Finite Group Databasebeta

← #6400493

Subgroups — #6400493

Order 512.

2104 conjugacy classes of subgroups.

The subgroup lattice has 544 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2984C211yesmaximal, characteristic, complemented
2C2×C8×#41C211yesmaximal, characteristic, complemented
3C4×#548C211yesmaximal, characteristic
4#23209C213yesmaximal, complemented
5#23084C213yesmaximal, complemented
6#23054C213yesmaximal, complemented
7#22797C213yesmaximal, complemented
8Q8×C4⋊C8C211yesmaximal, characteristic, complemented
9#22530C211yesmaximal, characteristic, complemented
10Q8×C22⋊C8C211yesmaximal, characteristic, complemented
11#22212C213yesmaximal, complemented
12#22135C213yesmaximal, complemented
13#21597C213yesmaximal, complemented
14#21101C213yesmaximal, complemented
15#20572C211yesmaximal, characteristic, complemented
16C2×#548C411yescharacteristic
17C2×C4×#41C2211yescharacteristic
18C22×C4×Q8C2211yescharacteristic
19#2984—28no—
20C8×#41—28no—
21C2×#446C2213yes—
22C2×#446C2213yes—
23C2×C8×Q8C2211yescharacteristic
24C2×C8×Q8C2211yescharacteristic
25C2×C8×Q8C2211yescharacteristic
26C2×C8×Q8C2211yescharacteristic
27C2×C8×D4C2213yes—
28C2×C8×D4C2213yes—
29C2×#432C2213yes—
30C2×#431C2213yes—
31C2×#431C2213yes—
32C22×C4×C8C2213yes—
33#2367C413yes—
34#2362C413yes—
35#2361C413yes—
36#2352C413yes—
37#2351C411yescharacteristic
38#2341C411yescharacteristic
39Q8×C22⋊C4C411yescharacteristic
40C4×#124C2213yes—
41C4×C22⋊Q8C2213yes—
42C4×#121C2213yes—
43C42×Q8C2211yescharacteristic
44C42×D4C2213yes—
45#1984C2213yes—
46#1984C2213yes—
47#1955C2216yes—
48#1954C2213yes—
49#1954C2213yes—
50#1939C2213yes—
51#1939C2213yes—
52#1924C2216yes—
53#1924C2216yes—
54#1924C2216yes—
55#1923C2213yes—
56#1923C2211yescharacteristic
57#1916C2213yes—
58#1916C2213yes—
59#1869C2213yes—
60#1869C2213yes—
61#1850C2216yes—
62#1850C2213yes—
63#1850C2213yes—
64#1842C2213yes—
65#1797C2213yes—
66#1797C2213yes—
67#1771C2216yes—
68#1770C2213yes—
69#1770C2213yes—
70#1767C2213yes—
71C4×C4⋊C8C2213yes—
72#1752C2216yes—
73C4×C22⋊C8C2213yes—
74C4×C22⋊C8C2213yes—
75#1744C2213yes—
76C22×#41C2×C411yescharacteristic
77C23×Q8C2×C411yescharacteristic
78C23×C8C2313yes—
79C23×C8C2313yes—
80#548—28no—
81C2×#124C2×C413yes—
82C2×C22⋊Q8C2×C413yes—
83C2×#121C2×C413yes—
84C4×#41—24no—
85C4×#41—24no—
86C2×C4×Q8—24no—
87C2×C4×Q8—24no—
88C2×C4×Q8C2×C411yescharacteristic
89C2×C4×Q8C2×C411yescharacteristic
90C2×C4×Q8C2×C411yescharacteristic
91C2×C4×Q8C2×C411yescharacteristic
92C2×C4×Q8C2×C413yes—
93C2×C4×Q8C2313yes—
94C2×C4×Q8C2311yescharacteristic
95C2×C4×D4C2×C413yes—
96C2×C4×D4C2×C413yes—
97C2×C4×D4C2×C413yes—
98C2×C4×D4C2313yes—
99C2×C4×D4C2313yes—
100C2×#117C2×C413yes—
101C22×C4⋊C4C2×C413yes—
102C22×C42C2313yes—
103C22×C42C2313yes—
104#446—212no—
105#446—212no—
106C8×Q8—24no—
107C8×Q8—24no—
108C8×Q8—24no—
109C8×Q8D418yescomplemented
110C8×D4—212no—
111C8×D4—212no—
112#432—212no—
113#431—212no—
114#431—212no—
115C2×C4⋊C8C2313yes—
116C2×C4⋊C8C2313yes—
117C2×C4⋊C8C2313yes—
118C2×C4⋊C8C2313yes—
119C2×C4⋊C8C2311yescharacteristic
120C2×C4⋊C8C2313yes—
121C2×C4⋊C8C2313yes—
122C2×C4⋊C8C2316yes—
123C2×C4⋊C8C2313yes—
124C2×C22⋊C8C2311yescharacteristic
125C2×C22⋊C8C2311yescharacteristic
126C2×C22⋊C8C2313yes—
127C2×C22⋊C8C2316yes—
128C2×C22⋊C8C2313yes—
129C2×C22⋊C8C2316yes—
130C2×C22⋊C8C2313yes—
131C2×C22⋊C8C2313yes—
132C2×#97C2316yes—
133C2×#97C2316yes—
134C2×C4×C8—212no—
135C2×C4×C8C2311yescharacteristic
136C2×C4×C8C2313yes—
137C2×C4×C8C2313yes—
138C2×C4×C8C2313yes—
139C2×C4×C8C2313yes—
140C2×C4×C8C2313yes—
141#391C2×C413yes—
142#391C2×C413yes—
143#390C2×C413yes—
144#390C2×C413yes—
145#389C2×C413yes—
146#389C2×C413yes—
147#388C2×C416yes—
148#388C2×C416yes—
149#388C2×C416yes—
150#387C2×C416yes—
151#386C2×C416yes—
152#385C2×C416yes—
153#382C2×C413yes—
154#380C2×C413yes—
155#380C2×C413yes—
156C4×C4⋊C4C2×C413yes—
157C4×C4⋊C4C2×C413yes—
158C4×C4⋊C4C2×C413yes—
159C4×C4⋊C4C2×C413yes—
160C4×C4⋊C4C2313yes—
161C4×C4⋊C4C2313yes—
162C4×C22⋊C4C2×C413yes—
163C4×C22⋊C4C2×C413yes—
164C4×C22⋊C4C2×C413yes—
165C4×C22⋊C4C2×C413yes—
166C4×C22⋊C4C2313yes—
167C4×C22⋊C4C2316yes—
168C4×C22⋊C4C2311yescharacteristic
169C43C2313yes—
170#336C2313yes—
171#336C2316yes—
172#336C2311yescharacteristic
173#336C2313yes—
174#336C2313yes—
175#336C2313yes—
176#336C2313yes—
177#336C2313yes—
178#336C2313yes—
179#336C2313yes—
180#336C2311yescharacteristic
181C2×#41—24no—
182C2×#41—24no—
183C2×#41M1618yes—
184C22×Q8—24no—
185C22×Q8—24no—
186C22×Q8—24no—
187C22×Q8C22×C413yes—
188C22×Q8C22×C411yescharacteristic
189C22×D4C22×C413yes—
190C22×D4C22×C413yes—
191C23×C4C22×C413yes—
192C23×C4C22×C413yes—
193C23×C4C22×C413yes—
194C23×C4C22×C413yes—
195C23×C4C2413yes—
196C23×C4C2411yescharacteristic
197C22×C8—26no—
198C22×C8—26no—
199C22×C8—26no—
200C22×C8—43no—
201C22×C8—43no—
202C22×C8—43no—
203C22×C8C2413yes—
204C22×C8C2411yescharacteristic
205C22×C8C2413yes—
206C22×C8C2411yescharacteristic
207C22×C8C2413yes—
208C22×C8C2411yescharacteristic
209C22×C8C2413yes—
210C22×C8C2411yescharacteristic
211#124—212no—
212C22⋊Q8—212no—
213#121—212no—
214C4×Q8—24no—
215C4×Q8—24no—
216C4×Q8—24no—
217C4×Q8—24no—
218C4×Q8—212no—
219C4×Q8—26no—
220C4×Q8—26no—
221C4×Q8C2×D414yes—
222C4×Q8#4114yes—
223C4×D4—212no—
224C4×D4—212no—
225C4×D4—212no—
226C4×D4—26no—
227C4×D4—26no—
228C4×D4—43no—
229C4×D4—43no—
230#117—212no—
231C2×C4⋊C4—212no—
232C2×C4⋊C4C22×C413yes—
233C2×C4⋊C4C22×C416yes—
234C2×C4⋊C4C22×C413yes—
235C2×C4⋊C4C22×C413yes—
236C2×C4⋊C4C22×C413yes—
237C2×C4⋊C4C22×C413yes—
238C2×C4⋊C4C22×C413yes—
239C2×C4⋊C4C22×C413yes—
240C2×C4⋊C4C22×C413yes—
241C2×C22⋊C4C22×C416yes—
242C2×C22⋊C4C22×C411yescharacteristic
243C2×C22⋊C4C22×C413yes—
244C2×C22⋊C4C22×C413yes—
245C2×C22⋊C4C22×C416yes—
246C2×C22⋊C4C22×C413yes—
247C2×C22⋊C4C22×C413yes—
248C2×C22⋊C4C22×C416yes—
249C2×C22⋊C4C22×C411yescharacteristic
250C2×C42—26no—
251C2×C42—26no—
252C2×C42—26no—
253C2×C42—26no—
254C2×C42C22×C413yes—
255C2×C42C22×C413yes—
256C2×C42C22×C413yes—
257C2×C42C22×C413yes—
258C2×C42C22×C413yes—
259C2×C42C22×C413yes—
260C2×C42C22×C413yes—
261C2×C42C22×C413yes—
262C2×C42C2413yes—
263C2×C42C2413yes—
264C2×C42C2411yescharacteristic
265C2×C42C2411yescharacteristic
266C2×C42C2413yes—
267C4⋊C8—26no—
268C4⋊C8—26no—
269C4⋊C8—22no—
270C4⋊C8—26no—
271C4⋊C8—26no—
272C4⋊C8—26no—
273C4⋊C8—43no—
274C4⋊C8—46no—
275C4⋊C8C2×D4112yes—
276C22⋊C8—22no—
277C22⋊C8—26no—
278C22⋊C8—212no—
279C22⋊C8—22no—
280C22⋊C8—43no—
281C22⋊C8—46no—
282C22⋊C8—43no—
283C22⋊C8—43no—
284#97—212no—
285#97—46no—
286C4×C8—22no—
287C4×C8—26no—
288C4×C8—26no—
289C4×C8—26no—
290C4×C8—26no—
291C4×C8C2×D4112yes—
292#95C22×C411yescharacteristic
293#95C22×C411yescharacteristic
294#95C22×C413yes—
295#95C22×C413yes—
296#95C22×C413yes—
297#95C22×C413yes—
298#95C22×C416yes—
299#95C22×C416yes—
300#95C22×C416yes—
301C24C23×C413yes—
302C24C23×C411yescharacteristic
303#41—232no—
304C2×Q8—26no—
305C2×Q8—26no—
306C2×Q8—26no—
307C2×Q8—416no—
308C2×Q8C4×D414yes—
309C2×Q8C2×M1614yes—
310C2×Q8#13114yes—
311C2×D4—26no—
312C2×D4—26no—
313C2×D4—26no—
314C2×D4—26no—
315C2×D4—43no—
316C2×D4C2×M16112yes—
317C22×C4—26no—
318C22×C4—26no—
319C22×C4—26no—
320C22×C4—26no—
321C22×C4—26no—
322C22×C4—26no—
323C22×C4—26no—
324C22×C4—26no—
325C22×C4—26no—
326C22×C4—21no—
327C22×C4—23no—
328C22×C4—21no—
329C22×C4—23no—
330C22×C4—21no—
331C22×C4—23no—
332C22×C4—21no—
333C22×C4—23no—
334C22×C4—23no—
335C22×C4—23no—
336C22×C4—21no—
337C22×C4—21no—
338C22×C4—43no—
339C22×C4—43no—
340C22×C4C2×M16112yes—
341C22×C4C23×C413yes—
342C22×C4C23×C413yes—
343C22×C4C23×C411yescharacteristic
344C22×C4C23×C413yes—
345C22×C4C23×C413yes—
346C22×C4C23×C413yes—
347C22×C4C23×C411yescharacteristic
348C22×C4C23×C413yes—
349C22×C4C23×C411yescharacteristic
350C22×C4C23×C411yescharacteristic
351C22×C4C23×C411yescharacteristic
352C22×C4C23×C413yes—
353C22×C4C2511yescharacteristic
354C2×C8—21no—
355C2×C8—23no—
356C2×C8—23no—
357C2×C8—21no—
358C2×C8—21no—
359C2×C8—21no—
360C2×C8—21no—
361C2×C8—23no—
362C2×C8—23no—
363C2×C8—21no—
364C2×C8—23no—
365C2×C8—21no—
366C2×C8—21no—
367C2×C8—23no—
368C2×C8—21no—
369C2×C8—21no—
370C2×C8—23no—
371C2×C8—23no—
372C2×C8—23no—
373C2×C8—23no—
374C2×C8—23no—
375C2×C8—412no—
376C2×C8—412no—
377C2×C8C22×D412yes—
378C2×C8C22×D416yes—
379C2×C82−1+412yes—
380C4⋊C4—26no—
381C4⋊C4—212no—
382C4⋊C4—26no—
383C4⋊C4—26no—
384C4⋊C4—26no—
385C4⋊C4—26no—
386C4⋊C4—26no—
387C4⋊C4—43no—
388C4⋊C4C4×D4112yes—
389C22⋊C4—26no—
390C22⋊C4—212no—
391C22⋊C4—212no—
392C22⋊C4—22no—
393C22⋊C4—26no—
394C22⋊C4—46no—
395C22⋊C4—43no—
396C22⋊C4—43no—
397C22⋊C4#13114yes—
398C42—26no—
399C42—26no—
400C42—26no—
401C42—26no—
402C42—26no—
403C42—26no—
404C42—26no—
405C42—21no—
406C42—23no—
407C42—21no—
408C42—23no—
409C42—21no—
410C42—23no—
411C42—21no—
412C42—23no—
413C42—43no—
414C42C22×D416yes—
415C42C2×#4116yes—
416C23—23no—
417C23—21no—
418C23—21no—
419C23—23no—
420C23—21no—
421C23—23no—
422C23—21no—
423C23—23no—
424C23—21no—
425C23—23no—
426C23—23no—
427C23—21no—
428C23C22×M1616yes—
429C23C2×#13112yes—
430C23C24×C411yescharacteristic
431Q8—412no—
432Q8#435116yes—
433D4—224no—
434D4—412no—
435C2×C4—21no—
436C2×C4—21no—
437C2×C4—21no—
438C2×C4—21no—
439C2×C4—23no—
440C2×C4—21no—
441C2×C4—23no—
442C2×C4—21no—
443C2×C4—224no—
444C2×C4—21no—
445C2×C4—23no—
446C2×C4—23no—
447C2×C4—23no—
448C2×C4—23no—
449C2×C4—23no—
450C2×C4—21no—
451C2×C4—23no—
452C2×C4—21no—
453C2×C4—21no—
454C2×C4—23no—
455C2×C4—21no—
456C2×C4—23no—
457C2×C4—23no—
458C2×C4—23no—
459C2×C4—23no—
460C2×C4—23no—
461C2×C4—23no—
462C2×C4—21no—
463C2×C4—23no—
464C2×C4—23no—
465C2×C4—412no—
466C2×C4—412no—
467C2×C4—412no—
468C2×C4—43no—
469C2×C4—41no—
470C2×C4—43no—
471C2×C4—41no—
472C2×C4—41no—
473C2×C4—43no—
474C2×C4—41no—
475C2×C4—43no—
476C2×C4C2×C4×D416yes—
477C2×C4C2×C4×D416yes—
478C2×C4C22×M1612yes—
479C2×C4C22×M1616yes—
480C2×C4C2×#13112yes—
481C2×C4C2×#13112yes—
482C2×C4C2×#13116yes—
483C2×C4C23×D411yescharacteristic
484C2×C4C22×#4111yescharacteristic
485C2×C4C2×2−1+411yescharacteristic
486C2×C4#58511yescharacteristic
487C2×C4#58511yescharacteristic
488C2×C4#58511yescharacteristic
489C8—22no—
490C8—22no—
491C8—26no—
492C8—43no—
493C8—43no—
494C8—41no—
495C8—41no—
496C8—43no—
497C22—22no—
498C22—26no—
499C22—22no—
500C22—26no—
501C22—26no—
502C22—22no—
503C22—41no—
504C22—43no—
505C22—43no—
506C22—41no—
507C22—41no—
508C22—43no—
509C22—43no—
510C22—41no—
511C22C22×C4×D411yescharacteristic
512C22#342811yescharacteristic
513C22C4×2−1+411yescharacteristic
514C22C23×M1611yescharacteristic
515C22C22×#13111yescharacteristic
516C22C22×#13111yescharacteristic
517C22#357311yescharacteristic
518C4—22no—
519C4—22no—
520C4—22no—
521C4—26no—
522C4—43no—
523C4—43no—
524C4—41no—
525C4—43no—
526C4—43no—
527C4—41no—
528C4—43no—
529C4C2×#435112yes—
530C4#348711yescharacteristic
531C4#348911yescharacteristic
532C4#351811yescharacteristic
533C4#352411yescharacteristic
534C2—42no—
535C2—46no—
536C2#3429311yescharacteristic
537C2C22×#43511yescharacteristic
538C2D4×#13111yescharacteristic
539C2#6027811yescharacteristic
540C2#6030111yescharacteristic
541C2#6033111yescharacteristic
542C2#6034311yescharacteristic
subgroupG/H
center, Frattini, cosocleC22×C4C25
commutator, socleC23C24×C4
prime radicalC2×#548C4
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C22×C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#2984C2semidirect
2. C2×C8×#41C2semidirect
4. #23209C2semidirect
5. #23084C2semidirect
6. #23054C2semidirect
7. #22797C2semidirect
8. Q8×C4⋊C8C2semidirect
9. #22530C2semidirect
10. Q8×C22⋊C8C2semidirect
11. #22212C2semidirect
12. #22135C2semidirect
13. #21597C2semidirect
14. #21101C2semidirect
15. #20572C2semidirect
109. C8×Q8D4semidirect

Maximal subgroups

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

subgroupindex
C2×#29842
C2×C8×#412
C4×#5482
#232092
#230842
#230542
#227972
Q8×C4⋊C82
#225302
Q8×C22⋊C82
#222122
#221352
#215972
#211012
#205722