Finite Group Databasebeta

← #613352

Subgroups — #613352

Order 512.

4080 conjugacy classes of subgroups.

The subgroup lattice has 658 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2854C211yesmaximal, characteristic
2C2×#2501C211yesmaximal, characteristic
3C2×#2413C214yesmaximal, complemented
4C22×#389C211yesmaximal, characteristic, complemented
5C2×C4×#117C211yesmaximal, characteristic, complemented
6#15151C212yesmaximal, complemented
7#15134C212yesmaximal, complemented
8#14910C214yesmaximal, complemented
9#14870C214yesmaximal, complemented
10#14712C212yesmaximal, complemented
11#14700C212yesmaximal, complemented
12#14137C211yesmaximal, characteristic
13#13996C214yesmaximal, complemented
14#13904C211yesmaximal, characteristic, complemented
15#13880C211yesmaximal, characteristic, complemented
16C22×#123C2212yes—
17C22×#123C2214yes—
18C22×#121C2211yescharacteristic
19C22×#117C2212yes—
20C23×C4⋊C4C2211yescharacteristic
21C23×C4⋊C4C2211yescharacteristic
22C23×C4⋊C4C2212yes—
23C23×C42C2211yescharacteristic
24#2854—28no—
25#2501—28no—
26#2413—232no—
27C2×#399C2214yes—
28C2×#398C2218yes—
29C2×#396C2218yes—
30C2×#390C2211yescharacteristic
31C2×#390C2211yescharacteristic
32C2×#390C2212yes—
33C2×#390C2212yes—
34C2×#389—28no—
35C2×#389C2211yescharacteristic
36C2×#389C2211yescharacteristic
37C2×#389C2212yes—
38C2×#389C2212yes—
39C2×#389C2212yes—
40C2×#389C2218yes—
41C2×#389C2212yes—
42C2×#388C2212yes—
43C2×#388C2212yes—
44C2×#388C2218yes—
45C2×#386C2212yes—
46C2×#386C2214yes—
47C2×#385C2214yes—
48C2×#384C2212yes—
49C2×#384C2212yes—
50C2×#381C2214yes—
51C2×#380C2211yescharacteristic
52C4×#117—28no—
53C2×C4×C4⋊C4C2212yes—
54C2×C4×C4⋊C4C2212yes—
55C2×C4×C22⋊C4C2211yescharacteristic
56C2×C4×C22⋊C4C2212yes—
57C2×C4×C22⋊C4C2211yescharacteristic
58C2×#376C2212yes—
59C22×#95C2212yes—
60C2×C43C2212yes—
61#1445C2214yes—
62#1445C2218yes—
63#1444C2214yes—
64#1439C2214yes—
65#1439C2218yes—
66#1437C2218yes—
67C4⋊#95C2218yes—
68C4⋊#95C2212yes—
69C4⋊#95C2212yes—
70C4⋊#95C2212yes—
71C4⋊#95C2212yes—
72C23×D4C2312yes—
73C23×D4C2311yescharacteristic
74C24×C4C2312yes—
75C24×C4C2312yes—
76C24×C4C2311yescharacteristic
77C24×C4C2311yescharacteristic
78C24×C4C2311yescharacteristic
79C2×#123—28no—
80C2×#123—216no—
81C2×#123—216no—
82C2×#123—216no—
83C2×#123—216no—
84C2×#121—24no—
85C2×#121—24no—
86C2×#121—24no—
87C2×#117—28no—
88C2×#117—28no—
89C2×#117D4116yescomplemented
90C22×C4⋊C4—24no—
91C22×C4⋊C4—28no—
92C22×C4⋊C4—216no—
93C22×C4⋊C4—24no—
94C22×C4⋊C4—24no—
95C22×C4⋊C4—28no—
96C22×C4⋊C4—24no—
97C22×C4⋊C4C2314yes—
98C22×C4⋊C4C2312yes—
99C22×C4⋊C4C2318yes—
100C22×C4⋊C4C2312yes—
101C22×C4⋊C4C2312yes—
102C22×C4⋊C4C2312yes—
103C22×C4⋊C4C2314yes—
104C22×C4⋊C4C2311yescharacteristic
105C22×C4⋊C4C2312yes—
106C22×C4⋊C4C2311yescharacteristic
107C22×C4⋊C4C2311yescharacteristic
108C22×C4⋊C4C2318yes—
109C22×C4⋊C4C2314yes—
110C22×C4⋊C4C2312yes—
111C22×C22⋊C4C2312yes—
112C22×C22⋊C4C2318yes—
113C22×C22⋊C4C2312yes—
114C22×C22⋊C4C2312yes—
115C22×C22⋊C4C2312yes—
116C22×C22⋊C4C2312yes—
117C22×C22⋊C4C2318yes—
118C22×C22⋊C4C2312yes—
119C22×C22⋊C4C2312yes—
120C22×C22⋊C4C2312yes—
121C22×C42—24no—
122C22×C42—24no—
123C22×C42D418yes—
124C22×C42C2312yes—
125C22×C42C2314yes—
126C22×C42C2311yescharacteristic
127C22×C42C2311yescharacteristic
128C22×C42C2311yescharacteristic
129C22×C42C2311yescharacteristic
130C22×C42C2314yes—
131C22×C42C2312yes—
132#399—216no—
133#398—232no—
134#396—232no—
135#390—28no—
136#390—24no—
137#390—24no—
138#390—28no—
139#389—24no—
140#389—24no—
141#389—232no—
142#389—28no—
143#389—28no—
144#389—28no—
145#389—28no—
146#388—28no—
147#388—28no—
148#388—232no—
149#386—28no—
150#386—216no—
151#385—216no—
152#384—28no—
153#384—28no—
154#381—216no—
155#380—24no—
156C4×C4⋊C4—28no—
157C4×C4⋊C4—28no—
158C4×C22⋊C4—24no—
159C4×C22⋊C4—28no—
160C4×C22⋊C4D418yes—
161#376—28no—
162C2×#95—28no—
163C2×#95C2318yes—
164C2×#95C2318yes—
165C2×#95C2318yes—
166C2×#95C2318yes—
167C2×#95C2312yes—
168C2×#95C2314yes—
169C2×#95C2314yes—
170C2×#95C2312yes—
171C2×#95C2314yes—
172C2×#95C2312yes—
173C2×#95C2312yes—
174C2×#95C2312yes—
175C43—28no—
176C25C2412yes—
177C25C2411yescharacteristic
178C22×D4—24no—
179C22×D4—24no—
180C22×D4—28no—
181C22×D4—24no—
182C22×D4—28no—
183C22×D4—22no—
184C22×D4—22no—
185C22×D4—22no—
186C22×D4—24no—
187C22×D4—24no—
188C23×C4—22no—
189C23×C4—24no—
190C23×C4—22no—
191C23×C4—24no—
192C23×C4—24no—
193C23×C4—24no—
194C23×C4—22no—
195C23×C4—22no—
196C23×C4—28no—
197C23×C4—24no—
198C23×C4—24no—
199C23×C4—24no—
200C23×C4—28no—
201C23×C4—28no—
202C23×C4—24no—
203C23×C4—28no—
204C23×C4—42no—
205C23×C4—44no—
206C23×C4C2×D414yes—
207C23×C4C2×D414yes—
208C23×C4C2×D414yes—
209C23×C4C2×D418yes—
210C23×C4C2×D414yes—
211C23×C4#4114yes—
212C23×C4C2412yes—
213C23×C4C2412yes—
214C23×C4C2414yes—
215C23×C4C2412yes—
216C23×C4C2414yes—
217C23×C4C2411yescharacteristic
218C23×C4C2414yes—
219C23×C4C2411yescharacteristic
220C23×C4C2411yescharacteristic
221C23×C4C2414yes—
222C23×C4C2411yescharacteristic
223C23×C4C2411yescharacteristic
224C23×C4C2411yescharacteristic
225#123—432no—
226#123—464no—
227#121—416no—
228#117—264no—
229C2×C4⋊C4—232no—
230C2×C4⋊C4—24no—
231C2×C4⋊C4—24no—
232C2×C4⋊C4—28no—
233C2×C4⋊C4—28no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—28no—
236C2×C4⋊C4—24no—
237C2×C4⋊C4—216no—
238C2×C4⋊C4—24no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—28no—
241C2×C4⋊C4—28no—
242C2×C4⋊C4—216no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—216no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—28no—
250C2×C4⋊C4—216no—
251C2×C4⋊C4—28no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—216no—
255C2×C4⋊C4—24no—
256C2×C4⋊C4—24no—
257C2×C4⋊C4—22no—
258C2×C4⋊C4—216no—
259C2×C4⋊C4—216no—
260C2×C4⋊C4—24no—
261C2×C4⋊C4—24no—
262C2×C4⋊C4—48no—
263C2×C4⋊C4—416no—
264C2×C4⋊C4—432no—
265C2×C4⋊C4C2×D4116yes—
266C2×C4⋊C4#4118yes—
267C2×C22⋊C4—24no—
268C2×C22⋊C4—24no—
269C2×C22⋊C4—24no—
270C2×C22⋊C4—28no—
271C2×C22⋊C4—24no—
272C2×C22⋊C4—24no—
273C2×C22⋊C4—28no—
274C2×C22⋊C4—24no—
275C2×C22⋊C4—24no—
276C2×C22⋊C4—216no—
277C2×C22⋊C4—216no—
278C2×C22⋊C4—28no—
279C2×C22⋊C4—24no—
280C2×C22⋊C4—24no—
281C2×C22⋊C4—216no—
282C2×C22⋊C4—24no—
283C2×C22⋊C4—216no—
284C2×C22⋊C4—216no—
285C2×C22⋊C4—28no—
286C2×C22⋊C4—48no—
287C2×C22⋊C4—44no—
288C2×C22⋊C4C2×D418yes—
289C2×C22⋊C4C2×D418yes—
290C2×C22⋊C4C2×D418yes—
291C2×C22⋊C4#4118yes—
292C2×C42—232no—
293C2×C42—28no—
294C2×C42—22no—
295C2×C42—22no—
296C2×C42—22no—
297C2×C42—22no—
298C2×C42—28no—
299C2×C42—24no—
300C2×C42—22no—
301C2×C42—22no—
302C2×C42—22no—
303C2×C42—22no—
304C2×C42—24no—
305C2×C42—28no—
306C2×C42—24no—
307C2×C42—24no—
308C2×C42—28no—
309C2×C42—24no—
310C2×C42—44no—
311C2×C42C2×D4116yes—
312C2×C42C2×D418yes—
313C2×C42C2×D414yes—
314C2×C42C2×D414yes—
315C2×C42C2×D414yes—
316C2×C42C2×D414yes—
317#95—28no—
318#95—28no—
319#95—28no—
320#95—24no—
321#95—24no—
322#95—216no—
323#95—216no—
324#95—216no—
325#95—24no—
326#95—24no—
327#95—216no—
328#95C2×D418yes—
329C24—22no—
330C24—22no—
331C24—22no—
332C24—22no—
333C24—24no—
334C24—22no—
335C24—22no—
336C24—24no—
337C24—24no—
338C24—22no—
339C24—24no—
340C24—24no—
341C24—24no—
342C24C22×D412yes—
343C24C22×D414yes—
344C24C22×D412yes—
345C24C2×#4114yes—
346C24C2×#4112yes—
347C24C2511yescharacteristic
348C2×D4—48no—
349C2×D4—416no—
350C2×D4—48no—
351C2×D4—48no—
352C2×D4—416no—
353C2×D4—44no—
354C2×D4—44no—
355C2×D4—44no—
356C2×D4—48no—
357C2×D4—48no—
358C22×C4—28no—
359C22×C4—216no—
360C22×C4—28no—
361C22×C4—216no—
362C22×C4—21no—
363C22×C4—21no—
364C22×C4—21no—
365C22×C4—21no—
366C22×C4—21no—
367C22×C4—22no—
368C22×C4—21no—
369C22×C4—24no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—24no—
373C22×C4—24no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—24no—
377C22×C4—28no—
378C22×C4—28no—
379C22×C4—216no—
380C22×C4—216no—
381C22×C4—216no—
382C22×C4—28no—
383C22×C4—21no—
384C22×C4—24no—
385C22×C4—28no—
386C22×C4—24no—
387C22×C4—21no—
388C22×C4—24no—
389C22×C4—24no—
390C22×C4—24no—
391C22×C4—24no—
392C22×C4—24no—
393C22×C4—22no—
394C22×C4—24no—
395C22×C4—22no—
396C22×C4—28no—
397C22×C4—28no—
398C22×C4—24no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—24no—
402C22×C4—28no—
403C22×C4—22no—
404C22×C4—24no—
405C22×C4—24no—
406C22×C4—24no—
407C22×C4—22no—
408C22×C4—28no—
409C22×C4—22no—
410C22×C4—24no—
411C22×C4—24no—
412C22×C4—24no—
413C22×C4—24no—
414C22×C4—24no—
415C22×C4—24no—
416C22×C4—44no—
417C22×C4—48no—
418C22×C4—48no—
419C22×C4—432no—
420C22×C4—48no—
421C22×C4—48no—
422C22×C4—48no—
423C22×C4—416no—
424C22×C4—48no—
425C22×C4—48no—
426C22×C4—416no—
427C22×C4#121116yes—
428C22×C4#127116yes—
429C22×C4C22×D412yes—
430C22×C4C22×D418yes—
431C22×C4C22×D412yes—
432C22×C4C22×D412yes—
433C22×C4C22×D418yes—
434C22×C4C22×D418yes—
435C22×C4C22×D412yes—
436C22×C4C22×D412yes—
437C22×C4C22×D414yes—
438C22×C4C22×D414yes—
439C22×C4C22×D412yes—
440C22×C4C22×D412yes—
441C22×C4C22×D412yes—
442C22×C4C2×#4118yes—
443C22×C4C2×#4112yes—
444C22×C4C2×#4118yes—
445C22×C4C2×#4112yes—
446C22×C4C2×#4114yes—
447C22×C42−1+414yes—
448C22×C42−1+414yes—
449C4⋊C4—216no—
450C4⋊C4—232no—
451C4⋊C4—216no—
452C4⋊C4—432no—
453C4⋊C4—48no—
454C4⋊C4—48no—
455C4⋊C4—44no—
456C4⋊C4—432no—
457C4⋊C4—44no—
458C4⋊C4—416no—
459C4⋊C4—416no—
460C4⋊C4—48no—
461C4⋊C4—48no—
462C4⋊C4—44no—
463C22⋊C4—216no—
464C22⋊C4—432no—
465C22⋊C4—48no—
466C22⋊C4—48no—
467C22⋊C4—48no—
468C22⋊C4—432no—
469C22⋊C4—48no—
470C22⋊C4—48no—
471C22⋊C4—48no—
472C22⋊C4#121132yes—
473C42—28no—
474C42—28no—
475C42—28no—
476C42—28no—
477C42—232no—
478C42—216no—
479C42—416no—
480C42—48no—
481C23—22no—
482C23—22no—
483C23—22no—
484C23—22no—
485C23—24no—
486C23—28no—
487C23—24no—
488C23—28no—
489C23—24no—
490C23—24no—
491C23—28no—
492C23—28no—
493C23—44no—
494C23—42no—
495C23—42no—
496C23—42no—
497C23—42no—
498C23—44no—
499C23—42no—
500C23—42no—
501C23—42no—
502C23—44no—
503C23—48no—
504C23—44no—
505C23—48no—
506C23—44no—
507C23—44no—
508C23—48no—
509C23—48no—
510C23C2×#12118yes—
511C23C2×#12114yes—
512C23C2×#12714yes—
513C23#53214yes—
514C23#53218yes—
515C23C23×D411yescharacteristic
516C23C23×D411yescharacteristic
517C23C23×D412yes—
518C23C22×#4111yescharacteristic
519C23C22×#4112yes—
520C23C22×#4112yes—
521C23C2×2−1+412yes—
522C23C2×2−1+412yes—
523C23#58512yes—
524D4—816no—
525D4—88no—
526C2×C4—232no—
527C2×C4—24no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—24no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—24no—
536C2×C4—24no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—28no—
540C2×C4—24no—
541C2×C4—28no—
542C2×C4—24no—
543C2×C4—28no—
544C2×C4—24no—
545C2×C4—28no—
546C2×C4—24no—
547C2×C4—28no—
548C2×C4—28no—
549C2×C4—232no—
550C2×C4—22no—
551C2×C4—28no—
552C2×C4—28no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—28no—
556C2×C4—216no—
557C2×C4—28no—
558C2×C4—416no—
559C2×C4—432no—
560C2×C4—44no—
561C2×C4—44no—
562C2×C4—44no—
563C2×C4—48no—
564C2×C4—44no—
565C2×C4—44no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—44no—
569C2×C4—48no—
570C2×C4—42no—
571C2×C4—44no—
572C2×C4—44no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—44no—
577C2×C4—42no—
578C2×C4—44no—
579C2×C4—48no—
580C2×C4—42no—
581C2×C4—44no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—48no—
586C2×C4—48no—
587C2×C4—816no—
588C2×C4C2×#12114yes—
589C2×C4C2×#121116yes—
590C2×C4C2×#12114yes—
591C2×C4#52514yes—
592C2×C4C2×#12714yes—
593C2×C4C2×#12714yes—
594C2×C4#53614yes—
595C2×C4#53614yes—
596C2×C4#54718yes—
597C2×C4#55218yes—
598C22—24no—
599C22—24no—
600C22—24no—
601C22—216no—
602C22—216no—
603C22—24no—
604C22—24no—
605C22—24no—
606C22—82no—
607C22—82no—
608C22—82no—
609C22—82no—
610C22—88no—
611C22—88no—
612C22—82no—
613C22—82no—
614C22—82no—
615C22#285618yes—
616C22C22×#12112yes—
617C22C22×#12111yescharacteristic
618C22C2×#52511yescharacteristic
619C22C22×#12711yescharacteristic
620C22C2×#53211yescharacteristic
621C22C2×#53212yes—
622C22C2×#53611yescharacteristic
623C22C2×#53611yescharacteristic
624C22#344911yescharacteristic
625C22C2×#54712yes—
626C22D4×#4114yes—
627C22C2×#55212yes—
628C22#348214yes—
629C22#348912yes—
630C22#349512yes—
631C22#349914yes—
632C22#351514yes—
633C4—24no—
634C4—24no—
635C4—216no—
636C4—24no—
637C4—24no—
638C4—48no—
639C4—48no—
640C4—42no—
641C4—84no—
642C4—82no—
643C4—82no—
644C4—82no—
645C4#241318yescomplemented
646C2—28no—
647C2—162no—
648C2C2×#241311yescharacteristic
649C2#3493611yescharacteristic
650C2#3497211yescharacteristic
651C2#3513911yescharacteristic
652C2#3530611yescharacteristic
653C2#3531711yescharacteristic
654C2#3625914yes—
655C2#3626514yes—
656C2C2×#285611yescharacteristic
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC22×#121C22
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×#2413C2semidirect
4. C22×#389C2semidirect
5. C2×C4×#117C2semidirect
6. #15151C2semidirect
7. #15134C2semidirect
8. #14910C2semidirect
9. #14870C2semidirect
10. #14712C2semidirect
11. #14700C2semidirect
13. #13996C2semidirect
14. #13904C2semidirect
15. #13880C2semidirect
89. C2×#117D4semidirect
645. C4#2413semidirect

Maximal subgroups

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

subgroupindex
C2×#28542
C2×#25012
C2×#24132
C22×#3892
C2×C4×#1172
#151512
#151342
#149102
#148702
#147122
#147002
#141372
#139962
#139042
#138802