Finite Group Databasebeta

← #225508

Subgroups — #225508

Order 512.

557 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#17898C211yesmaximal, characteristic
3#17858C211yesmaximal, characteristic
4#17812C211yesmaximal, characteristic, complemented
5#17811C211yesmaximal, characteristic, complemented
6#9608C211yesmaximal, characteristic, complemented
7#9607C211yesmaximal, characteristic, complemented
8#8842C211yesmaximal, characteristic, complemented
9#8841C211yesmaximal, characteristic, complemented
10#8840C211yesmaximal, characteristic, complemented
11#8839C211yesmaximal, characteristic, complemented
12#8808C211yesmaximal, characteristic, complemented
13#8607C211yesmaximal, characteristic, complemented
14#8606C211yesmaximal, characteristic, complemented
15#8268C211yesmaximal, 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#1592—21no—
38#1592—21no—
39#1591—21no—
40#1589—21no—
41#1580—21no—
42#1579—21no—
43C8×SD16—21no—
44C8×SD16—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#1450C2211yescharacteristic
68#1450C2211yescharacteristic
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—
188C82—21no—
189C82—21no—
190C2×#41C22×C411yescharacteristic
191C2×#41C22×C411yescharacteristic
192#135—22no—
193#135—22no—
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—22no—
238#108—22no—
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—22no—
300#97—22no—
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—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—22no—
348SD16—22no—
349SD16—22no—
350D8—22no—
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—41no—
376C2×C8—41no—
377C2×C8—41no—
378C2×C8—41no—
379C2×C8—41no—
380C2×C8—41no—
381C2×C8—81no—
382C2×C8—81no—
383C2×C8—81no—
384C2×C8—81no—
385C2×C8—161no—
386C2×C8—161no—
387C2×C8C2×C4⋊C411yescharacteristic
388C2×C8C2×C4⋊C411yescharacteristic
389C2×C8C2×C4⋊C411yescharacteristic
390C2×C8C2×C4⋊C411yescharacteristic
391C2×C8C4×D411yescharacteristic
392C2×C8C4×D411yescharacteristic
393C2×C8C4×D411yescharacteristic
394C2×C8C4×D411yescharacteristic
395C2×C8#13112yes—
396C2×C8#13112yes—
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×C4C2×#13111yescharacteristic
476C2×C4C2×#13111yescharacteristic
477C2×C4#56811yescharacteristic
478C8—21no—
479C8—21no—
480C8—21no—
481C8—21no—
482C8—21no—
483C8—21no—
484C8—41no—
485C8—41no—
486C8—41no—
487C8—41no—
488C8—41no—
489C8—41no—
490C8—41no—
491C8—41no—
492C8—81no—
493C8—81no—
494C8—81no—
495C8—81no—
496C8—81no—
497C8—81no—
498C8#42412yes—
499C8#42412yes—
500C22—21no—
501C22—21no—
502C22—41no—
503C22—41no—
504C22—41no—
505C22—41no—
506C22—41no—
507C22—41no—
508C22D4×C4⋊C411yescharacteristic
509C22#299311yescharacteristic
510C22#299411yescharacteristic
511C4—21no—
512C4—21no—
513C4—21no—
514C4—42no—
515C4—41no—
516C4—41no—
517C4—41no—
518C4—41no—
519C4—41no—
520C4—41no—
521C4—81no—
522C4C2×#42411yescharacteristic
523C4#290611yescharacteristic
524C4#293111yescharacteristic
525C4#293511yescharacteristic
526C4#328811yescharacteristic
527C4#328911yescharacteristic
528C2—21no—
529C2—21no—
530C2—81no—
531C2—81no—
532C2#2265111yescharacteristic
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. #9608C2semidirect
7. #9607C2semidirect
8. #8842C2semidirect
9. #8841C2semidirect
10. #8840C2semidirect
11. #8839C2semidirect
12. #8808C2semidirect
13. #8607C2semidirect
14. #8606C2semidirect
15. #8268C2semidirect

Maximal subgroups

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

subgroupindex
#212932
#178982
#178582
#178122
#178112
#96082
#96072
#88422
#88412
#88402
#88392
#88082
#86072
#86062
#82682