Finite Group Databasebeta

← #227249

Subgroups — #227249

Order 512.

746 conjugacy classes of subgroups.

The subgroup lattice has 550 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23102C211yesmaximal, characteristic, complemented
2#22962C211yesmaximal, characteristic
3#22519C211yesmaximal, characteristic
4#22487C211yesmaximal, characteristic
5C2×#1644C211yesmaximal, characteristic
6#9826C211yesmaximal, characteristic, complemented
7#9576C211yesmaximal, characteristic, complemented
8#9567C211yesmaximal, characteristic, complemented
9#9219C211yesmaximal, characteristic
10#9218C211yesmaximal, characteristic, complemented
11#9065C211yesmaximal, characteristic, complemented
12#8887C211yesmaximal, characteristic, complemented
13#8886C211yesmaximal, characteristic
14#8202C211yesmaximal, characteristic
15#8201C211yesmaximal, characteristic, complemented
16C2×#554C2211yescharacteristic
17C2×#487C2211yescharacteristic
18C2×#474C2211yescharacteristic
19C2×#462C2211yescharacteristic
20C2×C8×Q8C2211yescharacteristic
21C2×C8×D4C2211yescharacteristic
22#2365C2211yescharacteristic
23#2355C2211yescharacteristic
24D4×C4⋊C4C2211yescharacteristic
25#1991—21no—
26#1990C2211yescharacteristic
27#1990C2211yescharacteristic
28#1990C2211yescharacteristic
29#1989—21no—
30#1987C2211yescharacteristic
31#1987C2211yescharacteristic
32#1971—21no—
33#1956C2211yescharacteristic
34#1948C2211yescharacteristic
35#1947C2211yescharacteristic
36#1942—21no—
37#1927—21no—
38#1918—21no—
39#1892—21no—
40#1878—21no—
41#1872—21no—
42#1862—21no—
43#1844C2211yescharacteristic
44#1824—21no—
45#1804—21no—
46#1803C2211yescharacteristic
47#1802C2211yescharacteristic
48#1770C2211yescharacteristic
49C8×C4⋊C4C2211yescharacteristic
50#1763—21no—
51C4×#103C2211yescharacteristic
52C4×#102—21no—
53C4×#102—21no—
54C4×#102C2211yescharacteristic
55C2×#340—21no—
56#1644C418yescomplemented
57C2×#333C2211yescharacteristic
58C2×#332C2211yescharacteristic
59C2×Q8⋊C8C2211yescharacteristic
60C2×D4⋊C8C2211yescharacteristic
61C2×#322C2211yescharacteristic
62#1296C2211yescharacteristic
63#1296C2211yescharacteristic
64#1279C2211yescharacteristic
65#1279C2211yescharacteristic
66#1277C2211yescharacteristic
67#1277C2211yescharacteristic
68C22×Q16—21no—
69C23×C8—21no—
70#554—22no—
71#554—22no—
72#554—41no—
73#554C2×C414yes—
74C2×C4⋊Q8C2311yescharacteristic
75C2×C4⋊Q8C2311yescharacteristic
76C2×#125—21no—
77C2×C22⋊Q8—21no—
78C2×C22⋊Q8—41no—
79C2×C4×Q8—21no—
80C2×C4×Q8C2311yescharacteristic
81C2×C4×D4—21no—
82C2×C4×D4—21no—
83C2×C4×D4C2311yescharacteristic
84C22×C4⋊C4—21no—
85C22×C4⋊C4—21no—
86#487—41no—
87#487C2×C414yescomplemented
88#474—22no—
89#474C2×C414yes—
90#462—22no—
91#462C2×C414yescomplemented
92C8×Q8—22no—
93C8×Q8—41no—
94C8×D4—22no—
95C8×D4—41no—
96C2×#107—21no—
97C2×#107—21no—
98C2×#106—21no—
99C2×#106—41no—
100C2×C4⋊C8—21no—
101C2×C4⋊C8C2311yescharacteristic
102C2×C4⋊C8C2311yescharacteristic
103C2×C4⋊C8C2311yescharacteristic
104C2×C4⋊C8C2311yescharacteristic
105C2×#103—21no—
106C2×#103—21no—
107C2×#103—21no—
108C2×#103—21no—
109C2×#103—41no—
110C2×#103D411yescharacteristic
111C2×#103D411yescharacteristic
112C2×#102—21no—
113C2×#102—21no—
114C2×#102D411yescharacteristic
115C2×#102D411yescharacteristic
116C2×C22⋊C8—21no—
117C2×#97—21no—
118C2×C4×C8—21no—
119C2×C4×C8—21no—
120C2×C4×C8C2311yescharacteristic
121C2×C4×C8C2311yescharacteristic
122C2×C4×C8C2311yescharacteristic
123#391—21no—
124#391—21no—
125#389—21no—
126#389—21no—
127#387—41no—
128#387—41no—
129#385—41no—
130#384C2311yescharacteristic
131#384C2311yescharacteristic
132#382—21no—
133#380—21no—
134C4×C4⋊C4—21no—
135C4×C4⋊C4—21no—
136C4×C4⋊C4—21no—
137C4×C4⋊C4C2311yescharacteristic
138C4×C4⋊C4C2311yescharacteristic
139#340—21no—
140#340—21no—
141#340—21no—
142#340—21no—
143#340—21no—
144#340—21no—
145#340—21no—
146#340—21no—
147#340—21no—
148#340—21no—
149#340—21no—
150#340—21no—
151#340—21no—
152#340—21no—
153#340D411yescharacteristic
154#340D411yescharacteristic
155#340D411yescharacteristic
156#340D411yescharacteristic
157#336—21no—
158#336—21no—
159#333C2×C414yescomplemented
160#332—41no—
161Q8⋊C8C2×C414yescomplemented
162D4⋊C8C2×C414yes—
163#322—41no—
164C22×Q8—21no—
165C22×Q8—21no—
166C22×Q8C2×D411yescharacteristic
167C22×D4C2×D411yescharacteristic
168C23×C4—21no—
169C23×C4—41no—
170C23×C4—41no—
171C2×Q16—24no—
172C2×Q16—81no—
173C2×Q16—81no—
174C22×C8—21no—
175C22×C8—21no—
176C22×C8—21no—
177C22×C8—21no—
178C22×C8—21no—
179C22×C8—42no—
180C22×C8—42no—
181C22×C8—81no—
182C22×C8C2×D411yescharacteristic
183C22×C8C2×D411yescharacteristic
184C22×C8C2×D411yescharacteristic
185C22×C8C2×D411yescharacteristic
186C4⋊Q8—21no—
187C4⋊Q8—21no—
188C4⋊Q8—21no—
189C4⋊Q8—21no—
190C4⋊Q8—21no—
191C4⋊Q8—21no—
192C4⋊Q8C22×C412yes—
193C4⋊Q8C22×C412yes—
194#125—24no—
195#125—42no—
196C22⋊Q8—24no—
197C22⋊Q8—44no—
198C22⋊Q8—44no—
199C22⋊Q8—42no—
200C4×Q8—21no—
201C4×Q8—21no—
202C4×Q8—42no—
203C4×Q8—81no—
204C4×Q8C22×C412yes—
205C4×Q8#4112yes—
206C4×D4—24no—
207C4×D4—21no—
208C4×D4—21no—
209C4×D4—42no—
210C4×D4—42no—
211C4×D4—81no—
212C4×D4C22×C412yes—
213C4×D4#4112yes—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—24no—
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—42no—
227C2×C4⋊C4—41no—
228C2×C4⋊C4—41no—
229C2×C4⋊C4—42no—
230C2×C4⋊C4—41no—
231C2×C4⋊C4—41no—
232C2×C4⋊C4—81no—
233C2×C4⋊C4C2×D411yescharacteristic
234C2×C4⋊C4C2×D411yescharacteristic
235C2×C4⋊C4C2×D411yescharacteristic
236C2×C4⋊C4C2×D411yescharacteristic
237C2×C4⋊C4C2×D411yescharacteristic
238C2×C4⋊C4C2×D411yescharacteristic
239C2×C22⋊C4—21no—
240C2×C22⋊C4—41no—
241C2×C22⋊C4—41no—
242C2×C42—21no—
243C2×C42—21no—
244C2×C42—21no—
245C2×C42—41no—
246C2×C42—41no—
247C2×C42C2411yescharacteristic
248#107—24no—
249#107—81no—
250#106—81no—
251#106—82no—
252C4⋊C8—21no—
253C4⋊C8—21no—
254C4⋊C8—21no—
255C4⋊C8—21no—
256C4⋊C8—21no—
257C4⋊C8—21no—
258C4⋊C8—42no—
259C4⋊C8C22×C412yes—
260C4⋊C8C22×C412yes—
261#103—22no—
262#103—24no—
263#103—22no—
264#103—44no—
265#103—42no—
266#103—81no—
267#103—81no—
268#102—24no—
269#102—22no—
270#102—22no—
271#102—42no—
272C22⋊C8—42no—
273#97—81no—
274C4×C8—21no—
275C4×C8—21no—
276C4×C8—21no—
277C4×C8—21no—
278C4×C8—21no—
279C4×C8—42no—
280C4×C8—81no—
281C4×C8C22×C412yes—
282#95—21no—
283#95—21no—
284#95—41no—
285#95—41no—
286#95—41no—
287#95—41no—
288#95—41no—
289#95—41no—
290C24—21no—
291C2×Q8—21no—
292C2×Q8—22no—
293C2×Q8—22no—
294C2×Q8—21no—
295C2×Q8—21no—
296C2×Q8—21no—
297C2×Q8—21no—
298C2×Q8—41no—
299C2×Q8—41no—
300C2×Q8—41no—
301C2×Q8—41no—
302C2×Q8—41no—
303C2×Q8—41no—
304C2×Q8—81no—
305C2×Q8—81no—
306C2×Q8C4×D412yes—
307C2×D4—21no—
308C2×D4—21no—
309C2×D4—21no—
310C2×D4C4×D412yes—
311C2×D4#13712yes—
312C2×D4#13712yes—
313C22×C4—21no—
314C22×C4—21no—
315C22×C4—21no—
316C22×C4—21no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—21no—
320C22×C4—21no—
321C22×C4—21no—
322C22×C4—44no—
323C22×C4—41no—
324C22×C4—41no—
325C22×C4—41no—
326C22×C4—41no—
327C22×C4—41no—
328C22×C4—41no—
329C22×C4—41no—
330C22×C4—41no—
331C22×C4—82no—
332C22×C4—82no—
333C22×C4—161no—
334C22×C4—161no—
335C22×C4—161no—
336C22×C4C22×D411yescharacteristic
337C22×C4C22×D411yescharacteristic
338C22×C4H(5,𝔽2)11yescharacteristic
339Q16—84no—
340C2×C8—21no—
341C2×C8—21no—
342C2×C8—22no—
343C2×C8—22no—
344C2×C8—22no—
345C2×C8—21no—
346C2×C8—21no—
347C2×C8—21no—
348C2×C8—21no—
349C2×C8—21no—
350C2×C8—21no—
351C2×C8—21no—
352C2×C8—21no—
353C2×C8—21no—
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—84no—
367C2×C8#13512yes—
368C4⋊C4—22no—
369C4⋊C4—21no—
370C4⋊C4—21no—
371C4⋊C4—22no—
372C4⋊C4—21no—
373C4⋊C4—22no—
374C4⋊C4—22no—
375C4⋊C4—22no—
376C4⋊C4—22no—
377C4⋊C4—21no—
378C4⋊C4—21no—
379C4⋊C4—21no—
380C4⋊C4—21no—
381C4⋊C4—21no—
382C4⋊C4—21no—
383C4⋊C4—44no—
384C4⋊C4—44no—
385C4⋊C4—41no—
386C4⋊C4—41no—
387C4⋊C4—41no—
388C4⋊C4—41no—
389C4⋊C4—41no—
390C4⋊C4—41no—
391C4⋊C4—41no—
392C4⋊C4—41no—
393C4⋊C4—41no—
394C4⋊C4—41no—
395C4⋊C4—41no—
396C4⋊C4—41no—
397C4⋊C4—82no—
398C4⋊C4—82no—
399C4⋊C4—81no—
400C4⋊C4—81no—
401C4⋊C4C4×D412yes—
402C4⋊C4C4×D412yes—
403C4⋊C4#13512yes—
404C22⋊C4—22no—
405C22⋊C4—22no—
406C22⋊C4—44no—
407C22⋊C4—82no—
408C42—22no—
409C42—22no—
410C42—41no—
411C42—41no—
412C42—41no—
413C42—41no—
414C42—82no—
415C42—161no—
416C42C23×C411yescharacteristic
417C42C2×#4111yescharacteristic
418C42C2×#4111yescharacteristic
419C42H(5,𝔽2)11yescharacteristic
420C23—22no—
421C23—22no—
422C23—22no—
423C23—41no—
424C23—41no—
425C23—41no—
426C23—41no—
427C23D4211yescharacteristic
428Q8—22no—
429Q8—22no—
430Q8—41no—
431Q8—41no—
432Q8—42no—
433Q8—42no—
434Q8—82no—
435Q8—81no—
436Q8—81no—
437Q8—161no—
438D4—22no—
439D4—22no—
440D4—22no—
441D4#44114yes—
442C2×C4—21no—
443C2×C4—21no—
444C2×C4—22no—
445C2×C4—22no—
446C2×C4—22no—
447C2×C4—22no—
448C2×C4—41no—
449C2×C4—41no—
450C2×C4—41no—
451C2×C4—41no—
452C2×C4—41no—
453C2×C4—41no—
454C2×C4—41no—
455C2×C4—41no—
456C2×C4—42no—
457C2×C4—42no—
458C2×C4—42no—
459C2×C4—42no—
460C2×C4—42no—
461C2×C4—42no—
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×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—164no—
482C2×C4—164no—
483C2×C4C2×C4×D411yescharacteristic
484C2×C4C2×C4×D411yescharacteristic
485C2×C4#51811yescharacteristic
486C2×C4#54611yescharacteristic
487C2×C4#54611yescharacteristic
488C2×C4#55111yescharacteristic
489C2×C4C2×#13511yescharacteristic
490C2×C4C2×#13511yescharacteristic
491C2×C4C2×#13711yescharacteristic
492C2×C4C2×#13711yescharacteristic
493C2×C4#57611yescharacteristic
494C2×C4#57711yescharacteristic
495C2×C4#57711yescharacteristic
496C2×C4#57811yescharacteristic
497C8—22no—
498C8—42no—
499C8—42no—
500C8—42no—
501C8—41no—
502C8—41no—
503C8—41no—
504C8—81no—
505C8—81no—
506C22—24no—
507C22—42no—
508C22—42no—
509C22—42no—
510C22—42no—
511C22—42no—
512C22—42no—
513C22#235811yescharacteristic
514C22#326611yescharacteristic
515C22#327011yescharacteristic
516C22#328011yescharacteristic
517C22#328411yescharacteristic
518C22#328811yescharacteristic
519C22#328911yescharacteristic
520C4—21no—
521C4—21no—
522C4—42no—
523C4—42no—
524C4—42no—
525C4—42no—
526C4—81no—
527C4—81no—
528C4—82no—
529C4—82no—
530C4—81no—
531C4—161no—
532C4—161no—
533C4C4×#13511yescharacteristic
534C4C2×#44111yescharacteristic
535C4#294611yescharacteristic
536C4#294711yescharacteristic
537C4#329711yescharacteristic
538C4#330311yescharacteristic
539C4#331211yescharacteristic
540C4#331611yescharacteristic
541C2—44no—
542C2#1977411yescharacteristic
543C2#1978211yescharacteristic
544C2#1985311yescharacteristic
545C2#1985711yescharacteristic
546C2#2285911yescharacteristic
547C2#2287211yescharacteristic
548C2#2289011yescharacteristic
subgroupG/H
center, socleC23D42
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radicalC22×D4C2×D4
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C2×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #23102C2semidirect
6. #9826C2semidirect
7. #9576C2semidirect
8. #9567C2semidirect
10. #9218C2semidirect
11. #9065C2semidirect
12. #8887C2semidirect
15. #8201C2semidirect
56. #1644C4semidirect
87. #487C2×C4semidirect
91. #462C2×C4semidirect
159. #333C2×C4semidirect
161. Q8⋊C8C2×C4semidirect

Maximal subgroups

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

subgroupindex
#231022
#229622
#225192
#224872
C2×#16442
#98262
#95762
#95672
#92192
#92182
#90652
#88872
#88862
#82022
#82012