Finite Group Databasebeta

← #4416784

Subgroups — #4416784

Order 512.

3716 conjugacy classes of subgroups.

The subgroup lattice has 525 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2848C211yesmaximal, characteristic, complemented
2C2×C4×#120C211yesmaximal, characteristic
3#16894C216yesmaximal, complemented
4#16893C216yesmaximal, complemented
5#15183C213yesmaximal, complemented
6#15181C213yesmaximal, complemented
7#15179C213yesmaximal, complemented
8#15177C213yesmaximal, complemented
9#14680C211yesmaximal, characteristic, complemented
10#14673C212yesmaximal, complemented
11#14657C212yesmaximal
12C25×C4C2211yescharacteristic
13C22×C22⋊Q8C2213yes—
14C22×C22⋊Q8C2211yescharacteristic, complemented
15C22×#120C2211yescharacteristic
16C22×C4×D4C2213yes—
17#2848—28no—
18C4×#120—28no—
19C2×#391C2213yescomplemented
20C2×#389C2213yescomplemented
21C2×#386C2213yes—
22C2×#385C2216yes—
23C2×#385C2213yes—
24C2×#380C2213yes—
25C2×#380C2213yes—
26C2×#380C2213yes—
27C2×#380C2213yescomplemented
28C2×#379C2212yes—
29C2×#379C2212yes—
30C2×#379C2211yescharacteristic
31C2×C4×C22⋊C4C2213yes—
32#1445C2216yescomplemented
33#1444C2216yescomplemented
34#1443C2216yescomplemented
35#1442C2216yescomplemented
36#1442C2216yescomplemented
37#1441C2216yescomplemented
38#1440C2216yes—
39#1440C2216yes—
40#1439C2216yes—
41#1439C2216yes—
42#1439C2216yes—
43#1439C2216yes—
44#1439C2216yescomplemented
45#1439C2216yescomplemented
46#1438C2216yes—
47#1438C2216yes—
48C4⋊#95C2213yescomplemented
49C4⋊#95C2213yescomplemented
50C22⋊#95C2216yes—
51C26C2311yescharacteristic
52C23×Q8C2311yescharacteristic
53C23×D4C2313yes—
54C24×C4—46no—
55C24×C4—46no—
56C24×C4D418yes—
57C24×C4C2311yescharacteristic
58C24×C4C2313yes—
59C24×C4C2313yes—
60C2×C22⋊Q8—212no—
61C2×C22⋊Q8—46no—
62C2×C22⋊Q8—46no—
63C2×C22⋊Q8—46no—
64C2×#120—24no—
65C2×#120—24no—
66C2×#120Q818yescomplemented
67C2×C4×D4—212no—
68C2×C4×D4—46no—
69C2×C4×D4—46no—
70C22×C4⋊C4C2313yes—
71C22×C4⋊C4C2311yescharacteristic
72C22×C4⋊C4C2313yes—
73C22×C4⋊C4C2313yes—
74C22×C4⋊C4C2316yes—
75C22×C4⋊C4C2312yes—
76C22×C4⋊C4C2316yes—
77C22×C22⋊C4C2313yes—
78C22×C22⋊C4C2313yes—
79C22×C22⋊C4C2313yes—
80C22×C22⋊C4C2316yes—
81C22×C22⋊C4C2316yes—
82C22×C22⋊C4C2316yes—
83C22×C22⋊C4C2316yes—
84C22×C22⋊C4C2313yes—
85C22×C22⋊C4C2312yes—
86C22×C22⋊C4C2316yes—
87C22×C42C2313yes—
88#391—46no—
89#389—46no—
90#386—212no—
91#385—224no—
92#385—212no—
93#380—212no—
94#380—46no—
95#380—46no—
96#380—46no—
97#379—28no—
98#379—24no—
99#379—28no—
100C4×C22⋊C4—212no—
101C2×#95C2316yes—
102C2×#95C2313yes—
103C2×#95C2313yes—
104C2×#95C2316yes—
105C2×#95C2316yes—
106C2×#95C2316yes—
107C2×#95C2316yes—
108C2×#95C2316yes—
109C2×#95C2316yes—
110C2×#95C2316yes—
111C2×#95C2316yes—
112C2×#95C2316yes—
113C2×#95C2316yes—
114C25—43no—
115C25—43no—
116C25—43no—
117C25—43no—
118C25C2×D414yes—
119C25C2×Q814yes—
120C25#4114yes—
121C25C2411yescharacteristic
122C25C2413yes—
123C22×Q8—22no—
124C22×Q8—26no—
125C22×Q8—26no—
126C22×D4—26no—
127C22×D4—26no—
128C22×D4—43no—
129C22×D4—43no—
130C22×D4—43no—
131C22×D4—43no—
132C22×D4C2×Q8112yes—
133C23×C4—26no—
134C23×C4—26no—
135C23×C4—26no—
136C23×C4—26no—
137C23×C4—26no—
138C23×C4—26no—
139C23×C4—22no—
140C23×C4—26no—
141C23×C4—26no—
142C23×C4—448no—
143C23×C4—43no—
144C23×C4—43no—
145C23×C4—43no—
146C23×C4—43no—
147C23×C4—43no—
148C23×C4—43no—
149C23×C4—448no—
150C23×C4—416no—
151C23×C4C2×D4112yes—
152C23×C4C2×D4112yes—
153C23×C4C2412yes—
154C23×C4C2416yes—
155C23×C4C2416yes—
156C23×C4C2412yes—
157C23×C4C2413yes—
158C23×C4C2411yescharacteristic
159C23×C4C2413yes—
160C23×C4C2413yes—
161C23×C4C2411yescharacteristic
162C22⋊Q8—448no—
163C22⋊Q8—416no—
164#120—232no—
165C4×D4—448no—
166C2×C4⋊C4—26no—
167C2×C4⋊C4—26no—
168C2×C4⋊C4—212no—
169C2×C4⋊C4—26no—
170C2×C4⋊C4—26no—
171C2×C4⋊C4—26no—
172C2×C4⋊C4—26no—
173C2×C4⋊C4—26no—
174C2×C4⋊C4—26no—
175C2×C4⋊C4—43no—
176C2×C4⋊C4—43no—
177C2×C4⋊C4—46no—
178C2×C4⋊C4—46no—
179C2×C4⋊C4—46no—
180C2×C4⋊C4—46no—
181C2×C4⋊C4—46no—
182C2×C4⋊C4—46no—
183C2×C22⋊C4—26no—
184C2×C22⋊C4—26no—
185C2×C22⋊C4—26no—
186C2×C22⋊C4—212no—
187C2×C22⋊C4—26no—
188C2×C22⋊C4—212no—
189C2×C22⋊C4—26no—
190C2×C22⋊C4—212no—
191C2×C22⋊C4—212no—
192C2×C22⋊C4—43no—
193C2×C22⋊C4—46no—
194C2×C22⋊C4—46no—
195C2×C22⋊C4—43no—
196C2×C22⋊C4—43no—
197C2×C22⋊C4—43no—
198C2×C22⋊C4—43no—
199C2×C22⋊C4—43no—
200C2×C22⋊C4—46no—
201C2×C22⋊C4—46no—
202C2×C22⋊C4—46no—
203C2×C22⋊C4—46no—
204C2×C22⋊C4—46no—
205C2×C22⋊C4—46no—
206C2×C22⋊C4—46no—
207C2×C22⋊C4—46no—
208C2×C22⋊C4—46no—
209C2×C22⋊C4—46no—
210C2×C22⋊C4C2×Q8112yes—
211C2×C42—26no—
212C2×C42—26no—
213C2×C42—26no—
214#95—26no—
215#95—26no—
216#95—212no—
217#95—46no—
218#95—46no—
219#95—46no—
220#95—46no—
221#95—46no—
222#95—46no—
223#95—46no—
224#95—46no—
225#95—46no—
226#95—46no—
227C24—23no—
228C24—23no—
229C24—23no—
230C24—23no—
231C24—23no—
232C24—23no—
233C24—23no—
234C24—23no—
235C24—23no—
236C24—23no—
237C24—23no—
238C24—23no—
239C24—21no—
240C24—21no—
241C24—23no—
242C24—23no—
243C24—23no—
244C24—23no—
245C24—412no—
246C24—412no—
247C24—412no—
248C24—412no—
249C24—412no—
250C24—412no—
251C24—412no—
252C24—44no—
253C24—412no—
254C24—412no—
255C24—412no—
256C24—412no—
257C24C22⋊Q8116yescomplemented
258C24C22×D416yes—
259C24C22×Q812yes—
260C24C22×Q816yes—
261C24C2×#4116yes—
262C24C2511yescharacteristic
263C2×Q8—412no—
264C2×Q8—412no—
265C2×Q8—44no—
266C2×D4—224no—
267C2×D4—412no—
268C2×D4—412no—
269C2×D4—412no—
270C2×D4—412no—
271C2×D4—412no—
272C2×D4—412no—
273C22×C4—23no—
274C22×C4—23no—
275C22×C4—23no—
276C22×C4—23no—
277C22×C4—23no—
278C22×C4—23no—
279C22×C4—23no—
280C22×C4—23no—
281C22×C4—23no—
282C22×C4—23no—
283C22×C4—23no—
284C22×C4—23no—
285C22×C4—224no—
286C22×C4—224no—
287C22×C4—224no—
288C22×C4—23no—
289C22×C4—26no—
290C22×C4—21no—
291C22×C4—22no—
292C22×C4—26no—
293C22×C4—22no—
294C22×C4—26no—
295C22×C4—23no—
296C22×C4—26no—
297C22×C4—26no—
298C22×C4—23no—
299C22×C4—26no—
300C22×C4—26no—
301C22×C4—26no—
302C22×C4—26no—
303C22×C4—23no—
304C22×C4—26no—
305C22×C4—23no—
306C22×C4—26no—
307C22×C4—23no—
308C22×C4—23no—
309C22×C4—26no—
310C22×C4—26no—
311C22×C4—26no—
312C22×C4—26no—
313C22×C4—26no—
314C22×C4—26no—
315C22×C4—26no—
316C22×C4—23no—
317C22×C4—23no—
318C22×C4—412no—
319C22×C4—412no—
320C22×C4—412no—
321C22×C4—412no—
322C22×C4—412no—
323C22×C4—412no—
324C22×C4—412no—
325C22×C4—4128no—
326C22×C4—412no—
327C22×C4—412no—
328C22×C4—412no—
329C22×C4—412no—
330C22×C4—412no—
331C22×C4—412no—
332C22×C4—44no—
333C22×C4C22×D416yes—
334C22×C4C22×D412yes—
335C22×C4C22×Q816yes—
336C22×C4H(5,𝔽2)16yes—
337C22×C4H(5,𝔽2)16yes—
338C4⋊C4—412no—
339C4⋊C4—424no—
340C4⋊C4—48no—
341C4⋊C4—412no—
342C4⋊C4—412no—
343C4⋊C4—44no—
344C4⋊C4—812no—
345C22⋊C4—224no—
346C22⋊C4—412no—
347C22⋊C4—424no—
348C22⋊C4—412no—
349C22⋊C4—412no—
350C22⋊C4—424no—
351C22⋊C4—48no—
352C22⋊C4—812no—
353C22⋊C4—812no—
354C22⋊C4—812no—
355C42—412no—
356C23—26no—
357C23—26no—
358C23—26no—
359C23—26no—
360C23—26no—
361C23—26no—
362C23—26no—
363C23—26no—
364C23—26no—
365C23—26no—
366C23—26no—
367C23—26no—
368C23—26no—
369C23—26no—
370C23—26no—
371C23—26no—
372C23—26no—
373C23—26no—
374C23—22no—
375C23—26no—
376C23—26no—
377C23—448no—
378C23—416no—
379C23—448no—
380C23—448no—
381C23—416no—
382C23—448no—
383C23—416no—
384C23—43no—
385C23—43no—
386C23—43no—
387C23—43no—
388C23—43no—
389C23—43no—
390C23—43no—
391C23—43no—
392C23—43no—
393C23—43no—
394C23—43no—
395C23—43no—
396C23—43no—
397C23—43no—
398C23—43no—
399C23—43no—
400C23—43no—
401C23—43no—
402C23—43no—
403C23—43no—
404C23—43no—
405C23—41no—
406C23—43no—
407C23—43no—
408C23—43no—
409C23—43no—
410C23C2×C22⋊Q8112yes—
411C23C23×D411yescharacteristic
412C23C23×Q811yescharacteristic
413C23C22×#4111yescharacteristic
414C23C2×H(5,𝔽2)13yes—
415C23C2×H(5,𝔽2)13yes—
416C23#58513yes—
417C23#58513yes—
418Q8—88no—
419D4—448no—
420C2×C4—26no—
421C2×C4—26no—
422C2×C4—26no—
423C2×C4—26no—
424C2×C4—26no—
425C2×C4—26no—
426C2×C4—448no—
427C2×C4—43no—
428C2×C4—43no—
429C2×C4—43no—
430C2×C4—43no—
431C2×C4—43no—
432C2×C4—43no—
433C2×C4—43no—
434C2×C4—43no—
435C2×C4—448no—
436C2×C4—43no—
437C2×C4—46no—
438C2×C4—43no—
439C2×C4—46no—
440C2×C4—43no—
441C2×C4—43no—
442C2×C4—46no—
443C2×C4—46no—
444C2×C4—46no—
445C2×C4—46no—
446C2×C4—46no—
447C2×C4—46no—
448C2×C4—46no—
449C2×C4—43no—
450C2×C4—46no—
451C2×C4—42no—
452C2×C4—42no—
453C2×C4—41no—
454C2×C4—43no—
455C2×C4—46no—
456C2×C4—46no—
457C2×C4—43no—
458C2×C4—46no—
459C2×C4—46no—
460C2×C4—46no—
461C2×C4—46no—
462C2×C4—43no—
463C2×C4—46no—
464C2×C4—46no—
465C2×C4—43no—
466C2×C4—88no—
467C2×C4#534112yes—
468C2×C4#560112yes—
469C2×C4H(3,𝔽4)14yes—
470C22—212no—
471C22—212no—
472C22—212no—
473C22—464no—
474C22—46no—
475C22—46no—
476C22—46no—
477C22—46no—
478C22—46no—
479C22—46no—
480C22—46no—
481C22—46no—
482C22—46no—
483C22—46no—
484C22—46no—
485C22—46no—
486C22—46no—
487C22—42no—
488C22—46no—
489C22—81no—
490C22—83no—
491C22—83no—
492C22—81no—
493C22C22×C22⋊Q811yescharacteristic
494C22C2×#53413yes—
495C22#344913yes—
496C22#350113yes—
497C22#350113yes—
498C22#350513yes—
499C22#350513yes—
500C22C2×#56013yes—
501C22C2×H(3,𝔽4)11yescharacteristic
502C22#353913yes—
503C22#353913yes—
504C22#354013yes—
505C22#354013yes—
506C4—46no—
507C4—46no—
508C4—81no—
509C4—82no—
510C4—83no—
511C4—86no—
512C4—86no—
513C4—82no—
514C4#281218yes—
515C2—412no—
516C2—82no—
517C2#3501013yes—
518C2C2×#281211yescharacteristic
519C2#3744513yes—
520C2#4549313yes—
521C2#4549411yescharacteristic
522C2#4550813yes—
523C2#4550911yescharacteristic
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC22×#120C22
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#2848C2semidirect
3. #16894C2semidirect
4. #16893C2semidirect
5. #15183C2semidirect
6. #15181C2semidirect
7. #15179C2semidirect
8. #15177C2semidirect
9. #14680C2semidirect
10. #14673C2semidirect
14. C22×C22⋊Q8C22semidirect
19. C2×#391C22semidirect
20. C2×#389C22semidirect
27. C2×#380C22semidirect
32. #1445C22semidirect
33. #1444C22semidirect
34. #1443C22semidirect
35. #1442C22semidirect
36. #1442C22semidirect
37. #1441C22semidirect
44. #1439C22semidirect
45. #1439C22semidirect
48. C4⋊#95C22semidirect
49. C4⋊#95C22semidirect
66. C2×#120Q8semidirect
257. C24C22⋊Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#28482
C2×C4×#1202
#168942
#168932
#151832
#151812
#151792
#151772
#146802
#146732
#146572