Finite Group Databasebeta

← #4307615

Subgroups — #4307615

Order 512.

2308 conjugacy classes of subgroups.

The subgroup lattice has 661 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2362C211yesmaximal, characteristic
2C2×#2355C211yesmaximal, characteristic, complemented
3#15242C211yesmaximal, characteristic, complemented
4#15236C212yesmaximal
5#15195C211yesmaximal, characteristic
6#15192C212yesmaximal, complemented
7#15175C212yesmaximal
8#15156C211yesmaximal, characteristic, complemented
9#15086C212yesmaximal, complemented
10#15025C212yesmaximal
11#14990C214yesmaximal, complemented
12#14952C214yesmaximal
13C4×#400C212yesmaximal, complemented
14C4×#399C211yesmaximal, characteristic
15#13817C211yesmaximal, characteristic, complemented
16#13804C212yesmaximal, complemented
17#13801C212yesmaximal
18C22×C22⋊Q8C2211yescharacteristic
19C22×C4×Q8C2211yescharacteristic
20C22×#117C2211yescharacteristic
21C2×#400C2212yes—
22C2×#399C2211yescharacteristic
23#2362—28no—
24#2355—28no—
25C2×#391C2212yes—
26C2×#391C2211yescharacteristic
27C2×#391C2212yes—
28C2×#389C2211yescharacteristic
29C2×#388C2212yes—
30C2×#388C2212yes—
31C2×#387C2214yes—
32C2×#387C2212yes—
33C2×#384C2211yescharacteristic
34C2×#382C2212yes—
35C2×#382C2212yes—
36C2×#380C2212yes—
37C2×C4×C4⋊C4C2211yescharacteristic
38C2×C4×C4⋊C4C2212yes—
39C2×C4×C4⋊C4C2212yes—
40C2×C4×C4⋊C4C2212yes—
41C2×C4×C4⋊C4C2212yes—
42C2×C4×C4⋊C4C2211yescharacteristic
43C2×C4×C4⋊C4C2212yes—
44C2×C4×C22⋊C4C2211yescharacteristic
45C2×C4×C22⋊C4C2212yes—
46#1445C2211yescharacteristic
47#1445C2212yes—
48#1445C2214yes—
49#1445C2214yes—
50#1444C2211yescharacteristic
51#1443C2214yes—
52#1443C2214yes—
53#1443C2214yes—
54#1442C2212yes—
55#1442C2214yes—
56#1442C2214yes—
57#1442C2212yes—
58#1442C2214yes—
59#1441C2212yes—
60#1441C2212yes—
61#1441C2212yes—
62#1440C2214yes—
63#1440C2214yes—
64#1439C2212yes—
65#1439C2211yescharacteristic
66#1439C2212yes—
67#1439C2212yes—
68#1437C2214yes—
69#1437C2214yes—
70#1436C2214yes—
71#1436C2214yes—
72#1436C2214yes—
73C4⋊#95C2212yes—
74C4⋊#95C2211yescharacteristic
75C4⋊#95C2212yes—
76C4⋊#95C2212yes—
77C4⋊#95C2212yes—
78#1434C2212yes—
79#1434C2214yes—
80#1434C2214yes—
81C4×#95C2211yescharacteristic
82C4×#95C2214yes—
83C4×#95C2214yes—
84C4×#95C2212yes—
85C23×Q8C2311yescharacteristic
86C24×C4C2311yescharacteristic
87C2×C22⋊Q8—24no—
88C2×C22⋊Q8—24no—
89C2×C22⋊Q8—24no—
90C2×C4×Q8—28no—
91C2×C4×Q8D418yes—
92C2×#117—24no—
93C2×#117—28no—
94C22×C4⋊C4C2312yes—
95C22×C4⋊C4C2311yescharacteristic
96C22×C4⋊C4C2312yes—
97C22×C4⋊C4C2312yes—
98C22×C4⋊C4C2312yes—
99C22×C4⋊C4C2312yes—
100C22×C4⋊C4C2312yes—
101C22×C4⋊C4C2312yes—
102C22×C4⋊C4C2311yescharacteristic
103C22×C4⋊C4C2311yescharacteristic
104C22×C4⋊C4C2312yes—
105C22×C4⋊C4C2311yescharacteristic
106C22×C4⋊C4C2312yes—
107C22×C4⋊C4C2311yescharacteristic
108C22×C22⋊C4C2311yescharacteristic
109C22×C22⋊C4C2312yes—
110C22×C22⋊C4C2312yes—
111C22×C22⋊C4C2312yes—
112C22×C22⋊C4C2311yescharacteristic
113C22×C22⋊C4C2312yes—
114C22×C22⋊C4C2314yes—
115C22×C42C2312yes—
116C22×C42C2311yescharacteristic
117C22×C42C2312yes—
118C22×C42C2312yes—
119C22×C42C2312yes—
120C22×C42C2311yescharacteristic
121C22×C42C2312yes—
122C22×C42C2314yes—
123C22×C42C2314yes—
124#400—28no—
125#399—24no—
126#391—28no—
127#391—24no—
128#391—28no—
129#389—24no—
130#388—28no—
131#388—28no—
132#387—216no—
133#387—28no—
134#384—24no—
135#382—28no—
136#382—28no—
137#380—28no—
138C4×C4⋊C4—28no—
139C4×C4⋊C4—24no—
140C4×C4⋊C4—28no—
141C4×C4⋊C4—24no—
142C4×C4⋊C4—28no—
143C4×C4⋊C4—28no—
144C4×C4⋊C4—28no—
145C4×C22⋊C4—24no—
146C4×C22⋊C4—28no—
147C2×#95C2311yescharacteristic
148C2×#95C2312yes—
149C2×#95C2314yes—
150C2×#95C2314yes—
151C2×#95C2311yescharacteristic
152C2×#95C2312yes—
153C2×#95C2312yes—
154C2×#95C2314yes—
155C2×#95C2314yes—
156C2×#95C2314yes—
157C2×#95C2314yes—
158C2×#95C2312yes—
159C2×#95C2314yes—
160C2×#95C2312yes—
161C2×#95C2312yes—
162C2×#95C2314yes—
163C2×#95C2314yes—
164C2×#95C2312yes—
165C2×#95C2314yes—
166C2×#95C2314yes—
167C2×#95C2312yes—
168C2×#95C2311yescharacteristic
169C2×#95C2311yescharacteristic
170C2×#95C2312yes—
171C2×#95C2312yes—
172C2×#95C2314yes—
173C2×#95C2314yes—
174C2×#95C2312yes—
175C2×#95C2314yes—
176C2×#95C2314yes—
177C2×#95C2314yes—
178C2×#95C2314yes—
179C2×#95C2312yes—
180C25C2411yescharacteristic
181C22×Q8—24no—
182C22×Q8—22no—
183C22×Q8—22no—
184C22×Q8—24no—
185C22×Q8C2×D414yes—
186C23×C4—22no—
187C23×C4—24no—
188C23×C4—22no—
189C23×C4—24no—
190C23×C4#4114yes—
191C23×C4C2414yes—
192C23×C4C2411yescharacteristic
193C23×C4C2412yes—
194C23×C4C2412yes—
195C23×C4C2414yes—
196C23×C4C2411yescharacteristic
197C23×C4C2412yes—
198C23×C4C2412yes—
199C23×C4C2412yes—
200C23×C4C2412yes—
201C23×C4C2412yes—
202C23×C4C2411yescharacteristic
203C23×C4C2411yescharacteristic
204C23×C4C2412yes—
205C23×C4C2411yescharacteristic
206C23×C4C2411yescharacteristic
207C22⋊Q8—232no—
208C4×Q8—232no—
209#117—416no—
210C2×C4⋊C4—24no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—28no—
214C2×C4⋊C4—24no—
215C2×C4⋊C4—24no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—24no—
221C2×C4⋊C4—24no—
222C2×C4⋊C4—24no—
223C2×C4⋊C4—24no—
224C2×C4⋊C4—24no—
225C2×C4⋊C4—24no—
226C2×C4⋊C4—24no—
227C2×C4⋊C4—24no—
228C2×C4⋊C4—24no—
229C2×C4⋊C4—24no—
230C2×C4⋊C4—24no—
231C2×C4⋊C4—24no—
232C2×C4⋊C4—24no—
233C2×C4⋊C4—24no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—24no—
236C2×C4⋊C4—42no—
237C2×C4⋊C4—42no—
238C2×C4⋊C4—44no—
239C2×C4⋊C4—42no—
240C2×C4⋊C4—44no—
241C2×C4⋊C4C2×D418yes—
242C2×C4⋊C4C2×D414yes—
243C2×C4⋊C4#4114yes—
244C2×C4⋊C4#4114yes—
245C2×C4⋊C4#4118yes—
246C2×C22⋊C4—24no—
247C2×C22⋊C4—24no—
248C2×C22⋊C4—24no—
249C2×C22⋊C4—24no—
250C2×C22⋊C4—22no—
251C2×C22⋊C4—22no—
252C2×C22⋊C4—28no—
253C2×C22⋊C4—24no—
254C2×C22⋊C4—24no—
255C2×C22⋊C4—24no—
256C2×C22⋊C4—22no—
257C2×C22⋊C4—22no—
258C2×C22⋊C4—22no—
259C2×C22⋊C4—28no—
260C2×C22⋊C4—41no—
261C2×C22⋊C4—42no—
262C2×C22⋊C4—42no—
263C2×C22⋊C4—42no—
264C2×C22⋊C4—44no—
265C2×C22⋊C4—44no—
266C2×C42—24no—
267C2×C42—28no—
268C2×C42—24no—
269C2×C42—24no—
270C2×C42—24no—
271C2×C42—22no—
272C2×C42—24no—
273C2×C42—24no—
274C2×C42—28no—
275C2×C42—24no—
276C2×C42—24no—
277C2×C42—24no—
278C2×C42—24no—
279C2×C42—28no—
280C2×C42—28no—
281C2×C42—42no—
282C2×C42—44no—
283C2×C42—44no—
284C2×C42—44no—
285C2×C42—44no—
286C2×C42C2×D414yes—
287C2×C42C2×D418yes—
288C2×C42#4118yes—
289#95—22no—
290#95—22no—
291#95—28no—
292#95—24no—
293#95—24no—
294#95—28no—
295#95—24no—
296#95—28no—
297#95—24no—
298#95—24no—
299#95—24no—
300#95—24no—
301#95—28no—
302#95—24no—
303#95—24no—
304#95—28no—
305#95—28no—
306#95—28no—
307#95—28no—
308#95—28no—
309#95—22no—
310#95—28no—
311#95—24no—
312#95—28no—
313#95—22no—
314#95—44no—
315#95—44no—
316#95—42no—
317#95—44no—
318#95—44no—
319#95—42no—
320#95—44no—
321#95—44no—
322C24—21no—
323C24—21no—
324C24—21no—
325C24—22no—
326C24—22no—
327C24—21no—
328C24—21no—
329C24—21no—
330C24—22no—
331C24—22no—
332C24C2×#4112yes—
333C24C2511yescharacteristic
334C2×Q8—28no—
335C2×Q8—28no—
336C2×Q8—28no—
337C2×Q8—48no—
338C2×Q8—48no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—24no—
343C22×C4—22no—
344C22×C4—21no—
345C22×C4—22no—
346C22×C4—21no—
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—24no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—21no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—21no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—21no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—21no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—28no—
385C22×C4—216no—
386C22×C4—21no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—24no—
390C22×C4—24no—
391C22×C4—24no—
392C22×C4—24no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—24no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—24no—
399C22×C4—24no—
400C22×C4—24no—
401C22×C4—21no—
402C22×C4—21no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—24no—
406C22×C4—24no—
407C22×C4—24no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—24no—
412C22×C4—24no—
413C22×C4—24no—
414C22×C4—24no—
415C22×C4—24no—
416C22×C4—22no—
417C22×C4—44no—
418C22×C4—48no—
419C22×C4—44no—
420C22×C4C22×D414yes—
421C22×C4C22×D412yes—
422C22×C4C22×D414yes—
423C22×C4C22×D412yes—
424C22×C4C22×D412yes—
425C22×C4C2×#4114yes—
426C22×C4C2×#4114yes—
427C22×C4C2×#4114yes—
428C22×C4C2×#4112yes—
429C22×C4C2×#4114yes—
430C22×C4C2×#4118yes—
431C22×C4C2×#4112yes—
432C22×C4C2×#4114yes—
433C22×C4C2×#4112yes—
434C22×C4C2×#4112yes—
435C22×C4C2×#4114yes—
436C22×C42−1+412yes—
437C22×C42−1+412yes—
438C4⋊C4—216no—
439C4⋊C4—216no—
440C4⋊C4—28no—
441C4⋊C4—28no—
442C4⋊C4—48no—
443C4⋊C4—48no—
444C4⋊C4—44no—
445C4⋊C4—48no—
446C4⋊C4—44no—
447C4⋊C4—48no—
448C4⋊C4—48no—
449C4⋊C4—48no—
450C4⋊C4—84no—
451C4⋊C4#123116yes—
452C22⋊C4—216no—
453C22⋊C4—48no—
454C22⋊C4—44no—
455C22⋊C4—48no—
456C22⋊C4—48no—
457C22⋊C4—44no—
458C22⋊C4—88no—
459C42—28no—
460C42—216no—
461C42—216no—
462C42—48no—
463C42—48no—
464C42—44no—
465C42—48no—
466C42—88no—
467C42—88no—
468C23—22no—
469C23—24no—
470C23—22no—
471C23—22no—
472C23—24no—
473C23—41no—
474C23—42no—
475C23—41no—
476C23—41no—
477C23—42no—
478C23—42no—
479C23—41no—
480C23—41no—
481C23—41no—
482C23—42no—
483C23—42no—
484C23—41no—
485C23—42no—
486C23—41no—
487C23—42no—
488C23—42no—
489C23—42no—
490C23—42no—
491C23C23×D411yescharacteristic
492C23C22×#4112yes—
493C23C22×#4111yescharacteristic
494C23C22×#4111yescharacteristic
495C23C22×#4112yes—
496C23C22×#4111yescharacteristic
497C23C2×2−1+411yescharacteristic
498C23C2×2−1+411yescharacteristic
499C23#58512yes—
500C23#58512yes—
501C23#58511yescharacteristic
502Q8—416no—
503C2×C4—22no—
504C2×C4—24no—
505C2×C4—22no—
506C2×C4—24no—
507C2×C4—24no—
508C2×C4—28no—
509C2×C4—24no—
510C2×C4—28no—
511C2×C4—22no—
512C2×C4—24no—
513C2×C4—24no—
514C2×C4—24no—
515C2×C4—22no—
516C2×C4—24no—
517C2×C4—24no—
518C2×C4—24no—
519C2×C4—24no—
520C2×C4—24no—
521C2×C4—24no—
522C2×C4—24no—
523C2×C4—24no—
524C2×C4—24no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—24no—
528C2×C4—24no—
529C2×C4—24no—
530C2×C4—22no—
531C2×C4—24no—
532C2×C4—24no—
533C2×C4—41no—
534C2×C4—42no—
535C2×C4—44no—
536C2×C4—44no—
537C2×C4—42no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—41no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—416no—
560C2×C4—42no—
561C2×C4—44no—
562C2×C4—44no—
563C2×C4—41no—
564C2×C4—42no—
565C2×C4—41no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—44no—
569C2×C4—44no—
570C2×C4—44no—
571C2×C4—44no—
572C2×C4—44no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—44no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—44no—
582C2×C4—41no—
583C2×C4—42no—
584C2×C4—44no—
585C2×C4—44no—
586C2×C4—44no—
587C2×C4—44no—
588C2×C4—44no—
589C2×C4—44no—
590C2×C4—44no—
591C2×C4C2×#12318yes—
592C2×C4C2×#12314yes—
593C2×C4#52514yes—
594C2×C4#52918yes—
595C2×C4#52918yes—
596C2×C4#53214yes—
597C2×C4#54814yes—
598C22—42no—
599C22—44no—
600C22—42no—
601C22—44no—
602C22—42no—
603C22—82no—
604C22—81no—
605C22—82no—
606C22—81no—
607C22—81no—
608C22—81no—
609C22C22×#12311yescharacteristic
610C22C2×#52511yescharacteristic
611C22C2×#52912yes—
612C22C2×#52912yes—
613C22C2×#53211yescharacteristic
614C22C2×#54811yescharacteristic
615C22#347512yes—
616C22#348011yescharacteristic
617C22#348012yes—
618C22#348211yescharacteristic
619C22#348711yescharacteristic
620C22#348911yescharacteristic
621C22#349312yes—
622C22#349512yes—
623C22#350011yescharacteristic
624C22#350712yes—
625C22#350911yescharacteristic
626C22#351012yes—
627C22#351511yescharacteristic
628C22#351512yes—
629C22#351612yes—
630C22#352112yes—
631C22#352412yes—
632C4—24no—
633C4—28no—
634C4—44no—
635C4—44no—
636C4—44no—
637C4—44no—
638C4—42no—
639C4—42no—
640C4—42no—
641C4—44no—
642C4—84no—
643C4—81no—
644C4—84no—
645C4—82no—
646C4—82no—
647C4#239518yes—
648C2—82no—
649C2C2×#239511yescharacteristic
650C2#3580312yes—
651C2#3581812yes—
652C2#3604111yescharacteristic
653C2#3606511yescharacteristic
654C2#3606711yescharacteristic
655C2#3650511yescharacteristic
656C2#3650612yes—
657C2#3651512yes—
658C2#3653411yescharacteristic
659C2#3657811yescharacteristic
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
2. C2×#2355C2semidirect
3. #15242C2semidirect
6. #15192C2semidirect
8. #15156C2semidirect
9. #15086C2semidirect
11. #14990C2semidirect
13. C4×#400C2semidirect
15. #13817C2semidirect
16. #13804C2semidirect

Maximal subgroups

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

subgroupindex
C2×#23622
C2×#23552
#152422
#152362
#151952
#151922
#151752
#151562
#150862
#150252
#149902
#149522
C4×#4002
C4×#3992
#138172
#138042
#138012