Finite Group Databasebeta

← #5956146

Subgroups — #5956146

Order 512.

4122 conjugacy classes of subgroups.

The subgroup lattice has 708 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2317C211yesmaximal, characteristic, complemented
2C2×#2311C211yesmaximal, characteristic
3C2×#2311C211yesmaximal, characteristic
4C2×#2297C211yesmaximal, characteristic
5C2×#2296C211yesmaximal, characteristic, complemented
6C22×#389C211yesmaximal, characteristic, complemented
7#15223C214yesmaximal, complemented
8#15193C214yesmaximal, complemented
9#15148C214yesmaximal, complemented
10#15135C214yesmaximal, complemented
11#14946C214yesmaximal, complemented
12#14777C214yesmaximal, complemented
13#13795C211yesmaximal, characteristic, complemented
14C23×#41C2211yescharacteristic
15C22×#117C2211yescharacteristic
16C22×#117C2211yescharacteristic
17C22×#117C2211yescharacteristic
18C23×C4⋊C4C2211yescharacteristic
19C23×C4⋊C4C2211yescharacteristic
20C23×C4⋊C4C2212yes—
21C23×C42C2211yescharacteristic
22C2×#399C2218yes—
23#2317—28no—
24#2311—28no—
25#2311—28no—
26#2297—28no—
27#2296—28no—
28C2×#391C2212yes—
29C2×#391C2212yes—
30C2×#391C2212yes—
31C2×#390C2212yes—
32C2×#390C2212yes—
33C2×#390C2212yes—
34C2×#389—28no—
35C2×#389C2214yes—
36C2×#389C2212yes—
37C2×#389C2214yes—
38C2×#389C2214yes—
39C2×#389C2212yes—
40C2×#386C2212yes—
41C2×#386C2214yes—
42C2×#386C2214yes—
43C2×#386C2212yes—
44C2×#385C2214yes—
45C2×#385C2212yes—
46C2×#385C2212yes—
47C2×#384C2212yes—
48C2×#382C2212yes—
49C2×#380C2212yes—
50C2×#380C2214yes—
51C2×#380C2212yes—
52C2×C4×C4⋊C4C2214yes—
53C2×C4×C22⋊C4C2214yes—
54C22×#95C2212yes—
55#1445C2218yes—
56#1445C2218yes—
57#1445C2218yes—
58#1444C2218yes—
59#1439C2218yes—
60#1439C2218yes—
61#1439C2218yes—
62C4⋊#95C2214yes—
63C4⋊#95C2214yes—
64C22⋊#95C2214yes—
65C22×#41—24no—
66C22×#41—24no—
67C22×#41—24no—
68C22×#41—24no—
69C22×#41—24no—
70C22×#41—24no—
71C22×#41Q818yes—
72C23×Q8C2311yescharacteristic
73C23×D4C2312yes—
74C23×D4C2311yescharacteristic
75C24×C4C2312yes—
76C24×C4C2311yescharacteristic
77C24×C4C2311yescharacteristic
78C24×C4C2312yes—
79C24×C4C2312yes—
80C24×C4C2311yescharacteristic
81C2×#117—24no—
82C2×#117—24no—
83C2×#117—24no—
84C2×#117—24no—
85C2×#117—24no—
86C2×#117—24no—
87C2×#117—24no—
88C2×#117D418yes—
89C2×#117D418yes—
90C22×C4⋊C4—24no—
91C22×C4⋊C4—24no—
92C22×C4⋊C4—24no—
93C22×C4⋊C4—24no—
94C22×C4⋊C4—28no—
95C22×C4⋊C4—24no—
96C22×C4⋊C4—48no—
97C22×C4⋊C4D418yes—
98C22×C4⋊C4C2312yes—
99C22×C4⋊C4C2312yes—
100C22×C4⋊C4C2311yescharacteristic
101C22×C4⋊C4C2311yescharacteristic
102C22×C4⋊C4C2314yes—
103C22×C4⋊C4C2318yes—
104C22×C4⋊C4C2314yes—
105C22×C4⋊C4C2312yes—
106C22×C4⋊C4C2312yes—
107C22×C4⋊C4C2312yes—
108C22×C4⋊C4C2312yes—
109C22×C4⋊C4C2318yes—
110C22×C22⋊C4C2312yes—
111C22×C22⋊C4C2314yes—
112C22×C22⋊C4C2314yes—
113C22×C22⋊C4C2312yes—
114C22×C22⋊C4C2312yes—
115C22×C22⋊C4C2314yes—
116C22×C22⋊C4C2318yes—
117C22×C22⋊C4C2314yes—
118C22×C42—24no—
119C22×C42—24no—
120C22×C42—24no—
121C22×C42C2312yes—
122C22×C42C2312yes—
123C22×C42C2312yes—
124C22×C42C2312yes—
125C22×C42C2312yes—
126#399—416no—
127#391—28no—
128#391—28no—
129#391—28no—
130#390—28no—
131#390—28no—
132#390—28no—
133#389—216no—
134#389—216no—
135#389—216no—
136#389—28no—
137#389—28no—
138#386—216no—
139#386—216no—
140#386—28no—
141#386—28no—
142#385—216no—
143#385—28no—
144#385—28no—
145#384—28no—
146#382—28no—
147#380—216no—
148#380—28no—
149#380—28no—
150C4×C4⋊C4—216no—
151C4×C22⋊C4—216no—
152C2×#95—28no—
153C2×#95C2318yes—
154C2×#95C2314yes—
155C2×#95C2314yes—
156C2×#95C2318yes—
157C2×#95C2314yes—
158C2×#95C2314yes—
159C2×#95C2318yes—
160C2×#95C2314yes—
161C2×#95C2314yes—
162C2×#95C2318yes—
163C2×#95C2318yes—
164C25C2412yes—
165C25C2411yescharacteristic
166C2×#41—232no—
167C2×#41—232no—
168C2×#41—232no—
169C2×#41—416no—
170C2×#41—416no—
171C2×#41—416no—
172C2×#41—416no—
173C22×Q8—22no—
174C22×Q8—22no—
175C22×Q8—22no—
176C22×Q8—22no—
177C22×Q8—22no—
178C22×Q8—22no—
179C22×Q8C2×Q814yes—
180C22×D4—24no—
181C22×D4—24no—
182C22×D4—24no—
183C22×D4—24no—
184C22×D4—24no—
185C22×D4—24no—
186C22×D4—22no—
187C22×D4—22no—
188C22×D4—22no—
189C22×D4—22no—
190C22×D4—22no—
191C22×D4—22no—
192C22×D4C2×Q818yes—
193C22×D4C2×Q814yes—
194C23×C4—22no—
195C23×C4—22no—
196C23×C4—24no—
197C23×C4—24no—
198C23×C4—24no—
199C23×C4—24no—
200C23×C4—24no—
201C23×C4—24no—
202C23×C4—24no—
203C23×C4—24no—
204C23×C4—24no—
205C23×C4—24no—
206C23×C4—24no—
207C23×C4—22no—
208C23×C4—22no—
209C23×C4—22no—
210C23×C4—22no—
211C23×C4—24no—
212C23×C4—24no—
213C23×C4—44no—
214C23×C4—44no—
215C23×C4—42no—
216C23×C4—42no—
217C23×C4—44no—
218C23×C4—44no—
219C23×C4C2×D418yes—
220C23×C4C2×D414yes—
221C23×C4C2×D414yes—
222C23×C4C2×D414yes—
223C23×C4C2×Q818yes—
224C23×C4C2×Q814yes—
225C23×C4#4114yes—
226C23×C4#4114yes—
227C23×C4#4114yes—
228C23×C4C2412yes—
229C23×C4C2412yes—
230C23×C4C2414yes—
231C23×C4C2414yes—
232C23×C4C2414yes—
233C23×C4C2414yes—
234C23×C4C2412yes—
235C23×C4C2412yes—
236C23×C4C2412yes—
237C23×C4C2411yescharacteristic
238C23×C4C2411yescharacteristic
239#117—232no—
240#117—232no—
241#117—416no—
242C2×C4⋊C4—232no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—24no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—216no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—24no—
256C2×C4⋊C4—24no—
257C2×C4⋊C4—24no—
258C2×C4⋊C4—22no—
259C2×C4⋊C4—22no—
260C2×C4⋊C4—22no—
261C2×C4⋊C4—22no—
262C2×C4⋊C4—22no—
263C2×C4⋊C4—22no—
264C2×C4⋊C4—28no—
265C2×C4⋊C4—28no—
266C2×C4⋊C4—216no—
267C2×C4⋊C4—28no—
268C2×C4⋊C4—28no—
269C2×C4⋊C4—216no—
270C2×C4⋊C4—48no—
271C2×C4⋊C4—44no—
272C2×C4⋊C4—44no—
273C2×C4⋊C4—432no—
274C2×C4⋊C4—416no—
275C2×C4⋊C4—48no—
276C2×C4⋊C4—48no—
277C2×C4⋊C4C2×D418yes—
278C2×C4⋊C4C2×D418yes—
279C2×C4⋊C4C2×D418yes—
280C2×C4⋊C4C2×D418yes—
281C2×C22⋊C4—24no—
282C2×C22⋊C4—24no—
283C2×C22⋊C4—24no—
284C2×C22⋊C4—24no—
285C2×C22⋊C4—24no—
286C2×C22⋊C4—24no—
287C2×C22⋊C4—24no—
288C2×C22⋊C4—28no—
289C2×C22⋊C4—28no—
290C2×C22⋊C4—28no—
291C2×C22⋊C4—28no—
292C2×C22⋊C4—28no—
293C2×C22⋊C4—28no—
294C2×C22⋊C4—216no—
295C2×C22⋊C4—44no—
296C2×C22⋊C4—44no—
297C2×C22⋊C4—48no—
298C2×C22⋊C4—48no—
299C2×C22⋊C4—48no—
300C2×C22⋊C4—48no—
301C2×C22⋊C4C2×D418yes—
302C2×C22⋊C4C2×D418yes—
303C2×C42—24no—
304C2×C42—24no—
305C2×C42—24no—
306C2×C42—24no—
307C2×C42—24no—
308C2×C42—24no—
309C2×C42—24no—
310C2×C42—24no—
311C2×C42—24no—
312C2×C42—24no—
313C2×C42—24no—
314C2×C42—24no—
315C2×C42—24no—
316C2×C42—416no—
317C2×C42C2×D418yes—
318C2×C42C2×D418yes—
319#95—216no—
320#95—28no—
321#95—28no—
322#95—28no—
323#95—28no—
324#95—28no—
325#95—28no—
326#95—216no—
327#95—216no—
328#95—48no—
329#95—48no—
330C24—22no—
331C24—22no—
332C24—22no—
333C24—22no—
334C24—22no—
335C24—22no—
336C24—22no—
337C24—22no—
338C24—22no—
339C24—22no—
340C24—24no—
341C24—24no—
342C24—24no—
343C24—24no—
344C24C22×D412yes—
345C24C22×D412yes—
346C24C22×D412yes—
347C24C22×Q814yes—
348C24C22×Q812yes—
349C24C2×#4112yes—
350C24C2×#4112yes—
351C24C2×#4112yes—
352C24C2511yescharacteristic
353#41—4128no—
354C2×Q8—28no—
355C2×Q8—28no—
356C2×Q8—28no—
357C2×Q8—44no—
358C2×Q8—44no—
359C2×Q8—44no—
360C2×Q8—44no—
361C2×D4—216no—
362C2×D4—216no—
363C2×D4—216no—
364C2×D4—28no—
365C2×D4—28no—
366C2×D4—28no—
367C2×D4—48no—
368C2×D4—48no—
369C2×D4—48no—
370C2×D4—48no—
371C2×D4—44no—
372C2×D4—44no—
373C2×D4—44no—
374C2×D4—44no—
375C22×C4—28no—
376C22×C4—28no—
377C22×C4—216no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
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—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—24no—
397C22×C4—24no—
398C22×C4—28no—
399C22×C4—22no—
400C22×C4—24no—
401C22×C4—24no—
402C22×C4—22no—
403C22×C4—24no—
404C22×C4—24no—
405C22×C4—28no—
406C22×C4—24no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—24no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—216no—
413C22×C4—216no—
414C22×C4—216no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—24no—
421C22×C4—24no—
422C22×C4—24no—
423C22×C4—22no—
424C22×C4—24no—
425C22×C4—28no—
426C22×C4—24no—
427C22×C4—24no—
428C22×C4—28no—
429C22×C4—24no—
430C22×C4—48no—
431C22×C4—48no—
432C22×C4—48no—
433C22×C4—48no—
434C22×C4—48no—
435C22×C4—48no—
436C22×C4—416no—
437C22×C4—44no—
438C22×C4—44no—
439C22×C4—416no—
440C22×C4—44no—
441C22×C4—44no—
442C22×C4—48no—
443C22×C4—48no—
444C22×C4—48no—
445C22×C4—48no—
446C22×C4—48no—
447C22×C4—48no—
448C22×C4—48no—
449C22×C4—88no—
450C22×C4—88no—
451C22×C4C22⋊Q8116yes—
452C22×C4C22⋊Q8116yes—
453C22×C4C22⋊Q8116yes—
454C22×C4#123116yes—
455C22×C4#123116yes—
456C22×C4#123116yes—
457C22×C4#127116yes—
458C22×C4C22×D418yes—
459C22×C4C22×D418yes—
460C22×C4C22×D418yes—
461C22×C4C22×D412yes—
462C22×C4C22×D412yes—
463C22×C4C22×D412yes—
464C22×C4C22×D412yes—
465C22×C4C22×D412yes—
466C22×C4C22×D412yes—
467C22×C4C22×Q814yes—
468C22×C4C22×Q812yes—
469C22×C4C22×Q812yes—
470C22×C4C2×#4112yes—
471C22×C4C2×#4112yes—
472C22×C4C2×#4112yes—
473C22×C4C2×#4112yes—
474C22×C4C2×#4112yes—
475C22×C4C2×#4112yes—
476C22×C4H(5,𝔽2)14yes—
477C22×C42−1+414yes—
478C4⋊C4—216no—
479C4⋊C4—216no—
480C4⋊C4—216no—
481C4⋊C4—216no—
482C4⋊C4—416no—
483C4⋊C4—416no—
484C4⋊C4—432no—
485C4⋊C4—48no—
486C4⋊C4—48no—
487C4⋊C4—44no—
488C4⋊C4—44no—
489C4⋊C4—816no—
490C22⋊C4—216no—
491C22⋊C4—216no—
492C22⋊C4—416no—
493C22⋊C4—416no—
494C22⋊C4—48no—
495C22⋊C4—88no—
496C22⋊C4—88no—
497C22⋊C4—816no—
498C42—216no—
499C42—216no—
500C42—48no—
501C42—48no—
502C42—48no—
503C23—24no—
504C23—24no—
505C23—22no—
506C23—22no—
507C23—22no—
508C23—22no—
509C23—24no—
510C23—24no—
511C23—24no—
512C23—24no—
513C23—24no—
514C23—24no—
515C23—24no—
516C23—24no—
517C23—24no—
518C23—24no—
519C23—24no—
520C23—24no—
521C23—24no—
522C23—28no—
523C23—28no—
524C23—42no—
525C23—42no—
526C23—42no—
527C23—42no—
528C23—44no—
529C23—44no—
530C23—42no—
531C23—42no—
532C23—42no—
533C23—42no—
534C23—44no—
535C23—44no—
536C23—44no—
537C23—44no—
538C23—44no—
539C23—44no—
540C23—44no—
541C23—44no—
542C23C2×C22⋊Q814yes—
543C23C2×C22⋊Q814yes—
544C23C2×C22⋊Q814yes—
545C23C2×#12314yes—
546C23C2×#12314yes—
547C23C2×#12314yes—
548C23C2×#12714yes—
549C23C23×D411yescharacteristic
550C23C23×D411yescharacteristic
551C23C23×D411yescharacteristic
552C23C23×Q811yescharacteristic
553C23C22×#4111yescharacteristic
554C23C22×#4111yescharacteristic
555C23C22×#4111yescharacteristic
556C23C2×H(5,𝔽2)12yes—
557C23C2×2−1+412yes—
558C23#58512yes—
559C23#58512yes—
560Q8—416no—
561D4—432no—
562D4—416no—
563C2×C4—24no—
564C2×C4—24no—
565C2×C4—24no—
566C2×C4—28no—
567C2×C4—28no—
568C2×C4—28no—
569C2×C4—28no—
570C2×C4—24no—
571C2×C4—28no—
572C2×C4—24no—
573C2×C4—24no—
574C2×C4—28no—
575C2×C4—28no—
576C2×C4—24no—
577C2×C4—24no—
578C2×C4—28no—
579C2×C4—24no—
580C2×C4—28no—
581C2×C4—416no—
582C2×C4—432no—
583C2×C4—416no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—44no—
591C2×C4—48no—
592C2×C4—44no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—44no—
600C2×C4—48no—
601C2×C4—44no—
602C2×C4—44no—
603C2×C4—44no—
604C2×C4—44no—
605C2×C4—44no—
606C2×C4—432no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—44no—
610C2×C4—42no—
611C2×C4—44no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—44no—
616C2×C4—48no—
617C2×C4—44no—
618C2×C4—44no—
619C2×C4—48no—
620C2×C4—44no—
621C2×C4—44no—
622C2×C4—816no—
623C2×C4#399164yescomplemented
624C2×C4C2×C22⋊Q814yes—
625C2×C4C2×C22⋊Q814yes—
626C2×C4C2×C22⋊Q814yes—
627C2×C4C2×C22⋊Q814yes—
628C2×C4C2×C22⋊Q814yes—
629C2×C4C2×C22⋊Q814yes—
630C2×C4C2×#12314yes—
631C2×C4C2×#12314yes—
632C2×C4C2×#12314yes—
633C2×C4C2×#12314yes—
634C2×C4C2×#12314yes—
635C2×C4C2×#12314yes—
636C2×C4C2×#12714yes—
637C2×C4C2×#12714yes—
638C22—24no—
639C22—24no—
640C22—24no—
641C22—24no—
642C22—28no—
643C22—28no—
644C22—28no—
645C22—28no—
646C22—24no—
647C22—24no—
648C22—44no—
649C22—48no—
650C22—48no—
651C22—44no—
652C22—44no—
653C22—82no—
654C22—82no—
655C22—82no—
656C22—84no—
657C22—84no—
658C22—82no—
659C22C2×#39918yes—
660C22C22×C22⋊Q811yescharacteristic
661C22C22×C22⋊Q811yescharacteristic
662C22C22×C22⋊Q811yescharacteristic
663C22C22×#12311yescharacteristic
664C22C22×#12311yescharacteristic
665C22C22×#12311yescharacteristic
666C22C22×#12711yescharacteristic
667C22#346012yes—
668C22#346112yes—
669C22#348411yescharacteristic
670C22#348511yescharacteristic
671C22#348611yescharacteristic
672C22#348711yescharacteristic
673C22#348812yes—
674C22#348812yes—
675C22#348912yes—
676C22#348912yes—
677C22#350312yes—
678C22#350412yes—
679C22#350612yes—
680C22#350712yes—
681C22#351112yes—
682C22#351212yes—
683C4—44no—
684C4—48no—
685C4—48no—
686C4—48no—
687C4—44no—
688C4—44no—
689C4—82no—
690C4—82no—
691C4—84no—
692C4C2×#39918yes—
693C4C2×#39918yes—
694C2—28no—
695C2—84no—
696C2C22×#39911yescharacteristic
697C2#3515211yescharacteristic
698C2#3518211yescharacteristic
699C2#3525512yes—
700C2#3525712yes—
701C2#3551712yes—
702C2#3551912yes—
703C2#3556511yescharacteristic
704C2#3557411yescharacteristic
705C2#3563411yescharacteristic
706C2#3563911yescharacteristic
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC23×#41C22
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×#2317C2semidirect
5. C2×#2296C2semidirect
6. C22×#389C2semidirect
7. #15223C2semidirect
8. #15193C2semidirect
9. #15148C2semidirect
10. #15135C2semidirect
11. #14946C2semidirect
12. #14777C2semidirect
13. #13795C2semidirect
623. C2×C4#399semidirect

Maximal subgroups

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

subgroupindex
C2×#23172
C2×#23112
C2×#22972
C2×#22962
C22×#3892
#152232
#151932
#151482
#151352
#149462
#147772
#137952