Finite Group Databasebeta

← #946807

Subgroups — #946807

Order 512.

2414 conjugacy classes of subgroups.

The subgroup lattice has 758 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2567C211yesmaximal, characteristic, complemented
2C2×#2580C211yesmaximal, characteristic
3#17104C214yesmaximal, complemented
4#16609C214yesmaximal, complemented
5#16318C214yesmaximal, complemented
6#15392C214yesmaximal, complemented
7#15239C211yesmaximal, characteristic, complemented
8#15231C211yesmaximal, characteristic, complemented
9#15151C211yesmaximal, characteristic, complemented
10#15144C211yesmaximal, characteristic
11#15143C211yesmaximal, characteristic, complemented
12#15134C211yesmaximal, characteristic
13#14277C211yesmaximal, characteristic, complemented
14#14260C211yesmaximal, characteristic
15#14048C211yesmaximal, characteristic, complemented
16#14002C211yesmaximal, characteristic
17#13977C211yesmaximal, characteristic
18#13977C211yesmaximal, characteristic
19#13850C211yesmaximal, characteristic, complemented
20C22×C4⋊Q8C2211yescharacteristic
21C22×C4×Q8C2211yescharacteristic
22C22×C4×D4C2211yescharacteristic
23#2580—28no—
24#2567—28no—
25C2×#399C2212yes—
26C2×#398C2214yes—
27C2×#396C2214yes—
28C2×#395C2212yes—
29C2×#395C2212yes—
30C2×#395C2214yes—
31C2×#393C2212yes—
32C2×#393C2214yes—
33C2×#391C2212yes—
34C2×#390C2211yescharacteristic
35C2×#390C2211yescharacteristic
36C2×#390C2211yescharacteristic
37C2×#390C2211yescharacteristic
38C2×#390C2211yescharacteristic
39C2×#390C2211yescharacteristic
40C2×#389C2211yescharacteristic
41C2×#389C2211yescharacteristic
42C2×#389C2211yescharacteristic
43C2×#389C2211yescharacteristic
44C2×#389C2211yescharacteristic
45C2×#389C2211yescharacteristic
46C2×#385C2212yes—
47C2×#385C2212yes—
48C2×#385C2214yes—
49C2×#385C2214yes—
50C2×#384C2211yescharacteristic
51C2×#384C2212yes—
52C2×#384C2211yescharacteristic
53C2×#384C2211yescharacteristic
54C2×#382C2211yescharacteristic
55C2×#382C2211yescharacteristic
56C2×C4×C4⋊C4C2211yescharacteristic
57C2×C4×C4⋊C4C2211yescharacteristic
58C2×#376C2211yescharacteristic
59#1446C2212yes—
60#1445C2212yes—
61#1445C2214yes—
62#1444C2214yes—
63#1444C2214yes—
64#1443C2214yes—
65#1442C2214yes—
66#1442C2214yes—
67#1442C2214yes—
68#1441C2212yes—
69#1441C2212yes—
70#1441C2214yes—
71#1441C2214yes—
72#1440C2214yes—
73#1440C2214yes—
74#1439C2214yes—
75#1439C2214yes—
76#1439C2214yes—
77#1439C2214yes—
78#1439C2214yes—
79#1437C2214yes—
80#1436C2214yes—
81C4⋊#95C2211yescharacteristic
82C4⋊#95C2211yescharacteristic
83C4⋊#95C2211yescharacteristic
84C4⋊#95C2211yescharacteristic
85C4⋊#95C2211yescharacteristic
86C4⋊#95C2211yescharacteristic
87C4⋊#95C2211yescharacteristic
88C4⋊#95C2211yescharacteristic
89C4×#95C2211yescharacteristic
90C4×#95C2211yescharacteristic
91C4×#95C2211yescharacteristic
92C4×#95C2211yescharacteristic
93C23×Q8C2311yescharacteristic
94C23×Q8C2312yes—
95C23×D4C2311yescharacteristic
96C24×C4C2312yes—
97C2×C4⋊Q8—24no—
98C2×C4⋊Q8—42no—
99C2×C4⋊Q8—42no—
100C2×C4×Q8—42no—
101C2×C4×Q8—42no—
102C2×C4×Q8D418yes—
103C2×C4×D4—24no—
104C2×C4×D4—42no—
105C2×C4×D4—42no—
106C22×C4⋊C4C2311yescharacteristic
107C22×C4⋊C4C2311yescharacteristic
108C22×C4⋊C4C2311yescharacteristic
109C22×C4⋊C4C2311yescharacteristic
110C22×C4⋊C4C2311yescharacteristic
111C22×C4⋊C4C2312yes—
112C22×C4⋊C4C2311yescharacteristic
113C22×C4⋊C4C2311yescharacteristic
114C22×C4⋊C4C2312yes—
115C22×C4⋊C4C2312yes—
116C22×C4⋊C4C2312yes—
117C22×C4⋊C4C2314yes—
118C22×C4⋊C4C2312yes—
119C22×C4⋊C4C2312yes—
120C22×C4⋊C4C2314yes—
121C22×C4⋊C4C2312yes—
122C22×C4⋊C4C2312yes—
123C22×C4⋊C4C2314yes—
124C22×C4⋊C4C2311yescharacteristic
125C22×C4⋊C4C2311yescharacteristic
126C22×C4⋊C4C2311yescharacteristic
127C22×C22⋊C4C2312yes—
128C22×C22⋊C4C2314yes—
129C22×C22⋊C4C2312yes—
130C22×C22⋊C4C2314yes—
131C22×C22⋊C4C2312yes—
132C22×C22⋊C4C2312yes—
133C22×C22⋊C4C2312yes—
134C22×C22⋊C4C2314yes—
135C22×C22⋊C4C2314yes—
136C22×C42C2311yescharacteristic
137C22×C42C2311yescharacteristic
138C22×C42C2311yescharacteristic
139C22×C42C2311yescharacteristic
140C22×C42C2311yescharacteristic
141C22×C42C2311yescharacteristic
142C22×C42C2311yescharacteristic
143C22×C42C2312yes—
144#399—28no—
145#398—48no—
146#396—48no—
147#395—28no—
148#395—28no—
149#395—48no—
150#393—28no—
151#393—48no—
152#391—28no—
153#390—24no—
154#390—24no—
155#390—24no—
156#390—42no—
157#390—42no—
158#390Q818yescomplemented
159#389—24no—
160#389—24no—
161#389—24no—
162#389—24no—
163#389—24no—
164#389—24no—
165#385—28no—
166#385—28no—
167#385—48no—
168#385—48no—
169#384—28no—
170#384—24no—
171#384—42no—
172#384—42no—
173#382—24no—
174#382—24no—
175C4×C4⋊C4—24no—
176C4×C4⋊C4—42no—
177#376—42no—
178C2×#95C2312yes—
179C2×#95C2314yes—
180C2×#95C2312yes—
181C2×#95C2312yes—
182C2×#95C2312yes—
183C2×#95C2312yes—
184C2×#95C2312yes—
185C2×#95C2314yes—
186C2×#95C2312yes—
187C2×#95C2312yes—
188C2×#95C2312yes—
189C2×#95C2312yes—
190C2×#95C2314yes—
191C2×#95C2312yes—
192C2×#95C2312yes—
193C2×#95C2312yes—
194C2×#95C2312yes—
195C2×#95C2314yes—
196C2×#95C2312yes—
197C2×#95C2314yes—
198C2×#95C2314yes—
199C2×#95C2314yes—
200C2×#95C2312yes—
201C2×#95C2312yes—
202C2×#95C2314yes—
203C2×#95C2314yes—
204C2×#95C2312yes—
205C2×#95C2312yes—
206C2×#95C2312yes—
207C25C2412yes—
208C22×Q8—24no—
209C22×Q8—24no—
210C22×Q8—41no—
211C22×Q8—41no—
212C22×Q8—42no—
213C22×Q8—42no—
214C22×Q8—42no—
215C22×Q8—44no—
216C22×Q8—44no—
217C22×Q8C2×D414yes—
218C22×D4—22no—
219C22×D4—22no—
220C22×D4—22no—
221C22×D4—22no—
222C22×D4—22no—
223C22×D4—22no—
224C22×D4C2×Q814yes—
225C23×C4—24no—
226C23×C4—42no—
227C23×C4—42no—
228C23×C4—44no—
229C23×C4—44no—
230C23×C4C2412yes—
231C23×C4C2412yes—
232C23×C4C2412yes—
233C23×C4C2412yes—
234C23×C4C2412yes—
235C23×C4C2412yes—
236C23×C4C2411yescharacteristic
237C23×C4C2411yescharacteristic
238C23×C4C2411yescharacteristic
239C23×C4C2411yescharacteristic
240C23×C4C2411yescharacteristic
241C23×C4C2412yes—
242C23×C4C2412yes—
243C23×C4C2412yes—
244C23×C4C2412yes—
245C23×C4C2412yes—
246C23×C4C2411yescharacteristic
247C23×C4C2411yescharacteristic
248C4⋊Q8—416no—
249C4×Q8—416no—
250C4×D4—416no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—24no—
257C2×C4⋊C4—22no—
258C2×C4⋊C4—22no—
259C2×C4⋊C4—22no—
260C2×C4⋊C4—22no—
261C2×C4⋊C4—22no—
262C2×C4⋊C4—22no—
263C2×C4⋊C4—22no—
264C2×C4⋊C4—22no—
265C2×C4⋊C4—22no—
266C2×C4⋊C4—22no—
267C2×C4⋊C4—22no—
268C2×C4⋊C4—22no—
269C2×C4⋊C4—24no—
270C2×C4⋊C4—24no—
271C2×C4⋊C4—22no—
272C2×C4⋊C4—22no—
273C2×C4⋊C4—24no—
274C2×C4⋊C4—24no—
275C2×C4⋊C4—24no—
276C2×C4⋊C4—24no—
277C2×C4⋊C4—24no—
278C2×C4⋊C4—24no—
279C2×C4⋊C4—28no—
280C2×C4⋊C4—28no—
281C2×C4⋊C4—41no—
282C2×C4⋊C4—41no—
283C2×C4⋊C4—41no—
284C2×C4⋊C4—41no—
285C2×C4⋊C4—41no—
286C2×C4⋊C4—41no—
287C2×C4⋊C4—42no—
288C2×C4⋊C4—42no—
289C2×C4⋊C4—42no—
290C2×C4⋊C4—44no—
291C2×C4⋊C4—44no—
292C2×C4⋊C4—44no—
293C2×C4⋊C4—44no—
294C2×C4⋊C4—44no—
295C2×C4⋊C4—44no—
296C2×C4⋊C4—44no—
297C2×C4⋊C4—44no—
298C2×C4⋊C4—44no—
299C2×C4⋊C4—44no—
300C2×C4⋊C4—44no—
301C2×C4⋊C4—44no—
302C2×C4⋊C4—44no—
303C2×C4⋊C4C2×D414yes—
304C2×C4⋊C4C2×D414yes—
305C2×C4⋊C4C2×D414yes—
306C2×C4⋊C4C2×Q814yes—
307C2×C4⋊C4#4114yes—
308C2×C22⋊C4—24no—
309C2×C22⋊C4—24no—
310C2×C22⋊C4—24no—
311C2×C22⋊C4—28no—
312C2×C22⋊C4—28no—
313C2×C22⋊C4—28no—
314C2×C22⋊C4—24no—
315C2×C22⋊C4—24no—
316C2×C22⋊C4—28no—
317C2×C22⋊C4—44no—
318C2×C22⋊C4—44no—
319C2×C22⋊C4—44no—
320C2×C22⋊C4—42no—
321C2×C22⋊C4—42no—
322C2×C22⋊C4—44no—
323C2×C22⋊C4—44no—
324C2×C22⋊C4—44no—
325C2×C22⋊C4—44no—
326C2×C22⋊C4—44no—
327C2×C22⋊C4—44no—
328C2×C22⋊C4—44no—
329C2×C22⋊C4—44no—
330C2×C22⋊C4C2×Q8116yes—
331C2×C42—22no—
332C2×C42—22no—
333C2×C42—22no—
334C2×C42—22no—
335C2×C42—22no—
336C2×C42—22no—
337C2×C42—22no—
338C2×C42—22no—
339C2×C42—24no—
340C2×C42—22no—
341C2×C42—22no—
342C2×C42—22no—
343C2×C42—22no—
344C2×C42—22no—
345C2×C42—22no—
346C2×C42—22no—
347C2×C42—22no—
348C2×C42—22no—
349C2×C42—44no—
350C2×C42C2×D414yes—
351C2×C42C2×D414yes—
352C2×C42C2×D414yes—
353C2×C42C2×Q814yes—
354#95—24no—
355#95—24no—
356#95—24no—
357#95—24no—
358#95—24no—
359#95—24no—
360#95—24no—
361#95—24no—
362#95—24no—
363#95—28no—
364#95—24no—
365#95—24no—
366#95—28no—
367#95—24no—
368#95—28no—
369#95—24no—
370#95—28no—
371#95—24no—
372#95—24no—
373#95—42no—
374#95—42no—
375#95—44no—
376#95—42no—
377#95—42no—
378#95—42no—
379#95—44no—
380#95—44no—
381#95—44no—
382#95—44no—
383C24—22no—
384C24—22no—
385C24—24no—
386C24—24no—
387C24—22no—
388C24—22no—
389C24—24no—
390C24—22no—
391C24—22no—
392C24—24no—
393C24C22×Q814yes—
394C24C2511yescharacteristic
395C2×Q8—28no—
396C2×Q8—44no—
397C2×Q8—44no—
398C2×Q8—48no—
399C2×Q8—48no—
400C2×Q8—48no—
401C2×Q8—84no—
402C2×Q8—84no—
403C2×Q8—88no—
404C2×Q8—88no—
405C2×D4—28no—
406C2×D4—28no—
407C2×D4—28no—
408C2×D4—44no—
409C2×D4—44no—
410C2×D4—44no—
411C2×D4—44no—
412C22×C4—21no—
413C22×C4—21no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—24no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—21no—
428C22×C4—22no—
429C22×C4—21no—
430C22×C4—21no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—24no—
435C22×C4—21no—
436C22×C4—21no—
437C22×C4—21no—
438C22×C4—21no—
439C22×C4—21no—
440C22×C4—21no—
441C22×C4—21no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—24no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—24no—
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—22no—
470C22×C4—24no—
471C22×C4—24no—
472C22×C4—24no—
473C22×C4—24no—
474C22×C4—24no—
475C22×C4—22no—
476C22×C4—24no—
477C22×C4—24no—
478C22×C4—24no—
479C22×C4—22no—
480C22×C4—22no—
481C22×C4—22no—
482C22×C4—22no—
483C22×C4—22no—
484C22×C4—22no—
485C22×C4—22no—
486C22×C4—22no—
487C22×C4—22no—
488C22×C4—24no—
489C22×C4—22no—
490C22×C4—22no—
491C22×C4—22no—
492C22×C4—24no—
493C22×C4—48no—
494C22×C4—84no—
495C22×C4—84no—
496C22×C4—88no—
497C22×C4—88no—
498C22×C4C22×D412yes—
499C22×C4C22×D412yes—
500C22×C4C22×D412yes—
501C22×C4C22×D412yes—
502C22×C4C22×D412yes—
503C22×C4C22×D412yes—
504C22×C4C22×D412yes—
505C22×C4C22×Q814yes—
506C22×C4C22×Q814yes—
507C22×C4C22×Q812yes—
508C22×C4C2×#4114yes—
509C22×C4C2×#4112yes—
510C22×C4H(5,𝔽2)14yes—
511C22×C4H(5,𝔽2)14yes—
512C22×C4H(5,𝔽2)12yes—
513C22×C4H(5,𝔽2)12yes—
514C22×C42−1+414yes—
515C22×C42−1+412yes—
516C22×C42−1+412yes—
517C4⋊C4—28no—
518C4⋊C4—28no—
519C4⋊C4—44no—
520C4⋊C4—44no—
521C4⋊C4—44no—
522C4⋊C4—48no—
523C4⋊C4—48no—
524C4⋊C4—44no—
525C4⋊C4—44no—
526C4⋊C4—44no—
527C4⋊C4—44no—
528C4⋊C4—44no—
529C4⋊C4—84no—
530C4⋊C4—84no—
531C4⋊C4—84no—
532C4⋊C4—84no—
533C4⋊C4—84no—
534C4⋊C4—84no—
535C4⋊C4—88no—
536C4⋊C4—88no—
537C4⋊C4—88no—
538C22⋊C4—416no—
539C22⋊C4—416no—
540C22⋊C4—48no—
541C22⋊C4—48no—
542C22⋊C4—48no—
543C22⋊C4—84no—
544C22⋊C4—84no—
545C22⋊C4—88no—
546C22⋊C4—88no—
547C42—28no—
548C42—28no—
549C42—28no—
550C42—28no—
551C42—48no—
552C42—44no—
553C42—44no—
554C42—44no—
555C23—24no—
556C23—24no—
557C23—24no—
558C23—28no—
559C23—28no—
560C23—42no—
561C23—42no—
562C23—42no—
563C23—42no—
564C23—42no—
565C23—42no—
566C23—42no—
567C23—44no—
568C23—44no—
569C23—42no—
570C23—44no—
571C23—44no—
572C23—44no—
573C23—44no—
574C23—44no—
575C23—44no—
576C23—44no—
577C23—44no—
578C23C23×D411yescharacteristic
579C23C23×Q811yescharacteristic
580C23C22×#4111yescharacteristic
581C23C2×H(5,𝔽2)11yescharacteristic
582C23C2×H(5,𝔽2)11yescharacteristic
583C23C2×H(5,𝔽2)12yes—
584C23C2×H(5,𝔽2)12yes—
585C23C2×2−1+412yes—
586C23C2×2−1+411yescharacteristic
587C23C2×2−1+411yescharacteristic
588C23#58512yes—
589Q8—416no—
590Q8—816no—
591D4—416no—
592C2×C4—22no—
593C2×C4—22no—
594C2×C4—24no—
595C2×C4—24no—
596C2×C4—22no—
597C2×C4—22no—
598C2×C4—22no—
599C2×C4—22no—
600C2×C4—22no—
601C2×C4—22no—
602C2×C4—22no—
603C2×C4—22no—
604C2×C4—22no—
605C2×C4—24no—
606C2×C4—24no—
607C2×C4—24no—
608C2×C4—22no—
609C2×C4—22no—
610C2×C4—22no—
611C2×C4—22no—
612C2×C4—22no—
613C2×C4—22no—
614C2×C4—24no—
615C2×C4—28no—
616C2×C4—24no—
617C2×C4—24no—
618C2×C4—28no—
619C2×C4—24no—
620C2×C4—24no—
621C2×C4—24no—
622C2×C4—24no—
623C2×C4—24no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—42no—
630C2×C4—42no—
631C2×C4—42no—
632C2×C4—42no—
633C2×C4—42no—
634C2×C4—42no—
635C2×C4—42no—
636C2×C4—42no—
637C2×C4—42no—
638C2×C4—42no—
639C2×C4—42no—
640C2×C4—42no—
641C2×C4—42no—
642C2×C4—42no—
643C2×C4—42no—
644C2×C4—42no—
645C2×C4—42no—
646C2×C4—42no—
647C2×C4—42no—
648C2×C4—42no—
649C2×C4—44no—
650C2×C4—42no—
651C2×C4—44no—
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—44no—
669C2×C4—42no—
670C2×C4—44no—
671C2×C4—44no—
672C2×C4—44no—
673C2×C4—42no—
674C2×C4—42no—
675C2×C4—42no—
676C2×C4—44no—
677C2×C4—44no—
678C2×C4—44no—
679C2×C4—44no—
680C2×C4—42no—
681C2×C4—42no—
682C2×C4—44no—
683C2×C4—42no—
684C2×C4—816no—
685C2×C4#53414yes—
686C2×C4#53614yes—
687C2×C4#54214yes—
688C2×C4#54414yes—
689C2×C4#54814yes—
690C2×C4D4×Q814yes—
691C2×C4#55414yes—
692C2×C4#56314yes—
693C22—48no—
694C22—48no—
695C22—44no—
696C22—44no—
697C22—44no—
698C22—82no—
699C22—84no—
700C22—82no—
701C22—84no—
702C22—82no—
703C22—82no—
704C22C2×#53411yescharacteristic
705C22C2×#53611yescharacteristic
706C22#344911yescharacteristic
707C22C2×#54211yescharacteristic
708C22C2×#54411yescharacteristic
709C22C2×#54811yescharacteristic
710C22C2×D4×Q811yescharacteristic
711C22C2×#55411yescharacteristic
712C22#349811yescharacteristic
713C22#349811yescharacteristic
714C22#350112yes—
715C22#350312yes—
716C22#351312yes—
717C22#352212yes—
718C22C2×#56311yescharacteristic
719C22#353712yes—
720C22#354112yes—
721C22#354112yes—
722C22#354412yes—
723C22#354912yes—
724C22#355012yes—
725C22#355412yes—
726C22#355912yes—
727C4—24no—
728C4—44no—
729C4—44no—
730C4—42no—
731C4—42no—
732C4—44no—
733C4—42no—
734C4—42no—
735C4—42no—
736C4—44no—
737C4—44no—
738C4—82no—
739C4—82no—
740C4—82no—
741C4—82no—
742C4—82no—
743C4—82no—
744C4#272918yes—
745C2—84no—
746C2#3729711yescharacteristic
747C2#3744911yescharacteristic
748C2#3916111yescharacteristic
749C2#3916811yescharacteristic
750C2#3961111yescharacteristic
751C2#3961711yescharacteristic
752C2#4249612yes—
753C2#4253712yes—
754C2#4494612yes—
755C2C2×#272911yescharacteristic
756C2#5310912yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC23×D4C23
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#2567C2semidirect
3. #17104C2semidirect
4. #16609C2semidirect
5. #16318C2semidirect
6. #15392C2semidirect
7. #15239C2semidirect
8. #15231C2semidirect
9. #15151C2semidirect
11. #15143C2semidirect
13. #14277C2semidirect
15. #14048C2semidirect
19. #13850C2semidirect
158. #390Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#25672
C2×#25802
#171042
#166092
#163182
#153922
#152392
#152312
#151512
#151442
#151432
#151342
#142772
#142602
#140482
#140022
#139772
#138502