Finite Group Databasebeta

← #6399851

Subgroups — #6399851

Order 512.

2054 conjugacy classes of subgroups.

The subgroup lattice has 542 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2984C211yesmaximal, characteristic, complemented
2C22×C8×Q8C211yesmaximal, characteristic, complemented
3C4×D4×Q8C211yesmaximal, characteristic
4#23150C213yesmaximal, complemented
5#23051C213yesmaximal, complemented
6#22956C213yesmaximal, complemented
7#22750C213yesmaximal
8#22734C211yesmaximal, characteristic, complemented
9Q8×C22⋊C8C211yesmaximal, characteristic, complemented
10Q8×C22⋊C8C211yesmaximal, characteristic, complemented
11#22240C213yesmaximal, complemented
12#22100C213yesmaximal, complemented
13#21597C213yesmaximal, complemented
14#21101C213yesmaximal
15#20572C211yesmaximal, characteristic, complemented
16C2×D4×Q8C411yescharacteristic
17C22×C4×Q8C2211yescharacteristic
18C22×C4×Q8C2211yescharacteristic
19#2984—28no—
20C2×#446C2213yes—
21C2×#446C2213yes—
22C2×C8×Q8—28no—
23C2×C8×Q8C2213yes—
24C2×C8×Q8C2213yes—
25C2×C8×Q8C2211yescharacteristic
26C2×C8×Q8C2211yescharacteristic
27C2×C8×Q8C2211yescharacteristic
28C2×C8×Q8C2211yescharacteristic
29C2×#432C2213yes—
30C2×#431C2213yes—
31C22×C4⋊C8C2213yes—
32C22×C4×C8C2213yes—
33#2365C413yes—
34#2360C413yes—
35#2360C413yes—
36Q8×C4⋊C4C411yescharacteristic
37D4×C4⋊C4C413yes—
38Q8×C22⋊C4C411yescharacteristic
39Q8×C22⋊C4C411yescharacteristic
40C4×C4⋊Q8C2213yes—
41C4×C22⋊Q8C2213yes—
42C4×C22⋊Q8C2213yes—
43C42×Q8C2211yescharacteristic
44C42×D4C2213yes—
45#1984C2213yes—
46#1984C2213yes—
47#1984C2213yes—
48#1984C2213yes—
49#1940C2216yes—
50#1939C2213yes—
51#1939C2213yes—
52#1924C2216yes—
53#1923C2213yes—
54#1923C2213yes—
55#1917C2216yes—
56#1917C2216yes—
57#1916C2216yes—
58#1916C2213yes—
59#1916C2213yes—
60#1869C2211yescharacteristic
61#1869C2213yes—
62#1850C2213yes—
63#1850C2213yes—
64#1831C2213yes—
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—
76C23×Q8C2×C411yescharacteristic
77C23×Q8C2×C411yescharacteristic
78C23×C8C2311yescharacteristic
79C23×C8C2313yes—
80D4×Q8—28no—
81C2×C4⋊Q8C2×C413yes—
82C2×C22⋊Q8C2×C413yes—
83C2×C22⋊Q8C2×C413yes—
84C2×C4×Q8—24no—
85C2×C4×Q8—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×Q8C2×C413yes—
94C2×C4×Q8C2313yes—
95C2×C4×Q8C2313yes—
96C2×C4×Q8C2311yescharacteristic
97C2×C4×D4C2×C413yes—
98C2×C4×D4C2×C413yes—
99C2×C4×D4C2311yescharacteristic
100C22×C4⋊C4C2×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×Q8—212no—
110C8×Q8—212no—
111C8×Q8D418yescomplemented
112#432—212no—
113#431—212no—
114C2×C4⋊C8—212no—
115C2×C4⋊C8C2313yes—
116C2×C4⋊C8C2313yes—
117C2×C4⋊C8C2313yes—
118C2×C4⋊C8C2313yes—
119C2×C4⋊C8C2313yes—
120C2×C4⋊C8C2313yes—
121C2×C4⋊C8C2316yes—
122C2×C4⋊C8C2313yes—
123C2×C4⋊C8C2313yes—
124C2×C22⋊C8C2313yes—
125C2×C22⋊C8C2311yescharacteristic
126C2×C22⋊C8C2311yescharacteristic
127C2×C22⋊C8C2313yes—
128C2×C22⋊C8C2311yescharacteristic
129C2×C22⋊C8C2313yes—
130C2×#97C2316yes—
131C2×#97C2316yes—
132C2×C4×C8—212no—
133C2×C4×C8C2313yes—
134C2×C4×C8C2313yes—
135C2×C4×C8C2313yes—
136C2×C4×C8C2313yes—
137C2×C4×C8C2313yes—
138C2×C4×C8C2313yes—
139#391C2×C413yes—
140#391C2×C413yes—
141#391C2×C413yes—
142#391C2×C413yes—
143#389C2×C416yes—
144#389C2×C416yes—
145#389C2×C413yes—
146#389C2×C413yes—
147#387C2×C416yes—
148#387C2×C416yes—
149#385C2×C416yes—
150#385C2×C416yes—
151#384C2×C413yes—
152#380C2×C413yes—
153#380C2×C413yes—
154C4×C4⋊C4C2×C413yes—
155C4×C4⋊C4C2×C413yes—
156C4×C4⋊C4C2×C413yes—
157C4×C4⋊C4C2×C413yes—
158C4×C4⋊C4C2313yes—
159C4×C4⋊C4C2316yes—
160C4×C4⋊C4C2313yes—
161C4×C22⋊C4C2×C413yes—
162C4×C22⋊C4C2×C413yes—
163C4×C22⋊C4C2×C413yes—
164C4×C22⋊C4C2×C413yes—
165C4×C22⋊C4C2313yes—
166C4×C22⋊C4C2313yes—
167C43C2313yes—
168#336C2316yes—
169#336C2316yes—
170#336C2313yes—
171#336C2316yes—
172#336C2313yes—
173#336C2313yes—
174#336C2313yes—
175#336C2313yes—
176#336C2311yescharacteristic
177#336C2311yescharacteristic
178#336C2313yes—
179#336C2313yes—
180#336C2313yes—
181C22×Q8—24no—
182C22×Q8—24no—
183C22×Q8—24no—
184C22×Q8—24no—
185C22×Q8—24no—
186C22×Q8M1618yes—
187C22×Q8C22×C413yes—
188C22×Q8C22×C413yes—
189C22×Q8C22×C411yescharacteristic
190C22×D4C22×C411yescharacteristic
191C23×C4C22×C413yes—
192C23×C4C22×C413yes—
193C23×C4C22×C413yes—
194C23×C4C22×C413yes—
195C23×C4C2411yescharacteristic
196C23×C4C2411yescharacteristic
197C22×C8—22no—
198C22×C8—22no—
199C22×C8—22no—
200C22×C8—26no—
201C22×C8—26no—
202C22×C8—43no—
203C22×C8C2413yes—
204C22×C8C2411yescharacteristic
205C22×C8C2413yes—
206C22×C8C2413yes—
207C22×C8C2411yescharacteristic
208C22×C8C2411yescharacteristic
209C22×C8C2413yes—
210C22×C8C2411yescharacteristic
211C4⋊Q8—212no—
212C22⋊Q8—212no—
213C22⋊Q8—212no—
214C4×Q8—24no—
215C4×Q8—24no—
216C4×Q8—24no—
217C4×Q8—24no—
218C4×Q8—212no—
219C4×Q8—212no—
220C4×Q8—26no—
221C4×Q8—26no—
222C4×Q8—26no—
223C4×Q8—26no—
224C4×Q8C2×D414yes—
225C4×Q8#4114yes—
226C4×D4—212no—
227C4×D4—212no—
228C4×D4—22no—
229C4×D4—22no—
230C2×C4⋊C4—212no—
231C2×C4⋊C4—212no—
232C2×C4⋊C4C22×C413yes—
233C2×C4⋊C4C22×C413yes—
234C2×C4⋊C4C22×C416yes—
235C2×C4⋊C4C22×C416yes—
236C2×C4⋊C4C22×C413yes—
237C2×C4⋊C4C22×C413yes—
238C2×C4⋊C4C22×C411yescharacteristic
239C2×C4⋊C4C22×C413yes—
240C2×C4⋊C4C22×C413yes—
241C2×C4⋊C4C22×C416yes—
242C2×C4⋊C4C22×C416yes—
243C2×C4⋊C4C22×C413yes—
244C2×C22⋊C4C22×C413yes—
245C2×C22⋊C4C22×C413yes—
246C2×C22⋊C4C22×C413yes—
247C2×C22⋊C4C22×C413yes—
248C2×C22⋊C4C22×C411yescharacteristic
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×C42C2411yescharacteristic
263C2×C42C2413yes—
264C2×C42C2413yes—
265C2×C42C2413yes—
266C2×C42C2413yes—
267C4⋊C8—212no—
268C4⋊C8—26no—
269C4⋊C8—26no—
270C4⋊C8—26no—
271C4⋊C8—26no—
272C4⋊C8—26no—
273C4⋊C8—26no—
274C4⋊C8—43no—
275C4⋊C8C2×D4112yes—
276C22⋊C8—26no—
277C22⋊C8—22no—
278C22⋊C8—22no—
279C22⋊C8—26no—
280C22⋊C8—22no—
281C22⋊C8—43no—
282#97—212no—
283#97—46no—
284C4×C8—26no—
285C4×C8—26no—
286C4×C8—26no—
287C4×C8—26no—
288C4×C8—26no—
289C4×C8C2×D4112yes—
290#95C22×C413yes—
291#95C22×C413yes—
292#95C22×C413yes—
293#95C22×C413yes—
294#95C22×C413yes—
295#95C22×C413yes—
296#95C22×C416yes—
297#95C22×C416yes—
298#95C22×C416yes—
299C24C23×C411yescharacteristic
300C24C23×C411yescharacteristic
301C2×Q8—232no—
302C2×Q8—26no—
303C2×Q8—26no—
304C2×Q8—26no—
305C2×Q8—26no—
306C2×Q8—26no—
307C2×Q8—416no—
308C2×Q8C4×D414yes—
309C2×Q8C2×M16112yes—
310C2×Q8C2×M1614yes—
311C2×Q8#13114yes—
312C2×D4—22no—
313C2×D4—22no—
314C2×D4#13114yes—
315C22×C4—26no—
316C22×C4—26no—
317C22×C4—21no—
318C22×C4—21no—
319C22×C4—21no—
320C22×C4—21no—
321C22×C4—26no—
322C22×C4—26no—
323C22×C4—26no—
324C22×C4—26no—
325C22×C4—26no—
326C22×C4—26no—
327C22×C4—26no—
328C22×C4—21no—
329C22×C4—21no—
330C22×C4—21no—
331C22×C4—21no—
332C22×C4—21no—
333C22×C4—21no—
334C22×C4—43no—
335C22×C4—43no—
336C22×C4C2×M16112yes—
337C22×C4C23×C413yes—
338C22×C4C23×C411yescharacteristic
339C22×C4C23×C411yescharacteristic
340C22×C4C23×C413yes—
341C22×C4C23×C413yes—
342C22×C4C23×C413yes—
343C22×C4C23×C413yes—
344C22×C4C23×C413yes—
345C22×C4C23×C411yescharacteristic
346C22×C4C23×C413yes—
347C22×C4C23×C411yescharacteristic
348C22×C4C23×C413yes—
349C22×C42−1+412yes—
350C22×C42−1+412yes—
351C22×C4C2511yescharacteristic
352C2×C8—28no—
353C2×C8—21no—
354C2×C8—21no—
355C2×C8—21no—
356C2×C8—21no—
357C2×C8—23no—
358C2×C8—23no—
359C2×C8—21no—
360C2×C8—23no—
361C2×C8—23no—
362C2×C8—21no—
363C2×C8—23no—
364C2×C8—23no—
365C2×C8—21no—
366C2×C8—23no—
367C2×C8—21no—
368C2×C8—21no—
369C2×C8—23no—
370C2×C8—23no—
371C2×C8—23no—
372C2×C8—23no—
373C2×C8—21no—
374C2×C8—412no—
375C2×C8C22×D412yes—
376C2×C8C22×D416yes—
377C2×C82−1+412yes—
378C4⋊C4—212no—
379C4⋊C4—26no—
380C4⋊C4—26no—
381C4⋊C4—212no—
382C4⋊C4—22no—
383C4⋊C4—26no—
384C4⋊C4—26no—
385C4⋊C4—212no—
386C4⋊C4—26no—
387C4⋊C4—26no—
388C4⋊C4—46no—
389C4⋊C4C4×D4112yes—
390C22⋊C4—22no—
391C22⋊C4—26no—
392C22⋊C4—26no—
393C22⋊C4—26no—
394C22⋊C4—22no—
395C22⋊C4—26no—
396C42—26no—
397C42—26no—
398C42—26no—
399C42—26no—
400C42—26no—
401C42—26no—
402C42—26no—
403C42—23no—
404C42—23no—
405C42—23no—
406C42—23no—
407C42—23no—
408C42—21no—
409C42—21no—
410C42—23no—
411C42—43no—
412C42C22×D416yes—
413C42C2×#4116yes—
414C23—21no—
415C23—21no—
416C23—21no—
417C23—21no—
418C23—21no—
419C23—21no—
420C23—21no—
421C23—21no—
422C23—21no—
423C23—21no—
424C23#51912yes—
425C23C22×M1612yes—
426C23C2×#13112yes—
427C23C2×#13112yes—
428C23C24×C411yescharacteristic
429Q8—224no—
430Q8—412no—
431Q8#435116yes—
432D4—28no—
433C2×C4—21no—
434C2×C4—23no—
435C2×C4—21no—
436C2×C4—21no—
437C2×C4—21no—
438C2×C4—23no—
439C2×C4—23no—
440C2×C4—23no—
441C2×C4—224no—
442C2×C4—22no—
443C2×C4—22no—
444C2×C4—22no—
445C2×C4—22no—
446C2×C4—21no—
447C2×C4—23no—
448C2×C4—23no—
449C2×C4—23no—
450C2×C4—21no—
451C2×C4—21no—
452C2×C4—23no—
453C2×C4—23no—
454C2×C4—21no—
455C2×C4—23no—
456C2×C4—23no—
457C2×C4—21no—
458C2×C4—23no—
459C2×C4—23no—
460C2×C4—23no—
461C2×C4—23no—
462C2×C4—23no—
463C2×C4—21no—
464C2×C4—23no—
465C2×C4—23no—
466C2×C4—23no—
467C2×C4—412no—
468C2×C4—412no—
469C2×C4—412no—
470C2×C4—41no—
471C2×C4—41no—
472C2×C4—41no—
473C2×C4—41no—
474C2×C4C2×C4×D416yes—
475C2×C4C2×C4×D416yes—
476C2×C4#51912yes—
477C2×C4C22×M1616yes—
478C2×C4C22×M1616yes—
479C2×C4C2×#13112yes—
480C2×C4C2×#13116yes—
481C2×C4C23×D411yescharacteristic
482C2×C4C22×#4111yescharacteristic
483C2×C4C2×2−1+411yescharacteristic
484C2×C4C2×2−1+411yescharacteristic
485C2×C4C2×2−1+411yescharacteristic
486C2×C4#58511yescharacteristic
487C8—22no—
488C8—22no—
489C8—26no—
490C8—43no—
491C8—41no—
492C8—43no—
493C8—43no—
494C8—41no—
495C22—22no—
496C22—22no—
497C22—22no—
498C22—22no—
499C22—22no—
500C22—22no—
501C22—22no—
502C22—22no—
503C22—22no—
504C22—41no—
505C22—41no—
506C22—41no—
507C22—41no—
508C22#298414yescomplemented
509C22C22×C4×D411yescharacteristic
510C22C2×#51911yescharacteristic
511C22C4×2−1+411yescharacteristic
512C22C23×M1611yescharacteristic
513C22C22×#13111yescharacteristic
514C22C22×#13111yescharacteristic
515C22#357311yescharacteristic
516C4—22no—
517C4—26no—
518C4—26no—
519C4—22no—
520C4—41no—
521C4—43no—
522C4—43no—
523C4—43no—
524C4—43no—
525C4—43no—
526C4—41no—
527C4C2×#435112yes—
528C4#348611yescharacteristic
529C4#348911yescharacteristic
530C4#351611yescharacteristic
531C4#352311yescharacteristic
532C2—24no—
533C2—42no—
534C2#3429111yescharacteristic
535C2C22×#43511yescharacteristic
536C2D4×#13111yescharacteristic
537C2C2×#298411yescharacteristic
538C2#6025311yescharacteristic
539C2#6033111yescharacteristic
540C2#6033311yescharacteristic
subgroupG/H
center, Frattini, cosocleC22×C4C25
commutator, socleC23C24×C4
prime radicalC22×D4C22×C4
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. C22×C8×Q8C2semidirect
4. #23150C2semidirect
5. #23051C2semidirect
6. #22956C2semidirect
8. #22734C2semidirect
9. Q8×C22⋊C8C2semidirect
10. Q8×C22⋊C8C2semidirect
11. #22240C2semidirect
12. #22100C2semidirect
13. #21597C2semidirect
15. #20572C2semidirect
111. C8×Q8D4semidirect
508. C22#2984semidirect

Maximal subgroups

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

subgroupindex
C2×#29842
C22×C8×Q82
C4×D4×Q82
#231502
#230512
#229562
#227502
#227342
Q8×C22⋊C82
#222402
#221002
#215972
#211012
#205722