Finite Group Databasebeta

← #678512

Subgroups — #678512

Order 512.

2174 conjugacy classes of subgroups.

The subgroup lattice has 641 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#16171C214yesmaximal, complemented
2#15689C214yesmaximal
3#15183C212yesmaximal, complemented
4#15150C212yesmaximal, complemented
5#14859C214yesmaximal, complemented
6#14707C211yesmaximal, characteristic, complemented
7#14193C214yesmaximal, complemented
8#13993C212yesmaximal, complemented
9#13955C211yesmaximal, characteristic, complemented
10#13905C211yesmaximal, characteristic
11#13905C211yesmaximal, characteristic
12#13876C212yesmaximal, complemented
13#13804C211yesmaximal, characteristic, complemented
14C4×#384C211yesmaximal, characteristic, complemented
15C4×#390C211yesmaximal, characteristic
16C22×C4×D4C2211yescharacteristic
17C2×#397C2214yes—
18C2×#397C2214yes—
19C2×#390C2212yes—
20C2×#389C2212yes—
21C2×#388C2214yes—
22C2×#384C2211yescharacteristic
23C2×#384C2211yescharacteristic
24C2×#382C2211yescharacteristic
25C2×#382C2211yescharacteristic
26C2×#380C2212yes—
27C2×#380C2212yes—
28C2×#380C2212yes—
29C2×#380C2212yes—
30C2×C4×C4⋊C4C2211yescharacteristic
31C2×C4×C4⋊C4C2211yescharacteristic
32C2×C4×C4⋊C4C2211yescharacteristic
33C2×C4×C4⋊C4C2214yes—
34C2×C4×C4⋊C4C2212yes—
35C2×C4×C22⋊C4C2212yes—
36C2×C4×C22⋊C4C2212yes—
37C2×C43C2211yescharacteristic
38#1445C2214yes—
39#1445C2214yes—
40#1445C2212yes—
41#1445C2212yes—
42#1445C2214yes—
43#1443C2214yes—
44#1443C2214yes—
45#1442C2214yes—
46#1442C2214yes—
47#1440C2214yes—
48#1439C2214yes—
49#1439C2214yes—
50#1439C2214yes—
51#1438C2212yes—
52#1438C2212yes—
53#1437C2212yes—
54#1437C2214yes—
55#1437C2214yes—
56#1437C2214yes—
57#1437C2212yes—
58#1436C2214yes—
59#1436C2214yes—
60#1436C2214yes—
61#1436C2214yes—
62#1436C2214yes—
63C4⋊#95C2212yes—
64C4⋊#95C2214yes—
65C4⋊#95C2212yes—
66C4⋊#95C2212yes—
67C4⋊#95C2212yes—
68C4⋊#95C2212yes—
69C4⋊#95C2211yescharacteristic
70C4⋊#95C2211yescharacteristic
71#1434C2212yes—
72#1434C2212yes—
73#1434C2212yes—
74C4×#95C2212yes—
75C23×D4C2311yescharacteristic
76C24×C4C2312yes—
77C2×C4×D4—24no—
78C2×C4×D4—24no—
79C2×C4×D4—24no—
80C22×C4⋊C4C2314yes—
81C22×C4⋊C4C2311yescharacteristic
82C22×C4⋊C4C2311yescharacteristic
83C22×C4⋊C4C2311yescharacteristic
84C22×C4⋊C4C2311yescharacteristic
85C22×C4⋊C4C2312yes—
86C22×C4⋊C4C2312yes—
87C22×C4⋊C4C2312yes—
88C22×C4⋊C4C2314yes—
89C22×C4⋊C4C2311yescharacteristic
90C22×C4⋊C4C2314yes—
91C22×C22⋊C4C2312yes—
92C22×C22⋊C4C2312yes—
93C22×C22⋊C4C2312yes—
94C22×C22⋊C4C2312yes—
95C22×C22⋊C4C2312yes—
96C22×C22⋊C4C2314yes—
97C22×C22⋊C4C2314yes—
98C22×C22⋊C4C2314yes—
99C22×C22⋊C4C2314yes—
100C22×C42C2312yes—
101C22×C42C2311yescharacteristic
102C22×C42C2312yes—
103C22×C42C2312yes—
104C22×C42C2311yescharacteristic
105C22×C42C2312yes—
106C22×C42C2312yes—
107C22×C42C2314yes—
108C22×C42C2311yescharacteristic
109#397—216no—
110#397—48no—
111#390—28no—
112#389—44no—
113#388—48no—
114#384—24no—
115#384—24no—
116#382—24no—
117#382—24no—
118#380—28no—
119#380—44no—
120#380—44no—
121#380—44no—
122C4×C4⋊C4—24no—
123C4×C4⋊C4—24no—
124C4×C4⋊C4—24no—
125C4×C4⋊C4—48no—
126C4×C4⋊C4—44no—
127C4×C22⋊C4—28no—
128C4×C22⋊C4—44no—
129C2×#95C2314yes—
130C2×#95C2314yes—
131C2×#95C2314yes—
132C2×#95C2314yes—
133C2×#95C2311yescharacteristic
134C2×#95C2312yes—
135C2×#95C2311yescharacteristic
136C2×#95C2312yes—
137C2×#95C2312yes—
138C2×#95C2312yes—
139C2×#95C2312yes—
140C2×#95C2312yes—
141C2×#95C2312yes—
142C2×#95C2314yes—
143C2×#95C2312yes—
144C2×#95C2314yes—
145C2×#95C2314yes—
146C2×#95C2314yes—
147C2×#95C2314yes—
148C2×#95C2314yes—
149C2×#95C2314yes—
150C2×#95C2314yes—
151C2×#95C2312yes—
152C2×#95C2312yes—
153C2×#95C2314yes—
154C2×#95C2314yes—
155C2×#95C2314yes—
156C2×#95C2314yes—
157C43—24no—
158C25C2412yes—
159C22×D4—24no—
160C22×D4—22no—
161C22×D4—22no—
162C22×D4—22no—
163C22×D4—42no—
164C23×C4—24no—
165C23×C4—24no—
166C23×C4—24no—
167C23×C4—24no—
168C23×C4—24no—
169C23×C4—24no—
170C23×C4—24no—
171C23×C4C2412yes—
172C23×C4C2412yes—
173C23×C4C2412yes—
174C23×C4C2414yes—
175C23×C4C2414yes—
176C23×C4C2412yes—
177C23×C4C2412yes—
178C23×C4C2412yes—
179C23×C4C2412yes—
180C23×C4C2412yes—
181C23×C4C2411yescharacteristic
182C23×C4C2411yescharacteristic
183C23×C4C2412yes—
184C23×C4C2411yescharacteristic
185C4×D4—416no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—24no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—24no—
196C2×C4⋊C4—24no—
197C2×C4⋊C4—24no—
198C2×C4⋊C4—28no—
199C2×C4⋊C4—28no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—24no—
204C2×C4⋊C4—28no—
205C2×C4⋊C4—24no—
206C2×C4⋊C4—28no—
207C2×C4⋊C4—28no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—24no—
212C2×C4⋊C4—24no—
213C2×C4⋊C4—28no—
214C2×C4⋊C4—44no—
215C2×C4⋊C4—42no—
216C2×C4⋊C4—44no—
217C2×C4⋊C4—44no—
218C2×C4⋊C4#4114yes—
219C2×C22⋊C4—24no—
220C2×C22⋊C4—24no—
221C2×C22⋊C4—24no—
222C2×C22⋊C4—24no—
223C2×C22⋊C4—24no—
224C2×C22⋊C4—24no—
225C2×C22⋊C4—24no—
226C2×C22⋊C4—24no—
227C2×C22⋊C4—24no—
228C2×C22⋊C4—24no—
229C2×C22⋊C4—24no—
230C2×C22⋊C4—24no—
231C2×C22⋊C4—28no—
232C2×C22⋊C4—28no—
233C2×C22⋊C4—28no—
234C2×C22⋊C4—42no—
235C2×C22⋊C4—42no—
236C2×C22⋊C4—44no—
237C2×C22⋊C4—44no—
238C2×C22⋊C4—44no—
239C2×C22⋊C4—44no—
240C2×C22⋊C4—44no—
241C2×C22⋊C4—44no—
242C2×C22⋊C4—44no—
243C2×C22⋊C4—44no—
244C2×C22⋊C4—44no—
245C2×C22⋊C4#4118yes—
246C2×C42—24no—
247C2×C42—22no—
248C2×C42—22no—
249C2×C42—24no—
250C2×C42—24no—
251C2×C42—24no—
252C2×C42—22no—
253C2×C42—24no—
254C2×C42—24no—
255C2×C42—24no—
256C2×C42—22no—
257C2×C42—24no—
258C2×C42—22no—
259C2×C42—24no—
260C2×C42—24no—
261C2×C42—42no—
262C2×C42—44no—
263C2×C42—44no—
264C2×C42—44no—
265C2×C42—44no—
266C2×C42—44no—
267C2×C42#4114yes—
268C2×C42#4114yes—
269C2×C42#4114yes—
270C2×C42#4114yes—
271#95—24no—
272#95—24no—
273#95—22no—
274#95—24no—
275#95—22no—
276#95—24no—
277#95—28no—
278#95—24no—
279#95—24no—
280#95—24no—
281#95—24no—
282#95—28no—
283#95—28no—
284#95—28no—
285#95—28no—
286#95—44no—
287#95—42no—
288#95—42no—
289#95—44no—
290#95—44no—
291#95—44no—
292#95—44no—
293#95—44no—
294#95—44no—
295#95—44no—
296#95—44no—
297#95—44no—
298#95—44no—
299C24—22no—
300C24—22no—
301C24—22no—
302C24—22no—
303C24—22no—
304C24—22no—
305C24—22no—
306C24—22no—
307C24—22no—
308C24—22no—
309C24—22no—
310C24—22no—
311C24—22no—
312C24—22no—
313C24C2×#4114yes—
314C24C2511yescharacteristic
315C2×D4—28no—
316C2×D4—48no—
317C2×D4—48no—
318C2×D4—44no—
319C2×D4—44no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—22no—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—24no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—24no—
343C22×C4—22no—
344C22×C4—24no—
345C22×C4—24no—
346C22×C4—24no—
347C22×C4—24no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—24no—
356C22×C4—24no—
357C22×C4—24no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—24no—
366C22×C4—24no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—24no—
372C22×C4—24no—
373C22×C4—24no—
374C22×C4—24no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—24no—
379C22×C4—24no—
380C22×C4—22no—
381C22×C4—24no—
382C22×C4—24no—
383C22×C4—22no—
384C22×C4—24no—
385C22×C4—24no—
386C22×C4—24no—
387C22×C4—48no—
388C22×C4—48no—
389C22×C4—48no—
390C22×C4—48no—
391C22×C4—48no—
392C22×C4—48no—
393C22×C4—48no—
394C22×C4C2×#4114yes—
395C22×C4C2×#4114yes—
396C22×C4C2×#4112yes—
397C22×C4C2×#4112yes—
398C22×C4C2×#4112yes—
399C22×C4C2×#4112yes—
400C22×C4C2×#4112yes—
401C22×C4C2×#4112yes—
402C22×C4C2×#4114yes—
403C22×C4C2×#4114yes—
404C22×C4C2×#4112yes—
405C22×C4C2×#4112yes—
406C22×C4C2×#4112yes—
407C22×C4C2×#4112yes—
408C22×C4C2×#4112yes—
409C22×C4H(5,𝔽2)14yes—
410C22×C4H(5,𝔽2)12yes—
411C22×C4H(5,𝔽2)12yes—
412C22×C42−1+414yes—
413C4⋊C4—28no—
414C4⋊C4—48no—
415C4⋊C4—416no—
416C4⋊C4—44no—
417C4⋊C4—48no—
418C4⋊C4—416no—
419C4⋊C4—44no—
420C4⋊C4—44no—
421C4⋊C4—44no—
422C4⋊C4—48no—
423C4⋊C4—88no—
424C22⋊C4—216no—
425C22⋊C4—48no—
426C22⋊C4—48no—
427C22⋊C4—48no—
428C22⋊C4—48no—
429C22⋊C4—88no—
430C22⋊C4—88no—
431C22⋊C4—88no—
432C22⋊C4—88no—
433C42—216no—
434C42—28no—
435C42—48no—
436C42—44no—
437C42—48no—
438C42—84no—
439C42—84no—
440C42—88no—
441C42#126116yes—
442C23—24no—
443C23—24no—
444C23—24no—
445C23—24no—
446C23—24no—
447C23—24no—
448C23—24no—
449C23—42no—
450C23—42no—
451C23—42no—
452C23—42no—
453C23—42no—
454C23—42no—
455C23—42no—
456C23—42no—
457C23—42no—
458C23—42no—
459C23—42no—
460C23—42no—
461C23—42no—
462C23—42no—
463C23—42no—
464C23—42no—
465C23—42no—
466C23—42no—
467C23—42no—
468C23—42no—
469C23—42no—
470C23—42no—
471C23—42no—
472C23—42no—
473C23—42no—
474C23—42no—
475C23—42no—
476C23—42no—
477C23C22×#4111yescharacteristic
478C23C22×#4111yescharacteristic
479C23C22×#4111yescharacteristic
480C23C22×#4111yescharacteristic
481C23C22×#4111yescharacteristic
482C23C22×#4112yes—
483C23C2×H(5,𝔽2)11yescharacteristic
484C23C2×H(5,𝔽2)11yescharacteristic
485C23C2×H(5,𝔽2)12yes—
486C23C2×2−1+412yes—
487C23#58512yes—
488D4—416no—
489C2×C4—24no—
490C2×C4—22no—
491C2×C4—22no—
492C2×C4—24no—
493C2×C4—24no—
494C2×C4—24no—
495C2×C4—24no—
496C2×C4—24no—
497C2×C4—24no—
498C2×C4—24no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—24no—
502C2×C4—24no—
503C2×C4—24no—
504C2×C4—24no—
505C2×C4—24no—
506C2×C4—24no—
507C2×C4—28no—
508C2×C4—24no—
509C2×C4—24no—
510C2×C4—24no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—44no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—44no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—44no—
532C2×C4—44no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—44no—
539C2×C4—44no—
540C2×C4—42no—
541C2×C4—44no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—44no—
545C2×C4—44no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—44no—
551C2×C4—44no—
552C2×C4—44no—
553C2×C4—44no—
554C2×C4—42no—
555C2×C4—44no—
556C2×C4—44no—
557C2×C4—44no—
558C2×C4—42no—
559C2×C4—44no—
560C2×C4—44no—
561C2×C4—44no—
562C2×C4—42no—
563C2×C4—44no—
564C2×C4—44no—
565C2×C4—816no—
566C2×C4C2×#12614yes—
567C2×C4C2×#12614yes—
568C2×C4C2×#12614yes—
569C2×C4#52914yes—
570C2×C4#54014yes—
571C2×C4#55318yes—
572C2×C4#55314yes—
573C2×C4#55514yes—
574C2×C4#55914yes—
575C2×C4#55914yes—
576C22—44no—
577C22—44no—
578C22—44no—
579C22—44no—
580C22—44no—
581C22—44no—
582C22—44no—
583C22—82no—
584C22—82no—
585C22—82no—
586C22—82no—
587C22—82no—
588C22—82no—
589C22—82no—
590C22—82no—
591C22C22×#12611yescharacteristic
592C22C2×#52911yescharacteristic
593C22#344411yescharacteristic
594C22C2×#54011yescharacteristic
595C22#345511yescharacteristic
596C22C2×#55311yescharacteristic
597C22C2×#55312yes—
598C22C2×#55511yescharacteristic
599C22#347512yes—
600C22#347512yes—
601C22C2×#55911yescharacteristic
602C22C2×#55911yescharacteristic
603C22#348212yes—
604C22#349212yes—
605C22#350712yes—
606C22#350811yescharacteristic
607C22#350812yes—
608C22#350911yescharacteristic
609C22#351112yes—
610C22#351312yes—
611C22#352012yes—
612C22#354612yes—
613C22#356112yes—
614C4—28no—
615C4—24no—
616C4—24no—
617C4—44no—
618C4—44no—
619C4—44no—
620C4—82no—
621C4—82no—
622C4—82no—
623C4—82no—
624C4—82no—
625C4—84no—
626C4—84no—
627C4#251318yes—
628C2—84no—
629C2C2×#251311yescharacteristic
630C2#3481411yescharacteristic
631C2#3555711yescharacteristic
632C2#3621111yescharacteristic
633C2#3648612yes—
634C2#3650911yescharacteristic
635C2#3918511yescharacteristic
636C2#3950111yescharacteristic
637C2#3992612yes—
638C2#5964312yes—
639C2#5967812yes—
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
1. #16171C2semidirect
3. #15183C2semidirect
4. #15150C2semidirect
5. #14859C2semidirect
6. #14707C2semidirect
7. #14193C2semidirect
8. #13993C2semidirect
9. #13955C2semidirect
12. #13876C2semidirect
13. #13804C2semidirect
14. C4×#384C2semidirect

Maximal subgroups

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

subgroupindex
#161712
#156892
#151832
#151502
#148592
#147072
#141932
#139932
#139552
#139052
#138762
#138042
C4×#3842
C4×#3902