Finite Group Databasebeta

← #874314

Subgroups — #874314

Order 512.

2446 conjugacy classes of subgroups.

The subgroup lattice has 749 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2497C211yesmaximal, characteristic
2C2×#2440C211yesmaximal, characteristic
3#16524C214yesmaximal, complemented
4#15614C214yesmaximal
5#15517C214yesmaximal, complemented
6#15197C211yesmaximal, characteristic, complemented
7#15196C211yesmaximal, characteristic, complemented
8#15195C211yesmaximal, characteristic
9#15195C211yesmaximal, characteristic
10#15145C211yesmaximal, characteristic, complemented
11#15143C211yesmaximal, characteristic, complemented
12#14712C211yesmaximal, characteristic, complemented
13#14687C211yesmaximal, characteristic, complemented
14#14403C214yesmaximal
15#14161C211yesmaximal, characteristic, complemented
16#14048C211yesmaximal, characteristic, complemented
17#13924C211yesmaximal, characteristic
18#13880C211yesmaximal, characteristic
19C4×#388C211yesmaximal, characteristic
20C22×C4⋊Q8C2211yescharacteristic
21C22×C22⋊Q8C2212yes—
22C23×C4⋊C4C2211yescharacteristic
23#2497—28no—
24#2440—28no—
25C2×#400C2212yes—
26C2×#398C2212yes—
27C2×#398C2214yes—
28C2×#396C2212yes—
29C2×#395C2212yes—
30C2×#395C2214yes—
31C2×#389C2211yescharacteristic
32C2×#389C2211yescharacteristic
33C2×#389C2211yescharacteristic
34C2×#389C2211yescharacteristic
35C2×#389C2214yes—
36C2×#388C2212yes—
37C2×#388C2212yes—
38C2×#387C2212yes—
39C2×#385C2212yes—
40C2×#385C2214yes—
41C2×#384C2211yescharacteristic
42C2×#384C2211yescharacteristic
43C2×#384C2211yescharacteristic
44C2×#384C2211yescharacteristic
45C2×#384C2211yescharacteristic
46C2×#384C2211yescharacteristic
47C2×#382C2214yes—
48C2×#380C2211yescharacteristic
49C2×#380C2211yescharacteristic
50C2×#380C2211yescharacteristic
51C2×C4×C4⋊C4C2211yescharacteristic
52C2×C4×C4⋊C4C2211yescharacteristic
53C2×C4×C22⋊C4C2211yescharacteristic
54C2×C4×C22⋊C4C2211yescharacteristic
55C2×C4×C22⋊C4C2211yescharacteristic
56C2×C43C2211yescharacteristic
57#1446C2214yes—
58#1445C2214yes—
59#1444C2214yes—
60#1444C2214yes—
61#1444C2214yes—
62#1444C2214yes—
63#1442C2214yes—
64#1442C2214yes—
65#1442C2214yes—
66#1441C2214yes—
67#1441C2214yes—
68#1441C2214yes—
69#1439C2214yes—
70#1439C2214yes—
71#1437C2212yes—
72#1436C2214yes—
73#1436C2212yes—
74C4⋊#95C2211yescharacteristic
75C4⋊#95C2211yescharacteristic
76C4⋊#95C2211yescharacteristic
77C4⋊#95C2211yescharacteristic
78C4⋊#95C2211yescharacteristic
79C4⋊#95C2211yescharacteristic
80C4⋊#95C2214yes—
81C4⋊#95C2214yes—
82C4⋊#95C2211yescharacteristic
83C4⋊#95C2211yescharacteristic
84C4⋊#95C2211yescharacteristic
85C4⋊#95C2211yescharacteristic
86C4⋊#95C2211yescharacteristic
87C4⋊#95C2211yescharacteristic
88C4⋊#95C2214yes—
89#1434C2212yes—
90C22⋊#95C2214yes—
91C4×#95C2211yescharacteristic
92C4×#95C2211yescharacteristic
93C23×Q8C2312yes—
94C24×C4C2311yescharacteristic
95C24×C4C2312yes—
96C2×C4⋊Q8—24no—
97C2×C4⋊Q8—24no—
98C2×C4⋊Q8—42no—
99C2×C22⋊Q8—28no—
100C2×C22⋊Q8—48no—
101C22×C4⋊C4—24no—
102C22×C4⋊C4—24no—
103C22×C4⋊C4—42no—
104C22×C4⋊C4C2312yes—
105C22×C4⋊C4C2312yes—
106C22×C4⋊C4C2312yes—
107C22×C4⋊C4C2311yescharacteristic
108C22×C4⋊C4C2311yescharacteristic
109C22×C4⋊C4C2311yescharacteristic
110C22×C4⋊C4C2311yescharacteristic
111C22×C4⋊C4C2311yescharacteristic
112C22×C4⋊C4C2311yescharacteristic
113C22×C4⋊C4C2311yescharacteristic
114C22×C4⋊C4C2312yes—
115C22×C4⋊C4C2314yes—
116C22×C4⋊C4C2312yes—
117C22×C4⋊C4C2311yescharacteristic
118C22×C4⋊C4C2312yes—
119C22×C4⋊C4C2312yes—
120C22×C4⋊C4C2312yes—
121C22×C4⋊C4C2312yes—
122C22×C4⋊C4C2312yes—
123C22×C4⋊C4C2314yes—
124C22×C4⋊C4C2314yes—
125C22×C4⋊C4C2314yes—
126C22×C4⋊C4C2312yes—
127C22×C22⋊C4C2311yescharacteristic
128C22×C22⋊C4C2312yes—
129C22×C22⋊C4C2312yes—
130C22×C22⋊C4C2312yes—
131C22×C22⋊C4C2312yes—
132C22×C22⋊C4C2312yes—
133C22×C22⋊C4C2311yescharacteristic
134C22×C42C2311yescharacteristic
135C22×C42C2311yescharacteristic
136C22×C42C2311yescharacteristic
137C22×C42C2312yes—
138C22×C42C2311yescharacteristic
139C22×C42C2311yescharacteristic
140C22×C42C2312yes—
141C22×C42C2311yescharacteristic
142C22×C42C2312yes—
143#400—28no—
144#398—28no—
145#398—48no—
146#396—28no—
147#395—28no—
148#395—48no—
149#389—24no—
150#389—24no—
151#389—24no—
152#389—24no—
153#389—48no—
154#388—28no—
155#388Q8116yescomplemented
156#387—28no—
157#385—28no—
158#385—48no—
159#384—24no—
160#384—24no—
161#384—24no—
162#384—24no—
163#384—42no—
164#384—42no—
165#382—48no—
166#380—24no—
167#380—24no—
168#380—24no—
169C4×C4⋊C4—24no—
170C4×C4⋊C4—42no—
171C4×C22⋊C4—24no—
172C4×C22⋊C4—42no—
173C4×C22⋊C4D418yes—
174C2×#95C2312yes—
175C2×#95C2312yes—
176C2×#95C2314yes—
177C2×#95C2312yes—
178C2×#95C2312yes—
179C2×#95C2312yes—
180C2×#95C2312yes—
181C2×#95C2312yes—
182C2×#95C2312yes—
183C2×#95C2312yes—
184C2×#95C2312yes—
185C2×#95C2314yes—
186C2×#95C2312yes—
187C2×#95C2314yes—
188C2×#95C2314yes—
189C2×#95C2314yes—
190C2×#95C2314yes—
191C2×#95C2314yes—
192C2×#95C2312yes—
193C2×#95C2312yes—
194C2×#95C2314yes—
195C2×#95C2314yes—
196C2×#95C2314yes—
197C2×#95C2314yes—
198C2×#95C2314yes—
199C2×#95C2311yescharacteristic
200C2×#95C2311yescharacteristic
201C2×#95C2312yes—
202C2×#95C2312yes—
203C43—24no—
204C25C2411yescharacteristic
205C22×Q8—24no—
206C22×Q8—24no—
207C22×Q8—44no—
208C22×Q8—42no—
209C22×Q8—44no—
210C23×C4—22no—
211C23×C4—22no—
212C23×C4—22no—
213C23×C4—22no—
214C23×C4—22no—
215C23×C4—22no—
216C23×C4—24no—
217C23×C4—24no—
218C23×C4—42no—
219C23×C4—44no—
220C23×C4—44no—
221C23×C4C2×D414yes—
222C23×C4C2412yes—
223C23×C4C2412yes—
224C23×C4C2412yes—
225C23×C4C2412yes—
226C23×C4C2412yes—
227C23×C4C2412yes—
228C23×C4C2412yes—
229C23×C4C2411yescharacteristic
230C23×C4C2411yescharacteristic
231C23×C4C2412yes—
232C23×C4C2412yes—
233C23×C4C2412yes—
234C23×C4C2412yes—
235C23×C4C2411yescharacteristic
236C23×C4C2411yescharacteristic
237C23×C4C2411yescharacteristic
238C23×C4C2412yes—
239C23×C4C2411yescharacteristic
240C4⋊Q8—416no—
241C22⋊Q8—432no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—24no—
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—24no—
258C2×C4⋊C4—22no—
259C2×C4⋊C4—24no—
260C2×C4⋊C4—22no—
261C2×C4⋊C4—28no—
262C2×C4⋊C4—22no—
263C2×C4⋊C4—22no—
264C2×C4⋊C4—22no—
265C2×C4⋊C4—22no—
266C2×C4⋊C4—28no—
267C2×C4⋊C4—28no—
268C2×C4⋊C4—22no—
269C2×C4⋊C4—22no—
270C2×C4⋊C4—28no—
271C2×C4⋊C4—28no—
272C2×C4⋊C4—22no—
273C2×C4⋊C4—22no—
274C2×C4⋊C4—22no—
275C2×C4⋊C4—22no—
276C2×C4⋊C4—22no—
277C2×C4⋊C4—22no—
278C2×C4⋊C4—24no—
279C2×C4⋊C4—24no—
280C2×C4⋊C4—24no—
281C2×C4⋊C4—24no—
282C2×C4⋊C4—28no—
283C2×C4⋊C4—416no—
284C2×C4⋊C4—42no—
285C2×C4⋊C4—42no—
286C2×C4⋊C4—44no—
287C2×C4⋊C4—42no—
288C2×C4⋊C4—42no—
289C2×C4⋊C4—44no—
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⋊C4C2×Q818yes—
303C2×C4⋊C4#4114yes—
304C2×C22⋊C4—24no—
305C2×C22⋊C4—24no—
306C2×C22⋊C4—24no—
307C2×C22⋊C4—24no—
308C2×C22⋊C4—24no—
309C2×C22⋊C4—24no—
310C2×C22⋊C4—42no—
311C2×C22⋊C4—44no—
312C2×C22⋊C4—42no—
313C2×C22⋊C4—44no—
314C2×C22⋊C4—44no—
315C2×C22⋊C4C2×D414yes—
316C2×C22⋊C4C2×D414yes—
317C2×C22⋊C4C2×Q818yes—
318C2×C22⋊C4C2×Q818yes—
319C2×C22⋊C4C2×Q818yes—
320C2×C42—22no—
321C2×C42—22no—
322C2×C42—24no—
323C2×C42—22no—
324C2×C42—22no—
325C2×C42—24no—
326C2×C42—22no—
327C2×C42—22no—
328C2×C42—24no—
329C2×C42—22no—
330C2×C42—22no—
331C2×C42—22no—
332C2×C42—22no—
333C2×C42—22no—
334C2×C42—24no—
335C2×C42—22no—
336C2×C42—22no—
337C2×C42—22no—
338C2×C42—22no—
339C2×C42—24no—
340C2×C42—42no—
341C2×C42—44no—
342C2×C42C2×D414yes—
343C2×C42C2×D414yes—
344C2×C42C2×Q818yes—
345C2×C42#4114yes—
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—28no—
357#95—24no—
358#95—24no—
359#95—44no—
360#95—44no—
361#95—42no—
362#95—42no—
363#95—44no—
364#95—44no—
365#95—44no—
366#95—44no—
367#95—44no—
368#95—44no—
369#95—44no—
370#95—44no—
371#95C2×D414yes—
372#95C2×D414yes—
373#95C2×Q818yes—
374#95C2×Q818yes—
375C24—21no—
376C24—21no—
377C24—22no—
378C24—22no—
379C24—21no—
380C24—21no—
381C24—21no—
382C24—21no—
383C24—22no—
384C24C22×D412yes—
385C24C22×Q814yes—
386C24C2511yescharacteristic
387C2×Q8—48no—
388C2×Q8—48no—
389C2×Q8—84no—
390C2×Q8—88no—
391C2×Q8—88no—
392C22×C4—21no—
393C22×C4—21no—
394C22×C4—21no—
395C22×C4—22no—
396C22×C4—21no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—21no—
400C22×C4—21no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—21no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—21no—
410C22×C4—21no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—21no—
419C22×C4—22no—
420C22×C4—24no—
421C22×C4—21no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—24no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—21no—
432C22×C4—21no—
433C22×C4—21no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—28no—
438C22×C4—28no—
439C22×C4—28no—
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—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—24no—
453C22×C4—24no—
454C22×C4—24no—
455C22×C4—24no—
456C22×C4—24no—
457C22×C4—24no—
458C22×C4—24no—
459C22×C4—24no—
460C22×C4—24no—
461C22×C4—24no—
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—22no—
471C22×C4—24no—
472C22×C4—24no—
473C22×C4—24no—
474C22×C4—24no—
475C22×C4—48no—
476C22×C4—44no—
477C22×C4—44no—
478C22×C4—44no—
479C22×C4—44no—
480C22×C4—48no—
481C22×C4—88no—
482C22×C4—84no—
483C22×C4—88no—
484C22×C4C22×D412yes—
485C22×C4C22×D412yes—
486C22×C4C22×D412yes—
487C22×C4C22×D412yes—
488C22×C4C22×D412yes—
489C22×C4C22×D412yes—
490C22×C4C22×Q814yes—
491C22×C4C22×Q814yes—
492C22×C4C22×Q814yes—
493C22×C4C22×Q814yes—
494C22×C4C22×Q814yes—
495C22×C4C22×Q814yes—
496C22×C4C2×#4114yes—
497C22×C4C2×#4114yes—
498C22×C4C2×#4112yes—
499C22×C4C2×#4112yes—
500C22×C4H(5,𝔽2)14yes—
501C22×C4H(5,𝔽2)14yes—
502C22×C4H(5,𝔽2)12yes—
503C22×C42−1+412yes—
504C22×C42−1+412yes—
505C22×C42−1+412yes—
506C4⋊C4—28no—
507C4⋊C4—48no—
508C4⋊C4—416no—
509C4⋊C4—48no—
510C4⋊C4—48no—
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—44no—
522C4⋊C4—88no—
523C4⋊C4—84no—
524C4⋊C4—84no—
525C4⋊C4—84no—
526C4⋊C4—84no—
527C4⋊C4—88no—
528C4⋊C4—88no—
529C22⋊C4—28no—
530C22⋊C4—216no—
531C22⋊C4—48no—
532C22⋊C4—48no—
533C22⋊C4—84no—
534C22⋊C4—84no—
535C22⋊C4C4⋊Q8116yes—
536C42—28no—
537C42—28no—
538C42—28no—
539C42—216no—
540C42—48no—
541C42—44no—
542C42—44no—
543C42—44no—
544C42—84no—
545C23—22no—
546C23—22no—
547C23—22no—
548C23—22no—
549C23—24no—
550C23—24no—
551C23—22no—
552C23—22no—
553C23—24no—
554C23—24no—
555C23—24no—
556C23—24no—
557C23—41no—
558C23—41no—
559C23—41no—
560C23—42no—
561C23—42no—
562C23—41no—
563C23—42no—
564C23—42no—
565C23—42no—
566C23—42no—
567C23C2×C4⋊Q814yes—
568C23C23×D411yescharacteristic
569C23C23×Q812yes—
570C23C22×#4111yescharacteristic
571C23C22×#4111yescharacteristic
572C23C2×H(5,𝔽2)12yes—
573C23C2×H(5,𝔽2)11yescharacteristic
574C23C2×H(5,𝔽2)12yes—
575C23C2×2−1+411yescharacteristic
576C23C2×2−1+411yescharacteristic
577C23C2×2−1+411yescharacteristic
578C23#58512yes—
579Q8—816no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—24no—
584C2×C4—24no—
585C2×C4—24no—
586C2×C4—24no—
587C2×C4—24no—
588C2×C4—24no—
589C2×C4—22no—
590C2×C4—22no—
591C2×C4—22no—
592C2×C4—24no—
593C2×C4—24no—
594C2×C4—24no—
595C2×C4—24no—
596C2×C4—24no—
597C2×C4—24no—
598C2×C4—22no—
599C2×C4—22no—
600C2×C4—22no—
601C2×C4—24no—
602C2×C4—22no—
603C2×C4—22no—
604C2×C4—24no—
605C2×C4—24no—
606C2×C4—22no—
607C2×C4—24no—
608C2×C4—24no—
609C2×C4—24no—
610C2×C4—24no—
611C2×C4—24no—
612C2×C4—24no—
613C2×C4—24no—
614C2×C4—28no—
615C2×C4—42no—
616C2×C4—41no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—41no—
620C2×C4—41no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—42no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—41no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—42no—
630C2×C4—42no—
631C2×C4—42no—
632C2×C4—44no—
633C2×C4—42no—
634C2×C4—42no—
635C2×C4—42no—
636C2×C4—416no—
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—44no—
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—44no—
661C2×C4—44no—
662C2×C4—44no—
663C2×C4—44no—
664C2×C4—44no—
665C2×C4—44no—
666C2×C4—44no—
667C2×C4—44no—
668C2×C4—44no—
669C2×C4—44no—
670C2×C4—44no—
671C2×C4—44no—
672C2×C4—816no—
673C2×C4C2×C4⋊Q814yes—
674C2×C4C2×C4⋊Q814yes—
675C2×C4#53514yes—
676C2×C4#53614yes—
677C2×C4#54114yes—
678C2×C4#54214yes—
679C2×C4#54614yes—
680C2×C4#54714yes—
681C2×C4#55214yes—
682C2×C4#55418yes—
683C2×C4#56314yes—
684C22—24no—
685C22—24no—
686C22—24no—
687C22—42no—
688C22—44no—
689C22—42no—
690C22—44no—
691C22—42no—
692C22—44no—
693C22—42no—
694C22—44no—
695C22C22×C4⋊Q811yescharacteristic
696C22C2×#53511yescharacteristic
697C22C2×#53611yescharacteristic
698C22C2×#54111yescharacteristic
699C22C2×#54211yescharacteristic
700C22C2×#54611yescharacteristic
701C22C2×#54711yescharacteristic
702C22C2×#55211yescharacteristic
703C22C2×#55412yes—
704C22Q8×#4112yes—
705C22#349412yes—
706C22#349811yescharacteristic
707C22#350011yescharacteristic
708C22#350112yes—
709C22#350312yes—
710C22#350312yes—
711C22#350312yes—
712C22#351912yes—
713C22C2×#56311yescharacteristic
714C22#354712yes—
715C22#354912yes—
716C22#355912yes—
717C22#355912yes—
718C4—24no—
719C4—28no—
720C4—24no—
721C4—24no—
722C4—44no—
723C4—42no—
724C4—44no—
725C4—42no—
726C4—44no—
727C4—44no—
728C4—82no—
729C4—82no—
730C4—82no—
731C4—82no—
732C4—82no—
733C4—82no—
734C4—82no—
735C4#269818yes—
736C2—44no—
737C2#3528111yescharacteristic
738C2#3618011yescharacteristic
739C2#3627611yescharacteristic
740C2#3730411yescharacteristic
741C2#3786411yescharacteristic
742C2#3895311yescharacteristic
743C2#3960812yes—
744C2#3966112yes—
745C2#4615212yes—
746C2C2×#269811yescharacteristic
747C2#5929112yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC25C24
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #16524C2semidirect
5. #15517C2semidirect
6. #15197C2semidirect
7. #15196C2semidirect
10. #15145C2semidirect
11. #15143C2semidirect
12. #14712C2semidirect
13. #14687C2semidirect
15. #14161C2semidirect
16. #14048C2semidirect
155. #388Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#24972
C2×#24402
#165242
#156142
#155172
#151972
#151962
#151952
#151452
#151432
#147122
#146872
#144032
#141612
#140482
#139242
#138802
C4×#3882