Finite Group Databasebeta

← #2737234

Subgroups — #2737234

Order 512.

2048 conjugacy classes of subgroups.

The subgroup lattice has 655 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2568C211yesmaximal, characteristic, complemented
2C2×#2423C211yesmaximal, characteristic
3#17212C214yesmaximal, complemented
4#17183C214yesmaximal
5#16898C212yesmaximal, complemented
6#16889C212yesmaximal, complemented
7#16805C212yesmaximal
8#15236C212yesmaximal
9#15231C212yesmaximal, complemented
10#15230C211yesmaximal, characteristic, complemented
11#15220C212yesmaximal
12#15194C211yesmaximal, characteristic, complemented
13#15186C212yesmaximal, complemented
14#15125C211yesmaximal, characteristic
15#14625C212yesmaximal
16#13935C211yesmaximal, characteristic, complemented
17C4×#397C211yesmaximal, characteristic
18C22×C4⋊Q8C2211yescharacteristic
19C22×C22⋊Q8C2211yescharacteristic
20C22×C4×Q8C2211yescharacteristic
21#2568—28no—
22#2423—28no—
23C2×#400C2214yes—
24C2×#399C2212yes—
25C2×#398C2212yes—
26C2×#397C2211yescharacteristic
27C2×#393C2212yes—
28C2×#391C2212yes—
29C2×#391C2211yescharacteristic
30C2×#391C2212yes—
31C2×#391C2212yes—
32C2×#391C2212yes—
33C2×#389C2211yescharacteristic
34C2×#389C2212yes—
35C2×#389C2212yes—
36C2×#388C2212yes—
37C2×#387C2212yes—
38C2×#384C2211yescharacteristic
39C2×#382C2212yes—
40C2×#382C2212yes—
41C2×#380C2211yescharacteristic
42C2×#380C2212yes—
43C2×C4×C4⋊C4C2211yescharacteristic
44C2×C4×C22⋊C4C2211yescharacteristic
45C2×C4×C22⋊C4C2212yes—
46#1446C2212yes—
47#1446C2212yes—
48#1446C2212yes—
49#1445C2214yes—
50#1445C2211yescharacteristic
51#1445C2212yes—
52#1444C2212yes—
53#1443C2214yes—
54#1443C2212yes—
55#1443C2214yes—
56#1443C2214yes—
57#1442C2211yescharacteristic
58#1442C2214yes—
59#1442C2212yes—
60#1442C2214yes—
61#1442C2214yes—
62#1442C2214yes—
63#1442C2214yes—
64#1441C2214yes—
65#1441C2214yes—
66#1441C2214yes—
67#1441C2214yes—
68#1441C2214yes—
69#1441C2214yes—
70#1440C2212yes—
71#1440C2214yes—
72#1440C2214yes—
73#1439C2212yes—
74#1439C2214yes—
75#1439C2214yes—
76#1439C2211yescharacteristic
77C4⋊#95C2212yes—
78C4⋊#95C2212yes—
79C4⋊#95C2211yescharacteristic
80C4⋊#95C2212yes—
81C4⋊#95C2212yes—
82#1434C2212yes—
83C4×#95C2212yes—
84C4×#95C2211yescharacteristic
85C23×Q8C2311yescharacteristic
86C23×Q8C2312yes—
87C24×C4C2311yescharacteristic
88C2×C4⋊Q8—24no—
89C2×C4⋊Q8—44no—
90C2×C22⋊Q8—24no—
91C2×C22⋊Q8—44no—
92C2×C4×Q8—44no—
93C2×C4×Q8D418yes—
94C22×C4⋊C4C2311yescharacteristic
95C22×C4⋊C4C2311yescharacteristic
96C22×C4⋊C4C2312yes—
97C22×C4⋊C4C2312yes—
98C22×C4⋊C4C2311yescharacteristic
99C22×C4⋊C4C2312yes—
100C22×C4⋊C4C2311yescharacteristic
101C22×C4⋊C4C2312yes—
102C22×C4⋊C4C2312yes—
103C22×C4⋊C4C2312yes—
104C22×C4⋊C4C2312yes—
105C22×C4⋊C4C2311yescharacteristic
106C22×C4⋊C4C2314yes—
107C22×C4⋊C4C2314yes—
108C22×C4⋊C4C2312yes—
109C22×C22⋊C4C2311yescharacteristic
110C22×C22⋊C4C2311yescharacteristic
111C22×C22⋊C4C2314yes—
112C22×C22⋊C4C2312yes—
113C22×C22⋊C4C2312yes—
114C22×C22⋊C4C2312yes—
115C22×C22⋊C4C2312yes—
116C22×C42C2311yescharacteristic
117C22×C42C2312yes—
118C22×C42C2312yes—
119C22×C42C2311yescharacteristic
120C22×C42C2312yes—
121#400—216no—
122#399—28no—
123#398—28no—
124#397—24no—
125#393—28no—
126#391—28no—
127#391—24no—
128#391—28no—
129#391—28no—
130#391—44no—
131#389—24no—
132#389—44no—
133#389—44no—
134#388—28no—
135#387—28no—
136#384—24no—
137#382—28no—
138#382—44no—
139#380—24no—
140#380—44no—
141C4×C4⋊C4—24no—
142C4×C22⋊C4—24no—
143C4×C22⋊C4—44no—
144C2×#95C2312yes—
145C2×#95C2314yes—
146C2×#95C2311yescharacteristic
147C2×#95C2312yes—
148C2×#95C2311yescharacteristic
149C2×#95C2312yes—
150C2×#95C2312yes—
151C2×#95C2312yes—
152C2×#95C2312yes—
153C2×#95C2314yes—
154C2×#95C2314yes—
155C2×#95C2314yes—
156C2×#95C2314yes—
157C2×#95C2311yescharacteristic
158C2×#95C2312yes—
159C2×#95C2311yescharacteristic
160C2×#95C2312yes—
161C2×#95C2312yes—
162C2×#95C2312yes—
163C2×#95C2312yes—
164C2×#95C2314yes—
165C2×#95C2314yes—
166C2×#95C2312yes—
167C2×#95C2314yes—
168C2×#95C2314yes—
169C2×#95C2314yes—
170C2×#95C2314yes—
171C2×#95C2314yes—
172C2×#95C2314yes—
173C2×#95C2314yes—
174C2×#95C2314yes—
175C2×#95C2314yes—
176C2×#95C2312yes—
177C2×#95C2314yes—
178C2×#95C2312yes—
179C25C2411yescharacteristic
180C22×Q8—24no—
181C22×Q8—24no—
182C22×Q8—24no—
183C22×Q8—22no—
184C22×Q8—24no—
185C22×Q8—22no—
186C22×Q8—24no—
187C22×Q8—44no—
188C22×Q8—44no—
189C22×Q8C2×D414yes—
190C23×C4—22no—
191C23×C4—24no—
192C23×C4—22no—
193C23×C4—24no—
194C23×C4—22no—
195C23×C4C2412yes—
196C23×C4C2412yes—
197C23×C4C2414yes—
198C23×C4C2412yes—
199C23×C4C2414yes—
200C23×C4C2412yes—
201C23×C4C2412yes—
202C23×C4C2412yes—
203C23×C4C2411yescharacteristic
204C23×C4C2412yes—
205C23×C4C2411yescharacteristic
206C23×C4C2412yes—
207C23×C4C2411yescharacteristic
208C23×C4C2411yescharacteristic
209C23×C4C2411yescharacteristic
210C23×C4C2411yescharacteristic
211C4⋊Q8—416no—
212C22⋊Q8—416no—
213C4×Q8—416no—
214C2×C4⋊C4—24no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—24no—
217C2×C4⋊C4—24no—
218C2×C4⋊C4—24no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—24no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—24no—
225C2×C4⋊C4—24no—
226C2×C4⋊C4—24no—
227C2×C4⋊C4—24no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—24no—
230C2×C4⋊C4—24no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—24no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—24no—
236C2×C4⋊C4—42no—
237C2×C4⋊C4—42no—
238C2×C4⋊C4—44no—
239C2×C4⋊C4—44no—
240C2×C4⋊C4—44no—
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⋊C4—44no—
249C2×C4⋊C4C2×D414yes—
250C2×C4⋊C4C2×D418yes—
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—28no—
258C2×C22⋊C4—24no—
259C2×C22⋊C4—41no—
260C2×C22⋊C4—42no—
261C2×C22⋊C4—42no—
262C2×C22⋊C4—42no—
263C2×C22⋊C4—44no—
264C2×C22⋊C4—42no—
265C2×C22⋊C4—42no—
266C2×C22⋊C4—44no—
267C2×C22⋊C4—44no—
268C2×C22⋊C4—44no—
269C2×C22⋊C4#4114yes—
270C2×C42—24no—
271C2×C42—22no—
272C2×C42—24no—
273C2×C42—24no—
274C2×C42—24no—
275C2×C42—24no—
276C2×C42—24no—
277C2×C42—24no—
278C2×C42—24no—
279C2×C42—44no—
280C2×C42C2×D414yes—
281C2×C42C2×D418yes—
282#95—22no—
283#95—22no—
284#95—24no—
285#95—24no—
286#95—28no—
287#95—28no—
288#95—24no—
289#95—24no—
290#95—24no—
291#95—28no—
292#95—28no—
293#95—24no—
294#95—28no—
295#95—22no—
296#95—24no—
297#95—24no—
298#95—22no—
299#95—42no—
300#95—44no—
301#95—42no—
302#95—42no—
303#95—42no—
304#95—44no—
305#95—44no—
306#95—44no—
307#95—44no—
308#95—42no—
309#95—42no—
310#95—44no—
311#95—44no—
312#95—44no—
313#95—44no—
314#95—44no—
315#95—44no—
316#95—44no—
317C24—21no—
318C24—21no—
319C24—22no—
320C24—21no—
321C24—21no—
322C24—21no—
323C24—22no—
324C24—21no—
325C24—22no—
326C24—22no—
327C24C2×#4112yes—
328C24C2511yescharacteristic
329C2×Q8—28no—
330C2×Q8—28no—
331C2×Q8—28no—
332C2×Q8—48no—
333C2×Q8—48no—
334C2×Q8—48no—
335C2×Q8—48no—
336C2×Q8—48no—
337C2×Q8—88no—
338C2×Q8—88no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—21no—
342C22×C4—21no—
343C22×C4—22no—
344C22×C4—24no—
345C22×C4—22no—
346C22×C4—24no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—21no—
350C22×C4—21no—
351C22×C4—22no—
352C22×C4—24no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—21no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—21no—
359C22×C4—24no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—21no—
365C22×C4—24no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—24no—
369C22×C4—22no—
370C22×C4—24no—
371C22×C4—22no—
372C22×C4—21no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—24no—
377C22×C4—22no—
378C22×C4—24no—
379C22×C4—24no—
380C22×C4—22no—
381C22×C4—24no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—24no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—24no—
396C22×C4—22no—
397C22×C4—21no—
398C22×C4—24no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—24no—
405C22×C4—22no—
406C22×C4—24no—
407C22×C4—24no—
408C22×C4—22no—
409C22×C4—24no—
410C22×C4—24no—
411C22×C4—24no—
412C22×C4—22no—
413C22×C4—24no—
414C22×C4—22no—
415C22×C4—24no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—44no—
419C22×C4—44no—
420C22×C4—44no—
421C22×C4—48no—
422C22×C4—48no—
423C22×C4C22×D414yes—
424C22×C4C22×D414yes—
425C22×C4C22×D412yes—
426C22×C4C22×D412yes—
427C22×C4C22×D412yes—
428C22×C4C2×#4112yes—
429C22×C4C2×#4112yes—
430C22×C4H(5,𝔽2)14yes—
431C22×C4H(5,𝔽2)12yes—
432C22×C42−1+412yes—
433C22×C42−1+412yes—
434C22×C42−1+414yes—
435C22×C42−1+412yes—
436C4⋊C4—28no—
437C4⋊C4—48no—
438C4⋊C4—48no—
439C4⋊C4—44no—
440C4⋊C4—48no—
441C4⋊C4—48no—
442C4⋊C4—44no—
443C4⋊C4—48no—
444C4⋊C4—44no—
445C4⋊C4—44no—
446C4⋊C4—48no—
447C4⋊C4—84no—
448C4⋊C4—84no—
449C4⋊C4—88no—
450C4⋊C4—88no—
451C22⋊C4—28no—
452C22⋊C4—44no—
453C22⋊C4—48no—
454C22⋊C4—48no—
455C22⋊C4—48no—
456C22⋊C4—84no—
457C22⋊C4—88no—
458C42—28no—
459C42—48no—
460C42—44no—
461C42—48no—
462C42—48no—
463C23—22no—
464C23—22no—
465C23—22no—
466C23—24no—
467C23—24no—
468C23—41no—
469C23—41no—
470C23—41no—
471C23—41no—
472C23—41no—
473C23—41no—
474C23—42no—
475C23—42no—
476C23—42no—
477C23—41no—
478C23—42no—
479C23—42no—
480C23—41no—
481C23—42no—
482C23—42no—
483C23—42no—
484C23—42no—
485C23—42no—
486C23C23×D411yescharacteristic
487C23C22×#4111yescharacteristic
488C23C2×H(5,𝔽2)12yes—
489C23C2×H(5,𝔽2)11yescharacteristic
490C23C2×2−1+412yes—
491C23C2×2−1+411yescharacteristic
492C23C2×2−1+411yescharacteristic
493C23C2×2−1+411yescharacteristic
494C23#58512yes—
495C23#58511yescharacteristic
496C23#58512yes—
497Q8—416no—
498Q8—816no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—24no—
502C2×C4—24no—
503C2×C4—22no—
504C2×C4—24no—
505C2×C4—24no—
506C2×C4—24no—
507C2×C4—24no—
508C2×C4—22no—
509C2×C4—22no—
510C2×C4—22no—
511C2×C4—24no—
512C2×C4—24no—
513C2×C4—22no—
514C2×C4—24no—
515C2×C4—24no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—44no—
521C2×C4—44no—
522C2×C4—44no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—44no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—41no—
530C2×C4—44no—
531C2×C4—41no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—41no—
536C2×C4—42no—
537C2×C4—44no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—44no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—44no—
549C2×C4—42no—
550C2×C4—44no—
551C2×C4—44no—
552C2×C4—42no—
553C2×C4—44no—
554C2×C4—42no—
555C2×C4—44no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—44no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—44no—
565C2×C4—44no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—44no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—44no—
575C2×C4—44no—
576C2×C4—42no—
577C2×C4—44no—
578C2×C4—44no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—44no—
583C2×C4—42no—
584C2×C4—44no—
585C2×C4—88no—
586C2×C4#53414yes—
587C2×C4#53614yes—
588C2×C4#53714yes—
589C2×C4#54814yes—
590C2×C4#56218yes—
591C2×C4#56318yes—
592C22—42no—
593C22—42no—
594C22—44no—
595C22—42no—
596C22—44no—
597C22—81no—
598C22—82no—
599C22—82no—
600C22—81no—
601C22—81no—
602C22—81no—
603C22C2×#53411yescharacteristic
604C22C2×#53611yescharacteristic
605C22C2×#53711yescharacteristic
606C22C2×#54811yescharacteristic
607C22#348711yescharacteristic
608C22#348911yescharacteristic
609C22#350011yescharacteristic
610C22#350611yescharacteristic
611C22#350911yescharacteristic
612C22#351412yes—
613C22#351512yes—
614C22C2×#56212yes—
615C22C2×#56312yes—
616C22#354012yes—
617C22#355011yescharacteristic
618C22#355412yes—
619C22#355512yes—
620C22#355511yescharacteristic
621C22#356112yes—
622C22#356212yes—
623C22#356312yes—
624C22#356512yes—
625C22#356512yes—
626C4—24no—
627C4—42no—
628C4—44no—
629C4—42no—
630C4—42no—
631C4—44no—
632C4—81no—
633C4—82no—
634C4—82no—
635C4—82no—
636C4—82no—
637C4—82no—
638C4—84no—
639C4—82no—
640C4—84no—
641C4#281418yes—
642C2—82no—
643C2C2×#281411yescharacteristic
644C2#3912311yescharacteristic
645C2#3917311yescharacteristic
646C2#3972711yescharacteristic
647C2#4288211yescharacteristic
648C2#4288811yescharacteristic
649C2#4291311yescharacteristic
650C2#4571112yes—
651C2#4606912yes—
652C2#5307012yes—
653C2#5309312yes—
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×#2568C2semidirect
3. #17212C2semidirect
5. #16898C2semidirect
6. #16889C2semidirect
9. #15231C2semidirect
10. #15230C2semidirect
12. #15194C2semidirect
13. #15186C2semidirect
16. #13935C2semidirect

Maximal subgroups

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

subgroupindex
C2×#25682
C2×#24232
#172122
#171832
#168982
#168892
#168052
#152362
#152312
#152302
#152202
#151942
#151862
#151252
#146252
#139352
C4×#3972