Finite Group Databasebeta

← #274141

Subgroups — #274141

Order 512.

666 conjugacy classes of subgroups.

The subgroup lattice has 551 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33154C211yesmaximal, characteristic
2#25600C211yesmaximal, characteristic
3#22632C211yesmaximal, characteristic
4#21311C211yesmaximal, characteristic
5#10593C211yesmaximal, characteristic, complemented
6#10592C211yesmaximal, characteristic, complemented
7#10541C211yesmaximal, characteristic, complemented
8#10538C211yesmaximal, characteristic, complemented
9#9737C211yesmaximal, characteristic, complemented
10#9733C211yesmaximal, characteristic, complemented
11#9567C211yesmaximal, characteristic, complemented
12#9562C211yesmaximal, characteristic, complemented
13#9129C211yesmaximal, characteristic
14#8829C211yesmaximal, characteristic
15#8343C211yesmaximal, characteristic
16C2×#533C2211yescharacteristic
17C2×#494C2211yescharacteristic
18C2×#487C2211yescharacteristic
19C22×C4⋊C8C2211yescharacteristic
20#2857C2211yescharacteristic
21C4×#117C2211yescharacteristic
22#2001—21no—
23#2001—21no—
24#1990C2211yescharacteristic
25#1989—21no—
26#1989—21no—
27#1988C2211yescharacteristic
28#1988C2211yescharacteristic
29#1988C2211yescharacteristic
30#1988C2211yescharacteristic
31#1987C2211yescharacteristic
32#1987C2211yescharacteristic
33#1956C2211yescharacteristic
34#1948C2211yescharacteristic
35#1947C2211yescharacteristic
36#1939C2211yescharacteristic
37#1939C2211yescharacteristic
38#1923—21no—
39#1923—21no—
40#1921—21no—
41#1920—21no—
42#1878—21no—
43#1878—21no—
44#1854—21no—
45#1831—21no—
46#1825—21no—
47#1797C2211yescharacteristic
48#1797C2211yescharacteristic
49C8×C4⋊C4C2211yescharacteristic
50C8×C4⋊C4C2211yescharacteristic
51C4×C22⋊C8C2211yescharacteristic
52#1744—21no—
53#1735—21no—
54#1735—21no—
55C2×#335C2211yescharacteristic
56C2×#335C2211yescharacteristic
57C2×#333C2211yescharacteristic
58C2×#333C2211yescharacteristic
59C2×#332C2211yescharacteristic
60C2×#332C2211yescharacteristic
61C2×#332C2211yescharacteristic
62C2×#332C2211yescharacteristic
63#1296C2211yescharacteristic
64#1296C2211yescharacteristic
65#1277C2211yescharacteristic
66#1277C2211yescharacteristic
67C22×Q16—21no—
68C22×Q16—21no—
69C23×C8—21no—
70#533—22no—
71#533—22no—
72#533—41no—
73#533—41no—
74C2×C4⋊Q8C2311yescharacteristic
75C2×C4⋊Q8C2311yescharacteristic
76C2×C4⋊Q8C2311yescharacteristic
77C2×#125—21no—
78C2×C22⋊Q8—21no—
79C2×C22⋊Q8—21no—
80C2×C22⋊Q8—21no—
81C2×C4×Q8—21no—
82C2×C4×Q8—21no—
83C2×#117—21no—
84C2×#117D411yescharacteristic
85C2×#117D411yescharacteristic
86C22×C42C2311yescharacteristic
87#494—22no—
88#494—41no—
89#487—22no—
90#487—41no—
91C2×#107—21no—
92C2×#106—21no—
93C2×C4⋊C8—21no—
94C2×C4⋊C8—21no—
95C2×C4⋊C8—21no—
96C2×C4⋊C8—41no—
97C2×C4⋊C8—41no—
98C2×C4⋊C8C2311yescharacteristic
99C2×C4⋊C8C2311yescharacteristic
100C2×C4⋊C8C2311yescharacteristic
101C2×C4⋊C8C2311yescharacteristic
102C2×C4⋊C8C2311yescharacteristic
103C2×C4⋊C8C2311yescharacteristic
104C2×#103—21no—
105C2×#103—21no—
106C2×#103—21no—
107C2×#103—21no—
108C2×C22⋊C8—21no—
109C2×C22⋊C8—21no—
110C2×C22⋊C8D411yescharacteristic
111C2×C22⋊C8D411yescharacteristic
112C2×C4×C8—21no—
113C2×C4×C8—21no—
114C2×C4×C8C2311yescharacteristic
115C2×C4×C8C2311yescharacteristic
116#391—21no—
117#389—21no—
118#389—21no—
119#384C2311yescharacteristic
120#380—21no—
121#380—21no—
122C4×C4⋊C4D411yescharacteristic
123C4×C4⋊C4D411yescharacteristic
124C4×C4⋊C4C2311yescharacteristic
125C4×C4⋊C4C2311yescharacteristic
126C4×C22⋊C4—21no—
127C4×C22⋊C4—21no—
128#376—21no—
129C43—21no—
130#340—21no—
131#340—21no—
132#340—21no—
133#340—21no—
134#340—21no—
135#340—21no—
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#336—21no—
152#336D411yescharacteristic
153#336D411yescharacteristic
154#335—41no—
155#335—41no—
156#333—22no—
157#333—41no—
158#332—22no—
159#332—41no—
160#332—41no—
161#332—41no—
162C22×Q8—21no—
163C22×Q8—21no—
164C22×Q8—21no—
165C23×C4C2×D411yescharacteristic
166C23×C4C2×D411yescharacteristic
167C23×C4#4111yescharacteristic
168C2×Q16—42no—
169C2×Q16—42no—
170C2×Q16—42no—
171C2×Q16—42no—
172C2×Q16—81no—
173C2×Q16—81no—
174C22×C8—21no—
175C22×C8—21no—
176C22×C8—21no—
177C22×C8—21no—
178C22×C8—21no—
179C22×C8—21no—
180C22×C8—42no—
181C22×C8—42no—
182C22×C8—42no—
183C22×C8C2×D411yescharacteristic
184C22×C8C2×D411yescharacteristic
185C22×C8C2×D411yescharacteristic
186C22×C8C2×D411yescharacteristic
187C4⋊Q8—21no—
188C4⋊Q8—21no—
189C4⋊Q8—21no—
190C4⋊Q8—21no—
191C4⋊Q8—21no—
192C4⋊Q8—21no—
193C4⋊Q8—21no—
194C4⋊Q8—21no—
195C4⋊Q8—21no—
196C4⋊Q8—21no—
197C4⋊Q8#4112yes—
198C4⋊Q8#4112yes—
199#125—24no—
200#125—42no—
201C22⋊Q8—42no—
202C22⋊Q8—42no—
203C22⋊Q8—42no—
204C22⋊Q8—42no—
205C22⋊Q8—81no—
206C22⋊Q8—81no—
207C4×Q8—42no—
208C4×Q8—42no—
209C4×Q8—42no—
210C4×Q8—42no—
211#117—22no—
212#117—24no—
213#117—22no—
214#117—41no—
215#117—42no—
216#117Q1614yescomplemented
217C2×C4⋊C4—21no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—21no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4—21no—
224C2×C4⋊C4—21no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4—41no—
227C2×C4⋊C4—41no—
228C2×C4⋊C4—41no—
229C2×C4⋊C4C2×D411yescharacteristic
230C2×C4⋊C4C2×D411yescharacteristic
231C2×C4⋊C4C2×D411yescharacteristic
232C2×C4⋊C4C2×D411yescharacteristic
233C2×C22⋊C4—21no—
234C2×C22⋊C4—21no—
235C2×C22⋊C4—41no—
236C2×C22⋊C4—41no—
237C2×C42—21no—
238C2×C42—21no—
239C2×C42—21no—
240C2×C42—21no—
241C2×C42—21no—
242C2×C42—21no—
243C2×C42—21no—
244C2×C42—21no—
245C2×C42—21no—
246C2×C42—21no—
247C2×C42—41no—
248C2×C42C2×D411yescharacteristic
249C2×C42C2×D411yescharacteristic
250C2×C42#4111yescharacteristic
251C2×C42C2411yescharacteristic
252#107—42no—
253#106—42no—
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—21no—
266C4⋊C8—42no—
267C4⋊C8—81no—
268C4⋊C8—81no—
269#103—42no—
270#103—42no—
271#103—81no—
272#103—81no—
273C22⋊C8—22no—
274C22⋊C8—41no—
275C22⋊C8—42no—
276C22⋊C8—81no—
277C4×C8—21no—
278C4×C8—21no—
279C4×C8—21no—
280C4×C8—21no—
281C4×C8—42no—
282C4×C8—42no—
283#95—21no—
284#95—21no—
285#95—21no—
286#95—21no—
287#95—41no—
288#95—41no—
289C24#12111yescharacteristic
290C2×Q8—22no—
291C2×Q8—22no—
292C2×Q8—41no—
293C2×Q8—41no—
294C2×Q8—41no—
295C2×Q8—41no—
296C2×Q8—41no—
297C2×Q8—41no—
298C2×Q8—41no—
299C2×Q8—41no—
300C2×Q8—41no—
301C2×Q8—41no—
302C2×Q8—41no—
303C2×Q8—41no—
304C2×Q8—81no—
305C2×Q8—81no—
306C22×C4—21no—
307C22×C4—21no—
308C22×C4—21no—
309C22×C4—21no—
310C22×C4—21no—
311C22×C4—21no—
312C22×C4—21no—
313C22×C4—21no—
314C22×C4—21no—
315C22×C4—21no—
316C22×C4—21no—
317C22×C4—21no—
318C22×C4—21no—
319C22×C4—21no—
320C22×C4—21no—
321C22×C4—21no—
322C22×C4—41no—
323C22×C4—41no—
324C22×C4—41no—
325C22×C4—41no—
326C22×C4—41no—
327C22×C4—41no—
328C22×C4#12111yescharacteristic
329C22×C4#12111yescharacteristic
330C22×C4#12111yescharacteristic
331C22×C4C2×Q1612yes—
332C22×C4#13512yes—
333C22×C4C22×D411yescharacteristic
334C22×C4C22×D411yescharacteristic
335C22×C4C2×#4111yescharacteristic
336Q16—84no—
337Q16—84no—
338C2×C8—21no—
339C2×C8—21no—
340C2×C8—21no—
341C2×C8—21no—
342C2×C8—21no—
343C2×C8—21no—
344C2×C8—21no—
345C2×C8—21no—
346C2×C8—21no—
347C2×C8—21no—
348C2×C8—21no—
349C2×C8—41no—
350C2×C8—41no—
351C2×C8—41no—
352C2×C8—41no—
353C2×C8—41no—
354C2×C8—41no—
355C2×C8—41no—
356C2×C8—41no—
357C2×C8—41no—
358C2×C8—41no—
359C2×C8—41no—
360C2×C8—41no—
361C2×C8—41no—
362C2×C8—41no—
363C2×C8—41no—
364C2×C8—41no—
365C2×C8—41no—
366C2×C8—41no—
367C2×C8—84no—
368C2×C8#13512yes—
369C4⋊C4—21no—
370C4⋊C4—21no—
371C4⋊C4—21no—
372C4⋊C4—22no—
373C4⋊C4—22no—
374C4⋊C4—22no—
375C4⋊C4—22no—
376C4⋊C4—21no—
377C4⋊C4—21no—
378C4⋊C4—44no—
379C4⋊C4—44no—
380C4⋊C4—41no—
381C4⋊C4—41no—
382C4⋊C4—41no—
383C4⋊C4—41no—
384C4⋊C4—41no—
385C4⋊C4—41no—
386C4⋊C4—41no—
387C4⋊C4—41no—
388C4⋊C4—41no—
389C4⋊C4—41no—
390C4⋊C4—41no—
391C4⋊C4—41no—
392C4⋊C4—81no—
393C4⋊C4—82no—
394C4⋊C4C2×Q1612yes—
395C4⋊C4C2×Q1612yes—
396C4⋊C4#13712yes—
397C22⋊C4—24no—
398C22⋊C4—42no—
399C22⋊C4—44no—
400C22⋊C4—82no—
401C42—21no—
402C42—21no—
403C42—21no—
404C42—21no—
405C42—21no—
406C42—21no—
407C42—24no—
408C42—42no—
409C42—42no—
410C42—44no—
411C42—41no—
412C42—41no—
413C42—41no—
414C42—41no—
415C42—81no—
416C42C2×#4111yescharacteristic
417C42C2×#4111yescharacteristic
418C42H(5,𝔽2)11yescharacteristic
419C42H(5,𝔽2)11yescharacteristic
420C23—21no—
421C23—21no—
422C23—21no—
423C23—21no—
424C23—21no—
425C23—21no—
426C23#46712yes—
427C23C2×#12111yescharacteristic
428Q8—42no—
429Q8—42no—
430Q8—82no—
431Q8—82no—
432Q8—81no—
433Q8—81no—
434Q8—81no—
435Q8—81no—
436Q8—81no—
437Q8—81no—
438C2×C4—21no—
439C2×C4—21no—
440C2×C4—21no—
441C2×C4—21no—
442C2×C4—22no—
443C2×C4—22no—
444C2×C4—22no—
445C2×C4—22no—
446C2×C4—21no—
447C2×C4—21no—
448C2×C4—21no—
449C2×C4—21no—
450C2×C4—22no—
451C2×C4—22no—
452C2×C4—22no—
453C2×C4—21no—
454C2×C4—21no—
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—42no—
466C2×C4—42no—
467C2×C4—41no—
468C2×C4—41no—
469C2×C4—41no—
470C2×C4—81no—
471C2×C4—81no—
472C2×C4—81no—
473C2×C4—81no—
474C2×C4—81no—
475C2×C4—81no—
476C2×C4—81no—
477C2×C4—81no—
478C2×C4—81no—
479C2×C4—81no—
480C2×C4—81no—
481C2×C4—81no—
482C2×C4#46712yes—
483C2×C4#52511yescharacteristic
484C2×C4#53511yescharacteristic
485C2×C4#54611yescharacteristic
486C2×C4#54611yescharacteristic
487C2×C4#55111yescharacteristic
488C2×C4#55111yescharacteristic
489C2×C4C22×Q1611yescharacteristic
490C2×C4C2×#13511yescharacteristic
491C2×C4C2×#13511yescharacteristic
492C2×C4C2×#13711yescharacteristic
493C2×C4#57511yescharacteristic
494C2×C4#57511yescharacteristic
495C2×C4#57611yescharacteristic
496C2×C4#57711yescharacteristic
497C8—22no—
498C8—41no—
499C8—41no—
500C8—41no—
501C8—81no—
502C8—81no—
503C8—81no—
504C8—81no—
505C8—81no—
506C8—81no—
507C22—22no—
508C22—22no—
509C22—22no—
510C22—41no—
511C22—41no—
512C22—41no—
513C22—41no—
514C22#241111yescharacteristic
515C22#303511yescharacteristic
516C22#304511yescharacteristic
517C22C2×#46711yescharacteristic
518C22#305611yescharacteristic
519C22#306711yescharacteristic
520C22#306911yescharacteristic
521C4—22no—
522C4—21no—
523C4—21no—
524C4—41no—
525C4—42no—
526C4—41no—
527C4—42no—
528C4—42no—
529C4—81no—
530C4—82no—
531C4—82no—
532C4—161no—
533C4—161no—
534C4#310411yescharacteristic
535C4#310811yescharacteristic
536C4#320911yescharacteristic
537C4#321111yescharacteristic
538C4#329611yescharacteristic
539C4#330511yescharacteristic
540C4#331711yescharacteristic
541C4#331911yescharacteristic
542C2—42no—
543C2#1912611yescharacteristic
544C2#1918111yescharacteristic
545C2#1941111yescharacteristic
546C2#1949311yescharacteristic
547C2#2448611yescharacteristic
548C2#2450211yescharacteristic
549C2#2452511yescharacteristic
subgroupG/H
center, socleC23C2×#121
commutator, Frattini, cosocleC2×C42C24
prime radicalC24#121
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
5. #10593C2semidirect
6. #10592C2semidirect
7. #10541C2semidirect
8. #10538C2semidirect
9. #9737C2semidirect
10. #9733C2semidirect
11. #9567C2semidirect
12. #9562C2semidirect
216. #117Q16semidirect

Maximal subgroups

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

subgroupindex
#331542
#256002
#226322
#213112
#105932
#105922
#105412
#105382
#97372
#97332
#95672
#95622
#91292
#88292
#83432