Finite Group Databasebeta

← #2746549

Subgroups — #2746549

Order 512.

1874 conjugacy classes of subgroups.

The subgroup lattice has 597 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#17300C212yesmaximal, complemented
2#17268C214yesmaximal, complemented
3#17112C212yesmaximal, complemented
4#16417C214yesmaximal
5#16318C212yesmaximal
6#15233C212yesmaximal
7#15232C212yesmaximal, complemented
8#15232C212yesmaximal, complemented
9#15195C211yesmaximal, characteristic
10#15158C212yesmaximal, complemented
11#15146C212yesmaximal, complemented
12#14669C212yesmaximal
13#14130C211yesmaximal, characteristic
14#14130C212yesmaximal
15C4×#393C211yesmaximal, characteristic
16C22×C4×Q8C2211yescharacteristic
17C23×C4⋊C4C2211yescharacteristic
18C2×#400C2214yes—
19C2×#395C2214yes—
20C2×#393C2211yescharacteristic
21C2×#389C2212yes—
22C2×#389C2212yes—
23C2×#389C2212yes—
24C2×#385C2212yes—
25C2×#385C2214yes—
26C2×#384C2211yescharacteristic
27C2×#382C2212yes—
28C2×#380C2212yes—
29C2×#380C2211yescharacteristic
30C2×C4×C22⋊C4C2211yescharacteristic
31C2×C4×C22⋊C4C2212yes—
32#1446C2211yescharacteristic
33#1445C2214yes—
34#1445C2214yes—
35#1444C2214yes—
36#1444C2212yes—
37#1443C2214yes—
38#1443C2214yes—
39#1443C2214yes—
40#1443C2214yes—
41#1443C2214yes—
42#1443C2214yes—
43#1442C2214yes—
44#1442C2214yes—
45#1442C2212yes—
46#1442C2211yescharacteristic
47#1442C2214yes—
48#1442C2212yes—
49#1442C2214yes—
50#1442C2212yes—
51#1442C2212yes—
52#1442C2214yes—
53#1441C2214yes—
54#1441C2214yes—
55#1441C2214yes—
56#1441C2214yes—
57#1440C2214yes—
58#1440C2214yes—
59#1439C2212yes—
60#1439C2212yes—
61#1439C2211yescharacteristic
62#1439C2212yes—
63C4⋊#95C2212yes—
64C4⋊#95C2212yes—
65C4⋊#95C2212yes—
66C4⋊#95C2212yes—
67C4⋊#95C2212yes—
68C4⋊#95C2212yes—
69C4⋊#95C2212yes—
70C4⋊#95C2211yescharacteristic
71C22⋊#95C2212yes—
72C22⋊#95C2214yes—
73C4×#95C2212yes—
74C4×#95C2211yescharacteristic
75C23×Q8C2311yescharacteristic
76C24×C4C2311yescharacteristic
77C24×C4C2312yes—
78C2×C4×Q8—42no—
79C2×C4×Q8—44no—
80C22×C4⋊C4—42no—
81C22×C4⋊C4—44no—
82C22×C4⋊C4C2312yes—
83C22×C4⋊C4C2312yes—
84C22×C4⋊C4C2312yes—
85C22×C4⋊C4C2312yes—
86C22×C4⋊C4C2314yes—
87C22×C4⋊C4C2314yes—
88C22×C4⋊C4C2314yes—
89C22×C4⋊C4C2312yes—
90C22×C4⋊C4C2312yes—
91C22×C4⋊C4C2311yescharacteristic
92C22×C4⋊C4C2312yes—
93C22×C4⋊C4C2312yes—
94C22×C22⋊C4C2312yes—
95C22×C22⋊C4C2312yes—
96C22×C22⋊C4C2311yescharacteristic
97C22×C22⋊C4C2312yes—
98C22×C22⋊C4C2314yes—
99C22×C22⋊C4C2311yescharacteristic
100C22×C42C2311yescharacteristic
101C22×C42C2312yes—
102C22×C42C2312yes—
103C22×C42C2311yescharacteristic
104#400—48no—
105#395—48no—
106#393—24no—
107#389—44no—
108#389—44no—
109#389—44no—
110#385—28no—
111#385—48no—
112#384—42no—
113#382—44no—
114#380—28no—
115#380—24no—
116C4×C22⋊C4—44no—
117C4×C22⋊C4—42no—
118C2×#95C2312yes—
119C2×#95C2311yescharacteristic
120C2×#95C2314yes—
121C2×#95C2312yes—
122C2×#95C2312yes—
123C2×#95C2312yes—
124C2×#95C2314yes—
125C2×#95C2314yes—
126C2×#95C2312yes—
127C2×#95C2314yes—
128C2×#95C2314yes—
129C2×#95C2314yes—
130C2×#95C2314yes—
131C2×#95C2314yes—
132C2×#95C2312yes—
133C2×#95C2312yes—
134C2×#95C2311yescharacteristic
135C2×#95C2314yes—
136C2×#95C2314yes—
137C2×#95C2314yes—
138C2×#95C2314yes—
139C2×#95C2314yes—
140C2×#95C2312yes—
141C2×#95C2314yes—
142C2×#95C2314yes—
143C2×#95C2314yes—
144C2×#95C2314yes—
145C2×#95C2314yes—
146C2×#95C2314yes—
147C2×#95C2314yes—
148C2×#95C2311yescharacteristic
149C2×#95C2312yes—
150C2×#95C2311yescharacteristic
151C2×#95C2312yes—
152C25C2411yescharacteristic
153C22×Q8—22no—
154C22×Q8—24no—
155C22×Q8—41no—
156C22×Q8—42no—
157C22×Q8—41no—
158C23×C4—22no—
159C23×C4—24no—
160C23×C4—22no—
161C23×C4—24no—
162C23×C4—24no—
163C23×C4—42no—
164C23×C4—44no—
165C23×C4—42no—
166C23×C4—44no—
167C23×C4#4114yes—
168C23×C4C2414yes—
169C23×C4C2414yes—
170C23×C4C2414yes—
171C23×C4C2412yes—
172C23×C4C2411yescharacteristic
173C23×C4C2412yes—
174C23×C4C2412yes—
175C23×C4C2414yes—
176C23×C4C2411yescharacteristic
177C23×C4C2411yescharacteristic
178C23×C4C2411yescharacteristic
179C23×C4C2412yes—
180C23×C4C2411yescharacteristic
181C23×C4C2411yescharacteristic
182C4×Q8—416no—
183C2×C4⋊C4—24no—
184C2×C4⋊C4—24no—
185C2×C4⋊C4—24no—
186C2×C4⋊C4—24no—
187C2×C4⋊C4—24no—
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—28no—
196C2×C4⋊C4—28no—
197C2×C4⋊C4—28no—
198C2×C4⋊C4—42no—
199C2×C4⋊C4—44no—
200C2×C4⋊C4—41no—
201C2×C4⋊C4—42no—
202C2×C4⋊C4—42no—
203C2×C4⋊C4—42no—
204C2×C4⋊C4—42no—
205C2×C4⋊C4—416no—
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×C4⋊C4—44no—
213C2×C4⋊C4—44no—
214C2×C4⋊C4—44no—
215C2×C4⋊C4—44no—
216C2×C22⋊C4—28no—
217C2×C22⋊C4—24no—
218C2×C22⋊C4—24no—
219C2×C22⋊C4—24no—
220C2×C22⋊C4—22no—
221C2×C22⋊C4—24no—
222C2×C22⋊C4—24no—
223C2×C22⋊C4—24no—
224C2×C22⋊C4—22no—
225C2×C22⋊C4—41no—
226C2×C22⋊C4—42no—
227C2×C22⋊C4—44no—
228C2×C22⋊C4—41no—
229C2×C22⋊C4—41no—
230C2×C22⋊C4—42no—
231C2×C22⋊C4—42no—
232C2×C22⋊C4—44no—
233C2×C22⋊C4#4114yes—
234C2×C42—22no—
235C2×C42—24no—
236C2×C42—24no—
237C2×C42—24no—
238C2×C42—24no—
239C2×C42—24no—
240C2×C42—22no—
241C2×C42—24no—
242C2×C42—24no—
243C2×C42—24no—
244#95—28no—
245#95—24no—
246#95—28no—
247#95—24no—
248#95—22no—
249#95—28no—
250#95—24no—
251#95—24no—
252#95—24no—
253#95—42no—
254#95—41no—
255#95—42no—
256#95—42no—
257#95—44no—
258#95—44no—
259#95—44no—
260#95—44no—
261#95—44no—
262#95—41no—
263#95—41no—
264#95—42no—
265#95—42no—
266#95—44no—
267#95—44no—
268#95—44no—
269#95—44no—
270#95—44no—
271#95—44no—
272#95—44no—
273#95—44no—
274#95—44no—
275#95—44no—
276#95—44no—
277#95—44no—
278C24—22no—
279C24—21no—
280C24—22no—
281C24—21no—
282C24—21no—
283C24—22no—
284C24—22no—
285C24—21no—
286C24C2×#4112yes—
287C24C2×#4112yes—
288C242−1+412yes—
289C24C2511yescharacteristic
290C2×Q8—48no—
291C2×Q8—44no—
292C2×Q8—44no—
293C2×Q8—48no—
294C2×Q8—44no—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—21no—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—22no—
301C22×C4—21no—
302C22×C4—21no—
303C22×C4—21no—
304C22×C4—22no—
305C22×C4—22no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—21no—
311C22×C4—21no—
312C22×C4—22no—
313C22×C4—21no—
314C22×C4—21no—
315C22×C4—22no—
316C22×C4—22no—
317C22×C4—216no—
318C22×C4—28no—
319C22×C4—22no—
320C22×C4—24no—
321C22×C4—24no—
322C22×C4—24no—
323C22×C4—24no—
324C22×C4—24no—
325C22×C4—22no—
326C22×C4—24no—
327C22×C4—24no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—24no—
332C22×C4—24no—
333C22×C4—24no—
334C22×C4—24no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—24no—
339C22×C4—21no—
340C22×C4—22no—
341C22×C4—24no—
342C22×C4—24no—
343C22×C4—24no—
344C22×C4—24no—
345C22×C4—24no—
346C22×C4—24no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—24no—
354C22×C4—21no—
355C22×C4—21no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—24no—
359C22×C4—24no—
360C22×C4—24no—
361C22×C4—24no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—24no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—24no—
370C22×C4—24no—
371C22×C4—24no—
372C22×C4—24no—
373C22×C4—44no—
374C22×C4—48no—
375C22×C4—44no—
376C22×C4—48no—
377C22×C4—84no—
378C22×C4—88no—
379C22×C4—84no—
380C22×C4—88no—
381C22×C4C2×#4112yes—
382C22×C4C2×#4112yes—
383C22×C4C2×#4112yes—
384C22×C4C2×#4112yes—
385C22×C4H(5,𝔽2)14yes—
386C22×C4H(5,𝔽2)12yes—
387C22×C4H(5,𝔽2)14yes—
388C22×C42−1+412yes—
389C22×C42−1+412yes—
390C22×C42−1+414yes—
391C4⋊C4—48no—
392C4⋊C4—44no—
393C4⋊C4—48no—
394C4⋊C4—48no—
395C4⋊C4—48no—
396C4⋊C4—48no—
397C4⋊C4—48no—
398C4⋊C4—88no—
399C4⋊C4—88no—
400C4⋊C4—88no—
401C4⋊C4—84no—
402C4⋊C4—84no—
403C22⋊C4—44no—
404C22⋊C4—48no—
405C22⋊C4—44no—
406C22⋊C4—416no—
407C22⋊C4—48no—
408C22⋊C4—48no—
409C42—44no—
410C42—48no—
411C42—44no—
412C42—48no—
413C23—24no—
414C23—22no—
415C23—22no—
416C23—24no—
417C23—22no—
418C23—22no—
419C23—24no—
420C23—24no—
421C23—22no—
422C23—24no—
423C23—24no—
424C23—22no—
425C23—41no—
426C23—42no—
427C23—41no—
428C23—41no—
429C23—42no—
430C23—42no—
431C23—42no—
432C23—41no—
433C23—42no—
434C23—42no—
435C23#55614yes—
436C23C22×#4111yescharacteristic
437C23C22×#4111yescharacteristic
438C23C2×H(5,𝔽2)11yescharacteristic
439C23C2×H(5,𝔽2)12yes—
440C23C2×H(5,𝔽2)12yes—
441C23C2×2−1+411yescharacteristic
442C23C2×2−1+412yes—
443C23C2×2−1+411yescharacteristic
444C23C2×2−1+411yescharacteristic
445C23#58511yescharacteristic
446C23#58512yes—
447Q8—416no—
448C2×C4—24no—
449C2×C4—22no—
450C2×C4—22no—
451C2×C4—22no—
452C2×C4—24no—
453C2×C4—24no—
454C2×C4—24no—
455C2×C4—24no—
456C2×C4—22no—
457C2×C4—24no—
458C2×C4—24no—
459C2×C4—42no—
460C2×C4—42no—
461C2×C4—41no—
462C2×C4—42no—
463C2×C4—41no—
464C2×C4—41no—
465C2×C4—41no—
466C2×C4—42no—
467C2×C4—41no—
468C2×C4—41no—
469C2×C4—41no—
470C2×C4—42no—
471C2×C4—42no—
472C2×C4—41no—
473C2×C4—42no—
474C2×C4—42no—
475C2×C4—416no—
476C2×C4—44no—
477C2×C4—44no—
478C2×C4—44no—
479C2×C4—44no—
480C2×C4—44no—
481C2×C4—44no—
482C2×C4—44no—
483C2×C4—44no—
484C2×C4—44no—
485C2×C4—44no—
486C2×C4—42no—
487C2×C4—44no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—44no—
491C2×C4—44no—
492C2×C4—44no—
493C2×C4—41no—
494C2×C4—42no—
495C2×C4—44no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—44no—
499C2×C4—44no—
500C2×C4—41no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—44no—
507C2×C4—44no—
508C2×C4—44no—
509C2×C4—42no—
510C2×C4—44no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—44no—
515C2×C4—44no—
516C2×C4—44no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—44no—
521C2×C4—42no—
522C2×C4—41no—
523C2×C4—44no—
524C2×C4—44no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—44no—
529C2×C4—42no—
530C2×C4—816no—
531C2×C4#53918yes—
532C2×C4#53914yes—
533C2×C4H(3,𝔽4)14yes—
534C2×C4#56318yes—
535C2×C4#56314yes—
536C22—28no—
537C22—24no—
538C22—42no—
539C22—44no—
540C22—42no—
541C22—42no—
542C22—42no—
543C22—44no—
544C22—42no—
545C22—44no—
546C22—42no—
547C22C2×#53912yes—
548C22C2×#53911yescharacteristic
549C22C2×#55611yescharacteristic
550C22#350511yescharacteristic
551C22#350612yes—
552C22#350711yescharacteristic
553C22#350911yescharacteristic
554C22#351512yes—
555C22#351912yes—
556C22C2×H(3,𝔽4)11yescharacteristic
557C22C2×#56312yes—
558C22C2×#56311yescharacteristic
559C22#353711yescharacteristic
560C22#353811yescharacteristic
561C22#354112yes—
562C22#354212yes—
563C22#354712yes—
564C22#354712yes—
565C22#355111yescharacteristic
566C22#355212yes—
567C22#355212yes—
568C22#356311yescharacteristic
569C22#356512yes—
570C4—42no—
571C4—42no—
572C4—44no—
573C4—42no—
574C4—42no—
575C4—81no—
576C4—82no—
577C4—84no—
578C4—84no—
579C4—84no—
580C4—84no—
581C4—82no—
582C4—82no—
583C4#283518yes—
584C2—44no—
585C2C2×#283511yescharacteristic
586C2#3898212yes—
587C2#3951211yescharacteristic
588C2#4251612yes—
589C2#4252211yescharacteristic
590C2#4564511yescharacteristic
591C2#4618012yes—
592C2#4619011yescharacteristic
593C2#5298711yescharacteristic
594C2#5300511yescharacteristic
595C2#5311912yes—
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. #17300C2semidirect
2. #17268C2semidirect
3. #17112C2semidirect
7. #15232C2semidirect
8. #15232C2semidirect
10. #15158C2semidirect
11. #15146C2semidirect

Maximal subgroups

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

subgroupindex
#173002
#172682
#171122
#164172
#163182
#152332
#152322
#151952
#151582
#151462
#146692
#141302
C4×#3932