic
118#424C2×C411yescharacteristic
119#424C2×C411yescharacteristic
120#424C2×C411yescharacteristic
121#423—21no—
122#423—21no—
123C2×C4⋊C8—21no—
124C2×C4⋊C8—21no—
125C2×C4⋊C8—41no—
126C2×C4⋊C8C2311yescharacteristic
127C2×C4⋊C8C2311yescharacteristic
128C2×C4⋊C8C2311yescharacteristic
129C2×C4⋊C8C2311yescharacteristic
130C2×#104C2×C411yescharacteristic
131C2×#104C2×C411yescharacteristic
132C2×#104C2×C411yescharacteristic
133C2×#104C2×C411yescharacteristic
134#416C2×C411yescharacteristic
135#416C2×C411yescharacteristic
136#408C2×C411yescharacteristic
137#408C2×C411yescharacteristic
138#408C2×C411yescharacteristic
139#408C2×C411yescharacteristic
140C4×M16C2311yescharacteristic
141C4×M16C2311yescharacteristic
142C2×#97C2×C411yescharacteristic
143C2×#97C2×C411yescharacteristic
144C2×#97C2×C411yescharacteristic
145C2×#97C2×C411yescharacteristic
146C2×C4×C8C2311yescharacteristic
147C2×C4×C8C2311yescharacteristic
148C43C2311yescharacteristic
149#341C2×C411yescharacteristic
150#341C2×C411yescharacteristic
151#339C2×C411yescharacteristic
152#339C2×C411yescharacteristic
153#339C2×C411yescharacteristic
154#339C2×C411yescharacteristic
155#336—41no—
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#335D411yescharacteristic
166#335D411yescharacteristic
167#334—21no—
168#334—21no—
169#334—21no—
170#334—21no—
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—
189#322—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
198C2×M16—41no—
199C2×M16—41no—
200C2×M16C22×C411yescharacteristic
201C2×M16C22×C411yescharacteristic
202C2×M16C22×C411yescharacteristic
203C2×M16C22×C411yescharacteristic
204C22×C8—21no—
205C22×C8—21no—
206C22×C8—21no—
207C22×C8—21no—
208C22×C8—41no—
209C22×C8—41no—
210C22×C8C22×C411yescharacteristic
211C22×C8C22×C411yescharacteristic
212C22×C8C22×C411yescharacteristic
213C22×C8C22×C411yescharacteristic
214C4×Q8C2×D411yescharacteristic
215C4×Q8C2×D411yescharacteristic
216C4×Q8C2×Q811yescharacteristic
217C4×Q8C2×Q811yescharacteristic
218C4×D4—21no—
219C4×D4—21no—
220C4×D4—21no—
221C4×D4—21no—
222C4×D4C2×D411yescharacteristic
223C4×D4C2×D411yescharacteristic
224C4×D4C2×Q811yescharacteristic
225C4×D4C2×Q811yescharacteristic
226#117—21no—
227#117—21no—
228#117C22×C411yescharacteristic
229#117C22×C411yescharacteristic
230C2×C42—21no—
231C2×C42—21no—
232C2×C42—21no—
233C2×C42—21no—
234C2×C42C22×C411yescharacteristic
235C2×C42C22×C411yescharacteristic
236C2×C42C2411yescharacteristic
237#108—41no—
238#108—41no—
239#107C4⋊C411yescharacteristic
240#107C4⋊C411yescharacteristic
241#106C4⋊C411yescharacteristic
242#106C4⋊C411yescharacteristic
243C4⋊C8—21no—
244C4⋊C8—21no—
245C4⋊C8—21no—
246C4⋊C8—21no—
247C4⋊C8—21no—
248C4⋊C8—21no—
249C4⋊C8—21no—
250C4⋊C8—21no—
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—41no—
262C4⋊C8—41no—
263C4⋊C8—41no—
264C4⋊C8—41no—
265C4⋊C8—82no—
266C4⋊C8C2×D411yescharacteristic
267C4⋊C8C2×D411yescharacteristic
268C4⋊C8C2×D411yescharacteristic
269C4⋊C8C2×D411yescharacteristic
270#104—22no—
271#104—22no—
272#104—41no—
273#104—41no—
274#103C4⋊C411yescharacteristic
275#103C4⋊C411yescharacteristic
276#103C4⋊C411yescharacteristic
277#103C4⋊C411yescharacteristic
278#102C4⋊C411yescharacteristic
279#102C4⋊C411yescharacteristic
280#102C4⋊C411yescharacteristic
281#102C4⋊C411yescharacteristic
282C22⋊C8—21no—
283C22⋊C8—21no—
284C22⋊C8—21no—
285C22⋊C8—21no—
286C22⋊C8—21no—
287C22⋊C8—21no—
288C22⋊C8—21no—
289C22⋊C8—21no—
290C22⋊C8—21no—
291C22⋊C8—21no—
292C22⋊C8—21no—
293C22⋊C8—21no—
294C22⋊C8—41no—
295C22⋊C8—41no—
296C22⋊C8—41no—
297C22⋊C8—41no—
298#97—22no—
299#97—21no—
300#97—21no—
301#97—41no—
302#97—41no—
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—41no—
313C4×C8—41no—
314C4×C8C2×D411yescharacteristic
315C4×C8C2×D411yescharacteristic
316C4×C8C2×D411yescharacteristic
317C4×C8C2×D411yescharacteristic
318C4×C8C2×Q811yescharacteristic
319C4×C8C2×Q811yescharacteristic
320C4×C8#4111yescharacteristic
321C4×C8#4111yescharacteristic
322#41—21no—
323#41—21no—
324#41—21no—
325#41—21no—
326#41—21no—
327#41—21no—
328#41—21no—
329#41—21no—
330C2×Q8C2×C4⋊C411yescharacteristic
331C2×Q8C2×C4⋊C411yescharacteristic
332C2×D4—21no—
333C2×D4—21no—
334C2×D4C2×C4⋊C411yescharacteristic
335C2×D4C2×C4⋊C411yescharacteristic
336C22×C4—21no—
337C22×C4—21no—
338C22×C4—21no—
339C22×C4—21no—
340C22×C4—21no—
341C22×C4—21no—
342C22×C4—21no—
343C22×C4—21no—
344C22×C4C23×C411yescharacteristic
345C22×C4H(5,𝔽2)11yescharacteristic
346C22×C42−1+411yescharacteristic
347Q16—41no—
348SD16—41no—
349SD16—41no—
350D8—41no—
351M16—21no—
352M16—21no—
353M16—21no—
354M16—21no—
355M16—21no—
356M16—21no—
357M16—21no—
358M16—21no—
359M16—161no—
360M16—161no—
361C2×C8—21no—
362C2×C8—21no—
363C2×C8—21no—
364C2×C8—21no—
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—41no—
378C2×C8—41no—
379C2×C8—41no—
380C2×C8—41no—
381C2×C8—41no—
382C2×C8—41no—
383C2×C8—81no—
384C2×C8—81no—
385C2×C8—81no—
386C2×C8—81no—
387C2×C8—161no—
388C2×C8—161no—
389C2×C8C2×C4⋊C411yescharacteristic
390C2×C8C2×C4⋊C411yescharacteristic
391C2×C8C2×C4⋊C411yescharacteristic
392C2×C8C2×C4⋊C411yescharacteristic
393C2×C8C4×D411yescharacteristic
394C2×C8C4×D411yescharacteristic
395C2×C8C4×D411yescharacteristic
396C2×C8C4×D411yescharacteristic
397C4⋊C4—21no—
398C4⋊C4—21no—
399C4⋊C4—21no—
400C4⋊C4—21no—
401C4⋊C4C2×C4⋊C411yescharacteristic
402C4⋊C4C2×C4⋊C411yescharacteristic
403C4⋊C4C2×C4⋊C411yescharacteristic
404C4⋊C4C2×C4⋊C411yescharacteristic
405C22⋊C4—21no—
406C22⋊C4—21no—
407C22⋊C4—21no—
408C22⋊C4—21no—
409C22⋊C4—21no—
410C22⋊C4—21no—
411C42—21no—
412C42—21no—
413C42—21no—
414C42—21no—
415C42—21no—
416C42—21no—
417C42—21no—
418C42—21no—
419C42—21no—
420C42—41no—
421C42—41no—
422C42—81no—
423C42#13112yes—
424C42C22×D411yescharacteristic
425C42C22×D411yescharacteristic
426C42C22×Q811yescharacteristic
427C42C2×#4111yescharacteristic
428C23—21no—
429C23—21no—
430C23#52011yescharacteristic
431Q8—21no—
432Q8—21no—
433Q8—21no—
434Q8—21no—
435D4—21no—
436D4—21no—
437D4—21no—
438D4—21no—
439D4—41no—
440D4—41no—
441D4—41no—
442D4—41no—
443C2×C4—21no—
444C2×C4—21no—
445C2×C4—21no—
446C2×C4—21no—
447C2×C4—21no—
448C2×C4—21no—
449C2×C4—21no—
450C2×C4—21no—
451C2×C4—22no—
452C2×C4—41no—
453C2×C4—41no—
454C2×C4—41no—
455C2×C4—41no—
456C2×C4—41no—
457C2×C4—41no—
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—81no—
467C2×C4—81no—
468C2×C4C22×C4⋊C411yescharacteristic
469C2×C4C2×C4×D411yescharacteristic
470C2×C4D4211yescharacteristic
471C2×C4#54711yescharacteristic
472C2×C4D4×Q811yescharacteristic
473C2×C4#55411yescharacteristic
474C2×C4C2×#13111yescharacteristic
475C2×C4#56811yescharacteristic
476C2×C4#56811yescharacteristic
477C2×C4#56811yescharacteristic
478C8—21no—
479C8—21no—
480C8—21no—
481C8—21no—
482C8—21no—
483C8—21no—
484C8—21no—
485C8—21no—
486C8—41no—
487C8—41no—
488C8—41no—
489C8—41no—
490C8—41no—
491C8—41no—
492C8—41no—
493C8—41no—
494C8—81no—
495C8—81no—
496C8—81no—
497C8—81no—
498C8—81no—
499C8—81no—
500C22—21no—
501C22—21no—
502C22—41no—
503C22—41no—
504C22—41no—
505C22—41no—
506C22—41no—
507C22—41no—
508C22D4×C4⋊C411yescharacteristic
509C22#299211yescharacteristic
510C22#299511yescharacteristic
511C4—21no—
512C4—21no—
513C4—21no—
514C4—42no—
515C4—41no—
516C4—41no—
517C4—41no—
518C4—41no—
519C4—41no—
520C4—41no—
521C4—81no—
522C4#290511yescharacteristic
523C4#290611yescharacteristic
524C4#293211yescharacteristic
525C4#293511yescharacteristic
526C4#328811yescharacteristic
527C4#328911yescharacteristic
528C2—21no—
529C2—21no—
530C2—81no—
531C2—81no—
532C2#2265211yescharacteristic
533C2#2269311yescharacteristic
534C2#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. #17811C2semidirect
6. #9610C2semidirect
7. #9609C2semidirect
8. #8841C2semidirect
9. #8841C2semidirect
10. #8840C2semidirect
11. #8840C2semidirect
12. #8808C2semidirect
13. #8607C2semidirect
14. #8606C2semidirect
15. #8269C2semidirect

Maximal subgroups

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

subgroupindex
#212932
#178822
#178812
#178122
#178112
#96102
#96092
#88412
#88402
#88082
#86072
#86062
#82692