Finite Group Databasebeta

← #4463836

Subgroups — #4463836

Order 512.

2468 conjugacy classes of subgroups.

The subgroup lattice has 651 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#17255C214yesmaximal, complemented
2#17117C214yesmaximal, complemented
3#17115C214yesmaximal, complemented
4#17095C214yesmaximal, complemented
5#15229C212yesmaximal, complemented
6#15216C211yesmaximal, characteristic, complemented
7#15156C212yesmaximal, complemented
8#15138C212yesmaximal, complemented
9#15127C212yesmaximal, complemented
10#15126C212yesmaximal, complemented
11#15124C211yesmaximal, characteristic
12#14257C212yesmaximal, complemented
13#13844C211yesmaximal, characteristic, complemented
14C22×#127C2211yescharacteristic
15C22×#125C2211yescharacteristic
16C22×#121C2212yes—
17C22×#117C2211yescharacteristic
18C2×#400C2214yes—
19C2×#397C2214yes—
20C2×#394C2214yes—
21C2×#394C2214yes—
22C2×#394C2214yes—
23C2×#392C2214yes—
24C2×#390C2212yes—
25C2×#390C2212yes—
26C2×#390C2212yes—
27C2×#390C2212yes—
28C2×#390C2212yes—
29C2×#389C2212yes—
30C2×#380C2212yes—
31C2×C4×C4⋊C4C2212yes—
32C2×C4×C22⋊C4C2211yescharacteristic
33C2×C4×C22⋊C4C2211yescharacteristic
34#1444C2212yes—
35#1443C2214yes—
36#1443C2214yes—
37#1442C2212yes—
38#1442C2214yes—
39#1441C2214yes—
40#1440C2214yes—
41#1440C2214yes—
42#1440C2214yes—
43#1439C2212yes—
44#1439C2212yes—
45#1439C2214yes—
46#1439C2214yes—
47#1439C2214yes—
48#1439C2214yes—
49#1439C2214yes—
50#1439C2214yes—
51#1439C2214yes—
52#1439C2214yes—
53#1439C2214yes—
54#1438C2214yes—
55#1438C2214yes—
56#1438C2214yes—
57#1438C2214yes—
58#1438C2214yes—
59#1438C2214yes—
60C4⋊#95C2211yescharacteristic
61C4⋊#95C2212yes—
62C4⋊#95C2212yes—
63C4⋊#95C2211yescharacteristic
64C4⋊#95C2212yes—
65C4⋊#95C2212yes—
66C4×#95C2212yes—
67C23×D4C2312yes—
68C23×D4C2314yes—
69C23×D4C2314yes—
70C24×C4C2311yescharacteristic
71C2×#127—42no—
72C2×#127—44no—
73C2×#125—42no—
74C2×#125—44no—
75C2×#121—28no—
76C2×#121—44no—
77C2×#121—44no—
78C2×#117—44no—
79C2×#117—42no—
80C22×C4⋊C4C2312yes—
81C22×C4⋊C4C2312yes—
82C22×C4⋊C4C2312yes—
83C22×C4⋊C4C2312yes—
84C22×C4⋊C4C2312yes—
85C22×C4⋊C4C2314yes—
86C22×C4⋊C4C2312yes—
87C22×C4⋊C4C2314yes—
88C22×C22⋊C4C2312yes—
89C22×C22⋊C4C2312yes—
90C22×C22⋊C4C2314yes—
91C22×C22⋊C4C2314yes—
92C22×C22⋊C4C2314yes—
93C22×C22⋊C4C2314yes—
94C22×C22⋊C4C2314yes—
95C22×C22⋊C4C2314yes—
96C22×C22⋊C4C2314yes—
97C22×C22⋊C4C2314yes—
98C22×C22⋊C4C2314yes—
99C22×C22⋊C4C2314yes—
100C22×C22⋊C4C2314yes—
101C22×C22⋊C4C2311yescharacteristic
102C22×C22⋊C4C2311yescharacteristic
103C22×C22⋊C4C2314yes—
104C22×C42C2312yes—
105C22×C42C2311yescharacteristic
106C22×C42C2311yescharacteristic
107C22×C42C2312yes—
108#400—48no—
109#397—48no—
110#394—48no—
111#394—48no—
112#394—48no—
113#392—48no—
114#390—44no—
115#390—44no—
116#390—44no—
117#390—44no—
118#390—44no—
119#389—44no—
120#380—44no—
121C4×C4⋊C4—44no—
122C4×C22⋊C4—24no—
123C4×C22⋊C4—42no—
124C2×#95C2314yes—
125C2×#95C2314yes—
126C2×#95C2312yes—
127C2×#95C2311yescharacteristic
128C2×#95C2311yescharacteristic
129C2×#95C2314yes—
130C2×#95C2314yes—
131C2×#95C2314yes—
132C2×#95C2314yes—
133C2×#95C2314yes—
134C2×#95C2314yes—
135C2×#95C2314yes—
136C2×#95C2312yes—
137C2×#95C2312yes—
138C2×#95C2312yes—
139C2×#95C2314yes—
140C2×#95C2312yes—
141C2×#95C2312yes—
142C2×#95C2312yes—
143C2×#95C2314yes—
144C2×#95C2314yes—
145C25C2414yes—
146C25C2411yescharacteristic
147C22×D4—28no—
148C22×D4—24no—
149C22×D4—24no—
150C22×D4—24no—
151C22×D4—24no—
152C22×D4—24no—
153C22×D4—24no—
154C22×D4—28no—
155C22×D4—24no—
156C22×D4—44no—
157C22×D4—44no—
158C22×D4—44no—
159C22×D4—44no—
160C22×D4—44no—
161C22×D4—44no—
162C22×D4—44no—
163C22×D4—44no—
164C22×D4—44no—
165C22×D4—44no—
166C22×D4—44no—
167C22×D4—44no—
168C23×C4—24no—
169C23×C4—22no—
170C23×C4—24no—
171C23×C4—22no—
172C23×C4#4114yes—
173C23×C4C2414yes—
174C23×C4C2414yes—
175C23×C4C2412yes—
176C23×C4C2412yes—
177C23×C4C2412yes—
178C23×C4C2412yes—
179C23×C4C2414yes—
180C23×C4C2411yescharacteristic
181C23×C4C2412yes—
182C23×C4C2412yes—
183C23×C4C2411yescharacteristic
184#127—416no—
185#125—416no—
186#121—432no—
187#117—416no—
188C2×C4⋊C4—24no—
189C2×C4⋊C4—24no—
190C2×C4⋊C4—24no—
191C2×C4⋊C4—24no—
192C2×C4⋊C4—24no—
193C2×C4⋊C4—24no—
194C2×C4⋊C4—24no—
195C2×C4⋊C4—24no—
196C2×C4⋊C4—24no—
197C2×C4⋊C4—24no—
198C2×C4⋊C4—24no—
199C2×C4⋊C4—24no—
200C2×C4⋊C4—28no—
201C2×C4⋊C4—42no—
202C2×C4⋊C4—42no—
203C2×C4⋊C4—42no—
204C2×C4⋊C4—42no—
205C2×C4⋊C4—44no—
206C2×C4⋊C4—44no—
207C2×C4⋊C4—44no—
208C2×C4⋊C4—42no—
209C2×C4⋊C4—42no—
210C2×C4⋊C4—44no—
211C2×C4⋊C4—44no—
212C2×C22⋊C4—24no—
213C2×C22⋊C4—28no—
214C2×C22⋊C4—24no—
215C2×C22⋊C4—24no—
216C2×C22⋊C4—28no—
217C2×C22⋊C4—28no—
218C2×C22⋊C4—42no—
219C2×C22⋊C4—42no—
220C2×C22⋊C4—42no—
221C2×C22⋊C4—42no—
222C2×C22⋊C4—41no—
223C2×C22⋊C4—41no—
224C2×C22⋊C4—44no—
225C2×C22⋊C4—44no—
226C2×C22⋊C4—42no—
227C2×C22⋊C4—44no—
228C2×C22⋊C4—44no—
229C2×C22⋊C4—44no—
230C2×C22⋊C4—44no—
231C2×C22⋊C4—44no—
232C2×C22⋊C4—44no—
233C2×C22⋊C4—44no—
234C2×C22⋊C4—44no—
235C2×C22⋊C4—44no—
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—44no—
246C2×C22⋊C4—44no—
247C2×C22⋊C4—44no—
248C2×C22⋊C4—44no—
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×C42—24no—
259C2×C42—24no—
260C2×C42—24no—
261C2×C42—24no—
262C2×C42—22no—
263C2×C42—24no—
264C2×C42—22no—
265C2×C42—24no—
266C2×C42—24no—
267C2×C42—24no—
268#95—22no—
269#95—22no—
270#95—28no—
271#95—28no—
272#95—24no—
273#95—42no—
274#95—42no—
275#95—42no—
276#95—44no—
277#95—42no—
278#95—42no—
279#95—42no—
280#95—44no—
281#95—44no—
282#95—44no—
283#95—44no—
284#95—44no—
285#95—44no—
286#95—44no—
287#95—44no—
288#95—44no—
289C24—22no—
290C24—22no—
291C24—21no—
292C24—22no—
293C24—22no—
294C24—21no—
295C24—22no—
296C24—22no—
297C24—24no—
298C24—24no—
299C24—24no—
300C24—24no—
301C24—24no—
302C24—24no—
303C24—24no—
304C24—24no—
305C24—24no—
306C24—24no—
307C24—24no—
308C24—24no—
309C24—24no—
310C24—24no—
311C24—24no—
312C24C2×#4112yes—
313C24C2511yescharacteristic
314C2×D4—416no—
315C2×D4—416no—
316C2×D4—48no—
317C2×D4—48no—
318C2×D4—48no—
319C2×D4—48no—
320C2×D4—48no—
321C2×D4—48no—
322C2×D4—48no—
323C2×D4—88no—
324C2×D4—88no—
325C2×D4—88no—
326C2×D4—88no—
327C2×D4—88no—
328C2×D4—88no—
329C2×D4—88no—
330C2×D4—88no—
331C2×D4—88no—
332C2×D4—88no—
333C2×D4—88no—
334C2×D4—88no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—21no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—21no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—216no—
352C22×C4—28no—
353C22×C4—22no—
354C22×C4—24no—
355C22×C4—22no—
356C22×C4—24no—
357C22×C4—22no—
358C22×C4—24no—
359C22×C4—24no—
360C22×C4—24no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—24no—
364C22×C4—22no—
365C22×C4—24no—
366C22×C4—24no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—24no—
371C22×C4—24no—
372C22×C4—24no—
373C22×C4—24no—
374C22×C4—24no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—24no—
386C22×C4—24no—
387C22×C4—24no—
388C22×C4—24no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—24no—
398C22×C4—24no—
399C22×C4—22no—
400C22×C4—24no—
401C22×C4—24no—
402C22×C4—44no—
403C22×C4—44no—
404C22×C4—48no—
405C22×C4C2×#4112yes—
406C22×C4C2×#4112yes—
407C22×C4H(5,𝔽2)14yes—
408C22×C4H(5,𝔽2)14yes—
409C22×C4H(5,𝔽2)14yes—
410C22×C4H(5,𝔽2)14yes—
411C22×C4H(5,𝔽2)12yes—
412C22×C42−1+412yes—
413C4⋊C4—48no—
414C4⋊C4—48no—
415C4⋊C4—48no—
416C4⋊C4—416no—
417C4⋊C4—48no—
418C4⋊C4—48no—
419C4⋊C4—48no—
420C4⋊C4—88no—
421C22⋊C4—48no—
422C22⋊C4—48no—
423C22⋊C4—416no—
424C22⋊C4—44no—
425C22⋊C4—44no—
426C22⋊C4—88no—
427C22⋊C4—88no—
428C22⋊C4—88no—
429C22⋊C4—88no—
430C22⋊C4—88no—
431C22⋊C4—88no—
432C22⋊C4—88no—
433C22⋊C4—88no—
434C22⋊C4—88no—
435C22⋊C4—88no—
436C22⋊C4—88no—
437C42—48no—
438C42—44no—
439C42—48no—
440C42—44no—
441C23—24no—
442C23—22no—
443C23—24no—
444C23—22no—
445C23—22no—
446C23—41no—
447C23—41no—
448C23—42no—
449C23—42no—
450C23—42no—
451C23—42no—
452C23—42no—
453C23—42no—
454C23—42no—
455C23—42no—
456C23—42no—
457C23—41no—
458C23—42no—
459C23—42no—
460C23—41no—
461C23—42no—
462C23—44no—
463C23—44no—
464C23—44no—
465C23—44no—
466C23—44no—
467C23—44no—
468C23—44no—
469C23—44no—
470C23—44no—
471C23—44no—
472C23—44no—
473C23—44no—
474C23—44no—
475C23—44no—
476C23—44no—
477C23—44no—
478C23—44no—
479C23—44no—
480C23—44no—
481C23—44no—
482C23—44no—
483C23—44no—
484C23—44no—
485C23—44no—
486C23—44no—
487C23—44no—
488C23—44no—
489C23—44no—
490C23—44no—
491C23—44no—
492C23—44no—
493C23—44no—
494C23—44no—
495C23—44no—
496C23—44no—
497C23C22×#4111yescharacteristic
498C23C2×H(5,𝔽2)12yes—
499C23C2×H(5,𝔽2)12yes—
500C23C2×H(5,𝔽2)11yescharacteristic
501C23C2×H(5,𝔽2)12yes—
502C23C2×H(5,𝔽2)12yes—
503C23C2×2−1+411yescharacteristic
504C23#58512yes—
505C23#58512yes—
506D4—832no—
507D4—832no—
508D4—816no—
509C2×C4—24no—
510C2×C4—22no—
511C2×C4—24no—
512C2×C4—24no—
513C2×C4—24no—
514C2×C4—24no—
515C2×C4—24no—
516C2×C4—24no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—41no—
520C2×C4—42no—
521C2×C4—41no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—416no—
529C2×C4—42no—
530C2×C4—44no—
531C2×C4—42no—
532C2×C4—44no—
533C2×C4—44no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—44no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—44no—
540C2×C4—42no—
541C2×C4—44no—
542C2×C4—42no—
543C2×C4—44no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—44no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—44no—
551C2×C4—42no—
552C2×C4—44no—
553C2×C4—44no—
554C2×C4—44no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—44no—
560C2×C4—44no—
561C2×C4—44no—
562C2×C4—44no—
563C2×C4—44no—
564C2×C4—44no—
565C2×C4—42no—
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—42no—
576C2×C4—44no—
577C2×C4—42no—
578C2×C4#53818yes—
579C2×C4#53914yes—
580C2×C4#56014yes—
581C2×C4H(3,𝔽4)14yes—
582C2×C4#56218yes—
583C22—42no—
584C22—42no—
585C22—42no—
586C22—44no—
587C22—44no—
588C22—82no—
589C22—82no—
590C22—82no—
591C22—82no—
592C22—84no—
593C22—84no—
594C22—84no—
595C22—84no—
596C22—84no—
597C22—84no—
598C22—84no—
599C22—84no—
600C22—84no—
601C22—84no—
602C22—84no—
603C22—84no—
604C22—84no—
605C22—84no—
606C22—84no—
607C22C2×#53812yes—
608C22C2×#53911yescharacteristic
609C22#349412yes—
610C22#350811yescharacteristic
611C22#351211yescharacteristic
612C22C2×#56011yescharacteristic
613C22C2×H(3,𝔽4)11yescharacteristic
614C22C2×#56212yes—
615C22#353712yes—
616C22#354212yes—
617C22#354212yes—
618C22#354312yes—
619C22#354612yes—
620C22#354612yes—
621C22#354612yes—
622C22#354612yes—
623C22#354811yescharacteristic
624C22#355112yes—
625C22#355412yes—
626C22#355711yescharacteristic
627C22#356212yes—
628C4—42no—
629C4—44no—
630C4—44no—
631C4—82no—
632C4—84no—
633C4—84no—
634C4—82no—
635C4—82no—
636C4—82no—
637C4—84no—
638C4#282418yes—
639C2—82no—
640C2—164no—
641C2C2×#282411yescharacteristic
642C2#3728611yescharacteristic
643C2#3731811yescharacteristic
644C2#3733112yes—
645C2#3735612yes—
646C2#5296712yes—
647C2#5297312yes—
648C2#5305912yes—
649C2#5311012yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radical#15124C2
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. #17255C2semidirect
2. #17117C2semidirect
3. #17115C2semidirect
4. #17095C2semidirect
5. #15229C2semidirect
6. #15216C2semidirect
7. #15156C2semidirect
8. #15138C2semidirect
9. #15127C2semidirect
10. #15126C2semidirect
12. #14257C2semidirect
13. #13844C2semidirect

Maximal subgroups

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

subgroupindex
#172552
#171172
#171152
#170952
#152292
#152162
#151562
#151382
#151272
#151262
#151242
#142572
#138442