Finite Group Databasebeta

← #545393

Subgroups — #545393

Order 512.

2786 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×#2580C211yesmaximal, characteristic
2C2×Q8×C4⋊C4C211yesmaximal, characteristic, complemented
3C2×#2319C211yesmaximal, characteristic
4#14628C212yesmaximal, complemented
5#14402C214yesmaximal, complemented
6#14382C214yesmaximal, complemented
7#14322C212yesmaximal, complemented
8#14301C212yesmaximal, complemented
9#14277C211yesmaximal, characteristic, complemented
10#14188C212yesmaximal, complemented
11#14141C212yesmaximal
12#14132C212yesmaximal, complemented
13#14005C212yesmaximal, complemented
14#14001C212yesmaximal
15#13906C211yesmaximal, characteristic
16#13850C211yesmaximal, characteristic, complemented
17#13804C211yesmaximal, characteristic, complemented
18C22×C4×Q8C2211yescharacteristic
19C22×C4×Q8C2212yes—
20C22×C4×D4C2211yescharacteristic
21#2580C4116yescomplemented
22C2×#399C2212yes—
23C2×#393C2212yes—
24Q8×C4⋊C4—28no—
25#2319—28no—
26C2×#390C2211yescharacteristic
27C2×#390C2211yescharacteristic
28C2×#390C2212yes—
29C2×#390C2212yes—
30C2×#389C2211yescharacteristic
31C2×#389C2211yescharacteristic
32C2×#389C2212yes—
33C2×#389C2214yes—
34C2×#389C2212yes—
35C2×#389C2214yes—
36C2×#389C2212yes—
37C2×#388C2212yes—
38C2×#388C2212yes—
39C2×#388C2214yes—
40C2×#388C2214yes—
41C2×#388C2214yes—
42C2×#385C2214yes—
43C2×#384C2211yescharacteristic
44C2×#384C2212yes—
45C2×#382C2211yescharacteristic
46C2×#381C2212yes—
47C2×#380C2212yes—
48C2×#380C2214yes—
49C2×C4×C4⋊C4C2211yescharacteristic
50C2×C4×C4⋊C4C2212yes—
51C2×C4×C4⋊C4C2212yes—
52C2×C4×C4⋊C4C2211yescharacteristic
53C2×#376C2211yescharacteristic
54#1446C2211yescharacteristic
55#1446C2211yescharacteristic
56#1445C2211yescharacteristic
57#1445C2211yescharacteristic
58#1445C2214yes—
59#1444C2212yes—
60#1444C2214yes—
61#1444C2212yes—
62#1442C2212yes—
63#1442C2212yes—
64#1441C2212yes—
65#1441C2212yes—
66#1441C2214yes—
67#1441C2214yes—
68#1440C2214yes—
69#1439C2214yes—
70#1439C2212yes—
71#1439C2214yes—
72#1439C2214yes—
73#1439C2212yes—
74#1437C2214yes—
75#1436C2214yes—
76#1436C2214yes—
77#1436C2214yes—
78C4⋊#95C2212yes—
79C4⋊#95C2212yes—
80C4⋊#95C2212yes—
81#1434C2212yes—
82#1434C2214yes—
83C4×#95C2212yes—
84C4×#95C2212yes—
85C4×#95C2212yes—
86C23×Q8C2311yescharacteristic
87C23×D4C2311yescharacteristic
88C24×C4C2312yes—
89C2×C4×Q8—24no—
90C2×C4×Q8—28no—
91C2×C4×Q8—24no—
92C2×C4×Q8—48no—
93C2×C4×Q8C2×C418yes—
94C2×C4×D4—24no—
95C2×C4×D4—24no—
96C2×C4×D4C2×C418yes—
97C22×C4⋊C4C2312yes—
98C22×C4⋊C4C2314yes—
99C22×C4⋊C4C2312yes—
100C22×C4⋊C4C2311yescharacteristic
101C22×C4⋊C4C2311yescharacteristic
102C22×C4⋊C4C2312yes—
103C22×C4⋊C4C2312yes—
104C22×C4⋊C4C2311yescharacteristic
105C22×C4⋊C4C2311yescharacteristic
106C22×C4⋊C4C2312yes—
107C22×C4⋊C4C2312yes—
108C22×C4⋊C4C2314yes—
109C22×C4⋊C4C2312yes—
110C22×C4⋊C4C2312yes—
111C22×C4⋊C4C2314yes—
112C22×C4⋊C4C2314yes—
113C22×C22⋊C4C2312yes—
114C22×C22⋊C4C2314yes—
115C22×C22⋊C4C2312yes—
116C22×C22⋊C4C2314yes—
117C22×C22⋊C4C2312yes—
118C22×C22⋊C4C2312yes—
119C22×C22⋊C4C2312yes—
120C22×C22⋊C4C2314yes—
121C22×C22⋊C4C2314yes—
122C22×C42C2311yescharacteristic
123C22×C42C2312yes—
124C22×C42C2311yescharacteristic
125C22×C42C2314yes—
126C22×C42C2312yes—
127C22×C42C2314yes—
128C22×C42C2312yes—
129C22×C42C2311yescharacteristic
130#399C2×C4116yes—
131#393C2×C4116yes—
132#390—24no—
133#390—24no—
134#390—44no—
135#390C2×C4116yes—
136#389—24no—
137#389—24no—
138#389—28no—
139#389—216no—
140#389—48no—
141#389—44no—
142#389C2×C4116yes—
143#388—28no—
144#388—28no—
145#388—48no—
146#388—48no—
147#388—48no—
148#385C2×C4132yes—
149#384—28no—
150#384—24no—
151#382C2×C418yes—
152#381—28no—
153#380—28no—
154#380—48no—
155C4×C4⋊C4—28no—
156C4×C4⋊C4—24no—
157C4×C4⋊C4—44no—
158C4×C4⋊C4D418yes—
159#376—24no—
160C2×#95C2311yescharacteristic
161C2×#95C2312yes—
162C2×#95C2312yes—
163C2×#95C2312yes—
164C2×#95C2312yes—
165C2×#95C2312yes—
166C2×#95C2312yes—
167C2×#95C2311yescharacteristic
168C2×#95C2312yes—
169C2×#95C2314yes—
170C2×#95C2312yes—
171C2×#95C2314yes—
172C2×#95C2311yescharacteristic
173C2×#95C2311yescharacteristic
174C2×#95C2311yescharacteristic
175C2×#95C2311yescharacteristic
176C2×#95C2314yes—
177C2×#95C2312yes—
178C2×#95C2312yes—
179C2×#95C2314yes—
180C2×#95C2314yes—
181C2×#95C2314yes—
182C2×#95C2314yes—
183C2×#95C2314yes—
184C2×#95C2314yes—
185C2×#95C2312yes—
186C2×#95C2314yes—
187C2×#95C2312yes—
188C2×#95C2312yes—
189C25C2412yes—
190C22×Q8—22no—
191C22×Q8—22no—
192C22×Q8—24no—
193C22×Q8—24no—
194C22×Q8C22×C414yes—
195C22×D4—22no—
196C22×D4—22no—
197C22×D4—22no—
198C22×D4—22no—
199C22×D4—22no—
200C22×D4—22no—
201C22×D4C22×C414yes—
202C23×C4—24no—
203C23×C4—24no—
204C23×C4—44no—
205C23×C4—44no—
206C23×C4C22×C418yes—
207C23×C4C2411yescharacteristic
208C23×C4C2414yes—
209C23×C4C2411yescharacteristic
210C23×C4C2412yes—
211C23×C4C2411yescharacteristic
212C23×C4C2411yescharacteristic
213C23×C4C2412yes—
214C23×C4C2414yes—
215C23×C4C2412yes—
216C23×C4C2412yes—
217C23×C4C2412yes—
218C23×C4C2412yes—
219C23×C4C2411yescharacteristic
220C23×C4C2411yescharacteristic
221C23×C4C2412yes—
222C23×C4C2411yescharacteristic
223C4×Q8—232no—
224C4×Q8—432no—
225C4×D4—232no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—24no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—24no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—24no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—28no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—28no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—24no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—28no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—44no—
251C2×C4⋊C4—42no—
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⋊C4—44no—
262C2×C4⋊C4—44no—
263C2×C4⋊C4C22×C414yes—
264C2×C4⋊C4C22×C418yes—
265C2×C4⋊C4C22×C414yes—
266C2×C4⋊C4C22×C418yes—
267C2×C4⋊C4C22×C4116yes—
268C2×C4⋊C4C2×D414yes—
269C2×C4⋊C4C2×D414yes—
270C2×C22⋊C4—24no—
271C2×C22⋊C4—24no—
272C2×C22⋊C4—28no—
273C2×C22⋊C4—28no—
274C2×C22⋊C4—24no—
275C2×C22⋊C4—24no—
276C2×C22⋊C4—24no—
277C2×C22⋊C4—24no—
278C2×C22⋊C4—28no—
279C2×C22⋊C4—44no—
280C2×C22⋊C4—44no—
281C2×C22⋊C4—44no—
282C2×C22⋊C4—44no—
283C2×C22⋊C4—44no—
284C2×C22⋊C4—44no—
285C2×C22⋊C4—44no—
286C2×C22⋊C4—44no—
287C2×C22⋊C4—44no—
288C2×C22⋊C4—44no—
289C2×C22⋊C4—44no—
290C2×C22⋊C4C22×C418yes—
291C2×C22⋊C4C22×C4116yes—
292C2×C22⋊C4C22×C4116yes—
293C2×C42—24no—
294C2×C42—24no—
295C2×C42—28no—
296C2×C42—22no—
297C2×C42—22no—
298C2×C42—24no—
299C2×C42—28no—
300C2×C42—24no—
301C2×C42—22no—
302C2×C42—22no—
303C2×C42—22no—
304C2×C42—22no—
305C2×C42—24no—
306C2×C42—28no—
307C2×C42—44no—
308C2×C42—44no—
309C2×C42—44no—
310C2×C42—44no—
311C2×C42C22×C418yes—
312C2×C42C22×C414yes—
313C2×C42C2×D418yes—
314C2×C42C2×D414yes—
315C2×C42#4114yes—
316#95—22no—
317#95—22no—
318#95—22no—
319#95—22no—
320#95—28no—
321#95—24no—
322#95—24no—
323#95—22no—
324#95—22no—
325#95—28no—
326#95—24no—
327#95—24no—
328#95—24no—
329#95—24no—
330#95—24no—
331#95—24no—
332#95—44no—
333#95—42no—
334#95—42no—
335#95—44no—
336#95—44no—
337#95—44no—
338#95—44no—
339#95—44no—
340#95—44no—
341#95C22×C418yes—
342#95C22×C418yes—
343#95C22×C4116yes—
344#95C2×D418yes—
345C24—22no—
346C24—22no—
347C24—24no—
348C24—24no—
349C24—22no—
350C24—24no—
351C24—22no—
352C24—22no—
353C24—22no—
354C24—24no—
355C24C23×C414yes—
356C24C2511yescharacteristic
357C2×Q8—28no—
358C2×Q8—28no—
359C2×Q8—216no—
360C2×Q8—28no—
361C2×Q8—216no—
362C2×D4—28no—
363C2×D4—28no—
364C2×D4—28no—
365C2×D4—44no—
366C2×D4—44no—
367C2×D4—44no—
368C2×D4—44no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—24no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—24no—
388C22×C4—24no—
389C22×C4—24no—
390C22×C4—24no—
391C22×C4—22no—
392C22×C4—24no—
393C22×C4—24no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—24no—
397C22×C4—24no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—24no—
404C22×C4—24no—
405C22×C4—22no—
406C22×C4—24no—
407C22×C4—216no—
408C22×C4—21no—
409C22×C4—21no—
410C22×C4—21no—
411C22×C4—21no—
412C22×C4—24no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—24no—
419C22×C4—22no—
420C22×C4—21no—
421C22×C4—22no—
422C22×C4—24no—
423C22×C4—22no—
424C22×C4—24no—
425C22×C4—24no—
426C22×C4—24no—
427C22×C4—21no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—21no—
432C22×C4—21no—
433C22×C4—21no—
434C22×C4—21no—
435C22×C4—21no—
436C22×C4—21no—
437C22×C4—22no—
438C22×C4—24no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—416no—
443C22×C4—88no—
444C22×C4—84no—
445C22×C4—84no—
446C22×C4C23×C418yes—
447C22×C4C23×C414yes—
448C22×C4C23×C414yes—
449C22×C4C23×C412yes—
450C22×C4C23×C412yes—
451C22×C4C23×C414yes—
452C22×C4C23×C412yes—
453C22×C4C22×D414yes—
454C22×C4C22×D414yes—
455C22×C4C22×D412yes—
456C22×C4C22×D412yes—
457C22×C4C22×D412yes—
458C22×C4C2×#4112yes—
459C22×C4C2×#4112yes—
460C22×C4C2×#4112yes—
461C22×C4H(5,𝔽2)14yes—
462C22×C4H(5,𝔽2)14yes—
463C22×C4H(5,𝔽2)12yes—
464C22×C42−1+414yes—
465C22×C42−1+412yes—
466C22×C42−1+412yes—
467C22×C42−1+412yes—
468C4⋊C4—28no—
469C4⋊C4—216no—
470C4⋊C4—28no—
471C4⋊C4—216no—
472C4⋊C4—28no—
473C4⋊C4—28no—
474C4⋊C4—416no—
475C4⋊C4—48no—
476C4⋊C4—416no—
477C4⋊C4—48no—
478C4⋊C4—48no—
479C4⋊C4—84no—
480C4⋊C4—88no—
481C4⋊C4—84no—
482C4⋊C4—84no—
483C4⋊C4—88no—
484C22⋊C4—216no—
485C22⋊C4—232no—
486C22⋊C4—416no—
487C22⋊C4—84no—
488C22⋊C4—84no—
489C22⋊C4—84no—
490C22⋊C4—84no—
491C22⋊C4—88no—
492C22⋊C4—88no—
493C42—216no—
494C42—48no—
495C42—416no—
496C42—44no—
497C42—44no—
498C42—48no—
499C42—88no—
500C42C4×D4116yes—
501C23—24no—
502C23—24no—
503C23—28no—
504C23—24no—
505C23—28no—
506C23—42no—
507C23—42no—
508C23—42no—
509C23—42no—
510C23—42no—
511C23—42no—
512C23—44no—
513C23—44no—
514C23—42no—
515C23—42no—
516C23—44no—
517C23—44no—
518C23—44no—
519C23—44no—
520C23—44no—
521C23—44no—
522C23—44no—
523C23—44no—
524C23C24×C411yescharacteristic
525C23C23×D411yescharacteristic
526C23C22×#4111yescharacteristic
527C23C2×H(5,𝔽2)11yescharacteristic
528C23C2×H(5,𝔽2)12yes—
529C23C2×H(5,𝔽2)12yes—
530C23C2×2−1+411yescharacteristic
531C23C2×2−1+411yescharacteristic
532C23C2×2−1+411yescharacteristic
533C23C2×2−1+412yes—
534C23#58512yes—
535Q8—232no—
536D4—416no—
537C2×C4—24no—
538C2×C4—24no—
539C2×C4—24no—
540C2×C4—24no—
541C2×C4—22no—
542C2×C4—24no—
543C2×C4—22no—
544C2×C4—24no—
545C2×C4—24no—
546C2×C4—24no—
547C2×C4—24no—
548C2×C4—24no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—24no—
552C2×C4—24no—
553C2×C4—28no—
554C2×C4—28no—
555C2×C4—24no—
556C2×C4—28no—
557C2×C4—24no—
558C2×C4—28no—
559C2×C4—24no—
560C2×C4—24no—
561C2×C4—24no—
562C2×C4—24no—
563C2×C4—24no—
564C2×C4—24no—
565C2×C4—24no—
566C2×C4—24no—
567C2×C4—24no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—44no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—44no—
578C2×C4—44no—
579C2×C4—42no—
580C2×C4—44no—
581C2×C4—44no—
582C2×C4—44no—
583C2×C4—42no—
584C2×C4—44no—
585C2×C4—44no—
586C2×C4—44no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—44no—
592C2×C4—42no—
593C2×C4—44no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—41no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—41no—
601C2×C4—41no—
602C2×C4—44no—
603C2×C4—41no—
604C2×C4—41no—
605C2×C4—42no—
606C2×C4—44no—
607C2×C4—44no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—44no—
615C2×C4—44no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—816no—
622C2×C4C2×C4×D414yes—
623C2×C4C2×C4×D414yes—
624C2×C4C2×C4×D414yes—
625C2×C4#51914yes—
626C2×C4#52018yes—
627C2×C4#52014yes—
628C2×C4#53614yes—
629C2×C4#53714yes—
630C2×C4#54114yes—
631C2×C4#54214yes—
632C22—44no—
633C22—44no—
634C22—48no—
635C22—48no—
636C22—44no—
637C22—84no—
638C22—84no—
639C22—82no—
640C22—82no—
641C22—82no—
642C22—82no—
643C22C22×C4×D411yescharacteristic
644C22C2×#51911yescharacteristic
645C22C2×#52011yescharacteristic
646C22C2×#52012yes—
647C22C4×H(5,𝔽2)12yes—
648C22C2×#53611yescharacteristic
649C22C2×#53711yescharacteristic
650C22C2×#54111yescharacteristic
651C22C2×#54211yescharacteristic
652C22#348411yescharacteristic
653C22#348611yescharacteristic
654C22#349212yes—
655C22#349412yes—
656C22#349811yescharacteristic
657C22#350011yescharacteristic
658C22#354312yes—
659C22#354412yes—
660C22#354712yes—
661C22#354812yes—
662C22#355012yes—
663C22#355412yes—
664C22#355412yes—
665C22#355512yes—
666C4—24no—
667C4—28no—
668C4—24no—
669C4—44no—
670C4—44no—
671C4—44no—
672C4—48no—
673C4—44no—
674C4—44no—
675C4—82no—
676C4—81no—
677C4—81no—
678C4—81no—
679C4—84no—
680C4—81no—
681C4#232318yes—
682C2—84no—
683C2C2×#232311yescharacteristic
684C2#3420011yescharacteristic
685C2#3427511yescharacteristic
686C2#3448412yes—
687C2#3449812yes—
688C2#4242511yescharacteristic
689C2#4243111yescharacteristic
690C2#4250912yes—
691C2#4251512yes—
692C2#4278911yescharacteristic
693C2#4279511yescharacteristic
subgroupG/H
center, Frattini, socle, cosocleC24C25
commutatorC23C24×C4
prime radicalC23×D4C23
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×Q8×C4⋊C4C2semidirect
4. #14628C2semidirect
5. #14402C2semidirect
6. #14382C2semidirect
7. #14322C2semidirect
8. #14301C2semidirect
9. #14277C2semidirect
10. #14188C2semidirect
12. #14132C2semidirect
13. #14005C2semidirect
16. #13850C2semidirect
17. #13804C2semidirect
21. #2580C4semidirect

Maximal subgroups

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

subgroupindex
C2×#25802
C2×Q8×C4⋊C42
C2×#23192
#146282
#144022
#143822
#143222
#143012
#142772
#141882
#141412
#141322
#140052
#140012
#139062
#138502
#138042