Finite Group Databasebeta

← #624142

Subgroups — #624142

Order 512.

3356 conjugacy classes of subgroups.

The subgroup lattice has 616 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2853C211yesmaximal, characteristic
2C2×#2614C211yesmaximal, characteristic
3C2×#2322C211yesmaximal, characteristic
4C22×#391C211yesmaximal, characteristic, complemented
5#15867C214yesmaximal, complemented
6#15860C214yesmaximal, complemented
7#15832C214yesmaximal, complemented
8#15824C214yesmaximal, complemented
9#14935C212yesmaximal, complemented
10#14927C212yesmaximal
11#14684C211yesmaximal, characteristic, complemented
12#14339C211yesmaximal, characteristic, complemented
13#14336C211yesmaximal, characteristic, complemented
14#13992C212yesmaximal, complemented
15#13968C212yesmaximal
16C24×Q8C2211yescharacteristic
17C22×#123C2212yes—
18C22×#123C2211yescharacteristic
19C22×C4×D4C2211yescharacteristic
20C23×C4⋊C4C2211yescharacteristic
21C23×C42C2211yescharacteristic
22#2853—28no—
23#2614—28no—
24C2×#395C2214yes—
25C2×#393C2214yes—
26#2322—28no—
27C2×#391—28no—
28C2×#391C2214yes—
29C2×#391C2214yes—
30C2×#389C2212yes—
31C2×#388C2214yes—
32C2×#388C2212yes—
33C2×#387C2214yes—
34C2×#387C2212yes—
35C2×#386C2212yes—
36C2×#386C2211yescharacteristic
37C2×#385C2212yes—
38C2×#385C2211yescharacteristic
39C2×#385C2212yes—
40C2×#381C2211yescharacteristic
41C2×#381C2211yescharacteristic
42C2×#381C2212yes—
43C2×#381C2211yescharacteristic
44C2×#381C2212yes—
45C2×#381C2212yes—
46C2×#380C2212yes—
47C2×C4×C22⋊C4C2211yescharacteristic
48C22×#95C2212yes—
49C22×#95C2212yes—
50#1446C2214yes—
51#1446C2214yes—
52#1443C2214yes—
53#1443C2214yes—
54#1442C2214yes—
55#1442C2214yes—
56#1441C2214yes—
57#1441C2214yes—
58#1441C2214yes—
59#1441C2214yes—
60#1441C2214yes—
61#1441C2214yes—
62#1441C2214yes—
63#1440C2214yes—
64#1440C2214yes—
65#1439C2214yes—
66#1436C2214yes—
67#1436C2214yes—
68#1434C2214yes—
69#1434C2214yes—
70#1434C2214yes—
71C22⋊#95C2212yes—
72C22⋊#95C2212yes—
73C22⋊#95C2212yes—
74C22⋊#95C2214yes—
75C23×Q8—24no—
76C23×Q8—24no—
77C23×Q8—28no—
78C23×Q8—24no—
79C23×Q8—44no—
80C23×Q8C2314yes—
81C23×D4C2311yescharacteristic
82C24×C4C2312yes—
83C24×C4C2312yes—
84C24×C4C2311yescharacteristic
85C24×C4C2311yescharacteristic
86C24×C4C2311yescharacteristic
87C24×C4C2311yescharacteristic
88C2×#123—24no—
89C2×#123—24no—
90C2×#123—28no—
91C2×#123—28no—
92C2×#123—44no—
93C2×#123D418yes—
94C2×C4×D4—24no—
95C2×C4×D4—44no—
96C22×C4⋊C4—44no—
97C22×C4⋊C4D418yes—
98C22×C4⋊C4C2314yes—
99C22×C4⋊C4C2314yes—
100C22×C4⋊C4C2312yes—
101C22×C4⋊C4C2311yescharacteristic
102C22×C4⋊C4C2312yes—
103C22×C4⋊C4C2314yes—
104C22×C4⋊C4C2312yes—
105C22×C4⋊C4C2312yes—
106C22×C22⋊C4C2311yescharacteristic
107C22×C22⋊C4C2312yes—
108C22×C22⋊C4C2311yescharacteristic
109C22×C22⋊C4C2312yes—
110C22×C22⋊C4C2314yes—
111C22×C22⋊C4C2314yes—
112C22×C22⋊C4C2312yes—
113C22×C22⋊C4C2311yescharacteristic
114C22×C22⋊C4C2311yescharacteristic
115C22×C22⋊C4C2312yes—
116C22×C42—24no—
117C22×C42—24no—
118C22×C42—24no—
119C22×C42C2314yes—
120C22×C42C2311yescharacteristic
121C22×C42C2312yes—
122#395—216no—
123#393—216no—
124#391—216no—
125#391—216no—
126#389—28no—
127#388—28no—
128#388—216no—
129#387—28no—
130#387—216no—
131#386—28no—
132#386—24no—
133#385—24no—
134#385—28no—
135#385—28no—
136#381—24no—
137#381—24no—
138#381—24no—
139#381—28no—
140#381—28no—
141#381—44no—
142#380—44no—
143C4×C22⋊C4—24no—
144C2×#95—28no—
145C2×#95—28no—
146C2×#95C2314yes—
147C2×#95C2314yes—
148C2×#95C2314yes—
149C2×#95C2314yes—
150C2×#95C2314yes—
151C2×#95C2314yes—
152C2×#95C2312yes—
153C2×#95C2312yes—
154C2×#95C2314yes—
155C2×#95C2314yes—
156C2×#95C2314yes—
157C2×#95C2314yes—
158C2×#95C2314yes—
159C2×#95C2314yes—
160C2×#95C2314yes—
161C2×#95C2312yes—
162C2×#95C2312yes—
163C2×#95C2314yes—
164C2×#95C2312yes—
165C2×#95C2314yes—
166C2×#95C2314yes—
167C2×#95C2314yes—
168C2×#95C2314yes—
169C2×#95C2314yes—
170C2×#95C2314yes—
171C2×#95C2314yes—
172C25C2411yescharacteristic
173C25C2411yescharacteristic
174C22×Q8—232no—
175C22×Q8—28no—
176C22×Q8—28no—
177C22×Q8—28no—
178C22×Q8—28no—
179C22×Q8—28no—
180C22×Q8—44no—
181C22×Q8—44no—
182C22×Q8—432no—
183C22×Q8—432no—
184C22×Q8—416no—
185C22×Q8—416no—
186C22×D4—22no—
187C22×D4—22no—
188C22×D4—24no—
189C22×D4—24no—
190C22×D4C2×D414yes—
191C23×C4—24no—
192C23×C4—24no—
193C23×C4—24no—
194C23×C4—24no—
195C23×C4—24no—
196C23×C4—24no—
197C23×C4—24no—
198C23×C4—24no—
199C23×C4—24no—
200C23×C4—24no—
201C23×C4—24no—
202C23×C4—28no—
203C23×C4—24no—
204C23×C4—24no—
205C23×C4—22no—
206C23×C4—44no—
207C23×C4—42no—
208C23×C4—44no—
209C23×C4—44no—
210C23×C4—44no—
211C23×C4C2×D414yes—
212C23×C4C2×D414yes—
213C23×C4C2×D414yes—
214C23×C4C2×D414yes—
215C23×C4#4118yes—
216C23×C4#4114yes—
217C23×C4C2414yes—
218C23×C4C2414yes—
219C23×C4C2412yes—
220C23×C4C2411yescharacteristic
221C23×C4C2411yescharacteristic
222C23×C4C2412yes—
223C23×C4C2412yes—
224C23×C4C2411yescharacteristic
225C23×C4C2411yescharacteristic
226C23×C4C2411yescharacteristic
227C23×C4C2414yes—
228C23×C4C2414yes—
229C23×C4C2412yes—
230#123—232no—
231#123—432no—
232C4×D4—416no—
233C2×C4⋊C4—24no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—28no—
237C2×C4⋊C4—28no—
238C2×C4⋊C4—28no—
239C2×C4⋊C4—28no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—416no—
242C2×C4⋊C4—42no—
243C2×C4⋊C4—42no—
244C2×C4⋊C4—42no—
245C2×C4⋊C4—42no—
246C2×C4⋊C4—44no—
247C2×C4⋊C4—42no—
248C2×C4⋊C4—44no—
249C2×C4⋊C4—44no—
250C2×C4⋊C4—44no—
251C2×C4⋊C4—44no—
252C2×C4⋊C4—44no—
253C2×C4⋊C4C2×D418yes—
254C2×C4⋊C4C2×D418yes—
255C2×C4⋊C4C2×D418yes—
256C2×C22⋊C4—22no—
257C2×C22⋊C4—22no—
258C2×C22⋊C4—24no—
259C2×C22⋊C4—24no—
260C2×C22⋊C4—22no—
261C2×C22⋊C4—24no—
262C2×C22⋊C4—28no—
263C2×C22⋊C4—22no—
264C2×C22⋊C4—24no—
265C2×C22⋊C4—28no—
266C2×C22⋊C4—24no—
267C2×C22⋊C4—24no—
268C2×C22⋊C4—24no—
269C2×C22⋊C4—24no—
270C2×C22⋊C4—24no—
271C2×C22⋊C4—24no—
272C2×C22⋊C4—22no—
273C2×C22⋊C4—28no—
274C2×C22⋊C4—28no—
275C2×C22⋊C4—24no—
276C2×C22⋊C4—44no—
277C2×C22⋊C4—42no—
278C2×C22⋊C4—44no—
279C2×C22⋊C4—44no—
280C2×C22⋊C4C2×D414yes—
281C2×C22⋊C4C2×D418yes—
282C2×C42—22no—
283C2×C42—28no—
284C2×C42—28no—
285C2×C42—28no—
286C2×C42—24no—
287C2×C42—42no—
288C2×C42—416no—
289C2×C42—44no—
290#95—28no—
291#95—28no—
292#95—28no—
293#95—28no—
294#95—24no—
295#95—24no—
296#95—24no—
297#95—28no—
298#95—28no—
299#95—28no—
300#95—28no—
301#95—28no—
302#95—24no—
303#95—28no—
304#95—24no—
305#95—28no—
306#95—28no—
307#95—28no—
308#95—28no—
309#95—44no—
310#95—44no—
311#95—44no—
312#95—44no—
313#95—44no—
314#95—44no—
315#95—44no—
316C24—24no—
317C24—24no—
318C24—21no—
319C24—21no—
320C24—22no—
321C24—22no—
322C24—24no—
323C24—24no—
324C24C22×D412yes—
325C24C22×D412yes—
326C24C22×D412yes—
327C24C2×#4112yes—
328C24C2×#4114yes—
329C24H(5,𝔽2)14yes—
330C24C2511yescharacteristic
331C2×Q8—232no—
332C2×Q8—416no—
333C2×Q8—416no—
334C2×Q8—416no—
335C2×Q8—416no—
336C2×Q8—416no—
337C2×Q8—4128no—
338C2×Q8—416no—
339C2×D4—28no—
340C2×D4—216no—
341C2×D4—44no—
342C2×D4—44no—
343C2×D4—48no—
344C22×C4—28no—
345C22×C4—28no—
346C22×C4—216no—
347C22×C4—216no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—216no—
353C22×C4—216no—
354C22×C4—216no—
355C22×C4—216no—
356C22×C4—24no—
357C22×C4—24no—
358C22×C4—24no—
359C22×C4—22no—
360C22×C4—21no—
361C22×C4—24no—
362C22×C4—24no—
363C22×C4—22no—
364C22×C4—21no—
365C22×C4—22no—
366C22×C4—24no—
367C22×C4—24no—
368C22×C4—22no—
369C22×C4—24no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—24no—
374C22×C4—22no—
375C22×C4—24no—
376C22×C4—24no—
377C22×C4—24no—
378C22×C4—24no—
379C22×C4—24no—
380C22×C4—24no—
381C22×C4—24no—
382C22×C4—24no—
383C22×C4—24no—
384C22×C4—28no—
385C22×C4—28no—
386C22×C4—22no—
387C22×C4—21no—
388C22×C4—24no—
389C22×C4—24no—
390C22×C4—21no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—24no—
394C22×C4—24no—
395C22×C4—24no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—24no—
399C22×C4—24no—
400C22×C4—24no—
401C22×C4—24no—
402C22×C4—24no—
403C22×C4—24no—
404C22×C4—24no—
405C22×C4—24no—
406C22×C4—24no—
407C22×C4—24no—
408C22×C4—24no—
409C22×C4—24no—
410C22×C4—24no—
411C22×C4—48no—
412C22×C4—48no—
413C22×C4—416no—
414C22×C4—416no—
415C22×C4—44no—
416C22×C4—48no—
417C22×C4—48no—
418C22×C4—48no—
419C22×C4—48no—
420C22×C4—48no—
421C22×C4—48no—
422C22×C4—84no—
423C22×C4—88no—
424C22×C4—88no—
425C22×C4—88no—
426C22×C4#121116yes—
427C22×C4C22×D412yes—
428C22×C4C22×D414yes—
429C22×C4C22×D414yes—
430C22×C4C22×D412yes—
431C22×C4C22×D412yes—
432C22×C4C22×D414yes—
433C22×C4C22×D414yes—
434C22×C4C2×#4118yes—
435C22×C4C2×#4114yes—
436C22×C42−1+418yes—
437C4⋊C4—216no—
438C4⋊C4—48no—
439C4⋊C4—48no—
440C4⋊C4—416no—
441C4⋊C4—416no—
442C4⋊C4—44no—
443C4⋊C4—84no—
444C4⋊C4—88no—
445C22⋊C4—28no—
446C22⋊C4—216no—
447C22⋊C4—416no—
448C22⋊C4—416no—
449C22⋊C4—44no—
450C22⋊C4—44no—
451C22⋊C4—44no—
452C22⋊C4—48no—
453C22⋊C4—48no—
454C22⋊C4—84no—
455C42—44no—
456C42—416no—
457C42—84no—
458C23—28no—
459C23—24no—
460C23—24no—
461C23—28no—
462C23—28no—
463C23—28no—
464C23—28no—
465C23—28no—
466C23—22no—
467C23—24no—
468C23—28no—
469C23—41no—
470C23—41no—
471C23—44no—
472C23—42no—
473C23—42no—
474C23—42no—
475C23—44no—
476C23—44no—
477C23—44no—
478C23—44no—
479C23C2×#12114yes—
480C23#52514yes—
481C23#53414yes—
482C23#54618yes—
483C23#55118yes—
484C23C23×D411yescharacteristic
485C23C23×D411yescharacteristic
486C23C22×#4111yescharacteristic
487C23C22×#4112yes—
488C23C2×H(5,𝔽2)12yes—
489C23C2×2−1+414yes—
490C23#58514yes—
491Q8—464no—
492D4—416no—
493C2×C4—24no—
494C2×C4—24no—
495C2×C4—24no—
496C2×C4—24no—
497C2×C4—24no—
498C2×C4—24no—
499C2×C4—232no—
500C2×C4—24no—
501C2×C4—22no—
502C2×C4—28no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—24no—
506C2×C4—28no—
507C2×C4—24no—
508C2×C4—28no—
509C2×C4—24no—
510C2×C4—24no—
511C2×C4—24no—
512C2×C4—24no—
513C2×C4—28no—
514C2×C4—28no—
515C2×C4—28no—
516C2×C4—28no—
517C2×C4—432no—
518C2×C4—44no—
519C2×C4—44no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—44no—
523C2×C4—44no—
524C2×C4—44no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—44no—
528C2×C4—42no—
529C2×C4—44no—
530C2×C4—44no—
531C2×C4—44no—
532C2×C4—44no—
533C2×C4—44no—
534C2×C4—44no—
535C2×C4—44no—
536C2×C4—432no—
537C2×C4—42no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—44no—
541C2×C4—44no—
542C2×C4—44no—
543C2×C4—44no—
544C2×C4—44no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—44no—
550C2×C4—44no—
551C2×C4—44no—
552C2×C4—44no—
553C2×C4—44no—
554C2×C4—44no—
555C2×C4—44no—
556C2×C4—44no—
557C2×C4—44no—
558C2×C4—44no—
559C2×C4—44no—
560C2×C4—44no—
561C2×C4—44no—
562C2×C4—88no—
563C2×C4—88no—
564C2×C4—88no—
565C2×C4C2×#12118yes—
566C22—28no—
567C22—28no—
568C22—216no—
569C22—24no—
570C22—216no—
571C22—24no—
572C22—42no—
573C22—44no—
574C22—48no—
575C22—82no—
576C22—84no—
577C22—81no—
578C22—81no—
579C22#253418yes—
580C22C22×#12111yescharacteristic
581C22C2×#52511yescharacteristic
582C22C2×#53411yescharacteristic
583C22#346114yes—
584C22C2×#54612yes—
585C22C2×#55112yes—
586C22#348512yes—
587C22#348612yes—
588C22#348914yes—
589C22#349514yes—
590C22#350714yes—
591C22#354414yes—
592C22#354514yes—
593C4—28no—
594C4—48no—
595C4—42no—
596C4—44no—
597C4—48no—
598C4—42no—
599C4—44no—
600C4—42no—
601C4—82no—
602C4—81no—
603C4—84no—
604C4—84no—
605C4—81no—
606C2—28no—
607C2—82no—
608C2C2×#253411yescharacteristic
609C2#3509212yes—
610C2#3563312yes—
611C2#3711911yescharacteristic
612C2#3725811yescharacteristic
613C2#3777914yes—
614C2#3796714yes—
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
4. C22×#391C2semidirect
5. #15867C2semidirect
6. #15860C2semidirect
7. #15832C2semidirect
8. #15824C2semidirect
9. #14935C2semidirect
11. #14684C2semidirect
12. #14339C2semidirect
13. #14336C2semidirect
14. #13992C2semidirect

Maximal subgroups

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

subgroupindex
C2×#28532
C2×#26142
C2×#23222
C22×#3912
#158672
#158602
#158322
#158242
#149352
#149272
#146842
#143392
#143362
#139922
#139682