Finite Group Databasebeta

← #629356

Subgroups — #629356

Order 512.

3138 conjugacy classes of subgroups.

The subgroup lattice has 659 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2626C211yesmaximal, characteristic
2C2×#2315C211yesmaximal, characteristic
3#15470C214yesmaximal, complemented
4#15469C214yesmaximal, complemented
5#15385C214yesmaximal, complemented
6#15371C214yesmaximal, complemented
7#14826C214yesmaximal, complemented
8#14702C211yesmaximal, characteristic, complemented
9#14682C211yesmaximal, characteristic, complemented
10#14347C211yesmaximal, characteristic, complemented
11#14281C211yesmaximal, characteristic, complemented
12#13869C214yesmaximal
13C2×#1437C211yesmaximal, characteristic
14C22×C22⋊Q8C2211yescharacteristic
15C23×C22⋊C4C2212yes—
16C23×C22⋊C4C2211yescharacteristic
17C23×C22⋊C4C2212yes—
18C23×C42C2211yescharacteristic
19#2626—28no—
20C2×#400C2214yescomplemented
21C2×#399C2212yes—
22C2×#396C2214yes—
23C2×#393C2212yes—
24#2315—28no—
25C2×#391C2212yescomplemented
26C2×#385C2214yes—
27C2×#385C2214yes—
28C2×#385C2212yes—
29C2×#382C2212yescomplemented
30C2×#381C2214yes—
31C2×#381C2214yes—
32C2×#380C2211yescharacteristic
33C2×#380C2211yescharacteristic
34C2×#380C2212yes—
35C2×#380C2211yescharacteristic
36C2×#379C2212yes—
37C22×#95C2211yescharacteristic
38C22×#95C2212yes—
39#1446C2212yescomplemented
40#1445C2214yescomplemented
41#1445C2212yescomplemented
42#1443C2214yescomplemented
43#1443C2214yescomplemented
44#1442C2214yescomplemented
45#1442C2214yescomplemented
46#1442C2214yescomplemented
47#1442C2214yescomplemented
48#1442C2214yescomplemented
49#1441C2214yescomplemented
50#1441C2214yescomplemented
51#1440C2214yes—
52#1440C2214yes—
53#1440C2214yes—
54#1440C2214yes—
55#1437C2214yes—
56#1437C2212yes—
57#1437C2212yes—
58#1436C2212yescomplemented
59#1436C2212yescomplemented
60C4⋊#95C2212yescomplemented
61C4⋊#95C2212yescomplemented
62#1434C2214yescomplemented
63C22⋊#95C2214yes—
64C22⋊#95C2214yes—
65C22⋊#95C2214yes—
66C22⋊#95C2214yes—
67C22⋊#95C2214yes—
68C22⋊#95C2214yes—
69C26C2311yescharacteristic
70C23×Q8C2311yescharacteristic
71C24×C4C2312yes—
72C24×C4C2312yes—
73C24×C4C2314yes—
74C24×C4C2311yescharacteristic
75C24×C4C2311yescharacteristic
76C2×C22⋊Q8—24no—
77C2×C22⋊Q8—24no—
78C2×C22⋊Q8—24no—
79C22×C4⋊C4C2312yes—
80C22×C4⋊C4C2311yescharacteristic
81C22×C4⋊C4C2312yes—
82C22×C4⋊C4C2314yes—
83C22×C4⋊C4C2314yes—
84C22×C4⋊C4C2312yes—
85C22×C4⋊C4C2312yes—
86C22×C22⋊C4—24no—
87C22×C22⋊C4—24no—
88C22×C22⋊C4—28no—
89C22×C22⋊C4—28no—
90C22×C22⋊C4—216no—
91C22×C22⋊C4—44no—
92C22×C22⋊C4—44no—
93C22×C22⋊C4D418yes—
94C22×C22⋊C4C2312yes—
95C22×C22⋊C4C2312yes—
96C22×C22⋊C4C2314yes—
97C22×C22⋊C4C2312yes—
98C22×C22⋊C4C2312yes—
99C22×C22⋊C4C2314yes—
100C22×C22⋊C4C2314yes—
101C22×C22⋊C4C2312yes—
102C22×C22⋊C4C2312yes—
103C22×C22⋊C4C2314yes—
104C22×C22⋊C4C2314yes—
105C22×C42—24no—
106C22×C42—44no—
107C22×C42C2311yescharacteristic
108C22×C42C2312yes—
109C22×C42C2311yescharacteristic
110#400—48no—
111#399—28no—
112#396—216no—
113#393—28no—
114#391—44no—
115#385—28no—
116#385—216no—
117#385—216no—
118#382—44no—
119#381—48no—
120#381—48no—
121#380—28no—
122#380—24no—
123#380—24no—
124#380—24no—
125#379—28no—
126C2×#95—24no—
127C2×#95—44no—
128C2×#95C2314yes—
129C2×#95C2312yes—
130C2×#95C2314yes—
131C2×#95C2314yes—
132C2×#95C2314yes—
133C2×#95C2314yes—
134C2×#95C2314yes—
135C2×#95C2312yes—
136C2×#95C2312yes—
137C2×#95C2312yes—
138C2×#95C2312yes—
139C2×#95C2314yes—
140C2×#95C2314yes—
141C2×#95C2314yes—
142C2×#95C2314yes—
143C2×#95C2314yes—
144C2×#95C2314yes—
145C2×#95C2314yes—
146C2×#95C2314yes—
147C2×#95C2314yes—
148C2×#95C2314yes—
149C2×#95C2314yes—
150C2×#95C2314yes—
151C2×#95C2314yes—
152C2×#95C2314yes—
153C25—22no—
154C25—22no—
155C25—24no—
156C25—42no—
157C25—42no—
158C25—42no—
159C25—42no—
160C25C2×D414yes—
161C25#4118yes—
162C25C2412yes—
163C25C2411yescharacteristic
164C22×Q8—22no—
165C22×Q8—22no—
166C22×Q8—22no—
167C22×Q8—42no—
168C22×Q8—42no—
169C23×C4—24no—
170C23×C4—22no—
171C23×C4—22no—
172C23×C4—24no—
173C23×C4—24no—
174C23×C4—24no—
175C23×C4—28no—
176C23×C4—24no—
177C23×C4—24no—
178C23×C4—28no—
179C23×C4—28no—
180C23×C4—24no—
181C23×C4—24no—
182C23×C4—42no—
183C23×C4—42no—
184C23×C4—42no—
185C23×C4—42no—
186C23×C4—42no—
187C23×C4—42no—
188C23×C4—42no—
189C23×C4—42no—
190C23×C4—44no—
191C23×C4—44no—
192C23×C4—42no—
193C23×C4—42no—
194C23×C4—44no—
195C23×C4—44no—
196C23×C4C2×D414yes—
197C23×C4C2×D414yes—
198C23×C4#4114yes—
199C23×C4#4114yes—
200C23×C4C2412yes—
201C23×C4C2414yes—
202C23×C4C2412yes—
203C23×C4C2414yes—
204C23×C4C2414yes—
205C23×C4C2414yes—
206C23×C4C2412yes—
207C23×C4C2412yes—
208C23×C4C2412yes—
209C23×C4C2411yescharacteristic
210C23×C4C2411yescharacteristic
211C22⋊Q8—416no—
212C2×C4⋊C4—24no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—24no—
217C2×C4⋊C4—24no—
218C2×C4⋊C4—28no—
219C2×C4⋊C4—24no—
220C2×C4⋊C4—24no—
221C2×C4⋊C4—24no—
222C2×C4⋊C4—28no—
223C2×C4⋊C4—42no—
224C2×C4⋊C4—42no—
225C2×C4⋊C4—42no—
226C2×C4⋊C4—44no—
227C2×C4⋊C4—44no—
228C2×C4⋊C4—42no—
229C2×C4⋊C4—42no—
230C2×C4⋊C4—42no—
231C2×C4⋊C4—44no—
232C2×C4⋊C4—44no—
233C2×C22⋊C4—232no—
234C2×C22⋊C4—24no—
235C2×C22⋊C4—24no—
236C2×C22⋊C4—24no—
237C2×C22⋊C4—24no—
238C2×C22⋊C4—24no—
239C2×C22⋊C4—24no—
240C2×C22⋊C4—28no—
241C2×C22⋊C4—24no—
242C2×C22⋊C4—24no—
243C2×C22⋊C4—28no—
244C2×C22⋊C4—28no—
245C2×C22⋊C4—28no—
246C2×C22⋊C4—28no—
247C2×C22⋊C4—28no—
248C2×C22⋊C4—28no—
249C2×C22⋊C4—24no—
250C2×C22⋊C4—24no—
251C2×C22⋊C4—28no—
252C2×C22⋊C4—28no—
253C2×C22⋊C4—28no—
254C2×C22⋊C4—28no—
255C2×C22⋊C4—28no—
256C2×C22⋊C4—44no—
257C2×C22⋊C4—44no—
258C2×C22⋊C4—432no—
259C2×C22⋊C4—432no—
260C2×C22⋊C4—44no—
261C2×C22⋊C4—44no—
262C2×C22⋊C4—44no—
263C2×C22⋊C4—44no—
264C2×C22⋊C4C2×D418yes—
265C2×C22⋊C4C2×D418yes—
266C2×C42—22no—
267C2×C42—24no—
268C2×C42—24no—
269C2×C42—22no—
270C2×C42—24no—
271C2×C42—416no—
272C2×C42—42no—
273C2×C42—42no—
274#95—28no—
275#95—24no—
276#95—24no—
277#95—24no—
278#95—28no—
279#95—24no—
280#95—24no—
281#95—28no—
282#95—28no—
283#95—28no—
284#95—28no—
285#95—28no—
286#95—44no—
287#95—44no—
288#95—44no—
289#95—44no—
290#95—44no—
291#95—44no—
292#95—44no—
293#95—44no—
294#95—44no—
295#95—44no—
296#95—44no—
297#95—44no—
298#95—44no—
299C24—22no—
300C24—22no—
301C24—22no—
302C24—22no—
303C24—22no—
304C24—22no—
305C24—24no—
306C24—24no—
307C24—24no—
308C24—24no—
309C24—24no—
310C24—28no—
311C24—28no—
312C24—216no—
313C24—216no—
314C24—48no—
315C24—48no—
316C24—48no—
317C24—48no—
318C24—48no—
319C24—48no—
320C24—82no—
321C24—82no—
322C24—84no—
323C24—84no—
324C24—82no—
325C24—82no—
326C24—84no—
327C24—84no—
328C24—84no—
329C24—84no—
330C24#124116yes—
331C24C22×D414yes—
332C24C22×D412yes—
333C24C2×#4118yes—
334C24C2×#4112yes—
335C24C2×#4112yes—
336C24C2×#4114yes—
337C242−1+414yes—
338C24C2511yescharacteristic
339C2×Q8—48no—
340C2×Q8—44no—
341C2×Q8—82no—
342C2×Q8—82no—
343C2×Q8—84no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—216no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—24no—
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—28no—
366C22×C4—24no—
367C22×C4—22no—
368C22×C4—24no—
369C22×C4—24no—
370C22×C4—22no—
371C22×C4—24no—
372C22×C4—24no—
373C22×C4—24no—
374C22×C4—24no—
375C22×C4—24no—
376C22×C4—24no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—24no—
382C22×C4—24no—
383C22×C4—24no—
384C22×C4—24no—
385C22×C4—24no—
386C22×C4—24no—
387C22×C4—24no—
388C22×C4—24no—
389C22×C4—24no—
390C22×C4—24no—
391C22×C4—24no—
392C22×C4—24no—
393C22×C4—24no—
394C22×C4—24no—
395C22×C4—24no—
396C22×C4—24no—
397C22×C4—24no—
398C22×C4—24no—
399C22×C4—24no—
400C22×C4—24no—
401C22×C4—24no—
402C22×C4—24no—
403C22×C4—24no—
404C22×C4—44no—
405C22×C4—48no—
406C22×C4—44no—
407C22×C4—48no—
408C22×C4—48no—
409C22×C4—48no—
410C22×C4—48no—
411C22×C4—48no—
412C22×C4—48no—
413C22×C4—48no—
414C22×C4—48no—
415C22×C4—48no—
416C22×C4—48no—
417C22×C4—416no—
418C22×C4—416no—
419C22×C4—416no—
420C22×C4—48no—
421C22×C4—84no—
422C22×C4—82no—
423C22×C4—82no—
424C22×C4—88no—
425C22×C4—88no—
426C22×C4—88no—
427C22×C4—88no—
428C22×C4—84no—
429C22×C4—84no—
430C22×C4—84no—
431C22×C4—84no—
432C22×C4#123116yes—
433C22×C4C22×D414yes—
434C22×C4C22×D412yes—
435C22×C4C22×D412yes—
436C22×C4C2×#4112yes—
437C22×C4C2×#4112yes—
438C22×C4C2×#4112yes—
439C22×C4C2×#4112yes—
440C22×C4H(5,𝔽2)14yes—
441C22×C4H(5,𝔽2)14yes—
442C4⋊C4—44no—
443C4⋊C4—48no—
444C4⋊C4—84no—
445C4⋊C4—84no—
446C4⋊C4—84no—
447C4⋊C4—88no—
448C4⋊C4—88no—
449C22⋊C4—216no—
450C22⋊C4—216no—
451C22⋊C4—416no—
452C22⋊C4—416no—
453C22⋊C4—48no—
454C22⋊C4—416no—
455C22⋊C4—48no—
456C22⋊C4—48no—
457C22⋊C4—48no—
458C22⋊C4—88no—
459C22⋊C4—88no—
460C42—28no—
461C42—48no—
462C42—44no—
463C23—24no—
464C23—24no—
465C23—24no—
466C23—22no—
467C23—22no—
468C23—24no—
469C23—24no—
470C23—24no—
471C23—24no—
472C23—24no—
473C23—22no—
474C23—22no—
475C23—24no—
476C23—24no—
477C23—24no—
478C23—24no—
479C23—24no—
480C23—24no—
481C23—28no—
482C23—28no—
483C23—28no—
484C23—28no—
485C23—28no—
486C23—28no—
487C23—28no—
488C23—28no—
489C23—232no—
490C23—432no—
491C23—42no—
492C23—42no—
493C23—44no—
494C23—44no—
495C23—42no—
496C23—42no—
497C23—44no—
498C23—44no—
499C23—44no—
500C23—44no—
501C23—88no—
502C23—816no—
503C23—816no—
504C23—816no—
505C23—816no—
506C23—816no—
507C23—88no—
508C23C2×#12314yes—
509C23C2×#12418yes—
510C23C2×#12414yes—
511C23#53614yes—
512C23#55218yes—
513C23#55218yes—
514C23C23×D411yescharacteristic
515C23C22×#4111yescharacteristic
516C23C22×#4111yescharacteristic
517C23C22×#4112yes—
518C23C2×H(5,𝔽2)12yes—
519C23C2×H(5,𝔽2)12yes—
520C23C2×2−1+412yes—
521C23#58512yes—
522C23#58512yes—
523Q8—88no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—24no—
529C2×C4—24no—
530C2×C4—24no—
531C2×C4—24no—
532C2×C4—24no—
533C2×C4—24no—
534C2×C4—24no—
535C2×C4—24no—
536C2×C4—24no—
537C2×C4—28no—
538C2×C4—24no—
539C2×C4—24no—
540C2×C4—416no—
541C2×C4—432no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—44no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—44no—
556C2×C4—44no—
557C2×C4—44no—
558C2×C4—44no—
559C2×C4—44no—
560C2×C4—44no—
561C2×C4—44no—
562C2×C4—44no—
563C2×C4—44no—
564C2×C4—44no—
565C2×C4—44no—
566C2×C4—44no—
567C2×C4—42no—
568C2×C4—44no—
569C2×C4—44no—
570C2×C4—42no—
571C2×C4—44no—
572C2×C4—44no—
573C2×C4—44no—
574C2×C4—44no—
575C2×C4—44no—
576C2×C4—44no—
577C2×C4—44no—
578C2×C4—44no—
579C2×C4—44no—
580C2×C4—44no—
581C2×C4—44no—
582C2×C4—42no—
583C2×C4—44no—
584C2×C4—42no—
585C2×C4—44no—
586C2×C4—44no—
587C2×C4—44no—
588C2×C4—44no—
589C2×C4—44no—
590C2×C4—44no—
591C2×C4—88no—
592C2×C4—816no—
593C2×C4—832no—
594C2×C4C2×#12314yes—
595C2×C4C2×#12314yes—
596C22—24no—
597C22—28no—
598C22—28no—
599C22—28no—
600C22—28no—
601C22—28no—
602C22—28no—
603C22—24no—
604C22—216no—
605C22—28no—
606C22—44no—
607C22—48no—
608C22—48no—
609C22—48no—
610C22—48no—
611C22—48no—
612C22—44no—
613C22—832no—
614C22#250318yes—
615C22C22×#12311yescharacteristic
616C22C22×#12411yescharacteristic
617C22C2×#53611yescharacteristic
618C22#345912yes—
619C22#345912yes—
620C22C2×#55212yes—
621C22C2×#55212yes—
622C22#349011yescharacteristic
623C22#349012yes—
624C22#349111yescharacteristic
625C22#349112yes—
626C22#349412yes—
627C22#349612yes—
628C22#349612yes—
629C22#350512yes—
630C22#350811yescharacteristic
631C22#351011yescharacteristic
632C22#353512yes—
633C22#353612yes—
634C22#356112yes—
635C22#356412yes—
636C4—24no—
637C4—24no—
638C4—44no—
639C4—44no—
640C4—44no—
641C4—82no—
642C4—84no—
643C4—84no—
644C4—84no—
645C4—84no—
646C4—82no—
647C2—28no—
648C2—48no—
649C2C2×#250311yescharacteristic
650C2#3499312yes—
651C2#3536011yescharacteristic
652C2#3554111yescharacteristic
653C2#3674012yes—
654C2#3793212yes—
655C2#3793412yes—
656C2#3954012yes—
657C2#3954212yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC26C23
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #15470C2semidirect
4. #15469C2semidirect
5. #15385C2semidirect
6. #15371C2semidirect
7. #14826C2semidirect
8. #14702C2semidirect
9. #14682C2semidirect
10. #14347C2semidirect
11. #14281C2semidirect
20. C2×#400C22semidirect
25. C2×#391C22semidirect
29. C2×#382C22semidirect
39. #1446C22semidirect
40. #1445C22semidirect
41. #1445C22semidirect
42. #1443C22semidirect
43. #1443C22semidirect
44. #1442C22semidirect
45. #1442C22semidirect
46. #1442C22semidirect
47. #1442C22semidirect
48. #1442C22semidirect
49. #1441C22semidirect
50. #1441C22semidirect
58. #1436C22semidirect
59. #1436C22semidirect
60. C4⋊#95C22semidirect
61. C4⋊#95C22semidirect
62. #1434C22semidirect

Maximal subgroups

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

subgroupindex
C2×#26262
C2×#23152
#154702
#154692
#153852
#153712
#148262
#147022
#146822
#143472
#142812
#138692
C2×#14372