Finite Group Databasebeta

← #4457641

Subgroups — #4457641

Order 512.

2258 conjugacy classes of subgroups.

The subgroup lattice has 744 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2629C211yesmaximal, characteristic
2C2×#2317C211yesmaximal, characteristic
3#16687C214yesmaximal, complemented
4#16523C214yesmaximal, complemented
5#16324C214yesmaximal
6#16322C214yesmaximal
7#15234C211yesmaximal, characteristic, complemented
8#15234C211yesmaximal, characteristic, complemented
9#15228C211yesmaximal, characteristic, complemented
10#15228C211yesmaximal, characteristic, complemented
11#15197C211yesmaximal, characteristic, complemented
12#15196C211yesmaximal, characteristic, complemented
13#15153C211yesmaximal, characteristic, complemented
14#15151C211yesmaximal, characteristic, complemented
15#13939C211yesmaximal, characteristic
16#13939C211yesmaximal, characteristic
17#13881C211yesmaximal, characteristic
18#13880C211yesmaximal, characteristic
19#13852C211yesmaximal, characteristic
20C22×#125C2211yescharacteristic
21C22×#117C2211yescharacteristic
22C22×#117C2211yescharacteristic
23C23×C4⋊C4C2211yescharacteristic
24#2629—28no—
25C2×#400C2214yes—
26C2×#398C2212yes—
27C2×#398C2212yes—
28C2×#398C2214yes—
29C2×#396C2212yes—
30C2×#396C2212yes—
31#2317—28no—
32C2×#389C2211yescharacteristic
33C2×#389C2211yescharacteristic
34C2×#389C2211yescharacteristic
35C2×#389C2211yescharacteristic
36C2×#389C2211yescharacteristic
37C2×#389C2211yescharacteristic
38C2×#385C2212yes—
39C2×#385C2212yes—
40C2×#385C2214yes—
41C2×#384C2212yes—
42C2×#384C2211yescharacteristic
43C2×#384C2211yescharacteristic
44C2×#384C2211yescharacteristic
45C2×#382C2212yes—
46C2×#382C2211yescharacteristic
47C2×#382C2211yescharacteristic
48C2×#380C2211yescharacteristic
49C2×#380C2211yescharacteristic
50C2×C4×C4⋊C4C2211yescharacteristic
51C2×C4×C4⋊C4C2211yescharacteristic
52C2×C4×C22⋊C4C2211yescharacteristic
53C2×C4×C22⋊C4C2211yescharacteristic
54#1445C2214yes—
55#1445C2214yes—
56#1444C2214yes—
57#1444C2214yes—
58#1444C2214yes—
59#1444C2214yes—
60#1442C2214yes—
61#1442C2214yes—
62#1442C2214yes—
63#1442C2214yes—
64#1441C2212yes—
65#1441C2212yes—
66#1441C2214yes—
67#1441C2214yes—
68#1441C2214yes—
69#1441C2214yes—
70#1441C2214yes—
71#1441C2214yes—
72#1440C2214yes—
73#1440C2214yes—
74#1439C2212yes—
75#1439C2212yes—
76C4⋊#95C2211yescharacteristic
77C4⋊#95C2211yescharacteristic
78C4⋊#95C2211yescharacteristic
79C4⋊#95C2211yescharacteristic
80C4⋊#95C2211yescharacteristic
81C4⋊#95C2211yescharacteristic
82C4⋊#95C2211yescharacteristic
83C4⋊#95C2211yescharacteristic
84C4⋊#95C2211yescharacteristic
85C4⋊#95C2211yescharacteristic
86C4⋊#95C2211yescharacteristic
87C4⋊#95C2211yescharacteristic
88#1434C2214yes—
89#1434C2214yes—
90C22⋊#95C2214yes—
91C4×#95C2211yescharacteristic
92C4×#95C2211yescharacteristic
93C24×C4C2311yescharacteristic
94C24×C4C2312yes—
95C2×#125—24no—
96C2×#125—42no—
97C2×#125—42no—
98C2×#117—24no—
99C2×#117—24no—
100C2×#117—42no—
101C2×#117—42no—
102C2×#117D418yes—
103C2×#117Q818yes—
104C22×C4⋊C4—24no—
105C22×C4⋊C4—42no—
106C22×C4⋊C4—42no—
107C22×C4⋊C4C2311yescharacteristic
108C22×C4⋊C4C2311yescharacteristic
109C22×C4⋊C4C2311yescharacteristic
110C22×C4⋊C4C2311yescharacteristic
111C22×C4⋊C4C2311yescharacteristic
112C22×C4⋊C4C2311yescharacteristic
113C22×C4⋊C4C2311yescharacteristic
114C22×C4⋊C4C2311yescharacteristic
115C22×C4⋊C4C2312yes—
116C22×C4⋊C4C2312yes—
117C22×C4⋊C4C2312yes—
118C22×C4⋊C4C2312yes—
119C22×C4⋊C4C2312yes—
120C22×C4⋊C4C2312yes—
121C22×C4⋊C4C2312yes—
122C22×C4⋊C4C2312yes—
123C22×C4⋊C4C2314yes—
124C22×C4⋊C4C2312yes—
125C22×C4⋊C4C2312yes—
126C22×C4⋊C4C2312yes—
127C22×C4⋊C4C2312yes—
128C22×C4⋊C4C2312yes—
129C22×C4⋊C4C2314yes—
130C22×C4⋊C4C2311yescharacteristic
131C22×C4⋊C4C2311yescharacteristic
132C22×C22⋊C4C2312yes—
133C22×C22⋊C4C2312yes—
134C22×C22⋊C4C2312yes—
135C22×C22⋊C4C2312yes—
136C22×C22⋊C4C2312yes—
137C22×C22⋊C4C2311yescharacteristic
138C22×C22⋊C4C2311yescharacteristic
139C22×C42C2311yescharacteristic
140C22×C42C2311yescharacteristic
141C22×C42C2311yescharacteristic
142C22×C42C2311yescharacteristic
143C22×C42C2311yescharacteristic
144C22×C42C2311yescharacteristic
145C22×C42C2312yes—
146#400—48no—
147#398—28no—
148#398—28no—
149#398—48no—
150#396—28no—
151#396—28no—
152#389—24no—
153#389—24no—
154#389—24no—
155#389—24no—
156#389—24no—
157#389—24no—
158#385—28no—
159#385—28no—
160#385—48no—
161#384—28no—
162#384—24no—
163#384—42no—
164#384—42no—
165#382—28no—
166#382—42no—
167#382—42no—
168#380—24no—
169#380—24no—
170C4×C4⋊C4—42no—
171C4×C4⋊C4—42no—
172C4×C22⋊C4—24no—
173C4×C22⋊C4—24no—
174C2×#95C2312yes—
175C2×#95C2312yes—
176C2×#95C2312yes—
177C2×#95C2312yes—
178C2×#95C2314yes—
179C2×#95C2312yes—
180C2×#95C2314yes—
181C2×#95C2312yes—
182C2×#95C2312yes—
183C2×#95C2312yes—
184C2×#95C2314yes—
185C2×#95C2312yes—
186C2×#95C2312yes—
187C2×#95C2312yes—
188C2×#95C2312yes—
189C2×#95C2314yes—
190C2×#95C2314yes—
191C2×#95C2314yes—
192C2×#95C2314yes—
193C2×#95C2312yes—
194C2×#95C2312yes—
195C2×#95C2314yes—
196C2×#95C2314yes—
197C2×#95C2314yes—
198C2×#95C2314yes—
199C2×#95C2314yes—
200C2×#95C2314yes—
201C2×#95C2314yes—
202C2×#95C2312yes—
203C2×#95C2312yes—
204C25C2411yescharacteristic
205C23×C4—22no—
206C23×C4—22no—
207C23×C4—22no—
208C23×C4—22no—
209C23×C4—24no—
210C23×C4—42no—
211C23×C4—42no—
212C23×C4—44no—
213C23×C4—44no—
214C23×C4C2×D414yes—
215C23×C4C2×Q814yes—
216C23×C4#4114yes—
217C23×C4C2412yes—
218C23×C4C2412yes—
219C23×C4C2412yes—
220C23×C4C2412yes—
221C23×C4C2412yes—
222C23×C4C2412yes—
223C23×C4C2412yes—
224C23×C4C2412yes—
225C23×C4C2411yescharacteristic
226C23×C4C2411yescharacteristic
227C23×C4C2411yescharacteristic
228C23×C4C2411yescharacteristic
229C23×C4C2412yes—
230C23×C4C2412yes—
231C23×C4C2412yes—
232C23×C4C2412yes—
233C23×C4C2411yescharacteristic
234C23×C4C2411yescharacteristic
235#125—416no—
236#117—232no—
237#117—416no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—22no—
257C2×C4⋊C4—22no—
258C2×C4⋊C4—22no—
259C2×C4⋊C4—24no—
260C2×C4⋊C4—24no—
261C2×C4⋊C4—22no—
262C2×C4⋊C4—22no—
263C2×C4⋊C4—22no—
264C2×C4⋊C4—22no—
265C2×C4⋊C4—24no—
266C2×C4⋊C4—24no—
267C2×C4⋊C4—24no—
268C2×C4⋊C4—24no—
269C2×C4⋊C4—24no—
270C2×C4⋊C4—24no—
271C2×C4⋊C4—24no—
272C2×C4⋊C4—24no—
273C2×C4⋊C4—28no—
274C2×C4⋊C4—42no—
275C2×C4⋊C4—42no—
276C2×C4⋊C4—42no—
277C2×C4⋊C4—42no—
278C2×C4⋊C4—41no—
279C2×C4⋊C4—41no—
280C2×C4⋊C4—41no—
281C2×C4⋊C4—41no—
282C2×C4⋊C4—42no—
283C2×C4⋊C4—42no—
284C2×C4⋊C4—42no—
285C2×C4⋊C4—42no—
286C2×C4⋊C4—42no—
287C2×C4⋊C4—42no—
288C2×C4⋊C4—416no—
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×D414yes—
303C2×C4⋊C4C2×D414yes—
304C2×C4⋊C4C2×Q814yes—
305C2×C4⋊C4C2×Q814yes—
306C2×C22⋊C4—24no—
307C2×C22⋊C4—24no—
308C2×C22⋊C4—24no—
309C2×C22⋊C4—24no—
310C2×C22⋊C4—24no—
311C2×C22⋊C4—24no—
312C2×C22⋊C4—24no—
313C2×C22⋊C4—24no—
314C2×C22⋊C4—42no—
315C2×C22⋊C4—42no—
316C2×C22⋊C4—41no—
317C2×C22⋊C4—41no—
318C2×C22⋊C4—41no—
319C2×C22⋊C4—41no—
320C2×C22⋊C4—44no—
321C2×C22⋊C4—44no—
322C2×C22⋊C4C2×D414yes—
323C2×C22⋊C4C2×D414yes—
324C2×C22⋊C4C2×Q818yes—
325C2×C42—24no—
326C2×C42—24no—
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—22no—
336C2×C42—22no—
337C2×C42—22no—
338C2×C42—22no—
339C2×C42—22no—
340C2×C42—22no—
341C2×C42—22no—
342C2×C42—22no—
343C2×C42C2×D414yes—
344C2×C42C2×D414yes—
345C2×C42C2×Q818yes—
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—24no—
356#95—28no—
357#95—28no—
358#95—24no—
359#95—24no—
360#95—28no—
361#95—44no—
362#95—44no—
363#95—42no—
364#95—44no—
365#95—42no—
366#95—44no—
367#95—42no—
368#95—42no—
369#95—44no—
370#95—44no—
371#95—44no—
372#95—44no—
373#95—44no—
374#95—44no—
375#95—44no—
376C24—22no—
377C24—21no—
378C24—21no—
379C24—22no—
380C24—21no—
381C24—21no—
382C24—22no—
383C24—22no—
384C24C22×D412yes—
385C24C22×Q812yes—
386C24C2×#4112yes—
387C24C2511yescharacteristic
388C22×C4—21no—
389C22×C4—21no—
390C22×C4—21no—
391C22×C4—21no—
392C22×C4—28no—
393C22×C4—28no—
394C22×C4—28no—
395C22×C4—28no—
396C22×C4—28no—
397C22×C4—28no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—24no—
405C22×C4—22no—
406C22×C4—21no—
407C22×C4—21no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—21no—
412C22×C4—21no—
413C22×C4—21no—
414C22×C4—21no—
415C22×C4—21no—
416C22×C4—21no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—24no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—24no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—24no—
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—24no—
445C22×C4—24no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—24no—
454C22×C4—24no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—24no—
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—22no—
471C22×C4—22no—
472C22×C4—24no—
473C22×C4—24no—
474C22×C4—24no—
475C22×C4—24no—
476C22×C4—24no—
477C22×C4—24no—
478C22×C4—48no—
479C22×C4—84no—
480C22×C4—84no—
481C22×C4—88no—
482C22×C4—88no—
483C22×C4C22⋊Q8116yes—
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×Q812yes—
493C22×C4C22×Q812yes—
494C22×C4C2×#4112yes—
495C22×C4C2×#4112yes—
496C22×C4H(5,𝔽2)14yes—
497C22×C4H(5,𝔽2)12yes—
498C22×C4H(5,𝔽2)12yes—
499C22×C42−1+414yes—
500C22×C42−1+412yes—
501C22×C42−1+412yes—
502C4⋊C4—28no—
503C4⋊C4—28no—
504C4⋊C4—44no—
505C4⋊C4—44no—
506C4⋊C4—48no—
507C4⋊C4—48no—
508C4⋊C4—48no—
509C4⋊C4—48no—
510C4⋊C4—48no—
511C4⋊C4—44no—
512C4⋊C4—44no—
513C4⋊C4—44no—
514C4⋊C4—44no—
515C4⋊C4—44no—
516C4⋊C4—44no—
517C4⋊C4—84no—
518C4⋊C4—84no—
519C4⋊C4—84no—
520C4⋊C4—84no—
521C4⋊C4—84no—
522C4⋊C4—84no—
523C4⋊C4—84no—
524C4⋊C4—84no—
525C4⋊C4—88no—
526C4⋊C4—88no—
527C22⋊C4—216no—
528C22⋊C4—216no—
529C22⋊C4—44no—
530C22⋊C4—44no—
531C22⋊C4—48no—
532C22⋊C4—84no—
533C22⋊C4—84no—
534C42—216no—
535C42—28no—
536C42—28no—
537C42—44no—
538C42—44no—
539C42—44no—
540C42—44no—
541C23—22no—
542C23—22no—
543C23—22no—
544C23—22no—
545C23—22no—
546C23—22no—
547C23—24no—
548C23—24no—
549C23—22no—
550C23—22no—
551C23—22no—
552C23—22no—
553C23—24no—
554C23—24no—
555C23—42no—
556C23—42no—
557C23—42no—
558C23—42no—
559C23—42no—
560C23—42no—
561C23—42no—
562C23—42no—
563C23C2×C22⋊Q814yes—
564C23C23×D411yescharacteristic
565C23C23×Q811yescharacteristic
566C23C22×#4111yescharacteristic
567C23C2×H(5,𝔽2)11yescharacteristic
568C23C2×H(5,𝔽2)11yescharacteristic
569C23C2×H(5,𝔽2)12yes—
570C23C2×2−1+411yescharacteristic
571C23C2×2−1+411yescharacteristic
572C23C2×2−1+412yes—
573C23#58512yes—
574C23#58512yes—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—232no—
582C2×C4—24no—
583C2×C4—24no—
584C2×C4—22no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—24no—
589C2×C4—24no—
590C2×C4—22no—
591C2×C4—22no—
592C2×C4—22no—
593C2×C4—22no—
594C2×C4—22no—
595C2×C4—22no—
596C2×C4—22no—
597C2×C4—22no—
598C2×C4—28no—
599C2×C4—28no—
600C2×C4—24no—
601C2×C4—24no—
602C2×C4—24no—
603C2×C4—24no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—44no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—44no—
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—42no—
645C2×C4—42no—
646C2×C4—42no—
647C2×C4—42no—
648C2×C4—42no—
649C2×C4—42no—
650C2×C4—42no—
651C2×C4—44no—
652C2×C4—42no—
653C2×C4—44no—
654C2×C4—44no—
655C2×C4—44no—
656C2×C4—44no—
657C2×C4—44no—
658C2×C4—42no—
659C2×C4—44no—
660C2×C4—42no—
661C2×C4—42no—
662C2×C4—44no—
663C2×C4—44no—
664C2×C4—44no—
665C2×C4—42no—
666C2×C4—42no—
667C2×C4—44no—
668C2×C4—44no—
669C2×C4—42no—
670C2×C4—816no—
671C2×C4C2×C22⋊Q814yes—
672C2×C4C2×C22⋊Q814yes—
673C2×C4#53414yes—
674C2×C4#53614yes—
675C2×C4#53914yes—
676C2×C4#53914yes—
677C2×C4#54414yes—
678C2×C4#54414yes—
679C22—24no—
680C22—24no—
681C22—24no—
682C22—44no—
683C22—44no—
684C22—42no—
685C22—42no—
686C22—42no—
687C22—44no—
688C22—42no—
689C22—44no—
690C22C22×C22⋊Q811yescharacteristic
691C22C2×#53411yescharacteristic
692C22C2×#53611yescharacteristic
693C22C2×#53911yescharacteristic
694C22C2×#53911yescharacteristic
695C22C2×#54411yescharacteristic
696C22C2×#54411yescharacteristic
697C22#349212yes—
698C22#349312yes—
699C22#349811yescharacteristic
700C22#349911yescharacteristic
701C22#350011yescharacteristic
702C22#350011yescharacteristic
703C22#350312yes—
704C22#350412yes—
705C22#354112yes—
706C22#354212yes—
707C22#354612yes—
708C22#354712yes—
709C22#354712yes—
710C22#355512yes—
711C22#355912yes—
712C22#356312yes—
713C4—24no—
714C4—44no—
715C4—44no—
716C4—42no—
717C4—42no—
718C4—44no—
719C4—42no—
720C4—42no—
721C4—44no—
722C4—82no—
723C4—82no—
724C4—82no—
725C4—82no—
726C4—82no—
727C4—82no—
728C4—82no—
729C4—82no—
730C4#262718yes—
731C2—44no—
732C2C2×#262711yescharacteristic
733C2#3553211yescharacteristic
734C2#3562011yescharacteristic
735C2#3730011yescharacteristic
736C2#3730411yescharacteristic
737C2#3738811yescharacteristic
738C2#3739211yescharacteristic
739C2#4250112yes—
740C2#4250712yes—
741C2#4293212yes—
742C2#4297212yes—
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. #16687C2semidirect
4. #16523C2semidirect
7. #15234C2semidirect
8. #15234C2semidirect
9. #15228C2semidirect
10. #15228C2semidirect
11. #15197C2semidirect
12. #15196C2semidirect
13. #15153C2semidirect
14. #15151C2semidirect

Maximal subgroups

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

subgroupindex
C2×#26292
C2×#23172
#166872
#165232
#163242
#163222
#152342
#152282
#151972
#151962
#151532
#151512
#139392
#138812
#138802
#138522