Finite Group Databasebeta

← #4453326

Subgroups — #4453326

Order 512.

2890 conjugacy classes of subgroups.

The subgroup lattice has 676 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#396C211yesmaximal, characteristic
2C2×#2313C211yesmaximal, characteristic, complemented
3#16445C214yesmaximal, complemented
4#16441C214yesmaximal, complemented
5#16317C214yesmaximal, complemented
6#16315C214yesmaximal, complemented
7#15073C212yesmaximal, complemented
8#15019C211yesmaximal, characteristic, complemented
9#14936C211yesmaximal, characteristic, complemented
10#14918C212yesmaximal
11#14820C211yesmaximal, characteristic
12#14734C211yesmaximal, characteristic
13#14339C211yesmaximal, characteristic, complemented
14#14130C212yesmaximal, complemented
15#13903C212yesmaximal
16C23×C4⋊C4C2211yescharacteristic
17C23×C4⋊C4C2211yescharacteristic
18C23×C4⋊C4C2211yescharacteristic
19C23×C22⋊C4C2211yescharacteristic
20C23×C22⋊C4C2211yescharacteristic
21C23×C22⋊C4C2211yescharacteristic
22C2×#399C2212yes—
23C2×#397C2212yes—
24C2×#396—28no—
25C2×#396C2214yes—
26C2×#396C2214yes—
27#2313—28no—
28C2×#389C2214yescomplemented
29C2×#387C2212yescomplemented
30C2×#387C2212yescomplemented
31C2×#387C2212yescomplemented
32C2×#387C2212yescomplemented
33C2×#385C2212yes—
34C2×#385C2212yes—
35C2×#381C2212yes—
36C2×#381C2212yes—
37C2×#381C2212yes—
38C2×#381C2212yes—
39C2×#380C2212yes—
40C2×#380C2211yescharacteristic
41C2×#380C2211yescharacteristic
42C2×#379C2212yes—
43C2×#379C2212yes—
44C2×C4×C22⋊C4C2212yes—
45C22×#95C2211yescharacteristic
46#1445C2212yescomplemented
47#1445C2214yescomplemented
48#1445C2212yescomplemented
49#1444C2214yescomplemented
50#1444C2212yescomplemented
51#1443C2214yescomplemented
52#1442C2214yescomplemented
53#1442C2214yescomplemented
54#1442C2214yescomplemented
55#1442C2212yescomplemented
56#1441C2214yescomplemented
57#1441C2214yescomplemented
58#1441C2214yescomplemented
59#1441C2214yescomplemented
60#1440C2214yes—
61#1440C2214yes—
62#1440C2214yes—
63#1439C2212yes—
64#1439C2214yes—
65#1439C2214yes—
66#1439C2214yes—
67#1439C2212yes—
68#1439C2214yes—
69#1439C2214yes—
70#1437C2214yes—
71#1437C2214yes—
72C4⋊#95C2214yescomplemented
73C22⋊#95C2211yescharacteristic
74C22⋊#95C2211yescharacteristic
75C22⋊#95C2212yes—
76C22⋊#95C2212yes—
77C26C2311yescharacteristic
78C24×C4C2311yescharacteristic
79C24×C4C2311yescharacteristic
80C24×C4C2311yescharacteristic
81C24×C4C2311yescharacteristic
82C24×C4C2311yescharacteristic
83C24×C4C2311yescharacteristic
84C22×C4⋊C4—28no—
85C22×C4⋊C4—24no—
86C22×C4⋊C4—24no—
87C22×C4⋊C4—44no—
88C22×C4⋊C4—44no—
89C22×C4⋊C4D418yes—
90C22×C4⋊C4C2314yes—
91C22×C4⋊C4C2312yes—
92C22×C4⋊C4C2312yes—
93C22×C4⋊C4C2314yes—
94C22×C4⋊C4C2311yescharacteristic
95C22×C4⋊C4C2311yescharacteristic
96C22×C4⋊C4C2314yes—
97C22×C4⋊C4C2314yes—
98C22×C4⋊C4C2311yescharacteristic
99C22×C4⋊C4C2311yescharacteristic
100C22×C4⋊C4C2314yes—
101C22×C22⋊C4—24no—
102C22×C22⋊C4—24no—
103C22×C22⋊C4—24no—
104C22×C22⋊C4—28no—
105C22×C22⋊C4—44no—
106C22×C22⋊C4—44no—
107C22×C22⋊C4C2314yes—
108C22×C22⋊C4C2312yes—
109C22×C22⋊C4C2312yes—
110C22×C22⋊C4C2312yes—
111C22×C22⋊C4C2312yes—
112C22×C22⋊C4C2312yes—
113C22×C22⋊C4C2312yes—
114C22×C22⋊C4C2314yes—
115C22×C22⋊C4C2314yes—
116C22×C22⋊C4C2314yes—
117C22×C22⋊C4C2314yes—
118C22×C22⋊C4C2314yes—
119C22×C42C2312yes—
120C22×C42C2312yes—
121#399—44no—
122#397—44no—
123#396—216no—
124#396—216no—
125#389—48no—
126#387—28no—
127#387—28no—
128#387—28no—
129#387—28no—
130#385—44no—
131#385—44no—
132#381—28no—
133#381—44no—
134#381—44no—
135#381—44no—
136#380—24no—
137#380—24no—
138#380—28no—
139#379—44no—
140#379—44no—
141C4×C22⋊C4—28no—
142C2×#95—24no—
143C2×#95C2312yes—
144C2×#95C2312yes—
145C2×#95C2314yes—
146C2×#95C2314yes—
147C2×#95C2312yes—
148C2×#95C2312yes—
149C2×#95C2312yes—
150C2×#95C2312yes—
151C2×#95C2312yes—
152C2×#95C2312yes—
153C2×#95C2312yes—
154C2×#95C2312yes—
155C2×#95C2312yes—
156C2×#95C2312yes—
157C2×#95C2314yes—
158C2×#95C2314yes—
159C2×#95C2312yes—
160C2×#95C2314yes—
161C2×#95C2312yes—
162C2×#95C2314yes—
163C2×#95C2314yes—
164C2×#95C2314yes—
165C2×#95C2314yes—
166C2×#95C2314yes—
167C2×#95C2314yes—
168C2×#95C2314yes—
169C2×#95C2314yes—
170C25—22no—
171C25—24no—
172C25—24no—
173C25—24no—
174C25—22no—
175C25—22no—
176C25—42no—
177C25—42no—
178C25—42no—
179C25C2412yes—
180C25C2411yescharacteristic
181C23×C4—22no—
182C23×C4—24no—
183C23×C4—24no—
184C23×C4—22no—
185C23×C4—22no—
186C23×C4—24no—
187C23×C4—24no—
188C23×C4—22no—
189C23×C4—22no—
190C23×C4—24no—
191C23×C4—24no—
192C23×C4—24no—
193C23×C4—22no—
194C23×C4—22no—
195C23×C4—24no—
196C23×C4—42no—
197C23×C4—42no—
198C23×C4—42no—
199C23×C4—42no—
200C23×C4—42no—
201C23×C4—42no—
202C23×C4—42no—
203C23×C4C2×D414yes—
204C23×C4C2×D414yes—
205C23×C4C2×D414yes—
206C23×C4#4118yes—
207C23×C4C2412yes—
208C23×C4C2414yes—
209C23×C4C2414yes—
210C23×C4C2412yes—
211C23×C4C2412yes—
212C23×C4C2412yes—
213C23×C4C2412yes—
214C23×C4C2412yes—
215C23×C4C2412yes—
216C23×C4C2411yescharacteristic
217C23×C4C2411yescharacteristic
218C23×C4C2412yes—
219C23×C4C2412yes—
220C2×C4⋊C4—232no—
221C2×C4⋊C4—24no—
222C2×C4⋊C4—24no—
223C2×C4⋊C4—24no—
224C2×C4⋊C4—24no—
225C2×C4⋊C4—28no—
226C2×C4⋊C4—28no—
227C2×C4⋊C4—28no—
228C2×C4⋊C4—28no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—28no—
234C2×C4⋊C4—44no—
235C2×C4⋊C4—44no—
236C2×C4⋊C4—44no—
237C2×C4⋊C4—44no—
238C2×C4⋊C4—44no—
239C2×C4⋊C4—416no—
240C2×C4⋊C4—416no—
241C2×C4⋊C4—44no—
242C2×C4⋊C4—42no—
243C2×C4⋊C4—42no—
244C2×C4⋊C4—44no—
245C2×C4⋊C4—44no—
246C2×C4⋊C4—44no—
247C2×C4⋊C4—44no—
248C2×C4⋊C4—42no—
249C2×C4⋊C4—42no—
250C2×C4⋊C4C2×D418yes—
251C2×C4⋊C4C2×D418yes—
252C2×C22⋊C4—24no—
253C2×C22⋊C4—24no—
254C2×C22⋊C4—24no—
255C2×C22⋊C4—24no—
256C2×C22⋊C4—24no—
257C2×C22⋊C4—24no—
258C2×C22⋊C4—24no—
259C2×C22⋊C4—24no—
260C2×C22⋊C4—24no—
261C2×C22⋊C4—24no—
262C2×C22⋊C4—24no—
263C2×C22⋊C4—24no—
264C2×C22⋊C4—24no—
265C2×C22⋊C4—24no—
266C2×C22⋊C4—28no—
267C2×C22⋊C4—28no—
268C2×C22⋊C4—28no—
269C2×C22⋊C4—28no—
270C2×C22⋊C4—24no—
271C2×C22⋊C4—24no—
272C2×C22⋊C4—28no—
273C2×C22⋊C4—28no—
274C2×C22⋊C4—42no—
275C2×C22⋊C4—42no—
276C2×C22⋊C4—44no—
277C2×C22⋊C4—44no—
278C2×C22⋊C4—416no—
279C2×C22⋊C4—416no—
280C2×C22⋊C4—416no—
281C2×C22⋊C4—44no—
282C2×C22⋊C4—44no—
283C2×C22⋊C4—44no—
284C2×C22⋊C4—44no—
285C2×C22⋊C4—44no—
286C2×C22⋊C4—44no—
287C2×C22⋊C4—44no—
288C2×C22⋊C4—44no—
289C2×C22⋊C4—44no—
290C2×C22⋊C4—44no—
291C2×C42—24no—
292C2×C42—24no—
293C2×C42—44no—
294C2×C42—44no—
295#95—24no—
296#95—24no—
297#95—24no—
298#95—24no—
299#95—24no—
300#95—24no—
301#95—24no—
302#95—24no—
303#95—24no—
304#95—24no—
305#95—28no—
306#95—28no—
307#95—44no—
308#95—42no—
309#95—42no—
310#95—44no—
311#95—44no—
312#95—44no—
313#95—44no—
314#95—44no—
315#95—44no—
316#95—44no—
317#95—44no—
318#95—44no—
319#95—44no—
320#95—42no—
321#95—42no—
322C24—22no—
323C24—22no—
324C24—22no—
325C24—22no—
326C24—28no—
327C24—22no—
328C24—24no—
329C24—22no—
330C24—22no—
331C24—24no—
332C24—22no—
333C24—22no—
334C24—22no—
335C24—24no—
336C24—24no—
337C24—416no—
338C24—416no—
339C24—48no—
340C24—48no—
341C24—44no—
342C24—44no—
343C24—44no—
344C24—44no—
345C24—44no—
346C24—44no—
347C24—48no—
348C24—48no—
349C24—88no—
350C24—88no—
351C24—84no—
352C24—84no—
353C24C22×D412yes—
354C24C2×#4114yes—
355C24H(5,𝔽2)14yes—
356C24H(5,𝔽2)12yes—
357C242−1+414yes—
358C242−1+412yes—
359C24C2511yescharacteristic
360C22×C4—24no—
361C22×C4—24no—
362C22×C4—28no—
363C22×C4—28no—
364C22×C4—28no—
365C22×C4—28no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
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—24no—
382C22×C4—28no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—24no—
390C22×C4—24no—
391C22×C4—24no—
392C22×C4—24no—
393C22×C4—24no—
394C22×C4—22no—
395C22×C4—24no—
396C22×C4—22no—
397C22×C4—24no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—24no—
403C22×C4—24no—
404C22×C4—24no—
405C22×C4—24no—
406C22×C4—24no—
407C22×C4—24no—
408C22×C4—24no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—24no—
413C22×C4—24no—
414C22×C4—24no—
415C22×C4—24no—
416C22×C4—22no—
417C22×C4—24no—
418C22×C4—22no—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—24no—
422C22×C4—24no—
423C22×C4—24no—
424C22×C4—24no—
425C22×C4—24no—
426C22×C4—22no—
427C22×C4—44no—
428C22×C4—44no—
429C22×C4—416no—
430C22×C4—44no—
431C22×C4—44no—
432C22×C4—416no—
433C22×C4—44no—
434C22×C4—44no—
435C22×C4—48no—
436C22×C4—48no—
437C22×C4—44no—
438C22×C4—44no—
439C22×C4—44no—
440C22×C4—44no—
441C22×C4—44no—
442C22×C4—44no—
443C22×C4—48no—
444C22×C4—48no—
445C22×C4—88no—
446C22×C4—88no—
447C22×C4#123116yes—
448C22×C4C22×D414yes—
449C22×C4C22×D412yes—
450C22×C4C22×D412yes—
451C22×C4C22×D414yes—
452C22×C4C2×#4114yes—
453C22×C4C2×#4114yes—
454C22×C4H(5,𝔽2)14yes—
455C4⋊C4—216no—
456C4⋊C4—216no—
457C4⋊C4—416no—
458C4⋊C4—416no—
459C4⋊C4—416no—
460C4⋊C4—82no—
461C4⋊C4—82no—
462C4⋊C4—82no—
463C4⋊C4—82no—
464C4⋊C4—88no—
465C4⋊C4—88no—
466C22⋊C4—48no—
467C22⋊C4—48no—
468C22⋊C4—416no—
469C22⋊C4—48no—
470C22⋊C4—48no—
471C22⋊C4—48no—
472C22⋊C4—48no—
473C22⋊C4—416no—
474C22⋊C4—88no—
475C22⋊C4—88no—
476C22⋊C4—88no—
477C22⋊C4—88no—
478C42—84no—
479C42—84no—
480C23—22no—
481C23—22no—
482C23—22no—
483C23—22no—
484C23—28no—
485C23—28no—
486C23—28no—
487C23—28no—
488C23—24no—
489C23—24no—
490C23—24no—
491C23—24no—
492C23—24no—
493C23—28no—
494C23—28no—
495C23—24no—
496C23—24no—
497C23—416no—
498C23—416no—
499C23—416no—
500C23—42no—
501C23—42no—
502C23—42no—
503C23—44no—
504C23—44no—
505C23—42no—
506C23—42no—
507C23—42no—
508C23—44no—
509C23—44no—
510C23—44no—
511C23—44no—
512C23—44no—
513C23—44no—
514C23—42no—
515C23—42no—
516C23—44no—
517C23—44no—
518C23—832no—
519C23—832no—
520C23—88no—
521C23—88no—
522C23—88no—
523C23—88no—
524C23C2×#12314yes—
525C23#53514yes—
526C23#53714yes—
527C23#54018yes—
528C23#54218yes—
529C23C23×D411yescharacteristic
530C23C22×#4112yes—
531C23C2×H(5,𝔽2)12yes—
532C23C2×H(5,𝔽2)11yescharacteristic
533C23C2×H(5,𝔽2)12yes—
534C23C2×2−1+411yescharacteristic
535C23C2×2−1+412yes—
536C23#58512yes—
537C23#58512yes—
538C2×C4—24no—
539C2×C4—24no—
540C2×C4—28no—
541C2×C4—28no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—24no—
545C2×C4—24no—
546C2×C4—24no—
547C2×C4—24no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—28no—
553C2×C4—416no—
554C2×C4—416no—
555C2×C4—416no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—44no—
563C2×C4—44no—
564C2×C4—44no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—44no—
574C2×C4—44no—
575C2×C4—44no—
576C2×C4—44no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—44no—
588C2×C4—44no—
589C2×C4—44no—
590C2×C4—44no—
591C2×C4—44no—
592C2×C4—44no—
593C2×C4—44no—
594C2×C4—44no—
595C2×C4—44no—
596C2×C4—44no—
597C2×C4—44no—
598C2×C4—44no—
599C2×C4—44no—
600C2×C4—44no—
601C2×C4—44no—
602C2×C4—44no—
603C2×C4—44no—
604C2×C4—44no—
605C2×C4—44no—
606C2×C4—44no—
607C2×C4—44no—
608C2×C4—44no—
609C2×C4—88no—
610C2×C4—88no—
611C2×C4—88no—
612C2×C4C2×#12318yes—
613C22—24no—
614C22—24no—
615C22—24no—
616C22—24no—
617C22—24no—
618C22—216no—
619C22—216no—
620C22—24no—
621C22—44no—
622C22—44no—
623C22—48no—
624C22—48no—
625C22—44no—
626C22—832no—
627C22—82no—
628C22—82no—
629C22—82no—
630C22—82no—
631C22—84no—
632C22—84no—
633C22#262418yes—
634C22C22×#12311yescharacteristic
635C22C2×#53511yescharacteristic
636C22C2×#53711yescharacteristic
637C22C2×#54012yes—
638C22C2×#54212yes—
639C22#348411yescharacteristic
640C22#348511yescharacteristic
641C22#348611yescharacteristic
642C22#348711yescharacteristic
643C22#349514yes—
644C22#350514yes—
645C22#353414yes—
646C22#353514yes—
647C22#353612yes—
648C22#353812yes—
649C22#356012yes—
650C22#356412yes—
651C4—28no—
652C4—42no—
653C4—42no—
654C4—44no—
655C4—44no—
656C4—82no—
657C4—82no—
658C4—82no—
659C4—82no—
660C4—82no—
661C4—84no—
662C4—84no—
663C4—82no—
664C2—28no—
665C2—84no—
666C2C2×#262411yescharacteristic
667C2#3535111yescharacteristic
668C2#3567211yescharacteristic
669C2#3671911yescharacteristic
670C2#3697111yescharacteristic
671C2#3711811yescharacteristic
672C2#3718511yescharacteristic
673C2#4056514yes—
674C2#4056714yes—
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
2. C2×#2313C2semidirect
3. #16445C2semidirect
4. #16441C2semidirect
5. #16317C2semidirect
6. #16315C2semidirect
7. #15073C2semidirect
8. #15019C2semidirect
9. #14936C2semidirect
13. #14339C2semidirect
14. #14130C2semidirect
28. C2×#389C22semidirect
29. C2×#387C22semidirect
30. C2×#387C22semidirect
31. C2×#387C22semidirect
32. C2×#387C22semidirect
46. #1445C22semidirect
47. #1445C22semidirect
48. #1445C22semidirect
49. #1444C22semidirect
50. #1444C22semidirect
51. #1443C22semidirect
52. #1442C22semidirect
53. #1442C22semidirect
54. #1442C22semidirect
55. #1442C22semidirect
56. #1441C22semidirect
57. #1441C22semidirect
58. #1441C22semidirect
59. #1441C22semidirect
72. C4⋊#95C22semidirect

Maximal subgroups

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

subgroupindex
C22×#3962
C2×#23132
#164452
#164412
#163172
#163152
#150732
#150192
#149362
#149182
#148202
#147342
#143392
#141302
#139032