Finite Group Databasebeta

← #449641

Subgroups — #449641

Order 512.

1590 conjugacy classes of subgroups.

The subgroup lattice has 621 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#484C211yesmaximal, characteristic
2C2×#2421C211yesmaximal, characteristic
3C2×#1865C212yesmaximal, complemented
4C2×#1853C211yesmaximal, characteristic
5#14130C211yesmaximal, characteristic
6#11328C212yesmaximal, complemented
7#11325C212yesmaximal, complemented
8#11199C211yesmaximal, characteristic
9#11190C212yesmaximal, complemented
10#11147C211yesmaximal, characteristic
11#10970C211yesmaximal, characteristic
12C22×C22⋊Q8C2211yescharacteristic
13C22×C4×Q8C2211yescharacteristic
14C23×C4⋊C4C2211yescharacteristic
15C2×#484—28no—
16C2×#484—24no—
17C22×#107C2212yes—
18C22×#106C2212yes—
19C22×#103C2212yes—
20C22×#103C2212yes—
21C22×C22⋊C8C2211yescharacteristic
22C22×C22⋊C8C2211yescharacteristic
23#2421—24no—
24#2421—24no—
25C2×#400—21no—
26C2×#397—21no—
27C2×#391C2211yescharacteristic
28C2×#389C2211yescharacteristic
29C2×#389C2212yes—
30C2×#388—21no—
31C2×#387—21no—
32C2×#385—21no—
33C2×#382C2211yescharacteristic
34C2×#380C2211yescharacteristic
35C2×#380C2211yescharacteristic
36C2×C4×C22⋊C4C2211yescharacteristic
37#1865—28no—
38#1853—24no—
39C2×#340C2212yes—
40C2×#340C2212yes—
41C2×#340C2212yes—
42C2×#340C2212yes—
43C2×#336C2212yes—
44C2×#336C2212yes—
45#1444—21no—
46#1442—21no—
47C4⋊#95C2212yes—
48C22⋊#95—21no—
49C23×Q8C2311yescharacteristic
50C24×C4—21no—
51C24×C4C2311yescharacteristic
52C23×C8C2312yes—
53C23×C8C2312yes—
54C2×C22⋊Q8—22no—
55C2×C22⋊Q8—24no—
56C2×C22⋊Q8—42no—
57C2×C22⋊Q8D414yes—
58C2×C4×Q8—22no—
59C2×C4×Q8—22no—
60C2×C4×Q8—24no—
61C2×C4×Q8—42no—
62C22×C4⋊C4—22no—
63C22×C4⋊C4—24no—
64C22×C4⋊C4—21no—
65C22×C4⋊C4—21no—
66C22×C4⋊C4—21no—
67C22×C4⋊C4—22no—
68C22×C4⋊C4—21no—
69C22×C4⋊C4—22no—
70C22×C4⋊C4—42no—
71C22×C4⋊C4C2312yes—
72C22×C4⋊C4C2311yescharacteristic
73C22×C4⋊C4C2311yescharacteristic
74C22×C4⋊C4C2311yescharacteristic
75C22×C4⋊C4C2312yes—
76C22×C22⋊C4—21no—
77C22×C22⋊C4—21no—
78C22×C22⋊C4—21no—
79C22×C22⋊C4D411yescharacteristic
80C22×C22⋊C4D411yescharacteristic
81C22×C42—21no—
82C22×C42C2311yescharacteristic
83C22×C42C2311yescharacteristic
84#484—232no—
85C2×#107—24no—
86C2×#107—24no—
87C2×#107—24no—
88C2×#106—24no—
89C2×#106—24no—
90C2×#106—42no—
91C2×#103—24no—
92C2×#103—24no—
93C2×#103—24no—
94C2×#103—24no—
95C2×#103—24no—
96C2×#103D418yes—
97C2×C22⋊C8—22no—
98C2×C22⋊C8—22no—
99C2×C22⋊C8—24no—
100C2×C22⋊C8—24no—
101#400—28no—
102#397—28no—
103#391—22no—
104#391—22no—
105#389—22no—
106#389—22no—
107#389—24no—
108#389—42no—
109#388—28no—
110#387—28no—
111#385—44no—
112#382—22no—
113#382—22no—
114#380—22no—
115#380—24no—
116#380D414yes—
117C4×C22⋊C4—22no—
118C4×C22⋊C4—22no—
119C2×#95—21no—
120C2×#95—21no—
121C2×#95—21no—
122C2×#95—21no—
123C2×#95—21no—
124C2×#95—21no—
125C2×#95—21no—
126C2×#95—22no—
127C2×#95—22no—
128C2×#95—22no—
129C2×#95D411yescharacteristic
130C2×#95D411yescharacteristic
131#340—24no—
132#340—24no—
133#340—24no—
134#340—42no—
135#336—24no—
136#336—24no—
137C25C2×D411yescharacteristic
138C22×Q8—21no—
139C22×Q8—22no—
140C22×Q8—22no—
141C22×Q8—22no—
142C22×Q8—22no—
143C22×Q8—22no—
144C22×Q8C2×D414yes—
145C22×Q8C2×D412yes—
146C23×C4—21no—
147C23×C4—22no—
148C23×C4—22no—
149C23×C4—22no—
150C23×C4—22no—
151C23×C4—22no—
152C23×C4—22no—
153C23×C4—21no—
154C23×C4—21no—
155C23×C4—22no—
156C23×C4—21no—
157C23×C4—22no—
158C23×C4—44no—
159C23×C4—42no—
160C23×C4—42no—
161C23×C4—42no—
162C23×C4—82no—
163C23×C4C2×D411yescharacteristic
164C23×C4C2×D411yescharacteristic
165C23×C4C2×D411yescharacteristic
166C23×C4C2×D411yescharacteristic
167C23×C4C2×D412yes—
168C23×C4C2×D412yes—
169C23×C4C2×D411yescharacteristic
170C23×C4C2411yescharacteristic
171C22×C8—22no—
172C22×C8—22no—
173C22×C8—22no—
174C22×C8—22no—
175C22×C8—22no—
176C22×C8—22no—
177C22×C8—22no—
178C22×C8—22no—
179C22×C8—22no—
180C22×C8—22no—
181C22×C8—22no—
182C22×C8—22no—
183C22×C8C2×D414yes—
184C22×C8#4114yes—
185C22⋊Q8—28no—
186C22⋊Q8—44no—
187C22⋊Q8—48no—
188C4×Q8—48no—
189C4×Q8—44no—
190C4×Q8—44no—
191C2×C4⋊C4—28no—
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—22no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—24no—
207C2×C4⋊C4—24no—
208C2×C4⋊C4—24no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—48no—
215C2×C4⋊C4—44no—
216C2×C4⋊C4—44no—
217C2×C4⋊C4—44no—
218C2×C4⋊C4—42no—
219C2×C4⋊C4—44no—
220C2×C4⋊C4—42no—
221C2×C4⋊C4—42no—
222C2×C4⋊C4—44no—
223C2×C4⋊C4—44no—
224C2×C4⋊C4—82no—
225C2×C4⋊C4C2×D414yes—
226C2×C4⋊C4C2×D412yes—
227C2×C4⋊C4C2×D412yes—
228C2×C4⋊C4C2×D412yes—
229C2×C4⋊C4#4114yes—
230C2×C22⋊C4—24no—
231C2×C22⋊C4—24no—
232C2×C22⋊C4—24no—
233C2×C22⋊C4—22no—
234C2×C22⋊C4—22no—
235C2×C22⋊C4—22no—
236C2×C22⋊C4—22no—
237C2×C22⋊C4—22no—
238C2×C22⋊C4—41no—
239C2×C22⋊C4—44no—
240C2×C22⋊C4—42no—
241C2×C22⋊C4—42no—
242C2×C22⋊C4—42no—
243C2×C22⋊C4—42no—
244C2×C42—24no—
245C2×C42—22no—
246C2×C42—21no—
247C2×C42—21no—
248C2×C42—21no—
249C2×C42—21no—
250C2×C42—22no—
251C2×C42—22no—
252C2×C42—22no—
253C2×C42—44no—
254#107—216no—
255#106—48no—
256#103—216no—
257#103—216no—
258C22⋊C8—28no—
259C22⋊C8—28no—
260#95—24no—
261#95—24no—
262#95—24no—
263#95—24no—
264#95—24no—
265#95—24no—
266#95—24no—
267#95—22no—
268#95—22no—
269#95—44no—
270#95—44no—
271#95—82no—
272C24—21no—
273C24—21no—
274C24—22no—
275C24—22no—
276C24—21no—
277C24—21no—
278C24—22no—
279C24—22no—
280C24#12012yes—
281C24#13512yes—
282C24#13712yes—
283C24C22×D411yescharacteristic
284C2×Q8—22no—
285C2×Q8—24no—
286C2×Q8—24no—
287C2×Q8—24no—
288C2×Q8—24no—
289C2×Q8—24no—
290C2×Q8—24no—
291C2×Q8—24no—
292C2×Q8—41no—
293C2×Q8—41no—
294C2×Q8—42no—
295C2×Q8—42no—
296C2×Q8—42no—
297C2×Q8—42no—
298C2×Q8—42no—
299C2×Q8#12014yes—
300C22×C4—22no—
301C22×C4—21no—
302C22×C4—21no—
303C22×C4—21no—
304C22×C4—22no—
305C22×C4—24no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—21no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—24no—
315C22×C4—24no—
316C22×C4—24no—
317C22×C4—24no—
318C22×C4—21no—
319C22×C4—21no—
320C22×C4—22no—
321C22×C4—21no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—24no—
325C22×C4—22no—
326C22×C4—24no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—21no—
330C22×C4—21no—
331C22×C4—21no—
332C22×C4—21no—
333C22×C4—21no—
334C22×C4—21no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—41no—
339C22×C4—41no—
340C22×C4—42no—
341C22×C4—42no—
342C22×C4—42no—
343C22×C4—42no—
344C22×C4—42no—
345C22×C4—42no—
346C22×C4—42no—
347C22×C4—42no—
348C22×C4—42no—
349C22×C4—48no—
350C22×C4—42no—
351C22×C4—41no—
352C22×C4—41no—
353C22×C4—42no—
354C22×C4—42no—
355C22×C4—42no—
356C22×C4—42no—
357C22×C4—42no—
358C22×C4—42no—
359C22×C4—41no—
360C22×C4—41no—
361C22×C4—42no—
362C22×C4—42no—
363C22×C4—41no—
364C22×C4—41no—
365C22×C4—41no—
366C22×C4—41no—
367C22×C4—41no—
368C22×C4—41no—
369C22×C4—42no—
370C22×C4—42no—
371C22×C4—42no—
372C22×C4—42no—
373C22×C4—42no—
374C22×C4—42no—
375C22×C4—42no—
376C22×C4—88no—
377C22×C4—88no—
378C22×C4—84no—
379C22×C4—84no—
380C22×C4#12012yes—
381C22×C4AGL(1,ℤ/8)12yes—
382C22×C4C22×D412yes—
383C22×C4C22×D411yescharacteristic
384C22×C4C22×D411yescharacteristic
385C22×C4C2×#4112yes—
386C22×C4C2×#4112yes—
387C22×C4H(5,𝔽2)12yes—
388C22×C42−1+412yes—
389C22×C42−1+412yes—
390C2×C8—24no—
391C2×C8—24no—
392C2×C8—24no—
393C2×C8—24no—
394C2×C8—24no—
395C2×C8—24no—
396C2×C8—42no—
397C2×C8—42no—
398C2×C8—42no—
399C2×C8—42no—
400C2×C8—42no—
401C2×C8—42no—
402C2×C8—42no—
403C2×C8—42no—
404C4⋊C4—24no—
405C4⋊C4—22no—
406C4⋊C4—24no—
407C4⋊C4—22no—
408C4⋊C4—24no—
409C4⋊C4—24no—
410C4⋊C4—24no—
411C4⋊C4—24no—
412C4⋊C4—48no—
413C4⋊C4—48no—
414C4⋊C4—48no—
415C4⋊C4—48no—
416C4⋊C4—41no—
417C4⋊C4—42no—
418C4⋊C4—41no—
419C4⋊C4—42no—
420C4⋊C4—42no—
421C4⋊C4—42no—
422C4⋊C4—42no—
423C4⋊C4—42no—
424C4⋊C4—88no—
425C22⋊C4—44no—
426C22⋊C4—48no—
427C22⋊C4—48no—
428C22⋊C4—44no—
429C22⋊C4—48no—
430C42—48no—
431C42—41no—
432C42—41no—
433C42—41no—
434C42—41no—
435C42—42no—
436C42—42no—
437C23—24no—
438C23—24no—
439C23—22no—
440C23—22no—
441C23—22no—
442C23—22no—
443C23—22no—
444C23—22no—
445C23—22no—
446C23—22no—
447C23—22no—
448C23—22no—
449C23—24no—
450C23—24no—
451C23—42no—
452C23—42no—
453C23—42no—
454C23—42no—
455C23—42no—
456C23—42no—
457C23—42no—
458C23—42no—
459C23#45214yes—
460C23C2×#12011yescharacteristic
461C23#52511yescharacteristic
462C23#53611yescharacteristic
463C23D4212yes—
464C23#54712yes—
465C23C2×#13511yescharacteristic
466C23C2×AGL(1,ℤ/8)11yescharacteristic
467C23C2×#13711yescharacteristic
468C23#57511yescharacteristic
469C23#57712yes—
470C23#57812yes—
471Q8—28no—
472Q8—44no—
473Q8—44no—
474Q8—44no—
475C2×C4—24no—
476C2×C4—22no—
477C2×C4—22no—
478C2×C4—22no—
479C2×C4—22no—
480C2×C4—24no—
481C2×C4—41no—
482C2×C4—42no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—44no—
486C2×C4—42no—
487C2×C4—44no—
488C2×C4—41no—
489C2×C4—42no—
490C2×C4—44no—
491C2×C4—44no—
492C2×C4—44no—
493C2×C4—44no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—44no—
499C2×C4—44no—
500C2×C4—42no—
501C2×C4—41no—
502C2×C4—42no—
503C2×C4—44no—
504C2×C4—41no—
505C2×C4—41no—
506C2×C4—42no—
507C2×C4—44no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—41no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—44no—
516C2×C4—41no—
517C2×C4—42no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—44no—
523C2×C4—42no—
524C2×C4—816no—
525C2×C4—82no—
526C2×C4—82no—
527C2×C4—81no—
528C2×C4—81no—
529C2×C4—81no—
530C2×C4—81no—
531C2×C4—82no—
532C2×C4—82no—
533C2×C4—82no—
534C2×C4—82no—
535C2×C4—82no—
536C2×C4—82no—
537C2×C4—82no—
538C2×C4—82no—
539C2×C4—82no—
540C2×C4—82no—
541C2×C4C2×#12011yescharacteristic
542C2×C4#52512yes—
543C2×C4#52511yescharacteristic
544C2×C4#53411yescharacteristic
545C2×C4#53612yes—
546C2×C4#53611yescharacteristic
547C2×C4#53912yes—
548C2×C4#53912yes—
549C2×C4D4212yes—
550C2×C4#54712yes—
551C2×C4#54712yes—
552C2×C4#54712yes—
553C2×C4#54712yes—
554C2×C4#54712yes—
555C2×C4#55912yes—
556C2×C4#56312yes—
557C8—44no—
558C8—44no—
559C22—24no—
560C22—24no—
561C22—24no—
562C22—42no—
563C22—42no—
564C22—42no—
565C22—42no—
566C22—44no—
567C22—44no—
568C22—44no—
569C22—44no—
570C22#240311yescharacteristic
571C22#240611yescharacteristic
572C22#240611yescharacteristic
573C22#245512yes—
574C22#268012yes—
575C22C2×#45211yescharacteristic
576C22#300611yescharacteristic
577C22#301011yescharacteristic
578C22#301111yescharacteristic
579C22#310511yescharacteristic
580C22#311711yescharacteristic
581C22#312012yes—
582C22#323211yescharacteristic
583C22#323911yescharacteristic
584C22#324212yes—
585C22#326512yes—
586C22#327012yes—
587C22#327512yes—
588C22#328512yes—
589C22#333712yes—
590C22#334412yes—
591C22#334912yes—
592C22#335312yes—
593C4—42no—
594C4—42no—
595C4—44no—
596C4—82no—
597C4—81no—
598C4—81no—
599C4—81no—
600C4—82no—
601C4—82no—
602C4—84no—
603C4—84no—
604C4#240612yes—
605C4#245512yes—
606C4#268012yes—
607C4#272412yes—
608C2—44no—
609C2#1594911yescharacteristic
610C2#2385911yescharacteristic
611C2#2386611yescharacteristic
612C2#2406711yescharacteristic
613C2#2407111yescharacteristic
614C2#2409511yescharacteristic
615C2#2410711yescharacteristic
616C2#2655512yes—
617C2#2657512yes—
618C2#3107912yes—
619C2#3110312yes—
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. C2×#1865C2semidirect
6. #11328C2semidirect
7. #11325C2semidirect
9. #11190C2semidirect

Maximal subgroups

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

subgroupindex
C22×#4842
C2×#24212
C2×#18652
C2×#18532
#141302
#113282
#113252
#111992
#111902
#111472
#109702