Finite Group Databasebeta

← #229937

Subgroups — #229937

Order 512.

687 conjugacy classes of subgroups.

The subgroup lattice has 580 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23700C211yesmaximal, characteristic
2#23083C211yesmaximal, characteristic
3#23051C211yesmaximal, characteristic, complemented
4C2×SD16⋊C8C211yesmaximal, characteristic, complemented
5C2×#1572C211yesmaximal, characteristic, complemented
6#9666C211yesmaximal, characteristic, complemented
7#9596C211yesmaximal, characteristic, complemented
8#9546C211yesmaximal, characteristic, complemented
9#9543C211yesmaximal, characteristic, complemented
10#8943C211yesmaximal, characteristic
11#8938C211yesmaximal, characteristic, complemented
12#8846C211yesmaximal, characteristic, complemented
13#8845C211yesmaximal, characteristic, complemented
14#8289C211yesmaximal, characteristic, complemented
15#8149C211yesmaximal, characteristic, complemented
16C2×C8×Q8C2211yescharacteristic
17C2×C4×SD16C2211yescharacteristic
18C2×C8×D4C2211yescharacteristic
19C2×#432C2211yescharacteristic
20C2×#431C2211yescharacteristic
21C2×#431C2211yescharacteristic
22C4×C22⋊Q8C2211yescharacteristic
23C4×#121C2211yescharacteristic
24#2088C411yescharacteristic
25#2077C411yescharacteristic
26#2066C411yescharacteristic
27#2065C411yescharacteristic
28#2055C411yescharacteristic
29#2035C411yescharacteristic
30#2034C411yescharacteristic
31#2000C411yescharacteristic
32#1984C2211yescharacteristic
33#1954—21no—
34#1954C2211yescharacteristic
35#1939C2211yescharacteristic
36#1939C2211yescharacteristic
37#1924—21no—
38#1923—21no—
39#1923—21no—
40#1917—21no—
41#1869—21no—
42C4×#106C2211yescharacteristic
43#1771—21no—
44#1770C2211yescharacteristic
45C8×C4⋊C4C2211yescharacteristic
46C4×#103C2211yescharacteristic
47C4×#102C2211yescharacteristic
48C8×C22⋊C4—21no—
49C4×C22⋊C8C2211yescharacteristic
50C4×C22⋊C8C2211yescharacteristic
51SD16⋊C8—24no—
52#1572—24no—
53C2×#335C2211yescharacteristic
54C2×#335C2211yescharacteristic
55C2×#334C2211yescharacteristic
56C2×#334C2211yescharacteristic
57C2×Q8⋊C8C2211yescharacteristic
58C2×Q8⋊C8C2211yescharacteristic
59C2×D4⋊C8C2211yescharacteristic
60C2×D4⋊C8C2211yescharacteristic
61C2×#322C2211yescharacteristic
62C2×#322C2211yescharacteristic
63#1279C2211yescharacteristic
64#1279C2211yescharacteristic
65#1279C2211yescharacteristic
66#1279C2211yescharacteristic
67#1273C2211yescharacteristic
68#1273C2211yescharacteristic
69C22×SD16C2×C411yescharacteristic
70C23×C8—21no—
71C2×C22⋊Q8C2×C411yescharacteristic
72C2×#121C2×C411yescharacteristic
73C2×C4×Q8C2311yescharacteristic
74C2×C4×D4—21no—
75C2×C4×D4C2311yescharacteristic
76C22×C42C2311yescharacteristic
77C8×Q8—22no—
78C8×Q8—41no—
79C4×SD16—22no—
80C4×SD16—22no—
81C8×D4—22no—
82C8×D4—41no—
83#432—22no—
84#432—41no—
85#431—22no—
86#431—22no—
87#431—22no—
88#431—41no—
89C2×#106C2×C411yescharacteristic
90C2×#106C2×C411yescharacteristic
91C2×#106C2×C411yescharacteristic
92C2×C4⋊C8—21no—
93C2×C4⋊C8—21no—
94C2×C4⋊C8—21no—
95C2×C4⋊C8C2×C411yescharacteristic
96C2×C4⋊C8C2×C411yescharacteristic
97C2×C4⋊C8C2311yescharacteristic
98C2×C4⋊C8C2311yescharacteristic
99C2×C4⋊C8C2311yescharacteristic
100C2×C4⋊C8C2311yescharacteristic
101C2×#103C2×C411yescharacteristic
102C2×#103C2×C411yescharacteristic
103C2×#103C2×C411yescharacteristic
104C2×#102C2×C411yescharacteristic
105C2×#102C2×C411yescharacteristic
106C2×#102C2×C411yescharacteristic
107C2×C22⋊C8—21no—
108C2×C22⋊C8—21no—
109C2×C22⋊C8—21no—
110C2×C22⋊C8—21no—
111C2×C22⋊C8—21no—
112C2×C22⋊C8—21no—
113C2×C22⋊C8C2×C411yescharacteristic
114C2×C22⋊C8C2×C411yescharacteristic
115C2×C22⋊C8D411yescharacteristic
116C2×C22⋊C8D411yescharacteristic
117C2×#97—21no—
118C2×#97—21no—
119C2×C4×C8—21no—
120C2×C4×C8—21no—
121C2×C4×C8—21no—
122C2×C4×C8C2311yescharacteristic
123C2×C4×C8C2311yescharacteristic
124C2×C4×C8C2311yescharacteristic
125C2×C4×C8C2311yescharacteristic
126C2×C4×C8C2311yescharacteristic
127C2×C4×C8C2311yescharacteristic
128#391C2×C411yescharacteristic
129#390C2×C411yescharacteristic
130#389C2×C411yescharacteristic
131#389C2×C411yescharacteristic
132#388—21no—
133#387—21no—
134#386—21no—
135#385—21no—
136#380C2×C411yescharacteristic
137#380C2×C411yescharacteristic
138C4×C4⋊C4—21no—
139C4×C4⋊C4C2311yescharacteristic
140C4×C4⋊C4C2311yescharacteristic
141C4×C22⋊C4—21no—
142C4×C22⋊C4—21no—
143#340C2×C411yescharacteristic
144#340C2×C411yescharacteristic
145#340C2×C411yescharacteristic
146#340C2×C411yescharacteristic
147#340C2×C411yescharacteristic
148#340C2×C411yescharacteristic
149#336—21no—
150#336—21no—
151#336—21no—
152#336—21no—
153#336—21no—
154#336—21no—
155#336—21no—
156#336D411yescharacteristic
157#336D411yescharacteristic
158#335—22no—
159#335D414yescomplemented
160#334—22no—
161#334—22no—
162Q8⋊C8—22no—
163Q8⋊C8—41no—
164D4⋊C8—22no—
165D4⋊C8—41no—
166#322—22no—
167#322—22no—
168C22×Q8C22×C411yescharacteristic
169C22×D4—21no—
170C22×D4C22×C411yescharacteristic
171C23×C4—21no—
172C23×C4C22×C411yescharacteristic
173C23×C4C2×D411yescharacteristic
174C23×C4#4111yescharacteristic
175C2×SD16—22no—
176C2×SD16—22no—
177C2×SD16—22no—
178C22×C8—21no—
179C22×C8—21no—
180C22×C8—21no—
181C22×C8—21no—
182C22×C8—21no—
183C22×C8—21no—
184C22×C8—42no—
185C22×C8—42no—
186C22×C8—81no—
187C22×C8C22×C411yescharacteristic
188C22×C8C22×C411yescharacteristic
189C22×C8C22×C411yescharacteristic
190C22×C8C22×C411yescharacteristic
191C22×C8C2×D411yescharacteristic
192C22×C8C2×D411yescharacteristic
193C22×C8C2×D411yescharacteristic
194C22×C8C2×D411yescharacteristic
195C22⋊Q8—22no—
196C22⋊Q8—41no—
197#121—22no—
198#121—22no—
199C4×Q8—21no—
200C4×Q8—21no—
201C4×Q8—21no—
202C4×Q8—21no—
203C4×D4—21no—
204C4×D4—21no—
205C4×D4—21no—
206C4×D4—21no—
207C4×D4—81no—
208C4×D4—81no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4C22×C411yescharacteristic
214C2×C4⋊C4C22×C411yescharacteristic
215C2×C4⋊C4C22×C411yescharacteristic
216C2×C4⋊C4C22×C411yescharacteristic
217C2×C4⋊C4C22×C411yescharacteristic
218C2×C4⋊C4C22×C411yescharacteristic
219C2×C22⋊C4—21no—
220C2×C22⋊C4—21no—
221C2×C22⋊C4—21no—
222C2×C22⋊C4—21no—
223C2×C22⋊C4—21no—
224C2×C22⋊C4—21no—
225C2×C22⋊C4—21no—
226C2×C42—21no—
227C2×C42—21no—
228C2×C42—21no—
229C2×C42—21no—
230C2×C42—21no—
231C2×C42—21no—
232C2×C42—21no—
233C2×C42C22×C411yescharacteristic
234C2×C42C2×D411yescharacteristic
235C2×C42#4111yescharacteristic
236C2×C42C2411yescharacteristic
237#106—22no—
238#106—22no—
239#106—41no—
240C4⋊C8—22no—
241C4⋊C8—22no—
242C4⋊C8—21no—
243C4⋊C8—21no—
244C4⋊C8—21no—
245C4⋊C8—21no—
246C4⋊C8—21no—
247C4⋊C8—21no—
248C4⋊C8—42no—
249C4⋊C8—42no—
250C4⋊C8—81no—
251C4⋊C8C2×D412yes—
252C4⋊C8C2×D412yes—
253#103—22no—
254#103—22no—
255#103—22no—
256#102—22no—
257#102—22no—
258#102—41no—
259C22⋊C8—22no—
260C22⋊C8—22no—
261C22⋊C8—41no—
262C22⋊C8—41no—
263C22⋊C8—42no—
264C22⋊C8—42no—
265C22⋊C8—42no—
266C22⋊C8—42no—
267C22⋊C8—42no—
268C22⋊C8—81no—
269#97—42no—
270#97—42no—
271C4×C8—21no—
272C4×C8—21no—
273C4×C8—21no—
274C4×C8—21no—
275C4×C8—21no—
276C4×C8—21no—
277C4×C8—21no—
278C4×C8—21no—
279C4×C8—21no—
280C4×C8—21no—
281C4×C8—42no—
282C4×C8—42no—
283C4×C8—81no—
284C4×C8C2×D412yes—
285C4×C8#4112yes—
286#95—21no—
287#95—21no—
288#95—21no—
289#95—21no—
290#95—21no—
291#95—21no—
292#95—21no—
293#95—21no—
294C24—21no—
295C24C4×D411yescharacteristic
296C2×Q8—21no—
297C2×Q8—21no—
298C2×Q8—21no—
299C2×Q8—21no—
300C2×Q8—21no—
301C2×Q8#13112yes—
302C2×D4—24no—
303C2×D4—21no—
304C2×D4—21no—
305C2×D4—21no—
306C2×D4—21no—
307C2×D4—21no—
308C2×D4—81no—
309C2×D4—81no—
310C2×D4#13112yes—
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—21no—
323C22×C4—21no—
324C22×C4—21no—
325C22×C4—21no—
326C22×C4—21no—
327C22×C4—21no—
328C22×C4—21no—
329C22×C4—21no—
330C22×C4—21no—
331C22×C4—21no—
332C22×C4—41no—
333C22×C4—41no—
334C22×C4—41no—
335C22×C4—41no—
336C22×C4—41no—
337C22×C4—41no—
338C22×C4—41no—
339C22×C4—41no—
340C22×C4C4×D411yescharacteristic
341C22×C4C4×D411yescharacteristic
342C22×C4C4×D411yescharacteristic
343C22×C4C23×C411yescharacteristic
344C22×C4C22×D411yescharacteristic
345C22×C4C2×#4111yescharacteristic
346SD16—28no—
347C2×C8—21no—
348C2×C8—21no—
349C2×C8—21no—
350C2×C8—21no—
351C2×C8—22no—
352C2×C8—21no—
353C2×C8—21no—
354C2×C8—21no—
355C2×C8—21no—
356C2×C8—22no—
357C2×C8—22no—
358C2×C8—21no—
359C2×C8—21no—
360C2×C8—21no—
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—41no—
373C2×C8—41no—
374C2×C8—41no—
375C2×C8—41no—
376C2×C8—41no—
377C2×C8—41no—
378C2×C8—41no—
379C2×C8—41no—
380C2×C8—41no—
381C2×C8—41no—
382C2×C8—41no—
383C2×C8—41no—
384C2×C8—41no—
385C2×C8—41no—
386C2×C8—41no—
387C2×C8—84no—
388C2×C8C4×D412yes—
389C2×C8C4×D412yes—
390C4⋊C4—21no—
391C4⋊C4—21no—
392C4⋊C4—21no—
393C4⋊C4—21no—
394C4⋊C4—21no—
395C4⋊C4—21no—
396C4⋊C4—21no—
397C4⋊C4—21no—
398C4⋊C4—21no—
399C4⋊C4—21no—
400C4⋊C4—21no—
401C4⋊C4—42no—
402C4⋊C4—41no—
403C4⋊C4—41no—
404C4⋊C4—81no—
405C4⋊C4—81no—
406C4⋊C4#13112yes—
407C22⋊C4—24no—
408C22⋊C4—42no—
409C22⋊C4—42no—
410C22⋊C4—42no—
411C22⋊C4—41no—
412C22⋊C4—41no—
413C22⋊C4—81no—
414C22⋊C4—81no—
415C42—21no—
416C42—21no—
417C42—21no—
418C42—21no—
419C42—21no—
420C42—21no—
421C42—41no—
422C42—41no—
423C42—41no—
424C42—41no—
425C42—41no—
426C42—41no—
427C42C22×D411yescharacteristic
428C42C2×#4111yescharacteristic
429C42H(5,𝔽2)11yescharacteristic
430C422−1+411yescharacteristic
431C23—21no—
432C23—22no—
433C23—21no—
434C23—21no—
435C23—21no—
436C23—21no—
437C23—21no—
438C23—21no—
439C23—41no—
440C23—41no—
441C23—41no—
442C23—41no—
443C23—41no—
444C23—41no—
445C23C2×C4×D411yescharacteristic
446Q8—22no—
447Q8—22no—
448Q8—41no—
449Q8—41no—
450D4—22no—
451D4—22no—
452D4—41no—
453D4—41no—
454D4—84no—
455C2×C4—21no—
456C2×C4—21no—
457C2×C4—21no—
458C2×C4—21no—
459C2×C4—21no—
460C2×C4—21no—
461C2×C4—22no—
462C2×C4—22no—
463C2×C4—21no—
464C2×C4—21no—
465C2×C4—21no—
466C2×C4—21no—
467C2×C4—21no—
468C2×C4—41no—
469C2×C4—41no—
470C2×C4—41no—
471C2×C4—41no—
472C2×C4—41no—
473C2×C4—41no—
474C2×C4—41no—
475C2×C4—41no—
476C2×C4—41no—
477C2×C4—41no—
478C2×C4—41no—
479C2×C4—41no—
480C2×C4—41no—
481C2×C4—41no—
482C2×C4—41no—
483C2×C4—41no—
484C2×C4—41no—
485C2×C4—41no—
486C2×C4—41no—
487C2×C4—41no—
488C2×C4—41no—
489C2×C4—41no—
490C2×C4—41no—
491C2×C4—41no—
492C2×C4—41no—
493C2×C4—41no—
494C2×C4—41no—
495C2×C4—81no—
496C2×C4—81no—
497C2×C4—81no—
498C2×C4—81no—
499C2×C4—81no—
500C2×C4—81no—
501C2×C4C2×C4×D411yescharacteristic
502C2×C4#52011yescharacteristic
503C2×C4D4211yescharacteristic
504C2×C4#54711yescharacteristic
505C2×C4#54811yescharacteristic
506C2×C4#55311yescharacteristic
507C2×C4C2×#13111yescharacteristic
508C2×C4C2×#13111yescharacteristic
509C2×C4C2×#13111yescharacteristic
510C2×C4#56811yescharacteristic
511C2×C4#57511yescharacteristic
512C2×C4#57511yescharacteristic
513C2×C4#57711yescharacteristic
514C2×C4#57711yescharacteristic
515C8—22no—
516C8—22no—
517C8—42no—
518C8—41no—
519C8—41no—
520C8—41no—
521C8—41no—
522C8—41no—
523C8—42no—
524C8—42no—
525C8—41no—
526C8—81no—
527C8—81no—
528C8—81no—
529C22—41no—
530C22—42no—
531C22—41no—
532C22—41no—
533C22—41no—
534C22—42no—
535C22—41no—
536C22—41no—
537C22—42no—
538C22—41no—
539C22—81no—
540C22—81no—
541C22—81no—
542C22—81no—
543C22#236111yescharacteristic
544C22#292911yescharacteristic
545C22#293311yescharacteristic
546C22#294311yescharacteristic
547C22#294911yescharacteristic
548C22#295511yescharacteristic
549C22#295611yescharacteristic
550C4—21no—
551C4—21no—
552C4—42no—
553C4—41no—
554C4—42no—
555C4—41no—
556C4—41no—
557C4—81no—
558C4—81no—
559C4—81no—
560C4—81no—
561C4—81no—
562C4#292911yescharacteristic
563C4#293511yescharacteristic
564C4#299211yescharacteristic
565C4#299411yescharacteristic
566C4#326711yescharacteristic
567C4#327111yescharacteristic
568C4#334811yescharacteristic
569C4#335011yescharacteristic
570C2—81no—
571C2—82no—
572C2#1790911yescharacteristic
573C2#1792411yescharacteristic
574C2#1800511yescharacteristic
575C2#1803011yescharacteristic
576C2#2299111yescharacteristic
577C2#2302111yescharacteristic
578C2#2303311yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC23C2×C4×D4
prime radicalC2×#121C2×C4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C4×C22⋊C8 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #23051C2semidirect
4. C2×SD16⋊C8C2semidirect
5. C2×#1572C2semidirect
6. #9666C2semidirect
7. #9596C2semidirect
8. #9546C2semidirect
9. #9543C2semidirect
11. #8938C2semidirect
12. #8846C2semidirect
13. #8845C2semidirect
14. #8289C2semidirect
15. #8149C2semidirect
159. #335D4semidirect

Maximal subgroups

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

subgroupindex
#237002
#230832
#230512
C2×SD16⋊C82
C2×#15722
#96662
#95962
#95462
#95432
#89432
#89382
#88462
#88452
#82892
#81492