Finite Group Databasebeta

← #4452808

Subgroups — #4452808

Order 512.

2989 conjugacy classes of subgroups.

The subgroup lattice has 595 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2623C211yesmaximal, characteristic
2C22×#395C211yesmaximal, characteristic, complemented
3#16874C214yesmaximal, complemented
4#16553C214yesmaximal, complemented
5#16546C214yesmaximal, complemented
6#16327C214yesmaximal, complemented
7#15026C211yesmaximal, characteristic, complemented
8#14946C211yesmaximal, characteristic
9#14919C212yesmaximal, complemented
10#14912C212yesmaximal
11#14874C211yesmaximal, characteristic, complemented
12#14873C211yesmaximal, characteristic
13#14043C212yesmaximal, complemented
14#14019C212yesmaximal
15#13776C211yesmaximal, characteristic, complemented
16C24×Q8C2211yescharacteristic
17C22×#123C2211yescharacteristic
18C23×C4⋊C4C2212yes—
19C23×C4⋊C4C2211yescharacteristic
20#2623—28no—
21C2×#400C2212yes—
22C2×#399C2212yes—
23C2×#398C2212yes—
24C2×#397C2212yes—
25C2×#395—28no—
26C2×#395C2214yes—
27C2×#395C2214yes—
28C2×#395C2214yes—
29C2×#393C2214yes—
30C2×#389C2214yes—
31C2×#388C2214yes—
32C2×#386C2211yescharacteristic
33C2×#386C2211yescharacteristic
34C2×#386C2212yes—
35C2×#386C2212yes—
36C2×#385C2211yescharacteristic
37C2×#385C2211yescharacteristic
38C2×#385C2212yes—
39C2×#385C2212yes—
40C2×#381C2211yescharacteristic
41C2×#381C2212yes—
42C2×C4×C22⋊C4C2212yes—
43C22×#95C2212yes—
44C22×#95C2211yescharacteristic
45#1446C2214yes—
46#1446C2214yes—
47#1445C2214yes—
48#1445C2214yes—
49#1444C2214yes—
50#1443C2214yes—
51#1443C2214yes—
52#1443C2214yes—
53#1442C2214yes—
54#1442C2214yes—
55#1442C2214yes—
56#1442C2214yes—
57#1441C2214yes—
58#1441C2214yes—
59#1440C2214yes—
60#1439C2214yes—
61#1439C2214yes—
62#1439C2214yes—
63#1436C2214yes—
64#1436C2214yes—
65#1436C2214yes—
66#1436C2214yes—
67C22⋊#95C2212yes—
68C22⋊#95C2211yescharacteristic
69C22⋊#95C2212yes—
70C22⋊#95C2211yescharacteristic
71C22⋊#95C2212yes—
72C22⋊#95C2212yes—
73C23×Q8—24no—
74C23×Q8—44no—
75C23×Q8—48no—
76C23×Q8C2314yes—
77C23×D4C2311yescharacteristic
78C24×C4C2311yescharacteristic
79C24×C4C2312yes—
80C24×C4C2312yes—
81C24×C4C2311yescharacteristic
82C24×C4C2311yescharacteristic
83C2×#123—24no—
84C2×#123—44no—
85C22×C4⋊C4—28no—
86C22×C4⋊C4—48no—
87C22×C4⋊C4—44no—
88C22×C4⋊C4D418yes—
89C22×C4⋊C4C2314yes—
90C22×C4⋊C4C2314yes—
91C22×C4⋊C4C2314yes—
92C22×C4⋊C4C2312yes—
93C22×C4⋊C4C2314yes—
94C22×C4⋊C4C2311yescharacteristic
95C22×C4⋊C4C2312yes—
96C22×C4⋊C4C2311yescharacteristic
97C22×C4⋊C4C2312yes—
98C22×C4⋊C4C2312yes—
99C22×C22⋊C4C2311yescharacteristic
100C22×C22⋊C4C2312yes—
101C22×C22⋊C4C2311yescharacteristic
102C22×C22⋊C4C2311yescharacteristic
103C22×C22⋊C4C2314yes—
104C22×C22⋊C4C2312yes—
105C22×C22⋊C4C2312yes—
106C22×C22⋊C4C2314yes—
107C22×C22⋊C4C2311yescharacteristic
108C22×C22⋊C4C2312yes—
109C22×C22⋊C4C2311yescharacteristic
110C22×C42C2314yes—
111#400—28no—
112#399—28no—
113#398—28no—
114#397—28no—
115#395—216no—
116#395—216no—
117#395—48no—
118#393—48no—
119#389—48no—
120#388—48no—
121#386—24no—
122#386—24no—
123#386—28no—
124#386—44no—
125#385—24no—
126#385—24no—
127#385—28no—
128#385—44no—
129#381—24no—
130#381—28no—
131C4×C22⋊C4—44no—
132C2×#95—28no—
133C2×#95—24no—
134C2×#95C2314yes—
135C2×#95C2314yes—
136C2×#95C2314yes—
137C2×#95C2314yes—
138C2×#95C2314yes—
139C2×#95C2312yes—
140C2×#95C2312yes—
141C2×#95C2312yes—
142C2×#95C2312yes—
143C2×#95C2312yes—
144C2×#95C2312yes—
145C2×#95C2314yes—
146C2×#95C2314yes—
147C2×#95C2314yes—
148C2×#95C2314yes—
149C2×#95C2314yes—
150C2×#95C2314yes—
151C2×#95C2314yes—
152C2×#95C2314yes—
153C2×#95C2314yes—
154C2×#95C2314yes—
155C2×#95C2314yes—
156C2×#95C2314yes—
157C2×#95C2314yes—
158C2×#95C2314yes—
159C2×#95C2314yes—
160C25C2411yescharacteristic
161C25C2411yescharacteristic
162C22×Q8—28no—
163C22×Q8—28no—
164C22×Q8—28no—
165C22×Q8—44no—
166C22×Q8—44no—
167C22×Q8—44no—
168C22×Q8—44no—
169C22×Q8—432no—
170C22×Q8—416no—
171C22×Q8—464no—
172C22×D4—22no—
173C22×D4—24no—
174C22×D4—24no—
175C22×D4—22no—
176C22×D4—22no—
177C23×C4—24no—
178C23×C4—24no—
179C23×C4—24no—
180C23×C4—24no—
181C23×C4—24no—
182C23×C4—22no—
183C23×C4—22no—
184C23×C4—24no—
185C23×C4—28no—
186C23×C4—28no—
187C23×C4—24no—
188C23×C4—24no—
189C23×C4—24no—
190C23×C4—24no—
191C23×C4—44no—
192C23×C4—44no—
193C23×C4—44no—
194C23×C4—44no—
195C23×C4C2×D414yes—
196C23×C4C2×D414yes—
197C23×C4C2×D414yes—
198C23×C4C2411yescharacteristic
199C23×C4C2412yes—
200C23×C4C2412yes—
201C23×C4C2411yescharacteristic
202C23×C4C2414yes—
203C23×C4C2411yescharacteristic
204C23×C4C2412yes—
205C23×C4C2412yes—
206C23×C4C2411yescharacteristic
207C23×C4C2411yescharacteristic
208C23×C4C2414yes—
209C23×C4C2414yes—
210C23×C4C2414yes—
211#123—416no—
212C2×C4⋊C4—28no—
213C2×C4⋊C4—28no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—28no—
217C2×C4⋊C4—28no—
218C2×C4⋊C4—28no—
219C2×C4⋊C4—28no—
220C2×C4⋊C4—24no—
221C2×C4⋊C4—24no—
222C2×C4⋊C4—28no—
223C2×C4⋊C4—416no—
224C2×C4⋊C4—42no—
225C2×C4⋊C4—42no—
226C2×C4⋊C4—42no—
227C2×C4⋊C4—42no—
228C2×C4⋊C4—44no—
229C2×C4⋊C4—44no—
230C2×C4⋊C4—42no—
231C2×C4⋊C4—42no—
232C2×C4⋊C4—44no—
233C2×C4⋊C4—432no—
234C2×C4⋊C4—44no—
235C2×C4⋊C4—44no—
236C2×C4⋊C4—44no—
237C2×C4⋊C4—44no—
238C2×C4⋊C4C2×D418yes—
239C2×C4⋊C4C2×D418yes—
240C2×C22⋊C4—24no—
241C2×C22⋊C4—24no—
242C2×C22⋊C4—22no—
243C2×C22⋊C4—22no—
244C2×C22⋊C4—22no—
245C2×C22⋊C4—24no—
246C2×C22⋊C4—24no—
247C2×C22⋊C4—24no—
248C2×C22⋊C4—24no—
249C2×C22⋊C4—22no—
250C2×C22⋊C4—22no—
251C2×C22⋊C4—24no—
252C2×C22⋊C4—22no—
253C2×C22⋊C4—22no—
254C2×C22⋊C4—28no—
255C2×C22⋊C4—28no—
256C2×C22⋊C4—24no—
257C2×C22⋊C4—24no—
258C2×C22⋊C4—28no—
259C2×C22⋊C4—42no—
260C2×C22⋊C4—42no—
261C2×C22⋊C4—44no—
262C2×C22⋊C4—44no—
263C2×C22⋊C4—44no—
264C2×C22⋊C4—44no—
265C2×C22⋊C4—44no—
266C2×C22⋊C4#4118yes—
267C2×C42—44no—
268C2×C42—44no—
269C2×C42—44no—
270#95—28no—
271#95—28no—
272#95—28no—
273#95—24no—
274#95—24no—
275#95—24no—
276#95—24no—
277#95—24no—
278#95—24no—
279#95—28no—
280#95—28no—
281#95—28no—
282#95—28no—
283#95—28no—
284#95—28no—
285#95—28no—
286#95—28no—
287#95—44no—
288#95—44no—
289#95—44no—
290#95—44no—
291#95—44no—
292#95—44no—
293#95—44no—
294#95—44no—
295#95—44no—
296C24—24no—
297C24—24no—
298C24—21no—
299C24—22no—
300C24—24no—
301C24—22no—
302C24—24no—
303C24—21no—
304C24—21no—
305C24C22×D412yes—
306C24C2×#4114yes—
307C24H(5,𝔽2)14yes—
308C242−1+412yes—
309C242−1+412yes—
310C24C2511yescharacteristic
311C2×Q8—416no—
312C2×Q8—416no—
313C2×Q8—416no—
314C2×Q8—416no—
315C2×Q8—416no—
316C2×Q8—416no—
317C2×Q8—4128no—
318C2×Q8—416no—
319C2×D4—44no—
320C2×D4—44no—
321C2×D4—44no—
322C2×D4—48no—
323C2×D4—48no—
324C22×C4—216no—
325C22×C4—216no—
326C22×C4—28no—
327C22×C4—28no—
328C22×C4—28no—
329C22×C4—28no—
330C22×C4—24no—
331C22×C4—24no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—24no—
335C22×C4—24no—
336C22×C4—24no—
337C22×C4—24no—
338C22×C4—24no—
339C22×C4—22no—
340C22×C4—24no—
341C22×C4—24no—
342C22×C4—24no—
343C22×C4—24no—
344C22×C4—24no—
345C22×C4—24no—
346C22×C4—24no—
347C22×C4—24no—
348C22×C4—22no—
349C22×C4—24no—
350C22×C4—24no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—24no—
356C22×C4—24no—
357C22×C4—24no—
358C22×C4—24no—
359C22×C4—21no—
360C22×C4—21no—
361C22×C4—21no—
362C22×C4—21no—
363C22×C4—24no—
364C22×C4—28no—
365C22×C4—22no—
366C22×C4—21no—
367C22×C4—22no—
368C22×C4—21no—
369C22×C4—21no—
370C22×C4—22no—
371C22×C4—24no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—24no—
375C22×C4—24no—
376C22×C4—24no—
377C22×C4—24no—
378C22×C4—24no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—24no—
382C22×C4—22no—
383C22×C4—24no—
384C22×C4—24no—
385C22×C4—24no—
386C22×C4—24no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—24no—
390C22×C4—24no—
391C22×C4—21no—
392C22×C4—21no—
393C22×C4—48no—
394C22×C4—48no—
395C22×C4—416no—
396C22×C4—416no—
397C22×C4—416no—
398C22×C4—416no—
399C22×C4—48no—
400C22×C4—48no—
401C22×C4—416no—
402C22×C4—48no—
403C22×C4—48no—
404C22×C4—48no—
405C22×C4—48no—
406C22×C4—88no—
407C22×C4C22×D412yes—
408C22×C4C22×D412yes—
409C22×C4C22×D414yes—
410C22×C4C22×D414yes—
411C22×C4C2×#4118yes—
412C22×C4H(5,𝔽2)18yes—
413C22×C42−1+418yes—
414C4⋊C4—48no—
415C4⋊C4—48no—
416C4⋊C4—416no—
417C4⋊C4—416no—
418C4⋊C4—416no—
419C4⋊C4—44no—
420C4⋊C4—44no—
421C4⋊C4—84no—
422C4⋊C4—84no—
423C4⋊C4—88no—
424C22⋊C4—216no—
425C22⋊C4—416no—
426C22⋊C4—44no—
427C22⋊C4—44no—
428C22⋊C4—44no—
429C22⋊C4—44no—
430C22⋊C4—44no—
431C22⋊C4—48no—
432C22⋊C4—84no—
433C22⋊C4—84no—
434C22⋊C4—88no—
435C42—88no—
436C23—24no—
437C23—24no—
438C23—28no—
439C23—28no—
440C23—28no—
441C23—28no—
442C23—28no—
443C23—28no—
444C23—41no—
445C23—41no—
446C23—41no—
447C23—42no—
448C23—42no—
449C23—42no—
450C23—42no—
451C23—44no—
452C23—44no—
453C23—44no—
454C23—44no—
455C23—44no—
456C23—44no—
457C23#53214yes—
458C23#53414yes—
459C23#53614yes—
460C23#53918yes—
461C23#54218yes—
462C23C23×D411yescharacteristic
463C23C22×#4112yes—
464C23C2×H(5,𝔽2)12yes—
465C23C2×H(5,𝔽2)14yes—
466C23C2×2−1+411yescharacteristic
467C23C2×2−1+411yescharacteristic
468C23C2×2−1+414yes—
469Q8—464no—
470D4—88no—
471C2×C4—22no—
472C2×C4—22no—
473C2×C4—22no—
474C2×C4—22no—
475C2×C4—24no—
476C2×C4—24no—
477C2×C4—22no—
478C2×C4—22no—
479C2×C4—28no—
480C2×C4—24no—
481C2×C4—24no—
482C2×C4—24no—
483C2×C4—28no—
484C2×C4—24no—
485C2×C4—28no—
486C2×C4—28no—
487C2×C4—28no—
488C2×C4—28no—
489C2×C4—28no—
490C2×C4—28no—
491C2×C4—28no—
492C2×C4—432no—
493C2×C4—432no—
494C2×C4—416no—
495C2×C4—416no—
496C2×C4—44no—
497C2×C4—44no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—44no—
503C2×C4—44no—
504C2×C4—44no—
505C2×C4—44no—
506C2×C4—44no—
507C2×C4—44no—
508C2×C4—44no—
509C2×C4—44no—
510C2×C4—44no—
511C2×C4—44no—
512C2×C4—44no—
513C2×C4—44no—
514C2×C4—44no—
515C2×C4—44no—
516C2×C4—42no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—42no—
522C2×C4—41no—
523C2×C4—42no—
524C2×C4—44no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—44no—
533C2×C4—44no—
534C2×C4—44no—
535C2×C4—44no—
536C2×C4—44no—
537C2×C4—44no—
538C2×C4—44no—
539C2×C4—44no—
540C2×C4—44no—
541C2×C4—44no—
542C2×C4—44no—
543C2×C4—44no—
544C2×C4—88no—
545C22—28no—
546C22—24no—
547C22—28no—
548C22—216no—
549C22—216no—
550C22—24no—
551C22—81no—
552C22—82no—
553C22—81no—
554C22—82no—
555C22—84no—
556C22—84no—
557C22—81no—
558C22#266318yes—
559C22C2×#53211yescharacteristic
560C22C2×#53411yescharacteristic
561C22C2×#53611yescharacteristic
562C22C2×#53912yes—
563C22C2×#54212yes—
564C22#348412yes—
565C22#348612yes—
566C22#352014yes—
567C22#352014yes—
568C22#353214yes—
569C22#354414yes—
570C22#355214yes—
571C22#355214yes—
572C4—42no—
573C4—42no—
574C4—48no—
575C4—44no—
576C4—48no—
577C4—44no—
578C4—48no—
579C4—81no—
580C4—82no—
581C4—81no—
582C4—82no—
583C4—84no—
584C4—81no—
585C2—28no—
586C2—161no—
587C2C2×#266311yescharacteristic
588C2#3539712yes—
589C2#3670211yescharacteristic
590C2#3723312yes—
591C2#3725811yescharacteristic
592C2#4083614yes—
593C2#4083814yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC23×D4C23
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C22×#395C2semidirect
3. #16874C2semidirect
4. #16553C2semidirect
5. #16546C2semidirect
6. #16327C2semidirect
7. #15026C2semidirect
9. #14919C2semidirect
11. #14874C2semidirect
13. #14043C2semidirect
15. #13776C2semidirect

Maximal subgroups

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

subgroupindex
C2×#26232
C22×#3952
#168742
#165532
#165462
#163272
#150262
#149462
#149192
#149122
#148742
#148732
#140432
#140192
#137762