Finite Group Databasebeta

← #616477

Subgroups — #616477

Order 512.

2750 conjugacy classes of subgroups.

The subgroup lattice has 639 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2859C211yesmaximal, characteristic, complemented
2C2×#2851C211yesmaximal, characteristic
3C2×#2851C211yesmaximal, characteristic
4C2×C4×#125C211yesmaximal, characteristic, complemented
5#15220C212yesmaximal
6#15137C212yesmaximal, complemented
7#15005C214yesmaximal, complemented
8#14753C214yesmaximal
9#14709C211yesmaximal, characteristic, complemented
10#14707C211yesmaximal, characteristic, complemented
11#14208C214yesmaximal, complemented
12#14112C214yesmaximal
13#14006C212yesmaximal, complemented
14#13876C212yesmaximal
15C42×C22⋊C4C211yesmaximal, characteristic
16C22×#125C2211yescharacteristic
17C22×#125C2211yescharacteristic
18C23×C42C2211yescharacteristic
19#2859—28no—
20#2851—28no—
21#2851—28no—
22C2×#398C2214yes—
23C4×#125—28no—
24C2×#389C2212yes—
25C2×#389C2212yes—
26C2×#389C2214yes—
27C2×#389C2214yes—
28C2×#389C2212yes—
29C2×#389C2212yes—
30C2×#388C2214yes—
31C2×#387C2214yes—
32C2×#387C2214yes—
33C2×#387C2214yes—
34C2×#385C2214yes—
35C2×#385C2214yes—
36C2×#384C2212yes—
37C2×#382C2211yescharacteristic
38C2×#382C2212yes—
39C2×#382C2211yescharacteristic
40C2×#380C2212yes—
41C2×#380C2212yes—
42C2×C4×C4⋊C4C2214yes—
43C2×C4×C4⋊C4C2212yes—
44C2×C4×C22⋊C4C2212yes—
45C2×C4×C22⋊C4C2211yescharacteristic
46C2×C4×C22⋊C4C2212yes—
47C2×C4×C22⋊C4C2211yescharacteristic
48C2×C43C2211yescharacteristic
49C2×C43C2211yescharacteristic
50#1444C2214yes—
51#1443C2214yes—
52#1442C2214yes—
53#1442C2214yes—
54#1442C2214yes—
55#1441C2214yes—
56#1441C2214yes—
57#1441C2214yes—
58#1439C2214yes—
59#1439C2214yes—
60#1439C2214yes—
61#1436C2214yes—
62#1436C2214yes—
63#1436C2214yes—
64#1436C2214yes—
65#1436C2214yes—
66#1436C2214yes—
67C4⋊#95C2214yes—
68C4⋊#95C2212yes—
69C4⋊#95C2212yes—
70C4⋊#95C2212yes—
71C4×#95C2211yescharacteristic
72C4×#95C2212yes—
73C4×#95C2212yes—
74C4×#95C2211yescharacteristic
75C24×C4C2311yescharacteristic
76C24×C4C2312yes—
77C2×#125—24no—
78C2×#125—24no—
79C2×#125—24no—
80C2×#125—24no—
81C2×#125—24no—
82C2×#125D418yes—
83C22×C4⋊C4C2312yes—
84C22×C4⋊C4C2314yes—
85C22×C4⋊C4C2312yes—
86C22×C4⋊C4C2314yes—
87C22×C4⋊C4C2314yes—
88C22×C4⋊C4C2314yes—
89C22×C4⋊C4C2314yes—
90C22×C4⋊C4C2314yes—
91C22×C4⋊C4C2312yes—
92C22×C4⋊C4C2314yes—
93C22×C4⋊C4C2312yes—
94C22×C4⋊C4C2314yes—
95C22×C22⋊C4C2312yes—
96C22×C22⋊C4C2314yes—
97C22×C22⋊C4C2314yes—
98C22×C22⋊C4C2311yescharacteristic
99C22×C22⋊C4C2311yescharacteristic
100C22×C42—24no—
101C22×C42—24no—
102C22×C42—24no—
103C22×C42C2312yes—
104C22×C42C2311yescharacteristic
105C22×C42C2311yescharacteristic
106C22×C42C2312yes—
107C22×C42C2312yes—
108C22×C42C2312yes—
109C22×C42C2312yes—
110C22×C42C2314yes—
111C22×C42C2311yescharacteristic
112C22×C42C2311yescharacteristic
113C22×C42C2311yescharacteristic
114C22×C42C2311yescharacteristic
115#398—216no—
116#389—28no—
117#389—28no—
118#389—28no—
119#389—216no—
120#389—216no—
121#389—28no—
122#388—216no—
123#387—216no—
124#387—216no—
125#387—216no—
126#385—216no—
127#385—216no—
128#384—28no—
129#382—28no—
130#382—24no—
131#382D418yes—
132#380—28no—
133#380—28no—
134C4×C4⋊C4—28no—
135C4×C4⋊C4—216no—
136C4×C22⋊C4—28no—
137C4×C22⋊C4—24no—
138C4×C22⋊C4—28no—
139C4×C22⋊C4Q818yes—
140C2×#95C2314yes—
141C2×#95C2314yes—
142C2×#95C2312yes—
143C2×#95C2312yes—
144C2×#95C2312yes—
145C2×#95C2312yes—
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—
160C2×#95C2314yes—
161C2×#95C2311yescharacteristic
162C2×#95C2311yescharacteristic
163C2×#95C2312yes—
164C43—24no—
165C43—24no—
166C25C2411yescharacteristic
167C23×C4—24no—
168C23×C4—22no—
169C23×C4—22no—
170C23×C4—24no—
171C23×C4—22no—
172C23×C4—24no—
173C23×C4—22no—
174C23×C4—22no—
175C23×C4—22no—
176C23×C4—24no—
177C23×C4—24no—
178C23×C4—42no—
179C23×C4—42no—
180C23×C4C2×Q814yes—
181C23×C4C2414yes—
182C23×C4C2414yes—
183C23×C4C2414yes—
184C23×C4C2414yes—
185C23×C4C2411yescharacteristic
186C23×C4C2411yescharacteristic
187C23×C4C2412yes—
188C23×C4C2412yes—
189C23×C4C2412yes—
190C23×C4C2412yes—
191C23×C4C2411yescharacteristic
192C23×C4C2411yescharacteristic
193C23×C4C2411yescharacteristic
194C23×C4C2411yescharacteristic
195#125—232no—
196#125—416no—
197C2×C4⋊C4—28no—
198C2×C4⋊C4—24no—
199C2×C4⋊C4—28no—
200C2×C4⋊C4—24no—
201C2×C4⋊C4—24no—
202C2×C4⋊C4—24no—
203C2×C4⋊C4—24no—
204C2×C4⋊C4—24no—
205C2×C4⋊C4—28no—
206C2×C4⋊C4—28no—
207C2×C4⋊C4—28no—
208C2×C4⋊C4—24no—
209C2×C4⋊C4—28no—
210C2×C4⋊C4—28no—
211C2×C4⋊C4—28no—
212C2×C4⋊C4—28no—
213C2×C4⋊C4—28no—
214C2×C4⋊C4—24no—
215C2×C4⋊C4—28no—
216C2×C4⋊C4—24no—
217C2×C4⋊C4—28no—
218C2×C4⋊C4—28no—
219C2×C4⋊C4—28no—
220C2×C4⋊C4—28no—
221C2×C4⋊C4—24no—
222C2×C4⋊C4—28no—
223C2×C4⋊C4—44no—
224C2×C4⋊C4—44no—
225C2×C4⋊C4—44no—
226C2×C4⋊C4—44no—
227C2×C4⋊C4—44no—
228C2×C4⋊C4—44no—
229C2×C4⋊C4—44no—
230C2×C4⋊C4C2×D418yes—
231C2×C4⋊C4C2×D4116yes—
232C2×C4⋊C4C2×D418yes—
233C2×C22⋊C4—24no—
234C2×C22⋊C4—24no—
235C2×C22⋊C4—24no—
236C2×C22⋊C4—24no—
237C2×C22⋊C4—28no—
238C2×C22⋊C4—28no—
239C2×C22⋊C4—28no—
240C2×C22⋊C4—28no—
241C2×C22⋊C4—44no—
242C2×C22⋊C4—44no—
243C2×C22⋊C4C2×Q814yes—
244C2×C22⋊C4C2×Q814yes—
245C2×C22⋊C4#4118yes—
246C2×C42—22no—
247C2×C42—22no—
248C2×C42—22no—
249C2×C42—22no—
250C2×C42—22no—
251C2×C42—22no—
252C2×C42—24no—
253C2×C42—24no—
254C2×C42—24no—
255C2×C42—24no—
256C2×C42—22no—
257C2×C42—28no—
258C2×C42—24no—
259C2×C42—24no—
260C2×C42—22no—
261C2×C42—22no—
262C2×C42—24no—
263C2×C42—24no—
264C2×C42—24no—
265C2×C42—24no—
266C2×C42—22no—
267C2×C42—28no—
268C2×C42—24no—
269C2×C42—22no—
270C2×C42—22no—
271C2×C42—42no—
272C2×C42—44no—
273C2×C42—42no—
274C2×C42—42no—
275C2×C42—42no—
276C2×C42—416no—
277C2×C42C2×D414yes—
278C2×C42C2×D414yes—
279C2×C42C2×Q814yes—
280C2×C42C2×Q814yes—
281C2×C42#4114yes—
282C2×C42#4114yes—
283#95—28no—
284#95—28no—
285#95—28no—
286#95—28no—
287#95—28no—
288#95—28no—
289#95—28no—
290#95—28no—
291#95—28no—
292#95—24no—
293#95—24no—
294#95—24no—
295#95—24no—
296#95—28no—
297#95—28no—
298#95—28no—
299#95—28no—
300#95—28no—
301#95—44no—
302#95—44no—
303#95C2×D4116yes—
304#95C2×Q814yes—
305#95C2×Q814yes—
306#95#4118yes—
307C24—21no—
308C24—21no—
309C24—22no—
310C24—21no—
311C24—21no—
312C24—21no—
313C24—21no—
314C24—22no—
315C24—22no—
316C24C22×Q812yes—
317C24C2×#4114yes—
318C24C2511yescharacteristic
319C22×C4—21no—
320C22×C4—21no—
321C22×C4—21no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—21no—
325C22×C4—21no—
326C22×C4—21no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—216no—
333C22×C4—22no—
334C22×C4—21no—
335C22×C4—21no—
336C22×C4—21no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—24no—
341C22×C4—21no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—24no—
349C22×C4—24no—
350C22×C4—24no—
351C22×C4—21no—
352C22×C4—22no—
353C22×C4—24no—
354C22×C4—24no—
355C22×C4—24no—
356C22×C4—22no—
357C22×C4—21no—
358C22×C4—22no—
359C22×C4—24no—
360C22×C4—21no—
361C22×C4—21no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—24no—
365C22×C4—22no—
366C22×C4—24no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—24no—
370C22×C4—24no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—28no—
374C22×C4—28no—
375C22×C4—28no—
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—24no—
387C22×C4—24no—
388C22×C4—24no—
389C22×C4—24no—
390C22×C4—48no—
391C22×C4—48no—
392C22×C4—48no—
393C22×C4—48no—
394C22×C4—48no—
395C22×C4—48no—
396C22×C4—44no—
397C22×C4—44no—
398C22×C4—44no—
399C22×C4—44no—
400C22×C4C22×D418yes—
401C22×C4C22×D418yes—
402C22×C4C22×D414yes—
403C22×C4C22×D414yes—
404C22×C4C22×D412yes—
405C22×C4C22×D412yes—
406C22×C4C22×Q812yes—
407C22×C4C22×Q812yes—
408C22×C4C22×Q812yes—
409C22×C4C22×Q812yes—
410C22×C4C22×Q812yes—
411C22×C4C22×Q812yes—
412C22×C4C2×#4114yes—
413C22×C4C2×#4114yes—
414C22×C4C2×#4114yes—
415C22×C4C2×#4114yes—
416C22×C4C2×#4114yes—
417C22×C4C2×#4114yes—
418C22×C4C2×#4114yes—
419C22×C4C2×#4112yes—
420C22×C4C2×#4112yes—
421C22×C42−1+412yes—
422C22×C42−1+412yes—
423C4⋊C4—232no—
424C4⋊C4—216no—
425C4⋊C4—216no—
426C4⋊C4—416no—
427C4⋊C4—416no—
428C4⋊C4—416no—
429C4⋊C4—416no—
430C4⋊C4—416no—
431C4⋊C4—416no—
432C4⋊C4—48no—
433C4⋊C4—48no—
434C4⋊C4—88no—
435C22⋊C4—216no—
436C22⋊C4—28no—
437C22⋊C4—416no—
438C22⋊C4—416no—
439C22⋊C4#125116yescomplemented
440C42—28no—
441C42—28no—
442C42—28no—
443C42—48no—
444C42—416no—
445C42—48no—
446C42—48no—
447C42—48no—
448C42—48no—
449C42—44no—
450C42—44no—
451C42#121116yes—
452C23—22no—
453C23—22no—
454C23—24no—
455C23—24no—
456C23—22no—
457C23—22no—
458C23—24no—
459C23—24no—
460C23—22no—
461C23—22no—
462C23—24no—
463C23—24no—
464C23—41no—
465C23—41no—
466C23—42no—
467C23—41no—
468C23—41no—
469C23—42no—
470C23—42no—
471C23—42no—
472C23—42no—
473C23—42no—
474C23C2×#12514yes—
475C23C23×D411yescharacteristic
476C23C23×D411yescharacteristic
477C23C23×Q811yescharacteristic
478C23C22×#4111yescharacteristic
479C23C22×#4112yes—
480C23C22×#4111yescharacteristic
481C23C22×#4112yes—
482C23C2×2−1+411yescharacteristic
483C23C2×2−1+411yescharacteristic
484C23#58512yes—
485C23#58512yes—
486C2×C4—24no—
487C2×C4—24no—
488C2×C4—22no—
489C2×C4—22no—
490C2×C4—22no—
491C2×C4—24no—
492C2×C4—24no—
493C2×C4—24no—
494C2×C4—24no—
495C2×C4—24no—
496C2×C4—24no—
497C2×C4—24no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—24no—
502C2×C4—24no—
503C2×C4—24no—
504C2×C4—28no—
505C2×C4—22no—
506C2×C4—22no—
507C2×C4—24no—
508C2×C4—24no—
509C2×C4—24no—
510C2×C4—28no—
511C2×C4—24no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—28no—
516C2×C4—24no—
517C2×C4—28no—
518C2×C4—24no—
519C2×C4—24no—
520C2×C4—28no—
521C2×C4—28no—
522C2×C4—22no—
523C2×C4—24no—
524C2×C4—24no—
525C2×C4—432no—
526C2×C4—41no—
527C2×C4—42no—
528C2×C4—44no—
529C2×C4—44no—
530C2×C4—41no—
531C2×C4—44no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—44no—
541C2×C4—44no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—44no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—44no—
548C2×C4—44no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—416no—
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—44no—
563C2×C4—44no—
564C2×C4—44no—
565C2×C4—44no—
566C2×C4—44no—
567C2×C4C2×#12118yes—
568C2×C4C2×#12114yes—
569C2×C4C2×#12514yes—
570C2×C4C2×#12514yes—
571C2×C4#53314yes—
572C2×C4#53314yes—
573C2×C4#54114yes—
574C2×C4#54714yes—
575C2×C4#54814yes—
576C2×C4D4×Q814yes—
577C2×C4D4×Q814yes—
578C22—24no—
579C22—24no—
580C22—24no—
581C22—42no—
582C22—42no—
583C22—42no—
584C22—44no—
585C22—44no—
586C22—44no—
587C22—42no—
588C22—44no—
589C22C22×#12111yescharacteristic
590C22C22×#12511yescharacteristic
591C22C2×#53311yescharacteristic
592C22C2×#53311yescharacteristic
593C22#344412yes—
594C22#344412yes—
595C22C2×#54111yescharacteristic
596C22#345512yes—
597C22C2×#54711yescharacteristic
598C22C2×#54811yescharacteristic
599C22C2×D4×Q811yescharacteristic
600C22C2×D4×Q811yescharacteristic
601C22D4×#4112yes—
602C22D4×#4112yes—
603C22D4×#4112yes—
604C22D4×#4112yes—
605C22#350411yescharacteristic
606C22#350411yescharacteristic
607C22#351012yes—
608C22#351512yes—
609C22#351512yes—
610C22#351512yes—
611C22#351812yes—
612C4—24no—
613C4—24no—
614C4—24no—
615C4—28no—
616C4—42no—
617C4—44no—
618C4—48no—
619C4—44no—
620C4—42no—
621C4—48no—
622C4—44no—
623C4—84no—
624C4—84no—
625C4#242918yes—
626C2—44no—
627C2C2×#242911yescharacteristic
628C2#3474312yes—
629C2#3514212yes—
630C2#3533711yescharacteristic
631C2D4×#12511yescharacteristic
632C2D4×#12511yescharacteristic
633C2#3613111yescharacteristic
634C2#3613211yescharacteristic
635C2#3613512yes—
636C2#3614212yes—
637C2#3664011yescharacteristic
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC25C24
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×#2859C2semidirect
4. C2×C4×#125C2semidirect
6. #15137C2semidirect
7. #15005C2semidirect
9. #14709C2semidirect
10. #14707C2semidirect
11. #14208C2semidirect
13. #14006C2semidirect
439. C22⋊C4#125semidirect

Maximal subgroups

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

subgroupindex
C2×#28592
C2×#28512
C2×C4×#1252
#152202
#151372
#150052
#147532
#147092
#147072
#142082
#141122
#140062
#138762
C42×C22⋊C42