Finite Group Databasebeta

← #2926240

Subgroups — #2926240

Order 512.

2654 conjugacy classes of subgroups.

The subgroup lattice has 590 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2724C211yesmaximal, characteristic, complemented
2C2×#2501C211yesmaximal, characteristic
3#16701C214yesmaximal, complemented
4#16563C212yesmaximal, complemented
5#15983C212yesmaximal, complemented
6#15950C212yesmaximal
7#15669C214yesmaximal, complemented
8#15545C214yesmaximal, complemented
9#15242C212yesmaximal, complemented
10#14821C212yesmaximal, complemented
11#14359C214yesmaximal, complemented
12#14230C212yesmaximal, complemented
13#14157C211yesmaximal, characteristic, complemented
14C22×#123C2214yes—
15C22×#123C2212yes—
16C22×#121C2211yescharacteristic
17C23×C4⋊C4C2211yescharacteristic
18#2724—28no—
19#2501—28no—
20C2×#399C2212yes—
21C2×#399C2214yes—
22C2×#398C2212yes—
23C2×#398C2212yes—
24C2×#397C2214yes—
25C2×#390C2211yescharacteristic
26C2×#390C2211yescharacteristic
27C2×#389C2212yes—
28C2×#388C2214yes—
29C2×#387C2214yes—
30C2×#386C2212yes—
31C2×#385C2214yes—
32C2×#385C2214yes—
33C2×#385C2212yes—
34C2×#384C2211yescharacteristic
35C2×#384C2211yescharacteristic
36C2×#382C2214yes—
37C2×#380C2212yes—
38C2×C4×C22⋊C4C2211yescharacteristic
39#1445C2214yes—
40#1445C2214yes—
41#1445C2212yes—
42#1444C2212yes—
43#1441C2214yes—
44#1441C2214yes—
45#1441C2214yes—
46#1441C2214yes—
47#1441C2212yes—
48#1441C2214yes—
49#1441C2212yes—
50#1441C2214yes—
51#1441C2214yes—
52#1440C2214yes—
53#1440C2212yes—
54#1439C2212yes—
55#1439C2214yes—
56#1439C2214yes—
57#1439C2214yes—
58#1438C2214yes—
59#1438C2212yes—
60#1438C2212yes—
61#1437C2214yes—
62#1437C2214yes—
63C4⋊#95C2214yes—
64C4⋊#95C2214yes—
65C22⋊#95C2214yes—
66C22⋊#95C2214yes—
67C22⋊#95C2212yes—
68C4×#95C2212yes—
69C23×D4C2312yes—
70C23×D4C2311yescharacteristic
71C24×C4C2311yescharacteristic
72C24×C4C2312yes—
73C24×C4C2312yes—
74C2×#123—28no—
75C2×#123—216no—
76C2×#123—216no—
77C2×#123—48no—
78C2×#123—48no—
79C2×#121—24no—
80C2×#121—42no—
81C2×#121—42no—
82C22×C4⋊C4—24no—
83C22×C4⋊C4—42no—
84C22×C4⋊C4—42no—
85C22×C4⋊C4C2312yes—
86C22×C4⋊C4C2314yes—
87C22×C4⋊C4C2314yes—
88C22×C4⋊C4C2312yes—
89C22×C4⋊C4C2314yes—
90C22×C4⋊C4C2314yes—
91C22×C4⋊C4C2312yes—
92C22×C4⋊C4C2314yes—
93C22×C4⋊C4C2311yescharacteristic
94C22×C4⋊C4C2311yescharacteristic
95C22×C4⋊C4C2312yes—
96C22×C4⋊C4C2312yes—
97C22×C4⋊C4C2311yescharacteristic
98C22×C22⋊C4C2312yes—
99C22×C22⋊C4C2314yes—
100C22×C22⋊C4C2314yes—
101C22×C22⋊C4C2312yes—
102C22×C22⋊C4C2314yes—
103C22×C22⋊C4C2312yes—
104C22×C22⋊C4C2312yes—
105C22×C22⋊C4C2314yes—
106C22×C22⋊C4C2314yes—
107C22×C22⋊C4C2314yes—
108C22×C22⋊C4C2312yes—
109C22×C42C2311yescharacteristic
110C22×C42C2314yes—
111C22×C42C2311yescharacteristic
112#399—216no—
113#399—28no—
114#398—28no—
115#398—44no—
116#397—216no—
117#390—24no—
118#390D418yes—
119#389—28no—
120#388—48no—
121#387—48no—
122#386—44no—
123#385—216no—
124#385—48no—
125#385—44no—
126#384—24no—
127#384—24no—
128#382—48no—
129#380—44no—
130C4×C22⋊C4—24no—
131C2×#95C2312yes—
132C2×#95C2314yes—
133C2×#95C2314yes—
134C2×#95C2312yes—
135C2×#95C2312yes—
136C2×#95C2314yes—
137C2×#95C2312yes—
138C2×#95C2314yes—
139C2×#95C2312yes—
140C2×#95C2314yes—
141C2×#95C2314yes—
142C2×#95C2314yes—
143C2×#95C2314yes—
144C2×#95C2314yes—
145C2×#95C2314yes—
146C2×#95C2314yes—
147C2×#95C2314yes—
148C2×#95C2314yes—
149C2×#95C2314yes—
150C2×#95C2314yes—
151C2×#95C2314yes—
152C25C2412yes—
153C25C2411yescharacteristic
154C22×D4—24no—
155C22×D4—28no—
156C22×D4—24no—
157C22×D4—24no—
158C22×D4—41no—
159C22×D4—41no—
160C22×D4—42no—
161C22×D4—44no—
162C22×D4—42no—
163C22×D4C2×D414yes—
164C23×C4—22no—
165C23×C4—22no—
166C23×C4—22no—
167C23×C4—28no—
168C23×C4—24no—
169C23×C4—24no—
170C23×C4—24no—
171C23×C4—24no—
172C23×C4—42no—
173C23×C4—42no—
174C23×C4—44no—
175C23×C4—44no—
176C23×C4—44no—
177C23×C4—42no—
178C23×C4—42no—
179C23×C4C2414yes—
180C23×C4C2414yes—
181C23×C4C2414yes—
182C23×C4C2412yes—
183C23×C4C2412yes—
184C23×C4C2412yes—
185C23×C4C2412yes—
186C23×C4C2411yescharacteristic
187C23×C4C2414yes—
188C23×C4C2412yes—
189C23×C4C2411yescharacteristic
190#123—464no—
191#123—432no—
192#121—416no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—24no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—28no—
197C2×C4⋊C4—28no—
198C2×C4⋊C4—28no—
199C2×C4⋊C4—24no—
200C2×C4⋊C4—28no—
201C2×C4⋊C4—28no—
202C2×C4⋊C4—28no—
203C2×C4⋊C4—24no—
204C2×C4⋊C4—24no—
205C2×C4⋊C4—28no—
206C2×C4⋊C4—28no—
207C2×C4⋊C4—28no—
208C2×C4⋊C4—24no—
209C2×C4⋊C4—28no—
210C2×C4⋊C4—24no—
211C2×C4⋊C4—24no—
212C2×C4⋊C4—28no—
213C2×C4⋊C4—41no—
214C2×C4⋊C4—41no—
215C2×C4⋊C4—42no—
216C2×C4⋊C4—41no—
217C2×C4⋊C4—41no—
218C2×C4⋊C4—41no—
219C2×C4⋊C4—41no—
220C2×C4⋊C4—44no—
221C2×C4⋊C4—44no—
222C2×C4⋊C4—416no—
223C2×C4⋊C4—44no—
224C2×C4⋊C4—44no—
225C2×C4⋊C4—42no—
226C2×C4⋊C4—42no—
227C2×C4⋊C4—44no—
228C2×C4⋊C4—44no—
229C2×C4⋊C4—42no—
230C2×C4⋊C4C2×D414yes—
231C2×C22⋊C4—24no—
232C2×C22⋊C4—28no—
233C2×C22⋊C4—24no—
234C2×C22⋊C4—28no—
235C2×C22⋊C4—24no—
236C2×C22⋊C4—28no—
237C2×C22⋊C4—28no—
238C2×C22⋊C4—24no—
239C2×C22⋊C4—28no—
240C2×C22⋊C4—28no—
241C2×C22⋊C4—24no—
242C2×C22⋊C4—28no—
243C2×C22⋊C4—28no—
244C2×C22⋊C4—28no—
245C2×C22⋊C4—28no—
246C2×C22⋊C4—42no—
247C2×C22⋊C4—44no—
248C2×C22⋊C4—44no—
249C2×C22⋊C4—44no—
250C2×C22⋊C4—44no—
251C2×C22⋊C4—42no—
252C2×C22⋊C4—44no—
253C2×C22⋊C4—44no—
254C2×C22⋊C4—44no—
255C2×C22⋊C4—44no—
256C2×C22⋊C4—42no—
257C2×C22⋊C4—44no—
258C2×C22⋊C4—44no—
259C2×C22⋊C4C2×D4116yes—
260C2×C22⋊C4#4118yes—
261C2×C42—22no—
262C2×C42—28no—
263C2×C42—24no—
264C2×C42—28no—
265C2×C42—42no—
266C2×C42—44no—
267C2×C42C2×D414yes—
268#95—24no—
269#95—28no—
270#95—28no—
271#95—24no—
272#95—24no—
273#95—28no—
274#95—24no—
275#95—44no—
276#95—44no—
277#95—44no—
278#95—44no—
279#95—44no—
280#95—44no—
281#95—44no—
282#95—44no—
283#95—44no—
284#95—44no—
285#95—44no—
286#95—44no—
287#95—42no—
288#95—44no—
289C24—21no—
290C24—22no—
291C24—22no—
292C24—21no—
293C24—24no—
294C24—22no—
295C24—24no—
296C24—24no—
297C24—22no—
298C24—24no—
299C24—22no—
300C24—24no—
301C24—24no—
302C24—24no—
303C24C22×D414yes—
304C24C2×#4114yes—
305C24H(5,𝔽2)12yes—
306C24C2511yescharacteristic
307C2×D4—28no—
308C2×D4—44no—
309C2×D4—44no—
310C2×D4—48no—
311C2×D4—416no—
312C2×D4—48no—
313C2×D4—84no—
314C2×D4—84no—
315C2×D4—88no—
316C2×D4—84no—
317C22×C4—21no—
318C22×C4—21no—
319C22×C4—24no—
320C22×C4—24no—
321C22×C4—22no—
322C22×C4—28no—
323C22×C4—21no—
324C22×C4—22no—
325C22×C4—24no—
326C22×C4—24no—
327C22×C4—24no—
328C22×C4—24no—
329C22×C4—24no—
330C22×C4—24no—
331C22×C4—22no—
332C22×C4—24no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—21no—
336C22×C4—22no—
337C22×C4—24no—
338C22×C4—24no—
339C22×C4—24no—
340C22×C4—22no—
341C22×C4—24no—
342C22×C4—24no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—24no—
346C22×C4—24no—
347C22×C4—24no—
348C22×C4—24no—
349C22×C4—24no—
350C22×C4—24no—
351C22×C4—24no—
352C22×C4—24no—
353C22×C4—24no—
354C22×C4—22no—
355C22×C4—24no—
356C22×C4—24no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—24no—
360C22×C4—24no—
361C22×C4—22no—
362C22×C4—24no—
363C22×C4—24no—
364C22×C4—24no—
365C22×C4—24no—
366C22×C4—24no—
367C22×C4—24no—
368C22×C4—24no—
369C22×C4—24no—
370C22×C4—24no—
371C22×C4—24no—
372C22×C4—48no—
373C22×C4—48no—
374C22×C4—44no—
375C22×C4—48no—
376C22×C4—416no—
377C22×C4—44no—
378C22×C4—48no—
379C22×C4—48no—
380C22×C4—88no—
381C22×C4—84no—
382C22×C4—84no—
383C22×C4—84no—
384C22×C4—88no—
385C22×C4—88no—
386C22×C4C22×D414yes—
387C22×C4C22×D412yes—
388C22×C4C22×D414yes—
389C22×C4C2×#4114yes—
390C22×C4C2×#4114yes—
391C22×C4H(5,𝔽2)12yes—
392C22×C4H(5,𝔽2)14yes—
393C22×C4H(5,𝔽2)14yes—
394C22×C42−1+418yes—
395C4⋊C4—44no—
396C4⋊C4—416no—
397C4⋊C4—48no—
398C4⋊C4—416no—
399C4⋊C4—44no—
400C4⋊C4—44no—
401C4⋊C4—416no—
402C4⋊C4—48no—
403C4⋊C4—416no—
404C4⋊C4—84no—
405C4⋊C4—84no—
406C4⋊C4—88no—
407C4⋊C4—84no—
408C22⋊C4—232no—
409C22⋊C4—216no—
410C22⋊C4—48no—
411C22⋊C4—416no—
412C22⋊C4—48no—
413C22⋊C4—416no—
414C22⋊C4—48no—
415C22⋊C4—416no—
416C22⋊C4—88no—
417C22⋊C4—88no—
418C22⋊C4—84no—
419C42—44no—
420C42—416no—
421C42—44no—
422C23—24no—
423C23—22no—
424C23—24no—
425C23—28no—
426C23—28no—
427C23—24no—
428C23—22no—
429C23—24no—
430C23—24no—
431C23—28no—
432C23—24no—
433C23—24no—
434C23—28no—
435C23—41no—
436C23—41no—
437C23—42no—
438C23—42no—
439C23—44no—
440C23—42no—
441C23—44no—
442C23—42no—
443C23—44no—
444C23—44no—
445C23—44no—
446C23—44no—
447C23—44no—
448C23—44no—
449C23—44no—
450C23—44no—
451C23—44no—
452C23—42no—
453C23—42no—
454C23—42no—
455C23—44no—
456C23—44no—
457C23—44no—
458C23#55914yes—
459C23C23×D411yescharacteristic
460C23C22×#4112yes—
461C23C2×H(5,𝔽2)12yes—
462C23C2×H(5,𝔽2)12yes—
463C23C2×H(5,𝔽2)11yescharacteristic
464C23C2×H(5,𝔽2)11yescharacteristic
465C23C2×2−1+414yes—
466C23#58512yes—
467D4—88no—
468D4—816no—
469C2×C4—22no—
470C2×C4—24no—
471C2×C4—22no—
472C2×C4—24no—
473C2×C4—28no—
474C2×C4—24no—
475C2×C4—24no—
476C2×C4—28no—
477C2×C4—24no—
478C2×C4—28no—
479C2×C4—22no—
480C2×C4—24no—
481C2×C4—28no—
482C2×C4—28no—
483C2×C4—24no—
484C2×C4—28no—
485C2×C4—28no—
486C2×C4—416no—
487C2×C4—42no—
488C2×C4—41no—
489C2×C4—42no—
490C2×C4—41no—
491C2×C4—42no—
492C2×C4—44no—
493C2×C4—44no—
494C2×C4—44no—
495C2×C4—44no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—44no—
499C2×C4—44no—
500C2×C4—44no—
501C2×C4—44no—
502C2×C4—44no—
503C2×C4—44no—
504C2×C4—44no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—44no—
510C2×C4—44no—
511C2×C4—44no—
512C2×C4—44no—
513C2×C4—44no—
514C2×C4—44no—
515C2×C4—44no—
516C2×C4—44no—
517C2×C4—44no—
518C2×C4—44no—
519C2×C4—44no—
520C2×C4—44no—
521C2×C4—44no—
522C2×C4—44no—
523C2×C4—44no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—44no—
527C2×C4—44no—
528C2×C4—44no—
529C2×C4—44no—
530C2×C4—44no—
531C2×C4—816no—
532C2×C4—816no—
533C2×C4#53514yes—
534C2×C4#54618yes—
535C22—24no—
536C22—28no—
537C22—44no—
538C22—44no—
539C22—42no—
540C22—42no—
541C22—48no—
542C22—44no—
543C22—44no—
544C22—48no—
545C22—44no—
546C22—44no—
547C22—48no—
548C22—84no—
549C22—84no—
550C22—84no—
551C22—84no—
552C22C2×#53511yescharacteristic
553C22C2×#54612yes—
554C22C2×#55911yescharacteristic
555C22#348411yescharacteristic
556C22#348511yescharacteristic
557C22#349912yes—
558C22#350812yes—
559C22#351514yes—
560C22#352014yes—
561C22#353412yes—
562C22#353412yes—
563C22#353514yes—
564C22#353812yes—
565C22#354311yescharacteristic
566C22#354412yes—
567C22#354814yes—
568C4—24no—
569C4—28no—
570C4—44no—
571C4—44no—
572C4—42no—
573C4—48no—
574C4—82no—
575C4—82no—
576C4—84no—
577C4—84no—
578C4—84no—
579C2—44no—
580C2—84no—
581C2#3672111yescharacteristic
582C2#3895112yes—
583C2#4328912yes—
584C2#4349612yes—
585C2#4989314yes—
586C2#5154211yescharacteristic
587C2#5485012yes—
588C2#5914311yescharacteristic
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
1. C2×#2724C2semidirect
3. #16701C2semidirect
4. #16563C2semidirect
5. #15983C2semidirect
7. #15669C2semidirect
8. #15545C2semidirect
9. #15242C2semidirect
10. #14821C2semidirect
11. #14359C2semidirect
12. #14230C2semidirect
13. #14157C2semidirect

Maximal subgroups

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

subgroupindex
C2×#27242
C2×#25012
#167012
#165632
#159832
#159502
#156692
#155452
#152422
#148212
#143592
#142302
#141572