Finite Group Databasebeta

← #545517

Subgroups — #545517

Order 512.

2976 conjugacy classes of subgroups.

The subgroup lattice has 757 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2542C211yesmaximal, characteristic
2C2×#2365C211yesmaximal, characteristic, complemented
3C2×#2352C211yesmaximal, characteristic
4#14628C212yesmaximal, complemented
5#14431C214yesmaximal
6#14425C214yesmaximal, complemented
7#14317C212yesmaximal, complemented
8#14305C212yesmaximal, complemented
9#14274C211yesmaximal, characteristic, complemented
10#14240C212yesmaximal, complemented
11#14139C212yesmaximal, complemented
12#14138C212yesmaximal, complemented
13#13969C212yesmaximal, complemented
14#13969C212yesmaximal, complemented
15#13955C211yesmaximal, characteristic, complemented
16#13847C211yesmaximal, characteristic
17#13797C211yesmaximal, characteristic, complemented
18C22×C4⋊Q8C2211yescharacteristic
19C22×C22⋊Q8C2212yes—
20C22×C22⋊Q8C2212yes—
21C22×C4×Q8C2212yes—
22C22×C4×D4C2211yescharacteristic
23C22×#117C2211yescharacteristic
24C23×C4⋊C4C2211yescharacteristic
25#2542C4116yescomplemented
26C2×#399C2212yes—
27C2×#399C2212yes—
28#2365—28no—
29#2352—28no—
30C2×#391C2212yes—
31C2×#391C2212yes—
32C2×#391C2212yes—
33C2×#390C2212yes—
34C2×#389C2212yes—
35C2×#388C2212yes—
36C2×#387C2212yes—
37C2×#387C2214yes—
38C2×#386C2212yes—
39C2×#384C2212yes—
40C2×#382C2211yescharacteristic
41C2×#380C2211yescharacteristic
42C2×#380C2211yescharacteristic
43C2×C4×C4⋊C4C2211yescharacteristic
44C2×C4×C4⋊C4C2212yes—
45C2×C4×C4⋊C4C2211yescharacteristic
46C2×C4×C4⋊C4C2211yescharacteristic
47C2×C4×C22⋊C4C2212yes—
48C2×C4×C22⋊C4C2212yes—
49C2×C4×C22⋊C4C2212yes—
50C2×#376C2211yescharacteristic
51#1445C2212yes—
52#1445C2212yes—
53#1445C2214yes—
54#1445C2212yes—
55#1442C2212yes—
56#1442C2212yes—
57#1441C2214yes—
58#1441C2214yes—
59#1441C2214yes—
60#1441C2214yes—
61#1440C2214yes—
62#1440C2214yes—
63#1439C2211yescharacteristic
64#1439C2211yescharacteristic
65#1439C2211yescharacteristic
66#1439C2211yescharacteristic
67#1439C2214yes—
68#1439C2214yes—
69#1438C2212yes—
70#1437C2214yes—
71#1437C2214yes—
72#1436C2214yes—
73#1436C2214yes—
74#1436C2214yes—
75#1436C2214yes—
76C4⋊#95C2212yes—
77C4⋊#95C2212yes—
78C4⋊#95C2212yes—
79C4⋊#95C2212yes—
80#1434C2212yes—
81#1434C2214yes—
82#1434C2214yes—
83#1434C2212yes—
84C22⋊#95C2214yes—
85C22⋊#95C2214yes—
86C23×Q8C2312yes—
87C23×D4C2311yescharacteristic
88C24×C4C2311yescharacteristic
89C24×C4C2311yescharacteristic
90C24×C4C2312yes—
91C2×C4⋊Q8—24no—
92C2×C4⋊Q8—24no—
93C2×C4⋊Q8C2×C418yescomplemented
94C2×C22⋊Q8—44no—
95C2×C22⋊Q8—44no—
96C2×C22⋊Q8—44no—
97C2×C22⋊Q8—44no—
98C2×C22⋊Q8C2×C4116yes—
99C2×C22⋊Q8C2×C4116yescomplemented
100C2×C4×Q8—28no—
101C2×C4×Q8—44no—
102C2×C4×Q8—44no—
103C2×C4×D4—24no—
104C2×C4×D4—24no—
105C2×C4×D4C2×C418yes—
106C2×#117—24no—
107C2×#117—44no—
108C22×C4⋊C4—24no—
109C22×C4⋊C4—44no—
110C22×C4⋊C4C2311yescharacteristic
111C22×C4⋊C4C2311yescharacteristic
112C22×C4⋊C4C2311yescharacteristic
113C22×C4⋊C4C2312yes—
114C22×C4⋊C4C2311yescharacteristic
115C22×C4⋊C4C2312yes—
116C22×C4⋊C4C2312yes—
117C22×C4⋊C4C2311yescharacteristic
118C22×C4⋊C4C2311yescharacteristic
119C22×C4⋊C4C2312yes—
120C22×C4⋊C4C2314yes—
121C22×C4⋊C4C2311yescharacteristic
122C22×C4⋊C4C2314yes—
123C22×C4⋊C4C2314yes—
124C22×C22⋊C4C2312yes—
125C22×C22⋊C4C2312yes—
126C22×C22⋊C4C2311yescharacteristic
127C22×C22⋊C4C2311yescharacteristic
128C22×C22⋊C4C2312yes—
129C22×C22⋊C4C2312yes—
130C22×C22⋊C4C2312yes—
131C22×C22⋊C4C2312yes—
132C22×C22⋊C4C2312yes—
133C22×C22⋊C4C2314yes—
134C22×C22⋊C4C2314yes—
135C22×C42C2312yes—
136C22×C42C2312yes—
137C22×C42C2312yes—
138C22×C42C2314yes—
139C22×C42C2312yes—
140C22×C42C2312yes—
141C22×C42C2311yescharacteristic
142#399C2×C4116yes—
143#399C2×C4116yescomplemented
144#391—28no—
145#391—44no—
146#391—44no—
147#390C2×C4116yes—
148#389—28no—
149#388—28no—
150#387—28no—
151#387—216no—
152#386—28no—
153#384C2×C4116yescomplemented
154#382—24no—
155#380—24no—
156#380—24no—
157C4×C4⋊C4—24no—
158C4×C4⋊C4—28no—
159C4×C4⋊C4C2×C418yescomplemented
160C4×C4⋊C4D418yes—
161C4×C22⋊C4—28no—
162C4×C22⋊C4—44no—
163C4×C22⋊C4—44no—
164#376—24no—
165C2×#95C2314yes—
166C2×#95C2314yes—
167C2×#95C2312yes—
168C2×#95C2312yes—
169C2×#95C2311yescharacteristic
170C2×#95C2311yescharacteristic
171C2×#95C2312yes—
172C2×#95C2314yes—
173C2×#95C2312yes—
174C2×#95C2312yes—
175C2×#95C2312yes—
176C2×#95C2314yes—
177C2×#95C2314yes—
178C2×#95C2314yes—
179C2×#95C2314yes—
180C2×#95C2314yes—
181C2×#95C2314yes—
182C2×#95C2314yes—
183C2×#95C2314yes—
184C2×#95C2314yes—
185C2×#95C2312yes—
186C2×#95C2314yes—
187C2×#95C2312yes—
188C2×#95C2312yes—
189C2×#95C2314yes—
190C2×#95C2312yes—
191C2×#95C2312yes—
192C2×#95C2312yes—
193C25C2411yescharacteristic
194C25C2411yescharacteristic
195C22×Q8—24no—
196C22×Q8—24no—
197C22×Q8—24no—
198C22×Q8—24no—
199C22×Q8—24no—
200C22×Q8—24no—
201C22×Q8C22×C418yes—
202C22×D4—22no—
203C22×D4—22no—
204C22×D4—22no—
205C22×D4—22no—
206C22×D4—41no—
207C22×D4—41no—
208C22×D4C22×C414yes—
209C23×C4—22no—
210C23×C4—22no—
211C23×C4—22no—
212C23×C4—22no—
213C23×C4—24no—
214C23×C4—24no—
215C23×C4—24no—
216C23×C4—24no—
217C23×C4—24no—
218C23×C4—24no—
219C23×C4—44no—
220C23×C4—42no—
221C23×C4—44no—
222C23×C4C22×C414yes—
223C23×C4C22×C414yes—
224C23×C4C2412yes—
225C23×C4C2412yes—
226C23×C4C2412yes—
227C23×C4C2412yes—
228C23×C4C2412yes—
229C23×C4C2414yes—
230C23×C4C2414yes—
231C23×C4C2411yescharacteristic
232C23×C4C2411yescharacteristic
233C23×C4C2412yes—
234C23×C4C2412yes—
235C23×C4C2412yes—
236C23×C4C2411yescharacteristic
237C23×C4C2411yescharacteristic
238C23×C4C2411yescharacteristic
239C4⋊Q8—232no—
240C22⋊Q8—432no—
241C22⋊Q8—432no—
242C4×Q8—432no—
243C4×D4—232no—
244#117—416no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—24no—
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—22no—
258C2×C4⋊C4—24no—
259C2×C4⋊C4—22no—
260C2×C4⋊C4—28no—
261C2×C4⋊C4—22no—
262C2×C4⋊C4—24no—
263C2×C4⋊C4—24no—
264C2×C4⋊C4—416no—
265C2×C4⋊C4—44no—
266C2×C4⋊C4—42no—
267C2×C4⋊C4—44no—
268C2×C4⋊C4—44no—
269C2×C4⋊C4—44no—
270C2×C4⋊C4—44no—
271C2×C4⋊C4—44no—
272C2×C4⋊C4—44no—
273C2×C4⋊C4C22×C414yes—
274C2×C4⋊C4C22×C418yes—
275C2×C4⋊C4C22×C418yes—
276C2×C4⋊C4C22×C418yes—
277C2×C4⋊C4C22×C4116yes—
278C2×C4⋊C4C22×C4116yes—
279C2×C4⋊C4C2×D414yes—
280C2×C4⋊C4C2×D414yes—
281C2×C22⋊C4—22no—
282C2×C22⋊C4—22no—
283C2×C22⋊C4—22no—
284C2×C22⋊C4—22no—
285C2×C22⋊C4—24no—
286C2×C22⋊C4—24no—
287C2×C22⋊C4—24no—
288C2×C22⋊C4—24no—
289C2×C22⋊C4—24no—
290C2×C22⋊C4—24no—
291C2×C22⋊C4—24no—
292C2×C22⋊C4—24no—
293C2×C22⋊C4—42no—
294C2×C22⋊C4—42no—
295C2×C22⋊C4—42no—
296C2×C22⋊C4—42no—
297C2×C22⋊C4—42no—
298C2×C22⋊C4—44no—
299C2×C22⋊C4—44no—
300C2×C22⋊C4—44no—
301C2×C22⋊C4—44no—
302C2×C22⋊C4—44no—
303C2×C22⋊C4—42no—
304C2×C22⋊C4—42no—
305C2×C22⋊C4—42no—
306C2×C22⋊C4—42no—
307C2×C22⋊C4—44no—
308C2×C22⋊C4C22×C414yes—
309C2×C22⋊C4C22×C414yes—
310C2×C22⋊C4C22×C4116yes—
311C2×C22⋊C4C22×C4116yes—
312C2×C42—24no—
313C2×C42—28no—
314C2×C42—24no—
315C2×C42—24no—
316C2×C42—24no—
317C2×C42—24no—
318C2×C42—24no—
319C2×C42—24no—
320C2×C42—44no—
321C2×C42—44no—
322C2×C42—44no—
323C2×C42—42no—
324C2×C42—42no—
325C2×C42—44no—
326C2×C42C22×C418yes—
327C2×C42C22×C414yes—
328C2×C42C2×D418yes—
329C2×C42C2×D414yes—
330C2×C42#4114yes—
331#95—24no—
332#95—24no—
333#95—24no—
334#95—28no—
335#95—24no—
336#95—24no—
337#95—28no—
338#95—22no—
339#95—22no—
340#95—24no—
341#95—28no—
342#95—44no—
343#95—44no—
344#95—44no—
345#95—44no—
346#95—44no—
347#95—44no—
348#95—44no—
349#95—42no—
350#95—42no—
351#95—42no—
352#95—42no—
353#95—44no—
354#95—44no—
355#95—44no—
356#95—44no—
357#95C22×C418yes—
358#95C2×D418yes—
359C24—21no—
360C24—21no—
361C24—22no—
362C24—22no—
363C24—22no—
364C24—21no—
365C24—22no—
366C24—21no—
367C24—21no—
368C24—21no—
369C24—22no—
370C24—22no—
371C24—22no—
372C24—21no—
373C24—21no—
374C24—22no—
375C24—21no—
376C24—21no—
377C24C23×C412yes—
378C24C23×C412yes—
379C242−1+412yes—
380C242−1+412yes—
381C24C2511yescharacteristic
382C2×Q8—216no—
383C2×Q8—216no—
384C2×Q8—216no—
385C2×Q8—48no—
386C2×Q8—48no—
387C2×Q8—48no—
388C2×Q8—48no—
389C2×D4—28no—
390C2×D4—28no—
391C2×D4—44no—
392C2×D4—44no—
393C2×D4—44no—
394C2×D4—44no—
395C2×D4—44no—
396C22×C4—22no—
397C22×C4—21no—
398C22×C4—21no—
399C22×C4—22no—
400C22×C4—21no—
401C22×C4—21no—
402C22×C4—22no—
403C22×C4—21no—
404C22×C4—24no—
405C22×C4—21no—
406C22×C4—21no—
407C22×C4—21no—
408C22×C4—24no—
409C22×C4—24no—
410C22×C4—24no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—24no—
416C22×C4—22no—
417C22×C4—24no—
418C22×C4—24no—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—24no—
422C22×C4—22no—
423C22×C4—24no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—24no—
429C22×C4—24no—
430C22×C4—24no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—24no—
435C22×C4—24no—
436C22×C4—28no—
437C22×C4—28no—
438C22×C4—216no—
439C22×C4—216no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—24no—
450C22×C4—24no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—24no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—22no—
469C22×C4—24no—
470C22×C4—22no—
471C22×C4—48no—
472C22×C4—48no—
473C22×C4—44no—
474C22×C4—44no—
475C22×C4—44no—
476C22×C4—44no—
477C22×C4—48no—
478C22×C4—48no—
479C22×C4—84no—
480C22×C4—88no—
481C22×C4—88no—
482C22×C4C23×C418yes—
483C22×C4C23×C418yes—
484C22×C4C23×C412yes—
485C22×C4C23×C412yes—
486C22×C4C23×C412yes—
487C22×C4C23×C412yes—
488C22×C4C23×C412yes—
489C22×C4C22×D414yes—
490C22×C4C22×D414yes—
491C22×C4C22×D412yes—
492C22×C4C22×D412yes—
493C22×C4C22×D412yes—
494C22×C4C2×#4112yes—
495C22×C4C2×#4112yes—
496C22×C4C2×#4112yes—
497C22×C4H(5,𝔽2)14yes—
498C22×C4H(5,𝔽2)14yes—
499C22×C42−1+414yes—
500C4⋊C4—28no—
501C4⋊C4—28no—
502C4⋊C4—216no—
503C4⋊C4—216no—
504C4⋊C4—216no—
505C4⋊C4—28no—
506C4⋊C4—28no—
507C4⋊C4—416no—
508C4⋊C4—416no—
509C4⋊C4—44no—
510C4⋊C4—48no—
511C4⋊C4—416no—
512C4⋊C4—44no—
513C4⋊C4—44no—
514C22⋊C4—28no—
515C22⋊C4—28no—
516C22⋊C4—416no—
517C22⋊C4—416no—
518C22⋊C4—48no—
519C22⋊C4—48no—
520C22⋊C4—48no—
521C22⋊C4—48no—
522C22⋊C4—48no—
523C22⋊C4—48no—
524C22⋊C4—84no—
525C42—48no—
526C42—48no—
527C42—48no—
528C42—416no—
529C42—48no—
530C42—48no—
531C42C4×D4116yes—
532C23—22no—
533C23—24no—
534C23—22no—
535C23—24no—
536C23—22no—
537C23—22no—
538C23—22no—
539C23—22no—
540C23—24no—
541C23—24no—
542C23—24no—
543C23—24no—
544C23—22no—
545C23—24no—
546C23—22no—
547C23—22no—
548C23—24no—
549C23—41no—
550C23—41no—
551C23—42no—
552C23—41no—
553C23—41no—
554C23—42no—
555C23—42no—
556C23—42no—
557C23—42no—
558C23—42no—
559C23—41no—
560C23—41no—
561C23—42no—
562C23—41no—
563C23—41no—
564C23—41no—
565C23—41no—
566C23—41no—
567C23—41no—
568C23—42no—
569C23—42no—
570C23—42no—
571C23—42no—
572C23—42no—
573C23—42no—
574C23—42no—
575C23—42no—
576C23—42no—
577C23#51914yes—
578C23C24×C411yescharacteristic
579C23C23×D411yescharacteristic
580C23C22×#4111yescharacteristic
581C23C2×H(5,𝔽2)12yes—
582C23C2×H(5,𝔽2)12yes—
583C23C2×2−1+411yescharacteristic
584C23C2×2−1+411yescharacteristic
585C23C2×2−1+412yes—
586C23#58511yescharacteristic
587C23#58511yescharacteristic
588C23#58512yes—
589Q8—432no—
590D4—416no—
591C2×C4—22no—
592C2×C4—22no—
593C2×C4—24no—
594C2×C4—22no—
595C2×C4—22no—
596C2×C4—22no—
597C2×C4—22no—
598C2×C4—22no—
599C2×C4—22no—
600C2×C4—24no—
601C2×C4—22no—
602C2×C4—22no—
603C2×C4—28no—
604C2×C4—28no—
605C2×C4—28no—
606C2×C4—28no—
607C2×C4—24no—
608C2×C4—24no—
609C2×C4—28no—
610C2×C4—28no—
611C2×C4—24no—
612C2×C4—24no—
613C2×C4—24no—
614C2×C4—24no—
615C2×C4—24no—
616C2×C4—24no—
617C2×C4—24no—
618C2×C4—24no—
619C2×C4—24no—
620C2×C4—42no—
621C2×C4—41no—
622C2×C4—44no—
623C2×C4—41no—
624C2×C4—41no—
625C2×C4—41no—
626C2×C4—44no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—44no—
630C2×C4—44no—
631C2×C4—42no—
632C2×C4—42no—
633C2×C4—44no—
634C2×C4—44no—
635C2×C4—42no—
636C2×C4—44no—
637C2×C4—44no—
638C2×C4—44no—
639C2×C4—44no—
640C2×C4—44no—
641C2×C4—44no—
642C2×C4—416no—
643C2×C4—416no—
644C2×C4—42no—
645C2×C4—42no—
646C2×C4—42no—
647C2×C4—42no—
648C2×C4—42no—
649C2×C4—42no—
650C2×C4—42no—
651C2×C4—42no—
652C2×C4—42no—
653C2×C4—42no—
654C2×C4—42no—
655C2×C4—42no—
656C2×C4—42no—
657C2×C4—42no—
658C2×C4—42no—
659C2×C4—42no—
660C2×C4—42no—
661C2×C4—42no—
662C2×C4—42no—
663C2×C4—42no—
664C2×C4—42no—
665C2×C4—42no—
666C2×C4—42no—
667C2×C4—44no—
668C2×C4—42no—
669C2×C4—44no—
670C2×C4—42no—
671C2×C4—42no—
672C2×C4—44no—
673C2×C4—42no—
674C2×C4—44no—
675C2×C4—816no—
676C2×C4C2×C4×D414yes—
677C2×C4C2×C4×D414yes—
678C2×C4C2×C4×D414yes—
679C2×C4#51818yes—
680C2×C4#53514yes—
681C2×C4#53514yes—
682C2×C4#54018yes—
683C22—24no—
684C22—28no—
685C22—42no—
686C22—42no—
687C22—42no—
688C22—44no—
689C22—44no—
690C22—44no—
691C22—42no—
692C22—42no—
693C22—42no—
694C22—42no—
695C22—42no—
696C22—44no—
697C22—44no—
698C22—42no—
699C22—81no—
700C22—81no—
701C22—82no—
702C22—82no—
703C22—81no—
704C22—81no—
705C22C22×C4×D411yescharacteristic
706C22C2×#51812yes—
707C22C2×#51911yescharacteristic
708C22#342811yescharacteristic
709C22C4×2−1+412yes—
710C22C2×#53511yescharacteristic
711C22C2×#53511yescharacteristic
712C22C2×#54012yes—
713C22#348511yescharacteristic
714C22#348611yescharacteristic
715C22#348911yescharacteristic
716C22#348911yescharacteristic
717C22#349511yescharacteristic
718C22#349511yescharacteristic
719C22#350712yes—
720C22#353812yes—
721C22#353912yes—
722C22#354112yes—
723C22#354112yes—
724C22#355212yes—
725C22#355412yes—
726C22#356112yes—
727C22#356212yes—
728C4—24no—
729C4—24no—
730C4—42no—
731C4—42no—
732C4—48no—
733C4—44no—
734C4—48no—
735C4—44no—
736C4—44no—
737C4—82no—
738C4—82no—
739C4—82no—
740C4—82no—
741C4—82no—
742C4#232818yes—
743C2—44no—
744C2—82no—
745C2C2×#232811yescharacteristic
746C2#3420311yescharacteristic
747C2#3426611yescharacteristic
748C2#3447312yes—
749C2#3447512yes—
750C2#4266812yes—
751C2#4266912yes—
752C2#4267911yescharacteristic
753C2#4268011yescharacteristic
754C2#4280811yescharacteristic
755C2#4281511yescharacteristic
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×#2365C2semidirect
4. #14628C2semidirect
6. #14425C2semidirect
7. #14317C2semidirect
8. #14305C2semidirect
9. #14274C2semidirect
10. #14240C2semidirect
11. #14139C2semidirect
12. #14138C2semidirect
13. #13969C2semidirect
14. #13969C2semidirect
15. #13955C2semidirect
17. #13797C2semidirect
25. #2542C4semidirect
93. C2×C4⋊Q8C2×C4semidirect
99. C2×C22⋊Q8C2×C4semidirect
143. #399C2×C4semidirect
153. #384C2×C4semidirect
159. C4×C4⋊C4C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#25422
C2×#23652
C2×#23522
#146282
#144312
#144252
#143172
#143052
#142742
#142402
#141392
#141382
#139692
#139552
#138472
#137972