Finite Group Databasebeta

← #626968

Subgroups — #626968

Order 512.

2744 conjugacy classes of subgroups.

The subgroup lattice has 672 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2868C211yesmaximal, characteristic, complemented
2C2×#2628C211yesmaximal, characteristic
3C2×#2628C211yesmaximal, characteristic
4C2×#2331C211yesmaximal, characteristic, complemented
5#15789C214yesmaximal, complemented
6#15765C214yesmaximal
7#15322C214yesmaximal
8#15318C214yesmaximal, complemented
9#15137C212yesmaximal, complemented
10#15131C212yesmaximal
11#14707C211yesmaximal, characteristic, complemented
12#14282C211yesmaximal, characteristic, complemented
13#14005C212yesmaximal, complemented
14#13876C212yesmaximal
15#13761C211yesmaximal, characteristic
16C22×C4⋊Q8C2211yescharacteristic
17C22×C4⋊Q8C2211yescharacteristic
18C22×C4⋊Q8C2214yes—
19C22×#117C2211yescharacteristic
20#2868—28no—
21#2628—28no—
22#2628—28no—
23C2×#400C2212yes—
24C2×#400C2212yes—
25C2×#399C2212yes—
26C2×#399C2212yes—
27C2×#398C2214yes—
28C2×#396C2214yes—
29C2×#395C2212yes—
30C2×#395C2212yes—
31C2×#393C2212yes—
32C2×#393C2212yes—
33#2331—28no—
34C2×#391C2214yes—
35C2×#389C2212yes—
36C2×#389C2212yes—
37C2×#389C2212yes—
38C2×#389C2212yes—
39C2×#389C2214yes—
40C2×#388C2214yes—
41C2×#387C2214yes—
42C2×#384C2212yes—
43C2×#384C2212yes—
44C2×#384C2214yes—
45C2×#382C2211yescharacteristic
46C2×#382C2211yescharacteristic
47C2×#380C2212yes—
48C2×#380C2212yes—
49C2×C4×C4⋊C4C2212yes—
50C2×C4×C22⋊C4C2211yescharacteristic
51C2×C4×C22⋊C4C2211yescharacteristic
52C2×#376C2211yescharacteristic
53C2×C43C2211yescharacteristic
54#1446C2214yes—
55#1445C2214yes—
56#1444C2214yes—
57#1444C2214yes—
58#1442C2214yes—
59#1442C2214yes—
60#1441C2214yes—
61#1441C2214yes—
62#1441C2214yes—
63#1441C2214yes—
64#1439C2214yes—
65#1439C2214yes—
66#1437C2212yes—
67#1437C2212yes—
68#1436C2214yes—
69#1436C2214yes—
70#1436C2214yes—
71#1436C2212yes—
72#1436C2214yes—
73#1436C2212yes—
74C4⋊#95C2212yes—
75C4⋊#95C2212yes—
76C4⋊#95C2212yes—
77C4×#95C2211yescharacteristic
78C4×#95C2211yescharacteristic
79C23×Q8C2312yes—
80C23×Q8C2312yes—
81C24×C4C2311yescharacteristic
82C2×C4⋊Q8—216no—
83C2×C4⋊Q8—24no—
84C2×C4⋊Q8—24no—
85C2×C4⋊Q8—24no—
86C2×C4⋊Q8—24no—
87C2×C4⋊Q8—24no—
88C2×C4⋊Q8—48no—
89C2×C4⋊Q8—48no—
90C2×C4⋊Q8D418yes—
91C2×#117—24no—
92C2×#117—24no—
93C2×#117—24no—
94C22×C4⋊C4C2311yescharacteristic
95C22×C4⋊C4C2314yes—
96C22×C4⋊C4C2312yes—
97C22×C4⋊C4C2311yescharacteristic
98C22×C4⋊C4C2312yes—
99C22×C4⋊C4C2312yes—
100C22×C4⋊C4C2314yes—
101C22×C4⋊C4C2314yes—
102C22×C4⋊C4C2314yes—
103C22×C4⋊C4C2314yes—
104C22×C4⋊C4C2312yes—
105C22×C4⋊C4C2312yes—
106C22×C4⋊C4C2312yes—
107C22×C4⋊C4C2314yes—
108C22×C4⋊C4C2314yes—
109C22×C4⋊C4C2314yes—
110C22×C22⋊C4C2311yescharacteristic
111C22×C22⋊C4C2312yes—
112C22×C22⋊C4C2311yescharacteristic
113C22×C22⋊C4C2312yes—
114C22×C22⋊C4C2314yes—
115C22×C22⋊C4C2314yes—
116C22×C42C2311yescharacteristic
117C22×C42C2312yes—
118C22×C42C2311yescharacteristic
119C22×C42C2312yes—
120C22×C42C2311yescharacteristic
121C22×C42C2311yescharacteristic
122C22×C42C2311yescharacteristic
123C22×C42C2314yes—
124C22×C42C2311yescharacteristic
125#400—28no—
126#400—28no—
127#399—28no—
128#399—28no—
129#398—48no—
130#396—48no—
131#395—28no—
132#395—28no—
133#393—28no—
134#393—28no—
135#391—216no—
136#389—28no—
137#389—28no—
138#389—28no—
139#389—28no—
140#389—48no—
141#388—48no—
142#387—216no—
143#384—28no—
144#384—28no—
145#384—216no—
146#382—24no—
147#382D418yes—
148#380—28no—
149#380—28no—
150C4×C4⋊C4—28no—
151C4×C22⋊C4—24no—
152C4×C22⋊C4Q818yescomplemented
153#376—24no—
154C2×#95C2312yes—
155C2×#95C2314yes—
156C2×#95C2314yes—
157C2×#95C2314yes—
158C2×#95C2314yes—
159C2×#95C2312yes—
160C2×#95C2314yes—
161C2×#95C2312yes—
162C2×#95C2312yes—
163C2×#95C2312yes—
164C2×#95C2312yes—
165C2×#95C2314yes—
166C2×#95C2314yes—
167C2×#95C2312yes—
168C2×#95C2312yes—
169C2×#95C2312yes—
170C2×#95C2314yes—
171C2×#95C2314yes—
172C2×#95C2314yes—
173C2×#95C2314yes—
174C2×#95C2314yes—
175C2×#95C2314yes—
176C2×#95C2312yes—
177C2×#95C2312yes—
178C2×#95C2312yes—
179C43—24no—
180C25C2411yescharacteristic
181C22×Q8—24no—
182C22×Q8—24no—
183C22×Q8—24no—
184C22×Q8—24no—
185C22×Q8—24no—
186C22×Q8—44no—
187C22×Q8—44no—
188C22×Q8—44no—
189C22×Q8—44no—
190C22×Q8C2×D418yes—
191C23×C4—22no—
192C23×C4—22no—
193C23×C4—22no—
194C23×C4—22no—
195C23×C4—22no—
196C23×C4—22no—
197C23×C4C2×Q814yes—
198C23×C4C2411yescharacteristic
199C23×C4C2414yes—
200C23×C4C2414yes—
201C23×C4C2412yes—
202C23×C4C2411yescharacteristic
203C23×C4C2414yes—
204C23×C4C2414yes—
205C23×C4C2412yes—
206C23×C4C2411yescharacteristic
207C23×C4C2411yescharacteristic
208C23×C4C2411yescharacteristic
209C23×C4C2412yes—
210C23×C4C2412yes—
211C23×C4C2411yescharacteristic
212C4⋊Q8—232no—
213C4⋊Q8—464no—
214C4⋊Q8—416no—
215#117—416no—
216C2×C4⋊C4—24no—
217C2×C4⋊C4—24no—
218C2×C4⋊C4—24no—
219C2×C4⋊C4—24no—
220C2×C4⋊C4—24no—
221C2×C4⋊C4—24no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—24no—
227C2×C4⋊C4—28no—
228C2×C4⋊C4—28no—
229C2×C4⋊C4—28no—
230C2×C4⋊C4—28no—
231C2×C4⋊C4—24no—
232C2×C4⋊C4—24no—
233C2×C4⋊C4—24no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—24no—
239C2×C4⋊C4—28no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—28no—
242C2×C4⋊C4—28no—
243C2×C4⋊C4—28no—
244C2×C4⋊C4—28no—
245C2×C4⋊C4—28no—
246C2×C4⋊C4—28no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—28no—
249C2×C4⋊C4—28no—
250C2×C4⋊C4—28no—
251C2×C4⋊C4—44no—
252C2×C4⋊C4—44no—
253C2×C4⋊C4—44no—
254C2×C4⋊C4—44no—
255C2×C4⋊C4—44no—
256C2×C4⋊C4—44no—
257C2×C4⋊C4—44no—
258C2×C4⋊C4—44no—
259C2×C4⋊C4—44no—
260C2×C4⋊C4—44no—
261C2×C4⋊C4C2×D418yes—
262C2×C4⋊C4C2×D418yes—
263C2×C4⋊C4C2×D418yes—
264C2×C22⋊C4—24no—
265C2×C22⋊C4—24no—
266C2×C22⋊C4—22no—
267C2×C22⋊C4—22no—
268C2×C22⋊C4—22no—
269C2×C22⋊C4—22no—
270C2×C22⋊C4—22no—
271C2×C22⋊C4—22no—
272C2×C22⋊C4—24no—
273C2×C22⋊C4—24no—
274C2×C22⋊C4—24no—
275C2×C22⋊C4—28no—
276C2×C22⋊C4—28no—
277C2×C22⋊C4—44no—
278C2×C22⋊C4—44no—
279C2×C22⋊C4—44no—
280C2×C22⋊C4—44no—
281C2×C22⋊C4C2×Q818yes—
282C2×C42—24no—
283C2×C42—24no—
284C2×C42—22no—
285C2×C42—22no—
286C2×C42—22no—
287C2×C42—24no—
288C2×C42—24no—
289C2×C42—24no—
290C2×C42—24no—
291C2×C42—22no—
292C2×C42—22no—
293C2×C42—22no—
294C2×C42—22no—
295C2×C42—22no—
296C2×C42—22no—
297C2×C42—22no—
298C2×C42—22no—
299C2×C42—22no—
300C2×C42—28no—
301C2×C42—44no—
302C2×C42—44no—
303C2×C42C2×D414yes—
304C2×C42C2×D414yes—
305C2×C42C2×Q814yes—
306C2×C42C2×Q814yes—
307C2×C42#4114yes—
308C2×C42#4114yes—
309#95—24no—
310#95—24no—
311#95—28no—
312#95—28no—
313#95—28no—
314#95—24no—
315#95—24no—
316#95—28no—
317#95—28no—
318#95—28no—
319#95—24no—
320#95—28no—
321#95—28no—
322#95—24no—
323#95—24no—
324#95—24no—
325#95—28no—
326#95—24no—
327#95—44no—
328#95—44no—
329#95—44no—
330#95—44no—
331#95C2×D418yes—
332#95C2×D418yes—
333#95C2×Q818yes—
334C24—21no—
335C24—21no—
336C24—21no—
337C24—21no—
338C24—22no—
339C24—22no—
340C24—22no—
341C24—22no—
342C24—21no—
343C24—21no—
344C24C22×Q812yes—
345C24C2511yescharacteristic
346C2×Q8—216no—
347C2×Q8—416no—
348C2×Q8—416no—
349C2×Q8—48no—
350C2×Q8—88no—
351C2×Q8—88no—
352C2×Q8—84no—
353C2×Q8—84no—
354C2×Q8—84no—
355C2×Q8—84no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—21no—
370C22×C4—21no—
371C22×C4—24no—
372C22×C4—21no—
373C22×C4—24no—
374C22×C4—21no—
375C22×C4—24no—
376C22×C4—22no—
377C22×C4—24no—
378C22×C4—22no—
379C22×C4—24no—
380C22×C4—24no—
381C22×C4—24no—
382C22×C4—24no—
383C22×C4—24no—
384C22×C4—24no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—24no—
389C22×C4—24no—
390C22×C4—21no—
391C22×C4—21no—
392C22×C4—28no—
393C22×C4—28no—
394C22×C4—28no—
395C22×C4—21no—
396C22×C4—21no—
397C22×C4—22no—
398C22×C4—24no—
399C22×C4—24no—
400C22×C4—21no—
401C22×C4—21no—
402C22×C4—24no—
403C22×C4—22no—
404C22×C4—24no—
405C22×C4—21no—
406C22×C4—21no—
407C22×C4—24no—
408C22×C4—24no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—24no—
414C22×C4—24no—
415C22×C4—24no—
416C22×C4—24no—
417C22×C4—24no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—24no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—24no—
425C22×C4—24no—
426C22×C4—44no—
427C22×C4—44no—
428C22×C4—44no—
429C22×C4—44no—
430C22×C4C22×D418yes—
431C22×C4C22×D418yes—
432C22×C4C22×D412yes—
433C22×C4C22×D412yes—
434C22×C4C22×D412yes—
435C22×C4C22×D412yes—
436C22×C4C22×D412yes—
437C22×C4C22×D412yes—
438C22×C4C22×Q812yes—
439C22×C4C22×Q814yes—
440C22×C4C22×Q814yes—
441C22×C4C22×Q812yes—
442C22×C4C2×#4112yes—
443C22×C4C2×#4112yes—
444C22×C4C2×#4114yes—
445C22×C4C2×#4112yes—
446C22×C4C2×#4112yes—
447C22×C4H(5,𝔽2)14yes—
448C22×C4H(5,𝔽2)14yes—
449C22×C42−1+414yes—
450C22×C42−1+414yes—
451C22×C42−1+412yes—
452C22×C42−1+412yes—
453C4⋊C4—216no—
454C4⋊C4—216no—
455C4⋊C4—216no—
456C4⋊C4—416no—
457C4⋊C4—416no—
458C4⋊C4—416no—
459C4⋊C4—416no—
460C4⋊C4—44no—
461C4⋊C4—44no—
462C4⋊C4—48no—
463C4⋊C4—48no—
464C4⋊C4—416no—
465C4⋊C4—48no—
466C4⋊C4—88no—
467C4⋊C4—88no—
468C4⋊C4—88no—
469C22⋊C4—216no—
470C22⋊C4—44no—
471C22⋊C4—44no—
472C22⋊C4—48no—
473C22⋊C4—88no—
474C22⋊C4—88no—
475C42—216no—
476C42—216no—
477C42—28no—
478C42—28no—
479C42—28no—
480C42—44no—
481C42—44no—
482C42—88no—
483C42#121116yes—
484C23—22no—
485C23—24no—
486C23—22no—
487C23—22no—
488C23—24no—
489C23—41no—
490C23—41no—
491C23—42no—
492C23—41no—
493C23—41no—
494C23—42no—
495C23—42no—
496C23—42no—
497C23—42no—
498C23—41no—
499C23—41no—
500C23—41no—
501C23—41no—
502C23—42no—
503C23—42no—
504C23—42no—
505C23—42no—
506C23—42no—
507C23C23×D411yescharacteristic
508C23C23×D411yescharacteristic
509C23C23×Q811yescharacteristic
510C23C22×#4111yescharacteristic
511C23C22×#4111yescharacteristic
512C23C2×H(5,𝔽2)12yes—
513C23C2×H(5,𝔽2)12yes—
514C23C2×2−1+411yescharacteristic
515C23C2×2−1+411yescharacteristic
516C23C2×2−1+412yes—
517C23C2×2−1+412yes—
518Q8—816no—
519Q8—816no—
520C2×C4—24no—
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—22no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—24no—
536C2×C4—24no—
537C2×C4—24no—
538C2×C4—24no—
539C2×C4—24no—
540C2×C4—28no—
541C2×C4—22no—
542C2×C4—24no—
543C2×C4—28no—
544C2×C4—24no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—28no—
548C2×C4—28no—
549C2×C4—28no—
550C2×C4—28no—
551C2×C4—24no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—44no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—42no—
558C2×C4—44no—
559C2×C4—44no—
560C2×C4—42no—
561C2×C4—44no—
562C2×C4—44no—
563C2×C4—44no—
564C2×C4—42no—
565C2×C4—44no—
566C2×C4—42no—
567C2×C4—44no—
568C2×C4—416no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—44no—
572C2×C4—42no—
573C2×C4—44no—
574C2×C4—42no—
575C2×C4—44no—
576C2×C4—44no—
577C2×C4—44no—
578C2×C4—44no—
579C2×C4—42no—
580C2×C4—44no—
581C2×C4—42no—
582C2×C4—44no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—44no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—44no—
591C2×C4—44no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—44no—
595C2×C4—42no—
596C2×C4—44no—
597C2×C4—42no—
598C2×C4—44no—
599C2×C4—42no—
600C2×C4C2×#12114yes—
601C2×C4C2×#12114yes—
602C2×C4C2×#12114yes—
603C2×C4#53314yes—
604C2×C4#54114yes—
605C2×C4#54218yes—
606C2×C4#54418yes—
607C2×C4#54714yes—
608C2×C4#54814yes—
609C2×C4D4×Q814yes—
610C2×C4D4×Q814yes—
611C22—44no—
612C22—42no—
613C22—44no—
614C22—42no—
615C22—42no—
616C22—82no—
617C22—81no—
618C22—81no—
619C22—81no—
620C22—81no—
621C22—82no—
622C22C22×#12111yescharacteristic
623C22C2×#53311yescharacteristic
624C22C2×#54111yescharacteristic
625C22C2×#54212yes—
626C22C2×#54412yes—
627C22C2×#54711yescharacteristic
628C22C2×#54811yescharacteristic
629C22C2×D4×Q811yescharacteristic
630C22C2×D4×Q811yescharacteristic
631C22#348412yes—
632C22#348612yes—
633C22#349812yes—
634C22#349912yes—
635C22#350312yes—
636C22#350811yescharacteristic
637C22#350811yescharacteristic
638C22#350911yescharacteristic
639C22#350911yescharacteristic
640C22#351112yes—
641C22#354412yes—
642C22#354412yes—
643C22#355212yes—
644C22#355212yes—
645C4—28no—
646C4—28no—
647C4—24no—
648C4—24no—
649C4—42no—
650C4—48no—
651C4—44no—
652C4—48no—
653C4—84no—
654C4—81no—
655C4—81no—
656C4—84no—
657C4—82no—
658C4#242918yes—
659C2—82no—
660C2C2×#242911yescharacteristic
661C2#3528311yescharacteristic
662C2#3529411yescharacteristic
663C2#3552811yescharacteristic
664C2#3566711yescharacteristic
665C2#3732011yescharacteristic
666C2#3732011yescharacteristic
667C2#3768612yes—
668C2#3770412yes—
669C2#3936912yes—
670C2#3937712yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC25C24
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×#2868C2semidirect
4. C2×#2331C2semidirect
5. #15789C2semidirect
8. #15318C2semidirect
9. #15137C2semidirect
11. #14707C2semidirect
12. #14282C2semidirect
13. #14005C2semidirect
152. C4×C22⋊C4Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#28682
C2×#26282
C2×#23312
#157892
#157652
#153222
#153182
#151372
#151312
#147072
#142822
#140052
#138762
#137612