Finite Group Databasebeta

← #719949

Subgroups — #719949

Order 512.

2626 conjugacy classes of subgroups.

The subgroup lattice has 665 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2552C211yesmaximal, characteristic, complemented
2C2×#2441C212yesmaximal, complemented
3C2×#2354C211yesmaximal, characteristic
4#16555C214yesmaximal
5#15396C214yesmaximal, complemented
6#15353C214yesmaximal, complemented
7#15220C212yesmaximal, complemented
8#15216C212yesmaximal, complemented
9#14779C214yesmaximal, complemented
10#14276C211yesmaximal, characteristic, complemented
11#14150C211yesmaximal, characteristic, complemented
12#14129C212yesmaximal, complemented
13#13905C211yesmaximal, characteristic
14#13847C211yesmaximal, characteristic
15#13779C211yesmaximal, characteristic, complemented
16C22×#125C2212yes—
17C22×C4×D4C2211yescharacteristic
18C22×#117C2212yes—
19#2552—28no—
20#2441—216no—
21C2×#399C2214yes—
22C2×#398C2214yes—
23C2×#396C2214yes—
24#2354—28no—
25C2×#390C2212yes—
26C2×#389C2212yes—
27C2×#389C2212yes—
28C2×#389C2212yes—
29C2×#389C2212yes—
30C2×#389C2214yes—
31C2×#388C2214yes—
32C2×#387C2214yes—
33C2×#384C2212yes—
34C2×#384C2211yescharacteristic
35C2×#384C2211yescharacteristic
36C2×#382C2211yescharacteristic
37C2×#382C2211yescharacteristic
38C2×#380C2212yes—
39C2×#380C2212yes—
40C2×#380C2212yes—
41C2×C4×C4⋊C4C2211yescharacteristic
42C2×C4×C4⋊C4C2211yescharacteristic
43C2×C4×C4⋊C4C2212yes—
44C2×C4×C4⋊C4C2211yescharacteristic
45C2×C4×C22⋊C4C2212yes—
46C2×C4×C22⋊C4C2212yes—
47C2×#376C2211yescharacteristic
48#1445C2212yes—
49#1445C2212yes—
50#1444C2212yes—
51#1443C2214yes—
52#1442C2212yes—
53#1442C2214yes—
54#1442C2214yes—
55#1441C2214yes—
56#1441C2214yes—
57#1441C2214yes—
58#1439C2212yes—
59#1439C2212yes—
60#1439C2214yes—
61#1439C2214yes—
62#1439C2214yes—
63#1439C2214yes—
64#1439C2214yes—
65#1439C2214yes—
66#1438C2212yes—
67#1438C2212yes—
68#1437C2214yes—
69#1436C2214yes—
70#1436C2214yes—
71#1436C2214yes—
72#1436C2214yes—
73#1436C2214yes—
74C4⋊#95C2211yescharacteristic
75C4⋊#95C2212yes—
76C4⋊#95C2212yes—
77C4⋊#95C2211yescharacteristic
78C23×D4C2311yescharacteristic
79C24×C4C2312yes—
80C2×#125—28no—
81C2×#125—28no—
82C2×#125—44no—
83C2×C4×D4—24no—
84C2×C4×D4—24no—
85C2×C4×D4Q818yes—
86C2×#117—28no—
87C2×#117—28no—
88C2×#117—44no—
89C22×C4⋊C4C2311yescharacteristic
90C22×C4⋊C4C2312yes—
91C22×C4⋊C4C2314yes—
92C22×C4⋊C4C2314yes—
93C22×C4⋊C4C2311yescharacteristic
94C22×C4⋊C4C2312yes—
95C22×C4⋊C4C2311yescharacteristic
96C22×C4⋊C4C2311yescharacteristic
97C22×C4⋊C4C2311yescharacteristic
98C22×C4⋊C4C2314yes—
99C22×C4⋊C4C2314yes—
100C22×C4⋊C4C2314yes—
101C22×C4⋊C4C2311yescharacteristic
102C22×C4⋊C4C2311yescharacteristic
103C22×C4⋊C4C2314yes—
104C22×C4⋊C4C2312yes—
105C22×C4⋊C4C2312yes—
106C22×C22⋊C4C2314yes—
107C22×C22⋊C4C2312yes—
108C22×C22⋊C4C2312yes—
109C22×C22⋊C4C2314yes—
110C22×C22⋊C4C2312yes—
111C22×C22⋊C4C2314yes—
112C22×C22⋊C4C2314yes—
113C22×C22⋊C4C2312yes—
114C22×C22⋊C4C2312yes—
115C22×C42C2312yes—
116C22×C42C2311yescharacteristic
117C22×C42C2314yes—
118C22×C42C2311yescharacteristic
119C22×C42C2312yes—
120C22×C42C2312yes—
121C22×C42C2311yescharacteristic
122#399—216no—
123#398—216no—
124#396—216no—
125#390—28no—
126#389—216no—
127#389—28no—
128#389—28no—
129#389—28no—
130#389—28no—
131#388—216no—
132#387—216no—
133#384—28no—
134#384D418yes—
135#384D418yescomplemented
136#382—24no—
137#382—24no—
138#380—28no—
139#380—28no—
140#380—28no—
141C4×C4⋊C4—28no—
142C4×C4⋊C4—24no—
143C4×C4⋊C4—24no—
144C4×C4⋊C4—24no—
145C4×C22⋊C4—28no—
146C4×C22⋊C4—44no—
147#376—24no—
148C2×#95C2314yes—
149C2×#95C2312yes—
150C2×#95C2312yes—
151C2×#95C2314yes—
152C2×#95C2314yes—
153C2×#95C2314yes—
154C2×#95C2312yes—
155C2×#95C2312yes—
156C2×#95C2312yes—
157C2×#95C2314yes—
158C2×#95C2312yes—
159C2×#95C2312yes—
160C2×#95C2314yes—
161C2×#95C2312yes—
162C2×#95C2312yes—
163C2×#95C2314yes—
164C2×#95C2314yes—
165C2×#95C2314yes—
166C2×#95C2314yes—
167C2×#95C2314yes—
168C2×#95C2314yes—
169C2×#95C2314yes—
170C2×#95C2312yes—
171C2×#95C2312yes—
172C25C2412yes—
173C22×D4—22no—
174C22×D4—22no—
175C22×D4—42no—
176C22×D4—42no—
177C22×D4C2×Q814yes—
178C23×C4—24no—
179C23×C4—24no—
180C23×C4—24no—
181C23×C4—24no—
182C23×C4—24no—
183C23×C4—24no—
184C23×C4C2×Q818yes—
185C23×C4C2412yes—
186C23×C4C2412yes—
187C23×C4C2414yes—
188C23×C4C2412yes—
189C23×C4C2412yes—
190C23×C4C2414yes—
191C23×C4C2412yes—
192C23×C4C2412yes—
193C23×C4C2412yes—
194C23×C4C2412yes—
195C23×C4C2412yes—
196C23×C4C2411yescharacteristic
197C23×C4C2411yescharacteristic
198C23×C4C2411yescharacteristic
199#125—432no—
200C4×D4—232no—
201#117—432no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—24no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—24no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—28no—
213C2×C4⋊C4—28no—
214C2×C4⋊C4—28no—
215C2×C4⋊C4—28no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—24no—
220C2×C4⋊C4—28no—
221C2×C4⋊C4—24no—
222C2×C4⋊C4—24no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—28no—
226C2×C4⋊C4—28no—
227C2×C4⋊C4—44no—
228C2×C4⋊C4—44no—
229C2×C4⋊C4—44no—
230C2×C4⋊C4—44no—
231C2×C4⋊C4—42no—
232C2×C4⋊C4—42no—
233C2×C4⋊C4—44no—
234C2×C4⋊C4—44no—
235C2×C4⋊C4—44no—
236C2×C4⋊C4—44no—
237C2×C4⋊C4—44no—
238C2×C4⋊C4—44no—
239C2×C4⋊C4—44no—
240C2×C4⋊C4—44no—
241C2×C4⋊C4C2×D414yes—
242C2×C4⋊C4C2×D418yes—
243C2×C4⋊C4C2×D414yes—
244C2×C4⋊C4C2×D418yes—
245C2×C4⋊C4C2×D418yes—
246C2×C4⋊C4C2×D4116yes—
247C2×C4⋊C4C2×Q814yes—
248C2×C4⋊C4#4114yes—
249C2×C22⋊C4—24no—
250C2×C22⋊C4—24no—
251C2×C22⋊C4—28no—
252C2×C22⋊C4—28no—
253C2×C22⋊C4—24no—
254C2×C22⋊C4—24no—
255C2×C22⋊C4—28no—
256C2×C22⋊C4—28no—
257C2×C22⋊C4—28no—
258C2×C22⋊C4—42no—
259C2×C22⋊C4—42no—
260C2×C22⋊C4—42no—
261C2×C22⋊C4—42no—
262C2×C22⋊C4—42no—
263C2×C22⋊C4—42no—
264C2×C22⋊C4—42no—
265C2×C22⋊C4—42no—
266C2×C22⋊C4—44no—
267C2×C22⋊C4—42no—
268C2×C22⋊C4—42no—
269C2×C22⋊C4—44no—
270C2×C22⋊C4—44no—
271C2×C22⋊C4—44no—
272C2×C22⋊C4—44no—
273C2×C22⋊C4—44no—
274C2×C22⋊C4—44no—
275C2×C22⋊C4C2×Q818yes—
276C2×C42—22no—
277C2×C42—22no—
278C2×C42—28no—
279C2×C42—28no—
280C2×C42—24no—
281C2×C42—24no—
282C2×C42—24no—
283C2×C42—24no—
284C2×C42—22no—
285C2×C42—24no—
286C2×C42—22no—
287C2×C42—22no—
288C2×C42—44no—
289C2×C42—44no—
290C2×C42—44no—
291C2×C42C2×D414yes—
292C2×C42C2×D414yes—
293C2×C42C2×Q814yes—
294C2×C42#4114yes—
295#95—24no—
296#95—24no—
297#95—28no—
298#95—24no—
299#95—24no—
300#95—24no—
301#95—28no—
302#95—28no—
303#95—28no—
304#95—28no—
305#95—24no—
306#95—24no—
307#95—28no—
308#95—28no—
309#95—24no—
310#95—24no—
311#95—44no—
312#95—44no—
313#95—44no—
314#95—44no—
315#95—44no—
316#95—44no—
317#95—42no—
318#95—42no—
319C24—22no—
320C24—22no—
321C24—22no—
322C24—22no—
323C24—22no—
324C24—22no—
325C24—22no—
326C24—22no—
327C24—22no—
328C24—22no—
329C24—22no—
330C24—22no—
331C24—22no—
332C24—22no—
333C24C22×Q814yes—
334C24C2511yescharacteristic
335C2×D4—28no—
336C2×D4—48no—
337C2×D4—44no—
338C2×D4—44no—
339C2×D4—48no—
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—24no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—24no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—24no—
361C22×C4—24no—
362C22×C4—24no—
363C22×C4—24no—
364C22×C4—24no—
365C22×C4—24no—
366C22×C4—24no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—24no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—24no—
373C22×C4—24no—
374C22×C4—216no—
375C22×C4—216no—
376C22×C4—216no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—24no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—24no—
385C22×C4—22no—
386C22×C4—24no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—24no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—24no—
400C22×C4—24no—
401C22×C4—24no—
402C22×C4—24no—
403C22×C4—22no—
404C22×C4—24no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—48no—
411C22×C4—48no—
412C22×C4—48no—
413C22×C4—48no—
414C22×C4C22×D414yes—
415C22×C4C22×D414yes—
416C22×C4C22×D418yes—
417C22×C4C22×D414yes—
418C22×C4C22×D412yes—
419C22×C4C22×D412yes—
420C22×C4C22×D412yes—
421C22×C4C22×D412yes—
422C22×C4C22×Q814yes—
423C22×C4C22×Q812yes—
424C22×C4C22×Q812yes—
425C22×C4C22×Q812yes—
426C22×C4C2×#4114yes—
427C22×C4C2×#4112yes—
428C22×C4C2×#4112yes—
429C22×C4C2×#4112yes—
430C22×C4C2×#4112yes—
431C22×C4H(5,𝔽2)12yes—
432C22×C42−1+412yes—
433C4⋊C4—28no—
434C4⋊C4—28no—
435C4⋊C4—216no—
436C4⋊C4—28no—
437C4⋊C4—28no—
438C4⋊C4—48no—
439C4⋊C4—48no—
440C4⋊C4—416no—
441C4⋊C4—416no—
442C4⋊C4—416no—
443C4⋊C4—44no—
444C4⋊C4—44no—
445C4⋊C4—416no—
446C4⋊C4—48no—
447C4⋊C4—44no—
448C4⋊C4—88no—
449C4⋊C4—88no—
450C22⋊C4—216no—
451C22⋊C4—416no—
452C22⋊C4—48no—
453C22⋊C4—48no—
454C22⋊C4—48no—
455C22⋊C4—48no—
456C22⋊C4—88no—
457C22⋊C4—88no—
458C22⋊C4—88no—
459C42—28no—
460C42—416no—
461C42—48no—
462C42—48no—
463C42—44no—
464C42—48no—
465C42C22⋊Q8116yes—
466C23—24no—
467C23—24no—
468C23—24no—
469C23—24no—
470C23—24no—
471C23—24no—
472C23—24no—
473C23—42no—
474C23—42no—
475C23—42no—
476C23—42no—
477C23—42no—
478C23—42no—
479C23—42no—
480C23—42no—
481C23—42no—
482C23—42no—
483C23—42no—
484C23—42no—
485C23—42no—
486C23—42no—
487C23—42no—
488C23—42no—
489C23—42no—
490C23—42no—
491C23—42no—
492C23—42no—
493C23—42no—
494C23—42no—
495C23—42no—
496C23—42no—
497C23—42no—
498C23—42no—
499C23—42no—
500C23—42no—
501C23C23×D411yescharacteristic
502C23C23×D411yescharacteristic
503C23C23×Q811yescharacteristic
504C23C22×#4111yescharacteristic
505C23C22×#4111yescharacteristic
506C23C2×H(5,𝔽2)11yescharacteristic
507C23C2×2−1+411yescharacteristic
508C23#58512yes—
509C23#58512yes—
510C23#58512yes—
511C23#58512yes—
512D4—416no—
513C2×C4—24no—
514C2×C4—22no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—24no—
519C2×C4—24no—
520C2×C4—24no—
521C2×C4—24no—
522C2×C4—24no—
523C2×C4—24no—
524C2×C4—28no—
525C2×C4—24no—
526C2×C4—24no—
527C2×C4—24no—
528C2×C4—24no—
529C2×C4—24no—
530C2×C4—28no—
531C2×C4—28no—
532C2×C4—28no—
533C2×C4—28no—
534C2×C4—24no—
535C2×C4—28no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—44no—
545C2×C4—42no—
546C2×C4—44no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—44no—
553C2×C4—44no—
554C2×C4—44no—
555C2×C4—44no—
556C2×C4—44no—
557C2×C4—432no—
558C2×C4—44no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—44no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—44no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—44no—
577C2×C4—42no—
578C2×C4—44no—
579C2×C4—44no—
580C2×C4—44no—
581C2×C4—44no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—44no—
591C2×C4C2×C22⋊Q814yes—
592C2×C4C2×C22⋊Q814yes—
593C2×C4C2×C22⋊Q814yes—
594C2×C4#53214yes—
595C2×C4#54214yes—
596C2×C4D4214yes—
597C2×C4#54814yes—
598C2×C4D4×Q814yes—
599C2×C4#55414yes—
600C22—44no—
601C22—44no—
602C22—44no—
603C22—44no—
604C22—44no—
605C22—44no—
606C22—44no—
607C22—82no—
608C22—82no—
609C22—82no—
610C22—82no—
611C22—82no—
612C22—82no—
613C22—82no—
614C22—82no—
615C22C22×C22⋊Q811yescharacteristic
616C22C2×#53211yescharacteristic
617C22C2×#54211yescharacteristic
618C22#345812yes—
619C22C2×D4211yescharacteristic
620C22C2×#54811yescharacteristic
621C22C2×D4×Q811yescharacteristic
622C22C2×#55411yescharacteristic
623C22#348511yescharacteristic
624C22#348512yes—
625C22#348511yescharacteristic
626C22#348711yescharacteristic
627C22#348712yes—
628C22#348711yescharacteristic
629C22#349312yes—
630C22#350412yes—
631C22#351212yes—
632C22#352112yes—
633C22#352412yes—
634C22#353912yes—
635C22#353912yes—
636C22#356512yes—
637C22#356512yes—
638C4—24no—
639C4—24no—
640C4—44no—
641C4—48no—
642C4—44no—
643C4—44no—
644C4—44no—
645C4—44no—
646C4—82no—
647C4—82no—
648C4—82no—
649C4—84no—
650C4—82no—
651C4#245218yes—
652C2—84no—
653C2C2×#245211yescharacteristic
654C2#3515111yescharacteristic
655C2#3563911yescharacteristic
656C2#3601512yes—
657C2#3769112yes—
658C2#3810712yes—
659C2#4266212yes—
660C2#4514211yescharacteristic
661C2#4527011yescharacteristic
662C2#4538711yescharacteristic
663C2#4538711yescharacteristic
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
1. C2×#2552C2semidirect
2. C2×#2441C2semidirect
5. #15396C2semidirect
6. #15353C2semidirect
7. #15220C2semidirect
8. #15216C2semidirect
9. #14779C2semidirect
10. #14276C2semidirect
11. #14150C2semidirect
12. #14129C2semidirect
15. #13779C2semidirect
135. #384D4semidirect

Maximal subgroups

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

subgroupindex
C2×#25522
C2×#24412
C2×#23542
#165552
#153962
#153532
#152202
#152162
#147792
#142762
#141502
#141292
#139052
#138472
#137792