Finite Group Databasebeta

← #4930462

Subgroups — #4930462

Order 512.

4402 conjugacy classes of subgroups.

The subgroup lattice has 716 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#519C211yesmaximal, characteristic
2C2×#2441C211yesmaximal, characteristic
3C2×#2453C212yesmaximal, complemented
4#15214C212yesmaximal, complemented
5#15202C212yesmaximal, complemented
6#14893C214yesmaximal
7#14891C214yesmaximal
8#14884C212yesmaximal, complemented
9#14878C212yesmaximal, complemented
10#14264C211yesmaximal, characteristic
11#14017C212yesmaximal, complemented
12#13976C212yesmaximal, complemented
13#13951C211yesmaximal, characteristic
14#13935C212yesmaximal, complemented
15#13922C211yesmaximal, characteristic
16#13922C211yesmaximal, characteristic
17#13844C211yesmaximal, characteristic
18C24×Q8C2211yescharacteristic
19C22×#125C2211yescharacteristic
20C2×#519—216no—
21C2×#519—28no—
22C22×C4×Q8C2212yes—
23C22×C4×Q8C2212yes—
24C22×C4×Q8C2212yes—
25C22×C4×Q8C2212yes—
26C22×#117C2211yescharacteristic
27C22×#117C2214yes—
28C22×#117C2211yescharacteristic
29#2453—216no—
30#2441—28no—
31C2×#400C2214yes—
32C2×#399C2212yes—
33C2×#398C2214yes—
34C2×#396C2212yes—
35C2×#391C2212yes—
36C2×#391C2212yes—
37C2×#391C2212yes—
38C2×#391C2212yes—
39C2×#389C2212yes—
40C2×#389C2214yes—
41C2×#389C2211yescharacteristic
42C2×#389C2212yes—
43C2×#389C2212yes—
44C2×#389C2211yescharacteristic
45C2×#387C2212yes—
46C2×#387C2214yes—
47C2×#384C2211yescharacteristic
48C2×#380C2211yescharacteristic
49C2×#380C2211yescharacteristic
50C2×C4×C4⋊C4C2212yes—
51C2×C4×C22⋊C4C2211yescharacteristic
52C2×C4×C22⋊C4C2211yescharacteristic
53#1446C2212yes—
54#1446C2212yes—
55#1446C2212yes—
56#1446C2212yes—
57#1444C2211yescharacteristic
58#1444C2211yescharacteristic
59#1444C2214yes—
60#1444C2214yes—
61#1444C2214yes—
62#1444C2212yes—
63#1442C2211yescharacteristic
64#1442C2211yescharacteristic
65#1442C2212yes—
66#1442C2214yes—
67#1441C2214yes—
68#1441C2214yes—
69#1439C2212yes—
70#1439C2212yes—
71#1439C2214yes—
72#1439C2214yes—
73#1436C2214yes—
74#1436C2214yes—
75#1436C2214yes—
76#1436C2214yes—
77C4⋊#95C2212yes—
78C4⋊#95C2214yes—
79C4⋊#95C2212yes—
80C4⋊#95C2212yes—
81#1434C2212yes—
82#1434C2214yes—
83C22⋊#95C2212yes—
84C22⋊#95C2214yes—
85C22⋊#95C2212yes—
86C4×#95C2212yes—
87C23×Q8—28no—
88C23×Q8—28no—
89C23×Q8—24no—
90C23×Q8—24no—
91C23×Q8Q818yes—
92C23×Q8C2312yes—
93C23×Q8C2312yes—
94C24×C4C2312yes—
95C24×C4C2311yescharacteristic
96C2×#125—24no—
97C2×#125—28no—
98#519—2128no—
99C2×C4×Q8—28no—
100C2×C4×Q8—216no—
101C2×C4×Q8—28no—
102C2×C4×Q8—28no—
103C2×C4×Q8—28no—
104C2×C4×Q8—28no—
105C2×C4×Q8—216no—
106C2×C4×Q8—28no—
107C2×C4×Q8—28no—
108C2×C4×Q8D4116yes—
109C2×#117—24no—
110C2×#117—216no—
111C2×#117—28no—
112C2×#117—216no—
113C2×#117—28no—
114C2×#117—216no—
115C2×#117D418yes—
116C22×C4⋊C4C2312yes—
117C22×C4⋊C4C2312yes—
118C22×C4⋊C4C2311yescharacteristic
119C22×C4⋊C4C2311yescharacteristic
120C22×C4⋊C4C2314yes—
121C22×C4⋊C4C2311yescharacteristic
122C22×C4⋊C4C2311yescharacteristic
123C22×C4⋊C4C2312yes—
124C22×C4⋊C4C2312yes—
125C22×C4⋊C4C2312yes—
126C22×C4⋊C4C2314yes—
127C22×C4⋊C4C2312yes—
128C22×C4⋊C4C2314yes—
129C22×C4⋊C4C2311yescharacteristic
130C22×C4⋊C4C2312yes—
131C22×C4⋊C4C2312yes—
132C22×C4⋊C4C2312yes—
133C22×C4⋊C4C2314yes—
134C22×C4⋊C4C2311yescharacteristic
135C22×C22⋊C4C2312yes—
136C22×C22⋊C4C2311yescharacteristic
137C22×C22⋊C4C2311yescharacteristic
138C22×C22⋊C4C2312yes—
139C22×C22⋊C4C2312yes—
140C22×C22⋊C4C2312yes—
141C22×C22⋊C4C2312yes—
142C22×C42C2314yes—
143C22×C42C2311yescharacteristic
144C22×C42C2311yescharacteristic
145C22×C42C2312yes—
146C22×C42C2312yes—
147C22×C42C2314yes—
148C22×C42C2312yes—
149#400—216no—
150#399—28no—
151#398—216no—
152#396—28no—
153#391—28no—
154#391—28no—
155#391—28no—
156#391—28no—
157#389—24no—
158#389—24no—
159#389—28no—
160#389—28no—
161#389—216no—
162#389—28no—
163#387—28no—
164#387—216no—
165#384—24no—
166#380—24no—
167#380—24no—
168C4×C4⋊C4—28no—
169C4×C22⋊C4—24no—
170C4×C22⋊C4—24no—
171C2×#95C2312yes—
172C2×#95C2314yes—
173C2×#95C2314yes—
174C2×#95C2314yes—
175C2×#95C2312yes—
176C2×#95C2312yes—
177C2×#95C2314yes—
178C2×#95C2314yes—
179C2×#95C2314yes—
180C2×#95C2312yes—
181C2×#95C2312yes—
182C2×#95C2312yes—
183C2×#95C2314yes—
184C2×#95C2314yes—
185C2×#95C2314yes—
186C2×#95C2314yes—
187C2×#95C2311yescharacteristic
188C2×#95C2312yes—
189C2×#95C2311yescharacteristic
190C2×#95C2312yes—
191C2×#95C2311yescharacteristic
192C2×#95C2311yescharacteristic
193C2×#95C2312yes—
194C2×#95C2312yes—
195C2×#95C2314yes—
196C2×#95C2314yes—
197C2×#95C2314yes—
198C2×#95C2314yes—
199C25C2411yescharacteristic
200C22×Q8—232no—
201C22×Q8—232no—
202C22×Q8—264no—
203C22×Q8—264no—
204C22×Q8—232no—
205C22×Q8—28no—
206C22×Q8—24no—
207C22×Q8—24no—
208C22×Q8—24no—
209C22×Q8—28no—
210C22×Q8—24no—
211C22×Q8—24no—
212C22×Q8—24no—
213C22×Q8C2×D418yes—
214C22×Q8C2×Q818yes—
215C22×Q8C2×Q818yes—
216C22×Q8#4118yes—
217C23×C4—24no—
218C23×C4—24no—
219C23×C4—28no—
220C23×C4—24no—
221C23×C4—28no—
222C23×C4—24no—
223C23×C4C2×D414yes—
224C23×C4C2×Q818yes—
225C23×C4C2×Q814yes—
226C23×C4#4114yes—
227C23×C4C2412yes—
228C23×C4C2411yescharacteristic
229C23×C4C2411yescharacteristic
230C23×C4C2414yes—
231C23×C4C2412yes—
232C23×C4C2412yes—
233C23×C4C2412yes—
234C23×C4C2412yes—
235C23×C4C2412yes—
236C23×C4C2412yes—
237C23×C4C2411yescharacteristic
238C23×C4C2411yescharacteristic
239C23×C4C2412yes—
240C23×C4C2414yes—
241C23×C4C2411yescharacteristic
242C23×C4C2411yescharacteristic
243#125—416no—
244C4×Q8—264no—
245C4×Q8—264no—
246C4×Q8—264no—
247C4×Q8—264no—
248#117—2128no—
249#117—232no—
250#117—232no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—28no—
253C2×C4⋊C4—24no—
254C2×C4⋊C4—28no—
255C2×C4⋊C4—28no—
256C2×C4⋊C4—22no—
257C2×C4⋊C4—22no—
258C2×C4⋊C4—28no—
259C2×C4⋊C4—28no—
260C2×C4⋊C4—24no—
261C2×C4⋊C4—22no—
262C2×C4⋊C4—24no—
263C2×C4⋊C4—28no—
264C2×C4⋊C4—24no—
265C2×C4⋊C4—24no—
266C2×C4⋊C4—28no—
267C2×C4⋊C4—24no—
268C2×C4⋊C4—24no—
269C2×C4⋊C4—22no—
270C2×C4⋊C4—24no—
271C2×C4⋊C4—28no—
272C2×C4⋊C4—24no—
273C2×C4⋊C4—24no—
274C2×C4⋊C4—28no—
275C2×C4⋊C4—24no—
276C2×C4⋊C4—24no—
277C2×C4⋊C4—24no—
278C2×C4⋊C4—24no—
279C2×C4⋊C4—28no—
280C2×C4⋊C4—24no—
281C2×C4⋊C4—24no—
282C2×C4⋊C4—24no—
283C2×C4⋊C4—24no—
284C2×C4⋊C4—42no—
285C2×C4⋊C4—44no—
286C2×C4⋊C4—42no—
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⋊C4C2×D4116yes—
295C2×C4⋊C4C2×D414yes—
296C2×C4⋊C4C2×D418yes—
297C2×C4⋊C4C2×D414yes—
298C2×C22⋊C4—24no—
299C2×C22⋊C4—24no—
300C2×C22⋊C4—24no—
301C2×C22⋊C4—28no—
302C2×C22⋊C4—22no—
303C2×C22⋊C4—22no—
304C2×C22⋊C4—24no—
305C2×C22⋊C4—22no—
306C2×C22⋊C4—22no—
307C2×C22⋊C4—24no—
308C2×C22⋊C4—24no—
309C2×C22⋊C4—24no—
310C2×C22⋊C4—42no—
311C2×C22⋊C4—41no—
312C2×C22⋊C4—41no—
313C2×C22⋊C4—42no—
314C2×C22⋊C4—42no—
315C2×C22⋊C4—44no—
316C2×C22⋊C4C2×D418yes—
317C2×C42—28no—
318C2×C42—28no—
319C2×C42—28no—
320C2×C42—28no—
321C2×C42—28no—
322C2×C42—28no—
323C2×C42—24no—
324C2×C42—24no—
325C2×C42—22no—
326C2×C42—24no—
327C2×C42—22no—
328C2×C42—24no—
329C2×C42—24no—
330C2×C42—24no—
331C2×C42—24no—
332C2×C42C2×D418yes—
333C2×C42C2×D4116yes—
334C2×C42C2×D418yes—
335#95—22no—
336#95—22no—
337#95—24no—
338#95—24no—
339#95—24no—
340#95—24no—
341#95—24no—
342#95—28no—
343#95—28no—
344#95—28no—
345#95—28no—
346#95—28no—
347#95—28no—
348#95—22no—
349#95—22no—
350#95—24no—
351#95—28no—
352#95—28no—
353#95—28no—
354#95—28no—
355#95—24no—
356#95—24no—
357#95—24no—
358#95—42no—
359#95—44no—
360#95—44no—
361#95—44no—
362#95—44no—
363C24—22no—
364C24—22no—
365C24—21no—
366C24—21no—
367C24—22no—
368C24—21no—
369C24—21no—
370C24—22no—
371C24C22×D412yes—
372C24C22×Q812yes—
373C24C2×#4112yes—
374C24C2511yescharacteristic
375C2×Q8—216no—
376C2×Q8—2256no—
377C2×Q8—232no—
378C2×Q8—216no—
379C2×Q8—216no—
380C2×Q8—232no—
381C2×Q8—216no—
382C2×Q8—216no—
383C2×Q8—216no—
384C2×Q8—216no—
385C2×Q8—216no—
386C2×Q8—216no—
387C2×Q8C22⋊Q8132yes—
388C22×C4—216no—
389C22×C4—232no—
390C22×C4—216no—
391C22×C4—216no—
392C22×C4—24no—
393C22×C4—22no—
394C22×C4—24no—
395C22×C4—24no—
396C22×C4—24no—
397C22×C4—22no—
398C22×C4—24no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—28no—
402C22×C4—232no—
403C22×C4—216no—
404C22×C4—28no—
405C22×C4—216no—
406C22×C4—21no—
407C22×C4—22no—
408C22×C4—21no—
409C22×C4—24no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—24no—
430C22×C4—22no—
431C22×C4—21no—
432C22×C4—21no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—24no—
438C22×C4—21no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—24no—
443C22×C4—21no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—24no—
447C22×C4—24no—
448C22×C4—22no—
449C22×C4—24no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—21no—
453C22×C4—21no—
454C22×C4—22no—
455C22×C4—24no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—24no—
461C22×C4—22no—
462C22×C4—24no—
463C22×C4—24no—
464C22×C4—24no—
465C22×C4—21no—
466C22×C4—24no—
467C22×C4—21no—
468C22×C4—22no—
469C22×C4—22no—
470C22×C4C22⋊Q8116yes—
471C22×C4C22×D414yes—
472C22×C4C22×D414yes—
473C22×C4C22×D414yes—
474C22×C4C22×D418yes—
475C22×C4C22×D414yes—
476C22×C4C22×D418yes—
477C22×C4C22×D412yes—
478C22×C4C22×D412yes—
479C22×C4C22×D414yes—
480C22×C4C22×Q818yes—
481C22×C4C22×Q812yes—
482C22×C4C22×Q812yes—
483C22×C4C2×#4118yes—
484C22×C4C2×#4112yes—
485C22×C4C2×#4114yes—
486C22×C4C2×#4112yes—
487C22×C42−1+412yes—
488C22×C42−1+412yes—
489C22×C42−1+414yes—
490C4⋊C4—232no—
491C4⋊C4—28no—
492C4⋊C4—28no—
493C4⋊C4—232no—
494C4⋊C4—28no—
495C4⋊C4—48no—
496C4⋊C4—416no—
497C4⋊C4—48no—
498C4⋊C4—48no—
499C4⋊C4—48no—
500C4⋊C4—416no—
501C4⋊C4—48no—
502C4⋊C4—48no—
503C4⋊C4—44no—
504C4⋊C4—44no—
505C4⋊C4—48no—
506C4⋊C4—84no—
507C4⋊C4—84no—
508C4⋊C4#120116yes—
509C22⋊C4—216no—
510C22⋊C4—216no—
511C22⋊C4—44no—
512C22⋊C4—44no—
513C22⋊C4—48no—
514C22⋊C4—84no—
515C22⋊C4—84no—
516C42—232no—
517C42—232no—
518C42—216no—
519C42—216no—
520C42—44no—
521C42—44no—
522C42—48no—
523C23—22no—
524C23—24no—
525C23—22no—
526C23—24no—
527C23—22no—
528C23—22no—
529C23—24no—
530C23—22no—
531C23—22no—
532C23—22no—
533C23—22no—
534C23—24no—
535C23—22no—
536C23—22no—
537C23—42no—
538C23—42no—
539C23—42no—
540C23—42no—
541C23—42no—
542C23—42no—
543C23—42no—
544C23—42no—
545C23C2×C22⋊Q814yes—
546C23C23×D412yes—
547C23C23×D411yescharacteristic
548C23C23×Q811yescharacteristic
549C23C22×#4111yescharacteristic
550C23C22×#4112yes—
551C23C2×2−1+412yes—
552C23C2×2−1+411yescharacteristic
553C23C2×2−1+411yescharacteristic
554C23#58511yescharacteristic
555C23#58512yes—
556C23#58511yescharacteristic
557Q8—264no—
558Q8—264no—
559C2×C4—264no—
560C2×C4—28no—
561C2×C4—22no—
562C2×C4—28no—
563C2×C4—24no—
564C2×C4—28no—
565C2×C4—28no—
566C2×C4—22no—
567C2×C4—28no—
568C2×C4—24no—
569C2×C4—28no—
570C2×C4—24no—
571C2×C4—28no—
572C2×C4—28no—
573C2×C4—24no—
574C2×C4—232no—
575C2×C4—24no—
576C2×C4—24no—
577C2×C4—24no—
578C2×C4—24no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—24no—
582C2×C4—24no—
583C2×C4—24no—
584C2×C4—24no—
585C2×C4—24no—
586C2×C4—24no—
587C2×C4—24no—
588C2×C4—24no—
589C2×C4—24no—
590C2×C4—24no—
591C2×C4—42no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—44no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—44no—
602C2×C4—41no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—41no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—41no—
612C2×C4—41no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—44no—
616C2×C4—42no—
617C2×C4—44no—
618C2×C4—42no—
619C2×C4—44no—
620C2×C4—41no—
621C2×C4—41no—
622C2×C4—44no—
623C2×C4—42no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—44no—
628C2×C4—42no—
629C2×C4—42no—
630C2×C4—44no—
631C2×C4—42no—
632C2×C4—42no—
633C2×C4—41no—
634C2×C4—42no—
635C2×C4—42no—
636C2×C4—41no—
637C2×C4—42no—
638C2×C4—44no—
639C2×C4—44no—
640C2×C4—42no—
641C2×C4—44no—
642C2×C4—42no—
643C2×C4—44no—
644C2×C4—44no—
645C2×C4C2×#12018yes—
646C2×C4C2×#12014yes—
647C2×C4C2×C22⋊Q8116yes—
648C2×C4C2×C22⋊Q814yes—
649C2×C4C2×C22⋊Q818yes—
650C2×C4C2×C22⋊Q814yes—
651C2×C4#52518yes—
652C2×C4#52514yes—
653C22—24no—
654C22—24no—
655C22—24no—
656C22—44no—
657C22—44no—
658C22—44no—
659C22—42no—
660C22—42no—
661C22—42no—
662C22—44no—
663C22—42no—
664C22C22×#12011yescharacteristic
665C22C22×C22⋊Q812yes—
666C22C22×C22⋊Q811yescharacteristic
667C22C2×#52512yes—
668C22C2×#52511yescharacteristic
669C22#348712yes—
670C22#348911yescharacteristic
671C22#348911yescharacteristic
672C22#348912yes—
673C22#349512yes—
674C22#349712yes—
675C22#349912yes—
676C22#349911yescharacteristic
677C22#350011yescharacteristic
678C22#350211yescharacteristic
679C22#350212yes—
680C22#350211yescharacteristic
681C22#350911yescharacteristic
682C22#351011yescharacteristic
683C22#351612yes—
684C22#351612yes—
685C22#352312yes—
686C22#352412yes—
687C4—216no—
688C4—24no—
689C4—28no—
690C4—44no—
691C4—42no—
692C4—42no—
693C4—44no—
694C4—44no—
695C4—44no—
696C4—84no—
697C4—82no—
698C4—82no—
699C4—81no—
700C4—82no—
701C4—81no—
702C4#284818yes—
703C2—44no—
704C2#3566211yescharacteristic
705C2#3566412yes—
706C2#3566811yescharacteristic
707C2#3567112yes—
708C2#3575711yescharacteristic
709C2#3578612yes—
710C2#3584512yes—
711C2#3598511yescharacteristic
712C2#3599711yescharacteristic
713C2#3608511yescharacteristic
714C2C2×#284811yescharacteristic
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. C2×#2453C2semidirect
4. #15214C2semidirect
5. #15202C2semidirect
8. #14884C2semidirect
9. #14878C2semidirect
11. #14017C2semidirect
12. #13976C2semidirect
14. #13935C2semidirect

Maximal subgroups

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

subgroupindex
C22×#5192
C2×#24412
C2×#24532
#152142
#152022
#148932
#148912
#148842
#148782
#142642
#140172
#139762
#139512
#139352
#139222
#138442