Finite Group Databasebeta

← #517281

Subgroups — #517281

Order 512.

2902 conjugacy classes of subgroups.

The subgroup lattice has 648 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2851C211yesmaximal, characteristic
2C2×#2330C211yesmaximal, characteristic, complemented
3C2×C4×#117C211yesmaximal, characteristic
4#14699C212yesmaximal, complemented
5#14686C211yesmaximal, characteristic
6#14283C211yesmaximal, characteristic, complemented
7#14277C211yesmaximal, characteristic, complemented
8#14264C212yesmaximal
9#14120C214yesmaximal, complemented
10#14111C214yesmaximal, complemented
11#14011C214yesmaximal
12#13987C214yesmaximal
13#13779C211yesmaximal, characteristic, complemented
14#13777C212yesmaximal
15C4×#389C212yesmaximal, complemented
16C22×#125C2211yescharacteristic
17C22×#117C2211yescharacteristic
18C22×#117C2211yescharacteristic
19C23×C42C2211yescharacteristic
20#2851C4116yescomplemented
21#2330—28no—
22C2×#389C2214yes—
23C2×#389C2212yes—
24C2×#389C2212yes—
25C2×#389C2212yes—
26C2×#387C2214yes—
27C2×#387C2214yes—
28C2×#387C2214yes—
29C2×#385C2214yes—
30C2×#384C2211yescharacteristic
31C2×#384C2211yescharacteristic
32C2×#382C2211yescharacteristic
33C2×#380C2212yes—
34C2×#380C2214yes—
35C2×#380C2212yes—
36C4×#117—28no—
37C2×C4×C4⋊C4C2212yes—
38C2×C4×C4⋊C4C2212yes—
39C2×C4×C4⋊C4C2212yes—
40C2×C4×C4⋊C4C2212yes—
41C2×C4×C4⋊C4C2214yes—
42C2×C4×C22⋊C4C2212yes—
43C2×C4×C22⋊C4C2212yes—
44C2×#376C2211yescharacteristic
45C2×#376C2211yescharacteristic
46C2×C43C2211yescharacteristic
47C2×C43C2211yescharacteristic
48#1444C2214yes—
49#1444C2214yes—
50#1442C2214yes—
51#1442C2214yes—
52#1441C2214yes—
53#1441C2214yes—
54#1439C2214yes—
55#1439C2214yes—
56#1437C2214yes—
57#1437C2214yes—
58#1436C2214yes—
59#1436C2214yes—
60#1436C2214yes—
61#1436C2214yes—
62#1436C2214yes—
63#1436C2214yes—
64#1436C2214yes—
65#1436C2214yes—
66C4⋊#95C2214yes—
67C4⋊#95C2212yes—
68C4⋊#95C2212yes—
69C4⋊#95C2212yes—
70C4⋊#95C2212yes—
71C4⋊#95C2212yes—
72#1434C2214yes—
73C4×#95C2214yes—
74C24×C4C2312yes—
75C24×C4C2311yescharacteristic
76C2×#125—24no—
77C2×#125—24no—
78C2×#125C2×C418yes—
79C2×#117—24no—
80C2×#117—24no—
81C2×#117—24no—
82C2×#117—24no—
83C2×#117—24no—
84C2×#117Q818yes—
85C22×C4⋊C4C2311yescharacteristic
86C22×C4⋊C4C2311yescharacteristic
87C22×C4⋊C4C2312yes—
88C22×C4⋊C4C2312yes—
89C22×C4⋊C4C2312yes—
90C22×C4⋊C4C2314yes—
91C22×C4⋊C4C2312yes—
92C22×C4⋊C4C2314yes—
93C22×C4⋊C4C2312yes—
94C22×C4⋊C4C2314yes—
95C22×C4⋊C4C2314yes—
96C22×C4⋊C4C2314yes—
97C22×C4⋊C4C2314yes—
98C22×C22⋊C4C2311yescharacteristic
99C22×C22⋊C4C2311yescharacteristic
100C22×C22⋊C4C2312yes—
101C22×C22⋊C4C2314yes—
102C22×C22⋊C4C2314yes—
103C22×C42—24no—
104C22×C42—24no—
105C22×C42C2×C418yes—
106C22×C42C2312yes—
107C22×C42C2312yes—
108C22×C42C2312yes—
109C22×C42C2312yes—
110C22×C42C2311yescharacteristic
111C22×C42C2312yes—
112C22×C42C2311yescharacteristic
113C22×C42C2311yescharacteristic
114C22×C42C2311yescharacteristic
115C22×C42C2314yes—
116C22×C42C2314yes—
117C22×C42C2311yescharacteristic
118C22×C42C2311yescharacteristic
119#389—28no—
120#389—28no—
121#389—216no—
122#389C2×C4116yes—
123#387—216no—
124#387—48no—
125#387C2×C4132yes—
126#385C2×C4132yes—
127#384—24no—
128#384—24no—
129#382C2×C418yes—
130#380—28no—
131#380—48no—
132#380C2×C4116yes—
133C4×C4⋊C4—28no—
134C4×C4⋊C4—28no—
135C4×C4⋊C4—28no—
136C4×C4⋊C4—28no—
137C4×C4⋊C4—48no—
138C4×C22⋊C4—28no—
139C4×C22⋊C4—28no—
140#376—24no—
141#376D418yes—
142C2×#95C2314yes—
143C2×#95C2314yes—
144C2×#95C2314yes—
145C2×#95C2314yes—
146C2×#95C2314yes—
147C2×#95C2314yes—
148C2×#95C2314yes—
149C2×#95C2312yes—
150C2×#95C2312yes—
151C2×#95C2314yes—
152C2×#95C2314yes—
153C2×#95C2314yes—
154C2×#95C2314yes—
155C2×#95C2314yes—
156C2×#95C2314yes—
157C2×#95C2314yes—
158C2×#95C2314yes—
159C2×#95C2314yes—
160C2×#95C2314yes—
161C2×#95C2314yes—
162C2×#95C2312yes—
163C2×#95C2312yes—
164C43—24no—
165C43—24no—
166C25C2411yescharacteristic
167C23×C4—22no—
168C23×C4—22no—
169C23×C4—22no—
170C23×C4—22no—
171C23×C4—24no—
172C23×C4—24no—
173C23×C4—24no—
174C23×C4—24no—
175C23×C4—42no—
176C23×C4—42no—
177C23×C4C22×C418yes—
178C23×C4C22×C414yes—
179C23×C4C2×Q814yes—
180C23×C4#4114yes—
181C23×C4C2414yes—
182C23×C4C2414yes—
183C23×C4C2414yes—
184C23×C4C2414yes—
185C23×C4C2411yescharacteristic
186C23×C4C2411yescharacteristic
187C23×C4C2412yes—
188C23×C4C2411yescharacteristic
189C23×C4C2411yescharacteristic
190C23×C4C2412yes—
191C23×C4C2411yescharacteristic
192C23×C4C2412yes—
193C23×C4C2412yes—
194C23×C4C2411yescharacteristic
195#125—232no—
196#117—232no—
197#117—232no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—24no—
201C2×C4⋊C4—24no—
202C2×C4⋊C4—28no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—24no—
208C2×C4⋊C4—28no—
209C2×C4⋊C4—24no—
210C2×C4⋊C4—24no—
211C2×C4⋊C4—24no—
212C2×C4⋊C4—28no—
213C2×C4⋊C4—28no—
214C2×C4⋊C4—24no—
215C2×C4⋊C4—24no—
216C2×C4⋊C4—24no—
217C2×C4⋊C4—24no—
218C2×C4⋊C4—24no—
219C2×C4⋊C4—28no—
220C2×C4⋊C4—24no—
221C2×C4⋊C4—28no—
222C2×C4⋊C4—28no—
223C2×C4⋊C4—28no—
224C2×C4⋊C4—44no—
225C2×C4⋊C4—44no—
226C2×C4⋊C4—44no—
227C2×C4⋊C4—44no—
228C2×C4⋊C4—44no—
229C2×C4⋊C4—44no—
230C2×C4⋊C4—44no—
231C2×C4⋊C4—44no—
232C2×C4⋊C4C22×C418yes—
233C2×C4⋊C4C22×C4116yes—
234C2×C4⋊C4C22×C418yes—
235C2×C4⋊C4C22×C4116yes—
236C2×C4⋊C4C2×Q818yes—
237C2×C22⋊C4—22no—
238C2×C22⋊C4—22no—
239C2×C22⋊C4—22no—
240C2×C22⋊C4—22no—
241C2×C22⋊C4—24no—
242C2×C22⋊C4—22no—
243C2×C22⋊C4—22no—
244C2×C22⋊C4—24no—
245C2×C22⋊C4—28no—
246C2×C22⋊C4—44no—
247C2×C22⋊C4—44no—
248C2×C22⋊C4—44no—
249C2×C22⋊C4—44no—
250C2×C22⋊C4C22×C4116yes—
251C2×C22⋊C4C2×Q818yes—
252C2×C42—22no—
253C2×C42—22no—
254C2×C42—22no—
255C2×C42—24no—
256C2×C42—22no—
257C2×C42—22no—
258C2×C42—22no—
259C2×C42—22no—
260C2×C42—24no—
261C2×C42—22no—
262C2×C42—232no—
263C2×C42—24no—
264C2×C42—24no—
265C2×C42—24no—
266C2×C42—24no—
267C2×C42—28no—
268C2×C42—24no—
269C2×C42—24no—
270C2×C42—24no—
271C2×C42—24no—
272C2×C42—28no—
273C2×C42—44no—
274C2×C42—42no—
275C2×C42—42no—
276C2×C42—42no—
277C2×C42—42no—
278C2×C42—44no—
279C2×C42—44no—
280C2×C42—44no—
281C2×C42C22×C418yes—
282C2×C42C22×C414yes—
283C2×C42C22×C414yes—
284C2×C42C2×D414yes—
285C2×C42C2×D414yes—
286C2×C42C2×D414yes—
287C2×C42C2×Q814yes—
288C2×C42C2×Q814yes—
289C2×C42#4114yes—
290C2×C42#4114yes—
291C2×C42#4114yes—
292#95—24no—
293#95—24no—
294#95—28no—
295#95—28no—
296#95—28no—
297#95—28no—
298#95—28no—
299#95—28no—
300#95—28no—
301#95—44no—
302#95—44no—
303#95—44no—
304#95—44no—
305#95—44no—
306#95—44no—
307#95—44no—
308#95—44no—
309#95C22×C4116yes—
310#95C22×C4116yes—
311#95C22×C4116yes—
312#95C2×D418yes—
313#95C2×D418yes—
314C24—21no—
315C24—21no—
316C24—21no—
317C24—21no—
318C24—22no—
319C24—22no—
320C24—22no—
321C24—22no—
322C24C23×C412yes—
323C24C22×Q812yes—
324C24C2×#4112yes—
325C24C2511yescharacteristic
326C22×C4—22no—
327C22×C4—21no—
328C22×C4—21no—
329C22×C4—22no—
330C22×C4—21no—
331C22×C4—21no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—28no—
343C22×C4—28no—
344C22×C4—28no—
345C22×C4—28no—
346C22×C4—28no—
347C22×C4—28no—
348C22×C4—22no—
349C22×C4—21no—
350C22×C4—21no—
351C22×C4—24no—
352C22×C4—24no—
353C22×C4—24no—
354C22×C4—22no—
355C22×C4—21no—
356C22×C4—21no—
357C22×C4—21no—
358C22×C4—21no—
359C22×C4—21no—
360C22×C4—21no—
361C22×C4—24no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—24no—
366C22×C4—24no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—24no—
371C22×C4—24no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—24no—
377C22×C4—24no—
378C22×C4—24no—
379C22×C4—24no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—216no—
383C22×C4—216no—
384C22×C4—24no—
385C22×C4—24no—
386C22×C4—24no—
387C22×C4—24no—
388C22×C4—24no—
389C22×C4—24no—
390C22×C4—24no—
391C22×C4—24no—
392C22×C4—24no—
393C22×C4—24no—
394C22×C4—24no—
395C22×C4—24no—
396C22×C4—24no—
397C22×C4—24no—
398C22×C4—48no—
399C22×C4—48no—
400C22×C4—48no—
401C22×C4—48no—
402C22×C4—48no—
403C22×C4C4×Q8116yes—
404C22×C4C23×C418yes—
405C22×C4C23×C418yes—
406C22×C4C23×C414yes—
407C22×C4C23×C412yes—
408C22×C4C23×C414yes—
409C22×C4C23×C412yes—
410C22×C4C22×D412yes—
411C22×C4C22×D412yes—
412C22×C4C22×D414yes—
413C22×C4C22×D414yes—
414C22×C4C22×D412yes—
415C22×C4C22×Q814yes—
416C22×C4C22×Q812yes—
417C22×C4C22×Q814yes—
418C22×C4C22×Q812yes—
419C22×C4C2×#4112yes—
420C22×C4C2×#4112yes—
421C22×C4C2×#4114yes—
422C22×C4C2×#4112yes—
423C22×C4C2×#4114yes—
424C22×C4C2×#4112yes—
425C22×C4C2×#4112yes—
426C22×C4C2×#4112yes—
427C22×C4C2×#4112yes—
428C22×C4H(5,𝔽2)14yes—
429C22×C42−1+414yes—
430C4⋊C4—28no—
431C4⋊C4—28no—
432C4⋊C4—232no—
433C4⋊C4—216no—
434C4⋊C4—216no—
435C4⋊C4—232no—
436C4⋊C4—216no—
437C4⋊C4—416no—
438C4⋊C4—48no—
439C4⋊C4—48no—
440C4⋊C4—88no—
441C4⋊C4—88no—
442C4⋊C4—88no—
443C22⋊C4—28no—
444C22⋊C4—28no—
445C22⋊C4—216no—
446C22⋊C4—416no—
447C22⋊C4—88no—
448C42—216no—
449C42—28no—
450C42—28no—
451C42—28no—
452C42—28no—
453C42—48no—
454C42—48no—
455C42—48no—
456C42—48no—
457C42—416no—
458C42—88no—
459C42C4×D4116yes—
460C42C22⋊Q8116yes—
461C23—22no—
462C23—22no—
463C23—24no—
464C23—24no—
465C23—22no—
466C23—22no—
467C23—22no—
468C23—22no—
469C23—24no—
470C23—24no—
471C23—22no—
472C23—22no—
473C23—22no—
474C23—22no—
475C23—42no—
476C23—42no—
477C23—42no—
478C23—42no—
479C23—42no—
480C23—42no—
481C23—42no—
482C23—42no—
483C23C2×C4×Q814yes—
484C23C24×C411yescharacteristic
485C23C23×D411yescharacteristic
486C23C23×Q811yescharacteristic
487C23C22×#4111yescharacteristic
488C23C22×#4111yescharacteristic
489C23C22×#4111yescharacteristic
490C23C22×#4111yescharacteristic
491C23C2×H(5,𝔽2)12yes—
492C23C2×2−1+412yes—
493C23#58512yes—
494C23#58512yes—
495C2×C4—24no—
496C2×C4—22no—
497C2×C4—22no—
498C2×C4—24no—
499C2×C4—24no—
500C2×C4—24no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—24no—
506C2×C4—24no—
507C2×C4—24no—
508C2×C4—24no—
509C2×C4—24no—
510C2×C4—24no—
511C2×C4—24no—
512C2×C4—24no—
513C2×C4—24no—
514C2×C4—24no—
515C2×C4—24no—
516C2×C4—24no—
517C2×C4—24no—
518C2×C4—24no—
519C2×C4—232no—
520C2×C4—28no—
521C2×C4—28no—
522C2×C4—28no—
523C2×C4—28no—
524C2×C4—28no—
525C2×C4—28no—
526C2×C4—24no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—24no—
532C2×C4—24no—
533C2×C4—24no—
534C2×C4—24no—
535C2×C4—24no—
536C2×C4—24no—
537C2×C4—432no—
538C2×C4—44no—
539C2×C4—44no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—44no—
543C2×C4—44no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—44no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—44no—
552C2×C4—44no—
553C2×C4—44no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—44no—
563C2×C4—44no—
564C2×C4—44no—
565C2×C4—44no—
566C2×C4—44no—
567C2×C4—44no—
568C2×C4—44no—
569C2×C4—44no—
570C2×C4—44no—
571C2×C4—44no—
572C2×C4—44no—
573C2×C4—44no—
574C2×C4—44no—
575C2×C4—44no—
576C2×C4C2×C4×D418yes—
577C2×C4C2×C4×D414yes—
578C2×C4C2×C4×Q814yes—
579C2×C4C2×C4×Q814yes—
580C2×C4C4×#4114yes—
581C2×C4#52018yes—
582C2×C4C2×C22⋊Q818yes—
583C2×C4C2×C22⋊Q814yes—
584C2×C4#52514yes—
585C2×C4#52914yes—
586C2×C4#53314yes—
587C22—24no—
588C22—24no—
589C22—24no—
590C22—44no—
591C22—44no—
592C22—44no—
593C22—42no—
594C22—44no—
595C22—42no—
596C22—42no—
597C22—42no—
598C22C22×C4×D411yescharacteristic
599C22C22×C4×Q811yescharacteristic
600C22C2×C4×#4111yescharacteristic
601C22C2×#52012yes—
602C22#342812yes—
603C22C22×C22⋊Q811yescharacteristic
604C22C2×#52511yescharacteristic
605C22C2×#52911yescharacteristic
606C22C2×#53311yescharacteristic
607C22#346012yes—
608C22#346012yes—
609C22#346112yes—
610C22#346112yes—
611C22#348812yes—
612C22#348912yes—
613C22#349412yes—
614C22#349512yes—
615C22#350312yes—
616C22#350412yes—
617C22#350811yescharacteristic
618C22#350911yescharacteristic
619C22#351211yescharacteristic
620C22#351211yescharacteristic
621C4—28no—
622C4—28no—
623C4—28no—
624C4—24no—
625C4—48no—
626C4—48no—
627C4—44no—
628C4—42no—
629C4—42no—
630C4—42no—
631C4—42no—
632C4—84no—
633C4—84no—
634C4C4×C22⋊Q818yes—
635C2—44no—
636C2C2×C4×C22⋊Q811yescharacteristic
637C2#3408412yes—
638C2#3420212yes—
639C2#3421411yescharacteristic
640C2#3421711yescharacteristic
641C2#3525712yes—
642C2#3525911yescharacteristic
643C2#3528212yes—
644C2#3529511yescharacteristic
645C2#3530911yescharacteristic
646C2#3532311yescharacteristic
subgroupG/H
center, Frattini, socle, cosocleC24C25
commutatorC23C24×C4
prime radicalC25C24
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. C2×#2330C2semidirect
4. #14699C2semidirect
6. #14283C2semidirect
7. #14277C2semidirect
9. #14120C2semidirect
10. #14111C2semidirect
13. #13779C2semidirect
15. C4×#389C2semidirect
20. #2851C4semidirect

Maximal subgroups

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

subgroupindex
C2×#28512
C2×#23302
C2×C4×#1172
#146992
#146862
#142832
#142772
#142642
#141202
#141112
#140112
#139872
#137792
#137772
C4×#3892