Finite Group Databasebeta

← #620501

Subgroups — #620501

Order 512.

3012 conjugacy classes of subgroups.

The subgroup lattice has 601 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2322C211yesmaximal, characteristic
2C22×#384C211yesmaximal, characteristic, complemented
3#16095C214yesmaximal, complemented
4#15438C214yesmaximal, complemented
5#15396C214yesmaximal, complemented
6#15392C214yesmaximal, complemented
7#14729C212yesmaximal
8#14728C212yesmaximal, complemented
9#14706C211yesmaximal, characteristic
10#14705C211yesmaximal, characteristic, complemented
11#14339C211yesmaximal, characteristic, complemented
12#14263C211yesmaximal, characteristic, complemented
13#14256C211yesmaximal, characteristic
14#14039C212yesmaximal, complemented
15#14015C212yesmaximal
16C22×#123C2211yescharacteristic
17C22×C4×D4C2211yescharacteristic
18C23×C4⋊C4C2212yes—
19C23×C4⋊C4C2211yescharacteristic
20C23×C4⋊C4C2212yes—
21C23×C4⋊C4C2211yescharacteristic
22C23×C42C2211yescharacteristic
23C2×#398C2214yes—
24C2×#396C2214yes—
25#2322—28no—
26C2×#389C2212yes—
27C2×#388C2212yes—
28C2×#387C2212yes—
29C2×#387C2214yes—
30C2×#386C2211yescharacteristic
31C2×#386C2212yes—
32C2×#386C2212yes—
33C2×#385C2212yes—
34C2×#385C2211yescharacteristic
35C2×#385C2212yes—
36C2×#385C2214yes—
37C2×#385C2212yes—
38C2×#384—28no—
39C2×#384C2214yes—
40C2×#384C2214yes—
41C2×#381C2211yescharacteristic
42C2×#381C2211yescharacteristic
43C2×#381C2211yescharacteristic
44C2×#380C2212yes—
45C2×C4×C22⋊C4C2211yescharacteristic
46#1444C2214yes—
47#1444C2214yes—
48#1444C2214yes—
49#1442C2214yes—
50#1442C2214yes—
51#1442C2214yes—
52#1441C2214yes—
53#1441C2214yes—
54#1441C2214yes—
55#1441C2214yes—
56#1441C2214yes—
57#1441C2214yes—
58#1441C2214yes—
59#1439C2214yes—
60#1439C2214yes—
61#1439C2214yes—
62#1437C2214yes—
63#1436C2214yes—
64#1436C2214yes—
65#1436C2214yes—
66#1436C2214yes—
67#1436C2214yes—
68C22⋊#95C2212yes—
69C22⋊#95C2212yes—
70C22⋊#95C2212yes—
71C22⋊#95C2212yes—
72C22⋊#95C2212yes—
73C23×D4C2311yescharacteristic
74C24×C4C2312yes—
75C24×C4C2311yescharacteristic
76C24×C4C2312yes—
77C24×C4C2311yescharacteristic
78C24×C4C2311yescharacteristic
79C24×C4C2311yescharacteristic
80C2×#123—24no—
81C2×#123—24no—
82C2×#123Q818yes—
83C2×C4×D4—24no—
84C2×C4×D4—44no—
85C22×C4⋊C4—28no—
86C22×C4⋊C4—28no—
87C22×C4⋊C4—24no—
88C22×C4⋊C4—48no—
89C22×C4⋊C4—48no—
90C22×C4⋊C4—44no—
91C22×C4⋊C4—44no—
92C22×C4⋊C4D418yes—
93C22×C4⋊C4C2314yes—
94C22×C4⋊C4C2314yes—
95C22×C4⋊C4C2314yes—
96C22×C4⋊C4C2314yes—
97C22×C4⋊C4C2314yes—
98C22×C4⋊C4C2312yes—
99C22×C4⋊C4C2312yes—
100C22×C4⋊C4C2314yes—
101C22×C4⋊C4C2311yescharacteristic
102C22×C4⋊C4C2314yes—
103C22×C4⋊C4C2314yes—
104C22×C4⋊C4C2312yes—
105C22×C4⋊C4C2312yes—
106C22×C22⋊C4C2314yes—
107C22×C22⋊C4C2311yescharacteristic
108C22×C22⋊C4C2312yes—
109C22×C22⋊C4C2312yes—
110C22×C22⋊C4C2311yescharacteristic
111C22×C22⋊C4C2311yescharacteristic
112C22×C22⋊C4C2312yes—
113C22×C22⋊C4C2312yes—
114C22×C22⋊C4C2311yescharacteristic
115C22×C22⋊C4C2314yes—
116C22×C42—24no—
117C22×C42—28no—
118C22×C42C2314yes—
119C22×C42C2311yescharacteristic
120C22×C42C2312yes—
121#398—48no—
122#396—48no—
123#389—28no—
124#388—28no—
125#387—28no—
126#387—48no—
127#386—24no—
128#386—28no—
129#386—44no—
130#385—24no—
131#385—28no—
132#385—28no—
133#385—48no—
134#385—44no—
135#384—216no—
136#384—216no—
137#381—24no—
138#381—24no—
139#381—24no—
140#380—44no—
141C4×C22⋊C4—24no—
142C2×#95C2314yes—
143C2×#95C2314yes—
144C2×#95C2314yes—
145C2×#95C2314yes—
146C2×#95C2314yes—
147C2×#95C2314yes—
148C2×#95C2314yes—
149C2×#95C2312yes—
150C2×#95C2312yes—
151C2×#95C2314yes—
152C2×#95C2312yes—
153C2×#95C2312yes—
154C2×#95C2312yes—
155C2×#95C2314yes—
156C2×#95C2314yes—
157C2×#95C2314yes—
158C2×#95C2314yes—
159C2×#95C2314yes—
160C2×#95C2314yes—
161C2×#95C2314yes—
162C2×#95C2314yes—
163C2×#95C2314yes—
164C25C2411yescharacteristic
165C25C2411yescharacteristic
166C22×D4—22no—
167C22×D4—22no—
168C22×D4—24no—
169C22×D4—24no—
170C22×D4C2×Q814yes—
171C23×C4—24no—
172C23×C4—24no—
173C23×C4—24no—
174C23×C4—24no—
175C23×C4—28no—
176C23×C4—28no—
177C23×C4—24no—
178C23×C4—24no—
179C23×C4—24no—
180C23×C4—24no—
181C23×C4—24no—
182C23×C4—22no—
183C23×C4—44no—
184C23×C4—44no—
185C23×C4—44no—
186C23×C4—44no—
187C23×C4—42no—
188C23×C4—44no—
189C23×C4C2×D414yes—
190C23×C4C2×D414yes—
191C23×C4C2×D414yes—
192C23×C4C2×Q814yes—
193C23×C4#4118yes—
194C23×C4#4114yes—
195C23×C4C2414yes—
196C23×C4C2414yes—
197C23×C4C2412yes—
198C23×C4C2411yescharacteristic
199C23×C4C2411yescharacteristic
200C23×C4C2414yes—
201C23×C4C2412yes—
202C23×C4C2411yescharacteristic
203C23×C4C2411yescharacteristic
204C23×C4C2412yes—
205C23×C4C2411yescharacteristic
206C23×C4C2412yes—
207C23×C4C2414yes—
208#123—232no—
209C4×D4—416no—
210C2×C4⋊C4—24no—
211C2×C4⋊C4—24no—
212C2×C4⋊C4—28no—
213C2×C4⋊C4—28no—
214C2×C4⋊C4—28no—
215C2×C4⋊C4—28no—
216C2×C4⋊C4—28no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—28no—
219C2×C4⋊C4—28no—
220C2×C4⋊C4—28no—
221C2×C4⋊C4—28no—
222C2×C4⋊C4—28no—
223C2×C4⋊C4—28no—
224C2×C4⋊C4—28no—
225C2×C4⋊C4—24no—
226C2×C4⋊C4—416no—
227C2×C4⋊C4—42no—
228C2×C4⋊C4—42no—
229C2×C4⋊C4—42no—
230C2×C4⋊C4—42no—
231C2×C4⋊C4—44no—
232C2×C4⋊C4—44no—
233C2×C4⋊C4—44no—
234C2×C4⋊C4—44no—
235C2×C4⋊C4—44no—
236C2×C4⋊C4—44no—
237C2×C4⋊C4—42no—
238C2×C4⋊C4—432no—
239C2×C4⋊C4—432no—
240C2×C4⋊C4—416no—
241C2×C4⋊C4—44no—
242C2×C4⋊C4—44no—
243C2×C4⋊C4—44no—
244C2×C4⋊C4—44no—
245C2×C4⋊C4—44no—
246C2×C4⋊C4—44no—
247C2×C4⋊C4—44no—
248C2×C4⋊C4C2×D418yes—
249C2×C4⋊C4C2×D418yes—
250C2×C4⋊C4C2×Q818yes—
251C2×C22⋊C4—22no—
252C2×C22⋊C4—22no—
253C2×C22⋊C4—24no—
254C2×C22⋊C4—24no—
255C2×C22⋊C4—22no—
256C2×C22⋊C4—22no—
257C2×C22⋊C4—24no—
258C2×C22⋊C4—24no—
259C2×C22⋊C4—24no—
260C2×C22⋊C4—24no—
261C2×C22⋊C4—24no—
262C2×C22⋊C4—24no—
263C2×C22⋊C4—24no—
264C2×C22⋊C4—24no—
265C2×C22⋊C4—22no—
266C2×C22⋊C4—28no—
267C2×C22⋊C4—28no—
268C2×C22⋊C4—28no—
269C2×C22⋊C4—24no—
270C2×C22⋊C4—42no—
271C2×C22⋊C4—44no—
272C2×C22⋊C4—44no—
273C2×C22⋊C4—44no—
274C2×C22⋊C4—44no—
275C2×C22⋊C4C2×Q814yes—
276C2×C22⋊C4C2×Q818yes—
277C2×C42—232no—
278C2×C42—28no—
279C2×C42—28no—
280C2×C42—28no—
281C2×C42—22no—
282C2×C42—24no—
283C2×C42—42no—
284C2×C42—44no—
285#95—24no—
286#95—24no—
287#95—24no—
288#95—24no—
289#95—24no—
290#95—28no—
291#95—28no—
292#95—28no—
293#95—28no—
294#95—28no—
295#95—28no—
296#95—28no—
297#95—28no—
298#95—44no—
299#95—44no—
300#95—44no—
301#95—44no—
302#95—44no—
303#95—44no—
304#95—44no—
305#95—44no—
306#95—44no—
307C24—24no—
308C24—24no—
309C24—21no—
310C24—21no—
311C24—22no—
312C24—24no—
313C24—22no—
314C24—24no—
315C24C22×D412yes—
316C24C22×Q812yes—
317C24C22×Q812yes—
318C24C2×#4112yes—
319C24C2×#4114yes—
320C24H(5,𝔽2)14yes—
321C24C2511yescharacteristic
322C2×D4—28no—
323C2×D4—216no—
324C2×D4—44no—
325C2×D4—44no—
326C2×D4—48no—
327C22×C4—216no—
328C22×C4—216no—
329C22×C4—232no—
330C22×C4—28no—
331C22×C4—216no—
332C22×C4—28no—
333C22×C4—216no—
334C22×C4—216no—
335C22×C4—24no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—24no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—24no—
342C22×C4—24no—
343C22×C4—232no—
344C22×C4—22no—
345C22×C4—21no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—21no—
349C22×C4—24no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—24no—
353C22×C4—24no—
354C22×C4—22no—
355C22×C4—24no—
356C22×C4—22no—
357C22×C4—24no—
358C22×C4—22no—
359C22×C4—24no—
360C22×C4—28no—
361C22×C4—28no—
362C22×C4—22no—
363C22×C4—21no—
364C22×C4—21no—
365C22×C4—24no—
366C22×C4—22no—
367C22×C4—24no—
368C22×C4—24no—
369C22×C4—24no—
370C22×C4—24no—
371C22×C4—22no—
372C22×C4—24no—
373C22×C4—24no—
374C22×C4—24no—
375C22×C4—22no—
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—24no—
385C22×C4—24no—
386C22×C4—22no—
387C22×C4—24no—
388C22×C4—24no—
389C22×C4—24no—
390C22×C4—24no—
391C22×C4—24no—
392C22×C4—24no—
393C22×C4—24no—
394C22×C4—48no—
395C22×C4—48no—
396C22×C4—44no—
397C22×C4—48no—
398C22×C4—48no—
399C22×C4—48no—
400C22×C4—84no—
401C22×C4—88no—
402C22×C4—88no—
403C22×C4—88no—
404C22×C4—88no—
405C22×C4—88no—
406C22×C4C22⋊Q8116yes—
407C22×C4C22×D412yes—
408C22×C4C22×D414yes—
409C22×C4C22×D412yes—
410C22×C4C22×D414yes—
411C22×C4C22×Q814yes—
412C22×C4C22×Q812yes—
413C22×C4C22×Q814yes—
414C22×C4C2×#4114yes—
415C22×C4C2×#4118yes—
416C22×C4H(5,𝔽2)18yes—
417C22×C42−1+418yes—
418C4⋊C4—216no—
419C4⋊C4—48no—
420C4⋊C4—48no—
421C4⋊C4—416no—
422C4⋊C4—416no—
423C4⋊C4—416no—
424C4⋊C4—416no—
425C4⋊C4—416no—
426C4⋊C4—44no—
427C4⋊C4—416no—
428C4⋊C4—84no—
429C4⋊C4—88no—
430C4⋊C4—88no—
431C22⋊C4—28no—
432C22⋊C4—216no—
433C22⋊C4—44no—
434C22⋊C4—44no—
435C22⋊C4—44no—
436C22⋊C4—48no—
437C22⋊C4—48no—
438C22⋊C4—416no—
439C22⋊C4—84no—
440C22⋊C4—88no—
441C42—232no—
442C42—44no—
443C42—84no—
444C23—28no—
445C23—24no—
446C23—24no—
447C23—28no—
448C23—28no—
449C23—28no—
450C23—28no—
451C23—28no—
452C23—22no—
453C23—24no—
454C23—28no—
455C23—41no—
456C23—41no—
457C23—44no—
458C23—42no—
459C23—42no—
460C23—42no—
461C23—44no—
462C23—44no—
463C23—44no—
464C23—44no—
465C23C2×C22⋊Q814yes—
466C23#52514yes—
467C23#53414yes—
468C23#55318yes—
469C23#55418yes—
470C23C23×D411yescharacteristic
471C23C23×Q811yescharacteristic
472C23C22×#4111yescharacteristic
473C23C22×#4112yes—
474C23C2×H(5,𝔽2)12yes—
475C23C2×H(5,𝔽2)14yes—
476C23C2×2−1+414yes—
477D4—416no—
478C2×C4—264no—
479C2×C4—232no—
480C2×C4—24no—
481C2×C4—28no—
482C2×C4—24no—
483C2×C4—28no—
484C2×C4—24no—
485C2×C4—24no—
486C2×C4—24no—
487C2×C4—24no—
488C2×C4—28no—
489C2×C4—28no—
490C2×C4—28no—
491C2×C4—28no—
492C2×C4—24no—
493C2×C4—22no—
494C2×C4—22no—
495C2×C4—22no—
496C2×C4—24no—
497C2×C4—24no—
498C2×C4—24no—
499C2×C4—24no—
500C2×C4—24no—
501C2×C4—24no—
502C2×C4—28no—
503C2×C4—41no—
504C2×C4—41no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—44no—
509C2×C4—44no—
510C2×C4—44no—
511C2×C4—44no—
512C2×C4—44no—
513C2×C4—42no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—44no—
517C2×C4—44no—
518C2×C4—44no—
519C2×C4—44no—
520C2×C4—44no—
521C2×C4—44no—
522C2×C4—42no—
523C2×C4—44no—
524C2×C4—44no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—44no—
529C2×C4—44no—
530C2×C4—44no—
531C2×C4—44no—
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—44no—
545C2×C4—88no—
546C2×C4—88no—
547C2×C4—88no—
548C2×C4—816no—
549C2×C4C2×C22⋊Q818yes—
550C2×C4#539116yes—
551C22—216no—
552C22—24no—
553C22—24no—
554C22—28no—
555C22—216no—
556C22—28no—
557C22—42no—
558C22—48no—
559C22—44no—
560C22—84no—
561C22—81no—
562C22—81no—
563C22—82no—
564C22#257918yes—
565C22C22×C22⋊Q811yescharacteristic
566C22C2×#52511yescharacteristic
567C22C2×#53411yescharacteristic
568C22C2×#53914yes—
569C22C2×#55312yes—
570C22C2×#55412yes—
571C22#348412yes—
572C22#348612yes—
573C22#349214yes—
574C22#350314yes—
575C22#351114yes—
576C22#354314yes—
577C22#354414yes—
578C4—28no—
579C4—216no—
580C4—42no—
581C4—42no—
582C4—44no—
583C4—42no—
584C4—44no—
585C4—84no—
586C4—84no—
587C4—81no—
588C4—84no—
589C4—81no—
590C4—82no—
591C2—28no—
592C2—82no—
593C2C2×#257911yescharacteristic
594C2#3498912yes—
595C2#3557412yes—
596C2#3723211yescharacteristic
597C2#3725811yescharacteristic
598C2#3960414yes—
599C2#3973214yes—
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×#384C2semidirect
3. #16095C2semidirect
4. #15438C2semidirect
5. #15396C2semidirect
6. #15392C2semidirect
8. #14728C2semidirect
10. #14705C2semidirect
11. #14339C2semidirect
12. #14263C2semidirect
14. #14039C2semidirect

Maximal subgroups

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

subgroupindex
C2×#23222
C22×#3842
#160952
#154382
#153962
#153922
#147292
#147282
#147062
#147052
#143392
#142632
#142562
#140392
#140152