Finite Group Databasebeta

← #517279

Subgroups — #517279

Order 512.

3170 conjugacy classes of subgroups.

The subgroup lattice has 666 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2855C211yesmaximal, characteristic
2C2×#2329C211yesmaximal, characteristic
3C22×C4×C4⋊C4C211yesmaximal, characteristic
4#14695C212yesmaximal
5#14685C211yesmaximal, characteristic
6#14281C211yesmaximal, characteristic
7#14275C211yesmaximal, characteristic
8#14263C212yesmaximal
9#14120C214yesmaximal, complemented
10#14111C214yesmaximal, complemented
11#14059C214yesmaximal, complemented
12#14035C214yesmaximal, complemented
13#13778C211yesmaximal, characteristic
14#13776C212yesmaximal
15C4×#385C212yesmaximal
16C22×C22⋊Q8C2211yescharacteristic
17C23×C4⋊C4C2211yescharacteristic
18C23×C4⋊C4C2211yescharacteristic
19C23×C42C2211yescharacteristic
20C23×C42C2211yescharacteristic
21C23×C42C2211yescharacteristic
22#2855C4116yescomplemented
23#2329—28no—
24C2×#391C2212yes—
25C2×#391C2212yes—
26C2×#389C2212yes—
27C2×#389C2212yes—
28C2×#387C2212yes—
29C2×#387C2212yes—
30C2×#387C2214yes—
31C2×#387C2214yes—
32C2×#385C2212yes—
33C2×#385C2212yes—
34C2×#385C2212yes—
35C2×#381C2212yes—
36C2×#381C2212yes—
37C2×#380C2211yescharacteristic
38C2×#380C2211yescharacteristic
39C2×#380C2211yescharacteristic
40C2×C4×C4⋊C4—28no—
41C2×C4×C4⋊C4C2214yes—
42C2×C4×C4⋊C4C2214yes—
43C2×C4×C4⋊C4C2214yes—
44C2×C4×C22⋊C4C2212yes—
45C2×C4×C22⋊C4C2212yes—
46C22×#95C2211yescharacteristic
47C22×#95C2211yescharacteristic
48#1446C2212yes—
49#1445C2212yes—
50#1444C2212yes—
51#1444C2214yes—
52#1444C2214yes—
53#1442C2212yes—
54#1442C2214yes—
55#1442C2214yes—
56#1441C2214yes—
57#1441C2214yes—
58#1436C2214yes—
59#1436C2214yes—
60#1436C2214yes—
61#1436C2214yes—
62#1436C2214yes—
63#1436C2214yes—
64#1436C2214yes—
65#1436C2214yes—
66#1436C2214yes—
67#1436C2214yes—
68C4⋊#95C2212yes—
69C4⋊#95C2212yes—
70#1434C2214yes—
71#1434C2214yes—
72C22⋊#95C2212yes—
73C22⋊#95C2212yes—
74C22⋊#95C2212yes—
75C22⋊#95C2212yes—
76C22⋊#95C2212yes—
77C4×#95C2214yes—
78C23×Q8C2311yescharacteristic
79C24×C4C2311yescharacteristic
80C24×C4C2311yescharacteristic
81C24×C4C2311yescharacteristic
82C24×C4C2311yescharacteristic
83C24×C4C2311yescharacteristic
84C24×C4C2311yescharacteristic
85C24×C4C2311yescharacteristic
86C2×C22⋊Q8—24no—
87C2×C22⋊Q8—24no—
88C2×C22⋊Q8C2×C418yes—
89C22×C4⋊C4—24no—
90C22×C4⋊C4—24no—
91C22×C4⋊C4—24no—
92C22×C4⋊C4—24no—
93C22×C4⋊C4—24no—
94C22×C4⋊C4Q818yescomplemented
95C22×C4⋊C4C2314yes—
96C22×C4⋊C4C2312yes—
97C22×C4⋊C4C2311yescharacteristic
98C22×C4⋊C4C2314yes—
99C22×C4⋊C4C2314yes—
100C22×C4⋊C4C2312yes—
101C22×C4⋊C4C2311yescharacteristic
102C22×C4⋊C4C2311yescharacteristic
103C22×C4⋊C4C2312yes—
104C22×C4⋊C4C2314yes—
105C22×C4⋊C4C2312yes—
106C22×C4⋊C4C2312yes—
107C22×C22⋊C4C2312yes—
108C22×C22⋊C4C2312yes—
109C22×C22⋊C4C2312yes—
110C22×C22⋊C4C2312yes—
111C22×C42—24no—
112C22×C42—24no—
113C22×C42—24no—
114C22×C42—24no—
115C22×C42—44no—
116C22×C42—44no—
117C22×C42C2×C418yes—
118C22×C42C2314yes—
119C22×C42C2314yes—
120C22×C42C2312yes—
121C22×C42C2312yes—
122C22×C42C2312yes—
123C22×C42C2314yes—
124C22×C42C2312yes—
125#391—28no—
126#391C2×C4116yescomplemented
127#389—28no—
128#389C2×C4116yescomplemented
129#387—28no—
130#387—28no—
131#387—48no—
132#387C2×C4132yescomplemented
133#385—28no—
134#385—28no—
135#385C2×C4116yes—
136#381—28no—
137#381C2×C4116yes—
138#380—24no—
139#380—24no—
140#380C2×C418yes—
141C4×C4⋊C4—216no—
142C4×C4⋊C4—216no—
143C4×C4⋊C4—48no—
144C4×C22⋊C4—28no—
145C4×C22⋊C4—28no—
146C2×#95—24no—
147C2×#95D418yes—
148C2×#95C2314yes—
149C2×#95C2314yes—
150C2×#95C2314yes—
151C2×#95C2312yes—
152C2×#95C2312yes—
153C2×#95C2312yes—
154C2×#95C2314yes—
155C2×#95C2314yes—
156C2×#95C2314yes—
157C2×#95C2312yes—
158C2×#95C2312yes—
159C2×#95C2312yes—
160C2×#95C2312yes—
161C2×#95C2312yes—
162C2×#95C2314yes—
163C2×#95C2314yes—
164C2×#95C2314yes—
165C2×#95C2314yes—
166C2×#95C2314yes—
167C2×#95C2314yes—
168C2×#95C2314yes—
169C2×#95C2314yes—
170C2×#95C2314yes—
171C2×#95C2312yes—
172C2×#95C2314yes—
173C2×#95C2314yes—
174C2×#95C2312yes—
175C2×#95C2312yes—
176C25C2411yescharacteristic
177C22×Q8—22no—
178C22×Q8—22no—
179C22×Q8—42no—
180C22×Q8—42no—
181C22×Q8C22×C414yes—
182C23×C4—22no—
183C23×C4—22no—
184C23×C4—22no—
185C23×C4—22no—
186C23×C4—22no—
187C23×C4—22no—
188C23×C4—24no—
189C23×C4—24no—
190C23×C4—22no—
191C23×C4—22no—
192C23×C4—22no—
193C23×C4—22no—
194C23×C4—22no—
195C23×C4—22no—
196C23×C4—42no—
197C23×C4—42no—
198C23×C4—42no—
199C23×C4—42no—
200C23×C4—42no—
201C23×C4—42no—
202C23×C4—42no—
203C23×C4—42no—
204C23×C4—42no—
205C23×C4—42no—
206C23×C4C22×C414yes—
207C23×C4C22×C414yes—
208C23×C4C22×C414yes—
209C23×C4C2×D414yes—
210C23×C4C2×D414yes—
211C23×C4C2×D414yes—
212C23×C4C2×Q814yes—
213C23×C4C2×Q814yes—
214C23×C4C2×Q814yes—
215C23×C4#4114yes—
216C23×C4#4114yes—
217C23×C4#4114yes—
218C23×C4#4114yes—
219C23×C4C2412yes—
220C23×C4C2412yes—
221C23×C4C2414yes—
222C23×C4C2412yes—
223C23×C4C2414yes—
224C23×C4C2412yes—
225C23×C4C2412yes—
226C23×C4C2412yes—
227C23×C4C2412yes—
228C23×C4C2411yescharacteristic
229C23×C4C2411yescharacteristic
230C23×C4C2412yes—
231C23×C4C2412yes—
232C23×C4C2412yes—
233C22⋊Q8—232no—
234C2×C4⋊C4—232no—
235C2×C4⋊C4—232no—
236C2×C4⋊C4—28no—
237C2×C4⋊C4—28no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—28no—
241C2×C4⋊C4—28no—
242C2×C4⋊C4—28no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—24no—
245C2×C4⋊C4—28no—
246C2×C4⋊C4—28no—
247C2×C4⋊C4—28no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—42no—
254C2×C4⋊C4—42no—
255C2×C4⋊C4—44no—
256C2×C4⋊C4—44no—
257C2×C4⋊C4—42no—
258C2×C4⋊C4—42no—
259C2×C4⋊C4—42no—
260C2×C4⋊C4—42no—
261C2×C4⋊C4—44no—
262C2×C4⋊C4—44no—
263C2×C4⋊C4C22×C418yes—
264C2×C4⋊C4C22×C414yes—
265C2×C4⋊C4C22×C4116yes—
266C2×C4⋊C4C22×C418yes—
267C2×C4⋊C4C2×Q8116yes—
268C2×C22⋊C4—24no—
269C2×C22⋊C4—24no—
270C2×C22⋊C4—24no—
271C2×C22⋊C4—24no—
272C2×C22⋊C4—24no—
273C2×C22⋊C4—24no—
274C2×C22⋊C4—24no—
275C2×C22⋊C4—24no—
276C2×C22⋊C4—24no—
277C2×C22⋊C4—24no—
278C2×C22⋊C4C22×C418yes—
279C2×C22⋊C4C22×C418yes—
280C2×C42—232no—
281C2×C42—24no—
282C2×C42—24no—
283C2×C42—24no—
284C2×C42—24no—
285C2×C42—28no—
286C2×C42—28no—
287C2×C42—28no—
288C2×C42—28no—
289C2×C42—28no—
290C2×C42—24no—
291C2×C42—24no—
292C2×C42—44no—
293C2×C42—44no—
294C2×C42—44no—
295C2×C42—44no—
296C2×C42—416no—
297C2×C42—416no—
298C2×C42—44no—
299C2×C42—44no—
300C2×C42C22×C418yes—
301C2×C42C22×C418yes—
302#95—24no—
303#95—24no—
304#95—24no—
305#95—24no—
306#95—24no—
307#95—24no—
308#95—24no—
309#95—28no—
310#95—24no—
311#95—28no—
312#95—28no—
313#95—28no—
314#95—28no—
315#95—28no—
316#95—44no—
317#95—44no—
318#95—44no—
319#95—44no—
320#95—44no—
321#95—44no—
322#95—44no—
323#95—44no—
324#95C22×C418yes—
325#95C22×C4116yes—
326#95C22×C4116yes—
327#95C22×C4116yes—
328#95C2×D418yes—
329#95C2×D418yes—
330C24—22no—
331C24—22no—
332C24—22no—
333C24—22no—
334C24C23×C412yes—
335C24C22×D412yes—
336C24C22×Q812yes—
337C24C2×#4112yes—
338C24C2×#4112yes—
339C24C2×#4112yes—
340C24C2×#4112yes—
341C24C2511yescharacteristic
342C2×Q8—28no—
343C2×Q8—48no—
344C2×Q8—82no—
345C2×Q8—82no—
346C2×Q8—84no—
347C22×C4—28no—
348C22×C4—28no—
349C22×C4—216no—
350C22×C4—28no—
351C22×C4—28no—
352C22×C4—28no—
353C22×C4—28no—
354C22×C4—28no—
355C22×C4—28no—
356C22×C4—28no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—24no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—24no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—216no—
372C22×C4—28no—
373C22×C4—28no—
374C22×C4—28no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—24no—
380C22×C4—24no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
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—24no—
393C22×C4—24no—
394C22×C4—24no—
395C22×C4—24no—
396C22×C4—24no—
397C22×C4—24no—
398C22×C4—24no—
399C22×C4—24no—
400C22×C4—24no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—24no—
407C22×C4—24no—
408C22×C4—24no—
409C22×C4—22no—
410C22×C4—24no—
411C22×C4—24no—
412C22×C4—24no—
413C22×C4—24no—
414C22×C4—24no—
415C22×C4—24no—
416C22×C4—24no—
417C22×C4—24no—
418C22×C4—44no—
419C22×C4—44no—
420C22×C4—48no—
421C22×C4—44no—
422C22×C4—44no—
423C22×C4—44no—
424C22×C4—44no—
425C22×C4—48no—
426C22×C4—48no—
427C22×C4—48no—
428C22×C4—48no—
429C22×C4—48no—
430C22×C4—44no—
431C22×C4—44no—
432C22×C4—48no—
433C22×C4—48no—
434C22×C4—44no—
435C22×C4—44no—
436C22×C4—48no—
437C22×C4—48no—
438C22×C4C4×D4116yes—
439C22×C4C4×Q8116yes—
440C22×C4C22⋊Q8116yes—
441C22×C4C23×C418yes—
442C22×C4C23×C414yes—
443C22×C4C23×C414yes—
444C22×C4C23×C412yes—
445C22×C4C23×C412yes—
446C22×C4C23×C414yes—
447C22×C4C23×C414yes—
448C22×C4C22×D414yes—
449C22×C4C22×D412yes—
450C22×C4C22×D412yes—
451C22×C4C22×D414yes—
452C22×C4C22×Q814yes—
453C22×C4C22×Q814yes—
454C22×C4C22×Q814yes—
455C22×C4C2×#4114yes—
456C22×C4C2×#4112yes—
457C22×C4C2×#4112yes—
458C22×C4C2×#4114yes—
459C22×C4C2×#4114yes—
460C22×C4H(5,𝔽2)14yes—
461C22×C42−1+414yes—
462C4⋊C4—232no—
463C4⋊C4—28no—
464C4⋊C4—216no—
465C4⋊C4—232no—
466C4⋊C4—232no—
467C4⋊C4—48no—
468C4⋊C4—416no—
469C4⋊C4—82no—
470C4⋊C4—82no—
471C4⋊C4—84no—
472C4⋊C4—84no—
473C4⋊C4—84no—
474C22⋊C4—216no—
475C22⋊C4—216no—
476C22⋊C4—48no—
477C22⋊C4—48no—
478C42—216no—
479C42—216no—
480C42—416no—
481C42—416no—
482C42—416no—
483C42—84no—
484C42—84no—
485C23—24no—
486C23—24no—
487C23—24no—
488C23—24no—
489C23—24no—
490C23—24no—
491C23—22no—
492C23—22no—
493C23—22no—
494C23—22no—
495C23—24no—
496C23—24no—
497C23—24no—
498C23—24no—
499C23—24no—
500C23—24no—
501C23C2×C4×D414yes—
502C23C2×C4×Q814yes—
503C23C4×#4114yes—
504C23C2×C22⋊Q814yes—
505C23#52514yes—
506C23#52914yes—
507C23#53314yes—
508C23C24×C411yescharacteristic
509C23C23×D411yescharacteristic
510C23C23×Q811yescharacteristic
511C23C22×#4111yescharacteristic
512C23C22×#4111yescharacteristic
513C23C22×#4111yescharacteristic
514C23C22×#4111yescharacteristic
515C23C2×H(5,𝔽2)12yes—
516C23C2×2−1+412yes—
517C23#58512yes—
518C23#58512yes—
519Q8—88no—
520C2×C4—232no—
521C2×C4—24no—
522C2×C4—24no—
523C2×C4—24no—
524C2×C4—24no—
525C2×C4—24no—
526C2×C4—24no—
527C2×C4—24no—
528C2×C4—24no—
529C2×C4—24no—
530C2×C4—24no—
531C2×C4—28no—
532C2×C4—24no—
533C2×C4—24no—
534C2×C4—24no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—24no—
540C2×C4—24no—
541C2×C4—24no—
542C2×C4—24no—
543C2×C4—24no—
544C2×C4—28no—
545C2×C4—232no—
546C2×C4—24no—
547C2×C4—24no—
548C2×C4—28no—
549C2×C4—28no—
550C2×C4—28no—
551C2×C4—28no—
552C2×C4—28no—
553C2×C4—24no—
554C2×C4—24no—
555C2×C4—24no—
556C2×C4—28no—
557C2×C4—24no—
558C2×C4—24no—
559C2×C4—28no—
560C2×C4—416no—
561C2×C4—416no—
562C2×C4—416no—
563C2×C4—416no—
564C2×C4—416no—
565C2×C4—44no—
566C2×C4—44no—
567C2×C4—44no—
568C2×C4—44no—
569C2×C4—44no—
570C2×C4—44no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—44no—
580C2×C4—44no—
581C2×C4—44no—
582C2×C4—44no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—44no—
586C2×C4—44no—
587C2×C4—44no—
588C2×C4—44no—
589C2×C4—44no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—44no—
595C2×C4—44no—
596C2×C4—44no—
597C2×C4—44no—
598C2×C4—44no—
599C2×C4—44no—
600C2×C4C2×C4×D414yes—
601C2×C4C2×C4×D414yes—
602C2×C4C2×C4×Q818yes—
603C2×C4#52018yes—
604C2×C4C2×C22⋊Q818yes—
605C22—24no—
606C22—28no—
607C22—24no—
608C22—24no—
609C22—24no—
610C22—28no—
611C22—24no—
612C22—24no—
613C22—28no—
614C22—28no—
615C22C4×C22⋊Q818yes—
616C22C22×C4×D411yescharacteristic
617C22C22×C4×Q811yescharacteristic
618C22C2×C4×#4111yescharacteristic
619C22C2×#52012yes—
620C22#342812yes—
621C22C22×C22⋊Q811yescharacteristic
622C22C2×#52511yescharacteristic
623C22C2×#52911yescharacteristic
624C22C2×#53311yescharacteristic
625C22#345912yes—
626C22#345912yes—
627C22#346112yes—
628C22#346112yes—
629C22#348411yescharacteristic
630C22#348511yescharacteristic
631C22#348511yescharacteristic
632C22#348611yescharacteristic
633C22#349112yes—
634C22#349212yes—
635C22#350112yes—
636C22#350212yes—
637C22#350612yes—
638C22#350712yes—
639C4—24no—
640C4—24no—
641C4—28no—
642C4—28no—
643C4—28no—
644C4—48no—
645C4—44no—
646C4—44no—
647C4—44no—
648C4—44no—
649C4—44no—
650C4—82no—
651C4—82no—
652C4—84no—
653C2—28no—
654C2C2×C4×C22⋊Q811yescharacteristic
655C2#3408312yes—
656C2#3419911yescharacteristic
657C2#3420011yescharacteristic
658C2#3421312yes—
659C2#3498511yescharacteristic
660C2#3498611yescharacteristic
661C2#3501312yes—
662C2#3504212yes—
663C2#3509211yescharacteristic
664C2#3515411yescharacteristic
subgroupG/H
center, Frattini, socle, cosocleC24C25
commutatorC23C24×C4
prime radicalC25C24
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
9. #14120C2semidirect
10. #14111C2semidirect
11. #14059C2semidirect
12. #14035C2semidirect
22. #2855C4semidirect
94. C22×C4⋊C4Q8semidirect
126. #391C2×C4semidirect
128. #389C2×C4semidirect
132. #387C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#28552
C2×#23292
C22×C4×C4⋊C42
#146952
#146852
#142812
#142752
#142632
#141202
#141112
#140592
#140352
#137782
#137762
C4×#3852