Finite Group Databasebeta

← #629422

Subgroups — #629422

Order 512.

2144 conjugacy classes of subgroups.

The subgroup lattice has 630 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2864C211yesmaximal, characteristic, complemented
2C2×#2628C211yesmaximal, characteristic
3#15752C214yesmaximal, complemented
4#15638C214yesmaximal, complemented
5#15584C214yesmaximal
6#15459C214yesmaximal
7#15183C212yesmaximal
8#15139C212yesmaximal, complemented
9#14714C211yesmaximal, characteristic, complemented
10#14340C211yesmaximal, characteristic
11#14282C211yesmaximal, characteristic, complemented
12#13994C212yesmaximal
13#13933C212yesmaximal, complemented
14#13779C211yesmaximal, characteristic, complemented
15#13761C211yesmaximal, characteristic
16C22×C4⋊Q8C2211yescharacteristic
17C22×#117C2211yescharacteristic
18#2864—28no—
19#2628—28no—
20C2×#400C2212yes—
21C2×#400C2214yes—
22C2×#399C2212yes—
23C2×#395C2212yes—
24C2×#393C2212yes—
25C2×#391C2214yes—
26C2×#389C2212yes—
27C2×#389C2212yes—
28C2×#389C2212yes—
29C2×#388C2214yes—
30C2×#387C2214yes—
31C2×#387C2214yes—
32C2×#384C2211yescharacteristic
33C2×#382C2211yescharacteristic
34C2×#382C2211yescharacteristic
35C2×#380C2212yes—
36C2×#380C2212yes—
37C2×C4×C4⋊C4C2212yes—
38C2×C4×C22⋊C4C2211yescharacteristic
39C2×C4×C22⋊C4C2211yescharacteristic
40C2×#376C2211yescharacteristic
41C2×C43C2211yescharacteristic
42#1446C2214yes—
43#1445C2214yes—
44#1445C2214yes—
45#1443C2214yes—
46#1443C2214yes—
47#1442C2214yes—
48#1442C2214yes—
49#1442C2214yes—
50#1442C2214yes—
51#1442C2214yes—
52#1441C2212yes—
53#1441C2212yes—
54#1440C2214yes—
55#1440C2214yes—
56#1439C2212yes—
57#1439C2212yes—
58#1439C2214yes—
59#1437C2212yes—
60#1437C2212yes—
61#1436C2212yes—
62#1436C2214yes—
63#1436C2212yes—
64#1436C2214yes—
65C4⋊#95C2212yes—
66C4⋊#95C2214yes—
67C4⋊#95C2212yes—
68C4⋊#95C2212yes—
69C4⋊#95C2212yes—
70C4⋊#95C2212yes—
71C4⋊#95C2212yes—
72C4⋊#95C2214yes—
73#1434C2214yes—
74#1434C2214yes—
75C4×#95C2211yescharacteristic
76C4×#95C2211yescharacteristic
77C23×Q8C2312yes—
78C24×C4C2311yescharacteristic
79C2×C4⋊Q8—24no—
80C2×C4⋊Q8—24no—
81C2×C4⋊Q8—24no—
82C2×#117—24no—
83C2×#117—24no—
84C2×#117—24no—
85C22×C4⋊C4C2311yescharacteristic
86C22×C4⋊C4C2312yes—
87C22×C4⋊C4C2311yescharacteristic
88C22×C4⋊C4C2312yes—
89C22×C4⋊C4C2312yes—
90C22×C4⋊C4C2312yes—
91C22×C4⋊C4C2314yes—
92C22×C4⋊C4C2314yes—
93C22×C4⋊C4C2312yes—
94C22×C4⋊C4C2314yes—
95C22×C4⋊C4C2312yes—
96C22×C4⋊C4C2312yes—
97C22×C22⋊C4C2311yescharacteristic
98C22×C22⋊C4C2311yescharacteristic
99C22×C22⋊C4C2312yes—
100C22×C22⋊C4C2312yes—
101C22×C22⋊C4C2314yes—
102C22×C22⋊C4C2314yes—
103C22×C42C2311yescharacteristic
104C22×C42C2311yescharacteristic
105C22×C42C2311yescharacteristic
106C22×C42C2311yescharacteristic
107C22×C42C2312yes—
108C22×C42C2312yes—
109C22×C42C2311yescharacteristic
110C22×C42C2314yes—
111C22×C42C2311yescharacteristic
112#400—28no—
113#400—48no—
114#399—28no—
115#395—28no—
116#393—28no—
117#391—48no—
118#389—28no—
119#389—28no—
120#389—28no—
121#388—48no—
122#387—216no—
123#387—216no—
124#384—24no—
125#382—24no—
126#382D418yes—
127#380—28no—
128#380—28no—
129C4×C4⋊C4—28no—
130C4×C22⋊C4—24no—
131C4×C22⋊C4—24no—
132#376—24no—
133C2×#95C2314yes—
134C2×#95C2314yes—
135C2×#95C2312yes—
136C2×#95C2314yes—
137C2×#95C2312yes—
138C2×#95C2312yes—
139C2×#95C2314yes—
140C2×#95C2314yes—
141C2×#95C2314yes—
142C2×#95C2314yes—
143C2×#95C2312yes—
144C2×#95C2312yes—
145C2×#95C2314yes—
146C2×#95C2312yes—
147C2×#95C2312yes—
148C2×#95C2312yes—
149C2×#95C2314yes—
150C2×#95C2314yes—
151C2×#95C2314yes—
152C2×#95C2312yes—
153C2×#95C2312yes—
154C2×#95C2314yes—
155C2×#95C2314yes—
156C2×#95C2314yes—
157C2×#95C2314yes—
158C2×#95C2314yes—
159C2×#95C2314yes—
160C2×#95C2314yes—
161C2×#95C2312yes—
162C2×#95C2312yes—
163C43—24no—
164C25C2411yescharacteristic
165C22×Q8—24no—
166C22×Q8—24no—
167C22×Q8—24no—
168C22×Q8—44no—
169C22×Q8—44no—
170C23×C4—22no—
171C23×C4—22no—
172C23×C4—22no—
173C23×C4—22no—
174C23×C4—22no—
175C23×C4—22no—
176C23×C4#4114yes—
177C23×C4C2411yescharacteristic
178C23×C4C2414yes—
179C23×C4C2414yes—
180C23×C4C2414yes—
181C23×C4C2411yescharacteristic
182C23×C4C2412yes—
183C23×C4C2414yes—
184C23×C4C2412yes—
185C23×C4C2411yescharacteristic
186C23×C4C2412yes—
187C23×C4C2412yes—
188C23×C4C2411yescharacteristic
189C23×C4C2411yescharacteristic
190C23×C4C2411yescharacteristic
191C4⋊Q8—416no—
192#117—416no—
193C2×C4⋊C4—24no—
194C2×C4⋊C4—24no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—24no—
198C2×C4⋊C4—24no—
199C2×C4⋊C4—24no—
200C2×C4⋊C4—28no—
201C2×C4⋊C4—28no—
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—24no—
210C2×C4⋊C4—24no—
211C2×C4⋊C4—24no—
212C2×C4⋊C4—24no—
213C2×C4⋊C4—24no—
214C2×C4⋊C4—28no—
215C2×C4⋊C4—28no—
216C2×C4⋊C4—24no—
217C2×C4⋊C4—28no—
218C2×C4⋊C4—24no—
219C2×C4⋊C4—24no—
220C2×C4⋊C4—44no—
221C2×C4⋊C4—44no—
222C2×C4⋊C4—44no—
223C2×C4⋊C4—44no—
224C2×C4⋊C4—44no—
225C2×C4⋊C4—44no—
226C2×C4⋊C4C2×D418yes—
227C2×C22⋊C4—24no—
228C2×C22⋊C4—28no—
229C2×C22⋊C4—22no—
230C2×C22⋊C4—22no—
231C2×C22⋊C4—22no—
232C2×C22⋊C4—22no—
233C2×C22⋊C4—22no—
234C2×C22⋊C4—22no—
235C2×C22⋊C4—24no—
236C2×C22⋊C4—24no—
237C2×C22⋊C4—24no—
238C2×C22⋊C4—28no—
239C2×C22⋊C4—42no—
240C2×C22⋊C4—42no—
241C2×C22⋊C4—44no—
242C2×C22⋊C4—44no—
243C2×C22⋊C4—44no—
244C2×C22⋊C4—44no—
245C2×C42—22no—
246C2×C42—22no—
247C2×C42—22no—
248C2×C42—24no—
249C2×C42—24no—
250C2×C42—22no—
251C2×C42—22no—
252C2×C42—22no—
253C2×C42—22no—
254C2×C42—22no—
255C2×C42—22no—
256C2×C42—22no—
257C2×C42—22no—
258C2×C42—24no—
259C2×C42—24no—
260C2×C42—22no—
261C2×C42—22no—
262C2×C42—22no—
263C2×C42—28no—
264C2×C42—42no—
265C2×C42—42no—
266C2×C42—44no—
267C2×C42—44no—
268C2×C42C2×D414yes—
269C2×C42#4114yes—
270C2×C42#4114yes—
271C2×C42#4114yes—
272#95—28no—
273#95—24no—
274#95—24no—
275#95—28no—
276#95—28no—
277#95—28no—
278#95—24no—
279#95—24no—
280#95—24no—
281#95—24no—
282#95—28no—
283#95—28no—
284#95—28no—
285#95—28no—
286#95—24no—
287#95—24no—
288#95—42no—
289#95—44no—
290#95—42no—
291#95—44no—
292#95—44no—
293#95—44no—
294#95—44no—
295#95—44no—
296#95—44no—
297#95—44no—
298#95—44no—
299#95—44no—
300#95C2×D418yes—
301#95C2×D418yes—
302C24—21no—
303C24—21no—
304C24—21no—
305C24—21no—
306C24—22no—
307C24—22no—
308C24—21no—
309C24—21no—
310C24—22no—
311C24—22no—
312C24C2×#4112yes—
313C24C2511yescharacteristic
314C2×Q8—416no—
315C2×Q8—48no—
316C2×Q8—88no—
317C2×Q8—84no—
318C2×Q8—84no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—21no—
324C22×C4—22no—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—21no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—24no—
331C22×C4—21no—
332C22×C4—21no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—24no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—24no—
342C22×C4—24no—
343C22×C4—21no—
344C22×C4—21no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—24no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—24no—
353C22×C4—28no—
354C22×C4—28no—
355C22×C4—28no—
356C22×C4—21no—
357C22×C4—24no—
358C22×C4—24no—
359C22×C4—21no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—24no—
363C22×C4—24no—
364C22×C4—24no—
365C22×C4—21no—
366C22×C4—21no—
367C22×C4—21no—
368C22×C4—21no—
369C22×C4—24no—
370C22×C4—24no—
371C22×C4—24no—
372C22×C4—22no—
373C22×C4—24no—
374C22×C4—24no—
375C22×C4—22no—
376C22×C4—24no—
377C22×C4—24no—
378C22×C4—24no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—24no—
383C22×C4—24no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—24no—
388C22×C4—24no—
389C22×C4—24no—
390C22×C4—24no—
391C22×C4—24no—
392C22×C4—22no—
393C22×C4—24no—
394C22×C4—44no—
395C22×C4—44no—
396C22×C4—44no—
397C22×C4—44no—
398C22×C4C22×D418yes—
399C22×C4C22×D412yes—
400C22×C4C22×D412yes—
401C22×C4C22×D412yes—
402C22×C4C2×#4112yes—
403C22×C4C2×#4114yes—
404C22×C4C2×#4112yes—
405C22×C4C2×#4112yes—
406C22×C4C2×#4112yes—
407C22×C4C2×#4112yes—
408C22×C4C2×#4112yes—
409C22×C4C2×#4112yes—
410C22×C4C2×#4112yes—
411C22×C4C2×#4112yes—
412C22×C4H(5,𝔽2)14yes—
413C22×C42−1+414yes—
414C22×C42−1+412yes—
415C22×C42−1+412yes—
416C4⋊C4—216no—
417C4⋊C4—416no—
418C4⋊C4—416no—
419C4⋊C4—44no—
420C4⋊C4—44no—
421C4⋊C4—48no—
422C4⋊C4—48no—
423C4⋊C4—48no—
424C4⋊C4—48no—
425C4⋊C4—84no—
426C4⋊C4—84no—
427C4⋊C4—88no—
428C22⋊C4—48no—
429C22⋊C4—48no—
430C22⋊C4—416no—
431C22⋊C4—44no—
432C22⋊C4—44no—
433C22⋊C4—88no—
434C42—28no—
435C42—48no—
436C42—48no—
437C42—44no—
438C42—44no—
439C42—44no—
440C42—44no—
441C42—88no—
442C42#124116yes—
443C23—22no—
444C23—24no—
445C23—22no—
446C23—22no—
447C23—24no—
448C23—41no—
449C23—41no—
450C23—42no—
451C23—41no—
452C23—41no—
453C23—42no—
454C23—42no—
455C23—42no—
456C23—42no—
457C23—42no—
458C23—41no—
459C23—41no—
460C23—41no—
461C23—41no—
462C23—42no—
463C23—42no—
464C23—42no—
465C23—42no—
466C23C23×D411yescharacteristic
467C23C22×#4111yescharacteristic
468C23C22×#4111yescharacteristic
469C23C22×#4111yescharacteristic
470C23C22×#4111yescharacteristic
471C23C2×H(5,𝔽2)12yes—
472C23C2×2−1+411yescharacteristic
473C23C2×2−1+411yescharacteristic
474C23C2×2−1+412yes—
475C23#58512yes—
476C23#58512yes—
477Q8—816no—
478C2×C4—24no—
479C2×C4—24no—
480C2×C4—24no—
481C2×C4—24no—
482C2×C4—22no—
483C2×C4—22no—
484C2×C4—22no—
485C2×C4—22no—
486C2×C4—22no—
487C2×C4—24no—
488C2×C4—28no—
489C2×C4—22no—
490C2×C4—24no—
491C2×C4—22no—
492C2×C4—24no—
493C2×C4—28no—
494C2×C4—24no—
495C2×C4—28no—
496C2×C4—24no—
497C2×C4—24no—
498C2×C4—24no—
499C2×C4—24no—
500C2×C4—24no—
501C2×C4—41no—
502C2×C4—41no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—44no—
506C2×C4—42no—
507C2×C4—41no—
508C2×C4—42no—
509C2×C4—41no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—44no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—44no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—44no—
521C2×C4—416no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—44no—
525C2×C4—44no—
526C2×C4—44no—
527C2×C4—44no—
528C2×C4—42no—
529C2×C4—44no—
530C2×C4—44no—
531C2×C4—42no—
532C2×C4—44no—
533C2×C4—44no—
534C2×C4—42no—
535C2×C4—44no—
536C2×C4—44no—
537C2×C4—42no—
538C2×C4—44no—
539C2×C4—44no—
540C2×C4—42no—
541C2×C4—44no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—44no—
545C2×C4—44no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—44no—
549C2×C4—41no—
550C2×C4—44no—
551C2×C4—41no—
552C2×C4—44no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—44no—
556C2×C4—42no—
557C2×C4—44no—
558C2×C4—42no—
559C2×C4—44no—
560C2×C4C2×#12414yes—
561C2×C4C2×#12414yes—
562C2×C4C2×#12414yes—
563C2×C4#52914yes—
564C2×C4#54114yes—
565C2×C4#54714yes—
566C2×C4#54814yes—
567C2×C4#55214yes—
568C2×C4#55214yes—
569C22—44no—
570C22—44no—
571C22—42no—
572C22—42no—
573C22—42no—
574C22—81no—
575C22—81no—
576C22—82no—
577C22—82no—
578C22—81no—
579C22—81no—
580C22C22×#12411yescharacteristic
581C22C2×#52911yescharacteristic
582C22C2×#54111yescharacteristic
583C22#346012yes—
584C22#346112yes—
585C22C2×#54711yescharacteristic
586C22C2×#54811yescharacteristic
587C22C2×#55211yescharacteristic
588C22C2×#55211yescharacteristic
589C22#348812yes—
590C22#348912yes—
591C22#349412yes—
592C22#349512yes—
593C22#350612yes—
594C22#350712yes—
595C22#350811yescharacteristic
596C22#350911yescharacteristic
597C22#351011yescharacteristic
598C22#351011yescharacteristic
599C22#354412yes—
600C22#355212yes—
601C22#356512yes—
602C22#356512yes—
603C4—24no—
604C4—24no—
605C4—44no—
606C4—44no—
607C4—48no—
608C4—44no—
609C4—42no—
610C4—81no—
611C4—82no—
612C4—84no—
613C4—84no—
614C4—81no—
615C4—84no—
616C4#248918yes—
617C2—82no—
618C2C2×#248911yescharacteristic
619C2#3526811yescharacteristic
620C2#3532411yescharacteristic
621C2#3555611yescharacteristic
622C2#3562811yescharacteristic
623C2#3732011yescharacteristic
624C2#3743411yescharacteristic
625C2#3775912yes—
626C2#3776612yes—
627C2#3954412yes—
628C2#3954512yes—
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. C2×#2864C2semidirect
3. #15752C2semidirect
4. #15638C2semidirect
8. #15139C2semidirect
9. #14714C2semidirect
11. #14282C2semidirect
13. #13933C2semidirect
14. #13779C2semidirect

Maximal subgroups

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

subgroupindex
C2×#28642
C2×#26282
#157522
#156382
#155842
#154592
#151832
#151392
#147142
#143402
#142822
#139942
#139332
#137792
#137612