Finite Group Databasebeta

← #5956394

Subgroups — #5956394

Order 512.

2428 conjugacy classes of subgroups.

The subgroup lattice has 726 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×C4⋊Q8C211yesmaximal, characteristic, complemented
2C2×#2298C211yesmaximal, characteristic
3#15221C212yesmaximal, complemented
4#15220C212yesmaximal
5#15152C211yesmaximal, characteristic, complemented
6#15135C211yesmaximal, characteristic
7#15128C211yesmaximal, characteristic, complemented
8#15125C211yesmaximal, characteristic
9#15102C212yesmaximal, complemented
10#15100C212yesmaximal
11#15068C212yesmaximal, complemented
12#15065C212yesmaximal
13#15053C212yesmaximal, complemented
14#15047C212yesmaximal
15#14876C212yesmaximal, complemented
16#14875C212yesmaximal
17#13783C212yesmaximal, complemented
18#13777C212yesmaximal
19C4×#382C211yesmaximal, characteristic, complemented
20C22×C4⋊Q8C2211yescharacteristic
21C22×C4×Q8C2212yes—
22C22×#117C2211yescharacteristic
23C2×#400C2212yes—
24C2×#399C2212yes—
25C2×#395C2212yes—
26C2×#393C2212yes—
27C4×C4⋊Q8—28no—
28#2298—28no—
29C2×#391C2212yes—
30C2×#389C2212yes—
31C2×#389C2212yes—
32C2×#389C2212yes—
33C2×#389C2212yes—
34C2×#388C2211yescharacteristic
35C2×#388C2211yescharacteristic
36C2×#388C2211yescharacteristic
37C2×#388C2211yescharacteristic
38C2×#387C2211yescharacteristic
39C2×#387C2211yescharacteristic
40C2×#387C2211yescharacteristic
41C2×#387C2211yescharacteristic
42C2×#384C2211yescharacteristic
43C2×#382C2212yes—
44C2×#380C2212yes—
45C2×#380C2212yes—
46C2×C4×C4⋊C4C2211yescharacteristic
47C2×C4×C4⋊C4C2211yescharacteristic
48C2×C4×C4⋊C4C2212yes—
49C2×C4×C4⋊C4C2212yes—
50C2×C4×C4⋊C4C2212yes—
51C2×C4×C22⋊C4C2211yescharacteristic
52C2×C4×C22⋊C4C2211yescharacteristic
53C2×#376C2211yescharacteristic
54C2×C43C2211yescharacteristic
55#1446C2212yes—
56#1446C2212yes—
57#1445C2212yes—
58#1445C2212yes—
59#1445C2214yes—
60#1444C2212yes—
61#1444C2214yes—
62#1444C2212yes—
63#1443C2214yes—
64#1442C2212yes—
65#1442C2214yes—
66#1442C2212yes—
67#1442C2214yes—
68#1441C2212yes—
69#1441C2214yes—
70#1441C2212yes—
71#1439C2212yes—
72#1439C2214yes—
73#1439C2212yes—
74#1439C2214yes—
75#1437C2214yes—
76#1437C2214yes—
77#1436C2214yes—
78#1436C2214yes—
79#1436C2214yes—
80#1436C2214yes—
81C4⋊#95C2211yescharacteristic
82C4⋊#95C2212yes—
83C4⋊#95C2211yescharacteristic
84C4⋊#95C2212yes—
85C4⋊#95C2212yes—
86C4⋊#95C2212yes—
87C4⋊#95C2214yes—
88#1434C2212yes—
89#1434C2214yes—
90#1434C2212yes—
91C4×#95C2212yes—
92C4×#95C2212yes—
93C23×Q8C2312yes—
94C24×C4C2311yescharacteristic
95C2×C4⋊Q8—28no—
96C2×C4⋊Q8—24no—
97C2×C4×Q8—28no—
98C2×C4×Q8—44no—
99C2×C4×Q8—44no—
100C2×#117—24no—
101C2×#117—24no—
102C2×#117Q818yescomplemented
103C22×C4⋊C4C2311yescharacteristic
104C22×C4⋊C4C2312yes—
105C22×C4⋊C4C2312yes—
106C22×C4⋊C4C2312yes—
107C22×C4⋊C4C2312yes—
108C22×C4⋊C4C2311yescharacteristic
109C22×C4⋊C4C2311yescharacteristic
110C22×C4⋊C4C2311yescharacteristic
111C22×C4⋊C4C2312yes—
112C22×C4⋊C4C2312yes—
113C22×C4⋊C4C2314yes—
114C22×C4⋊C4C2314yes—
115C22×C4⋊C4C2314yes—
116C22×C4⋊C4C2312yes—
117C22×C4⋊C4C2311yescharacteristic
118C22×C4⋊C4C2311yescharacteristic
119C22×C22⋊C4C2311yescharacteristic
120C22×C22⋊C4C2311yescharacteristic
121C22×C22⋊C4C2311yescharacteristic
122C22×C22⋊C4C2311yescharacteristic
123C22×C22⋊C4C2314yes—
124C22×C22⋊C4C2314yes—
125C22×C22⋊C4C2311yescharacteristic
126C22×C22⋊C4C2311yescharacteristic
127C22×C42C2311yescharacteristic
128C22×C42C2311yescharacteristic
129C22×C42C2311yescharacteristic
130C22×C42C2311yescharacteristic
131C22×C42C2311yescharacteristic
132C22×C42C2311yescharacteristic
133C22×C42C2311yescharacteristic
134C22×C42C2314yes—
135C22×C42C2314yes—
136C22×C42C2311yescharacteristic
137C22×C42C2311yescharacteristic
138C22×C42C2311yescharacteristic
139#400—44no—
140#399—44no—
141#395—44no—
142#393—44no—
143#391—28no—
144#389—28no—
145#389—28no—
146#389—28no—
147#389—28no—
148#388—24no—
149#388—24no—
150#388—24no—
151#388—24no—
152#387—24no—
153#387—24no—
154#387—24no—
155#387—24no—
156#384—24no—
157#382—28no—
158#380—28no—
159#380—28no—
160C4×C4⋊C4—28no—
161C4×C4⋊C4—28no—
162C4×C4⋊C4—24no—
163C4×C4⋊C4—24no—
164C4×C4⋊C4—28no—
165C4×C22⋊C4—24no—
166C4×C22⋊C4—24no—
167#376—24no—
168C2×#95C2312yes—
169C2×#95C2312yes—
170C2×#95C2312yes—
171C2×#95C2312yes—
172C2×#95C2314yes—
173C2×#95C2314yes—
174C2×#95C2311yescharacteristic
175C2×#95C2314yes—
176C2×#95C2314yes—
177C2×#95C2314yes—
178C2×#95C2314yes—
179C2×#95C2311yescharacteristic
180C2×#95C2311yescharacteristic
181C2×#95C2311yescharacteristic
182C2×#95C2314yes—
183C2×#95C2314yes—
184C2×#95C2314yes—
185C2×#95C2311yescharacteristic
186C2×#95C2311yescharacteristic
187C2×#95C2311yescharacteristic
188C2×#95C2311yescharacteristic
189C2×#95C2311yescharacteristic
190C2×#95C2311yescharacteristic
191C2×#95C2311yescharacteristic
192C2×#95C2311yescharacteristic
193C2×#95C2314yes—
194C2×#95C2314yes—
195C2×#95C2314yes—
196C2×#95C2314yes—
197C2×#95C2314yes—
198C2×#95C2314yes—
199C2×#95C2314yes—
200C2×#95C2314yes—
201C43D418yes—
202C25C2411yescharacteristic
203C22×Q8—24no—
204C22×Q8—28no—
205C22×Q8—24no—
206C22×Q8—24no—
207C22×Q8—44no—
208C23×C4—22no—
209C23×C4—22no—
210C23×C4—24no—
211C23×C4—24no—
212C23×C4C2×Q814yes—
213C23×C4C2411yescharacteristic
214C23×C4C2411yescharacteristic
215C23×C4C2414yes—
216C23×C4C2411yescharacteristic
217C23×C4C2411yescharacteristic
218C23×C4C2414yes—
219C23×C4C2411yescharacteristic
220C23×C4C2411yescharacteristic
221C23×C4C2411yescharacteristic
222C23×C4C2411yescharacteristic
223C23×C4C2411yescharacteristic
224C23×C4C2414yes—
225C23×C4C2414yes—
226C23×C4C2411yescharacteristic
227C23×C4C2411yescharacteristic
228C23×C4C2411yescharacteristic
229C23×C4C2411yescharacteristic
230C23×C4C2411yescharacteristic
231C4⋊Q8—232no—
232C4×Q8—432no—
233#117—232no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—24no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—28no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—24no—
245C2×C4⋊C4—28no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—24no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—24no—
256C2×C4⋊C4—24no—
257C2×C4⋊C4—24no—
258C2×C4⋊C4—24no—
259C2×C4⋊C4—22no—
260C2×C4⋊C4—22no—
261C2×C4⋊C4—22no—
262C2×C4⋊C4—22no—
263C2×C4⋊C4—28no—
264C2×C4⋊C4—28no—
265C2×C4⋊C4—44no—
266C2×C4⋊C4—42no—
267C2×C4⋊C4—42no—
268C2×C4⋊C4—44no—
269C2×C4⋊C4—44no—
270C2×C4⋊C4—44no—
271C2×C4⋊C4—44no—
272C2×C4⋊C4—44no—
273C2×C4⋊C4—44no—
274C2×C4⋊C4C2×Q814yes—
275C2×C4⋊C4C2×Q814yes—
276C2×C4⋊C4#4118yes—
277C2×C4⋊C4#4118yes—
278C2×C22⋊C4—22no—
279C2×C22⋊C4—22no—
280C2×C22⋊C4—22no—
281C2×C22⋊C4—22no—
282C2×C22⋊C4—22no—
283C2×C22⋊C4—22no—
284C2×C22⋊C4—22no—
285C2×C22⋊C4—22no—
286C2×C22⋊C4—22no—
287C2×C22⋊C4—22no—
288C2×C22⋊C4—22no—
289C2×C22⋊C4—22no—
290C2×C22⋊C4—28no—
291C2×C22⋊C4—28no—
292C2×C22⋊C4—42no—
293C2×C22⋊C4—42no—
294C2×C22⋊C4—44no—
295C2×C22⋊C4—44no—
296C2×C22⋊C4—44no—
297C2×C22⋊C4—44no—
298C2×C22⋊C4C2×Q814yes—
299C2×C22⋊C4C2×Q814yes—
300C2×C42—22no—
301C2×C42—22no—
302C2×C42—24no—
303C2×C42—24no—
304C2×C42—22no—
305C2×C42—22no—
306C2×C42—22no—
307C2×C42—22no—
308C2×C42—28no—
309C2×C42—22no—
310C2×C42—22no—
311C2×C42—24no—
312C2×C42—24no—
313C2×C42—28no—
314C2×C42—44no—
315C2×C42—44no—
316C2×C42—42no—
317C2×C42—42no—
318C2×C42—44no—
319C2×C42—44no—
320C2×C42C2×D414yes—
321C2×C42C2×D414yes—
322C2×C42C2×D414yes—
323C2×C42C2×D414yes—
324C2×C42C2×D414yes—
325C2×C42C2×D414yes—
326C2×C42C2×D414yes—
327C2×C42C2×Q814yes—
328C2×C42C2×Q814yes—
329C2×C42#4114yes—
330#95—22no—
331#95—22no—
332#95—22no—
333#95—22no—
334#95—22no—
335#95—22no—
336#95—22no—
337#95—22no—
338#95—22no—
339#95—22no—
340#95—22no—
341#95—22no—
342#95—28no—
343#95—28no—
344#95—28no—
345#95—28no—
346#95—28no—
347#95—28no—
348#95—28no—
349#95—28no—
350#95—28no—
351#95—24no—
352#95—24no—
353#95—24no—
354#95—24no—
355#95—44no—
356#95—44no—
357#95—44no—
358#95—44no—
359#95—44no—
360#95—44no—
361#95—44no—
362#95—44no—
363C24—21no—
364C24—21no—
365C24—22no—
366C24—22no—
367C24—21no—
368C24—21no—
369C24—21no—
370C24—21no—
371C24—22no—
372C24—22no—
373C24C22×Q812yes—
374C24C2511yescharacteristic
375C2×Q8—216no—
376C2×Q8—416no—
377C2×Q8—416no—
378C2×Q8—48no—
379C2×Q8—48no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—21no—
383C22×C4—21no—
384C22×C4—21no—
385C22×C4—21no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—21no—
391C22×C4—21no—
392C22×C4—21no—
393C22×C4—24no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—24no—
399C22×C4—24no—
400C22×C4—21no—
401C22×C4—21no—
402C22×C4—21no—
403C22×C4—24no—
404C22×C4—24no—
405C22×C4—21no—
406C22×C4—21no—
407C22×C4—24no—
408C22×C4—24no—
409C22×C4—21no—
410C22×C4—21no—
411C22×C4—24no—
412C22×C4—21no—
413C22×C4—21no—
414C22×C4—21no—
415C22×C4—21no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—24no—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—24no—
424C22×C4—28no—
425C22×C4—216no—
426C22×C4—21no—
427C22×C4—21no—
428C22×C4—21no—
429C22×C4—21no—
430C22×C4—24no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—24no—
434C22×C4—24no—
435C22×C4—24no—
436C22×C4—24no—
437C22×C4—24no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—24no—
441C22×C4—21no—
442C22×C4—21no—
443C22×C4—24no—
444C22×C4—24no—
445C22×C4—21no—
446C22×C4—21no—
447C22×C4—24no—
448C22×C4—21no—
449C22×C4—21no—
450C22×C4—21no—
451C22×C4—21no—
452C22×C4—24no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—24no—
456C22×C4—24no—
457C22×C4—24no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—44no—
461C22×C4—48no—
462C22×C4—44no—
463C22×C4C22×D412yes—
464C22×C4C22×D412yes—
465C22×C4C22×D412yes—
466C22×C4C22×D412yes—
467C22×C4C22×D412yes—
468C22×C4C22×D412yes—
469C22×C4C22×D412yes—
470C22×C4C22×Q812yes—
471C22×C4C22×Q812yes—
472C22×C4C22×Q812yes—
473C22×C4C22×Q812yes—
474C22×C4C22×Q812yes—
475C22×C4C22×Q812yes—
476C22×C4C2×#4118yes—
477C22×C4C2×#4118yes—
478C22×C4C2×#4112yes—
479C22×C4C2×#4112yes—
480C22×C4C2×#4112yes—
481C22×C4C2×#4114yes—
482C22×C4C2×#4114yes—
483C22×C4H(5,𝔽2)12yes—
484C22×C4H(5,𝔽2)12yes—
485C22×C42−1+412yes—
486C22×C42−1+412yes—
487C4⋊C4—216no—
488C4⋊C4—28no—
489C4⋊C4—28no—
490C4⋊C4—216no—
491C4⋊C4—216no—
492C4⋊C4—48no—
493C4⋊C4—44no—
494C4⋊C4—44no—
495C4⋊C4—416no—
496C4⋊C4—48no—
497C4⋊C4—48no—
498C4⋊C4—44no—
499C4⋊C4—48no—
500C4⋊C4—44no—
501C4⋊C4—88no—
502C4⋊C4—88no—
503C22⋊C4—28no—
504C22⋊C4—28no—
505C22⋊C4—44no—
506C22⋊C4—44no—
507C22⋊C4—44no—
508C22⋊C4—44no—
509C22⋊C4—88no—
510C22⋊C4—88no—
511C42—28no—
512C42—28no—
513C42—28no—
514C42—28no—
515C42—28no—
516C42—44no—
517C42—44no—
518C42—416no—
519C42—44no—
520C42—44no—
521C42—88no—
522C42C22⋊Q8116yes—
523C23—22no—
524C23—24no—
525C23—22no—
526C23—22no—
527C23—24no—
528C23—41no—
529C23—42no—
530C23—41no—
531C23—41no—
532C23—41no—
533C23—41no—
534C23—41no—
535C23—42no—
536C23—41no—
537C23—41no—
538C23—42no—
539C23—42no—
540C23—42no—
541C23—42no—
542C23—42no—
543C23—42no—
544C23—42no—
545C23—42no—
546C23C23×D411yescharacteristic
547C23C23×Q811yescharacteristic
548C23C22×#4111yescharacteristic
549C23C22×#4112yes—
550C23C22×#4112yes—
551C23C2×H(5,𝔽2)11yescharacteristic
552C23C2×H(5,𝔽2)11yescharacteristic
553C23C2×2−1+411yescharacteristic
554C23C2×2−1+411yescharacteristic
555C23#58512yes—
556C23#58512yes—
557Q8—432no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—24no—
569C2×C4—24no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—24no—
573C2×C4—24no—
574C2×C4—24no—
575C2×C4—24no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—28no—
580C2×C4—28no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—24no—
584C2×C4—22no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—24no—
591C2×C4—28no—
592C2×C4—28no—
593C2×C4—28no—
594C2×C4—28no—
595C2×C4—41no—
596C2×C4—42no—
597C2×C4—41no—
598C2×C4—44no—
599C2×C4—41no—
600C2×C4—41no—
601C2×C4—41no—
602C2×C4—41no—
603C2×C4—44no—
604C2×C4—44no—
605C2×C4—42no—
606C2×C4—44no—
607C2×C4—41no—
608C2×C4—41no—
609C2×C4—44no—
610C2×C4—44no—
611C2×C4—44no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—44no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—416no—
618C2×C4—41no—
619C2×C4—41no—
620C2×C4—44no—
621C2×C4—41no—
622C2×C4—41no—
623C2×C4—44no—
624C2×C4—44no—
625C2×C4—41no—
626C2×C4—41no—
627C2×C4—41no—
628C2×C4—41no—
629C2×C4—41no—
630C2×C4—41no—
631C2×C4—41no—
632C2×C4—44no—
633C2×C4—41no—
634C2×C4—44no—
635C2×C4—44no—
636C2×C4—44no—
637C2×C4—42no—
638C2×C4—42no—
639C2×C4—44no—
640C2×C4—42no—
641C2×C4—44no—
642C2×C4—42no—
643C2×C4—44no—
644C2×C4—42no—
645C2×C4—42no—
646C2×C4—44no—
647C2×C4—44no—
648C2×C4—44no—
649C2×C4—42no—
650C2×C4—44no—
651C2×C4—42no—
652C2×C4C2×C22⋊Q814yes—
653C2×C4C2×C22⋊Q814yes—
654C2×C4C2×C22⋊Q814yes—
655C2×C4#52514yes—
656C2×C4#52918yes—
657C2×C4#53214yes—
658C2×C4#53318yes—
659C2×C4#53514yes—
660C2×C4#53614yes—
661C22—42no—
662C22—42no—
663C22—42no—
664C22—44no—
665C22—44no—
666C22—82no—
667C22—82no—
668C22—81no—
669C22—81no—
670C22—81no—
671C22—81no—
672C22C22×C22⋊Q811yescharacteristic
673C22C2×#52511yescharacteristic
674C22C2×#52912yes—
675C22C2×#53211yescharacteristic
676C22C2×#53312yes—
677C22C2×#53511yescharacteristic
678C22C2×#53611yescharacteristic
679C22#344911yescharacteristic
680C22#344911yescharacteristic
681C22#350311yescharacteristic
682C22#350311yescharacteristic
683C22#350412yes—
684C22#351012yes—
685C22#351111yescharacteristic
686C22#351111yescharacteristic
687C22#351312yes—
688C22#351312yes—
689C22#351412yes—
690C22#351512yes—
691C22#351612yes—
692C22#351612yes—
693C22#351712yes—
694C22#351812yes—
695C4—24no—
696C4—24no—
697C4—24no—
698C4—24no—
699C4—42no—
700C4—48no—
701C4—42no—
702C4—42no—
703C4—48no—
704C4—42no—
705C4—42no—
706C4—84no—
707C4—81no—
708C4—84no—
709C4—81no—
710C4—81no—
711C4—81no—
712C4#239618yes—
713C2—82no—
714C2C2×#239611yescharacteristic
715C2#3519111yescharacteristic
716C2#3520111yescharacteristic
717C2#3523211yescharacteristic
718C2#3524511yescharacteristic
719C2#3527911yescharacteristic
720C2#3533411yescharacteristic
721C2#3557712yes—
722C2#3557912yes—
723C2#3559312yes—
724C2#3559512yes—
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
1. C2×C4×C4⋊Q8C2semidirect
3. #15221C2semidirect
5. #15152C2semidirect
7. #15128C2semidirect
9. #15102C2semidirect
11. #15068C2semidirect
13. #15053C2semidirect
15. #14876C2semidirect
17. #13783C2semidirect
19. C4×#382C2semidirect
102. C2×#117Q8semidirect

Maximal subgroups

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

subgroupindex
C2×C4×C4⋊Q82
C2×#22982
#152212
#152202
#151522
#151352
#151282
#151252
#151022
#151002
#150682
#150652
#150532
#150472
#148762
#148752
#137832
#137772
C4×#3822