Finite Group Databasebeta

← #4453403

Subgroups — #4453403

Order 512.

4034 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
1C22×#539C211yesmaximal, characteristic
2C2×#2486C212yesmaximal, complemented
3C22×#398C211yesmaximal, characteristic, complemented
4#16673C214yesmaximal, complemented
5#16607C214yesmaximal, complemented
6#16365C214yesmaximal, complemented
7#16359C214yesmaximal, complemented
8#15069C214yesmaximal, complemented
9#14934C212yesmaximal, complemented
10#14926C212yesmaximal
11#14339C211yesmaximal, characteristic, complemented
12#13960C212yesmaximal, complemented
13C2×#539—216no—
14C2×#539—28no—
15C22×#126C2212yes—
16C22×#125C2212yes—
17C22×#123C2212yes—
18C22×#123C2212yes—
19C22×#123C2214yes—
20C22×C22⋊Q8C2211yescharacteristic
21C22×C22⋊Q8C2212yes—
22C22×#121C2211yescharacteristic
23C22×C4×D4C2212yes—
24C23×C4⋊C4C2211yescharacteristic
25C23×C4⋊C4C2211yescharacteristic
26C23×C4⋊C4C2212yes—
27#2486—216no—
28C2×#398—28no—
29C2×#398C2214yes—
30C2×#398C2214yes—
31C2×#398C2214yes—
32C2×#397C2214yes—
33C2×#389C2214yes—
34C2×#386C2214yes—
35C2×#381C2214yes—
36C2×#381C2212yes—
37C2×#380C2212yes—
38C2×#380C2214yes—
39C22×#95C2212yes—
40C22×#95C2211yescharacteristic
41#1446C2214yes—
42#1444C2218yes—
43#1444C2214yes—
44#1444C2218yes—
45#1442C2214yes—
46#1442C2218yes—
47#1441C2214yes—
48#1441C2214yes—
49#1441C2214yes—
50#1439C2218yes—
51#1439C2214yes—
52#1439C2218yes—
53#1439C2218yes—
54#1438C2214yes—
55#1434C2214yes—
56C22⋊#95C2212yes—
57C22⋊#95C2214yes—
58C22⋊#95C2214yes—
59C23×Q8C2311yescharacteristic
60C23×D4C2312yes—
61C23×D4C2311yescharacteristic
62C24×C4C2312yes—
63C24×C4C2311yescharacteristic
64C24×C4C2312yes—
65C24×C4C2311yescharacteristic
66C24×C4C2312yes—
67C24×C4C2311yescharacteristic
68#539—2128no—
69C2×#126—28no—
70C2×#126—28no—
71C2×#126—28no—
72C2×#125—28no—
73C2×#125—28no—
74C2×#125—28no—
75C2×#123—28no—
76C2×#123—28no—
77C2×#123—28no—
78C2×#123—216no—
79C2×#123—28no—
80C2×#123—28no—
81C2×#123—28no—
82C2×#123—48no—
83C2×#123—48no—
84C2×C22⋊Q8—24no—
85C2×C22⋊Q8—28no—
86C2×C22⋊Q8—28no—
87C2×C22⋊Q8—28no—
88C2×C22⋊Q8—28no—
89C2×#121—28no—
90C2×#121—24no—
91C2×C4×D4—28no—
92C2×C4×D4—28no—
93C2×C4×D4D4116yes—
94C22×C4⋊C4—28no—
95C22×C4⋊C4—28no—
96C22×C4⋊C4—24no—
97C22×C4⋊C4—28no—
98C22×C4⋊C4—48no—
99C22×C4⋊C4D418yes—
100C22×C4⋊C4C2314yes—
101C22×C4⋊C4C2311yescharacteristic
102C22×C4⋊C4C2312yes—
103C22×C4⋊C4C2312yes—
104C22×C4⋊C4C2311yescharacteristic
105C22×C4⋊C4C2312yes—
106C22×C4⋊C4C2312yes—
107C22×C4⋊C4C2312yes—
108C22×C4⋊C4C2312yes—
109C22×C4⋊C4C2318yes—
110C22×C4⋊C4C2318yes—
111C22×C4⋊C4C2314yes—
112C22×C22⋊C4C2312yes—
113C22×C22⋊C4C2312yes—
114C22×C22⋊C4C2312yes—
115C22×C22⋊C4C2314yes—
116C22×C22⋊C4C2312yes—
117C22×C22⋊C4C2312yes—
118C22×C22⋊C4C2318yes—
119C22×C22⋊C4C2318yes—
120C22×C42C2312yes—
121#398—216no—
122#398—216no—
123#398—216no—
124#397—216no—
125#389—216no—
126#386—48no—
127#381—216no—
128#381—44no—
129#380—216no—
130#380—28no—
131C2×#95—24no—
132C2×#95—28no—
133C2×#95C2314yes—
134C2×#95C2318yes—
135C2×#95C2314yes—
136C2×#95C2314yes—
137C2×#95C2314yes—
138C2×#95C2318yes—
139C2×#95C2318yes—
140C2×#95C2318yes—
141C2×#95C2314yes—
142C2×#95C2314yes—
143C2×#95C2318yes—
144C2×#95C2318yes—
145C25C2412yes—
146C25C2411yescharacteristic
147C22×Q8—22no—
148C22×Q8—24no—
149C22×Q8—44no—
150C22×D4—24no—
151C22×D4—24no—
152C22×D4—24no—
153C22×D4—24no—
154C22×D4—24no—
155C22×D4—24no—
156C22×D4—24no—
157C22×D4—22no—
158C22×D4—44no—
159C22×D4C2×D418yes—
160C23×C4—24no—
161C23×C4—24no—
162C23×C4—24no—
163C23×C4—22no—
164C23×C4—22no—
165C23×C4—24no—
166C23×C4—28no—
167C23×C4—24no—
168C23×C4—24no—
169C23×C4—24no—
170C23×C4—28no—
171C23×C4—28no—
172C23×C4—28no—
173C23×C4—24no—
174C23×C4—24no—
175C23×C4—44no—
176C23×C4—44no—
177C23×C4—48no—
178C23×C4—44no—
179C23×C4C2×D414yes—
180C23×C4C2×D414yes—
181C23×C4C2×D418yes—
182C23×C4C2×D418yes—
183C23×C4C2×D414yes—
184C23×C4C2418yes—
185C23×C4C2414yes—
186C23×C4C2412yes—
187C23×C4C2412yes—
188C23×C4C2411yescharacteristic
189C23×C4C2412yes—
190C23×C4C2412yes—
191C23×C4C2411yescharacteristic
192C23×C4C2414yes—
193C23×C4C2412yes—
194#126—264no—
195#125—264no—
196#123—264no—
197#123—264no—
198#123—464no—
199C22⋊Q8—264no—
200C22⋊Q8—232no—
201#121—232no—
202C4×D4—264no—
203C2×C4⋊C4—232no—
204C2×C4⋊C4—24no—
205C2×C4⋊C4—24no—
206C2×C4⋊C4—24no—
207C2×C4⋊C4—24no—
208C2×C4⋊C4—24no—
209C2×C4⋊C4—24no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—24no—
212C2×C4⋊C4—24no—
213C2×C4⋊C4—24no—
214C2×C4⋊C4—24no—
215C2×C4⋊C4—24no—
216C2×C4⋊C4—24no—
217C2×C4⋊C4—24no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—24no—
220C2×C4⋊C4—24no—
221C2×C4⋊C4—24no—
222C2×C4⋊C4—24no—
223C2×C4⋊C4—28no—
224C2×C4⋊C4—28no—
225C2×C4⋊C4—216no—
226C2×C4⋊C4—28no—
227C2×C4⋊C4—216no—
228C2×C4⋊C4—28no—
229C2×C4⋊C4—48no—
230C2×C4⋊C4—48no—
231C2×C4⋊C4—48no—
232C2×C4⋊C4—48no—
233C2×C4⋊C4—48no—
234C2×C4⋊C4—432no—
235C2×C4⋊C4—416no—
236C2×C4⋊C4C2×D418yes—
237C2×C4⋊C4C2×D418yes—
238C2×C4⋊C4C2×D418yes—
239C2×C22⋊C4—24no—
240C2×C22⋊C4—24no—
241C2×C22⋊C4—24no—
242C2×C22⋊C4—24no—
243C2×C22⋊C4—24no—
244C2×C22⋊C4—24no—
245C2×C22⋊C4—24no—
246C2×C22⋊C4—24no—
247C2×C22⋊C4—24no—
248C2×C22⋊C4—24no—
249C2×C22⋊C4—24no—
250C2×C22⋊C4—24no—
251C2×C22⋊C4—24no—
252C2×C22⋊C4—28no—
253C2×C22⋊C4—216no—
254C2×C22⋊C4—28no—
255C2×C22⋊C4—28no—
256C2×C22⋊C4—216no—
257C2×C22⋊C4—48no—
258C2×C22⋊C4—48no—
259C2×C22⋊C4—48no—
260C2×C22⋊C4—48no—
261C2×C22⋊C4C2×D418yes—
262C2×C22⋊C4C2×D418yes—
263C2×C42—24no—
264C2×C42—24no—
265C2×C42C2×D418yes—
266#95—28no—
267#95—28no—
268#95—28no—
269#95—216no—
270#95—28no—
271#95—216no—
272#95—216no—
273#95—48no—
274#95—44no—
275#95—44no—
276#95—48no—
277#95—48no—
278C24—24no—
279C24—24no—
280C24—22no—
281C24—22no—
282C24—22no—
283C24—22no—
284C24—22no—
285C24—28no—
286C24—22no—
287C24—28no—
288C24C22×D414yes—
289C24C22×D414yes—
290C24C22×D412yes—
291C24H(5,𝔽2)12yes—
292C24H(5,𝔽2)14yes—
293C242−1+412yes—
294C24C2511yescharacteristic
295C2×Q8—28no—
296C2×Q8—84no—
297C2×Q8—88no—
298C2×D4—216no—
299C2×D4—216no—
300C2×D4—216no—
301C2×D4—28no—
302C2×D4—48no—
303C2×D4—48no—
304C2×D4—48no—
305C2×D4—48no—
306C2×D4—84no—
307C2×D4—88no—
308C22×C4—28no—
309C22×C4—216no—
310C22×C4—28no—
311C22×C4—28no—
312C22×C4—28no—
313C22×C4—24no—
314C22×C4—22no—
315C22×C4—22no—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—22no—
325C22×C4—22no—
326C22×C4—28no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—28no—
332C22×C4—22no—
333C22×C4—24no—
334C22×C4—22no—
335C22×C4—28no—
336C22×C4—24no—
337C22×C4—24no—
338C22×C4—28no—
339C22×C4—28no—
340C22×C4—216no—
341C22×C4—28no—
342C22×C4—28no—
343C22×C4—24no—
344C22×C4—28no—
345C22×C4—28no—
346C22×C4—24no—
347C22×C4—28no—
348C22×C4—28no—
349C22×C4—28no—
350C22×C4—24no—
351C22×C4—28no—
352C22×C4—28no—
353C22×C4—416no—
354C22×C4—48no—
355C22×C4—416no—
356C22×C4—48no—
357C22×C4—432no—
358C22×C4—48no—
359C22×C4—48no—
360C22×C4—48no—
361C22×C4—48no—
362C22×C4—48no—
363C22×C4—48no—
364C22×C4—416no—
365C22×C4—48no—
366C22×C4—416no—
367C22×C4—88no—
368C22×C4—88no—
369C22×C4—88no—
370C22×C4#120116yes—
371C22×C4C22×D414yes—
372C22×C4C22×D412yes—
373C22×C4C22×D414yes—
374C22×C4C22×D412yes—
375C22×C4C22×D414yes—
376C22×C4C22×D414yes—
377C22×C4C22×D414yes—
378C22×C4C22×D414yes—
379C22×C4C22×D414yes—
380C22×C42−1+418yes—
381C4⋊C4—28no—
382C4⋊C4—216no—
383C4⋊C4—216no—
384C4⋊C4—28no—
385C4⋊C4—216no—
386C4⋊C4—216no—
387C4⋊C4—216no—
388C4⋊C4—216no—
389C4⋊C4—416no—
390C4⋊C4—432no—
391C4⋊C4—432no—
392C4⋊C4—416no—
393C22⋊C4—216no—
394C22⋊C4—216no—
395C22⋊C4—216no—
396C22⋊C4—216no—
397C22⋊C4—216no—
398C22⋊C4—432no—
399C22⋊C4—416no—
400C22⋊C4—816no—
401C42—216no—
402C23—24no—
403C23—28no—
404C23—24no—
405C23—28no—
406C23—216no—
407C23—28no—
408C23—28no—
409C23—24no—
410C23—24no—
411C23—24no—
412C23—28no—
413C23—28no—
414C23—42no—
415C23—42no—
416C23—42no—
417C23—42no—
418C23—44no—
419C23—44no—
420C23—44no—
421C23—44no—
422C23—48no—
423C23—48no—
424C23—48no—
425C23—48no—
426C23C2×#12014yes—
427C23#53518yes—
428C23#53514yes—
429C23#53714yes—
430C23#53718yes—
431C23C23×D411yescharacteristic
432C23C23×D412yes—
433C23C2×H(5,𝔽2)12yes—
434C23C2×H(5,𝔽2)11yescharacteristic
435C23C2×2−1+411yescharacteristic
436C23C2×2−1+414yes—
437C23#58514yes—
438Q8—88no—
439D4—432no—
440D4—88no—
441C2×C4—28no—
442C2×C4—28no—
443C2×C4—28no—
444C2×C4—24no—
445C2×C4—24no—
446C2×C4—24no—
447C2×C4—22no—
448C2×C4—24no—
449C2×C4—22no—
450C2×C4—24no—
451C2×C4—24no—
452C2×C4—24no—
453C2×C4—24no—
454C2×C4—24no—
455C2×C4—24no—
456C2×C4—28no—
457C2×C4—28no—
458C2×C4—28no—
459C2×C4—216no—
460C2×C4—416no—
461C2×C4—432no—
462C2×C4—416no—
463C2×C4—44no—
464C2×C4—42no—
465C2×C4—42no—
466C2×C4—42no—
467C2×C4—42no—
468C2×C4—42no—
469C2×C4—42no—
470C2×C4—42no—
471C2×C4—42no—
472C2×C4—44no—
473C2×C4—44no—
474C2×C4—44no—
475C2×C4—44no—
476C2×C4—44no—
477C2×C4—44no—
478C2×C4—44no—
479C2×C4—48no—
480C2×C4—44no—
481C2×C4—48no—
482C2×C4—48no—
483C2×C4—44no—
484C2×C4—44no—
485C2×C4—48no—
486C2×C4—48no—
487C2×C4—48no—
488C2×C4—48no—
489C2×C4—48no—
490C2×C4—48no—
491C2×C4—48no—
492C2×C4—816no—
493C2×C4—88no—
494C2×C4—816no—
495C2×C4C2×#12018yes—
496C22—24no—
497C22—28no—
498C22—24no—
499C22—216no—
500C22—216no—
501C22—28no—
502C22—44no—
503C22—44no—
504C22—44no—
505C22—48no—
506C22—48no—
507C22—84no—
508C22—84no—
509C22—84no—
510C22—84no—
511C22#261518yes—
512C22C22×#12011yescharacteristic
513C22C2×#53512yes—
514C22C2×#53511yescharacteristic
515C22C2×#53711yescharacteristic
516C22C2×#53712yes—
517C22#348512yes—
518C22#348514yes—
519C22#348612yes—
520C22#349914yes—
521C22#353518yes—
522C22#355914yes—
523C22#356314yes—
524C4—28no—
525C4—44no—
526C4—44no—
527C4—44no—
528C4—42no—
529C4—44no—
530C4—42no—
531C4—48no—
532C4—84no—
533C4—88no—
534C2—28no—
535C2—84no—
536C2C2×#261511yescharacteristic
537C2#3544611yescharacteristic
538C2#3566611yescharacteristic
539C2#3696812yes—
540C2#3711812yes—
541C2#4137314yes—
542C2#4137514yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC22×#121C22
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2486C2semidirect
3. C22×#398C2semidirect
4. #16673C2semidirect
5. #16607C2semidirect
6. #16365C2semidirect
7. #16359C2semidirect
8. #15069C2semidirect
9. #14934C2semidirect
11. #14339C2semidirect
12. #13960C2semidirect

Maximal subgroups

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

subgroupindex
C22×#5392
C2×#24862
C22×#3982
#166732
#166072
#163652
#163592
#150692
#149342
#149262
#143392
#139602