Finite Group Databasebeta

← #6384493

Subgroups — #6384493

Order 512.

2198 conjugacy classes of subgroups.

The subgroup lattice has 554 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C8×#41C211yesmaximal, characteristic, complemented
2C2×Q8×M16C211yesmaximal, characteristic, complemented
3C4×#548C211yesmaximal, characteristic
4#23209C213yesmaximal, complemented
5#23054C213yesmaximal, complemented
6#22986C213yesmaximal, complemented
7#22749C213yesmaximal, complemented
8#22734C211yesmaximal, characteristic, complemented
9#22530C211yesmaximal, characteristic, complemented
10Q8×C22⋊C8C211yesmaximal, characteristic, complemented
11#21633C213yesmaximal, complemented
12#21597C213yesmaximal, complemented
13#21486C213yesmaximal, complemented
14Q8×#97C211yesmaximal, characteristic, complemented
15C4×#435C213yesmaximal, complemented
16C2×#548C411yescharacteristic
17C2×C4×#41C2211yescharacteristic
18C22×C4×Q8C2211yescharacteristic
19C8×#41—28no—
20Q8×M16—28no—
21C2×#446C2213yes—
22C2×#446C2213yes—
23C2×C8×Q8C2211yescharacteristic
24C2×C8×Q8C2211yescharacteristic
25C2×C8×Q8C2211yescharacteristic
26C2×C8×Q8C2211yescharacteristic
27C2×#435C2213yes—
28C2×#435C2213yes—
29C2×C8×D4C2213yes—
30C2×C8×D4C2213yes—
31C2×#431C2213yes—
32C2×#423C2213yes—
33C2×C4×M16C2213yes—
34C22×C4×C8C2213yes—
35#2367C413yes—
36#2362C413yes—
37#2361C413yes—
38#2352C413yes—
39#2351C411yescharacteristic
40#2341C411yescharacteristic
41Q8×C22⋊C4C411yescharacteristic
42C4×#124C2213yes—
43C4×C22⋊Q8C2213yes—
44C4×#121C2213yes—
45C42×Q8C2211yescharacteristic
46C42×D4C2213yes—
47#1984C2213yes—
48#1984C2213yes—
49#1954C2213yes—
50#1954C2213yes—
51#1939C2213yes—
52#1939C2213yes—
53#1924C2216yes—
54#1924C2216yes—
55#1924C2216yes—
56#1916C2216yes—
57#1870C2216yes—
58#1850C2216yes—
59#1831C2213yes—
60#1797C2213yes—
61#1797C2213yes—
62#1771C2213yes—
63#1771C2213yes—
64#1770C2213yes—
65#1770C2213yes—
66#1767C2213yes—
67C4×C4⋊C8C2213yes—
68#1753C2211yescharacteristic
69#1753C2213yes—
70#1752C2213yes—
71#1752C2213yes—
72#1752C2216yes—
73C4×C22⋊C8C2213yes—
74C4×C22⋊C8C2213yes—
75C4×#97C2213yes—
76C22×#41C2×C411yescharacteristic
77C23×Q8C2×C411yescharacteristic
78C22×M16C2313yes—
79C22×M16C2311yescharacteristic
80C23×C8C2313yes—
81C23×C8C2313yes—
82#548—28no—
83C2×#124C2×C413yes—
84C2×C22⋊Q8C2×C413yes—
85C2×#121C2×C413yes—
86C4×#41—24no—
87C4×#41—24no—
88C2×C4×Q8—24no—
89C2×C4×Q8—24no—
90C2×C4×Q8C2×C411yescharacteristic
91C2×C4×Q8C2×C411yescharacteristic
92C2×C4×Q8C2×C411yescharacteristic
93C2×C4×Q8C2×C411yescharacteristic
94C2×C4×Q8C2×C413yes—
95C2×C4×Q8C2313yes—
96C2×C4×Q8C2311yescharacteristic
97C2×C4×D4C2×C413yes—
98C2×C4×D4C2×C413yes—
99C2×C4×D4C2×C413yes—
100C2×C4×D4C2313yes—
101C2×C4×D4C2313yes—
102C2×#117C2×C413yes—
103C22×C4⋊C4C2×C413yes—
104C22×C42C2313yes—
105C22×C42C2313yes—
106#446—212no—
107#446—212no—
108C8×Q8—24no—
109C8×Q8—24no—
110C8×Q8—24no—
111C8×Q8D418yescomplemented
112#435—212no—
113#435—212no—
114C8×D4—212no—
115C8×D4—212no—
116#431—212no—
117#423—212no—
118C2×C4⋊C8C2313yes—
119C2×C4⋊C8C2313yes—
120C2×C4⋊C8C2313yes—
121C2×C4⋊C8C2313yes—
122C2×C4⋊C8C2316yes—
123C2×C4⋊C8C2313yes—
124C2×C4⋊C8C2313yes—
125C2×C22⋊C8C2313yes—
126C2×C22⋊C8C2316yes—
127C2×C22⋊C8C2313yes—
128C2×C22⋊C8C2313yes—
129C2×C22⋊C8C2316yes—
130C2×C22⋊C8C2311yescharacteristic
131C4×M16—212no—
132C2×#97C2313yes—
133C2×#97C2313yes—
134C2×#97C2313yes—
135C2×#97C2313yes—
136C2×#97C2311yescharacteristic
137C2×#97C2313yes—
138C2×C4×C8—212no—
139C2×C4×C8C2311yescharacteristic
140C2×C4×C8C2313yes—
141C2×C4×C8C2313yes—
142C2×C4×C8C2313yes—
143C2×C4×C8C2313yes—
144C2×C4×C8C2313yes—
145#391C2×C413yes—
146#391C2×C413yes—
147#390C2×C413yes—
148#390C2×C413yes—
149#389C2×C413yes—
150#389C2×C413yes—
151#388C2×C416yes—
152#388C2×C416yes—
153#388C2×C416yes—
154#387C2×C416yes—
155#386C2×C416yes—
156#385C2×C416yes—
157#382C2×C413yes—
158#380C2×C413yes—
159#380C2×C413yes—
160C4×C4⋊C4C2×C413yes—
161C4×C4⋊C4C2×C413yes—
162C4×C4⋊C4C2×C413yes—
163C4×C4⋊C4C2×C413yes—
164C4×C4⋊C4C2313yes—
165C4×C4⋊C4C2313yes—
166C4×C22⋊C4C2×C413yes—
167C4×C22⋊C4C2×C413yes—
168C4×C22⋊C4C2×C413yes—
169C4×C22⋊C4C2×C413yes—
170C4×C22⋊C4C2316yes—
171C4×C22⋊C4C2313yes—
172C4×C22⋊C4C2311yescharacteristic
173C43C2313yes—
174#336C2313yes—
175#336C2316yes—
176#336C2313yes—
177#336C2316yes—
178#336C2313yes—
179#336C2316yes—
180#336C2313yes—
181#336C2311yescharacteristic
182#336C2311yescharacteristic
183C2×#41—24no—
184C2×#41—24no—
185C2×#41M1618yescomplemented
186C22×Q8—24no—
187C22×Q8—24no—
188C22×Q8—24no—
189C22×Q8C22×C413yes—
190C22×Q8C22×C411yescharacteristic
191C22×D4C22×C413yes—
192C22×D4C22×C413yes—
193C23×C4C22×C413yes—
194C23×C4C22×C413yes—
195C23×C4C22×C413yes—
196C23×C4C22×C413yes—
197C23×C4C2413yes—
198C23×C4C2411yescharacteristic
199C2×M16—26no—
200C2×M16—22no—
201C2×M16—26no—
202C2×M16—22no—
203C2×M16—43no—
204C2×M16#4114yescomplemented
205C22×C8—26no—
206C22×C8—26no—
207C22×C8—26no—
208C22×C8—26no—
209C22×C8—26no—
210C22×C8—43no—
211C22×C8C2413yes—
212C22×C8C2413yes—
213C22×C8C2413yes—
214C22×C8C2413yes—
215C22×C8C2411yescharacteristic
216C22×C8C2411yescharacteristic
217C22×C8C2411yescharacteristic
218C22×C8C2411yescharacteristic
219#124—212no—
220C22⋊Q8—212no—
221#121—212no—
222C4×Q8—24no—
223C4×Q8—24no—
224C4×Q8—24no—
225C4×Q8—24no—
226C4×Q8—212no—
227C4×Q8—26no—
228C4×Q8—26no—
229C4×Q8C2×D414yes—
230C4×Q8#4114yes—
231C4×D4—212no—
232C4×D4—26no—
233C4×D4—26no—
234C4×D4—212no—
235C4×D4—26no—
236C4×D4—26no—
237C4×D4—46no—
238#117—212no—
239C2×C4⋊C4—212no—
240C2×C4⋊C4C22×C413yes—
241C2×C4⋊C4C22×C416yes—
242C2×C4⋊C4C22×C413yes—
243C2×C4⋊C4C22×C413yes—
244C2×C4⋊C4C22×C413yes—
245C2×C4⋊C4C22×C413yes—
246C2×C4⋊C4C22×C413yes—
247C2×C4⋊C4C22×C413yes—
248C2×C4⋊C4C22×C413yes—
249C2×C22⋊C4C22×C416yes—
250C2×C22⋊C4C22×C411yescharacteristic
251C2×C22⋊C4C22×C411yescharacteristic
252C2×C22⋊C4C22×C413yes—
253C2×C22⋊C4C22×C413yes—
254C2×C22⋊C4C22×C413yes—
255C2×C22⋊C4C22×C416yes—
256C2×C22⋊C4C22×C413yes—
257C2×C22⋊C4C22×C416yes—
258C2×C42—26no—
259C2×C42—26no—
260C2×C42—26no—
261C2×C42—26no—
262C2×C42C22×C413yes—
263C2×C42C22×C413yes—
264C2×C42C22×C413yes—
265C2×C42C22×C413yes—
266C2×C42C22×C413yes—
267C2×C42C22×C413yes—
268C2×C42C22×C413yes—
269C2×C42C22×C413yes—
270C2×C42C2413yes—
271C2×C42C2413yes—
272C2×C42C2411yescharacteristic
273C2×C42C2411yescharacteristic
274C2×C42C2413yes—
275C4⋊C8—212no—
276C4⋊C8—26no—
277C4⋊C8—26no—
278C4⋊C8—26no—
279C4⋊C8—26no—
280C4⋊C8—26no—
281C4⋊C8C2×D4112yes—
282C22⋊C8—26no—
283C22⋊C8—212no—
284C22⋊C8—26no—
285C22⋊C8—26no—
286C22⋊C8—46no—
287C22⋊C8#4114yescomplemented
288#97—26no—
289#97—26no—
290#97—26no—
291#97—26no—
292#97—22no—
293#97—43no—
294C4×C8—22no—
295C4×C8—26no—
296C4×C8—26no—
297C4×C8—26no—
298C4×C8—26no—
299C4×C8C2×D4112yes—
300#95C22×C411yescharacteristic
301#95C22×C411yescharacteristic
302#95C22×C413yes—
303#95C22×C413yes—
304#95C22×C413yes—
305#95C22×C413yes—
306#95C22×C416yes—
307#95C22×C416yes—
308#95C22×C416yes—
309C24C23×C411yescharacteristic
310C24C23×C413yes—
311#41—232no—
312C2×Q8—26no—
313C2×Q8—26no—
314C2×Q8—26no—
315C2×Q8—416no—
316C2×Q8C4×D414yes—
317C2×Q8C2×M1614yes—
318C2×Q8#13114yes—
319C2×D4—26no—
320C2×D4—26no—
321C2×D4—26no—
322C2×D4—26no—
323C2×D4—43no—
324C2×D4C2×M16112yes—
325C22×C4—26no—
326C22×C4—26no—
327C22×C4—21no—
328C22×C4—21no—
329C22×C4—21no—
330C22×C4—21no—
331C22×C4—26no—
332C22×C4—26no—
333C22×C4—26no—
334C22×C4—26no—
335C22×C4—26no—
336C22×C4—23no—
337C22×C4—23no—
338C22×C4—23no—
339C22×C4—23no—
340C22×C4—23no—
341C22×C4—23no—
342C22×C4—26no—
343C22×C4—43no—
344C22×C4—43no—
345C22×C4—43no—
346C22×C4C2×M16112yes—
347C22×C4C23×C413yes—
348C22×C4C23×C413yes—
349C22×C4C23×C411yescharacteristic
350C22×C4C23×C411yescharacteristic
351C22×C4C23×C413yes—
352C22×C4C23×C413yes—
353C22×C4C23×C413yes—
354C22×C4C23×C413yes—
355C22×C4C23×C411yescharacteristic
356C22×C4C23×C411yescharacteristic
357C22×C4C23×C411yescharacteristic
358C22×C4C23×C413yes—
359C22×C4C2×#4112yes—
360C22×C4C2×#4112yes—
361C22×C4C2511yescharacteristic
362M16—28no—
363M16—412no—
364C2×C8—21no—
365C2×C8—21no—
366C2×C8—21no—
367C2×C8—23no—
368C2×C8—21no—
369C2×C8—21no—
370C2×C8—23no—
371C2×C8—21no—
372C2×C8—23no—
373C2×C8—23no—
374C2×C8—23no—
375C2×C8—23no—
376C2×C8—23no—
377C2×C8—23no—
378C2×C8—23no—
379C2×C8—23no—
380C2×C8—23no—
381C2×C8—412no—
382C2×C8—412no—
383C2×C8C22×D416yes—
384C2×C8C22×D412yes—
385C2×C8C2×#4112yes—
386C2×C8C2×#4112yes—
387C2×C8C2×#4112yes—
388C2×C8C2×#4112yes—
389C2×C82−1+412yes—
390C4⋊C4—26no—
391C4⋊C4—212no—
392C4⋊C4—26no—
393C4⋊C4—26no—
394C4⋊C4—26no—
395C4⋊C4—26no—
396C4⋊C4—26no—
397C4⋊C4—26no—
398C4⋊C4C4×D4112yes—
399C22⋊C4—26no—
400C22⋊C4—26no—
401C22⋊C4—212no—
402C22⋊C4—212no—
403C22⋊C4—26no—
404C22⋊C4—22no—
405C22⋊C4—22no—
406C22⋊C4—46no—
407C22⋊C4—43no—
408C42—26no—
409C42—26no—
410C42—26no—
411C42—26no—
412C42—26no—
413C42—26no—
414C42—21no—
415C42—23no—
416C42—23no—
417C42—23no—
418C42—21no—
419C42—23no—
420C42—21no—
421C42—21no—
422C42—43no—
423C42—43no—
424C42C22×D416yes—
425C42C2×#4116yes—
426C23—23no—
427C23—21no—
428C23—23no—
429C23—21no—
430C23—23no—
431C23—21no—
432C23—21no—
433C23—23no—
434C23—21no—
435C23—23no—
436C23—23no—
437C23—21no—
438C23C4×#4112yes—
439C23C22×M1616yes—
440C23C24×C411yescharacteristic
441Q8—412no—
442Q8#435116yescomplemented
443D4—224no—
444D4—412no—
445C2×C4—21no—
446C2×C4—21no—
447C2×C4—21no—
448C2×C4—21no—
449C2×C4—21no—
450C2×C4—23no—
451C2×C4—23no—
452C2×C4—21no—
453C2×C4—224no—
454C2×C4—22no—
455C2×C4—22no—
456C2×C4—22no—
457C2×C4—22no—
458C2×C4—23no—
459C2×C4—23no—
460C2×C4—23no—
461C2×C4—23no—
462C2×C4—23no—
463C2×C4—23no—
464C2×C4—21no—
465C2×C4—21no—
466C2×C4—23no—
467C2×C4—21no—
468C2×C4—23no—
469C2×C4—23no—
470C2×C4—21no—
471C2×C4—23no—
472C2×C4—23no—
473C2×C4—21no—
474C2×C4—23no—
475C2×C4—23no—
476C2×C4—21no—
477C2×C4—23no—
478C2×C4—23no—
479C2×C4—412no—
480C2×C4—412no—
481C2×C4—43no—
482C2×C4—43no—
483C2×C4—43no—
484C2×C4—43no—
485C2×C4—86no—
486C2×C4C2×C4×D416yes—
487C2×C4C2×C4×D416yes—
488C2×C4C4×#4112yes—
489C2×C4C22×M1612yes—
490C2×C4C22×M1616yes—
491C2×C4C2×#13116yes—
492C2×C4#56812yes—
493C2×C4C23×D411yescharacteristic
494C2×C4C22×#4111yescharacteristic
495C2×C4C22×#4111yescharacteristic
496C2×C4C22×#4111yescharacteristic
497C2×C4C2×2−1+411yescharacteristic
498C2×C4#58511yescharacteristic
499C8—22no—
500C8—22no—
501C8—26no—
502C8—22no—
503C8—43no—
504C8—43no—
505C8—43no—
506C8#54814yescomplemented
507C22—22no—
508C22—26no—
509C22—22no—
510C22—22no—
511C22—26no—
512C22—26no—
513C22—43no—
514C22—43no—
515C22—41no—
516C22—41no—
517C22—43no—
518C22—41no—
519C22—43no—
520C22—41no—
521C22C22×C4×D411yescharacteristic
522C22C2×C4×#4111yescharacteristic
523C22C4×2−1+411yescharacteristic
524C22C23×M1611yescharacteristic
525C22C22×#13111yescharacteristic
526C22C2×#56811yescharacteristic
527C22#357311yescharacteristic
528C4—22no—
529C4—22no—
530C4—22no—
531C4—26no—
532C4—43no—
533C4—43no—
534C4—43no—
535C4—41no—
536C4—41no—
537C4—43no—
538C4—43no—
539C4C2×#435112yes—
540C4C2×#54811yescharacteristic
541C4D4×#4111yescharacteristic
542C4#348011yescharacteristic
543C4#348211yescharacteristic
544C2—42no—
545C2—46no—
546C2C4×#54811yescharacteristic
547C2C22×#43511yescharacteristic
548C2#6014611yescharacteristic
549C2M16×#4111yescharacteristic
550C2#6021811yescharacteristic
551C2#6034311yescharacteristic
552C2#6034511yescharacteristic
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×C8×#41C2semidirect
2. C2×Q8×M16C2semidirect
4. #23209C2semidirect
5. #23054C2semidirect
6. #22986C2semidirect
7. #22749C2semidirect
8. #22734C2semidirect
9. #22530C2semidirect
10. Q8×C22⋊C8C2semidirect
11. #21633C2semidirect
12. #21597C2semidirect
13. #21486C2semidirect
14. Q8×#97C2semidirect
15. C4×#435C2semidirect
111. C8×Q8D4semidirect
185. C2×#41M16semidirect
204. C2×M16#41semidirect
287. C22⋊C8#41semidirect
442. Q8#435semidirect
506. C8#548semidirect

Maximal subgroups

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

subgroupindex
C2×C8×#412
C2×Q8×M162
C4×#5482
#232092
#230542
#229862
#227492
#227342
#225302
Q8×C22⋊C82
#216332
#215972
#214862
Q8×#972
C4×#4352