Finite Group Databasebeta

← #210767

Subgroups — #210767

Order 512.

764 conjugacy classes of subgroups.

The subgroup lattice has 566 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21597C211yesmaximal, characteristic
2C8×C22⋊Q8C211yesmaximal, characteristic
3C4×#467C211yesmaximal, characteristic
4C2×#1584C211yesmaximal, characteristic, complemented
5C22×#322C211yesmaximal, characteristic
6#9662C211yesmaximal, characteristic, complemented
7#9598C211yesmaximal, characteristic, complemented
8#9557C211yesmaximal, characteristic, complemented
9#9557C211yesmaximal, characteristic, complemented
10#9071C211yesmaximal, characteristic
11#9071C211yesmaximal, characteristic
12#8596C211yesmaximal, characteristic
13#8291C211yesmaximal, characteristic, complemented
14#8211C211yesmaximal, characteristic, complemented
15#8203C211yesmaximal, characteristic, complemented
16C2×#467C411yescharacteristic
17C2×#446C2211yescharacteristic
18C2×C8×Q8C2211yescharacteristic
19C2×C4×Q16C2211yescharacteristic
20C22×C4×C8C2211yescharacteristic
21C22×C4×C8C2211yescharacteristic
22C22×C4×C8C2211yescharacteristic
23C4×C22⋊Q8C2211yescharacteristic
24C4×C22⋊Q8C2211yescharacteristic
25#1984C2211yescharacteristic
26#1984C2211yescharacteristic
27#1969C411yescharacteristic
28#1942C411yescharacteristic
29#1939C2211yescharacteristic
30#1939C2211yescharacteristic
31#1933C411yescharacteristic
32#1933C411yescharacteristic
33#1916—21no—
34#1916—21no—
35#1892C411yescharacteristic
36#1892C411yescharacteristic
37#1850—21no—
38#1850—21no—
39#1808C411yescharacteristic
40#1797C2211yescharacteristic
41#1797C2211yescharacteristic
42C4×#107C2211yescharacteristic
43#1771—21no—
44#1770C2211yescharacteristic
45C8×C4⋊C4—21no—
46C8×C4⋊C4C2211yescharacteristic
47C4×#103C2211yescharacteristic
48C4×#103C2211yescharacteristic
49#1752—21no—
50C8×C22⋊C4—21no—
51#1584—24no—
52C2×#335C2211yescharacteristic
53C2×#335C2211yescharacteristic
54C2×Q8⋊C8C2211yescharacteristic
55C2×Q8⋊C8C2211yescharacteristic
56C2×Q8⋊C8C2211yescharacteristic
57C2×Q8⋊C8C2211yescharacteristic
58C2×#322—24no—
59C2×#322C2211yescharacteristic
60C2×#322C2211yescharacteristic
61C2×#322C2211yescharacteristic
62C2×#322C2211yescharacteristic
63#1279C2211yescharacteristic
64#1279C2211yescharacteristic
65#1279C2211yescharacteristic
66#1279C2211yescharacteristic
67#1277C2211yescharacteristic
68#1277C2211yescharacteristic
69C22×Q16C2×C411yescharacteristic
70C23×C8—21no—
71C23×C8C2×C411yescharacteristic
72C23×C8C2×C411yescharacteristic
73C23×C8D411yescharacteristic
74C23×C8D411yescharacteristic
75C2×C22⋊Q8C2×C411yescharacteristic
76C2×C22⋊Q8C2×C411yescharacteristic
77C2×C4×Q8C2311yescharacteristic
78C2×C4×Q8C2311yescharacteristic
79C22×C42C2311yescharacteristic
80#467—24no—
81#446—22no—
82#446—22no—
83C8×Q8—22no—
84C8×Q8—22no—
85C4×Q16—22no—
86C4×Q16—22no—
87C2×#107C2×C411yescharacteristic
88C2×#107C2×C411yescharacteristic
89C2×#107C2×C411yescharacteristic
90C2×C4⋊C8—21no—
91C2×C4⋊C8—21no—
92C2×C4⋊C8—21no—
93C2×C4⋊C8—21no—
94C2×C4⋊C8C2311yescharacteristic
95C2×C4⋊C8C2311yescharacteristic
96C2×C4⋊C8C2311yescharacteristic
97C2×C4⋊C8C2311yescharacteristic
98C2×#103C2×C411yescharacteristic
99C2×#103C2×C411yescharacteristic
100C2×#103C2×C411yescharacteristic
101C2×#103C2×C411yescharacteristic
102C2×#103C2×C411yescharacteristic
103C2×#103C2×C411yescharacteristic
104C2×C22⋊C8—21no—
105C2×C22⋊C8—21no—
106C2×#97—21no—
107C2×#97—21no—
108C2×C4×C8—21no—
109C2×C4×C8—22no—
110C2×C4×C8—22no—
111C2×C4×C8—22no—
112C2×C4×C8—22no—
113C2×C4×C8—22no—
114C2×C4×C8—22no—
115C2×C4×C8—21no—
116C2×C4×C8—21no—
117C2×C4×C8C2×C411yescharacteristic
118C2×C4×C8C2×C411yescharacteristic
119C2×C4×C8D411yescharacteristic
120C2×C4×C8D411yescharacteristic
121C2×C4×C8C2311yescharacteristic
122C2×C4×C8C2311yescharacteristic
123C2×C4×C8C2311yescharacteristic
124C2×C4×C8C2311yescharacteristic
125C2×C4×C8C2311yescharacteristic
126C2×C4×C8C2311yescharacteristic
127#391C2×C411yescharacteristic
128#391C2×C411yescharacteristic
129#389C2×C411yescharacteristic
130#389C2×C411yescharacteristic
131#387—21no—
132#387—21no—
133#385—21no—
134#385—21no—
135#380C2×C411yescharacteristic
136#380C2×C411yescharacteristic
137C4×C4⋊C4—21no—
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#336—21no—
157#336—21no—
158#336—21no—
159#335—22no—
160#335—22no—
161Q8⋊C8—22no—
162Q8⋊C8—22no—
163Q8⋊C8—22no—
164Q8⋊C8—22no—
165#322—22no—
166#322—22no—
167#322—22no—
168#322D414yes—
169C22×Q8C22×C411yescharacteristic
170C22×Q8C22×C411yescharacteristic
171C23×C4C22×C411yescharacteristic
172C23×C4C2×D411yescharacteristic
173C23×C4#4111yescharacteristic
174C2×Q16—22no—
175C2×Q16—22no—
176C2×Q16—41no—
177C22×C8—22no—
178C22×C8—22no—
179C22×C8—22no—
180C22×C8—22no—
181C22×C8—22no—
182C22×C8—24no—
183C22×C8—22no—
184C22×C8—22no—
185C22×C8—21no—
186C22×C8—21no—
187C22×C8—22no—
188C22×C8—22no—
189C22×C8—21no—
190C22×C8—21no—
191C22×C8—21no—
192C22×C8—21no—
193C22×C8—41no—
194C22×C8—41no—
195C22×C8—42no—
196C22×C8—42no—
197C22×C8Q1614yescomplemented
198C22×C8C22×C411yescharacteristic
199C22×C8C22×C411yescharacteristic
200C22×C8C22×C411yescharacteristic
201C22×C8C22×C411yescharacteristic
202C22×C8C2×D411yescharacteristic
203C22×C8C2×D411yescharacteristic
204C22×C8C2×D411yescharacteristic
205C22×C8C2×D411yescharacteristic
206C22⋊Q8—22no—
207C22⋊Q8—22no—
208C22⋊Q8—22no—
209C22⋊Q8M1614yes—
210C4×Q8—21no—
211C4×Q8—21no—
212C4×Q8—21no—
213C4×Q8—21no—
214C4×Q8—21no—
215C4×Q8—21no—
216C4×Q8—21no—
217C4×Q8—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⋊C4C22×C411yescharacteristic
225C2×C4⋊C4C22×C411yescharacteristic
226C2×C4⋊C4C22×C411yescharacteristic
227C2×C4⋊C4C22×C411yescharacteristic
228C2×C4⋊C4C22×C411yescharacteristic
229C2×C4⋊C4C22×C411yescharacteristic
230C2×C22⋊C4—21no—
231C2×C22⋊C4—21no—
232C2×C22⋊C4—21no—
233C2×C22⋊C4—21no—
234C2×C42—21no—
235C2×C42—21no—
236C2×C42—21no—
237C2×C42—21no—
238C2×C42—21no—
239C2×C42—21no—
240C2×C42—21no—
241C2×C42—21no—
242C2×C42C22×C411yescharacteristic
243C2×C42C2×D411yescharacteristic
244C2×C42#4111yescharacteristic
245C2×C42C2411yescharacteristic
246#107—22no—
247#107—22no—
248#107—41no—
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—42no—
258C4⋊C8—42no—
259C4⋊C8—42no—
260C4⋊C8—42no—
261#103—22no—
262#103—22no—
263#103—22no—
264#103—22no—
265#103—41no—
266#103—41no—
267C22⋊C8—42no—
268C22⋊C8—42no—
269#97—42no—
270#97—42no—
271C4×C8—24no—
272C4×C8—22no—
273C4×C8—22no—
274C4×C8—22no—
275C4×C8—22no—
276C4×C8—21no—
277C4×C8—21no—
278C4×C8—21no—
279C4×C8—21no—
280C4×C8—21no—
281C4×C8—21no—
282C4×C8—42no—
283C4×C8—42no—
284C4×C8C2×D412yes—
285C4×C8C2×D412yes—
286C4×C8C2×D412yes—
287C4×C8#4112yes—
288C4×C8#4112yes—
289C4×C8#4112yes—
290#95—21no—
291#95—21no—
292#95—21no—
293#95—21no—
294#95—21no—
295#95—21no—
296#95—21no—
297#95—21no—
298#95—21no—
299#95—21no—
300C24C4×D411yescharacteristic
301C2×Q8—21no—
302C2×Q8—21no—
303C2×Q8—21no—
304C2×Q8—21no—
305C2×Q8—21no—
306C2×Q8—21no—
307C2×Q8—21no—
308C2×Q8—21no—
309C2×Q8—21no—
310C2×Q8—21no—
311C2×Q8—21no—
312C2×Q8C2×M1612yes—
313C22×C4—22no—
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—22no—
325C22×C4—21no—
326C22×C4—21no—
327C22×C4—21no—
328C22×C4—21no—
329C22×C4—21no—
330C22×C4—21no—
331C22×C4—21no—
332C22×C4—21no—
333C22×C4C4×D411yescharacteristic
334C22×C4C4×D411yescharacteristic
335C22×C4C4×D411yescharacteristic
336C22×C4C2×M1612yes—
337C22×C4#13112yes—
338C22×C4C2×Q1612yes—
339C22×C4#13512yes—
340C22×C4C23×C411yescharacteristic
341C22×C4C22×D411yescharacteristic
342C22×C4C2×#4111yescharacteristic
343Q16—44no—
344C2×C8—28no—
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—21no—
355C2×C8—28no—
356C2×C8—21no—
357C2×C8—21no—
358C2×C8—22no—
359C2×C8—22no—
360C2×C8—21no—
361C2×C8—21no—
362C2×C8—21no—
363C2×C8—21no—
364C2×C8—44no—
365C2×C8—44no—
366C2×C8—48no—
367C2×C8—41no—
368C2×C8—41no—
369C2×C8—41no—
370C2×C8—41no—
371C2×C8—41no—
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×C8C4×D412yes—
384C2×C8C4×D412yes—
385C2×C8#12112yes—
386C2×C8#12112yes—
387C2×C8C2×Q1612yes—
388C2×C8C2×Q1612yes—
389C2×C8#13512yes—
390C2×C8#13512yes—
391C4⋊C4—21no—
392C4⋊C4—24no—
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—21no—
402C4⋊C4—21no—
403C4⋊C4—41no—
404C4⋊C4—41no—
405C4⋊C4—41no—
406C4⋊C4—41no—
407C4⋊C4—42no—
408C4⋊C4—42no—
409C4⋊C4—42no—
410C4⋊C4C2×M1612yes—
411C22⋊C4—24no—
412C22⋊C4—42no—
413C22⋊C4—42no—
414C22⋊C4—42no—
415C42—21no—
416C42—21no—
417C42—21no—
418C42—21no—
419C42—21no—
420C42—21no—
421C42—41no—
422C42—41no—
423C42—41no—
424C42—41no—
425C42—41no—
426C42—41no—
427C42—41no—
428C42—41no—
429C42C22×D411yescharacteristic
430C42C2×#4111yescharacteristic
431C42C2×#4111yescharacteristic
432C42C2×#4111yescharacteristic
433C23—21no—
434C23—21no—
435C23—21no—
436C23—21no—
437C23#43512yes—
438C23C4×Q1612yes—
439C23#44412yes—
440C23C2×C4×D411yescharacteristic
441Q8—22no—
442Q8—22no—
443Q8—41no—
444Q8—41no—
445Q8—41no—
446Q8—41no—
447Q8—41no—
448Q8—41no—
449C2×C4—24no—
450C2×C4—24no—
451C2×C4—21no—
452C2×C4—21no—
453C2×C4—21no—
454C2×C4—21no—
455C2×C4—21no—
456C2×C4—21no—
457C2×C4—21no—
458C2×C4—21no—
459C2×C4—22no—
460C2×C4—22no—
461C2×C4—22no—
462C2×C4—22no—
463C2×C4—22no—
464C2×C4—22no—
465C2×C4—22no—
466C2×C4—22no—
467C2×C4—22no—
468C2×C4—22no—
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#43512yes—
492C2×C4C4×Q1612yes—
493C2×C4#44412yes—
494C2×C4C2×C4×D411yescharacteristic
495C2×C4C4×#4111yescharacteristic
496C2×C4C2×#12111yescharacteristic
497C2×C4#52511yescharacteristic
498C2×C4#52911yescharacteristic
499C2×C4#53211yescharacteristic
500C2×C4C22×M1611yescharacteristic
501C2×C4C2×#13111yescharacteristic
502C2×C4#56811yescharacteristic
503C2×C4#56811yescharacteristic
504C2×C4C22×Q1611yescharacteristic
505C2×C4C2×#13511yescharacteristic
506C2×C4C2×#13511yescharacteristic
507C2×C4C2×#13511yescharacteristic
508C8—22no—
509C8—22no—
510C8—22no—
511C8—22no—
512C8—22no—
513C8—42no—
514C8—42no—
515C8—41no—
516C8—41no—
517C8—81no—
518C8—81no—
519C8—81no—
520C8—81no—
521C8#46714yescomplemented
522C22—22no—
523C22—22no—
524C22—22no—
525C22—22no—
526C22—22no—
527C22—22no—
528C22#158414yescomplemented
529C22C4×#12111yescharacteristic
530C22C2×#43511yescharacteristic
531C22#293211yescharacteristic
532C22C2×C4×Q1611yescharacteristic
533C22C4×#13511yescharacteristic
534C22C2×#44411yescharacteristic
535C22C2×#44411yescharacteristic
536C4—22no—
537C4—21no—
538C4—21no—
539C4—22no—
540C4—22no—
541C4—42no—
542C4—42no—
543C4—81no—
544C4—81no—
545C4—81no—
546C4—81no—
547C4—81no—
548C4—81no—
549C4#293311yescharacteristic
550C4D4×M1611yescharacteristic
551C4#296811yescharacteristic
552C4#297011yescharacteristic
553C4C2×#46711yescharacteristic
554C4#305011yescharacteristic
555C4#316811yescharacteristic
556C4#317011yescharacteristic
557C2—24no—
558C2C2×#158411yescharacteristic
559C2#1786311yescharacteristic
560C2#1797711yescharacteristic
561C2#1797811yescharacteristic
562C2#2148911yescharacteristic
563C2C4×#46711yescharacteristic
564C2#2154611yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC23C2×C4×D4
prime radicalC24C4×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C22×C4×C8 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. C2×#1584C2semidirect
6. #9662C2semidirect
7. #9598C2semidirect
8. #9557C2semidirect
9. #9557C2semidirect
13. #8291C2semidirect
14. #8211C2semidirect
15. #8203C2semidirect
197. C22×C8Q16semidirect
521. C8#467semidirect
528. C22#1584semidirect

Maximal subgroups

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

subgroupindex
#215972
C8×C22⋊Q82
C4×#4672
C2×#15842
C22×#3222
#96622
#95982
#95572
#90712
#85962
#82912
#82112
#82032