Finite Group Databasebeta

← #723283

Subgroups — #723283

Order 512.

3466 conjugacy classes of subgroups.

The subgroup lattice has 755 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2538C211yesmaximal, characteristic, complemented
2C2×#2580C211yesmaximal, characteristic, complemented
3C2×D4×C4⋊C4C211yesmaximal, characteristic, complemented
4C2×#2328C211yesmaximal, characteristic
5#16562C214yesmaximal, complemented
6#15439C214yesmaximal, complemented
7#15323C214yesmaximal, complemented
8#15225C212yesmaximal, complemented
9#15141C212yesmaximal, complemented
10#14729C214yesmaximal, complemented
11#14276C211yesmaximal, characteristic, complemented
12#14025C212yesmaximal
13#13941C211yesmaximal, characteristic, complemented
14#13920C212yesmaximal, complemented
15#13850C211yesmaximal, characteristic, complemented
16C22×#127C2211yescharacteristic
17C22×C22⋊Q8C2212yes—
18C22×C4×Q8C2211yescharacteristic
19C22×C4×D4C2211yescharacteristic
20C22×C4×D4C2212yes—
21C22×C4×D4C2211yescharacteristic
22C23×C4⋊C4C2212yes—
23#2580—28no—
24#2538—28no—
25C2×#399C2212yes—
26C2×#396C2214yes—
27C2×#396C2214yes—
28C2×#393C2212yes—
29D4×C4⋊C4—28no—
30#2328—28no—
31C2×#390C2211yescharacteristic
32C2×#390C2211yescharacteristic
33C2×#390C2212yes—
34C2×#390C2212yes—
35C2×#390C2212yes—
36C2×#389C2212yes—
37C2×#389C2212yes—
38C2×#389C2212yes—
39C2×#389C2212yes—
40C2×#386C2214yes—
41C2×#386C2214yes—
42C2×#386C2214yes—
43C2×#386C2214yes—
44C2×#385C2214yes—
45C2×#385C2214yes—
46C2×#384C2211yescharacteristic
47C2×#382C2211yescharacteristic
48C2×#382C2211yescharacteristic
49C2×#380C2212yes—
50C2×#380C2212yes—
51C2×#380C2214yes—
52C2×C4×C4⋊C4C2211yescharacteristic
53C2×#376C2211yescharacteristic
54#1446C2212yes—
55#1445C2212yes—
56#1445C2214yes—
57#1442C2214yes—
58#1441C2212yes—
59#1441C2212yes—
60#1441C2214yes—
61#1441C2214yes—
62#1441C2214yes—
63#1441C2214yes—
64#1439C2214yes—
65#1439C2214yes—
66#1439C2214yes—
67#1439C2214yes—
68#1439C2214yes—
69#1439C2212yes—
70#1439C2212yes—
71#1438C2214yes—
72#1438C2214yes—
73C4⋊#95C2212yes—
74C4⋊#95C2212yes—
75C22⋊#95C2214yes—
76C22⋊#95C2214yes—
77C4×#95C2212yes—
78C4×#95C2212yes—
79C23×Q8C2311yescharacteristic
80C23×D4C2311yescharacteristic
81C23×D4C2311yescharacteristic
82C23×D4C2314yes—
83C24×C4C2314yes—
84C24×C4C2312yes—
85C24×C4C2312yes—
86C2×#127—24no—
87C2×#127—24no—
88C2×#127Q818yescomplemented
89C2×C22⋊Q8—28no—
90C2×C22⋊Q8—48no—
91C2×C4×Q8—24no—
92C2×C4×Q8—24no—
93C2×C4×Q8—24no—
94C2×C4×D4—28no—
95C2×C4×D4—28no—
96C2×C4×D4—24no—
97C2×C4×D4—24no—
98C2×C4×D4—24no—
99C2×C4×D4—24no—
100C2×C4×D4—24no—
101C2×C4×D4—44no—
102C2×C4×D4D418yes—
103C22×C4⋊C4—28no—
104C22×C4⋊C4—48no—
105C22×C4⋊C4C2312yes—
106C22×C4⋊C4C2311yescharacteristic
107C22×C4⋊C4C2311yescharacteristic
108C22×C4⋊C4C2311yescharacteristic
109C22×C4⋊C4C2312yes—
110C22×C4⋊C4C2314yes—
111C22×C4⋊C4C2314yes—
112C22×C4⋊C4C2312yes—
113C22×C4⋊C4C2311yescharacteristic
114C22×C4⋊C4C2314yes—
115C22×C4⋊C4C2311yescharacteristic
116C22×C4⋊C4C2314yes—
117C22×C4⋊C4C2312yes—
118C22×C4⋊C4C2312yes—
119C22×C22⋊C4C2312yes—
120C22×C22⋊C4C2314yes—
121C22×C22⋊C4C2314yes—
122C22×C22⋊C4C2314yes—
123C22×C22⋊C4C2312yes—
124C22×C22⋊C4C2314yes—
125C22×C22⋊C4C2314yes—
126C22×C22⋊C4C2314yes—
127C22×C22⋊C4C2314yes—
128C22×C22⋊C4C2314yes—
129C22×C22⋊C4C2314yes—
130C22×C42C2312yes—
131C22×C42C2312yes—
132C22×C42C2311yescharacteristic
133C22×C42C2311yescharacteristic
134C22×C42C2311yescharacteristic
135#399—28no—
136#396—216no—
137#396—48no—
138#393—28no—
139#390—28no—
140#390—24no—
141#390—24no—
142#390—28no—
143#390—44no—
144#389—28no—
145#389—28no—
146#389—28no—
147#389—44no—
148#386—216no—
149#386—216no—
150#386—48no—
151#386—48no—
152#385—216no—
153#385—216no—
154#384D418yescomplemented
155#382—24no—
156#382—24no—
157#380—28no—
158#380—28no—
159#380—48no—
160C4×C4⋊C4—24no—
161#376—24no—
162C2×#95C2312yes—
163C2×#95C2314yes—
164C2×#95C2312yes—
165C2×#95C2314yes—
166C2×#95C2312yes—
167C2×#95C2312yes—
168C2×#95C2312yes—
169C2×#95C2312yes—
170C2×#95C2312yes—
171C2×#95C2314yes—
172C2×#95C2312yes—
173C2×#95C2314yes—
174C2×#95C2312yes—
175C2×#95C2312yes—
176C2×#95C2312yes—
177C2×#95C2312yes—
178C2×#95C2312yes—
179C2×#95C2314yes—
180C2×#95C2314yes—
181C2×#95C2314yes—
182C2×#95C2314yes—
183C2×#95C2314yes—
184C25C2412yes—
185C25C2412yes—
186C22×Q8—22no—
187C22×Q8—22no—
188C22×Q8—22no—
189C22×Q8—24no—
190C22×Q8—42no—
191C22×D4—28no—
192C22×D4—28no—
193C22×D4—22no—
194C22×D4—22no—
195C22×D4—22no—
196C22×D4—22no—
197C22×D4—22no—
198C22×D4—22no—
199C22×D4—22no—
200C22×D4—22no—
201C22×D4—22no—
202C22×D4—22no—
203C22×D4—44no—
204C22×D4—44no—
205C22×D4—44no—
206C22×D4—44no—
207C22×D4C2×D414yes—
208C22×D4C2×Q814yes—
209C22×D4C2×Q814yes—
210C22×D4C2×Q8116yes—
211C22×D4#4114yes—
212C23×C4—24no—
213C23×C4—24no—
214C23×C4—28no—
215C23×C4—24no—
216C23×C4—28no—
217C23×C4—28no—
218C23×C4—24no—
219C23×C4—24no—
220C23×C4—44no—
221C23×C4—44no—
222C23×C4—44no—
223C23×C4—44no—
224C23×C4—44no—
225C23×C4—44no—
226C23×C4—44no—
227C23×C4—44no—
228C23×C4C2×D418yes—
229C23×C4C2412yes—
230C23×C4C2414yes—
231C23×C4C2412yes—
232C23×C4C2412yes—
233C23×C4C2412yes—
234C23×C4C2414yes—
235C23×C4C2412yes—
236C23×C4C2412yes—
237C23×C4C2412yes—
238C23×C4C2412yes—
239C23×C4C2411yescharacteristic
240C23×C4C2411yescharacteristic
241C23×C4C2411yescharacteristic
242#127—232no—
243C22⋊Q8—432no—
244C4×Q8—416no—
245C4×D4—232no—
246C4×D4—432no—
247C4×D4—416no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—24no—
254C2×C4⋊C4—28no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—22no—
257C2×C4⋊C4—22no—
258C2×C4⋊C4—24no—
259C2×C4⋊C4—24no—
260C2×C4⋊C4—22no—
261C2×C4⋊C4—28no—
262C2×C4⋊C4—22no—
263C2×C4⋊C4—24no—
264C2×C4⋊C4—22no—
265C2×C4⋊C4—24no—
266C2×C4⋊C4—22no—
267C2×C4⋊C4—28no—
268C2×C4⋊C4—28no—
269C2×C4⋊C4—24no—
270C2×C4⋊C4—24no—
271C2×C4⋊C4—28no—
272C2×C4⋊C4—432no—
273C2×C4⋊C4—44no—
274C2×C4⋊C4—44no—
275C2×C4⋊C4—42no—
276C2×C4⋊C4—44no—
277C2×C4⋊C4—44no—
278C2×C4⋊C4—44no—
279C2×C4⋊C4—44no—
280C2×C4⋊C4—44no—
281C2×C4⋊C4—44no—
282C2×C4⋊C4—44no—
283C2×C4⋊C4C2×D418yes—
284C2×C4⋊C4C2×D414yes—
285C2×C4⋊C4C2×D418yes—
286C2×C4⋊C4C2×D418yes—
287C2×C22⋊C4—24no—
288C2×C22⋊C4—24no—
289C2×C22⋊C4—28no—
290C2×C22⋊C4—28no—
291C2×C22⋊C4—28no—
292C2×C22⋊C4—24no—
293C2×C22⋊C4—24no—
294C2×C22⋊C4—28no—
295C2×C22⋊C4—24no—
296C2×C22⋊C4—28no—
297C2×C22⋊C4—28no—
298C2×C22⋊C4—28no—
299C2×C22⋊C4—28no—
300C2×C22⋊C4—28no—
301C2×C22⋊C4—28no—
302C2×C22⋊C4—28no—
303C2×C22⋊C4—44no—
304C2×C22⋊C4—44no—
305C2×C22⋊C4—44no—
306C2×C22⋊C4—44no—
307C2×C22⋊C4—44no—
308C2×C22⋊C4—44no—
309C2×C22⋊C4—44no—
310C2×C22⋊C4—44no—
311C2×C22⋊C4—44no—
312C2×C22⋊C4—44no—
313C2×C22⋊C4—44no—
314C2×C22⋊C4—44no—
315C2×C22⋊C4—44no—
316C2×C22⋊C4—44no—
317C2×C22⋊C4—44no—
318C2×C22⋊C4—44no—
319C2×C22⋊C4C2×D418yes—
320C2×C42—22no—
321C2×C42—22no—
322C2×C42—24no—
323C2×C42—24no—
324C2×C42—24no—
325C2×C42—22no—
326C2×C42—24no—
327C2×C42—24no—
328C2×C42—24no—
329C2×C42—22no—
330C2×C42—22no—
331C2×C42C2×D414yes—
332C2×C42C2×D414yes—
333C2×C42C2×Q814yes—
334C2×C42#4114yes—
335#95—24no—
336#95—24no—
337#95—24no—
338#95—28no—
339#95—28no—
340#95—24no—
341#95—28no—
342#95—24no—
343#95—28no—
344#95—24no—
345#95—24no—
346#95—24no—
347#95—24no—
348#95—42no—
349#95—42no—
350#95—42no—
351#95—44no—
352#95—42no—
353#95—44no—
354#95—44no—
355#95—44no—
356#95—44no—
357C24—22no—
358C24—22no—
359C24—24no—
360C24—22no—
361C24—24no—
362C24—22no—
363C24—24no—
364C24—24no—
365C24—22no—
366C24—22no—
367C24—22no—
368C24—22no—
369C24—24no—
370C24—24no—
371C24—22no—
372C24—22no—
373C24—24no—
374C24—24no—
375C24C22×D414yes—
376C24C22×Q814yes—
377C24C22×Q814yes—
378C24C2×#4114yes—
379C24C2511yescharacteristic
380C2×Q8—28no—
381C2×Q8—48no—
382C2×Q8—48no—
383C2×Q8—44no—
384C2×Q8—44no—
385C2×D4—232no—
386C2×D4—28no—
387C2×D4—28no—
388C2×D4—28no—
389C2×D4—28no—
390C2×D4—28no—
391C2×D4—28no—
392C2×D4—28no—
393C2×D4—28no—
394C2×D4—28no—
395C2×D4—416no—
396C2×D4—416no—
397C2×D4—44no—
398C2×D4—44no—
399C2×D4—44no—
400C2×D4—44no—
401C2×D4—88no—
402C2×D4—88no—
403C2×D4—88no—
404C2×D4—88no—
405C2×D4C22⋊Q8116yescomplemented
406C22×C4—22no—
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—22no—
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—24no—
426C22×C4—24no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—24no—
430C22×C4—24no—
431C22×C4—24no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—24no—
438C22×C4—216no—
439C22×C4—22no—
440C22×C4—24no—
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—22no—
453C22×C4—22no—
454C22×C4—24no—
455C22×C4—24no—
456C22×C4—24no—
457C22×C4—24no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—24no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—24no—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—24no—
469C22×C4—22no—
470C22×C4—22no—
471C22×C4—24no—
472C22×C4—416no—
473C22×C4—416no—
474C22×C4—48no—
475C22×C4—416no—
476C22×C4—48no—
477C22×C4—416no—
478C22×C4—416no—
479C22×C4—48no—
480C22×C4—48no—
481C22×C4—48no—
482C22×C4—88no—
483C22×C4—88no—
484C22×C4—88no—
485C22×C4—88no—
486C22×C4—88no—
487C22×C4—88no—
488C22×C4C22×D414yes—
489C22×C4C22×D418yes—
490C22×C4C22×D412yes—
491C22×C4C22×D412yes—
492C22×C4C22×D412yes—
493C22×C4C22×D412yes—
494C22×C4C22×D412yes—
495C22×C4C22×D412yes—
496C22×C4C22×Q814yes—
497C22×C4C22×Q812yes—
498C22×C4C2×#4112yes—
499C22×C4C2×#4112yes—
500C22×C4C2×#4112yes—
501C22×C4C2×#4112yes—
502C22×C4H(5,𝔽2)14yes—
503C22×C4H(5,𝔽2)14yes—
504C22×C4H(5,𝔽2)12yes—
505C22×C42−1+412yes—
506C4⋊C4—28no—
507C4⋊C4—28no—
508C4⋊C4—216no—
509C4⋊C4—48no—
510C4⋊C4—48no—
511C4⋊C4—416no—
512C4⋊C4—416no—
513C4⋊C4—44no—
514C4⋊C4—44no—
515C4⋊C4—48no—
516C4⋊C4—44no—
517C4⋊C4—48no—
518C4⋊C4—88no—
519C4⋊C4—88no—
520C22⋊C4—216no—
521C22⋊C4—416no—
522C22⋊C4—416no—
523C22⋊C4—416no—
524C22⋊C4—48no—
525C22⋊C4—88no—
526C22⋊C4—88no—
527C22⋊C4—88no—
528C22⋊C4—88no—
529C22⋊C4—88no—
530C22⋊C4—88no—
531C42—28no—
532C42—48no—
533C42—44no—
534C42—48no—
535C42#121116yes—
536C23—24no—
537C23—24no—
538C23—24no—
539C23—24no—
540C23—24no—
541C23—24no—
542C23—28no—
543C23—28no—
544C23—28no—
545C23—28no—
546C23—28no—
547C23—28no—
548C23—24no—
549C23—24no—
550C23—24no—
551C23—28no—
552C23—28no—
553C23—44no—
554C23—42no—
555C23—42no—
556C23—42no—
557C23—42no—
558C23—44no—
559C23—44no—
560C23—44no—
561C23—44no—
562C23—44no—
563C23—42no—
564C23—42no—
565C23—44no—
566C23—42no—
567C23—42no—
568C23—42no—
569C23—42no—
570C23—42no—
571C23—42no—
572C23—44no—
573C23—44no—
574C23—44no—
575C23—44no—
576C23—44no—
577C23—44no—
578C23—44no—
579C23—44no—
580C23—44no—
581C23C2×C22⋊Q818yes—
582C23C23×D411yescharacteristic
583C23C23×D411yescharacteristic
584C23C23×Q811yescharacteristic
585C23C22×#4111yescharacteristic
586C23C22×#4111yescharacteristic
587C23C2×H(5,𝔽2)11yescharacteristic
588C23C2×H(5,𝔽2)12yes—
589C23C2×H(5,𝔽2)12yes—
590C23C2×2−1+411yescharacteristic
591C23#58512yes—
592C23#58512yes—
593Q8—416no—
594D4—232no—
595D4—416no—
596D4—832no—
597C2×C4—24no—
598C2×C4—22no—
599C2×C4—22no—
600C2×C4—22no—
601C2×C4—22no—
602C2×C4—24no—
603C2×C4—24no—
604C2×C4—24no—
605C2×C4—24no—
606C2×C4—24no—
607C2×C4—28no—
608C2×C4—24no—
609C2×C4—28no—
610C2×C4—28no—
611C2×C4—24no—
612C2×C4—24no—
613C2×C4—24no—
614C2×C4—28no—
615C2×C4—24no—
616C2×C4—24no—
617C2×C4—24no—
618C2×C4—24no—
619C2×C4—432no—
620C2×C4—42no—
621C2×C4—44no—
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—44no—
632C2×C4—44no—
633C2×C4—42no—
634C2×C4—42no—
635C2×C4—42no—
636C2×C4—44no—
637C2×C4—44no—
638C2×C4—42no—
639C2×C4—42no—
640C2×C4—42no—
641C2×C4—42no—
642C2×C4—44no—
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—42no—
652C2×C4—44no—
653C2×C4—42no—
654C2×C4—44no—
655C2×C4—44no—
656C2×C4—42no—
657C2×C4—42no—
658C2×C4—44no—
659C2×C4—42no—
660C2×C4—42no—
661C2×C4—42no—
662C2×C4—44no—
663C2×C4—42no—
664C2×C4—42no—
665C2×C4—42no—
666C2×C4—44no—
667C2×C4—44no—
668C2×C4—42no—
669C2×C4—44no—
670C2×C4—42no—
671C2×C4—42no—
672C2×C4—816no—
673C2×C4—832no—
674C2×C4C2×#12114yes—
675C2×C4C2×#12114yes—
676C2×C4C2×#12114yes—
677C2×C4C2×C22⋊Q814yes—
678C2×C4#53714yes—
679C2×C4#54614yes—
680C2×C4D4×Q814yes—
681C2×C4#55214yes—
682C2×C4#55414yes—
683C22—28no—
684C22—216no—
685C22—44no—
686C22—44no—
687C22—48no—
688C22—44no—
689C22—44no—
690C22—48no—
691C22—44no—
692C22—44no—
693C22—48no—
694C22—44no—
695C22—44no—
696C22—44no—
697C22—48no—
698C22—48no—
699C22—84no—
700C22—82no—
701C22—82no—
702C22—84no—
703C22—82no—
704C22—82no—
705C22C22×#12111yescharacteristic
706C22C22×C22⋊Q811yescharacteristic
707C22C2×#53711yescharacteristic
708C22#346012yes—
709C22C2×#54611yescharacteristic
710C22C2×D4×Q811yescharacteristic
711C22C2×#55211yescharacteristic
712C22C2×#55411yescharacteristic
713C22#348411yescharacteristic
714C22#348411yescharacteristic
715C22#348711yescharacteristic
716C22#348711yescharacteristic
717C22#348812yes—
718C22#349412yes—
719C22#349612yes—
720C22#350112yes—
721C22#350112yes—
722C22#351912yes—
723C22#352112yes—
724C22#354312yes—
725C22#355412yes—
726C22#355412yes—
727C22#356212yes—
728C4—24no—
729C4—24no—
730C4—44no—
731C4—48no—
732C4—44no—
733C4—44no—
734C4—44no—
735C4—82no—
736C4—82no—
737C4—84no—
738C4—82no—
739C4—82no—
740C4#241718yes—
741C2—48no—
742C2—84no—
743C2C2×#241711yescharacteristic
744C2#3516811yescharacteristic
745C2#3556311yescharacteristic
746C2#3577612yes—
747C2#3822612yes—
748C2#3967212yes—
749C2#4244412yes—
750C2#4493611yescharacteristic
751C2#4526911yescharacteristic
752C2#4535711yescharacteristic
753C2#4535711yescharacteristic
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC22×#127C22
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×#2538C2semidirect
2. C2×#2580C2semidirect
3. C2×D4×C4⋊C4C2semidirect
5. #16562C2semidirect
6. #15439C2semidirect
7. #15323C2semidirect
8. #15225C2semidirect
9. #15141C2semidirect
10. #14729C2semidirect
11. #14276C2semidirect
13. #13941C2semidirect
14. #13920C2semidirect
15. #13850C2semidirect
88. C2×#127Q8semidirect
154. #384D4semidirect
405. C2×D4C22⋊Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#25382
C2×#25802
C2×D4×C4⋊C42
C2×#23282
#165622
#154392
#153232
#152252
#151412
#147292
#142762
#140252
#139412
#139202
#138502