Finite Group Databasebeta

← #6400686

Subgroups — #6400686

Order 512.

2132 conjugacy classes of subgroups.

The subgroup lattice has 548 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2988C211yesmaximal, characteristic, complemented
2C2×Q8×M16C211yesmaximal, characteristic, complemented
3C4×#548C211yesmaximal, characteristic
4#23209C213yesmaximal, complemented
5#22986C213yesmaximal, complemented
6#22956C213yesmaximal, complemented
7Q8×C4⋊C8C211yesmaximal, characteristic, complemented
8#22648C213yesmaximal, complemented
9#22530C211yesmaximal, characteristic, complemented
10Q8×C22⋊C8C211yesmaximal, characteristic, complemented
11#22212C213yesmaximal, complemented
12#22104C213yesmaximal, complemented
13#21486C213yesmaximal, complemented
14#21102C213yesmaximal, complemented
15#20572C211yesmaximal, characteristic, complemented
16C2×#548C411yescharacteristic
17C2×C4×#41C2211yescharacteristic
18C22×C4×Q8C2211yescharacteristic
19#2988—28no—
20Q8×M16—28no—
21C2×#446C2213yes—
22C2×#446C2213yes—
23C2×C8×Q8C2211yescharacteristic
24C2×C8×Q8C2211yescharacteristic
25C2×C8×Q8C2211yescharacteristic
26C2×C8×Q8C2211yescharacteristic
27C2×#435C2216yes—
28C2×C8×D4C2213yes—
29C2×C8×D4C2213yes—
30C2×#431C2213yes—
31C2×#423C2213yes—
32C22×C4⋊C8C2213yes—
33C2×C4×M16C2213yes—
34#2367C413yes—
35#2362C413yes—
36#2361C413yes—
37#2352C413yes—
38#2351C411yescharacteristic
39#2341C411yescharacteristic
40Q8×C22⋊C4C411yescharacteristic
41C4×#124C2213yes—
42C4×C22⋊Q8C2213yes—
43C4×#121C2213yes—
44C42×Q8C2211yescharacteristic
45C42×D4C2213yes—
46#1984C2213yes—
47#1984C2213yes—
48#1954C2213yes—
49#1954C2213yes—
50#1940C2216yes—
51#1939C2213yes—
52#1939C2213yes—
53#1924C2216yes—
54#1924C2216yes—
55#1924C2213yes—
56#1924C2213yes—
57#1917C2213yes—
58#1917C2213yes—
59#1870C2213yes—
60#1870C2213yes—
61#1851C2211yescharacteristic
62#1851C2213yes—
63#1850C2216yes—
64#1850C2216yes—
65#1842C2213yes—
66#1797C2213yes—
67#1797C2213yes—
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
78C22×M16C2311yescharacteristic
79C22×M16C2313yes—
80C23×C8C2316yes—
81#548—28no—
82C2×#124C2×C413yes—
83C2×C22⋊Q8C2×C413yes—
84C2×#121C2×C413yes—
85C4×#41—24no—
86C4×#41—24no—
87C2×C4×Q8—24no—
88C2×C4×Q8—24no—
89C2×C4×Q8C2×C411yescharacteristic
90C2×C4×Q8C2×C411yescharacteristic
91C2×C4×Q8C2×C411yescharacteristic
92C2×C4×Q8C2×C411yescharacteristic
93C2×C4×Q8C2×C413yes—
94C2×C4×Q8C2313yes—
95C2×C4×Q8C2311yescharacteristic
96C2×C4×D4C2×C413yes—
97C2×C4×D4C2×C413yes—
98C2×C4×D4C2×C413yes—
99C2×C4×D4C2313yes—
100C2×C4×D4C2313yes—
101C2×#117C2×C413yes—
102C22×C4⋊C4C2×C413yes—
103C22×C42C2313yes—
104C22×C42C2313yes—
105#446—212no—
106#446—212no—
107C8×Q8—24no—
108C8×Q8—24no—
109C8×Q8—24no—
110C8×Q8D418yescomplemented
111#435—412no—
112C8×D4—212no—
113C8×D4—212no—
114#431—212no—
115#423—212no—
116C2×C4⋊C8—212no—
117C2×C4⋊C8C2313yes—
118C2×C4⋊C8C2313yes—
119C2×C4⋊C8C2313yes—
120C2×C4⋊C8C2316yes—
121C2×C4⋊C8C2313yes—
122C2×C4⋊C8C2311yescharacteristic
123C2×C4⋊C8C2311yescharacteristic
124C2×C4⋊C8C2313yes—
125C2×C4⋊C8C2316yes—
126C2×C4⋊C8C2313yes—
127C2×C22⋊C8C2313yes—
128C2×C22⋊C8C2311yescharacteristic
129C2×C22⋊C8C2313yes—
130C2×C22⋊C8C2313yes—
131C2×C22⋊C8C2313yes—
132C2×C22⋊C8C2316yes—
133C2×C22⋊C8C2313yes—
134C4×M16—212no—
135C2×#97C2313yes—
136C2×#97C2313yes—
137C2×#97C2316yes—
138C2×C4×C8C2313yes—
139C2×C4×C8C2313yes—
140C2×C4×C8C2313yes—
141C2×C4×C8C2313yes—
142#391C2×C413yes—
143#391C2×C413yes—
144#390C2×C413yes—
145#390C2×C413yes—
146#389C2×C413yes—
147#389C2×C413yes—
148#388C2×C416yes—
149#388C2×C416yes—
150#388C2×C416yes—
151#387C2×C416yes—
152#386C2×C416yes—
153#385C2×C416yes—
154#382C2×C413yes—
155#380C2×C413yes—
156#380C2×C413yes—
157C4×C4⋊C4C2×C413yes—
158C4×C4⋊C4C2×C413yes—
159C4×C4⋊C4C2×C413yes—
160C4×C4⋊C4C2×C413yes—
161C4×C4⋊C4C2313yes—
162C4×C4⋊C4C2313yes—
163C4×C22⋊C4C2×C413yes—
164C4×C22⋊C4C2×C413yes—
165C4×C22⋊C4C2×C413yes—
166C4×C22⋊C4C2×C413yes—
167C4×C22⋊C4C2313yes—
168C4×C22⋊C4C2316yes—
169C4×C22⋊C4C2311yescharacteristic
170C43C2313yes—
171#336C2313yes—
172#336C2316yes—
173#336C2313yes—
174#336C2313yes—
175#336C2313yes—
176#336C2311yescharacteristic
177#336C2311yescharacteristic
178#336C2313yes—
179#336C2313yes—
180#336C2313yes—
181#336C2313yes—
182C2×#41—24no—
183C2×#41—24no—
184C2×#41M1618yescomplemented
185C22×Q8—24no—
186C22×Q8—24no—
187C22×Q8—24no—
188C22×Q8C22×C413yes—
189C22×Q8C22×C411yescharacteristic
190C22×D4C22×C413yes—
191C22×D4C22×C413yes—
192C23×C4C22×C413yes—
193C23×C4C22×C413yes—
194C23×C4C22×C413yes—
195C23×C4C22×C413yes—
196C23×C4C2413yes—
197C23×C4C2411yescharacteristic
198C2×M16—22no—
199C2×M16—26no—
200C2×M16—22no—
201C2×M16—26no—
202C2×M16—22no—
203C2×M16—43no—
204C22×C8—212no—
205C22×C8—46no—
206C22×C8—46no—
207C22×C8C2413yes—
208C22×C8C2413yes—
209C22×C8C2411yescharacteristic
210C22×C8C2411yescharacteristic
211C22×C8C2413yes—
212C22×C8C2413yes—
213C22×C8C2411yescharacteristic
214C22×C8C2411yescharacteristic
215#124—212no—
216C22⋊Q8—212no—
217#121—212no—
218C4×Q8—24no—
219C4×Q8—24no—
220C4×Q8—24no—
221C4×Q8—24no—
222C4×Q8—212no—
223C4×Q8—26no—
224C4×Q8—26no—
225C4×Q8C2×D414yes—
226C4×Q8#4114yes—
227C4×D4—212no—
228C4×D4—26no—
229C4×D4—26no—
230C4×D4—212no—
231C4×D4—26no—
232C4×D4—26no—
233C4×D4—46no—
234#117—212no—
235C2×C4⋊C4—212no—
236C2×C4⋊C4C22×C413yes—
237C2×C4⋊C4C22×C416yes—
238C2×C4⋊C4C22×C413yes—
239C2×C4⋊C4C22×C413yes—
240C2×C4⋊C4C22×C413yes—
241C2×C4⋊C4C22×C413yes—
242C2×C4⋊C4C22×C413yes—
243C2×C4⋊C4C22×C413yes—
244C2×C4⋊C4C22×C413yes—
245C2×C22⋊C4C22×C416yes—
246C2×C22⋊C4C22×C411yescharacteristic
247C2×C22⋊C4C22×C411yescharacteristic
248C2×C22⋊C4C22×C413yes—
249C2×C22⋊C4C22×C413yes—
250C2×C22⋊C4C22×C413yes—
251C2×C22⋊C4C22×C413yes—
252C2×C22⋊C4C22×C416yes—
253C2×C22⋊C4C22×C416yes—
254C2×C42—26no—
255C2×C42—26no—
256C2×C42—26no—
257C2×C42—26no—
258C2×C42C22×C413yes—
259C2×C42C22×C413yes—
260C2×C42C22×C413yes—
261C2×C42C22×C413yes—
262C2×C42C22×C413yes—
263C2×C42C22×C413yes—
264C2×C42C22×C413yes—
265C2×C42C22×C413yes—
266C2×C42C2413yes—
267C2×C42C2413yes—
268C2×C42C2411yescharacteristic
269C2×C42C2411yescharacteristic
270C2×C42C2413yes—
271C4⋊C8—26no—
272C4⋊C8—212no—
273C4⋊C8—22no—
274C4⋊C8—22no—
275C4⋊C8—26no—
276C4⋊C8—26no—
277C4⋊C8—26no—
278C4⋊C8—43no—
279C4⋊C8—46no—
280C4⋊C8C2×D4112yes—
281C22⋊C8—26no—
282C22⋊C8—22no—
283C22⋊C8—26no—
284C22⋊C8—26no—
285C22⋊C8—26no—
286C22⋊C8—43no—
287C22⋊C8—46no—
288#97—26no—
289#97—26no—
290#97—46no—
291C4×C8—26no—
292C4×C8—26no—
293C4×C8—26no—
294C4×C8C2×D4112yes—
295#95C22×C411yescharacteristic
296#95C22×C411yescharacteristic
297#95C22×C413yes—
298#95C22×C413yes—
299#95C22×C413yes—
300#95C22×C413yes—
301#95C22×C416yes—
302#95C22×C416yes—
303#95C22×C416yes—
304C24C23×C411yescharacteristic
305C24C23×C413yes—
306#41—232no—
307C2×Q8—26no—
308C2×Q8—26no—
309C2×Q8—26no—
310C2×Q8—416no—
311C2×Q8C4×D414yes—
312C2×Q8C2×M1614yes—
313C2×Q8#13114yes—
314C2×D4—26no—
315C2×D4—26no—
316C2×D4—26no—
317C2×D4—26no—
318C2×D4—43no—
319C2×D4C2×M16112yes—
320C22×C4—26no—
321C22×C4—26no—
322C22×C4—21no—
323C22×C4—21no—
324C22×C4—21no—
325C22×C4—21no—
326C22×C4—26no—
327C22×C4—26no—
328C22×C4—26no—
329C22×C4—23no—
330C22×C4—23no—
331C22×C4—23no—
332C22×C4—23no—
333C22×C4—23no—
334C22×C4—23no—
335C22×C4—26no—
336C22×C4—26no—
337C22×C4—43no—
338C22×C4—43no—
339C22×C4—43no—
340C22×C4—43no—
341C22×C4C2×M16112yes—
342C22×C4C23×C413yes—
343C22×C4C23×C413yes—
344C22×C4C23×C411yescharacteristic
345C22×C4C23×C411yescharacteristic
346C22×C4C23×C413yes—
347C22×C4C23×C413yes—
348C22×C4C23×C413yes—
349C22×C4C23×C413yes—
350C22×C4C23×C411yescharacteristic
351C22×C4C23×C411yescharacteristic
352C22×C4C23×C411yescharacteristic
353C22×C4C23×C413yes—
354C22×C4H(5,𝔽2)12yes—
355C22×C42−1+412yes—
356C22×C4C2511yescharacteristic
357M16—28no—
358M16—412no—
359C2×C8—21no—
360C2×C8—23no—
361C2×C8—21no—
362C2×C8—21no—
363C2×C8—21no—
364C2×C8—23no—
365C2×C8—21no—
366C2×C8—23no—
367C2×C8—23no—
368C2×C8—21no—
369C2×C8—23no—
370C2×C8—23no—
371C2×C8—23no—
372C2×C8—23no—
373C2×C8—21no—
374C2×C8—21no—
375C2×C8—23no—
376C2×C8—21no—
377C2×C8—21no—
378C2×C8—23no—
379C2×C8—23no—
380C2×C8—812no—
381C2×C8C22×D412yes—
382C2×C8C22×D416yes—
383C2×C82−1+412yes—
384C4⋊C4—26no—
385C4⋊C4—212no—
386C4⋊C4—26no—
387C4⋊C4—26no—
388C4⋊C4—26no—
389C4⋊C4—26no—
390C4⋊C4—26no—
391C4⋊C4—43no—
392C4⋊C4C4×D4112yes—
393C22⋊C4—26no—
394C22⋊C4—26no—
395C22⋊C4—212no—
396C22⋊C4—212no—
397C22⋊C4—26no—
398C22⋊C4—22no—
399C22⋊C4—26no—
400C22⋊C4—22no—
401C22⋊C4—46no—
402C42—26no—
403C42—26no—
404C42—26no—
405C42—26no—
406C42—26no—
407C42—26no—
408C42—26no—
409C42—23no—
410C42—21no—
411C42—21no—
412C42—23no—
413C42—23no—
414C42—21no—
415C42—23no—
416C42—21no—
417C42—43no—
418C42C22×D416yes—
419C42C2×#4116yes—
420C23—23no—
421C23—21no—
422C23—23no—
423C23—23no—
424C23—21no—
425C23—21no—
426C23—21no—
427C23—23no—
428C23—21no—
429C23—23no—
430C23—23no—
431C23—21no—
432C23#52012yes—
433C23C22×M1616yes—
434C23C24×C411yescharacteristic
435Q8—412no—
436Q8#435116yescomplemented
437D4—224no—
438D4—412no—
439C2×C4—21no—
440C2×C4—21no—
441C2×C4—21no—
442C2×C4—21no—
443C2×C4—21no—
444C2×C4—21no—
445C2×C4—23no—
446C2×C4—23no—
447C2×C4—224no—
448C2×C4—22no—
449C2×C4—22no—
450C2×C4—22no—
451C2×C4—22no—
452C2×C4—21no—
453C2×C4—23no—
454C2×C4—23no—
455C2×C4—23no—
456C2×C4—23no—
457C2×C4—23no—
458C2×C4—23no—
459C2×C4—21no—
460C2×C4—23no—
461C2×C4—21no—
462C2×C4—23no—
463C2×C4—23no—
464C2×C4—23no—
465C2×C4—23no—
466C2×C4—23no—
467C2×C4—21no—
468C2×C4—23no—
469C2×C4—21no—
470C2×C4—23no—
471C2×C4—23no—
472C2×C4—21no—
473C2×C4—412no—
474C2×C4—43no—
475C2×C4—43no—
476C2×C4—43no—
477C2×C4—43no—
478C2×C4—86no—
479C2×C4—86no—
480C2×C4C2×C4×D416yes—
481C2×C4C2×C4×D416yes—
482C2×C4#52012yes—
483C2×C4C22×M1612yes—
484C2×C4C22×M1616yes—
485C2×C4C2×#13116yes—
486C2×C4#56812yes—
487C2×C4C23×D411yescharacteristic
488C2×C4C22×#4111yescharacteristic
489C2×C4C2×H(5,𝔽2)11yescharacteristic
490C2×C4C2×2−1+411yescharacteristic
491C2×C4C2×2−1+411yescharacteristic
492C2×C4#58511yescharacteristic
493C8—22no—
494C8—22no—
495C8—26no—
496C8—41no—
497C8—43no—
498C8—43no—
499C8—41no—
500C8—43no—
501C22—22no—
502C22—26no—
503C22—22no—
504C22—26no—
505C22—26no—
506C22—22no—
507C22—43no—
508C22—41no—
509C22—43no—
510C22—41no—
511C22—41no—
512C22—41no—
513C22—43no—
514C22—43no—
515C22C22×C4×D411yescharacteristic
516C22C2×#52011yescharacteristic
517C22C4×2−1+411yescharacteristic
518C22C23×M1611yescharacteristic
519C22C22×#13111yescharacteristic
520C22C2×#56811yescharacteristic
521C22#357311yescharacteristic
522C4—22no—
523C4—22no—
524C4—26no—
525C4—22no—
526C4—43no—
527C4—43no—
528C4—41no—
529C4—43no—
530C4—43no—
531C4—41no—
532C4—43no—
533C4C2×#435112yes—
534C4#348611yescharacteristic
535C4#348811yescharacteristic
536C4#351611yescharacteristic
537C4#352211yescharacteristic
538C2—42no—
539C2—46no—
540C2#3429411yescharacteristic
541C2C22×#43511yescharacteristic
542C2#6014711yescharacteristic
543C2#6026311yescharacteristic
544C2#6027411yescharacteristic
545C2#6033511yescharacteristic
546C2#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×#2988C2semidirect
2. C2×Q8×M16C2semidirect
4. #23209C2semidirect
5. #22986C2semidirect
6. #22956C2semidirect
7. Q8×C4⋊C8C2semidirect
8. #22648C2semidirect
9. #22530C2semidirect
10. Q8×C22⋊C8C2semidirect
11. #22212C2semidirect
12. #22104C2semidirect
13. #21486C2semidirect
14. #21102C2semidirect
15. #20572C2semidirect
110. C8×Q8D4semidirect
184. C2×#41M16semidirect
436. Q8#435semidirect

Maximal subgroups

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

subgroupindex
C2×#29882
C2×Q8×M162
C4×#5482
#232092
#229862
#229562
Q8×C4⋊C82
#226482
#225302
Q8×C22⋊C82
#222122
#221042
#214862
#211022
#205722