Finite Group Databasebeta

← #681127

Subgroups — #681127

Order 512.

2618 conjugacy classes of subgroups.

The subgroup lattice has 645 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2580C211yesmaximal, characteristic
2C2×#2349C211yesmaximal, characteristic
3#15469C214yesmaximal, complemented
4#15371C214yesmaximal, complemented
5#15187C212yesmaximal, complemented
6#15139C212yesmaximal, complemented
7#14841C214yesmaximal, complemented
8#14689C211yesmaximal, characteristic, complemented
9#13972C214yesmaximal, complemented
10#13955C211yesmaximal, characteristic, complemented
11#13925C211yesmaximal, characteristic, complemented
12#13916C212yesmaximal
13#13916C212yesmaximal
14C4×#391C211yesmaximal, characteristic, complemented
15C4×#390C211yesmaximal, characteristic
16C22×C4×Q8C2211yescharacteristic
17C22×C4×D4C2212yes—
18C22×C4×D4C2211yescharacteristic
19#2580—28no—
20C2×#400C2214yes—
21C2×#399C2212yes—
22C2×#393C2212yes—
23#2349—28no—
24C2×#391C2212yes—
25C2×#390C2212yes—
26C2×#390C2211yescharacteristic
27C2×#390C2211yescharacteristic
28C2×#389C2212yes—
29C2×#389C2212yes—
30C2×#385C2214yes—
31C2×#385C2214yes—
32C2×#382C2211yescharacteristic
33C2×#382C2211yescharacteristic
34C2×#381C2214yes—
35C2×#381C2214yes—
36C2×#380C2212yes—
37C2×C4×C4⋊C4C2211yescharacteristic
38C2×C4×C4⋊C4C2211yescharacteristic
39C2×C4×C22⋊C4C2214yes—
40C22×#95C2212yes—
41C2×C43C2211yescharacteristic
42#1446C2212yes—
43#1445C2214yes—
44#1445C2212yes—
45#1443C2214yes—
46#1443C2214yes—
47#1442C2214yes—
48#1442C2214yes—
49#1442C2214yes—
50#1442C2214yes—
51#1441C2214yes—
52#1440C2214yes—
53#1440C2214yes—
54#1440C2214yes—
55#1440C2214yes—
56#1437C2214yes—
57#1436C2214yes—
58#1436C2212yes—
59#1436C2212yes—
60C4⋊#95C2211yescharacteristic
61C4⋊#95C2212yes—
62C4⋊#95C2212yes—
63C4⋊#95C2212yes—
64C4⋊#95C2211yescharacteristic
65C4⋊#95C2212yes—
66C4⋊#95C2212yes—
67C4⋊#95C2212yes—
68C4⋊#95C2212yes—
69#1434C2214yes—
70#1434C2214yes—
71#1434C2212yes—
72#1434C2212yes—
73C22⋊#95C2214yes—
74C22⋊#95C2214yes—
75C4×#95C2212yes—
76C4×#95C2212yes—
77C23×Q8C2311yescharacteristic
78C23×D4C2311yescharacteristic
79C24×C4C2314yes—
80C24×C4C2312yes—
81C2×C4×Q8—24no—
82C2×C4×Q8—24no—
83C2×C4×Q8—24no—
84C2×C4×D4—28no—
85C2×C4×D4—28no—
86C2×C4×D4—28no—
87C2×C4×D4—24no—
88C2×C4×D4—24no—
89C2×C4×D4D418yes—
90C22×C4⋊C4C2311yescharacteristic
91C22×C4⋊C4C2312yes—
92C22×C4⋊C4C2312yes—
93C22×C4⋊C4C2311yescharacteristic
94C22×C4⋊C4C2311yescharacteristic
95C22×C4⋊C4C2312yes—
96C22×C4⋊C4C2312yes—
97C22×C4⋊C4C2311yescharacteristic
98C22×C4⋊C4C2314yes—
99C22×C4⋊C4C2314yes—
100C22×C22⋊C4C2314yes—
101C22×C22⋊C4C2314yes—
102C22×C22⋊C4C2314yes—
103C22×C22⋊C4C2314yes—
104C22×C22⋊C4C2312yes—
105C22×C22⋊C4C2314yes—
106C22×C42C2312yes—
107C22×C42C2312yes—
108C22×C42C2311yescharacteristic
109C22×C42C2311yescharacteristic
110C22×C42C2312yes—
111C22×C42C2312yes—
112C22×C42C2312yes—
113C22×C42C2311yescharacteristic
114#400—48no—
115#399—28no—
116#393—28no—
117#391—28no—
118#390—28no—
119#390—24no—
120#390—24no—
121#389—28no—
122#389—28no—
123#385—216no—
124#385—216no—
125#382—24no—
126#382—24no—
127#381—216no—
128#381—48no—
129#380—28no—
130C4×C4⋊C4—24no—
131C4×C4⋊C4—24no—
132C4×C22⋊C4—48no—
133C2×#95—28no—
134C2×#95C2312yes—
135C2×#95C2314yes—
136C2×#95C2312yes—
137C2×#95C2314yes—
138C2×#95C2314yes—
139C2×#95C2312yes—
140C2×#95C2312yes—
141C2×#95C2312yes—
142C2×#95C2312yes—
143C2×#95C2314yes—
144C2×#95C2312yes—
145C2×#95C2314yes—
146C2×#95C2311yescharacteristic
147C2×#95C2312yes—
148C2×#95C2311yescharacteristic
149C2×#95C2312yes—
150C2×#95C2314yes—
151C2×#95C2312yes—
152C2×#95C2314yes—
153C2×#95C2314yes—
154C2×#95C2314yes—
155C2×#95C2312yes—
156C2×#95C2312yes—
157C2×#95C2314yes—
158C2×#95C2314yes—
159C2×#95C2314yes—
160C2×#95C2314yes—
161C2×#95C2314yes—
162C2×#95C2314yes—
163C2×#95C2314yes—
164C2×#95C2312yes—
165C43—24no—
166C25C2412yes—
167C22×Q8—22no—
168C22×Q8—22no—
169C22×Q8—24no—
170C22×Q8—22no—
171C22×Q8—42no—
172C22×D4—22no—
173C22×D4—22no—
174C22×D4—22no—
175C22×D4—22no—
176C22×D4C2×D414yes—
177C22×D4#4114yes—
178C22×D4#4114yes—
179C23×C4—24no—
180C23×C4—24no—
181C23×C4—28no—
182C23×C4—28no—
183C23×C4—28no—
184C23×C4—28no—
185C23×C4—44no—
186C23×C4—44no—
187C23×C4—44no—
188C23×C4—44no—
189C23×C4—44no—
190C23×C4C2×D418yes—
191C23×C4C2412yes—
192C23×C4C2412yes—
193C23×C4C2414yes—
194C23×C4C2412yes—
195C23×C4C2412yes—
196C23×C4C2414yes—
197C23×C4C2412yes—
198C23×C4C2412yes—
199C23×C4C2412yes—
200C23×C4C2412yes—
201C23×C4C2412yes—
202C23×C4C2411yescharacteristic
203C23×C4C2411yescharacteristic
204C23×C4C2411yescharacteristic
205C4×Q8—416no—
206C4×D4—232no—
207C4×D4—432no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—24no—
211C2×C4⋊C4—24no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—24no—
214C2×C4⋊C4—24no—
215C2×C4⋊C4—24no—
216C2×C4⋊C4—24no—
217C2×C4⋊C4—24no—
218C2×C4⋊C4—24no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—24no—
226C2×C4⋊C4—24no—
227C2×C4⋊C4—24no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—28no—
231C2×C4⋊C4—28no—
232C2×C4⋊C4—42no—
233C2×C4⋊C4—44no—
234C2×C4⋊C4—44no—
235C2×C4⋊C4—44no—
236C2×C4⋊C4—44no—
237C2×C4⋊C4C2×D414yes—
238C2×C22⋊C4—24no—
239C2×C22⋊C4—24no—
240C2×C22⋊C4—28no—
241C2×C22⋊C4—28no—
242C2×C22⋊C4—28no—
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—28no—
250C2×C22⋊C4—44no—
251C2×C22⋊C4—44no—
252C2×C22⋊C4—44no—
253C2×C22⋊C4—44no—
254C2×C22⋊C4—44no—
255C2×C22⋊C4C2×D418yes—
256C2×C42—24no—
257C2×C42—24no—
258C2×C42—24no—
259C2×C42—24no—
260C2×C42—24no—
261C2×C42—24no—
262C2×C42—24no—
263C2×C42—24no—
264C2×C42—22no—
265C2×C42—22no—
266C2×C42—22no—
267C2×C42—22no—
268C2×C42—24no—
269C2×C42—24no—
270C2×C42—22no—
271C2×C42—22no—
272C2×C42—24no—
273C2×C42—42no—
274C2×C42—42no—
275C2×C42—44no—
276C2×C42C2×D414yes—
277C2×C42#4114yes—
278C2×C42#4114yes—
279#95—22no—
280#95—24no—
281#95—28no—
282#95—22no—
283#95—24no—
284#95—24no—
285#95—24no—
286#95—28no—
287#95—24no—
288#95—28no—
289#95—28no—
290#95—28no—
291#95—24no—
292#95—24no—
293#95—24no—
294#95—28no—
295#95—24no—
296#95—28no—
297#95—24no—
298#95—42no—
299#95—44no—
300#95—42no—
301#95—44no—
302#95—44no—
303#95—44no—
304#95—44no—
305#95—44no—
306#95—44no—
307#95—44no—
308#95—44no—
309#95#4118yes—
310C24—24no—
311C24—22no—
312C24—22no—
313C24—24no—
314C24—22no—
315C24—22no—
316C24—24no—
317C24—24no—
318C24C22×D414yes—
319C24C2×#4114yes—
320C24C2×#4114yes—
321C24C2511yescharacteristic
322C2×Q8—28no—
323C2×Q8—48no—
324C2×Q8—48no—
325C2×Q8—44no—
326C2×Q8—44no—
327C2×D4—28no—
328C2×D4—28no—
329C2×D4—28no—
330C2×D4—28no—
331C2×D4—28no—
332C2×D4—28no—
333C2×D4#124116yes—
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—22no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—24no—
354C22×C4—22no—
355C22×C4—22no—
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—22no—
366C22×C4—216no—
367C22×C4—24no—
368C22×C4—24no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—24no—
372C22×C4—24no—
373C22×C4—22no—
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—24no—
384C22×C4—24no—
385C22×C4—24no—
386C22×C4—22no—
387C22×C4—24no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—24no—
391C22×C4—24no—
392C22×C4—24no—
393C22×C4—24no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—24no—
397C22×C4—24no—
398C22×C4—22no—
399C22×C4—24no—
400C22×C4—22no—
401C22×C4—24no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—24no—
405C22×C4—416no—
406C22×C4—416no—
407C22×C4—416no—
408C22×C4—416no—
409C22×C4—416no—
410C22×C4—88no—
411C22×C4—84no—
412C22×C4—84no—
413C22×C4—88no—
414C22×C4—88no—
415C22×C4—88no—
416C22×C4C22×D414yes—
417C22×C4C22×D412yes—
418C22×C4C22×D412yes—
419C22×C4C22×D412yes—
420C22×C4C2×#4114yes—
421C22×C4C2×#4114yes—
422C22×C4C2×#4114yes—
423C22×C4C2×#4112yes—
424C22×C4C2×#4112yes—
425C22×C4C2×#4112yes—
426C22×C4C2×#4112yes—
427C22×C4C2×#4112yes—
428C22×C4C2×#4112yes—
429C22×C4C2×#4112yes—
430C22×C4C2×#4112yes—
431C22×C4H(5,𝔽2)14yes—
432C22×C42−1+412yes—
433C22×C42−1+412yes—
434C4⋊C4—28no—
435C4⋊C4—48no—
436C4⋊C4—48no—
437C4⋊C4—44no—
438C4⋊C4—48no—
439C4⋊C4—48no—
440C4⋊C4—44no—
441C4⋊C4—44no—
442C4⋊C4—88no—
443C4⋊C4—88no—
444C22⋊C4—216no—
445C22⋊C4—416no—
446C22⋊C4—416no—
447C22⋊C4—416no—
448C22⋊C4—416no—
449C22⋊C4—88no—
450C42—48no—
451C42—48no—
452C42—48no—
453C42—48no—
454C42—44no—
455C42—44no—
456C42—84no—
457C42#123116yes—
458C23—24no—
459C23—24no—
460C23—28no—
461C23—24no—
462C23—24no—
463C23—24no—
464C23—24no—
465C23—28no—
466C23—28no—
467C23—28no—
468C23—28no—
469C23—28no—
470C23—42no—
471C23—42no—
472C23—44no—
473C23—44no—
474C23—44no—
475C23—42no—
476C23—42no—
477C23—44no—
478C23—44no—
479C23—44no—
480C23C2×#12418yes—
481C23C23×D411yescharacteristic
482C23C22×#4111yescharacteristic
483C23C22×#4111yescharacteristic
484C23C22×#4111yescharacteristic
485C23C22×#4111yescharacteristic
486C23C22×#4112yes—
487C23C2×H(5,𝔽2)12yes—
488C23C2×2−1+411yescharacteristic
489C23C2×2−1+411yescharacteristic
490C23#58512yes—
491C23#58512yes—
492Q8—416no—
493D4—232no—
494C2×C4—22no—
495C2×C4—22no—
496C2×C4—24no—
497C2×C4—24no—
498C2×C4—24no—
499C2×C4—24no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—24no—
503C2×C4—24no—
504C2×C4—24no—
505C2×C4—24no—
506C2×C4—24no—
507C2×C4—24no—
508C2×C4—24no—
509C2×C4—24no—
510C2×C4—24no—
511C2×C4—24no—
512C2×C4—28no—
513C2×C4—24no—
514C2×C4—24no—
515C2×C4—24no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—44no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—44no—
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—42no—
547C2×C4—44no—
548C2×C4—44no—
549C2×C4—42no—
550C2×C4—44no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—44no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—44no—
558C2×C4—44no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—44no—
564C2×C4—42no—
565C2×C4—44no—
566C2×C4—42no—
567C2×C4—44no—
568C2×C4—44no—
569C2×C4—44no—
570C2×C4—44no—
571C2×C4—44no—
572C2×C4—44no—
573C2×C4—816no—
574C2×C4—832no—
575C2×C4C2×#12314yes—
576C2×C4C2×#12314yes—
577C2×C4C2×#12314yes—
578C2×C4C2×#12414yes—
579C2×C4#53614yes—
580C2×C4#55214yes—
581C2×C4#55214yes—
582C2×C4#55214yes—
583C2×C4#55214yes—
584C22—216no—
585C22—28no—
586C22—48no—
587C22—44no—
588C22—44no—
589C22—44no—
590C22—44no—
591C22—48no—
592C22—48no—
593C22—44no—
594C22—44no—
595C22C22×#12311yescharacteristic
596C22C22×#12411yescharacteristic
597C22#344411yescharacteristic
598C22C2×#53611yescharacteristic
599C22#345511yescharacteristic
600C22D4×#4112yes—
601C22C2×#55211yescharacteristic
602C22C2×#55211yescharacteristic
603C22C2×#55211yescharacteristic
604C22C2×#55211yescharacteristic
605C22#347512yes—
606C22#347512yes—
607C22#348112yes—
608C22#349412yes—
609C22#349612yes—
610C22#350712yes—
611C22#350811yescharacteristic
612C22#351011yescharacteristic
613C22#351312yes—
614C22#352112yes—
615C22#352212yes—
616C22#355412yes—
617C22#355712yes—
618C4—24no—
619C4—24no—
620C4—44no—
621C4—44no—
622C4—44no—
623C4—44no—
624C4—44no—
625C4—84no—
626C4—82no—
627C4—82no—
628C4—82no—
629C4—84no—
630C4—82no—
631C4#250318yes—
632C2—48no—
633C2#3474211yescharacteristic
634C2C2×#250311yescharacteristic
635C2#3554111yescharacteristic
636C2#3623812yes—
637C2#3637011yescharacteristic
638C2#3639411yescharacteristic
639C2#3896911yescharacteristic
640C2#3897111yescharacteristic
641C2#3994912yes—
642C2#5926012yes—
643C2#5926412yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC23×D4C23
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #15469C2semidirect
4. #15371C2semidirect
5. #15187C2semidirect
6. #15139C2semidirect
7. #14841C2semidirect
8. #14689C2semidirect
9. #13972C2semidirect
10. #13955C2semidirect
11. #13925C2semidirect
14. C4×#391C2semidirect

Maximal subgroups

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

subgroupindex
C2×#25802
C2×#23492
#154692
#153712
#151872
#151392
#148412
#146892
#139722
#139552
#139252
#139162
C4×#3912
C4×#3902