Finite Group Databasebeta

← #517238

Subgroups — #517238

Order 512.

2878 conjugacy classes of subgroups.

The subgroup lattice has 703 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2625C211yesmaximal, characteristic
2C2×C4×C22⋊Q8C211yesmaximal, characteristic
3C2×C4×#117C211yesmaximal, characteristic
4#14703C212yesmaximal, complemented
5#14699C212yesmaximal, complemented
6#14343C211yesmaximal, characteristic
7#14264C211yesmaximal, characteristic
8#14258C211yesmaximal, characteristic
9#14250C212yesmaximal, complemented
10#14248C214yesmaximal
11#14244C212yesmaximal, complemented
12#14239C212yesmaximal, complemented
13#14223C212yesmaximal, complemented
14#14202C214yesmaximal
15#13786C212yesmaximal, complemented
16#13783C212yesmaximal, complemented
17C4×#380C211yesmaximal, characteristic
18C22×C22⋊Q8C2211yescharacteristic
19C22×C4×Q8C2211yescharacteristic
20C22×#117C2211yescharacteristic
21C22×#117C2211yescharacteristic
22C23×C42C2211yescharacteristic
23#2625C4116yescomplemented
24C2×#400C2212yes—
25C2×#398C2212yes—
26C2×#397C2214yes—
27C4×C22⋊Q8—28no—
28C2×#391C2211yescharacteristic
29C2×#391C2212yes—
30C2×#391C2212yes—
31C2×#389C2211yescharacteristic
32C2×#389C2212yes—
33C2×#389C2212yes—
34C2×#387C2212yes—
35C2×#387C2214yes—
36C2×#385C2212yes—
37C2×#380C2211yescharacteristic
38C2×#380C2211yescharacteristic
39C2×#380C2211yescharacteristic
40C4×#117—28no—
41C2×C4×C4⋊C4C2211yescharacteristic
42C2×C4×C4⋊C4C2212yes—
43C2×C4×C4⋊C4C2211yescharacteristic
44C2×C4×C4⋊C4C2211yescharacteristic
45C2×C4×C4⋊C4C2211yescharacteristic
46C2×C4×C4⋊C4C2212yes—
47C2×C4×C4⋊C4C2211yescharacteristic
48C2×C4×C22⋊C4C2212yes—
49C2×C4×C22⋊C4C2212yes—
50C2×C4×C22⋊C4C2212yes—
51C2×C4×C22⋊C4C2212yes—
52C2×#376C2212yes—
53C2×C43C2212yes—
54#1445C2212yes—
55#1445C2212yes—
56#1444C2212yes—
57#1442C2212yes—
58#1441C2212yes—
59#1441C2212yes—
60#1440C2214yes—
61#1440C2214yes—
62#1439C2214yes—
63#1436C2214yes—
64#1436C2214yes—
65#1436C2214yes—
66#1436C2214yes—
67#1436C2214yes—
68#1436C2214yes—
69#1436C2214yes—
70#1436C2214yes—
71C4⋊#95C2212yes—
72C4⋊#95C2212yes—
73C4⋊#95C2212yes—
74C4⋊#95C2212yes—
75C4⋊#95C2212yes—
76C4⋊#95C2214yes—
77#1434C2214yes—
78#1434C2214yes—
79#1434C2214yes—
80#1434C2214yes—
81#1434C2212yes—
82#1434C2214yes—
83#1434C2214yes—
84C22⋊#95C2212yes—
85C4×#95C2212yes—
86C23×Q8C2311yescharacteristic
87C24×C4C2311yescharacteristic
88C24×C4C2311yescharacteristic
89C24×C4C2311yescharacteristic
90C2×C22⋊Q8—24no—
91C2×C22⋊Q8—24no—
92C2×C22⋊Q8C2×C418yes—
93C2×C4×Q8—24no—
94C2×C4×Q8—28no—
95C2×#117—24no—
96C2×#117—24no—
97C2×#117—24no—
98C2×#117—28no—
99C2×#117C2×C418yes—
100C22×C4⋊C4C2311yescharacteristic
101C22×C4⋊C4C2311yescharacteristic
102C22×C4⋊C4C2311yescharacteristic
103C22×C4⋊C4C2312yes—
104C22×C4⋊C4C2312yes—
105C22×C4⋊C4C2311yescharacteristic
106C22×C4⋊C4C2312yes—
107C22×C4⋊C4C2311yescharacteristic
108C22×C4⋊C4C2312yes—
109C22×C4⋊C4C2311yescharacteristic
110C22×C4⋊C4C2311yescharacteristic
111C22×C4⋊C4C2312yes—
112C22×C4⋊C4C2314yes—
113C22×C4⋊C4C2314yes—
114C22×C22⋊C4C2312yes—
115C22×C22⋊C4C2312yes—
116C22×C22⋊C4C2312yes—
117C22×C22⋊C4C2312yes—
118C22×C22⋊C4C2312yes—
119C22×C22⋊C4C2312yes—
120C22×C42—28no—
121C22×C42D418yes—
122C22×C42C2312yes—
123C22×C42C2312yes—
124C22×C42C2312yes—
125C22×C42C2312yes—
126C22×C42C2311yescharacteristic
127C22×C42C2311yescharacteristic
128C22×C42C2311yescharacteristic
129C22×C42C2312yes—
130C22×C42C2314yes—
131C22×C42C2312yes—
132C22×C42C2312yes—
133C22×C42C2312yes—
134C22×C42C2311yescharacteristic
135#400C2×C4116yescomplemented
136#398C2×C4116yescomplemented
137#397C2×C4132yes—
138#391—24no—
139#391—28no—
140#391C2×C4116yescomplemented
141#389—24no—
142#389—28no—
143#389C2×C4116yescomplemented
144#387—28no—
145#387—48no—
146#385—28no—
147#380—24no—
148#380—24no—
149#380C2×C418yes—
150C4×C4⋊C4—24no—
151C4×C4⋊C4—24no—
152C4×C4⋊C4—24no—
153C4×C4⋊C4—24no—
154C4×C4⋊C4—28no—
155C4×C4⋊C4—24no—
156C4×C4⋊C4—28no—
157C4×C22⋊C4—28no—
158C4×C22⋊C4—28no—
159C4×C22⋊C4—28no—
160C4×C22⋊C4—44no—
161#376—28no—
162C2×#95C2314yes—
163C2×#95C2312yes—
164C2×#95C2314yes—
165C2×#95C2314yes—
166C2×#95C2314yes—
167C2×#95C2312yes—
168C2×#95C2314yes—
169C2×#95C2312yes—
170C2×#95C2312yes—
171C2×#95C2312yes—
172C2×#95C2312yes—
173C2×#95C2312yes—
174C2×#95C2312yes—
175C2×#95C2314yes—
176C2×#95C2312yes—
177C2×#95C2312yes—
178C2×#95C2314yes—
179C2×#95C2314yes—
180C2×#95C2314yes—
181C2×#95C2314yes—
182C2×#95C2314yes—
183C2×#95C2314yes—
184C2×#95C2314yes—
185C2×#95C2314yes—
186C2×#95C2314yes—
187C2×#95C2314yes—
188C2×#95C2314yes—
189C2×#95C2312yes—
190C43—28no—
191C25C2411yescharacteristic
192C22×Q8—22no—
193C22×Q8—24no—
194C22×Q8—22no—
195C22×Q8—24no—
196C22×Q8C22×C414yes—
197C23×C4—24no—
198C23×C4—24no—
199C23×C4—24no—
200C23×C4—24no—
201C23×C4—22no—
202C23×C4—24no—
203C23×C4—22no—
204C23×C4—41no—
205C23×C4—41no—
206C23×C4—42no—
207C23×C4C22×C414yes—
208C23×C4C2×D414yes—
209C23×C4C2×D414yes—
210C23×C4C2×D414yes—
211C23×C4#4114yes—
212C23×C4C2412yes—
213C23×C4C2412yes—
214C23×C4C2414yes—
215C23×C4C2412yes—
216C23×C4C2414yes—
217C23×C4C2412yes—
218C23×C4C2412yes—
219C23×C4C2412yes—
220C23×C4C2412yes—
221C23×C4C2412yes—
222C23×C4C2411yescharacteristic
223C23×C4C2411yescharacteristic
224C23×C4C2411yescharacteristic
225C23×C4C2411yescharacteristic
226C23×C4C2411yescharacteristic
227C23×C4C2411yescharacteristic
228C22⋊Q8—232no—
229C4×Q8—416no—
230#117—232no—
231#117—232no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—24no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—24no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—24no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—28no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—24no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—24no—
256C2×C4⋊C4—24no—
257C2×C4⋊C4—24no—
258C2×C4⋊C4—28no—
259C2×C4⋊C4—44no—
260C2×C4⋊C4—44no—
261C2×C4⋊C4—44no—
262C2×C4⋊C4—44no—
263C2×C4⋊C4C22×C414yes—
264C2×C4⋊C4C22×C414yes—
265C2×C4⋊C4C22×C414yes—
266C2×C4⋊C4C22×C418yes—
267C2×C4⋊C4C22×C418yes—
268C2×C4⋊C4C22×C4116yes—
269C2×C22⋊C4—24no—
270C2×C22⋊C4—24no—
271C2×C22⋊C4—24no—
272C2×C22⋊C4—24no—
273C2×C22⋊C4—24no—
274C2×C22⋊C4—24no—
275C2×C22⋊C4—24no—
276C2×C22⋊C4—24no—
277C2×C22⋊C4—24no—
278C2×C22⋊C4—42no—
279C2×C22⋊C4—42no—
280C2×C22⋊C4—44no—
281C2×C22⋊C4—42no—
282C2×C22⋊C4—42no—
283C2×C22⋊C4C22×C418yes—
284C2×C22⋊C4C22×C418yes—
285C2×C22⋊C4C22×C418yes—
286C2×C42—24no—
287C2×C42—24no—
288C2×C42—24no—
289C2×C42—24no—
290C2×C42—24no—
291C2×C42—24no—
292C2×C42—24no—
293C2×C42—232no—
294C2×C42—24no—
295C2×C42—24no—
296C2×C42—24no—
297C2×C42—24no—
298C2×C42—24no—
299C2×C42—24no—
300C2×C42—28no—
301C2×C42—24no—
302C2×C42—24no—
303C2×C42—24no—
304C2×C42—24no—
305C2×C42—28no—
306C2×C42—42no—
307C2×C42—42no—
308C2×C42—42no—
309C2×C42—42no—
310C2×C42—42no—
311C2×C42—42no—
312C2×C42—44no—
313C2×C42—44no—
314C2×C42C22×C418yes—
315C2×C42C2×D414yes—
316C2×C42C2×D414yes—
317C2×C42C2×D414yes—
318C2×C42C2×D414yes—
319C2×C42#4118yes—
320C2×C42#4118yes—
321#95—24no—
322#95—24no—
323#95—24no—
324#95—24no—
325#95—24no—
326#95—24no—
327#95—24no—
328#95—24no—
329#95—24no—
330#95—28no—
331#95—24no—
332#95—28no—
333#95—28no—
334#95—28no—
335#95—28no—
336#95—28no—
337#95—44no—
338#95—44no—
339#95—44no—
340#95—44no—
341#95—44no—
342#95—44no—
343#95—44no—
344#95—44no—
345#95C22×C418yes—
346#95C22×C4116yes—
347#95C22×C4116yes—
348#95C22×C4116yes—
349C24—22no—
350C24—22no—
351C24—21no—
352C24—22no—
353C24—21no—
354C24—21no—
355C24—22no—
356C24—21no—
357C24C23×C412yes—
358C24C22×D412yes—
359C24C2×#4112yes—
360C24C2511yescharacteristic
361C2×Q8—28no—
362C2×Q8—216no—
363C2×Q8—44no—
364C2×Q8—44no—
365C2×Q8—48no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—21no—
371C22×C4—22no—
372C22×C4—21no—
373C22×C4—22no—
374C22×C4—21no—
375C22×C4—21no—
376C22×C4—21no—
377C22×C4—21no—
378C22×C4—22no—
379C22×C4—216no—
380C22×C4—28no—
381C22×C4—28no—
382C22×C4—216no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—24no—
390C22×C4—24no—
391C22×C4—24no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—24no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—24no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—24no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—24no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—24no—
423C22×C4—28no—
424C22×C4—28no—
425C22×C4—28no—
426C22×C4—28no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—24no—
430C22×C4—24no—
431C22×C4—24no—
432C22×C4—24no—
433C22×C4—24no—
434C22×C4—24no—
435C22×C4—22no—
436C22×C4—24no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—24no—
440C22×C4—24no—
441C22×C4—24no—
442C22×C4—24no—
443C22×C4—22no—
444C22×C4—44no—
445C22×C4—44no—
446C22×C4—48no—
447C22×C4—48no—
448C22×C4—48no—
449C22×C4—48no—
450C22×C4C4×D4116yes—
451C22×C4C23×C414yes—
452C22×C4C23×C418yes—
453C22×C4C23×C414yes—
454C22×C4C23×C414yes—
455C22×C4C23×C414yes—
456C22×C4C23×C412yes—
457C22×C4C23×C412yes—
458C22×C4C22×D412yes—
459C22×C4C22×D412yes—
460C22×C4C22×D412yes—
461C22×C4C22×D412yes—
462C22×C4C22×D412yes—
463C22×C4C22×D412yes—
464C22×C4C2×#4114yes—
465C22×C4C2×#4114yes—
466C22×C4C2×#4112yes—
467C22×C4C2×#4114yes—
468C22×C4C2×#4114yes—
469C22×C4C2×#4112yes—
470C22×C4C2×#4114yes—
471C22×C4C2×#4114yes—
472C22×C4H(5,𝔽2)12yes—
473C22×C42−1+414yes—
474C22×C42−1+412yes—
475C4⋊C4—28no—
476C4⋊C4—28no—
477C4⋊C4—28no—
478C4⋊C4—28no—
479C4⋊C4—28no—
480C4⋊C4—216no—
481C4⋊C4—216no—
482C4⋊C4—416no—
483C4⋊C4—48no—
484C4⋊C4—44no—
485C4⋊C4—44no—
486C4⋊C4—48no—
487C4⋊C4—48no—
488C4⋊C4—88no—
489C22⋊C4—216no—
490C22⋊C4—216no—
491C22⋊C4—216no—
492C22⋊C4—48no—
493C22⋊C4—48no—
494C22⋊C4—48no—
495C42—216no—
496C42—28no—
497C42—28no—
498C42—28no—
499C42—216no—
500C42—48no—
501C42—48no—
502C42—48no—
503C42—48no—
504C42—48no—
505C42—48no—
506C42—88no—
507C42#123116yes—
508C23—24no—
509C23—22no—
510C23—24no—
511C23—22no—
512C23—22no—
513C23—22no—
514C23—24no—
515C23—22no—
516C23—22no—
517C23—22no—
518C23—22no—
519C23—22no—
520C23—24no—
521C23—22no—
522C23—42no—
523C23—42no—
524C23—42no—
525C23—42no—
526C23—42no—
527C23—42no—
528C23—42no—
529C23—42no—
530C23C2×C4×D414yes—
531C23C24×C411yescharacteristic
532C23C23×D411yescharacteristic
533C23C22×#4111yescharacteristic
534C23C22×#4112yes—
535C23C22×#4112yes—
536C23C2×H(5,𝔽2)11yescharacteristic
537C23C2×2−1+412yes—
538C23C2×2−1+411yescharacteristic
539C23#58511yescharacteristic
540C23#58511yescharacteristic
541C23#58512yes—
542Q8—416no—
543C2×C4—24no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—24no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—24no—
556C2×C4—24no—
557C2×C4—24no—
558C2×C4—24no—
559C2×C4—24no—
560C2×C4—24no—
561C2×C4—24no—
562C2×C4—24no—
563C2×C4—232no—
564C2×C4—24no—
565C2×C4—24no—
566C2×C4—24no—
567C2×C4—24no—
568C2×C4—28no—
569C2×C4—28no—
570C2×C4—28no—
571C2×C4—24no—
572C2×C4—24no—
573C2×C4—24no—
574C2×C4—24no—
575C2×C4—24no—
576C2×C4—24no—
577C2×C4—28no—
578C2×C4—24no—
579C2×C4—24no—
580C2×C4—24no—
581C2×C4—24no—
582C2×C4—24no—
583C2×C4—24no—
584C2×C4—416no—
585C2×C4—416no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—44no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—42no—
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—42no—
608C2×C4—44no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—44no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—44no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—42no—
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×C4C2×C4×D414yes—
631C2×C4C2×C4×D414yes—
632C2×C4C4×#4118yes—
633C2×C4C2×#12314yes—
634C2×C4C2×#12314yes—
635C2×C4C2×#12314yes—
636C2×C4#52514yes—
637C2×C4#52918yes—
638C2×C4#53614yes—
639C2×C4#53714yes—
640C22—24no—
641C22—24no—
642C22—24no—
643C22—42no—
644C22—42no—
645C22—44no—
646C22—42no—
647C22—44no—
648C22—44no—
649C22—42no—
650C22—44no—
651C22C22×C4×D411yescharacteristic
652C22C2×C4×#4112yes—
653C22C4×H(5,𝔽2)11yescharacteristic
654C22C4×2−1+412yes—
655C22C4×2−1+411yescharacteristic
656C22C22×#12311yescharacteristic
657C22C2×#52511yescharacteristic
658C22C2×#52912yes—
659C22C2×#53611yescharacteristic
660C22C2×#53711yescharacteristic
661C22#344911yescharacteristic
662C22#344911yescharacteristic
663C22#346011yescharacteristic
664C22#346112yes—
665C22#346111yescharacteristic
666C22#349412yes—
667C22#349512yes—
668C22#349512yes—
669C22#349512yes—
670C22#350912yes—
671C22#351012yes—
672C22#351112yes—
673C22#351212yes—
674C4—24no—
675C4—24no—
676C4—24no—
677C4—24no—
678C4—24no—
679C4—44no—
680C4—44no—
681C4—44no—
682C4—44no—
683C4—44no—
684C4—48no—
685C4—44no—
686C4—82no—
687C4—82no—
688C4—84no—
689C4C4×#12318yes—
690C2—44no—
691C2C2×C4×#12311yescharacteristic
692C2C4×#53611yescharacteristic
693C2C4×#53711yescharacteristic
694C2#3427912yes—
695C2#3428112yes—
696C2#3526311yescharacteristic
697C2#3526912yes—
698C2#3529711yescharacteristic
699C2#3531111yescharacteristic
700C2#3531412yes—
701C2#3534211yescharacteristic
subgroupG/H
center, Frattini, socle, cosocleC24C25
commutatorC23C24×C4
prime radicalC25C24
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #14703C2semidirect
5. #14699C2semidirect
9. #14250C2semidirect
11. #14244C2semidirect
12. #14239C2semidirect
13. #14223C2semidirect
15. #13786C2semidirect
16. #13783C2semidirect
23. #2625C4semidirect
135. #400C2×C4semidirect
136. #398C2×C4semidirect
140. #391C2×C4semidirect
143. #389C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#26252
C2×C4×C22⋊Q82
C2×C4×#1172
#147032
#146992
#143432
#142642
#142582
#142502
#142482
#142442
#142392
#142232
#142022
#137862
#137832
C4×#3802