Finite Group Databasebeta

← #657636

Subgroups — #657636

Order 512.

2426 conjugacy classes of subgroups.

The subgroup lattice has 742 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2439C211yesmaximal, characteristic
2C2×#2346C211yesmaximal, characteristic
3#15827C214yesmaximal
4#15824C214yesmaximal
5#15239C211yesmaximal, characteristic, complemented
6#15234C211yesmaximal, characteristic, complemented
7#15230C211yesmaximal, characteristic, complemented
8#15196C211yesmaximal, characteristic, complemented
9#14941C214yesmaximal, complemented
10#14682C211yesmaximal, characteristic
11#14155C211yesmaximal, characteristic, complemented
12#14090C214yesmaximal, complemented
13#14051C211yesmaximal, characteristic, complemented
14#14050C211yesmaximal, characteristic, complemented
15#13939C211yesmaximal, characteristic
16#13909C211yesmaximal, characteristic, complemented
17#13879C211yesmaximal, characteristic
18C4×#385C211yesmaximal, characteristic
19#13756C211yesmaximal, characteristic
20C22×C4⋊Q8C2211yescharacteristic
21C22×#117C2211yescharacteristic
22C22×#117C2211yescharacteristic
23C23×C42C2211yescharacteristic
24#2439—28no—
25C2×#396C2212yes—
26C2×#393C2212yes—
27#2346—28no—
28C2×#391C2212yes—
29C2×#391C2214yes—
30C2×#389C2211yescharacteristic
31C2×#389C2211yescharacteristic
32C2×#389C2211yescharacteristic
33C2×#389C2211yescharacteristic
34C2×#388C2212yes—
35C2×#388C2212yes—
36C2×#387C2212yes—
37C2×#385C2212yes—
38C2×#384C2211yescharacteristic
39C2×#382C2211yescharacteristic
40C2×#382C2212yes—
41C2×#382C2211yescharacteristic
42C2×#382C2211yescharacteristic
43C2×#382C2211yescharacteristic
44C2×#380C2211yescharacteristic
45C2×#380C2211yescharacteristic
46C2×C4×C4⋊C4C2211yescharacteristic
47C2×C4×C4⋊C4C2211yescharacteristic
48C2×C4×C4⋊C4C2211yescharacteristic
49C2×C4×C4⋊C4C2211yescharacteristic
50C2×C4×C22⋊C4C2211yescharacteristic
51C2×C4×C22⋊C4C2211yescharacteristic
52C2×#376C2212yes—
53#1446C2214yes—
54#1445C2214yes—
55#1444C2214yes—
56#1443C2214yes—
57#1443C2214yes—
58#1442C2214yes—
59#1442C2214yes—
60#1442C2214yes—
61#1441C2214yes—
62#1441C2214yes—
63#1441C2214yes—
64#1441C2214yes—
65#1440C2214yes—
66#1439C2214yes—
67#1436C2212yes—
68#1436C2214yes—
69#1436C2214yes—
70#1436C2214yes—
71#1436C2212yes—
72C4⋊#95C2211yescharacteristic
73C4⋊#95C2211yescharacteristic
74C4⋊#95C2211yescharacteristic
75C4⋊#95C2211yescharacteristic
76C4⋊#95C2211yescharacteristic
77C4⋊#95C2211yescharacteristic
78C4⋊#95C2211yescharacteristic
79C4⋊#95C2211yescharacteristic
80C4⋊#95C2211yescharacteristic
81C4⋊#95C2211yescharacteristic
82C4⋊#95C2211yescharacteristic
83C4⋊#95C2211yescharacteristic
84#1434C2214yes—
85#1434C2214yes—
86#1434C2214yes—
87C22⋊#95C2214yes—
88C22⋊#95C2214yes—
89C22⋊#95C2212yes—
90C4×#95C2214yes—
91C4×#95C2211yescharacteristic
92C4×#95C2211yescharacteristic
93C23×Q8C2312yes—
94C24×C4C2311yescharacteristic
95C24×C4C2312yes—
96C2×C4⋊Q8—24no—
97C2×C4⋊Q8—24no—
98C2×C4⋊Q8—42no—
99C2×#117—24no—
100C2×#117—24no—
101C2×#117—24no—
102C2×#117—24no—
103C2×#117—42no—
104C2×#117Q818yescomplemented
105C22×C4⋊C4C2311yescharacteristic
106C22×C4⋊C4C2311yescharacteristic
107C22×C4⋊C4C2311yescharacteristic
108C22×C4⋊C4C2311yescharacteristic
109C22×C4⋊C4C2311yescharacteristic
110C22×C4⋊C4C2312yes—
111C22×C4⋊C4C2312yes—
112C22×C4⋊C4C2311yescharacteristic
113C22×C4⋊C4C2311yescharacteristic
114C22×C4⋊C4C2311yescharacteristic
115C22×C4⋊C4C2312yes—
116C22×C4⋊C4C2312yes—
117C22×C4⋊C4C2312yes—
118C22×C4⋊C4C2312yes—
119C22×C4⋊C4C2314yes—
120C22×C4⋊C4C2312yes—
121C22×C22⋊C4C2312yes—
122C22×C22⋊C4C2312yes—
123C22×C22⋊C4C2311yescharacteristic
124C22×C22⋊C4C2312yes—
125C22×C22⋊C4C2312yes—
126C22×C22⋊C4C2312yes—
127C22×C22⋊C4C2311yescharacteristic
128C22×C42—24no—
129C22×C42—24no—
130C22×C42—42no—
131C22×C42C2311yescharacteristic
132C22×C42C2311yescharacteristic
133C22×C42C2312yes—
134C22×C42C2311yescharacteristic
135C22×C42C2312yes—
136C22×C42C2312yes—
137C22×C42C2311yescharacteristic
138C22×C42C2311yescharacteristic
139C22×C42C2312yes—
140C22×C42C2311yescharacteristic
141C22×C42C2312yes—
142#396—28no—
143#393—28no—
144#391—216no—
145#391—28no—
146#389—24no—
147#389—24no—
148#389—24no—
149#389—24no—
150#388—28no—
151#388—28no—
152#387—28no—
153#385—28no—
154#384—42no—
155#382—24no—
156#382—24no—
157#382—24no—
158#382—28no—
159#382—42no—
160#380—24no—
161#380—24no—
162C4×C4⋊C4—24no—
163C4×C4⋊C4—24no—
164C4×C4⋊C4—24no—
165C4×C4⋊C4—42no—
166C4×C22⋊C4—24no—
167C4×C22⋊C4D418yes—
168#376—28no—
169C2×#95C2312yes—
170C2×#95C2312yes—
171C2×#95C2312yes—
172C2×#95C2312yes—
173C2×#95C2314yes—
174C2×#95C2312yes—
175C2×#95C2312yes—
176C2×#95C2314yes—
177C2×#95C2314yes—
178C2×#95C2312yes—
179C2×#95C2312yes—
180C2×#95C2312yes—
181C2×#95C2312yes—
182C2×#95C2314yes—
183C2×#95C2314yes—
184C2×#95C2314yes—
185C2×#95C2312yes—
186C2×#95C2312yes—
187C2×#95C2312yes—
188C2×#95C2312yes—
189C2×#95C2314yes—
190C2×#95C2314yes—
191C2×#95C2314yes—
192C2×#95C2312yes—
193C2×#95C2312yes—
194C2×#95C2314yes—
195C2×#95C2314yes—
196C2×#95C2312yes—
197C2×#95C2314yes—
198C2×#95C2314yes—
199C2×#95C2314yes—
200C2×#95C2314yes—
201C2×#95C2311yescharacteristic
202C2×#95C2311yescharacteristic
203C25C2411yescharacteristic
204C22×Q8—24no—
205C22×Q8—24no—
206C22×Q8—28no—
207C22×Q8—42no—
208C22×Q8—44no—
209C23×C4—22no—
210C23×C4—22no—
211C23×C4—22no—
212C23×C4—22no—
213C23×C4—24no—
214C23×C4—24no—
215C23×C4—24no—
216C23×C4—42no—
217C23×C4—42no—
218C23×C4—42no—
219C23×C4—42no—
220C23×C4C2×D414yes—
221C23×C4C2×Q814yes—
222C23×C4#4114yes—
223C23×C4C2412yes—
224C23×C4C2412yes—
225C23×C4C2412yes—
226C23×C4C2412yes—
227C23×C4C2412yes—
228C23×C4C2412yes—
229C23×C4C2411yescharacteristic
230C23×C4C2411yescharacteristic
231C23×C4C2411yescharacteristic
232C23×C4C2412yes—
233C23×C4C2412yes—
234C23×C4C2412yes—
235C23×C4C2412yes—
236C23×C4C2412yes—
237C23×C4C2412yes—
238C23×C4C2411yescharacteristic
239C23×C4C2411yescharacteristic
240C23×C4C2411yescharacteristic
241C4⋊Q8—416no—
242#117—232no—
243#117—416no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—24no—
257C2×C4⋊C4—22no—
258C2×C4⋊C4—24no—
259C2×C4⋊C4—22no—
260C2×C4⋊C4—24no—
261C2×C4⋊C4—22no—
262C2×C4⋊C4—22no—
263C2×C4⋊C4—22no—
264C2×C4⋊C4—24no—
265C2×C4⋊C4—22no—
266C2×C4⋊C4—22no—
267C2×C4⋊C4—22no—
268C2×C4⋊C4—22no—
269C2×C4⋊C4—22no—
270C2×C4⋊C4—24no—
271C2×C4⋊C4—24no—
272C2×C4⋊C4—24no—
273C2×C4⋊C4—42no—
274C2×C4⋊C4—42no—
275C2×C4⋊C4—44no—
276C2×C4⋊C4—42no—
277C2×C4⋊C4—42no—
278C2×C4⋊C4—44no—
279C2×C4⋊C4—44no—
280C2×C4⋊C4—44no—
281C2×C4⋊C4—44no—
282C2×C4⋊C4—44no—
283C2×C4⋊C4—44no—
284C2×C4⋊C4C2×Q814yes—
285C2×C4⋊C4C2×Q814yes—
286C2×C4⋊C4#4114yes—
287C2×C4⋊C4#4118yes—
288C2×C22⋊C4—24no—
289C2×C22⋊C4—24no—
290C2×C22⋊C4—24no—
291C2×C22⋊C4—24no—
292C2×C22⋊C4—24no—
293C2×C22⋊C4—24no—
294C2×C22⋊C4—24no—
295C2×C22⋊C4—24no—
296C2×C22⋊C4—24no—
297C2×C22⋊C4—42no—
298C2×C22⋊C4—42no—
299C2×C22⋊C4—42no—
300C2×C22⋊C4—44no—
301C2×C22⋊C4—44no—
302C2×C22⋊C4C2×D414yes—
303C2×C22⋊C4C2×D414yes—
304C2×C22⋊C4C2×Q818yes—
305C2×C42—24no—
306C2×C42—24no—
307C2×C42—22no—
308C2×C42—22no—
309C2×C42—22no—
310C2×C42—22no—
311C2×C42—22no—
312C2×C42—22no—
313C2×C42—24no—
314C2×C42—24no—
315C2×C42—24no—
316C2×C42—24no—
317C2×C42—22no—
318C2×C42—22no—
319C2×C42—22no—
320C2×C42—22no—
321C2×C42—28no—
322C2×C42—24no—
323C2×C42—24no—
324C2×C42—22no—
325C2×C42—22no—
326C2×C42—22no—
327C2×C42—22no—
328C2×C42—22no—
329C2×C42—416no—
330C2×C42—42no—
331C2×C42—42no—
332C2×C42—44no—
333C2×C42C2×D414yes—
334C2×C42C2×D414yes—
335C2×C42C2×Q818yes—
336C2×C42#4114yes—
337#95—24no—
338#95—24no—
339#95—24no—
340#95—24no—
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—28no—
350#95—28no—
351#95—24no—
352#95—24no—
353#95—28no—
354#95—28no—
355#95—28no—
356#95—28no—
357#95—44no—
358#95—44no—
359#95—44no—
360#95—44no—
361#95—44no—
362#95—44no—
363#95—44no—
364#95—42no—
365#95—42no—
366#95—44no—
367#95—42no—
368#95—44no—
369#95C2×D414yes—
370#95C2×D414yes—
371C24—22no—
372C24—21no—
373C24—21no—
374C24—21no—
375C24—21no—
376C24—22no—
377C24—22no—
378C24—22no—
379C24C22×D412yes—
380C24C22×Q812yes—
381C24C2×#4112yes—
382C24C2511yescharacteristic
383C2×Q8—48no—
384C2×Q8—416no—
385C2×Q8—48no—
386C2×Q8—84no—
387C2×Q8—88no—
388C22×C4—21no—
389C22×C4—21no—
390C22×C4—21no—
391C22×C4—21no—
392C22×C4—21no—
393C22×C4—21no—
394C22×C4—22no—
395C22×C4—28no—
396C22×C4—28no—
397C22×C4—28no—
398C22×C4—28no—
399C22×C4—28no—
400C22×C4—28no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—21no—
405C22×C4—24no—
406C22×C4—24no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—21no—
417C22×C4—21no—
418C22×C4—22no—
419C22×C4—21no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—21no—
424C22×C4—21no—
425C22×C4—24no—
426C22×C4—24no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—24no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—24no—
438C22×C4—22no—
439C22×C4—22no—
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—24no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—24no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—24no—
468C22×C4—24no—
469C22×C4—24no—
470C22×C4—24no—
471C22×C4—24no—
472C22×C4—24no—
473C22×C4—24no—
474C22×C4—24no—
475C22×C4—48no—
476C22×C4—48no—
477C22×C4—48no—
478C22×C4—48no—
479C22×C4—48no—
480C22×C4—48no—
481C22×C4—48no—
482C22×C4C22⋊Q8116yes—
483C22×C4C22×D412yes—
484C22×C4C22×D412yes—
485C22×C4C22×D412yes—
486C22×C4C22×D412yes—
487C22×C4C22×D412yes—
488C22×C4C22×D412yes—
489C22×C4C22×Q814yes—
490C22×C4C22×Q814yes—
491C22×C4C22×Q812yes—
492C22×C4C22×Q812yes—
493C22×C4C2×#4114yes—
494C22×C4C2×#4114yes—
495C22×C4C2×#4114yes—
496C22×C4C2×#4114yes—
497C22×C4C2×#4112yes—
498C22×C4C2×#4114yes—
499C22×C4C2×#4112yes—
500C22×C4C2×#4112yes—
501C22×C4C2×#4112yes—
502C22×C4H(5,𝔽2)12yes—
503C22×C42−1+412yes—
504C4⋊C4—28no—
505C4⋊C4—28no—
506C4⋊C4—28no—
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—84no—
517C4⋊C4—84no—
518C4⋊C4—84no—
519C4⋊C4—88no—
520C22⋊C4—216no—
521C22⋊C4—28no—
522C22⋊C4—44no—
523C22⋊C4—48no—
524C22⋊C4—48no—
525C22⋊C4—84no—
526C22⋊C4—84no—
527C42—216no—
528C42—28no—
529C42—28no—
530C42—28no—
531C42—48no—
532C42—48no—
533C42—48no—
534C42—48no—
535C42—44no—
536C42—44no—
537C42—44no—
538C23—22no—
539C23—22no—
540C23—22no—
541C23—22no—
542C23—22no—
543C23—22no—
544C23—22no—
545C23—22no—
546C23—24no—
547C23—24no—
548C23—24no—
549C23—22no—
550C23—22no—
551C23—24no—
552C23—42no—
553C23—42no—
554C23—42no—
555C23—42no—
556C23—42no—
557C23—42no—
558C23—42no—
559C23—42no—
560C23C2×C22⋊Q814yes—
561C23C23×D411yescharacteristic
562C23C23×Q811yescharacteristic
563C23C22×#4111yescharacteristic
564C23C22×#4112yes—
565C23C22×#4111yescharacteristic
566C23C22×#4111yescharacteristic
567C23C2×H(5,𝔽2)11yescharacteristic
568C23C2×2−1+411yescharacteristic
569C23#58512yes—
570C23#58512yes—
571C23#58512yes—
572Q8—816no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—24no—
581C2×C4—22no—
582C2×C4—24no—
583C2×C4—232no—
584C2×C4—24no—
585C2×C4—24no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—24no—
591C2×C4—22no—
592C2×C4—22no—
593C2×C4—24no—
594C2×C4—24no—
595C2×C4—24no—
596C2×C4—24no—
597C2×C4—24no—
598C2×C4—24no—
599C2×C4—28no—
600C2×C4—28no—
601C2×C4—24no—
602C2×C4—22no—
603C2×C4—22no—
604C2×C4—22no—
605C2×C4—28no—
606C2×C4—24no—
607C2×C4—24no—
608C2×C4—432no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—44no—
616C2×C4—42no—
617C2×C4—44no—
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—44no—
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—44no—
649C2×C4—42no—
650C2×C4—44no—
651C2×C4—42no—
652C2×C4—44no—
653C2×C4—42no—
654C2×C4—42no—
655C2×C4—42no—
656C2×C4—42no—
657C2×C4—42no—
658C2×C4—44no—
659C2×C4—44no—
660C2×C4—42no—
661C2×C4—42no—
662C2×C4—44no—
663C2×C4—44no—
664C2×C4—44no—
665C2×C4—44no—
666C2×C4—42no—
667C2×C4—44no—
668C2×C4C2×C22⋊Q814yes—
669C2×C4C2×C22⋊Q814yes—
670C2×C4#53214yes—
671C2×C4#53314yes—
672C2×C4#54414yes—
673C2×C4#54614yes—
674C2×C4#54714yes—
675C2×C4#55114yes—
676C2×C4#55214yes—
677C22—24no—
678C22—24no—
679C22—24no—
680C22—42no—
681C22—44no—
682C22—44no—
683C22—42no—
684C22—44no—
685C22—44no—
686C22—42no—
687C22—42no—
688C22C22×C22⋊Q811yescharacteristic
689C22C2×#53211yescharacteristic
690C22C2×#53311yescharacteristic
691C22#344911yescharacteristic
692C22C2×#54411yescharacteristic
693C22C2×#54611yescharacteristic
694C22C2×#54711yescharacteristic
695C22C2×#55111yescharacteristic
696C22C2×#55211yescharacteristic
697C22#347512yes—
698C22#347512yes—
699C22#347512yes—
700C22Q8×#4112yes—
701C22#349312yes—
702C22#349312yes—
703C22#350011yescharacteristic
704C22#350011yescharacteristic
705C22#350412yes—
706C22#350512yes—
707C22#351812yes—
708C22#352412yes—
709C22#354012yes—
710C22#356512yes—
711C4—24no—
712C4—24no—
713C4—44no—
714C4—44no—
715C4—44no—
716C4—44no—
717C4—44no—
718C4—42no—
719C4—44no—
720C4—42no—
721C4—42no—
722C4—82no—
723C4—82no—
724C4—82no—
725C4—82no—
726C4—82no—
727C4—82no—
728C4#243818yes—
729C2—44no—
730C2C2×#243811yescharacteristic
731C2#3494711yescharacteristic
732C2#3562011yescharacteristic
733C2#3618511yescharacteristic
734C2#3628412yes—
735C2#3638611yescharacteristic
736C2#3787111yescharacteristic
737C2#3809511yescharacteristic
738C2#3998712yes—
739C2#5920212yes—
740C2#5929712yes—
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
5. #15239C2semidirect
6. #15234C2semidirect
7. #15230C2semidirect
8. #15196C2semidirect
9. #14941C2semidirect
11. #14155C2semidirect
12. #14090C2semidirect
13. #14051C2semidirect
14. #14050C2semidirect
16. #13909C2semidirect
104. C2×#117Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#24392
C2×#23462
#158272
#158242
#152392
#152342
#152302
#151962
#149412
#146822
#141552
#140902
#140512
#140502
#139392
#139092
#138792
C4×#3852
#137562