Finite Group Databasebeta

← #629323

Subgroups — #629323

Order 512.

2830 conjugacy classes of subgroups.

The subgroup lattice has 683 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2864C211yesmaximal, characteristic, complemented
2C2×#2328C211yesmaximal, characteristic
3#15962C214yesmaximal, complemented
4#15355C214yesmaximal, complemented
5#15335C214yesmaximal, complemented
6#15334C214yesmaximal, complemented
7#15148C212yesmaximal, complemented
8#15141C212yesmaximal, complemented
9#14713C211yesmaximal, characteristic
10#14277C211yesmaximal, characteristic, complemented
11#14276C211yesmaximal, characteristic, complemented
12#14272C211yesmaximal, characteristic
13#14018C212yesmaximal, complemented
14#13920C212yesmaximal, complemented
15#13759C211yesmaximal, characteristic, complemented
16C22×#127C2211yescharacteristic
17C22×C4×D4C2212yes—
18#2864—28no—
19C2×#399C2214yes—
20C2×#396C2214yes—
21#2328—28no—
22C2×#390C2212yes—
23C2×#390C2212yes—
24C2×#390C2212yes—
25C2×#390C2212yes—
26C2×#390C2212yes—
27C2×#390C2214yes—
28C2×#390C2214yes—
29C2×#389C2212yes—
30C2×#389C2212yes—
31C2×#389C2212yes—
32C2×#389C2212yes—
33C2×#387C2214yes—
34C2×#387C2214yes—
35C2×#386C2214yes—
36C2×#386C2214yes—
37C2×#385C2214yes—
38C2×#384C2211yescharacteristic
39C2×#382C2211yescharacteristic
40C2×#382C2211yescharacteristic
41C2×#380C2214yes—
42C2×C4×C4⋊C4C2211yescharacteristic
43C2×C4×C4⋊C4C2211yescharacteristic
44C2×#376C2211yescharacteristic
45C2×#376C2211yescharacteristic
46C2×C43C2211yescharacteristic
47#1444C2212yes—
48#1444C2212yes—
49#1442C2212yes—
50#1442C2212yes—
51#1441C2214yes—
52#1441C2214yes—
53#1439C2214yes—
54#1439C2214yes—
55#1439C2214yes—
56#1439C2214yes—
57#1439C2214yes—
58#1439C2214yes—
59#1439C2214yes—
60#1439C2214yes—
61#1439C2214yes—
62#1439C2214yes—
63#1438C2212yes—
64#1438C2212yes—
65#1438C2212yes—
66#1438C2212yes—
67#1436C2212yes—
68#1436C2212yes—
69#1436C2212yes—
70#1436C2212yes—
71C4⋊#95C2212yes—
72C4⋊#95C2212yes—
73C22⋊#95C2214yes—
74C22⋊#95C2214yes—
75C4×#95C2211yescharacteristic
76C4×#95C2211yescharacteristic
77C23×D4C2312yes—
78C23×D4C2312yes—
79C23×D4C2312yes—
80C24×C4C2314yes—
81C2×#127—24no—
82C2×#127—24no—
83C2×#127Q818yes—
84C2×C4×D4—28no—
85C2×C4×D4—28no—
86C2×C4×D4—44no—
87C22×C4⋊C4C2312yes—
88C22×C4⋊C4C2312yes—
89C22×C4⋊C4C2314yes—
90C22×C4⋊C4C2312yes—
91C22×C4⋊C4C2314yes—
92C22×C4⋊C4C2312yes—
93C22×C4⋊C4C2312yes—
94C22×C4⋊C4C2312yes—
95C22×C4⋊C4C2312yes—
96C22×C4⋊C4C2314yes—
97C22×C4⋊C4C2314yes—
98C22×C22⋊C4C2314yes—
99C22×C22⋊C4C2314yes—
100C22×C22⋊C4C2314yes—
101C22×C22⋊C4C2314yes—
102C22×C22⋊C4C2314yes—
103C22×C22⋊C4C2314yes—
104C22×C22⋊C4C2314yes—
105C22×C22⋊C4C2314yes—
106C22×C22⋊C4C2314yes—
107C22×C22⋊C4C2314yes—
108C22×C22⋊C4C2314yes—
109C22×C42C2311yescharacteristic
110C22×C42C2311yescharacteristic
111C22×C42C2311yescharacteristic
112C22×C42C2311yescharacteristic
113C22×C42C2312yes—
114C22×C42C2312yes—
115C22×C42C2311yescharacteristic
116C22×C42C2311yescharacteristic
117C22×C42C2311yescharacteristic
118#399—48no—
119#396—48no—
120#390—28no—
121#390—28no—
122#390—48no—
123#390—48no—
124#390—44no—
125#390—44no—
126#390—44no—
127#389—28no—
128#389—28no—
129#389—44no—
130#389—44no—
131#387—216no—
132#387—216no—
133#386—216no—
134#386—216no—
135#385—48no—
136#384—24no—
137#382—24no—
138#382D418yes—
139#380—48no—
140C4×C4⋊C4—24no—
141C4×C4⋊C4—24no—
142#376—24no—
143#376—24no—
144C2×#95C2312yes—
145C2×#95C2312yes—
146C2×#95C2312yes—
147C2×#95C2314yes—
148C2×#95C2314yes—
149C2×#95C2311yescharacteristic
150C2×#95C2311yescharacteristic
151C2×#95C2311yescharacteristic
152C2×#95C2311yescharacteristic
153C2×#95C2312yes—
154C2×#95C2312yes—
155C2×#95C2312yes—
156C2×#95C2312yes—
157C2×#95C2312yes—
158C2×#95C2314yes—
159C2×#95C2312yes—
160C2×#95C2312yes—
161C2×#95C2314yes—
162C2×#95C2314yes—
163C2×#95C2314yes—
164C2×#95C2314yes—
165C2×#95C2314yes—
166C2×#95C2312yes—
167C2×#95C2312yes—
168C43—24no—
169C25C2414yes—
170C22×D4—24no—
171C22×D4—24no—
172C22×D4—24no—
173C22×D4—24no—
174C22×D4—24no—
175C22×D4—24no—
176C22×D4—24no—
177C22×D4—24no—
178C22×D4—24no—
179C22×D4—24no—
180C22×D4—44no—
181C22×D4—44no—
182C22×D4—44no—
183C22×D4—44no—
184C22×D4C2×Q818yes—
185C22×D4C2×Q818yes—
186C22×D4C2×Q818yes—
187C23×C4—28no—
188C23×C4—28no—
189C23×C4—44no—
190C23×C4—44no—
191C23×C4—44no—
192C23×C4—44no—
193C23×C4—44no—
194C23×C4C2414yes—
195C23×C4C2414yes—
196C23×C4C2411yescharacteristic
197C23×C4C2412yes—
198C23×C4C2411yescharacteristic
199C23×C4C2414yes—
200C23×C4C2411yescharacteristic
201C23×C4C2411yescharacteristic
202C23×C4C2412yes—
203C23×C4C2412yes—
204C23×C4C2412yes—
205C23×C4C2411yescharacteristic
206C23×C4C2411yescharacteristic
207C23×C4C2411yescharacteristic
208#127—232no—
209C4×D4—432no—
210C2×C4⋊C4—24no—
211C2×C4⋊C4—24no—
212C2×C4⋊C4—24no—
213C2×C4⋊C4—28no—
214C2×C4⋊C4—28no—
215C2×C4⋊C4—24no—
216C2×C4⋊C4—24no—
217C2×C4⋊C4—24no—
218C2×C4⋊C4—24no—
219C2×C4⋊C4—28no—
220C2×C4⋊C4—24no—
221C2×C4⋊C4—28no—
222C2×C4⋊C4—24no—
223C2×C4⋊C4—24no—
224C2×C4⋊C4—24no—
225C2×C4⋊C4—24no—
226C2×C4⋊C4—24no—
227C2×C4⋊C4—24no—
228C2×C4⋊C4—42no—
229C2×C4⋊C4—42no—
230C2×C4⋊C4—44no—
231C2×C4⋊C4—44no—
232C2×C4⋊C4—44no—
233C2×C4⋊C4—44no—
234C2×C4⋊C4—44no—
235C2×C4⋊C4—44no—
236C2×C4⋊C4—44no—
237C2×C4⋊C4—44no—
238C2×C4⋊C4—44no—
239C2×C4⋊C4—44no—
240C2×C4⋊C4C2×D418yes—
241C2×C22⋊C4—28no—
242C2×C22⋊C4—28no—
243C2×C22⋊C4—28no—
244C2×C22⋊C4—28no—
245C2×C22⋊C4—28no—
246C2×C22⋊C4—28no—
247C2×C22⋊C4—28no—
248C2×C22⋊C4—28no—
249C2×C22⋊C4—28no—
250C2×C22⋊C4—28no—
251C2×C22⋊C4—44no—
252C2×C22⋊C4—44no—
253C2×C22⋊C4—44no—
254C2×C22⋊C4—44no—
255C2×C22⋊C4—44no—
256C2×C22⋊C4—44no—
257C2×C22⋊C4—44no—
258C2×C22⋊C4—44no—
259C2×C22⋊C4—44no—
260C2×C22⋊C4—44no—
261C2×C22⋊C4—44no—
262C2×C22⋊C4—44no—
263C2×C22⋊C4—44no—
264C2×C22⋊C4—44no—
265C2×C22⋊C4—44no—
266C2×C22⋊C4—44no—
267C2×C22⋊C4—44no—
268C2×C22⋊C4—44no—
269C2×C22⋊C4—44no—
270C2×C22⋊C4—44no—
271C2×C22⋊C4—44no—
272C2×C22⋊C4—44no—
273C2×C22⋊C4—44no—
274C2×C42—22no—
275C2×C42—22no—
276C2×C42—22no—
277C2×C42—22no—
278C2×C42—24no—
279C2×C42—24no—
280C2×C42—22no—
281C2×C42—22no—
282C2×C42—22no—
283C2×C42—22no—
284C2×C42—22no—
285C2×C42—22no—
286C2×C42—22no—
287C2×C42—22no—
288C2×C42—22no—
289C2×C42—24no—
290C2×C42—24no—
291C2×C42—22no—
292C2×C42—22no—
293C2×C42—22no—
294C2×C42—42no—
295C2×C42—42no—
296C2×C42C2×D414yes—
297C2×C42C2×Q814yes—
298C2×C42#4114yes—
299C2×C42#4114yes—
300C2×C42#4114yes—
301#95—24no—
302#95—24no—
303#95—24no—
304#95—24no—
305#95—28no—
306#95—28no—
307#95—22no—
308#95—22no—
309#95—22no—
310#95—22no—
311#95—24no—
312#95—24no—
313#95—24no—
314#95—24no—
315#95—42no—
316#95—42no—
317#95—44no—
318#95—44no—
319#95—44no—
320#95—44no—
321#95—44no—
322#95—44no—
323#95C2×D418yes—
324#95C2×D418yes—
325C24—24no—
326C24—24no—
327C24—24no—
328C24—24no—
329C24—24no—
330C24—24no—
331C24—24no—
332C24—24no—
333C24—24no—
334C24—24no—
335C24—24no—
336C24—24no—
337C24—24no—
338C24—24no—
339C24C22×Q818yes—
340C24C2511yescharacteristic
341C2×D4—216no—
342C2×D4—216no—
343C2×D4—216no—
344C2×D4—216no—
345C2×D4—216no—
346C2×D4—416no—
347C2×D4—416no—
348C2×D4—48no—
349C2×D4—48no—
350C2×D4—48no—
351C2×D4—48no—
352C2×D4—88no—
353C2×D4—88no—
354C2×D4—84no—
355C2×D4—84no—
356C2×D4—84no—
357C2×D4—84no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—21no—
366C22×C4—21no—
367C22×C4—21no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—21no—
371C22×C4—24no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—24no—
377C22×C4—24no—
378C22×C4—24no—
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—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—24no—
392C22×C4—24no—
393C22×C4—22no—
394C22×C4—24no—
395C22×C4—22no—
396C22×C4—21no—
397C22×C4—21no—
398C22×C4—22no—
399C22×C4—24no—
400C22×C4—24no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—21no—
405C22×C4—22no—
406C22×C4—24no—
407C22×C4—24no—
408C22×C4—24no—
409C22×C4—24no—
410C22×C4—24no—
411C22×C4—24no—
412C22×C4—24no—
413C22×C4—24no—
414C22×C4—21no—
415C22×C4—21no—
416C22×C4—21no—
417C22×C4—24no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—24no—
423C22×C4—22no—
424C22×C4—24no—
425C22×C4—416no—
426C22×C4—416no—
427C22×C4—88no—
428C22×C4—88no—
429C22×C4—88no—
430C22×C4—88no—
431C22×C4—88no—
432C22×C4C22×D418yes—
433C22×C4C22×D412yes—
434C22×C4C22×D412yes—
435C22×C4C22×D412yes—
436C22×C4C22×Q812yes—
437C22×C4C22×Q812yes—
438C22×C4C22×Q812yes—
439C22×C4C2×#4112yes—
440C22×C4C2×#4114yes—
441C22×C4C2×#4112yes—
442C22×C4C2×#4112yes—
443C22×C4C2×#4112yes—
444C22×C4C2×#4112yes—
445C22×C4C2×#4112yes—
446C22×C4C2×#4112yes—
447C22×C4H(5,𝔽2)14yes—
448C22×C4H(5,𝔽2)14yes—
449C22×C4H(5,𝔽2)12yes—
450C22×C4H(5,𝔽2)12yes—
451C4⋊C4—216no—
452C4⋊C4—416no—
453C4⋊C4—416no—
454C4⋊C4—48no—
455C4⋊C4—48no—
456C4⋊C4—48no—
457C4⋊C4—48no—
458C4⋊C4—84no—
459C4⋊C4—84no—
460C4⋊C4—88no—
461C4⋊C4—88no—
462C22⋊C4—416no—
463C22⋊C4—416no—
464C22⋊C4—88no—
465C22⋊C4—88no—
466C22⋊C4—88no—
467C22⋊C4—88no—
468C22⋊C4—88no—
469C22⋊C4—88no—
470C22⋊C4—88no—
471C22⋊C4—88no—
472C22⋊C4—88no—
473C42—28no—
474C42—28no—
475C42—48no—
476C42—48no—
477C42—44no—
478C42—44no—
479C42—44no—
480C42—44no—
481C42C22⋊Q8116yes—
482C23—28no—
483C23—28no—
484C23—28no—
485C23—28no—
486C23—28no—
487C23—28no—
488C23—28no—
489C23—44no—
490C23—44no—
491C23—44no—
492C23—44no—
493C23—44no—
494C23—44no—
495C23—44no—
496C23—44no—
497C23—44no—
498C23—44no—
499C23—44no—
500C23—44no—
501C23—44no—
502C23—44no—
503C23—44no—
504C23—44no—
505C23—44no—
506C23—44no—
507C23—44no—
508C23—44no—
509C23—44no—
510C23—44no—
511C23—44no—
512C23—44no—
513C23—44no—
514C23—44no—
515C23—44no—
516C23—44no—
517C23C23×D411yescharacteristic
518C23C23×Q811yescharacteristic
519C23C22×#4111yescharacteristic
520C23C22×#4111yescharacteristic
521C23C22×#4111yescharacteristic
522C23C2×H(5,𝔽2)11yescharacteristic
523C23C2×H(5,𝔽2)11yescharacteristic
524C23C2×H(5,𝔽2)12yes—
525C23C2×H(5,𝔽2)12yes—
526C23#58512yes—
527C23#58512yes—
528D4—432no—
529D4—816no—
530D4—816no—
531C2×C4—24no—
532C2×C4—24no—
533C2×C4—24no—
534C2×C4—24no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—24no—
542C2×C4—24no—
543C2×C4—24no—
544C2×C4—28no—
545C2×C4—24no—
546C2×C4—24no—
547C2×C4—28no—
548C2×C4—28no—
549C2×C4—24no—
550C2×C4—24no—
551C2×C4—24no—
552C2×C4—24no—
553C2×C4—24no—
554C2×C4—24no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—44no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—44no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—44no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—44no—
576C2×C4—42no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—44no—
580C2×C4—44no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—44no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—44no—
587C2×C4—44no—
588C2×C4—44no—
589C2×C4—42no—
590C2×C4—44no—
591C2×C4—44no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—42no—
597C2×C4—44no—
598C2×C4—44no—
599C2×C4—44no—
600C2×C4—44no—
601C2×C4—44no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—44no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—832no—
609C2×C4C2×C22⋊Q814yes—
610C2×C4C2×C22⋊Q814yes—
611C2×C4C2×C22⋊Q814yes—
612C2×C4#52914yes—
613C2×C4#54014yes—
614C2×C4#54614yes—
615C2×C4#55014yes—
616C2×C4#55414yes—
617C2×C4#55414yes—
618C22—48no—
619C22—48no—
620C22—48no—
621C22—48no—
622C22—48no—
623C22—48no—
624C22—48no—
625C22—84no—
626C22—84no—
627C22—84no—
628C22—84no—
629C22—84no—
630C22—84no—
631C22—84no—
632C22—84no—
633C22C22×C22⋊Q811yescharacteristic
634C22C2×#52911yescharacteristic
635C22C2×#54011yescharacteristic
636C22#346012yes—
637C22#346012yes—
638C22C2×#54611yescharacteristic
639C22C2×#55011yescharacteristic
640C22C2×#55411yescharacteristic
641C22C2×#55411yescharacteristic
642C22#348812yes—
643C22#348812yes—
644C22#349412yes—
645C22#349412yes—
646C22#350112yes—
647C22#350112yes—
648C22#350811yescharacteristic
649C22#350811yescharacteristic
650C22#351011yescharacteristic
651C22#351011yescharacteristic
652C22#354312yes—
653C22#354312yes—
654C22#356212yes—
655C22#356212yes—
656C4—24no—
657C4—24no—
658C4—44no—
659C4—48no—
660C4—44no—
661C4—44no—
662C4—42no—
663C4—42no—
664C4—81no—
665C4—84no—
666C4—82no—
667C4—81no—
668C4—84no—
669C4#245418yes—
670C2—88no—
671C2C2×#245411yescharacteristic
672C2#3526811yescharacteristic
673C2#3529611yescharacteristic
674C2#3551511yescharacteristic
675C2#3551511yescharacteristic
676C2#3731911yescharacteristic
677C2#3743211yescharacteristic
678C2#3775112yes—
679C2#3777512yes—
680C2#3967212yes—
681C2#3967212yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC22×#127C22
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×#2864C2semidirect
3. #15962C2semidirect
4. #15355C2semidirect
5. #15335C2semidirect
6. #15334C2semidirect
7. #15148C2semidirect
8. #15141C2semidirect
10. #14277C2semidirect
11. #14276C2semidirect
13. #14018C2semidirect
14. #13920C2semidirect
15. #13759C2semidirect

Maximal subgroups

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

subgroupindex
C2×#28642
C2×#23282
#159622
#153552
#153352
#153342
#151482
#151412
#147132
#142772
#142762
#142722
#140182
#139202
#137592