Finite Group Databasebeta

← #759999

Subgroups — #759999

Order 512.

2394 conjugacy classes of subgroups.

The subgroup lattice has 716 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2567C211yesmaximal, characteristic
2C2×C4×C4⋊Q8C211yesmaximal, characteristic
3#16610C214yesmaximal
4#15193C212yesmaximal
5#15187C212yesmaximal
6#15137C212yesmaximal
7#15133C212yesmaximal
8#14879C214yesmaximal
9#14161C211yesmaximal, characteristic
10#14142C212yesmaximal
11#14132C212yesmaximal
12#14029C212yesmaximal
13#13981C212yesmaximal
14#13926C211yesmaximal, characteristic
15#13910C211yesmaximal, characteristic
16C4×#391C211yesmaximal, characteristic
17C4×#389C211yesmaximal, characteristic
18C22×C4⋊Q8C2211yescharacteristic
19C22×C4×Q8C2211yescharacteristic
20C22×C4×Q8C2212yes—
21#2567—28no—
22C2×#398C2214yes—
23C2×#395C2214yes—
24C4×C4⋊Q8—28no—
25C2×#391C2212yes—
26C2×#391C2211yescharacteristic
27C2×#391C2212yes—
28C2×#391C2212yes—
29C2×#391C2211yescharacteristic
30C2×#389C2212yes—
31C2×#389C2211yescharacteristic
32C2×#389C2212yes—
33C2×#389C2212yes—
34C2×#389C2212yes—
35C2×#389C2212yes—
36C2×#389C2212yes—
37C2×#389C2211yescharacteristic
38C2×#387C2214yes—
39C2×#384C2212yes—
40C2×#384C2212yes—
41C2×#384C2211yescharacteristic
42C2×C4×C4⋊C4C2212yes—
43C2×C4×C4⋊C4C2212yes—
44C2×C4×C4⋊C4C2211yescharacteristic
45C2×C4×C4⋊C4C2211yescharacteristic
46C2×C4×C4⋊C4C2211yescharacteristic
47C2×C4×C4⋊C4C2212yes—
48C2×C43C2211yescharacteristic
49#1446C2214yes—
50#1445C2214yes—
51#1445C2214yes—
52#1445C2214yes—
53#1444C2214yes—
54#1444C2214yes—
55#1443C2214yes—
56#1442C2214yes—
57#1442C2214yes—
58#1441C2214yes—
59#1441C2214yes—
60#1441C2214yes—
61#1441C2214yes—
62#1441C2214yes—
63#1436C2212yes—
64#1436C2212yes—
65#1436C2214yes—
66#1436C2214yes—
67#1436C2212yes—
68#1436C2212yes—
69C4⋊#95C2211yescharacteristic
70C4⋊#95C2212yes—
71C4⋊#95C2212yes—
72C4⋊#95C2212yes—
73C4⋊#95C2212yes—
74C4⋊#95C2212yes—
75C4⋊#95C2212yes—
76C4⋊#95C2212yes—
77C4⋊#95C2211yescharacteristic
78C4⋊#95C2212yes—
79C4⋊#95C2211yescharacteristic
80C4⋊#95C2211yescharacteristic
81#1434C2214yes—
82C4×#95C2212yes—
83C4×#95C2212yes—
84C4×#95C2212yes—
85C23×Q8C2311yescharacteristic
86C23×Q8C2312yes—
87C2×C4⋊Q8—24no—
88C2×C4⋊Q8—24no—
89C2×C4⋊Q8D418yes—
90C2×C4×Q8—28no—
91C2×C4×Q8—24no—
92C2×C4×Q8—24no—
93C2×C4×Q8—24no—
94C2×C4×Q8—44no—
95C2×C4×Q8—44no—
96C22×C4⋊C4C2312yes—
97C22×C4⋊C4C2311yescharacteristic
98C22×C4⋊C4C2311yescharacteristic
99C22×C4⋊C4C2312yes—
100C22×C4⋊C4C2312yes—
101C22×C4⋊C4C2312yes—
102C22×C4⋊C4C2314yes—
103C22×C4⋊C4C2314yes—
104C22×C4⋊C4C2312yes—
105C22×C4⋊C4C2312yes—
106C22×C4⋊C4C2311yescharacteristic
107C22×C4⋊C4C2311yescharacteristic
108C22×C4⋊C4C2311yescharacteristic
109C22×C4⋊C4C2311yescharacteristic
110C22×C4⋊C4C2311yescharacteristic
111C22×C4⋊C4C2314yes—
112C22×C4⋊C4C2314yes—
113C22×C4⋊C4C2314yes—
114C22×C4⋊C4C2312yes—
115C22×C4⋊C4C2312yes—
116C22×C42C2311yescharacteristic
117C22×C42C2312yes—
118C22×C42C2312yes—
119C22×C42C2311yescharacteristic
120C22×C42C2312yes—
121C22×C42C2312yes—
122C22×C42C2311yescharacteristic
123C22×C42C2314yes—
124C22×C42C2311yescharacteristic
125C22×C42C2311yescharacteristic
126C22×C42C2311yescharacteristic
127C22×C42C2311yescharacteristic
128#398—48no—
129#395—216no—
130#391—28no—
131#391—28no—
132#391—24no—
133#391—44no—
134#391Q818yescomplemented
135#389—28no—
136#389—28no—
137#389—28no—
138#389—28no—
139#389—24no—
140#389—28no—
141#389—44no—
142#389Q818yescomplemented
143#387—48no—
144#384—28no—
145#384—28no—
146#384—24no—
147C4×C4⋊C4—28no—
148C4×C4⋊C4—28no—
149C4×C4⋊C4—24no—
150C4×C4⋊C4—24no—
151C4×C4⋊C4—24no—
152C4×C4⋊C4—44no—
153C2×#95C2312yes—
154C2×#95C2314yes—
155C2×#95C2314yes—
156C2×#95C2312yes—
157C2×#95C2312yes—
158C2×#95C2312yes—
159C2×#95C2312yes—
160C2×#95C2312yes—
161C2×#95C2312yes—
162C2×#95C2314yes—
163C2×#95C2312yes—
164C2×#95C2312yes—
165C2×#95C2314yes—
166C2×#95C2314yes—
167C2×#95C2314yes—
168C2×#95C2314yes—
169C2×#95C2312yes—
170C2×#95C2312yes—
171C2×#95C2312yes—
172C2×#95C2314yes—
173C2×#95C2314yes—
174C2×#95C2314yes—
175C2×#95C2314yes—
176C2×#95C2314yes—
177C2×#95C2314yes—
178C2×#95C2312yes—
179C2×#95C2312yes—
180C2×#95C2314yes—
181C2×#95C2312yes—
182C2×#95C2312yes—
183C2×#95C2312yes—
184C43—24no—
185C22×Q8—22no—
186C22×Q8—22no—
187C22×Q8—22no—
188C22×Q8—24no—
189C22×Q8—24no—
190C22×Q8—22no—
191C22×Q8—22no—
192C22×Q8—22no—
193C22×Q8—42no—
194C22×Q8—42no—
195C22×Q8—42no—
196C22×Q8—42no—
197C22×Q8C2×D418yes—
198C22×Q8C2×Q814yes—
199C23×C4C2414yes—
200C23×C4C2412yes—
201C23×C4C2412yes—
202C23×C4C2414yes—
203C23×C4C2412yes—
204C23×C4C2412yes—
205C23×C4C2412yes—
206C23×C4C2411yescharacteristic
207C23×C4C2412yes—
208C23×C4C2411yescharacteristic
209C23×C4C2412yes—
210C23×C4C2412yes—
211C23×C4C2411yescharacteristic
212C23×C4C2411yescharacteristic
213C23×C4C2411yescharacteristic
214C23×C4C2411yescharacteristic
215C23×C4C2411yescharacteristic
216C4⋊Q8—232no—
217C4×Q8—432no—
218C4×Q8—416no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—24no—
222C2×C4⋊C4—24no—
223C2×C4⋊C4—24no—
224C2×C4⋊C4—24no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—28no—
236C2×C4⋊C4—24no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—24no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—28no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—28no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—28no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—28no—
256C2×C4⋊C4—28no—
257C2×C4⋊C4—28no—
258C2×C4⋊C4—42no—
259C2×C4⋊C4—44no—
260C2×C4⋊C4—42no—
261C2×C4⋊C4—42no—
262C2×C4⋊C4—42no—
263C2×C4⋊C4—42no—
264C2×C4⋊C4—44no—
265C2×C4⋊C4—44no—
266C2×C4⋊C4—44no—
267C2×C4⋊C4—44no—
268C2×C4⋊C4—44no—
269C2×C4⋊C4—44no—
270C2×C4⋊C4—44no—
271C2×C4⋊C4C2×D418yes—
272C2×C4⋊C4C2×D418yes—
273C2×C4⋊C4C2×Q814yes—
274C2×C4⋊C4C2×Q814yes—
275C2×C4⋊C4C2×Q814yes—
276C2×C4⋊C4C2×Q818yes—
277C2×C4⋊C4#4114yes—
278C2×C4⋊C4#4114yes—
279C2×C42—22no—
280C2×C42—22no—
281C2×C42—22no—
282C2×C42—22no—
283C2×C42—22no—
284C2×C42—22no—
285C2×C42—22no—
286C2×C42—28no—
287C2×C42—24no—
288C2×C42—22no—
289C2×C42—22no—
290C2×C42—22no—
291C2×C42—24no—
292C2×C42—24no—
293C2×C42—24no—
294C2×C42—24no—
295C2×C42—24no—
296C2×C42—24no—
297C2×C42—22no—
298C2×C42—24no—
299C2×C42—22no—
300C2×C42—28no—
301C2×C42—22no—
302C2×C42—24no—
303C2×C42—24no—
304C2×C42—24no—
305C2×C42—22no—
306C2×C42—24no—
307C2×C42—44no—
308C2×C42—41no—
309C2×C42—41no—
310C2×C42C2×D414yes—
311C2×C42C2×Q814yes—
312C2×C42C2×Q814yes—
313C2×C42#4114yes—
314C2×C42#4114yes—
315#95—28no—
316#95—28no—
317#95—24no—
318#95—28no—
319#95—28no—
320#95—24no—
321#95—24no—
322#95—24no—
323#95—28no—
324#95—24no—
325#95—24no—
326#95—24no—
327#95—24no—
328#95—24no—
329#95—42no—
330#95—42no—
331#95—44no—
332#95—44no—
333#95—42no—
334#95—42no—
335#95—42no—
336#95—44no—
337#95—44no—
338#95—44no—
339#95—44no—
340#95—44no—
341#95—44no—
342#95—44no—
343#95C2×Q818yes—
344#95C2×Q818yes—
345#95C2×Q818yes—
346C24C2511yescharacteristic
347C2×Q8—216no—
348C2×Q8—28no—
349C2×Q8—28no—
350C2×Q8—28no—
351C2×Q8—48no—
352C2×Q8—48no—
353C2×Q8—48no—
354C2×Q8—48no—
355C2×Q8—48no—
356C2×Q8—48no—
357C2×Q8—44no—
358C2×Q8—44no—
359C2×Q8—44no—
360C2×Q8—44no—
361C22×C4—21no—
362C22×C4—21no—
363C22×C4—21no—
364C22×C4—21no—
365C22×C4—21no—
366C22×C4—21no—
367C22×C4—21no—
368C22×C4—21no—
369C22×C4—21no—
370C22×C4—21no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—21no—
375C22×C4—21no—
376C22×C4—22no—
377C22×C4—21no—
378C22×C4—21no—
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—21no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—24no—
391C22×C4—24no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—21no—
399C22×C4—22no—
400C22×C4—21no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—24no—
404C22×C4—24no—
405C22×C4—21no—
406C22×C4—22no—
407C22×C4—21no—
408C22×C4—21no—
409C22×C4—21no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—21no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—21no—
421C22×C4—21no—
422C22×C4—21no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—24no—
426C22×C4—24no—
427C22×C4—21no—
428C22×C4—22no—
429C22×C4—24no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—24no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—24no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—24no—
441C22×C4—24no—
442C22×C4—24no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—24no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4C22×D418yes—
451C22×C4C22×D412yes—
452C22×C4C22×D412yes—
453C22×C4C22×D412yes—
454C22×C4C22×Q812yes—
455C22×C4C22×Q814yes—
456C22×C4C22×Q814yes—
457C22×C4C22×Q814yes—
458C22×C4C22×Q814yes—
459C22×C4C22×Q812yes—
460C22×C4C22×Q812yes—
461C22×C4C22×Q812yes—
462C22×C4C22×Q812yes—
463C22×C4C22×Q812yes—
464C22×C4C2×#4114yes—
465C22×C4C2×#4114yes—
466C22×C4C2×#4112yes—
467C22×C4C2×#4112yes—
468C22×C4C2×#4112yes—
469C22×C4C2×#4112yes—
470C22×C4C2×#4112yes—
471C22×C4C2×#4112yes—
472C22×C4C2×#4112yes—
473C22×C4C2×#4112yes—
474C22×C4H(5,𝔽2)12yes—
475C22×C4H(5,𝔽2)12yes—
476C22×C4H(5,𝔽2)12yes—
477C22×C42−1+412yes—
478C22×C42−1+412yes—
479C22×C42−1+412yes—
480C4⋊C4—216no—
481C4⋊C4—216no—
482C4⋊C4—28no—
483C4⋊C4—28no—
484C4⋊C4—28no—
485C4⋊C4—28no—
486C4⋊C4—28no—
487C4⋊C4—48no—
488C4⋊C4—416no—
489C4⋊C4—48no—
490C4⋊C4—416no—
491C4⋊C4—48no—
492C4⋊C4—48no—
493C4⋊C4—44no—
494C4⋊C4—44no—
495C4⋊C4—48no—
496C4⋊C4—48no—
497C4⋊C4—88no—
498C4⋊C4—88no—
499C4⋊C4—88no—
500C42—28no—
501C42—28no—
502C42—28no—
503C42—416no—
504C42—44no—
505C42—48no—
506C42—48no—
507C42—44no—
508C42—44no—
509C42—48no—
510C42—48no—
511C42#123116yes—
512C23C23×D411yescharacteristic
513C23C23×Q811yescharacteristic
514C23C23×Q811yescharacteristic
515C23C22×#4111yescharacteristic
516C23C22×#4111yescharacteristic
517C23C22×#4111yescharacteristic
518C23C22×#4111yescharacteristic
519C23C2×H(5,𝔽2)11yescharacteristic
520C23C2×H(5,𝔽2)11yescharacteristic
521C23C2×H(5,𝔽2)11yescharacteristic
522C23C2×2−1+411yescharacteristic
523C23C2×2−1+411yescharacteristic
524C23C2×2−1+411yescharacteristic
525C23#58511yescharacteristic
526C23#58511yescharacteristic
527Q8—432no—
528Q8—416no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—28no—
542C2×C4—24no—
543C2×C4—24no—
544C2×C4—24no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—24no—
548C2×C4—24no—
549C2×C4—24no—
550C2×C4—24no—
551C2×C4—24no—
552C2×C4—24no—
553C2×C4—22no—
554C2×C4—24no—
555C2×C4—22no—
556C2×C4—24no—
557C2×C4—24no—
558C2×C4—22no—
559C2×C4—28no—
560C2×C4—28no—
561C2×C4—22no—
562C2×C4—24no—
563C2×C4—24no—
564C2×C4—22no—
565C2×C4—24no—
566C2×C4—24no—
567C2×C4—24no—
568C2×C4—22no—
569C2×C4—24no—
570C2×C4—22no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—44no—
576C2×C4—44no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—41no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—41no—
588C2×C4—42no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—44no—
595C2×C4—44no—
596C2×C4—41no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—41no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—41no—
608C2×C4—42no—
609C2×C4—41no—
610C2×C4—41no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—44no—
618C2×C4—44no—
619C2×C4—44no—
620C2×C4—44no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—42no—
624C2×C4—42no—
625C2×C4—44no—
626C2×C4—44no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—42no—
630C2×C4—44no—
631C2×C4—42no—
632C2×C4C2×#12314yes—
633C2×C4C2×#12314yes—
634C2×C4C2×#12314yes—
635C2×C4#53314yes—
636C2×C4#53314yes—
637C2×C4#54114yes—
638C2×C4#54214yes—
639C2×C4#54814yes—
640C2×C4D4×Q814yes—
641C2×C4D4×Q814yes—
642C2×C4#55014yes—
643C2×C4#55214yes—
644C2×C4#55414yes—
645C2×C4#55614yes—
646C2×C4#55714yes—
647C2×C4Q8214yes—
648C22C22×#12311yescharacteristic
649C22C2×#53311yescharacteristic
650C22C2×#53311yescharacteristic
651C22C2×#54111yescharacteristic
652C22C2×#54211yescharacteristic
653C22#346011yescharacteristic
654C22#346011yescharacteristic
655C22C2×#54811yescharacteristic
656C22C2×D4×Q811yescharacteristic
657C22C2×D4×Q811yescharacteristic
658C22C2×#55011yescharacteristic
659C22C2×#55211yescharacteristic
660C22C2×#55411yescharacteristic
661C22C2×#55611yescharacteristic
662C22C2×#55711yescharacteristic
663C22C2×Q8211yescharacteristic
664C22Q8×#4111yescharacteristic
665C22Q8×#4111yescharacteristic
666C22#348111yescharacteristic
667C22#348411yescharacteristic
668C22#348511yescharacteristic
669C22#350111yescharacteristic
670C22#350211yescharacteristic
671C22#350311yescharacteristic
672C22#350311yescharacteristic
673C22#350611yescharacteristic
674C22#350611yescharacteristic
675C22#351511yescharacteristic
676C22#351811yescharacteristic
677C22#351911yescharacteristic
678C22#352211yescharacteristic
679C22#354411yescharacteristic
680C22#354711yescharacteristic
681C22#354811yescharacteristic
682C22#355511yescharacteristic
683C4—24no—
684C4—24no—
685C4—44no—
686C4—44no—
687C4—48no—
688C4—44no—
689C4—44no—
690C4—42no—
691C4—44no—
692C4—44no—
693C4—42no—
694C4—42no—
695C4—82no—
696C4—82no—
697C4—84no—
698C4#242318yes—
699C4#244618yes—
700C2C2×#242311yescharacteristic
701C2C2×#244611yescharacteristic
702C2#3495011yescharacteristic
703C2#3515211yescharacteristic
704C2#3517711yescharacteristic
705C2#3520811yescharacteristic
706C2#3763711yescharacteristic
707C2#5905211yescharacteristic
708C2#5905711yescharacteristic
709C2#5908111yescharacteristic
710C2#5915111yescharacteristic
711C2#5942911yescharacteristic
712C2#5943611yescharacteristic
713C2#5944511yescharacteristic
714C2#5987111yescharacteristic
subgroupG/H
center, commutator, Frattini, socle, prime radical, cosocleC24C25
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
134. #391Q8semidirect
142. #389Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#25672
C2×C4×C4⋊Q82
#166102
#151932
#151872
#151372
#151332
#148792
#141612
#141422
#141322
#140292
#139812
#139262
#139102
C4×#3912
C4×#3892