Finite Group Databasebeta

← #413305

Subgroups — #413305

Order 512.

1488 conjugacy classes of subgroups.

The subgroup lattice has 618 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2552C211yesmaximal, characteristic, complemented
2C2×#2026C212yesmaximal, complemented
3C2×#2025C212yesmaximal
4C2×#1983C211yesmaximal, characteristic
5C2×#1959C211yesmaximal, characteristic
6#14153C211yesmaximal, characteristic
7#11067C212yesmaximal, complemented
8#11066C212yesmaximal
9#10849C211yesmaximal, characteristic, complemented
10#10835C211yesmaximal, characteristic, complemented
11#10703C211yesmaximal, characteristic, complemented
12C22×#125C2212yes—
13C22×#124C2211yescharacteristic
14C22×C4⋊C8C2211yescharacteristic
15C22×#103C2211yescharacteristic
16C22×#103C2211yescharacteristic
17C22×#103C2212yes—
18C22×#102C2211yescharacteristic
19C22×#102C2211yescharacteristic
20C22×#102C2212yes—
21C22×#97C2211yescharacteristic
22#2552—24no—
23#2552—24no—
24C2×#398—22no—
25C2×#391C2212yes—
26C2×#390C2212yes—
27C2×#389—21no—
28C2×#389C2212yes—
29C2×#384—21no—
30C2×#384C2211yescharacteristic
31C2×C4×C4⋊C4C2211yescharacteristic
32C2×C4×C4⋊C4C2211yescharacteristic
33C2×C4×C4⋊C4C2211yescharacteristic
34#2026—28no—
35#2025—28no—
36#1983—24no—
37#1959—24no—
38C2×#340C2212yes—
39C2×#340C2212yes—
40C2×#340C2212yes—
41C2×#340C2212yes—
42C2×#336C2212yes—
43C2×#336C2212yes—
44#1439—21no—
45#1439—21no—
46#1437—22no—
47C23×Q8C2311yescharacteristic
48C23×D4C2311yescharacteristic
49C23×C8C2311yescharacteristic
50C23×C8C2311yescharacteristic
51C23×C8C2312yes—
52C2×#125—24no—
53C2×#125—24no—
54C2×#125—24no—
55C2×#125—24no—
56C2×#125—42no—
57C2×#125—42no—
58C2×#124—22no—
59C2×#124—22no—
60C2×#124—22no—
61C2×#124—22no—
62C2×#124—22no—
63C2×#124Q814yes—
64C22×C4⋊C4—22no—
65C22×C4⋊C4—22no—
66C22×C4⋊C4—21no—
67C22×C4⋊C4—21no—
68C22×C4⋊C4—22no—
69C22×C4⋊C4D411yescharacteristic
70C22×C4⋊C4D411yescharacteristic
71C22×C4⋊C4C2312yes—
72C22×C4⋊C4C2312yes—
73C22×C4⋊C4C2311yescharacteristic
74C22×C4⋊C4C2311yescharacteristic
75C22×C22⋊C4—21no—
76C22×C22⋊C4—21no—
77C22×C22⋊C4—22no—
78C22×C22⋊C4—22no—
79C22×C42—21no—
80C22×C42—21no—
81C22×C42C2312yes—
82C22×C42C2311yescharacteristic
83C2×C4⋊C8—22no—
84C2×C4⋊C8—22no—
85C2×C4⋊C8—22no—
86C2×#103—24no—
87C2×#103—24no—
88C2×#103—22no—
89C2×#103—22no—
90C2×#103—24no—
91C2×#103—24no—
92C2×#103—42no—
93C2×#102—24no—
94C2×#102—24no—
95C2×#102—22no—
96C2×#102—22no—
97C2×#102—24no—
98C2×#102—24no—
99C2×#102—42no—
100C2×#97—22no—
101C2×#97—22no—
102C2×#97—22no—
103#398—216no—
104#391—24no—
105#391—42no—
106#390—24no—
107#390—42no—
108#389—28no—
109#389—24no—
110#389—24no—
111#384—28no—
112#384—22no—
113#384D414yes—
114C4×C4⋊C4—22no—
115C4×C4⋊C4—22no—
116C4×C4⋊C4—22no—
117C4×C4⋊C4—22no—
118C4×C4⋊C4—22no—
119C4×C4⋊C4D414yes—
120C2×#95—21no—
121C2×#95—22no—
122C2×#95—22no—
123C2×#95—22no—
124C2×#95—21no—
125C2×#95D412yes—
126#340—24no—
127#340—24no—
128#340—24no—
129#340D418yes—
130#336—24no—
131#336—24no—
132C25—21no—
133C22×Q8—21no—
134C22×Q8—21no—
135C22×Q8—21no—
136C22×Q8—22no—
137C22×Q8—21no—
138C22×Q8—21no—
139C22×Q8—22no—
140C22×Q8—22no—
141C22×Q8—22no—
142C22×Q8C2×Q812yes—
143C22×D4—21no—
144C22×D4—21no—
145C22×D4—22no—
146C22×D4—21no—
147C22×D4—21no—
148C22×D4—22no—
149C22×D4—21no—
150C22×D4—22no—
151C22×D4—22no—
152C22×D4C2×Q812yes—
153C23×C4—21no—
154C23×C4—21no—
155C23×C4—22no—
156C23×C4—21no—
157C23×C4—22no—
158C23×C4C2×D412yes—
159C23×C4C2×D412yes—
160C23×C4C2×D411yescharacteristic
161C23×C4C2×D411yescharacteristic
162C23×C4C2411yescharacteristic
163C22×C8—22no—
164C22×C8—22no—
165C22×C8—22no—
166C22×C8—22no—
167C22×C8—22no—
168C22×C8—22no—
169C22×C8—22no—
170C22×C8—22no—
171C22×C8—21no—
172C22×C8—21no—
173C22×C8—22no—
174C22×C8—22no—
175C22×C8—21no—
176C22×C8—21no—
177C22×C8—21no—
178C22×C8—21no—
179C22×C8C2×D414yes—
180#125—48no—
181#125—48no—
182#125—48no—
183#125—48no—
184#124—28no—
185#124—28no—
186#124—44no—
187#124—44no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—28no—
201C2×C4⋊C4—28no—
202C2×C4⋊C4—24no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—28no—
208C2×C4⋊C4—24no—
209C2×C4⋊C4—42no—
210C2×C4⋊C4—42no—
211C2×C4⋊C4—44no—
212C2×C4⋊C4—44no—
213C2×C4⋊C4—44no—
214C2×C4⋊C4—44no—
215C2×C4⋊C4—44no—
216C2×C4⋊C4—81no—
217C2×C4⋊C4—81no—
218C2×C4⋊C4—82no—
219C2×C4⋊C4—82no—
220C2×C4⋊C4C2×D414yes—
221C2×C4⋊C4C2×D414yes—
222C2×C4⋊C4C2×D412yes—
223C2×C4⋊C4C2×D414yes—
224C2×C4⋊C4C2×D412yes—
225C2×C4⋊C4#4114yes—
226C2×C22⋊C4—24no—
227C2×C22⋊C4—24no—
228C2×C22⋊C4—42no—
229C2×C22⋊C4—42no—
230C2×C22⋊C4—42no—
231C2×C22⋊C4—42no—
232C2×C22⋊C4—44no—
233C2×C22⋊C4—44no—
234C2×C22⋊C4—44no—
235C2×C22⋊C4—44no—
236C2×C22⋊C4—82no—
237C2×C22⋊C4—82no—
238C2×C42—24no—
239C2×C42—22no—
240C2×C42—22no—
241C2×C42—22no—
242C2×C42—22no—
243C2×C42—22no—
244C2×C42—22no—
245C2×C42—44no—
246C2×C42—42no—
247C2×C42—81no—
248C2×C42—81no—
249C2×C42C2×D412yes—
250C2×C42C2×D412yes—
251C2×C42C2×Q812yes—
252C2×C42#4112yes—
253C2×C42#4112yes—
254C2×C42#4112yes—
255C4⋊C8—44no—
256#103—48no—
257#103—44no—
258#103—44no—
259#102—48no—
260#102—44no—
261#102—44no—
262#97—44no—
263#95—28no—
264#95—24no—
265#95—24no—
266#95—44no—
267#95—44no—
268#95—42no—
269C24—22no—
270C24—41no—
271C24—41no—
272C24—42no—
273C24—42no—
274C24—42no—
275C24—42no—
276C24—41no—
277C24—41no—
278C24—41no—
279C24—41no—
280C24C22×D411yescharacteristic
281C2×Q8—22no—
282C2×Q8—22no—
283C2×Q8—24no—
284C2×Q8—24no—
285C2×Q8—41no—
286C2×Q8—41no—
287C2×Q8—42no—
288C2×Q8—42no—
289C2×Q8—41no—
290C2×Q8—41no—
291C2×Q8—42no—
292C2×Q8—41no—
293C2×Q8—41no—
294C2×Q8—42no—
295C2×Q8—42no—
296C2×Q8—42no—
297C2×Q8—42no—
298C2×Q8—42no—
299C2×D4—22no—
300C2×D4—22no—
301C2×D4—24no—
302C2×D4—24no—
303C2×D4—41no—
304C2×D4—41no—
305C2×D4—41no—
306C2×D4—41no—
307C2×D4—42no—
308C2×D4—42no—
309C2×D4—41no—
310C2×D4—41no—
311C2×D4—42no—
312C2×D4—42no—
313C2×D4—42no—
314C2×D4—42no—
315C2×D4—42no—
316C2×D4—42no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—21no—
320C22×C4—21no—
321C22×C4—21no—
322C22×C4—21no—
323C22×C4—22no—
324C22×C4—22no—
325C22×C4—24no—
326C22×C4—24no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—24no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—24no—
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—42no—
345C22×C4—42no—
346C22×C4—41no—
347C22×C4—41no—
348C22×C4—41no—
349C22×C4—41no—
350C22×C4—41no—
351C22×C4—42no—
352C22×C4—41no—
353C22×C4—42no—
354C22×C4—42no—
355C22×C4—42no—
356C22×C4—42no—
357C22×C4—42no—
358C22×C4—42no—
359C22×C4—42no—
360C22×C4—42no—
361C22×C4—42no—
362C22×C4—42no—
363C22×C4—42no—
364C22×C4—42no—
365C22×C4—42no—
366C22×C4#12012yes—
367C22×C4#12012yes—
368C22×C4C22⋊Q812yes—
369C22×C4C22⋊Q812yes—
370C22×C4#12412yes—
371C22×C4#12412yes—
372C22×C4C22×D411yescharacteristic
373C22×C4C22×D411yescharacteristic
374C22×C4C22×D412yes—
375C22×C4C22×Q811yescharacteristic
376C22×C4C2×#4111yescharacteristic
377C22×C4C2×#4111yescharacteristic
378C22×C4C2×#4111yescharacteristic
379C22×C4C2×#4112yes—
380C22×C4H(5,𝔽2)12yes—
381C22×C42−1+412yes—
382C2×C8—24no—
383C2×C8—24no—
384C2×C8—24no—
385C2×C8—42no—
386C2×C8—42no—
387C2×C8—42no—
388C2×C8—42no—
389C2×C8—41no—
390C2×C8—41no—
391C2×C8—41no—
392C2×C8—41no—
393C2×C8—41no—
394C2×C8—41no—
395C2×C8—42no—
396C2×C8—42no—
397C2×C8—42no—
398C2×C8—42no—
399C4⋊C4—22no—
400C4⋊C4—22no—
401C4⋊C4—24no—
402C4⋊C4—24no—
403C4⋊C4—24no—
404C4⋊C4—24no—
405C4⋊C4—24no—
406C4⋊C4—24no—
407C4⋊C4—44no—
408C4⋊C4—44no—
409C4⋊C4—48no—
410C4⋊C4—416no—
411C4⋊C4—416no—
412C4⋊C4—42no—
413C4⋊C4—42no—
414C4⋊C4—42no—
415C4⋊C4—42no—
416C4⋊C4—84no—
417C4⋊C4—88no—
418C4⋊C4#12414yes—
419C4⋊C4#12714yes—
420C22⋊C4—48no—
421C22⋊C4—48no—
422C22⋊C4—88no—
423C22⋊C4—88no—
424C42—48no—
425C42—42no—
426C42—42no—
427C42—42no—
428C42—42no—
429C42—84no—
430C42C22⋊Q814yes—
431C42C22⋊Q814yes—
432C42#12414yes—
433C42#12414yes—
434C23—44no—
435C23—42no—
436C23—42no—
437C23—42no—
438C23—44no—
439C23—82no—
440C23—82no—
441C23—82no—
442C23—82no—
443C23—81no—
444C23—81no—
445C23—82no—
446C23—81no—
447C23—81no—
448C23—82no—
449C23—81no—
450C23—81no—
451C23—82no—
452C23—81no—
453C23—82no—
454C23—81no—
455C23—82no—
456C23—82no—
457C23C2×#12011yescharacteristic
458C23C2×C22⋊Q811yescharacteristic
459C23C2×#12411yescharacteristic
460C23D4212yes—
461C23#54812yes—
462C23#57512yes—
463C23#57512yes—
464C23#57611yescharacteristic
465C23#57711yescharacteristic
466C23#57711yescharacteristic
467C23#57811yescharacteristic
468Q8—42no—
469Q8—44no—
470Q8—42no—
471Q8—81no—
472Q8—81no—
473Q8—82no—
474D4—42no—
475D4—44no—
476D4—42no—
477D4—81no—
478D4—81no—
479D4—82no—
480C2×C4—24no—
481C2×C4—22no—
482C2×C4—22no—
483C2×C4—22no—
484C2×C4—22no—
485C2×C4—24no—
486C2×C4—22no—
487C2×C4—22no—
488C2×C4—22no—
489C2×C4—22no—
490C2×C4—24no—
491C2×C4—24no—
492C2×C4—44no—
493C2×C4—44no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—44no—
497C2×C4—42no—
498C2×C4—42no—
499C2×C4—44no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—44no—
504C2×C4—44no—
505C2×C4—44no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—44no—
509C2×C4—42no—
510C2×C4—44no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—44no—
514C2×C4—42no—
515C2×C4—44no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—82no—
523C2×C4—82no—
524C2×C4—82no—
525C2×C4—82no—
526C2×C4—81no—
527C2×C4—81no—
528C2×C4—82no—
529C2×C4—82no—
530C2×C4—82no—
531C2×C4—82no—
532C2×C4—82no—
533C2×C4#39314yes—
534C2×C4#39314yes—
535C2×C4C2×C22⋊Q811yescharacteristic
536C2×C4C2×C22⋊Q811yescharacteristic
537C2×C4C2×#12411yescharacteristic
538C2×C4C2×#12411yescharacteristic
539C2×C4C2×#12411yescharacteristic
540C2×C4C2×#12711yescharacteristic
541C2×C4#53212yes—
542C2×C4#53212yes—
543C2×C4#53511yescharacteristic
544C2×C4#53611yescharacteristic
545C2×C4#54212yes—
546C2×C4#54212yes—
547C2×C4D4212yes—
548C2×C4#54712yes—
549C2×C4#54812yes—
550C2×C4D4×Q812yes—
551C2×C4#55412yes—
552C2×C4#55512yes—
553C8—44no—
554C8—81no—
555C8—81no—
556C22—82no—
557C22—84no—
558C22—84no—
559C22—82no—
560C22—82no—
561C22—161no—
562C22—161no—
563C22—162no—
564C22—162no—
565C22—161no—
566C22—161no—
567C22C2×#39311yescharacteristic
568C22#243311yescharacteristic
569C22#243611yescharacteristic
570C22#245212yes—
571C22#249112yes—
572C22#300511yescharacteristic
573C22#300511yescharacteristic
574C22#301211yescharacteristic
575C22#301311yescharacteristic
576C22#308012yes—
577C22#308311yescharacteristic
578C22#308411yescharacteristic
579C22#313611yescharacteristic
580C22#313711yescharacteristic
581C22#314212yes—
582C22#326712yes—
583C22#326812yes—
584C22#326912yes—
585C22#327112yes—
586C22#337012yes—
587C22#337112yes—
588C22#337612yes—
589C22#337712yes—
590C4—24no—
591C4—24no—
592C4—44no—
593C4—44no—
594C4—82no—
595C4—84no—
596C4—82no—
597C4—82no—
598C4—84no—
599C4#245212yes—
600C4#249112yes—
601C4#254211yescharacteristic
602C4#256311yescharacteristic
603C4#264011yescharacteristic
604C4#285811yescharacteristic
605C2—162no—
606C2#1514411yescharacteristic
607C2#2345311yescharacteristic
608C2#2345811yescharacteristic
609C2#2576411yescharacteristic
610C2#2576511yescharacteristic
611C2#2594011yescharacteristic
612C2#2594111yescharacteristic
613C2#2647712yes—
614C2#2647912yes—
615C2#2777912yes—
616C2#2778112yes—
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC23×D4C23
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#2552C2semidirect
2. C2×#2026C2semidirect
7. #11067C2semidirect
9. #10849C2semidirect
10. #10835C2semidirect
11. #10703C2semidirect

Maximal subgroups

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

subgroupindex
C2×#25522
C2×#20262
C2×#20252
C2×#19832
C2×#19592
#141532
#110672
#110662
#108492
#108352
#107032