Finite Group Databasebeta

← #515376

Subgroups — #515376

Order 512.

3510 conjugacy classes of subgroups.

The subgroup lattice has 701 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2312C211yesmaximal, characteristic
2C22×#388C211yesmaximal, characteristic
3C2×#2274C211yesmaximal, characteristic, complemented
4#14085C214yesmaximal, complemented
5#14079C214yesmaximal, complemented
6#14037C214yesmaximal, complemented
7#14031C214yesmaximal, complemented
8#13883C212yesmaximal, complemented
9#13878C212yesmaximal, complemented
10#13873C212yesmaximal
11#13868C212yesmaximal
12#13776C211yesmaximal, characteristic, complemented
13#13775C211yesmaximal, characteristic
14C4×#380C211yesmaximal, characteristic, complemented
15#13753C211yesmaximal, characteristic, complemented
16C22×#117C2212yes—
17C23×C4⋊C4C2211yescharacteristic
18C23×C22⋊C4C2211yescharacteristic
19C23×C22⋊C4C2211yescharacteristic
20C23×C22⋊C4C2211yescharacteristic
21C23×C42C2211yescharacteristic
22#2312—28no—
23C2×#388C4116yescomplemented
24C2×#388C2214yes—
25C2×#388C2214yes—
26C2×#388C2214yes—
27C2×#388C2212yes—
28C2×#388C2212yes—
29C2×#388C2212yes—
30C2×#388C2212yes—
31C2×#381C2212yes—
32C2×#380C2212yes—
33C2×#380C2212yes—
34C2×#379C2212yes—
35C2×#379C2212yes—
36#2274—28no—
37C2×C4×C4⋊C4C2214yescomplemented
38C2×C4×C4⋊C4C2214yescomplemented
39C2×C4×C4⋊C4C2212yescomplemented
40C2×C4×C22⋊C4C2212yes—
41C2×C4×C22⋊C4C2211yescharacteristic
42C2×C4×C22⋊C4C2211yescharacteristic
43C2×C4×C22⋊C4C2211yescharacteristic
44C2×C4×C22⋊C4C2211yescharacteristic
45C2×#376C2212yescomplemented
46C22×#95C2211yescharacteristic
47C22×#95C2211yescharacteristic
48#1444C2214yescomplemented
49#1443C2214yescomplemented
50#1442C2214yescomplemented
51#1442C2214yescomplemented
52#1440C2214yes—
53#1439C2214yes—
54#1439C2214yes—
55#1439C2214yes—
56#1437C2214yes—
57#1437C2214yes—
58#1437C2214yes—
59#1437C2214yes—
60#1437C2214yes—
61#1436C2214yescomplemented
62#1436C2214yescomplemented
63#1436C2214yescomplemented
64#1436C2214yescomplemented
65#1436C2214yescomplemented
66C4⋊#95C2212yescomplemented
67#1434C2212yescomplemented
68#1434C2214yescomplemented
69#1434C2212yescomplemented
70#1434C2212yescomplemented
71C22⋊#95C2212yes—
72C22⋊#95C2212yes—
73C22⋊#95C2212yes—
74C22⋊#95C2212yes—
75C22⋊#95C2212yes—
76C4×#95C2212yescomplemented
77C4×#95C2212yescomplemented
78C26C2311yescharacteristic
79C24×C4C2311yescharacteristic
80C24×C4C2311yescharacteristic
81C24×C4C2311yescharacteristic
82C24×C4C2311yescharacteristic
83C24×C4C2311yescharacteristic
84C24×C4C2311yescharacteristic
85C2×#117—28no—
86C2×#117—28no—
87C2×#117—28no—
88C22×C4⋊C4—24no—
89C22×C4⋊C4—24no—
90C22×C4⋊C4C2×C418yes—
91C22×C4⋊C4C2314yes—
92C22×C4⋊C4C2314yes—
93C22×C4⋊C4C2312yes—
94C22×C4⋊C4C2312yes—
95C22×C4⋊C4C2314yes—
96C22×C22⋊C4—24no—
97C22×C22⋊C4—28no—
98C22×C22⋊C4—24no—
99C22×C22⋊C4—44no—
100C22×C22⋊C4C2×C418yes—
101C22×C22⋊C4C2×C418yes—
102C22×C22⋊C4C2×C418yes—
103C22×C22⋊C4C2312yes—
104C22×C22⋊C4C2312yes—
105C22×C22⋊C4C2312yes—
106C22×C22⋊C4C2312yes—
107C22×C22⋊C4C2312yes—
108C22×C22⋊C4C2312yes—
109C22×C22⋊C4C2312yes—
110C22×C22⋊C4C2312yes—
111C22×C22⋊C4C2312yes—
112C22×C22⋊C4C2314yes—
113C22×C22⋊C4C2314yes—
114C22×C22⋊C4C2314yes—
115C22×C22⋊C4C2314yes—
116C22×C22⋊C4C2312yes—
117C22×C42—44no—
118C22×C42C2×C418yes—
119C22×C42C2314yes—
120C22×C42C2311yescharacteristic
121C22×C42C2311yescharacteristic
122C22×C42C2312yes—
123C22×C42C2312yes—
124C22×C42C2312yes—
125C22×C42C2314yes—
126C22×C42C2311yescharacteristic
127C22×C42C2311yescharacteristic
128C22×C42C2312yes—
129#388—216no—
130#388—216no—
131#388—48no—
132#388C2×C4116yes—
133#388C2×C4116yes—
134#388C2×C4116yes—
135#388C2×C4116yes—
136#381—28no—
137#380—28no—
138#380—28no—
139#379—28no—
140#379—28no—
141C4×C4⋊C4—216no—
142C4×C4⋊C4—216no—
143C4×C4⋊C4—44no—
144C4×C22⋊C4—24no—
145C4×C22⋊C4—24no—
146C4×C22⋊C4—24no—
147C4×C22⋊C4—28no—
148C4×C22⋊C4D418yes—
149#376—44no—
150C2×#95C2×C418yes—
151C2×#95C2×C418yes—
152C2×#95C2314yes—
153C2×#95C2314yes—
154C2×#95C2312yes—
155C2×#95C2312yes—
156C2×#95C2312yes—
157C2×#95C2314yes—
158C2×#95C2314yes—
159C2×#95C2314yes—
160C2×#95C2314yes—
161C2×#95C2314yes—
162C2×#95C2314yes—
163C2×#95C2314yes—
164C2×#95C2314yes—
165C2×#95C2314yes—
166C2×#95C2314yes—
167C2×#95C2314yes—
168C2×#95C2314yes—
169C2×#95C2314yes—
170C2×#95C2312yes—
171C2×#95C2312yes—
172C2×#95C2312yes—
173C2×#95C2312yes—
174C2×#95C2312yes—
175C25—24no—
176C25—22no—
177C25—22no—
178C25—24no—
179C25—42no—
180C25—42no—
181C25—42no—
182C25—42no—
183C25C22×C414yes—
184C25C2412yes—
185C25C2411yescharacteristic
186C23×C4—22no—
187C23×C4—22no—
188C23×C4—24no—
189C23×C4—24no—
190C23×C4—24no—
191C23×C4—22no—
192C23×C4—22no—
193C23×C4—22no—
194C23×C4—22no—
195C23×C4—24no—
196C23×C4—42no—
197C23×C4—42no—
198C23×C4—42no—
199C23×C4—42no—
200C23×C4—42no—
201C23×C4—42no—
202C23×C4—42no—
203C23×C4—42no—
204C23×C4—42no—
205C23×C4C22×C414yes—
206C23×C4C22×C414yes—
207C23×C4C22×C414yes—
208C23×C4C22×C414yes—
209C23×C4C22×C414yes—
210C23×C4C22×C414yes—
211C23×C4C2×D414yes—
212C23×C4#4118yes—
213C23×C4#4114yes—
214C23×C4C2414yes—
215C23×C4C2414yes—
216C23×C4C2412yes—
217C23×C4C2412yes—
218C23×C4C2411yescharacteristic
219C23×C4C2411yescharacteristic
220C23×C4C2411yescharacteristic
221C23×C4C2414yes—
222C23×C4C2414yes—
223C23×C4C2411yescharacteristic
224C23×C4C2411yescharacteristic
225C23×C4C2412yes—
226C23×C4C2411yescharacteristic
227#117—264no—
228C2×C4⋊C4—232no—
229C2×C4⋊C4—24no—
230C2×C4⋊C4—24no—
231C2×C4⋊C4—28no—
232C2×C4⋊C4—24no—
233C2×C4⋊C4—24no—
234C2×C4⋊C4—28no—
235C2×C4⋊C4—28no—
236C2×C4⋊C4—28no—
237C2×C4⋊C4—28no—
238C2×C4⋊C4—28no—
239C2×C4⋊C4—44no—
240C2×C4⋊C4—44no—
241C2×C4⋊C4—44no—
242C2×C4⋊C4C22×C418yes—
243C2×C4⋊C4C22×C418yes—
244C2×C22⋊C4—24no—
245C2×C22⋊C4—24no—
246C2×C22⋊C4—24no—
247C2×C22⋊C4—24no—
248C2×C22⋊C4—232no—
249C2×C22⋊C4—232no—
250C2×C22⋊C4—24no—
251C2×C22⋊C4—24no—
252C2×C22⋊C4—24no—
253C2×C22⋊C4—24no—
254C2×C22⋊C4—24no—
255C2×C22⋊C4—24no—
256C2×C22⋊C4—24no—
257C2×C22⋊C4—24no—
258C2×C22⋊C4—28no—
259C2×C22⋊C4—24no—
260C2×C22⋊C4—24no—
261C2×C22⋊C4—24no—
262C2×C22⋊C4—24no—
263C2×C22⋊C4—24no—
264C2×C22⋊C4—24no—
265C2×C22⋊C4—28no—
266C2×C22⋊C4—28no—
267C2×C22⋊C4—28no—
268C2×C22⋊C4—28no—
269C2×C22⋊C4—416no—
270C2×C22⋊C4—44no—
271C2×C22⋊C4—44no—
272C2×C22⋊C4—44no—
273C2×C22⋊C4—44no—
274C2×C22⋊C4—44no—
275C2×C22⋊C4—44no—
276C2×C22⋊C4—44no—
277C2×C22⋊C4—44no—
278C2×C22⋊C4—44no—
279C2×C22⋊C4C22×C418yes—
280C2×C22⋊C4C22×C418yes—
281C2×C22⋊C4C22×C418yes—
282C2×C22⋊C4C22×C418yes—
283C2×C22⋊C4C22×C418yes—
284C2×C22⋊C4C22×C418yes—
285C2×C22⋊C4C2×D418yes—
286C2×C22⋊C4#4118yes—
287C2×C42—28no—
288C2×C42—28no—
289C2×C42—28no—
290C2×C42—24no—
291C2×C42—24no—
292C2×C42—24no—
293C2×C42—24no—
294C2×C42—28no—
295C2×C42—24no—
296C2×C42—416no—
297C2×C42—42no—
298C2×C42—42no—
299C2×C42—42no—
300C2×C42—42no—
301C2×C42—42no—
302C2×C42—42no—
303C2×C42—42no—
304C2×C42—42no—
305C2×C42—44no—
306C2×C42—42no—
307C2×C42—42no—
308C2×C42—44no—
309C2×C42C22×C414yes—
310C2×C42C22×C414yes—
311C2×C42C22×C414yes—
312C2×C42C22×C414yes—
313C2×C42C2×D418yes—
314#95—24no—
315#95—28no—
316#95—24no—
317#95—28no—
318#95—24no—
319#95—28no—
320#95—28no—
321#95—28no—
322#95—28no—
323#95—44no—
324#95—44no—
325#95—44no—
326#95—44no—
327#95—44no—
328#95—44no—
329#95—44no—
330#95—44no—
331#95—44no—
332#95C22×C418yes—
333#95C22×C418yes—
334#95C22×C418yes—
335#95C22×C418yes—
336#95C2×D418yes—
337C24—22no—
338C24—22no—
339C24—22no—
340C24—22no—
341C24—28no—
342C24—22no—
343C24—22no—
344C24—22no—
345C24—22no—
346C24—22no—
347C24—22no—
348C24—24no—
349C24—24no—
350C24—24no—
351C24—24no—
352C24—28no—
353C24—28no—
354C24—44no—
355C24—44no—
356C24—44no—
357C24—44no—
358C24—416no—
359C24—416no—
360C24—48no—
361C24—48no—
362C24—84no—
363C24—84no—
364C24—84no—
365C24—84no—
366C24—88no—
367C24—88no—
368C24C23×C414yes—
369C24C23×C412yes—
370C24C22×D412yes—
371C24C2×#4114yes—
372C24C2×#4112yes—
373C24C2×#4114yes—
374C24C2511yescharacteristic
375C22×C4—28no—
376C22×C4—28no—
377C22×C4—216no—
378C22×C4—216no—
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—28no—
390C22×C4—28no—
391C22×C4—24no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—24no—
396C22×C4—24no—
397C22×C4—24no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—24no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—24no—
406C22×C4—24no—
407C22×C4—22no—
408C22×C4—24no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—24no—
413C22×C4—24no—
414C22×C4—24no—
415C22×C4—24no—
416C22×C4—24no—
417C22×C4—24no—
418C22×C4—24no—
419C22×C4—24no—
420C22×C4—28no—
421C22×C4—28no—
422C22×C4—28no—
423C22×C4—24no—
424C22×C4—24no—
425C22×C4—24no—
426C22×C4—24no—
427C22×C4—24no—
428C22×C4—24no—
429C22×C4—24no—
430C22×C4—24no—
431C22×C4—24no—
432C22×C4—24no—
433C22×C4—24no—
434C22×C4—24no—
435C22×C4—24no—
436C22×C4—24no—
437C22×C4—44no—
438C22×C4—44no—
439C22×C4—44no—
440C22×C4—44no—
441C22×C4—44no—
442C22×C4—48no—
443C22×C4—48no—
444C22×C4—44no—
445C22×C4—44no—
446C22×C4—44no—
447C22×C4—44no—
448C22×C4—416no—
449C22×C4—416no—
450C22×C4—84no—
451C22×C4—84no—
452C22×C4—84no—
453C22×C4—84no—
454C22×C4#117116yes—
455C22×C4C4×D4116yes—
456C22×C4C23×C414yes—
457C22×C4C23×C412yes—
458C22×C4C23×C412yes—
459C22×C4C23×C412yes—
460C22×C4C23×C414yes—
461C22×C4C23×C414yes—
462C22×C4C23×C412yes—
463C22×C4C23×C412yes—
464C22×C4C23×C412yes—
465C22×C4C22×D418yes—
466C22×C4C22×D412yes—
467C22×C4C22×D412yes—
468C22×C4C2×#4118yes—
469C22×C4C2×#4114yes—
470C22×C4C2×#4114yes—
471C22×C4C2×#4112yes—
472C22×C4C2×#4112yes—
473C22×C4H(5,𝔽2)14yes—
474C4⋊C4—232no—
475C4⋊C4—216no—
476C4⋊C4—216no—
477C4⋊C4—416no—
478C4⋊C4—88no—
479C22⋊C4—216no—
480C22⋊C4—216no—
481C22⋊C4—216no—
482C22⋊C4—216no—
483C22⋊C4—216no—
484C22⋊C4—216no—
485C22⋊C4—216no—
486C22⋊C4—48no—
487C22⋊C4—48no—
488C22⋊C4—48no—
489C22⋊C4—88no—
490C22⋊C4—88no—
491C22⋊C4—88no—
492C22⋊C4—88no—
493C42—232no—
494C42—28no—
495C42—28no—
496C42—44no—
497C42—44no—
498C42—48no—
499C42—48no—
500C42—84no—
501C42—84no—
502C42—88no—
503C23—22no—
504C23—22no—
505C23—22no—
506C23—22no—
507C23—28no—
508C23—24no—
509C23—24no—
510C23—24no—
511C23—28no—
512C23—24no—
513C23—28no—
514C23—28no—
515C23—24no—
516C23—24no—
517C23—24no—
518C23—24no—
519C23—24no—
520C23—24no—
521C23—24no—
522C23—416no—
523C23—416no—
524C23—42no—
525C23—42no—
526C23—42no—
527C23—42no—
528C23—44no—
529C23—44no—
530C23—44no—
531C23—44no—
532C23—44no—
533C23—44no—
534C23—44no—
535C23—44no—
536C23—44no—
537C23—44no—
538C23—44no—
539C23—44no—
540C23—416no—
541C23—88no—
542C23—88no—
543C23—88no—
544C23—88no—
545C23—832no—
546C23—832no—
547C23C2×#11714yes—
548C23C2×C4×D414yes—
549C23C4×#4114yes—
550C23#52518yes—
551C23#52918yes—
552C23C24×C411yescharacteristic
553C23C23×D411yescharacteristic
554C23C22×#4112yes—
555C23C22×#4111yescharacteristic
556C23C22×#4112yes—
557C23C2×H(5,𝔽2)12yes—
558C23#58512yes—
559C23#58512yes—
560C23#58512yes—
561C2×C4—24no—
562C2×C4—22no—
563C2×C4—24no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—24no—
570C2×C4—24no—
571C2×C4—24no—
572C2×C4—24no—
573C2×C4—24no—
574C2×C4—24no—
575C2×C4—24no—
576C2×C4—24no—
577C2×C4—28no—
578C2×C4—232no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—24no—
582C2×C4—24no—
583C2×C4—22no—
584C2×C4—22no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—24no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—24no—
591C2×C4—24no—
592C2×C4—24no—
593C2×C4—22no—
594C2×C4—28no—
595C2×C4—28no—
596C2×C4—28no—
597C2×C4—28no—
598C2×C4—28no—
599C2×C4—28no—
600C2×C4—416no—
601C2×C4—416no—
602C2×C4—416no—
603C2×C4—44no—
604C2×C4—44no—
605C2×C4—44no—
606C2×C4—44no—
607C2×C4—44no—
608C2×C4—44no—
609C2×C4—44no—
610C2×C4—44no—
611C2×C4—44no—
612C2×C4—44no—
613C2×C4—44no—
614C2×C4—44no—
615C2×C4—44no—
616C2×C4—44no—
617C2×C4—44no—
618C2×C4—44no—
619C2×C4—44no—
620C2×C4—44no—
621C2×C4—44no—
622C2×C4—44no—
623C2×C4—44no—
624C2×C4—44no—
625C2×C4—44no—
626C2×C4—44no—
627C2×C4—44no—
628C2×C4—44no—
629C2×C4—44no—
630C2×C4—44no—
631C2×C4—44no—
632C2×C4—88no—
633C2×C4—88no—
634C2×C4C2×#11718yes—
635C2×C4C2×C4×D414yes—
636C2×C4C2×C4×D414yes—
637C2×C4#51814yes—
638C22—216no—
639C22—216no—
640C22—24no—
641C22—24no—
642C22—24no—
643C22—24no—
644C22—24no—
645C22—24no—
646C22—44no—
647C22—44no—
648C22—44no—
649C22—44no—
650C22—44no—
651C22—44no—
652C22—44no—
653C22—832no—
654C22—84no—
655C22—84no—
656C22—84no—
657C22—84no—
658C22#229018yes—
659C22C22×#11711yescharacteristic
660C22C22×C4×D411yescharacteristic
661C22C2×C4×#4111yescharacteristic
662C22C2×#51811yescharacteristic
663C22#342811yescharacteristic
664C22#342811yescharacteristic
665C22#342811yescharacteristic
666C22C2×#52512yes—
667C22C2×#52912yes—
668C22#349314yes—
669C22#349612yes—
670C22#350012yes—
671C22#350512yes—
672C22#350614yes—
673C22#351012yes—
674C22#351314yes—
675C22#352114yes—
676C4—24no—
677C4—24no—
678C4—24no—
679C4—28no—
680C4—48no—
681C4—48no—
682C4—42no—
683C4—42no—
684C4—44no—
685C4—44no—
686C4—42no—
687C4—84no—
688C4—84no—
689C2—28no—
690C2—84no—
691C2C2×#229011yescharacteristic
692C2#3417911yescharacteristic
693C2#3418611yescharacteristic
694C2#3420511yescharacteristic
695C2#3421111yescharacteristic
696C2#3423811yescharacteristic
697C2#3424711yescharacteristic
698C2#3536414yes—
699C2#3538114yes—
subgroupG/H
center, Frattini, socle, cosocleC24C25
commutatorC23C24×C4
prime radicalC26C23
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
3. C2×#2274C2semidirect
4. #14085C2semidirect
5. #14079C2semidirect
6. #14037C2semidirect
7. #14031C2semidirect
8. #13883C2semidirect
9. #13878C2semidirect
12. #13776C2semidirect
14. C4×#380C2semidirect
15. #13753C2semidirect
23. C2×#388C4semidirect
37. C2×C4×C4⋊C4C22semidirect
38. C2×C4×C4⋊C4C22semidirect
39. C2×C4×C4⋊C4C22semidirect
45. C2×#376C22semidirect
48. #1444C22semidirect
49. #1443C22semidirect
50. #1442C22semidirect
51. #1442C22semidirect
61. #1436C22semidirect
62. #1436C22semidirect
63. #1436C22semidirect
64. #1436C22semidirect
65. #1436C22semidirect
66. C4⋊#95C22semidirect
67. #1434C22semidirect
68. #1434C22semidirect
69. #1434C22semidirect
70. #1434C22semidirect
76. C4×#95C22semidirect
77. C4×#95C22semidirect

Maximal subgroups

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

subgroupindex
C2×#23122
C22×#3882
C2×#22742
#140852
#140792
#140372
#140312
#138832
#138782
#138732
#138682
#137762
#137752
C4×#3802
#137532