Finite Group Databasebeta

← #726119

Subgroups — #726119

Order 512.

2554 conjugacy classes of subgroups.

The subgroup lattice has 647 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2552C211yesmaximal, characteristic, complemented
2C2×#2354C211yesmaximal, characteristic
3#16521C214yesmaximal, complemented
4#15846C214yesmaximal, complemented
5#15425C214yesmaximal, complemented
6#15148C212yesmaximal
7#15134C212yesmaximal
8#14891C214yesmaximal, complemented
9#14276C211yesmaximal, characteristic, complemented
10#14150C211yesmaximal, characteristic, complemented
11#14045C212yesmaximal, complemented
12#13941C211yesmaximal, characteristic
13#13884C212yesmaximal, complemented
14#13856C211yesmaximal, characteristic, complemented
15#13785C211yesmaximal, characteristic
16C22×#125C2212yes—
17C22×C4×D4C2211yescharacteristic
18C22×#117C2212yes—
19#2552—28no—
20C2×#399C2214yes—
21C2×#398C2214yes—
22C2×#396C2214yes—
23C2×#396C2214yes—
24#2354—28no—
25C2×#390C2211yescharacteristic
26C2×#390C2212yes—
27C2×#390C2211yescharacteristic
28C2×#390C2212yes—
29C2×#389C2212yes—
30C2×#389C2212yes—
31C2×#389C2212yes—
32C2×#389C2212yes—
33C2×#388C2214yes—
34C2×#387C2214yes—
35C2×#384C2212yes—
36C2×#384C2211yescharacteristic
37C2×#382C2211yescharacteristic
38C2×#380C2212yes—
39C2×C4×C4⋊C4C2212yes—
40C2×C4×C4⋊C4C2211yescharacteristic
41C2×C4×C4⋊C4C2211yescharacteristic
42C2×C4×C4⋊C4C2212yes—
43C2×C4×C4⋊C4C2211yescharacteristic
44C2×C4×C4⋊C4C2212yes—
45C2×#376C2211yescharacteristic
46#1444C2212yes—
47#1444C2212yes—
48#1444C2212yes—
49#1444C2212yes—
50#1444C2212yes—
51#1444C2214yes—
52#1442C2212yes—
53#1442C2214yes—
54#1441C2214yes—
55#1441C2214yes—
56#1441C2214yes—
57#1441C2214yes—
58#1439C2214yes—
59#1439C2214yes—
60#1439C2214yes—
61#1439C2214yes—
62#1439C2214yes—
63#1439C2212yes—
64#1439C2212yes—
65#1437C2214yes—
66#1436C2214yes—
67#1436C2214yes—
68#1436C2214yes—
69C4⋊#95C2212yes—
70C4⋊#95C2214yes—
71C4⋊#95C2212yes—
72C4⋊#95C2211yescharacteristic
73C4⋊#95C2212yes—
74C4⋊#95C2211yescharacteristic
75C22⋊#95C2214yes—
76C22⋊#95C2214yes—
77C23×D4C2311yescharacteristic
78C24×C4C2312yes—
79C2×#125—28no—
80C2×#125—48no—
81C2×C4×D4—24no—
82C2×C4×D4—24no—
83C2×C4×D4Q818yes—
84C2×#117—28no—
85C2×#117—48no—
86C22×C4⋊C4C2311yescharacteristic
87C22×C4⋊C4C2311yescharacteristic
88C22×C4⋊C4C2311yescharacteristic
89C22×C4⋊C4C2312yes—
90C22×C4⋊C4C2311yescharacteristic
91C22×C4⋊C4C2312yes—
92C22×C4⋊C4C2311yescharacteristic
93C22×C4⋊C4C2312yes—
94C22×C4⋊C4C2314yes—
95C22×C4⋊C4C2314yes—
96C22×C4⋊C4C2312yes—
97C22×C4⋊C4C2312yes—
98C22×C4⋊C4C2311yescharacteristic
99C22×C4⋊C4C2314yes—
100C22×C4⋊C4C2312yes—
101C22×C4⋊C4C2314yes—
102C22×C4⋊C4C2311yescharacteristic
103C22×C4⋊C4C2314yes—
104C22×C4⋊C4C2312yes—
105C22×C4⋊C4C2312yes—
106C22×C22⋊C4C2314yes—
107C22×C22⋊C4C2312yes—
108C22×C22⋊C4C2314yes—
109C22×C22⋊C4C2314yes—
110C22×C22⋊C4C2314yes—
111C22×C22⋊C4C2314yes—
112C22×C22⋊C4C2314yes—
113C22×C42C2312yes—
114C22×C42C2312yes—
115C22×C42C2314yes—
116C22×C42C2311yescharacteristic
117C22×C42C2311yescharacteristic
118C22×C42C2312yes—
119C22×C42C2311yescharacteristic
120#399—48no—
121#398—216no—
122#396—48no—
123#396—48no—
124#390—28no—
125#390—24no—
126#390—24no—
127#390—44no—
128#389—28no—
129#389—28no—
130#389—28no—
131#389—44no—
132#388—216no—
133#387—48no—
134#384—28no—
135#384D418yes—
136#382—24no—
137#380—28no—
138C4×C4⋊C4—28no—
139C4×C4⋊C4—24no—
140C4×C4⋊C4—24no—
141C4×C4⋊C4—24no—
142C4×C4⋊C4—28no—
143C4×C4⋊C4—44no—
144#376—24no—
145C2×#95C2314yes—
146C2×#95C2312yes—
147C2×#95C2312yes—
148C2×#95C2314yes—
149C2×#95C2312yes—
150C2×#95C2312yes—
151C2×#95C2312yes—
152C2×#95C2312yes—
153C2×#95C2312yes—
154C2×#95C2314yes—
155C2×#95C2312yes—
156C2×#95C2312yes—
157C2×#95C2314yes—
158C2×#95C2314yes—
159C2×#95C2314yes—
160C2×#95C2314yes—
161C2×#95C2314yes—
162C2×#95C2314yes—
163C2×#95C2314yes—
164C2×#95C2312yes—
165C2×#95C2314yes—
166C2×#95C2314yes—
167C2×#95C2312yes—
168C25C2412yes—
169C22×D4—22no—
170C22×D4—22no—
171C22×D4—22no—
172C22×D4—22no—
173C22×D4—22no—
174C22×D4—22no—
175C22×D4C2×Q814yes—
176C23×C4—24no—
177C23×C4—24no—
178C23×C4—44no—
179C23×C4—44no—
180C23×C4C2×Q818yes—
181C23×C4C2412yes—
182C23×C4C2412yes—
183C23×C4C2412yes—
184C23×C4C2414yes—
185C23×C4C2414yes—
186C23×C4C2412yes—
187C23×C4C2412yes—
188C23×C4C2412yes—
189C23×C4C2412yes—
190C23×C4C2412yes—
191C23×C4C2412yes—
192C23×C4C2411yescharacteristic
193C23×C4C2411yescharacteristic
194C23×C4C2411yescharacteristic
195#125—432no—
196C4×D4—232no—
197#117—432no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—24no—
204C2×C4⋊C4—24no—
205C2×C4⋊C4—24no—
206C2×C4⋊C4—24no—
207C2×C4⋊C4—24no—
208C2×C4⋊C4—24no—
209C2×C4⋊C4—24no—
210C2×C4⋊C4—24no—
211C2×C4⋊C4—28no—
212C2×C4⋊C4—28no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—24no—
217C2×C4⋊C4—24no—
218C2×C4⋊C4—24no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—28no—
222C2×C4⋊C4—28no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—24no—
225C2×C4⋊C4—24no—
226C2×C4⋊C4—24no—
227C2×C4⋊C4—24no—
228C2×C4⋊C4—24no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—28no—
231C2×C4⋊C4—28no—
232C2×C4⋊C4—42no—
233C2×C4⋊C4—42no—
234C2×C4⋊C4—42no—
235C2×C4⋊C4—44no—
236C2×C4⋊C4—44no—
237C2×C4⋊C4—44no—
238C2×C4⋊C4—44no—
239C2×C4⋊C4—42no—
240C2×C4⋊C4—42no—
241C2×C4⋊C4—44no—
242C2×C4⋊C4—44no—
243C2×C4⋊C4—44no—
244C2×C4⋊C4—44no—
245C2×C4⋊C4—44no—
246C2×C4⋊C4—44no—
247C2×C4⋊C4—44no—
248C2×C4⋊C4C2×D418yes—
249C2×C4⋊C4C2×D418yes—
250C2×C4⋊C4C2×D418yes—
251C2×C4⋊C4C2×Q814yes—
252C2×C22⋊C4—24no—
253C2×C22⋊C4—24no—
254C2×C22⋊C4—28no—
255C2×C22⋊C4—28no—
256C2×C22⋊C4—28no—
257C2×C22⋊C4—28no—
258C2×C22⋊C4—28no—
259C2×C22⋊C4—28no—
260C2×C22⋊C4—28no—
261C2×C22⋊C4—44no—
262C2×C22⋊C4—44no—
263C2×C22⋊C4—44no—
264C2×C22⋊C4—44no—
265C2×C22⋊C4—44no—
266C2×C22⋊C4—44no—
267C2×C22⋊C4—44no—
268C2×C22⋊C4—44no—
269C2×C22⋊C4—44no—
270C2×C22⋊C4—44no—
271C2×C22⋊C4—44no—
272C2×C22⋊C4C2×Q818yes—
273C2×C42—22no—
274C2×C42—22no—
275C2×C42—22no—
276C2×C42—22no—
277C2×C42—28no—
278C2×C42—24no—
279C2×C42—24no—
280C2×C42—24no—
281C2×C42—24no—
282C2×C42—24no—
283C2×C42—28no—
284C2×C42—24no—
285C2×C42—24no—
286C2×C42—44no—
287C2×C42—44no—
288C2×C42C2×D414yes—
289C2×C42C2×Q814yes—
290C2×C42#4114yes—
291C2×C42#4114yes—
292C2×C42#4114yes—
293#95—28no—
294#95—24no—
295#95—24no—
296#95—24no—
297#95—28no—
298#95—24no—
299#95—24no—
300#95—24no—
301#95—24no—
302#95—24no—
303#95—24no—
304#95—28no—
305#95—28no—
306#95—44no—
307#95—44no—
308#95—42no—
309#95—42no—
310#95—44no—
311#95—44no—
312#95—44no—
313#95—44no—
314#95—44no—
315#95—44no—
316C24—22no—
317C24—22no—
318C24—22no—
319C24—22no—
320C24—24no—
321C24—24no—
322C24—22no—
323C24—24no—
324C24—22no—
325C24—24no—
326C24C22×Q814yes—
327C24C2511yescharacteristic
328C2×D4—28no—
329C2×D4—28no—
330C2×D4—28no—
331C2×D4—44no—
332C2×D4—44no—
333C2×D4—44no—
334C2×D4—44no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—24no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—24no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—24no—
359C22×C4—24no—
360C22×C4—22no—
361C22×C4—24no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—24no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—24no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—24no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—216no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—24no—
384C22×C4—22no—
385C22×C4—24no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—24no—
389C22×C4—24no—
390C22×C4—24no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—24no—
395C22×C4—24no—
396C22×C4—24no—
397C22×C4—22no—
398C22×C4—24no—
399C22×C4—22no—
400C22×C4—24no—
401C22×C4—24no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—416no—
405C22×C4—48no—
406C22×C4—416no—
407C22×C4—48no—
408C22×C4C22×D418yes—
409C22×C4C22×D412yes—
410C22×C4C22×D412yes—
411C22×C4C22×D412yes—
412C22×C4C22×Q814yes—
413C22×C4C22×Q812yes—
414C22×C4C22×Q812yes—
415C22×C4C22×Q812yes—
416C22×C4C2×#4114yes—
417C22×C4C2×#4114yes—
418C22×C4C2×#4112yes—
419C22×C4C2×#4112yes—
420C22×C4C2×#4112yes—
421C22×C4C2×#4112yes—
422C22×C4C2×#4112yes—
423C22×C4H(5,𝔽2)14yes—
424C22×C4H(5,𝔽2)14yes—
425C22×C4H(5,𝔽2)12yes—
426C22×C42−1+414yes—
427C22×C42−1+412yes—
428C4⋊C4—28no—
429C4⋊C4—216no—
430C4⋊C4—28no—
431C4⋊C4—28no—
432C4⋊C4—28no—
433C4⋊C4—216no—
434C4⋊C4—48no—
435C4⋊C4—48no—
436C4⋊C4—48no—
437C4⋊C4—48no—
438C4⋊C4—416no—
439C4⋊C4—416no—
440C4⋊C4—44no—
441C4⋊C4—48no—
442C4⋊C4—44no—
443C4⋊C4—44no—
444C4⋊C4—48no—
445C4⋊C4—88no—
446C4⋊C4—88no—
447C4⋊C4—88no—
448C22⋊C4—216no—
449C22⋊C4—416no—
450C22⋊C4—88no—
451C22⋊C4—88no—
452C22⋊C4—88no—
453C22⋊C4—88no—
454C22⋊C4—88no—
455C42—28no—
456C42—28no—
457C42—416no—
458C42—48no—
459C42—48no—
460C42—48no—
461C42C22⋊Q8116yes—
462C23—24no—
463C23—24no—
464C23—24no—
465C23—28no—
466C23—28no—
467C23—44no—
468C23—42no—
469C23—42no—
470C23—42no—
471C23—42no—
472C23—42no—
473C23—42no—
474C23—42no—
475C23—42no—
476C23—44no—
477C23—44no—
478C23—44no—
479C23—44no—
480C23—44no—
481C23—44no—
482C23—44no—
483C23—44no—
484C23—44no—
485C23C23×D411yescharacteristic
486C23C23×Q811yescharacteristic
487C23C22×#4111yescharacteristic
488C23C22×#4111yescharacteristic
489C23C22×#4111yescharacteristic
490C23C2×H(5,𝔽2)11yescharacteristic
491C23C2×H(5,𝔽2)12yes—
492C23C2×H(5,𝔽2)12yes—
493C23C2×2−1+411yescharacteristic
494C23C2×2−1+412yes—
495C23#58512yes—
496D4—416no—
497C2×C4—24no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—24no—
503C2×C4—24no—
504C2×C4—24no—
505C2×C4—24no—
506C2×C4—28no—
507C2×C4—24no—
508C2×C4—24no—
509C2×C4—24no—
510C2×C4—28no—
511C2×C4—28no—
512C2×C4—24no—
513C2×C4—24no—
514C2×C4—24no—
515C2×C4—24no—
516C2×C4—24no—
517C2×C4—24no—
518C2×C4—24no—
519C2×C4—28no—
520C2×C4—24no—
521C2×C4—24no—
522C2×C4—24no—
523C2×C4—24no—
524C2×C4—24no—
525C2×C4—432no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—44no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—44no—
541C2×C4—44no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—44no—
551C2×C4—44no—
552C2×C4—42no—
553C2×C4—44no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—44no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—44no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—44no—
568C2×C4—44no—
569C2×C4—44no—
570C2×C4—44no—
571C2×C4—44no—
572C2×C4—44no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—44no—
576C2×C4—44no—
577C2×C4C2×C22⋊Q814yes—
578C2×C4C2×C22⋊Q814yes—
579C2×C4C2×C22⋊Q814yes—
580C2×C4#53314yes—
581C2×C4#54414yes—
582C2×C4#54714yes—
583C2×C4#55014yes—
584C2×C4#55214yes—
585C2×C4#55314yes—
586C22—48no—
587C22—44no—
588C22—44no—
589C22—44no—
590C22—48no—
591C22—82no—
592C22—82no—
593C22—84no—
594C22—82no—
595C22—82no—
596C22—84no—
597C22C22×C22⋊Q811yescharacteristic
598C22C2×#53311yescharacteristic
599C22C2×#54411yescharacteristic
600C22#346012yes—
601C22C2×#54711yescharacteristic
602C22C2×#55011yescharacteristic
603C22C2×#55211yescharacteristic
604C22C2×#55311yescharacteristic
605C22#348411yescharacteristic
606C22#348411yescharacteristic
607C22#348611yescharacteristic
608C22#348711yescharacteristic
609C22#349212yes—
610C22#349412yes—
611C22#350112yes—
612C22#350312yes—
613C22#350712yes—
614C22#350812yes—
615C22#351112yes—
616C22#353612yes—
617C22#354212yes—
618C22#354812yes—
619C22#355512yes—
620C4—24no—
621C4—24no—
622C4—44no—
623C4—44no—
624C4—44no—
625C4—44no—
626C4—48no—
627C4—44no—
628C4—44no—
629C4—82no—
630C4—82no—
631C4—82no—
632C4—84no—
633C4#244118yes—
634C2—84no—
635C2C2×#244111yescharacteristic
636C2#3517411yescharacteristic
637C2#3557411yescharacteristic
638C2#3593412yes—
639C2#3798312yes—
640C2#3981712yes—
641C2#4249412yes—
642C2#4493111yescharacteristic
643C2#4493311yescharacteristic
644C2#4535611yescharacteristic
645C2#4539411yescharacteristic
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC23×D4C23
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×#2552C2semidirect
3. #16521C2semidirect
4. #15846C2semidirect
5. #15425C2semidirect
8. #14891C2semidirect
9. #14276C2semidirect
10. #14150C2semidirect
11. #14045C2semidirect
13. #13884C2semidirect
14. #13856C2semidirect

Maximal subgroups

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

subgroupindex
C2×#25522
C2×#23542
#165212
#158462
#154252
#151482
#151342
#148912
#142762
#141502
#140452
#139412
#138842
#138562
#137852