Finite Group Databasebeta

← #695790

Subgroups — #695790

Order 512.

3712 conjugacy classes of subgroups.

The subgroup lattice has 804 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2504C211yesmaximal, characteristic, complemented
2C2×#2615C211yesmaximal, characteristic
3C2×#2344C211yesmaximal, characteristic, complemented
4C2×#2311C211yesmaximal, characteristic, complemented
5#16660C214yesmaximal, complemented
6#15737C214yesmaximal, complemented
7#15640C214yesmaximal, complemented
8#15220C212yesmaximal, complemented
9#15126C212yesmaximal, complemented
10#14886C214yesmaximal, complemented
11#14136C211yesmaximal, characteristic, complemented
12#14129C211yesmaximal, characteristic, complemented
13#13998C212yesmaximal, complemented
14#13875C212yesmaximal, complemented
15#13851C211yesmaximal, characteristic, complemented
16C23×#41C2211yescharacteristic
17C22×#121C2211yescharacteristic
18C22×#121C2212yes—
19C22×#117C2212yes—
20C22×#117C2211yescharacteristic
21C23×C4⋊C4C2211yescharacteristic
22#2615—28no—
23#2504—28no—
24C2×#397C2214yes—
25C2×#394C2214yes—
26C2×#393C2212yes—
27C2×#392C2214yes—
28C2×#392C2212yes—
29#2344—28no—
30#2311—28no—
31C2×#391C2212yes—
32C2×#390C2211yescharacteristic
33C2×#390C2212yes—
34C2×#390C2211yescharacteristic
35C2×#390C2212yes—
36C2×#390C2214yes—
37C2×#389C2212yes—
38C2×#389C2212yes—
39C2×#387C2214yes—
40C2×#386C2214yes—
41C2×#386C2214yes—
42C2×#382C2211yescharacteristic
43C2×#382C2211yescharacteristic
44C2×#381C2212yes—
45C2×#380C2212yes—
46C2×#380C2212yes—
47C2×#380C2212yes—
48C2×#380C2211yescharacteristic
49C2×C4×C4⋊C4C2212yes—
50C2×C4×C22⋊C4C2212yes—
51C2×C4×C22⋊C4C2211yescharacteristic
52C2×C4×C22⋊C4C2211yescharacteristic
53C2×C4×C22⋊C4C2212yes—
54#1446C2212yes—
55#1443C2214yes—
56#1443C2214yes—
57#1441C2214yes—
58#1440C2214yes—
59#1440C2214yes—
60#1439C2214yes—
61#1439C2214yes—
62#1439C2214yes—
63#1439C2214yes—
64#1439C2214yes—
65#1438C2212yes—
66#1438C2212yes—
67#1438C2214yes—
68#1438C2214yes—
69#1438C2212yes—
70#1438C2212yes—
71#1438C2212yes—
72#1437C2214yes—
73#1437C2214yes—
74#1437C2214yes—
75#1436C2214yes—
76C4⋊#95C2212yes—
77C4⋊#95C2212yes—
78#1434C2214yes—
79C22×#41—24no—
80C22×#41—24no—
81C22×#41—28no—
82C22×#41—24no—
83C22×#41—44no—
84C23×Q8C2311yescharacteristic
85C23×D4C2311yescharacteristic
86C23×D4C2312yes—
87C23×D4C2311yescharacteristic
88C23×D4C2312yes—
89C23×D4C2314yes—
90C24×C4C2312yes—
91C24×C4C2311yescharacteristic
92C24×C4C2312yes—
93C2×#121—24no—
94C2×#121—24no—
95C2×#121—28no—
96C2×#121—24no—
97C2×#121—28no—
98C2×#121—44no—
99C2×#117—24no—
100C2×#117—28no—
101C2×#117—48no—
102C2×#117—44no—
103C22×C4⋊C4—24no—
104C22×C4⋊C4—24no—
105C22×C4⋊C4D418yescomplemented
106C22×C4⋊C4C2312yes—
107C22×C4⋊C4C2311yescharacteristic
108C22×C4⋊C4C2314yes—
109C22×C4⋊C4C2312yes—
110C22×C4⋊C4C2311yescharacteristic
111C22×C4⋊C4C2311yescharacteristic
112C22×C4⋊C4C2312yes—
113C22×C4⋊C4C2314yes—
114C22×C22⋊C4C2312yes—
115C22×C22⋊C4C2311yescharacteristic
116C22×C22⋊C4C2314yes—
117C22×C22⋊C4C2312yes—
118C22×C22⋊C4C2314yes—
119C22×C22⋊C4C2314yes—
120C22×C22⋊C4C2314yes—
121C22×C22⋊C4C2314yes—
122C22×C22⋊C4C2314yes—
123C22×C22⋊C4C2312yes—
124C22×C22⋊C4C2314yes—
125C22×C22⋊C4C2312yes—
126C22×C22⋊C4C2311yescharacteristic
127C22×C22⋊C4C2312yes—
128C22×C22⋊C4C2312yes—
129C22×C22⋊C4C2314yes—
130C22×C22⋊C4C2312yes—
131C22×C22⋊C4C2312yes—
132C22×C42C2311yescharacteristic
133C22×C42C2312yes—
134C22×C42C2311yescharacteristic
135C22×C42C2314yes—
136C22×C42C2312yes—
137#397—216no—
138#394—48no—
139#393—28no—
140#392—216no—
141#392—28no—
142#391—28no—
143#390—28no—
144#390—24no—
145#390—24no—
146#390—48no—
147#390—44no—
148#389—28no—
149#389—44no—
150#387—216no—
151#386—216no—
152#386—216no—
153#382—24no—
154#382—24no—
155#381—28no—
156#380—28no—
157#380—28no—
158#380—28no—
159#380D418yes—
160C4×C4⋊C4—28no—
161C4×C22⋊C4—24no—
162C4×C22⋊C4—24no—
163C4×C22⋊C4—28no—
164C4×C22⋊C4—44no—
165C2×#95C2314yes—
166C2×#95C2312yes—
167C2×#95C2314yes—
168C2×#95C2314yes—
169C2×#95C2312yes—
170C2×#95C2314yes—
171C2×#95C2312yes—
172C2×#95C2314yes—
173C2×#95C2314yes—
174C2×#95C2312yes—
175C2×#95C2314yes—
176C2×#95C2314yes—
177C2×#95C2314yes—
178C2×#95C2314yes—
179C2×#95C2314yes—
180C2×#95C2314yes—
181C2×#95C2312yes—
182C2×#95C2312yes—
183C2×#95C2312yes—
184C25C2412yes—
185C25C2412yes—
186C25C2411yescharacteristic
187C2×#41—232no—
188C2×#41—432no—
189C2×#41—432no—
190C2×#41—416no—
191C2×#41—416no—
192C22×Q8—22no—
193C22×Q8—22no—
194C22×Q8—24no—
195C22×Q8—22no—
196C22×Q8—42no—
197C22×D4—24no—
198C22×D4—24no—
199C22×D4—24no—
200C22×D4—24no—
201C22×D4—24no—
202C22×D4—24no—
203C22×D4—28no—
204C22×D4—24no—
205C22×D4—24no—
206C22×D4—22no—
207C22×D4—22no—
208C22×D4—22no—
209C22×D4—22no—
210C22×D4—28no—
211C22×D4—24no—
212C22×D4—22no—
213C22×D4—24no—
214C22×D4—22no—
215C22×D4—24no—
216C22×D4—42no—
217C22×D4—42no—
218C22×D4—42no—
219C22×D4—44no—
220C22×D4—44no—
221C22×D4—44no—
222C22×D4—44no—
223C22×D4—44no—
224C22×D4—44no—
225C22×D4—44no—
226C23×C4—24no—
227C23×C4—24no—
228C23×C4—24no—
229C23×C4—24no—
230C23×C4—24no—
231C23×C4—24no—
232C23×C4—24no—
233C23×C4—24no—
234C23×C4—24no—
235C23×C4—24no—
236C23×C4—44no—
237C23×C4—44no—
238C23×C4C2×D414yes—
239C23×C4C2×D414yes—
240C23×C4C2×D418yes—
241C23×C4#4118yes—
242C23×C4#4114yes—
243C23×C4C2412yes—
244C23×C4C2414yes—
245C23×C4C2412yes—
246C23×C4C2412yes—
247C23×C4C2412yes—
248C23×C4C2412yes—
249C23×C4C2412yes—
250C23×C4C2414yes—
251C23×C4C2412yes—
252C23×C4C2412yes—
253C23×C4C2411yescharacteristic
254C23×C4C2411yescharacteristic
255#121—432no—
256#121—416no—
257#117—432no—
258#117—416no—
259C2×C4⋊C4—232no—
260C2×C4⋊C4—22no—
261C2×C4⋊C4—24no—
262C2×C4⋊C4—22no—
263C2×C4⋊C4—24no—
264C2×C4⋊C4—22no—
265C2×C4⋊C4—22no—
266C2×C4⋊C4—24no—
267C2×C4⋊C4—24no—
268C2×C4⋊C4—24no—
269C2×C4⋊C4—28no—
270C2×C4⋊C4—22no—
271C2×C4⋊C4—22no—
272C2×C4⋊C4—22no—
273C2×C4⋊C4—24no—
274C2×C4⋊C4—24no—
275C2×C4⋊C4—28no—
276C2×C4⋊C4—44no—
277C2×C4⋊C4—44no—
278C2×C4⋊C4—44no—
279C2×C4⋊C4—44no—
280C2×C4⋊C4C2×D414yes—
281C2×C4⋊C4C2×D414yes—
282C2×C4⋊C4C2×D418yes—
283C2×C4⋊C4C2×D418yes—
284C2×C22⋊C4—22no—
285C2×C22⋊C4—24no—
286C2×C22⋊C4—24no—
287C2×C22⋊C4—22no—
288C2×C22⋊C4—24no—
289C2×C22⋊C4—24no—
290C2×C22⋊C4—24no—
291C2×C22⋊C4—28no—
292C2×C22⋊C4—24no—
293C2×C22⋊C4—24no—
294C2×C22⋊C4—28no—
295C2×C22⋊C4—24no—
296C2×C22⋊C4—24no—
297C2×C22⋊C4—24no—
298C2×C22⋊C4—28no—
299C2×C22⋊C4—28no—
300C2×C22⋊C4—28no—
301C2×C22⋊C4—28no—
302C2×C22⋊C4—28no—
303C2×C22⋊C4—28no—
304C2×C22⋊C4—42no—
305C2×C22⋊C4—42no—
306C2×C22⋊C4—42no—
307C2×C22⋊C4—42no—
308C2×C22⋊C4—42no—
309C2×C22⋊C4—42no—
310C2×C22⋊C4—42no—
311C2×C22⋊C4—42no—
312C2×C22⋊C4—42no—
313C2×C22⋊C4—44no—
314C2×C22⋊C4—44no—
315C2×C22⋊C4—44no—
316C2×C22⋊C4—42no—
317C2×C22⋊C4—42no—
318C2×C22⋊C4—44no—
319C2×C22⋊C4—44no—
320C2×C22⋊C4—44no—
321C2×C22⋊C4—44no—
322C2×C22⋊C4—44no—
323C2×C22⋊C4—44no—
324C2×C22⋊C4—44no—
325C2×C22⋊C4—44no—
326C2×C22⋊C4—44no—
327C2×C22⋊C4—44no—
328C2×C22⋊C4—44no—
329C2×C22⋊C4—44no—
330C2×C22⋊C4—44no—
331C2×C22⋊C4—44no—
332C2×C22⋊C4—44no—
333C2×C22⋊C4C2×D418yes—
334C2×C42—28no—
335C2×C42—24no—
336C2×C42—24no—
337C2×C42—22no—
338C2×C42—22no—
339C2×C42—24no—
340C2×C42—24no—
341C2×C42—28no—
342C2×C42—24no—
343C2×C42—24no—
344C2×C42—44no—
345C2×C42—44no—
346#95—24no—
347#95—24no—
348#95—28no—
349#95—28no—
350#95—28no—
351#95—28no—
352#95—28no—
353#95—28no—
354#95—28no—
355#95—24no—
356#95—24no—
357#95—44no—
358#95—44no—
359#95—44no—
360#95—42no—
361#95—42no—
362#95—44no—
363#95—44no—
364#95C2×D418yes—
365C24—22no—
366C24—22no—
367C24—22no—
368C24—22no—
369C24—22no—
370C24—22no—
371C24—22no—
372C24—22no—
373C24—22no—
374C24—22no—
375C24—22no—
376C24—22no—
377C24—22no—
378C24—22no—
379C24—22no—
380C24—22no—
381C24—22no—
382C24—22no—
383C24—22no—
384C24—22no—
385C24—22no—
386C24—22no—
387C24—22no—
388C24—24no—
389C24—24no—
390C24—24no—
391C24—24no—
392C24—24no—
393C24—24no—
394C24C22×D412yes—
395C24C22×D412yes—
396C24C2×#4114yes—
397C24C2×#4112yes—
398C24C2511yescharacteristic
399#41—4128no—
400C2×Q8—28no—
401C2×Q8—48no—
402C2×Q8—48no—
403C2×Q8—44no—
404C2×Q8—44no—
405C2×D4—28no—
406C2×D4—216no—
407C2×D4—48no—
408C2×D4—48no—
409C2×D4—48no—
410C2×D4—48no—
411C2×D4—44no—
412C2×D4—416no—
413C2×D4—48no—
414C2×D4—48no—
415C2×D4—48no—
416C2×D4—48no—
417C2×D4—416no—
418C2×D4—416no—
419C2×D4—48no—
420C2×D4—44no—
421C2×D4—44no—
422C2×D4—44no—
423C2×D4—44no—
424C2×D4—48no—
425C2×D4—48no—
426C2×D4—88no—
427C2×D4—88no—
428C2×D4—88no—
429C2×D4—88no—
430C2×D4—84no—
431C2×D4—84no—
432C2×D4—88no—
433C2×D4—88no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—28no—
437C22×C4—28no—
438C22×C4—216no—
439C22×C4—216no—
440C22×C4—216no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—24no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—24no—
455C22×C4—24no—
456C22×C4—24no—
457C22×C4—24no—
458C22×C4—22no—
459C22×C4—24no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—24no—
463C22×C4—22no—
464C22×C4—24no—
465C22×C4—24no—
466C22×C4—24no—
467C22×C4—216no—
468C22×C4—216no—
469C22×C4—216no—
470C22×C4—22no—
471C22×C4—22no—
472C22×C4—22no—
473C22×C4—22no—
474C22×C4—24no—
475C22×C4—22no—
476C22×C4—22no—
477C22×C4—22no—
478C22×C4—22no—
479C22×C4—22no—
480C22×C4—22no—
481C22×C4—22no—
482C22×C4—22no—
483C22×C4—22no—
484C22×C4—24no—
485C22×C4—24no—
486C22×C4—24no—
487C22×C4—22no—
488C22×C4—22no—
489C22×C4—24no—
490C22×C4—22no—
491C22×C4—24no—
492C22×C4—24no—
493C22×C4—24no—
494C22×C4—22no—
495C22×C4—22no—
496C22×C4—22no—
497C22×C4—22no—
498C22×C4—22no—
499C22×C4—22no—
500C22×C4—22no—
501C22×C4—24no—
502C22×C4—22no—
503C22×C4—416no—
504C22×C4—48no—
505C22×C4—416no—
506C22×C4—48no—
507C22×C4—48no—
508C22×C4—48no—
509C22×C4—48no—
510C22×C4—48no—
511C22×C4#121116yes—
512C22×C4C22×D414yes—
513C22×C4C22×D414yes—
514C22×C4C22×D418yes—
515C22×C4C22×D412yes—
516C22×C4C22×D412yes—
517C22×C4C22×D412yes—
518C22×C4C22×D412yes—
519C22×C4C2×#4114yes—
520C22×C4C2×#4112yes—
521C22×C4C2×#4112yes—
522C22×C4C2×#4114yes—
523C22×C4H(5,𝔽2)14yes—
524C22×C4H(5,𝔽2)14yes—
525C4⋊C4—216no—
526C4⋊C4—28no—
527C4⋊C4—28no—
528C4⋊C4—216no—
529C4⋊C4—416no—
530C4⋊C4—44no—
531C4⋊C4—48no—
532C4⋊C4—88no—
533C22⋊C4—28no—
534C22⋊C4—48no—
535C22⋊C4—416no—
536C22⋊C4—48no—
537C22⋊C4—48no—
538C22⋊C4—44no—
539C22⋊C4—48no—
540C22⋊C4—48no—
541C22⋊C4—48no—
542C22⋊C4—48no—
543C22⋊C4—48no—
544C22⋊C4—88no—
545C22⋊C4—88no—
546C22⋊C4—88no—
547C22⋊C4—88no—
548C22⋊C4—88no—
549C22⋊C4—88no—
550C22⋊C4—88no—
551C42—416no—
552C42—48no—
553C42—44no—
554C42—44no—
555C42—48no—
556C23—22no—
557C23—22no—
558C23—24no—
559C23—24no—
560C23—24no—
561C23—22no—
562C23—22no—
563C23—24no—
564C23—24no—
565C23—24no—
566C23—22no—
567C23—22no—
568C23—24no—
569C23—24no—
570C23—24no—
571C23—24no—
572C23—24no—
573C23—24no—
574C23—24no—
575C23—42no—
576C23—42no—
577C23—42no—
578C23—42no—
579C23—42no—
580C23—42no—
581C23—42no—
582C23—42no—
583C23—42no—
584C23—42no—
585C23—42no—
586C23—42no—
587C23—42no—
588C23—42no—
589C23—42no—
590C23—42no—
591C23—42no—
592C23—42no—
593C23—42no—
594C23—42no—
595C23—42no—
596C23—42no—
597C23—42no—
598C23—42no—
599C23—42no—
600C23—42no—
601C23—42no—
602C23—42no—
603C23—42no—
604C23—42no—
605C23—42no—
606C23—42no—
607C23—42no—
608C23—42no—
609C23—42no—
610C23—42no—
611C23—42no—
612C23—42no—
613C23—42no—
614C23—44no—
615C23—44no—
616C23—44no—
617C23—42no—
618C23—44no—
619C23—42no—
620C23—44no—
621C23—44no—
622C23—44no—
623C23—44no—
624C23—44no—
625C23—42no—
626C23—42no—
627C23—44no—
628C23—44no—
629C23—44no—
630C23—44no—
631C23—44no—
632C23C2×#12114yes—
633C23C23×D411yescharacteristic
634C23C23×D411yescharacteristic
635C23C22×#4111yescharacteristic
636C23C22×#4112yes—
637C23C2×H(5,𝔽2)12yes—
638C23C2×H(5,𝔽2)12yes—
639C23#58512yes—
640C23#58512yes—
641C23#58512yes—
642Q8—416no—
643D4—416no—
644D4—432no—
645D4—832no—
646D4—816no—
647D4—88no—
648C2×C4—22no—
649C2×C4—22no—
650C2×C4—24no—
651C2×C4—24no—
652C2×C4—232no—
653C2×C4—24no—
654C2×C4—24no—
655C2×C4—24no—
656C2×C4—24no—
657C2×C4—28no—
658C2×C4—24no—
659C2×C4—24no—
660C2×C4—28no—
661C2×C4—24no—
662C2×C4—28no—
663C2×C4—28no—
664C2×C4—28no—
665C2×C4—24no—
666C2×C4—432no—
667C2×C4—42no—
668C2×C4—42no—
669C2×C4—44no—
670C2×C4—44no—
671C2×C4—42no—
672C2×C4—42no—
673C2×C4—42no—
674C2×C4—42no—
675C2×C4—42no—
676C2×C4—42no—
677C2×C4—42no—
678C2×C4—44no—
679C2×C4—42no—
680C2×C4—42no—
681C2×C4—44no—
682C2×C4—44no—
683C2×C4—44no—
684C2×C4—42no—
685C2×C4—432no—
686C2×C4—42no—
687C2×C4—42no—
688C2×C4—44no—
689C2×C4—42no—
690C2×C4—42no—
691C2×C4—42no—
692C2×C4—42no—
693C2×C4—42no—
694C2×C4—44no—
695C2×C4—44no—
696C2×C4—42no—
697C2×C4—42no—
698C2×C4—42no—
699C2×C4—42no—
700C2×C4—42no—
701C2×C4—44no—
702C2×C4—44no—
703C2×C4—42no—
704C2×C4—42no—
705C2×C4—42no—
706C2×C4—42no—
707C2×C4—42no—
708C2×C4—42no—
709C2×C4—42no—
710C2×C4—42no—
711C2×C4—44no—
712C2×C4—42no—
713C2×C4—44no—
714C2×C4—42no—
715C2×C4—44no—
716C2×C4—42no—
717C2×C4—44no—
718C2×C4—42no—
719C2×C4C2×#12114yes—
720C2×C4C2×#12114yes—
721C2×C4#52514yes—
722C2×C4#53514yes—
723C2×C4#54618yes—
724C2×C4#55118yes—
725C22—24no—
726C22—24no—
727C22—24no—
728C22—44no—
729C22—44no—
730C22—44no—
731C22—44no—
732C22—44no—
733C22—44no—
734C22—44no—
735C22—44no—
736C22—44no—
737C22—44no—
738C22—44no—
739C22—44no—
740C22—44no—
741C22—82no—
742C22—82no—
743C22—82no—
744C22—82no—
745C22—82no—
746C22—82no—
747C22—82no—
748C22—82no—
749C22—84no—
750C22—84no—
751C22—82no—
752C22—82no—
753C22—84no—
754C22—84no—
755C22—82no—
756C22—84no—
757C22—84no—
758C22C22×#12111yescharacteristic
759C22C2×#52511yescharacteristic
760C22C2×#53511yescharacteristic
761C22#346012yes—
762C22C2×#54612yes—
763C22C2×#55112yes—
764C22#348411yescharacteristic
765C22#348712yes—
766C22#348711yescharacteristic
767C22#349312yes—
768C22#349412yes—
769C22#349612yes—
770C22#350012yes—
771C22#350612yes—
772C22#351011yescharacteristic
773C22#351011yescharacteristic
774C22#351212yes—
775C22#353912yes—
776C22#354212yes—
777C22#354612yes—
778C22#355112yes—
779C4—24no—
780C4—44no—
781C4—48no—
782C4—44no—
783C4—44no—
784C4—44no—
785C4—82no—
786C4—82no—
787C4—82no—
788C4—84no—
789C4—82no—
790C4#241118yes—
791C2—44no—
792C2—84no—
793C2—162no—
794C2C2×#241111yescharacteristic
795C2#3516811yescharacteristic
796C2#3567711yescharacteristic
797C2#3575212yes—
798C2#3797512yes—
799C2#3956712yes—
800C2#4243312yes—
801C2#4487112yes—
802C2#4523612yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC2×#2615C2
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×#2504C2semidirect
3. C2×#2344C2semidirect
4. C2×#2311C2semidirect
5. #16660C2semidirect
6. #15737C2semidirect
7. #15640C2semidirect
8. #15220C2semidirect
9. #15126C2semidirect
10. #14886C2semidirect
11. #14136C2semidirect
12. #14129C2semidirect
13. #13998C2semidirect
14. #13875C2semidirect
15. #13851C2semidirect
105. C22×C4⋊C4D4semidirect

Maximal subgroups

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

subgroupindex
C2×#25042
C2×#26152
C2×#23442
C2×#23112
#166602
#157372
#156402
#152202
#151262
#148862
#141362
#141292
#139982
#138752
#138512