Finite Group Databasebeta

← #515539

Subgroups — #515539

Order 512.

4182 conjugacy classes of subgroups.

The subgroup lattice has 695 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2311C211yesmaximal, characteristic
2C2×#2311C211yesmaximal, characteristic
3C2×#2297C211yesmaximal, characteristic
4C22×#389C211yesmaximal, characteristic, complemented
5C2×#2274C211yesmaximal, characteristic, complemented
6#14142C214yesmaximal, complemented
7#14141C214yesmaximal, complemented
8#14020C214yesmaximal, complemented
9#14019C214yesmaximal, complemented
10#13994C214yesmaximal, complemented
11#13993C214yesmaximal, complemented
12#13795C211yesmaximal, characteristic, complemented
13#13756C211yesmaximal, characteristic, complemented
14C23×#41C2211yescharacteristic
15C22×#117C2211yescharacteristic
16C22×#117C2211yescharacteristic
17C22×#117C2211yescharacteristic
18C23×C4⋊C4C2211yescharacteristic
19C23×C4⋊C4C2211yescharacteristic
20C23×C4⋊C4C2212yes—
21C23×C42C2211yescharacteristic
22#2311—28no—
23#2311C4116yescomplemented
24#2297—28no—
25C2×#391C2212yes—
26C2×#391C2212yes—
27C2×#391C2212yes—
28C2×#390C2212yes—
29C2×#390C2212yes—
30C2×#390C2212yes—
31C2×#389—28no—
32C2×#389C2214yes—
33C2×#389C2214yes—
34C2×#388C2218yes—
35C2×#386C2212yes—
36C2×#386C2214yes—
37C2×#386C2212yes—
38C2×#386C2214yes—
39C2×#380C2212yes—
40C2×#380C2212yes—
41C2×#380C2214yes—
42#2274—28no—
43C2×C4×C4⋊C4C2214yes—
44C2×C4×C4⋊C4C2214yes—
45C2×C4×C22⋊C4C2214yes—
46C2×C4×C22⋊C4C2214yes—
47C22×#95C2212yes—
48#1443C2218yes—
49#1442C2218yes—
50#1440C2218yes—
51#1439C2218yes—
52#1437C2218yes—
53#1436C2218yes—
54C4⋊#95C2214yes—
55C4⋊#95C2212yes—
56C4⋊#95C2212yes—
57C4⋊#95C2212yes—
58C4⋊#95C2212yes—
59#1434C2218yes—
60C22⋊#95C2212yes—
61C22⋊#95C2212yes—
62C22⋊#95C2214yes—
63C4×#95C2214yes—
64C22×#41—24no—
65C22×#41—24no—
66C22×#41—24no—
67C22×#41—24no—
68C22×#41—24no—
69C22×#41—24no—
70C22×#41C2×C418yes—
71C23×Q8C2311yescharacteristic
72C23×D4C2312yes—
73C23×D4C2311yescharacteristic
74C24×C4C2312yes—
75C24×C4C2311yescharacteristic
76C24×C4C2311yescharacteristic
77C24×C4C2312yes—
78C24×C4C2312yes—
79C24×C4C2311yescharacteristic
80C2×#117—24no—
81C2×#117—24no—
82C2×#117—24no—
83C2×#117—24no—
84C2×#117—24no—
85C2×#117—24no—
86C2×#117—24no—
87C2×#117C2×C418yes—
88C2×#117D418yes—
89C22×C4⋊C4—24no—
90C22×C4⋊C4—24no—
91C22×C4⋊C4—24no—
92C22×C4⋊C4—24no—
93C22×C4⋊C4—28no—
94C22×C4⋊C4—48no—
95C22×C4⋊C4C2×C418yes—
96C22×C4⋊C4D418yes—
97C22×C4⋊C4C2311yescharacteristic
98C22×C4⋊C4C2311yescharacteristic
99C22×C4⋊C4C2318yes—
100C22×C4⋊C4C2312yes—
101C22×C4⋊C4C2312yes—
102C22×C4⋊C4C2312yes—
103C22×C4⋊C4C2312yes—
104C22×C4⋊C4C2312yes—
105C22×C4⋊C4C2312yes—
106C22×C22⋊C4C2312yes—
107C22×C22⋊C4C2314yes—
108C22×C22⋊C4C2314yes—
109C22×C22⋊C4C2312yes—
110C22×C22⋊C4C2312yes—
111C22×C22⋊C4C2314yes—
112C22×C22⋊C4C2314yes—
113C22×C22⋊C4C2318yes—
114C22×C42—24no—
115C22×C42—24no—
116C22×C42—24no—
117C22×C42C2312yes—
118C22×C42C2312yes—
119C22×C42C2312yes—
120C22×C42C2312yes—
121C22×C42C2312yes—
122C22×C42C2318yes—
123#391—28no—
124#391—28no—
125#391C2×C4116yescomplemented
126#390—28no—
127#390—28no—
128#390C2×C4116yescomplemented
129#389—216no—
130#389—216no—
131#388—416no—
132#386—216no—
133#386—28no—
134#386—28no—
135#386C2×C4132yes—
136#380—28no—
137#380—28no—
138#380C2×C4132yescomplemented
139C4×C4⋊C4—216no—
140C4×C4⋊C4—216no—
141C4×C22⋊C4—216no—
142C4×C22⋊C4—216no—
143C2×#95—28no—
144C2×#95C2314yes—
145C2×#95C2314yes—
146C2×#95C2314yes—
147C2×#95C2314yes—
148C2×#95C2318yes—
149C2×#95C2314yes—
150C2×#95C2314yes—
151C2×#95C2318yes—
152C2×#95C2314yes—
153C2×#95C2314yes—
154C2×#95C2318yes—
155C2×#95C2318yes—
156C2×#95C2318yes—
157C25C2412yes—
158C25C2411yescharacteristic
159C2×#41—232no—
160C2×#41—232no—
161C2×#41—232no—
162C2×#41—416no—
163C2×#41—416no—
164C2×#41—416no—
165C2×#41—416no—
166C22×Q8—22no—
167C22×Q8—22no—
168C22×Q8—22no—
169C22×Q8—22no—
170C22×Q8—22no—
171C22×Q8—22no—
172C22×Q8C22×C414yes—
173C22×D4—24no—
174C22×D4—24no—
175C22×D4—24no—
176C22×D4—24no—
177C22×D4—24no—
178C22×D4—24no—
179C22×D4—22no—
180C22×D4—22no—
181C22×D4—22no—
182C22×D4—22no—
183C22×D4—22no—
184C22×D4—22no—
185C22×D4C22×C418yes—
186C22×D4C22×C414yes—
187C23×C4—24no—
188C23×C4—22no—
189C23×C4—22no—
190C23×C4—24no—
191C23×C4—24no—
192C23×C4—24no—
193C23×C4—24no—
194C23×C4—24no—
195C23×C4—24no—
196C23×C4—24no—
197C23×C4—24no—
198C23×C4—24no—
199C23×C4—24no—
200C23×C4—24no—
201C23×C4—22no—
202C23×C4—22no—
203C23×C4—22no—
204C23×C4—22no—
205C23×C4—44no—
206C23×C4—42no—
207C23×C4—42no—
208C23×C4—44no—
209C23×C4—44no—
210C23×C4—44no—
211C23×C4C22×C418yes—
212C23×C4C22×C418yes—
213C23×C4C22×C414yes—
214C23×C4C2×D418yes—
215C23×C4C2×D414yes—
216C23×C4C2×D414yes—
217C23×C4#4114yes—
218C23×C4#4114yes—
219C23×C4#4114yes—
220C23×C4#4114yes—
221C23×C4C2412yes—
222C23×C4C2412yes—
223C23×C4C2414yes—
224C23×C4C2414yes—
225C23×C4C2412yes—
226C23×C4C2414yes—
227C23×C4C2414yes—
228C23×C4C2412yes—
229C23×C4C2412yes—
230C23×C4C2411yescharacteristic
231C23×C4C2411yescharacteristic
232#117—232no—
233#117—232no—
234#117—416no—
235C2×C4⋊C4—232no—
236C2×C4⋊C4—232no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—24no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—24no—
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—216no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—216no—
257C2×C4⋊C4—48no—
258C2×C4⋊C4—432no—
259C2×C4⋊C4C22×C418yes—
260C2×C4⋊C4C22×C418yes—
261C2×C4⋊C4C22×C418yes—
262C2×C4⋊C4C2×D418yes—
263C2×C4⋊C4C2×D418yes—
264C2×C4⋊C4C2×D418yes—
265C2×C22⋊C4—24no—
266C2×C22⋊C4—24no—
267C2×C22⋊C4—24no—
268C2×C22⋊C4—24no—
269C2×C22⋊C4—24no—
270C2×C22⋊C4—24no—
271C2×C22⋊C4—24no—
272C2×C22⋊C4—28no—
273C2×C22⋊C4—28no—
274C2×C22⋊C4—28no—
275C2×C22⋊C4—28no—
276C2×C22⋊C4—28no—
277C2×C22⋊C4—28no—
278C2×C22⋊C4—48no—
279C2×C22⋊C4—44no—
280C2×C22⋊C4—44no—
281C2×C22⋊C4—48no—
282C2×C22⋊C4—48no—
283C2×C22⋊C4—48no—
284C2×C22⋊C4C22×C418yes—
285C2×C22⋊C4C22×C4132yes—
286C2×C22⋊C4C2×D418yes—
287C2×C42—24no—
288C2×C42—24no—
289C2×C42—24no—
290C2×C42—24no—
291C2×C42—24no—
292C2×C42—24no—
293C2×C42—24no—
294C2×C42—24no—
295C2×C42—24no—
296C2×C42—24no—
297C2×C42—24no—
298C2×C42—24no—
299C2×C42—24no—
300C2×C42—216no—
301C2×C42—416no—
302C2×C42—48no—
303C2×C42—48no—
304C2×C42C22×C418yes—
305C2×C42C2×D418yes—
306#95—216no—
307#95—216no—
308#95—28no—
309#95—28no—
310#95—28no—
311#95—28no—
312#95—28no—
313#95—28no—
314#95—48no—
315#95—44no—
316#95—44no—
317#95—48no—
318#95C22×C4132yes—
319C24—22no—
320C24—22no—
321C24—22no—
322C24—22no—
323C24—22no—
324C24—22no—
325C24—22no—
326C24—22no—
327C24—22no—
328C24—22no—
329C24—24no—
330C24—24no—
331C24—24no—
332C24—24no—
333C24C23×C414yes—
334C24C23×C412yes—
335C24C22×D412yes—
336C24C22×D412yes—
337C24C2×#4112yes—
338C24C2×#4112yes—
339C24C2×#4112yes—
340C24C2×#4112yes—
341C24C2511yescharacteristic
342#41—4128no—
343C2×Q8—28no—
344C2×Q8—28no—
345C2×Q8—28no—
346C2×Q8—44no—
347C2×Q8—44no—
348C2×Q8—44no—
349C2×Q8—44no—
350C2×D4—216no—
351C2×D4—216no—
352C2×D4—216no—
353C2×D4—28no—
354C2×D4—28no—
355C2×D4—28no—
356C2×D4—48no—
357C2×D4—48no—
358C2×D4—48no—
359C2×D4—48no—
360C2×D4—44no—
361C2×D4—44no—
362C2×D4—44no—
363C2×D4—44no—
364C22×C4—28no—
365C22×C4—28no—
366C22×C4—216no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—24no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—24no—
374C22×C4—24no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—24no—
379C22×C4—24no—
380C22×C4—24no—
381C22×C4—24no—
382C22×C4—24no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—24no—
388C22×C4—24no—
389C22×C4—28no—
390C22×C4—28no—
391C22×C4—22no—
392C22×C4—24no—
393C22×C4—24no—
394C22×C4—22no—
395C22×C4—24no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—24no—
399C22×C4—24no—
400C22×C4—28no—
401C22×C4—24no—
402C22×C4—28no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—216no—
406C22×C4—216no—
407C22×C4—216no—
408C22×C4—216no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—24no—
416C22×C4—24no—
417C22×C4—24no—
418C22×C4—24no—
419C22×C4—416no—
420C22×C4—48no—
421C22×C4—48no—
422C22×C4—48no—
423C22×C4—48no—
424C22×C4—48no—
425C22×C4—48no—
426C22×C4—48no—
427C22×C4—44no—
428C22×C4—48no—
429C22×C4—44no—
430C22×C4—416no—
431C22×C4—44no—
432C22×C4—44no—
433C22×C4—48no—
434C22×C4—48no—
435C22×C4—48no—
436C22×C4—48no—
437C22×C4—88no—
438C22×C4—88no—
439C22×C4#117116yes—
440C22×C4C4×D4116yes—
441C22×C4C4×D4116yes—
442C22×C4#121116yes—
443C22×C4#123116yes—
444C22×C4#124116yes—
445C22×C4#126116yes—
446C22×C4C23×C414yes—
447C22×C4C23×C418yes—
448C22×C4C23×C418yes—
449C22×C4C23×C412yes—
450C22×C4C23×C412yes—
451C22×C4C22×D418yes—
452C22×C4C22×D418yes—
453C22×C4C22×D412yes—
454C22×C4C22×D412yes—
455C22×C4C22×D412yes—
456C22×C4C22×D412yes—
457C22×C4C2×#4112yes—
458C22×C4C2×#4112yes—
459C22×C4C2×#4112yes—
460C22×C4C2×#4112yes—
461C22×C4C2×#4112yes—
462C22×C4C2×#4112yes—
463C22×C4C2×#4112yes—
464C22×C4C2×#4112yes—
465C22×C4H(5,𝔽2)14yes—
466C22×C42−1+414yes—
467C4⋊C4—216no—
468C4⋊C4—216no—
469C4⋊C4—216no—
470C4⋊C4—216no—
471C4⋊C4—216no—
472C4⋊C4—216no—
473C4⋊C4—432no—
474C4⋊C4—44no—
475C4⋊C4—44no—
476C22⋊C4—216no—
477C22⋊C4—216no—
478C22⋊C4—432no—
479C22⋊C4—416no—
480C22⋊C4—416no—
481C22⋊C4—48no—
482C22⋊C4—88no—
483C22⋊C4—88no—
484C42—216no—
485C42—216no—
486C42—48no—
487C42—48no—
488C42—48no—
489C42—816no—
490C23—24no—
491C23—24no—
492C23—22no—
493C23—22no—
494C23—22no—
495C23—22no—
496C23—24no—
497C23—24no—
498C23—24no—
499C23—24no—
500C23—24no—
501C23—24no—
502C23—24no—
503C23—24no—
504C23—24no—
505C23—24no—
506C23—24no—
507C23—24no—
508C23—24no—
509C23—28no—
510C23—28no—
511C23—42no—
512C23—42no—
513C23—42no—
514C23—42no—
515C23—42no—
516C23—42no—
517C23—42no—
518C23—42no—
519C23—44no—
520C23—44no—
521C23—44no—
522C23—44no—
523C23—44no—
524C23—44no—
525C23—44no—
526C23—44no—
527C23—44no—
528C23—44no—
529C23C2×#11714yes—
530C23C2×C4×D414yes—
531C23C2×C4×D414yes—
532C23C2×#12114yes—
533C23C2×#12314yes—
534C23C2×#12414yes—
535C23C2×#12614yes—
536C23C24×C411yescharacteristic
537C23C23×D411yescharacteristic
538C23C23×D411yescharacteristic
539C23C22×#4111yescharacteristic
540C23C22×#4111yescharacteristic
541C23C22×#4111yescharacteristic
542C23C22×#4111yescharacteristic
543C23C2×H(5,𝔽2)12yes—
544C23C2×2−1+412yes—
545C23#58512yes—
546C23#58512yes—
547Q8—416no—
548D4—432no—
549D4—416no—
550C2×C4—24no—
551C2×C4—28no—
552C2×C4—28no—
553C2×C4—24no—
554C2×C4—24no—
555C2×C4—28no—
556C2×C4—216no—
557C2×C4—28no—
558C2×C4—28no—
559C2×C4—28no—
560C2×C4—28no—
561C2×C4—24no—
562C2×C4—28no—
563C2×C4—24no—
564C2×C4—28no—
565C2×C4—24no—
566C2×C4—24no—
567C2×C4—28no—
568C2×C4—24no—
569C2×C4—24no—
570C2×C4—416no—
571C2×C4—416no—
572C2×C4—432no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—44no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—44no—
583C2×C4—44no—
584C2×C4—48no—
585C2×C4—42no—
586C2×C4—44no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—48no—
590C2×C4—42no—
591C2×C4—44no—
592C2×C4—48no—
593C2×C4—44no—
594C2×C4—44no—
595C2×C4—44no—
596C2×C4—44no—
597C2×C4—44no—
598C2×C4—432no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—44no—
606C2×C4—44no—
607C2×C4—44no—
608C2×C4—44no—
609C2×C4—816no—
610C2×C4#388164yescomplemented
611C2×C4C2×#11714yes—
612C2×C4C2×#11714yes—
613C2×C4C2×C4×D414yes—
614C2×C4C2×C4×D414yes—
615C2×C4C2×C4×D414yes—
616C2×C4C2×C4×D414yes—
617C2×C4C2×#12114yes—
618C2×C4C2×#12114yes—
619C2×C4C2×#12314yes—
620C2×C4C2×#12314yes—
621C2×C4C2×#12414yes—
622C2×C4C2×#12414yes—
623C2×C4C2×#12614yes—
624C2×C4C2×#12614yes—
625C22—24no—
626C22—24no—
627C22—24no—
628C22—24no—
629C22—28no—
630C22—28no—
631C22—24no—
632C22—28no—
633C22—24no—
634C22—28no—
635C22—48no—
636C22—44no—
637C22—44no—
638C22—48no—
639C22—44no—
640C22—82no—
641C22—82no—
642C22—82no—
643C22—82no—
644C22—84no—
645C22—84no—
646C22C2×#38818yes—
647C22C22×#11711yescharacteristic
648C22C22×C4×D411yescharacteristic
649C22C22×C4×D411yescharacteristic
650C22C4×H(5,𝔽2)12yes—
651C22C4×2−1+412yes—
652C22C22×#12111yescharacteristic
653C22C22×#12311yescharacteristic
654C22C22×#12411yescharacteristic
655C22C22×#12611yescharacteristic
656C22#348411yescharacteristic
657C22#348511yescharacteristic
658C22#348611yescharacteristic
659C22#348711yescharacteristic
660C22#348812yes—
661C22#348912yes—
662C22#349412yes—
663C22#349512yes—
664C22#350612yes—
665C22#350712yes—
666C22#351112yes—
667C22#351112yes—
668C22#351212yes—
669C22#351212yes—
670C4—44no—
671C4—44no—
672C4—48no—
673C4—48no—
674C4—48no—
675C4—48no—
676C4—44no—
677C4—82no—
678C4—82no—
679C4C2×#38818yes—
680C4C2×#38818yes—
681C2—28no—
682C2—84no—
683C2C22×#38811yescharacteristic
684C2#3425312yes—
685C2#3426511yescharacteristic
686C2#3426611yescharacteristic
687C2#3426712yes—
688C2#3551411yescharacteristic
689C2#3551712yes—
690C2#3554512yes—
691C2#3559211yescharacteristic
692C2#3564911yescharacteristic
693C2#3566111yescharacteristic
subgroupG/H
center, Frattini, socle, cosocleC24C25
commutatorC23C24×C4
prime radicalC23×#41C22
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
4. C22×#389C2semidirect
5. C2×#2274C2semidirect
6. #14142C2semidirect
7. #14141C2semidirect
8. #14020C2semidirect
9. #14019C2semidirect
10. #13994C2semidirect
11. #13993C2semidirect
12. #13795C2semidirect
13. #13756C2semidirect
23. #2311C4semidirect
125. #391C2×C4semidirect
128. #390C2×C4semidirect
138. #380C2×C4semidirect
610. C2×C4#388semidirect

Maximal subgroups

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

subgroupindex
C2×#23112
C2×#22972
C22×#3892
C2×#22742
#141422
#141412
#140202
#140192
#139942
#139932
#137952
#137562