Finite Group Databasebeta

← #732490

Subgroups — #732490

Order 512.

2494 conjugacy classes of subgroups.

The subgroup lattice has 745 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2571C211yesmaximal, characteristic, complemented
2C2×#2349C211yesmaximal, characteristic
3#16304C214yesmaximal, complemented
4#16057C214yesmaximal, complemented
5#15462C214yesmaximal, complemented
6#15239C211yesmaximal, characteristic, complemented
7#15239C211yesmaximal, characteristic, complemented
8#15234C211yesmaximal, characteristic, complemented
9#15151C211yesmaximal, characteristic, complemented
10#14767C214yesmaximal, complemented
11#14348C211yesmaximal, characteristic, complemented
12#13979C211yesmaximal, characteristic
13#13977C211yesmaximal, characteristic
14#13941C211yesmaximal, characteristic
15#13941C211yesmaximal, characteristic
16#13909C211yesmaximal, characteristic, complemented
17#13906C211yesmaximal, characteristic
18#13848C211yesmaximal, characteristic, complemented
19#13785C211yesmaximal, characteristic
20C22×C4×D4C2211yescharacteristic
21C22×C4×D4C2211yescharacteristic
22C22×C4×D4C2211yescharacteristic
23#2571—28no—
24C2×#400C2212yes—
25C2×#400C2212yes—
26C2×#398C2214yes—
27#2349—28no—
28C2×#390C2211yescharacteristic
29C2×#390C2211yescharacteristic
30C2×#390C2211yescharacteristic
31C2×#390C2211yescharacteristic
32C2×#389C2211yescharacteristic
33C2×#389C2211yescharacteristic
34C2×#389C2211yescharacteristic
35C2×#389C2212yes—
36C2×#389C2211yescharacteristic
37C2×#385C2212yes—
38C2×#385C2212yes—
39C2×#385C2214yes—
40C2×#384C2211yescharacteristic
41C2×#384C2211yescharacteristic
42C2×#382C2211yescharacteristic
43C2×#382C2211yescharacteristic
44C2×#382C2211yescharacteristic
45C2×#382C2212yes—
46C2×#382C2211yescharacteristic
47C2×#380C2212yes—
48C2×#380C2214yes—
49C2×C4×C4⋊C4C2211yescharacteristic
50C2×C4×C4⋊C4C2211yescharacteristic
51C2×C4×C4⋊C4C2211yescharacteristic
52C2×C4×C4⋊C4C2211yescharacteristic
53C2×C4×C4⋊C4C2211yescharacteristic
54C2×#376C2211yescharacteristic
55C22×#95C2212yes—
56#1445C2214yes—
57#1444C2212yes—
58#1443C2214yes—
59#1443C2214yes—
60#1442C2214yes—
61#1442C2214yes—
62#1442C2212yes—
63#1442C2214yes—
64#1442C2214yes—
65#1442C2214yes—
66#1441C2212yes—
67#1441C2212yes—
68#1441C2214yes—
69#1441C2214yes—
70#1439C2214yes—
71#1439C2214yes—
72#1439C2214yes—
73#1439C2214yes—
74#1439C2214yes—
75#1437C2214yes—
76#1436C2214yes—
77C4⋊#95C2211yescharacteristic
78C4⋊#95C2211yescharacteristic
79C4⋊#95C2211yescharacteristic
80C4⋊#95C2211yescharacteristic
81C4⋊#95C2211yescharacteristic
82C4⋊#95C2211yescharacteristic
83C4⋊#95C2211yescharacteristic
84C4⋊#95C2211yescharacteristic
85C4⋊#95C2211yescharacteristic
86C4⋊#95C2211yescharacteristic
87C22⋊#95C2214yes—
88C22⋊#95C2214yes—
89C22⋊#95C2214yes—
90C22⋊#95C2214yes—
91C4×#95C2211yescharacteristic
92C4×#95C2211yescharacteristic
93C23×D4C2311yescharacteristic
94C24×C4C2312yes—
95C24×C4C2312yes—
96C24×C4C2312yes—
97C2×C4×D4—24no—
98C2×C4×D4—24no—
99C2×C4×D4—24no—
100C2×C4×D4—24no—
101C2×C4×D4—24no—
102C2×C4×D4—24no—
103C2×C4×D4—42no—
104C2×C4×D4—42no—
105C2×C4×D4Q818yes—
106C22×C4⋊C4C2311yescharacteristic
107C22×C4⋊C4C2311yescharacteristic
108C22×C4⋊C4C2311yescharacteristic
109C22×C4⋊C4C2311yescharacteristic
110C22×C4⋊C4C2311yescharacteristic
111C22×C4⋊C4C2311yescharacteristic
112C22×C4⋊C4C2312yes—
113C22×C4⋊C4C2311yescharacteristic
114C22×C4⋊C4C2312yes—
115C22×C4⋊C4C2312yes—
116C22×C4⋊C4C2312yes—
117C22×C4⋊C4C2314yes—
118C22×C4⋊C4C2312yes—
119C22×C4⋊C4C2312yes—
120C22×C4⋊C4C2312yes—
121C22×C4⋊C4C2312yes—
122C22×C4⋊C4C2312yes—
123C22×C4⋊C4C2311yescharacteristic
124C22×C4⋊C4C2311yescharacteristic
125C22×C22⋊C4C2312yes—
126C22×C22⋊C4C2312yes—
127C22×C22⋊C4C2314yes—
128C22×C22⋊C4C2312yes—
129C22×C22⋊C4C2314yes—
130C22×C22⋊C4C2314yes—
131C22×C22⋊C4C2314yes—
132C22×C42C2311yescharacteristic
133C22×C42C2311yescharacteristic
134C22×C42C2311yescharacteristic
135C22×C42C2311yescharacteristic
136C22×C42C2311yescharacteristic
137C22×C42C2311yescharacteristic
138C22×C42C2312yes—
139C22×C42C2311yescharacteristic
140#400—28no—
141#400—28no—
142#398—48no—
143#390—24no—
144#390—24no—
145#390—24no—
146#390—24no—
147#389—28no—
148#389—24no—
149#389—24no—
150#389—24no—
151#389—24no—
152#385—28no—
153#385—28no—
154#385—48no—
155#384—24no—
156#384—42no—
157#382—28no—
158#382—24no—
159#382—24no—
160#382—24no—
161#382—42no—
162#380—28no—
163#380—48no—
164C4×C4⋊C4—24no—
165C4×C4⋊C4—24no—
166C4×C4⋊C4—42no—
167C4×C4⋊C4—42no—
168C4×C4⋊C4D418yes—
169#376—24no—
170C2×#95—28no—
171C2×#95C2314yes—
172C2×#95C2312yes—
173C2×#95C2312yes—
174C2×#95C2312yes—
175C2×#95C2312yes—
176C2×#95C2312yes—
177C2×#95C2312yes—
178C2×#95C2312yes—
179C2×#95C2312yes—
180C2×#95C2312yes—
181C2×#95C2312yes—
182C2×#95C2312yes—
183C2×#95C2312yes—
184C2×#95C2312yes—
185C2×#95C2312yes—
186C2×#95C2312yes—
187C2×#95C2314yes—
188C2×#95C2314yes—
189C2×#95C2312yes—
190C2×#95C2312yes—
191C2×#95C2312yes—
192C2×#95C2312yes—
193C2×#95C2312yes—
194C2×#95C2314yes—
195C2×#95C2314yes—
196C2×#95C2314yes—
197C2×#95C2314yes—
198C2×#95C2314yes—
199C2×#95C2314yes—
200C2×#95C2314yes—
201C2×#95C2314yes—
202C2×#95C2311yescharacteristic
203C2×#95C2311yescharacteristic
204C25C2412yes—
205C22×D4—22no—
206C22×D4—22no—
207C22×D4—22no—
208C22×D4—22no—
209C22×D4C2×Q814yes—
210C22×D4#4114yes—
211C22×D4#4114yes—
212C23×C4—24no—
213C23×C4—24no—
214C23×C4—24no—
215C23×C4—24no—
216C23×C4—24no—
217C23×C4—24no—
218C23×C4—42no—
219C23×C4—42no—
220C23×C4—44no—
221C23×C4—44no—
222C23×C4—44no—
223C23×C4—44no—
224C23×C4—44no—
225C23×C4—44no—
226C23×C4C2×Q818yes—
227C23×C4C2412yes—
228C23×C4C2412yes—
229C23×C4C2412yes—
230C23×C4C2412yes—
231C23×C4C2412yes—
232C23×C4C2412yes—
233C23×C4C2412yes—
234C23×C4C2412yes—
235C23×C4C2412yes—
236C23×C4C2411yescharacteristic
237C23×C4C2411yescharacteristic
238C23×C4C2411yescharacteristic
239C23×C4C2411yescharacteristic
240C23×C4C2412yes—
241C23×C4C2412yes—
242C23×C4C2411yescharacteristic
243C23×C4C2411yescharacteristic
244C23×C4C2411yescharacteristic
245C4×D4—232no—
246C4×D4—416no—
247C4×D4—416no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—22no—
257C2×C4⋊C4—22no—
258C2×C4⋊C4—22no—
259C2×C4⋊C4—24no—
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—24no—
267C2×C4⋊C4—22no—
268C2×C4⋊C4—24no—
269C2×C4⋊C4—22no—
270C2×C4⋊C4—22no—
271C2×C4⋊C4—22no—
272C2×C4⋊C4—24no—
273C2×C4⋊C4—24no—
274C2×C4⋊C4—24no—
275C2×C4⋊C4—24no—
276C2×C4⋊C4—24no—
277C2×C4⋊C4—28no—
278C2×C4⋊C4—42no—
279C2×C4⋊C4—42no—
280C2×C4⋊C4—42no—
281C2×C4⋊C4—44no—
282C2×C4⋊C4—42no—
283C2×C4⋊C4—42no—
284C2×C4⋊C4—42no—
285C2×C4⋊C4—42no—
286C2×C4⋊C4—42no—
287C2×C4⋊C4—44no—
288C2×C4⋊C4—44no—
289C2×C4⋊C4—44no—
290C2×C4⋊C4—44no—
291C2×C4⋊C4—44no—
292C2×C4⋊C4—44no—
293C2×C4⋊C4—44no—
294C2×C4⋊C4C2×D414yes—
295C2×C4⋊C4C2×D414yes—
296C2×C4⋊C4C2×Q814yes—
297C2×C22⋊C4—24no—
298C2×C22⋊C4—24no—
299C2×C22⋊C4—24no—
300C2×C22⋊C4—24no—
301C2×C22⋊C4—28no—
302C2×C22⋊C4—24no—
303C2×C22⋊C4—24no—
304C2×C22⋊C4—28no—
305C2×C22⋊C4—28no—
306C2×C22⋊C4—28no—
307C2×C22⋊C4—28no—
308C2×C22⋊C4—44no—
309C2×C22⋊C4—42no—
310C2×C22⋊C4—42no—
311C2×C22⋊C4—44no—
312C2×C22⋊C4—44no—
313C2×C22⋊C4—44no—
314C2×C22⋊C4—44no—
315C2×C22⋊C4—44no—
316C2×C22⋊C4—44no—
317C2×C22⋊C4C2×Q818yes—
318C2×C42—22no—
319C2×C42—22no—
320C2×C42—22no—
321C2×C42—22no—
322C2×C42—24no—
323C2×C42—22no—
324C2×C42—22no—
325C2×C42—22no—
326C2×C42—22no—
327C2×C42—22no—
328C2×C42—22no—
329C2×C42—22no—
330C2×C42—22no—
331C2×C42—22no—
332C2×C42—22no—
333C2×C42—22no—
334C2×C42—22no—
335C2×C42—44no—
336C2×C42C2×D414yes—
337C2×C42C2×D414yes—
338C2×C42C2×D414yes—
339C2×C42C2×Q814yes—
340C2×C42#4114yes—
341#95—24no—
342#95—24no—
343#95—24no—
344#95—24no—
345#95—24no—
346#95—24no—
347#95—24no—
348#95—24no—
349#95—24no—
350#95—24no—
351#95—24no—
352#95—24no—
353#95—24no—
354#95—24no—
355#95—28no—
356#95—24no—
357#95—44no—
358#95—42no—
359#95—42no—
360#95—44no—
361#95—42no—
362#95—44no—
363#95—44no—
364#95—44no—
365#95—42no—
366#95—42no—
367#95—44no—
368#95—44no—
369#95—44no—
370#95—44no—
371#95—44no—
372#95C2×D414yes—
373#95C2×D414yes—
374C24—22no—
375C24—22no—
376C24—24no—
377C24—22no—
378C24—22no—
379C24—24no—
380C24—24no—
381C24—24no—
382C24C22×Q814yes—
383C24C2×#4114yes—
384C24C2×#4114yes—
385C24C2511yescharacteristic
386C2×D4—28no—
387C2×D4—28no—
388C2×D4—28no—
389C2×D4—28no—
390C2×D4—28no—
391C2×D4—28no—
392C2×D4#125116yes—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—21no—
400C22×C4—21no—
401C22×C4—22no—
402C22×C4—21no—
403C22×C4—21no—
404C22×C4—22no—
405C22×C4—24no—
406C22×C4—22no—
407C22×C4—21no—
408C22×C4—21no—
409C22×C4—22no—
410C22×C4—21no—
411C22×C4—21no—
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—22no—
423C22×C4—216no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—24no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—24no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—24no—
452C22×C4—24no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—24no—
462C22×C4—22no—
463C22×C4—24no—
464C22×C4—24no—
465C22×C4—22no—
466C22×C4—24no—
467C22×C4—24no—
468C22×C4—22no—
469C22×C4—24no—
470C22×C4—24no—
471C22×C4—24no—
472C22×C4—22no—
473C22×C4—22no—
474C22×C4—22no—
475C22×C4—416no—
476C22×C4—48no—
477C22×C4—48no—
478C22×C4—48no—
479C22×C4—48no—
480C22×C4—88no—
481C22×C4—84no—
482C22×C4—84no—
483C22×C4—84no—
484C22×C4—84no—
485C22×C4—88no—
486C22×C4—88no—
487C22×C4—88no—
488C22×C4—88no—
489C22×C4C22×D412yes—
490C22×C4C22×D412yes—
491C22×C4C22×D412yes—
492C22×C4C22×D412yes—
493C22×C4C22×D412yes—
494C22×C4C22×D412yes—
495C22×C4C22×D412yes—
496C22×C4C22×Q814yes—
497C22×C4C22×Q812yes—
498C22×C4C22×Q812yes—
499C22×C4C22×Q812yes—
500C22×C4C2×#4112yes—
501C22×C4C2×#4112yes—
502C22×C4C2×#4112yes—
503C22×C4C2×#4112yes—
504C22×C4C2×#4112yes—
505C22×C4H(5,𝔽2)14yes—
506C22×C4H(5,𝔽2)12yes—
507C22×C42−1+412yes—
508C4⋊C4—28no—
509C4⋊C4—44no—
510C4⋊C4—44no—
511C4⋊C4—48no—
512C4⋊C4—48no—
513C4⋊C4—48no—
514C4⋊C4—48no—
515C4⋊C4—44no—
516C4⋊C4—44no—
517C4⋊C4—44no—
518C4⋊C4—44no—
519C4⋊C4—44no—
520C4⋊C4—44no—
521C4⋊C4—84no—
522C4⋊C4—84no—
523C4⋊C4—84no—
524C4⋊C4—84no—
525C4⋊C4—84no—
526C4⋊C4—88no—
527C22⋊C4—216no—
528C22⋊C4—416no—
529C22⋊C4—48no—
530C22⋊C4—48no—
531C22⋊C4—88no—
532C22⋊C4—88no—
533C22⋊C4—88no—
534C42—28no—
535C42—28no—
536C42—48no—
537C42—44no—
538C42—44no—
539C42—44no—
540C42—44no—
541C42C22⋊Q8116yes—
542C23—24no—
543C23—24no—
544C23—24no—
545C23—24no—
546C23—24no—
547C23—24no—
548C23—28no—
549C23—28no—
550C23—28no—
551C23—28no—
552C23—28no—
553C23—28no—
554C23—42no—
555C23—42no—
556C23—44no—
557C23—42no—
558C23—42no—
559C23—44no—
560C23—44no—
561C23—44no—
562C23—44no—
563C23—44no—
564C23C2×#12518yes—
565C23C23×D411yescharacteristic
566C23C23×Q811yescharacteristic
567C23C22×#4111yescharacteristic
568C23C22×#4111yescharacteristic
569C23C22×#4111yescharacteristic
570C23C2×H(5,𝔽2)11yescharacteristic
571C23C2×H(5,𝔽2)12yes—
572C23C2×2−1+411yescharacteristic
573C23#58512yes—
574C23#58512yes—
575C23#58512yes—
576D4—232no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—24no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—24no—
583C2×C4—24no—
584C2×C4—24no—
585C2×C4—24no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—22no—
591C2×C4—22no—
592C2×C4—22no—
593C2×C4—22no—
594C2×C4—24no—
595C2×C4—22no—
596C2×C4—22no—
597C2×C4—22no—
598C2×C4—22no—
599C2×C4—28no—
600C2×C4—24no—
601C2×C4—24no—
602C2×C4—42no—
603C2×C4—44no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—42no—
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—44no—
645C2×C4—42no—
646C2×C4—44no—
647C2×C4—42no—
648C2×C4—42no—
649C2×C4—44no—
650C2×C4—44no—
651C2×C4—42no—
652C2×C4—44no—
653C2×C4—42no—
654C2×C4—42no—
655C2×C4—44no—
656C2×C4—42no—
657C2×C4—42no—
658C2×C4—44no—
659C2×C4—42no—
660C2×C4—44no—
661C2×C4—44no—
662C2×C4—42no—
663C2×C4—42no—
664C2×C4—44no—
665C2×C4—44no—
666C2×C4—42no—
667C2×C4—44no—
668C2×C4—816no—
669C2×C4—816no—
670C2×C4—816no—
671C2×C4C2×C22⋊Q814yes—
672C2×C4C2×C22⋊Q814yes—
673C2×C4C2×C22⋊Q814yes—
674C2×C4C2×#12514yes—
675C2×C4#54414yes—
676C2×C4#54614yes—
677C2×C4#54814yes—
678C2×C4#55214yes—
679C2×C4#55314yes—
680C22—216no—
681C22—28no—
682C22—44no—
683C22—48no—
684C22—48no—
685C22—44no—
686C22—44no—
687C22—44no—
688C22—44no—
689C22—44no—
690C22—48no—
691C22C22×C22⋊Q811yescharacteristic
692C22C22×#12511yescharacteristic
693C22#344911yescharacteristic
694C22#344911yescharacteristic
695C22C2×#54411yescharacteristic
696C22C2×#54611yescharacteristic
697C22C2×#54811yescharacteristic
698C22C2×#55211yescharacteristic
699C22C2×#55311yescharacteristic
700C22#349412yes—
701C22#349811yescharacteristic
702C22#350011yescharacteristic
703C22#350112yes—
704C22#350412yes—
705C22#351212yes—
706C22#351312yes—
707C22#351412yes—
708C22#352112yes—
709C22#352112yes—
710C22#354012yes—
711C22#354212yes—
712C22#355412yes—
713C22#355712yes—
714C4—24no—
715C4—24no—
716C4—44no—
717C4—44no—
718C4—42no—
719C4—42no—
720C4—42no—
721C4—42no—
722C4—82no—
723C4—82no—
724C4—82no—
725C4—82no—
726C4—82no—
727C4—82no—
728C4—82no—
729C4—82no—
730C4—82no—
731C4#244218yes—
732C2—48no—
733C2C2×#244211yescharacteristic
734C2#3533311yescharacteristic
735C2#3552911yescharacteristic
736C2#3596112yes—
737C2#3809011yescharacteristic
738C2#3809411yescharacteristic
739C2#3916111yescharacteristic
740C2#3916311yescharacteristic
741C2#4249312yes—
742C2#4506612yes—
743C2#4520612yes—
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×#2571C2semidirect
3. #16304C2semidirect
4. #16057C2semidirect
5. #15462C2semidirect
6. #15239C2semidirect
7. #15239C2semidirect
8. #15234C2semidirect
9. #15151C2semidirect
10. #14767C2semidirect
11. #14348C2semidirect
16. #13909C2semidirect
18. #13848C2semidirect

Maximal subgroups

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

subgroupindex
C2×#25712
C2×#23492
#163042
#160572
#154622
#152392
#152342
#151512
#147672
#143482
#139792
#139772
#139412
#139092
#139062
#138482
#137852