Finite Group Databasebeta

← #636437

Subgroups — #636437

Order 512.

3218 conjugacy classes of subgroups.

The subgroup lattice has 640 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2546C211yesmaximal, characteristic, complemented
2C2×#2542C211yesmaximal, characteristic
3C2×#2538C211yesmaximal, characteristic
4C2×#2621C212yesmaximal, complemented
5C2×#2293C211yesmaximal, characteristic
6#16469C218yesmaximal, complemented
7#15236C212yesmaximal, complemented
8#15178C212yesmaximal, complemented
9#15138C214yesmaximal
10#15137C214yesmaximal, complemented
11#14714C211yesmaximal, characteristic, complemented
12#14258C212yesmaximal, complemented
13#13911C211yesmaximal, characteristic, complemented
14C4×#391C211yesmaximal, characteristic, complemented
15C22×C4⋊Q8C2211yescharacteristic
16C22×C4⋊Q8C2212yes—
17C22×C22⋊Q8C2212yes—
18C22×C22⋊Q8C2212yes—
19C22×C22⋊Q8C2212yes—
20C22×C4×Q8C2211yescharacteristic
21C22×C4×D4C2211yescharacteristic
22C22×#117C2212yes—
23#2621—216no—
24#2546—28no—
25#2542—28no—
26#2538—28no—
27C2×#399C2214yes—
28C2×#398C2218yes—
29C2×#398C2214yes—
30C2×#397C2218yes—
31C2×#396C2214yes—
32C2×#391C2212yes—
33C2×#391C2211yescharacteristic
34C2×#391C2212yes—
35C2×#391C2212yes—
36C2×#391C2211yescharacteristic
37#2293—28no—
38C2×#390C2212yes—
39C2×#390C2212yes—
40C2×#390C2212yes—
41C2×#389C2212yes—
42C2×#389C2212yes—
43C2×#389C2214yes—
44C2×#388C2214yes—
45C2×#384C2212yes—
46C2×#384C2211yescharacteristic
47C2×#382C2212yes—
48C2×#380C2212yes—
49C2×C4×C4⋊C4C2211yescharacteristic
50C2×C4×C4⋊C4C2211yescharacteristic
51C2×C4×C22⋊C4C2212yes—
52C2×C43C2211yescharacteristic
53#1444C2218yes—
54#1442C2218yes—
55#1442C2214yes—
56#1441C2218yes—
57#1440C2218yes—
58#1440C2218yes—
59#1439C2218yes—
60#1436C2214yes—
61C4⋊#95C2214yes—
62C4⋊#95C2214yes—
63C4⋊#95C2212yes—
64C4⋊#95C2214yes—
65C4⋊#95C2211yescharacteristic
66C4⋊#95C2211yescharacteristic
67C4×#95C2214yes—
68C23×Q8C2312yes—
69C23×Q8C2311yescharacteristic
70C23×D4C2311yescharacteristic
71C24×C4C2312yes—
72C2×C4⋊Q8—28no—
73C2×C4⋊Q8—24no—
74C2×C4⋊Q8—28no—
75C2×C4⋊Q8—24no—
76C2×C4⋊Q8—24no—
77C2×C4⋊Q8—44no—
78C2×C22⋊Q8—28no—
79C2×C22⋊Q8—28no—
80C2×C22⋊Q8—28no—
81C2×C22⋊Q8—28no—
82C2×C22⋊Q8—28no—
83C2×C22⋊Q8—28no—
84C2×C22⋊Q8—44no—
85C2×C22⋊Q8—44no—
86C2×C22⋊Q8—44no—
87C2×C4×Q8—24no—
88C2×C4×Q8—24no—
89C2×C4×Q8—24no—
90C2×C4×D4—24no—
91C2×C4×D4—24no—
92C2×C4×D4D418yes—
93C2×#117—28no—
94C2×#117—28no—
95C2×#117—44no—
96C22×C4⋊C4C2312yes—
97C22×C4⋊C4C2312yes—
98C22×C4⋊C4C2311yescharacteristic
99C22×C4⋊C4C2312yes—
100C22×C4⋊C4C2312yes—
101C22×C4⋊C4C2312yes—
102C22×C4⋊C4C2314yes—
103C22×C4⋊C4C2318yes—
104C22×C4⋊C4C2314yes—
105C22×C4⋊C4C2314yes—
106C22×C4⋊C4C2314yes—
107C22×C4⋊C4C2311yescharacteristic
108C22×C4⋊C4C2311yescharacteristic
109C22×C4⋊C4C2312yes—
110C22×C4⋊C4C2311yescharacteristic
111C22×C4⋊C4C2312yes—
112C22×C22⋊C4C2314yes—
113C22×C22⋊C4C2314yes—
114C22×C22⋊C4C2314yes—
115C22×C22⋊C4C2314yes—
116C22×C22⋊C4C2314yes—
117C22×C22⋊C4C2312yes—
118C22×C22⋊C4C2314yes—
119C22×C42C2312yes—
120C22×C42C2312yes—
121C22×C42C2312yes—
122C22×C42C2311yescharacteristic
123C22×C42C2312yes—
124C22×C42C2311yescharacteristic
125C22×C42C2312yes—
126C22×C42C2311yescharacteristic
127#399—216no—
128#398—232no—
129#398—216no—
130#397—232no—
131#396—216no—
132#391—24no—
133#391—28no—
134#391—28no—
135#391—44no—
136#391D418yescomplemented
137#390—28no—
138#390—28no—
139#390—28no—
140#389—28no—
141#389—216no—
142#389—28no—
143#388—216no—
144#384—28no—
145#384—24no—
146#382—28no—
147#380—28no—
148C4×C4⋊C4—24no—
149C4×C4⋊C4D418yes—
150C4×C22⋊C4—28no—
151C2×#95C2314yes—
152C2×#95C2318yes—
153C2×#95C2318yes—
154C2×#95C2318yes—
155C2×#95C2312yes—
156C2×#95C2314yes—
157C2×#95C2314yes—
158C2×#95C2314yes—
159C2×#95C2314yes—
160C2×#95C2318yes—
161C2×#95C2318yes—
162C2×#95C2314yes—
163C2×#95C2312yes—
164C43—24no—
165C25C2412yes—
166C22×Q8—22no—
167C22×Q8—22no—
168C22×Q8—24no—
169C22×Q8—24no—
170C22×Q8—24no—
171C22×Q8—22no—
172C22×Q8—22no—
173C22×Q8—24no—
174C22×Q8—24no—
175C22×Q8—22no—
176C22×Q8—22no—
177C22×Q8—24no—
178C22×Q8—24no—
179C22×Q8C2×D414yes—
180C22×D4—22no—
181C22×D4—22no—
182C22×D4—24no—
183C22×D4—24no—
184C22×D4C2×D414yes—
185C23×C4—24no—
186C23×C4—24no—
187C23×C4—24no—
188C23×C4—24no—
189C23×C4—24no—
190C23×C4—24no—
191C23×C4C2×D418yes—
192C23×C4C2414yes—
193C23×C4C2414yes—
194C23×C4C2412yes—
195C23×C4C2412yes—
196C23×C4C2414yes—
197C23×C4C2412yes—
198C23×C4C2412yes—
199C23×C4C2412yes—
200C23×C4C2412yes—
201C23×C4C2412yes—
202C23×C4C2411yescharacteristic
203C23×C4C2411yescharacteristic
204C23×C4C2411yescharacteristic
205C4⋊Q8—432no—
206C4⋊Q8—416no—
207C22⋊Q8—432no—
208C22⋊Q8—432no—
209C22⋊Q8—432no—
210C4×Q8—416no—
211C4×D4—232no—
212#117—432no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—24no—
220C2×C4⋊C4—24no—
221C2×C4⋊C4—28no—
222C2×C4⋊C4—28no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—24no—
226C2×C4⋊C4—24no—
227C2×C4⋊C4—24no—
228C2×C4⋊C4—24no—
229C2×C4⋊C4—28no—
230C2×C4⋊C4—28no—
231C2×C4⋊C4—28no—
232C2×C4⋊C4—28no—
233C2×C4⋊C4—28no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—24no—
236C2×C4⋊C4—216no—
237C2×C4⋊C4—216no—
238C2×C4⋊C4—24no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—28no—
244C2×C4⋊C4—24no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—28no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—44no—
250C2×C4⋊C4—44no—
251C2×C4⋊C4—44no—
252C2×C4⋊C4—44no—
253C2×C4⋊C4—48no—
254C2×C4⋊C4—44no—
255C2×C4⋊C4C2×D414yes—
256C2×C4⋊C4C2×D414yes—
257C2×C4⋊C4C2×D418yes—
258C2×C4⋊C4#4114yes—
259C2×C22⋊C4—24no—
260C2×C22⋊C4—24no—
261C2×C22⋊C4—28no—
262C2×C22⋊C4—28no—
263C2×C22⋊C4—28no—
264C2×C22⋊C4—28no—
265C2×C22⋊C4—28no—
266C2×C22⋊C4—28no—
267C2×C22⋊C4—28no—
268C2×C22⋊C4—28no—
269C2×C22⋊C4—28no—
270C2×C22⋊C4—28no—
271C2×C22⋊C4—28no—
272C2×C22⋊C4—44no—
273C2×C22⋊C4—44no—
274C2×C22⋊C4—44no—
275C2×C22⋊C4—44no—
276C2×C22⋊C4—44no—
277C2×C22⋊C4—48no—
278C2×C22⋊C4C2×D418yes—
279C2×C42—22no—
280C2×C42—24no—
281C2×C42—22no—
282C2×C42—24no—
283C2×C42—24no—
284C2×C42—24no—
285C2×C42—22no—
286C2×C42—24no—
287C2×C42—24no—
288C2×C42—24no—
289C2×C42—24no—
290C2×C42—22no—
291C2×C42—22no—
292C2×C42—24no—
293C2×C42—24no—
294C2×C42—28no—
295C2×C42—24no—
296C2×C42—42no—
297C2×C42C2×D414yes—
298C2×C42C2×D418yes—
299C2×C42C2×D414yes—
300C2×C42C2×D414yes—
301C2×C42#4114yes—
302#95—28no—
303#95—24no—
304#95—28no—
305#95—216no—
306#95—216no—
307#95—216no—
308#95—28no—
309#95—28no—
310#95—44no—
311#95—48no—
312#95—48no—
313#95C2×D418yes—
314#95C2×D4116yes—
315C24—22no—
316C24—22no—
317C24—22no—
318C24—22no—
319C24—22no—
320C24—22no—
321C24—24no—
322C24—24no—
323C24—24no—
324C24—24no—
325C24C22×D414yes—
326C24C2511yescharacteristic
327C2×Q8—28no—
328C2×Q8—28no—
329C2×Q8—28no—
330C2×Q8—44no—
331C2×Q8—48no—
332C2×Q8—48no—
333C2×Q8—44no—
334C2×Q8—48no—
335C2×Q8—48no—
336C2×Q8—48no—
337C2×Q8—44no—
338C2×Q8—48no—
339C2×Q8—44no—
340C2×Q8—48no—
341C2×D4—28no—
342C2×D4—28no—
343C2×D4—28no—
344C2×D4—48no—
345C2×D4—48no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—24no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—24no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—24no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—24no—
375C22×C4—22no—
376C22×C4—24no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—24no—
380C22×C4—24no—
381C22×C4—24no—
382C22×C4—24no—
383C22×C4—22no—
384C22×C4—24no—
385C22×C4—22no—
386C22×C4—216no—
387C22×C4—216no—
388C22×C4—216no—
389C22×C4—22no—
390C22×C4—24no—
391C22×C4—22no—
392C22×C4—24no—
393C22×C4—24no—
394C22×C4—24no—
395C22×C4—24no—
396C22×C4—22no—
397C22×C4—24no—
398C22×C4—24no—
399C22×C4—24no—
400C22×C4—24no—
401C22×C4—24no—
402C22×C4—24no—
403C22×C4—28no—
404C22×C4—24no—
405C22×C4—28no—
406C22×C4—48no—
407C22×C4—48no—
408C22×C4—48no—
409C22×C4—48no—
410C22×C4C22×D414yes—
411C22×C4C22×D414yes—
412C22×C4C22×D418yes—
413C22×C4C22×D414yes—
414C22×C4C22×D414yes—
415C22×C4C22×D412yes—
416C22×C4C22×D412yes—
417C22×C4C22×D412yes—
418C22×C4C22×D412yes—
419C22×C4C22×D412yes—
420C22×C4C22×D412yes—
421C22×C4C22×D412yes—
422C22×C4C2×#4114yes—
423C22×C4C2×#4112yes—
424C22×C4C2×#4112yes—
425C22×C4C2×#4112yes—
426C22×C4C2×#4112yes—
427C22×C4H(5,𝔽2)12yes—
428C22×C4H(5,𝔽2)12yes—
429C22×C42−1+414yes—
430C22×C42−1+414yes—
431C22×C42−1+414yes—
432C4⋊C4—28no—
433C4⋊C4—216no—
434C4⋊C4—28no—
435C4⋊C4—28no—
436C4⋊C4—416no—
437C4⋊C4—416no—
438C4⋊C4—416no—
439C4⋊C4—416no—
440C4⋊C4—432no—
441C4⋊C4—48no—
442C4⋊C4—44no—
443C4⋊C4—48no—
444C4⋊C4—48no—
445C4⋊C4—48no—
446C4⋊C4—48no—
447C4⋊C4—84no—
448C22⋊C4—216no—
449C22⋊C4—416no—
450C22⋊C4—416no—
451C22⋊C4—416no—
452C22⋊C4—416no—
453C22⋊C4—416no—
454C22⋊C4—88no—
455C42—28no—
456C42—216no—
457C42—48no—
458C42—48no—
459C42—48no—
460C42—48no—
461C42—44no—
462C42#121116yes—
463C23—24no—
464C23—24no—
465C23—24no—
466C23—24no—
467C23—24no—
468C23—24no—
469C23—24no—
470C23—42no—
471C23—44no—
472C23—42no—
473C23—44no—
474C23—44no—
475C23—42no—
476C23—42no—
477C23—44no—
478C23—44no—
479C23—44no—
480C23—44no—
481C23—44no—
482C23—44no—
483C23—44no—
484C23—44no—
485C23—44no—
486C23C23×D411yescharacteristic
487C23C23×D411yescharacteristic
488C23C23×D411yescharacteristic
489C23C22×#4111yescharacteristic
490C23C22×#4111yescharacteristic
491C23C2×H(5,𝔽2)11yescharacteristic
492C23C2×H(5,𝔽2)11yescharacteristic
493C23C2×2−1+412yes—
494C23C2×2−1+412yes—
495C23C2×2−1+412yes—
496C23#58512yes—
497Q8—416no—
498Q8—816no—
499D4—416no—
500C2×C4—24no—
501C2×C4—24no—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—24no—
505C2×C4—24no—
506C2×C4—28no—
507C2×C4—24no—
508C2×C4—28no—
509C2×C4—24no—
510C2×C4—24no—
511C2×C4—24no—
512C2×C4—24no—
513C2×C4—24no—
514C2×C4—24no—
515C2×C4—24no—
516C2×C4—28no—
517C2×C4—24no—
518C2×C4—24no—
519C2×C4—28no—
520C2×C4—28no—
521C2×C4—24no—
522C2×C4—24no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—42no—
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—44no—
538C2×C4—44no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—44no—
543C2×C4—42no—
544C2×C4—44no—
545C2×C4—42no—
546C2×C4—44no—
547C2×C4—44no—
548C2×C4—42no—
549C2×C4—432no—
550C2×C4—42no—
551C2×C4—44no—
552C2×C4—44no—
553C2×C4—44no—
554C2×C4—44no—
555C2×C4—44no—
556C2×C4—44no—
557C2×C4—44no—
558C2×C4—44no—
559C2×C4—44no—
560C2×C4—44no—
561C2×C4—44no—
562C2×C4—48no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—44no—
566C2×C4—48no—
567C2×C4C2×#12114yes—
568C2×C4C2×#12114yes—
569C2×C4C2×#12114yes—
570C2×C4#53214yes—
571C2×C4#53614yes—
572C2×C4#53618yes—
573C2×C4#53614yes—
574C2×C4#54014yes—
575C2×C4#54118yes—
576C2×C4D4214yes—
577C2×C4D4214yes—
578C2×C4#54614yes—
579C2×C4#55014yes—
580C22—44no—
581C22—44no—
582C22—44no—
583C22—44no—
584C22—44no—
585C22—44no—
586C22—44no—
587C22—84no—
588C22—84no—
589C22—84no—
590C22—84no—
591C22C22×#12111yescharacteristic
592C22C2×#53211yescharacteristic
593C22C2×#53611yescharacteristic
594C22C2×#53611yescharacteristic
595C22C2×#53612yes—
596C22C2×#54011yescharacteristic
597C22C2×#54112yes—
598C22#346112yes—
599C22C2×D4211yescharacteristic
600C22C2×D4211yescharacteristic
601C22C2×#54611yescharacteristic
602C22C2×#55011yescharacteristic
603C22#348912yes—
604C22#349712yes—
605C22#349911yescharacteristic
606C22#349912yes—
607C22#350011yescharacteristic
608C22#351512yes—
609C22#352312yes—
610C22#354712yes—
611C22#355012yes—
612C22#355212yes—
613C22#355412yes—
614C4—24no—
615C4—44no—
616C4—44no—
617C4—44no—
618C4—48no—
619C4—44no—
620C4—44no—
621C4—44no—
622C4—84no—
623C4—84no—
624C4—82no—
625C4#242818yes—
626C4#262118yescomplemented
627C2—84no—
628C2C2×#262111yescharacteristic
629C2C2×#242811yescharacteristic
630C2#3520912yes—
631C2#3530811yescharacteristic
632C2#3762812yes—
633C2#3895411yescharacteristic
634C2#3895511yescharacteristic
635C2#3913912yes—
636C2#3934312yes—
637C2#3936011yescharacteristic
638C2#3936311yescharacteristic
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×#2546C2semidirect
4. C2×#2621C2semidirect
6. #16469C2semidirect
7. #15236C2semidirect
8. #15178C2semidirect
10. #15137C2semidirect
11. #14714C2semidirect
12. #14258C2semidirect
13. #13911C2semidirect
14. C4×#391C2semidirect
136. #391D4semidirect
626. C4#2621semidirect

Maximal subgroups

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

subgroupindex
C2×#25462
C2×#25422
C2×#25382
C2×#26212
C2×#22932
#164692
#152362
#151782
#151382
#151372
#147142
#142582
#139112
C4×#3912