Finite Group Databasebeta

← #211287

Subgroups — #211287

Order 512.

721 conjugacy classes of subgroups.

The subgroup lattice has 530 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22221C211yesmaximal, characteristic, complemented
2#21613C211yesmaximal, characteristic
3C4×#476C211yesmaximal, characteristic
4#20555C211yesmaximal, characteristic
5C2×#1683C211yesmaximal, characteristic
6#9737C211yesmaximal, characteristic, complemented
7#9615C211yesmaximal, characteristic, complemented
8#9567C211yesmaximal, characteristic, complemented
9#9562C211yesmaximal, characteristic, complemented
10#9219C211yesmaximal, characteristic
11#9215C211yesmaximal, characteristic
12#8644C211yesmaximal, characteristic
13#8278C211yesmaximal, characteristic, complemented
14#8235C211yesmaximal, characteristic, complemented
15C4×#332C211yesmaximal, characteristic, complemented
16C2×#487C2211yescharacteristic
17C2×#476C2211yescharacteristic
18C2×#474C2211yescharacteristic
19C2×C8×D4C2211yescharacteristic
20C2×C8×D4C2211yescharacteristic
21C4×C4⋊Q8C2211yescharacteristic
22C4×C4⋊Q8C2211yescharacteristic
23C42×D4C2211yescharacteristic
24#1990C2211yescharacteristic
25#1988C2211yescharacteristic
26#1987C2211yescharacteristic
27#1956C2211yescharacteristic
28#1948C2211yescharacteristic
29#1948C2211yescharacteristic
30#1928—21no—
31#1928—21no—
32#1921—21no—
33#1920—21no—
34#1864—21no—
35#1864—21no—
36#1863—21no—
37#1863—21no—
38#1845C2211yescharacteristic
39#1802C2211yescharacteristic
40#1802C2211yescharacteristic
41#1776—21no—
42C4×#106—21no—
43C8×C4⋊C4C2211yescharacteristic
44C8×C4⋊C4C2211yescharacteristic
45#1767C2211yescharacteristic
46C4×C4⋊C8C2211yescharacteristic
47C4×C4⋊C8C2211yescharacteristic
48#1762—21no—
49C4×#102—21no—
50C8×C22⋊C4—21no—
51C8×C22⋊C4—21no—
52C4×C22⋊C8—21no—
53#1744C2211yescharacteristic
54C2×#336—21no—
55#1683C418yescomplemented
56C2×#335C2211yescharacteristic
57C2×#335C2211yescharacteristic
58C2×#333C2211yescharacteristic
59C2×#333C2211yescharacteristic
60C2×#332C2211yescharacteristic
61C2×#332C2211yescharacteristic
62#1296C2211yescharacteristic
63#1296C2211yescharacteristic
64#1296C2211yescharacteristic
65#1296C2211yescharacteristic
66#1277C2211yescharacteristic
67#1277C2211yescharacteristic
68C23×C8—21no—
69C23×C8—21no—
70C2×C4⋊Q8C2311yescharacteristic
71C2×C4⋊Q8C2311yescharacteristic
72C2×C4×Q8—21no—
73C2×C4×Q8—21no—
74C2×C4×D4D411yescharacteristic
75C2×C4×D4D411yescharacteristic
76C2×C4×D4C2311yescharacteristic
77C22×C42—21no—
78#487—22no—
79#487C2×C414yescomplemented
80#476—22no—
81#476C2×C414yes—
82#474—22no—
83#474C2×C414yes—
84C8×D4—22no—
85C8×D4—22no—
86C8×D4—22no—
87C8×D4C2×C414yes—
88C2×#107—21no—
89C2×#107—21no—
90C2×#107—21no—
91C2×#106—21no—
92C2×#106—21no—
93C2×#106—21no—
94C2×C4⋊C8C2311yescharacteristic
95C2×C4⋊C8C2311yescharacteristic
96C2×C4⋊C8C2311yescharacteristic
97C2×C4⋊C8C2311yescharacteristic
98C2×C4⋊C8C2311yescharacteristic
99C2×C4⋊C8C2311yescharacteristic
100C2×#103—21no—
101C2×#103—21no—
102C2×#103—21no—
103C2×#103—21no—
104C2×#102—21no—
105C2×#102—21no—
106C2×#102—21no—
107C2×#102—21no—
108C2×C22⋊C8—21no—
109C2×C22⋊C8—21no—
110C2×#97—21no—
111C2×C4×C8—21no—
112C2×C4×C8—21no—
113C2×C4×C8—21no—
114C2×C4×C8—21no—
115C2×C4×C8C2311yescharacteristic
116C2×C4×C8C2311yescharacteristic
117#391—21no—
118#391—21no—
119#389—21no—
120#389—21no—
121#389—21no—
122#389—21no—
123#384C2311yescharacteristic
124#384C2311yescharacteristic
125C4×C4⋊C4—21no—
126C4×C4⋊C4—21no—
127C4×C4⋊C4—21no—
128C4×C4⋊C4—21no—
129C4×C4⋊C4D411yescharacteristic
130C4×C4⋊C4D411yescharacteristic
131C4×C4⋊C4C2311yescharacteristic
132C4×C22⋊C4—21no—
133C4×C22⋊C4—21no—
134C4×C22⋊C4—21no—
135C43C2311yescharacteristic
136#340—21no—
137#340—21no—
138#340—21no—
139#340—21no—
140#340—21no—
141#340—21no—
142#340—21no—
143#340—21no—
144#340—21no—
145#340—21no—
146#336—21no—
147#336—21no—
148#336—21no—
149#336—21no—
150#336—21no—
151#336D411yescharacteristic
152#336D411yescharacteristic
153#336D411yescharacteristic
154#336D411yescharacteristic
155#335—22no—
156#335C2×C414yescomplemented
157#333—22no—
158#333C2×C414yescomplemented
159#332—22no—
160#332C2×C414yescomplemented
161C22×Q8—21no—
162C22×Q8—21no—
163C22×D4C2×D411yescharacteristic
164C23×C4—21no—
165C23×C4—21no—
166C23×C4—21no—
167C22×C8—21no—
168C22×C8—21no—
169C22×C8—21no—
170C22×C8—21no—
171C22×C8—24no—
172C22×C8—24no—
173C22×C8—21no—
174C22×C8—21no—
175C22×C8—81no—
176C22×C8—81no—
177C22×C8—81no—
178C22×C8—81no—
179C22×C8C2×D411yescharacteristic
180C22×C8C2×D411yescharacteristic
181C22×C8C2×D411yescharacteristic
182C22×C8C2×D411yescharacteristic
183C4⋊Q8—21no—
184C4⋊Q8—21no—
185C4⋊Q8—21no—
186C4⋊Q8—21no—
187C4⋊Q8—21no—
188C4⋊Q8—21no—
189C4⋊Q8C22×C412yes—
190C4⋊Q8C22×C412yes—
191C4×Q8—81no—
192C4×Q8—81no—
193C4×Q8—82no—
194C4×D4—22no—
195C4×D4—22no—
196C4×D4—22no—
197C4×D4—21no—
198C4×D4—21no—
199C4×D4Q1614yes—
200C4×D4C22×C412yes—
201C4×D4#4112yes—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—41no—
213C2×C4⋊C4—41no—
214C2×C4⋊C4C2×D411yescharacteristic
215C2×C4⋊C4C2×D411yescharacteristic
216C2×C4⋊C4C2×D411yescharacteristic
217C2×C22⋊C4—21no—
218C2×C22⋊C4—21no—
219C2×C22⋊C4—21no—
220C2×C42—21no—
221C2×C42—21no—
222C2×C42—21no—
223C2×C42—41no—
224C2×C42—41no—
225C2×C42—81no—
226C2×C42—81no—
227C2×C42C2×D411yescharacteristic
228C2×C42C2×D411yescharacteristic
229C2×C42C2×D411yescharacteristic
230C2×C42C2×D411yescharacteristic
231C2×C42#4111yescharacteristic
232C2×C42#4111yescharacteristic
233C2×C42C2411yescharacteristic
234#107—24no—
235#107—81no—
236#107—81no—
237#106—24no—
238#106—42no—
239#106—42no—
240C4⋊C8—21no—
241C4⋊C8—21no—
242C4⋊C8—21no—
243C4⋊C8—21no—
244C4⋊C8—21no—
245C4⋊C8—21no—
246C4⋊C8—21no—
247C4⋊C8—21no—
248C4⋊C8C22×C412yes—
249C4⋊C8C22×C412yes—
250C4⋊C8C22×C412yes—
251C4⋊C8#4112yes—
252#103—24no—
253#103—24no—
254#103—81no—
255#103—81no—
256#102—24no—
257#102—24no—
258#102—42no—
259#102—42no—
260C22⋊C8—24no—
261C22⋊C8—24no—
262#97—42no—
263C4×C8—21no—
264C4×C8—21no—
265C4×C8—42no—
266C4×C8—81no—
267C4×C8—81no—
268C4×C8—81no—
269C4×C8C22×C412yes—
270C4×C8#4112yes—
271#95—21no—
272#95—21no—
273#95—21no—
274#95—41no—
275#95—41no—
276#95—41no—
277#95—41no—
278C24—21no—
279C2×Q8—22no—
280C2×Q8—22no—
281C2×Q8—41no—
282C2×Q8—41no—
283C2×Q8—41no—
284C2×Q8—41no—
285C2×Q8—41no—
286C2×Q8—41no—
287C2×Q8—81no—
288C2×Q8—81no—
289C2×D4—21no—
290C2×D4—21no—
291C2×D4—21no—
292C2×D4C4×D412yes—
293C2×D4C2×Q1612yes—
294C2×D4#13512yes—
295C22×C4—21no—
296C22×C4—21no—
297C22×C4—21no—
298C22×C4—24no—
299C22×C4—22no—
300C22×C4—22no—
301C22×C4—24no—
302C22×C4—41no—
303C22×C4—41no—
304C22×C4—41no—
305C22×C4—41no—
306C22×C4—41no—
307C22×C4—41no—
308C22×C4—41no—
309C22×C4—41no—
310C22×C4—81no—
311C22×C4—81no—
312C22×C4—81no—
313C22×C4—81no—
314C22×C4#12111yescharacteristic
315C22×C4#12111yescharacteristic
316C22×C4#12111yescharacteristic
317C22×C4#12111yescharacteristic
318C22×C4C22×D411yescharacteristic
319C22×C4C22×D411yescharacteristic
320C22×C4C2×#4111yescharacteristic
321C2×C8—21no—
322C2×C8—21no—
323C2×C8—21no—
324C2×C8—21no—
325C2×C8—22no—
326C2×C8—22no—
327C2×C8—22no—
328C2×C8—21no—
329C2×C8—21no—
330C2×C8—22no—
331C2×C8—21no—
332C2×C8—21no—
333C2×C8—21no—
334C2×C8—21no—
335C2×C8—41no—
336C2×C8—41no—
337C2×C8—41no—
338C2×C8—41no—
339C2×C8—41no—
340C2×C8—41no—
341C2×C8—41no—
342C2×C8—41no—
343C2×C8—41no—
344C2×C8—41no—
345C2×C8—41no—
346C2×C8—41no—
347C2×C8—41no—
348C2×C8—41no—
349C2×C8—84no—
350C2×C8—84no—
351C2×C8C4×D412yes—
352C2×C8C4×D412yes—
353C4⋊C4—21no—
354C4⋊C4—21no—
355C4⋊C4—21no—
356C4⋊C4—22no—
357C4⋊C4—22no—
358C4⋊C4—22no—
359C4⋊C4—22no—
360C4⋊C4—41no—
361C4⋊C4—41no—
362C4⋊C4—41no—
363C4⋊C4—41no—
364C4⋊C4—41no—
365C4⋊C4—41no—
366C4⋊C4—41no—
367C4⋊C4—41no—
368C4⋊C4—41no—
369C4⋊C4—41no—
370C4⋊C4—41no—
371C4⋊C4—41no—
372C4⋊C4—81no—
373C4⋊C4—81no—
374C4⋊C4—82no—
375C4⋊C4—82no—
376C4⋊C4C4×D412yes—
377C4⋊C4C2×Q1612yes—
378C4⋊C4#13512yes—
379C22⋊C4—24no—
380C22⋊C4—22no—
381C22⋊C4—22no—
382C22⋊C4—24no—
383C42—21no—
384C42—21no—
385C42—21no—
386C42—21no—
387C42—21no—
388C42—21no—
389C42—41no—
390C42—41no—
391C42—41no—
392C42—41no—
393C42—81no—
394C42—81no—
395C42—82no—
396C42—82no—
397C42C2×Q1612yes—
398C42#13512yes—
399C42C23×C411yescharacteristic
400C42C2×#4111yescharacteristic
401C42C2×#4111yescharacteristic
402C42C2×#4111yescharacteristic
403C23—22no—
404C23—22no—
405C23—22no—
406C23—41no—
407C23—41no—
408C23—41no—
409C23—41no—
410C23C2×#12111yescharacteristic
411Q8—42no—
412Q8—42no—
413Q8—82no—
414Q8—82no—
415Q8—81no—
416Q8—81no—
417D4—22no—
418D4—22no—
419D4—22no—
420D4C4×Q1614yes—
421C2×C4—21no—
422C2×C4—21no—
423C2×C4—21no—
424C2×C4—21no—
425C2×C4—21no—
426C2×C4—21no—
427C2×C4—21no—
428C2×C4—21no—
429C2×C4—22no—
430C2×C4—21no—
431C2×C4—21no—
432C2×C4—22no—
433C2×C4—22no—
434C2×C4—21no—
435C2×C4—21no—
436C2×C4—41no—
437C2×C4—41no—
438C2×C4—41no—
439C2×C4—41no—
440C2×C4—41no—
441C2×C4—41no—
442C2×C4—42no—
443C2×C4—42no—
444C2×C4—42no—
445C2×C4—42no—
446C2×C4—42no—
447C2×C4—42no—
448C2×C4—84no—
449C2×C4—84no—
450C2×C4—81no—
451C2×C4—81no—
452C2×C4—81no—
453C2×C4—81no—
454C2×C4—81no—
455C2×C4—81no—
456C2×C4—81no—
457C2×C4—81no—
458C2×C4—81no—
459C2×C4—81no—
460C2×C4#46712yes—
461C2×C4#46712yes—
462C2×C4C2×C4×D411yescharacteristic
463C2×C4C2×C4×D411yescharacteristic
464C2×C4C4×#4111yescharacteristic
465C2×C4#52511yescharacteristic
466C2×C4#52911yescharacteristic
467C2×C4#53211yescharacteristic
468C2×C4C22×Q1611yescharacteristic
469C2×C4C2×#13511yescharacteristic
470C2×C4C2×#13511yescharacteristic
471C2×C4C2×#13511yescharacteristic
472C2×C4#57611yescharacteristic
473C2×C4#57711yescharacteristic
474C2×C4#57711yescharacteristic
475C2×C4#57711yescharacteristic
476C8—22no—
477C8—22no—
478C8—41no—
479C8—42no—
480C8—41no—
481C8—42no—
482C8—42no—
483C8—42no—
484C8—81no—
485C8—81no—
486C22—24no—
487C22—42no—
488C22—42no—
489C22—42no—
490C22—42no—
491C22—42no—
492C22—42no—
493C22C4×#12111yescharacteristic
494C22#304211yescharacteristic
495C22#304511yescharacteristic
496C22C2×#46711yescharacteristic
497C22#305011yescharacteristic
498C22#306811yescharacteristic
499C22#306911yescharacteristic
500C4—21no—
501C4—21no—
502C4—22no—
503C4—22no—
504C4—42no—
505C4—42no—
506C4—41no—
507C4—42no—
508C4—41no—
509C4—82no—
510C4—82no—
511C4—161no—
512C4—161no—
513C4C2×C4×Q1611yescharacteristic
514C4C4×#13511yescharacteristic
515C4#294611yescharacteristic
516C4#295211yescharacteristic
517C4#311811yescharacteristic
518C4#312511yescharacteristic
519C4#316811yescharacteristic
520C4#317011yescharacteristic
521C2—44no—
522C2#1939311yescharacteristic
523C2#1939511yescharacteristic
524C2#1941111yescharacteristic
525C2#1941211yescharacteristic
526C2#2151011yescharacteristic
527C2C4×#46711yescharacteristic
528C2#2155711yescharacteristic
subgroupG/H
center, socleC23C2×#121
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radicalC22×D4C2×D4
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C43 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. #22221C2semidirect
6. #9737C2semidirect
7. #9615C2semidirect
8. #9567C2semidirect
9. #9562C2semidirect
13. #8278C2semidirect
14. #8235C2semidirect
15. C4×#332C2semidirect
55. #1683C4semidirect
79. #487C2×C4semidirect
156. #335C2×C4semidirect
158. #333C2×C4semidirect
160. #332C2×C4semidirect

Maximal subgroups

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

subgroupindex
#222212
#216132
C4×#4762
#205552
C2×#16832
#97372
#96152
#95672
#95622
#92192
#92152
#86442
#82782
#82352
C4×#3322