Finite Group Databasebeta

← #613349

Subgroups — #613349

Order 512.

3401 conjugacy classes of subgroups.

The subgroup lattice has 707 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2853C211yesmaximal, characteristic
2C2×#2498C211yesmaximal, characteristic
3C2×#2296C211yesmaximal, characteristic, complemented
4C22×C4×C4⋊C4C211yesmaximal, characteristic, complemented
5#14870C212yesmaximal, complemented
6#14869C212yesmaximal, complemented
7#14821C212yesmaximal, complemented
8#14818C212yesmaximal, complemented
9#14808C212yesmaximal, complemented
10#14729C212yesmaximal
11#14697C212yesmaximal, complemented
12#14696C212yesmaximal
13#14203C214yesmaximal, complemented
14#14200C214yesmaximal, complemented
15#14135C211yesmaximal, characteristic
16#13958C211yesmaximal, characteristic, complemented
17#13878C211yesmaximal, characteristic, complemented
18C22×#123C2211yescharacteristic
19C22×#117C2211yescharacteristic
20C22×#117C2211yescharacteristic
21C23×C4⋊C4C2212yes—
22C23×C42C2211yescharacteristic
23C23×C42C2212yes—
24#2853—28no—
25#2498—28no—
26C2×#400C2211yescharacteristic
27C2×#399C2211yescharacteristic
28C2×#398C2214yes—
29C2×#398C2211yescharacteristic
30C2×#397C2211yescharacteristic
31C2×#396C2214yes—
32#2296—28no—
33C2×#389C2212yes—
34C2×#389C2212yes—
35C2×#389C2214yes—
36C2×#388C2212yes—
37C2×#388C2212yes—
38C2×#388C2212yes—
39C2×#388C2214yes—
40C2×#387C2212yes—
41C2×#387C2212yes—
42C2×#387C2212yes—
43C2×#386C2211yescharacteristic
44C2×#386C2211yescharacteristic
45C2×#386C2212yes—
46C2×#386C2212yes—
47C2×#385C2211yescharacteristic
48C2×#385C2211yescharacteristic
49C2×#385C2212yes—
50C2×#385C2212yes—
51C2×#381C2211yescharacteristic
52C2×#380C2212yes—
53C2×#380C2212yes—
54C2×C4×C4⋊C4—28no—
55C2×C4×C4⋊C4C2212yes—
56C2×C4×C4⋊C4C2212yes—
57C2×C4×C4⋊C4C2214yes—
58C2×C4×C4⋊C4C2214yes—
59C22×#95C2211yescharacteristic
60C22×#95C2211yescharacteristic
61#1445C2212yes—
62#1445C2212yes—
63#1444C2212yes—
64#1442C2212yes—
65#1441C2212yes—
66#1441C2214yes—
67#1441C2214yes—
68#1441C2214yes—
69#1441C2214yes—
70#1441C2214yes—
71#1439C2212yes—
72#1439C2214yes—
73#1437C2214yes—
74#1437C2212yes—
75#1436C2214yes—
76#1436C2214yes—
77#1436C2212yes—
78#1436C2214yes—
79#1436C2214yes—
80C4⋊#95C2214yes—
81#1434C2214yes—
82#1434C2214yes—
83C22⋊#95C2212yes—
84C22⋊#95C2212yes—
85C22⋊#95C2212yes—
86C22⋊#95C2212yes—
87C23×D4C2311yescharacteristic
88C24×C4C2311yescharacteristic
89C24×C4C2311yescharacteristic
90C24×C4C2312yes—
91C24×C4C2312yes—
92C24×C4C2311yescharacteristic
93C2×#123—24no—
94C2×#123—24no—
95C2×#123—24no—
96C2×#117—24no—
97C2×#117—24no—
98C2×#117—24no—
99C2×#117—24no—
100C2×#117—28no—
101C22×C4⋊C4—28no—
102C22×C4⋊C4—216no—
103C22×C4⋊C4C2314yes—
104C22×C4⋊C4C2311yescharacteristic
105C22×C4⋊C4C2312yes—
106C22×C4⋊C4C2312yes—
107C22×C4⋊C4C2311yescharacteristic
108C22×C4⋊C4C2311yescharacteristic
109C22×C4⋊C4C2314yes—
110C22×C4⋊C4C2312yes—
111C22×C4⋊C4C2311yescharacteristic
112C22×C4⋊C4C2314yes—
113C22×C4⋊C4C2312yes—
114C22×C4⋊C4C2314yes—
115C22×C4⋊C4C2314yes—
116C22×C4⋊C4C2312yes—
117C22×C22⋊C4C2312yes—
118C22×C22⋊C4C2312yes—
119C22×C22⋊C4C2311yescharacteristic
120C22×C22⋊C4C2312yes—
121C22×C22⋊C4C2311yescharacteristic
122C22×C22⋊C4C2311yescharacteristic
123C22×C22⋊C4C2311yescharacteristic
124C22×C22⋊C4C2311yescharacteristic
125C22×C22⋊C4C2312yes—
126C22×C22⋊C4C2314yes—
127C22×C22⋊C4C2312yes—
128C22×C22⋊C4C2312yes—
129C22×C42—24no—
130C22×C42—24no—
131C22×C42—28no—
132C22×C42—48no—
133C22×C42D418yes—
134C22×C42C2312yes—
135C22×C42C2312yes—
136C22×C42C2314yes—
137C22×C42C2314yes—
138C22×C42C2312yes—
139C22×C42C2312yes—
140#400—24no—
141#399—24no—
142#398—216no—
143#398—24no—
144#397—24no—
145#396—216no—
146#389—28no—
147#389—28no—
148#389—48no—
149#388—28no—
150#388—28no—
151#388—28no—
152#388—48no—
153#387—28no—
154#387—28no—
155#387—28no—
156#386—24no—
157#386—24no—
158#386—28no—
159#386—28no—
160#385—24no—
161#385—24no—
162#385—28no—
163#385—28no—
164#381—24no—
165#380—28no—
166#380—28no—
167C4×C4⋊C4—216no—
168C4×C4⋊C4—216no—
169C4×C4⋊C4—28no—
170C4×C4⋊C4—28no—
171C2×#95—24no—
172C2×#95D418yes—
173C2×#95C2314yes—
174C2×#95C2312yes—
175C2×#95C2314yes—
176C2×#95C2314yes—
177C2×#95C2314yes—
178C2×#95C2312yes—
179C2×#95C2312yes—
180C2×#95C2312yes—
181C2×#95C2312yes—
182C2×#95C2312yes—
183C2×#95C2312yes—
184C2×#95C2314yes—
185C2×#95C2314yes—
186C2×#95C2312yes—
187C2×#95C2314yes—
188C2×#95C2314yes—
189C2×#95C2312yes—
190C2×#95C2312yes—
191C2×#95C2312yes—
192C2×#95C2312yes—
193C2×#95C2314yes—
194C2×#95C2314yes—
195C2×#95C2314yes—
196C2×#95C2314yes—
197C2×#95C2311yescharacteristic
198C2×#95C2311yescharacteristic
199C2×#95C2312yes—
200C25C2411yescharacteristic
201C25C2411yescharacteristic
202C22×D4—22no—
203C22×D4—22no—
204C22×D4—22no—
205C22×D4—24no—
206C22×D4—24no—
207C23×C4—22no—
208C23×C4—22no—
209C23×C4—24no—
210C23×C4—22no—
211C23×C4—22no—
212C23×C4—24no—
213C23×C4—28no—
214C23×C4—24no—
215C23×C4—24no—
216C23×C4—24no—
217C23×C4—24no—
218C23×C4—24no—
219C23×C4—28no—
220C23×C4—28no—
221C23×C4—24no—
222C23×C4—24no—
223C23×C4—44no—
224C23×C4—44no—
225C23×C4C2×D414yes—
226C23×C4C2×D414yes—
227C23×C4C2×D418yes—
228C23×C4C2×D414yes—
229C23×C4C2×D414yes—
230C23×C4#4114yes—
231C23×C4C2411yescharacteristic
232C23×C4C2411yescharacteristic
233C23×C4C2412yes—
234C23×C4C2412yes—
235C23×C4C2412yes—
236C23×C4C2412yes—
237C23×C4C2411yescharacteristic
238C23×C4C2412yes—
239C23×C4C2414yes—
240C23×C4C2414yes—
241C23×C4C2411yescharacteristic
242C23×C4C2411yescharacteristic
243C23×C4C2412yes—
244C23×C4C2412yes—
245C23×C4C2412yes—
246#123—416no—
247#117—232no—
248#117—416no—
249C2×C4⋊C4—264no—
250C2×C4⋊C4—28no—
251C2×C4⋊C4—28no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—28no—
254C2×C4⋊C4—28no—
255C2×C4⋊C4—28no—
256C2×C4⋊C4—28no—
257C2×C4⋊C4—22no—
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—22no—
265C2×C4⋊C4—28no—
266C2×C4⋊C4—28no—
267C2×C4⋊C4—24no—
268C2×C4⋊C4—24no—
269C2×C4⋊C4—24no—
270C2×C4⋊C4—22no—
271C2×C4⋊C4—22no—
272C2×C4⋊C4—22no—
273C2×C4⋊C4—22no—
274C2×C4⋊C4—28no—
275C2×C4⋊C4—28no—
276C2×C4⋊C4—28no—
277C2×C4⋊C4—28no—
278C2×C4⋊C4—28no—
279C2×C4⋊C4—28no—
280C2×C4⋊C4—28no—
281C2×C4⋊C4—24no—
282C2×C4⋊C4—28no—
283C2×C4⋊C4—24no—
284C2×C4⋊C4—24no—
285C2×C4⋊C4—44no—
286C2×C4⋊C4—44no—
287C2×C4⋊C4#41116yes—
288C2×C22⋊C4—24no—
289C2×C22⋊C4—24no—
290C2×C22⋊C4—24no—
291C2×C22⋊C4—24no—
292C2×C22⋊C4—22no—
293C2×C22⋊C4—22no—
294C2×C22⋊C4—22no—
295C2×C22⋊C4—22no—
296C2×C22⋊C4—24no—
297C2×C22⋊C4—22no—
298C2×C22⋊C4—24no—
299C2×C22⋊C4—24no—
300C2×C22⋊C4—22no—
301C2×C22⋊C4—22no—
302C2×C22⋊C4—22no—
303C2×C22⋊C4—22no—
304C2×C22⋊C4—24no—
305C2×C22⋊C4—24no—
306C2×C22⋊C4—24no—
307C2×C22⋊C4—22no—
308C2×C22⋊C4—22no—
309C2×C22⋊C4—28no—
310C2×C22⋊C4—28no—
311C2×C22⋊C4—28no—
312C2×C22⋊C4—28no—
313C2×C22⋊C4—24no—
314C2×C22⋊C4—24no—
315C2×C22⋊C4—24no—
316C2×C22⋊C4—44no—
317C2×C22⋊C4—44no—
318C2×C22⋊C4#4118yes—
319C2×C42—232no—
320C2×C42—24no—
321C2×C42—24no—
322C2×C42—24no—
323C2×C42—28no—
324C2×C42—24no—
325C2×C42—24no—
326C2×C42—24no—
327C2×C42—24no—
328C2×C42—28no—
329C2×C42—28no—
330C2×C42—24no—
331C2×C42—28no—
332C2×C42—28no—
333C2×C42—44no—
334C2×C42—44no—
335C2×C42—432no—
336C2×C42C2×D418yes—
337C2×C42C2×D418yes—
338#95—24no—
339#95—24no—
340#95—28no—
341#95—28no—
342#95—24no—
343#95—24no—
344#95—24no—
345#95—24no—
346#95—24no—
347#95—24no—
348#95—24no—
349#95—24no—
350#95—24no—
351#95—24no—
352#95—28no—
353#95—28no—
354#95—28no—
355#95—28no—
356#95—28no—
357#95—28no—
358#95—44no—
359#95—44no—
360#95—44no—
361#95—44no—
362#95C2×D418yes—
363#95C2×D414yes—
364#95C2×D414yes—
365C24—24no—
366C24—22no—
367C24—22no—
368C24—21no—
369C24—21no—
370C24—24no—
371C24—21no—
372C24—22no—
373C24—22no—
374C24—22no—
375C24—22no—
376C24C22×D412yes—
377C24C22×D412yes—
378C24C2×#4112yes—
379C24C2×#4114yes—
380C242−1+414yes—
381C24C2511yescharacteristic
382C2×D4—44no—
383C2×D4—44no—
384C2×D4—44no—
385C2×D4—48no—
386C2×D4—48no—
387C22×C4—28no—
388C22×C4—216no—
389C22×C4—216no—
390C22×C4—28no—
391C22×C4—216no—
392C22×C4—28no—
393C22×C4—28no—
394C22×C4—28no—
395C22×C4—28no—
396C22×C4—216no—
397C22×C4—216no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—24no—
405C22×C4—22no—
406C22×C4—24no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—216no—
413C22×C4—216no—
414C22×C4—216no—
415C22×C4—216no—
416C22×C4—24no—
417C22×C4—24no—
418C22×C4—24no—
419C22×C4—22no—
420C22×C4—24no—
421C22×C4—24no—
422C22×C4—24no—
423C22×C4—24no—
424C22×C4—24no—
425C22×C4—24no—
426C22×C4—24no—
427C22×C4—24no—
428C22×C4—24no—
429C22×C4—24no—
430C22×C4—22no—
431C22×C4—24no—
432C22×C4—216no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—21no—
436C22×C4—21no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—21no—
442C22×C4—21no—
443C22×C4—21no—
444C22×C4—21no—
445C22×C4—24no—
446C22×C4—22no—
447C22×C4—24no—
448C22×C4—24no—
449C22×C4—24no—
450C22×C4—24no—
451C22×C4—24no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—21no—
455C22×C4—21no—
456C22×C4—21no—
457C22×C4—22no—
458C22×C4—24no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—24no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—24no—
466C22×C4—22no—
467C22×C4—48no—
468C22×C4—416no—
469C22×C4—416no—
470C22×C4—416no—
471C22×C4—416no—
472C22×C4—48no—
473C22×C4—48no—
474C22×C4#121116yes—
475C22×C4C22×D414yes—
476C22×C4C22×D412yes—
477C22×C4C22×D412yes—
478C22×C4C22×D412yes—
479C22×C4C22×D412yes—
480C22×C4C22×D414yes—
481C22×C4C22×D414yes—
482C22×C4C22×D414yes—
483C22×C4C2×#4118yes—
484C22×C4C2×#4118yes—
485C22×C4C2×#4112yes—
486C22×C4C2×#4112yes—
487C22×C4C2×#4118yes—
488C22×C4C2×#4118yes—
489C22×C4H(5,𝔽2)14yes—
490C22×C42−1+414yes—
491C4⋊C4—232no—
492C4⋊C4—216no—
493C4⋊C4—232no—
494C4⋊C4—416no—
495C4⋊C4—44no—
496C4⋊C4—44no—
497C4⋊C4—44no—
498C4⋊C4—44no—
499C4⋊C4—48no—
500C4⋊C4—416no—
501C4⋊C4—48no—
502C4⋊C4—416no—
503C4⋊C4—48no—
504C4⋊C4—84no—
505C22⋊C4—216no—
506C22⋊C4—416no—
507C22⋊C4—48no—
508C22⋊C4—44no—
509C22⋊C4—44no—
510C22⋊C4—44no—
511C22⋊C4—44no—
512C22⋊C4—44no—
513C22⋊C4—48no—
514C22⋊C4—48no—
515C22⋊C4—48no—
516C22⋊C4—84no—
517C42—216no—
518C42—216no—
519C42—216no—
520C42—416no—
521C42—416no—
522C42—48no—
523C23—24no—
524C23—24no—
525C23—24no—
526C23—24no—
527C23—28no—
528C23—28no—
529C23—22no—
530C23—22no—
531C23—24no—
532C23—28no—
533C23—28no—
534C23—41no—
535C23—42no—
536C23—41no—
537C23—42no—
538C23—42no—
539C23—41no—
540C23—42no—
541C23—42no—
542C23—42no—
543C23—44no—
544C23—44no—
545C23—41no—
546C23—41no—
547C23—42no—
548C23—44no—
549C23—44no—
550C23C2×#12114yes—
551C23#52514yes—
552C23#53614yes—
553C23#54718yes—
554C23#55218yes—
555C23C23×D411yescharacteristic
556C23C23×D411yescharacteristic
557C23C22×#4111yescharacteristic
558C23C22×#4114yes—
559C23C22×#4112yes—
560C23C2×H(5,𝔽2)12yes—
561C23C2×2−1+412yes—
562C23C2×2−1+412yes—
563D4—88no—
564C2×C4—232no—
565C2×C4—232no—
566C2×C4—24no—
567C2×C4—24no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—28no—
571C2×C4—232no—
572C2×C4—24no—
573C2×C4—28no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—24no—
577C2×C4—24no—
578C2×C4—24no—
579C2×C4—24no—
580C2×C4—24no—
581C2×C4—24no—
582C2×C4—28no—
583C2×C4—24no—
584C2×C4—24no—
585C2×C4—24no—
586C2×C4—24no—
587C2×C4—24no—
588C2×C4—28no—
589C2×C4—28no—
590C2×C4—24no—
591C2×C4—28no—
592C2×C4—24no—
593C2×C4—28no—
594C2×C4—28no—
595C2×C4—28no—
596C2×C4—24no—
597C2×C4—28no—
598C2×C4—432no—
599C2×C4—432no—
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—44no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—41no—
613C2×C4—41no—
614C2×C4—41no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—41no—
620C2×C4—42no—
621C2×C4—44no—
622C2×C4—42no—
623C2×C4—42no—
624C2×C4—44no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—41no—
628C2×C4—42no—
629C2×C4—42no—
630C2×C4—44no—
631C2×C4—42no—
632C2×C4—42no—
633C2×C4—42no—
634C2×C4—44no—
635C2×C4—44no—
636C2×C4—42no—
637C2×C4—41no—
638C2×C4—42no—
639C2×C4—41no—
640C2×C4—44no—
641C2×C4—42no—
642C2×C4—44no—
643C2×C4—44no—
644C2×C4C2×#12114yes—
645C2×C4C2×#12114yes—
646C2×C4#52518yes—
647C2×C4#53218yes—
648C2×C4#53718yes—
649C22—24no—
650C22—28no—
651C22—28no—
652C22—24no—
653C22—28no—
654C22—28no—
655C22—216no—
656C22—81no—
657C22—82no—
658C22—82no—
659C22—81no—
660C22—82no—
661C22—81no—
662C22—82no—
663C22—84no—
664C22#241318yes—
665C22C22×#12111yescharacteristic
666C22C2×#52512yes—
667C22C2×#52511yescharacteristic
668C22C2×#53212yes—
669C22#344412yes—
670C22C2×#53611yescharacteristic
671C22C2×#53712yes—
672C22C2×#54712yes—
673C22C2×#55212yes—
674C22#348114yes—
675C22#348214yes—
676C22#348411yescharacteristic
677C22#348611yescharacteristic
678C22#349212yes—
679C22#352014yes—
680C22#352314yes—
681C4—28no—
682C4—28no—
683C4—24no—
684C4—24no—
685C4—28no—
686C4—48no—
687C4—44no—
688C4—48no—
689C4—82no—
690C4—81no—
691C4—82no—
692C4—82no—
693C4—81no—
694C4—82no—
695C4—81no—
696C2—28no—
697C2—161no—
698C2C2×#241311yescharacteristic
699C2#3496412yes—
700C2#3522311yescharacteristic
701C2#3524112yes—
702C2#3631712yes—
703C2#3632914yes—
704C2#3642312yes—
705C2#3662411yescharacteristic
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
3. C2×#2296C2semidirect
4. C22×C4×C4⋊C4C2semidirect
5. #14870C2semidirect
6. #14869C2semidirect
7. #14821C2semidirect
8. #14818C2semidirect
9. #14808C2semidirect
11. #14697C2semidirect
13. #14203C2semidirect
14. #14200C2semidirect
16. #13958C2semidirect
17. #13878C2semidirect

Maximal subgroups

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

subgroupindex
C2×#28532
C2×#24982
C2×#22962
C22×C4×C4⋊C42
#148702
#148692
#148212
#148182
#148082
#147292
#146972
#146962
#142032
#142002
#141352
#139582
#138782