Finite Group Databasebeta

← #719565

Subgroups — #719565

Order 512.

2434 conjugacy classes of subgroups.

The subgroup lattice has 633 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2512C212yesmaximal
2#16422C214yesmaximal, complemented
3#15911C214yesmaximal, complemented
4#15433C214yesmaximal, complemented
5#15200C212yesmaximal, complemented
6#15147C212yesmaximal
7#14931C214yesmaximal, complemented
8#14276C211yesmaximal, characteristic, complemented
9#13941C211yesmaximal, characteristic
10#13910C211yesmaximal, characteristic, complemented
11#13906C211yesmaximal, characteristic
12#13884C212yesmaximal, complemented
13#13856C211yesmaximal, characteristic, complemented
14#13804C211yesmaximal, characteristic, complemented
15#13785C211yesmaximal, characteristic
16C22×#126C2214yes—
17C22×C4×D4C2211yescharacteristic
18#2512—216no—
19C2×#397C2214yes—
20C2×#397C2214yes—
21C2×#396C2214yes—
22C2×#390C2211yescharacteristic
23C2×#390C2212yes—
24C2×#390C2212yes—
25C2×#390C2211yescharacteristic
26C2×#389C2212yes—
27C2×#389C2214yes—
28C2×#388C2214yes—
29C2×#387C2214yes—
30C2×#384C2212yes—
31C2×#384C2211yescharacteristic
32C2×#382C2211yescharacteristic
33C2×#380C2214yes—
34C2×C4×C4⋊C4C2212yes—
35C2×C4×C4⋊C4C2211yescharacteristic
36C2×C4×C4⋊C4C2211yescharacteristic
37C2×C4×C4⋊C4C2212yes—
38C2×C4×C4⋊C4C2212yes—
39C2×C4×C4⋊C4C2211yescharacteristic
40C2×#376C2211yescharacteristic
41#1445C2214yes—
42#1444C2212yes—
43#1444C2212yes—
44#1444C2212yes—
45#1444C2212yes—
46#1444C2212yes—
47#1444C2212yes—
48#1442C2214yes—
49#1442C2214yes—
50#1442C2214yes—
51#1442C2214yes—
52#1441C2214yes—
53#1441C2214yes—
54#1441C2214yes—
55#1440C2214yes—
56#1440C2214yes—
57#1439C2212yes—
58#1439C2214yes—
59#1439C2214yes—
60#1439C2212yes—
61#1437C2214yes—
62#1436C2214yes—
63#1436C2214yes—
64C4⋊#95C2212yes—
65C4⋊#95C2212yes—
66C4⋊#95C2212yes—
67C4⋊#95C2211yescharacteristic
68C4⋊#95C2212yes—
69C4⋊#95C2212yes—
70C4⋊#95C2211yescharacteristic
71#1434C2212yes—
72C22⋊#95C2212yes—
73C22⋊#95C2214yes—
74C22⋊#95C2212yes—
75C4×#95C2212yes—
76C23×D4C2311yescharacteristic
77C24×C4C2312yes—
78C2×#126—216no—
79C2×#126—48no—
80C2×#126—48no—
81C2×C4×D4—24no—
82C2×C4×D4—24no—
83C2×C4×D4—24no—
84C22×C4⋊C4C2311yescharacteristic
85C22×C4⋊C4C2312yes—
86C22×C4⋊C4C2311yescharacteristic
87C22×C4⋊C4C2312yes—
88C22×C4⋊C4C2312yes—
89C22×C4⋊C4C2312yes—
90C22×C4⋊C4C2312yes—
91C22×C4⋊C4C2312yes—
92C22×C4⋊C4C2312yes—
93C22×C4⋊C4C2314yes—
94C22×C4⋊C4C2312yes—
95C22×C4⋊C4C2311yescharacteristic
96C22×C4⋊C4C2314yes—
97C22×C4⋊C4C2314yes—
98C22×C4⋊C4C2312yes—
99C22×C4⋊C4C2312yes—
100C22×C22⋊C4C2312yes—
101C22×C22⋊C4C2314yes—
102C22×C22⋊C4C2314yes—
103C22×C22⋊C4C2314yes—
104C22×C22⋊C4C2314yes—
105C22×C22⋊C4C2314yes—
106C22×C22⋊C4C2314yes—
107C22×C42C2312yes—
108C22×C42C2312yes—
109C22×C42C2314yes—
110C22×C42C2311yescharacteristic
111C22×C42C2311yescharacteristic
112C22×C42C2312yes—
113C22×C42C2311yescharacteristic
114#397—216no—
115#397—48no—
116#396—216no—
117#390—28no—
118#390—28no—
119#390—24no—
120#390D418yes—
121#389—28no—
122#389—48no—
123#388—216no—
124#387—48no—
125#384—28no—
126#384—24no—
127#382—24no—
128#380—48no—
129C4×C4⋊C4—28no—
130C4×C4⋊C4—28no—
131C4×C4⋊C4—28no—
132C4×C4⋊C4—24no—
133C4×C4⋊C4—24no—
134C4×C4⋊C4—24no—
135#376—24no—
136C2×#95C2314yes—
137C2×#95C2312yes—
138C2×#95C2314yes—
139C2×#95C2314yes—
140C2×#95C2311yescharacteristic
141C2×#95C2312yes—
142C2×#95C2312yes—
143C2×#95C2312yes—
144C2×#95C2312yes—
145C2×#95C2311yescharacteristic
146C2×#95C2311yescharacteristic
147C2×#95C2311yescharacteristic
148C2×#95C2314yes—
149C2×#95C2312yes—
150C2×#95C2312yes—
151C2×#95C2312yes—
152C2×#95C2312yes—
153C2×#95C2314yes—
154C2×#95C2314yes—
155C2×#95C2314yes—
156C2×#95C2314yes—
157C2×#95C2314yes—
158C2×#95C2314yes—
159C2×#95C2314yes—
160C2×#95C2314yes—
161C2×#95C2314yes—
162C2×#95C2314yes—
163C25C2412yes—
164C22×D4—22no—
165C22×D4—22no—
166C22×D4—22no—
167C22×D4—22no—
168C22×D4—22no—
169C22×D4—22no—
170C22×D4C2×D414yes—
171C23×C4—24no—
172C23×C4—24no—
173C23×C4—24no—
174C23×C4—44no—
175C23×C4—44no—
176C23×C4C2414yes—
177C23×C4C2412yes—
178C23×C4C2414yes—
179C23×C4C2412yes—
180C23×C4C2412yes—
181C23×C4C2412yes—
182C23×C4C2412yes—
183C23×C4C2412yes—
184C23×C4C2412yes—
185C23×C4C2412yes—
186C23×C4C2412yes—
187C23×C4C2411yescharacteristic
188C23×C4C2411yescharacteristic
189C23×C4C2411yescharacteristic
190#126—464no—
191C4×D4—416no—
192C2×C4⋊C4—24no—
193C2×C4⋊C4—24no—
194C2×C4⋊C4—24no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—24no—
197C2×C4⋊C4—24no—
198C2×C4⋊C4—24no—
199C2×C4⋊C4—24no—
200C2×C4⋊C4—24no—
201C2×C4⋊C4—24no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—28no—
204C2×C4⋊C4—28no—
205C2×C4⋊C4—24no—
206C2×C4⋊C4—24no—
207C2×C4⋊C4—24no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—24no—
212C2×C4⋊C4—24no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—24no—
215C2×C4⋊C4—24no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—28no—
218C2×C4⋊C4—24no—
219C2×C4⋊C4—24no—
220C2×C4⋊C4—24no—
221C2×C4⋊C4—24no—
222C2×C4⋊C4—24no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—28no—
225C2×C4⋊C4—24no—
226C2×C4⋊C4—24no—
227C2×C4⋊C4—42no—
228C2×C4⋊C4—42no—
229C2×C4⋊C4—42no—
230C2×C4⋊C4—44no—
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⋊C4C2×D414yes—
238C2×C22⋊C4—28no—
239C2×C22⋊C4—28no—
240C2×C22⋊C4—28no—
241C2×C22⋊C4—24no—
242C2×C22⋊C4—24no—
243C2×C22⋊C4—24no—
244C2×C22⋊C4—28no—
245C2×C22⋊C4—28no—
246C2×C22⋊C4—28no—
247C2×C22⋊C4—28no—
248C2×C22⋊C4—28no—
249C2×C22⋊C4—44no—
250C2×C22⋊C4—44no—
251C2×C22⋊C4—44no—
252C2×C22⋊C4—44no—
253C2×C22⋊C4—44no—
254C2×C22⋊C4—44no—
255C2×C22⋊C4—44no—
256C2×C22⋊C4—44no—
257C2×C22⋊C4—44no—
258C2×C22⋊C4C2×D4116yes—
259C2×C42—22no—
260C2×C42—22no—
261C2×C42—22no—
262C2×C42—22no—
263C2×C42—28no—
264C2×C42—24no—
265C2×C42—24no—
266C2×C42—24no—
267C2×C42—24no—
268C2×C42—24no—
269C2×C42—24no—
270C2×C42—28no—
271C2×C42—24no—
272C2×C42—44no—
273C2×C42—44no—
274C2×C42C2×D414yes—
275C2×C42#4114yes—
276C2×C42#4114yes—
277C2×C42#4114yes—
278C2×C42#4114yes—
279#95—24no—
280#95—28no—
281#95—24no—
282#95—28no—
283#95—24no—
284#95—24no—
285#95—24no—
286#95—22no—
287#95—22no—
288#95—22no—
289#95—24no—
290#95—24no—
291#95—22no—
292#95—24no—
293#95—24no—
294#95—28no—
295#95—28no—
296#95—44no—
297#95—44no—
298#95—44no—
299#95—44no—
300#95—44no—
301#95—44no—
302#95—44no—
303#95—44no—
304#95—44no—
305#95—44no—
306C24—22no—
307C24—22no—
308C24—22no—
309C24—22no—
310C24—24no—
311C24—24no—
312C24—24no—
313C24—22no—
314C24—22no—
315C24—24no—
316C24C22×D414yes—
317C24C2511yescharacteristic
318C2×D4—28no—
319C2×D4—28no—
320C2×D4—28no—
321C2×D4—44no—
322C2×D4—44no—
323C2×D4—44no—
324C2×D4—44no—
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—22no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—24no—
346C22×C4—22no—
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—22no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—24no—
365C22×C4—24no—
366C22×C4—24no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—24no—
375C22×C4—24no—
376C22×C4—24no—
377C22×C4—24no—
378C22×C4—24no—
379C22×C4—24no—
380C22×C4—24no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—24no—
384C22×C4—24no—
385C22×C4—24no—
386C22×C4—24no—
387C22×C4—24no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—24no—
391C22×C4—24no—
392C22×C4—24no—
393C22×C4—22no—
394C22×C4—416no—
395C22×C4—48no—
396C22×C4—88no—
397C22×C4—84no—
398C22×C4—84no—
399C22×C4C22×D414yes—
400C22×C4C22×D414yes—
401C22×C4C22×D412yes—
402C22×C4C2×#4114yes—
403C22×C4C2×#4112yes—
404C22×C4C2×#4112yes—
405C22×C4C2×#4112yes—
406C22×C4C2×#4112yes—
407C22×C4C2×#4112yes—
408C22×C4C2×#4112yes—
409C22×C4C2×#4112yes—
410C22×C4C2×#4112yes—
411C22×C4C2×#4112yes—
412C22×C4C2×#4112yes—
413C22×C4H(5,𝔽2)14yes—
414C22×C4H(5,𝔽2)14yes—
415C22×C42−1+414yes—
416C22×C42−1+412yes—
417C22×C42−1+412yes—
418C4⋊C4—216no—
419C4⋊C4—216no—
420C4⋊C4—216no—
421C4⋊C4—28no—
422C4⋊C4—48no—
423C4⋊C4—48no—
424C4⋊C4—48no—
425C4⋊C4—416no—
426C4⋊C4—416no—
427C4⋊C4—44no—
428C4⋊C4—48no—
429C4⋊C4—48no—
430C4⋊C4—44no—
431C4⋊C4—84no—
432C4⋊C4—84no—
433C4⋊C4—88no—
434C22⋊C4—416no—
435C22⋊C4—416no—
436C22⋊C4—416no—
437C22⋊C4—416no—
438C22⋊C4—48no—
439C22⋊C4—88no—
440C22⋊C4—88no—
441C42—28no—
442C42—28no—
443C42—416no—
444C42—48no—
445C42—48no—
446C42—84no—
447C42#126116yescomplemented
448C23—24no—
449C23—24no—
450C23—24no—
451C23—28no—
452C23—28no—
453C23—42no—
454C23—42no—
455C23—44no—
456C23—42no—
457C23—42no—
458C23—44no—
459C23—44no—
460C23—44no—
461C23—44no—
462C23—42no—
463C23—42no—
464C23—42no—
465C23—42no—
466C23—44no—
467C23—44no—
468C23—44no—
469C23—44no—
470C23—44no—
471C23C23×D411yescharacteristic
472C23C22×#4111yescharacteristic
473C23C22×#4111yescharacteristic
474C23C22×#4111yescharacteristic
475C23C22×#4111yescharacteristic
476C23C2×H(5,𝔽2)12yes—
477C23C2×H(5,𝔽2)12yes—
478C23C2×2−1+411yescharacteristic
479C23C2×2−1+411yescharacteristic
480C23C2×2−1+412yes—
481C23#58512yes—
482D4—416no—
483C2×C4—24no—
484C2×C4—24no—
485C2×C4—24no—
486C2×C4—22no—
487C2×C4—22no—
488C2×C4—22no—
489C2×C4—22no—
490C2×C4—24no—
491C2×C4—24no—
492C2×C4—24no—
493C2×C4—24no—
494C2×C4—24no—
495C2×C4—28no—
496C2×C4—24no—
497C2×C4—24no—
498C2×C4—24no—
499C2×C4—24no—
500C2×C4—24no—
501C2×C4—24no—
502C2×C4—24no—
503C2×C4—24no—
504C2×C4—24no—
505C2×C4—24no—
506C2×C4—24no—
507C2×C4—24no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—42no—
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—42no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—44no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—44no—
534C2×C4—44no—
535C2×C4—44no—
536C2×C4—44no—
537C2×C4—42no—
538C2×C4—44no—
539C2×C4—44no—
540C2×C4—42no—
541C2×C4—44no—
542C2×C4—44no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—44no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—44no—
550C2×C4—42no—
551C2×C4—44no—
552C2×C4—44no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—44no—
557C2×C4—44no—
558C2×C4—44no—
559C2×C4—44no—
560C2×C4—44no—
561C2×C4—44no—
562C2×C4—816no—
563C2×C4C2×#12614yes—
564C2×C4C2×#12614yes—
565C2×C4C2×#12614yes—
566C2×C4#53214yes—
567C2×C4#54114yes—
568C2×C4#54714yes—
569C2×C4#54714yes—
570C2×C4#55214yes—
571C2×C4#55614yes—
572C22—48no—
573C22—44no—
574C22—44no—
575C22—48no—
576C22—44no—
577C22—84no—
578C22—84no—
579C22—82no—
580C22—82no—
581C22—82no—
582C22—82no—
583C22C22×#12611yescharacteristic
584C22C2×#53211yescharacteristic
585C22C2×#54111yescharacteristic
586C22#346012yes—
587C22C2×#54711yescharacteristic
588C22C2×#54711yescharacteristic
589C22C2×#55211yescharacteristic
590C22C2×#55611yescharacteristic
591C22#348412yes—
592C22#348912yes—
593C22#349212yes—
594C22#349212yes—
595C22#349412yes—
596C22#349412yes—
597C22#350811yescharacteristic
598C22#350811yescharacteristic
599C22#350911yescharacteristic
600C22#351011yescharacteristic
601C22#351112yes—
602C22#354812yes—
603C22#355012yes—
604C22#355412yes—
605C22#355512yes—
606C4—24no—
607C4—24no—
608C4—44no—
609C4—44no—
610C4—44no—
611C4—44no—
612C4—44no—
613C4—44no—
614C4—82no—
615C4—82no—
616C4—84no—
617C4—84no—
618C4—82no—
619C4#249618yes—
620C2—84no—
621C2C2×#249611yescharacteristic
622C2#3526611yescharacteristic
623C2#3555711yescharacteristic
624C2#3592312yes—
625C2#3779412yes—
626C2#3798312yes—
627C2#4250912yes—
628C2#4493111yescharacteristic
629C2#4497911yescharacteristic
630C2#4542911yescharacteristic
631C2#4543011yescharacteristic
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. #16422C2semidirect
3. #15911C2semidirect
4. #15433C2semidirect
5. #15200C2semidirect
7. #14931C2semidirect
8. #14276C2semidirect
10. #13910C2semidirect
12. #13884C2semidirect
13. #13856C2semidirect
14. #13804C2semidirect
447. C42#126semidirect

Maximal subgroups

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

subgroupindex
C2×#25122
#164222
#159112
#154332
#152002
#151472
#149312
#142762
#139412
#139102
#139062
#138842
#138562
#138042
#137852